Virtual Holonomic Constraints for Euler-Lagrange Control ... - TSpace

Virtual Holonomic Constraints for Euler-Lagrange Control Systems

by

Alireza Mohammadi

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Electrical and Computer Engineering University of Toronto

c 2016 by Alireza Mohammadi Copyright

Abstract Virtual Holonomic Constraints for Euler-Lagrange Control Systems Alireza Mohammadi Doctor of Philosophy Graduate Department of Electrical and Computer Engineering University of Toronto 2016 In this thesis we investigate virtual holonomic constraints (VHCs) for mechanical control systems. A VHC is a relation among the configuration variables of a mechanical system that does not physically exist, but can be emulated via feedback in a precise sense that is defined within this thesis. An example of VHC is the requirement that the end effector of a robot should only move along a plane. Over the past decade, VHCs have raised to prominence in robotics as they have been successfully employed to induce stable walking motions in biped robots. There is a growing body of evidence that, for complex motion control problems, the VHC paradigm might be more appropriate than the traditional reference tracking approach. Motivated by the hope to establish a new paradigm for complex motion control problems in robotics, this thesis makes contributions in two main directions. First, the thesis presents a complete theory employing VHCs for the stabilization of repetitive “behaviors” in underactuated mechanical control systems with degree of underactuation one. The theory in question has a number of components, the development of which takes us along a journey that includes, among other things, the solution of an inverse Lagrangian problem, the development of an algorithm to implicitize parametric constraints, and the stabilization of closed orbits by means of VHCs parametrized by states of dynamic compensators. The second direction of this thesis is the exploration of the hypothesis that the design ii

of controllers enforcing VHCs might represent a viable paradigm for complex motion control problems. To this end, we solve a challenging open problem in snake robotics: make a planar snake approach and follow a path with desired speed.

iii

Symmetry is what we see at a glance; based on the fact that there is no reason for any difference ... —Blaise Pascal, Pens´ ees

iv

Dedication To my teachers: Past, present, and future.

v

Acknowledgements First and foremost, I owe my deepest gratitude to my academic advisor, Professor Manfredi Maggiore, a dear friend and a great mentor. I learned patience and perseverance from you and was honored to be your academic disciple Professor Maggiore. I would like to sincerely thank my supervisory committee members Professor Lacra Pavel and Professor Luca Scardovi for their encouragement, support, and invaluable feedback on my research during the course of my PhD studies. I sincerely thank Professor Jessy Grizzle, the external examiner of my dissertation, for providing me with his invaluable feedback on my PhD thesis. I would also like to sincerely thank Professor Raymond Kwong for serving on my examination committee. I express my sincere gratitude to Professor Kristin Y. Pettersen, Dr. Ehsan Rezapour, Ms. Anna Kohl, and Dr. Eleni Kelasidi at the Department of Engineering Cybernetics at the Norwegian University of Science and Technology (NTNU) for providing me with a unique academic collaboration opportunity and a warm friendship. I gratefully thank Professor Pettersen for her kind hospitality during my one week visit in Trondheim, Norway in January 2016. Thank you Professor Pettersen for this unique opportunity. The following individuals have interacted with me through my Ph.D studies and I wish to thank them. Murray Wonham, Bruce Francis, Edward Davison, Gabriele D’Eleuterio , Mireille Broucke, Karla Kvaternik, Florian Herzig, Luca Consolini, Alessandro Costalunga, Markus Bussmann, and Stefan Johansson. I am thankful to all my friends and colleagues in Toronto: Edoardo, Ashton, Zach, Farzad, Sina, Sepideh, Yasaman, Shuvo, Mohamed, Mario, Mehrdad, Miad, Saghar, Dr. Peter Agwa, Rev. Mark Kinghan, Rev. Steve Shaw, and Prof. Chul Park. I also would like to thank the Jakeway family for their warm and long-lasting friendship. Thank you Dr. Jakeway, Bruce, and Megan. This thesis never would have taken form without the loving support of my brother and parents. Thanks for your unconditional love and support Mom, Dad, and Brother. vi

Finally, I thank Natural Science and Engineering Research Council of Canada (NSERC), Ontario Ministry of Training, Colleges and Universities, and the University of Toronto ECE Department for providing financial support.

vii

Contents 1 Introduction

1

1.1

Motivating examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.2

VHC-based control paradigm . . . . . . . . . . . . . . . . . . . . . . . .

10

1.3

Applications of virtual holonomic constraints . . . . . . . . . . . . . . . .

11

1.4

Literature on virtual holonomic constraints . . . . . . . . . . . . . . . . .

13

1.5

Thesis objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

1.6

Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

1.7

A summary of thesis contributions

19

. . . . . . . . . . . . . . . . . . . . .

2 Virtual Holonomic Constraints Preliminaries

22

2.1

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

2.2

Virtual holonomic constraints: Definition and regularity . . . . . . . . . .

25

2.3

VHCs for mechanical systems with degree of underactuation one . . . . .

33

2.3.1

Global reduced dynamics . . . . . . . . . . . . . . . . . . . . . . .

34

2.3.2

Generation of regular and odd parametric VHCs . . . . . . . . . .

37

3 The Lagrangian Structure of Reduced Dynamics 3.1

45

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

3.1.1

Summary of chapter main contributions . . . . . . . . . . . . . .

46

3.1.2

Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

3.1.3

Relevance of ILP in VHC-based control paradigm . . . . . . . . .

48

viii

3.2

Introductory example . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

3.3

Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

3.4

Lift of ILP to R2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

3.5

Proofs of main results

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

3.6

Characterization of motion on the constraint manifold . . . . . . . . . . .

66

3.6.1

Effects of coordinate transformations . . . . . . . . . . . . . . . .

66

3.6.2

Qualitative properties of the reduced dynamics . . . . . . . . . . .

70

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

3.7

4 VHC Implicitization

79

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

4.2

Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

4.3

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

4.3.1

Preliminaries from Algebra . . . . . . . . . . . . . . . . . . . . . .

82

4.3.2

Properties of Bézier polynomials . . . . . . . . . . . . . . . . . . .

84

4.3.3

Preliminaries from Graph Theory . . . . . . . . . . . . . . . . . .

87

4.4

Step 1: Polynomial approximation . . . . . . . . . . . . . . . . . . . . . .

89

4.5

Step 2: Implicitization . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

4.6

Step 3: Regularity of implicit constraints . . . . . . . . . . . . . . . . . .

97

4.7

Solution to VHC Implicitization Problem . . . . . . . . . . . . . . . . . . 106

5 VHC-based Orbital Stabilization

110

5.1

Motivating example: Pendubot swing-up . . . . . . . . . . . . . . . . . . 111

5.2

Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.3

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.4

5.3.1

Basic notions of set stability . . . . . . . . . . . . . . . . . . . . . 115

5.3.2

Periodic Riccati differential equations . . . . . . . . . . . . . . . . 117

Step 1: Making the VHC dynamic . . . . . . . . . . . . . . . . . . . . . . 119

ix

5.4.1

Dynamic VHCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.4.2

Regularity and stabilizability of dynamic VHCs . . . . . . . . . . 122

5.4.3

Reduced dynamics induced by dynamic VHCs . . . . . . . . . . . 124

5.5

Reduction-based stabilization strategy . . . . . . . . . . . . . . . . . . . 125

5.6

Step 2: Transverse linearization and orbital stabilization . . . . . . . . . 127

5.7

Step 3: Enforcing the VHC and stabilizing the closed orbit . . . . . . . . 133 5.7.1

Solution to VHC-based orbital stabilization . . . . . . . . . . . . 133

5.7.2

Implementation details . . . . . . . . . . . . . . . . . . . . . . . . 135

5.8

Pendubot swing-up: Solution . . . . . . . . . . . . . . . . . . . . . . . . 139

5.9

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6 VHC-based Control of Planar Snake Robots

142

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.2

Model of the snake robot . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.3

Control specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

6.4

Body shape control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

6.5

Velocity control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 6.5.1

Head angle control . . . . . . . . . . . . . . . . . . . . . . . . . . 156

6.5.2

Speed control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

6.6

Path following control of snake robots . . . . . . . . . . . . . . . . . . . . 159

6.7

Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.8

Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

6.9

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

6.10 Proof of technical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 7 Conclusions and future research

181

7.1

Summary of contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 182

7.2

Relation to the VHC-based control of biped robots . . . . . . . . . . . . 183

x

7.3

Future research directions and open problems . . . . . . . . . . . . . . . 184

xi

List of Tables 6.1

The parameters of the snake robot . . . . . . . . . . . . . . . . . . . . . 146

6.2

Snake robot VHC-based controller parameters . . . . . . . . . . . . . . . 165

6.3

Robustness analysis for circle tracking

xii

. . . . . . . . . . . . . . . . . . . 170

List of Figures 1.1

Trajectory tracking versus VHC-based control framework.

. . . . . . . .

2

1.2

Path of the particle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.3

The pendubot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.4

VHC-induced pendulum-like motion of a pendubot. . . . . . . . . . . . .

7

1.5

Kinematic structure of a planar snake robot. . . . . . . . . . . . . . . . .

8

1.6

Conceptual flowchart of the thesis. . . . . . . . . . . . . . . . . . . . . .

20

2.1

A material particle in a central gravitational field. . . . . . . . . . . . . .

26

2.2

Geometrical interpretation of VHC regularity. . . . . . . . . . . . . . . .

28

2.3

Constraining the motion of a material particle to lie on a virtual line. . .

29

2.4

A brachiating acrobot. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

2.5

A planar three-link biped robot. . . . . . . . . . . . . . . . . . . . . . . .

30

2.6

Regularity of VHCs and transversality. . . . . . . . . . . . . . . . . . . .

32

2.7

Solutions to the pendubot VCG with different degrees. . . . . . . . . . .

40

2.8

VHC-induced pendubot configurations. . . . . . . . . . . . . . . . . . . .

44

3.1

A material particle immersed in a gravitational field. . . . . . . . . . . .

50

3.2

Example of an EL system. . . . . . . . . . . . . . . . . . . . . . . . . . .

75

3.3

Example of a SEL system. . . . . . . . . . . . . . . . . . . . . . . . . . .

76

3.4

Example of a non-EL system . . . . . . . . . . . . . . . . . . . . . . . . .

77

3.5

The particle mass motion arising from SEL dynamics. . . . . . . . . . . .

78

xiii

4.1

Undirected graphs

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

4.2

Spanning trees and path graphs. . . . . . . . . . . . . . . . . . . . . . . .

88

4.3

Equivalence relation between undirected graphs and implicit constraints.

93

4.4

The spanning trees corresponding to the implicit VHCs given by (4.17). .

96

4.5

The curve π23 ◦ σ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.6

Parametric VHCs and their approximations. . . . . . . . . . . . . . . . . 108

4.7

Configurations of the pendubot under VHCs and their approximations. . 108

4.8

Zero level sets of the implicit VHCs and the image of parametric VHCs. . 109

4.9

The norm of the Jacobian of the implicitized constraints. . . . . . . . . . 109

5.1

The pendubot and its low-high and high-high equilibrium configurations.

5.2

Configurations of the pendubot under an odd VHC. . . . . . . . . . . . . 113

5.3

The phase portraits of the pendubot VHC-induced reduced motion.

5.4

Geometric interpretation of making VHCs dynamic. . . . . . . . . . . . . 121

5.5

A tubular neighborhood of the closed orbit. . . . . . . . . . . . . . . . . 126

5.6

A well-defined transformation in a tubular neighborhood of the orbit. . . 128

5.7

Block diagram of the VHC-based orbital stabilizer. . . . . . . . . . . . . 135

5.8

The orthogonal projection of a configuration vector onto the VHC curve.

5.9

Constructing an orthonormal frame on the VHC curve. . . . . . . . . . . 136

5.10 Phase curve of the VHC-induced motion on the phase cylinder.

112

. . 113

136

. . . . . 140

5.11 Energy of the pendubot on the constraint manifold. . . . . . . . . . . . . 140 6.1

Snake robot kinematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.2

Forces and torques acting on individual links of the robotic snake. . . . . 147

6.3

The hierarchical VHC-based snake robot control system. . . . . . . . . . 153

6.4

Components of the path following reference vector. . . . . . . . . . . . . 160

6.5

Planar snake robot circle tracking simulation results– Part(a). . . . . . . 166

6.6

Planar snake robot circle tracking simulation results– Part (b). . . . . . . 167

xiv

Notation N

Set of natural numbers

R

Set of real numbers

R+

Set of positive real numbers

[R]T

The set of real numbers modulo T ; diffeomorphic to the unit circle S1

Tn

n-torus; diffeomorphic to [R]T × · · · × [R]T | {z } n times

Rn×m

Set of n × m matrices with real entries

X†

a right-inverse of the matrix X ∈ Rn×m with full row rank

kpkS

Point-to-set distance of point p to set S

dH (S1 , S2 )

Hausdorff distance of S1 to S2

cl(S)

Closure of set S

Br (x)

The neighborhood of point x: {y ∈ X : ky − xk < r}

Br (S)

The neighborhood of set S: {y : kykS < α}

N (S)

An open neighborhood of the set S

f −1 (y)

The level set {x : f (x) = y}

dfx

The Jacobian matrix of f at x

Hess(f )

Hessian matrix of C 2 function f

φu (t, x0 )

The solution of x˙ = f (x) + g(x)u at time t with initial condition x0 , where u is either a piecewise continuous signal, or a smooth feedback

φ(t, x0 )

The solution of x˙ = f (x) at time t with initial condition x0

TQ

The tangent bundle of Q: {(p, vp ) : p ∈ Q, vp ∈ Tp Q}

dh

The global differential of h: (p, vp ) 7→ (h(p), dhp (vp ))

kσkW 1, ∞

The Sobolev norm of the C 1 signal σ: max kσ(θ)k + max kσ ′ (θ)k θ∈[0, T ] xv

θ∈[0, T ]

DOF

Degrees-of-freedom

VHC

Virtual holonomic constraint

VCG

Virtual constraint generator

IPCV

Inverse problem of calculus of variations

IPLM

Inverse problem of Lagrangian mechanics

ILP

Inverse Lagrangian problem

LPTV

Linear periodically time-varying

PRDE

Periodic Riccati differential equation

VCP

Velocity control problem

PFP

Path following problem

xvi

Chapter 1 Introduction Over the past several decades, trajectory tracking has been the dominant paradigm for designing motion control algorithms in robotics. Trajectory tracking controllers have been successfully employed in industrial manufacturing settings where robots operate in highly structured environments. In this framework, timed reference trajectories are generated using motion planning algorithms (see, e.g., [1, 2]), and then the robotic system states are made to follow the reference trajectories using standard control schemes such as PID, computed torque, and sliding mode. The typical block diagram of a robot control system is depicted in Figure 1.1a. Notwithstanding its popularity, there are certain drawbacks associated with the trajectory tracking framework making it inadequate for certain emerging robotic applications. First and foremost, the presence of underactuation and interaction with uncertain environments in dynamic tasks such as walking make motion planning extremely challenging [3]. Furthermore, trajectory tracking controllers are not robust with respect to disturbances that tend to desynchronize motion prescribed by reference signals. In particular, if a disturbance perturbs the motion with respect to a reference signal, the controller attempts to regain synchrony with the provided signals. By forcing the system states to keep up with reference signals, the controller may destabilize the robotic sys-

1

2

Chapter 1. Introduction Virtual constraint h(q) Motion planner

yd (t)+ −

e

Controller

u

y

e

Controller

u

y

x = (q, q) ˙

x = (q, q) ˙

(b) Block diagram of a VHC-based con(a) Block diagram of a trajectory tracking

troller: The system states are enforced to

controller: The system states are enforced

satisfy the relation h(q) = 0. Coordination

to follow the reference signal yd (t) gener-

between various DOF is more crucial than

ated by a high-level motion planner.

following a strict “time-schedule” in VHCbased control framework.

Figure 1.1: Trajectory tracking versus VHC-based control framework. tem. For instance, if a walking biped robot is pushed backward, the trajectory tracking controller might cause the robot to lose its “balancing posture” and overturn. Following the work by J. Grizzle and collaborators [3, 4, 5, 6], the virtual holonomic constraint (VHC) framework has emerged as an alternative to the traditional trajectory tracking paradigm for solving arduous motion control problems such as walking for biped robots. In this framework, feedback is used to emulate the presence of constraints on the configuration variables of the robot. The adjective “virtual” indicates that the constraints do not physically exist, while the adjective “holonomic” indicates that the constraints involve only the configuration variables, and not their derivatives. The terminology “virtual holonomic constraint”, to the best of our knowledge, originated from [7]. Unlike trajectory tracking, in VHC-based control the coordination between various degrees-offreedom (DOF) is more crucial than following a strict time-schedule by joints and links of the robotic mechanism. The VHC approach relies on stabilizing relations among the configuration variables of the mechanical system, and thus there is no need for an a priori

3

Chapter 1. Introduction

time parameterization of system states. In other words, the robotic system is made to “generate and track its own reference motion” [8] (see Figure 1.1b). After a disturbance is exerted on the system, a VHC-based controller makes the robotic mechanism converge back to the desired posture without enforcing the system states to resynchronize with a timed reference trajectory [3]. To further illustrate the differences between the trajectory tracking and the VHCbased control frameworks, consider a point particle on the plane, with position q ∈ R2 and unit mass. The dynamics of the particle are

q¨ = f, where f is the control force. Suppose we wish to make the particle approach the unit circle centered at the origin and follow it with unit speed. In the trajectory tracking framework, one might define the reference signal qref (t) = [cos(t), sin(t)]⊤ and stabilize it using the PD control law

f = q¨ref (t) + kd q˙ref (t) − q˙ + kp qref (t) − q .

An alternative to the above approach is to consider the relation h(q) = q ⊤ q − 1, which is not parameterized by time, and stabilize its zero level set, which corresponds to the desired circular path. In the context of this thesis, the relation h(q) = 0 is a VHC. A technique for solving this particular path following problem was developed by Christopher Nielsen in [9, 10], and relies on the input transformation   q 0 −1 q f= f1 +  f2 .  kqk kqk 1 0 | {z } | {z } f⋔ fk

In the above, f ⋔ is the component of the control force perpendicular to the level sets of h(q), and f k is the component tangent to the level sets of h(q). We use the component f ⋔

4


to asymptotically stabilize the relation q ⊤ q − 1 = 0 while use the remaining component f k to make the particle traverse the circle with unit speed. We will not go in the details of the design which is simple but beyond the scope of this informal discussion (see [9, 10]). Rather, we present simulation results to illustrate the differences between the two philosophies. We consider the initial condition p(0) = [1, 0]⊤ and p(0) ˙ = [0, 1]⊤ . At t = 1, we stop the particle for π seconds and then release it to continue its motion. As it can be seen from Figure 1.2, with the trajectory tracking controller the particle leaves its circular path in order to “catch up” with the reference signals. On the other hand, the particle does not leave the circular path and the path invariance, when the relationship q ⊤ q − 1 = 0 is stabilized, is respected. As it is shown in this example, such a relationship stabilization approach to solving motion control problems is more desirable in applications where unwanted disturbances desynchronize the mechanical system motion with respect to reference signals. obstacle 0.8

0.6

0.4

y [m]

0.2

0

−0.2

−0.4

−0.6

−0.8

−1

−0.5

0

0.5

1

x [m]

Figure 1.2: Path of the particle. Trajectory tracking control (circles) versus the relation stabilization approach (triangles).


1.1

5

Motivating examples

In Chapter 2 of this thesis we give a precise definition of VHC. In this section, we provide the intuition behind this notion, by means of two examples. As we shall see, a VHC is a relation of the form h(q) = 0, where q is the vector of configuration variables of a mechanical system, that is stabilized via feedback control and made invariant. A controller enforcing a VHC in a mechanical system, after the stabilization transient has passed, causes the system to behave as if it were physically constrained through the relation h(q) = 0. By a suitable design of the function h(q), J. Grizzle and collaborators were able to induce a remarkable variety of walking motions in biped robots [3, 4, 5, 6]. The motivating hypothesis behind this thesis is that the VHC framework proposed by Grizzle can in fact be applied in great generality to a class of mechanical systems with the general objective of inducing complex behaviors without resorting to any exogenous reference signals.

Figure 1.3: The pendubot.

Example 1.1 Consider the pendubot system in Figure 1.3. This is a double-pendulum in which the shoulder is actuated while the elbow is not. The configuration variables (q1 , q2 ) are angles measured modulo 2π. In this thesis we use the notation [R]2π to indicate the set of real numbers modulo 2π, so we may write that (q1 , q2 ) ∈ [R]2π × [R]2π . Assuming that the masses and lengths of the two links are equal and unitary, the

6

Chapter 1. Introduction dynamical equations governing the motion of this mechanism read D(q)¨ q + C(q, q) ˙ q˙ + ∇P (q) = B(q)τ, where 



2 cos(q1 − q2 )  D(q) =  , cos(q1 − q2 ) 1   0 sin(q1 − q2 )q˙2   C(q, q) ˙ = , − sin(q1 − q2 )q˙1 0   1 P (q) = 2g cos q1 + g cos q2 , B =   . 0

The pendubot system is Euler-Lagrange with two degrees-of-freedom (q1 , q2 ) and one control input, the torque τ applied at the first joint (see Figure 1.4). We say that the degree of underactuation of the pendubot system is equal to one. We would like to control the pendubot such that it mimics the behavior of a simple pendulum. One way to achieve this control objective is to enforce the relation q1 = q2 on the pendubot configuration. The function h(q) := q1 − q2 will be our VHC. To stabilize this relation, we define the output e := q1 − q2 , for which it holds that e¨ = µ(q, q) ˙ + [1 − 1]D −1 (q)B(q)τ, where µ(·) is a smooth function. On the set h−1 (0), it can be shown that [1 −1]D −1 (q)B(q) = 2 and thus the coefficient of the control input τ in e¨ appears non-singularly. In other words, the output function e has relative degree two on the aforementioned set. The following input-output linearizing feedback

τ=

1 − k1 e − k2 e˙ − µ(q, q) ˙ , 2

7

Chapter 1. Introduction 5

4

3

2

2

1

θ˙

1.5

0

1

−1 0.5

−2 0

−3

−4

−0.5

−5

−1

0

1

2

3

4

5

6

θ −1.5

(b) Phase portrait of the reduced dynamics

−2 −2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(a) Configurations of the pendubot under

with the VHC q1 = q2 enforced on the pendubot. The red orbits in the gray area cor-

the VHC q1 − q2 = 0.

respond to oscillations of θ while the blue orbits correspond to full revolutions of θ.

Figure 1.4: VHC-induced pendulum-like motion of a pendubot.

where k1 , k2 > 0, gives e¨ + k2 e˙ + k1 e = 0, thus stabilizing the equilibrium (e, e) ˙ = (0, 0). In the language of nonlinear control, the feedback above stabilizes the zero dynamics manifold Γ = (q, q) ˙ : q1 = q2 , q˙1 = q˙2 , therefore enforcing the relation q1 − q2 = 0 (see

Figure 1.4a). As we mentioned earlier, the constraint q1 = q2 does not physically exist in the pendubot, but the feedback above emulates its presence.

Throughout this thesis, we call the zero dynamics manifold Γ the constraint manifold associated with the VHC h(q) = 0. The reduced dynamics are the dynamics that describe the evolution of the states of the pendubot system on the constraint manifold. In order to derive these dynamics, we multiply both sides of the pendubot dynamical equation by a left annihilator of B(q), e.g., B ⊥ (q) = [0, 1], and evaluate the result on Γ using the following parameterization h i⊤ h i⊤ ˙ θ˙ , q¨ = θ, ¨ θ¨ . q = [θ, θ]⊤ , q˙ = θ,

8

Chapter 1. Introduction By so doing, we obtain g θ¨ = sin(θ). 2

Not surprisingly the reduced dynamics is the equation of a simple pendulum of length two meters and mass one kilogram. Naturally, this is a Lagrangian system with Lagrangian (1/2)θ˙ 2 − (g/2)[cos(θ) − 1]. Due to its Lagrangian structure, the reduced motion has a wealth of closed orbits (see Figure 1.4b) that correspond to either full revolutions (rotations) of θ or oscillations of θ from which one may select and stabilize one that corresponds to a desired repetitive behavior. However, the reduced dynamics have no control inputs and stabilizing the desired motion on the constraint manifold Γ is impossible unless one breaks the invariance of Γ, therefore destroying the VHC. The pendubot example illustrates some important concepts associated with VHCs: their enforcement, the reduced dynamics they induce, and the presence of certain types of closed orbits (oscillations or rotations) on the constraint manifold.

△

Example 1.2 In Chapter 6 we investigate the problem of path following for snake robots on the plane. We consider a snake robot with n rigid links of equal lengths, masses, and moments of inertia. The snake robot has n + 2 DOF consisting of the n absolute link angles θ1 , · · · , θn , and the center of mass coordinates px , py in an inertial frame. The ith snake joint angle is φi = θi − θi+1 , 1 ≤ i ≤ n − 1. Figure 1.5 illustrates the kinematic structure of the snake robot.

Figure 1.5: Kinematic structure of a planar snake robot.

9


The propulsion mechanism is the interaction between the links and the ground through the friction forces. The control objective is to make the snake robot converge to a desired planar path and traverse the path with a desired velocity. Traditionally, to address this problem, researchers make each joint to track the following biologically-inspired sinusoidal reference signal φref,i (t) = α sin(ωt + (i − 1)δ) + φ0 (t),

(1.1)

where α denotes the amplitude of the sinusoid, ω denotes the frequency of the joint oscillations, δ denotes the phase shift between two consecutive joints, and φ0 (t) is a time-varying joint offset used to control the direction of snake locomotion. The main drawback associated with this approach is that there exists no analytically established method to choose the offset signal φ0 (t) and the frequency ω so that the path following control objective is achieved except for simplified linearized models of snake robots following straight lines. As an alternative to tracking the reference signals in (1.1), one can stabilize the following VHC

θi+1 − θi = α sin(λ + (i − 1)δ) + φ0 , 1 ≤ i ≤ n − 1.

(1.2)

In Chapter 6, we let the variables λ and φ0 , which parameterize the constraints above, be dynamic and their evolution be governed by the following compensators

¨ = uλ , φ¨0 = uφ . λ 0 We use the snake robot states, the desired path geometry, and the prescribed speed along the path to design the feedback control inputs uλ and uφ0 . The resulting closedloop control system is time invariant and amenable to hierarchical analysis.

△


1.2

10

VHC-based control paradigm

The notion of VHCs offers a broad paradigm for solving arduous motion control problems. This thesis is an initial step towards the systematization of the VHC framework whose overarching themes can be summarized as follows. • The aim of the VHC-based control paradigm is to solve challenging motion control problems arising from applications in engineering, e.g., the swing-up maneuver control of a pendubot, or biomimetics applications, e.g., the locomotion control of a snake robot. Due to the inherent limitations of the trajectory tracking framework highlighted earlier in the chapter, the motion control problem is required to be solved by removing the time parameterization from the control loop using the VHCbased approach. • The first step in solving the problem using the VHC-based control paradigm is to design appropriate constraints. At times, such constraints might arise naturally from the application domain, such as in Chapter 6 where they are inspired by biological considerations. Other times, they might be generated through a systematic mathematical procedure, as demonstrated in Chapter 2. • Once a suitable constraint is found, it is enforced by stabilizing a suitable set, which usually requires assigning all the physical controls, e.g., the motor torques. After enforcing the constraint, the motion is governed by a dynamic system with state-space dimensions that are often much smaller than the original underactuated system. • After enforcing the constraint, there are often additional control objectives to be met. For instance, in the case of the pendubot swing-up maneuver control we would like to stabilize specific closed orbits corresponding to repetitive maneuvers, while in the case of the snake robot path following control we would like to stabilize a


11

certain path on the plane. In order to address the lack of controls after assigning the physical ones, we parameterize the constraints using states of dynamic compensators. The theme of making constraints dynamic is explored in Chapter 5 for mechanical systems with degree of underactuation one, and in Chapter 6 for a mechanical system with degree of underactuation three. In the former case, we need to find the closed orbits that arise from the constraint which is, in general, a hard problem to solve; but, as it is shown in Chapter 3, the closed orbits are level-sets of a mechanical energy function under certain conditions.

1.3

Applications of virtual holonomic constraints

As mentioned earlier, some modern motion control problems cannot be adequately addressed by the traditional trajectory tracking framework, but instead can be solved using the VHC framework. These are some of them. • Biped robotics. In biped robotics, stable periodic walking motions can be induced by enforcing appropriately designed VHCs encoding stable walking gaits [3, 5, 6]. • Limbless robotic locomotion. In limbless biologically-inspired robots such as robotic snakes [11], VHCs can be used to make the robot follow desired paths. In this context, the VHC encodes a biologically-inspired gait whose parameters are dynamically adjusted to control the velocity vector of the robot in such a way that the center of mass converges to a desired path. VHC-based control of planar snake robots is presented in Chapter 6. • Humanoid robotics. A crucial objective in controlling anthropomorphic robotic devices is being able to mimic human motor patterns. Various studies suggest that there exist consistent geometric relations in human kinematic postures during different motions [12], such as single support phase of cyclic walking [13], sitting down


12

and chair-rise motions [14], arm movements during drawing [15], and static grasping by fingers [16]. Properly designed VHCs encoding human-like motor patterns can be used to imitate anthropomorphic robotic motions. In [17, 18], the authors extract motor data from various experimental human motion studies in the form of kinematic relations that display the same “universal” behavior. Driving the humanoid robot outputs to those of the universal human patterns would generate anthropomorphic motion. • Rehabilitation robotics. Rehabilitation robotic devices are utilized to train patients with impaired limbs in order to facilitate motor recovery [19]. To date, trajectory tracking has been the most pervasively control scheme used in rehabilitation robotics. Minimum-jerk point-to-point trajectories [20], pre-recorded trajectories from unimpaired subjects [21], and pre-recorded trajectories during therapist-guided assistance [22], are among the timed reference trajectories that are prescribed for patient training. The downside of this class of schemes is that they merely move patient limbs along predefined paths and impose a certain timing on their movement; consequently, employing such schemes might have negative impacts on motor recovery [19]. A more adequate physiotherapeutic control objective is to provide the patient with minimal support, sufficient for achieving the desired task in a physiologically correct way. A solution to achieve this goal is to create “virtual tunnels” in Cartesian or joint space of the robotic device around the desired trajectory [23, 24, 25]. Once out of the tunnel, the patient limbs are driven towards it, whereas the patient can move her limbs autonomously inside the tunnel. Instead of using ad-hoc control methods, the “virtual tunnel” controllers can be systematically designed using VHCs. • Teleoperation systems. Teleoperation systems project a human operator’s manipulation and sensing capabilities onto remote or virtual environments. They


13

consist of a master robot or a haptic interface with which the human operator interacts, a communication channel, and a slave robot interacting with a remote or virtual environment [26]. Stability and transparency are the two main control objectives that every teleoperation control system should achieve. In a transparent teleoperation system, the operator feels as if she is directly interacting with the remote environment. Thus far, control design for teleoperation systems has been mainly based on passivity and network theory formalisms originally proposed by Anderson and Spong [27]. An alternative approach might be to formulate control specifications in terms of virtual constraints emulating the presence of surfaces. In [28], VHCs are employed to solve the teleoperation control problem for a system consisting of two 2-DOF identical planar robots where synchronization and force control during the free motion and contact phase are achieved by enforcing proper VHCs. • Formation control of mobile robots. VHCs can be used to impose desired interagent distances such that the group of mobile robots behaves as a rigid body [29]. • Character animation in computer graphics. In physics-based computer graphics, VHC-based controllers have been recently employed for animation generation [30] as an alternative to motion capture-based and off-line optimization-based generation techniques where high level features of the character motion are described by VHCs.

1.4

Literature on virtual holonomic constraints

The notion of constraints that do not physically exist in a mechanical system but can be enforced through the application of external forces first appeared in the early 1900s [31, 32]. Modern low-cost computational capability for real-time implementation of control laws has revived the interest in virtual constraints. An early manifestation of the VHC


14

idea in robotics appeared in the work of Nakanishi et al. [8], where the authors enforced, via feedback control, a constraint on the angles of an acrobot to induce pendulum-like dynamics imitating the brachiating motion of a gibbon. Jessy Grizzle and collaborators brought the idea of virtual constraints to prominence (see, e.g., [5, 33, 34, 35]). In their body of work, VHCs are used to encode different walking gaits, without requiring the design of reference signals for the robot joints. They show that when a suitable VHC is enforced on the biped, the resulting constrained motion exhibits a stable hybrid limit cycle corresponding to a periodic walking motion. Recently, Shiriaev, Canudas-de-Wit, and collaborators have employed VHCs to aid the selection of closed orbits corresponding to desired repetitive behaviors of underactuated mechanical systems in [36, 37, 38, 7]. In [36], a system with n DOF and n − 1 controls is studied. It is demonstrated that an unforced second-order system possessing an integral of motion describes the constrained motion. Assuming that this unforced system has a closed orbit, a linear time-varying controller is designed that yields exponential stability of the closed orbit. In [39], Canudas-de-Wit and collaborators proposed a technique to stabilize a desired closed orbit that relies on enforcing a virtual constraint and on dynamically changing its geometry so as to impose that the reduced dynamics on the constraint manifold match the dynamics of a nonlinear oscillator. The approach in [39], however, cannot be used to stabilize an assigned orbit on the constraint manifold. In [38], the methodology in [36] is generalized to the case when the degree of underactuation is greater than one. In their body of work, Shiriaev et al. take the route of stabilizing individual trajectories on the constraint manifold, without stabilizing the constraint manifold itself. In essence, the approach of Shiriaev et al. only employs VHCs for the exploration of closed orbits, without enforcing the constraints via feedback. The reader is referred to [7] for a review of some of the ideas in the papers above. Motivated by Grizzle et al.’s methodology in biped robot control, Maggiore, Consolini, and collaborators have investigated VHCs for mechanical control systems in [40, 41, 42,


15

43]. In [40], an explicit definition of VHCs, which contains the feasibility requirement for the control system dynamics, is presented and sufficient conditions for VHC feasibility are found. In [40, 41], the VHC-induced dynamics, which govern the motion of underactuated mechanical systems under the influence of feasible VHCs, are derived. Furthermore, it is shown that the constrained dynamics induced by feasible VHCs are not necessarily Euler-Lagrange despite possessing an integral of motion. A systematic procedure for generating feasible VHCs in parametric form is provided in [42]. In Maggiore et al.’s framework, VHCs are viewed as relations that can be made invariant with respect to the system’s dynamics through application of suitable control inputs. This point of view on VHCs makes control design amenable to invoking reduction tools from dynamical systems theory and hierarchical design–analysis and is in line with Grizzle et al.’s approach to biped walking control. A discussion elaborating the differences between Shiriaev et al.’s and the approach adopted in this thesis is provided in Section 5.9. Maggiore et al.’s framework also possesses some philosophical similarities to the notion of path-following introduced in [44, 45], where the outputs of the system under study are forced to converge to a desired path while a desired dynamic behavior is satisfied along the path. However, in these works, the path invariance is not respected. As we shall see in Chapter 2, control invariance is a key property of VHCs in Maggiore et al.’s framework. As we shall see in Chapter 2, enforcing a VHC corresponds to stabilizing a set, the so-called constraint manifold. The VHC-based controllers developed in this thesis are, therefore, set stabilizing controllers. In [9, 46, 47], Christopher Nielsen and coauthors developed a general approach to set stabilization which bears some philosophical resemblance to the approach taken in this thesis, and in fact was one of the inspirations that led to the paper [40].


1.5

16

Thesis objectives

This thesis has two main objectives. The first, more prominent objective, is to provide a complete solution to the problem of using VHCs to stabilize repetitive “behaviors” for mechanical systems with degree of underactuation one. Some of the problems that we address in achieving the first objective are similar to the ones studied by Shiriaev et al. [36]. Our approach towards these problems, however, is inspired by Grizzle et al. [4, 5]. For this class of systems, the thesis solves the following problems: P1. Inverse Lagrangian problem for the reduced dynamics:

Find nec-

essary and sufficient conditions under which the reduced dynamics of mechanical control system subject to VHCs have a Lagrangian structure. This problem is important because Lagrangian systems have a wealth of closed orbits, representing repetitive “behaviors”. P2. Implicitization: In [42], the authors develop a procedure to generate VHCs in parametric form. However, in order to enforce VHCs, one needs to represent them in implicit form. Thus, given a VHC in parametric form enjoying certain properties, it is required to develop an algorithm to generate an implicit representation of the VHC enjoying the same properties. P3. VHC-based orbital stabilization: Once a closed orbit has been selected on the constraint manifold, design a controller that simultaneously enforces the VHC and stabilizes the closed orbit in question. The second objective of this thesis is to explore the hypothesis that VHCs can be used to solve complex motion control problems that go beyond the stabilization of walking gaits in biped robots. To this end, we explore the problem of making a snake robot follow an arbitrary path on the plane, a challenging open problem in snake robotics. We show that VHCs can indeed be used to great effect in developing a solution to this problem.


17

This part of the thesis is the result of a collaboration with Kristin Y. Pettersen and Ehsan Rezapour at the Norwegian University of Science and Technology (NTNU). The reader will find that the problems studied, on one hand, in Chapters 3–5 and the problem studied, on the other hand, in Chapter 6 lead to apparently disconnected developments in this thesis. Namely, while the objective in Chapters 3–5 is to stabilize a VHC-induced closed orbit in underactuated mechanical systems with degree of underactuation one, the objective in Chapter 6 is to solve a path-following problem for a specific mechanical system with degree of underactuation three. There are two fundamental conceptual differences between the framework in Chapters 3–5 and the one in Chapter 6. The control specifications in the two cases are different. Moreover, the models are different, not just for the different degree of underactuation, but also for the fact that the snake model in Chapter 6 is subject to external friction forces that are the key mechanism of locomotion. Despite the apparent differences mentioned above, the two parts of this thesis are motivated by the need, on one hand, to address a theoretical problem in full generality; and the need, on the other hand, to test the hypothesis that Grizzle et al.’s philosophy is applicable to general locomotion problems.

1.6

Thesis outline

The body of this thesis consists of six chapters. • Chapter 2: Virtual Holonomic Constraints Preliminaries We provide essential preliminaries needed in the thesis, including the notation, definition of VHCs, notion of regularity, deriving the dynamics induced by a regular VHC, and generation of regular parametric VHCs for systems with degree of underactuation one.


18

• Chapter 3: The Lagrangian Structure of Reduced Dynamics under Virtual Holonomic Constraints

This chapter provides a complete solution to Problem P1 for mechanical systems with degree of underactuation one. We characterize typical solutions of the reduced dynamics. • Chapter 4: VHC Implicitization This chapter solves Problem P2. We present a procedure for making parametric VHCs implicit and thus amenable to feedback implementation. The key ingredient used in developing the procedure is based on computing the resultants of polynomials, a well-known tool in classical algebra and widely used in computer graphics. • Chapter 5: VHC-based Orbital Stabilization This chapter provides a solution to Problem P3. We design a controller to stabilize a given closed orbit arising from the Lagrangian structure induced by a regular VHC. To do this, we rely on the idea of making the VHC dynamic. In the final chapter, we turn our attention to a robotic system where there are three control inputs less than the number of DOF. • Chapter 6: Maneuvering Control of Planar Snake Robots Using Virtual Holonomic Constraints

In this chapter, we propose a VHC-based controller to make the center of mass of a planar snake robot converge to a desired path and traverse the path with a desired velocity. The proposed feedback control strategy enforces virtual constraints encoding a lateral undulatory gait, parametrized by states of dynamic compensators


19

used to regulate the orientation and forward speed of the snake robot. This solution was developed in collaboration with Kristin Y. Pettersen and Ehsan Rezapour at the Norwegian University of Science and Technology (NTNU). Diagram 1.6 is a flowchart that shows the concepts contained in this thesis and their interdependence.

1.7

A summary of thesis contributions

The following list constitutes the major contributions that this thesis makes to the body of knowledge in the virtual constraints framework. 1. Chapter 3. • Checkable necessary and sufficient conditions for the reduced dynamics to have a well-defined global Lagrangian structure, Theorems 3.3.3, 3.3.5, and 3.3.9. • Invariance of the Lagrangian structure under VHC curve reparameterization, Propositions 3.6.2 and 3.6.1. • Characterization of the motion arising from the singular Lagrangian structure, Proposition 3.6.5. 2. Chapter 4. • Approximation of regular and odd parametric VHCs using Bézier polynomials, Proposition 4.4.2. • A procedure for making parametric VHCs in polynomial form implicit, Procedure 4.1. • Sufficient conditions for checking regularity of the implicit constraints generated by Procedure 4.1, Propositions 4.6.3, 4.6.4, and 4.6.5.

20

Chapter 1. Introduction Chapter 2 VHC definition, regularity, VHC-induced reduced dynamics, regular VHC generation Procedure 2.1 Chapter 6

Proposition 2.3.4

Path following control of snake robots, dynamic VHCs

Chapter 3 solution to ILP, charaterizing the VHC-induced motion Theorem 3.3.5 Proposition 3.3.7 Chapter 4

VHC implicitization, resultant of polynomials Approximation of parametric VHCs Procedure 4.2

Chapter 5 VHC-based orbital stabilization, dynamic VHCs

Notion of dynamic VHCs One-parameter dynamic VHCs for mechanical systems with degree of underactuation one (5.15) Two-parameter dynamic VHCs for snake robots (6.12)

Figure 1.6: Conceptual flowchart of the thesis. 3. Chapter 5. Generalizing the dynamic VHC-based orbital stabilization methodology in [43] to the setting of implicit VHCs. • Notion of dynamic implicit VHCs and their induced reduced dynamics, Section 5.4.


21

• Orbital stabilization of energy level-sets on the constraint manifold of dynamic VHCs using periodic Riccati differential equations (PRDEs), Section 5.4. • A solution to VHC-based orbital stabilization using PRDE solutions, Theorem 5.7.1. 4. Chapter 6. The first complete reported solution for maneuvering control of planar snake robots considering their full nonlinear model using dynamic VHCs. The main result is Theorem 6.7.1.

Chapter 2 Virtual Holonomic Constraints Preliminaries A virtual holonomic constraint (VHC) is a relation involving the configuration variables of a mechanical system that can be made invariant via feedback control. VHCs emulate the presence of physical constraints, and can be used to induce desired behaviors in a mechanical control system. In this chapter, we present the preliminary notions and theory of VHCs developed in [40, 41, 42]. In Section 2.1 we introduce the notation used in the thesis, which is summarized in the table on page xvi. Our exposition continues with the definition of a VHC for a mechanical control system, and the notion of regularity in Section 2.2. In Section 2.3 we restrict our attention to underactuated mechanical systems that have one control input less than the number of degrees of freedom. For these, we first find a global description of the constrained dynamics and we present a procedure for generating regular parametric VHCs with odd symmetry. Although this material is not an original contribution of this thesis, we present as it is needed for the developments presented in the subsequent chapters.

22

Chapter 2. Virtual Holonomic Constraints Preliminaries

2.1

23

Notation

We denote by N, R, and R+ the set of natural numbers, real numbers, and the positive real line (0, ∞), respectively. Given a natural number k ∈ N, Rk is the Cartesian product R × · · · × R, k times. Similarly, if n, k ∈ N and S is a set, S k denotes the k-fold Cartesian product S × · · · × S. Given n, m ∈ N, we denote by Rn×m the set of real-valued n × m matrices. If X ∈ Rn×m has full row rank, we denote by X † a right-inverse of X, e.g., X † = X ⊤ (XX ⊤ )−1 . We denote by n the set of integers {1, · · · , n}. We denote by col(x1 , · · · , xk ) the column vector [x1 · · · xk ]⊤ where ⊤ denotes transpose. If x and y are two column vectors then col(x, y) := [x⊤ y ⊤ ]⊤ . Given x ∈ R and T > 0, then [x]T := x modulo T . In other words, [x]T is the equivalence class of numbers that differ from x by an integer multiple of T . For instance, [ π3 ]2π = π3 + 2kπ : k ∈ Z . The set of real numbers modulo T is denoted by [R]T . There-

fore, [R]T = {[x]T : x ∈ R}. The set [R]T can be given the structure of a smooth manifold diffeomorphic to the unit circle S1 ⊂ C through the map [x]T 7→ exp(i(2π/T )[x]T ). The n-torus is denoted by Tn . It is diffeomorphic to [R]T × · · · × [R]T . {z } | n times

Given a nonempty set S ⊂ Rn , a point p ∈ Rn , and a vector norm k · k : Rn → R,

the point-to-set distance kpkS is defined as kpkS := inf{kp − xk : x ∈ S}. Given two subsets S1 and S2 of Rn , the Hausdorff distance of S1 to S2 , dH (S1 , S2 ), is defined as dH (S1 , S2 ) := max sup kp1 kS2 , sup kp2 kS1 . p1 ∈S1

p2 ∈S2

Given the set X an open subset or a smooth submanifold of Rn , we let k · k : X → R+

denote the restriction of the Euclidean norm on Rn to this set. Given a point x ∈ X , a set S ⊂ X , and a constant r > 0, we define the open sets Br (x) := {y ∈ X : ky − xk < r} and Br (S) := {y ∈ X : kykS < r}. We denote by cl(S) the closure of the set S, and by N (S) a generic open neighborhood of S. If y is a point in the image of f , we denote f −1 (y) := {x : f (x) = y}. Moreover, if f is real-valued, we denote by f −1 ([a, b]) the set {x : a ≤ f (x) ≤ b}. A function


24

f : Rn → Rm is said to be of class C k , or a C k function, if all the partial derivatives ∂ k f /∂xj1 · · · ∂xjk exist and are continuous, where each j1 , · · · , jk is an integer between 1 and n. Given a smooth C k function f , with k ≥ 1, we denote by ∂xk1 ···xk f the partial derivative ∂ k f /∂x1 · · · ∂xk , and by ∂xkk f the k-th partial derivative of f with respect to x. Given a real-valued function f of class C 2 , we denote by Hess(f ) its Hessian matrix, Hess(f ) ij = ∂x2i xj f . Given a function f : Rn → Rm and x ∈ Rn , dfx denotes the

Jacobian of f at x.

Given a smooth manifold Q, we denote by T Q its tangent bundle, T Q := {(p, vp ) : p ∈ Q, vp ∈ Tp Q}. If h : Q1 → Q2 is a smooth map between manifolds, and p ∈ Q1 , dhp : Tp Q1 → Th(p) Q2 denotes the differential of h at p, while dh : T Q1 → T Q2 denotes the global differential of h, defined as dh : (p, vp ) 7→ (h(p), dhp (vp )). If h : Q1 → Q2 is a diffeomorphism, then we say that Q1 , Q2 are diffeomorphic, and we write Q1 ≃ Q2 . In this case, the global differential dh : T Q1 → T Q2 is a diffeomorphism as well (see [48, Corollary 3.22]). Consider the control-affine system

x˙ = f (x) +

m X

gi (x)ui ,

i=1

y = h(x),

(2.1)

with state space X ⊂ Rn , set of output values Y ⊂ Rm and set of input values U ⊂ Rm . We assume that f and gi , 1 ≤ i ≤ m, are smooth vector fields on X , and that h : X → Y is a smooth mapping. Given either a smooth feedback u(x) or a piecewise-continuous open-loop control u(t) : R+ → U, we denote by φu (t, x0 ) the unique solution of (2.1) with initial condition x0 . By φ(t, x0 ) we denote the solution of the open-loop system x˙ = f (x) with initial condition x0 . Given an interval I of the real line and a set S ⊂ X , we denote by φu (I, S) the set φu (I, S) := {φu (t, x0 ) : t ∈ I, x0 ∈ S}. Similarly, we define the set φ(I, S) to be φ(I, S) := {φ(t, x0 ) : t ∈ I, x0 ∈ S}.


2.2

25

Virtual holonomic constraints: Definition and regularity

In this thesis we study mechanical systems whose configuration variables are either displacements in R (such is the case, e.g., for prismatic joints of the robots) or regular quantities in [R]T (such as angles of revolute joints). Accordingly, we define the configuration vector

q = q1 , . . . , qn ∈ Q, Q =

n Y

Θi ,

i=1

where Θi = R or Θi = [R]Ti , Ti > 0 (often, Ti is equal to 2π). The set Q is called the configuration manifold of the mechanical system. The state of the mechanical system is the pair (q, q) ˙ ∈ Q × Rn . The set Q × Rn is the so-called tangent bundle of Q, denoted T Q. We assume that the mechanical system under study has m control inputs (τ1 , . . . , τm ) ∈ Rm . We consider the class of Euler-Lagrange control systems modeled by the EulerLagrange equation

d ∂L ∂L − = B(q)τ, dt ∂ q˙ ∂q

(2.2)

where τ = col(τ1 , . . . , τm ) is the vector of control inputs, and the map B : Q → Rn×m is assumed to be smooth and of full rank for all q ∈ Q. Also, the Lagrangian L : T Q → R is assumed to be a smooth and of the special form

1 L(q, q) ˙ = q˙⊤ D(q)q˙ − P (q), 2

(2.3)

where D(q) = Dij (q) ∈ Rn×n , the generalized mass matrix, is symmetric and positive definite, and P (q), the potential energy function, is smooth for all q ∈ Q.


26

When the Lagrangian has the special form (2.3), the Euler-Lagrange control system (2.2) is referred to as a simple mechanical system. The Euler-Lagrange equations for simple mechanical systems can be rewritten in the standard form

D(q)¨ q + C(q, q) ˙ q˙ + ∇P (q) = B(q)τ.

(2.4)

In the above, the matrix C(q, q) ˙ = Ckj (q, q) ˙ ∈ Rn×n is formed in the following way [49,

50]

(∀k, j ∈ n) Ckj (q, q) ˙ =

n X

(Qi )jk (q)q˙i ,

(2.5)

i=1

where

(Qi )jk (q) =

o 1n ∂qi Dkj (q) + ∂qj Dki (q) − ∂qk Dij (q) , 2

(2.6)

are the so-called Christoffel symbols of the generalized mass matrix D. When m = n, system (2.2) is said to be fully actuated. When m < n, system (2.2) is said to have degree of underactuation n − m. In this case, we will assume that there exists a left-annihilator of B on Q, i.e., a smooth function B ⊥ : Q → R(n−m)×n such that B ⊥ (q)B(q) = 0 and rank B ⊥ (q) = n − m on Q.

Figure 2.1: A material particle in a central gravitational field.

27


Example 2.1 Consider a material particle on a plane with inertial coordinates q = [q1 q2 ]⊤ ∈ R2 and unit mass (see Figure 2.1). Assume the particle is subject to a planar gravitational central force with center at a = [a1 a2 ]⊤ ∈ R2 . Let the gravitational potential be given by P (q) = −1/kq − ak. Suppose a control force F = B(q)τ is exerted on the particle, where τ ∈ R is the control input. The particle model reads q¨ = −∇P (q) + B(q)τ, where

q1 − a1 q2 − a2 D(q) = diag([1 1]), C(q, q) ˙ = 0, ∇P (q) = kq − ak3 kq − ak3

⊤

.

This is an Euler-Lagrange control system of the form d ∂L ∂L − = B(q)τ, dt ∂ q˙ ∂q with L(q, q) ˙ = (1/2)kqk ˙ 2 −P (q). The material particle system has two degrees-of-freedom (DOF), one control input, and degree of underactuation one.

△

Definition 2.2.1. A virtual holonomic constraint (VHC) of order k ≤ m for system (2.4) is a relation h(q) = 0, where h : Q → Rk is smooth, rank dhq = k for all q ∈ h−1 (0), and the set

Γ = (q, q) ˙ ∈ T Q : h(q) = 0, dhq q˙ = 0 ,

(2.7)

is controlled invariant. That is, there exists a smooth feedback τ (q, q) ˙ defined on Γ such that Γ is positively invariant for the closed-loop system. The set Γ is called the constraint manifold associated with the VHC h(q) = 0. A VHC is stabilizable if there exists a smooth feedback τ (q, q) ˙ that asymptotically stabilizes Γ, and in this case τ (q, q) ˙ is said to enforce the VHC h(q) = 0.

△


28

The geometric implication of the condition rank(dhq ) h−1 (0) = k is that the zero level

set of h, on which the desired virtual relations hold, is an embedded submanifold of Q. At each q ∈ h−1 (0), the null space of the Jacobian matrix dhq is the tangent plane to h−1 (0), i.e., Tq h−1 (0) = v ∈ Rn : dhq v = 0 . In the light of this fact, the constraint

manifold Γ is the set of states (q, q) ˙ such that q satisfies the constraint h(q) = 0 and q˙ is tangent to the constraint surface h−1 (0). The control invariance of Γ in Definition 2.2.1 implies that if the control system is

initialized on Γ, then through the application of a suitable smooth feedback, the configuration trajectory q(t) can be made to satisfy the VHC h(q) = 0 for all t ≥ 0 (see Figure 2.2). Therefore, a properly initialized underactuated mechanical system (2.4) under the influence of a VHC will behave as if it had a physical constraint, h(q) = 0. Intuitively, for Γ to be controlled invariant there must exist a feedback that cancels out all accelerations of (2.4) transversal to h−1 (0).

Figure 2.2: Geometrical interpretation of VHC regularity.

Example 2.2 Consider the material particle system in Example 2.1. Suppose that B(q) = [1, 0]⊤ and a = [0, 0]⊤ . We would like to constrain the particle motion to the virtual line q2 = 1, so we consider the relation h(q) = q2 − 1. This is not a VHC because there does not exist any control input τ that would cancel out the acceleration due to gravity, which is transversal to the line q2 = 1. As a result, when q(0) = [0, 1]⊤ and


29

Figure 2.3: Constraining the motion of a material particle to lie on a virtual line.

q(0) ˙ = [0, 0]⊤ , the gravitational force will make the particle to leave the line. On the other hand, if B = [0, 1]⊤ , the applied control force acts along the q2 axis, then we can use the control input τ =

q2 kqk3

to make the set Γ = {(q1 , q2 , q˙1 , q˙2 ) : q2 = 1, q˙2 = 0}

invariant. We conclude that q2 = 1 is a VHC when B = [0, 1]⊤ (see Figure 2.3).

△

The next two examples represent the early ideas of VHC-based robotic locomotion control.

-

(a) Swinging motion of an acrobot under the influence of the VHC q1 + 12 q2 = 0.

(b) Brachiating motion of a gibbon.

Figure 2.4: A brachiating acrobot.

Example 2.3 Consider the acrobot, a double-pendulum with actuated elbow, in Figure 2.4a. In [8], the authors enforced, via feedback control, the VHC q1 + 12 q2 = 0, in order to induce pendulum-like dynamics imitating the brachiating motion observed in gibbons. Under the influence of this VHC, the acrobot is constrained to swing similar to a simple pendulum.

△

Example 2.4 Consider the planar three-link biped robot in Figure 2.5. This biped has


30

three DOF in swing phase, that is, the torso angle q1 , the swing leg angle q2 , and the stance leg angle q3 . There is actuation between each leg and the torso, and no actuation between the leg ends and ground. The first control law that analytically proved the stability of underactuated biped walking motion as an asymptotically stable periodic orbit was provided by Grizzle et al. in [4] considering the dynamics of this biped robot. It was shown that enforcing a VHC of the form   q1 − θd   = 0,  q2 + q3

on the biped configuration induces asymptotically stable walking. Intuitively, under the influence of the above VHC, the torso is maintained at a constant inclination angle, with constant angle θd , and the swing leg “mirrors” the stance leg behavior. Accordingly, the biped configuration under the VHC is entirely determined by the stance leg angle.

△

Figure 2.5: A planar three-link biped robot.

The focus of this thesis is on VHCs that enjoy the key property of regularity. Definition 2.2.2. A relation h(q) = 0 is a regular VHC if the output function e = h(q) yields vector relative degree {2, · · · , 2} everywhere on the set Γ = {(q, q) ˙ : h(q) = 0, dhq q˙ = 0}.

△

This definition implies that system (2.4) with output e = h(q) is input-output feedback linearizable, and the associated zero dynamics manifold is precisely Γ (see [51]).

31


Thus, Γ is controlled invariant, implying that h(q) = 0 is a VHC as per Definition 2.2.1. Under mild hypotheses, regular VHCs are stabilizable as stated in the following proposition. Proposition 2.2.3 (Lemma 2.2.4 in [28]). Let h(q) = 0 be a regular VHC of order k for system (2.4). If there exist strictly increasing functions α, β : [0, r) → [0, +∞), with r > 0, such that the map H : (q, q) ˙ 7→ col h(q), dhq q˙ is bounded as α(k(q, q)k ˙ Γ) ≤

kH(q, q)k ˙ ≤ β(k(q, q)k ˙ Γ ), then the input-output linearizing feedback

n o τ (q, q) ˙ = A† (q) dhq D −1 (q) C(q, q) ˙ q˙ + ∇P (q) − H(q, q) ˙ − k1 e − k2 e˙ ,

(2.8)

enforces the VHC, where A(q) = dhq D −1 (q)B(q), e = h(q), e˙ = dhq q, ˙ k1 , k2 > 0, and   q˙ Hess(h1 ) q q˙    , H(q, q) ˙ = · · ·     q˙⊤ Hess(hn−1 ) q q˙ 

⊤

provided that the closed-loop system does not have finite escape times. The following proposition provides necessary and sufficient conditions for a given relation h(q) = 0 to be a regular VHC. Proposition 2.2.4 (Proposition 2.2 in [41]). Let h : Q → Rk be smooth, with k ≤ m, and that rank dhq = k for all q ∈ h−1 (0). Then, h(q) = 0 is a regular VHC of order k if and only if one of the following equivalent conditions holds: (i) For all q ∈ h−1 (0), rank dhq D −1 (q)B(q) = k.

h i (ii) For all q ∈ h−1 (0), dim Tq h−1 (0) ∩ Im D −1 (q)B(q) = m − k.

(iii) For all q ∈ h−1 (0), if (W, ψ) is a coordinate chart of h−1 (0), then


for all q ∈ W .

32

rank B ⊥ (q)D(q)(dψ −1 )ψ(q) = n − m,

Remark 2.2.5. Condition (i) in Proposition 2.2.4 is simply the requirement that the output function e = h(q) yields a vector relative degree {2, · · · , 2}. Condition (ii) can be derived from (i) by means of straight forward manipulations, and it states that at each q ∈ h−1 (0), k of the m acceleration directions imparted by the control inputs are transversal to the tangent space Tq h−1 (0) (see Figure 2.6). Condition (iii) is yet another equivalent restatement of the relative degree requirement which is used to derive Corollary 2.3.3.

Figure 2.6: Regularity of VHCs and transversality.

Example 2.5 Consider the material particle system of Example 2.1 with B(q) = [b1 (q) b2 (q)]⊤ and a = [0 0]⊤ . According to condition (i) of Proposition 2.2.4, the relation h(q) = 0, where h(q) = q2 − 1, is a regular VHC if and only if rank dhq D −1 (q)B(q) = 1.

Since the mass matrix D(q) is the identity matrix and dhq = [0 1], the constraint q2 −1 = 0

is a regular VHC if and only if b2 (q) 6= 0 for all the points belonging to the line q2 = 1. The physical interpretation is that the motion of the material particle in the central gravitational field can be constrained to the line q2 = 1 if and only if the applied control force has a component transversal to this line.

△


33

Example 2.6 Consider the pendubot system of Example 1.1 and the constraint h(q) = 0, where h(q) = q2 − 2q1 . We let 



q1  ˆ 1 ) :=  φ(q  . 2q1

Condition (iii) in Proposition 2.2.4 states that h(q) = 0 is a regular VHC, if and only if

ˆ 1 ))D(φ(q ˆ 1 ))dφˆq 6= 0, B ⊥ (φ(q 1 ˆ 1 ))D(φ(q ˆ 1 ))dφˆq = for all q ∈ {q : q2 = 2q1 }. Letting B ⊥ = [0 1], we have B ⊥ (φ(q 1 2 + cos(q1 ) 6= 0 for all q1 ∈ [R]2π , and therefore the parametric relation q2 − 2q1 = 0 is a regular VHC.

2.3

△

VHCs for mechanical systems with degree of underactuation one

In this section we focus on VHCs for underactuated mechanical systems with degree of underactuation one. Examples for this class of systems include the pendubot [43], the acrobot [8], Getz’s bicycle model [52], and some planar biped robots in their swing phase, such as RABBIT with 7 degrees-of-freedom [3]. First, we investigate the underlying dynamics that govern the motion of the mechanical system (2.4) under the influence of a given regular VHC. In particular, we derive the dynamics of the mechanical system when the regular VHC h(q) = 0 is made invariant for the mechanical system configuration, that is, when the state (q, q) ˙ is constrained to evolve on the constraint manifold Γ. The dynamics of the mechanical system on Γ will be referred to as the reduced dynamics. Then, we present a procedure for generating regular parametric VHCs with odd symmetry for these systems.


2.3.1

34

Global reduced dynamics

The highest order of VHC that can be made invariant for a mechanical system with degree of underactuation one is k = n − 1. By the preimage theorem [53], if h(q) = 0 is a VHC of order n − 1, then the set h−1 (0) is a one-dimensional embedded submanifold of Q. In other words, h−1 (0) is a regular curve without self-intersections. As such, h−1 (0) is diffeomorphic to either the real line or the unit circle S1 . We call each connected component of h−1 (0) a VHC curve. Definition 2.3.1. Let h(q) = 0 be a VHC of order n−1 for the underactuated mechanical system (2.4). Let σ : Θ → Q be any regular parametrization of the VHC curve h−1 (0), where Θ = R if h−1 (0) ≃ R, while Θ = [R]T , T > 0, if h−1 (0) ≃ S1 . Then, a parametric representation of the VHC h(q) = 0 or, simply, a parametric VHC is a regular embedded curve

q = σ(θ), such that Im(σ) = h−1 (0), and θ is the parameterizing variable of the VHC curve h−1 (0). △ As mentioned in Chapter 1, in Chapters 3–5 we develop a theory for stabilizing repetitive behaviors in underactuated mechanical systems. Therefore, we focus on VHCs whose zero level sets are closed curves with Θ = [R]T , although most of the developed theory can also be applied to Θ = R. As we mentioned in Section 2.1, the set [R]T is diffeomorphic to S1 via the map [x]T 7→ exp(j(2π/T )[x]T ), so in principle we could replace [R]T by S1 . However, doing so would require a reparameterization of the VHC curve, which in practice would be undesirable. Therefore, we consider a general period T > 0 and use the set [R]T . Remark 2.3.2. The family of VHCs studied in [42, 40] are parametric VHCs that are parameterized by one of the configuration variables of the mechanical system and have,

35


⊤ by a suitable permutation of variables, the form q = φ(qn ), qn , where φ(·) is a C 2

function. The VHCs for the brachiating acrobot in Example 2.3 and the three-link biped

robot in Example 2.4 have this form. Condition (iii) of Proposition 2.2.4 can be employed to deduce the following necessary and sufficient condition for regularity of parametric VHCs of the form q = σ(θ). Corollary 2.3.3. The parametric VHC q = σ(θ) is a regular VHC of order n − 1 for the underactuated mechanical system (2.4) if and only if ∀θ ∈ [R]T

B ⊥ σ(θ) D σ(θ) σ ′ (θ) 6= 0.

It is always possible to find a global description of the reduced dynamics induced by a VHC of order n − 1. In order to find the reduced dynamics, we follow the procedure presented in [41]. We first pick a regular parametrization, e.g., σ, of the curve h−1 (0). Next, multiplying (2.4) on the left by B ⊥ (q) we obtain B ⊥ D q¨ + B ⊥ (C q˙ + ∇P ) = 0. The dynamics on Γ are found by restricting the equation above on Γ. To this end, we ˙ and q¨ = σ ′ (θ)θ¨ + σ ′′ (θ)θ˙2 . By so doing and dividing the above let q = σ(θ), q˙ = σ ′ (θ)θ, equation by B ⊥ Dσ ′ , which is bounded away from zero by Corollary 2.3.3, we arrive at the following proposition. Proposition 2.3.4 (Proposition 4.1 in [41]). Let h(q) = 0 be a regular VHC of order n − 1 for system (2.2). Assume that there are m = n − 1 control inputs and that h−1 (0) is a connected set. Let σ : Θ → Q, with Θ either R or [R]T , be a regular parametrization of h−1 (0). Then, the reduced dynamics on the set Γ in (2.7) are globally described by

θ¨ = Ψ1 (θ) + Ψ2 (θ)θ˙ 2 , where

(2.9)


Ψ1 (θ)

Ψ2 (θ)

B ⊥ ∇P = − ⊥ ′ , B Dσ q=σ(θ) n P T B ⊥ Dσ ′′ + Bi⊥ σ ′ Qi σ ′ i=1 =− ⊥ ′ B Dσ

36

,

q=σ(θ)

and where Bi⊥ is the i-th component of B ⊥ and (Qi )jk are the Christoffel symbols of the generalized mass matrix D given in (2.6). Example 2.7 Consider the material particle system of Example 2.1 with the center of gravitational field located at a = [a1 a2 ]⊤ , the radial control input vector B(q) = q, and B ⊥ (q) = B(q)⊤ J, where 



 0 1 J =  −1 0

Suppose that we would like to constrain the motion of the particle to a circle centered at b = [b1 b2 ]⊤ , kbk < 1, with radius 1. We consider the function h(q) = kq − bk2 − 1 and the parameterization of h−1 (0)   cos θ σ(θ) = b +  . sin θ

For each θ ∈ [R]2π , the vector Jσ ′ (θ) is orthogonal to the circle h−1 (0) at σ(θ), so it is proportional to (q − b)|q=σ(θ) . Since, on h−1 (0), the vectors B(q) and q − b are never orthogonal, we have that B ⊥ Dσ ′ = B ⊤ Jσ ′ 6= 0. By Corollary 2.3.3, the constraint h(q) = 0 is a regular VHC. Therefore, due to Proposition 2.3.4, this VHC induces the

Chapter 2. Virtual Holonomic Constraints Preliminaries following well-defined reduced dynamics ⊤ ′′ ⊤ B Jσ B J ∇P − ⊤ ′ θ¨ = − ⊤ ′ B Jσ B Jσ q=σ(θ)

37

θ˙2

q=σ(θ)

(b2 + sin(θ))(b1 − a1 + cos(θ)) − (b1 + cos(θ))(b2 − a2 + sin(θ)) = kσ(θ) − ak3/2 1 + b1 cos(θ) + b2 sin(θ) b2 cos(θ) − b1 sin(θ) ˙2 θ . (2.10) − 1 + b1 cos(θ) + b2 sin(θ) △

2.3.2

Generation of regular and odd parametric VHCs

In this section we present a procedure to generate regular parametric VHCs with odd symmetry. The material presented in this section is a basic modification of the VHC ⊤ generation procedure in [42] for parametric VHCs of the form q = φ(qn ), qn , where

φ(·) is a C 2 function. We start with an example to illustrate the procedure.

Example 2.8 Consider the pendubot system in Example 1.1. We would like to generate ⊤ a regular and odd parametric VHC of the form σ(θ) = σ1 (θ), σ2 (θ) such that when

the VHC is enforced, the second link cannot perform complete revolutions, and hence it does not fall over. A condition that guarantees the VHC regularity, according to Corollary 2.3.3, is B ⊥ (σ(θ))D(σ(θ))σ ′ (θ) 6= 0 for all θ ∈ [R]T . Thus, we consider B ⊥ (σ(θ))D(σ(θ))σ ′ (θ) = δ(θ), where δ(·) is some C 1 function which is bounded away from zero and needs to be designed. We have B ⊥ (q)D(q) = [cos(q1 − q2 ) 1], and the regularity condition becomes dσ1 dσ2 + cos σ1 (θ) − σ2 (θ) = δ(θ). dθ dθ

In order to find a regular VHC on which the second link does not fall over, we first pick σ1 (·). In this example, we choose σ1 (θ) = θ. After choosing σ1 (·), the above relation becomes an ODE in terms of the unknown function σ2 (·). We can choose the function

38


δ(·) arbitrarily as long as it is bounded away from zero. In this example, we choose δ(θ) = ǫ to be a constant function. Therefore, the regularity condition reads dσ2 + cos θ − σ2 (θ) = ǫ. dθ

We call the above ordinary differential equation the virtual constraint generator (VCG). Next, we need to determine ǫ so that σ2 is a 2π-periodic solution of the VCG above. The necessary and sufficient conditions for existence of periodic solutions to VCGs are given √ in Proposition 2.3.5. In this example, it can be shown that choosing ǫ = 1 − 2 gives rise √ to a periodic solution. In particular, we have σ2 (θ) = θ + 2 arctan[tan(−θ/2)(1 + 2)] which is a 2π-periodic function.1 This VCG solution is also odd. The conditions in △

Proposition 2.3.9 guarantee that the VCG solutions are odd functions. We investigate the following problem in this section.

Regular and Odd Parametric VHC Generation. Find a C 2 and odd function σ : [R]T → Q such that (∀θ ∈ [R]T ) B ⊥ σ(θ) D σ(θ) σ ′ (θ) = δ(θ),

(2.11)

for some C 1 function δ : [R]T → R\{0} that is bounded away from zero. The reason that we study the above problem is that the relations of the form q = σ(θ) that satisfy Condition (2.11) are guaranteed to be regular VHCs, due to Corollary 2.3.3. In the next chapter it is shown that regular and odd VHCs induce reduced dynamics with Lagrangian structures that give rise to a plethora of repetitive behaviors in underactuated mechanical systems. In order to solve this problem, we begin by arbitrarily selecting n−1 of the required n functions, σi2 (θ), . . . , σin (θ). Then, we will find the remaining function σi1 (θ) by solving a suitable differential equation. By permuting the configuration variable indices and 1

This function has 2π jumps θ = kπ, but its value modulo 2π is in fact smooth and 2π-periodic.


39

without loss of generality, we consider the case when the last n − 1 components of σ are chosen arbitrarily. After choosing σ2 , · · · , σn , the row vector B ⊥ (σ(θ))D(σ(θ)) depends on the function σ1 and on θ. To highlight these dependencies, we denote [b1 (σ1 , θ), · · · , bn (σ1 , θ)] := B ⊥ (q)D(q) q=(σ1 ,σ2 (θ),...,σn (θ),θ) . We may rewrite (2.11) as n−1 X dσ1 b1 (σ1 , θ) bk (σ1 , θ)σk′ (θ) + δ(σ1 , θ), =− dθ k=2

(2.12)

where we allow δ to depend on σ1 for greater flexibility. If we think of θ in (2.12) as “time,” then (2.12) is a scalar ODE which is T -periodic because θ ∈ [R]T . The fact that the “time variable” θ belongs to [R]T imposes the constraint that only T -periodic solutions of (2.12) are acceptable. Returning to the differential equation (2.12), the problem is to find δ(σ1 , θ) bounded away from zero (this is the regularity condition) so that (2.12) has a T -periodic solution σ1 (θ). Once this is done, σ1 (θ), . . . , σn (θ) will give rise to a parametric description of a regular VHC q = σ(θ). The ODE (2.12) is called a virtual constraint generator (VCG) [42]. We remark that when n > 2, VCGs are not unique because they depend on the initial choice of functions σ2 , . . . , σn−1 . In seeking a T -periodic solution σ1 (θ) of (2.12), we assume that b1 (σ1 , θ) is bounded away from zero and therefore the VCG (2.12) is a nonsingular T -periodic differential equation,

# " n−1 X 1 dσ1 bk (σk , θ)σk′ (θ) + δ(σ1 , θ) . − = dθ b1 (σ1 , θ) k=2

(2.13)

The state space of (2.13) is D = Θ1 , where Θ1 = R or Θ1 = [R]T1 , and any T -periodic solution of (2.13) is a map σ1 : [R]T → Θ1 . If Θ1 = [R]T1 , the degree of σ1 is the integer representing the number of revolutions of the angle σ1 (θ) around the axis q1 per each revolution of θ. More precisely, if q1 ∈ [R]T1 , then we say that the first component of σ, σ1 (θ), has degree d ∈ Z if

40


lim σ1 (θ) = σ1 (0) + d · T1 .

θ→T −

In other words, the integer d is the number of revolutions that the variable q1 performs while q moves once around the VHC curve Im(σ). If Θ1 = R, then we will set d = 0.

Figure 2.7: Solutions to the pendubot VCG with different degrees.

Example 2.9 Consider the VCG for the pendubot system in Example 2.8 dσ2 + cos θ − σ2 (θ) = ǫ. dθ

The VCG solutions with degrees 0, 1, -1 are σ2 (θ) = θ + 2 arctan[tan(−θ/2)(1 + √ θ + 2 arctan[tan(θ/2)( 2 − 1)], respectively, and depicted in Figure 2.7.

√

2)], θ, △

The next proposition gives necessary and sufficient conditions for the existence of a function δ(σ1 , θ) such that (2.13) has a T -periodic solution σ1 (θ) with a desired degree. Proposition 2.3.5 (Lemma 3.1 in [42]). Consider equation (2.13), suppose that b1 (σ1 , θ) is bounded away from zero, and that for each (q2 , . . . , qn−1 , θ) the function

q1 7→ B ⊥ (q)D(q)|q=(q1,...,qn ,θ) , is bounded. Fix an initial condition σ1 (θ0 ) = σ0 and a desired degree d. Then, there exists a C 1 function δ : [R]T1 × [R]T → R\{0} such that the solution σ1 (θ) of (2.13) is


41

T -periodic with degree d if, and only if, the solution σ ¯1 (θ) of (2.13) with the same initial condition and with δ = 0 satisfies σ ¯1 (θ0 + T ) − σ ¯1 (θ0 ) 6= d · T1 . Remark 2.3.6. The assumption that q1 7→ B ⊥ D is bounded for all (q2 , . . . , qn , θ) is very mild, for B ⊥ can always be chosen to have unit norm, and the inertia matrix is typically bounded. Once a T -periodic function σ1 (θ) is found according to Proposition 2.3.5, the relation q = (σ1 (θ), . . . , σn (θ)) is a regular VHC. Under the following symmetry assumption, the VCG (2.13) can be used to generate VHCs with odd symmetry. Assumption 2.3.7. For some q ⋆ ∈ Q, it holds that D(q), P (q), and B(q) in (2.4) are even with respect to q ⋆ , i.e., for all q ∈ Q, D(q ⋆ + q) = D(q ⋆ − q), P (q ⋆ + q) = P (q ⋆ − q), B(q ⋆ + q) = B(q ⋆ − q). Remark 2.3.8.

(i) Henceforth, for notational simplicity we will assume that q ⋆ = 0

so that D(q) = D(−q), P (q) = P (−q), and B(q) = B(−q). There is no loss of ˜ ˜ generality in this assumption, for by defining D(q) := D(q + q ⋆ ), one gets D(q) = ˜ D(−q). The same observation holds for P (·) and B(·). (ii) The pendubot in Example 1.1, the Furuta pendulum in [54], and the 5-DOF swing phase model of a biped robot in [33] (when the center of mass of the torso is on-axis) are among Euler-Lagrange systems that satisfy Assumption 2.3.7. Proposition 2.3.9 (Lemma 3.3 in [42]). Suppose that Assumption 2.3.7 holds and that σ2 (θ), · · · , σn−1 (θ) are chosen to be odd and T -periodic. Consider the VCG (2.12) and fix the initial condition σ1 (0) = 0. Then, the function δ : D × [R]T → R\{0} in Proposition 2.3.5 can be chosen such that the solution σ1 (θ) of (2.13) is odd.


42

The following procedure, a direct result of Propositions 2.3.5 and 2.3.9, can be used to generate regular and odd parametric VHCs for the underactuated mechanical system (2.4).

Procedure 2.1 (Generation of regular and odd parametric VHCs [42]) (i) Assume that the first coefficient of the row vector B ⊥ (q)D(q) is never zero, so that the VCG (2.13) is nonsingular. (ii) Choose arbitrary C 2 functions σ2 (θ), . . . , σn (θ) that are T -periodic. (iii) Write the scalar T -periodic differential equation (2.13). (iv) Fix the initial condition σ1 (0) = 0 and a desired degree d for the function σ1 (θ) (set d = 0 if q1 ∈ R). ¯1 (T ) 6= d · T1 . If it (v) Check whether the solution σ ¯1 (θ) of (2.13) with δ = 0 satisfies σ does not, change the functions σ2 (θ), . . . , σn (θ) and return to (iii). (vi) Set δ(σ1 , θ) = ǫµ(σ1 , θ), where µ(σ1 , θ) is any C 2 function which is bounded away from zero, T -periodic, and even with respect to θ. Find the unique ǫ ∈ R such that the solution σ1 (θ) of (2.13) with initial condition σ1 (0) = 0 satisfies σ1 (T ) = d · T1 . The corresponding solution σ1 (θ) will be T -periodic with degree d. (vii) The VHC q1 = σ1 (θ), . . . , qn = σn (θ), so obtained is regular and odd. Remark 2.3.10. In step (vi) the unique value of ǫ for which the solution σ1 (θ) of (2.13) is T -periodic with degree d can be easily found using numerical integration over the interval [0, T ] and a one-dimensional search. Example 2.10 Consider the pendubot VCG in Example 2.8. We would like to generate ⊤ regular and odd parametric VHCs of the form σ(θ) = σ1 (θ), σ2 (θ) such that when

the VHCs are enforced, the second link cannot perform complete revolutions, and hence


43

it does not fall over. According to Procedure 2.1, we pick σ1 and determine the function σ2 with degree d = 0. We choose

α0 =

2π , 9

σ1 (θ) = θ,

(2.14a)

σ1 (θ) = θ + α0 sin(θ),

(2.14b)

respectively.

We have B ⊥ (q)D(q) = [cos(q1 − q2 ) 1], and the virtual constraint generator (2.12) is the nonsingular differential equation, dσ2 = − cos σ1 (θ) − σ2 (θ) + δ(σ2 , θ). dθ

(2.15)

When δ = 0, the solution with initial condition σ2 (0) = 0 is σ ¯2 (θ) = θ − 2 arctan θ. Since σ ¯2 (2π) 6∈ {2πd : d ∈ Z}, we can find regular VHCs with any desired degree d. We choose µ(σ2 , θ) = 1 and µ(σ2 , θ) = 1 + a0 cos(θ)2 , a0 = 3.5, for the choices of σ1 in (2.14a) and (2.14b), respectively. We set δ = ǫµ(σ2 , θ), where µ(σ2 , θ) = 1. When d = 0, the unique values of ǫ, found using fminsearch in MATLAB, for which the solutions of (2.15) with initial condition σ2 (0) = 0 satisfy σ2 (2π) = 0 are √ √ ǫ = 1− 2 and ǫ = −0.1242, respectively. The VCG solution corresponding to ǫ = 1− 2 √ is σ2 (θ) = θ + 2 arctan[tan(−θ/2)(1 + 2)] while the VCG solution corresponding to ǫ = −0.1242 is a function σ2⋆ (θ) that does not have a closed form expression. The pendubot configurations under these VHCs are depicted in Figure 2.8. △

44


2

2

1.5

1.5

1

1

0.5

0.5

0

0

−0.5

−0.5

−1 −1.5

−1 −1

−0.5

0

0.5

1

1.5

−1.5

−1

−0.5

0

0.5

1

1.5

(a) The VHC generated by the choice

(b) The VHC generated by the choice

σ1 (θ) = θ in VCG (2.15).

σ1 (θ) = θ + α0 sin(θ) in VCG (2.15).

Figure 2.8: VHC-induced pendubot configurations.

Chapter 3 The Lagrangian Structure of Reduced Dynamics In Chapter 2 we presented the reduced dynamics that are induced by regular VHCs of order n − 1 for the underactuated mechanical system

D(q)¨ q + C(q, q) ˙ q˙ + ∇P (q) = B(q)τ,

(3.1)

with n DOF and n − 1 control inputs. Following the VHC-based control paradigm discussed in Section 1.2, in this chapter we investigate the existence of Lagrangian structures for the reduced dynamics. We present necessary and sufficient conditions under which the reduced dynamics are those of a mechanical system with one DOF and, more generally, under which they have a Lagrangian structure. In both cases, we show that typical solutions satisfying the virtual constraints lie in a restricted class which we completely characterize. A large portion of this chapter has appeared in [55, 56]. The chapter is organized as follows. We start our exposition with a brief review of the inverse Lagrangian problem (ILP) and its relevance in VHC-based control paradigm in Section 3.1. Next, in Section 3.2 we present an introductory example in which we show that arbitrarily small variations of the system parameters have drastic effects on 45

Chapter 3. The Lagrangian Structure of Reduced Dynamics

46

the Lagrangian properties of the reduced dynamics. In Section 3.3 we formulate and solve the main problem investigated in this chapter. Sections 3.4 and 3.5 present the proofs of the chapter main results. In Section 3.6 we investigate the qualitative properties of solutions of the reduced dynamics. Finally, in Section 3.7 we present some further examples.

3.1

Introduction

In classical mechanics, a Lagrangian system subject to an ideal holonomic constraint (one with the property that the constraint forces do not make work on virtual displacements), gives rise to Lagrangian reduced dynamics whose Lagrangian function is the restriction of the unconstrained Lagrangian to the constraint manifold. It is natural to ask whether an analogous property holds for Lagrangian control systems subject to virtual holonomic constraints. This chapter investigates this problem and solves it completely for the specific setup described below.

3.1.1

Summary of chapter main contributions

We consider Lagrangian control systems with n DOF and n − 1 controls. We assume that a regular VHC, h(q) = 0, of order n − 1 has been enforced via feedback control, and we investigate the resulting reduced dynamics. According to Proposition 2.3.4, these are given by a second-order unforced differential equation of the form θ¨ = Ψ1 (θ) + Ψ2 (θ)θ˙ 2 ,

(3.2)

˙ ∈ R × R or (θ, θ) ˙ ∈ S1 × R. where either (θ, θ) This chapter presents three main results. In Theorem 3.3.3, it is shown that when the state space of the reduced dynamics is R×R, the reduced dynamics always admit a global mechanical structure, i.e., equation (3.2) results from the Euler-Lagrange equation with


47

˙ = (1/2)M(θ)θ˙ 2 − V (θ), with M > 0. When the a Lagrangian function of the form L(θ, θ) state space of the reduced dynamics is the cylinder S1 ×R, a Lagrangian structure may not exist. In Theorem 3.3.5 we give explicit necessary and sufficient conditions guaranteeing that the reduced dynamics have a global mechanical structure. In Theorem 3.3.9 we go one step further, and give necessary and sufficient conditions under which the reduced dynamics possess any global Lagrangian structure, possibly not in mechanical form. A byproduct of Theorems 3.3.5 and 3.3.9 is that when the state space of (3.2) is S1 × R, generically there does not exist a global Lagrangian structure. In addition to these results, in Section 3.6 we characterize the qualitative properties of trajectories of the reduced dynamics.

3.1.2

Related work

The results presented in this chapter complement work in [41, 52], in which examples were given showing that the reduced dynamics may possess stable limit cycles, therefore ruling out the existence of a Lagrangian structure. In [41] sufficient conditions were provided guaranteeing the existence of a global mechanical structure, but their necessity was not investigated and more general Lagrangian structures were not considered. The inverse problem of calculus of variations (IPCV) is concerned with finding conditions under which a system of differential equations can be derived from a variational principle. Comprehensive historical surveys regarding this problem can be found in [57, 58, 59]. We will now give an account of some of the key findings in this field. In Section 3.3 (see Remark 3.3.12) we will comment on the fact that the results of this chapter are not contained in the existing literature. A special case of IPCV, namely, the inverse problem of Lagrangian mechanics (IPLM), can be traced back to the seminal work of Sonin in 1886 [60] and Helmholtz in 1887 [61]. The problem investigated in this chapter fits within the IPLM framework. Helmholtz found necessary conditions (today referred to as the “Helmholtz conditions,” [57]) under


48

which a given system of second-order ordinary differential equations is equivalent to a set of Euler-Lagrange equations derived from some Lagrangian function. In 1896, Mayer [62] showed that the Helmholtz conditions are sufficient as well for the local existence of a Lagrangian. The Helmholtz conditions are a mixed set of partial differential equations and algebraic equations in terms of a set of unknown functions. It is noteworthy that if these equations can be solved for a given system of second-order ODEs, the corresponding Lagrangian is given by the Tonti-Vainberg integral formula [63, 64]. Unfortunately, solving the equations is a nontrivial task. Indeed, the Helmholtz conditions, as shown by Henneaux [65], are in general strong and over-determined in the sense that if these conditions admit a solution, it will be generally unique. For the case of one DOF systems (i.e., given by one second-order ODE), Darboux [66] solved the IPLM in 1894, showing that such systems are always locally Lagrangian. In 1941, Douglas [67] could solve the IPLM for the case of two DOF. There was a revival of interest in the IPLM around the 1980’s thanks in part to the monograph by Santilli [57]. Using the tools of differential geometry and global analysis, researchers started to encode the Helmholtz conditions in geometric framework [57, 63, 68, 69, 70, 71, 72, 73]. The paper by Saunders [74] reviews the contributions to IPCV since 1979 to date.

3.1.3

Relevance of ILP in VHC-based control paradigm

Solving the ILP for the reduced dynamics is a crucial building block for later development of control laws for the class of mechanical systems that are studied in the next two chapters of this thesis. Indeed, if the constrained system is Lagrangian, then as we show in this chapter the generic trajectories of the mechanical system under the influence of VHCs is a trichotomy of oscillations, rotations, and helices (defined in Section 3.6.2). Based on the desired repetitive behavior that the mechanical system should perform, the designer can then choose from a plethora of closed orbits resulting from the Lagrangian structure. In the absence of a Lagrangian structure, closed orbits may not longer be a


49

generic feature of the constrained dynamics, making it hard or even impossible to enforce a repetitive behavior on the mechanical system.

3.2

Introductory example

Consider the material particle system in Example 2.1 with center of gravitational field at a = [a1 a2 ]⊤ . Pick b ∈ R2 such that kbk < 1, and consider the problem of constraining the motion of the particle on a unit circle centered at b, which corresponds to enforcing the VHC h(q) = kq − bk − 1 = 0 via feedback. As it is shown in Example 2.7, when kbk < 1, this VHC is regular. Using the parameterization   cos θ σ(θ) = b +  , sin θ

of h−1 (0), the reduced dynamics of our particle model (see Example 2.2) subject to the VHC h(q) = 0, read ⊤ B J ∇P θ¨ = − ⊤ ′ B Jσ

q=σ(θ)

B ⊤ Jσ ′′ − ⊤ ′ B Jσ

θ˙ 2 .

(3.3)

q=σ(θ)

In this chapter we investigate conditions under which the reduced dynamics possess a ˙ → R such that the Lagrangian structure, i.e., there exists a function L : Γ ∋ (θ, θ) reduced dynamics satisfy the Euler-Lagrange equation d ∂L ∂L − = 0. dt ∂ θ˙ ∂θ The second-order differential equation (3.3) describes the reduced dynamics on Γ. Its ˙ ∈ [R]2π × R}, which is diffeomorphic to Γ through state space is the cylinder C = {(θ, θ) ˙ 7→ (σ(θ), σ ′ (θ)θ). ˙ The results of this chapter will the diffeomorphism T : C → Γ, (θ, θ) show that small variations of the parameters a, b, and of the direction of the vector B(q), have major effects on the Lagrangian structure of the reduced dynamics, to the point

50


β

a, b, 0

(a)

b a, 0

(b)

a 0

b

a, b, 0

(c)

(d)

Figure 3.1: A material particle immersed in a gravitational field. The material particle is constrained via feedback control to lie on a unit circle. The figure depicts four situations corresponding to different values of the vectors a and b representing the center of the gravitational field and the center of the circle. Black arrows display the direction of the control force, while red arrows represent the gravitational force. In part (a), the control force is orthogonal to the VHC, and the VHC is equivalent to an ideal holonomic constraint. The reduced dynamics are Lagrangian and mechanical. In part (b), the control force is not orthogonal to the VHC, and the VHC is no longer equivalent to an ideal holonomic constraint. Yet, the reduced dynamics are still Lagrangian and mechanical. In part (c), the reduced dynamics are Lagrangian but not mechanical. In part (d), the control force imparts an acceleration on the particle as it moves along the circle, and the reduced dynamics are neither Lagrangian nor mechanical.


51

that the reduced dynamics may not admit a Lagrangian structure at all. In particular, we distinguish four cases. Case 1: a = b = 0. The gravity force and the control force are parallel to each other, and they are both orthogonal to the circle h−1 (0). See Figure 3.1(a). The gravity force is compensated by the control force, and it does not affect the reduced dynamics. Moreover, the work of the control force F on virtual displacements ξ ∈ Tq h−1 (0) is identically zero. Thus, the VHC h(q) = 0 is analogous to a holonomic constraint satisfying the Lagrange-d’Alembert principle of classical mechanics (see [75]). In mechanics, such holonomic constraint is said to be ideal. In this setting, we expect the reduced dynamics to be Lagrangian and, indeed, the reduced motion (3.3) is θ¨ = 0, which is a ˙ = (1/2)θ˙ 2 . Modulo a Lagrangian mechanical system with Lagrangian function L(θ, θ) constant, this function can be obtained by restricting the original Lagrangian L on Γ, ˙ = L(q, q) i.e., L(θ, θ) ˙ q=σ(θ),q=σ + c. This is precisely what happens in mechanics with ˙ ′ (θ)θ˙

ideal holonomic constraints.

Case 2: a = 0, b 6= 0. The gravity force is parallel to the control force, but the control force is no longer orthogonal to the circle h−1 (0). See Figure 3.1(b). Now the work of the control force on virtual displacements ξ ∈ Tq h−1 (0) is not zero, so one can no longer draw an analogy between the VHC h(q) = 0 and an ideal holonomic constraint. Nonetheless, the results of this chapter will show that the reduced dynamics ˙ = (1/2)M(θ)θ˙2 , are a Lagrangian mechanical system with Lagrangian function L(θ, θ) for a suitable smooth function M : [R]2π → R. Since the control force makes work on ˙ = L(q, q) virtual displacements, it is no longer true that L(θ, θ) ˙ q=σ(θ),q=σ + c. ˙ ′ (θ)θ˙ Case 3: a, b 6= 0. Now the gravity force is no longer parallel to the control force,

and the control force is not orthogonal to the circle h−1 (0). See Figure 3.1(c). In this case, the gravity force affects the reduced dynamics, and the work of the control force on virtual displacements ξ ∈ Tq h−1 (0) is not zero. We will see that for certain values of a, b, the reduced dynamics are Lagrangian, but not mechanical. In other words, the


52

Lagrangian function of the reduced dynamics cannot be written in the form kinetic minus potential energy. We will also see that the qualitative properties of the reduced motion are drastically different than in cases 1 and 2. Case 4: a = b = 0, B(q) = Rβ q, where Rβ is a counter-clockwise planar rotation by angle β ∈ (−π/2, π/2), β 6= 0. See Figure 3.1(d). In this case, the gravity force is orthogonal to the circle h−1 (0) and it does not affect the reduced dynamics, while the control force has a constant angle β to the normal vector to the circle. We shall show that the reduced dynamics are not Lagrangian. The example of a material particle on a plane illustrates that the reduced dynamics induced by VHCs can exhibit very different properties than the dynamics of a mechanical system subject to a holonomic constraint. A number of questions arise in this context: Q1 When are the reduced dynamics Lagrangian and mechanical (i.e., such that the Lagrangian has the form L = T − V )? Q2 When are the reduced dynamics Lagrangian but not mechanical? Q3 Can one expect a Lagrangian structure to exist generically for the reduced dynamics, or rather, is it an exceptional property? Q4 When a Lagrangian structure exists, what qualitative properties can one expect for the reduced dynamics? This chapter will provide answers to these questions. We will return to the particle example in Section 3.7.

3.3

Main results

In this section we formulate and solve the main problem investigated in this chapter for a two-dimensional system of the form (3.2), with state space X = T Θ, with Θ = R or


53

[R]T , T > 0. The functions Ψi : Θ → R, i = 1, 2, are assumed to be smooth. We begin by defining precisely the Lagrangian structures under consideration. Definition 3.3.1. System (3.2) is said to be: (a) Euler-Lagrange (EL) with Lagrangian L if there exists a smooth Lagrangian function L : X → R such that the following two properties hold: ˙ ∈ X. (i) The Lagrangian L is nondegenerate, i.e., ∂ 2 L/∂ θ˙2 > 0 for all (θ, θ) ˙ (ii) All solutions (θ(t), θ(t)) of (3.2) satisfy the Euler-Lagrange equation ∂L d ∂L ˙ ˙ (θ(t), θ(t)) − (θ(t), θ(t)) =0 dt ∂ θ˙ ∂θ

(3.4)

for all t in their maximal interval of definition. ˙ = (1/2)M(θ)θ˙ 2 − V (θ), where M : (b) Mechanical if it is EL with Lagrangian L(θ, θ) Θ → (0, ∞), V : Θ → R are smooth. (c) Singular Euler-Lagrange (SEL) with Lagrangian L if there exists a smooth Lagrangian function L : X → R such that property (ii) of part (a) holds. Moreover, if L is any function satisfying property (ii) of part (a) and such that ∂ 2 L/∂ θ˙2 is not identically zero, then (i)′ L is degenerate, i.e., ∂ 2 L/∂ θ˙2 has zeros. △ Remark 3.3.2. It is well-known that EL systems with Lagrangian L are Hamiltonian with Hamiltonian function given by the Legendre transform of L (see, e.g., [75]). On the other hand, while SEL systems have a Lagrangian structure, they are generally not Hamiltonian because the Legendre transform of L may not be well-defined. Moreover, SEL systems are not mechanical since, by definition, ∂ 2 L/∂ θ˙2 = M(θ) > 0 for a mechanical system. If L is the Lagrangian of an EL system of the form (3.2), the Euler-Lagrange


54

equation (3.4) defines a smooth vector field on X which coincides with (3.2). Indeed, requirement (i) in Definition 3.3.1(a) ensures that the coefficient of θ¨ in (3.4) is not zero, and therefore (3.4) defines a smooth vector field on X . Moreover, by uniqueness of solutions of (3.2) and requirement (ii) in Definition 3.3.1(a), the local phase flow of this vector field must coincide with the local phase flow of (3.2). Hence, the vector field arising from (3.4) must coincide with (3.2). On the other hand, we will show in the proof of Proposition 3.5.3 (see Remark 3.5.4) that, for a SEL system, the Euler-Lagrange equation (3.4) gives rise to the equation h i ˙ θ¨ − Ψ1 (θ) − Ψ2 (θ)θ˙2 = 0, α(θ, θ) where α is a smooth function with zeros. It follows from this identity that the EulerLagrange equation does not give rise to a well-defined vector field on X , and the collection of its solutions contains, but is not equal to the collection of solutions of (3.2). We will illustrate this fact with an example in Section 3.7. Finally, we remark that the requirement, in Definition 3.3.1(c), that ∂ 2 L/∂ θ˙2 is not identically zero guarantees that the Euler-Lagrange equation (3.4) gives rise to a second-order differential equation. Inverse Lagrangian Problem (ILP). Find necessary and sufficient conditions under which system (3.2) is, respectively, EL, mechanical, or SEL. ˜ i : R → R, i = 1, 2, be defined In order to present the solution of ILP, we let Ψ ˜ i (x) := Ψi ([x]T ), and we define the virtual mass M ˜ : R → (0, ∞) and virtual as Ψ potential V˜ : R → R as

Z x ˜ ˜ M (x) = exp − 2 Ψ2 (τ ) dτ , 0 Z x ˜ ˜ 1 (τ )M ˜ (τ ) dτ. V (x) = − Ψ

(3.5)

0

We now present the main results of this chapter. Theorem 3.3.3 (Solution to ILP - Part 1). If Θ = R, then system (3.2) with state ˜ and V = V˜ , where M ˜ , V˜ are defined in (3.5). space X = T Θ is mechanical, with M = M


55

Proof. By straightforward computation, the Euler-Lagrange equation with Lagrangian ˙ = (1/2)M(θ) ˜ θ˙2 − V (θ) produces equation (3.2). L(θ, θ) Remark 3.3.4. In [36, 37], the authors presented an integral of motion for a system of ˙ = (1/2)M(θ) ˜ θ˙2 + V˜ (θ), but the form (3.2) which is similar to the total energy E(θ, θ) depends on initial conditions. Theorem 3.3.5 (Solution to ILP - Part 2). If Θ = [R]T , then the following statements about system (3.2) with state space X = T Θ are equivalent: (i) System (3.2) is EL. (ii) System (3.2) is mechanical. ˜ and V˜ in (3.5) are T -periodic. (iii) The functions M Moreover, if (3.2) is EL, then the Lagrangian function L : T [R]T → R is given by ˙ = (1/2)M(θ)θ˙ 2 − V (θ), where M : [R]T → (0, ∞) and V : [R]T → R are the L(θ, θ) ˜ = M ◦ π and V˜ = V ◦ π. unique smooth functions such that M Remark 3.3.6. The sufficiency part of the theorem was proved in [40, 41], but we present it in Section 3.5 for completeness. Theorem 3.3.5 can be used to deduce the following proposition which states that VHCs with odd symmetry induce Euler-Lagrange reduced dynamics. This proposition is a basic modification of Proposition 4.4 in [40]. . Proposition 3.3.7 ([40]). Consider the mechanical system (3.1) and suppose that Assumption 2.3.7 holds. Let q = σ(θ) be a regular and odd parametric VHC such that (∀θ ∈ [R]T )(∀i ∈ {1, . . . , n}) σi (θ) = −σi (−θ). ˙ = (1/2)M(θ)θ˙ 2 − V (θ), Then, the reduced dynamics (3.2) are EL with Lagrangian L(θ, θ) ˜ = M ◦ π and potential V˜ = V ◦ π, where M, ˜ V˜ are defined in (3.5). with virtual mass M


56

Proof. We provide a sketch of the proof. According to Theorem 3.3.5, the reduced ˜ dynamics (3.2) have a globally well-defined Euler-Lagrange structure if and only if M(·) and V˜ (·) are T -periodic. Analogous to the proof of Proposition 4.4 in [40], it can be ˜ 2 and Ψ ˜1 · M ˜ have shown that under the given symmetry assumptions the functions Ψ ˜ (·) and V˜ (·) are T -periodic. This zero averages on the closed interval [0, T ]. Thus, M concludes the proof. Corollary 3.3.8. The regular parametric VHCs generated by Procedure 2.1 induce reduced dynamics with globally well-defined EL structure. Theorem 3.3.9 (Solution to ILP - Part 3). If Θ = [R]T , then the following statements about system (3.2) with state space X = T Θ are equivalent: (i) System (3.2) is SEL. ˜ is T -periodic, while V˜ is not T -periodic. (ii) The function M Moreover, if (3.2) is SEL, then the Lagrangian function L : T [R]T → R is the unique ˜ x) smooth function such that L(π(x), x) ˙ = L(x, ˙ for all (x, x) ˙ ∈ R × R, where q ˜ ˜ ˜ (x) π x˙ × L(x, x) ˙ = − sin(2πf0 E0 (x, x)) ˙ + 2f0 M " # q q ˜ (x) x˙ − sin(2πf0 V˜ (x)) S ˜ (x) x˙ , cos(2πf0 V˜ (x)) C 2f0 M 2f0 M

(3.6)

˜ where f0 = 1/V˜ (T ), E˜0 (x, x) ˙ = (1/2)M(x) x˙ 2 + V˜ (x), and C(·), S(·) are the Fresnel Rx Rx cosine and sine integrals, defined as C(x) = 0 cos(πt2 /2)dt, S(x) = 0 sin(πt2 /2)dt.

Remark 3.3.10. The periodicity conditions in Theorems 3.3.5 and 3.3.9 are coordinate invariant. In Proposition 3.6.1 we show that they are invariant under vector bundle ˙ 7→ (ϕ(θ), ϕ′ (θ)θ), ˙ where T1 , T2 > 0. isomorphisms T [R]T1 → T [R]T2 , (θ, θ) Remark 3.3.11. Theorems 3.3.5 and 3.3.9 show that, when Θ = [R]T which corresponds to the situation when the VHC h(q) = 0 is a Jordan curve, the property of (3.2)


57

being either EL or SEL is exceptional, in that it is not satisfied by a generic system of the form (3.6) with state space T Θ. Indeed, in order for (3.2) to be EL or SEL ˜ it is required at a minimum that M(x) be T -periodic, which corresponds to requiring ˜ 2 : R → R has zero average. In other words, the set that the T -periodic function Ψ R ˜ 2 : R → R| T Ψ ˜ 2 (τ )dτ = 0} has measure zero in the set of all smooth T -periodic and {Ψ 0 real-valued functions defined on the real line.

Remark 3.3.12. Having presented the main results of this chapter, we now return to the literature on the IPLM and place the theorems above in this context. First off, it is a matter of straightforward computation to check that the reduced dynamics (3.2) always satisfy the Helmholtz conditions and, as such, system (3.2) is automatically guaranteed to be locally Lagrangian. This fact is known since the work of Darboux [66]. For the existence of global Lagrangian structures, Theorem 5.8 in [72] indicates that when the state space of (3.2) is S1 × R, from the existence of a local Lagrangian structure one cannot deduce the existence of a global such structure. As a matter of fact, Theorems 3.3.5 and 3.3.9 show that a global Lagrangian structure generally does not exist. The work of Anderson and Duchamp [69, Theorem 4.2] provides necessary and sufficient conditions under which a locally variational source form (in our context, the reduced dynamics (3.2)) is globally variational (in our context, globally Lagrangian). The conditions are in terms of the vanishing of a cohomology class which is guaranteed to exist but for which there is no systematic construction method. The criterion in [69] is therefore indirect. It might be possible to use the methodology of [69] to obtain a different proof of some of the results presented above, the application of Theorem 4.2 in [69] to the context of this chapter is far from trivial, and it is unclear whether that formalism allows one to distinguish between the existence of EL and SEL structures. In this sense, to the best of our knowledge the results stated above are not contained in existing literature. Owing to the very specific form of the differential equation we investigate, we take a direct route to solving the inverse Lagrangian problem for the reduced dynamics arising from a VHC. The results


58

stated above present necessary and sufficient conditions which are explicit and checkable. In the next two sections we prove Theorems 3.3.5 and 3.3.9 assuming that Θ = [R]T . We now provide an outline of the arguments that follow. Outline of proofs of Theorems 3.3.5 and 3.3.9. ˜ 1 (x) + Ψ ˜ 2 (x)x˙ 2 , with state space Step 1 In Section 3.4, we define a lifted system, x¨ = Ψ R2 . In Lemma 3.4.1, we show that trajectories of the lifted system are related to trajectories of system (3.2) through the map dπ, where π(x) = [x]T . Step 2 In Lemma 3.4.2, we show that solutions of the Euler-Lagrange equation (3.4) are related through the map dπ to solutions of the Euler-Lagrange equation with ˜ = L ◦ dπ. Lagrangian L Step 3 Leveraging Lemmas 3.4.1 and 3.4.2, in Proposition 3.4.3 we show that (3.2) is EL ˜ x) or SEL if and only if the lifted system is EL or SEL with a Lagrangian L(x, ˙ which is T -periodic with respect to x. Step 4 In Section 3.5, we find necessary and sufficient conditions for the existence of ˜ for the lifted system which enjoys the periodicity property of a Lagrangian L ˜ x) Proposition 3.4.3. In Proposition 3.5.1 we show that in order for a function L(x, ˙ which is nondegenerate and T -periodic with respect to x to be a Lagrangian for ˜ and V˜ in (3.5) are T the lifted system, it is necessary and sufficient that M periodic. This result proves Theorem 3.3.5. ˜ (x + nT ), V˜ (x + nT ), n ∈ Z. Step 5 In Lemma 3.5.2, we find expressions for M Step 6 Using Lemma 3.5.2, in Proposition 3.5.3, we prove that the lifted system is SEL ˜ x) ˜ with a Lagrangian L(x, ˙ which is T -periodic with respect to x if and only if M in (3.5) is T -periodic, while V˜ is not. In light of Proposition 3.4.3, this proves Theorem 3.3.9.


3.4

59

Lift of ILP to R2

Let π : R → [R]T be defined as π(x) = [x]T , and let π ¯ : T R → T [R]T denote the global differential of π, π ¯ := dπ, so that π ¯ (x, x) ˙ = ([x]T , dπx x) ˙ = ([x]T , x). ˙ Given two functions f : [R]T → R and F : T [R]T → R, we define their lifts to be functions f˜ := f ◦π : R → R, and F˜ := F ◦ π¯ : T R → R, as in the following commutative diagrams: R f˜ R

π

[R]T

TR

π ¯ := dπ

F˜

f

R

T [R]T

F

˜ : T R → R is a smooth function, its associated Euler-Lagrange equation is If L ˜ ˜ ∂L d ∂L − = 0. dt ∂ x˙ ∂x

(3.7)

Finally, we define the lift of system (3.2) as ˜ 1 (x) + Ψ ˜ 2 (x)x˙ 2 , x¨ = Ψ

(3.8)

˜ 1 and Ψ ˜ 2 are the lifts of Ψ1 and Ψ2 . The state space of the above differential where Ψ equation is X˜ = T R. We will apply to system (3.8) the terminology of Definition 3.3.1, ˜ whereby L will be replaced by L. ¯ -related to the vector field of (3.8). Lemma 3.4.1. The vector field of equation (3.2) is π ˙ Therefore, pair (θ(t), θ(t)) is a solution of (3.2) if and only if there exists a solution ˙ (x(t), x(t)) ˙ of (3.8) such that (θ(t), θ(t)) =π ¯ (x(t), x(t)). ˙ Proof. The vector fields of system (3.2) and system (3.8) are given by ˙ 7→ θ˙ ∂ + Ψ1 (θ) + Ψ2 (θ)θ˙2 ∂ F : X → T X , (θ, θ) ∂θ ∂ θ˙ ∂ ˜ 1 (x) + Ψ ˜ 2 (x)x˙ 2 ∂ . F˜ : X˜ → T X˜ , (x, x) ˙ 7→ x˙ + Ψ ∂x ∂ x˙

Recall that π(x) = [x]T , and π ¯ (x, x) ˙ = ([x]T , dπx x) ˙ = ([x]T , x). ˙ For all (x, x) ˙ ∈ X˜ , the ˜ differential d¯ π(x,x) ˙ : T(x,x) ˙ X → Tπ ¯ (x,x) ˙ X is the identity map ∂ ∂ ∂ ∂ = v1 + v2 + v2 . d¯ π(x,x) v1 ˙ ∂x ∂ x˙ ∂θ ∂ θ˙


60

We thus have ∂ ˜ ˜ 1 (x) + Ψ ˜ 2 (x)x˙ 2 ∂ d¯ π(x,x) ˙ = x˙ + Ψ ˙ F (x, x) ∂θ ∂ θ˙ ∂ ∂ 2 = θ˙ + Ψ1 (θ) + Ψ2 (θ)θ˙ ˙ ∂θ ∂θ

˙ π (x,x) (θ,θ)=¯ ˙

=F ◦π ¯ (x, x), ˙

proving that F and F˜ are π ¯ -related. Since π ¯ is surjective, by [48, Proposition 9.6], a pair ˙ (θ(t), θ(t)) is a solution of (3.2) if and only if there exists a solution (x(t), x(t)) ˙ of (3.8) ˙ such that (θ(t), θ(t)) =π ¯ (x(t), x(t)). ˙ Lemma 3.4.2. Let I ⊂ R be an open interval, and θ : I → [R]T , x : I → R be C 1 signals ˙ ˙ such that (θ(t), θ(t)) =π ¯ (x(t), x(t)) ˙ for all t ∈ I. Then, the pair (θ(t), θ(t)) satisfies the Euler-Lagrange equation (3.4) with smooth Lagrangian L : T [R]T → R if and only if the pair (x(t), x(t)) ˙ satisfies the lifted Euler-Lagrange equation (3.7) with smooth Lagrangian ˜ = L◦π L ¯. Proof. We have ˜ (x(t),x(t)) dL = d(L ◦ π ¯ )(x(t),x(t)) = dLπ¯ (x(t),x(t)) ◦ d¯ π(x(t),x(t)) = dLπ¯ (x(t),x(t)) . ˙ ˙ ˙ ˙ ˙ ˜ and L are the components of dL ˜ (x,x) Using the fact that the partial derivatives of L ˙ and dL(θ,θ)˙ , respectively, we have ˜ ∂L ∂L (x(t), x(t)) ˙ = (¯ π (x(t), x(t))), ˙ ∂x ∂θ

˜ ∂L ∂L (¯ π (x(t), x(t))), ˙ (x(t), x(t)) ˙ = ∂ x˙ ∂ θ˙

˜ = L◦ from which it follows that the Euler-Lagrange equation (3.7) with Lagrangian L π ¯ is satisfied along (x(t), x(t)) ˙ if and only if the Euler-Lagrange equation (3.4) with ˙ Lagrangian L is satisfied along (θ(t), θ(t)) =π ¯ (x(t), x(t)). ˙ Proposition 3.4.3. The following statements are equivalent (i) System (3.2) with state space X = T [R]T is EL (resp., SEL) with Lagrangian L.


61

˜ = L◦¯ (ii) System (3.8) with state space X˜ = T R is EL (resp., SEL) with Lagrangian L π. ˜ = L◦π Proof. Let L ¯ . Then, by the reasoning used in the proof of Lemma 3.4.2, it is ˜ x˙ 2 )(x, x) easy to see that (∂ 2 L/∂ ˙ = (∂ 2 L/∂ θ˙2 )(¯ π (x, x)). ˙ Therefore, L is nondegenerate ˜ is nondegenerate (respectively, degenerate). (respectively, degenerate) if and only if L Now, suppose that system (3.2) is EL (respectively, SEL) with Lagrangian L. Consider an arbitrary solution of (3.8), namely, (x(t), x(t)), ˙ where x : I → R is C 1 and I ⊂ R is ˙ an open interval. By Lemma 3.4.1, (θ(t), θ(t)) := π ¯ (x(t), x(t)) ˙ is a solution of (3.2), and ˙ satisfies thus satisfies the Euler-Lagrange equation (3.4). By Lemma 3.4.2, (x(t), x(t)) ˜ = L◦ π the Euler-Lagrange equation with Lagrangian L ¯ . Since (x(t), x(t)) ˙ is an arbitrary solution of (3.8), and since π ¯ : T R → T [R]T is onto, system (3.8) is EL (respectively, ˜ = L◦π SEL) with Lagrangian L ¯ . The proof that if (3.8) is EL (respectively, SEL) ˜ = L◦π with Lagrangian L ¯ , then (3.2) is EL (respectively, SEL) with Lagrangian L is ˙ analogous. We consider an arbitrary solution (θ(t), θ(t)) of (3.2), and we let (x(t), x(t)) ˙ ˙ = π ¯ (x(t), x(t)). ˙ Such a solution exists by be a solution of (3.8) such that (θ(t), θ(t)) Lemma 3.4.1 and the fact that π¯ is onto. Thus, (x(t), x(t)) ˙ is a solution of the Euler˙ ˜ = L◦π Lagrange equation (3.7) with Lagrangian L ¯ . By Lemma 3.4.2, (θ(t), θ(t)) is a ˙ solution of the Euler-Lagrange equation (3.4) with Lagrangian L. Since (θ(t), θ(t)) is an arbitrary solution of (3.2), we conclude that (3.2) is EL (respectively, SEL).

3.5

Proofs of main results

By virtue of Proposition 3.4.3, solving ILP and finding a Lagrangian L for system (3.2) is ˜ for the lifted system (3.8) such that equivalent to solving ILP and finding a Lagrangian L ˜ = L◦π ˜ : T R → R, there L ¯ , for some smooth L : T [R]T → R. Given a smooth function L ˜ = L◦π ˜ is T -periodic exists a smooth function L : T [R]T → R satisfying L ¯ if and only if L ˜ + T, x) ˜ x) with respect to its first argument, i.e., L(x ˙ = L(x, ˙ for all (x, x) ˙ ∈ T R. In this section, we leverage this fact to prove Theorems 3.3.5 and 3.3.9.


62

˜ : TR → R Proposition 3.5.1. The lifted system (3.8) is EL with a smooth Lagrangian L ˜ + T, x) ˜ x) ˜ such that L(x ˙ = L(x, ˙ for all (x, x) ˙ ∈ T R, if and only if the virtual mass M and virtual potential V˜ in (3.8) are T -periodic. If this is the case, then system (3.4) is mechanical with Lagrangian L = (1/2)M(θ)θ˙2 − V (θ), where M and V are defined ˜ = M ◦ π, V˜ = V ◦ π. through M ˜ , V˜ are T -periodic, then L(x, ˜ x) ˜ Proof. (⇐) If M ˙ = (1/2)M(x) x˙ 2 − V˜ (x) is T -periodic with respect to x, and ˜ ∂L ˜ d ∂L ˜ (x) x¨ − Ψ ˜ 1 (x) − Ψ ˜ 2 (x)x˙ 2 . − =M dt ∂ x˙ ∂x

˜ > 0, the lifted system is mechanical with Lagrangian L. ˜ Since M

˜ : T R → R such that (⇒) Assume that system (3.8) is EL with smooth Lagrangian L ˜ ˜ x) ˜ is nondegenerate, L(x+ T, x) ˙ = L(x, ˙ for all (x, x) ˙ ∈ T R. By definition of EL system, L ˜ x˙ 2 6= 0. Define a smooth function E˜ : T R → R as i.e., ∂ 2 L/∂ ˜ ∂L ˜ x). ˜ x) (x, x) ˙ − L(x, ˙ E(x, ˙ := x˙ ∂ x˙ By differentiating the expression for E˜ above along the vector field of (3.8), it is readily seen that E˜ is an integral of motion for (3.8), i.e., E˜˙ = 0. Consequently, E˜ must satisfy the first-order linear PDE ∂ E˜ ∂ E˜ ˜ ˜ 2 (x)x˙ 2 = 0. Ψ1 (x) + Ψ x˙ + ∂x ∂ x˙

(3.9)

˜ x) Its general solution, obtained via the method of characteristics [76], is E(x, ˙ = F (E˜0 (x, x)), ˙ where F is a smooth function and 1 ˜ (x)x˙ 2 + V˜ (x). E˜0 (x, x) ˙ = M 2 ˜ we have Using the definition of E, ˜ ∂ E˜ ∂2L = x˙ 2 ∂ x˙ ∂ x˙


63

for all (x, x) ˙ ∈ T R. Therefore, ˜ ∂2L ˜ (x)F ′ (E˜0 (x, x)). =M ˙ ∂ x˙ 2 ˜ x˙ 2 > 0 and M ˜ > 0, it follows that F ′ (E˜0 (x, x)) Since ∂ 2 L/∂ ˙ > 0 for all (x, x) ˙ ∈ R2 , and ˜ + T, x) ˜ x) thus F is strictly increasing. Furthermore, we know that E(x ˙ = E(x, ˙ for all ˜0 (x+T, x)) ˜0 (x, x)), (x, x) ˙ ∈ R2 . Therefore, for all (x, x) ˙ ∈ T R, we have F (E ˙ = F (E ˙ which implies that E˜0 (x + T, x) ˙ = E˜0 (x, x). ˙ Since x˙ is arbitrary, this latter identity implies that ˜ and V˜ are T -periodic. Since M ˜ and V˜ are T -periodic, then (1/2)M(˜ ˜ x)x˜˙ 2 − V˜ (˜ M x) is ˙ = (1/2)M(θ)θ˙ 2 − a Lagrangian for the lifted system (3.8). By Proposition 3.4.3, L(θ, θ) V (θ) is a Lagrangian for the original system (3.2). Lemma 3.5.2. Consider the virtual mass and virtual potential in (3.5). For all n ∈ Z and all x ∈ R, the following holds: ˜ + nT ) = M ˜ (T )n M ˜ (x) M(x  n ˜   ˜ (T ) 6= 1, ˜ (T )n V˜ (x) + V˜ (T ) M (T ) − 1 , if M M ˜ M (T ) − 1 V˜ (x + nT ) =   V˜ (x) + nV˜ (T ), ˜ (T ) = 1. if M

(3.11)

˜ + T) = M ˜ (T )M(x). ˜ M(x

(3.12)

(3.10)

˜ 1 (x) and Ψ ˜ 2 (x), it is straightforward to verify that Proof. Using the T -periodicity of Ψ

˜ + kT ) = M ˜ (T )k M ˜ (x). On the other hand, the By induction, for k ≥ 0 it holds that M(x ˜ (x) = M ˜ (x − T + T ) = M ˜ (T )M ˜ (x − T ) results in M ˜ (x − T ) = M ˜ (T )−1 M ˜ (x). identity M ˜ (x − kT ) = M ˜ (T )−k M ˜ (x). This proves identity (3.10) By induction, for k ≥ 0 we have M ˜ 1 and identity (3.12), we have for all n ∈ Z. Turning to V˜ , using the T -periodicity of Ψ Z T Z T +x ˜ 1 (τ )M ˜ (τ ) dτ − ˜ 1 (τ )M ˜ (τ ) dτ V˜ (x + T ) = − Ψ Ψ 0 T Z x ˜ 1 (u + T )M ˜ (u + T )du = V˜ (T ) − Ψ 0

˜ (T )V˜ (x). = V˜ (T ) + M

64

Chapter 3. The Lagrangian Structure of Reduced Dynamics By induction, for k ≥ 0 we have ˜ )k V˜ (x) + V˜ (T ){1 + M ˜ (T ) + · · · + M ˜ (T )k−1 }. V˜ (x + kT ) = M(T

˜ (T ) 6= 1, by using the partial sum of the geometric series we obtain the first case If M ˜ ) = 1, then we obtain V˜ (x + kT ) = V˜ (x) + k V˜ (T ), which of identity (3.11). If M(T is the second case of identity (3.11). To prove the identity for negative n, we write ˜ )V˜ (x − T ), to get V˜ (x − T ) = M ˜ (T )−1 V˜ (x) − M ˜ (T )−1 V˜ (T ). V˜ (x − T + T ) = V˜ (T ) + M(T By induction, for all k ≥ 0 we have ˜ (T )−k V˜ (x) − M ˜ (T )−1V˜ (T ){1 + M ˜ (T )−1 + · · · + M(T ˜ )−(k−1) }. V˜ (x − kT ) = M ˜ (T ) = 1 we obtain the second case of identity (3.11). If M ˜ (T ) 6= 1, using the partial If M sum of the geometric series and elementary manipulations we arrive at the first case of identity (3.11). In conclusion, identity (3.11) holds for all n ∈ Z. ˜ : TR → Proposition 3.5.3. The lifted system (3.8) is SEL with a smooth Lagrangian L ˜ + T, x) ˜ x) R such that L(x ˙ = L(x, ˙ for all (x, x) ˙ ∈ T R, if and only if the virtual mass ˜ (x) in (3.5) is T -periodic, and the virtual potential V˜ (x) is not T -periodic. M ˜ (x) is T -periodic and the virtual potential Proof. (⇐) Suppose that the virtual mass M V˜ (x) is not T -periodic, so that V˜ (T ) 6= 0 and f0 = 1/V˜ (T ) is well-defined. Consider the ˜ : T R → R defined in (3.6). With our definition of f0 , L(x, ˜ x) function L ˙ is T -periodic with respect to x. Moreover, by direct computation, we have ˜ ∂L ˜ d ∂L 2 ˜ ˜ − = α(x, ˜ x) ˙ x¨ − Ψ1 (x) − Ψ2 (x)x˙ , dt ∂ x˙ ∂x

(3.13)

˜ ˜ (x) cos(2πf0 E ˜0 (x, x)). where α(x, ˜ x) ˙ = (∂ 2 L)/(∂ x˙ 2 ) = 2πf0 M ˙ Note first that α ˜ is not identically zero because V˜ is not identically zero (if it were, then V˜ would be T -periodic, contradicting our assumption). At the same time, we now show that α ˜ has zeros. By as˜ (T ) = M ˜ (0) = 1 and V˜ (T ) 6= V (0) = 0. By identity (3.11) in Lemma 3.5.2, sumption, M V˜ (x) → ±∞ as |x| → ∞, and the two limits as x → ±∞ have opposite signs, which


65

implies that the continuous map V˜ : R → R is onto. Thus, there exists x¯ ∈ R such that 2πf0 V˜ (¯ x) = π/2, implying that α(¯ ˜ x, 0) = 0. We have shown that α ˜ has zeros, which ˜ is degenerate. By definition, all solutions of the lifted system (3.8) satisfy implies that L ˜ 1 (x) + Ψ ˜ 2 (x)x˙ 2 . Therefore, by identity (3.13), any solution the differential equation x¨ = Ψ ˜ In order of (3.8) satisfies the Euler-Lagrange equation with a degenerate Lagrangian L. ˜ ′ is any other to complete the proof that system (3.8) is SEL, we need to show that if L ˜ is degenerate, i.e., ∂ 2 L ˜ ′ /∂ x˙ 2 has zeros. Suppose there Lagrangian for system (3.8), then L ˜ ′ for system (3.8). Then, system (3.8) is EL, which exists a nondegenerate Lagrangian L by Proposition 3.5.1 implies that V˜ is T -periodic, a contradiction. ˜ be a degenerate Lagrangian (⇒) Suppose that the lifted system (3.8) is SEL, and let L ˜ x) ˜ x˙ 2 has zeros, but it is not such that L(x, ˙ is T -periodic with respect to x, and ∂ 2 L/∂ ˜ ) = 1, so that M ˜ in (3.5) is T -periodic (this identically zero. We need to show that M(T fact will imply that V˜ is not T -periodic, because if it were so, then by Proposition 3.5.1 ˜ x˙ − L. ˜ the system would be EL). As in the proof of Proposition 3.5.1, let E˜ = x∂ ˙ L/∂ ˜ x) Then, E˜ satisfies the linear PDE (3.9), whose general solution is E(x, ˙ = F (E˜0 (x, x)), ˙ ˜ ˜ is T -periodic with respect to x, so is E. ˜ with E˜0 (x, x) ˙ = (1/2)M(x) x˙ 2 + V˜ (x). Since L ˜ x) ˜ + nT, x) Therefore, E(x, ˙ = E(x ˙ for all (x, x) ˙ ∈ T R and all n ∈ Z. Using Lemma 3.5.2, for all n ∈ Z we have ! ˜ )n − 1 M(T ˜ (T )n E˜0 (x, x) . F (E0 (x, x)) ˙ = F E˜0 (x + nT, x) ˙ =F M ˙ + V˜ (T ) ˜ (T ) − 1 M

˜ (T ) 6= 1, then F is a constant function. Indeed, for any p ∈ Im(E˜0 ) We claim that if M and any n ∈ Z, we have ! ˜ (T )n − 1 M ˜ (T )n p + V˜ (T ) . F (p) = F M ˜ )−1 M(T ˜ ) > 1, taking the limit as n → −∞ in both sides of the identity above we get If M(T ! −V˜ (T ) F (p) = F . ˜ (T ) − 1 M


66

˜ (T ) < 1, the same identity is obtained by taking the limit for n → +∞. Since the If M right-hand side of the identity above does not depend on p, F : Im(E˜0 ) → R is a constant map. Thus, for all (x, x) ˙ ∈ T R we have ˜ ∂ E˜ ∂2L = x˙ 2 = 0, ∂ x˙ ∂ x˙ ˜ x˙ 2 ≡ 0, contradicting our hypothesis on L. ˜ and so ∂ 2 L/∂ ˜ x) Remark 3.5.4. Since the degenerate Lagrangian L(x, ˙ in (3.6) is smooth and T periodic with respect to x, there exists a smooth function L : T [R]T → R such that ˜ By Lemma 3.4.2, since α(x, L◦π ¯ = L. ˜ x) ˙ is T -periodic with respect to x, (3.13) implies that L satisfies the identity d ∂L ∂L ˙ θ¨ − Ψ1 (θ) − Ψ2 (θ)θ˙ 2 , = α(θ, θ) − dt ∂ θ˙ ∂θ where α and α ˜ are related through α ˜ = α ◦ π¯ .

3.6

Characterization of motion on the constraint manifold

In this section we use the results of Section 3.3 to investigate the qualitative properties of solutions of the reduced dynamics (3.2) when h−1 (0) is a Jordan curve. In Section 3.6.1, we investigate the effect of coordinate transformations, and in Section 3.6.2 we investigate the qualitative properties of typical trajectories of EL and SEL systems.

3.6.1

Effects of coordinate transformations

When the set h−1 (0) is a Jordan curve, the state space of the reduced dynamics is a cylinder. The representation of the reduced dynamics in (3.2) was derived through a T -periodic regular parametrization of h−1 (0). In this section we investigate the effects of reparametrization of the curve h−1 (0). Reparametrizing h−1 (0) is equivalent to defining a

67


˙ 7→ (θ, θ) ˙ for system (3.2). More precisely, let T1 , T2 > 0, coordinate transformation (θ, θ) and let ϕ : [R]T1 → [R]T2 be a diffeomorphism. Let πi : R → [R]Ti , i = 1, 2, be defined as πi (x) = [x]Ti . Consider the smooth dynamical system with state space T [R]T1 , θ¨ = Ψ11 (θ) + Ψ12 (θ)θ˙ 2 ,

(3.14)

˙ 7→ (θ, θ) ˙ = and the vector bundle isomorphism T [R]T1 → T [R]T2 defined as (θ, θ) ˙ In (θ, θ) ˙ coordinates, system (3.14) reads (ϕ(θ), ϕ′ (θ)θ). θ¨ = Ψ21 (θ) + Ψ22 (θ)θ˙ 2 ,

(3.15)

where Ψ21 ◦ ϕ = ϕ′ Ψ11 Ψ22 ◦ ϕ =

ϕ′′ Ψ12 + . ϕ′ ϕ′2

Associated with the two dynamical systems above we have two lifted systems ˜ 1 (x) + Ψ ˜ 1 (x)x˙ 2 x¨ = Ψ 1 2

(3.16)

˜ 2 (y) + Ψ ˜ 2 (y)y˙ 2 y¨ = Ψ 1 2

(3.17)

˜ i := Ψi ◦πi , i, j = 1, 2. We also have virtual mass and virtual potential functions, where Ψ j j Z x ˜ ˜ i2 (τ )dτ , Mi (x) = exp − 2 Ψ 0 Z x ˜ i (τ )M ˜ i (τ )dτ, V˜i (x) = − Ψ 1

(3.18)

0

˜ 1 , V˜1 are T1 -periodic if and only if M ˜ 2, i = 1, 2. In Proposition 3.6.1 we prove that M V˜2 are T2 -periodic. This fact is important because the main results of this chapter in ˜ and V˜ Theorems 3.3.5 and 3.3.9 are stated in terms of the periodicity of the functions M ˜ 1 is T1 -periodic, there exists in (3.5). In Proposition 3.6.2, we show that if, and only if, M ϕ : [R]T1 → [R]T2 such that Ψ22 = 0, so that (3.15) is a one DOF conservative system.


68

Proposition 3.6.1. There exists a diffeomorphism ϕ˜ : R → R such that the following diagram commutes: R

ϕ˜

π1 [R]T1

ϕ

R π2

(3.19)

[R]T2

Moreover, the lifted systems (3.16), (3.17) are related through the coordinate transformation (x, x) ˙ 7→ (y, y) ˙ = (ϕ(x), ˜ ϕ˜′ (x)x), ˙ and the virtual masses and virtual potentials in (3.18) are related as follows: ˜ 1 (ϕ˜′ (ϕ˜−1 (0)))2 M ˜ , M2 ◦ ϕ˜ = ′ 2 ˜ 1 (ϕ˜−1 (0)) (ϕ˜ ) M

(ϕ˜′ (ϕ˜−1 (0)))2 ˜ −1 ˜ ˜ V2 = − V1 − V1 (ϕ˜ (0)) . ˜ 1 (ϕ˜−1 (0)) M

(3.20)

˜ 1 is T1 -periodic if and only if M ˜ 2 is T2 -periodic, and V˜1 is T1 -periodic if and Finally, M only if V˜2 is T2 -periodic. Proof. The function π1 : R → [R]T1 is a covering map [48]. Since ϕ : [R]T1 → [R]T2 is a diffeomorphism, the function ϕ ◦ π1 : R → [R]T2 is a covering map as well. By the path lifting property of the circle (see [48, Corollary 8.5]), there exists a map ϕ˜ : R → R such that π2 ◦ ϕ˜ = ϕ ◦ π1 , proving that the diagram (3.19) commutes. We claim that ϕ˜ is a diffeomorphism. Being covering maps, π1 , π2 are local diffeomorphisms, implying that ϕ˜ is a local diffeomorphism as well. ϕ˜ is surjective because ϕ and π1 are surjective. Suppose ϕ(x ˜ 1 ) = ϕ(x ˜ 2 ). Then, π2 ◦ ϕ(x ˜ 1 ) = π2 ◦ ϕ(x ˜ 2 ), and therefore ϕ ◦ π1 (x1 ) = ϕ ◦ π1 (x2 ). ϕ is a diffeomorphism, so π1 (x1 ) = π1 (x2 ), or x1 = x2 + lT1 , for some l ∈ Z. Since ϕ˜′ 6= 0 (because ϕ˜ is a local diffeomorphism), it must be that l = 0, since otherwise ϕ˜ would not be strictly monotonic. In conclusion, ϕ˜ is bijective, and therefore also a diffeomorphism. The diffeomorphisms ϕ and ϕ˜ induce the commutative diagram, TR

dϕ˜

dπ1 T [R]T1

TR dπ2

dϕ

T [R]T2

(3.21)


69

in which dϕ and dϕ˜ are vector bundle isomorphisms. Let F1 : [R]T1 → T [R]T1 and F2 : [R]T2 → T [R]T2 be the vector fields of systems (3.14) and (3.15), and let F˜1 : R → T R, F˜2 : R → T R be the vector fields of the lifted systems (3.16) and (3.17), respectively. By Lemma 3.4.1, dπ1 ◦ F˜1 = F1 ◦ π1 . Also, since dϕ is an isomorphism, dϕ ◦ F1 = F2 ◦ ϕ. Using these two identities, we have dπ1 ◦ F˜1 = F1 ◦ π1 = (dϕ)−1 ◦ F2 ◦ ϕ ◦ π1 .

Using the diagram (3.19) we have ϕ ◦ π1 = π2 ◦ ϕ, ˜ so

dϕ ◦ dπ1 ◦ F˜1 = F2 ◦ π2 ◦ ϕ. ˜ Using the diagram (3.21) and the fact that F2 and F˜2 are π2 -related, we have dπ2 ◦ dϕ˜ ◦ F˜1 = dπ2 ◦ F˜2 ◦ ϕ. ˜ Finally, since π2 is a local diffeomorphism, we get dϕ◦ ˜ F˜1 = F˜2 ◦ ϕ, ˜ proving that the vector fields of systems (3.16) and (3.17) are dϕ-related, ˜ i.e., the coordinate transformation ˜ 2 and V˜2 . Note first (y, y) ˙ = (ϕ(x), ˜ ϕ˜′ (x)x) ˙ maps (3.16) into (3.17). We now derive M ˜ 2 ◦ ϕ˜ = Ψ2 ◦ π2 ◦ ϕ˜ = Ψ2 ◦ ϕ ◦ π1 . Also, differentiating the identity ϕ ◦ π1 = π2 ◦ ϕ, that Ψ ˜ i i i and using the fact that π1′ = π2′ = 1, we have ϕ′ ◦ π1 = ϕ˜′ . Thus, ! Z ϕ(x) Z x ˜ 2 ˜ ˜ M2 (ϕ(x)) ˜ = exp − 2 Ψ2 (τ )dτ = exp − 2 (Ψ22 ◦ ϕ ◦ π1 (τ ))ϕ˜′ (τ )dτ ϕ ˜−1 (0)

0

= exp − 2

=

Z

x

ϕ ˜−1 (0)

˜ 1 (τ )dτ exp − 2 Ψ 2

˜ 1 (x) (ϕ˜′ (ϕ˜−1 (0)))2 M . ˜ 1 (ϕ˜−1 (0)) (ϕ˜′ (x))2 M

Z

x

ϕ ˜−1 (0)

ϕ˜′′ (τ ) dτ ϕ˜′(τ )

˜ 1 (ϕ˜−1 (0)), for V˜2 we have Similarly, letting C = (ϕ˜′ (ϕ˜−1 (0)))2 /M Z ϕ(x) ˜ ˜ ˜ 2 (τ )M ˜ 2 (τ )dτ V2 (ϕ(x)) ˜ =− Ψ 1 0 Z x ˜ 2 (ϕ(τ ˜ ˜ ))ϕ˜′ (τ )dτ =− Ψ 1 ˜ ))M2 (ϕ(τ ϕ ˜−1 (0) Z x ˜ 1 (τ )Ψ ˜ 1 (τ )dτ = −C V˜1 (x) + C V˜1 (ϕ˜−1 (0)). = −C M 1 ϕ ˜−1 (0)

70


Finally, since ϕ : [R]T1 → [R]T2 is a diffeomorphism, it has degree ±1. This implies that ˜ 2 , V˜2 imply that M ˜2 ϕ(x ˜ + T1 ) = ϕ(x) ˜ ± T2 . This fact and the above expressions for M ˜ 1 (resp., M ˜ 2 ) is T1 -periodic. (resp., V˜2 ) is T2 -periodic if and only if M Proposition 3.6.2. Let T2 > 0 be arbitrary. Then, there exists ϕ : [R]T1 → [R]T2 such ˜ 2 = 1 and Ψ2 = 0 if, and only if, M ˜ 1 is T1 -periodic. that M 2 ˜ 2 = 1, Proof. (⇒) Let T2 > 0 be arbitrary and ϕ : [R]T1 → [R]T2 be a diffeomorphism. If M ˜ 1 is T1 -periodic. ˜ 2 is T2 -periodic which by Proposition 3.6.1 implies that M then M (⇐) Let T2 > 0 be arbitrary, and let ϕ˜ : R → R be defined as Z xq T2 ˜ 1 (τ )dτ, λ := ϕ(x) ˜ =λ M . R T1 q 0 ˜ M1 (τ )dτ 0

Since inf ϕ˜′ > 0, ϕ˜ is a diffeomorphism R → R. Moreover, ϕ˜′ is T1 periodic, from which it is readily seen that ϕ(x ˜ + T1 ) = ϕ(x) ˜ + T2 . For all θ ∈ [R]T1 , letting x ∈ π1−1 (θ), we have π2 ◦ ϕ˜ ◦ π1−1 (θ) = π2 ◦ ϕ({x ˜ + lT1 : l ∈ Z}) = π2 ({ϕ(x) ˜ + lT2 : l ∈ Z}) = π2 (ϕ(x)). Thus, there exists a smooth function ϕ : [R]T1 → [R]T2 such that the diagram (3.19) commutes. This function is a diffeomorphism because ϕ˜ is such. By Proposition 3.6.1, we have ˜ 1 (x) λ2 M ˜ 1 (0) M ˜ 2 (ϕ(x)) M ˜ = = 1, ˜ 1 (x) M ˜ 1 (0) λ2 M ˜ 2 = 1. By (3.18), it follows that Ψ ˜ 2 = 0, and also Ψ2 = 0. proving that M 2 2

3.6.2

Qualitative properties of the reduced dynamics

Consider again the reduced dynamics θ¨ = Ψ1 (θ) + Ψ2 (θ)θ˙ 2 ,

(3.22)

with state space the cylinder T [R]T . We now characterize the qualitative properties of “typical” solutions of (3.22).


71

˙ Definition 3.6.3. A solution (θ(t), θ(t)) of (3.22) is said to be: ˙ (i) A rotation of (3.22) if the set γ = Im((θ(·), θ(·))) is homeomorphic to a circle ˙ ∈ T [R]T : θ˙ = constant} via a vector bundle isomorphism of the form {(θ, θ) ˙ 7→ (θ, µ(θ)θ), ˙ µ 6= 0. (θ, θ) ˙ ∈ T [R]T : (θ, θ) ˙ = (ii) An oscillation of (3.22) if γ is homeomorphic to a circle {(θ, θ) π ¯ (x, x), ˙ (x, x) ˙ ∈ T R, x2 + x˙ 2 = constant} via a vector bundle isomorphism of the form above. ˙ ∈ T [R]T : (θ, θ) ˙ = (iii) A helix of (3.22) if γ is homeomorphic to the set {(θ, θ) π ¯ (x, x), ˙ (x, x) ˙ ∈ T R, x˙ 2 + x = constant} via a vector bundle isomorphism of the form above. △ We now discuss the “typical” solutions of EL and SEL systems. The next result for EL systems is taken from [40, Proposition 4.7]. Proposition 3.6.4 ([40]). Suppose that the dynamical system (3.22) is EL and let V, M : ˜ = M ◦ π, with V˜ , M ˜ [R]T → R be the unique smooth functions such that V˜ = V ◦ π, M defined in (3.5). Let V = minx∈[0,T ] V˜ (x), V¯ = maxx∈[0,T ] V˜ (x). Then, all solutions of ˙ ∈ T [R]T : 1/2M(θ)θ˙2 + V (θ) > V } are rotations, and almost all (3.22) in the set {(θ, θ) ˙ ∈ T [R]T : V < 1/2M(θ)θ˙2 + (in the Lebesgue sense) solutions of (3.22) in the set {(θ, θ) V (θ) < V } are oscillations. Next, a new result concerning SEL systems. Proposition 3.6.5. Suppose that the dynamical system (3.22) is SEL. Then, almost all solutions of (3.22) are either oscillations or helices. Proof. Since (3.22) is a SEL system, by Proposition 3.6.2 it is diffeomorphic to a one DOF conservative system θ¨ = Ψ(θ)

(3.23)

72

Chapter 3. The Lagrangian Structure of Reduced Dynamics with state space T [R]T , whose associated virtual potential V˜ (x) = −

Rx 0

˜ )dτ (where Ψ(τ

˜ = Ψ ◦ π) is not T -periodic, i.e., V˜ (T ) 6= V˜ (0) = 0. The lifted system is given by Ψ ˜ x¨ = Ψ(x).

(3.24)

¯ -related, and to In light of Lemma 3.4.1, the solutions of systems (3.23) and (3.24) are π prove the proposition it suffices to show that almost all solutions of (3.24) are either closed curves homeomorphic to {(x, x) ˙ : x2 + x˙ 2 = constant} or open curves homeomorphic to parabolas {(x, x) ˙ : x + x˙ 2 = constant}. Without loss of generality, we assume that V˜ (T ) > 0. By Lemma 3.5.2, V˜ (x + nT ) = V˜ (x) + nV˜ (T ) for all x ∈ R and all n ∈ Z, implying that V˜ : R → R is onto. Each phase curve of (3.23) lies entirely in a level set of E˜0 (x, x) ˙ = 1/2x˙ 2 + V˜ (x). By Sard’s theorem [53], for almost all h ∈ R, V˜ ′ 6= 0 on the set V˜ −1 (h), which implies that the set E˜0−1 (h) does not contain equilibria. Moreover, since V˜ is onto, V˜ −1 (h) is non-empty. Let h be such that V˜ ′ 6= 0 on V˜ −1 (h), and consider the set Ωh = {x ∈ R : V˜ (x) ≤ h}. Let {x0 , . . . , xN } := V˜ −1 (h) be ordered so that xi < xi+1 . The sequence is finite since the continuity of V˜ and the fact that V˜ (x) → ±∞ as x → ±∞ imply that x0 = inf V˜ −1 (h) and xN = sup V˜ −1 (h) exist and are finite. For all x < x0 , V˜ (x) < h, for otherwise it would hold that inf V˜ −1 (h) < x0 . Moreover, since V˜ ′ 6= 0 on the set V˜ −1 (h), it follows that Ωh is the union of disjoint intervals with S S S nonzero measure. This latter fact implies that Ωh = (−∞, x0 ] [x1 , x2 ] · · · [xN −1 , xN ].

Now we apply the classical theory of one DOF conservative systems [75], from which we conclude that the energy level set E˜0−1 (h) is the union of N + 1 trajectories. On each band [x2i−1 , x2i ] × R, i = 1, . . . , N/2, the set E˜0−1 (h) ∩ [x2i−1 , x2i ] × R is a closed curve

homeomorphic to a circle x2 + x˙ 2 = constant (see also the proof of Lemma 3.12 in [77]).

On the band (−∞, x0 ]×R, the set E˜0−1 (h)∩((−∞, x0 ]×R) is homeomorphic to a parabola q {(x, x) ˙ : x + x˙ 2 = x0 } via the homeomorphism (x, x) ˙ 7→ x, x˙ (−x + x0 )/2(h − V˜ (x)) . Remark 3.6.6. By virtue of Propositions 3.6.4 and 3.6.5, EL and SEL systems cannot


73

possess limit cycles or asymptotically stable equilibria. Typical solutions of an EL system are rocking motions (oscillations) or complete revolutions of s (rotations). Typical solutions of a SEL system are complete revolutions of s with either a periodic speed profile (oscillations) or monotonically increasing or decreasing speed profiles (helices). We conclude this section with a slight extension of a result in [52, Proposition 4.1] which shows that certain systems of the form (3.22) which have no Lagrangian structure (i.e., they are neither EL nor SEL) possess exponentially stable limit cycles. Proposition 3.6.7 ([52]). Consider the dynamical system (3.22), and assume that either RT R ˜ 2 (τ )dτ < 0 or Ψ1 < 0 and T Ψ ˜ 2 (τ )dτ > 0. Define the T -periodic Ψ1 > 0 and 0 Ψ 0 smooth function ν˜ : R → R as

ν˜(x) = sgn(Ψ1 )

s

˜ −1 (x)[V˜ (x + T ) − V˜ (x)] −2M , ˜ (T ) − 1 M

and let ν : [R]T → R be the unique smooth function such that ν˜ = ν ◦ π. Then the closed ˙ ∈ T [R]T × : θ˙ = ν(θ)} is exponentially stable for (3.22), with domain of orbit R = {(θ, θ) ˙ ∈ T [R]T : sgn(Ψ1 )θ˙ ≥ 0}. attraction containing the set D = {(θ, θ) We omit the proof of this result, since it is almost identical to the proof of Proposition 4.1 in [52]. The element of novelty here is the explicit determination of the limit cycle θ˙ = ν(θ) which is made possible by Lemma 3.5.2. This latter result can also be used to show that ν˜(x) is a T -periodic function. Remark 3.6.8. Proposition 3.6.7 shows that, generally, the flow of the reduced dynamics induced by a VHC does not preserve volume. This is in contrast with the flow of Hamiltonian systems which, according to the Liouville-Arnold theorem [75], preserves volume. Moreover, the sufficient conditions of the proposition are expressed in terms of strict inequalities involving continuous functions and, as such, they persist under small perturbations of the vector field in (3.22). In other words, the existence of stable limit


74

cycles is not an “exceptional” phenomenon. In [52], it was shown that the reduced dynamics of a bicycle traveling along a closed curve and subject to a regular VHC meet the conditions of Proposition 3.6.7.

3.7

Examples

We now present a number of examples illustrating the results of this paper. Later, we return to the material particle example of Section 3.2 and analyze its Lagrangian structure using Theorems 3.3.5 and 3.3.9. Example 3.1 Consider the system 1 [sin(2θ) − sin(θ)θ˙2 ], 2 + cos(θ)

θ¨ =

˜ (x) = 9/(cos x + 2)2 and where θ ∈ [R]2π . The virtual mass and potential are given by M ˜ and V˜ are 2π-periodic, by Theorem 3.3.5 V˜ (x) = 4 − 18(cos x + 1)/(cos x + 2)2 . Since M the system is EL and mechanical. By Proposition 3.6.4, almost all solutions are either oscillations or rotations. Figure 3.2 shows the phase portrait of the system and two phase curves of the system on the phase cylinder [R]2π × R corresponding to an oscillation and △

a rotation. Example 3.2 For the system θ¨ = cos(θ) + 0.5 + cos(θ)θ˙2 , where θ ∈ [R]2π , we have Z ˜ M (x) = exp − 2

x

0

Z ˜ Ψ2 (τ )dτ = exp − 2

x

0

cos τ dτ = exp(−2 sin x),

is 2π-periodic. On the other hand, one can check that

V˜ (2π) = −

Z

2π 0

(cos τ + 0.5) exp(−2 sin τ )dτ ≃ 7.1615 6= 0,


75

Figure 3.2: Example of an EL system.

Left: Phase portrait of an EL system. Right: An oscillation (γ1 ) and a rotation (γ2 ) on the cylinder. so that V˜ is not 2π-periodic. By Theorem 3.3.9, the system is SEL. By Proposition 3.6.5, almost all its solutions are either oscillations or helices. Figure 3.3 shows the phase portrait and two typical phase curves on the cylinder, an oscillation and a helix.

△

˜ (x) = 1 and Example 3.3 For the system θ¨ = λ, with λ 6= 0 and θ ∈ [R]T , we have M ˜ is T periodic and V˜ isn’t, the system is SEL. By Theorem 3.3.9, V˜ (x) = −λx. Since M the Lagrangian is given by (3.6). The Euler-Lagrange equation with this Lagrangian reads 2π ˜ ∂L ˜ d ∂L 2π 2 − = cos (x˙ /2 − λx) (¨ x − λ) = 0. dt ∂ x˙ ∂x λT λT

We see that all solutions of the system θ¨ = λ satisfy the Euler-Lagrange equation, but there are signals (x(t), x(t)) ˙ = (T /4 + kT, 0), k ∈ Z satisfying the Euler-Lagrange equation which do not satisfy the equation θ¨ = λ. Thus, the collection of solutions of a SEL system is contained, but is not equal to, the collection of solutions of the associated Euler-Lagrange equation.

△


76

Figure 3.3: Example of a SEL system.

Left: Phase portrait of a SEL system. Right: An oscillation (γ1 ) and a helix (γ2 ) on the cylinder. Example 3.4 Consider the system θ˙ = − cos(θ) − 2 + (sin(θ) + 2)θ˙2 with θ ∈ [R]2π . We have Ψ1 (θ) = − cos(s) − 2 < 0 and

R 2π 0

˜ 2 (τ )dτ = Ψ

R 2π 0

(sin τ + 2)dτ =

˜ ˜ is not 2π-periodic, 4π > 0. This latter identity implies that M(2π) 6= 0, so that M and the system is neither EL nor SEL. Moreover, by Proposition 3.6.7 the system has ˙ ∈ an exponentially stable limit cycle with domain of attraction including D = {(θ, θ) T [R]2π : θ˙ ≤ 0}. Figure 3.4 depicts the phase portrait of the system along with the stable △

limit cycle.

Example 3.5 We return to the particle mass example of Section 3.2, in which θ ∈ [R]2π and Ψ1 (θ) = −

(a1 b2 + a2 b1 − a1 sin(θ) + a2 cos(θ)) (b1 cos(θ) + b2 sin(θ) + 1)

[(b1 − a1 + cos(θ))2 + (b2 − a2 + sin(θ))2 ]3/2 b1 sin(θ) + b2 cos(θ) , Ψ2 (θ) = − b1 cos(θ) + b2 sin(θ) + 1


77

Figure 3.4: Example of a non-EL system

Left: Phase portrait of a non-EL system with an attractive limit cycle. Right: Two phase curves and the stable limit cycle of the system on the phase cylinder. where ai , bi are the components of a, b ∈ R2 . We now revisit the four cases discussed in Section 3.2. Case 1: a = b = 0. In this case the reduced dynamics reads as θ¨ = 0, an EL system. Case 2: a = 0, b 6= 0. Here we have Ψ1 = 0, implying that V˜ is 2π-periodic. ˜ Moreover, one can check that M(x) = (4 + cos x)2 /25, a 2π-periodic function. Thus the ˙ = 1/2M(θ)θ˙2 is reduced dynamics are EL. In this case, the Lagrangian function L(θ, θ) not equal to the restriction of the Lagrangian of the particle mass, L(q, q) ˙ = (1/2)kqk ˙ 2− P (q) to the constraint manifold. Case 3: a = [1/4 3/4]⊤ , b = [3/4 0]⊤ . In this case Ψ2 (θ) is the same as in case 2, ˜ is 2π-periodic, one can check that V˜ (2π) = 0.2762 6= 0. but now Ψ1 (θ) 6= 0. While M The virtual potential is not 2π-periodic and thus the system is SEL. Figure 3.5 shows two typical solutions on the cylinder, an oscillation and a helix. Case 4: a = b = 0, B(q) = Rβ q, β ∈ (−π/2, π/2), β 6= 0. In this case, the reduced


78

4

3

2

1

θ˙

0

−1

−2

−3

−4

−3

−2

−1

0

1

2

3

θ Figure 3.5: The particle mass motion arising from SEL dynamics. Left: phase portrait of the particle mass example in case 3. Right: an oscillation and a helix on the phase cylinder. dynamics read as tan θ θ¨ = − (tan β)θ˙2 . 5 ˜ We have M(x) = exp(−2

Rx 0

− tan(β)dτ ) = exp(2(tan β)x). This is not 2π-periodic and

thus the reduced dynamics is neither EL nor SEL. In sum, arbitrarily small variations of the parameters a, b, β have drastic effects on the Lagrangian properties of the reduced dynamics of the particle.

△

Chapter 4 VHC Implicitization In this chapter, we present a procedure for making parametric VHCs implicit and thus amenable to feedback implementation. The key ingredient used in developing the procedure is based on computing the resultants of polynomials, a well-known tool in classical algebra. As it has been shown in Chapter 3, regular and odd VHCs will give rise to reduced dynamics with Lagrangian structure. We provide conditions that guarantee regularity and symmetry of the implicitized constraints generated by this procedure.

4.1

Introduction

In order to implement VHC-based controllers we need to find an implicit representation of a given regular parametric VHC for the Euler-Lagrange system

D(q)¨ q + C(q, q) ˙ q˙ + ∇P (q) = B(q)τ,

(4.1)

with n degrees-of-freedom (DOF) and n−1 actuators. We refer to the process of bringing a parametric VHC to its implicit form as VHC implicitization. In order to implicitize parametric VHCs, we first approximate them to an arbitrary degree of accuracy using the Bézier polynomial basis. Then, we implicitize the resulting constraints using an implicit79

Chapter 4. VHC Implicitization

80

ization method taken from classical algebra that is also widely used in computer graphics applications [78, 79]. As a result of the symbolic nature of this method, the implicitized constraints will be in polynomial form and thus particularly well-suited for feedback implementation. We show that under certain conditions, the implicitized constraints are regular VHCs. Moreover, the implicitization preserves the odd symmetry of the original parametric VHC, therefore preserving the Lagrangian structure of the reduced dynamics. This chapter is organized as follows. The formal problem statement and our strategy for solving it are presented in Section 4.2. Section 4.3 reviews preliminaries from algebra, basic properties of Bézier polynomials, and introductory graph theory. Then, we present an approximation method for regular and odd parametric VHCs using the Bézier polynomial basis in Section 4.4. In Section 4.5 we present Procedure 4.1 for implicitizing parametric VHCs with polynomial representation. Next, in Section 4.6, we give necessary and sufficient conditions, in Propositions 4.6.3 and 4.6.4, for the implicitized constraints to be regular when the number of degrees-of-freedom (DOF) are equal to two and three, respectively. Subsequently, we present sufficient conditions in Proposition 4.6.5 for regularity of the implicitized constraints when the number of DOF is greater than three. Finally, we present our solution to VHC implicitization problem in Section 4.7.

4.2

Problem Statement

In this chapter we investigate the following problem.

VHC Implicitization Problem. Given a regular and odd parametric VHC σ : [R]T → Q, and given ǫ > 0, find h : Q → Rn−1 such that (i) dH h−1 (0), σ([R]T ) < ǫ,

(ii) h(q) = 0 is a regular and odd implicit VHC.

81


Solution strategy. Our strategy for solving the VHC implicitization problem unfolds as follows: Step 1 Approximation: We find a regular and odd parametric VHC in the form of a Bézier polynomial σ ˜ : [R]T → Q such that k˜ σ − σkW 1,∞ < ǫ, where k · kW 1,∞ is a Sobolev norm, introduced in Section 4.4. Step 2 Implicitization: Using tools from classical algebra, we develop an implicitization procedure for σ˜ in Step 1. This is done in Section 4.5. Step 3 Investigating regularity: We find sufficient conditions for the implicitized constraints obtained from Step 2 to be regular. This is done in Section 4.6. Step 4 We use the results from Steps 1–3 to present a complete solution to the VHC implicitization problem in Section 4.7. Example 4.1 Consider an underactuated mechanical system with n = 4 degrees-offreedom, inertia matrix D(q) = I4×4 , and control input matrix  1   0  B(q) =   0  0

0 0



  1 0  .  0 1  0 0

A parametric relation q = σ(θ) is a regular parametric VHC for this system if and only if B ⊥ (σ(θ))D(σ(θ))σ ′ (θ) = σ4′ (θ) 6= 0 for all θ ∈ [R]T , due to Corollary 2.3.3. The parametric constraint σ(θ) = [σ1 (θ), · · · , σ4 (θ)]⊤ , σ1 (θ) = θ, σ2 (θ) = T 2 θ − 3T θ2 + 2θ3 , 3 2

2 3

4

5

σ3 (θ) = T θ − 4T θ + 5T θ − 2θ , σ4 (θ) = θ,

(4.2)

82


satisfies σ4′ (θ) = 1 for all θ ∈ [R]T and thus is a regular parametric VHC. In (4.2), σ1 , σ2 : [R]T → R have degree d = 1 while σ3 , σ4 : [R]T → R have degree d = 0. Also, the parametric VHC (4.2) is odd because σ(θ) = −σ(T − θ) for all θ ∈ [R]T . Having established that q = σ(θ) is a regular parametric VHC, the objective is to find an implicit representation of this VHC which preserves the regularity property. We will return to this example several times in this chapter to illustrate the ideas introduced △

at various points.

4.3

Preliminaries

In this section we review preliminaries from algebra and basic properties of Bézier polynomials.

4.3.1

Preliminaries from Algebra

Our approach to solving the VHC implicitization problem relies on computing resultants of polynomials. In this section, we present the definition of the resultant and some of its basic properties. The definitions and results presented in this section are standard and can be found in various textbooks on computer algebra or computer graphics (see, e.g., [79, 78, 80]). Given two polynomials in R[θ]

P1 = P2 =

N1 X

i=0 N2 X

a1i θi , a2i θi ,

(4.3)

i=0

with a1N 6= 0 and a2N 6= 0, their associated Sylvester matrix is a square matrix of 1

2

dimension (N1 + N2 ) defined as

83




1  aN1

a1N1 −1

a10



··· ··· ···      a1N1 a1N1 −1 · · · ··· ··· a10       ···  N2 rows      1 1 1   a a · · · · · · · · · a N1 N1 −1 0      (4.4) Syl(P1 , P2 ) =  − − − − − − − − − − − − − − − − − −     2   aN2 a2N2 −1 · · · ··· ··· a20     2 2 2  N1 rows  a a · · · · · · · · · a 0 N2 N2 −1       ···     2 2 2 aN2 aN2 −1 · · · · · · · · · a0 where the rest of its elements are equal to 0. Note that the first N2 and the last N1 rows of the Sylvester matrix are filled with the coefficients of P1 and P2 , respectively. The resultant of two polynomials P1 and P2 in (4.3) is defined as

Res(P1 , P2 ) := det Syl(P1 , P2 ) .

(4.5)

Res(P1 , P2 ) = (−1)N1 N2 Res(P2 , P1 ).

(4.6)

The following symmetry property follows immediately from the definition in (4.5)

Example 4.2 Consider the following two polynomials in R[θ] P1 = θ2 − q1 , P2 = θ − q2 , where q1 , q2 ∈ R. The resultant of P1 and P2 is 1 0 −q1 Res(P1 , P2 ) = 1 −q2 0 = q22 − q1 . 0 1 −q2

△

84


There exists an intrinsic relationship between the roots of two given polynomials and their resultant. Lemma 4.3.1 ([81]). Consider the two polynomials P1 and P2 in (4.3) of degrees N1 and N2 , respectively. Let Z(P1 ) and Z(P2 ) be the sets of real roots of P1 and P2 , respectively. Let rik be the multiplicity of the root θik ∈ Z(Pi ). Then, the resultant of P1 and P2 , defined in (4.5), satisfies

Res(P1 , P2 )

= aN1

N2

Y

P2 (θ1k )

θ1k ∈Z(P1 )

= (−1)N1 N2 aN2

N1

Y

r1k P1 (θ2k )

θ2k ∈Z(P2 )

r2k

.

(4.7)

Lemma 4.3.1 states that the resultant of any two given polynomials is equal to the product of the values that one polynomial assumes at the roots of the other one. This lemma has the following immediate corollary, which provides the necessary and sufficient condition for the resultant of two polynomials to vanish. Corollary 4.3.2. Consider the two polynomials P1 and P2 in (4.3). Their resultant vanishes if and only if the two have at least a common root. Example 4.3 Consider the two polynomials in Example 4.2. Suppose q1 > 0. Then, √ √ Z(P1 ) = { q1 , − q1 } and Z(P2 ) = {q2 }. We have Res(P1 , P2 ) =

Y

(θ − q2 ) = q22 − q1 .

√ √ θ∈{ q1 ,− q1 }

Therefore, Res(P1 , P2 ) = 0 if and only if q2 ∈ Z(P1 ).

4.3.2

△

Properties of B´ ezier polynomials

The Bézier polynomial basis is broadly used in computer graphics and computer aided geometric design because of its intrinsic numerical stability and wealth of available algorithms (see [82] for a detailed historical survey and a synopsis of the current algorithms

85


and applications). In the context of biped robots, J. Grizzle and collaborators have used Bézier polynomials extensively for parameterizing stable periodic walking gaits [5, 6]. Let N be a non-negative integer, T be a positive real number, and a ∈ RN +1 be a vector. The Bézier polynomial of degree N and vector of coefficients a is a polynomial in R[θ] defined as

N 1 X N k B(θ) = N θ (T − θ)N −k . ak k T k=0

(4.8)

The following lemma summarizes some of the basic properties of Bézier polynomials. These properties are standard and can be found in [82, 83]. Lemma 4.3.3. Let N be a non-negative integer, T be a positive real number, and a ∈ RN +1 be a vector. The Bézier polynomial with degree N and vector of coefficients a defined in (4.8) satisfies (i) End-point values: B(0) = a0 and B(T ) = aN . (ii) Symmetry: B(T − θ) = (iii) Derivatives:

1 TN

N P

k=0

aN −k

N k

k θ (T − θ)N −k .

N −1 N −1 k N X θ (T − θ)N −k , (ak+1 − ak ) B (θ) = N k T k=0 N −2 N(N − 1) X N −2 k ′′ B (θ) = θ (T − θ)N −k . (ak+2 − 2ak+1 + ak ) N k T k=0 ′

Given a set S and a function f : S → R, we say that the sequence of functions fN N ∈N , where each fN is a real-valued function with domain S, uniformly converges to f , if for every ǫ > 0, there exists a natural number N0 such that for all x ∈ S and all

N ≥ N0 , |fN (x) − f (x)| < ǫ. The following proposition states that Bézier polynomials can uniformly approximate a smooth function and its derivatives.

86


Proposition 4.3.4 (Theorem 7.1.6 in [84]). Let k be some non-negative integer and f be a real-valued class C k function on [0, T ]. Consider the sequence of Bézier polynomials BN (·) N ∈N with

where

N N k 1 X θ (T − θ)N −k , ak BN (θ) = N k T k=0

ak = T N f k Then, the sequence

di B N

dθ i

N ∈N

T . N

converges uniformly to

di f dθ i

(4.9)

on [0, T ] for each 0 ≤ i ≤ k.

The following lemma regarding the symmetry preserving feature of Bézier polynomials will be used in the sequel. Lemma 4.3.5. Let f : [R]T → R be a T -periodic, odd, and class C 2 function. Consider the sequence of Bézier polynomials BN (·) N ∈N given in (4.9). Then BN (·) is an odd and T -periodic function for each N ∈ N.

Proof. Suppose that f : [R]T → R is odd. Pick a positive integer N. Since ak = T T TNf kN , we have aN −k = T N f T −k N . The function f is odd, thus, f (θ) = −f (T −θ) for all θ ∈ [R]T . It follows that

aN −k = −ak .

(4.10)

By Property (i) of Bézier polynomials in Lemma 4.3.3, BN (0) = a0 and BN (T ) = aN . Since a0 = T N f (0) and f (0) = 0, we have BN (0) = BN (T ) = 0. Hence, BN (·) is a T -periodic function. Moreover, the symmetry property of Bézier polynomials in Lemma 4.3.3 and (4.10) imply that

B(θ) + B(T − θ) = 0,

87


for all θ ∈ [R]T . Therefore, BN (·) is an odd function. As it can be seen from Equation (4.10), odd symmetry of functions manifests itself as anti-symmetry of the coefficients of their Bézier approximations.

4.3.3

Preliminaries from Graph Theory

Outcomes of implicitization procedures in the chapter can be effectively represented by undirected graphs. In this section, we present the definition of an undirected graph and other standard related concepts and tools from introductory graph theory [85, 86]. An undirected graph (or just a graph) is an ordered pair G = (V, E), where V is a finite, non-empty set of elements called vertices, and E is a set of distinct pairs {(vi , vj ) : vi , vj ∈ V}. Each element (vi , vj ) ∈ E is called an edge and said to join the vertices vi and vj (see Figure 4.1). v1

v2

v1 v1

v4

v3

v3

v2

v2

Figure 4.1: Undirected graphs

The local structure of a graph can be described by the neighborhoods and degrees of its vertices. For a graph G = (V, E) and a vertex v in V, the neighborhood of the vertex v is defined as

Nv := u ∈ V − {v} : (u, v) ∈ E .

The degree of a node v is defined as cardinality of Nv .

Given a graph G = (V, E), a walk in G is an alternating sequence

88


W : v1 e1 v2 e2 · · · ek−1 vk of vertices vi ∈ V and edges ei = (vi , vi+1 ) ∈ E for every i = 1, 2, · · · , k − 1. If the vertices of a walk W are distinct, W is called a path joining vertices v1 and vk . If the nodes v1 , · · · , vk−1 are distinct and v1 = vk , W is a called a cycle. A graph without cycles is said to be acyclic. A graph G = (V, E), is said to be connected if for every distinct vertices vi , vj there exists a path joining vi and vj . A complete graph is a connected graph in which every pair of distinct vertices is connected by an edge. A spanning tree is a connected acyclic graph. A path graph is a spanning tree with two vertices of degree one, and the other vertices of degree two. Therefore, every path graph is a spanning tree but the converse is not necessarily true. The concepts of complete graph, spanning tree, and path graph are illustrated in Figure 4.2. v1

v2

v1

v2

v4

v3

v4

v3

(a) A spanning tree (in red) on a com-

(b) A path graph (in red) on a com-

plete graph (in dashed black) with

plete graph (in dashed black) with

n = 4 vertices.

n = 4 vertices.

Figure 4.2: Spanning trees and path graphs.

We have the following standard results from introductory graph theory [85]. Lemma 4.3.6. Given a positive integer n, • every complete graph has

n(n−1) 2

edges,

89

Chapter 4. VHC Implicitization • every connected graph has a spanning tree,

• the number of distinct spanning trees with n vertices is given by the Cayley’s formula and equal to nn−2 , • the number of distinct path graphs with n vertices is equal to

n! , 2

• a graph G with n vertices is a spanning tree if and only if it is a connected graph with n − 1 edges.

4.4

Step 1: Polynomial approximation

The formal problem that we study in this section can be stated as follows. Polynomial approximation of regular and odd parametric VHCs. Given ǫ > 0 and a regular and odd parametric VHC

σ : [R]T → Q, θ 7→ col σ1 (θ), · · · , σn (θ) , find a regular and odd parametric VHC σ˜ in polynomial form such that

k˜ σ − σkW 1,∞ < ǫ,

σ (θ) − σ(θ)k + max k˜ σ ′ (θ) − σ ′ (θ)k is the Sobolev norm where k˜ σ − σkW 1, ∞ := max k˜ θ∈[0, T ]

θ∈[0, T ]

of the signal σ ˜ − σ. Remark 4.4.1. The reason for using the Sobolev norm is that by guaranteeing the convergence of both σ ˜ to σ and σ ˜ ′ to σ ′ , we are able to guarantee that the approximation of the VHC preserves the regularity property B ⊥ (σ)D(σ)σ ′ 6= 0. This fact is used in Proposition 4.4.2 below.

90


Before proceeding with the approximation, we need to discuss the issue of degree. If the configuration variable qi belongs to [R]Ti (e.g., qi is an angle) then the curve q = σ(θ) may perform multiple revolutions around the i-th axis. More precisely, if qi ∈ [R]Ti , then we say that the i-th component of σ, σi (θ), has degree di ∈ Z if lim σi (θ) = σi (0) + di · Ti .

θ→T −

In other words, the integer di is the number of revolutions that the variable qi performs while q moves once around the curve Im(σ). If σi has degree di 6= 0, then we only approximate the quantity σi (θ) − di TTi · θ so as to deal with a T -periodic approximation. This idea is made precise below. Given a regular and odd parametric VHC σ : [R]T → Q with deg(σi ) = di , i ∈ n, we consider the following functions

i (θ) = di (∀i ∈ n) σÑ i

Ti i · θ + BN (θ), i T

(4.11)

i where Ni is a positive integer and BN (·) is the Bézier approximation of σi (θ) − di TTi · θ. i

By Proposition 4.3.4, we have

i BN (θ) i

Ni 1 X T i Ni = N θk (T − θ)Ni −k , aik = T Ni σi k , ak i T Ni k k=0

We have the new parametric constraint

1 n σ ˜ (·) : [R]T → Q, θ 7→ σ Ñ1 (θ), · · · , σ Ñn (θ) .

(4.12)

Proposition 4.4.2. Consider the underactuated mechanical system (4.1). Let σ : [R]T → Q be a C 2 regular and odd parametric VHC. For any ǫ > 0, there exist large enough positive integers N1 , · · · , Nn , such that the map σ ˜ : [R]T → Q given in (4.11) and (4.12) is a regular and odd parametric VHC and satisfies k˜ σ − σkW 1,∞ < ǫ.

91


i Proof. By Proposition 4.4.2 the sequence of odd Bézier polynomials σ Ñi (θ)−di TTi ·θ Ni ∈N ′i and the sequence of their derivatives σ Ñi (θ) − di TTi Ni ∈N converge uniformly to the

functions σi (θ) − di TTi · θ and σi′ (θ) − di TTi for each 1 ≤ i ≤ n on [R]T . Therefore, for sufficiently large N1 , · · · , Nn , the inequality k˜ σ − σkW 1,∞ < ǫ holds. In order to show regularity, we need to prove that for sufficiently large N1 , · · · , Nn , ∀θ ∈ [R]T

′ B⊥ σ ˜ (θ) D σ˜ (θ) σ ˜ (θ) 6= 0.

We claim that for sufficiently large N1 , · · · , Nn , and all θ ∈ [R]T , the pair σ ˜ (θ), σ ˜ ′ (θ) belongs to the compact set

S = {(x, y) ∈ Q × Rn : kxk ≤ max kσ(θ)k + ǫ, kyk ≤ max kσ ′ (θ)k + ǫ}, θ∈[0,T ]

θ∈[0,T ]

Indeed, if N1 , · · · , Nn , are sufficiently large, then k˜ σ − σkW 1,∞ < ǫ implies that max k˜ σ (θ) − σ(θ)k < ǫ, max k˜ σ ′ (θ) − σ ′ (θ)k < ǫ.

θ∈[0,T ]

θ∈[0,T ]

Let us define the continuous function f (x, y) := B ⊥ (x)D(x)y. Since f is continuous, it is uniformly continuous on the compact set S. Moreover, regularity of the VHC q = σ(θ) is equivalent to f σ(θ), σ ′ (θ) 6= 0 for all θ ∈ [R]T . Therefore, uniform continuity of

f (·) along with regularity of the VHC q = σ(θ) imply that for sufficiently large N1 , · · · , Nn , and all θ ∈ [R]T , f σ˜ (θ), σ ˜ ′ (θ) 6= 0. This is equivalent to regularity of the VHC q=σ ˜ (θ).

The parametric VHC (4.2) in Example 4.1 is already in polynomial form and there is no need for interpolation using Bézier polynomials.

92


4.5

Step 2: Implicitization

In this section, we find implicit representations of parametric VHCs in polynomial form. Given a point q ∈ Q and odd parametric VHCs σ ˜ : [R]T → Q with polynomial representation Ni X ∀i ∈ n σ ˜i (θ) = bik θk ,

(4.13)

k=0

we define the following polynomials

Piq (θ) := σ ˜i (θ) − qi , i ∈ n.

(4.14)

Using (4.14), we define the implicit constraints

˜ ij : Q → R, h ˜ ij (q) = Res(P q , P q ), i, j ∈ n, h i j

(4.15)

where Res(·, ·) is defined in (4.5). ˜ −1 (0) = h ˜ −1 (0) by the symmetry property of resultants in (4.6). Therefore, Note that h ij ji ˜ ji in the subsequent we will not distinguish between the implicit constraints ˜hij and h development. In the next definition we create an equivalence between polynomial implicit constraints and undirected graphs. Definition 4.5.1. (a) Let σ ˜ : [R]T → Q be a parametric VHC with polynomial representation (4.13). Con˜ i j (q), 1 ≤ k ≤ n − 1, sider the implicit VHC h : Q → Rn−1 with hk (q) = h k k ˜ i j is defined in (4.15). The implicitization graph of the ik , jk ∈ n, and where h k k implicit VHC h is the undirected graph Gh = (V, E) with V = {v1 , · · · , vn } and edge set E = (vik , vjk ) : k ∈ {1, · · · , n − 1}, ik , jk ∈ n , where each vertex vi ∈ V corre-

sponds to the polynomial Piq defined by (4.14) and each edge (vik , vjk ) ∈ E corresponds ˜ i j (q), 1 ≤ k ≤ n − 1. to the implicit constraint hk (q) = h k k

93


(b) Given an undirected graph G = (V, E), with V = {v1 , · · · , vn } and exactly n − 1 edges E = {(vik , vjk ) : 1 ≤ k ≤ n − 1, ik , jk ∈ n}, the implicit constraint induced by G ˜i j , · · · , h ˜ i j ), where h ˜ i j is defined in (4.15). is the function hG = col(h 1 1 n−1 n−1 k k △ Definition 4.5.1 states that vertex vi of the graph Gh corresponds to polynomial Piq , and an edge between vertex vi and vertex vj corresponds to having selected polynomials ˜ ij (q). Thus the structure of the graph gives information Piq and Pjq to form the resultant h about which polynomials Piq are selected to form the implicit constraint.

v1

˜ 12 (q) h

v2

˜ 23 (q) h v3

Figure 4.3: Equivalence relation between undirected graphs and implicit constraints.

Example 4.4 Let n = 3 and σ ˜ = (˜ σ1 , σ ˜2 , σ ˜3 ) be a parametric VHC with polynomial ˜ 12 (q) h ˜ 23 (q)]⊤ where representation (4.13). Consider the implicit constraint h(q) = [h ˜ 12 (q) and h ˜ 23 (q) are given in (4.15). The implicitization graph of h is the undirected h graph, G = (V, E), with three vertices v1 , v2 , and v3 and two edges that connect v1 , v2 , and v2 , v3 , respectively. The undirected graph G is shown in Figure 4.4. Conversely, the ˜ 12 (q) h ˜ 23 (q)]⊤ . △ undirected graph G = (V, E) induces the implicit constraint h(q) = [h The following lemma highlights an important property of implicit VHCs whose implicitization graphs are spanning trees. Proposition 4.5.2. Let σ ˜ : [R]T → Q be a parametric constraint with polynomial representation (4.13). Consider the implicit VHC

94


˜ i j (q), 1 ≤ k ≤ n − 1, ik , jk ∈ n, h(q) = col hk (q) , hk (q) = h k k

where the functions ˜hik jk are defined in (4.15). Let the implicitization graph of h be Gh = (V, E). If the implicitization graph of h, Gh , is a spanning tree, then \

Im(˜ σ) =

˜ −1 (0). h ij

(4.16)

{vi , vj }∈V

Proof. Pick q ∈ Im σ ˜ and {˜ σi , σ˜j } ∈ V, then (qi , qj ) = πij (q). We have (qi , qj ) = (˜ σi (θ0 ), σ ˜j (θ0 )), for some θ0 ∈ [R]T . Therefore, θ0 is a common root of the two polynomials Piq and Pjq defined by (4.14). By Corollary 4.3.2, each constraint hij vanishes if and only if the two ˜ ij (q) = 0 for all {vi , vj } ∈ V, and polynomials Piq and Pjq have common roots. Thus, h q∈

\

˜ −1 (0), h ij

{vi , vj }∈V

implying that

Next, we show that

Im σ ˜ ⊂

\

{vi , vj }∈V

Pick q ∈

T

{vi , vj }∈V

\

˜ −1 (0). h ij

{vi , vj }∈V

˜ −1 (0) ⊂ Im σ h ˜ . ij

˜ −1 (0). We need to show that qk ∈ Im(˜ h σk ) for all k ∈ n. Pick k ∈ n. ij

Since Gh is a spanning tree, it is a connected graph. Therefore, there exists an edge joining ˜ −1 (0) and Corollary 4.3.2 imply the vertex vk ∈ V to another vertex vl 6= vk . Hence, q ∈ h kl that the two polynomials Pkq and Plq have a common root θ0 ∈ [R]T . In particular, Pkq (θ0 ) = σ˜k (θ0 ) − qk = 0. Thus, qk ∈ Im(˜ σk ), as required.


95

The following procedure, a direct result of Proposition 4.5.2, can be used to find n − 1 implicit constraints that are candidate solutions for the VHC implicitization problem.

Procedure 4.1 (Implicitization) (i) Consider a parametric VHC σ ˜ : [R]T → Q with polynomial representation (4.13). (ii) Construct a spanning tree G = (V, E) with set of vertices V = {vi }i∈n and set of edges E = (vik , vjk ) k∈{1,··· ,n−1} . (iii) The implicit VHC hG associated with the spanning tree G is an implicit representation of the parametric VHC σ˜ . Step (ii) of Procedure 4.1 gives the designer freedom in choosing the implicit VHCs. Indeed, the designer has nn−2 choices according to the Cayley’s formula in Lemma 4.3.6. Example 4.5 Consider the parametric constraint σ ˜ (θ) = (θ2 , θ). Consider the two polynomials

˜2 (θ) − q2 , ˜1 (θ) − q1 , P2q = σ P1q = σ in Example 4.2. In this example, we construct the unique implicitization graph with vertex set V = {v1 , v2 } and edge set E = {(1, 2)}. Equation (4.15) yields ˜ 12 (q1 , q2 ) = q 2 − q1 . h 2 ˜ 12 (q). We want to verify that h−1 (0) = Im(˜ We set h(q) = h σ ). The curve Im(˜ σ ) is

Im(˜ σ ) = {(q1 , q2 ) : q1 = θ2 , q2 = θ} ˜ 12 is The zero level-set of the implicit constraint h

˜ −1 (0) = {(q1 , q2 ) : q 2 = q1 }. h 12 2


96

˜ −1 (0) = Im(˜ Therefore, h−1 (0) = h σ ). 12

△

Example 4.6 Consider the parametric VHC given by (4.2) in Example 4.1. Following Procedure 4.1, each choice of spanning tree with four vertices induces an implicit representation of the VHC. As we have seen, according to Cayley’s formula there are 42 distinct spanning trees Gi with four vertices. Figure 4.4 depicts three such spanning trees, which induce three different implicit representations of the VHC:

hG1

where

      ˜ ˜h12 ˜h12 h12            ˜  , hG3 = h  ˜ 24  = 23  ˜ 14  . h  , hG2 = h       ˜ 34 ˜h34 ˜h34 h

(4.17)

˜ 12 = −q2 + T 2 q1 − 3T q 2 + 2q 3 , h ˜ 13 = −q3 + T 3 q 2 − 4T 2 q 3 + 5T q 4 − 2q 5 , h 1 1 1 1 1 1 ˜ 14 = q1 − q4 , h

˜ 23 = −8q 3 + 2T 2 q2 q 2 − 2q 5 , h 3 3 2

˜ 24 = q2 − T 2 q4 + 3T q 2 − 2q 3 , h 4 4

˜ 34 = q3 − T 3 q 2 + 4T 2 q 3 − 5T q 4 + 2q 5 , h 4 4 4 4

Which of the above implicit representations preserves the regularity property? This is the subject of the next section. v1

v4

G1

v2

v1

v3

v4

G2

v2

v1

v3

v4

G3

v2

v3

Figure 4.4: The spanning trees corresponding to the implicit VHCs given by (4.17).

△

97


4.6

Step 3: Regularity of implicit constraints

In this section, we investigate conditions under which the implicit constraints generated by Procedure 4.1 are regular. We begin with two lemmas that are needed in the following development. Lemma 4.6.1. Let σ ˜ : [R]T → Q be a parametric VHC with polynomial representation (4.13). Suppose that for some i, j ∈ n, the projection of σ ˜ on the qi –qj coordinate ˜ ij defined plane, πij Im(˜ σ ) , is a smooth Jordan curve. Consider the implicit constraint h by (4.14) and (4.15), respectively. Then, there exists θ0 ∈ [R]T such that

σ ˜ (θ0 )

if and only if θ0 is a root of Pj

˜ ij ∂h = 0, ∂qi q=˜σ(θ0 )

(θ) with multiplicity greater than one.

˜ ij /∂qi ) Proof. Let θ0 ∈ [R]T be such that (∂ h = 0. Without loss of generality, suppose σ ˜ (θ0 ) that i = 1 and j = 2. Define

f (x) := Res(˜ σ1 (θ) − x, σ ˜2 (θ) − σ ˜2 (θ0 )), and note that ˜ 12 ∂h df . σ ˜1 (θ0 ) = dx ∂q1 q=˜σ(θ0 )

According to Lemma 4.3.1, we have

f (x) = α

Y

σ ˜ (θ ) θ2k ∈Z(P2 0 )

where α = a1N

N2

r k σ ˜1 (θ2k ) − x 2 , σ ˜ (θ0 )

(4.18)

σ ˜ (θ )

) is the set of real roots of P2 0 (θ). Since θ0 is a common root of σ˜1 (θ) − σ˜1 (θ0 ) and σ ˜2 (θ) − σ˜2 (θ0 ), f σ ˜1 (θ0 ) = Res σ ˜1 (θ) − σ ˜1 (θ0 ), σ ˜2 (θ) − σ ˜2 (θ0 ) = 0. Using the identity (4.18), 1

is a nonzero constant and Z(P2

98


∃θ2k

∈Z σ ˜2 (θ) − σ ˜2 (θ0 ) σ ˜ (θ0 )

Hence, σ ˜1 (θ2k ) = σ ˜1 (θ0 ). Also, θ2k ∈ Z(P2

σ ˜1 (θ2k ) − σ ˜1 (θ0 ) = 0.

) implies that σ ˜2 (θ2k ) = σ˜2 (θ0 ). Accordingly, (˜ σ1 (θ0 ), σ ˜2 (θ0 )) = (˜ σ1 (θ2k ), σ ˜2 (θ2k )). Since π12 Im(˜ σ ) does not have self-intersections, we

conclude that θ2k is unique and equal to θ0 . Denote r = r2k . Then, r f (x) = σ ˜1 (x) − σ ˜1 (θ0 ) Q(x),

where Q(˜ σ1 (θ0 )) 6= 0. Taking the derivative of f and using the fact that Q(˜ σ1 (θ0 )) 6= 0, we deduce that df σ1 (θ0 ) = 0 dx

if and only if r ≥ 2. Therefore,

σ ˜ (θ0 )

if and only if θ0 is a root of Pj

˜ ij ∂h = 0, ∂qi q=˜σ(θ0 )

=σ ˜j (θ) − σ ˜j (θ0 ) with multiplicity greater than one.

Lemma 4.6.2. Let i, j, k, l ∈ n be such that i 6= j, j 6= k, and k 6= l. Consider the parametric VHC σ ˜ : [R]T → Q with polynomial representation (4.13), and the implicit ˜ ij and h ˜ kl defined by (4.14). Suppose that πij Im(˜ constraints h σ ) , πjk Im(˜ σ ) , and πkl Im(˜ σ ) , are smooth Jordan curves with regular parameterizations πij ◦ σ ˜ , πjk ◦ σ ˜ , πkl ◦ σ ˜ : [R]T → R2 . Then,

˜ ij ∂h rank ∂qi

˜ kl ∂h = 1. ∂ql Im(˜σ)

Proof. Suppose, by way of contradiction, that there exists θ0 ∈ [R]T such that ˜ ij ∂h rank ∂qi

˜ kl ∂h = 1. ∂ql q=˜σ(θ0 )

(4.19)

99


Since πij Im(˜ σ) and πkl Im(˜ σ ) are smooth Jordan curves, Lemma 4.6.1 implies that σ ˜ (θ0 )

(4.19) holds if and only if θ0 is a common root of the two polynomials Pj σ ˜ (θ0 )

σ ˜j (θ) − σ ˜j (θ0 ) and Pk

(θ) =

(θ) = σ ˜k (θ) − σ ˜k (θ0 ) with multiplicity greater than one for both

of them. As a result, there exists θ0 ∈ [R]T such that σ ˜ (θ0 )

Pj

(θ) = (θ − θ0 )rj Qj (θ), σ ˜ (θ0 )

where rj ≥ 2 and Qj (θ0 ) 6= 0. But Pj

(θ) = σ ˜j (θ) − σ ˜j (θ0 ) by (4.14). Thus,

d σ˜ (θ0 ) P (θ0 ) = σ ˜j′ (θ0 ) = 0. dθ j σ ˜ (θ0 )

Applying the same argument to the polynomial Pk

,σ ˜ ′ (θ0 ) = 0. This is a contradiction

to the assumption that parameterization, πjk ◦ σ ˜ is regular. Using Lemma 4.6.2, we now give conditions under which Procedure 4.1 results in a regular VHC. The next three propositions concern the cases n = 2, n = 3, and general n when the spanning trees in Step (ii) of Procedure 4.1 are path graphs. The propositions can be used to check a priori regularity of

n! 2

out of the implicit constraints generated

by Procedure 4.1. If σ ˜ is a regular parametric VHC, then checking regularity of the implicitized constraint h obtained from Procedure 4.1 amounts to checking

rank(dh) Im(˜σ) = n − 1.

Proposition 4.6.3 (n = 2). Let dim(Q) = 2 and σ ˜ : [R]T → Q be the parametric VHC (4.13). Consider the spanning tree G = (V, E) with V = {v1 , v2 } and E = {(1, 2)}. The implicit VHC hG that is induced by G is a regular VHC if and only if σ ˜ is a regular VHC. ˜ 12 (q), where h ˜ 12 (q) Proof. The implicit VHC hG generated by Procedure 4.1 is hG (q) = h is given in (4.15).

100


(=⇒) Since σ ˜ is a regular parametric VHC, the VHC curve Im(˜ σ ) is smooth and Jordan. Therefore, sufficiency immediately follows from Lemma 4.6.2 by setting i = k = 1, j = l = 2. ˜ −1 (0) = Im(˜ ˜ 12 ) ˜ 12 (q) = (⇐=) By Proposition 4.5.2, h σ ). Also, rank(dh = 1 since h 12 Im(˜ σ)

0 is a regular VHC. Thus, Im(˜ σ ) is an embedded submanifold of Q due to the preimage theorem (see, e.g., [53]).

Proposition 4.6.4 (n = 3). Let dim(Q) = 3 and σ˜ : [R]T → Q be a regular parametric VHC with polynomial representation (4.13). Let G = (V, E) be a spanning tree with vertex set V = {v1 , v2 , v3 } and edge set E = (vik , vjk ) : k ∈ {1, 2}, ik , jk ∈ {1, 2, 3} . Let the

implicit constraint induced by G be hG . Then, hG is a regular VHC if and only if the curves πi1 j1 Im(˜ σ ) and πi2 j2 Im(˜ σ ) are smooth Jordan with regular parameterizations ˜ , πi2 j2 ◦ σ ˜ and σ ˜i2 is strictly monotonic. πi1 j1 ◦ σ

Proof. By reordering of indices, the implicit constraint obtained from Procedure 4.1 can ˜ 12 , h ˜ 23 ). By Proposition 4.5.2, h−1 (0) = Im σ be considered to be hG = col(h ˜ . G (=⇒) The Jacobian of the implicitized constraint is   ˜ ˜ 0   ∂q1 h12 ∂q2 h12 dhG =  . ˜ 23 ∂q h ˜ 23 0 ∂q2 h 3 The curves π12 Im(˜ σ ) and π23 Im(˜ σ ) are smooth Jordan with regular parameterizations π12 ◦ σ˜ and π23 ◦ σ ˜ . Hence, by setting i = k = 1 and j = l = 2 in Lemma 4.6.2, we

have

˜ 12 rank ∂q1 h

˜ 12 ∂q2 h

= 1.

(4.20)

Im(˜ σ)

Similarly, by setting i = k = 2 and j = l = 3 in Lemma 4.6.2, we get ˜ 23 ˜ 23 rank ∂q2 h ∂q3 h = 1.

(4.21)

Im(˜ σ)

Therefore, rank(dh) Im(˜σ) = 2 if and only if ˜ 12 rank ∂q1 h

˜ 23 ] ∂q3 h

Im(˜ σ)

= 1.

(4.22)

101


Suppose, by way of contradiction, that this is not the case. Then, there exists θ0 ∈ [R]T such that ˜ ∂q1 h12

q=˜ σ(θ0 )

˜ = ∂q3 h23

= 0.

(4.23)

q=˜ σ(θ0 )

σ ˜ (θ0 )

Lemma 4.6.1 implies that the polynomial P2

has a root at θ0 with multiplicity greater

than one. Since, d σ˜ (θ0 ) P (θ) = σ ˜2′ (θ), dθ 2

(4.24)

we conclude that, σ ˜2′ (θ0 ) = 0. This contradicts the assumption of strict monotonicity of σ ˜2 . (⇐=) Suppose that the implicit constraint hG obtained from Procedure 4.1 is a regu = 2, then (4.20), lar VHC. By Proposition 4.5.2, Im(˜ σ ) = h−1 G (0). Since rank(dhG ) Im(˜ σ) ˜ −1 (0) = (4.21), and (4.22) hold by necessity. Thus, the preimage theorem implies that π12 h 12 ˜ −1 (0) = π23 Im(˜ π12 Im(˜ σ ) and π23 h σ ) are smooth Jordan curves with regular pa23 rameterizations. Suppose, by way of contradiction, that σ ˜2 is not strictly monotonic. Then, there exists θ0 ∈ [R]T such that σ ˜2′ (θ0 ) = 0, and, by (4.24), we conclude that θ0 σ ˜ (θ0 )

is a root of P2

with multiplicity greater than one. Accordingly, Lemma 4.6.1 implies

that (4.23) holds. This contradicts the assumption of regularity of the implicit VHC hG . Therefore, σ ˜2 is strictly monotonic. Proposition 4.6.5 (n ≥ 4). Let dim(Q) = n, where n ≥ 4, and σ ˜ : [R]T → Q be a regular parametric VHC with polynomial representation (4.13). Let G = (V, E) be a path graph with vertex set V = {v1 , · · · , vn } and edge set E = (vik , vjk ) : k ∈ {1, · · · , n − 1}, ik , jk ∈ n . Let the implicit constraint induced by G be hG . Suppose that πik ,jk Im(˜ σ ) , πik +1 ,jk +1 Im(˜ σ ) , πik +2 ,jk +2 Im(˜ σ ) , 1 ≤ k ≤ n−1, 1 ≤ ik , jk ≤ n−2, are σ: σ , πik +2 ,jk +2 ◦˜ smooth Jordan curves with regular parameterizations πik ,jk ◦˜ σ , πik +1 ,jk +1 ◦˜

[R]T → R2 such that (vik , vjk ), (vik +1 , vjk +1 ), and (vik +2 , vjk +2 ) are edges of the path graph E. Suppose that at least one of the pairs (˜ σi2 , σ ˜i3 ), or (˜ σin−2 , σ ˜in−1 ) are strictly monotonic. Then, the implicit constraint generated by Procedure 4.1 is a regular VHC.

102


Proof. By reordering of indices, the implicit constraint obtained from Procedure 4.1 can ˜ 12 , · · · , h ˜ (n−1),n ). By Proposition 4.5.2, h−1 (0) = Im σ ˜ . be considered as hG = col(h G Since σ ˜ is a regular parametric VHC, we need to show that rank(dhG ) Im(˜σ ) = n − 1. The

(i + 1)th column of the Jacobian dhG , when 1 ≤ i ≤ n − 2, is equal to 



We claim that

0     . ..         ∂ h  qi+1 ˜ i,i+1 (q)  ith row    ˜ i+1,i+2 (q) vi+1 (q) := ∂qi+1 h i + 1th row .       0     ..   .     0

∀q ∈ Im(˜ σ)

rank [v2 (q), · · · , vn−1 (q)] = n − 2.

We show that vi+1 (q) and vi+2 (q) are linearly independent for all 1 ≤ i ≤ n − 2 and all q ∈ Im(˜ σ ). Without loss of generality, we show this for i = 1. We have 

 ˜ 0  ∂q2 h12 (q)     ∂ h  ˜ ˜ (q) ∂ h (q) q 23 q 23  2  3     ˜ 0 ∂q3 h34 (q)   . v2 (q), v3 (q) =      0 0     . .   .. ..     0 0

Since π23 Im(˜ σ ) is a smooth Jordan curve with regular parameterization π23 ◦ σ˜ , due to Lemma 4.6.2,

103


˜ 23 = 1. ∂q3 h Im(˜ σ) Let q ∈ Im(˜ σ ). Then, rank [v2 (q), v3 (q)] < 2 if and only if v2 (q) = 0, or v3 (q) = 0, ˜ 23 rank ∂q2 h

˜ ˜ 12 (q) = ∂q ˜h34 (q) = 0. Also, note that v2 (q) = 0 is equivalent to ∂q h or ∂q2 h 3 2 12 (q) = ˜ 23 (q) = 0, and v3 (q) = 0 is equivalent to ∂q h ˜ ˜ ∂q2 h 3 23 (q) = ∂q3 h34 (q) = 0. ˜ 12 (˜ ˜ 23 (˜ Suppose, by way of contradiction, that ∂q2 h σ (θ0 )) = ∂q2 h σ (θ0 )) = 0 for some ˜1 (θ) − σ ˜1 (θ0 ) and σ ˜3 (θ) − σ ˜3 (θ0 ) have roots with θ0 ∈ S1 . Then, due to Lemma 4.6.1, σ

multiplicity greater than one at θ0 . This assumption contradicts the earlier assumption that π13 Im(˜ σ ) is a smooth Jordan curve with regular parameterization π13 ◦ σ ˜ . Simi ˜ 23 = ∂q h ˜ ˜ ˜ larly, ∂q3 h σ) 3 34 = 0, and ∂q2 h12 = ∂q3 h34 = 0 lead to contradiction with π14 Im(˜ and π24 Im(˜ σ ) being smooth Jordan and having regular parameterizations, respectively. We deduce that rank [vi+1 (q), vi+2 (q)] = 2 for all q ∈ Im(˜ σ ) and all 1 ≤ i ≤ n − 2. Hence, rank(dh) Im(˜σ) = n − 1 if and only if rank[v1 (q), v2 (q)] = 2, or

rank[vn−1 (q), vn (q)] = 2, for all q ∈ Im(˜ σ ). Without loss of generality, we show that rank[v1 (q), v2 (q)] = 2 for all q ∈ Im(˜ σ ). We have 

˜ ˜  ∂q1 h12 (q) ∂q2 h12 (q)   ˜ 23 (q) 0 ∂q2 h    0 0  v1 (q), v2 (q) =    0 0   .. ..  . .   0 0



       .       

104


Since π12 Im(˜ σ ) is a smooth Jordan curve with regular parameterization π12 ◦ σ˜ , due to Lemma 4.6.2,

˜ 12 rank ∂q1 h

˜ 12 ∂q2 h

= 1.

Im(˜ σ)

Therefore, rank[v1 (q), v2 (q)] < 2 if and only if there exists q ∈ Im(˜ σ ) such that at least ˜ 12 (q) and ∂q h ˜ one of ∂q1 h ˜ (θ0 ) for some θ0 ∈ S1 . Let q = σ˜ (θ0 ) for some 2 23 (q) vanishes at σ ˜ 12 (q) = 0 implies that θ0 is a common root of polynomials θ0 ∈ [R]T . The condition ∂q1 h σ ˜ (θ0 )

P2

σ ˜ (θ0 )

and P3

with multiplicity greater than one for both of the two polynomials, due

to Lemma 4.6.1. This is in contradiction with the assumption that σ˜2 and σ ˜3 are strictly monotonic. We deduce that rank[v1 (q), v2 (q)] = 2. Therefore, rank(dh) Im(˜σ) = n − 1 and h is a regular VHC.

Remark 4.6.6. Any curve in an n-dimensional configuration space Q can be projected onto (n2 − n)/2 coordinate planes qi − qj . Not all of these projections need to be Jordan. Proposition 4.6.5 requires that, when n ≥ 4, n − 1 of (n2 − n)/2 possible projections be Jordan. Example 4.7 Consider the implicit VHCs given by (4.17) in Example 4.6 and their associated spanning trees G1 , G2 , and G3 . Note that G1 , G2 , and G3 are path graphs. All of the curves πij ◦ σ, i 6= j, are Jordan except the curve π23 ◦ σ (see Figure 4.5). Also, the functions σ1 and σ4 are strictly increasing. The Jacobian of the implicit constraints in (4.17) are

105

Chapter 4. VHC Implicitization 0.02

0.015

0.01

q3

0.005

0

−0.005

−0.01

−0.015

−0.02 −0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

q2

Figure 4.5: The curve π23 ◦ σ.

dhG1

   ˜ ˜ 0  ∂q1 h12 −1 ∂q1 h12 −1 0     ˜ 23 ˜ 24  = ∂2 h 1 0 ∂4 h  , dhG2 =  0  0    ˜ 34 0 0 0 0 1 ∂4 h  ˜ ∂q1 h12  dhG3 =   1  0

0



0   ∂3 ˜h23 0  ,  ˜ 24 1 ∂4 h  −1 0 0   0 0 −1  .  ˜ 24 0 1 ∂4 h

(4.25)

As it can be seen from (4.25), the implicit constraint hG3 is a regular VHC. Indeed, regularity of the VHC hG3 follows from Proposition 4.6.5 since the curves π12 ◦ σ, π14 ◦ σ, and π34 ◦ σ are smooth Jordan and σ1 and σ4 are strictly monotonic. On the other hand, ˜ 23 = ∂3 h ˜ 23 = 0 at the origin and the the implicit constraint hG2 is not regular because ∂2 h Jacobian dhG1 loses rank. Indeed, the curve π34 ◦ σ has a self-intersection at the origin which corresponds to rank(dhG1 ) < 3. Finally, the implicit constraint hG2 is not regular √ ˜ 12 = ∂4 h ˜ 24 = 0 and rank(dhG ) < 3. Indeed, if q1 = q3 = (3 ± 3)/6 resulting in ∂1 h 2 neither of the functions σ2 and σ3 , corresponding to the second and third nodes of the path graph G2 , are strictly monotonic.

△

106


4.7

Solution to VHC Implicitization Problem

The following procedure summarizes our solution to the VHC implicitization problem. Procedure 4.2 (VHC Implicitization Algorithm) (i) Let assumption 2.3.7 hold. Let σ : [R]T → Q be a regular and odd parametric VHC, and ǫ > 0. Using (4.11) and (4.12), find sufficiently large positive integers N1 , · · · , Nn such that the approximation σ ˜ : [R]T → Q of σ satisfy k˜ σ − σkW 1,∞ < ǫ, whose existence is guaranteed by Proposition 4.4.2. By Lemma 4.3.5, σ ˜ is odd. (ii) Using Procedure 4.1, implicitize the parametric VHC σ ˜ to obtain the implicit VHC h. (iii) Propositions 4.6.3, 4.6.4, and 4.6.5 give sufficient conditions for regularity of the implicit constraints h obtained from Procedure 4.1. Example 4.8 We return to the pendubot of Example 1.1. We consider the following two swing-up maneuvers





θ   σ(θ) =  , √ )(1 + 2) θ + 2 arctan tan( −θ 2   θ + α0 sin(θ) σ(θ) =  , ⋆ σ2 (θ)

(4.26a)

(4.26b)

√ where α0 = 40 ◦ , and the two parametric constraints θ + 2 arctan tan( −θ )(1 + 2) and 2 σ2⋆ (θ) are solutions to the VCG

dσ2 = − cos σ1 (θ) − σ2 (θ) + δ(σ1 , θ), dθ

(4.27)

when σ1 (θ) is chosen to be θ and θ + α0 sin(θ), respectively. Note that in the latter case the VCG has no closed-form solution.


107

We would like to implicitize the two parametric VHCs in the example using Procedure 4.2. We perform the following three steps. (i) We use the Bézier polynomials given in (4.11) and (4.12) to approximate the parametric VHCs in (4.26). In particular, we approximate the first parametric VHC using Bézier polynomials of order 0 and 8, with d1 = 1 and d2 = 0, and the second parametric VHC using Bézier polynomials of orders 6 and 10, with d1 = 1 and σ ) and Im(σ) on the d2 = 0, respectively. Figure 4.6 depicts the VHC curves Im(˜ cylinder S1 × R. Figure 4.8 depicts the configurations of the pendubot under the parametric VHCs in (4.26) and their approximations. (ii) We implicitize the two obtained parametric VHCs using the unique spanning tree G = (V, E) with 2 vertices in Procedure 4.1. The zero level sets of these two implicit constraints and Im(˜ σ ) are depicted in Figure 4.8. As it can be seen from the figure, h−1 σ ). G (0) = Im(˜ (iii) In the last step, we check the rank of dh. Since Im(˜ σ) is smooth Jordan, then rank dh Im σ˜ = 1, as depicted in Figure 4.9. △


108

(a) The parametric VHC (4.26a) (in blue)

(b) The parametric VHC (4.26b) (in blue)

and its approximation (in dash-dot red).

and its approximation (in dash-dot red).

Figure 4.6: Parametric VHCs and their approximations.

2

2

1.5

1.5

1

1

0.5

0.5

0

0

−0.5

−0.5

−1 −1.5

−1 −1

−0.5

0

0.5

1

1.5

−1.5

−1

−0.5

0

0.5

1

1.5

(a) Configurations of the pendubot under

(b) Configurations of the pendubot under

the VHC (4.26a) (in gray) and its approx-

the VHC (4.26b) (in gray) and its approx-

imation (in dash-dot black).

imation (in dash-dot black).

Figure 4.7: Configurations of the pendubot under VHCs and their approximations.

109

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

q2

q2


0

0

−0.2

−0.2

−0.4

−0.4

−0.6

−0.6

−0.8

−0.8 0

1

2

3

4

5

0

6

1

2

3

4

5

6

q1

q1

(a) The implicit constraint h−1 G (0) obtained

(b) The implicit constraint h−1 G (0) obtained

σ) from Procedure 4.1 (in blue) and Im(˜

σ) from Procedure 4.1 (in blue) and Im(˜

(red circles) for the VHC (4.26a).

(red circles) for the VHC (4.26b).

4

4

3

3

2

2

kdhk

kdhk

Figure 4.8: Zero level sets of the implicit VHCs and the image of parametric VHCs.

1

1

0

0

−1

−1

−2

−2 0

1

2

3

4

5

6

0

1

θ

2

3

4

5

6

θ

(a) The norm kdhkIm(˜σ) on the VHC

(b) The norm kdhkIm(˜σ) on the VHC

curve (4.26a).

curve (4.26b).

Figure 4.9: The norm of the Jacobian of the implicitized constraints.

Chapter 5 VHC-based Orbital Stabilization In this chapter, we return to the Euler-Lagrange system

D(q)¨ q + C(q, q) ˙ q˙ + ∇P (q) = B(q)τ,

(5.1)

with n degrees-of-freedom (DOFs) and n − 1 actuators that is under the influence of a regular VHC h(q) = 0 of order n − 1. We assume that this VHC induces a reduced dynamics with a mechanical Lagrangian structure (see part (b) of Definition 3.3.1). By Theorem 3.3.5, this assumption implies that there is an energy function E whose level sets are unions of orbits of the reduced dynamics. The problem studied in this chapter is to stabilize a level set of the energy E corresponding to a closed orbit of the reduced dynamics. The orbit in question is associated with a desired repetitive behavior that the mechanical system (5.1) should perform. The challenge that is encountered in solving this orbital stabilization problem is that the reduced dynamics have no control input. Our solution to this problem relies on the notion of dynamic VHCs, i.e., VHCs parameterized by the states of double integrators. The intuition is that by controlling the state of the double integrator, one dynamically changes the shape of the VHC so as to stabilize the desired closed orbit. This idea was first introduced in [43] for solving the same problem in a special case. In this chapter 110

Chapter 5. VHC-based Orbital Stabilization

111

we present a general theory. We mention that another approach to solving this problem is due to Shirieaev et al., presented in [36, 38], for stabilizing closed orbits induced by VHCs where the VHC invariance is not respected. This chapter is organized as follows. We begin our exposition with a motivating example in Section 5.1. The formal problem statement and our solution strategy are presented in Section 5.2. After reviewing preliminaries in Section 5.3, we continue our exposition by presenting dynamic VHCs and their associated reduced dynamics in Section 5.4. Then, we linearize the reduced dynamics that are induced by the dynamic VHC around the closed orbit in Section 5.6. We present an exponentially stabilizing control law for the linear transversal dynamics. In Section 5.7 we present the control law for solving the VHC-based orbital stabilization problem. Finally, in Section 5.9 we discuss the differences between the method presented in this chapter and the one in [36, 38].

5.1

Motivating example: Pendubot swing-up

Consider the pendubot system of Example 1.1 depicted in Figure 5.1. We would like to solve the following swing-up problem: 1. Low-high to high-high swing-up: For any neighborhood U of the high-high equilibrium (q1 , q2 , q˙1 , q˙2 ) = (0, 0, 0, 0), there exists a punctured neighborhood V of the low-high equilibrium (q1 , q2 , q˙1 , q˙2 ) = (π, 0, 0, 0) such that for each initial condition in V , the solution enters U in finite time. 2. Boundedness: For any initial condition in V , the solution has the property that q2 (t) ∈ (−π/2, π/2) for all t ≥ 0. In other words, the unactuated link does not fall over during transient. We consider the odd VHC obtained in Example 2.10 and implicitize it using Procedure 4.2 in Chapter 4. The pendubot configurations under this VHC are depicted in

112


(a) The pendubot

(b) low-high (c) high-high

Figure 5.1: The pendubot and its low-high and high-high equilibrium configurations.

Figure 5.2. This VHC has the property that on the constraint manifold the second link never falls over, and high-high and low-high configurations belong to it. This VHC is, therefore, a suitable candidate to meet the foregoing boundedness requirement. Since the VHC in Figure 5.2 was constructed to be regular and odd and since, by Lemma 4.3.5, the implicitization procedure (Procedure 4.2) preserves regularity and odd symmetry, by Corollary 3.3.8 we deduce that the reduced dynamics have a Lagrangian mechanical structure. By Proposition 3.6.4, almost all solutions of the reduced dynamics are closed orbits. The level sets of the energy of the reduced dynamics are depicted in Figure 5.3. In Figure 5.3 the gray areas are filled with oscillations of θ, while the white areas are filled with rotations of θ. Stabilizing the closed orbit that bounds the shaded region in the figure, which corresponds to setting E0 slightly larger than zero, solves the swing-up problem. A byproduct of the general theory developed in this chapter is a methodology to stabilize this energy level-set by enforcing the VHC while maintaining the VHC invariance.

113

Chapter 5. VHC-based Orbital Stabilization 2

1.5

1

0.5

0

−0.5

−1 −1.5

−1

−0.5

0

0.5

1

1.5

Figure 5.2: Configurations of the pendubot under an odd VHC. 10

8

6

4

θ˙

2

0

−2

−4

−6

−8

−10 0

1

2

3

4

5

6

θ

Figure 5.3: The phase portraits of the pendubot VHC-induced reduced motion.

5.2

Problem statement

Consider the Euler-Lagrange system (5.1) with n DOFs and n−1 actuators that is under the influence of the implicit VHC h(q) = 0 of order n−1. Assume that the VHC is regular (see Definition 2.2.2). The constraint manifold is

n o Γ = (q, q) ˙ ∈ T Q : h(q) = 0, dhq q˙ = 0 .

(5.2)

As we argued in Section 2.3.1, the set h−1 (0) is a union of disconnected curves, each one diffeomorphic to either R or S1 . We assume that h−1 (0) is a closed Jordan curve, and pick a regular parameterization σ : S1 → Q of it. By Proposition 2.3.4, the VHC h(q) = 0 induces the reduced dynamics

114


θ¨ = Ψ1 (θ) + Ψ2 (θ)θ˙ 2 ,

(5.3)

where

Ψ1 (θ)

Ψ2 (θ)

B ⊥ ∇P = − ⊥ ′ , B Dσ q=σ(θ) n P ⊤ B ⊥ Dσ ′′ + Bi⊥ σ ′ Qi σ ′ i=1 =− ⊥ ′ B Dσ

.

q=σ(θ)

We assume that the dynamics (5.3) have a EL structure (Theorem 3.3.5 gives necessary and sufficient conditions for this to be the case), so that its closed orbits are characterized by Proposition 3.6.4. In this chapter we investigate the following problem.

VHC-based orbital stabilization problem. Consider the Euler-Lagrange system (5.1) and the regular VHC h(q) = 0 with parametric representation q = σ(θ) inducing the reduced dynamics (5.3). Consider a closed orbit on the constraint manifold Γ

γ = (q, q) ˙ ∈ Γ : E θ(q), dθq q˙ = E0 ,

(5.4)

where the mapping θ(·) : Q → h−1 (0) is well-defined, smooth, and its differential has full ˙ = 1 M(ϑ)ϑ˙ 2 + V (ϑ), rank one in a neighborhood of the VHC curve h−1 (0). Also, E(ϑ, ϑ) 2 and M(·) and V (·) are given by Z

ϑ M(ϑ) = exp − 2 Ψ2 (τ ) dτ , 0 Z ϑ V (ϑ) = − Ψ1 (τ )M(τ ) dτ.

0

(5.5)


115

Find a control law that asymptotically stabilizes the closed orbit γ while preserving the invariance of a suitably modified constrained manifold. Remark 5.2.1. The underlying philosophy of willingness to preserve the invariance of the constraint manifold is that the mechanical system maintains a desired configuration on it, e.g., the second link of the pendubot does not fall over. Lack of controls on the original constraint manifold requires us to modify this manifold. In Section 5.4 we demonstrate how to perturb the original constraint manifold so as to introduce a new control, without affecting too much its “shape” during transient. Solution strategy. Our strategy for solving the VHC-based orbital stabilization problem consists of the following three steps Step 1 Making the VHC dynamic: in Section 5.4. Step 2 Transverse linearization and orbital stabilization: in Section 5.6. Step 3 Enforcing the VHC and stabilizing the closed orbit: in Section 5.7.

5.3

Preliminaries

In this section we review preliminaries from set stability and stabilization of linear periodically time-varying (LPTV) systems.

5.3.1

Basic notions of set stability

Here we review the basic notions of set stability and Seibert-Florio’s theorem for asymptotic stability of compact sets. A comprehensive treatment of the preliminaries here can be found in [87, 88]. Consider the dynamical system


Σ : x˙ = f (x), x ∈ X ⊂ Rn .

116

(5.6)

Let Γ ⊂ X be a closed and positively invariant set for Σ, that is, for all x0 ∈ Γ, and for all t in the maximal interval of existence associated with the initial condition x0 , φ(t, x0 ) ∈ Γ. The notions of stability, attractivity, and asymptotic stability for the set Γ can be defined as follows. (i) Γ is stable for the dynamical system Σ if for every ǫ > 0 there exists a neighborhood N (Γ) such that φ(R+ , N (Γ)) ⊂ Bǫ (Γ). (ii) Γ is an attractor for the dynamical system Σ if there exists a neighborhood N (Γ) such that, for all x0 ∈ N (Γ), limt→∞ kφ(t, x0 )kΓ = 0. (iii) Γ is [globally] asymptotically stable for a dynamical system Σ if it is stable and attractive [globally attractive] for Σ. Given l + 1 closed positively invariant subsets Γ1 ⊂ Γ2 ⊂ . . . ⊂ Γl+1 of X , we say that Γi is stable relative to Γi+1 , 1 ≤ i ≤ l, for Σ if, for any ǫ > 0, there exists a neighborhood N (Γi ) such that φ(R+ , N (Γi ) ∩ Γi+1 ) ⊂ Bǫ (Γi ). Similarly, the notions of relative attractivity and [global] asymptotic stability for Γi ⊂ Γi+1 ⊂ X can be defined as in the definition of attractivity and [global] asymptotic stability by restricting initial conditions to lie in Γi+1 . We now consider the dynamical system Σ:

x˙ = f (x, u), x ∈ X ⊂ Rn ,

(5.7)

where the vector field f is locally Lipschitz on X and u(x) is a locally Lipschitz feedback which makes the sets Γ1 ⊂ Γ2 ⊂ . . . ⊂ Γl positively invariant for the closed-loop system. The following theorem, which is stated in [87] and is a consequence of a result in [89], will be used in the sequel.

117


Proposition 5.3.1 ([87]). Consider system (5.7), and assume that there exists a locally Lipschitz feedback u(x) making the sets Γ1 ⊂ Γ2 ⊂ . . . ⊂ Γl ⊂ Γl+1 := X , positively invariant for the closed-loop system and such that, for i = 1, . . . , l, Γi is asymptotically stable relative to Γi+1 for the closed-loop system, and Γ1 is compact. Then, Γ1 is asymptotically stable for the closed-loop system x˙ = f (x, u(x)).

5.3.2

Periodic Riccati differential equations

In this section we present a family of stabilizing controllers for LPTV systems that are obtained by solving periodic Riccati differential equations (PRDEs). The results presented in this section are standard and can be found in [90, 91, 92]. Consider the following LPTV system

dx = A(θ)x(θ) + B(θ)v(θ), dθ

(5.8)

where A(·) : R → Rn×n and B(·) : R → Rn×m are given T -periodic functions. Given continuous T -periodic functions Q : R → Rn×n and R : R → Rm×m , where Q(θ)⊤ = Q(θ) ≥ 0 and R(θ)⊤ = R(θ) > 0, the linear quadratic regulator (LQR) cost function associated to LPTV system (5.8) is defined as

J(·) := min v(·)

Z

∞ 0

⊤

⊤

x(θ) Q(θ)x(θ) + v(θ) R(θ)v(θ) dθ.

(5.9)

The notion of stabilizability and detectability for LPTV systems can be formalized in various ways. As it is shown by Bittanti and Bolzern in [93], these definitions are equivalent. Here, we present the notions of W -stabilizability and W -detectability. The pair (A(θ), B(θ)) is W -stabilizable if there exists a T -periodic function K : R → Rn×m such that the LPTV system dx = A(θ) − B(θ)K(θ) x(θ), dθ


118

is asymptotically stable. Analogous to LTI systems, detectability is the dual notion 1

of stabilizability. Specifically, the pair (A(θ), Q(θ) 2 ) is W -detectable if and only if the 1 pair A⊤ (θ), Q⊤ (θ) 2 is W -stabilizable. In the sequel we refer to W -stabilizability and

W -detectability as stabilizability and detectability, respectively.

Consider the LPTV system (5.8) and the LQR cost function (5.9). The periodic Riccati differential equation (PRDE) associated to the 4-tuple A(·), B(·), Q(·), R(·) is defined as

−

∂X = A(θ)⊤ X(θ) + X(θ)A(θ) − X(θ)B(θ)R(θ)−1 B(θ)⊤ X(θ) + Q(θ). ∂θ

(5.10)

The following proposition highlights the relation between optimal stabilizing feedback control inputs for LPTV systems and the solution of PRDEs. Proposition 5.3.2 (Theorem 6.12 in [90]). Let A(·) : R → Rn×n and B(·) : R → Rn×m be continuous T -periodic functions. Let Q : R → Rn×n and R : R → Rm×m be continuous T -periodic functions, where Q(θ)⊤ = Q(θ) ≥ 0 and R(θ)⊤ = R(θ) > 0. If 1 the pair A(θ), B(θ) is W -stabilizable and the pair A(θ), Q(θ) 2 is W -detectable, the

optimal control input that exponentially stabilizes the origin of the LPTV system (5.8) and minimizes the LQR cost function (5.9) is

v ⋆ (θ) = −R(θ)−1 B(θ)⊤ X(θ)x(θ),

(5.11)

where X(θ) is the unique symmetric positive semi-definite T -periodic stabilizing solution of the PRDE (5.10). Several numerical algorithms have been proposed in the literature for finding stabilizing solutions of PRDEs. In this thesis we use the traditional one-shot generator method for computing stabilizing solutions of PRDEs [94]. A detailed treatment of existing numerical algorithms for solving PRDEs is presented in [95].


5.4

119

Step 1: Making the VHC dynamic

In this section we present the notion of dynamic VHCs and find their associated reduced dynamics.

5.4.1

Dynamic VHCs

Since the reduced dynamics in (5.3) have no control input, it is impossible to stabilize γ in (5.4) while at the same time preserving the invariance of the constraint manifold Γ in (5.2). A possible way to tackle this problem is to stabilize the closed orbit γ by dynamically changing the geometry of the VHC h(q) = 0 while preserving its invariance. In order to change the geometry of the VHC h(q) = 0, we modify the VHC making it depend on a dynamic scalar variable s whose evolution is governed by the double integrator

s¨ = v,

(5.12)

where v is a scalar control input. Consequently, variations of the variable s affect the dynamics on the constraint manifold. The VHC regularity condition in Definition 2.2.2 requires that the output function e = h(q) has well-defined vector relative degree {2, · · · , 2} everywhere on h−1 (0) for the system (5.1) and thus the control input τ appears after taking two derivatives of the output. Due to this vector relative degree requirement, we choose the evolution of the dynamic variable s be governed by a double integrator. As a result, the input to the double integrator (5.12), v, appears along with the control input τ after taking two derivatives of the output e = hs (q). Augmenting the dynamics of the Euler-Lagrange system in (5.1) with that of the variable s in (5.12), we obtain the following augmented control system


D(q)¨ q + C(q, q) ˙ q˙ + ∇P (q) = B(q)τ,

120

(5.13)

s¨ = v. Henceforth, we use overbars to distinguish objects associated to the augmented control ˙ s), ˙ system (5.13) from those associated to (5.1). Accordingly, we define q¯ := (q, s), q¯˙ := (q, ¯ := (q, s) : q ∈ Q, s ∈ I , and T Q ¯ := (¯ ¯ q¯˙ ∈ Rn+1 . Q q , q¯˙) : q¯ ∈ Q, Definition 5.4.1. Let h(q) = 0 be a regular VHC of order n − 1 for system (5.1).

A dynamic VHC based on h(q) = 0 is a relation hs (q) = 0 such that the map (s, q) 7→ hs (q) is C 1 , h0 (q) = h(q), and the parameter s is the output of the double integrator s¨ = v. The dynamic VHC hs (q) is regular if there exists an open interval I ⊂ R containing s = 0 such that, for all s ∈ I and all v ∈ R, the system (5.13) with input τ and output e = hs (q) has vector relative degree {2, · · · , 2}. The dynamic VHC hs (q) = 0 is stabilizable if there exists a smooth feedback τ (q, q, ˙ s, s, ˙ v) such that the manifold

¯ := {(q, q, ¯ : hs (q) = 0, ∂q hs q˙ + ∂s hs s˙ = 0}, Γ ˙ s, s) ˙ ∈ TQ is asymptotically stable for the closed-loop system.

(5.14) △

Remark 5.4.2. The regularity property of hs (q) in the foregoing definition means that, upon calculating the second derivative of e = hs (q) along the vector field in 5.13, one has that the control input τ appears nonsingularly, i.e., e¨ = (⋆) + As (q)τ + B s (q)v, where As and B s are suitable matrices and As is invertible for all q ∈ (hs )−1 (0) and all s ∈ I.


121

Given a regular VHC h(q) = 0, a possible way to generate a dynamic VHC based on h(q) = 0 is to translate the VHC curve h−1 (0) by an amount depending on s. In particular, we consider the following one-parameter family of mappings

hs : Q → Rn−1 , q 7→ h(q − Ls),

(5.15)

where s ∈ I is the variable parameterizing the family, I is an open interval of the real line containing zero, and L ∈ Rn is a non-zero constant vector. We have chosen the special family of dynamic VHCs (5.15) for two main reasons. First, the reduced dynamics that are induced by this family of dynamic VHCs have an analytical form given by (5.24) suitable for computing control laws. Second, dynamic VHCs in symbolic form can be easily obtained from the implicit VHCs that are generated by Procedure 4.1 through polynomial compositions. For instance, q12 + q22 − 1 = 0 can be made dynamic via the composition (q1 − L1 s)2 + (q2 − L2 s)2 − 1 = 0. The zero level set of each family member in (5.15) is

hs

−1

(0) = {q + Ls : q ∈ h−1 (0)},

which is a translation of the VHC curve h−1 (0) by the vector Ls (see Figure 5.4).

Figure 5.4: Geometric interpretation of making VHCs dynamic.

(5.16)


122

Thus, varying the parameter s in (5.15) corresponds to translating the VHC curve ¯ associated with the dynamic h−1 (0). In an analogous manner, the constraint manifold Γ VHC h(q − Ls) = 0 is the translation of Γ by the vector [Ls Ls] ˙ ⊤ . More precisely, it is possible to show that

¯ = {(q, q, ¯ : h(q − Ls) = 0, dhq−Ls (q˙ − Ls) Γ ˙ s, s) ˙ ∈ TQ ˙ = 0} ¯ : (q − Ls, q˙ − Ls) = {(q, q, ˙ s, s) ˙ ∈ TQ ˙ ∈ Γ}.

(5.17)

When s = 0, the original VHC h(q) = 0 is recovered.

5.4.2

Regularity and stabilizability of dynamic VHCs

If σ : [R]T → Q is a regular parameterization of the VHC curve h−1 (0), a regular parameterization of the zero level set of each family member in (5.15) is

σ s : [R]T → Q, θ 7→ σ(θ) + Ls.

(5.18)

The next two lemmas show that if h(q) = 0 is regular and stabilizable, so too is its dynamic counterpart h(q − Ls) = 0. Lemma 5.4.3. Let L ∈ Rn and h(q) = 0 be a regular VHC of order n − 1 for the EulerLagrange system (5.1) with regular parameterization σ : [R]T → Q. Then, the dynamic VHC h(q − Ls) = 0 is regular. ¯ q ) := h(q − Ls). Considering the Proof. For the augmented system (5.13), we define h(¯ output e = h(q − Ls) and taking two derivatives, we have e¨ = (⋆) − dh q−Ls Lv + As (q)τ,

123


where As (q) = dh q−Ls D −1 (q)B(q). Thus, the dynamic VHC h(q − Ls) = 0 is regular if and only if there exists some interval I ⊂ R containing s = 0 such that the decoupling

matrix As (q) is invertible for all q ∈ (hs )−1 (0). This is true if and only if dim

Im(D (q)B(q)) ∩ Ker dh q−Ls −1

= 0,

(5.19)

for all q ∈ (hs )−1 (0) and all s ∈ I. Also, σ s (θ) = σ(θ) + Ls is a regular parameterization for (hs )−1 (0) and Ker dh q−Ls = Im(∂θ σ s ) = Im(∂θ σ) for all q ∈ (hs )−1 (0) and all s ∈ I. Therefore, the condition (5.19) holds if and only if

Im(∂θ σ) ∩ Im D −1 (σ s (θ))B(σ s (θ)) = {0},

(5.20)

for all θ ∈ [R]T and all s ∈ I. Since B ⊥ (q) is a left-annihilator of B(q), then Im(B(q)) = Ker(B ⊥ (q)) for all q ∈ (hs )−1 (0) and all s ∈ I. Hence, the relation (5.20) holds if and only if

Im(D(σ s (θ))∂θ σ) ∩ Ker(B ⊥ (σ s (θ))) = 0, for all θ ∈ [R]T and all s ∈ I. In light of the fact that Im(∂θ σ s ) = Im(∂θ σ) for all q ∈ (hs )−1 (0) and all s ∈ I the above holds if and only if B ⊥ (σ s (θ))D(σ s (θ))∂θ σ s (θ) 6= 0 for all θ ∈ [R]T and all s ∈ I. Since q = σ(θ) is a regular VHC, and since σ 0 (θ) = σ(θ), by Corollary 2.3.3, we have that B ⊥ (σ 0 (θ))D(σ 0 (θ))∂θ σ 0 (θ) 6= 0. By continuity of the function (θ, s) 7→ B ⊥ (σ s (θ))D(σ s (θ))∂θ σ s (θ) with respect to s, and the fact that θ ∈ [R]T is a compact set, there exists an interval I ⊂ R containing s = 0 such that for all θ ∈ [R]T and all s ∈ I, B ⊥ (σ s (θ))D(σ s (θ))∂θ σ s (θ) 6= 0 holds. Lemma 5.4.4. Let L ∈ Rn and let h(q) = 0 be a regular VHC of order n − 1 which satisfies the stabilizability of Proposition 2.2.3: there exists strictly increasing functions α, β : [0, r] → [0, ∞) with r > 0, such that the map H : (q, q) ˙ 7→ (h(q), dhq q) ˙ is bounded


124

as α(k(q, q)k ˙ Γ) ≤ H(q, q) ˙ ≤ β(k(q, q)k ˙ Γ ). Then, the dynamic VHC h(q − Ls) = 0 is stabilizable, in the sense of Definition 5.4.1, by the feedback

τ =

† n ˙ q˙ + ∇P (q) + As (q) dh q−Ls D −1 (q) C(q, q) o ˙ s, s) ˙ − k1 e − k2 e˙ , dh q−Ls Lv − H(q, q,

(5.21)

˙ As (q) = dh q−Ls D −1 (q)B(q), k1 , k2 > 0, and where e = h(q − Ls), e˙ = dh q−Ls (q˙ − Ls),  ˙ Hess(h1 ) q−Ls (q˙ − Ls) ˙   (q˙ − Ls)   . H(q, q, ˙ s, s) ˙ = ···     ⊤ ˙ (q˙ − Ls) ˙ Hess(hn−1 ) q−Ls (q˙ − Ls) 

⊤

Proof. By (5.17), we have

¯ = {(q, q, ¯ : (q − Ls, q˙ − Ls) Γ ˙ s, s) ˙ ∈ TQ ˙ ∈ Γ}, from which it follows that

˙ Γ, k(q, q, ˙ s, s)k ˙ Γ¯ = k(q − Ls, q˙ − Ls)k

(5.22)

This fact and the hypothesis of stabilizability of the VHC h(q) = 0 imply that

˙ ≤ β(k(q, q, ˙ s, s)k ˙ Γ¯ ), α(k(q, q, ˙ s, s)k ˙ Γ¯ ) ≤ H(q − Ls, q˙ − Ls)

(5.23)

Letting e = h(q − Ls), the feedback (5.21) gives e¨ + k2 e˙ + k1 e = 0, so that the equilibrium (e, e) ˙ = (0, 0) is asymptotically stable. Since (e, e) ˙ = H(q − Ls, q˙ − Ls), ˙ ¯ is asymptotically stable. property (5.23) implies that Γ

5.4.3

Reduced dynamics induced by dynamic VHCs

Consider a regular dynamic VHC h(q − Ls) = 0 with associated constraint manifold ¯ in (5.17). In order to find the reduced dynamics of the augmented Euler-Lagrange Γ

125


¯ we left-multiply (5.13) by the left annihilator of B, B ⊥ , and evaluate system (5.13) on Γ, ¯ Performing this procedure, we obtain the following equations the resulting equation on Γ. ¯ when τ is chosen as in (5.21): describing the motion of the augmented system (5.13) on Γ θ¨ = Ψs1 (θ) + Ψs2 (θ)θ˙2 + Ψs3 (θ)θ˙s˙ + Ψs4 (θ)s˙ 2 + Ψs5 (θ)v, s¨ = v,

(5.24)

where

Ψs1 (θ)

Ψs2 (θ)

B ⊥ ∇P , = − ⊥ ′ B Dσ q=σ(θ)+Ls n P ⊤ B ⊥ Dσ ′′ + Bi⊥ σ ′ Qi σ ′ i=1 =− ⊥ ′ B Dσ

,

q=σ(θ)+Ls

Ψs4 (θ) Ψs5 (θ)

′⊤

Bi⊥ σ Qi L = − i=1 ⊥ ′ B Dσ 2

Ψs3 (θ)

n P

,

q=σ(θ)+Ls

n P

Bi⊥ L⊤ Qi L = − i=1 ⊥ ′ , B Dσ q=σ(θ)+Ls ⊥ B DL = − ⊥ ′ . B Dσ q=σ(θ)+Ls

It can be seen that Ψ01 = Ψ1 , Ψ02 = Ψ2 as in (5.3). Moreover, note that when (s, s) ˙ = (0, 0), the reduced dynamics (5.3) are recovered.

5.5

Reduction-based stabilization strategy

In this section we highlight our strategy for Steps 2 and 3. In Step 2 we design a feedback control input v for the reduced dynamics of the ˙ = E0 , s = 0, s˙ = 0} dynamic VHC h(q − Ls) = 0 in (5.24) such that the set {E(θ, θ) is asymptotically stabilized. This corresponds to asymptotic stabilization of the closed


126

¯ given in (5.17). To orbit γ, which is a compact set, relative to the constraint manifold Γ find v, we proceed according to the following parts.

Figure 5.5: A tubular neighborhood of the closed orbit.

Part (a)– Finding a diffeomorphism in a tubular neighborhood of the ˙ on the constraint manifold Γ, we find a diffeoclosed orbit In coordinates (θ, θ) ˙ 7→ (ν, E −E0 ) valid in a tubular neighborhood of γ (see Figure 5.5). morphism (θ, θ) Part (b)– Representing the reduced dynamics in new coordinates We find the representation of the reduced dynamics (5.24) in coordinates (ν, ρ) = (ν, E − E0 , s, s). ˙ Part (c)– Transverse linearization Using the classical theory of stability of periodic orbits exposed in Chapter VI of [96], we find the linearization of the reduced dynamics around the closed orbit. The result is an LPTV system, with variable ν as time. The transformed system in linear form is called the transverse linearization of (5.24) along the closed orbit γ. Part (d)– Stabilization on the constraint manifold By solving an appropriate periodic Riccati differential equation associated with the linearized system obtained in Part (c), we find the feedback control input v that asymptotically stabilizes the closed orbit on Γ.


Part (e)

127

Using Proposition 5.3.1, we deduce that the closed orbit is asymptoti-

¯ of (5.13). cally stable relative to the state space T Q

5.6

Step 2: Transverse linearization and orbital stabilization

In this section we consider the reduced dynamics of the dynamic VHC h(q − Ls) = 0 and design the control input v such that the closed orbit γ is stabilized relative to the ¯ given in (5.17). constraint manifold Γ Part (a)– Finding a diffeomorphism in a tubular neighborhood of the closed orbit:

Let ϕ : [R]T → [R]T × R be a regular parameterization of the closed orbit γ on the set ˙ ∈ [R]T × R. Therefore, γ = Im(ϕ) in (θ, θ) ˙ coordinates on Γ with coordinates (θ, θ) the constraint manifold Γ. We assume that T > 0 is the minimal period of ϕ (i.e., the smallest T > 0 such that ϕ(τ + T ) = ϕ(τ ) for all τ ). Then, ϕ : [R]T → [R]T × R is a diffeomorphism of [R]T onto its image Im(ϕ) ⊂ [R]T × R. Further, we define ˙ − ϕ(τ )k. ˙ := arg min k(θ, θ) ̟(θ, θ) τ ∈[R]T

(5.25)

¯ θ) ¯˙ near Im(ϕ), letting Figure 5.6 illustrates the mapping in (5.25). For a point (θ, ¯ θ), ¯˙ the point ϕ(ν) is the orthogonal projection of (θ, ¯ θ) ¯˙ onto the curve Im(ϕ). ν = ̟(θ,


128

Figure 5.6: A well-defined transformation in a tubular neighborhood of the orbit.

By the tubular neighborhood theorem (see [53]), there exists ǫ > 0 such that, letting ˙ : k(θ, θ)k ˙ Im(ϕ) < ǫ}, the map ̟ : Im(ϕ)ǫ → [R]T is well-defined, smooth, Im(ϕ)ǫ := {(θ, θ) and its differential has full rank one. ˙ is a Part (b)–Representing the reduced dynamics in new coordinates: If (θ, θ) ˙ s, s) ¯ Using set of local coordinates on Γ, then (θ, θ, ˙ is a set of local coordinates on Γ. the mapping ̟ in (5.25), we can define the coordinate transformation F : [R]T × R3 → ˙ s, s) [R]T × R3 , (θ, θ, ˙ 7→ (ν, ρ), as ˙ ν =̟(θ, θ),

(5.26)

˙ − E0 , s, s). ρ =(ρ1 , ρ2 , ρ3 ) = (E(θ, θ) ˙ Lemma 5.6.1. The coordinate transformation F in (5.26) is a diffeomorphism in a neighborhood of γ × {(0, 0)} onto a neighborhood of F γ × {(0, 0)} = (ν, ρ1 , s, s) ˙ : ρ1 = 0, s = 0, s˙ = 0 .

˙ → (̟(θ, θ), ˙ E(θ, θ)−E ˙ Remark 5.6.2. The portion (θ, θ) 0 ) of the mapping in Lemma 5.6.1 ˙ represents can be intuitively considered as a polar coordinate transformation where ̟(θ, θ) ˙ − E0 is a measure of distance to the closed orbit γ × {(0, 0)}. the polar angle and E(θ, θ)

Proof. The generalized inverse function theorem [53, p. 56] states that F is a diffeomorphism of a neighborhood of γ¯ := γ × {(0, 0)} onto its image provided that: (i) for all ˙ s, s) (θ, θ, ˙ ∈ γ¯ , dF(θ,θ,s, ¯ → F (¯ γ ) = [R]T × 0 × 0 × 0 ˙ s) ¯ : γ ˙ is an isomorphism, and (ii) F |γ

129


˙ 0, 0) 7→ (θ, 0, 0, 0), where θ = ̟ −1 (θ, θ). ˙ This is a diffeomorphism. We have Fγ¯ : (θ, θ, is a diffeomorphism because ̟ is a diffeomorphism of [R]T onto γ. Therefore, condition (ii) is satisfied. Now we turn to condition (i). First note that that the vector ˙ is zero at points (θ, θ) ˙ = (θ⋆ , 0) such that [∂θ E ∂θ˙ E] = [1/2M ′ (θ)θ˙ 2 + V ′ (θ) M(θ)θ] V ′ (θ⋆ ) = 0. These are equilibria of the reduced dynamics (5.3). Since γ does not contain ˙ ∈ γ or, equilibria of (5.3), it follows that the vector [∂θ E ∂θ˙ E] is never zero for all (θ, θ) ˙ 0, 0) ∈ γ¯ . The differential of F at a point what is the same, it is never zero for all (θ, θ, ˙ 0, 0) in γ¯ has matrix representation (θ, θ,  ∂ ̟ ∂θ˙ ̟  θ  ∂θ E ∂ ˙ E θ    0  0  0 0

0 0 1 0

 0   0    0  1

.

˙ (θ,θ,0,0)

The first two rows of the matrix above are linearly independent because they are vectors that are, respectively, tangent and orthogonal to γ¯ . Therefore, dF(θ,θ,s, ˙ s) ˙ is an isomor˙ s, s) phism for all (θ, θ, ˙ ∈ γ¯ , proving that condition (i) is satisfied. ˙ in terms of (ν, ρ) In light of this result, near γ we can express (θ, θ)

θ = a(ν, ρ), θ˙ = b(ν, ρ), and the reduced dynamics of the dynamic VHC, given in (5.24), in (ν, ρ) coordinates read as ν˙ = ω(ν, ρ) h ˙ 3 ρ˙ 1 = M θ˙ (Ψρ12 − Ψ01 ) + (Ψρ22 − Ψ02 )θ˙2 + Ψρ32 θρ i ρ2 2 ρ2 + Ψ4 ρ3 + Ψ5 v ˙ θ=a(ν,ρ),θ=b(ν,ρ)

ρ˙ 2 = ρ3 ρ˙ 3 = v,

(5.27)

130


where ω(ν, ρ) is a suitable C 1 function which is bounded away from zero near {(ν, ρ) : ρ = 0}. In the new coordinates (ν, ρ), the orbital stabilization problem amounts to stabilization of the set {(ν, ρ) : ρ = 0}. We can concisely rewrite (5.27) as ν˙ = ω(ν, ρ)

(5.28)

ρ˙ = f (ν, ρ) + g(ν, ρ)v, where f (ν, 0) = 0. Part (c)– Transverse linearization: In order to find the linearization of the reduced dynamics associated to the dynamic VHC, given in (5.28), transverseal to the set {(ν, ρ) : ρ = 0}, we proceed according to the classical theory of stability of periodic orbits exposed in Chapter VI of [96]. Suppose we have found a C 1 feedback v(ν, ρ) such that v(ν, 0) = 0 and the orbit {(ν, ρ) : ρ = 0} is exponentially stable for (5.28) with v = v(ν, ρ). We evaluate the Jacobian matrix of the vector field (5.28) on the set {(ν, ρ) : ρ = 0}. We have



∂ν ω

∂ρ ω

dx  = dν ∂ν f + (∂ν g)v + g(∂ν v) ∂ρ f + (∂ρ g)v + g(∂ρ v)   ∂ ω ∂ ω ν ρ   =  x. 0 ∂ρ f + (∂ρ g)v + g(∂ρ v) ρ=0



  x

ρ=0

This is a linear T -periodic system because ν ∈ [R]T . The second identity in the above expression follows from the fact that f (ν, 0) = 0 and v(ν, 0) = 0, and hence ∂ν f (ν, 0) = 0 and ∂ν v(ν, 0) = 0. Following [96], the orbit {(ν, ρ) : ρ = 0} is exponentially stable if and only if the origin of the T -periodic system dρ = (∂ρ f (ν, 0))ρ + g(ν, 0)(∂ρ v(ν, 0))ρ dν is exponentially stable. We see from the above that a feedback v(ν, ρ) affects the system above only through the term ∂ρ v(ν, 0). Therefore, there is no loss of generality in letting


131

v(ν, ρ) = K(ν)ρ. For, given any feedback v(ν, ρ), by defining K(ν) = ∂ρ v(ν, 0), we obtain ∂ρ [K(ν)ρ] = K(ν) = ∂ρ v(ν, 0). Returning to the synthesis problem, we see that in order to exponentially stabilize γ it is necessary and sufficient to design a feedback v = K(ν)ρ stabilizing the origin of the linear periodic control system dy = A(ν)y + b(ν)v, dν

(5.29)

where A(ν) = (∂ρ f (ν, 0)) and b(ν) = g(ν, 0). In [97], this subsystem is called the transverse linearization of (5.28) along γ. Since A(·) and b(·) are T -periodic functions, the system (5.29) is an LPTV system. Returning to system (5.27), its transverse linearization along γ is     b1 (ν) 0 a12 (ν) a13 (ν)      , b(ν) =  0  A(ν) =  0 0 1         1 0 0 0

(5.30)

where

a12 (ν) = M(ν)ω(ν, 0)[∂ρ2 Ψρ12 (ν) + ∂ρ2 Ψ2ρ2 (ν)ω 2 (ν, 0)] ν=a(ν,0),ρ2 =0 , , a13 (ν) = M(ν)ω 2 (ν, 0)Ψ3(ν) ν=a(ν,0) . b1 (ν) = M(ν)ω(ν, 0)Ψ5 (ν)

(5.31)

ν=a(ν,0)

If γ is a rotation of θ, i.e., if E0 > maxθ V (θ), then we have θ = ν, and thus θ˙ = ν. ˙ ˙ we obtain Solving (1/2)M(θ)θ˙2 + V (θ) = E0 for θ, s

ν˙ = ±

2 E0 − V (θ) , M(θ)

(5.32)

with positive and negative signs corresponding to counterclockwise and clockwise rotations, respectively. Therefore, when the orbit γ is a counterclockwise rotation, the functions in (5.31) in terms of the original variables (ν, s, s) ˙ read as follows: p a12 (θ) = M (θ) (2/M (θ))(E0 − V (θ))[∂s Ψs1 (θ) + ∂s Ψs2 (θ)(2/M (θ))(E0 − V (θ))] s=0 a13 (θ) = M (θ)(2/M (θ))(E0 − V (θ))Ψ3 (θ) p b1 (θ) = M (θ)( (2/M (θ))(E0 − V (θ)))Ψ5 (θ).

(5.33)


132

On the other hand, one should place a minus sign in front of the square roots for a clockwise rotation. Part (d)–Stabilization of the closed orbit on the constraint manifold: In this part we find a control law that is obtained from solving PRDEs for stabilizing the closed orbit γ. In particular, consider a symmetric positive semi-definite matrix Q ∈ R3×3 and a non-negative scalar R. The PRDE associated to the 4-tuple A(·), b(·), R, Q reads as −

∂X = A(ν)⊤ X(ν) + X(ν)A(ν) − X(ν)b(ν)R−1 b(ν)⊤ X(ν) + Q. ∂ν

(5.34) 1

Proposition 5.3.2 states that if the pair (A(ν), b(ν)) is stabilizable and the pair (A(ν), Q 2 ) is detectable, the optimal control input that exponentially stabilizes (5.29) is

v(ν) = −R−1 b(ν)⊤ X(ν)y,

(5.35)

where X(ν) is the unique symmetric positive semi-definite T -periodic stabilizing solution of the PRDE (5.34). Therefore, the periodic orbit γ in (5.4) can be exponentially ¯ in (5.17) using the feedback stabilized relative to the set Γ

 ˙ E(θ, θ) − E0    ˙ s, s) ˙  , v(θ, θ, ˙ = K ̟(θ, θ) s     s˙

(5.36)

K(ν) = −R−1 b(ν)⊤ X(ν).

(5.37)



where the T -periodic state feedback gain which is dependent on the angular variable ν is

133


5.7

Step 3: Enforcing the VHC and stabilizing the closed orbit

In this section we present a solution to the VHC-based orbital stabilization problem using the results from Steps 1 and 2. This solution relies on a nested set stabilization approach and the reduction theorem for asymptotic stability of compact sets in Proposition 5.3.1. In this nested set stabilization approach, the control law v in (5.36) asymptotically stabi ¯ relative to the constraint manifold Γ. ¯ At a higher lizes the closed orbit γ × (0, 0) ⊂ Γ

level of hierarchy, the input-output feedback linearizing controller τ in (5.21) asymptot¯ ⊂ T Q. ¯ ically stabilizes the constraint manifold Γ

5.7.1

Solution to VHC-based orbital stabilization

We propose the following control law for solving the VHC-based orbital stabilization problem

τ =

† n As (q) dh q−Ls D −1 (q) C(q, q) ˙ q˙ + ∇P (q) + o ˙ dh q−Ls Lv(θ, θ, s, s) ˙ − H(q, q, ˙ s, s) ˙ − k1 e − k2 e˙ ,

(5.38)

˙ As (q) = dh q−Ls D −1 (q)B(q), k1 , k2 > 0, where e = h(q − Ls), e˙ = dh q−Ls (q˙ − Ls),  ˙ Hess(h1 ) q−Ls (q˙ − Ls) ˙   (q˙ − Ls)   , H(q, q, ˙ s, s) ˙ = · · ·     (q˙ − Ls) ˙ ⊤ Hess(hn−1 ) q−Ls (q˙ − Ls) ˙ 

and

⊤

134






˙ E(θ, θ) − E0    ˙ s, s) ˙  , v(θ, θ, ˙ = K ̟(θ, θ) s     s˙

(5.39)

with state feedback gain K(·) given by (5.37).

The following theorem presents the solution to VHC-based orbital stabilization problem. Theorem 5.7.1. Consider the underactuated mechanical system (5.1). Let h(q) = 0 be a regular and stabilizable VHC with regular parameterization σ : [R]T → Q and constraint manifold Γ in (5.2). Let γ ⊂ Γ be a closed curve given by (5.4) and L ∈ Rn be a constant nonzero vector. Consider the augmented system (5.13) and the transverse linearization of the reduced dynamics (5.24) in (5.29) with A(·) and b(·) given by (5.30). Consider a symmetric positive semi-definite matrix Q ∈ R3×3 and a non-negative scalar 1

R. Assume that the pair (A(θ), b(θ)) is W -stabilizable1 and the pair (A(θ), Q 2 ) is W detectable. Then, the feedbacks (5.21) and (5.36) with state feedback gain given by (5.37) asymptotically stabilize the closed orbit γ × (0, 0) for the augmented system (5.13) while simultaneously enforcing the VHC h(q − Ls) = 0.

1

Proof. Since the pair (A(ν), b(ν)) is stabilizable and the pair (A(ν), Q 2 ) is detectable, X(ν) is the unique symmetric positive semi-definite T -periodic stabilizing solution of the PRDE (5.34). Therefore, a feedback of the form v(ν) = K(ν)y exponentially stabilizes the origin of (5.29). Consequently, the feedback (5.36), with feedback gain given ¯ given by (5.37), exponentially stabilizes the periodic orbit γ × (0, 0) relative to the set Γ

in (5.17). On the other hand, the input-output feedback linearizing controller (5.21) ex-

¯ in (5.17), due to Lemma 5.4.4. Since ponentially stabilizes the constraint manifold Γ ¯ invoking the reduction principle for γ × (0, 0) is a compact subset of the closed set Γ, asymptotic stability of compact sets given in Proposition 5.3.1 proves the theorem. 1

The notions of W -stabilizability (detectability) are defined in Section 5.3 on p. 117.


135

The block diagram of the VHC-based orbital stabilizer is depicted in Figure 5.7. In the block diagram, the input-output feedback linearizing control law τ is given by (5.38), ˙ the orbital stabilization control law v is given by (5.39). Also, θ(q) and θ(q) are given by (5.40) and (5.43).

Figure 5.7: Block diagram of the VHC-based orbital stabilizer.

Remark 5.7.2. The constant vector L in Theorem 5.7.1 should be chosen such that the transverse linearized dynamics in (5.29) is W -stabilizable and W -detectable. Choosing this vector in a systematic manner remains an open problem.

5.7.2

Implementation details

˙ s, s) ˙ needs to be computed in order As it can be seen from (5.38) the virtual input v(θ, θ, ˙ to implement the control law τ (q, q, ˙ s, s, ˙ v). Hence, we should compute numerically (θ, θ) from (q, q). ˙ A candidate mapping for computing θ is

θ : Q → [R]T , q 7→ arg min kq − σ(ϑ)k. ϑ∈[R]T

(5.40)

For a point q near the VHC curve h−1 (0), letting ϑ = θ(q), the point σ(ϑ) is the orthogonal projection of q onto the VHC curve (see Figure 5.8). Given q, θ(q) can be computed


136

as the result of a one-dimensional search over the compact set [R]T . For real-time implementation, one can solve a finite number of optimization problems and then interpolate the optima.

Figure 5.8: The orthogonal projection of a configuration vector onto the VHC curve.

Remark 5.7.3. In the computer graphics literature there are several numerical algorithms for computing approximation of an orthogonal projection of the form (5.40) (see, e.g., the survey in [98]) which are more suitable for real-time implementation.

Figure 5.9: Constructing an orthonormal frame on the VHC curve.

In order to find θ˙ = dθq q, ˙ we need the Jacobian of the mapping θ(·) in (5.40), which can be computed using geometrical considerations. We first build a moving orthonormal frame on the VHC curve Im(σ). Due to transversality of the VHC curve Im(σ) to the control acceleration directions D −1 (·)B(·), which results from regularity of the VHC


137

h(q) = 0, the columns of the matrix D −1 (σ(θ))B(σ(θ)), σ ′ (θ) are linearly independent

(this will be shown rigorously in the sequel). Applying the Gram-Schmidt process to the columns of this matrix at each θ ∈ [R]T , we obtain a collection of orthonormal vectors

n σ ′ (θ) o ξ1 (θ), · · · , ξn−1 (θ), ′ , kσ (θ)k

(5.41)

for every θ ∈ [R]T (see Figure 5.9). We can use this collection of orthonormal vectors to define the mapping

Ξ : [R]T → Rn×(n−1) , θ 7→ [ξ1 (θ), · · · , ξn−1 (θ)].

(5.42)

The Jacobian of θ(·) can be computed as follows:

σ ′ (ϑ)⊤ . dθq = ′ ⊤ ⊤ kσ (ϑ)k2 + σ ′ (ϑ)Ξ′ (ϑ) q − σ(ϑ) ϑ=θ(q)

(5.43)

Lemma 5.7.4. Let h(q) = 0 be a regular VHC for the underactuated mechanical system (5.1) with a regular parameterization σ : [R]T → Q. Consider the mapping (5.40), the moving orthonormal frame (5.41), and the mapping Ξ(·) in (5.42). Then, there exists ǫ > 0 such that, letting Im(σ)ǫ := q : kqkIm(σ) < ǫ , the map θ Im(σ)ǫ is well-defined,

smooth, and its differential has full rank one. Also, the Jacobian of θ is given by (5.43) and is well-defined on some open neighborhood N Im(σ) ⊂ Im(σ)ǫ .

Proof. Consider the mapping θ in (5.40). If q ∈ Im(σ), then θ(q) = σ −1 (q), so θ|Im(σ) =

σ −1 . By the tubular neighborhood theorem (see [53]), there exists ǫ > 0 such that θ(·) is well-defined, smooth, and its differential has full rank one on Im(σ)ǫ = q : kqkIm(σ) < ǫ . Since the VHC h(q) = 0 is regular, by Condition (ii) in Proposition 2.2.4, we have

∀q ∈ h−1 (0)

Im D −1 (q)B(q) ∩ Ker dhq = {0}.

138

Chapter 5. VHC-based Orbital Stabilization Also, Ker dhσ(θ) = Im σ ′ (θ) for all θ ∈ [R]T . Hence, rank for all θ ∈ [R]T .

D −1 (σ(θ))B(σ(θ)), σ ′ (θ) = n,

Let q ∈ Im(σ)ǫ and ϑ = θ(q). It can be seen that σ ′ (ϑ)⊤ q − σ(ϑ) = 0.

Hence, q − σ(ϑ) ∈ span ξ1 (ϑ), · · · , ξn−1(ϑ) and

q − σ(ϑ) = Ξ(ϑ)w(ϑ),

(5.44)

where w(ϑ) ∈ Rn−1 . Multiplying both sides of the above identity by Ξ⊤ (ϑ) and noting that the frame (5.41) is orthonormal, we obtain

w(ϑ) = Ξ⊤ (ϑ)(q − σ(ϑ)).

(5.45)

Taking the derivative of (5.44), we obtain

˙ q˙ − σ ′ (ϑ)dθq q˙ = Ξw˙ + Ξw. ⊤

⊤

Multiplying both sides of the above identity by σ ′ (ϑ) and noting that σ ′ (ϑ)Ξ(ϑ) = 0, we get

⊤

⊤

⊤

σ ′ (ϑ)q˙ − kσ ′ k2 dθq q˙ = σ ′ Ξ′ dθq qw. ˙

(5.46)

Since σ(·) is a regular parameterization of h−1 (0), kσ ′ (·)k is bounded away from zero. ⊤ ⊤ Thus, in a sufficiently small neighborhood N Im(σ) ⊂ Im(σ ǫ ), kσ ′ (ϑ)k2 +σ ′ (ϑ)Ξ′ (ϑ) q− σ(ϑ) is bounded away from zero. Using (5.45) in (5.46) concludes the proof.


5.8

139

Pendubot swing-up: Solution

We return to the pendubot swing-up problem introduced in Section 5.1. The configurations of the pendubot under the candidate VHC for solving the swing-up problem are depicted in Figure 5.2. The motion that is induced by this VHC on the constraint manifold is depicted in Figure 5.3. The figure gray areas are filled with oscillations of θ, while the white areas are filled with rotations of θ. To solve the swing-up problem, we stabilize the energy level set that bounds the shaded region in the figure, which corresponds to setting E0 = ǫ with ǫ ≪ 1, in this example, E0 = 0.01 corresponding to a rotation of θ. According to Step 1 (see Section 5.4) we make the VHC dynamic by setting hs (q) = h(q − Ls), with L = col(1, 1), where s is the state of the double integrator s¨ = v. In Step 2, we design the input v, which asymptotically stabilizes the closed orbit γ ×{(0, 0)} ¯ associated with the dynamic VHC, by solving the relative to the constraint manifold Γ ˙ s, s) PRDE (5.34), with R = 1 and Q = diag{1, 1, 1}. The input v(θ, θ, ˙ is given by (5.36) with state feedback gain given by (5.37), Having computed the input v, we use the feedback τ (q, q, ˙ s, s) ˙ in (5.21) to stabilize ¯ associated with the dynamic VHC h(q − Ls) = 0, with k1 = 10 the constraint manifold Γ ˙ on the cylinder S1 × R when the and k2 = 10. Figure 5.10 shows the curve (θ, θ) pendubot is initialized in a neighborhood of the low-high equilibrium, i.e., (q(0), q(0)) ˙ = ˙ (π + 0.1, 0, 0.1, 0.1), illustrating that E θ(t), θ(t) −→ 0.01, as depicted in Figure 5.11.

5.9

Discussion

In this section we briefly compare the chapter control methodology with [43], and [36, 38], respectively. Comparison with the work in [43]: This chapter makes two contributions with

140


Figure 5.10: Phase curve of the VHC-induced motion on the phase cylinder. 8

6

∆E

4

2

0

−2

−4 0

2

4

6

8

10

12

14

16

18

20

t [sec]

Figure 5.11: Energy of the pendubot on the constraint manifold. respect to the work in [43]. First, a general theory is developed that includes the special ˆ n ) investigated in [43]. Second, the work in [43] poscase of parametric VHCs q = φ(q tulates existence of constant feedback gains that can stabilize the transverse linearized dynamics. However, it does not present a synthesis for finding these gains. In this chapter, we synthesize periodically time-varying feedback gains for the transverse linearized dynamics using the well-established theory of PRDEs. Comparison with the work in [36, 38]: As we mentioned in Chapter 1, the approach presented in [36, 38] to stabilize the closed orbit γ does not enforce the VHC


141

that was used to generate γ, in that the constraint manifold is not invariant for the closed-loop system. Enforcing the VHC while simultaneously stabilizing a closed orbit is advantageous from a practical viewpoint, as the presence of an asymptotically stable constraint manifold allows one to have better control over the transient behavior of the system. There are other technical differences between the approach presented in the previous section and the one in [36]. In [36] transverse linearization along γ is applied to the original 2n-dimensional system (5.1), resulting in a 2n − 1-dimensional linearization. On the other hand, our transverse linearization along γ¯ is applied to ¯ in (5.24), resulting in a three-dimensional the reduced four-dimensional dynamics on Γ linearization, independent of n. Further, the control law in [36] is static, while we use a second-order dynamic compensator.

Chapter 6 VHC-based Control of Planar Snake Robots This chapter investigates the problem of planar snake robot path following control. The control objective is to make the center of mass (CM) of the snake robot converge to a desired path and traverse the path with a desired velocity. The proposed feedback control strategy enforces virtual constraints encoding a biologically-inspired gait, parameterized by states of dynamic compensators used to regulate the orientation and forward speed of the snake robot. The theory that has been developed in Chapters 3–5 does not apply to the problem investigated in this chapter because the snake robot model we consider has degree of underactuation three. Nonetheless, addressing this complicated underactuated motion control problem using VHCs serves two main purposes: First, we test the hypothesis that the VHC-based control paradigm can be applied for solving a variety of underactuated motion control problems. Second, the design methodology presented in this chapter follows the overarching themes of VHC-based control paradigm that are outlined in Section 1.2. We start our exposition with a brief review of snake robot control challenges and the

142

Chapter 6. VHC-based Control of Planar Snake Robots

143

proposed model-based approaches in the literature in Section 6.1. Next, in Section 6.2 we present the kinematic and dynamic model of the snake robot. In Section 6.3 we state the control design objectives and present our hierarchical approach for achieving the control objectives. We introduce our biologically-inspired dynamic VHCs for the snake robot and present the VHC-stabilization control law in Section 6.4. In Sections 6.5.1 and 6.5.2 we develop control strategies for the head angle and the speed of the robot, respectively. In Section 6.6 we develop a path following control strategy for the snake robot under the influence of dynamic VHCs. We present our solution to the snake robot path following control problem in Section 6.7. Finally, we verify the effectiveness of our control methodology via simulations in Section 6.8. A large portion of this chapter has appeared in [99, 100].

6.1

Introduction

Biologically-inspired snake robots are underactuated vehicle-manipulator systems with many degrees-of-freedom (DOF) that can effectively be used for operations in challenging environments. Snake robots pose significant motion control challenges arising from the fact that such robots typically have at least three degrees of underactuation. One of the basic gait patterns through which biological snakes achieve forward motion is called lateral undulation [101]. During lateral undulation, the snake undergoes periodic shape changes that resemble a wave traveling backward along its body, from head to tail. As a result of this motion, the snake body traces out a periodic curve on the plane, which Hirose [101] mathematically represented as a serpenoid. Thinking of a snake robot as a discrete approximation of a biological snake, researchers (see, e.g.,[101, 102, 103]) have observed that the serpenoid curve can be well-approximated by imposing the sinusoidal reference signal for the ith joint angle φref,i (t) = α sin(ωt + (i − 1)δ) + φ0 ,

(6.1)


144

where α denotes the amplitude of the sinusoid, ω denotes the frequency of the joint oscillations, δ denotes the phase shift between two consecutive joints, and φ0 is a joint offset used to control the direction of locomotion. In this chapter, we study holonomic snake robots, i.e., the class of snake robots that do not have passive wheels and exploit friction for locomotion. The motion of this class of snake robots mimics closely the motion of their biological counterparts [104]. This is due to the possibility of sideways motion, which enables the robot to perform various types of gait patterns used by biological snakes. However, unlike the snake robots with sideslip constraints, locomotion control of this class of snake robots has been considered in only a few previous works. One of the reasons might be that holonomic snake robots are harder to control. Indeed, for other classes of snake robots that use passive wheels for locomotion only the kinematics has to be considered when describing the snake robot motion because one may use velocity as the control input to the robot joints. On the other hand, in holonomic snake robots, both the dynamics and kinematics have to be considered in analysis and control design in that the propulsion mechanism is the complex interplay between joint friction forces and center of mass forces. The resulting dynamical model is underactuated, something which poses additional challenges for control design. Comparison with existing model-based snake robot control schemes. Previous research on position and path following control of holonomic snake robots is very limited but is considered in, e.g., [105], [106], [107], and [109]. The paper [107] was the first work to present a stability analysis of the locomotion along a straight path. Being based on a numerical Poincaré test, the analysis in [107] is only valid for a specific set of controller parameters. In [109], path following control of snake robots along straight paths is considered. Using cascaded systems theory, it is proved that the proposed path following controller exponentially stabilizes a snake robot to any desired straight path. A drawback of [109] is that the stability analysis is valid for a simplified model which is only valid for small joint angles. Another drawback of [107] and [109] is that they are


145

only valid for straight lines and not all curved paths. To the best of our knowledge, to date there is no proof of convergence of a path following controller for the complete nonlinear model of a holonomic snake robot. Moreover, the control schemes in the existing literature do not control speed of the snake robot along the path. This chapter presents the first control methodology applicable to the complete nonlinear model of a holonomic snake robot, with guaranteed stability properties. The methodology we propose is applicable to general paths, and in addition to path following capabilities, it regulates the speed of the center of mass. In particular, we establish a ˙ and speed. clear link between frequency of oscillations (λ) Comparison with the theory presented in Chapters 3–5.

As mentioned in

Chapter 1, the objective of this chapter is not to stabilize repetitive motions and thus there is no need for investigating the existence of Lagrangian structures for the VHCinduced reduced dynamics.

6.2

Model of the snake robot

In this section, we review the kinematic and dynamic model of a snake robot presented in [104]. We consider a snake robot with n rigid links each of length 2l. Each link is assumed to have uniformly distributed mass m and moment of inertia J. We denote the vector of absolute link angles by θ = [θ1 , . . . , θn ]⊤ , θi ∈ [R]2π , the vectors of inertial coordinates of the centers of mass of the links by X = [x1 , . . . , xn ]⊤ ∈ Rn , Y = [y1 , . . . , yn ]⊤ ∈ Rn , and the center of mass of the robot in inertial coordinates by p = [px , py ]⊤ ∈ R2 . Also, we denote the vector of joint angles by φ = [φ1 , . . . , φn−1 ]⊤ , where φi = θi −θi+1 denote the ith joint angle. In geometric mechanics, the joint angles of the snake robot are called the shape variables as they describe the internal configuration of the mechanism [110]. Figure 6.1a illustrates the kinematic parameters of the snake robot. Table 6.1 summarizes the snake robot parameters used in our simulations that


146

(a) Kinematic parameters of the

(b) CM velocity vector projected

snake robot.

onto the head link.

Figure 6.1: Snake robot kinematics.

Table 6.1: The parameters of the snake robot Symbol

Description

Numerical values in simulations

N

Number of links.

10

2l

Length of a link.

0.14 m

m

Mass of a link.

1 kg

θ

Vector of absolute link angles.

–

φ

Vector of joint angles.

–

p = [px , py ]⊤

CM position of the robot.

–

ct

Tangential viscous friction coefficient.

0.5

cn

Normal viscous friction coefficient.

3

are chosen based on the snake robot Wheeko in the NTNU snake robotics laboratory. The snake robot is essentially a serial chain mechanism which is subject to both joint motor torques and ground friction forces. Figure 6.2 depicts the ith link of the snake robot and forces/torques acting on it. In the figure, τi is the actuator torque applied to the ith joint, and fR,i is the ground friction force acting on the ith link. For simplicity, we assume that the friction forces acting on the robot are viscous. A snake robot which is subject to viscous friction qualitatively (although not quantitatively) behaves similarly to a snake robot which is subject to Coulomb friction force [104]. In

147


Figure 6.2: Forces and torques acting on individual links of the robotic snake. this chapter, inspired by biological snakes, we assume that cn > ct , where ct and cn denote the tangential and normal viscous friction coefficients of the links, respectively. This assumption implies that each link is subjected to an anisotropic viscous ground friction force, which means that the ground friction normal to the link is larger than the ground friction parallel to the link. It is shown in [104] that propulsion of a snake robot under viscous friction conditions requires the friction to be anisotropic. Considering the equations of motion of all of the individual links and using first principles, the dynamic equations of the snake robot can be written as follows (see Chapter 2 in [104] for detailed derivations)

˙ p) Mθ θ¨ + Wθ θ˙2 − lSCθT fR (θ, θ, ˙ = D T τ,

(6.2a)

˙ p), Nm¨ p = E ⊤ fR (θ, θ, ˙

(6.2b)

where τ ∈ Rn−1 is the vector of actuator torques, ⊤ fR (·) = fRx,1 , · · · , fRx,n , fRy,1 , · · · , fRy,n ∈ R2n ,

is the vector of ground friction forces, and the remaining quantities are defined according to the standard snake robotics literature [104]

Mθ = JIn + ml2 Sθ V Sθ + ml2 Cθ V Cθ ,

(6.3a)

Wθ = ml2 Sθ V Cθ − ml2 Cθ V Sθ ,

(6.3b)

148


 1   0  A=    0  1   0  D=    0

1

0

...

1

1

...

...

1

−1 . . .

0

... 0

1 ... 0

−1 . . .

 0   0   ∈ R(n−1)×n ,    1  0   0   ∈ R(n−1)×n ,    −1 

... 1  0n×1   e 2n×2 , e = [1, . . . , 1]⊤ ∈ Rn , E =  ∈R 0n×1 e e¯ = [1, . . . , 1]⊤ ∈ Rn−1 , θ = [θ1 , . . . , θn ]⊤ ∈ Rn ,

sin θ = [sin θ1 , . . . , sin θn ]⊤ ∈ Rn , cos θ = [cos θ1 , . . . , cos θn ]⊤ ∈ Rn Sθ = diag(sin θ) ∈ Rn×n , Cθ = diag(cos θ) ∈ Rn×n , h i⊤ θ˙2 = θ˙12 , . . . , θ˙n2 ∈ Rn , b = [0, . . . , 0, 1]⊤ ∈ Rn−1 ,   1 1 . . . 1   0 1 . . . 1       H =  ...  ∈ Rn×(n−1) ,     0 0 . . . 1     0 0 ... 0 

⊤



 K Sθ  V = A⊤ (DD ⊤ )−1 A, K = A⊤ (DD ⊤ )−1 D, SCθ =  . −K ⊤ Cθ

Finally, it can be shown that (see Chapter 2 in [104] for detailed derivations)






149



 ˙  fR,x   X  ˙ p) fR (θ, θ, ˙ =   = Qθ   fR,y Y˙   ⊤ ˙  lK Sθ θ + ep˙ x  = Qθ  ˙  = lQθ SCθ θ˙ + Qθ E p, ⊤ ˙ −lK Cθ θ + ep˙ y

(6.4)

where the matrix Qθ maps the inertial frame velocities of the centers of mass of the links to the inertial frame viscous friction forces acting on the links, and it is given by 



2 2 (ct − cn )Sθ Cθ   ct (Cθ ) + cn (Sθ ) Qθ = −  . 2 2 (ct − cn )Sθ Cθ ct (Sθ ) + cn (Cθ )

(6.5)

The mechanical system (6.2a)-(6.2b) is a dissipative Euler-Lagrange system, due to the presence of friction forces, with n + 2 configuration variables, q = [θ, p]⊤ ∈ Tn × R2 , and n − 1 controls. It therefore has three degrees of underactuation. The actuator torques have no direct effect on the center of mass dynamics (6.2b). The only coupling between the joint dynamics (6.2a) and center of mass dynamics (6.2b) occurs through the ground friction force fR . This coupling is the essential mechanism underlying snake locomotion, and it is what makes the motion control problem challenging. We mention that the underactuated mechanical system in (2.4), which was studied in Chapters 3– 5, is a non-dissipative Euler-Lagrange system that has a similar structure to the snake robot dynamical model (6.2) except that there is no dissipative force fR (·) acting on it. In comparison with the model in (2.4), the mass matrix of the snake robot is given by D(q) = diag Mθ , NmI2×2 and the vector of its Coriolis and centrifugal forces is given by C(q, q) ˙ q˙ = col Wθ θ˙2 , 02×1 .


150

Finally, letting uθn = [cos θn , sin θn ]⊤ and vθn = [− sin θn , cos θn ]⊤ , we define

˙ vt = u⊤ θn p,

(6.6a)

vn = vθ⊤n p. ˙

(6.6b)

The scalars vt and vn defined above are the components of the inertial velocity of the center of mass parallel and perpendicular to the angle of the head, respectively (see Figure 6.1b). According to (6.6a) and (6.6b), the map [vt , vn ]⊤ 7→ p˙ is a diffeomorphism given by 



 vt  p˙ = Rθn  . vn

6.3

(6.7)

Control specifications

In this section we present the blueprint of our control design. We begin by stating the control specifications. Velocity Control Problem (VCP): Given a desired velocity vector µ(p) with polar representation   cos(θref (p)) µ(p) = vref (p)  , sin(θref (p))

(6.8)

design a smooth feedback controller achieving the following specifications: (i) Practical stabilization1 of the head angle θn to θref (p). (ii) Practical stabilization of the tangential velocity vt = u⊤ θn p˙ to vref (p). 1

Practical stabilization of a variable means that by a suitable choice of controller parameters the variable is made to converge to an arbitrarily small neighborhood of its desired value.


151

(iii) Uniform ultimate boundedness of the normal velocity vn = vθ⊤n p˙ with a small ultimate bound, and ultimate boundedness of the solutions of the joint dynamics and all controller states. The above problem formulation relies on the observation that if θn = θref (p), then making p˙ → µ(p) is equivalent to making (vt , vn ) → (vref (p), 0). Path Following Problem (PFP): Given a desired continuously differentiable curve γ ⊂ R2 with implicit representation {p ∈ R2 : ℓ(p) = 0} with dℓp 6= 0 on γ, design a smooth feedback controller achieving the following specifications: (i) Path stabilization: make p(t) → γ. (ii) Velocity control: make kpk ˙ = v on γ, where v is the desired speed on the path γ. The first control specification, i.e., the VCP, will be used to achieve the second control specification, i.e., the PFP. Solution Methodology: In order to solve VCP and PFP, we create a hierarchy of three control specifications, highlighted in Figure 6.3, resulting in a three-stage control design. Stage 1: Body shape control (VHC stabilization). We use the controls τ in (6.2a) to stabilize a virtual constraint encoding a lateral undulatory gait similar to (6.1), in which ωt is replaced by a state λ, and φ0 affects only the head angle θn . ¨ = uλ . The evolution of λ, φ0 is governed by two compensators, φ¨0 = uφ0 and λ Stage 2: Velocity control. Given a desired velocity function µ(p) as in (6.8), this stage unfolds in two substages: A. Head Angle Control. Inspired by the biological observation that snakes keep their head pointed towards a target while their body undulates behind the head, we design uφ0 to practically stabilize θn → θref (p) while guaranteeing that (φ0 , φ˙ 0) is uniformly ultimately bounded.


152

B. Speed Control. We design uλ to practically stabilize vt → vref (p) while guaranteeing that vn settles into a small neighborhood of the origin and λ˙ is uniformly ultimately bounded. Stage 3: Path following control. Design the velocity function µ(p) in (6.8) such that when kp˙ − µ(p)k is sufficiently small, PFP is solved. Remark 6.3.1. The above design methodology is based on the following intuition. The inputs in Stages 1 and 2 of the hierarchical control system are used to create a point-mass abstraction of the snake robot. Indeed, the motor torque inputs, τ , which are determined at Stage 1, enforce a lateral undulatory gait on snake shape variables that induces forward propulsion. The lateral undulatory gait is dynamically changing according to the evolution of the dynamic variables φ0 and λ. Proper evolution of these two dynamic variables, governed by the inputs uφ0 and uλ at Stage 2, will control the orientation of the snake as well as its propulsion speed. Finally, in Stage 3 we design a proper reference vector, µ, for the point-mass abstraction that will drive the robot towards the target path and make it to traverse it with a desired speed. Figure 6.3 depicts the three stages of the hierarchical control system. In the sequel we will present our solution to the planar snake robot path following problem according to the aforementioned strategy. In order to improve readability, the technical details and proofs are presented in Section 6.10.

6.4

Body shape control

In this section, we use the control inputs τ in (6.2a) to stabilize a dynamic VHC encoding a lateral undulatory gait for the shape variables of the robot. Inspired by the lateral undulatory gait in (6.1), we replace ωt by the dynamic variable λ. Also, we let the offset joint angle φ0 be another dynamic variable that is only added to the head link. The


153

Speed controller

Path following controller

VHC stabilizer

Polar conversion

Snake robot

Head angle controller

Figure 6.3: The hierarchical VHC-based snake robot control system. result is the constraint

θi − θi+1 = α sin(λ + (i − 1)δ), i = 1, . . . , n − 2,

(6.9a)

θn−1 − θn = α sin(λ + (n − 2)δ) + φ0 ,

(6.9b)

where (α, δ) are positive constants referred to as gait parameters and (λ, φ0) ∈ [R]2π × [R]2π are the states of two double integrators

¨ = uλ , φ¨0 = uφ , λ 0

(6.10)

to be designed later. The relations in (6.9) are parametrized by the states of the dynamic compensators in (6.10) which will be used to control the planar orientation and position of the robot. Let Φi (λ) = α sin(λ + (i − 1)δ), i = 1, . . . , n − 1, and Φ(λ) = [Φ1 (λ), . . . , Φn−1 (λ)]⊤ . Since θ = HDθ + eθn , the relations in (6.9a)-(6.9b) can be expressed in vector form as follows θ = eθn + HΦ(λ) + Hbφ0 . Defining the two-parameter family of mappings

(6.11)

154


hλ,φ0 : Tn × R2 → Rn−1 , (θ, p) 7→ Dθ − Φ(λ) − bφ0 ,

(6.12)

which are parameterized by the states of two double integrators in (6.10), the relation (6.11) can also be written as

hλ,φ0 (θ, p) = 0, ¨ = uλ , φ¨0 = uφ . λ 0

(6.13)

The above is a dynamic VHC with two parameters, λ and φ0 . Considering the output function e := hλ,φ0 (θ, p) for the augmented system (6.2)–(6.10) and taking two derivatives, we obtain

e¨ = (⋆) + Φ′ (λ)uλ + buφ0 + DMθ−1 D ⊤ τ.

(6.14)

Since Mθ is a positive definite matrix and D has full row rank n − 1, rank(DMθ−1 D ⊤ ) = n − 1 for all (θ, p) ∈ Tn × R2 and all λ ∈ [R]2π , λ˙ ∈ R, uλ ∈ R, φ0 ∈ [R]2π , φ˙0 ∈ R, and uφ0 ∈ R. Therefore, the output above yields a vector relative degree {2, · · · , 2} everywhere , implying that the dynamic VHC (6.13) is regular. Similar to (5.12) in the previous chapter, the evolution of the dynamic variables λ and φ0 are chosen to be governed by double integrators. As a result, the input to the double integrators, i.e., uλ and uφ0 , appear after taking two derivatives of the output e = hλ,φ0 (q). The zero dynamics manifold associated with the output e = hλ,φ0 (θ, p) is the set ˙ p, p, ˙ φ0 , φ˙ 0 ) ∈ R2n+8 : Dθ = Φ(λ) + bφ0 , Γ3 = {(θ, θ, ˙ λ, λ, D θ˙ = Φ′ (λ)λ˙ + bφ˙ 0 }. This set can be stabilized by means of the input-output linearizing control law

(6.15)


155

τ = (DMθ−1 D ⊤ )−1 {DMθ−1 Wθ θ˙ 2 − lDMθ−1 SCθ⊤ fR + Φ (λ)λ˙ 2 + Φ′ (λ)uλ ′′

+buφ0 − KP [Dθ − Φ(λ) − bφ0 ] − KD [D θ˙ − Φ′ (λ)λ˙ − bφ˙ 0 ]},

(6.16)

where KD , KP are positive definite diagonal matrices containing the joint controller gains. To summarize, the relation (6.13) is a dynamic VHC for the augmented system (6.2)– (6.10) and its constraint manifold is Γ3 . After asymptotically stabilizing Γ3 , we are left with two control inputs, (uλ , uφ0 ) to solve the direction following problem. As described in Section 6.3 we will use the dynamic compensators to regulate the head angle and the velocity of the robot to desired values. To this end, we first derive the reduced dynamics of the robot, i.e., we reduce the system to the invariant manifold Γ3 . By left multiplying both sides of (6.2a) by e⊤ , which is a left annihilator of the control input matrix D ⊤ , and evaluating the result on the virtual constraint manifold Γ3 , the dynamics of the snake robot on the virtual constraint manifold Γ3 read as

θ¨n =

˙ φ0 , φ˙ 0 , p, p) Ψ1 (θn , θ˙n , λ, λ, ˙ + Ψ2 (θn , λ, φ0)uλ + Ψ3 (θn , λ, φ0)uφ0 ,

p¨ =

where

(6.17a)

Ψ4 (θn , λ, φ0)p˙ + Ψ5 (θn , λ, φ0)θ˙n + Ψ6 (θn , λ, φ0)λ˙ + Ψ7 (θn , λ, φ0)φ˙0 ,

(6.17b)

φ¨0 =

u φ0 ,

(6.17c)

¨= λ

uλ ,

(6.17d)


156

′′

e⊤ Mθ HΦ (λ) ˙ 2 1 Ψ1 (·) = − λ − ⊤ {Wθ θ˙2 − lSCθ⊤ fR (·)}, ⊤ e Mθ e e Mθ e ⊤ ′ e Mθ HΦ (λ) , Ψ2 (·) = − e⊤ Mθ e e⊤ Mθ Hb Ψ3 (·) = − ⊤ , e Mθ e 1 Ψ4 (·) = E ⊤ Qθ E, Nm l E ⊤ Qθ SCθ e, Ψ5 (·) = Nm l Ψ6 (·) = E ⊤ Qθ SCθ HΦ′ (λ), Nm l E ⊤ Qθ SCθ Hb. Ψ7 (·) = Nm

(6.18a) (6.18b) (6.18c) (6.18d) (6.18e) (6.18f) (6.18g)

In the above, each Ψi (·) is evaluated on the constraint manifold Γ3 . The equations in (6.17) describe a control system with two inputs, (uλ , uφ0 ). This system completely describes the motion of the snake once the VHC (6.9) has been enforced. The control specification for system (6.17) is to stabilize θn to an arbitrarily small neighborhood of θref ; to stabilize vt = u⊤ θn p˙ to an arbitrarily small neighborhood of vref ; and finally, to guarantee that vn = vθ⊤n p˙ converges to a neighborhood of the origin. ˙ φ0 , φ˙ 0 ) to remain bounded. Meanwhile, we also require (λ,

6.5 6.5.1

Velocity control Head angle control

In this section, we use the control input uφ0 to control the head angle of the robot in order to meet specification 2A. To this end, we consider the states (θn , θ˙n , φ0 , φ˙ 0 ) of the constrained system (6.17a)-(6.17c). We design a high-gain feedback uφ0 (θn , θ˙n , φ0 , φ˙ 0 ) that makes (θn − θref (p), θ˙n − θ˙ref (p)) converge to an arbitrarily small neighborhood of the origin and (φ0 , φ˙ 0 ) uniformly ultimately bounded. This analysis is made independent of the choice of uλ , using time scale separation. By (6.17a) and (6.17c), the dynamic


157

equations governing the states (θn , θ˙n , φ0, φ˙ 0 ) of the constrained system can be written as ˙ φ0 , φ˙ 0 , uλ ) + Ψ3 (·)uφ , θ¨n = f1 (θn , θ˙n , λ, λ, 0

(6.19)

φ¨0 = uφ0 . Proposition 6.5.1. Consider the head angle control law for system (6.19)

u φ0

1 = Ψ3 (·)

1 ˜˙ (θn + kn θñ ) − k1 φ0 − k2 φ˙ 0 . ǫ

(6.20)

˙ where θñ = θn − θref (p) where k1 , k2 , kn , ǫ are positive parameters. If uλ(t), λ(t), λ(t) are defined for all t ≥ 0, then for any k1 , k2 , ǫ1 > 0, there exist positive real constants ǫ⋆ , kn , ǫ2 and a positive definite function V (φ0 , φ˙ 0 ) such that for all ǫ ∈ (0, ǫ⋆ ), the set (θn , θ˙n , φ0, φ˙ 0 ) k(θñ , θ˜˙n + kn θñ )k ≤ ǫ1 , V (φ0 , φ˙ 0 ) ≤ ǫ2 is asymptotically stable. The proof can be found in Section 6.10.

Remark 6.5.2. The result of Proposition 6.5.1 can be interpreted as follows. Under (6.20), the head angle error can be made arbitrarily small provided that ǫ is chosen to be sufficiently small. Also, φ0 and φ˙ 0 remain uniformly ultimately bounded. In the next section we define a feedback controller uλ guaranteeing that for any initial condition, the closed-loop system has no finite escape time. This will guarantee that the above proposition is applicable.

6.5.2

Speed control

We now turn our attention to specification 2B. Consider the reduced dynamics (6.17). In the previous section, we controlled the states θN , θ˙n , φ0 , φ˙ 0. Now, we are left with ˙ The map p˙ 7→ (vt , vn ) is a diffeomorphism so for velocity control the states p, p, ˙ λ, λ. ˙ with ∆vt = vt − vref (p). In we may consider the subsystem with states (∆vt , vn , λ, λ), order to obtain the tangential and normal velocity dynamics evaluated on the constraint

158


manifold, we take the time derivatives of Equations (6.6a), (6.6b), which using (6.17b) yields

v˙ t =

⊤ ˙ u⊤ θn Ψ4 (·)uθn vt + uθn Ψ4 (·)vθn vn + θn vn + ⊤ ⊤ ˙ ˙ ˙ u⊤ θn Ψ5 (·)θn + uθn Ψ6 (·)λ + uθn Ψ7 (·)φ0

v˙ n =

(6.21a)

vθ⊤n Ψ4 (·)uθn vt + vθ⊤n Ψ4 (·)vθn vn − θ˙n vt + vθ⊤n Ψ5 (·)θ˙n + vθ⊤n Ψ6 (·)λ˙ + vθ⊤n Ψ7 (·)φ˙ 0 .

(6.21b)

Thus, the velocity error dynamics have the form ∆v˙ t =

f2 (θn , θ˙n , λ, φ0, φ˙ 0 , ∆vt , vn ) + ˙ ˙ u⊤ θn Ψ6 (·)λ − (dvref )p p,

v˙ n =

¨= λ

(6.22a)

˙ φ0 , φ˙ 0, ∆vt , vn ) + f3 (θn , θ˙n , λ, λ, vθ⊤n Ψ4 (·)vθn vn ,

(6.22b)

uλ .

(6.22c)

In order to stabilize the solutions of (6.22a), (6.22b) to a neighborhood of the origin, we use the following control input uλ = −Kz (λ˙ + Kλ ∆vt ) − Kλ [f2 (·) +

˙ u⊤ θn Ψ6 (·)λ

(6.23) − (dvref )p p]. ˙

where Kλ > 0 and Kz > 0 are positive constants. Note that u⊤ θn Ψ6 (·) is bounded away from zero by part (c) of Remark 6.10.1 provided that the ultimate bound on φ0 from Proposition 6.5.1 is small enough. We have the following proposition regarding the forward velocity control system. Proposition 6.5.3. Consider the control system (6.22a)-(6.22c) under the controller (6.23) with cn > ct . If the ultimate bound on φ0 from Proposition 6.5.1 is small enough that u⊤ θn Ψ6 (·) is bounded away from zero, then for all ǫ3 > 0 and for sufficiently large


159

˙ ∆vt , vn ) : controller gain Kλ > 0, there exists ǫ4 > 0 such that the compact set Λ1 = {(λ, λ, |∆vt | ≤ ǫ3 , λ˙ = −Kλ ∆vt , |vn | ≤ ǫ4 } is asymptotically stable. The proof is presented in Section 6.10. Remark 6.5.4. The result of Proposition 6.5.3 can be interpreted as follows. Under (6.23), the velocity error ∆vt can be made arbitrarily small provided that the gain Kλ is chosen to be sufficiently large. Also, the normal velocity vn remains uniformly ultimately bounded. Finally, we mention that we have not been able to analytically determine the effect of control gains on the ultimate bound of the dynamic variable φ0 and this bound needs to be determined a posteriori.

6.6

Path following control of snake robots

In this section we carry out the last design stage: the path following control. Thus far we have developed a velocity controller that asymptotically stabilizes the direction following manifold

˙ p, p, ˙ φ0 , φ˙ 0 ) ∈ Γ3 : k(θñ , θ˜˙n + kn θñ )k ≤ ǫ1 , V (φ0 , φ˙ 0 ) ≤ ǫ2 , Γ2 = {(θ, θ, ˙ λ, λ, |vt − vref (p)| ≤ ǫ3 , |vn | ≤ ǫ4 , λ˙ = −Kλ ∆vt }.

(6.24)

The next objective is to design θref (p) and vref (p) in (6.8) to stabilize an arbitrary small neighborhood of the planar curve {ℓ(p) = 0} while regulating the velocity along the curve. To this end, we define the path following manifold as follows

˙ p, p, ˙ φ0 , φ˙ 0 ) ∈ Γ2 : |ℓ(p)| ≤ ǫ5 }, Γ1 = {(θ, θ, ˙ λ, λ,

(6.25)

where ǫ5 is a small constant. Remark 6.6.1. The set Γ1 is compact. The reason is that the inequality |ℓ(p)| ≤ ǫ5 implies that p is bounded because ℓ(·) is a continuous function. Since θref (·) and vref (·) are


160

continuous functions, θref (p) and vref (p) are bounded. Therefore, θn and vt are bounded. Since vt and vn are bounded, p˙ is bounded. Since θn and φ0 are bounded and θ = eθn + ˙ HΦ(λ) + Hbφ0 on the set Γ1 , θ is bounded. Since θ˙ = eθ˙n + HΦ′ (λ)λ˙ + Hbφ˙ 0 , and θ˙n , λ, and φ˙ 0 are bounded, θ˙ is bounded. If we let y = ℓ(p), we want y → 0 to meet specification (i) of the PFP. On the direction following manifold Γ2 , we have

y˙ = dℓp p˙ = dℓp R∆1 µ + dℓp d(·)

(6.26)

We propose to use the following control law for determining the reference velocity 



dℓ⊤  0 1  ⊤ v p µ(p) = − Ktran ℓ(p) +   dℓp 2 kdℓp k kdℓp k −1 0 {z } | {z } | µ⊥ (p)

(6.27)

µk (p)

where Ktran is a positive constant.

Figure 6.4: Components of the path following reference vector.

Remark 6.6.2. Since dℓ⊤ p = ∇ℓ(p) is perpendicular to the level sets of ℓ(·), the control law (6.27) can be intuitively described as follows. The reference velocity µ(p) is composed of two components: (a) the component µ⊥ (p) is perpendicular to the level sets of ℓ(·) and decreases the distance of the center of mass to the curve γ = ℓ−1 (0); (b) the component

161


µk (p) is tangent to the level sets of ℓ(·) and regulates the velocity of the center of mass on the curve γ = ℓ−1 (0) (see Figure 6.4). We have the following proposition regarding the path following controller. Proposition 6.6.3. Consider system (6.26) where |∆1 | < ǫ1 . For sufficiently small ǫ1 the following property holds: for any ǫ5 > 0 there exists Ktran such that the set {|ℓ(p)| ≤ ǫ5 } is asymptotically stable for (6.26). Moreover, the velocity control specification, i.e., kpk ˙ = v, is approximately met on γ: kpk ˙ − v ≤ ǫ3 + ǫ4 . The proof is presented in Section 6.10.

6.7

Main Result

In this section we summarize the snake robot maneuvering controller and present our main result. The proposed VHC-based control system is comprised of three stages (see Figure 6.3). Stage 1, body shape control, stabilizes an undulatory gait pattern through the control inputs τ . This stage corresponds to the inner-most control loop, and it has the highest priority. For the snake robot model (6.2a)–(6.2b), we have

τ = (DMθ−1 D ⊤ )−1 {DMθ−1 Wθ θ˙ 2 − lDMθ−1 SCθ⊤ fR + Φ (λ)λ˙ 2 + Φ′ (λ)uλ + buφ0 ′′

−KP [Dθ − Φ(λ) − bφ0 ] − KD [D θ˙ − Φ′ (λ)λ˙ − bφ˙ 0 ]},

(6.28)

where φ0 , φ˙ 0 , λ, and λ˙ are the states of the following dynamic compensators

¨ = uλ , φ¨0 = uφ , λ 0

(6.29)

corresponding to the second stage, velocity control, which regulates the velocity vector of the snake center of mass to a reference. The control input uφ0 is given by


u φ0

1 = Ψ3 (·)

1 ˜˙ (θn + kn θñ ) − k1 φ0 − k2 φ˙ 0 , ǫ

162

(6.30)

where θñ = θn − θref (p). Also, the control input uλ is given by

˙ ˙ uλ = −Kz (λ˙ + Kλ ∆vt ) − Kλ [f2 (·) + u⊤ θn Ψ6 (·)λ − (dvref )p p],

(6.31)

where Kλ > 0 and Kz > 0 are positive constants. The reference signals θref (p)in (6.30) and vref (p) (6.31), required for Stage 2, are generated in Stage 3. These references are determined from the identity

µ(p) = vref (p)[cos θref (p), sin θref (p)]⊤ , where µ(p) is given by

µ(p) = −



dℓ⊤ p



 0 1  ⊤ v K ℓ(p) +  dℓp  tran kdℓp k2 kdℓp k −1 0

(6.32)

where Ktran is a positive constant. We have

θref (p) = atan(µ1 (p), µ2 (p)),

(6.33)

vref (p) = kµ(p)k.

(6.34)

Note that µ(p) = [vref (p) cos(θref (p)), vref (p) sin(θref (p))], i.e., θref (p) and vref (p) are generated according to Equations (6.33) and (6.34) where µ(p) is determined using (6.32). We have the following theorem regarding the snake robot control system. Theorem 6.7.1 (VHC-based control for planar snake robot maneuvering control). Consider the snake robot model (6.2a)–(6.2b) with feedback (6.28), (6.30), (6.31), and (6.32). Suppose that the ultimate bound on φ0 from Proposition 6.5.1 is small enough such that u⊤ θn Ψ6 (·) is bounded away from zero. For any ǫ5 > 0, there exist a sufficiently


163

small ǫ in (6.30), a sufficiently large Kλ in (6.31) and Ktran in (6.32) such that the path following manifold Γ1 in (6.25) is asymptotically stable and the velocity of the snake robot ˙ − v ≤ ǫ3 + ǫ4 . satisfies the asymptotic bound lim sup kpk

Proof. Consider the sets Γ1 , Γ2 , Γ3 defined in (6.25), (6.24), (6.15) and note that Γ1 ⊂

Γ2 ⊂ Γ3 . Also, by Proposition 6.6.3, Γ1 is asymptotically stable relative to Γ2 and by Propositions 6.5.1 and 6.5.3, Γ2 is asymptotically stable relative to Γ3 for the closed-loop system. On the other hand, Γ1 is a compact set (see Remark 6.6.1). Using Proposition 5.3.1, we conclude that the set Γ1 is asymptotically stable. Remark 6.7.2. (Implementation of the controller) The camera-based position measurement system of the robot Wheeko enables us to calculate the global frame coordinates of the head link, (xn , yn ), and the absolute angle of the head link θn (see [104] for more details). Also, the snake robot’s magnetic encoders measure the joint angles, i.e., the vector φ = [φ1 , . . . , φn−1]⊤ is available from measurements, instead of the absolute link angles. We can use the following kinematic relationships to calculate the center of mass position, p, and the vector of absolute link angles, θ

θ = Hφ + eθn , p =

6.8



⊤







−l  e HA cos(θ)   xn  . +  N ⊤ yn e HA sin(θ)

Simulation results

In this section we present the simulation results which illustrate the performance of the proposed path following controller. Simulation parameters. We consider a snake robot with N = 10 links with length 2l = 0.14 m, mass m = 1 kg, and moment of inertia J = 0.0016 kg.m2 . The friction coefficients are ct = 0.5 and cn = 3. The parameters of the VHC are chosen to be α = 30π/180 rad, and δ = 72π/180 rad. The model parameters are chosen based on the snake robot Wheeko in the NTNU snake robotics laboratory.


164

Circle tracking. We would like to follow a circular path with radius 2 m. The ˙ initial conditions are θ(0) = [0, . . . , 0]⊤ , θ(0) = [0, . . . , 0]⊤ , p(0) = [−4, 1]⊤ , p(0) ˙ = [0, 0]⊤ , ˙ ˙ λ(0) = λ(0) = φ(0) = φ(0) = 0. We run the simulation for 600 seconds. The controller parameters are listed in Table 6.2. Note that ǫ determines the ultimate bound on head angle error. Also, kn determines the rate of convergence of θn to θref . The gains k1 and k2 have influence on the ultimate bound of φ0 . The gains Kλ and Kz determine the rate of convergence and ultimate bound of ∆vt . Finally, Ktran controls the path following error. In order to show the performance of the proposed control scheme in the presence of angular position measurement noise, we subject every ith link angle θi to an additive noise by using Matlab function randn() which generates normally distributed pseudorandom numbers that can be considered as measurement noise for the joint angles. The root mean square (RMS) of the noise applied to the joint angle measurements was 0.1 rad. The simulation results show that the snake robot follows the desired path while the states of the compensators in (6.10) remain uniformly ultimately bounded. Figure 6.5a depicts the snake robot and the trajectory of the center of mass of the robot. Figure 6.5b depicts the path following error. The RMS value of the path following error in the steady state is 0.12 m. Figure 6.5c depicts the dynamic variable φ0 . As it is shown in Theorem 6.7.1, the variable φ0 remains uniformly ultimately bounded. Figure 6.5d ˙ and thus gives the frequency of the undulatory motion. depicts the dynamic variable λ, This is within acceptable range of frequency of oscillations of the existing snake robots at the NTNU snake robotics laboratory (up to 2 rad/sec). Figure 6.6a depicts the shape variable error. As it can be seen from the figure, the error converges exponentially to the origin. Figure 6.6b depicts the actual and the reference tangential velocities. The reason for the steady state error is that the gain Kλ in control law (6.23) is not large enough. Choosing Kλ sufficiently large causes the velocity error ∆vt to become arbitrarily small. However, such large gain values cause large oscillations and large control torques. Figure 6.6c depicts the head angle tracking error. As it is shown in Theorem 6.7.1,


165

the tracking error remains uniformly ultimately bounded. Finally, Figure 6.6d depicts the norm of the control torque vector, from which we can see that the control torques are within the physical limitations/saturation values of the existing snake robots at the NTNU snake robotics laboratory (up to 7 Nm). As it can be seen from (6.27), when the path following error kℓ(p)k is large, the reference speed kµ(p)k = vref (p) will be large. Tracking such a large reference speed will require very fast oscillations of the snake robot and large control torques. In order to avoid such large initial oscillations and joint control torques, if kℓ(p)k > 0.4 we set vref (p) = 0.05 m/sec. If kℓ(p)k < 0.3, we set vref (p) = kµ(p)k where µ(p) is determined from (46). Finally if 0.3 ≤ kℓ(p)k ≤ 0.4, we let the reference speed be determined from the smooth interpolation between 0.05 and kµ(p)k, i.e., from (0.5−10kµ(p)k)kℓ(p)k+(4kµ(p)k−0.15). Table 6.2: Snake robot VHC-based controller parameters Controller parameter

Controller expression

Numerical values in simulations

KP

(6.16)

10I2

KD

(6.16)

10I2

ǫ

(6.20)

10-1

kn

(6.20)

10

k1

(6.20)

1

k2

(6.20)

1

Kz

(6.23)

50

Kλ

(6.23)

50

Ktran

(6.27)

0.6

v

(6.27)

0.05

Robustness analysis for circle tracking. We now test the robustness of the path following controller (6.28)–(6.32) to uncertainties in the friction parameters and to noise in the joint angle and center of mass measurements. Specifically, we present three tests.

166

Chapter 6. VHC-based Control of Planar Snake Robots p reference path 14

3

12

2

10

8

0

h(p) [m]

y [m]

1

−1

6

4 −2

2 −3

0 −5

−4

−3

−2

−1

0

1

2

3

4

5

x [m] −2 0

50

100

150

200

250

300

Time [s]

(a) Plots of the snake robot with 10 links (b) The path following error. and the path of its center of mass. 0.8

20

0.6 15 0.4

0.2

λ˙ [rad/s]

φ0 [rad]

10

0

5 −0.2

−0.4 0 −0.6

−0.8

−5 0

50

100

150

200

250

300

Time [s]

0

50

100

150

200

250

300

Time [s]

(c) The dynamic variable φ0 remains uni-

(d) The dynamic variable λ˙ remains uni-

formly ultimately bounded.

formly ultimately bounded.

Figure 6.5: Planar snake robot circle tracking simulation results– Part(a). Test 1. To simulate the inaccuracy of the encoder measurements, we replace θ in the feedback law (6.28)–(6.32) by θ + n(t), where n(t) is a vector of zero average white Gaussian noise signals whose standard deviation σ is 10% of the maximum joint angle observed in steady-state under nominal operation, σ = 0.1 · maxi lim supt |θi (t) − θi+1 (t)|. In our simulations we found σ = 0.07 rad, or approximately 4 degrees. This is quite high, as in practical experiments with the snake robot Wheeko the angle measurements are seen to be accurate within 1-2 degrees. Test 2. Here we simulate errors in the measurement of the center of mass position, p. To this end, we replace p with p + n(t) in the feedback law (6.28)–(6.32), where n(t) is a vector of white Gaussian noise signals with standard deviation 0.1 m. While the camera

167


1

0.9

0.2

vref vt

0.18

0.7 0.16 0.6 0.14 0.5 0.12

v [m/s]

||Dθ − Φ(λ) − bφ0 || [rad]

0.8

0.4

0.3

0.1

0.08 0.2 0.06 0.1 0.04 0 0

1

2

3

4

5

6

7

8

9

0.02

10

Time [s] 0 0

50

100

150

200

250

300

Time [s]

(a) The lateral undulatory gait (6.13) is (b) Reference and actual tangential velocistabilized among the shape variables of the ties. snake robot. 0.2

5

4.5 0.15 4 0.1

0.05

3

||u|| [Nm]

θN − θref [rad]

3.5

0

2.5

2

−0.05 1.5 −0.1 1 −0.15

0.5

0

−0.2 0

50

100

150

200

250

300

Time [s]

(c) The head angle tracking error.

0

50

100

150

200

250

300

Time [s]

(d) Norm of the control torque vector.

Figure 6.6: Planar snake robot circle tracking simulation results– Part (b).


168

tracking system in the NTNU Snake Robotics Laboratory has sub-millimeter accuracy, we have chosen this large measurement error to reflect that in the absence of an indoor camera system, position estimation would have to rely on on-board camera data and suitable vision algorithms. In this setting, it is reasonable to assume a measurement error of the order of 0.1 m. Test 3. Since mass and moments of inertia can be determined with high accuracy, the most relevant parametric modelling uncertainty is in the friction coefficients cn and ct . In this test, we replace cn and ct by 0.9cn and 0.9ct in the feedback law (6.28)–(6.32). This corresponds to a 10% uncertainty in these parameters. Performance metrics. In order to assess the robustness of the feedback law (6.28)– (6.32) in the three tests above, we use four performance metrics: • The first performance metric, P1 , is the RMS value of the path following error ℓ(p(t)) = p2x (t) + p2y (t) − 4 in steady-state. • The second performance metric, P2 , is the settling time of the path following error: the largest time after which the absolute value of the path following error remains within one third of its initial value. • The third performance metric, P3 , is the largest torque magnitude applied to each individual joint, i.e, sup(max |ui (t)|), 1 ≤ i ≤ N − 1. This value should remain t

i

within the physical actuator limit of 7 Nm. • Finally, P4 represents the RMS value of the torque norm signal ku(t)k in steadystate. The results of the tests are found in Table 6.3. In order to examine the influence of the measurement noise on the performance of the controller in Tests 1 and 32, we ran the same tests for 10 times, and took the average of the results. In the table, Test 0 corresponds to the nominal situation where there are no noise and modelling uncertainty


169

in the simulation. The results in the table illustrate the robustness of the proposed controller. Specifically, the performance of the controller is only marginally affected by noise in the angle measurements and uncertainty in the friction coefficients (Tests 1 and 3). Also, the peak torque remains within the physical actuator limit of 7 Nm. Furthermore, the peak and the RMS torques decrease slightly with decreasing friction coefficients in Test 3. In Test 2, the large noise in the center of mass position measurement has no significant effect on the settling time, and it causes a gracious degradation of the RMS value of the path following error, which increases from 0.1 m to approximately 0.2 m. However, we note that the peak torque in this case exceeds the actuator limit. The degraded performance in Test 2 is not surprising, since the position measurement error is in the magnitude of the snake robot’s link length in this test. As we mentioned in the Introduction, the intrinsic robustness of the VHC controller is to be ascribed to the fact that this controller does not rely on any exogenous reference signal. This general principle can be roughly explained as follows. A feedback control loop aimed at tracking a reference signal reacts to errors between the system output and the reference. As such, it attempts to make the output conform to the timing of the reference signal. When the loop is affected by uncertainties or disturbances, it may happen that the time parametrization of the reference signal becomes unfeasible in that the system output cannot “keep up” with the reference. In such a situation, the loop will measure a large tracking error and the overall performance will be affected. On the other hand, if the time parametrization is removed from the loop and the control objective of tracking is replaced by the stabilization of a suitable relation (the VHC in this chapter), then the control loop no longer mandates a specific time parametrization for the output. It only requires the enforcement of the implicit relation. Such a loop, therefore, is completely insensitive to issues of timing of the reference signal, and typically displays a greater robustness to uncertainties or disturbances. We will illustrate this principle next.


170

Table 6.3: Robustness analysis for circle tracking Test No.

P1

P2

P3

P4

0

0.1 m

47.2895 sec

2.775 Nm

0.2887 Nm

1

0.1071 m

48.6097 sec

2.9136 Nm

1.9076 Nm

2

0.2056 m

47.4095 sec

11.8068 Nm

3.2157 Nm

3

0.1 m

47.2895 sec

2.6894 Nm

0.2708 Nm

Performance indicators for the VHC controller (6.28)–(6.32) in the presence of noise in the angle measurements (Test 1), noise in the center of mass position measurements (Test 2), and uncertainty in the friction coefficients (Test 3). Test 0 refers to the nominal performance of the robot in the absence of noise and uncertainties.

6.9

Discussion

The presented hierarchical approach in this chapter is to our best knowledge the first analytically established maneuvering controller for holonomic snake robots, which presents formal stability proofs for the controlled system. Removing the time parameterization from the biologically-inspired undulatory gaits and making them invariant via feedback enabled us to invoke reduction tools from nonlinear control theory. The simulation results presented in this chapter validate the performance of the proposed control strategy. We have also validated the performance of the controller in the presence of reasonably large measurement noise in the simulations. A main topic of future work is to implement the controllers on a robotic snake to validate the practical effectiveness of the approach. Our control design shows some degree of robustness to modeling errors and noise. In particular, our controller demonstrates a more robust performance in response to modeling errors and noise in comparison with the PD controller. In terms of controller implementation one might saturate the output of the two dynamic compensators and use anti-windup design. For practical implementations, actuator sat-


171

urations will present limitations on the achievable torques and torque rates, and tuning the controller parameters must be done accordingly. The parameter tuning will thus be a trade-off between making the ultimate bound on φ0 from Proposition 6.5.1 sufficiently small and staying within the actuator limitations.

6.10

Proof of technical results

We will require some knowledge of each function Ψi (·) appearing in (6.17) which is summarized in the following remark. Remark 6.10.1. We make some numerical observations that are important in some of the proofs. It can be numerically verified that for all gait parameters (α, δ): (a) Ψ3 (·) = −e⊤ Mθ Hb/e⊤ Mθ e is bounded away from zero and negative for all θn , λ, φ0 . (b) vθ⊤n Ψ4 (·)vθn ≈ −cn /m for all θn , λ, φ0 . (c) There exists γ6 > 0 such that −u⊤ θn Ψ6 (·) < −γ6 for all θn , λ and small values of φ0 and for cn > ct . (d) There exists ǫ0 > 0 such that we have |vθ⊤n Ψ6 (·)| ≤ αǫ0 for all θn , λ, φ0 where α denotes the amplitude of sinusoidal joint motion in (6.9a)–(6.9b). (e) kΨ4 (·)k ≤ cn /m for all θn , λ, φ0 . (f ) There exists γ7 > 0 such that kΨ7 (·)k ≤ γ7 for all θn , λ, φ0. (g) |vθ⊤n Ψ4 (·)uθn | < ct /m for all θn , λ, φ0. Note that the above observations are independent of the parameters N, m, l, J, because these parameters can be factored out of the Ψi ’s as it can be seen from (6.18). ˙ Proof of Proposition 6.5.1. Viewing the states λ(t), λ(t), and the input uλ (t) as exogenous signals, the control system (6.19) can be viewed as a time-varying system with


172

states (θn , θ˙n , φ0 , φ˙ 0 ). Under the control input (6.20), the closed-loop dynamics of system (6.19) in the standard singular perturbation form become

˙ θñ = ω ñ, ǫω ˜˙ n = ǫ θ¨ref + g1 (t, φ0 , φ˙ 0 , θn , θ˙n ) + Ψ3 (·)(k1φ0 + k2 φ˙ 0 ) − (˜ ωn + kn θñ ), (6.35) where

˙ g1 (t, φ0 , φ˙ 0 , θn , θ˙n ) = f1 (θn , θ˙n , λ(t), λ(t), φ0 , φ˙ 0, uλ (t)). Here we use time-scale separation to make the analysis independent of the choice of uλ . Note that (6.35) is a singularly perturbed system with reduced dynamics

˙ θñ = −kn θñ ,

(6.36)

dˆ y = −ˆ y, dτ

(6.37)

and boundary-layer dynamics

where yˆ = ω ˜ n + kn θñ . The origin is an exponentially stable equilibrium point of the reduced system. Also, the origin is an exponentially stable equilibrium point of the boundary-layer system. According to the singular perturbation theorem on an infinite interval (see Theorem 11.2 in [111]), there exist ǫ⋆1 such that for all ǫ ∈ (0, ǫ⋆1 ) and all θñ (0) ∈ R and t0 ≥ 0, the singularly perturbed system (6.35) has a unique solution (θñ (t, ǫ), ω ˜ n (t, ǫ)) such that

θñ (t, ǫ) − exp(−kn (t − t0 ))θñ (0) = O(ǫ),

t ω ˜ n (t, ǫ) + kn exp(−kn (t − t0 ))θñ (0) − exp(− )y0 = O(ǫ), ǫ

(6.38a) (6.38b)


173

for all t ∈ (0, ∞). Note that the closed-loop dynamics governing the states (φ0 , φ˙ 0 ) become 1 φ¨0 + k2 φ˙ 0 + k1 φ0 = Ψ (·) | 3

1 ˜˙ (θn + kn θñ ) . ǫ {z }

(6.39)

fn (t,ǫ)

From (6.38a)–(6.38b), it can be seen that fn (t, ǫ) is uniformly bounded and of order O(1), i.e., there exist positive constants k0 and c0 such that |fn (t, ǫ)| ≤ k0 for all |ǫ| < c0 . Since the unforced system φ¨0 + k2 φ˙ 0 + k1 φ0 = 0 is an LTI system and has a globally exponentially stable equilibrium point at the origin (φ0 , φ˙ 0 ) = (0, 0), the system (6.39) is input-to-state stable. Therefore, there exists an ISS-Lyapunov function V (.) and ǫ2 such that the set {V (φ0 , φ˙ 0 ) ≤ ǫ2 } is asymptotically stable (see Theorem 10.4.1 in [112]). Now, we consider the change of variable yˆ = ω ˜ n + kn θñ . The closed-loop dynamics become

˙ θñ = yˆ − kn θñ , ǫyˆ˙ = ǫ[θ¨ref + g1 (t, φ0 , φ˙ 0 , θn , θ˙n ) + Ψ3 (·)(k1φ0 + k2 φ˙ 0 ) + kn (ˆ y − kn θñ )] − yˆ.

(6.40)

Next,we consider the Lyapunov function candidate V1 = (1/2)θñ2 + (1/2)ˆ y 2. We have

V˙ 1 = θñ yˆ − kn θñ2 + yˆyˆ˙ .

(6.41)

It can be shown that there exists L3 > 0 such that yˆyˆ˙ ≤ −(1/2ǫ)ˆ y 2 + L3 yˆ (see proof of Theorem 11.1 in [111]). We have

1 V˙ 1 ≤ θñ yˆ − kn θñ2 − yˆ2 + L3 yˆ. 2ǫ Completing the squares, we get

(6.42)


174

1 1 1 V˙ 1 ≤ −(kn − )θñ2 − ( − 1)ˆ y 2 + L23 . (6.43) 2 2ǫ 2 We define ǫ⋆ := min ǫ⋆1 , 1/(L23 /4ǫ21 + 2) . Then, for kn > (1/2)(L23 /4ǫ21 + 1) and ǫ < ǫ⋆ we have

L2 1 V˙ 1 ≤ − 32 V + L23 . 4ǫ1 2

(6.44)

By the comparison lemma [111], we get

V1 (t) ≤ V1 (0) exp(−

L23 t) + 2ǫ21 . 2 4ǫ1

(6.45)

This implies that k[θñ , yˆ]⊤ k converges to a neighborhood of the origin given by ǫ1 . There fore, the set k[θñ , yˆ]⊤ k ≤ ǫ1 is asymptotically stable. Note that ǫ1 is a design parameter that we can choose arbitrarily.

˙ φ0 , φ˙ 0 ]⊤ . Under the control Remark 6.10.2. Consider the state vector x = [vt , vn , λ, λ, laws (6.20) and (6.23), we have x˙ = f (x) for the closed loop system. Because of the uniform bounds on Ψi , i = 2, . . . , 7, it can be seen that kf (x)k ≤ B(1 + kxk) for some constant B. Because of this linear growth condition, there is no finite escape time and ˙ the signals λ(t), uλ(t) are defined for all t ≥ 0 as required by Proposition 6.5.1. Proof of Proposition 6.5.3. The control law (6.23) is a feedback linearizing controller for system (6.22a) with output z = λ˙ + Kλ ∆vt , and it makes the set Λ3 = ˙ ∆vt , vn ) : λ˙ = −Kλ ∆vt } asymptotically stable. On the set Λ3 , the subsys{(λ, λ, tem (6.22a) becomes ∆v˙ t = f2 (·) − Kλ u⊤ θn Ψ6 (·)∆vt − (dvref )p p˙ Now, we find a positively invariant set

(6.46)


˙ ∆vt , vn ) : |∆vt | ≤ V¯1 , |vn | ≤ V¯2 }, Ω = {(λ, λ,

175

(6.47)

such that |f2 (θn , θ˙n , λ, φ0, φ˙ 0 , ∆vt , vn )| is uniformly bounded on Ω. Note that φ0 , φ˙ 0 have been proven to be uniformly ultimately bounded in Proposition 6.5.1. Therefore we need to show boundedness of ∆vt , vn . We pick V¯1 arbitrary and determine K3 such that |f3 (·)| ≤ K3 . Note that K3 depends on V¯1 . Next, we pick V¯2 > K3 /Kn . Finally, we choose

Kλ >

k1 + k2 V¯2 . γ6 V¯1

(6.48)

We claim that Ω is positively invariant. Note that

−Kλ γ6 ∆vt − k1 − k2 |vn | ≤ ∆v˙ t ≤ −Kλ γ6 ∆vt −k1 + k2 |vn |,

(6.49)

and

−Kn vn − K3 ≤ v˙ n ≤ −Kn vn + K3 .

(6.50)

On ∆vt = V¯1 , we have ∆v˙ t ≤ −Kλ γ6 V¯1 + k1 + k2 |vn | ≤ −Kλ γ6 V¯1 + k1 + k2 V¯2 ≤ 0. On ∆vt = −V¯1 , we have ∆v˙ t ≥ Kλ γ6 V¯1 − k1 − k2 |vn | ≥ Kλ γ6 V¯1 − k1 − k2 V¯2 ≥ 0. On vn = V¯2 , we have v˙ n ≤ −Kn V¯2 + K3 ≤ 0. On vn = −V¯2 , we have v˙ n ≥ Kn V¯2 − K3 ≥ 0. The inequalities above prove that on ∂Ω, the vector field given by (6.22a)–(6.22b) points inside Ω. Therefore, by Nagumo’s theorem [113], the set Ω is positively invariant. For all initial conditions in Ω, we have |f2 (·)| ≤ γ2 = k1 + k2 V¯2 . Now, we employ the Lyapunov function candidate V1 = 21 ∆vt2 , we have V˙ 1 < −Kλ γ6 ∆vt2 + γ2 ∆vt . Therefore we have


1 1 V˙ 1 < −(Kλ γ6 − )∆vt2 + γ22 2 2

176

(6.51)

Using the comparison lemma [111], we have, for all t ≥ 0

1 1 V1 (t) ≤ exp(−(Kλ γ6 − )t)V1 (0) + γ2 2 2(Kλ γ6 − 21 ) 2

Therefore, ∆vt converges to a ball of radius

q

(6.52)

γ22 /(Kλ γ6 − 21 ). Choosing Kλ large

enough makes the ultimate bound of ∆vt less than ǫ3 for any desired ǫ3 > 0. Letting q ˙ ∆vt , vn ) ∈ Λ3 : |∆vt | ≤ ǫ3 } is asymptotically ǫ3 = γ22 /(Kλ γ6 − 21 ), the set Λ2 = {(λ, λ,

stable relative to Λ3 . On the set Λ2 , the dynamics are described by subsystem (6.22b). The function f3 (·) is uniformly bounded on Λ2 , namely, there exists γ3 > 0 such that |f3 (·)| ≤ γ3 on Λ2 . Employing the Lyapunov function candidate V2 = 1/2vn2 and using part (b) of Remark 6.10.1 yields

−cn 2 cn V˙ 2 ≤ vn + γ3 vn ≤ − vn2 + m m 1 2 γ 2 v + γ , 2 n 2γ 3

(6.53)

where γ is some positive constant and we have used Young’s inequality, ab ≤ (γ/2)a2 + (1/2γ)b2 . We conclude that there exists a sufficiently small positive constant β such that 1 V˙ 2 ≤ −βV2 + γ32 . 2γ

(6.54)

Using the comparison lemma [111], we have, for all t ≥ 0, V2 (t) ≤ exp(−βt)V2 (0) + Therefore, vn converges to a ball of radius

1 2 γ . 2γβ 3

(6.55)

p p γ32 /(γβ). Letting ǫ4 = γ32 /(γβ), the set

˙ ∆vt , vn ) ∈ Λ2 : |vn | ≤ ǫ4 } is asymptotically stable relative to Λ2 . This set is Λ1 = {(λ, λ,


177

compact because λ ∈ S 1 , which is a compact set and on Λ1 , |∆vt | ≤ ǫ3 , and λ˙ = −Kλ ∆vt . In the above analysis, Λ1 ⊂ Λ2 ⊂ Λ3 . Also, Λi is asymptotically stable relative to Λi+1 for the closed-loop system for i = 1, 2. On the other hand, Λ1 is a compact set. Using Proposition 5.3.1, we conclude that the set Λ1 is asymptotically stable.

Before proceeding further, we define an operator which will be useful in proving an important kinematic relationship. Definition 6.10.3. The complexification operator is defined to be the map C : R2 → C, [x, y]⊤ 7→ x + jy, where j is the unit imaginary number.

△

According to the above definition, it can be easily seen that the operator C is a linear invertible map, i.e., an isomorphism from the real plane to the complex plane. We have the following lemma whose proof is omitted due to simplicity. Lemma 6.10.4. Given the counter-clockwise rotation matrix Rα ∈ R2×2 through an angle α and a vector [x, y]⊤ ∈ R2 , we have C(Rα [x, y]⊤ ) = exp(jα)(x + jy), where exp(jα) = cos(α) + j sin(α). Recalling that µ(p) = [vref (p) cos θref (p), vref (p) sin θref (p)]⊤ and using the complexification operator, we have the following kinematic result. Lemma 6.10.5. Let ∆1 = θn − θref (p). We have the following kinematic relationship between the velocity vector of the center of mass p˙ and the reference velocity vector µ(p)

p˙ = R∆1 µ(p) + d(vt , vn , θn , vref (p)), where kd(·)k ≤ ǫ3 + ǫ4 on the direction following manifold Γ2 . Proof. Applying the operator C to the vector p˙ − R∆1 µ(p), we have

(6.56)

178


     cos(θref (p))   vt  C(p˙ − R∆1 µ(p)) = C Rθn    − vref (p)R∆1  |{z} sin(θ (p)) v ref n Equation (6.7) = exp(jθn )(vt + jvn ) − vref (p) exp(j∆1 ) exp(jθref (p)) . |{z} | {z } Lemma 6.10.4 exp(jθn )

Applying C−1 to both sides of the above equality, we have 



 vt − vref (p)  p˙ − R∆1 µ(p) = Rθn  . vn | {z } d(vt ,vn ,θn ,vref (p))

p Therefore, we have kd(.)k ≤ (vt − vref (p))2 + vn2 . On the direction following manifold

Γ2 , we have |vt − vref (p)| < ǫ2 and |vn | < ǫ3 . It follows that kd(.)k ≤ ǫ2 + ǫ3 .

Proof of Proposition 6.6.3. Using the control input (6.27) in (6.26) the closed loop equation is obtained as follows

dℓp R∆1 dℓ⊤ p Ktran y + 2 kdℓp k    0 1  ⊤ v dℓp R∆1  + dℓp d(·)  dℓp kdℓp k −1 0

y˙ = −

(6.57)

Now, we consider the Lyapunov function candidate V = 21 y 2. We pick c > 0 and define Ωc = {|y| ≤ c}. By assumption, on {p : ℓ(p) = 0} it holds that dℓp 6= [0 0]. Therefore, there exists c > 0 such that dℓp 6= 0 for all p ∈ {p : |ℓ(p)| ≤ c}. Let Ωc = {p : |ℓ(p)| ≤ c}. We will now show that for sufficiently large Ktran , Ωc is positively invariant. To this end, it is enough to show that there exists K ⋆ > 0 such that for all Ktran ≥ K ⋆ , V˙ ≤ 0 for all p ∈ ∂Ωc . On ∂Ωc , dℓp is bounded. Therefore, |dℓp d(·)| ≤ K. By continuity, for small enough ǫ1 (note that ǫ1 can be made arbitrarily small by Proposition 6.5.1), there exist a1 , a2 > 0 such that

179


−

dℓp R∆1 dℓ⊤ p ≤ −a1 , 2 kdℓp k

(6.58)

and 



 0 1  ⊤ v |dℓp R∆1  | ≤ a2  dℓp kdℓ k p −1 0

We have

(6.59)

V˙ = y y˙ ≤ −Ktran a1 y 2 + a2 y + dℓp d(·)y ≤ −Ktran c2 + a2 |c| + K|c| Therefore, if Ktran ≥

a2 +K , |c|

(6.60)

we get V˙ ≤ 0 on ∂Ωc . This means that Ωc is positively

invariant. On Ωc , we have |dℓp d(·)| ≤ Kc because Ωc is compact. Therefore, we get

V˙ ≤ −Ktran a1 y 2 + (Kc + a2 )|y| ≤

1 1 −Ktran y 2 + y 2 + (Kc + a2 )2 =⇒ 2 2 1 1 V˙ ≤ −(Ktran − )y 2 + (Kc + a2 )2 2 2

(6.61)

Therefore for Ktran ≥ 21 , we have (by the comparison lemma) 1 (Kc + a2 )2 1 V (t) ≤ exp((−Ktran + )t)V (0) + 2 2 Ktran − 12

Therefore, y converges to a ball of radius

r

1 (Kc +a2 )2 2 Ktran − 12

(6.62)

. Choosing Ktran large enough makes

the ultimate bound of y less than ǫ5 for any desired ǫ5 > 0. Therefore, the path γ is practically stable with domain of attraction containing Ωc . On the path γ, ℓ(p) = 0, and we have




Therefore, we have



 0 1  ⊤ v p˙ = R∆1  + d(·)  dℓp kdℓp k −1 0

˙ − v ≤ kd(·)k ≤ ǫ3 + ǫ4 v − kd(·)k ≤ kpk ˙ ≤ v + kd(·)k =⇒ kpk

Therefore, we have approximate velocity control on γ.

180

(6.63)

(6.64)

Chapter 7 Conclusions and future research The VHC framework is motivated by the successful application of the VHC-based walking control algorithms for biped robots. This framework can be employed whenever traditional trajectory tracking schemes do not provide promising solutions, e.g., in the presence of underactuation in the robotic system. The most important themes in the VHC-based control paradigm can be summarized as follows. • Designing appropriate VHCs. The VHCs might be inspired by biology, as highlighted in Chapter 6, or might be generated by a mathematical algorithm, as shown in Chapter 2. • Enforcing the VHCs. The VHC is enforced by means of stabilizing a suitable set, which usually requires assigning all the physical controls. After enforcing the VHC, the motion is governed by a dynamic system with state-space dimensions smaller than those of the original underactuated system. • Achieving additional control objectives by means of dynamically changing the geometry of the VHC. In order to address the lack of controls after assigning the physical ones, we parameterize the constraints using states of dynamic compensators. The inputs to these compensators can then be designed in order to solve

181

Chapter 7. Conclusions and future research

182

the motion control problem under study. We enumerate the contributions of this thesis in Section 7.1. Next, we briefly highlight the relationship of the theory developed in Chapters 3– 5 with the VHC-based control of biped robots in Section 7.2. Finally, we present some of the future directions that might stem out of this work in Section 7.3.

7.1

Summary of contributions

The thesis makes contributions in two main directions. The more prominent direction, in Chapters 3–5, is to completely solve the problem of employing VHCs for the stabilization of repetitive “behaviors” for mechanical systems with degree of underactuation one. The contributions in this direction are as follows. • Characterizing the VHC-induced motion, in Chapter 3, where we have presented a complete solution for the inverse problem of Lagrangian mechanics for the reduced dynamics arising from regular VHCs. • VHC implicitization, in Chapter 4, where we have presented a procedure for making parametric VHCs implicit and thus amenable to feedback implementation. We have employed the resultant of polynomials, a tool from classical algebra, which is widely used in the field of computer graphics. • VHC-based orbital stabilization, in Chapter 5, where we have presented a stabilization methodology for energy level sets corresponding to closed orbits of the reduced dynamics. The challenge that is encountered in solving this orbital stabilization problem is that the reduced dynamics have no control input. Our solution to this problem relies on the notion of dynamic VHCs, i.e., VHCs parameterized by the states of double integrators. The intuition is that by controlling the state of the


183

double integrator, one dynamically changes the shape of the VHC so as to stabilize the desired closed orbit. The second contribution of this thesis is to explore the hypothesis that VHCs can be used to solve complex locomotion control problems that go beyond the stabilization of walking gaits in biped robots. To this end, in Chapter 6 we have presented a complete solution to the challenging problem of path following control of planar snake robots. The presented VHC-based approach enabled us to design a hierarchical control structure where a point-mass abstraction of the snake robot was created via enforcing biologicallyinspired dynamic VHCs.

7.2

Relation to the VHC-based control of biped robots

The relationship of the theory developed in Chapters 3–5 to the VHC-based control of biped robots can described as follows.

• Since the biped robots are hybrid dynamical systems that are subject to impulsive forces, the theory of Chapter 3 for investigating the existence of reduced dynamics with globally well-defined Lagrangian structures does not apply to bipeds with hybrid dynamics. However, the reduced dynamics during the swing phase are always locally Lagrangian as they satisfy the Helmholtz conditions. • The VHC implicitization techniques developed in Chapter 4 can be directly applied to biped walking gaits in order to obtain implicit constraints. Since the stable walking gaits designed by Grizzle et al. [4, 6, 3, 5, 114] are already in polynomial form, there is no need for polynomial approximation. • We conjecture that the idea of dynamic VHCs, presented in Chapter 5, can be employed for making smooth transitions between stable walking gaits in biped


184

robots. In particular, one can define a VHC homotopy1 of the form hs : [0, 1] → Q such that hs (q), 0 < s < 1, h0 (q), and h1 (q) represent the interim, initial, and final walking gaits, respectively.

7.3

Future research directions and open problems

A number of research directions stem out of this work. • Extending the theory developed in Chapters 3–5 to underactuated mechanical systems with higher degrees of underactuation. Here we enumerate some of the problems and challenges that might be encountered in this direction. – Solving the inverse Lagrangian problem (ILP) for systems with degrees of underactuation greater than one. In [41] the reduced dynamics, which are expressed in local coordinate charts and govern the motion of underactuated mechanical systems with degree of underactuation greater than one, have been found. In order to characterize the VHC-induced motion in this context, one needs to solve the ILP for the reduced dynamics. Even if such a structure exists, characterizing the VHC-induced motion is still a challenging task as the closed orbits are not the level-sets of the VHC-induced energy function anymore. Recently, sufficient conditions for existence of global Lagrangian structures for degree of underactuation two have been presented in [115]. – The VHC implicitization technique, presented in Chapter 4, is quite general and can be applied when the degree of underactuation is greater than one. However, new conditions for checking a priori the regularity of the implicitized constraints need to be found. 1

A homotopy is a continuous deformation of a function to another one.


185

– The idea of making VHCs dynamic, as presented in Chapter 5, can be generalized by making the VHCs dependent on more than one dynamic variable which are states of double integrators. The inputs to these dynamic compensators can be employed to achieve additional control objectives on the constraint manifold. • Finding alternative VHC generation algorithms. • A systematic investigation of the symmetry role in inducing reduced dynamics with Lagrangian structures. • Applying the theory to synchronization of underactuated mechanical systems. In [28, ˆ n ), which is a special case of the general 116], parametric VHCs of the form q = φ(q implicit VHCs studied in the thesis, have been employed to solve some mechanical synchronization problems such as synchronizing the master and slave robots in a teleoperation setup. We conjecture that the material presented in [28, 116] can be extended using the theory developed in Chapters 3–5. • Expanding the presented hierarchical framework for path following control of snake robots for underwater and three-dimensional locomotion. Considering the presented hierarchical control approach in Chapter 6, we conjecture that this approach might be expanded to locomotion control of underwater and three-dimensional robotic snake locomotion. In the former case, a preliminary investigation based on the VHC-based control paradigm presented in this thesis has been carried out for planar path following control of underwater snake robots in [117]. In the latter case, we conjecture that considering a 3D lateral undulatory gait and making it dynamic enables us to create a 3D point-mass abstraction of the snake robot and make the snake follow three-dimensional paths.

Bibliography [1] Y. Hwang and N. Ahuja, “A potential field approach to path planning,” IEEE Trans. Robot. Autom., vol. 8, no. 1, pp. 23–32, 1992. [2] S. Lindemann and S. LaValle, “Current issues in sampling-based motion planning,” Robotics Research, pp. 36–54, 2005. [3] C. Chevallereau, A. Gabriel, Y. Aoustin, F. Plestan, E. Westervelt, C. C. De Wit, and J. Grizzle, “Rabbit: A testbed for advanced control theory,” IEEE Control Systems Magazine, vol. 23, no. 5, pp. 57–79, 2003. [4] J. W. Grizzle, G. Abba, and F. Plestan, “Asymptotically stable walking for biped robots: Analysis via systems with impulse effects,” IEEE Trans. Automat. Contr., vol. 46, no. 1, pp. 51–64, 2001. [5] E. Westervelt, J. Grizzle, C. Chevallereau, J. Choi, and B. Morris, Feedback Control of Dynamic Bipedal Robot Locomotion. Taylor & Francis, CRC Press, 2007. [6] J. W. Grizzle, C. Chevallereau, R. W. Sinnet, and A. D. Ames, “Models, feedback control, and open problems of 3d bipedal robotic walking,” Automatica, vol. 50, no. 8, pp. 1955–1988, 2014. [7] C. Canudas-de Wit, “On the concept of virtual constraints as a tool for walking robot control and balancing,” Annual Reviews in Control, vol. 28, no. 2, pp. 157– 166, 2004. 186

Bibliography

187

[8] J. Nakanishi, T. Fukuda, and D. Koditschek, “A brachiating robot controller,” IEEE Trans. Robot. Autom., vol. 16, no. 2, pp. 109–123, 2000. [9] C. Nielsen, “Set stabilization using transverse feedback linearization,” Ph.D. dissertation, Dept. of Electrical and Computer Engineering, Univ. of Toronto, 2008. [10] C. Nielsen, C. Fulford, and M. Maggiore, “Path following using transverse feedback linearization: Application to a maglev positioning system,” Automatica, vol. 46, no. 3, pp. 585–590, 2010. [11] A. Transeth, K. Pettersen, and P. Liljebäck, “A survey on snake robot modeling and locomotion,” Robotica, vol. 27, no. 7, pp. 999–1015, 2009. [12] P. Holmes, R. J. Full, D. E. Koditschek, and J. Guckenheimer, “The dynamics of legged locomotion: Models, analyses, and challenges,” SIAM Rev., vol. 48, no. 2, pp. 207–304, 2006. [13] R. D. Gregg, E. J. Rouse, L. J. Hargrove, and J. W. Sensinger, “Evidence for a time-invariant phase variable in human ankle control,” PloS one, vol. 9, no. 2, p. e89163, 2014. [14] U. Mettin, P. La Hera, L. Freidovich, A. Shiriaev, and J. Helbo, “Motion planning for humanoid robots based on virtual constraints extracted from recorded human movements,” Intelligent Service Robotics, vol. 1, no. 4, pp. 289–301, 2008. [15] J. F. Soechting, F. Lacquaniti, and C. A. Terzuolo, “Coordination of arm movements in three-dimensional space. sensorimotor mapping during drawing movement,” Neuroscience, vol. 17, no. 2, pp. 295–311, 1986. [16] F. Valero-Cuevas, “An integrative approach to the biomechanical function and neuromuscular control of the fingers,” J. Biomech., vol. 38, no. 4, pp. 673–684, 2005.

Bibliography

188

[17] A. D. Ames, “First steps toward automatically generating bipedal robotic walking from human data,” in Robot Motion and Control 2011, K. R. Kozlowski, Ed. London: Springer-Verlag, 2012, pp. 89–116. [18] A. D. Ames, E. A. Cousineau, and M. J. Powell, “Dynamically stable bipedal robotic walking with NAO via human-inspired hybrid zero dynamics,” in Proc. ACM Int. Conf. Hybrid Systems: Computation and Control, 2012, pp. 135–144. [19] L. Marchal-Crespo and D. Reinkensmeyer, “Review of control strategies for robotic movement training after neurologic injury,” Journal of Neuroeng. Rehabil., vol. 6, no. 1, p. 20, 2009. [20] H. I. Krebs, J. J. Palazzolo, L. Dipietro, M. Ferraro, J. Krol, K. Rannekleiv, B. T. Volpe, and N. Hogan, “Rehabilitation robotics: Performance-based progressive robot-assisted therapy,” Auton. Robot., vol. 15, no. 1, pp. 7–20, 2003. [21] D. Aoyagi, W. Ichinose, S. Harkema, D. Reinkensmeyer, and J. Bobrow, “A robot and control algorithm that can synchronously assist in naturalistic motion during body-weight-supported gait training following neurologic injury,” IEEE. Trans. Neur. Sys. Rehab., vol. 15, no. 3, pp. 387–400, 2007. [22] T. Nef, M. Mihelj, and R. Riener, “ARMin: a robot for patient-cooperative arm therapy,” Med. Biol. Eng. Comput., vol. 45, no. 9, pp. 887–900, 2007. [23] A. Duschau-Wicke, J. von Zitzewitz, A. Caprez, L. Lunenburger, and R. Riener, “Path control: A method for patient-cooperative robot-aided gait rehabilitation,” IEEE. Trans. Neur. Sys. Rehab., vol. 18, no. 1, pp. 38–48, 2010. [24] L. Cai, A. Fong, C. Otoshi, Y. Liang, J. Burdick, R. Roy, and V. Edgerton, “Implications of assist-as-needed robotic step training after a complete spinal cord injury on intrinsic strategies of motor learning,” J. Neurosci., vol. 26, no. 41, pp. 10 564–10 568, 2006.

Bibliography

189

[25] Y. Mao and S. Agrawal, “Design of a cable-driven arm exoskeleton (carex) for neural rehabilitation,” IEEE Trans. Robot., vol. 28, no. 4, pp. 922–931, 2012. [26] T. Sheridan, “Telerobotics,” Automatica, vol. 25, no. 4, pp. 487–507, 1989. [27] R. Anderson and M. Spong, “Bilateral control of teleoperators with time delay,” IEEE Trans. Automat. Contr., vol. 34, no. 5, pp. 494–501, 1989. [28] D. Jankuloski, “Virtual holonomic constraints and the synchronization of EulerLagrange control systems,” Master’s thesis, University of Toronto, 2012. [29] S. Mastellone, J. Mej´ıa, D. Stipanović, and M. Spong, “Formation control and coordinated tracking via asymptotic decoupling for Lagrangian multi-agent systems,” Automatica, vol. 47, no. 11, pp. 2355–2363, 2011. [30] M. Al Borno, E. Fiume, A. Hertzmann, and M. de Lasa, “Feedback control for rotational movements in feature space,” in Computer Graphics Forum, vol. 33, no. 2, 2014, pp. 225–233. [31] P. Appell, “Exemple de mouvement d’un point assujetti a` une liaison exprimèe par une relation non-linéaire entre les composantes de la vitesse,” Rend. Circ. Mat. Palermo, vol. 32, pp. 48–50, 1911. ´ [32] H. Beghin, “Etude théorique des compas gyrostatiques Ansch¨ utz et sperry,” Ph.D. dissertation, Faculté des sciences de Paris, 1922. [33] F. Plestan, J. Grizzle, E. Westervelt, and G. Abba, “Stable walking of a 7-DOF biped robot,” IEEE Trans. Robot. Autom., vol. 19, no. 4, pp. 653–668, 2003. [34] E. Westervelt, J. Grizzle, and D. Koditschek, “Hybrid zero dynamics of planar biped robots,” IEEE Transactions on Automatic Control, vol. 48, no. 1, pp. 42–56, 2003.

Bibliography

190

[35] C. Chevallereau, J. Grizzle, and C. Shih, “Asymptotically stable walking of a fivelink underactuated 3D bipedal robot,” IEEE Transactions on Robotics, vol. 25, no. 1, pp. 37–50, 2008. [36] A. Shiriaev, J. Perram, and C. Canudas-de-Wit, “Constructive tool for orbital stabilization of underactuated nonlinear systems: Virtual constraints approach,” IEEE Transactions on Automatic Control., vol. 50, no. 8, pp. 1164–1176, August 2005. [37] A. Shiriaev, A. Robertsson, J. Perram, and A. Sandberg, “Periodic motion planning for virtually constrained Euler-Lagrange systems,” Systems & Control Letters, vol. 55, pp. 900–907, 2006. [38] A. Shiriaev, L. Freidovich, and S. Gusev, “Transverse linearization for controlled mechanical systems with several passive degrees of freedom,” IEEE Transactions on Automatic Control, vol. 55, no. 4, pp. 893–906, 2010. [39] C. Canudas-de Wit, B. Espiau, and C. Urrea, “Orbital stabilization of underactuated mechanical systems,” in Proceedings of the 15th IFAC World Congress. Barcelona, 2002. [40] M. Maggiore and L. Consolini, “Virtual holonomic constraints for Euler-Lagrange systems,” IEEE Trans. Automat. Contr., vol. 58, no. 4, pp. 1001–1008, 2013. [41] D. Jankuloski, M. Maggiore, and L. Consolini, “Further results on virtual holonomic constraints,” in the 4th IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, Bertinoro, Italy, 2012. [42] L. Consolini and M. Maggiore, “Virtual holonomic constraints for Euler-Lagrange systems,” in Symposium on Nonlinear Control Systems (NOLCOS), 2010.

191

Bibliography

[43] ——, “On the swing-up of the pendubot using virtual holonomic constraints,” in IFAC World Congress, Milano, Italy, 2011. [44] A. P. Aguiar, J. P. Hespanha, and P. V. Kokotović, “Path-following for nonminimum phase systems removes performance limitations,” IEEE Transactions on Automatic Control, vol. 50, no. 2, pp. 234–239, 2005. [45] R. Skjetne, T. I. Fossen, and P. V. Kokotović, “Robust output maneuvering for a class of nonlinear systems,” Automatica, vol. 40, no. 3, pp. 373–383, 2004. [46] C. Nielsen and M. Maggiore, “Output stabilization and maneuver regulation: A geometric approach,” Systems & control letters, vol. 55, no. 5, pp. 418–427, 2006. [47] ——, “On local transverse feedback linearization,” SIAM Journal on Control and Optimization, vol. 47, no. 5, pp. 2227–2250, 2008. [48] J. M. Lee, Introduction to Smooth Manifolds, 2nd ed. Springer, 2013. [49] M. W. Spong, S. Hutchinson, and M. Vidyasagar, Robot Modeling and Control. New York: Wiley, 2005. [50] R. M. Murray, Z. Li, S. S. Sastry, and S. S. Sastry, A mathematical introduction to robotic manipulation. CRC press, 1994. [51] A. Isidori, Nonlinear Control Systems, 3rd ed. New York: Springer, 1995. [52] L. Consolini and M. Maggiore, “Control of a bicycle using virtual holonomic constraints,” Automatica, vol. 49, no. 9, pp. 2831–2839, 2013. [53] V. Guillemin and A. Pollack, Differential Topology. 1974.

New Jersey: Prentice Hall,

Bibliography

192

[54] A. S. Shiriaev, L. B. Freidovich, A. Robertsson, R. Johansson, and A. Sandberg, “Virtual-holonomic-constraints-based design of stable oscillations of furuta pendulum: Theory and experiments,” IEEE Trans. Robot., vol. 23, no. 4, pp. 827–832, 2007. [55] A. Mohammadi, M. Maggiore, and L. Consolini, “When is a Lagrangian control system with virtual holonomic constraints Lagrangian?” in Proc. the 9th IFAC Symposium on Nonlinear Control Systems (NOLCOS), Toulouse, France, 2013. [56] ——, “On the Lagrangian structure of reduced dynamics under virtual holonomic constraints,” ESAIM: Control, Optimisation, and Calculus of Variations, 2016, doi: http://dx.doi.org/10.1051/cocv/2016020. [57] R. M. Santilli, Foundations of Theoretical Mechanics I, The Inverse Problem in Newtonian Mechanics. New York: Springer-Verlag, 1978. [58] E. Tonti, “Inverse problem: Its general solution,” in Differential Geometry, Calculus of Variations, and Their Applications, G. M. Rassias and T. M. Rassias, Eds. Marcel Dekker, Inc., 1985, pp. 497–510. [59] O. Krupková and G. E. Prince, “Second order ordinary differential equations in jet bundles and the inverse problem of the calculus of variations,” in Handbook of Global Analysis, D. Krupka and D. J. Saunders, Eds. Elsevier, 2008, pp. 837–904. [60] N. Sonin, “About determining maximal and minimal properties of plane curves,” Warsawskye Universitetskye Izvestiya, vol. 1, pp. 1–68, 1886, in Russian. ¨ [61] H. Helmholtz, “Uber der physikalische Bedeutung des Princips der kleinsten Wirkung,” J. Reine Angew. Math., vol. 100, pp. 137–166, 1887. [62] A. Mayer, “Die existenzbedingungen eines kinetischen potentiales,” Ber. Ver. Ges. d. Wiss. Leipzig, Math.-Phys. Cl., vol. 48, pp. 519–529, 1896.

Bibliography

193

[63] E. Tonti, “Variational formulation of nonlinear differential equations I, II,” Bull. Acad. Roy. Belg. Cl. Sci., vol. 55, pp. 137–165, 262–278, 1969. [64] M. M. Vainberg, Variational Methods in the Theory of Nonlinear Operators. Moscow: GITL, 1959, in Russian. [65] M. Henneaux, “Equation of motion, commutation relations and ambiguities in the Lagrangian formalism,” Ann. Phys., vol. 140, no. 1, pp. 45–64, 1982. [66] G. Darboux, Le¸cons sur la Théorie Générale des Surfaces. Paris: Gauthier-Villars, 1894. [67] J. Douglas, “Solution to the inverse problem of the calculus of variations,” Trans. Am. Math. Soc., vol. 50, pp. 71–129, 1941. [68] W. Sarlet, “The Helmholtz conditions revisited. A new approach to the inverse problem of Lagrangian dynamics,” J. Phys. A: Math. Gen., vol. 15, pp. 1503–1517, 1982. [69] I. Anderson and T. Duchamp, “On the existence of global variational principles,” Am. J. Math., vol. 102, pp. 781–867, 1980. [70] M. Crampin, “On the differential geometry of Euler-Lagrange equations, and the inverse problem of Lagrangian dynamics,” J. Phys. A: Math. Gen., vol. 14, pp. 2567–2575, 1981. [71] M. Crampin, G. E. Prince, and G. Thompson, “A geometric version of the Helmholtz conditions in time dependent Lagrangian dynamics,” J. Phys. A: Math. Gen., vol. 17, pp. 1437–1447, 1984. [72] F. Takens, “A global version of the inverse problem of the calculus of variations,” J. Diff. Geom., vol. 14, pp. 543–562, 1979.

Bibliography

194

[73] E. Mart´ı nez, J. F. Cari˜ nena, and W. Sarlet, “Derivations of differential forms along the tangent bundle projection II,” Diff. Geom. Appl., vol. 3, pp. 1–29, 1993. [74] D. J. Saunders, “Thirty years of the inverse problem of calculus of variations,” Reports on Mathematical Physics, vol. 66, no. 1, pp. 43–53, Aug. 2010. [75] V. I. Arnold, Mathematical Methods of Classical Mechanics (Graduate Texts in Mathematics, Vol. 60). Springer, 1989. [76] Y. Pinchover and J. Rubinstein, An Introduction to Partial Differential Equations. Cambridge University Press, 2005. [77] L. Consolini, M. Maggiore, C. Nielsen, and M. Tosques, “Path following for PVTOL aircraft,” Automatica, vol. 46, pp. 585–590, 2010. [78] T. W. Sederberg and J. Zheng, Algebraic Methods for Computer Aided Geometric Design. North-Holland, Amsterdam, 2002. [79] J. Von Zur Gathen and J. Gerhard, Modern Computer Algebra, 3rd ed. Cambridge university press, 2013. [80] B. L. van der Waerden, Modern Algebra. Frederick Ungar, New York, 1950. [81] C. E. Wee and R. N. Goldman, “Elimination and resultants. 1. elimination and bivariate resultants,” IEEE Transactions on Computer Graphics and Applications, vol. 15, no. 1, pp. 69–77, 1995. [82] R. T. Farouki, “The Bernstein polynomial basis: A centennial retrospective,” Computer Aided Geometric Design, vol. 29, no. 6, pp. 379–419, 2012. [83] P. Bézier, Numerical control; mathematics and applications. John Wiley & Sons, 1972.

Bibliography

195

[84] G. M. Phillips, Interpolation and Approximation by Polynomials. Springer Science & Business Media, 2003, vol. 14. [85] L. R. Foulds, Graph theory applications. Springer Science & Business Media, 2012. [86] J. L. Gross and J. Yellen, Graph theory and its applications. CRC press, 2005. [87] M. El-Hawwary and M. Maggiore, “Reduction theorems for stability of closed sets with application to backstepping control design,” Automatica, vol. 49, no. 1, pp. 214–222, 2013. [88] M. I. El-Hawwary, “Passivity methods for the stabilization of closed sets in nonlinear control systems,” Ph.D. dissertation, University of Toronto, 2011. [89] P. Seibert and J. S. Florio, “On the reduction to a subspace of stability properties of systems in metric spaces,” Annali di Matematica pura ed applicata, vol. 169, no. 1, pp. 291–320, 1995. [90] S. Bittanti, A. J. Laub, and J. C. Willems, The Riccati Equation. Springer Science & Business Media, 2012. [91] S. Bittanti, P. Colaneri, and G. De Nicolao, “The periodic Riccati equation,” in The Riccati Equation. Springer, 1991, pp. 127–162. [92] H. Abou-Kandil, G. Freiling, V. Ionescu, and G. Jank, Matrix Riccati Equations in Control and Systems Theory. Birkhäuser, 2012. [93] S. Bittanti and P. Bolzern, “Stabilizability and detectability of linear periodic systems,” Systems & control letters, vol. 6, no. 2, pp. 141–145, 1985. [94] J. J. Hench and A. J. Laub, “Numerical solution of the discrete-time periodic Riccati equation,” Automatic Control, IEEE Transactions on, vol. 39, no. 6, pp. 1197–1210, 1994.

Bibliography

196

[95] S. Johansson, “Tools for control system design: Stratification of matrix pairs and periodic Riccati differential equation solvers,” Ph.D. dissertation, Print & Media, Ume˚ a universitet, 2009. [96] J. K. Hale, Ordinary Differential Equations, 2nd ed. Robert E. Krieger Publishing Company, 1980. [97] A. Banaszuk and J. Hauser, “Feedback linearization of transverse dynamics for periodic orbits,” Systems and Control Letters, vol. 26, no. 2, pp. 95–105, Sept. 1995. [98] L. Kobbelt and M. Botsch, “A survey of point-based techniques in computer graphics,” Computers & Graphics, vol. 28, no. 6, pp. 801–814, 2004. [99] A. Mohammadi, E. Rezapour, M. Maggiore, and K. Y. Pettersen, “Direction following control of planar snake robots using virtual holonomic constraints,” in 53rd Conference on Decision and Control (CDC), 2014, pp. 3801–3808. [100] ——, “Maneuvering control of planar snake robots using virtual holonomic constraints,” IEEE Transactions on Control Systems Technology, vol. 24, no. 3, pp. 884–899, 2016. [101] S. Hirose, Biologically Inspired Robots: Snake-Like Locomotors and Manipulators. Oxford University Press, 1993. [102] C. Ye, S. Ma, B. Li, and Y. Wang, “Turning and side motion of snake-like robot,” in IEEE Int. Conf. Robot. Autom., vol. 5, 2004, pp. 5075–5080. [103] M. Sato, M. Fukaya, and T. Iwasaki, “Serpentine locomotion with robotic snakes,” IEEE Control Systems Magazine, vol. 22, no. 1, pp. 64–81, 2002. [104] P. Liljebäck, K. Y. Pettersen, Ø. Stavdahl, and J. T. Gravdahl, Snake Robots: Modelling, Mechatronics, and Control. Springer, 2013.

Bibliography

197

[105] G. Hicks and K. Ito, “A method for determination of optimal gaits with application to a snake-like serial-link structure,” IEEE Trans. Automat. Contr., vol. 50, no. 9, pp. 1291–1306, 2005. [106] K. A. McIsaac and J. P. Ostrowski, “Motion planning for anguilliform locomotion,” IEEE Trans. Robot. Autom., vol. 19, no. 4, pp. 637–652, 2003. [107] P. Liljebäck, K. Y. Pettersen, Ø. Stavdahl, and J. T. Gravdahl, “Controllability and stability analysis of planar snake robot locomotion,” IEEE Trans. Automat. Contr., vol. 56, no. 6, pp. 1365–1380, 2011. [108] E. Rezapour, A. Hofmann, K. Y. Pettersen, A. Mohammadi, and M. Maggiore, “Virtual holonomic constraint based direction following control of planar snake robots described by a simplified model,” in IEEE Conf. Control Applications, 2014, pp. 1064–1071. [109] P. Liljebäck, I. U. Haugstuen, and K. Y. Pettersen, “Path following control of planar snake robots using a cascaded approach,” IEEE Trans. Control Syst. Tech., vol. 20, no. 1, pp. 111–126, 2012. [110] J. Ostrowski and J. Burdick, “The geometric mechanics of undulatory robotic locomotion,” Int. J. Robot. Res., vol. 17, no. 7, pp. 683–701, 1998. [111] H. K. Khalil, Nonlinear Systems, 3rd ed. Prentice Hall, 2002. [112] A. Isidori, Nonlinear Control Systems II. Springer, 1999. ¨ [113] M. Nagumo, “Uber die lage der integralkurven gewöhnlicher differentialgleichungen,” Nippon Sugaku-Buturigakkwai Kizi Dai 3 Ki, vol. 24, pp. 551–559, 1942. [114] J. Grizzle, J. Hurst, B. Morris, H.-W. Park, and K. Sreenath, “Mabel, a new robotic bipedal walker and runner,” in American Control Conference, 2009. ACC’09., 2009, pp. 2030–2036.

Bibliography

198

[115] L. Consolini and A. Costalunga, “Induced connections on virtual holonomic constraints,” in 54th IEEE Conference on Decision and Control, 2015, pp. 139–144. [116] D. Jankuloski, M. Maggiore, and L. Consolini, “Synchronizing n cart-pendulums using virtual holonomic constraints.” in American Control Conference, 2012, pp. 1023–1028. [117] A. Kohl, E. Kelasidi, A. Mohammadi, M. Maggiore, and K. Y. Pettersen, “Planar maneuvering control of underwater snake robots using virtual holonomic constraints,” Bioinspiration & Biomimetics, 2016, under revision.

Virtual Holonomic Constraints for Euler-Lagrange Control ... - TSpace

Virtual Holonomic Constraints for Euler-Lagrange Control ... - TSpace

Suggest Documents

Virtual Holonomic Constraints for Euler-Lagrange Control Systems by ...

Control of a Bicycle Using Virtual Holonomic Constraints - CiteSeerX

Holonomic Constraints - MDPI

Quantum graphs as holonomic constraints

Quantum graphs as holonomic constraints

Nullspace Sampling with Holonomic Constraints

Path following control of planar snake robots using virtual holonomic ...

Non-Holonomic Control I

Nullspace Sampling with Holonomic Constraints ... - Semantic Scholar

Non-holonomic Control II: Non-holonomic Quantum Devices

Virtual holonomic constraint approach for planar bipedal ... - CiteSeerX

Human Gait Modeling: Dealing with Holonomic Constraints - CiteSeerX

Realizing holonomic constraints in classical and quantum mechanics

User Interface Constraints for Immersive Virtual ... - CiteSeerX

Utilization of Holonomic Distribution Control for ... - Dimitar Dimitrov

Holonomic systems for period mappings

Evaluation of virtual patient cases for teaching diagnostic ... - TSpace

Stability Constraints for Robust Model Predictive Control

Control Problems with State Constraints for the

Modelling and control of a non-holonomic pendulum ...

Control of an Pseudo-Omnidirectional, Non-Holonomic ... - CiteSeerX

THE EFFECTS OF MOTOR CONSTRAINTS ON INFANT ... - TSpace

Optical Holonomic Quantum Computation

Optical Holonomic Quantum Computer