A Unified Geometric Framework for Kinematics, Dynamics ... - TSpace

A Unified Geometric Framework for Kinematics, Dynamics and Concurrent Control of Free-base, Open-chain Multi-body Systems with Holonomic and Nonholonomic Constraints

by

Robin Chhabra

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

c Copyright 2014 by Robin Chhabra

Abstract A Unified Geometric Framework for Kinematics, Dynamics and Concurrent Control of Free-base, Open-chain Multi-body Systems with Holonomic and Nonholonomic Constraints Robin Chhabra Doctor of Philosophy Graduate Department of Aerospace Science and Engineering University of Toronto 2014 This thesis presents a geometric approach to studying kinematics, dynamics and controls of open-chain multi-body systems with non-zero momentum and multi-degreeof-freedom joints subject to holonomic and nonholonomic constraints. Some examples of such systems appear in space robotics, where mobile and free-base manipulators are developed. The proposed approach introduces a unified framework for considering holonomic and nonholonomic, multi-degree-of-freedom joints through: (i) generalization of the product of exponentials formula for kinematics, and (ii) aggregation of the dynamical reduction theories, using differential geometry. Further, this framework paves the ground for the input-output linearization and controller design for concurrent trajectory tracking of base-manipulator(s). In terms of kinematics, displacement subgroups are introduced, whose relative configuration manifolds are Lie groups and they are parametrized using the exponential map. Consequently, the product of exponentials formula for forward and differential kinematics is generalized to include multi-degree-of-freedom joints and nonholonomic constraints in open-chain multi-body systems. As for dynamics, it is observed that the action of the relative configuration manifold corresponding to the first joint of an open-chain multi-body system leaves Hamilton’s equation invariant. Using the symplectic reduction theorem, the dynamical equations ii

of such systems with constant momentum (not necessarily zero) are formulated in the reduced phase space, which present the system dynamics based on the internal parameters of the system. In the nonholonomic case, a three-step reduction process is presented for nonholonomic Hamiltonian mechanical systems. The Chaplygin reduction theorem eliminates the nonholonomic constraints in the first step, and an almost symplectic reduction procedure in the unconstrained phase space further reduces the dynamical equations. Consequently, the proposed approach is used to reduce the dynamical equations of nonholonomic openchain multi-body systems. Regarding the controls, it is shown that a generic free-base, holonomic or nonholonomic open-chain multi-body system is input-output linearizable in the reduced phase space. As a result, a feed-forward servo control law is proposed to concurrently control the base and the extremities of such systems. It is shown that the closed-loop system is exponentially stable, using a proper Lyapunov function. In each chapter of the thesis, the developed concepts are illustrated through various case studies.

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To my love, Fahimeh

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Acknowledgements First of all, I would like to thank my supervisors, M. Reza Emami and Yael Karshon. Reza showed me how to define practical problems and approach them in a scientific manner. He was the one who introduced me to the field of robotics, starting from the basics. Throughout my graduate studies, he was always inspiring and supportive, and he familiarized me with ethics in research. During the last four years of my Ph.D., Yael helped me to understand differential geometry and use it towards the final goals of my research. She was always patient to hear me and advise me in the theoretical aspects of my Ph.D. dissertation. She always encouraged me and reminded me that my research was a valuable piece of work. During my studies at the University of Toronto, I had the opportunity of knowing great professors who gave me constructive pieces of advice about my research. Amongst them, I particularly would like to thank Gabriele D’Eleuterio and Christopher J. Damaren, the members of my Doctorla Examination Committee. Further, I want to sincerely thank my friends in the Space Mechatronics group who made a very friendly and comfortable environment for me to perform my research. Specially, I would like to mention my amazing friends, Sina, Peter, Victor, Jason, Michael Anthony and Adrian. Finally, I would like to take a moment and appreciate my best friends and family who accompanied me in this journey. Special thanks go to Payman and Ali, my best friends, whose friendship and help has been endless. My parents and my brother Arvind have been always supportive in different perspectives of life. Without their help and support, I was not able to complete my Ph.D. degree. Thank you mama, thank you papa, and thank you Arvind! Last but not least, my sincere thanks go to Fahimeh and her beautiful smile. Since the first day we met, she has been encouraging and supporting me, as a friend and as my wife. She has been emotionally and technically supportive, and filled my life with happiness and joy. While I was writing this dissertation, she was the only one who was with me at all the moments, happy and sad. Thank you Fahimeh, and please keep smiling in the rest of our lives!

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Contents 1 Introduction

1

1.1

Kinematics of Open-chain Multi-body Systems . . . . . . . . . . . . . . .

1

1.2

Dynamical Reduction of Holonomic and Nonholonomic Hamiltonian Systems with Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3

Control of Free-base Multi-body Systems . . . . . . . . . . . . . . . . . .

7

1.4

Statement of Contributions

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8

1.4.1

Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.4.2

Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

1.4.3

Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

1.4.4

Produced Manuscripts . . . . . . . . . . . . . . . . . . . . . . . .

13

Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

1.5

2 A Generalized Exponential Formula for Kinematics 2.1

15

Holonomic and Nonholonomic Joints . . . . . . . . . . . . . . . . . . . .

16

2.1.1

Displacement Subgroups . . . . . . . . . . . . . . . . . . . . . . .

17

2.1.2

Nonholonomic Displacement Subgroups . . . . . . . . . . . . . . .

21

2.2

Forward Kinematics

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21

2.3

Differential Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

2.4

Coordinate Assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

2.5

Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

2.5.1

Forward Kinematics . . . . . . . . . . . . . . . . . . . . . . . . .

33

2.5.2

Differential Kinematics . . . . . . . . . . . . . . . . . . . . . . . .

36

3 Reduction of Holonomic Multi-body Systems

38

3.1

Hamilton-Pontryagin Principle and Hamilton’s Equation . . . . . . . . .

38

3.2

Hamiltonian Mechanical Systems with Symmetry . . . . . . . . . . . . .

45

3.3

Symplectic Reduction of Holonomic Open-chain Multi-body Systems with Displacement Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

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3.4

3.3.1 Indexing and Some Kinematics . . . . . . . . . . . . . . . . . . . 3.3.2 Lagrangian and Hamiltonian of an Open-chain Multi-body System 3.3.3 Reduction of Holonomic Open-chain Multi-body Systems . . . . Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54 58 59 69

4 Reduction of Nonholonomic Multi-body Systems 82 4.1 Nonholonomic Hamilton’s Equation and Lagrange-d’Alembert-Pontryagin principle . . . . . . . . . . . . . . . . . 83 4.2 Nonholonomic Hamiltonian Mechanical Systems with Symmetry . . . . . 87 4.3 Reduction of Nonholonomic Open-chain Multi-body Systems with Displacement Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.4 An Investigation on Further Symmetries of Open-chain Multi-body Systems113 4.4.1 Identifying Symmetry Groups using AP1 . . . . . . . . . . . . . . 114 4.4.2 Identifying Symmetry Groups using AP2 . . . . . . . . . . . . . . 115 4.5 Further Reduction of Nonholonomic Open-chain Multi-body Systems . . 118 4.6 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.6.1 Further Reduction of the System . . . . . . . . . . . . . . . . . . 141 5 Concurrent Control of Multi-body Systems 5.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Mathematical Formalization and Assumptions . . . . . . . 5.1.2 Reduced Hamilton’s Equation and Reconstruction . . . . . 5.2 End-effector Pose and Velocity Error . . . . . . . . . . . . . . . . 5.2.1 Error Function . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Velocity Error . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Input-output Linearization and Inverse Dynamics in the Reduced Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 An Output-tracking Feed-forward Servo Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusions 6.1 Summary of Contributions 6.2 Future Work . . . . . . . . 6.2.1 Kinematics . . . . 6.2.2 Dynamics . . . . . 6.2.3 Controls . . . . . .

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144 144 145 149 152 152 154

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176 176 178 178 179 179

List of Tables 2.1

Categories of displacement subgroups [38, 71] . . . . . . . . . . . . . . .

19

3.1

Displacement subgroups and their corresponding isotropy groups . . . . .

69

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List of Figures 2.1 2.2

A mobile manipulator on a six d.o.f. moving base . . . . . . . . . . . . . Coordinate frames assigned to A0 , ..., A6 at the initial configuration . . .

33 34

3.1 3.2

A six-d.o.f. manipulator mounted on a spacecraft . . . . . . . . . . . . . The coordinate frames attached to the bodies of the robot . . . . . . . .

70 71

4.1 4.2

An example of a mobile manipulator . . . . . . . . . . . . . . . . . . . . The coordinate frames attached to the bodies of the mobile manipulator (Note that, the Zi -axis (i = 0, · · · , 6) is normal to the plane) . . . . . . . An example of a crane . . . . . . . . . . . . . . . . . . . . . . . . . . . . The coordinate frames attached to the bodies of the crane . . . . . . . .

122

4.3 4.4 5.1 5.2

123 131 132

Feed-forward servo control for a generic free-base, open-chain multi-body system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Servo controller for concurrent control of a three-d.o.f. manipulator mounted on a two-wheeled rover . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

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Notation Lr Rr Kr Adr adξ [ξ, η] diag(A1 , ..., An ) v˜ R(θ) R(θ, v) ω e Tm f Tm∗ f Tm M TM Tm∗ M T ∗M exp(ξ) Lie(G) Lie∗ (G) Gµ n ·, · kvkh h·, ·i LX ξM ιX Ω X(M )

Left composition/translation by a Lie group element r Right composition/translation by a Lie group element r Conjugation by a Lie group element r Adjoint operator corresponding to a Lie group element r adjoint operator corresponding to a Lie algebra element ξ Lie bracket of two Lie algebra elements or matrix commutator of two matrices Block diagonal matrix of the matrices A1 , ..., An Skew-symmetric matrix corresponding to the vector v in R3 2 × 2 rotation matrix for the angle θ 3 × 3 rotation matrix of a rotation for the angle θ, about the vector v ∈ R3 The 3 × 3 anti-symmetric matrix corresponding to the vector ω in R3 Tangent map corresponding to the map f at m, an element of the domain manifold Cotangent map corresponding to the map f at m, an element of the target manifold Tangent space of the manifold M at the element m Tangent bundle of the manifold M Cotangent space of the manifold M at the element m Cotangent bundle of the manifold M Group/matrix exponential of a Lie algebra element ξ Lie algebra of the Lie group G Dual of the Lie algebra of the Lie group G Coadjoint isotropy group for µ ∈ Lie∗ (G) Semi-direct product of groups Euclidean metric Norm of the vector v with respect to the metric h Canonical pairing of the elements of tangent and cotangent space Lie derivative with respect to the vector field X Vector field on the manifold M induced by the infinitesimal action of ξ ∈ Lie(G) Interior product of the differential form Ω by the vector field X Space of all vector fields on the manifold M x

Ω2 (M ) dΩ dH M/G

Space of all differential 2-forms on the manifold M Exterior derivative of the differential form Ω Exterior derivative of the function H Quotient manifold corresponding to a free and proper action of the Lie group G

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Chapter 1 Introduction Holonomic and nonholonomic open-chain multi-body systems appear in the field of robotics. In the context of geometric mechanics, these systems can be considered as Hamiltonian mechanical systems. In this thesis, we have a geometric approach towards studying the kinematics, dynamics and controls of generic open-chain multi-body systems with holonomic and nonholonomic constraints. This study includes: revisiting the notion of lower kinematic pairs and generalizing it to define displacement subgroups, studying and unifying the reduction of Hamiltonian mechanical systems for holonomic and nonholonomic open-chain multi-body systems with symmetry, and deriving an output tracking, feed-forward servo controller for such systems. In the following, we first report the existing literature for different topics appearing in this thesis. Then, we list the main contributions of the thesis, and finally we give the outline of the thesis.

1.1

Kinematics of Open-chain Multi-body Systems

The product of exponentials formula for Forward Kinematics of serial-link multi-body systems with revolute and/or prismatic joints was first introduced by Brockett in 1984 [11]. This formulation was further developed and its roots in Lie group and screw theory were illustrated by Murray et al. in 1994 [57]. One of the most important contributions of this method of multi-body system modeling is the elimination of intermediate coordinate frames in the kinematic analysis of serial-link manipulators. Since then, a number of researchers have investigated the computational efficiency of this formulation [62], and have applied it to different robotic problems [64, 24, 37, 67, 68]. In 1995, Park et al. used this formulation to reformulate the dynamical equations of serial-link multi-body systems [63], and later in 2003 Muller et al. attempted to unify the kinematics and dynamics of open-chain multi-body systems with one degree-of-freedom (d.o.f.) joints [56]. 1

Chapter 1. Introduction

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The exponential map used in the product of exponentials formula is the exponential map of Lie groups, which maps an element of the corresponding Lie algebra to an element of the Lie group. For a rigid body the configuration manifold is the Lie group SE(3), and the elements of its Lie algebra se(3) are the screws associated with the possible motions of a rigid body in 3-dimensional space [57]. Screw theory, which was first introduced by Ball in 1900 [4] and also appeared in the work of Clifford [22, 23], has been extensively investigated as a powerful means for the kinematic modeling of mechanisms [47, 45, 32, 33, 38, 10] and robotic systems [24, 80, 92, 31], by defining the notion of screw systems [71]. Moreover, the relationship between screw theory, Lie groups and projective geometry in the study of rigid body motion was elaborated in a paper by Stramigioli in 2002 [82]. He subsequently defined the notions of relative configuration manifold and relative screw to study multi-body systems [81]. In 1999 Mladenova also applied Lie group theory to the modeling and control of multi-body systems [54]. As opposed to the geometric nature of most of the above-mentioned works, her approach was mainly algebraic. Based on a well-known theorem in the theory of Lie groups, any element of a connected Lie group can be written as product of exponentials of some elements of its Lie algebra. Accordingly, Wei and Norman introduced a product of exponentials representation for the elements of a connected Lie group [91], which was adopted by Liu [46] and Leonard et al. [44] to reformulate Kane’s equations for multi-body systems and solve nonholonomic control problems on Lie groups, respectively. On the other hand, surjectivity of the exponential map of SE(3) that is a direct consequence of Chasles’ Theorem [57] implies that any element of SE(3) can be written as the exponential of an element of se(3). However, not much work has been done on the exponential parametrization of the Lie subgroups of SE(3). Only for the one-parameter subgroups of SE(3), which correspond to one-d.o.f. joints, the exponential map has been used to parametrize the relative configuration manifold that leads to the standard product of exponentials formula. In fact, we will show that the Lie subgroups of SE(3) correspond to the relative configuration manifolds of displacement subgroups [38, 36]. These joints are generally multi-d.o.f. holonomic joints. For generic multi-d.o.f. joints, Stramigioli in [81] briefly mentions that at each point the exponential map can be used as a local diffeomorphism between the relative configuration manifold and its tangent space. He later used this local diffeomorphism to introduce singularity-free dynamic equations of a generic open-chain multi-body system with holonomic and nonholonomic joints [29]. In Chapter 2, we give the necessary and sufficient conditions for surjectivity of the exponential map of the relative configuration manifolds of displacement subgroups. Under those conditions the corresponding Lie subgroups are locally parametrized using the elements of their Lie algebras.


1.2

3

Dynamical Reduction of Holonomic and Nonholonomic Hamiltonian Systems with Symmetry

A symplectic manifold is a pair (M, Ω), where M is an even dimensional smooth manifold and Ω is a nondegenerate, closed 2-form. Such a 2-form is called a symplectic form. Consider the action of a Lie group G on M ; the G-action is called symplectic if it preserves the symplectic form Ω, i.e., ∀g ∈ G, Φ∗g Ω = Ω, where Φg : M → M is the action map. Now consider an Ad∗ -equivariant map M : M → Lie∗ (G) such that ∀ξ ∈ Lie(G) it satisfies the identity ιξM Ω = dhM, ξi, (1.2.1) where ξM is the vector field on M induced by the infinitesimal action of G in the direction of ξ. Such a map is called the momentum map. The symplectic reduction theorem states that in the presence of a free and proper G-action and an (Ad∗ -equivariant) momentum map, for any value µ ∈ Lie∗ (G) of the momentum map the quotient manifold Mµ := M−1 (µ)/Gµ inherits a symplectic form Ωµ , where Gµ is the coadjoint isotropy group for µ, Ωµ is identified by the equality i∗µ Ω = πµ∗ Ωµ , and where the maps iµ : M−1 (µ) ,→ M and πµ : M−1 (µ) → M−1 (µ)/Gµ are the canonical inclusion and quotient maps [53]. The pair (Mµ , Ωµ ) is called the symplectic reduced manifold. This theorem by Marsden and Weinstein made a huge impact on unifying the reduction methods that had been previously developed for holonomic dynamical systems, such as classical Routh method and the reduction of Lagrangian systems by cyclic parameters [70]. For mechanical systems, the space of momenta, or phase space, i.e., the cotangent bundle of the configuration manifold T ∗ Q, admits a canonical symplectic 2-form, which is the exterior derivative of the tautological 1-form Θcan defined by (Θcan )pq (Zpq ) := hpq , Tpq πQ (Zpq )i, ∀pq ∈ Tq∗ Q and ∀Zpq ∈ Tpq T ∗ Q and where πQ : T ∗ Q → Q is the cotangent bundle projection. That is, (T ∗ Q, Ωcan := −dΘcan ) is a symplectic manifold. Let H : T ∗ Q → R be the Hamiltonian of a mechanical system that is defined by a Riemannian metric and a function on Q. The solution curves of this system satisfy Hamilton’s equation ιX Ωcan = dH, where X ∈ X(T ∗ Q) is everywhere tangent to the solution curves. In general, for any function f ∈ C ∞ (T ∗ Q), the vector field Xf ∈ X(T ∗ Q) that satisfies Hamilton’s equation is called the Hamiltonian vector field of f . Let G be a group acting properly on the configuration manifold Q. The cotangent lifted action on the phase space is symplectic. In this case, if the Hamiltonian of the system is also invariant under the cotangent lift


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of the G-action, the group G is called the symmetry group of the mechanical system, and the system is called a mechanical system with symmetry [48, 50]. In the reduction process of mechanical systems with symmetry, we start with a Riemannian metric on Q, a symplectic structure on T ∗ Q, the Hamiltonian H, and a Lie group whose action preserves the above structures, and after the reduction, we have a mechanical system on the reduced phase space, which is a symplectic manifold, with the induced Riemannian metric and Hamiltonian. A Poisson manifold is a pair (P, {·, ·}), where P is a smooth manifold and {·, ·} : C (P ) × C ∞ (P ) → C ∞ (P ), called the Poisson bracket, satisfies the following properties: ∀f, g, h ∈ C ∞ (P ) and ∀λ ∈ R, ∞

i) {f, g} = − {g, f } (antisymmetry property) ii) {f + λh, g} = {f, g} + λ {h, g} (linearity property) iii) {hf, g} = h {f, g} + {h, g} f (Leibniz property) iv) {{f, g} , h} + {{h, f } , g} + {{g, h} , f } = 0. (Jacobi identity) For a mechanical system, the phase space T ∗ Q admits a canonical Poisson structure using the canonical symplectic form, given by {f, h} := −Ωcan (Xf , Xh ), ∀f, h ∈ C ∞ (T ∗ Q), where Xf and Xh satisfy the identities ιXf Ωcan = df and ιXh Ωcan = dh. Based on this definition of the Poisson bracket, one has {f, h} = LXf h, where LXf is the Lie derivative in the direction of the vector field Xf . For a mechanical system with symmetry, suppose that the symmetry group G acts freely and properly on Q, and hence on T ∗ Q. The Poisson bracket is invariant under the cotangent lifted action, i.e., the action is a Poisson action on (T ∗ Q, {·, ·}). The Poisson bracket on T ∗ Q descends to a Poisson bracket on the quotient manifold (T ∗ Q)/G, defined by {f, h}(T ∗ Q/G) ◦ π = {f ◦ π, h ◦ π} , where f and h are smooth functions on (T ∗ Q)/G, and π : T ∗ Q → (T ∗ Q)/G is the quotient map. This bracket is well-defined since f ◦ π, h ◦ π and {·, ·} are G invariant. This process, which has been introduced in [50, 8], is called Poisson reduction. The major difference between the Poisson reduction and the symplectic reduction is the concept of momentum map, which is not necessary for Poisson reduction, and as a result the induced Hamilton’s equation on the quotient phase space evolves in a bigger space. This approach unifies the Euler-Poincaré and Lagrange-Poincaré equations for mechanical systems with symmetry [50].


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Both of the abovementioned reduction theories were developed and extended to Lagrangian systems, in the 1990s [15, 52, 51]. Since the trivial behaviour of a mechanical system due to symmetry are eliminated during a reduction process, the behaviour of the system is more explicit in the reduced space. The reduction procedures are helpful for extracting coordinate-independent control laws for the mechanical systems with symmetry [8, 13], which is the subject of Chapter 5. A nonholonomic mechanical system with symmetry is a mechanical system with symmetry together with a G-invariant distribution D, i.e., a distribution D such that ∀g ∈ G and ∀q ∈ Q, Tq Φg (D(q)) = D(Φg (q)). The distribution D is a linear sub-bundle of T Q where the velocities of the physical trajectories of the system should lie. Generally, this distribution is non-involutive, and it is the result of kinematic nonholonomic constraints such as rolling without slipping. If D is involutive, we say that the constraints are holonomic. Although in general the physical constraints can be nonlinear, time dependant or affine, we only restrict our attention to the constraints that are linear in velocity. The distinguishing characteristics of nonholonomic systems from the holonomic ones are that i) they satisfy the Lagrange-d’Alembert principle instead of the Hamilton principle [9], and ii) the momentum is not generally conserved for them. A Chaplygin system is a nonholonomic mechanical system with symmetry such that the space of directions of the infinitesimal G-action is complementary to the distribution D. On the Lagrangian side, in [16] Chaplygin reduces such systems considering only abelian symmetry groups . Afterwards, Koiller generalizes his result to non-abelian symmetry groups [42]. He considers two cases: i) Nonholonomic systems whose configuration manifold is a total space of a G-principal bundle and the constraints are given by a connection, and ii) Nonholonomic systems whose configuration manifold is G itself with left invariant constraints and left invariant metric, which defines the Lagrangian. A more general reduction procedure for the tangent lifted symmetries of a nonholonomic system that results in Lagrange-d’Alembert-Poincaré equations [8, 14] is reported in [9]. This method is centred on defining a nonholonomic connection as the sum of an Ehresmann connection and the mechanical connection and introducing a nonholonomic momentum map. The analogue of this approach in Poisson formalism is also explained in [8] that is originated in a paper by van der Schaft and Maschke [86]. In this paper,


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the authors use an Ehresmann connection to project the canonical Poisson bracket of T ∗ Q to the image of the nonholonomic distribution under the Legendre transformation, and they show that the resulting bracket satisfies the Jacobi identity if and only if the original distribution is involutive.

´ On the Hamiltonian side, Bates and Sniatycki first show that the vector field representing the dynamics of a nonholonomic system, which is the solution of Hamilton’s equation for nonholonomic systems, indeed lies in the distribution T (FL(D)) ∩ {v ∈ T (T ∗ Q)| T πQ v ∈ D} ⊆ T (T ∗ Q). Here, the fibre-wise linear map FL : T Q → T ∗ Q is the Legendre transformation. Then under the symmetry hypotheses, after restricting Hamilton’s equation to this distribution, they show that the flow of the vector field, which is the solution of Hamilton’s equation, descends to the quotient manifold FL(D)/G [6, 25, 26, 27]. Later on, based on this method of reduction, which is called distributional Hamiltonian approach [27], the Noether theorem is extended to nonholonomic systems and accordingly a two-stage reduction procedure is introduced. In the first stage, the symplectic reduction theorem is applied to reduce Hamilton’s equation by a normal subgroup G0 ⊆ G, whose momentum is conserved, and yields another distributional Hamiltonian system. For the second stage, the method in [6] is used to reduce the equations by G/G0 [76]. This method is further extended to singular reduction of nonholonomic systems, and it is reformulated for almost Poisson manifolds in [77]. Here, an almost Poisson manifold is a manifold equipped with a bracket that satisfies the properties of the Poisson bracket except the Jacobi identity.

An extension of reduction of Chaplygin systems is also reported in the concept of nonholonomic Hamilton-Jacobi theory [59, 39], which uses the symplectic reduction theorem in the presence of further symmetries of the system to reduce a Chaplygin system in two steps. The first step is equivalent to the Chaplygin reduction in [42], which results in an almost symplectic 2-form to describe Hamilton’s equation in the reduced space. An almost symplectic 2-form is a non-degenerate differential 2-form (which is not necessarily closed). In the second step, under some assumptions an almost symplectic reduction [69] is performed. Based on this idea, a three-step reduction procedure for nonholonomic mechanical systems with symmetry is presented in Chapter 4 that generalizes the two-step reduction in [59] by trying to find constants of motion that are not necessarily correspond to the action of abelian Lie groups.


1.3

7

Control of Free-base Multi-body Systems

An example of a mechanical system with symmetry is a free-base multi-body system, which is mostly studied in the field of robotics and aerospace. Vafa and Dubowsky introduce the notion of Virtual Manipulator [85] (for a free-floating manipulator with zero total momentum), and they show that this approach decouples the system centre of mass translation and rotation. Dubowsky and Papadopoulos in [28] use this notion to solve for the inverse dynamics problem that yields to designing linear controllers in joint and task space. Since the trivial behaviour of a multi-body system due to momentum conservation is eliminated during a reduction process, the behaviour of the system is more explicit in the reduced space. The reduction procedures have been helpful for extracting control laws for space manipulators by restricting the dynamical equations to the submanifold of the phase space where the momentum of the system is constant (and usually equal to zero). Yoshida et al. investigate the kinematics of free-floating multibody systems utilizing the momentum conservation law. They derive a new Jacobian matrix in generalized form and develop a control method based on the resolved motion rate control concept [84, 58]. McClamroch et al. propose an articulated-body dynamical model for free-floating robots based on Hamilton’s equation, and implement it to derive an adaptive motion control law [90]. Based on the concept of Virtual manipulator, Parlaktuna and Ozkan also develop an adaptive controller for free-floating space manipulators [65]. Wang and Xie introduce an adaptive control law for position/force tracking of free-flying manipulators [87, 88], and later they use recursive Newton-Euler equations to derive a novel adaptive controller for position tracking of free-floating manipulators in their task space [89]. In this controller, they estimate the inertia tensor of the spacecraft (base body) by a parameter projection algorithm. As an application, Pazelli et al. investigate different nonlinear H∞ control schemes implemented to a free floating space manipulator, subject to parameter uncertainty and external disturbances [66]. In the case of underactuated space manipulators, Mukherjee and Chen in [55] show that even if the unactuated joints do not possess brakes, the manipulator can be brought to a complete rest provided that the system maintains zero momentum. In [83] an alternative path planning methodology is developed for underactuated manipulators using high order polynomials as arguments in cosine functions to specify the desired path directly in joint space. Note that all of the above mentioned control strategies were developed for holonomic multi-body systems with one-d.o.f. joints and for zero momentum of the system.


8

Geometric methods have also been used to reduce the dynamical model of free-base multi-body systems and introduce effective control laws. For example, in [78, 79] Sreenath reduces Hamilton’s equation by SO(2) for free-base planar multi-body systems with nonzero angular momentum. He uses the symplectic reduction theory to first reduce the dynamical equations and then derive a control law for reorienting the free-base system. Chen in his Ph.D. thesis [17] extends Sreenath’s approach to spatial multi-body systems with zero angular momentum. Duindam and Stramigioly derive the Boltzmann-Hamel equations for multi-body systems with generalized multi-d.o.f. holonomic and nonholonomic joints by restricting the dynamical equations to the nonholonomic distribution [29]. This is the first attempt to reduce the dynamical equations of a generic open-chain multibody systems with generalized holonomic and nonholonomic joints. Furthermore, Shen proposes a novel trajectory planning in shape space for nonlinear control of multi-body systems with symmetry [74, 72, 73]. In his work he performs symplectic reduction for zero momentum and assumes multi-body systems on trivial bundles. Then, in [75] he extends his results to include nonholonomic constraints. Hussein and Bloch study optimal control of nonholonomic mechanical systems, using an affine connection formulation [40]. Sliding mode control of underactuated multi-body systems is also studied in [3]. In the control community, Olfati-Saber in his Ph.D. thesis [60] studies the reduction of underactuated holonomic and nonholonomic Lagrangian mechanical systems with symmetry and its application to nonlinear control of such systems. He uses a feedback linearization method in the reduced phase space to extract control laws for such systems [61]. However, he only considers abelian symmetry groups, and he does not take into account non-zero momentum of the system in his approach. As a continuation of Olfati-Saber’s work, Grizzle et al. in [34] show that a planar mechanism with a cyclic unactuated parameter is always locally feedback linearizable, and they derive a nonlinear control law for such systems. Further, Bloch and Bullo extract coordinate-independent nonlinear control laws for holonomic and nonholonomic mechanical systems with symmetry [8, 12, 13].

1.4

Statement of Contributions

This section presents the contributions of this dissertation in different aspects of studying open-chain multi-body systems. In this work, we consider nonholonomic constraints as linear constraints on the joint velocities. Normally, systems with nonholonomic constraints are treated separately in the literature. This thesis is an attempt to use geometric tools to unify and extend the existing approaches for analyzing the kinematics and dynamics of open-chain multi-body systems with non-zero momentum and holo-


9

nomic/nonholonomic constraints. As a result, based on the developments in Chapters 2 to 4, we are able to derive a nonlinear control scheme in Chapter 5 for concurrent trajectory tracking of a generic freebase, open-chain multi-body system with multi-d.o.f. holonomic and/or nonholonomic joints. In the following sections, we elaborate on the contributions of this thesis in kinematics, dynamics and controls.

1.4.1

Kinematics

The main contributions of the thesis in kinematics can be listed as: i) group theoretic classification of multi-d.o.f. joints, and ii) development of a generalized exponential formula for forward and differential kinematics of open-chain multi-body systems with multi-d.o.f. holonomic and/or nonholonomic joints. In the following, we detail different steps of this phase of the research, which is the content of Chapter 2. Displacement Subgroups We start with the definition of joint as a distribution on the relative configuration manifold of a body with respect to another body. This configuration manifold is diffeomorphic to the Lie group SE(3). We observe that for a left invariant distribution (corresponding to a joint) the involutivity of the distribution and closedness of the Lie bracket (of the Lie algebra) coincide. Based on this observation, we show that the relative configuration manifolds of lower kinematic pairs are indeed Lie subgroups of SE(3), and in Section 2.1 we generalize this class of multi-d.o.f. holonomic joints by introducing the notion of displacement subgroups. In Table 2.1 we list different categories of displacement subgroups. Accordingly, we use the exponential map for Lie subgroups of SE(3) to introduce a new joint parametrization, called screw joint parameters. This joint parametrization is used in Chapter 3 and 4 to embed an open subset of a quotient manifold in the relative configuration manifold and in Chapter 5 to define the error function for the controller. We study the relationship between the screw and classic joint parameters in Theorem 2.1.5. We then define the nonholonomic constraints for a multi-d.o.f. joint in section 2.1.2. The contribution of this part of the thesis is stated in Proposition 2.1.3, in which we prove the surjectivity of the exponential map for all categories of displacement subgroups except for the 2-d.o.f. prismatic + helical category of joints.


10

Forward and Differential Kinematics The main contribution of this chapter is generalizing the existing product of exponential formula [57] for forward and differential kinematics of open-chain multi-body systems to include displacement subgroups, in Theorem 2.2.3 and 2.3.1. We accordingly derive a modified Jacobian for the screw joint parameters in (2.3.13), by considering the nonholonomic constraints. Finally in Section 2.4, we study different operators appearing in the developed differential kinematics formulation using the standard basis for the Lie algebra of SE(3). The results of this section are summarized in Proposition 2.4.5. To illustrate the contents of Chapter 2, we present a detailed example in Section 2.5.

1.4.2

Dynamics

The main contributions of this phase of research are: i) symplectic reduction of holonomic open-chain multi-body systems with multi-d.o.f. joints and non-zero momentum as a generalization of the existing reduction methods for free-base manipulators, which are for single-d.o.f. joints and zero momentum, ii) generalization of the existing approaches to the reduction of nonholonomic Hamiltonian mechanical systems and its application to dynamical reduction of nonholonomic open-chain multi-body systems with multi-d.o.f. joints, and iii) unification of the developed reduction methods for holonomic and nonholonomic cases. In addition, a new approach to the derivation of Hamilton’s equation for holonomic and nonholonomic Lagrangian systems is developed, using Lagrange-d’Alembert-Pontryagin principle on Pontryagin bundle. (See Section 3.1 and 4.1.) The study of the dynamical reduction of open-chain multi-body systems is the subject of Chapters 3 and 4. Different steps of this part of the research are detailed in the following. Holonomic Chapter 3 focuses on the case of holonomic Hamiltonian mechanical systems with symmetry. We denote the symmetry group by G and its coadjoint isotropy group corresponding to an element µ ∈ Lie∗ (G) by Gµ . The Hamiltonian function H of a Hamiltonian mechanical system consists of a quadratic term coming from the kinetic energy metric on the configuration manifold Q plus the potential energy function. We revisit the dynamical


11

reduction of Hamiltonian mechanical systems with symmetry, using the symplectic reduction theorem. We also use the mechanical connection, which is a principal connection compatible with the kinetic energy metric, to identify the symplectic reduced space with a vector sub-bundle of T ∗ (Q/Gµ ). One of the contributions of this chapter is identifying the relative configuration manifold of the first joint of a holonomic open-chain multi-body system with displacement subgroups as a symmetry group for the system (see Theorem 3.3.3). We then define the notion of a holonomic open-chain multi-body system with symmetry. Consequently, we apply the symplectic reduction procedure for Hamiltonian mechanical systems to holonomic open-chain multi-body systems with symmetry. The main contribution of this chapter is summarized in Theorem 3.3.6. In this theorem, we reduce the Hamilton’s equation for a holonomic open-chain multi-body system with symmetry in T ∗ Q to a Hamilton’s equation in the reduced phase space, which is a vector sub-bundle of T ∗ (Q/Gµ ). This theorem generalizes the existing reduction methods for holonomic openchain multi-body systems at zero momentum, e.g., used in [17, 28, 90, 72, 74]. Nonholonomic In Section 4.2, we consider nonholonomic Hamiltonian mechanical systems with symmetry, where the linear distribution corresponding to the nonholonomic constraints is denoted by D. In this section we restrict our attention to the nonholonomic systems with symmetry whose symmetry group has a Lie subgroup G that satisfies the Chaplygin assumption in (4.2.10). One of the contributions of this section is the proof of the Chaplygin reduction theorem [42]. In Theorem 4.2.4, we state the Chaplygin reduction theorem for the systems on T ∗ Q. And, we give a proof that is independent of the choice of local coordinate charts, and it illustrates the geometry behind the Chaplygin reduction theorem. Using this proof, we geometrically show the similarities and distinctions between this reduction procedure and the symplectic reduction of holonomic Hamiltonian mechanical systems with symmetry. The main difference between these two reduction methods is that in the holonomic case the reduced phase space is a symplectic manifold, as opposed to the almost symplectic manifold for the case of a Chaplygin system. This proof can be used to unify the reduction processes developed for holonomic and nonholonomic Hamiltonian mechanical systems with symmetry. Accordingly, we give a nonholonomic version of Noether’s theorem for reduced Chaplygin systems in Proposition 4.2.12, which is equivalent to the theorem presented in Section 3 of [76]. Another contribution of this section is using this proposition along with the almost symplectic reduction presented in [69] to perform a three-step reduction of nonholonomic Hamilto-


12

nian mechanical systems with symmetry. The main results of this section are presented in Proposition 4.2.14 and Theorem 4.2.18. Note that the three-step reduction process in this section is a generalization of the 2-step reduction of Chaplygin systems presented in [59]. To illustrate the contents of Chapter 3 and 4, we include three detailed case studies in Sections 3.4 and 4.6. In Section 4.3, we apply the developed reduction process to nonholonomic openchain multi-body systems with symmetry. We report the result of the first step of the reduction process in Theorem 4.3.1, which is one of the main contributions of Chapter 4. Before performing the next steps of the reduction process, in Section 4.4 we present a number of sufficient conditions, under which a nonholonomic open-chain multi-body system admits a symmetry group bigger than G = Q1 , which is one of the contributions of this dissertation. Then, Theorem 4.5.2 finalizes Chapter 4 by performing the second step of the reduction presented in Section 4.2 for nonholonomic open-chain multi-body systems with symmetry. This theorem is one of the main contribution of this dissertation.

1.4.3

Controls

The main contributions of this research in controls can be listed as: i) solving the input-output linearization problem in the reduced phase space of a freebase, holonomic (with non-zero momentum) or nonholonomic controlled open-chain multi-body system and multi-d.o.f. joints, and ii) deriving a coordinate-independent, trajectory tracking, feed-forward servo control law for concurrent control of the base and other extremities of a generic open-chain multi-body system with multi-d.o.f. joints, and proving the exponential stability of the closed-loop system by introducing a proper Lyapunov function. In the following, we detail different steps of this phase of research. Chapter 5 is devoted to the concurrent control of underactuated holonomic and nonholonomic open-chain multi-body systems with displacement subgroups. We only restrict our attention to systems in which there is no actuation in the directions of the group action and nonholonomic constraints. We call such systems free-base, open-chain multibody systems. The control problem considered in this thesis is a trajectory tracking problem for the extremities of an open-chain multi-body system. In order to formally define this problem, we need to make sense of pose and velocity error on the output manifold of a holonomic or nonholonomic open-chain multi-body system (see Section 5.2). For technical reasons we assume that the output manifold of the system can be


13

identified by a Lie subgroup of a Cartesian product of copies of SE(3). As a result, we use the exponential map of Lie groups and right trivialization of the tangent bundle of Lie groups to define an error function and connection on the output manifold. In order to control the pose of the extremities in the inertial coordinate frame, we need not only the reduced Hamilton’s equation but also the reconstruction equations. In Section 5.1.2, we derive the reconstruction equations for holonomic and nonholonomic open-chain multi-body systems. As mentioned above, one of the contributions of this dissertation is unification of the reduction of holonomic and nonholonomic open-chain multi-body systems. This enables us to develop a unified framework to derive control laws for both categories of multi-body systems. In Section 5.3, we first show that a controlled holonomic or nonholonomic open-chain multi-body system with symmetry is input-output linearizable in the reduced phase space. This result generalizes the existing linearization methods for underactuated, holonomic and nonholonomic mechanical systems presented, e.g., in [2, 5, 28, 34, 60, 61], to include non-abelian symmetry groups, non-zero momentum (of holonomic systems) and nonholonomic constraints. In addition, under a dimensional assumption and feasibility of the desired trajectory we solve the inverse dynamics problem for a generic holonomic or nonholonomic open-chain multi-body system with symmetry in the reduced phase space. Finally, in Theorem 5.4.2 (Section 5.4) we present a coordinate-independent, output tracking, feed-forward servo control law for a holonomic or nonholonomic open-chain multi-body system. And, using an appropriate Lyapunov function we prove that this controller exponentially stabilizes the closed-loop system for any feasible trajectory of the extremities. This control law depends only on the elements of the reduced phase space and the symmetry group, and it is independent of the velocity of the system in the directions of the group action.

1.4.4

Produced Manuscripts

Four manuscripts [20, 19, 18, 21] have been produced for publication (one is accepted for publication), as listed in the following: i) R. Chhabra and M.R. Emami. A Generalized Exponential Formula for Forward and Differential Kinematics of Open-chain Multi-body Systems. Accepted in Mechanism and Machine Theory, September 2013. ii) R. Chhabra and M.R. Emami. Symplectic Reduction of Holonomic Open-chain Multi-body Systems with Constant Momentum. Submitted to Multibody System


14

Dynamics, September 2013. iii) R. Chhabra and M.R. Emami. A Geometric Approach to Dynamical Reduction of Open-chain Multi-body Systems with Nonholonomic Constraints. Submission to Mechanism and Machine Theory, October 2013. iv) R. Chhabra and M.R. Emami. A Unified Approach to Input-output Linearization and Concurrent Control of Underactuated Holonomic and Nonholonomic Openchain Multi-body Systems. Submission to Journal of Dynamical and Control Systems, October 2013.

1.5

Outline of the Thesis

A brief outline of the content of different chapters of this dissertation is as follows: Chapter 2: In this chapter we study the kinematics of holonomic and nonholonomic open-chain multi-body systems with multi-d.o.f. joints. Chapter 3: This chapter is devoted to the study of the symplectic reduction of holonomic Hamiltonian mechanical systems with symmetry and its application to holonomic open-chain multi-body systems. Chapter 4: This chapter presents a three-step reduction method for nonholonomic Hamiltonian mechanical systems with symmetry and its application to nonholonomic open-chain multi-body systems. Chapter 5: An output tracking, feed-forward servo control law in the reduced phase space of a generic free-base holonomic or nonholonomic open-chain multi-body system is developed in this chapter, and exponential stability of the closed-loop system is proven. Chapter 6: This chapter includes some concluding remarks and states some future directions of the research presented in this dissertation.

Chapter 2 A Generalized Exponential Formula for Forward and Differential Kinematics of Open-chain Multi-body Systems

This chapter presents a generalized exponential formula for Forward and Differential Kinematics of open-chain multi-body systems with multi-degree-of-freedom, holonomic and nonholonomic joints. We revisit the notion of displacement subgroup, and show that the relative configuration manifolds of such joints are Lie groups. Accordingly, we categorize displacement subgroups, and prove that except for one class of displacement subgroups the exponential map is surjective. Screw joint parameters are defined to parametrize the relative configuration manifolds of displacement subgroups using the exponential map of Lie groups. For nonholonomic constraints, the admissible screw joint speeds are introduced, and the Jacobian of the open-chain multi-body system is modified accordingly. Then by assigning coordinate frames to the initial configuration of the multi-body system, employing the matrix representation of SE(3) and choosing a basis for se(3), we explore the computational aspects of the developed formulation for Forward and Differential Kinematics of open-chain multi-body systems. Finally, we study the developed formulation for an example of a mobile manipulator mounted on a spacecraft, i.e., on a six-degree-of-freedom moving base. 15

Chapter 2. A Generalized Exponential Formula for Kinematics

2.1

16

Holonomic and Nonholonomic Joints

A physical 3-dimensional (3D) space can be mathematically modelled as a 3D affine space, denoted by A, which is modelled on a vector space V , and a rigid body B is the closure of a bounded open subset of A. Let us fix a coordinate frame in the physical space. Considering a multi-body system M S(N ) = {(Ai , Bi )|Bi ⊂ Ai , i = 0, ..., N } and a body Bi in it, the space of all absolute poses (position and orientation) of Bi with respect to the fixed coordinate frame is then Gi = SE(3), Special Euclidean group. One can also introduce the relative pose of two bodies of a multi-body system. Let Bi and Bj be two bodies in M S(N ), the space of all relative poses of Bi with respect to Bj forms a smooth manifold Pij := gj−1 · gi gi ∈ Gi = SE(3), gj ∈ Gj = SE(3) ∼ = SE(3). When i = j this manifold, which is the space of all possible coordinate transformations of Ai , inherits Lie group structure isomorphic to Gi = SE(3) with the identity element ei and the Lie algebra denoted by Lie(Pii ). In the case of i = j, to simplify the notation only the lower index is used, e.g., Pi := Pii . A relative motion of Bi with respect to Bj is a smooth curve rij : [0, 1] → Pij , and the relative velocity at time t is the vector vij (t) = (drij /dt)(t) ∈ Trj (t) Pij , where Trj (t) Pij is the tangent space of Pij at the element i i rij (t). At each instant t, one can show that this vector induces a vector field Xt on Aj corresponding to the relative motion of Bi with respect to Bj such that ∀a ∈ Aj ,

Xt (a) = lim

δ→0

j exp δ Trj (t) Rrji (t) vi (t) (a) − (a) i

δ

;

(2.1.1)

where Rrji (t) : Pij → Pj denotes the right composition map by rji (t). If we identify the manifolds Pij and Pj by the Lie group SE(3), the right composition map becomes just −1 the right translation map by rji (t) = rji (t) ∈ SE(3). For a relative motion, if this vector field is independent of time, the relative motion is called relative screw motion. In other words, a relative screw motion is the curve on Pij corresponding to the flow of a left-invariant Killing vector field [82] on Pj . An interpretation of the Chasles’ Theorem indicates that from any initial relative pose, any relative pose of Bi with respect to Bj can j be reached by a relative screw motion. Therefore, the exponential map of SE(3) ∼ = Pi is onto [57]. Given two rigid bodies of a multi-body system, Bi and Bj , a joint is a mechanism that restricts the relative motion of Bi with respect to Bj , and specifies a subset Dij of T Pij . A joint may be time dependant, called rheonomic joint, or time independent, which is called scleronomic joint. A special type of scleronomic joints, which is mostly considered in the literature, is when we have Dij ⊆ T Pij being a distribution on Pij that


17

corresponds to admissible directions of the relative velocity of Bi with respect to Bj . We only consider this category of joints in this thesis. In particular, we assume that Dij has constant rank. If Dij is involutive, i.e. its space of sections is closed under the Lie bracket of vector fields, the joint is called holonomic; otherwise, it is a nonholonomic ¯ j be joint. For any non-involutive distribution Dij , under the existence assumption, let D i j j the involutive closure of Di . The involutive closure of a distribution Di is the smallest vector sub-bundle of T Pij containing Dij that is closed under the Lie bracket of vector ¯ j (for a holonomic fields. Based on the global Frobenius Theorem [43], either Dij or D i or nonholonomic joint) gives a foliation on Pij . The leaf Qji ⊆ Pij that contains the j initial relative pose of Bi with respect to Bj , ri,0 , is called the relative configuration j manifold. The manifold Qi is the space of all admissible relative poses considering the joint constraints. The dimension of this manifold, k, is called the number of d.o.f. of a joint, which is greater than or equal to the dimension of the joint distribution for a nonholonomic or holonomic joint, respectively. i One can define the submanifold Qi ⊆ Pi as the left composition of Qji by rj,0 , i.e., j j j i i i (Q ), where r Qi = Lrj,0 i i,0 ◦ rj,0 = ej and rj,0 ◦ ri,0 = ei . This submanifold contains j the identity element of Pi , which corresponds to ri,0 ∈ Qji . A local coordinate chart for a neighbourhood W ⊂ Qi of ei is a diffeomorphism ϕ : Rk ⊃ U → W such that ϕ([0, ..., 0]T ) = ei . Therefore, any element rij ∈ Lrj (W ) ⊆ Qji can be parametrized i,0 by a q ∈ U , which is called the classic joint parameter, through the diffeomorphism Lrj ◦ ϕ. A velocity vector vij ∈ Trj Qji can also be identified with a k-dimensional vector i,0 i j )(Tq ϕ). q˙ ∈ Tq U ∼ Note that the coordinate = Rk by the linear isomorphism (Tϕ(q) Lri,0 ∂ chart ϕ induces a basis {( ∂qb )|q |b = 1, ..., k} for Tϕ(q) W , where qb is the bth element of q, and in this basis Tq ϕ is the identity matrix, idk .

2.1.1

Displacement Subgroups

In this subsection, displacement subgroups are defined as a class of holonomic joints, and it is shown that their relative configuration manifolds are connected Lie groups. In Proposition 2.1.3, the necessary and sufficient conditions for the surjectivity of the exponential map of these relative configuration manifolds are given. Based on this identification of displacement subgroups, a set of new joint parameters, called screw joint parameters, is introduced. These joint parameters can be physically interpreted as the initial classic joint speeds for a screw motion on the corresponding relative configuration manifold. Finally, the relationship between the screw joint parameters and the classic joint parameters is formalized in Theorem 2.1.5.


18

j i (D ) ⊆ T Pj . Based on the For a holonomic joint, define the distribution Dj := Trj Rrj,0 i i definition of a holonomic joint, Dj is involutive, i.e., its space of sections is closed under the Lie bracket of vector fields on Pj . This bracket coincides with the definition of the Lie bracket [41] on Lie(Pj ) if Dj is left-invariant, i.e., Dj (rj ) = Tej Lrj (Dj (ej )), ∀rj ∈ Pj . We denote the integral manifold of Dj containing ej by Qj ⊆ Pj . Particularly, Dj (ej ), which is a linear subspace of Lie(Pj ), is closed under the Lie bracket of Lie(Pj ); hence Tej Qj = Dj (ej ) is a Lie sub-algebra of Lie(Pj ).

Proposition 2.1.1. For a holonomic joint, if Dj (defined above) is left-invariant, its integral manifold containing ej , i.e., Qj ⊆ Pj , is the unique k-dimensional connected Lie subgroup of Pj with the Lie algebra Lie(Qj ) = Dj (ej ). Note that conversely, for any Lie subgroup Q0j ⊆ Pj , there exists a unique involutive distribution corresponding to a holonomic joint, by left translating Lie(Q0j ) over Pj and j right composing it with ri,0 . Definition 2.1.2. A holonomic joint is called displacement subgroup if the corresponding distribution Dj (defined above) on Pj is left-invariant. Therefore, based on Proposition 2.1.1 and since Pj ∼ = SE(3), different types of displacement subgroups are identified by the connected Lie subgroups of SE(3), up to conjugation, which are tabulated in Table 2.1 [38, 71]. In this table, Hp is the Lie subgroup of SE(3) corresponding to a simultaneous rotation about and translation along a vector in R3 , where the ratio of translation to rotation is equal to the constant p. From this table, one can observe that the displacement subgroups consist of the six lower kinematic pairs, i.e., revolute, prismatic, helical, cylindrical, planar and spherical joints, and combinations of them. Therefore, in this joint categorization, the relative configuration manifolds of lower kinematic pairs are indeed subgroups of SE(3). There also exist other types of holonomic joints, e.g., universal joint and higher kinematic pairs, which are not included in the category of displacement subgroups. However, the relative configuration manifolds of these joints are not subgroups of SE(3). To parametrize the relative configuration manifolds of these joints one needs a product of exponentials of some elements of a basis for the tangent space of the relative configuration manifold at the identity element. Proposition 2.1.3. The group exponential map exp : Lie(Qj ) → Qj is surjective for all categories of displacement subgroups, except for a three-d.o.f. joint where a helical joint is combined with a two-d.o.f. prismatic joint such that the helical joint axis is perpendicular to the plane of the prismatic joint. This case is considered as two separate joints in this thesis.

19


Table 2.1: Categories of displacement subgroups [38, 71] Dim. 6 4 3 2 1 0 a b c d

Subgroups of SE(3)/displacement subgroups SE(3) = SO(3) n R3 freea SE(2) × R planar+prismaticb SE(2) = SO(2) n R2 planar SO(2) × R cylindricald SO(2) revolute {e} fixeda

SO(3) ball (spherical) R2 2-d.o.f. prismatic R prismatic

R3 3-d.o.f. prismatic

Hp n R2 helical + 2-d.o.f. prismaticc

Hp helical

These two subgroups are the trivial subgroups of SE(3). The axis of the prismatic joint is always perpendicular to the plane of the planar joint. The axis of the helical joint is always perpendicular to the plane of the 2-d.o.f. prismatic joint. The axis of the revolute and prismatic joints are always aligned.

Since this proposition is proved by coordinate chart assignment, its proof is presented in Section 2.4. Definition 2.1.4. Let ϕ be a coordinate chart for a neighbourhood of ei . By Proposition 2.1.3 any relative configuration manifold Qji of a displacement subgroup can be parametrized by vectors s ∈ Rk , called screw joint parameters, such that every rij ∈ Qji ⊆ Pij can be expressed as j j rij = exp(τij s) ◦ ri,0 := exp (Adrj )(Tei ι)(T0 ϕ)s ◦ ri,0 , i,0

(2.1.2)

where ι : Qi → Pi is the inclusion map. Therefore, for a relative motion rij : [0, 1] → Qji the relationship between (s, s), ˙ which are the screw joint parameters and their speeds, and (q, q), ˙ which are the classic joint parameters and their speeds, can be summarized in the following theorem. In this theorem, ∀η ∈ Lie(Qj ) adη : Lie(Qj ) → Lie(Qj ) is the endomorphism of Lie(Qj ) such that ∀ξ ∈ Lie(Qj ) we have adη (ξ) := [η, ξ] [41]. The linear map Z(s) (defined in Theorem 2.1.5) is an isomorphism between T0 Rk and Tq Rk if and only if adT0 ϕ(s) has no √ eigenvalue in 2πiZ, where i = −1. Theorem 2.1.5. For a displacement subgroup, consider a coordinate chart for Qi , ϕ : Rk ⊃ U → W such that ϕ([0, ..., 0]T ) = ei , and a relative motion rij : [0, 1] → Qji in the neighj j bourhood of ri,0 , denoted by W 0 := Lrj (W ) ⊆ Qji . Then, rij (t) = exp(τij s(t)) ◦ ri,0 where i,0


20

s(0) = 0, and q(s) = ϕ−1 ◦ exp(T0 ϕ s),

(2.1.3a)

q(s, ˙ s) ˙ = Z(s)s˙ −1

Z

:= (Tq(s) ϕ) Tej Lexp(T0 ϕs)

1

exp(−x adT0 ϕs ) dx T0 ϕ s. ˙

(2.1.3b)

0 i ◦ rij ⊂ W be the corresponding Proof. For the relative motion rij ⊂ W 0 , let ri = Lrj,0 i i curve on Qi . This curve on Pi is ι ◦ ϕ(q) = Lrj,0 ◦ Rrj ◦ exp(τij s) = Krj,0 ◦ exp(τij s). i,0 Based on (2.1.2) and the fact that exponential map is compatible with the Lie group i ◦K j ◦ homomorphisms [41], in this case conjugation and inclusion map, ι ◦ ϕ(q) = Krj,0 ri,0 ι ◦ exp(T0 ϕs) = ι ◦ exp(T0 ϕs). Therefore, (2.1.3a) is true since the inclusion map ι is an embedding, and ϕ is a diffeomorphism. Differentiating (2.1.3a) with respect to the curve parameter results in

q˙ = Texp(T0 ϕs) ϕ−1 (TT0 ϕs exp) T0 ϕs˙ = (Tq ϕ)−1 (TT0 ϕs exp) T0 ϕs. ˙ For a Lie group G, it can be shown that the differential of the exponential map at ξ ∈ Lie(G) is [30] Z 1 Tξ exp = Te Lexp(ξ) exp(−x adξ )dx. (2.1.4) 0

Hence, substituting (2.1.4) and (2.1.3a) in the above equation completes the proof for (2.1.3b). In (2.1.3b), Z(s) is defined as the composition of several linear operators, and it is invertible if and only if all of the linear operators are invertible. Since left translation is a global diffeomorphism and ϕ is a coordinate chart, it suffices to check the conditions R1 under which Θ := 0 exp(−x adT0 ϕs ) dx is invertible. For z ∈ C, consider the solution of R1 exp(−x z) dx that is equal to the entire holomorphic function f (z) = 1−exp(−z) such z 0 1−exp(−λi ) that f (0) = 1. Thus, the eigenvalues of Θ are equal to , where λi ’s are the λi eigenvalues of adT0 ϕs . The Lie algebra endomorphism Θ is invertible if and only if it has √ no eigenvalues equal to zero, i.e., λi 6∈ 2πiZ where i = −1. This theorem gives a condition for the size of the image of the coordinate chart associated with the screw joint parametrization. On Pj ∼ = SE(3) this condition dictates that the coordinate chart cannot include elements of Pj corresponding to 2π radian rotation about an axis in Aj . Also, note that the integral term in (2.1.4) is equal to the identity map for abelian Lie groups, and in general this term corresponds to the non-commutativity of ξ, ξ˙ ∈ Lie(Qj ) with respect to the Lie bracket.


2.1.2

21

Nonholonomic Displacement Subgroups

A nonholonomic displacement subgroup is a displacement subgroup together with k¯ linearly independent constraints in the space of the speeds of the classic joint parameters ¯ that are not integrable, i.e., C(q)q˙ = 0, where C(q) ∈ Rk×k , and C(q) is assumed to be a differentiable linear operator on Qi . In other words, for the neighbourhood W of the ¯ j initial relative pose ri,0 , ∀q ∈ U ⊂ Rk q˙ ∈ Tq Rk should lie in the ker(C(q)) ∼ = Rk−k that ¯ ¯ can be considered as the range of another linear operator C(q), i.e., C(q)C(q) = 0. The ¯ ¯ This ¯ C(q) ∈ Rk×(k−k) is a differentiable linear operator on Qi of constant rank k − k. linear operator identifies a smooth non-involutive distribution on Qji corresponding to the space of all admissible instantaneous relative velocities of the joint. Therefore, an admis¯ q¯˙ ∀q¯˙ ∈ Rk−k¯ . Note that the representation of C(q) ¯ sible joint speed has the form q˙ = C(q) in the local coordinates is not unique, and it could be chosen such that the admissible classic joint speeds are collocated with the joint control forces and torque to simplify the dynamic analysis. Based on (2.1.3b) in Theorem 2.1.5 and considering the screw joint parameters, the space of all admissible screw joint speeds at s can be identified by ¯

¯ s˙ = Σ(s)s¯˙ := Z −1 (s)C(q(s)) s¯˙ . ∀s¯˙ ∈ Rk−k

2.2

(2.1.5)

Forward Kinematics

Definition 2.2.1. An open-chain multi-body system is a multi-body system M S(N ) together with N − 1 joints between the bodies, such that there exists a unique path between any two bodies of the multi-body system. In an open-chain multi-body system, bodies with only one neighbouring body are called extremities. In robotics, the relative pose and velocity of the extremities with respect to a base body, labeled as B0 in M S(N ), is usually of interest. The base body is possibly an inertial observer. Definition 2.2.2. A branch of an open-chain multi-body system is a chain of m + 1 ≤ N bodies together with m joints that connects B0 to an extremity. In this chapter, an open-chain multi-body system is assumed to have n branches with both holonomic and nonholonomic multi-d.o.f. joints. In the branch i, joint j connects body Bj−1 to Bj . The branch configuration ri is defined as the collection of the relative mi −1 i −1 poses of rigid bodies, i.e., ri := r10 , ..., rm ∈ Q01 × ... × Qm mi . i Index the jth body of the branch i by ji . Let kji be the number of d.o.f. of the joint j in the ith branch, for an initial branch configuration, the set of all screw joint parameters

22


n o T of the branch is denoted by Gi := i s = i sT1 , ..., i sTmi |i sj ∈ Rkji , j = 1, ..., mi . Forward Kinematics of the ith branch of an open-chain multi-body system is a smooth map F Ki from the set of screw joint parameters of the branch to Pm0 i for an initial branch configuration that indicates the relative pose of the body Bmi with respect to B0 , i.e., mi −1 . F Ki : Gi → Pm0 i such that F Ki (i s) := r10 ◦ ... ◦ rm i Theorem 2.2.3. For an open-chain multi-body system M S(N ) along with N holonomic and nonholonomic displacement subgroups, the generalized exponential formula for the Forward Kinematics map corresponding to the ith branch can be formulated as F Ki (i s) = exp

0 0i τ1 s1

◦ ... ◦ exp

0 mi −1 i τmi smi

0 , ◦ rm i

(2.2.6)

0 )(Te ιj )(T0 ϕj ), ιj : Qj → Pj is the inclusion map, and ϕj is a coordinate where 0 τjj−1 = (Adrj,0 j chart for a neighbourhood of ej ∈ Pj ∀j = 1, ..., mi .

Proof. Using the screw joint parameters and the definition of the Forward Kinematics map, mi −1 0 mi −1 i F Ki (i s) = exp(τ10 i s1 ) ◦ r1,0 ◦ ... ◦ exp(τm smi ) ◦ rm . i ,0 i j−1 j−1 0 Due to the fact that rj,0 = r0,0 ◦ rj,0 , associativity of the composition operator, and compatibility of the exponential map with the conjugation map,

0 1 F Ki (i s) = exp(τ10 i s1 ) ◦ r1,0 ◦ exp(τ21 i s2 ) ◦ r0,0

mi −1

0 mi −1 i ◦ rm exp(τm smi ) ◦ r0,0 i −1,0 i

◦ ... 0 ◦ rm i ,0

1i mi −1 i 0 0 (τ 0 = exp(τ10 i s1 ) ◦ exp(Adr1,0 ( τm smi )) ◦ rm . 2 s2 )) ◦ ... ◦ exp(Adrm i i ,0 −1,0 i

Substituting the definition of τjj−1 , ∀j = 1, ..., mi , from (2.1.2) completes the proof. Note that since Forward Kinematics is only a function of the relative poses, nonholonomic constraints do not appear in (2.2.6). Forward Kinematics of an open chain multi-body system, F K, is defined as the collection of the relative poses of the extremities with respect to the base body B0 , i.e., F K : G1 × ... × Gn → Pm0 1 × ... × Pm0 n such that   F K1 (1 s)   .. , F K(s) :=  .   F Kn (n s) where s = [1 sT , ..., n sT ]T . For a serial-link multi-body system M S(N ) with one-d.o.f. revolute and/or prismatic joints, sj (t) ∀j = 1, ..., N is a real number function, instead of a vector function. Based


23

on the interpretation of the screw joint parameters given in the beginning of Subsection 2.1.1, sj (t) is the constant speed of a classic joint parameter during a screw motion from 0 to qj (t), in the interval of [0,1]. Therefore, its number is equal to the corresponding classic joint parameter. Moreover, since the joint has only one d.o.f., the linear operator 0 j−1 τj reduces to the joint screw at the initial configuration, which corresponds to the axis of rotation for a revolute joint or the direction of translation for a prismatic joint [57, 71]. Consequently, it can be shown that in this special case the formulation for Forward Kinematics of an open-chain multi-body system is equivalent to the product of exponentials formula suggested by Brockett [11]. This relationship is further illustrated in the case study in Section 2.5.

2.3

Differential Kinematics

For the ith branch of an open-chain multi-body system, Differential Kinematics is a linear map that relates the speed of the screw joint parameters of the branch to the instantaneous relative twist of Bmi with respect to B0 and observed in A0 , i.e., expressed 0 i in the vector space associated with A0 , V0 . The corresponding linear operator 0 Jm ( s), i 0 0 i called the Jacobian, for an initial branch configuration is Jmi ( s) : Ti s Gi → Lie(P0 ) such 0 i that 0 Jm ( s) := T i s) R(F K (i s))−1 Ti s F Ki . F K ( i i i Theorem 2.3.1. For an open-chain multi-body system M S(N ) along with N holonomic displacement subgroups, the generalized exponential formula for the Jacobian of the branch i can be formulated as 0

0 i Jm ( s) = i

h

··· ∆1 0 τ10 exp ad0 τ10 i s1 ∆2 0 τ21 0 mi −1 exp ad0 τ10 i s1 ... exp ad0 τ mi −2 i sm −1 ∆mi τmi , mi −1

where ∆j :=

R1 0

(2.3.7)

i

exp(x ad0 τ j−1 (i sj ) )dx is an endomorphism of Lie(P0 ). j

Proof. Consider a curve i s : [0, 1] → Gi , such that t 7→i s(t), in the set of screw joint parameters of the branch i. Let γj (t) := exp(0 τjj−1 i sj (t)) ∀j = 1, ..., mi . Using (2.2.6) and the product rule for Lie groups, d 0 F Ki (i s(t)) = Ti s(t) F Ki i s(t) ˙ = Tγ1 Rγ2 ◦...◦γmi ◦rm γ˙ 1 i ,0 dt 0 0 + Tγ2 ◦...◦γmi ◦rm Lγ1 Tγ2 Rγ3 ◦...◦γmi ◦rm γ˙ 2 + ... ,0 i ,0 i 0 0 + Tγmi ◦rm Lγ1 ◦...◦γmi −1 Tγmi Rrm γ˙ mi . ,0 ,0 i

i


24

By the definition of the Differential Kinematics map and rearranging the differential of the right and left composition maps, 0

Tγ1 Rγ1−1 γ˙ 1 + Tγ1 ◦γ2 R(γ1 ◦γ2 )−1 (Tγ2 Lγ1 ) γ˙ 2 + ... −1 + Tγ1 ◦...◦γmi R(γ1 ◦...◦γmi ) Tγmi Lγ1 ◦...◦γmi −1 γ˙ mi .

0 i ( s) i s˙ = Jm i

(2.3.8)

Now, use (2.1.4) for the exponential map exp : Lie(P0 ) → P0 , and the equality of operators [41] Adexp(ξ) = exp(adξ ), ∀ξ ∈ Lie(P0 ) (2.3.9) R 1 to calculate γ˙ j (t) = (Te0 Lγj ) 0 Adexp(−x 0 τ j−1 i sj ) dx 0 τjj−1 i s˙ j (t). Substitute γ˙ j and use j the identity Adr := Tr Rr−1 Te0 Lr ∀r ∈ P0 in (2.3.8) to achieve 0

0 i Jm ( s) i s˙ i

Z

1

Adexp(−x 0 τ10 i s1 ) dx 0 τ10 i s˙ 1 + ... 0 Z 1 mi −1 i Adexp(−x 0 τmmi −1 i sm ) dx 0 τm s˙ mi . + Adγ1 ◦...◦γmi i = Adγ1

(2.3.10)

i

i

0

Define ∆j ∀j = 1, ..., mi as Z ∆j := Adγj 0

1

Adexp(−x 0 τ j−1 i sj ) dx j

Z

1

Adexp((1−x) 0 τ j−1 i sj ) dx

=

j

0

Z = 0

1

exp x ad0 τ j−1 i sj dx, j

where the first equality holds since [x 0 τjj−1 i sj ,0 τjj−1 i sj ] = 0, and the second equality is the consequence of a change of variable and using (2.3.9). Finally, by substituting ∆j in (2.3.10) and employing the equality of operators in (2.3.9) one can show the desired expression for the Jacobian in (2.3.7) .

For a serial-link multi-body system with one-d.o.f. revolute and/or prismatic joints, since sj (t) is a real number function, 0 τjj−1 sj (t) ∈ Lie(P0 ) and 0 τjj−1 s˙ j (t) ∈ Lie(P0 ) commute, i.e., [0 τjj−1 sj ,0 τjj−1 s˙ j ] = 0, and hence ∆j becomes the identity map. In this case, the developed formulation simplifies to the existing product of exponentials formula for Differential Kinematics [57, 71]. 0 i i Based on the definition of the Differential Kinematics map, 0 Jm ( s) s˙ is the twist of i Bmi with respect to B0 and expressed in A0 . This twist can be viewed in the affine space


25

attached to the body j of the branch i, Aji , using the Adjoint operator, i.e., ji

0 i 0 i ( s), ( s) = Adrji (i s) 0 Jm Jm i i

(2.3.11)

0

where according to (2.2.6) rj0i (i s) = exp 0 τ10 i s1 ◦ ... ◦ exp 0 τjj−1 i sj ◦ rj0i ,0 . In addition, following the same calculations performed in the proof of Theorem 2.3.1, the Jacobian for the instantaneous relative twist of the body Bj with respect to Bl in the ith branch of M S(N ) and observed in A0 , i.e., 0 Jjl (i s) j > l > 0, can be determined to be the truncated version of the Jacobian in (2.3.7): 0

h l ··· Jjl (i s) = exp ad0 τ10 i s1 ... exp ad0 τ l−1 i sl ∆l+1 0 τl+1 l i exp ad0 τ10 i s1 ... exp ad0 τ j−2 i sj−1 ∆j 0 τjj−1 j−1

.

(2.3.12)

In order to include the nonholonomic constraints in the Jacobian of the ith branch of M S(N ), one can define admissible screw joint speeds according to (2.1.5). Therefore, the Jacobian in (2.3.7) can be modified to introduce the modified Jacobian for the ith branch of a multi-body system consisting of both holonomic and nonholonomic joints. 0

0 i 0 i J¯m ( s) := 0 Jm ( s)diag Σ1 (i s1 ), · · · , Σmi (i smi ) ; i i

(2.3.13)

where diag Σ1 (i s1 ), · · · , Σmi (i smi ) is the block diagonal matrix of its entries, and Σj = idkji for a holonomic joint. The modified Jacobian is a linear operator from the space of all n o ¯ i := i s¯˙ = i s¯˙ T , ...,i s¯˙ T T |i s¯˙ j ∈ Rkji −k¯ji , j = 1, ..., mi , admissible screw joint speeds, i.e., G 1

mi

to Lie(P0 ). For an open-chain multi-body system M S(N ), the modified Jacobian is defined as the collection of the modified Jacobians of the extremities with respect to ¯ s¯˙ := diag 0 J¯0 (1 s), ...,0 J¯0 (n s) s¯˙ , where the base body and observed in A0 , i.e., J(s) mn m1 T s¯˙ = 1 s¯˙ T , ...,n s¯˙ T .

2.4

Coordinate Assignment

At the computational level, consider a base point Oi for the affine space Ai in a multibody system M S(N ). Every point in this affine space can now be realized by a vector j in Vi ∼ = R3 through the action of (Vi , +) on Ai [7]. Therefore, any relative pose ri ∈ Pij can be represented by an orientation preserving isometry, Hij : Vi → Vj such that Hij := σOj ◦ rij ◦ (σOi )−1 ∈ SE(3), where σOl : Vl → Al for l = i, j is the map induced by the vector space action of Vl on Al . A matrix representation of SE(3) is the group of


26

orientation preserving linear isometries of R4 that preserve the plane x4 = 1 [7], i.e., (

# ) j j R p i i Hij = |Rij ∈ SO(3), pji ∈ R3 , 01×3 1

SE(3) ∼ =

"

where Rij is the rotation matrix whose columns are the elements of a basis for Vi expressed in terms of a basis for Vj and pji is the position of the point rij (Oi ) from Oj and expressed in Vj . In this representation, the Lie algebra of SE(3) is denoted by ( se(3) ∼ =

"

Tij

# ) ω ˜ ij wij j j = |˜ ωi ∈ so(3), wi ∈ R3 , 01×3 0

where wij is the relative velocity of the point rji (Oj ) with respect to Oj and expressed in Vj . The element ω ˜ ij ∈ so(3) corresponds to the relative angular velocity of Bi with respect to Bj and expressed in Vj , and it can be identified with the column vector ωij = [ω1 ω2 ω3 ]T ∈ R3 . This identification is through the following equality: 

 0 −ω3 ω2   ω ˜ ij =  ω3 0 −ω1  . −ω2 ω1 0 By choosing a basis for se(3) as       E1       0  0 E4 :=  0  0

 0  0 :=  0  0

  0 1 0    0 0  , E2 := 0 0 0 0   0 0 0   0 0 0 0 0    0 −1 0  , E5 :=  0 0 −1 0 1 0 0   0 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0

  0 0 0    1  , E3 := 0 0 0 0 0   0 0 0   0 0 −1    0  , E6 := 1 0 0 0 0   0 0 0

0 0 0 0

0 0 0 0 0 0 0 0

 0  0 , 1  0  0      0  , 0     0 

and using the propositions presented in the sequel, one can perform the computations for Forward and Differential Kinematics in the matrix representation of SE(3).

Proposition 2.4.1. For any element ξ = [wT , ω T ]T ∈ se(3), where ω, w ∈ R3 , ω 6= 0,


27

expressed in the basis {E1 , ..., E6 }, " exp(ξ) =

ω ˜w exp(˜ ω ) (id3 − exp(˜ ω )) kωk 2 +

01×3

1

ωω T w kωk2

# ,

(2.4.14)

wherek · k is the Euclidean norm of R3 and exp(˜ ω ) is evaluated using the Rodrigues’ formula for the exponential of skew-symmetric matrices, exp(˜ ω ) = id3 +

ω ˜2 ω ˜ sin(kωk) + (1 − cos(kωk)). kωk kωk2

(2.4.15)

"

# id3 w When ω = 0, exp(ξ) = . 01×3 1 Proof. See Appendix A in [57]. Now, using the matrix representation of SE(3) and the above proposition, the proof for Proposition 2.1.3 is presented. Proof. (Proposition 2.1.3) In the matrix representation, the exponential map for a connected Lie subgroup of SE(3) coincides with the restriction of the matrix exponential to the Lie sub-algebra corresponding to the subgroup. Up to conjugation, all of the connected Lie subgroups of SE(3) are listed in Table 2.1. Hence, to prove this proposition, it suffices to check the surjectivity of the exponential map for the matrix representation of each connected Lie subgroup, individually. Consider the following proposition and two lemmas. Proposition 2.4.2 (Chasles’ Theorem [57]). Every relative pose of a rigid body can be realized by a rotation about an axis combined with a translation parallel to that axis. In other words, the exponential map of the Lie group SE(3) is surjective. Lemma 2.4.3. The exponential map of a compact, connected Lie group is surjective [30]. Lemma 2.4.4. For a vector space V, Lie(V) = V with zero Lie bracket, and the exponential map is the identity map, i.e., exp(v) = v, ∀v ∈ V. Based on the Chasles’ Theorem and the above lemmas, the exponential maps of the subgroups SE(3), and SO(2), SO(3), R, R2 and R3 are surjective. In addition, since SO(2) × R is the direct product of two subgroups with surjective exponential maps, its own exponential map is also surjective.


28

In the following, we check the surjectivity of the exponential map for the remaining four non-trivial Lie subgroups of SE(3), i.e., Hp , SE(2), SE(2) × R and Hp n R2 , respectively. The subgroup Hp with p 6= 0 is a one dimensional subgroup of SE(3) that can be represented as      cos(θ) − sin(θ) 0 0           sin(θ) cos(θ) 0 0    . (2.4.16) Hp ∼ |θ ∈ R =     0 0 1 pθ           0 0 0 1 It is easy to check that the Lie algebra of Hp is      

  0 −1 0 0      1 0 0 0  Lie(Hp ) = Tid Hp = spanR Ep :=  0 0 0 p .          0 0 0 0 

(2.4.17)

Therefore, based on (2.4.14),  h11 h12  h21 h22 ∀H =   0 0  0 0

 0 0  0 0   ∈ Hp 1 h34   0 1

there exists θ = h34 /p such that exp(θEp ) = H. For   cos(θ) − sin(θ)      sin(θ) cos(θ) SE(2) = SO(2) n R2 ∼ =   0  0      0 0

0 0 1 0

   x      y 1  |θ ∈ S , x, y ∈ R ,  0      1

the corresponding Lie algebra is spanR {E1 , E2 , E6 }. Based on Lemma 2.4.4,  1  0 ∀H =  0  0

0 1 0 0

 0 h14  0 h24   ∈ SE(2), 1 0   0 1

(2.4.18)


29

exp(h14 E1 + h24 E2 ) = H, and otherwise for a general element of SE(2),  h11 h12  h21 h22 H=  0 0  0 0

 0 h14  0 h24   ∈ SE(2), 1 0   0 1

there exists θ = atan2(h21 , h11 ), where, based on (2.4.14), one has

θ θh24 θ θh14 θh24 θh14 + cot( ) E1 + cot( ) − E2 2 2 2 2 2 2   0  − sin(θ) 0 x   0  cos(θ) 0 y  , 0i 1 z0  

exp θE6 +  cos(θ)   sin(θ) =  0 h 

0 0 0

(2.4.19)

1

where    θh   0  24 + θh214 cot( 2θ ) 1 − cos(θ) sin(θ) 0 0 − 1θ 0 x 2     0  y  =  − sin(θ) 1 − cos(θ) 0  1θ 0 0  θh224 cot( 2θ ) − θh214  0 0 0 0 0 0 0 z0   h14   = h24  . (2.4.20) 00

Hence, the exponential map of SE(2) is surjective, and since SE(2) × R is the direct product of two subgroups with surjective exponential maps, its own exponential map is also surjective.

In the case of   cos(θ) − sin(θ)      sin(θ) cos(θ) Hp n R2 ∼ =   0  0      0 0

   0 x       0 y |θ, x, y ∈ R ,  1 pθ      0 1

(2.4.21)

30

Chapter 2. A Generalized Exponential Formula for Kinematics the Lie algebra is equal to spanR {Ep , E1 , E2 }. If θ ∈ {2πZ} \ {0}, then  1  0 H= 0  0

0 1 0 0

 0 x  0 y  ∈ Hp n R2 , 1 pθ  0 1

and there does not exist any τ ∈ spanR {Ep , E1 , E2 } such that exp(τ ) = H. Therefore, for Hp n R2 the exponential map is not surjective.

The following proposition presents closed form formulae for exp(adxi ), for any ξ ∈ se(3), and its integral that are used in the Differential Kinematics of open-chain multibody systems with displacement subgroups. Proposition 2.4.5. For any element ξ = [wT , ω T ]T ∈ se(3), where ω, w ∈ R3 and ω 6= 0, expressed in the basis {E1 , ..., E6 }, " adξ =

" exp(adξ ) =

1 kωk2

exp(˜ ω) 03×3

# w˜ , ω ˜

ω ˜

03×3

[[˜ ω , w], ˜ exp(˜ ω )] +

ω ˜ ωT w kωk2

exp

ω ˜ ωT w kωk2

# ,

(2.4.22)

exp(˜ ω)

where [·, ·] is the matrix commutator, exp(˜ ω ) is evaluated using (2.4.15) and, 1

Z 0

where, M1 = id3 +

"

# M1 M2 exp(x adξ ) dx = , 03×3 M1

ω ˜2 1 − cos(kωk)) + kωk 1 − sin(kωk) , and 2 kωk 1 ω ˜ ω ˜ ω ˜2 ωT w ω ˜ ω , w], ˜ M1 ] − ωT w + ωT w − kωk2 cos kωk + kωk + M2 = kωk2 [[˜ the case ω = 0, " # id3 w˜ exp(adξ ) = , 03×3 id3

(2.4.23)

ω ˜ (1 kωk2

and Z 0

1

"

# id3 w/2 ˜ exp(x adξ ) dx = . 03×3 id3

ω ˜2 kωkω T w

sin

ωT w kωk

. For


31

Proof. Case 1) When ω = 0, "

# 03×3 w˜ adξ = . 03×3 03×3 P i /i! , and ad Using the Taylor expansion of the matrix exponential, exp(adξ ) = ∞ ξ i=0 i the fact that adξ is nilpotent of degree two, i.e., adξ = 0 for i ≥ 2, it is easy to show the result. Case 2) To prove the result for ω 6= 0, the following lemma is required. Lemma 2.4.6. ∀ω, w ∈ R3 and ω ˜ ∈ so(3), (i) ω ˜ 2 = ωω T − kωk2 id3 [57], (ii) ω ˜ 3 = −kωk2 ω ˜ [57], (iii) ω ˜ w = −wω ˜ = ω × w, (iv) ω ˜f w = [˜ ω , w]. ˜ The proof for the above lemma is a straight forward computation. Now, consider the Adjoint operator corresponding to the element H, AdH , for " H=

id3

−˜ ωw kωk2

01×3

1

# ∈ SE(3),

and its action on ξ ∈ se(3). Based on Lemma 2.4.6, "

# #" # " ω ω w − ω ˜ w ˜ + w ˜ ω ˜ w kωk2 kωk2 ξ 0 : = AdH ξ = = ω 03×3 id3 ω " # " T # " # (ω w)ω w w + ωω T − kωk2 id3 kωk hω 2 2 = = kωk =: . ω ω ω id3

ω ,w] ˜ − [˜ kωk2

Hence, exp(adξ0 ) =

∞ X adiξ0 i=0

i!

" # " P∞ (h˜ω)i # ∞ X exp(˜ ω) ˜ i i(h˜ ω )i 1 ω i=1 (i−1)! = = i i! 03×3 ω ˜ 03×3 exp(˜ ω) i=0 " # " # ∂ |µ=1 exp(h˜ ω µ) exp(˜ ω ) ∂µ exp(˜ ω ) h˜ ω exp(h˜ ω) = . = 03×3 exp(˜ ω) 03×3 exp(˜ ω)


32

According to the definition of the adjoint operator, one has the following: exp(adξ ) = exp(adAdH −1 (ξ0 ) ) = Adexp(AdH −1 (ξ0 )) = Ad(H −1 exp(ξ0 )H) = AdH −1 exp(adξ0 )AdH . A straightforward calculation proves the first part of the proposition. For the second part of the proposition, Z 0

=

1

exp(x adξ ) dx Z 1" exp(x˜ ω)

1 x2 kωk2

# [[x˜ ω , xw], ˜ exp(x˜ ω )] + xh˜ ω exp(xh˜ ω)

03×3

0

exp(x˜ ω)

dx.

Since the matrix commutator is a bilinear operator, and the integral operator and partial derivative can commute, Z 0

=

1

exp(x adξ ) dx "R 1 exp(x˜ ω ) dx 0

1 kωk2

h

[˜ ω , w], ˜

03×3 Using (2.4.15) and substituting h =

2.5

ωT w , kωk2

R1 0

i R # 1 exp(x˜ ω ) dx + 0 xh˜ ω exp(xh˜ ω ) dx . R1 exp(x˜ ω ) dx 0

one can show the second part of the proposition.

Case Study

In this section, the kinematic analysis of a mobile manipulator moving on a spacecraft is performed to elaborate the computational aspects of the proposed formulation for Forward and Differential Kinematics of open-chain multi-body systems. The spacecraft can be considered as a six-d.o.f. moving base for the mobile manipulator that is shown in Figure 2.1. The multi-body system M S(6) = {(Bi , Ai )|i = 0, ..., 6, Bi ⊂ Ai } consists of two branches and six joints. The first branch consists of B0 to B5 . The second branch contains B6 and joint six is its last joint. Joint one is a free joint, the second joint is a nonholonomic three-d.o.f. planar joint, the next joint is a three-d.o.f. spherical joint and the rest of the joints are one-d.o.f. revolute joints. The coordinate frames assigned to A0 , ..., A6 at the initial configuration are shown in Figure 2.2. In the sequel, the joint parameters are specified, and Forward and Differential Kinematics maps of M S(6) are determined. Note that in the following, a basis for Vj at the initial configuration is denoted


33

Figure 2.1: A mobile manipulator on a six d.o.f. moving base ˆ j , Yˆj , Zˆj }, and the linear operator 0 τ j−1 in the chosen coordinates is represented by by {X j the matrix 0 Tjj−1 .

2.5.1

Forward Kinematics

The first joint is a six-d.o.f. holonomic joint between B0 and B1 . The classic joint parameters are q1 = [x1 , y1 , z1 , θ1,x , θ1,y , θ1,z ]T , where [x1 , y1 , z1 ]T is the position of H10 (t)(O1 ) 0 (O1 ) and expressed in V0 , and [θ1,x , θ1,y , θ1,z ]T is the rotation angles with respect to H1,0 of V1 with respect to the axes of V1 at the initial configuration. Therefore, the local coordinate chart ϕ1 for Q1 is "

# ˆ 1 )R(θ1,y , Yˆ1 )R(θ1,z , Zˆ1 ) [x1 , y1 , z1 ]T R(θ1,x , X ϕ1 (q1 ) = , 01×3 1 ˆ ) is the 3 × 3 rotation matrix corresponding to θ radian rotation about where R(θ, W ˆ . For this coordinate chart, any element of Lie(P0 ) corresponding to the the vector W relative pose of B1 with respect to B0 is parametrized with the screw joint parameters s1 = [s1,1 , ..., s1,6 ]T , such that 0

0 T10 s1 = AdH1,0 (Tid6 ι1 ) (T0 ϕ1 ) s1 .


34

Figure 2.2: Coordinate frames assigned to A0 , ..., A6 at the initial configuration With some basic calculations one can show that ∂ϕ1 ∂ϕ1 ∂ϕ1 ∂ϕ1 ∂ϕ1 ∂ϕ1 |0 = E1 , |0 = E2 , |0 = E3 , |0 = E4 , |0 = E5 , and |0 = E6 , ∂x1 ∂y1 ∂z1 ∂θ1,x ∂θ1,y ∂θ1,z which coincides with the basis selected for se(3) ∼ = Lie(P1 ). For this joint since Q1 = P1 , Tid6 ι1 and T0 ϕ1 are equal to the identity matrix. In the basis {E1 , ..., E6 }, "

j ∀Hi,0

j Ri,0 pji,0 = 01×3 1

#

the Adjoint operator can be represented by the matrix [81] "

AdH j

i,0

# j j Ri,0 p˜ji,0 Ri,0 = . j 03×3 Ri,0

0 s1 . Therefore, 0 T10 s1 = AdH1,0

Joint number two is a three-d.o.f. nonholonomic joint between B1 and B2 . The classic joint parameters can be chosen as q2 = [x2 , y2 , θ2,z ]T , where [x2 , y2 , 0]T is the position of 1 H21 (t)(O2 ) with respect to H2,0 (O2 ) and expressed in V2 , and θ2,z is the rotation angle of V2 about Zˆ2 . Hence, the local coordinate chart ϕ2 for Q2 is "

# R(θ2,z ) R(θ2,z )[x2 , y2 ]T ϕ2 (q2 ) = , 01×2 1 where R(θ2,z ) is the 2 × 2 rotation matrix for θ2,z . For this coordinate chart, any element


35

of Lie(P0 ) corresponding to the relative pose of B2 with respect to B1 is parametrized by the screw joint parameters s2 = [s2,1 , s2,2 , s2,3 ]T , such that 0

where Tid3 ι2

1 (Tid3 ι2 ) (T0 ϕ2 ) s2 , T21 s2 = AdH2,0

∂ϕ2 ∂ϕ2 ∂ϕ2 |0 = E1 , Tid3 ι2 |0 = E2 , and Tid3 ι2 |0 = E6 . ∂x2 ∂y2 ∂θ2,z

Thus,  1 0 ···  0 1 0 0 T2 s2 = AdH2,0 1 ··· 0 0 ···

T 0  0 s2 . 1

The third joint is a three-d.o.f. holonomic joint between B2 and B3 . The classic joint parameters are q3 = [θ3,x , θ3,y , θ3,z ]T , and the local coordinate chart for Q3 is ϕ3 (q3 ) = ˆ 3 )R(θ3,y , Yˆ3 )R(θ3,z , Zˆ3 ). The elements of Lie(P0 ) corresponding to the relative R(θ3,x , X poses of B3 with respect to B2 are parametrized by the screw joint parameters s3 = [s3,1 , s3,2 , s3,3 ]T , such that " # 03×3 0 2 0 T3 s3 = AdH3,0 s3 . id3 Joint 4 is a one-d.o.f. revolute joint, its classic joint parameter is q4 = θ4,z , and the local coordinate chart for Q4 is ϕ4 (q4 ) = R(θ4,z ).The line in Lie(P0 ) corresponding to the relative pose of B4 with respect to B5 is parametrized by the screw joint parameter s4 , such that 0 3 T 0 [0, ..., 1] s4 . T4 s4 = AdH4,0 By a simple calculation "

# 0 0 ˆ p × Z 4 0 3 T4 = 4,00 , ˆ Z4 where 0 Zˆ4 is the joint screw axis expressed in V0 . Hence, 0 T43 s4 coincides with the argument of the exponential map in the existing product of exponentials formula for a revolute joint [11, 57, 71]. Similarly, for the fifth and sixth joints 0

T 0 [0, ..., 1] s5 , T54 s5 = AdH5,0

0

T 0 [0, ..., 1] s6 , T64 s6 = AdH6,0

respectively. Therefore, based on (2.2.6), the Forward Kinematics map corresponding to M S(6) is

36


"

# 0 exp(0 T10 s1 )... exp(0 T54 s5 )H5,0 F K(s) = , 0 exp(0 T10 s1 )... exp(0 T64 s6 )H6,0 where exp is the matrix exponential for SE(3) that can be evaluated by (2.4.14) and s = [sT1 , ..., v6T ]T . According to the calculation performed in the case of joint four, for a serial-link multi-body system with revolute and/or prismatic joints, where the multi-body system consists of one branch, the above formulation for F K reduces to the existing product of exponentials formula.

2.5.2

Differential Kinematics

Based on Proposition 2.4.1 and 2.4.5, the Jacobian maps of B5 and B6 with respect to B0 and expressed in V0 , i.e., 0 J50 (s) and 0 J60 (s), can be determined as 6 × 14 matrices. The nonholonomic constraints at the second joint can be expressed in terms of the classical joint parameters as C2 (q2 )q˙2 = [0, 1, 0]q˙2 = 0, which indicates that the mobile base cannot drift side way. The annihilator of C2 can be selected to be " #T 1 0 0 C¯2 (q2 ) = , 0 0 1 and therefore using (2.1.3b) and (2.1.5)  Σ2 (s2 ) =

sin(s2,3 )  (cos(ss2,3 2,3 )−1)   s2,3

sin(s2,3 )−1) 2,3 )/s2,3 ) + s2,1 (cos(s2,3 )+sin(s s2,2 (cos(s2,3 )+ss2,3 2 s2,3



2,3  2,3 sin(s2,3 )) 2,3 )/s2,3 )  s2,1 (1−cos(s2,3 )−s + s2,2 (cos(s2,3 )−sin(s . s2,3 s2 2,3

0

1

Note that when s2,3 = 0, #T 1 0 0 Σ2 (s2 ) = . s2,2 /2 −s2,1 /2 1 "

Finally, according to (2.3.13) the modified Jacobian of the multi-body system M S(6) becomes " # 0 ¯0 J (s) 0 6×13 5 ¯ = J(s) , 0 ¯0 06×13 J6 (s)

Chapter 2. A Generalized Exponential Formula for Kinematics which can be calculated as a 12 × 26 matrix using Proposition 2.4.1 and 2.4.5.

37

Chapter 3 Symplectic Reduction of Holonomic Open-chain Multi-body Systems with Displacement Subgroups This Chapter presents a symplectic geometric approach to the reduction of Hamilton’s equation for holonomic open-chain multi-body systems with multi-degree-of-freedom displacement subgroups. First in Section 3.1, we revisit Hamilton’s principle for Lagrangian systems, and we use the Hamilton-Pontryagin principle to study the geometry of Hamiltonian systems. In Section 3.2 we use the symplectic reduction theorem to express Hamilton’s equation in the symplectic reduced manifold, for holonomic Hamiltonian mechanical systems. Then by identifying the symplectic reduced manifold with a cotangent bundle, we express the reduced Hamilton’s equation in that cotangent bundle. Consequently, in Section 3.3 we apply this procedure to open-chain multi-body systems with multi-degree-of-freedom displacement subgroups, for which the symmetry group is identified with the configuration manifold corresponding to the first joint. Then we derive their reduced dynamical equations in local coordinates, in Theorem 3.3.6.

3.1

Hamilton-Pontryagin Principle and Hamilton’s Equation

In this section we first explore the geometry of Hamilton’s principle for Lagrangian systems. Then we show how this principle leads to the Hamilton-Pontryagin principle on the Pontryagin bundle T Q ⊕ T ∗ Q. The Lagrangian systems that satisfy the Hamilton38

Chapter 3. Reduction of Holonomic Multi-body Systems

39

Pontryagin principle are called implicit Lagrangian systems, and the resulting equation of motion is called the implicit Euler-Lagrange equation [93, 94]. In addition, we show that for hyper-regular Lagrangian systems the implicit Euler-Lagrange equation is equivalent to Hamilton’s equation. In the next chapter, we use an analogous method to derive the equations of motion for nonholonomic systems, using Lagrange-d’Alembert-Pontryagin principle. Let T Q be the tangent bundle of the configuration manifold Q, and let L : T Q → R be a smooth function; we call L the Lagrangian. Let t 7→ vq(t) (t) ∈ Tq(t) Q be a smooth curve in T Q. This curve corresponds to a tangent lift of a curve in Q if vq(t) (t) = dq (t) =: dt q˙q(t) (t), ∀t. For a time interval [ts , tf ], let (t, ) 7→ q(t, ) ∈ Q, for ∈ R, be a variation of a smooth curve t 7→ q(t) ∈ Q with fixed end points qs , qf ∈ Q, i.e., q(ts , ) = qs and q(tf , ) = qf , along with the condition that q(t, 0) = q(t). Hamilton’s principle states that a Lagrangian system evolves on a curve t 7→ vq(t) (t) that is the tangent lift of the curve t 7→ q(t) and that makes the action functional stationary for any arbitrary variation of the curve t 7→ q(t) with fixed end points. That is, Z tf ∂ L(vq(t,) (t, ))dt = 0 ∂ =0 ts

(3.1.1)

for any variation as described above. This holds if and only if the curve t 7→ vq(t) (t) satisfies the Euler-Lagrange equation, which is written in coordinates as d ∂L ∂L ( (q˙q(t) (t))) − (q˙q(t) (t)) = 0. dt ∂ q˙ ∂q

(3.1.2)

We present the Euler-Lagrange equation (and upcoming dynamical equations) in the form of paired elements of cotangent and tangent bundles, for the sake of generalizing them to nonholonomic systems in the next chapter:

d ∂L ∂L ( (q˙q(t) (t))) − (q˙q(t) (t)) dq, wq(t) = 0, dt ∂ q˙ ∂q ∀t ∈ (ts , tf ) and ∀wq(t) ∈ Tq(t) Q,

As was mentioned above, in Hamilton’s principle the variational problem deals only with tangent lifted curves in T Q. One may implicitly impose this kinematic constraint in the variational problem, and form a variational problem in the Pontryagin bundle PQ := T Q ⊕ T ∗ Q. The Pontryagin bundle is a vector bundle over the configuration manifold Q with the canonical projection ΠQ : PQ → Q such that ∀vq ∈ Tq Q and ∀pq ∈ Tq∗ Q we write (vq , pq ) ∈ Pq Q and we have ΠQ (vq , pq ) = q. The resulting equivalent

40


principle is called Hamilton-Pontryagin principle. This principle states that an implicit Lagrangian system evolves on a curve t 7→ (vq(t) (t), pq(t) (t)) ∈ Pq(t) Q that makes the following functional stationary for any arbitrary variation of the curve in PQ with fixed end points in Q, i.e., q(ts , ) = qs and q(tf , ) = qf : Z tf

∂ L(vq(t,) (t, )) + pq(t,) (t, ), q˙q(t,) (t, ) − vq(t,) (t, ) dt = 0. ∂ =0 ts

(3.1.3)

For the time interval [ts , tf ], we denote any variation of the curve t 7→ (vq(t) (t), pq(t) (t)) ∈ Pq(t) Q by a function γ : [ts , tf ] × R → PQ: γ(t, ) = (vq(t,) (t, ), pq(t,) (t, )). For γ to be a variation with fixed end points in Q we assume that for all ∈ R, ΠQ (γ(ts , )) = qs and ΠQ (γ(tf , )) = qf . We denote the induced map by ΠQ on the tangent bundles by T ΠQ : T PQ → T Q and the projection map that projects the Pontryagin bundle onto T ∗ Q by ΠT ∗ Q : PQ → T ∗ Q. Let Θcan and Ωcan := −dΘcan be the ∈ Tγ(t,) (PQ) and tautological 1-form and the canonical 2-form on T ∗ Q, and let γ˙ := ∂γ ∂t ∂γ δγ := ∂ =0 ∈ Tγ(t,0) (PQ). We can write the left hand side of (3.1.3) as Z tf ∂ (L(vq ) + hpq , T ΠQ (γ) ˙ − vq i) ◦ γ dt ∂ =0 ts Z tf ∂ ∗ h(dL − dhpq , vq i) ◦ γ(t, 0), δγ(t)i + h(T ΠT ∗ Q (Θcan )) ◦ γ, γi = ˙ dt ∂ =0 ts Z tf ∂ h(T ∗ ΠT ∗ Q Θcan ) ◦ γ(t, 0), δγ(t)i dt = ∂t t Z stf

+ (dL − dhpq , vq i) ◦ γ(t, 0) + ιγ(t,0) ((T ∗ ΠT ∗ Q Ωcan ) ◦ γ(t, 0)) , δγ(t) dt ˙ ts

(by Lemma 3.1.1 bellow) t γ(t, 0), δγ(t)i]tfs

= [h(T ∗ ΠT ∗ Q Θcan ) ◦ Z tf

+ (dL − dhpq , vq i) ◦ γ(t, 0) + ιγ(t,0) ((T ∗ ΠT ∗ Q Ωcan ) ◦ γ(t, 0)) , δγ(t) dt ˙ t Z stf

= (dL − dhpq , vq i) ◦ γ(t, 0) + ιγ(t,0) ((T ∗ ΠT ∗ Q Ωcan ) ◦ γ(t, 0)) , δγ(t) dt ˙ ts

(since the variation in Q at the end points is zero) In the above calculation, the first equality follows from the definition of the tautological

41

Chapter 3. Reduction of Holonomic Multi-body Systems 1-form Θcan ∈ Ω1 (T ∗ Q) and from the following diagram: PQ

ΠT ∗ Q

/ T ∗Q

πQ

ΠQ

Q

Q

where πQ : T ∗ Q → Q is the canonical projection of the cotangent bundle. Since δγ(t) ∈ Tγ(t,0) PQ is arbitrary, we can write (3.1.3) as, ∀Wγ(t,0) ∈ Tγ(t,0) (PQ),

d(L − hpq , vq i) ◦ γ(t, 0) + ιγ(t,0) ((T ∗ ΠT ∗ Q Ωcan ) ◦ γ(t, 0)) , Wγ(t,0) = 0, ˙ or equivalently, d(L − hpq , vq i) ◦ γ(t, 0) + ιγ(t,0) ((T ∗ ΠT ∗ Q Ωcan ) ◦ γ(t, 0)) = 0. ˙

(3.1.4)

The 2-form T ∗ ΠT ∗ Q Ωcan is a closed degenerate 2-form on PQ. It is degenerate only in the direction of vq , that is, ∀(vq , pq ) ∈ Pq Q and ∀W(vq ,pq ) ∈ T(vq ,pq ) (PQ), we have ιW (T ∗ ΠT ∗ Q Ωcan ) = 0 if and only if T(vq ,pq ) ΠT ∗ Q (W(vq ,pq ) ) = 0.

Lemma 3.1.1. For any variation γ as described above and for all α ∈ Ω1 (PQ), we have the equality ∂ ∂ hα ◦ γ(t, ), γ(t, ˙ )i = hα ◦ γ(t, 0), δγ(t)i + hιγ(t,0) (−dα ◦ γ(t, 0)), δγ(t)i. ˙ ∂ =0 ∂t Proof. The proof is based on a straightforward calculation. ∂ ∂ ∂ ∂ ∗ hα ◦ γ(t, ), γ(t, ˙ )i = α ◦ γ, T γ( ) = L∂/∂ T γ(α), ∂ ∂ ∂t ∂t ∂ ∂ ∗ ∗ = L∂/∂ (T γ(α)), + T γ(α), L∂/∂ ( ) ∂t ∂t ∂ (t and are two independent variables) = L∂/∂ (T ∗ γ(α)), ∂t ∂ ∂ ∗ ∗ = ι∂/∂ (T γ(dα)) + d T γ(α), , (by Cartan’s formula) ∂ ∂t ∂ ∂ ∂ ∗ ∗ = −ι∂/∂t (T γ(dα)), + T γ(α), ∂ ∂t ∂


42

∂ ∂ ∂ ∗ ∗ = T γ (ιγ˙ (−dα ◦ γ)) , + T γ(α), ∂ ∂t ∂ ∂ ∂ ∂ α ◦ γ, T γ( ) = ιγ˙ (−dα ◦ γ), T γ( ) + ∂ ∂t ∂ Based on the definition of δγ(t), at = 0 we have the desired equality. We define the function E : PQ → R by E(vq , pq ) := hpq , vq i − L(vq ); it is called the energy function. We call the triple (PQ, T ∗ ΠT ∗ Q Ωcan ∈ Ω2 (PQ), E) an implicit Lagrangian system. In a coordinate chart, we have (γ(t, 0), γ(t, ˙ 0)) = (q(t), v(t), p(t), q(t), ˙ v(t), ˙ p(t)) ˙ and T ΠT ∗ Q Ωcan = −dp ∧ dq. We can write (3.1.4) as ∗

∂L ∂L (q, v)dq + (q, v)dv + qdp ˙ − pdq ˙ − vdp − pdv = 0, ∂q ∂v or equivalently, p˙ =

∂L ∂L (q, v), p = (q, v), q˙ = v. ∂q ∂v

(3.1.5)

This gives a bijection between the tangent lift of the curves in Q that satisfy the EulerLagrange equation (3.1.2) and the curves in PQ that satisfy the implicit Euler-Lagrange equation (3.1.5). The fibre derivative of the Lagrangian L induces a fibre preserving map FL : T Q → T Q, called Legendre transformation, ∗

∂L d hFLq (vq ), wq i := L(vq + wq ) = (vq ), wq . ∀wq ∈ Tq Q d =0 ∂v

(3.1.6)

For all vq ∈ Tq Q we can define the embedding grph : T Q → PQ by grphq (vq ) := (vq , FLq (vq )) ∈ Pq Q. By (3.1.5), we have that the solution curve of an implicit Lagrangian system is always in this submanifold. The Lagrangian L is called hyper-regular if FL is a diffeomorphism. Under the g : T ∗ Q → PQ assumption that L is hyper-regular, we also have the embedding grph that maps any element pq ∈ T ∗ Q to (FL−1 (pq ), pq ) ∈ Pq Q. In this case, we have g ∗ Q) = grph(T Q); hence in (3.1.4) the curve t 7→ γ(t, 0) is in the image of grph, g grph(T ∗ and it has a unique pre-image t 7→ λ(t) = pq(t) (t) ∈ Tq(t) Q, such that λ(t) = ΠT ∗ Q (γ(t, 0)), for all t. Assuming that L is hyper-regular, we can now rewrite (3.1.4) in T ∗ Q as g ιγ(t,0) T ∗ grph ((T ∗ ΠT ∗ Q Ωcan ) ◦ γ(t, 0)) − dE ◦ γ(t, 0) = 0, ˙

Chapter 3. Reduction of Holonomic Multi-body Systems ⇐⇒

43

g (Ωcan ◦ λ(t)) − dE ◦ grph(λ(t)) = 0, ιλ(t) ˙

since we have the following diagram: T ∗O Q ΠT ∗ Q

T ∗Q ˙ Here, λ(t) := bundle by

dλ (t). dt

g grph

/

PQ

We define the Hamiltonian function H : T ∗ Q → R on the cotangent

g q ) = hpq , FL−1 (pq )i − L(FL−1 (pq )). H(pq ) := E ◦ grph(p

(3.1.7)

For a hyper-regular Lagrangian, the solution curve of an implicit Lagrangian system, i.e., t 7→ γ(t, 0), satisfies (3.1.4) if and only if the curve t 7→ λ(t) satisfies Hamilton’s equation, defined by (Ωcan ◦ λ(t)) = dH ◦ λ(t). (3.1.8) ιλ(t) ˙ Let πQ : T ∗ Q → Q be the canonical projection map for the cotangent bundle, and (with some abuse of notation) denote a variation of the curve t 7→ λ(t) ∈ T ∗ Q by the function (t, ) 7→ λ(t, ) ∈ T ∗ Q. Under the assumptions considered to derive (3.1.4), Hamilton’s equation in T ∗ Q can also be derived from the Hamilton-Pontryagin principle, g That is, we once we restrict the variational problem to the image of the embedding grph. g ∗ Q) such that λ(t, ) = ΠT ∗ Q (γ(t, )): only consider the variations (t, ) 7→ γ(t, ) ∈ grph(T Z tf ∂ (L(vq ) + hpq , T ΠQ (γ) ˙ − vq i) ◦ γ dt = 0 ∂ =0 ts Z tf ∂ ˙ ⇐⇒ hλ(t, ), T π ( λ(t, ))i − H ◦ λ(t, )) dt = 0 Q ∂ =0 ts D E ⇐⇒ ιλ(t,0) (Ωcan ◦ λ(t, 0)) − dH ◦ λ(t, 0), δλ(t) = 0, ˙

∀δλ(t) ∈ Tλ(t,0) (T ∗ Q)

⇐⇒ ιλ(t) (Ωcan ◦ λ(t)) = dH ◦ λ(t). ˙ ˙ ) := ∂ λ(t, ) and δλ(t) := ∂ λ(t, ). Note that the details are omitted Here, λ(t, ∂t ∂ =0 here, since the derivation presented above is similar to the derivation of (3.1.4). ˙ Using any coordinate chart for T ∗ Q, we have (λ(t), λ(t)) = (q(t), p(t), q(t), ˙ p(t)), ˙ and


44

we can write (3.1.8) as q˙ =

∂H , ∂p

p˙ = −

∂H . ∂q

Now, let X be a vector field on the cotangent bundle T ∗ Q. It induces a vector field g ∗ Q) whose smooth extension to PQ is denoted by X . Note that X is not a on grph(T g pq ) = X g . unique vector field on PQ. In other words, ∀pq ∈ T ∗ Q we have Tpq grph(X grph(pq ) If the curve t 7→ γ(t) ∈ PQ is an integral curve of the vector field X and it satisfies (3.1.4), then ∀Wγ(t) ∈ Tγ(t) (PQ) we have

(−dE + ιX (T ∗ ΠT ∗ Q Ωcan )) ◦ γ(t), Wγ(t) = 0, ⇐⇒

(−dE + ιX (T ∗ ΠT ∗ Q Ωcan )) ◦ γ(t) = 0.

(3.1.9)

It is easy to show that the pull back of the 1-form ιX (T ∗ ΠT ∗ Q Ωcan ) − dE ∈ Ω1 (PQ) g by the embedding grph g is equal to (restricted to the image of grph) ιX Ωcan − dH ∈ Ω1 (T ∗ Q). g ∗ Q) such that Xγ(t) = dγ (t) satisfies (3.1.9) if Consequently, any curve t 7→ γ(t) ∈ grph(T dt ∗ and only if the curve t 7→ λ(t) = ΠT ∗ Q (γ(t)) ∈ T Q, which is the integral curve of the vector field X, satisfies

(ιX Ωcan − dH) ◦ λ(t), Yλ(t) = 0,

⇐⇒

(ιX Ωcan − dH) ◦ λ(t) = 0.

∀Yλ(t) ∈ Tλ(t) (T ∗ Q) (3.1.10)

If (3.1.10) holds for any integral curve of X ∈ X(T ∗ Q), we can define Hamilton’s equation as ιX Ωcan = dH. (3.1.11) In general, one can have a system satisfying Hamilton’s equation (3.1.11) on T ∗ Q for a Hamiltonian H ∈ C ∞ (T ∗ Q) that does not necessarily come from a Lagrangian. Such system is called a Hamiltonian system. The unique vector field that satisfies Hamilton’s equation for a Hamiltonian H is called the Hamiltonian vector field for the Hamiltonian H. We define a Hamiltonian system to be a triple (T ∗ Q, Ωcan , H), as above.

45


3.2

Hamiltonian Mechanical Systems with Symmetry

For a mechanical system, the Lagrangian is defined by L := 12 Kq (vq , vq ) − V (q), where Kq : Tq Q × Tq Q → R is a Riemannian metric, called the kinetic energy metric, and where V : Q → R is a smooth function, called the potential energy function. This Lagrangian is hyper-regular, and the corresponding Legendre transformation is equal to the fibre-wise linear isomorphism that is induced by the metric K: hFLq (vq ), wq i := Kq (vq , wq ).

∀vq , wq ∈ Tq Q

(3.2.12)

Likewise, the Hamiltonian of the system is 1 −1 H(pq ) = Kq (FL−1 q (pq ), FLq (pq )) + V (q), 2

(3.2.13)

which is the total energy of the mechanical system. A Hamiltonian mechanical system is such a quadruple (T ∗ Q, Ωcan , H, K). Let G be a Lie group with the Lie algebra Lie(G). Consider an action of G on Q, and denote the action by Φg : Q → Q, ∀g ∈ G. This action induces an action of G on T ∗ Q by the cotangent lift of Φg , which is denoted by T ∗ Φg : T ∗ Q → T ∗ Q. Lemma 3.2.1. For every g ∈ G, the map T ∗ Φg is a symplectomorphism, i.e., it preserves Ωcan [50]. Proof. The proof relies on the fact that T ∗ Φg preserves the tautological 1-form Θcan :

(T ∗ T ∗ Φg (Θcan ))pq , Ypq = T ∗ Φg (pq ), (T πQ )(T T ∗ Φg )(Ypq ) =

∗

T Φg (pq ), T (πQ ◦ T ∗ Φg )(Ypq ) = T ∗ Φg (pq ), T (Φg−1 ◦ πQ )(Ypq ) =

pq , T πQ (Ypq ) = (Θcan )pq , Ypq .

The third equality holds, since the following diagram commutes: T ∗Q

T ∗ Φg

/

T ∗Q πQ

πQ

Q

Φg−1

/Q

∀Ypq ∈ Tpq (T ∗ Q)

46

Chapter 3. Reduction of Holonomic Multi-body Systems Finally, we have T ∗ T ∗ Φg (Ωcan ) = T ∗ T ∗ Φg (−dΘcan ) = −d (T ∗ T ∗ Φg (Θcan )) = −dΘcan = Ωcan .

Consider the infinitesimal action of Lie(G) on Q. For any ξ ∈ Lie(G), this action induces a vector field ξQ ∈ X(Q) such that ∀q ∈ Q, ∂ ξQ (q) = Φ (q) . exp(ξ) ∂ =0

(3.2.14)

Denote the fibre-wise linear map corresponding to the infinitesimal action of Lie(G) by φq : Lie(G) → Tq Q, where φq (ξ) = ξQ (q). Likewise, we define ξT ∗ Q ∈ X(T ∗ Q) such that ∀pq ∈ Tq∗ Q, ∂ ∗ ξT ∗ Q (pq ) = T Φexp(−ξ) (pq ) . (3.2.15) ∂ =0 Φexp(ξ) (q) Now, consider the fibre-wise linear map M : T ∗ Q → Lie∗ (G), defined by hMq (pq ), ξi := hφ∗q (pq ), ξi = hpq , ξQ (q)i.

(3.2.16)

Lemma 3.2.2. The map M is an Ad∗ -equivariant momentum map corresponding to the cotangent lifted action T ∗ Φg . Proof. To prove that M is a momentum map, it suffices to show that M satisfies the momentum equation (1.2.1) for the G-action on T ∗ Q, ιξT ∗ Q Ωcan = dhM, ξi. Therefore, we have dhM(pq ), ξi = dhpq , ξQ i = d ιξT ∗ Q Θcan = LξT ∗ Q Θcan − ιξT ∗ Q dΘcan = ιξT ∗ Q Ωcan . The forth equality is true, since the cotangent lifted action preserves the tautological 1-form. To prove that M is Ad∗ -equivariant, we have to show M(T ∗ Φg (pq )) = Ad∗g M(pq ).


47

Using the definition of action and the map M, ∀ξ ∈ Lie(G) one has ∂ Φg exp(ξ)g−1 (q) i = hM(T Φg (pq )), ξi = hT Φg (pq ), ξQ (Φg−1 (q))i = hpq , ∂ =0 ∗

∗

hpq , (Adg ξ)Q (q))i = hAd∗g M(pq ), ξi.

Proposition 3.2.3 (Noether’s Theorem). Let H : T ∗ Q → R be the Hamiltonian of a mechanical system. If H is invariant under the cotangent lifted group action, the momentum map M is constant along the flow of the Hamiltonian vector field X for the Hamiltonian H. That is, ∀ξ ∈ Lie(G) we have LX (hM, ξi) = 0. Proof.

LX (hM, ξi) = hdhM, ξi, Xi = ιξT ∗ Q Ωcan , X = −Ωcan (X, ξT ∗ Q ) = − hιX Ωcan , ξT ∗ Q i = − hdH, ξT ∗ Q i = −LξT ∗ Q H = 0. The forth equality is true, since X is a Hamiltonian vector field for the function H, and the last equality is the consequence of the hypothesis that H is G-invariant. We define a Hamiltonian system with symmetry to be a quadruple (T ∗ Q, Ωcan , H, G), as above, where the Hamiltonian H is invariant under the cotangent lifted action of G. A Hamiltonian mechanical system with symmetry is defined by a quintuple (T ∗ Q, Ωcan , H, K, G), where K is the kinetic energy metric on Q, and in addition to H, K is invariant under the G-action. Theorem 3.2.4 (Symplectic Reduction Theorem [53]). Assume that the action of G on Q is free and proper, and let µ ∈ Lie∗ (G) be a regular value of its momentum map M. Also, let Gµ = {g ∈ G| Ad∗g µ = µ} be the coadjoint isotropy group for µ ∈ Lie∗ (G). Then the quotient manifold (T ∗ Q)µ := M−1 (µ)/Gµ is a symplectic manifold, called the symplectic reduced space, with the unique symplectic form Ωµ that is identified by the equality T ∗ πµ (Ωµ ) = T ∗ iµ (Ωcan ). Here, the maps πµ : M−1 (µ) → M−1 (µ)/Gµ and iµ : M−1 (µ) ,→ T ∗ Q are the projection map and inclusion map, respectively. This theorem was first stated and proved in a paper by Marsden and Weinstein in 1974 [53], and since then this result has been extended to non-free actions [27] and almost symplectic manifolds [39]. An almost symplectic manifold is a manifold equipped with a nondegenerate 2-form, which may not be closed. Based on the symplectic reduction


48

theorem, in the presence of a group action that preserves the symplectic structure and an Ad∗ -equivariant momentum map (corresponding to the symmetry group) we say that the phase space of a Hamiltonian system along with its symplectic 2-form can be reduced to the symplectic reduced space ((T ∗ Q)µ , Ωµ ). In order to have a well-defined projection of Hamilton’s equation onto the symplectic reduced space, the Hamiltonian of the system should be invariant under the group action, as well. Under these hypotheses, Hamilton’s equation can be written on (T ∗ Q)µ as ιXµ Ωµ = dHµ ,

(3.2.17)

where Hµ is defined by H ◦ iµ = Hµ ◦ πµ and Xµ ◦ πµ = T πµ (X ◦ iµ ). We say that the Hamiltonian system with symmetry (T ∗ Q, Ωcan , H, G) has been reduced to the Hamiltonian system ((T ∗ Q)µ , Ωµ , Hµ ). In the theory of cotangent bundle reduction, there exist two equivalent ways to describe the symplectic reduced space in terms cotangent bundles and coadjoint orbits [49]: i) Embedding version: in which the symplectic reduced space is identified with a e := Q/Gµ , called µ-shape space of a vector sub-bundle of the cotangent bundle of Q Hamiltonian system. ii) Bundle version: in which the symplectic reduced space is identified by a (locally trivial) fibre bundle over T ∗ Q, where Q := Q/G, and where the fibre is the coadjoint orbit through µ. The manifold Q is called the shape space of the Hamiltonian system. In this section, the embedding version of the cotangent bundle reduction is used to write Hamilton’s equation (3.2.17) in a sub-bundle of the cotangent bundle of the µ-shape e Prior to reporting the final result, we introduce a number space, i.e., a sub-bundle of T ∗ Q. of necessary objects. Note that since we consider multi-body systems for the application, from now on we only focus on Hamiltonian mechanical systems with symmetry, unless otherwise stated. Consider a Hamiltonian mechanical system with symmetry (T ∗ Q, Ωcan , K, H, G), and ∀g ∈ G denote the action map by Φg : Q → Q. Assume that the action is free and proper. The quotient manifold Q := Q/G gives rise to the principal bundle π : Q → Q with the base space Q, and the fibres of the bundle are isomorphic to the group G. A principal connection on the principle bundle π : Q → Q is a fibre-wise linear map A : T Q → Lie(G), such that A(ξQ (q)) = ξ (∀ξ ∈ Lie(G) and ∀q ∈ Q), and it is Ad-equivariant, i.e., A(Tq Φg (vq )) = Adg A(vq ) (∀vq ∈ Tq Q). Accordingly, for any base element q ∈ Q the


49

tangent space of Q can be written as the following direct sum Tq Q = ker(Tq π) ⊕ ker(Aq ).

(3.2.18)

Note that V := ker(T π) = { ξQ = φ(ξ)| ξ ∈ Lie(G)} is called the vertical vector subbundle of T Q, and H := ker(A) is called the horizontal vector sub-bundle of T Q. As a result, any vq ∈ Tq Q can be decomposed into the horizontal and vertical components such that vq = hor(vq ) + ver(vq ), where ver(vq ) := φq ◦ Aq (vq ) and hor(vq ) := vq − ver(vq ). For any q ∈ Q and q := π(q) ∈ Q the restriction of the tangent map Tq π : Tq Q → Tq Q to the horizontal subspace of Tq Q, namely Hq , is a linear isomorphism between Hq and Tq Q. Therefore, for any v q ∈ Tq Q it defines a horizontal lift map by hlq (v q ) := ( Tq π|Hq )−1 (v q ).

(3.2.19)

The choice of the principal connection A is arbitrary; however, for a Hamiltonian mechanical system, we can use the Legendre transformation, which is induced by the kinetic energy metric K, to define an appropriate principal connection. For any q ∈ Q consider the linear map Iq : Lie(G) → Lie∗ (G), defined by Iq := φ∗q ◦ FLq ◦ φq ,

(3.2.20)

such that the following diagram commutes: Lie(G)

φq

/ Tq Q

Iq

FLq

Lie∗ (G) o

φ∗q

Tq∗ Q

This map is a linear isomorphism for any q ∈ Q, and it is called the locked inertia tensor. For a Hamiltonian mechanical system with symmetry ∀ξ, η ∈ Lie(G) we have hIq (ξ), ηi = Kq (ξQ (q), ηQ (q)). The principal connection A can now be chosen to be the mechanical connection AM ech , which can be interpreted as the orthogonal projection with respect to the kinetic energy metric K, and defined by the following commuting diagram:

50


Tq Q

/

FLq

Tq∗ Q

ech AM q

Mq

Lie(G) o

I−1 q

Lie∗ (G)

Therefore, ∀q ∈ Q we have ech := I−1 Aq = AM q ◦ Mq ◦ FLq . q

(3.2.21)

For any µ ∈ Lie∗ (G), let the action of G restricted to the subgroup Gµ = {g ∈ G| Ad∗g µ = µ} ⊆ G be denoted by Φµh : Q → Q (∀h ∈ Gµ ). Similarly, for this action we have a prine := Q/Gµ . Using the same procedure detailed above, the locked cipal bundle π e: Q → Q inertia tensor Iµq : Lie(Gµ ) → Lie∗ (Gµ ) and the (mechanical) connection Aµq : Tq Q → Lie(Gµ ) (∀q ∈ Q) for the Gµ -action are defined by Iµq := (φµq )∗ ◦ FLq ◦ φµq ,

(3.2.22)

Aµq := (Iµq )−1 ◦ Mµq ◦ FLq ,

(3.2.23)

and

respectively. Here, the map φµq : Lie(Gµ ) → T Q corresponds to the infinitesimal Gµ action, and Mµ : T ∗ Q → Lie∗ (Gµ ) is the Ad∗ -equivariant momentum map for the cotangent lifted Gµ -action, which are defined based on (3.2.14) and (3.2.16). Let the map iµ : Gµ ,→ G be the canonical inclusion map. Denote the induced map in the Lie algebras by iµ∗ : Lie(Gµ ) ,→ Lie(G) and in the dual of the Lie algebras by (iµ )∗ : Lie∗ (G) → Lie∗ (Gµ ). The following diagrams commute: Lie∗ (G)

Lie(G) O

iµ ∗

?

Lie(Gµ )

b

(iµ )∗

φq φµ q

" / Tq Q

φ∗q

Lie∗ (Gµ ) o

∗ (φµ q)

Tq∗ Q


51

Based on these commuting diagrams, we have the following relations: Iµq = (iµ )∗ ◦ φ∗q ◦ FLq ◦ φq ◦ iµ∗ = (iµ )∗ ◦ Iq ◦ iµ∗ , Mµq = (iµ )∗ ◦ Mq , Aµq = (Iµq )−1 ◦ (iµ )∗ ◦ Mq ◦ FLq = (Iµq )−1 ◦ (iµ )∗ ◦ Iq ◦ Aq . e with the principal connection Aµ , the horizontal For the principal bundle π e: Q → Q and vertical sub-bundles are Hµ := ker(Aµ ) and V µ := ker(e π ) = {ηQ = φµ (η)| η ∈ Lie(Gµ )}, respectively. It is easy to check that V µ ⊆ V and H ⊆ Hµ as vector subbundles. The horizontal lift map corresponding to the connection Aµ can be defined as e q (e hl vqe) := (Tq π e|Hqµ )−1 (e vqe), e where qe := π e(q) and veqe ∈ TqeQ. Now, consider the 1-form αµ := A∗ µ ∈ Ω1 (Q). Lemma 3.2.5. The 1-form αµ takes values in M−1 (µ), and it is invariant under Gµ action. Proof. Using the definition of the momentum map and principal connection, we have ∀ξ ∈ Lie(G) hM(αµ ), ξi = hαµ , ξQ i = hA∗q µ, φq (ξ)i = hµ, (Aq ◦ φq )(ξ)i = hµ, ξi. As a result, αµ ∈ M−1 (µ). Finally, consider the action of an arbitrary element h ∈ Gµ , and denote the action simply by h · q := Φh (q) and h · vq := T Φh (vq ). Based on the Ad∗ -equivariance of A and the definition of Gµ , one can show that αµ is Gµ invariant. For all vq ∈ Tq Q, hαµ (h · q), h · vq i = hA∗h·q µ, h · vq i = hµ, Ah·q (h · vq )i = hµ, Adh−1 Aq (vq )i = hAd∗h−1 µ, Aq (vq )i = hµ, Aq (vq )i.

According to the Cartan Structure Equation derived in [49, Theorem 2.1.9] for principal connections, ∀Z, Y ∈ X(Q) the exterior derivative of αµ evaluated on Y and Z is equal to dαµ (Z, Y ) = hµ, dA(Z, Y )i = hµ, B(Z, Y ) + [A(Z), A(Y )]i,

(3.2.24)


52

where Bq (Zq , Yq ) := (dA)q (horq (Zq ), horq (Yq )) = −Aq ([hor(Z), hor(Y )]q ) is the curvature of the connection A, and [·, ·] in (3.2.24) corresponds to the Lie bracket in Lie(G). Lemma 3.2.6. For all η ∈ Lie(Gµ ), the interior product of the 2-form dαµ with ηQ is zero, i.e., ιηQ dαµ = 0. Proof. ιηQ dαµ = LηQ (αµ ) − d(ιηQ αµ ). The Lie derivative term is zero since αµ is invariant under the Gµ -action (see Lemma 3.2.5), and the exterior derivative term is zero since ιηQ αµ = hαµ , ηQ i = hµ, A ◦ φµ (η)i = hµ, ηi is a constant function on Q, since A ◦ φµ (η) = η, for all η ∈ Lie(Gµ ). By this lemma and Lemma 3.2.5 the 2-form dαµ is basic; hence, a closed 2-form e can be uniquely defined by the relation T ∗ π βµ ∈ Ω2 (Q) e(βµ ) = dαµ , and its pullback Ξµ e→Q e will be a closed 2-form on T ∗ Q, e by the cotangent bundle projection πQe : T ∗ Q Ξµ := T ∗ πQe (βµ ). e Ω e can −Ξµ ) Theorem 3.2.7. There is a symplectic embedding ϕµ : ((T ∗ Q)µ , Ωµ ) ,→ (T ∗ Q, e e that covers the base Q, e where Ω e can is the canonical 2-form on T ∗ Q onto [T π e(V)]0 ⊂ T ∗ Q and 0 indicates the annihilator with respect to the natural pairing between tangent and cotangent bundle. The map ϕµ is identified by hϕµ ([γq ]µ ), Tq π e(vq )i = hγq − αµ (q), vq i,

(3.2.25)

∀γq ∈ M−1 q (µ) and ∀vq ∈ Tq Q, where [·]µ refers to a class of elements in the quotient manifold M−1 (µ)/Gµ [49]. e Based on the above theorem, the inverse of the map ϕµ exists only on [T π e(V)]0 ⊂ T ∗ Q, and it is a diffeomorphism on this vector sub-bundle. Hence, one may rewrite the reduced e as Hamilton’s equation (3.2.17) in [T π e(V)]0 ⊂ T ∗ Q e can − Ξµ ) = dH, e ιXe (Ω

(3.2.26)

−1 e := Hµ ◦ ϕ−1 e ◦ ϕµ = where H e(V)]0 → (T ∗ Q)µ being the inverse of ϕµ , X µ for ϕµ : [T π e T ϕµ ◦Xµ , and Ξµ can be calculated as follows. Consider two vector fields Z, Y ∈ X(T ∗ Q),


53

e by qe := π e define Zqe := T π e Z(e denote an element of Q e(q), and ∀e αqe ∈ T ∗ Q αqe), Yqe := Q T πQe Y(e αqe): D E e e ))]q ) + [Aq (hl e q (Zqe)), Aq (hl e q (Yqe))] . (Ξµ )αeqe(Z(e αqe), Y(e αqe)) = µ, −Aq ([hor(hl(Z)), hor(hl(Y (3.2.27) µ For all h ∈ Gµ , we show the action of h at any q ∈ Q by h · q := Φh (q). The 2-form e is well-defined, since we have Ξµ ∈ Ω(T ∗ Q) D

E e e ))]h·q ) + [Ah·q (hl e h·q (Zqe)), Ah·q (hl e h·q (Yqe))] µ, −Ah·q ([hor(hl(Z)), hor(hl(Y D e e )))]h·q ) = µ, −Ah·q ([T Φµh (hor(hl(Z))), T Φµh (hor(hl(Y E e q (Zqe))), Ah·q (T Φµ (hl e q (Yqe)))] +[Ah·q (T Φµh (hl h D E e e ))]q ) + [Adh Aq (hl e q (Zqe)), Adh Aq (hl e q (Yqe))] = µ, −Ah·q (T Φµh [hor(hl(Z)), hor(hl(Y D E e e e e = µ, −Adh Aq ([hor(hl(Z)), hor(hl(Y ))]q ) + Adh [Aq (hlq (Zqe)), Aq (hlq (Yqe))] D E e e ))]q ) + [Aq (hl e q (Zqe)), Aq (hl e q (Yqe))] . = µ, −Aq ([hor(hl(Z)), hor(hl(Y

e and hor. The The first equality is the result of the definition of the bundle maps hl second and third equalities are true since the principal connection A is Ad-equivariant. And the last equality holds because h ∈ Gµ . If in Theorem 3.2.7 we assume Gµ = G, whose special examples are when G is Abelian or µ = 0, then the map ϕµ becomes a symplectomorphism. Under this assumption, since e and A ◦ hl = 0, Ξµ can be calculated by a simpler formulation hl = hl D

E e e (Ξµ )αeqe(Z(e αqe), Y(e αqe)) = µ, −Aq ([hl(Z), hl(Y )]q ) .

3.3

(3.2.28)

Symplectic Reduction of Holonomic Open-chain Multi-body Systems with Displacement Subgroups

In this section, we show that holonomic open-chain multi-body systems can be considered as Hamiltonian mechanical systems with symmetry. In Theorem 3.3.3 we prove that the configuration manifold of the first joint is a symmetry group, and the corresponding group action is left translation. We identify the spaces and maps introduced in the previous section. Accordingly, we apply the reduction theory for Hamiltonian mechanical systems with symmetry to holonomic open-chain multi-body systems with displacement subgroups.


3.3.1

54

Indexing and Some Kinematics

Based on Definition 2.2.1 in the previous chapter, a holonomic open-chain multi-body system is a multi-body system MS(N ) together with N holonomic displacement subgroups between the bodies, such that there exists a unique path between any two bodies of the multi-body system, where B0 (a fixed body) corresponds to the ground (inertial coordinate frame). In a holonomic open-chain multi-body system, bodies with only one neighbouring body are called extremities. We label the bodies starting from the inertial coordinate frame (ground), B0 , outwards. That is, we label the bodies connected to B0 by joints successively as B1 , · · · , BN0 (N0 ≤ N ), and we repeat the same procedure for all N0 bodies starting from B1 , e.g., all of the bodies connected to B1 by joints are labeled as BN0 +1 , · · · , BN0 +N1 and so on. P Thus, one has l=0 Nl = N . Joints are numbered using the larger body label, e.g., the joint between Bi and Bj , where i > j, is labeled as Ji . Considering the bodies and joints in an open-chain multi-body system as vertices and edges of a graph, respectively, we can encode the structure of the system in an N × (N + 1) matrix. We label this matrix by GM. The N rows of this matrix correspond to the joints, J1 , · · · , JN , and the columns represent the bodies, B0 , · · · , BN . Each row of this matrix consists of only two non-zero elements and the rest is equal to 0. The non-zero elements in the row i correspond to the two bodies that Ji connects. We put the element corresponding to the body with lower index equal to −1 and the one with the higher index is equal to 1. With the choice of numbering that was explained above, we have   −1 if Ji connects Bj to Bi GMij = , 1 if i = j   0 otherwise which is a lower triangular matrix. We have the following properties of the matrix GM. Corollary 3.3.1. Let GMj denote the j th column of the matrix GM. i) The summation of the columns of the matrix GM is equal to zero, i.e.,

 0 N +1 X .  ..   GMj = ..   . . j=1 JN 0 J1



ii) The summation of the rows corresponding to the edges (joints) that are connecting

55

Chapter 3. Reduction of Holonomic Multi-body Systems the vortex (body) Bj to Bi for i > j, has the following form B0

h

0

···

···

Bj−1

Bj

Bj+1

−1

0

···

···

0

Bi−1

0

Bi

1

Bi+1

0

···

BN

···

0

i

.

Denote the transpose of GM by GMT . For all i, j = 1, · · · , (N + 1) iii) ((GM)T (GM))ii = the number of neighbouring vortices (bodies) connected to Bi−1 . iv) if ((GM)T (GM))ij = −1 for i 6= j, then the vortex (body) Bi−1 is connected to Bj−1 , either with the edge (joint) Ji−1 if i > j, or with the edge (joint) Jj−1 if j > i. Note that for any i = 2, · · · , (N + 1), if ((GM)T (GM))ii = 1 then the body Bi−1 is an extremity. The body corresponding to the k th 1 is called the k th extremity. Accordingly, the path between B0 and the k th extremity is called the k th branch. Corollary 3.3.2. Let the row matrix Phi represent the path between the vertex (body) Bi (∀i = 1, · · · , N ) and B0 . The j th element of Phi is equal to 1 if the path crosses the edge (joint) Jj . Then we have

Phi × GM =

h

B0

B1

···

Bi−1

Bi

Bi+1

···

BN

−1

0

···

0

1

0

···

0

i

.

Hence, the matrix of all paths, i.e., 

 Ph1  .  .  Ph =   .  PhN −1

is equal to GM , where GM is the matrix GM when the first column is removed. For example, consider the following structure of an open-chain multi-body system B0

J1

B1 J2

B2

J3

B3

J4

B4

(3.3.29)


56

We have B0 J1

GM =

J2 J3

     

J4

B1

(GM)T (GM) =

B2 B3

       

B4

Ph1

Ph =

Ph2 Ph3 Ph4

B2

B3

B4

0 1 0 0

0 0 1 −1

0 0 0 1

−1 1 0 −1 0 −1 0 0 B0

B0

B1

     

B1

B2

B3

   ,   B4

 1 −1 0 0 0  −1 3 −1 −1 0   0 −1 1 0 0  ,  0 −1 0 2 −1  0 0 0 −1 1 J1

J2

J3

J4

1 1 1 1

0 1 0 0

0 0 1 1

0 0 0 1

   .  

From the matrix (GM)T (GM) one can see that the open-chain multi-body system represented by the above graph has two extremities, B2 and B4 . The body B2 is the first extremity and B4 is the second one.

Since only displacement subgroups are considered, the relative configuration manifold 0 Rr i Qi , where corresponding to the joint Ji is diffeomorphic to the Lie group Qi := Lri,0 0,0 0 0 Qi is defined in Section 2.1 and ri,0 ∈ Pi is the initial pose of Bi with respect to B0 , for i = 1, ..., N . Note that every Qi is a di dimensional Lie subgroup of P0 ∼ = SE(3), PN where di is the number of degrees of freedom of Ji , and D := i=1 di is the total number of degrees of freedom of the holonomic open-chain multi-body system. Accordingly, any state of the system can be realized by q := (q1 , · · · , qN ) ∈ Q := Q1 × · · · × QN , where Q is the configuration manifold. The manifold Q along with the group structure induced by Qi ’s is also a Lie group. Let rcm,i ∈ SE(3) be the initial pose of the centre of mass of Bi with respect to the inertial coordinate frame. Considering Qi ’s as subgroups of SE(3), we define the map F : Q → SE(3) × · · · × SE(3) =: P by F (q) := (q1 rcm,1 , q1 q2 rcm,2 , · · · , q1 · · · qN rcm,N ).

(3.3.30)


57

This map determines the position of the centre of mass of all bodies with respect to the inertial coordinate frame. Note that the ith component of this map consists of the joint parameters of all joints that connect B0 to Bi in the open-chain multi-body system. Also, in this thesis, we consider open-chain multi-body systems with only one joint (J1 ) attached to B0 , i.e., N0 = 1. Because, for any N0 > 1 we can split the open-chain multibody system to N0 decoupled open-chain multi-body systems and study each of them separately.

For any motion of the open-chain multi-body system, i.e., a curve t 7→ q(t) ∈ Q, the velocity of the centre of mass of the bodies with respect to the inertial coordinate frame ˙ Based on Corollary 3.3.2, (absolute velocity) is calculated by p˙ := dtd F (q(t)) = Tq F (q). we can explicitly write the tangent map Tq F using the matrix Ph. First, we substitute the zero elements in the matrix Ph by 6 × 6 block matrices of zero. Then, ∀i = 1, · · · , N we substitute all the elements equal to 1 in Phi by the linear maps that look like T (Rrcm,i )T (RQr qr )T (LQl ql ), where the maps L• : SE(3) → SE(3) and R• : SE(3) → SE(3) are the left and right Q Q translation maps on SE(3), respectively. Here, l ql and r qr are the product of some elements of the relative configuration manifolds Qi ⊆ P0 ∼ = SE(3), considered as elements of SE(3). In order to specify which joints contribute to the left or right translation maps, in Phi we look at the 1s that are on the left or right of the corresponding element, respectively. If there does not exist any element equal to 1 on left (right), then we put the argument of the left (right) translation map equal to the identity element of SE(3). Finally, Tq F is the right multiplication of the resulting matrix by   Tq1 ι1 · · · 0  . ..  .. . Tq1 ι1 ⊕ · · · ⊕ TqN ιN =  . .   . , 0 · · · TqN ιN where for all i = 1, · · · , N , ιi : Qi → SE(3) is the canonical inclusion map and T ιi : T Qi → T SE(3) is the induced map on the tangent bundles.

This simple procedure becomes clear in an example. Consider the structure of the


58

system in (3.3.29), we calculate Tq F to be the following matrix 

   T Rrcm,1 06×6 06×6 06×6 T ι · · · 0   q1 1  T Rrcm,2 T Rq2 T Rrcm,2 T Lq1  . 06×6 06×6 ..  ...    .. .   TR  . T R 0 T R T L 0 rcm,3 q3 6×6 rcm,3 q1 6×6   0 · · · TqN ιN T Rrcm,4 T Rq3 q4 06×6 T Rrcm,4 T Rq4 T Lq1 T Rrcm,4 T Lq1 q3

3.3.2

Lagrangian and Hamiltonian of an Open-chain Multi-body System

As mentioned in Section 3.2, the Lagrangian of an Open-chain Multi-body System L : T Q → R is L(vq ) = 21 Kq (vq , vq ) − V (q). In this section, we describe how the Lagrangian L and subsequently the Hamiltonian H of an Open-chain Multi-body System is calculated. Let hi for i = 1, · · · , N be the left-invariant kinetic energy metric for the rigid body Bi in the open-chain multi-body system. They induce h := h1 ⊕ · · · ⊕ hN as a left-invariant metric on P. For the open-chain multi-body system, the metric K := T ∗ F (h), where T ∗ F (h) refers to the pull back of the metric h by the map F . That is, ∀q ∈ Q and ∀vq , wq ∈ Tq Q we have Kq (vq , wq ) = hF (q) (Tq F (vq ), Tq F (wq )) = he TF (q) LF (q)−1 (Tq F (vq )), TF (q) LF (q)−1 (Tq F (wq )) ,

(3.3.31)

where e is the identity element of the Lie group P and Lp is the left translation map by any element p ∈ P. In this thesis, wherever we consider a non-zero potential energy function it is induced by a constant gravitational field g in A0 , which is defined in Section 2.1 as the 3-dimensional affine space corresponding to the inertial coordinate frame. Using the Euclidean inner product of R3 , which is denoted by ·, · , the potential energy function for an open-chain multi-body system is defined as V (q) :=

N X

mi g, O0 − Fi (q)(Oi ) ,

(3.3.32)

i=1

where mi is the mass of the rigid body Bi , and Fi (q) : Ai → A0 is the ith component of the map F that can be considered as an isometry between Ai and A0 with respect to the Euclidean norm of R3 . The points O0 ∈ A0 and Oi ∈ Ai are the base points for the affine


59

spaces A0 and Ai , where Oi is located at the centre of mass of Bi . Subsequently, using the Legendre transformation one can define the Hamiltonian H : T ∗ Q → R for an open-chain multi-body system by −1 H(pq ) := hpq , FL−1 q (pq )i − L(FLq (pq )).

(3.3.33)

Here, we remind the reader that FL : T Q → T ∗ Q is the fibre-wise invertible Legendre transformation induced by the kinetic energy metric, i.e., ∀vq , wq ∈ Tq Q, hFLq (vq ), wq i = Kq (vq , wq ). Accordingly, a holonomic open-chain multi-body system can be considered as a Hamiltonian mechanical system described by the quadruple (T ∗ Q, Ωcan , H, K). Here, the metric K and the Hamiltonian H are defined by (3.3.31) and (3.3.33), respectively.

3.3.3

Reduction of Holonomic Open-chain Multi-body Systems

Based on the definition of the kinetic energy metric K for a holonomic open-chain multibody system, we immediately find the following symmetry for K. Theorem 3.3.3. For a holonomic open-chain multi-body system, the action of G = Q1 on Q by left translation on the first component leaves the kinetic energy metric K invariant. Here,for any g ∈ G the action map Φg : Q → Q is given by Φg (q) = (gq1 , q2 , · · · , qN ), where q = (q1 , · · · , qN ) ∈ Q. Proof. For any g ∈ G, let T Φg : T Q → T Q be the induced action of G on the tangent bundle. For simplicity, ∀q ∈ Q and ∀vq ∈ Tq Q we respectively write Φg (q) and Tq Φg (vq ) as g · q and g · vq . Then, ∀wq ∈ Tq Q we have Kg·q (g · vq , g · wq ) = he (TF (g·q) LF (g·q)−1 )(Tg·q F )(g · vq ), (TF (g·q) LF (g·q)−1 )(Tg·q F )(g · wq ) = he (TF (g·q) LF (g·q)−1 )(Tq (F ◦ Φg ))(vq ), (TF (g·q) LF (g·q)−1 )(Tq (F ◦ Φg ))(wq ) = he (TF (g·q) LF (g·q)−1 )(Tq (∆g ◦ F ))(vq ), (TF (g·q) LF (g·q)−1 )(Tq (∆g ◦ F ))(wq ) = he (T∆g F (q) (LF (q)−1 ◦ ∆g−1 ))(Tq (∆g ◦ F ))(vq ) , (T∆g F (q) (LF (q)−1 ◦ ∆g−1 ))(Tq (∆g ◦ F ))(wq ) = he Tq (LF (q)−1 ◦ F )(vq ), Tq (LF (q)−1 ◦ F )(wq ) = he (TF (q) LF (q)−1 )(Tq F )(vq ), (TF (q) LF (q)−1 )(Tq F )(wq ) = Kq (vq , wq ). The first equality is based on the definition of the metric K, and the third and fourth equalities are true since the following diagram commutes. Note that ∆g = L(g,...,g) is the


60

diagonal action of G on P. Q

F

Φg

/P

∆g

Q

F

/P

For the special case of open-chain multi-body systems in space where the potential energy function is equal to zero, this theorem indicates that the Hamiltonian of the system is also invariant under the cotangent lifted action of G. In general, there exist joints for which the potential energy function V defined by (3.3.32) is also invariant under the G-action, e.g., if Q1 corresponds to a planar joint with the direction of the gravitational field g being perpendicular to the plane of the joint. For such first joints, the Hamiltonian of the system H becomes invariant under the cotangent lifted action of G. From here on, we always assume that V is also invariant under the G-action, unless otherwise stated. Accordingly, the quintuple (T ∗ Q, Ωcan , H, K, G) with the group action defined in Theorem 3.3.3 is called a holonomic open-chain multi-body system with symmetry. It is a mechanical system with symmetry. For a holonomic open-chain multi-body system with symmetry, the G-action is basically the left translation on Q1 . Therefore, the quotient manifolds Q = Q/G and e = Q/Gµ are equal to (Q2 × · · · × QN ) and (Q1 /Gµ × Q2 × · · · × QN ), respectively. Q We remind the reader that ∀µ ∈ Lie∗ (G) the subgroup Gµ ⊆ G is the coadjoint isotropy group corresponding to G. For any q1 ∈ Q1 , let qe1 ∈ Q1 /Gµ denote the equivalence class corresponding to q1 . Indeed, ∀q = (q1 , · · · , qN ) ∈ Q the quotient maps π : Q → Q and e are defined by q := π(q) = (q2 , · · · , qN ) and qe := π π e: Q → Q e(q) = (e q1 , q2 , · · · , qN ), respectively. For an open-chain multi-body system with symmetry, we then calculate the infinitesimal action of ξ ∈ Lie(G) on Q at q = (q1 , ..., qN ) by ∂ ξQ (q) = (exp(ξ)q1 , q2 , · · · , qN ) = (ξq1 , 0, · · · , 0), ∂ =0 where ξq1 corresponds to the right translation of ξ by q1 ∈ Q1 . This relation indicates

61

Chapter 3. Reduction of Holonomic Multi-body Systems that the map φ is the right translation of a Lie algebra element on Q1 , i.e.,   Te1 Rq1    0   φq :=  .  .  .. 

(3.3.34)

0 Accordingly, based on (3.2.16) ∀pq := (p1 , · · · , pN ) ∈ T ∗ Q the momentum map M : T ∗ Q → Lie∗ (G) for a holonomic open-chain multi-body system can be determined by the following calculation, hMq (pq ), ξi = h(p1 , · · · , pN ), (ξq1 , 0, · · · , 0)i = hp1 , ξq1 i = hTe∗1 Rq1 p1 , ξi. As a result, Mq =

φ∗q

h = Te∗1 Rq1 0 · · ·

i 0 .

(3.3.35)

Denote the block components of the kinetic energy tensor, which is equal to the Legendre transformation in the case of Hamiltonian mechanical systems, by Kij (q)dqi ⊗ dqj for P i, j = 1, · · · , N . Hence, we have FLq = N i,j=1 Kij (q)dqi ⊗ dqj or equivalently 



K11 (q) · · · K1N (q)  .  .. ..  . FLq =  .. . .  KN 1 (q) · · · KN N (q) Lemma 3.3.4. For all q ∈ Q we have the following equality:  ∗ (Tq1 Lq1−1 )(K 11 (q))(Tq1 Lq1−1 ) (Tq∗1 Lq1−1 )(K 12 (q))   K 22 (q) (K 21 (q))(Tq1 Lq1−1 ) FLq =  . ..  .. .  K N 2 (q) (K N 1 (q))(Tq1 Lq1−1 )

 (Tq∗1 Lq1−1 )(K 1N (q))   ··· K 2N (q) , . ..  .. .  ··· K N N (q) ···

where q = π(q) and K ij (q) = Kij ((e1 , q)).

Proof. By Theorem 3.3.3, ∀vq , wq ∈ Tq Q and q = π(q) ∈ Q we have Kq (vq , wq ) = K(e1 ,q) (Tq Φq1−1 vq , Tq Φq1−1 wq ).


62

By the definition of Legendre transformation in (3.2.12), we can rewrite this equation as E D E ∗ hFLq (vq ), wq i = FL(e1 ,q) (Tq Φq1−1 )(vq ), Tq Φq1−1 (wq ) = (Tq Φq1−1 )FL(e1 ,q) (Tq Φq1−1 )(vq ), wq . D

We prove the equality in the lemma, since we have " Tq Φq1−1 = Tq1 Lq1−1 ⊕ idTq Q =

Tq1 Lq1−1

0

0

idTq Q

# ,

where idTq Q is the identity map on Tq Q.

Based on this lemma we calculate the locked inertia tensor Iq = φ∗q ◦ FLq ◦ φq for a holonomic open-chain multi-body system by Iq = (Te∗1 Rq1 )K11 (q)(Te1 Rq1 ) = Ad∗q−1 K 11 (q)Adq1−1 .

(3.3.36)

1

Consequently, using (3.2.21) we determine the (mechanical) connection A corresponding to the G-action, for a holonomic open-chain multi-body system: Aq = I−1 q ◦ Mq ◦ FLq 

 K · · · K 11 1N h i .. ..  ... = (Adq1 )K 11 (q)−1 (Ad∗q1 ) Te∗1 Rq1 0 · · · 0  . .    KN 1 · · · KN N i i h h = Adq1 Tq1 Lq−1 K 11 (q)−1 K 12 (q) · · · K 11 (q)−1 K 1N (q) =: Adq1 Tq1 Lq−1 Aq , 1 1 (3.3.37) where the last line of (3.3.37) is the consequence of Lemma 3.3.4, and the fibre-wise linear map A : T Q → Lie(G) is defined by the last equality. According to (3.2.19), ∀q ∈ Q and ∀v q ∈ Tq Q the horizontal lift map hlq : Tq Q → Tq Q becomes " # −(Te1 Lq1 )Aq hlq = , idTq Q where q = (q1 , q). Using the decomposition T Q = H ⊕ V introduced in the previous section, we then show that ∀q ∈ Q the map horq : Tq Q → Hq , which maps any vector in the tangent space

63

Chapter 3. Reduction of Holonomic Multi-body Systems Tq Q to its horizontal component, is   Te1 Rq1   h i  0   horq = idTq Q − verq = idTq Q − φq ◦ Aq = idTq Q −  Ad −1 A T L q  ..  q1 q1 q1  .  0  0 ··· . . = . 0 ···



0 −Te1 Lq1 Aq  .. . .  0 idTq Q

(3.3.38)

We consider the principal bundle π e1 : Q1 → Q1 /Gµ to locally trivialize the Lie group Q1 . Let Uµ ⊆ Q1 /Gµ be an open neighbourhood of ee1 , where ee1 is the equivalence class corresponding to the identity element e1 ∈ Q1 . We denote the map corresponding to a local trivialization of the principal bundle π e1 by χ e : Gµ × Uµ → Q1 . This map can be defined by embedding Uµ in Q1 , for example by using the exponential map of Lie groups. We denote this embedding by χµ : Uµ ,→ Q1 such that ∀e q1 ∈ Q1 /Gµ we have e e χµ (e q1 ) = exp(ζ) for some ζ ∈ C, where C ⊂ Lie(Q1 ) is a complementary subspace to Lie(Gµ ) ⊂ Lie(G). Accordingly, ∀h ∈ Gµ we define the map χ e by the equality χ e((h, qe1 )) := hχµ (e q1 ). It is easy to show that the map χ e is a diffeomorphism onto its image [35]. Using this diffeomorphism, any element q1 ∈ π e1−1 (Uµ ) ⊆ Q1 can be uniquely identified χ((h, qe1 )), q). by an element (h, qe1 ) ∈ Gµ × Uµ . As a result, we have q = (q1 , q) = (e From now on, for brevity we write q = (h, qe1 , q). Accordingly, by Lemma 3.3.4, for all q = (h, qe1 , q) ∈ Gµ × Uµ × Q we can calculate Aµ as h i eqe , Aµq = Adh Th Lh−1 A eqe : Tqe(Uµ × Q) → Lie(Gµ ) is calculated by where qe = π e(q) = (e q1 , q) ∈ Uµ × Q and A h µ eqe := K e 1Gµ (e e 1Q1 /Gµ (e e 1Gµ (e e 12 A q )−1 K q) K q )−1 K (e q) · · ·

i Gµ e 11 e µ (e . K (e q )−1 K q ) 1N

Here, according to the local trivialization that we chose we have the following form for


64

the tensor FLq  Q /G G µ µ (q) · · · K1N (q) K1 µ (q) K1 1 µ (q) K12   Gµ   K2 (q) K2Q1 /Gµ (q)   .. ..   µ .. . FLq =  K21 (q) . . .   . . . .   . . . . . . . .   µ µ KN 1 (q) ··· ··· · · · KN N (q) 

G Q /G µ e 1Gµ (e e 1Q1 /Gµ (e e µ (e e)) And, we have K q ) = K1 µ ((eµ , qe)), K q ) = K1 1 µ ((eµ , qe)), and K 1i q ) = K1i ((eµ , q for all i = 2, · · · , N . Here, eµ ∈ Gµ is the identity element of the Lie group Gµ ⊆ G = Q1 .

Now, for any h ∈ Gµ and ∀q = (h, qe1 , q) ∈ Gµ × Uµ × Q, we calculate the horizontal e q : Tqe(Uµ × Q) → Tq Q for the principal bundle π e by e: Q → Q lift map hl "

# e L ) A −(T qe eµ h eq = hl , idTqe1 Uµ ⊕ idTq Q

(3.3.39)

where idTqe1 Uµ is the identity map on the tangent space Tqe1 Uµ . Let µ ∈ Lie∗ (G) be a regular value of the momentum map M. For a holonomic open-chain multi-body system with symmetry, the level set of the momentum map M at µ becomes M−1 (µ) = { pq = (p1 , · · · , pN ) ∈ T ∗ Q| p1 = Tq∗1 Rq1−1 µ} ⊂ T ∗ Q. Furthermore, we determine αµ = A∗ µ ∈ Ω1 (Q) in the local trivialization by " αµ (q) =

∗ T(h,e q1 )−1 q1 ) L(h,e

Aq∗

#

" Ad∗(h,eq1 ) µ =

∗ T(h,e q1 )−1 q1 ) L(h,e

# Ad∗(eµ ,eq1 ) µ,

A∗q

(3.3.40)

where (h, qe1 )−1 = χ e−1 ((e χ(h, qe1 ))−1 ), by definition. The second equality is true by the definition of the map χ e, and because h ∈ Gµ .

e is Lemma 3.3.5. Based on Theorem 3.2.7, the inverse of the map ϕµ : M−1 /Gµ → T ∗ Q defined on [T π e(V)]0 and in the local trivialization ∀e pqe = (e p1 , p) ∈ Tqe∗ (Uµ × Q), " ϕ−1 pqe) = µ (e

∗ T(h,e q1 )−1 (µ) q1 ) R(h,e

#

p + A∗q (Ad∗(eµ ,eq1 ) µ)

.

(3.3.41)

µ

Proof. First we show that pe ∈ [T π e(V)]0 if and only if pe1 = 0. For any pe ∈ [T π e(V)]0 and


65

∀ξ ∈ Lie(G) = Lie(Q1 ) we have

e(ξQ )i = φ∗q (0, pe1 , p), ξ = Te∗1 Rq1 (0, pe1 ), ξ = 0. h(e p1 , p), T π The first equality is true based on the definition of ξQ and the local trivialization that is chosen. The second equality is the consequence of the definition of the map φ in (3.3.34). Since the above equality should hold for every ξ ∈ Lie(G) and the right translation map is a diffeomorphism ∀q1 ∈ Q1 , we have pe1 = 0. Now, based on (3.3.40) and the definition of the map ϕµ in Theorem 3.2.7 we have the desired equation in the lemma. e pqe) := Hµ ◦ ϕ−1 pqe) and the above lemma, we calculate Based on the definition of H(e µ (e 0 e H on [T π e(V)] using the local trivialization: 1D e H(e pqe) = (Ad∗(eµ ,eq1 ) µ, p + A∗q (Ad∗(eµ ,eq1 ) µ)), 2 E ∗ ∗ ∗ (Ad µ, p + A (Ad µ)) + V (eµ , qe1 , q). , FL−1 (eµ ,e q1 ) q (eµ ,e q1 ) (e1 ,q)

(3.3.42)

Now we are ready to state the main result of this section in the following theorem. Theorem 3.3.6. Let µ ∈ Lie∗ (G) be a regular value of the momentum map M. A holonomic open-chain multi-body system with symmetry (T ∗ Q, Ωcan , H, K, G) is reduced e Ω e can |[T πe(V)]0 − Ξµ , H, e K), e where to a Hamiltonian mechanical system ([T π e(V)]0 ⊆ T ∗ Q, e can is the canonical 2-form on T ∗ Q, e H e is defined by (3.3.42) and K e is a metric on Q e Ω e we have such that ∀e uqe, w eqe ∈ TqeQ e q (e e q (w e qe(e K uqe, w eqe) = Kq (hl uqe), hl eqe)). e → Q e be the Here, in the local coordinates Ξµ is calculated as follows. Let πQe : T ∗ Q e → TQ e be its canonical projection map of the cotangent bundle and let T πQe : T (T ∗ Q) e and ∀U, e W f ∈ X(T ∗ Q) e we introduce u tangent map. For every α eqe ∈ T ∗ Q eqe = TαeqeπQe (Ueαeqe) fαe ). In the local trivialization, we have qe = (e and w eqe = TαeqeπQe (W q1 , q) ∈ Uµ × Q, u eqe = qe (e u1 , u) and w eqe = (w e1 , w): ∂A ∂A q q fαe ) = µ, −Adχµ (eq ) [Aq u, Aq w] + ( (Ξµ )αeqe(Ueαeqe, W w)u − ( u)w 1 qe ∂q ∂q h eqeu + −A e + (Tχµ (eq1 ) Rχµ (eq1 )−1 )(Tqe1 χµ )(e u1 ) + Adχµ (eq1 ) Aq u , iE eqew −A e + (Tχµ (eq1 ) Rχµ (eq1 )−1 )(Tqe1 χµ )(w e1 ) + Adχµ (eq1 ) Aq w , (3.3.43) where χµ : Uµ ,→ Q1 is the embedding that is used to define the local trivialization map

66

Chapter 3. Reduction of Holonomic Multi-body Systems χ e, using the exponential map of Q1 .

e = (qe˙ 1 , q˙ , p) ˙ as a vector field on [T π e(V)]0 . Finally, in local coordinates we have X Hamilton’s equation in the vector sub-bundle [T π e(V)]0 of the cotangent bundle of µ-shape space reads ι(qe˙ 1 ,q,˙ p) ˙ (−dp ∧ dq − Ξµ ) =

e e e ∂H ∂H ∂H dp + de q1 + dq, ∂p ∂e q1 ∂q

(3.3.44)

where Ξµ is calculated by (3.3.43). Proof. In order to prove (3.3.43), we start with (3.2.27): D E e u)), hor(hl( e w))] e q (e e q (w fαe ) = µ, −Aq ([hor(hl(e (Ξµ )αeqe(Ueαeqe, W e ) + [A ( hl u )), A ( hl e ))] . q q q e q q e qe Using the local trivialization, we write q = (h, qe1 , q) ∈ Gµ × Uµ × Q, and accordingly u e = (e u1 , u) and w e = (w e1 , w). By (3.3.39), the horizontal lift of u e and w e can be calculated as e q (e eqeu e, u e1 , u), hl uqe) = (−(Teµ Lh )A

e q (w eqew, e w e1 , w), hl eqe) = (−(Teµ Lh )A

e u)) and hor(hl( e w)) and using (3.3.38), the terms hor(hl(e e are e q (e horq (hl uqe)) = (−(T(eµ ,ee1 ) L(h,eq1 ) )Aq u, u),

e q (w horq (hl eqe)) = (−(T(eµ ,ee1 ) L(h,eq1 ) )Aq w, w).

Now, by (3.3.37) we have e q (e eqeu Aq (hl uqe)) = Ad(h,eq1 ) (T(h,eq1 ) L(h,eq1 )−1 ) −(Teµ Lh )A e, u e1 + Aq u .

(3.3.45)

Using the definition of the local trivialization map χ e we have T(h,eq1 ) L(h,eq1 )−1

eqeu −(Teµ Lh )A e, u e1

eqeu = Thχµ (eq1 ) Lχµ (eq1 )−1 h−1 Th Rχµ (eq1 ) (−(Te1 Lh )A e) + (Tχµ (eq1 ) Lh )(Tqe1 χµ )(e u1 ) eqeu u1 ), = Adχµ (eq1 )−1 (−A e) + (Tχµ (eq1 ) Lχµ (eq1 )−1 )(Tqe1 χµ )(e where χµ : Uµ ,→ Q1 is the embedding map that is defined using the exponential map. Therefore, we have e q (e eqeu Aq (hl uqe)) = Adh −A e + (Tχµ (eq1 ) Rχµ (eq1 )−1 )(Tqe1 χµ )(e u1 ) + Adχµ (eq1 ) Aq u .


67

Similarly, e e Aq (hlq (w eqe)) = Adh −Aqew e + (Tχµ (eq1 ) Rχµ (eq1 )−1 )(Tqe1 χµ )(w e1 ) + Adχµ (eq1 ) Aq w . Since for all g ∈ G and ξ, η ∈ Lie(G) we have the equality Adg [ξ, η] = [Adg ξ, Adg η]: e q (e e q (w [Aq (hl uqe)), Aq (hl eqe))] = Adh

h

eqeu −A e + (Tχµ (eq1 ) Rχµ (eq1 )−1 )(Tqe1 χµ )(e u1 ) + Adχµ (eq1 ) Aq u , i eqew −A e + (Tχµ (eq1 ) Rχµ (eq1 )−1 )(Tqe1 χµ )(w e1 ) + Adχµ (eq1 ) Aq w .

e u)), hor(hl( e w))] For all q ∈ Q, to calculate the Lie bracket [hor(hl(e e q , we express the vector e u)) and hor(hl( e w)) fields hor(hl(e e in coordinates: ∂ ∂ +u ∂(h, qe1 ) ∂q ∂ ∂ e q (w horq (hl eqe)) = −(T(eµ ,ee1 ) L(h,eq1 ) )Aq w +w . ∂(h, qe1 ) ∂q

e q (e horq (hl uqe)) = −(T(eµ ,ee1 ) L(h,eq1 ) )Aq u

In any coordinates chosen for Qi (i = 2, · · · , N ), Gµ and Q1 /Gµ we have ∂ ∂ , (T(eµ ,ee1 ) L(h,eq1 ) )Aq w ] ∂(h, qe1 ) ∂(h, qe1 ) ∂ ∂ ∂ ∂ + [u , w ] + [ (T(eµ ,ee1 ) L(h,eq1 ) )Aq w ,u ] ∂q ∂q ∂(h, qe1 ) ∂q ∂ ∂ ,w ] − [ (T(eµ ,ee1 ) L(h,eq1 ) )Aq u ∂(h, qe1 ) ∂q

e u)), hor(hl( e w))] [hor(hl(e e = [ (T(eµ ,ee1 ) L(h,eq1 ) )Aq u

Based on the definition of the Lie bracket for Lie groups, ∀q ∈ Q the first bracket on the right hand side can be written as [ (T(eµ ,ee1 ) L(h,eq1 ) )Aq u

∂ ∂ , (T(eµ ,ee1 ) L(h,eq1 ) )Aq w ] ∂(h, qe1 ) ∂(h, qe1 ) ∂ = (T(eµ ,ee1 ) L(h,eq1 ) )[Aq u, Aq w] ∂(h, qe1 ) ∂w ∂ (T(eµ ,ee1 ) L(h,eq1 ) )Aq u + (T(eµ ,ee1 ) L(h,eq1 ) )Aq ∂(h, qe1 ) ∂(h, qe1 ) ∂u ∂ − (T(eµ ,ee1 ) L(h,eq1 ) )Aq (T(eµ ,ee1 ) L(h,eq1 ) )Aq w . ∂(h, qe1 ) ∂(h, qe1 )

68

Chapter 3. Reduction of Holonomic Multi-body Systems The second bracket is equal to ∂ ∂ [u , w ] = ∂q ∂q

∂ ∂ ∂w ∂u u w − . ∂q ∂q ∂q ∂q

We calculate the third bracket as ∂ ∂ ∂u ∂ ,u ] = (T(eµ ,ee1 ) L(h,eq1 ) )Aq w [ (T(eµ ,ee1 ) L(h,eq1 ) )Aq w ∂(h, qe1 ) ∂q ∂(h, qe1 ) ∂q ∂Aq ∂ ∂w − (T(eµ ,ee1 ) L(h,eq1 ) ) u w + (T(eµ ,ee1 ) L(h,eq1 ) )Aq u . ∂q ∂q ∂(h, qe1 )

Similarly, the last bracket can be calculated. Accordingly, using (3.3.37), ∂A ∂A q q e u)), hor(hl( e w))] Aq ([hor(hl(e e q ) = Ad(h,eq1 ) [Aq u, Aq w] + w u− u w . ∂q ∂q Finally, knowing that h ∈ Gµ , we have the equation for Ξµ in the theorem. Regarding Hamilton’s equation, we should note that based on Lemma 3.3.5 the ree can to [T π striction of Ω e(V)]0 is equal to −dp ∧ dq, in coordinates.

Corollary 3.3.7. Let us assume that Gµ = G, in the above theorem. A holonomic openchain multi-body system with symmetry (T ∗ Q, Ωcan , H, K, G) is reduced to a Hamiltonian mechanical system (T ∗ Q, Ωcan − Ξµ , H, K), where Ωcan is the canonical 2-form on T ∗ Q, H(pq ) :=

E 1D ∗ (µ, p + A∗q µ), FL−1 (µ, p + A µ) + V (e1 , q), q (e1 ,q) 2

(3.3.46)

and K is a metric on Q such that ∀uq , wq ∈ Tq Q we have K q (uq , uq ) = Kq (hlq (uq ), hlq (wq )). Here, in the local coordinates Ξµ is calculated by a simpler formulation. Let πQ : T ∗ Q → Q be the canonical projection map of the cotangent bundle and let T πQ : T (T ∗ Q) → T Q be its tangent map. For every αq ∈ T ∗ Q and ∀U, W ∈ X(T ∗ Q) we introduce uq = Tαq πQ (U αq ) and wq = Tαq πQ (W αq ). We have ∂Aq ∂Aq (Ξµ )αq (U αq , W αq ) = µ, −[Aq u, Aq w] − ( w)u + ( u)w . ∂q ∂q

(3.3.47)

˙ as a vector field on T ∗ Q. Hamilton’s Finally, in local coordinates we have X = (q˙ , p)

69

Chapter 3. Reduction of Holonomic Multi-body Systems equation in the cotangent bundle of shape space reads ι(q,˙ p) ˙ (−dp ∧ dq − Ξµ ) =

∂H ∂H dp + dq, ∂p ∂q

where Ξµ is calculated by (3.3.47). We show the isotropy groups for different types of displacement subgroups in Table 3.1. Note that for different values of µ ∈ Lie∗ (G), the isotropy groups are isomorphic to the groups listed in the table, and the isomorphism map is conjugation by an element of SE(3). In this table we consider the configuration manifold of the first joint as a Lie sub-group of SE(3) whose Lie algebra is a vector space isomorphic to so(3) ⊕ R3 , where so(3) is the Lie algebra of SO(3). For any element ξ ∈ se(3), we call its component in R3 the linear and the one in so(3) the angular component of ξ, where se(3) denotes the Lie algebra of SE(3). Table 3.1: Displacement subgroups and their corresponding isotropy groups Displacement Subgroups Q1 ∼ =G

Gµ (µ = (µv , µω )a ) µv 6= 0, µω 6= 0

µv = 0, µω 6= 0

µv 6= 0, µω = 0

µv = µω = 0

SE(3) SO(2) × R SE(2) × R SO(2) × R SE(3) SE(2) × R R2 (SE(2) × R)b SE(2) × R R2 (SE(2) × R)b SE(2) × R SE(2) R SE(2) R SE(2) SO(3) SO(2) SO(3) R3 R3 R3 2 2 Hp n R R Hp n R R Hp n R2 SO(2) × R SO(2) × R SO(2) × R SO(2) × R SO(2) × R R2 R2 R2 SO(2) SO(2) SO(2) R R R Hp Hp Hp a µv is the linear component and µω is the angular component of the momentum. b If the linear momentum is in the direction of the allowed direction of rotation in the space.

3.4

Case Study

In this section we study the dynamics of an example of a holonomic open-chain multibody system. We derive the reduced dynamical equations of a six-d.o.f. manipulator mounted on top of a spacecraft whose initial configuration is shown in Figure 3.1. Using the indexing introduced in the previous section and starting with the spacecraft as B1 , we first number the bodies and joints. The following graph shows the structure of


70

Figure 3.1: A six-d.o.f. manipulator mounted on a spacecraft

the holonomic open-chain multi-body system. B4 J4

B0

J1

B1

J2

B2

J3

B3 J5

B5 We then identify the relative configuration manifolds corresponding to the joints of the robotic system. The relative pose of B1 with respect to the inertial coordinate frame is identified by the elements of the Special Euclidean group SE(3). We identify the elements of the relative configuration manifold corresponding to the first joint, which is a six-d.o.f. free joint, by      

    x        R  Y (θY )RX (θX )RZ (θZ )  y  0 0 1  x, y, z ∈ R, θX , θY , θZ ∈ S , Q1 = r1 =   z        h i       1 0 0 0 


71

Figure 3.2: The coordinate frames attached to the bodies of the robot

where we have   1 0 0   RX (θX ) = 0 cos(θX ) − sin(θX ) , 0 sin(θX ) cos(θX )   cos(θY ) 0 sin(θY )   RY (θY ) =  0 1 0 , − sin(θY ) 0 cos(θY )   cos(θZ ) − sin(θZ ) 0   RZ (θZ ) =  sin(θZ ) cos(θZ ) 0 . 0 0 1 The second joint is a three-d.o.f. spherical joint between B2 and B1 , and its corresponding relative configuration manifold is given by      

    0        R (ψ )R (ψ )R (ψ ) l     X X Y Y Z Z 1 1 1 1   Q2 = r2 =  ψX , ψY , ψZ ∈ S  . 0       h i       1 0 0 0 

The third joint is a one-d.o.f. revolute joint between B3 and B2 , and its relative config-

Chapter 3. Reduction of Holonomic Multi-body Systems uration manifolds is    1 0 0      0 cos(ψ1 ) − sin(ψ1 ) Q23 = r32 =  0 sin(ψ ) cos(ψ )  1 1      0 0 0

72

  0      l2   ∈ SE(3) ψ1 ∈ S1 .  0      1 

The forth and fifth joints are one-d.o.f. revolute joints whose axes of revolution are assumed to be the Xi -axis (i = 4, 5). The relative configuration manifolds of these joints are identified by      

 1 0 0  0 cos(ψ2 ) − sin(ψ2 ) Q34 = r43 =  0 sin(ψ ) cos(ψ )  2 2      0 0 0

   c       l3  1 ∈ SE(3) ψ2 ∈ S ,  0      1

   1 0 0 −c         0 cos(ψ ) − sin(ψ ) l 3 3 3 1 3 3   Q5 = r5 =  ∈ SE(3) ψ3 ∈ S .   0   0 sin(ψ3 ) cos(ψ3 )        0 0 0 1      



Here, we denote the distance between J4 and J5 by 2c, i.e., the origins of the coordinate frames V4 and V5 are located at c and −c in the x direction of V3 , respectively. Further, l1 , · · · , l5 are defined in Figure 3.2. We assume that the initial pose of B1 with respect to the inertial coordinate frame is the identity element of SE(3). We have located the coordinate frame attached to 0 B1 on its centre of mass. Hence, in matrix form we have r1,0 = rcm,1 = id4 , where id4 is the 4 × 4 identity matrix. For the second body, the initial relative pose with respect to B1 is   1 0 0 0   0 1 0 l1  1  r2,0 =  0 0 1 0  ,   0 0 0 1

0 r1,0

and we have



rcm,2

1  0 = 0  0

0 1 0 0

 0 0  0 l1 + l2 /2 .  1 0  0 1


73

The initial relative pose of B3 with respect to B2 is

2 r3,0

 1  0 = 0  0

0 1 0 0

0 0 1 0

 0  l2  , 0  1

and the relative pose of the centre of mass of B3 with respect to the inertial coordinate frame is   1 0 0 0   0 1 0 l1 + l2 + l3 /2 . rcm,3 =  0 0 1  0   0 0 0 1 Here we have assumed that the centre of mass of B2 and B3 are in the middle of the links. For the forth and fifth bodies we have (i = 4, 5)

3 ri,0

rcm,4

 1  0 = 0  0

0 1 0 0

 1  0 = 0  0

0 1 0 0

 0 ±c  0 l3  , 1 0  0 1

  1 0 c    0 l1 + l2 + l3 + l4   , rcm,5 = 0  0 1 0   0 1 0

0 1 0 0

 0 −c  0 l1 + l2 + l3 + l5  ,  1 0  0 1

where the plus and minus signs correspond to the body B4 and B5 , respectively.

With the above specifications of the system we identify the configuration manifold of the holonomic open-chain multi-body system in this case study by Q = Q1 × · · · × Q5 , where        x             RY (θY )RX (θX )RZ (θZ ) y   ∈ SE(3) , Q1 = q1 =   z        h i       1 0 0 0


Q2

Q3

Q4

Q5

     



 R  = q2 =     h i    0 0 0    1 0      0 cos(ψ1 ) = q3 =  0 sin(ψ )  1      0 0    1 0      0 cos(ψ2 ) = q4 =  0 sin(ψ )  2      0 0    1 0      0 cos(ψ3 ) = q5 =  0 sin(ψ )  3      0 0

74

      0 0         l1  − R l1   ∈ SE(3) R = RX (ψX )RY (ψY )RZ (ψZ ) , 0 0        1    0 0      2  − sin(ψ1 ) 2(l1 + l2 ) sin (ψ1 /2) ∈ SE(3) ,  cos(ψ1 ) −(l1 + l2 ) sin(ψ1 )       0 1    0 0      2  − sin(ψ2 ) 2(l1 + l2 + l3 ) sin (ψ2 /2) ∈ SE(3) ,  cos(ψ2 ) −(l1 + l2 + l3 ) sin(ψ2 )       0 1    0 0      2  − sin(ψ3 ) 2(l1 + l2 + l3 ) sin (ψ3 /2) ∈ SE(3) .  cos(ψ3 ) −(l1 + l2 + l3 ) sin(ψ3 )       0 1

In order to calculate the kinetic energy for the system under study, we need to first 5−times }| { z form the function F : Q → P = SE(3) × · · · × SE(3), which determines the pose of the coordinate frames attached to the centres of mass of the bodies with respect to the inertial coordinate frame. F (q1 , · · · , q5 ) = (q1 rcm,1 , q1 q2 rcm,2 , q1 q2 q3 rcm,3 , q1 q2 q3 q4 rcm,4 , q1 q2 q3 q5 rcm,5 ) Using (3.3.31), we can calculate the kinetic energy metric for the open-chain multibody system. In matrix form we have the following equation for the tangent map Tq (LF (q)−1 F ) : Tq Q → Lie(P)     −1 Adrcm,1 ··· 0 Tq1 (Lq1−1 ◦ ι1 ) · · · 0  .   ..  .. .. .. ..  Jq  , .. Tq (LF (q)−1 F ) =  . . . . .     −1 0 · · · Adrcm,5 0 · · · Tq5 (Lq5−1 ◦ ι5 )

75

Chapter 3. Reduction of Holonomic Multi-body Systems where we have 

id6 Adq2−1

06×6 id6

06×6 06×6

    Jq =  Ad(q2 q3 )−1 id6 Adq3−1  Ad(q q q )−1 Ad(q q )−1 Ad −1 2 3 4 3 4 q4 

Ad(q2 q3 q5 )−1 Ad(q3 q5 )−1 Adq5−1

 06×6 06×6  06×6 06×6    06×6 06×6  ,  id6 06×6   06×6 id6

and where id6 is the 6 × 6 identity matrix. Let us denote the standard basis for se(3) by {E1 , · · · , E6 }, such that  0  0 E1 =  0  0  0  0 E4 =  0  0

0 0 0 0 0 0 1 0

  1 0 0    0  , E2 = 0 0 0 0 0   0 0 0   0 0 0    −1 0  , E5 =  0 −1 0 0   0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

  0 0 0    1  , E3 = 0 0 0 0 0   0 0 0   1 0 0    0 0  , E6 = 1 0 0 0   0 0 0

0 0 0 0

 0  0  1  0

−1 0 0 0

0 0 0 0

 0  0  0  0

Using the introduced joint parameters, we have the following equalities:

Tq1 (Lq1−1

Tq2 (Lq2−1

Tq3 (Lq3−1 Tq4 (Lq4−1 Tq5 (Lq5−1

 −1 RZ−1 (θZ )RX (θX )RY−1 (θY ) 03×3     cos(θ ) cos(θ ) sin(θ ) 0 Z X Z ◦ ι1 ) =     03×3 − sin(θZ ) cos(θX ) cos(θZ ) 0  0 − sin(θX ) 1   −l1 sin(ψY ) 0 −l1    0 0 0     l cos(ψ ) cos(ψ ) −l sin(ψ ) 0   1 Y Z 1 Z ◦ ι2 ) =  , − cos(ψY ) cos(ψZ ) sin(ψZ ) 0     cos(ψ ) sin(ψ ) cos(ψZ ) 0  Y Z   − sin(ψY ) 0 1 h iT ◦ ι3 ) = 0 0 l1 + l2 1 0 0 , h iT ◦ ι4 ) = 0 0 l1 + l2 + l3 1 0 0 h iT ◦ ι5 ) = 0 0 l1 + l2 + l3 1 0 0 .

   ,  


76

Note that ∀r0 ∈ SE(3) that is in the following form (R0 ∈ SO(3) and p0 = [p0,1 , p0,2 , p0,3 ]T ∈ R3 ) # " R0 p0 r0 = , 01×3 1 we calculate the Adr0 operator by "

# R0 pe0 R0 Adr0 = , 03×3 R0 where 

 0 −p0,3 p0,2   pe0 =  p0,3 0 −p0,1  −p0,2 p0,1 0 is a skew-symmetric matrix. We choose the standard basis {E1 , · · · , E6 } for se(3). For this case study, the left-invariant metric h = h1 ⊕ · · · ⊕ h6 on P is identified, in the above basis, by the following metrics on the Lie algebras of copies of SE(3) corresponding to the bodies:   mi id3 03×3      j 0 0 x,i , he,i =    0  0 j 0   3×3 y,i   0 0 jz,i where i = 1, · · · , 5, id3 and 03×3 are the 3 × 3 identity and zero matrices, respectively, mi is the mass of Bi , and (jx,i , jy,i , jz,i ) are the moments of inertia of Bi about the X, Y and Z axes of the coordinate frame attached to the centre of mass of Bi . Note that we chose this coordinate frame such that its axes coincide with the principal axes of the body Bi . For the body Bi (i = 2, · · · , 5), since we assume a symmetric shapes with Yi -axis being the axis of symmetry, we have jx,i = jz,i . Finally, in the coordinates chosen to identify the configuration manifold (joint parameters), we have the following matrix form for FLq     he,1 · · · 0 K11 (q) · · · K15 (q)  .  ..  ..  .. ..  Tq (LF (q)−1 F ) =  ... .. FLq = Tq∗ (LF (q)−1 F )  . . . .     , 0 · · · he,5 K51 (q) · · · K55 (q) and the kinetic energy is calculated by 1 ˙ Kq (q, ˙ q) ˙ = q˙T FLq q, 2


77

where, with an abuse of notation, q˙ is the vector corresponding to the speed of the joint parameters. We assume zero potential energy for this holonomic open-chain multi-body system, Hence, we have the Hamiltonian of the system as 1 H(q, p) = pT FL−1 q p, 2 where p is the vector of generalized momenta corresponding to the joint parameters. In the following, we derive the reduced Hamilton’s equation for this system, with h iT the initial total momentum µ = 0 µ1 0 µ2 0 0 ∈ se∗ (3) represented in the dual of the standard basis for se(3). That is, the system has a constant linear momentum in the direction of Y0 , equal to µ1 , and a constant angular momentum in the direction of X0 , equal to µ2 . The kinetic energy (and hence the Hamiltonian) of the this multibody system is invariant under the action of G = Q1 = SE(3). The isotropy group corresponding to µ is      



cos(θY )

0 sin(θY )

µ2 µ1

  0 1 0 Gµ = h =   µ2   − sin(θY ) 0 cos(θY ) −2 µ1    0 0 0

        y  ∈ SE(3) ,  sin2 (θY /2)      1 sin(θY )



which is a Lie subgroup of G, and it is isomorphic to SO(2)×R. Now, consider the action of G = SE(3) by left translation on Q1 . Using the joint parameters, ∀(x0 , y0 , z0 , θX,0 , θY,0 , θZ,0 ) ∈ G we have h iT h iT Φ(x0 ,y0 ,z0 ,θX,0 ,θY,0 ,θZ,0 ) (q) = (RY (θY,0 )RX (θX,0 )RZ (θZ,0 ) x y z + x0 y0 z0 , RY (θY,0 )RX (θX,0 )RZ (θZ,0 )RY (θY )RX (θX )RZ (θZ ), q) where q = (ψX , ψY , ψZ , ψ1 , ψ2 , ψ3 ). We have the principal G-bundle π : Q → Q = Q2 × · · ·×Q5 , and using the joint parameters its corresponding principal connection A : T Q → se(3) is defined by (3.3.37) 

 g x  h  i RY (θY )RX (θX )RZ (θZ )   R (θ )R (θ )R (θ ) y   Y Y X X Z Z   Aq =   Tq1 Lq1−1 Aq , z   03×3 RY (θY )RX (θX )RZ (θZ )

78

Chapter 3. Reduction of Holonomic Multi-body Systems where we have g   x 0 −z y     0 −x , y  =  z z −y x 0

Tq1 Lq1−1

 −1 R−1 (θZ )RX (θX )RY−1 (θY ) 03×3    Z  cos(θ ) cos(θ ) sin(θ ) 0 Z X Z =    03×3 − sin(θZ ) cos(θX ) cos(θZ ) 0  0 − sin(θX ) 1 h i Aq = K 11 (q)−1 K 12 (q) · · · K 11 (q)−1 K 1N (q) ,

   ,  

where K 1i (q) = K1i (e1 , q) for i = 1, · · · , N , and consequently, the horizontal lift map hlq : Tq Q → Tq Q is       −   hlq =     

RY (θY )RX (θX )RZ (θZ )

03×3





 03×3

   cos(θZ ) − sin(θZ ) 0  Aq     sin(θZ )/ cos(θX ) cos(θZ )/ cos(θX ) 0  ,  sin(θZ ) tan(θX ) cos(θZ ) tan(θX ) 1  id6

where id6 is the 6 × 6 identity matrix. Then, we use the principal bundle π e : Q → Q/Gµ to introduce the local trivialization of Go = Q1 . The Lie algebra of Gµ as a vector subspace n µ2 of se(3) is spanned by E2 , µ1 E1 + E5 , and a complementary subspace to this subspace is spanned by {E1 , E3 , E4 , E6 }. Now, ∀e q1 ∈ Uµ ⊂ Q1 /Gµ we introduce the embedding χµ : Uµ ,→ Q1    x    RX (θX )RZ (θZ )  0    , χµ (e q1 ) =   z    01×3 1 which identifies the elements of Q1 /Gµ by elements of an embedded submanifold of Q1 ,


79

and in the local coordinates its induced map on the tangent bundles is  1  0  0  Tqe1 χµ =  0  0  0

0 0 1 0 0 0

0 0 0 1 0 0

 0  0  0  . 0  0  1

Subsequently, we define the local trivialization of the principal bundle π e : Q → Q/Gµ by χ e : Gµ × Uµ → Q1 χ e((h, qe1 )) = hχµ (e q1 ), and its induced map on the tangent bundles (in the local coordinates) is calculated as   cos(θY ) sin(θY ) 0 0 0 ( µµ12 + z) cos(θY ) − x sin(θY )   1 0 0 0 0 0   0 −( µ2 + z) sin(θ ) − x cos(θ ) − sin(θ ) cos(θ ) 0 0   Y Y Y Y µ1 T(h,eq1 ) χ e= , 0 0 0 0 1 0    0 1 0 0 0 0   0 0 0 0 0 1 where we use (y, θY ), (x, z, θX , θZ ), and (x, y, z, θX , θY , θZ ) as the local coordinates for the manifolds Gµ , Q1 /Gµ , and Q1 , respectively. Accordingly, we can calculate the map Aµqe : T(eq1 ,q) (Uµ × Q) → Lie(Gµ ) using the following equalities: h Gµ e 1Gµ (e e 1Q1 /Gµ (e e 1Gµ (e e 12 Aµqe := K (e q) · · · q) K q )−1 K q )−1 K

i e 1Gµ (e e Gµ (e , K q )−1 K q ) 1N

" G # Q /G K1 µ ((h, qe)) K1 1 µ ((h, qe)) ∗ = T(h,e e (K11 (e χ(h, qe))) T(h,eq1 ) χ e, q1 ) χ Q /G G K2 µ ((h, qe)) K2 1 µ ((h, qe)) " # G G h i K12µ ((h, qe)) · · · K1Nµ ((h, qe)) ∗ = T χ e . K (e χ (h, q e )) · · · K (e χ (h, q e )) 12 1N (h,e q1 ) Q /G Q /G K121 µ ((h, qe)) · · · K1N1 µ ((h, qe)) G Q /G e 1Gµ (e e 1Q1 /Gµ (e e Gµ (e And, we have K q ) = K1 µ ((eµ , qe)), K q ) = K1 1 µ ((eµ , qe)), and K 1i q ) =

80

Chapter 3. Reduction of Holonomic Multi-body Systems G

K1iµ ((eµ , qe)) for all i = 2, · · · , N . We also have the reduced Hamiltonian on [T π e(V)]0 : #T # " " T T µ µ Ad Ad 1 (eµ ,e q1 ) (eµ ,e q1 ) e pqe) = , FL−1 H(e (e1 ,q) 2 p + AqT AdT(eµ ,eq1 ) µ p + ATq AdT(eµ ,eq1 ) µ where

(3.4.48)

    0  T T RZ (θZ )RX (θX ) 03×3 µ1     g 0    x  AdT(eµ ,eq1 ) µ =  .     T T T T  −RZ (θZ )RX (θX ) 0  RZ (θZ )RX (θX ) µ2   0 z   0

In order to calculate the 2-form coordinates:  1  0  0  Tχµ (eq1 ) Rχµ (eq1 )−1 (Tqe1 χµ ) =  0  0  0 

 T 0   µ1    0   Fqe1 :=   µ2    0   0

Ξµ , we compute the following matrices in the local

 0 z sin(θX )  z −x cos(θX )  0 −x sin(θX )   ,  1 0  0 − sin(θX )   0 cos(θX )  g x    RX (θX )RZ (θZ )   0 RX (θX )RZ (θZ ) ,  Adχµ (eq1 ) =   z   03×3 RX (θX )RZ (θZ ) h i Dqe : = −Aµqe + Tχµ (eq1 ) Rχµ (eq1 )−1 (Tqe1 χµ ) Adχµ (eq1 ) Aq ,  T µ1 cos(θX ) sin(θZ )     µ1 cos(θX ) cos(θZ )     −µ1 sin(θX )   Adχµ (eq1 ) =   .  µ1 (z cos(θZ ) − x sin(θX ) sin(θZ )) + µ2 cos(θZ )    −µ (z sin(θ ) + x cos(θ ) sin(θ )) − µ sin(θ ) Z Z X 2 Z   1 −µ1 x cos(θX ) 0 0 1 0 0 0


81

Finally, we have the following expression for the 2-form Ξµ : 6 XX

Ξµ =

i

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