Modeling and Optimal Control of Hybrid Rigidbody

Modeling and Optimal Control of Hybrid Rigidbody Mechanical Systems Kerim Yunt and Christoph Glocker Swiss Federal Institute of Technology, Center of Mechanics Tannenstr. 3, 8092 Zurich, Switzerland

Abstract. A measure differential inclusion (MDI) based modeling approach for rigidbody mechanical systems will be introduced, that can exhibit autonomous or controlled mode transitions, accompanied by discontinuities on velocity and acceleration level. The hybrid optimal control necessitates the consideration of an uncommon concept of control, namely, controls of unbounded, impulsive and set-valued type. Examples to manipulators and wheeled robots are presented. Keywords: Impulsive Optimal Control, Impactive Systems, non-smooth analysis, hybrid, mechanical systems.

1

Introduction

A measure differential inclusion (MDI) based modeling approach for rigidbody mechanical systems will be introduced, that can exhibit autonomous or controlled mode transitions, accompanied by discontinuities on velocity and acceleration level. The introduced framework will have the ability to model and control hybrid mechanical systems with discontinuous transitions among different system modes. Modeling of rigidbody Lagrangian systems as Linear Complementarity Problem (LCP) will be presented, and the associated optimal control problem of hybrid Lagrangian systems will be stated. In recent years, several works have been presented in order to establish the relations between complementarity dynamical systems and hybrid systems. There are general results in literature that investigate the relation between different representations of dynamical systems and hybrid systems such as [8], [13]. The properties of the optimal control problem derive from the underlying modeling approach. Optimal Control of hybrid systems is addressed in several publications such as [4], [5], [23] and [25] on various field of applications, in which the modeling is based on approaches to the ones similar as in [3], [6]. The treatment of discontinuous transitions and the combinatorial nature of mode sequencing are partially treated in these publications about optimal control, due to the modeling approach chosen. In this work, the MDI modeling of mechanical hybrid systems will be proposed and the suitability from the viewpoints above presented. There are some works devoted to the complementarity modeling and optimal control of mechanical systems as given in references [26], [27], [28], [29]. The main issue in the optimal control of hybrid mechanical systems has been the blending of impact mechanics with A. Bemporad, A. Bicchi, and G. Buttazzo (Eds.): HSCC 2007, LNCS 4416, pp. 614–627, 2007. c Springer-Verlag Berlin Heidelberg 2007 

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impulsive optimal control. Indeed, the optimal control of such systems entails unavoidably impulsive control. References [18] and [2] consider impulsive systems and impulsive differential inclusions from the viewpoint of hybrid systems. There has been much interest in the research of modeling discontinuities and nonlinearities in multibody systems for which a compact overview is provided in [7]. The discontinuities arising from impacts and stick-slip transitions are primarily contact phenomena, which concurre temporally and spatially. The spatial concurrence of discontinuity is due to the fact that discontinuities on velocity level (e.g. collisions) can occur along with discontinuities on acceleration level (e.g.stick-slip transitions). Temporal concurrence is caused by collision, shock and impact phenomena occuring at multiple locations of the system at the same time as well as stick-slip transitions. Recent research showed that such rigidbody systems can best be described by variational inequalities which lead to nonlinear and linear complementarity type of systems to be solved in order to obtain the accelerations/velocities and forces. In the modeling considered in this work, impulsive forces can arise autonomously, due to effects such as collisions or controlled/nonautonomously, due to actions such as blocking some DOF. The hybrid optimal control requires the consideration of an uncommon concept of control, namely, controls of unbounded, impulsive and set-valued type. The existence of force and impulsive/discrete type of controls will through the solution of the complementarity problem take influence on the course of system trajectories. The presence of impulsive forces require to solve impact equations and constitutive laws that relate post- and pre-impact velocities of the system. The topic of impacts with and without friction is investigated in references [15], [16].

2

The Measure Differential Inclusion Representation of Hybrid Mechanical Systems

One of the cornerstones of non-smooth modeling of mechanical systems has been the introduction of the MDI concept that considers the equations of motion subject to variational inequalities as a balance of measures. The concept of MDI and its applications to mechanics stems from J. J. Moreau, and related works of him are given in [20], [21]. The application of the measure-differential inclusion approach to rigid-body mechanics can be overviewed in [14]. ˙ q ¨ represent the position, velocity and acceleration in the generalized Let q, q, coordinates of a scleronomic rigid body mechanical system with n degrees of freedom (DOF), respectively. The equations of motion (EOM) are obtained by using the well-known Lagrange II formalism for the smooth dynamics :       ∂T ∂V d ∂T − + −f =0 . (1) dt ∂ q˙ ∂q ∂q Here T denotes the total kinetic, and V the total potential energy of the system. The unilateral forces, which are non-potential in the classical sense, are incorporated by the appropriate generalized force directions in the generalized force

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vector f . The controls will also be introduced into the equations of motion by means of the structure of f . The tangential and normal local kinematics need to be defined in order to relate the contact distance to the set-valued force element. For the detection of the closing of a contact let the vector gu (q) entail the normal contact distances between the rigid bodies in the system which are always non-negative. The normal and tangential contact velocities γ u and γ s are defined as: ˙ γ u = WuT q,

˙ γ s = WsT q,

(2)

respectively, and γ u is obtained as the total time derivative of gu (q). The normal and tangential contact accelerations are given by the following equations: ¨ + ωu, γ˙ u = WuT q

¨ + ωs . γ˙ s = WsT q

(3)

Here Wu (q) and Ws (q) represent the generalized force directions of normal and tangential contact forces. In order to formulate the contact situations properly following index sets will be defined: IG = {1, 2, ..nG }, IS = {i ∈ IG | gui = 0}, IN = {i ∈ IS | γ ui = 0} .

(4)

IG denotes the set of all contacts that can occur on position level of the nonsmooth mechanical system. IS denotes the set of all contacts that are closed on position level. IN denotes the set of all contacts such that normal contact velocity and normal contact distance equal to zero. Further, the definition of following index sets are necessary: Cui = {λui | λui ≥ 0, ∀ i ∈ IG },

Csi = {λsi | |λsi | ≤ μi λui , ∀ i ∈ IS },

(5)

where the vectors λs , λu are the tangential and the normal contact forces, respectively and μi denotes the friction coefficient at contact i. The differential inclusion of a general non-autonomous mechanical system Sa subject to spatial Coulomb friction and unilateral contact forces in the absence of impacts can then be stated as: M u˙ − h − Ws λs − Wu λu − B τ = 0, ∀ i ∈ IN , γ˙ si ∈ −NCsi (λsi ),

γ˙ ui ∈ −NCui (λui ),

(6) ∀ i ∈ IN .

M(q) is the symmetric positive-definite (PD) mass matrix and h(q, u) represents the vector with gyroscopic and coriolis accelerations along with smooth potential forces such as gravity. B(q) entails columnwise the generalized force directions of controls. The vector τ ∈ IRd denotes the vector of controls. The normal cone to a set C at the point x ∈ C is given by NC (x) in the sense of convex analysis [24]. The normal cone representation in contact mechanics and friction problems are first treated in [21] and [1], respectively. In order to treat impacts in this framework constitutive laws will have to be introduced. In [17] a representation of Moreau’s impact law in local contact coordinates has been derived, showing that: 0 ≤ Λi

⊥

(γi+ + i γi− ) ≥ 0,

∀ i ∈ IS .

(7)

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Here i denotes the restitution coefficient. These concepts can be extended to the tangential and normal impact by introducing following variables ξsi and ξui , which vectorially are given by the following expressions if the Newton-like impact law is used: − − ξu = γ + (8) ξs = γ + s + s γ s , u + u γ u . The diagonal matrices s and u have as entries the tangential and normal restitution coefficients. The dynamics of a mechanical System Sv can be formulated on the measure-differential level: M du − h dt − Ws dΛs − Wu dΛu − B dΓ = 0, ξui ∈ −NCui (dΛui ), ∀ i ∈ IS , ξ si ∈ −NCsi (dΛsi ),

(9) ∀ i ∈ IS .

Here dΛs and dΛu are the differential measures of the tangential and normal contact forces, respectively. The vector dΓ denotes the differential measure of controls, which is unbounded, and can therefore induce impulsions. The differential measure of the generalized velocity is given by du. 2.1

The Linear Complementarity (LCP) Representation of MDI

If the normal cones in the force laws are finitely generated, then the determination of accelerations and forces can be represented in a linear complementarity form. This is the case, when the line of action of the friction forces are known, which is not the case for spatial friction. In the sequel between two types of frictional contacts is distinguished, because of their differing roles in the LCP. The set INA denotes the set of contacts at which the normal force, that induces the friction force is known a priori, whereas INC denote the set of contacts, where the value of the normal contact force depends on the friction force value at a given moment. The definition of following index sets are made: Cui = {λui | λui ≥ 0,

∀ i ∈ IG },

Csi = {λsci | |λsci | ≤ μi λui , ∀ i ∈ INC ⊂ IN }, Cai = {λsai | |λsai | ≤ aui , ∀ i ∈ INA ⊂ IN },

(10) (11) (12)

  such that INA INC = ∅, INA INC = IN , N (INA ) = w, N (INC ) = v, N (IN ) = m and N (IS ) = k. Here, the vectors λsa and λsc , denote tangential contact forces of Tresca-type and Coulomb-type, respectively. The entity aui denotes the apriori known sliding contact force at contact i. Further, for both type frictional contacts between sticking and sliding contacts will be distinguished,  which gives rise to  the definition of new index sets such that INA = H INA G INA and INC = H INC G INC are valid. The number of elements of these sets are related to each other by N (INA ) = w = N (H INA ) + N (G INA ) = p + s,

(13)

N (INC ) = v = N (H INC ) + N (G INC ) = r + t .

(14)

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Here subscript H refers to sticking and G refers to sliding. Based on this classifi T and the related relative cation, the vector γ s is decomposed as γ s = γ sa γ sc contact accelerations are given by: T ¨ + ω sa , q γ˙ sa = Wsa

T ¨ + ω sc . q γ˙ sc = Wsc

(15)

The force λsc can be decomposed into sliding contact forces G λsc and sticking contact forces H λsc at a given instant uniquely. The tangential contact forces and relative tangential contact velocities of contacts with Tresca-type friction can be decomposed analogously. The differential inclusion of a general non-autonomous mechanical system Sa subject to planar friction (known line of action) and unilateral contact forces is stated as: ¯ − H Wsc H λsc − H Wsa H λsa − Wu λu − B τ = 0, M u˙ − h

(16)

λui ∈ −Upr(γ˙ ui ), ∀ i ∈ IN ,  λscj ∈ −μj λuj Sgn γ˙ scj , ∀ j ∈ H INC , λsak ∈ −auk Sgn (γ˙ sak ) , ∀ k ∈ H INA . ¯ = h+ G Wsc G λsc + G Wsa G λsa , enables the incorporation of all sliding Defining h ¯ In the sequel for ease of notation the entities contact forces in the vector h. n×r r , H λsc ∈ IR , H Wsa ∈ IRn×p , H λsa ∈ IRp will be represented by H Wsc ∈ IR Wsc , λsc , Wsa , λsa , respectively. Here, the matrix Wu has dimensions n × m. In the sequel, the linear complementarity problem will be constructed. The setvalued signum type friction force characteristics can be decomposed into two unilateral force laws by introducing new variables as depicted in Fig. (1) and given below: γ˙ sc = γ˙ rc − γ˙ lc , γ˙ sa = γ˙ ra − γ˙ la ,

λlc = μ λu − λsc , λla = a − λsa ,

λrc = μ λu + λsc ,

λra = a + λsa ,

(17) (18)

along with following nonnegativity and complementarity conditions: γ˙ u ≥ 0,

γ˙ rc ≥ 0,

γ˙ lc ≥ 0,

γ˙ ra ≥ 0,

γ˙ la ≥ 0,

λu ≥ 0, λrc ≥ 0, λlc ≥ 0, λra ≥ 0, λla ≥ 0, γ˙ u λu = 0, γ˙ rc λrc = 0, γ˙ lc λlc = 0, γ˙ ra λra = 0, γ˙ la λla = 0 .

(19) (20) (21)

Here μ ∈ IRr×r is a diagonal matrix with friction coefficients. The vector a ∈ IRp denotes the normal force vector for frictional contacts of Tresca-type. Further λrc , λlc ∈ IRr ; λra , λla ∈ IRp ; γ rc , γ lc ∈ IRr and γ ra , γ la ∈ IRp and are related entities to sticking contacts. The generalized accelerations of the system Sa are given by:  ¯ u˙ = M−1 h(q, u) + Wsc (q)λsc + Wsa (q)λsa + Wu (q)λu + B(q) τ . (22) Insertion of (22) in the expressions for relative contact accelerations given in equations (3) and (15) reveals following set of equations:  ¯ + Wsc λsc + Wsa λsa + Wu λu + B τ + ω u , γ˙ u = WuT M−1 h (23)

Modeling and Optimal Control of Hybrid Rigidbody Mechanical Systems

 T ¯ + Wsc λsc + Wsa λsa + Wu λu + B τ + ωsc , γ˙ sc = Wsc M−1 h  T ¯ + Wsc λsc + Wsa λsa + Wu λu + B τ + ω sa , γ˙ sa = Wsa M−1 h

619

(24) (25)

which can be arranged in the form of a linear complementarity problem [10], [22]: y = A x + b, y 0, x 0, x y = 0 . (26) Where the complementarity vectors x and y are identified as: T T   x = λu λrc λra γ˙ lc γ˙ la , y = γ˙ u γ˙ rc γ˙ ra λlc λla ,

(27)

respectively. The matrix A ∈ IR(m+2p+2r)×(m+2p+2r) and vector b ∈ IR(m+2p+2r) are given by: ⎡ T −1 ⎤ Wu M (Wu − Wsc μ) WuT M−1 Wsc WuT M−1 Wsa 0m×r 0m×p T −1 T −1 T −1 ⎢ Wsc ⎥ ⎢ T M−1(Wu − Wsc μ) WTsc M−1Wsc WTsc M−1Wsa Ir×r 0r×p ⎥ ⎢ , A = ⎢ Wsa M (Wu − Wsc μ) Wsa M Wsc Wsa M Wsa 0p×r Ip×p ⎥ ⎥(28) ⎣ 2μ −Ir×r 0r×p 0r×r 0r×p ⎦ 0p×m 0p×r −Ip×p 0p×r 0p×p  T T T T fq¨ + ω sc Wsa fq¨ + ωsa 0r×1 2a , b = Wu fq¨ + ω u Wsc (29) ¯ where fq¨ is given by fq¨ = M−1 (h+B τ ) and I is an identity matrix of appropriate size. Making use of the formulation given in (7), these concepts can be extended

Fig. 1. Decomposition of the set-valued Sgn relation into two Unilateral Primitives

to the tangential and normal impact by introducing following variables ξsci , ξsai and ξui , which vectorially are given by the following expressions if the Newtonlike impact law is used: ξ sa = γ + sa ,

− ξ sc = γ + sc + sc γ sc ,

− ξu = γ + u + u γ u .

(30)

The matrices u and sc are diagonal matrices with normal and tangential restitution coefficients, respectively. The tangential impact in contacts with Coulomb type friction is induced by the dependence on the normal impactive force at the

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relevant contact. The dynamics of a mechanical System Sv can be formulated on the measure-differential level: M du − h dt − Wsa dΛsa − Wsc dΛsc − Wu dΛu − B dΓ = 0,

(31)

dΛui ∈ −Upr(ξui ), ∀ i ∈ IS , dΛsai ∈ −dAi Sgn(ξsai ), ∀ i ∈ ISA , dΛsci ∈ −μi dΛui Sgn(ξsci ), ∀ i ∈ ISC .   such that ISA ISC = ∅, ISA ISC = IS and N (ISA ) = l, N (ISC ) = z. As a consequence of the changing force laws in (31), the dimensionality of man×l is given by trices also change. In  equation (31) , the matrix Wsa ∈ IR Wsa = col{H Wsa } col{G Wsa }, col{·} denotes the column set of the matrix in argument. Analogously, the matrix Wsc ∈ IRn×z is given by Wsc =  col{H Wsc } col{G Wsc }. A linear complementarity problem on the measuredifferential level can be formulated in the form of (26). The set-valued signum relations in the MDI representation (31) will again be decomposed in analogy to Fig. (1). After introduction of slack variables, the complementarity vectors x ∈ IRk+2l+2z and y ∈ IRk+2l+2z are identified as:   T T x = dΛu dΛrc dΛra ξ lc ξ la , y = ξ u ξ rc ξra dΛlc dΛla , (32) respectively. The vector b ∈ IRk+2l+2z of the linear complementarity problem is given by: ⎡ ⎤ (I + u )WuT u− + WuT M−1 (h dt + B dΓ ) T − T ⎢ (I + sc )Wsc u + Wsc M−1 (h dt + B dΓ ) ⎥ ⎢ ⎥ T − T −1 ⎢ ⎥. b=⎢ Wsa u + Wsa M (h dt + B dΓ ) (33) ⎥ ⎣ ⎦ 0z×1 2dA The matrix A remains the same in the structure as in (28) but the dimensionality of the associated elements vary since the contact force laws are modified as in equation (31). As illustrated, for contacts which are closed in normal direction, the MDI representation removes the distinction between sliding and sticking, and considers all contacts closed on position level, which reduce the burden of management of index sets. 2.2

Statement of the Hybrid Optimal Control Problem

The non-smooth optimal control problem subject to a mechanical dynamical system described as a measure-differential inclusion can be stated as follows:  tf + min J (τ , dΓ , tf ) = Φ(q(tf ), u (tf ), tf ) + g(u+ , q, τ ) dt, (34) t0

du = f (u+ , q, τ , t) dt + pΛ (q, t) dΛ + pΓ (q, t) dΓ , (dΛ, dΓ ) ∈ Υ( dΛ, dΓ , q, u+ ),

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621

Π(u+ , q, τ ) ≤ 0, Ψ(q(t0 ), u− (t0 ), q(tf ), u+ (tf )) = 0, t0

fixed, tf

free,

t ∈ [t0 , tf ],

with absolutely continuous positions q ∈ IRn , right continuous bounded variation (RCBV) generalized velocities u ∈ IRn , control variables τ ∈ IRm , unilateral force differential measures dΛ ∈ IRp , impulsive and set-valued control differential measures dΓ ∈ IRr . Further, the Lebesgue measurable system dynamics is given by f : IRn × IRn × IRm × IR → IRn . The set-valued variational constraints on the measure variables Υ : IRp × IRr × IRn × IRn → IRk , influence matrix of contact force differential measures pΛ ∈ IRn×p , influence matrix of control differential measures pΓ ∈ IRn×r , state and control constraints Π : IRn × IRn × IRm → IRl and boundary constraints Ψ : IRn × IRn × IRn × IRn → IRq are incorporated in the optimal control problem. The end state cost Φ : IRn × IRn × IR → IR, integrand of the cost functional g : IRn × IRn × IRm → IR constitute the goal function to be minimized.

3 3.1

Examples Example: Two-DOF Underactuated Planar Manipulator with Impactively Blockable DOF

Fig. 2. The 2-DOF planar manipulator with 1 blockable DOF

The first application will deal with underactuated manipulators with blockable degrees of freedom (DOF). The example mechanical system can be seen in Fig. (2) and is treated in detail in [28]. The robot has two rotational degrees of freedom denoted by α and β. The DOF α is controlled continuously in a single valued manner, whereas DOF β can be blocked meaning that any given position the relative angular velocity β˙ − α˙ can be reduced to zero immediately. In the presented maneuver in Fig. (3), it is supposed to start at α0 = 0 rad and β0 = 0 rad from standstill, and to reach the final position αf = π rad and βf = π rad, time-optimally. The DOF β is blocked at times t1 = 0 s, t3 = 1.07 s, t5 = 2.39 s

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Fig. 3. The generalized and relative velocities, generalized positions of the optimal maneuver

Fig. 4. The differential measures of the controls

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and is released at times t2 = 0.41 s, t4 = 1.84 s, whereas the total maneuver takes 4.53 s. When the link is blocked, the whole system possesses one mechanical DOF, whereas when released it has two DOF. The transitions at times t3 , t5 are impactive as can be seen in Fig. (3). The control history has a time-optimal bang-bang character as can be seen in Fig. (4). The blocking control is set-valued because its value is within a set in a phase of blocking, unbounded; because it has to bring the link immediately to zero relative velocity and therefore impulsive as can be seen in Fig. (4). 3.2

Example: Three-Wheeled Differential Drive Robot

Fig. 5. Contact forces and motor moments on the simplified model

Fig. 6. The generalized coordinates of the wheeled robot

The second application will be about a three-wheeled robot with stick-slip transitions at the wheel contacts. The differential-drive robot is a three-wheeled actuated robot of which the rear wheels are actuated and controlled separately contrary to the front wheel which is neither actuated nor steered. The properties of the optimal control of this nonholonomically constrained system are studied in detail in [27] and [29]. A rigid-body mechanical model is used, in which the friction between wheels and ground is modeled as isotropic spatial Coulomb friction. The non-steered unactuated front wheel is replaced by a stick as a simplification, removing two degrees of freedom (DOF) to be modeled. The rotational inertia of the total actuation consisting of the components of motor rotors and transmissions are added to the rotational inertias of the wheels. Under the given assumptions there are five mechanical DOF necessary in order to model the mechanical system depicted in Figs. (5) and (6). The following set of generalized coordinates and velocities are used to describe the system:     ˙ ψ˙ L , ψ˙ R . (35) qT = x, y, φ, ψL , ψR , q˙ T = x, ˙ y, ˙ φ, These are the planar translational coordinates of center of mass (CM) of the chassis x and y as well as the planar orientation of the chassis φ, the angular

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Table 1. Relation of different modes to the contact state and relative contact velocities K γ Rx K γ Ry K γ Lx K γ Ly

= 0 =0 =0 =0

= 0 = 0 =0 =0

6

6

5

5

4

4 position in m

position in m

Modes 5-DOF mode 3R-DOF mode 3L-DOF mode 2-DOF mode

3 2

= 0 =0 = 0 =0

3 2 1

1 2−DOF mode 3−DOF mode 5−DOF mode

0 −1

= 0 =0 =0 =0

−3

−2

−1

0 1 2 position in m

3

4

Fig. 7. Maneuver A: Number of DOF during the maneuver

2−DOF mode 3−DOF mode 5−DOF mode

0 −1

−3

−2

−1

0 1 2 position in m

3

4

Fig. 8. Maneuver B: Number of DOF during the maneuver

positions ψR and ψL of the wheels with respect to the chassis frame. There are two coordinate systems, namely, the inertial coordinate system (subscript I ), a body attached coordinate system for chassis (subscript K ). The K-system of the chassis has its origin in the CM SK . When the axial slip constraints and rolling constraints are fulfilled, the non-actuated mechanical system possesses two DOF. If both wheels slide it is a mechanical system with five DOF. In the three-DOF mode one wheel contact sticks and the other wheel slides, meaning that the axial slip constraints are fulfilled but one wheel does not fulfill the rolling condition. If the actuated mechanical system is considered then one observes that the system is fully actuated when it moves in the two-DOF mode. In the other modes it is an underactuated system with less actuators than mechanical degrees of freedom. The transitions among all four operating modes are possible. The modes are classified according to relative contact velocities at the wheel contacts as represented in the chassis fixed frame in table 1. In Fig. (9) the contact velocities and forces for maneuver A are shown. In maneuver A the task is to reach the following end-point control-effort optimally. The sum of squares of the actuating torques is being minimised. The desired end state to be reached is   xf , yf , φf , ψLf , ψRf = 2, 5, − π2 , free, free . (36) The robot accomplishes this task in 4.05 seconds. In Maneuver B the robot is supposed to reach the same final end-state time-optimally. Maneuver B is characterized by a high dynamical activity in the orientation of the chassis.

Modeling and Optimal Control of Hybrid Rigidbody Mechanical Systems Normal contact forces

Magnitudes of relative contact velocities 8

vR

75

vL

6 [m/s]

70 [N]

625

65

vF

4

60 2 55 0

50 0

1 λ

NR

2 time λt NL[s]

3

4 λ

0

1

NF

Friction forces on the left wheel

2 time t [s]

3

Friction forces on the right wheel

30 20

20

10

10 [N]

[N]

4

0 −10

λ TRx λ TRy

0 −10

−20

λ TLx

−30

λ TLy

0

1

−20 2 time t [s]

3

4

0

1

2 time t [s]

3

4

Fig. 9. Maneuver A: Contact forces and contact relative velocities

The robot accomplishes this task in 4.03 seconds. These maneuvers are depicted in Figs. (7) and (8).

4

Discussion and Conclusion

The necessity to represent multibody systems as MDI’s emanates from several facts. The optimal control of hybrid rigidbody mechanical systems benefits from several aspects of the MDI approach, which are summarized below: a. The index sets that are used to take account of the behaviour of contacts on different levels such as position, velocity and acceleration for stick-slip transitions etc. is not managable for large systems with many contacts. The index-set reduction can be seen in going from the acceleration level representation in (6) to the measure-differential MDI representation in (9). b. The impacts, that may occur with or without collisions e.g. Painleve Paradox, velocity jumps due to C 0 constraints are a strong incentive to describe the mechanical systems as MDI. c. Systems which are zeno (e.g jumping ball on the ground) are problematic for event-driven schemes whereas MDI approach can handle them properly and realistically. d. The hybrid optimal control requires the consideration of an uncommon concept of control, namely, controls of unbounded, impulsive and set-valued type for non-autonomous impulsive transitions, such as sudden blocking of DOF, which can in a natural way be added to the MDI structure.

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e. As a novel property, the location and time of phase transitions where the system changes DOF is not prespecified but is determined as an outcome of the optimization. Though the underlying system might undergo structurevariant phase changes such as impactive phase transitions a mixed integer approach is not necessary. f. Numerical methods can be deviced that calculate the costate dynamics which itself is described as a MDI. The costate dynamics in the sense of optimal control is discontinuous and nonsmooth as well, as presented in [28] in the sense of optimal control.

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