The Power Dissipation Method and Kinematic Reducibility of

MURPHEY AND BURDICK. PREPRINT. SUBMITTED TOIEEE TRANSACTIONS ON ROBOTICS. 1

The Power Dissipation Method and KinematicReducibility of Multiple Model Robotic Systems

Submitted as a Regular paper.

Todd D. Murphey (corresponding author) and Joel W. BurdickEngineering and Applied ScienceCalifornia Institute of Technology

Pasadena, CA 91125murphey,[email protected]

Abstract—This paper develops a formal connection between the Power Dissipation Method and Lagrangian me-chanics, with specific application to robotic systems. Such a connection is necessary for understanding how someof the successes in motion planning and stabilization for smooth kinematic robotic systems can be extended tosystems with frictional interactions and overconstrained systems. We establish this connection using the idea of amultiple model system, and then show that multiple model systems arise naturally in a number of instances, in-cluding those arising in cases traditionally addressed using the Power Dissipation Method. We then give necessaryand sufficient conditions for a dynamic multiple model systems to be reducible to a kinematic multiple model sys-tem. We are particularly motivated by mechanical systems undergoing multiple intermittent frictional contacts,such as distributed manipulators, overconstrained wheeled vehicles, and objects that are manipulated by graspingor pushing. Examples illustrate how these results can provide insight into the analysis and control of physicalsystems.

Index Terms—quasi-static analysis, dynamics, contact modeling, frictional contacts, kinematic reducibility, mod-eling for control.


The Power Dissipation Method and KinematicReducibility of Multiple Model Robotic

SystemsT.D. Murphey and J.W. Burdick

Abstract—This paper develops a formal connection be-tween the Power Dissipation Method and Lagrangian me-chanics, with specific application to robotic systems. Such aconnection is necessary for understanding how some of thesuccesses in motion planning and stabilization for smoothkinematic robotic systems can be extended to systems withfrictional interactions and overconstrained systems. We es-tablish this connection using the idea of a multiple modelsystem, and then show that multiple model systems arisenaturally in a number of instances, including those arisingin cases traditionally addressed using the Power DissipationMethod. We then give necessary and sufficient conditionsfor a dynamic multiple model systems to be reducible to akinematic multiple model system. We are particularly moti-vated by mechanical systems undergoing multiple intermit-tent frictional contacts, such as distributed manipulators,overconstrained wheeled vehicles, and objects that are ma-nipulated by grasping or pushing. Examples illustrate howthese results can provide insight into the analysis and controlof physical systems.

I. I NTRODUCTION

Many mechanical systems, though intrinsically sec-ond order in their governing dynamics, can be adequatelydescribed by first order equations of motion. That is,one can often propose a “quasi-static” or “kinematic” ver-sion of the governing equations of motion for the pur-poses of system analysis or control design. The bene-fits of this simplification are numerous: the dimension ofthe state space drops by half, the control inputs go frombeing force inputs to being velocity inputs (which are of-ten more easily realized in practice), and the governingequations typically take a simpler form than the full dy-namic model. Additionally, kinematic systems, althoughpotentially nonlinear, do not typically involve drift terms.There is a greater quality and quantity of nonlinear con-trol results available for driftless systems, as compared tosystems with drift. See [3], [10], [16], [17], [22], [36],[41] for just a few examples.

This paper has several inter-related goals. One ofthe main technical goals of this paper is to determine the

This work was largely supported by the National Science Foundation(grant NSF9402726) through its Engineering Research Center (ERC)program.

formal conditions under which such reductions can beachieved formultiple model systems. In multiple modelsystems (see Section IV) the system’s governing equa-tions switch between several possible models that de-scribe the system’s evolution. This paper presents neces-sary and sufficient conditions for a multiple model systemto be kinematically reducible—i.e., the2nd-order dynam-ical models can be reduced to1st-order kinematic modelsof the form in Definition IV.1. The necessary and suffi-cient conditions for kinematic reducibility of smooth dy-namical systems were first developed by Lewis [23]. Oneof this paper’s contributions is the extension of kinematicreducibility theory to the multiple model case.

While our kinematic reducibility results can be ap-plied to a large class of problems, we are particularlymotivated by the multiple model systems that arise fre-quently in robotics practice. The multiple model frame-work has received an increasing amount of attention in thecontrol community recently [4], [19], [20], [18], so thereare many control results available for our use. There-fore, understanding the connection between problems inrobotics and the multiple model framework will be pro-ductive. Examples of multiple model systems includerobotic systems involving intermittent mechanical con-tacts, such as distributed manipulators, overconstrainedwheeled vehicles, and objects that are manipulated bygrasping or pushing (see Section X). A number of similarapproaches have been proposed or used to create quasi-static models of such systems. Most representative ofthese is the Power Dissipation Method (PDM) (see Sec-tion V) introduced by Alexander and Maddocks [3] inthe context of overconstrained wheeled vehicles. Peshkinalso used similar ideas in the study of pushed objects [39].Based on this method, one can develop first-order (orquasi-static) equations of motion for mechanical systemsthat undergo intermittent sliding contacts. We show inSection VII that solutions to the PDM are multiple modelsystems. We have used the PDM to model distributedmanipulation systems that generate motion via frictionalcontacts [34], [31]. The resulting multiple model descrip-tions are very amenable to control analysis [33], [30],


and the associated nonsmooth control laws worked wellin practice. See [34] for details.

As a second goal of this paper, we address a keyquestion: does the PDM produce models that are con-sistent with a complete dynamic (Lagrangian) analysis?The formalization of the PDM and the analysis of its rela-tionship to Lagrangian analysis are the other main contri-butions of this work. Formally, in Section VIII we showthat every solution to the power dissipation method is pre-cisely a reduction of a solution to the Lagrangian formu-lation. Moreover, this is true forall solutions, which isimportant, as solutions are not unique in either the powerdissipation method nor are they unique in the Lagrangianformulation (when nonsmooth interactions such as im-pacts and friction are taken into consideration).

The paper is organized as follows. To motivate ourresults, we first examine some examples of mechanismsthat naturally involve stick/slip phenomenon in Section II.Then we briefly review the classical Lagrangian approachin Section III before covering the basic ideas of the multi-ple model formalism in Section IV. We then specificallyaddress an example in Section VI using these ideas. InSection VII we cover characteristics of the power dissi-pation method and we then move on to reduction theoryfor multiple model systems in Section VIII. Section IXrelates solutions to the power dissipation method to so-lutions to the Lagrangian analysis. We end in Section Xwith a detailed look at several examples where we havefound our analysis practically useful.

II. EXAMPLES

To show the potential breadth of applications forour results, we summarize here four typical robotic andphysical systems to which our theory applies (Fig. 1):a wheeled bicycle, the Rocky 7 prototype of the NASAMars rover family, a distributed manipulation systemwhose function is to manipulate a planar object via roll-slide contacts, and a multi-fingered robotic hand. All ofthese systems are characterized by complex mechanicalinteractions involving contact mechanics and slip. Morespecifically, all of these systems can be modeled and an-alyzed using the multiple-model framework developed inthis paper.

Consider the bicycle of Fig. 1(a) . For simplicity,we assume that the bicycle is constrained to move alonga line, and that both wheels are actuated. (We will re-peatedly return to this example, as it exhibits many ofthe features that are relevant to our discussions). Apply-ing the exact same torque to both wheels is very diffi-cult task, and thus this bicycle would typically experiencesmall amounts of slipping in practice. More interestingly,this slipping is likely to change over time due to variabil-ity in contact friction characteristics, leading to a multiplemodel, or hybrid, mechanical system. The multiple model

NN

µµ

2 1

2 1

τ1τ2

Fig. 1. Here are a) a bicycle with both wheels driven, b) the Marsrover Rocky 7 Sojourner prototype, c) a distributed manipulation testbed developed at Caltech (see description below), and d) a hand capableof grasping objects

methodology introduced in this paper and companion pa-pers is well suited to analyze such systems.

The NASA Mars rover family members have six in-dependently driven wheels as well as two wheels inde-pendently steered. As discussed in [29] and reviewed inSection X, because this vehicle’s suspension is kinemat-ically overconstrained, some of these wheels are alwaysslipping, and it can be difficult to predict which wheelsslip at any given moment. There is already an exten-sive literature on wheeled vehicles, establishing control-lability based on a Lie Algebra Rank Condition (LARC)[21], [35], stability based on center manifold theory [41]


and hybrid systems theory [18], motion planning basedon Voronoi diagrams [9], and rapidly exploring randomtrees [10]. However, all of these methods assume thatthe vehicle motions are governed by smooth, kinematicequations of motion. Because of the inherent and unpre-dictable switches in slipping, the governing dynamics arenot smooth. Nevertheless, the methods developed in thispaper show that such vehicles are still kinematic systems,albeit nonsmooth ones. Moreover, in related work, wehave made progress on extending classical nonlinear con-trol concepts, such as the LARC, to the domain of mul-tiple model systems [32]. We will discuss this more inSection X-B.

Distributed manipulation has received recent atten-tion in the robotics community [6], [26]. Fig. 1(c) showsa distributed manipulation test-bed developed by the au-thors in which nine actuated wheels can be used to ma-nipulate planar objects set upon the manipulation sur-face. All of these wheels can be independently driven andsteered, giving the system 18 control inputs, with only theposition and orientation of the manipulated object as theoutput. Hence, this system is massively over-actuated.The idea of many actuated devices interacting with anobject to achieve some desired manipulation goal is ap-pealing partially because of its scalability and the possi-bility of using many inexpensive actuators rather than afew expensive ones. Moreover, micro-electromechanicalsystem (MEMS) fabrication technologies potentially en-able distributed manipulation to be a leading candidatefor micro-manipulation. We have shown in prior workhow distributed manipulators that employ frictional con-tacts fall into the multiple-model domain [34]. The multi-ple model kinematic reducibility theory developed in thispaper provides a simple but rigorous framework for thedesign of stabilizing control laws that take into accountthe non-smooth effects of friction. We have used kine-matic reductions both to show the potential shortcomingsof control laws based on smooth idealizations and to ex-plicitly compute stabilizing control laws that work wellexperimentally (see [34]).

Grasping and locomotion continue to be active ar-eas of robotics research. Current methods often use kine-matic models [16], [17] to represent the system dynamics,yet grasping implicitly contains many of the previouslymentioned difficulties. In particular, although stick/slipphenomena occur in a grasping problem, there are notvery convincing ways to show that the kinematic meth-ods typically used for grasping are robust with respect tothe variation in stick and slip states for a given contact.The analytical methods presented here create a methodfor analyzing these difficulties without resorting to dy-namic, second order analysis.

In Section X we will revisit these examples in orderto show how the kinematic reduction theory of this paper

can provide simplification or insight.

III. B ACKGROUND: LAGRANGIAN MODELS WITH

FRICTIONAL CONTACTS

This work has been largely motivated by the problemof modeling and controlling mechanical systems whichexperience multiple, possibly intermittent, contacts thatinvolve friction, particularly Coulomb friction. Clearly,the contacts place constraints on the system’s evolvingmotions. Constrained mechanical systems can be mod-eled using conventional Lagrangian mechanics throughthe use of Lagrange multipliers. Consider a generic me-chanical system with up ton frictional contacts betweenrigid body surfaces, where the contacts can be intermit-tently slide or stick. Such a system admits up to2n pos-sible contact states which represent all possible permuta-tions of sliding and sticking. LetL(q, q) denote the sys-tem’s Lagrangian (kinetic minus potential energy), whereq ∈ Q denotes the configuration of the mechanical sys-tem,Q is its the configuration space, which is assumedto be ann-dimensional manifold. If theith physical con-tact does not slip, the contact imposes a nonholonomicconstraint on the mechanical system’s motion. This con-straint can be expressed in the formωi(q)q = 0. If theith contact slips, the Coulomb friction law states thatthe tangential reaction force at that contact isFT

i =− vi

||vi||µiFNi , whereµi, FN

i , andvi are respectively theCoulomb friction coefficient, normal force to the contact-ing surface, and slipping velocity of the contact at theith

contact. Hence, the mechanical system’s overall equa-tions of motion can described by:

d

dt

(∂L

∂q

)− ∂L

∂q+

∑i∈S

FTi +

∑j 6∈S

λjωTj (q) = T (1)

whereS is theslipping contact set, the λj are unde-termined Lagrange multipliers, andT are the generalizedapplied forces. That is,k ∈ S if the kth contact is slip-ping. If thekth contact is not slipping,λk corresponds tothe tangential reaction force that is needed to maintain theno-slip constraint at thekth contact. We generally assumein this work that the contact normal forces,FN

i areknown. If this is not the case, then additional Lagrangemultipliers may typically be added to solve for these nor-mal forces. Note that this description involves a choiceof coordinates. The equivalent, coordinate independent,representation is the formalism in which we address theseproblems, and is briefly reviewed in the Appendix.

There are two primary practical problems with theLagrangian modeling approach. First, one must solve forthe Lagrange multipliers—a tedious task that often leadsto complex equations. Second, an additional (and oftensensitive) analysis is necessary to determine which con-tacts are slipping at any given instant. Consequently, the


practical need to analyze such systems in a tractable waymotivates the use of quasi-static or kinematic approxi-mations, and in particular the Power Dissipation Methodthat is reviewed in the Section V. A natural questionarises when using quasistatic analysis: what is the rela-tionship between the equations of motion predicted byquasi-static analysis and those generated by Lagrangiananalysis? Moreover, can the quasistatic equations prop-erly predict the motions of the true system? The nextsection briefly reviews the concept of a multiple-modelsystem, which is the appropriate mathematical setting forthis question in the case of intermittent frictional contacts.We describe a method for finding quasistatic equations ofmotion in Section V and we answer these questions inSection IX.

IV. BACKGROUND: MULTIPLE MODEL SYSTEMS

We use the formalism of multiple model systems toaddress kinematic reducibility of systems involving fric-tional and intermittent contact.

Definition IV.1: A control systemΣ evolving on asmoothn-dimensional manifold,Q, is said to be amul-tiple model driftless affine system (MMDA)if it can beexpressed in the form

Σ : q = f1(q)u1 + f2(q)u2 + · · ·+ fm(q)um (2)

whereq ∈ Q. For anyq andt, the vector fieldfi assumesa value in a finite set of vector fields:fi ∈ gαi

|αi ∈ Ii,with Ii an index set. The vector fieldsgαi

are assumedto be analytic in(q, t) for all αi, and the controlsui ∈ Rare piecewise constant and bounded for alli. Moreover,letting σi denote the “switching signals” associated withfi

σi : Q× R −→ N(q, t) −→ αi

theσi are measurable in(q, t).Definition IV.1 implies that the control vector fields

may change, or switch, among a finite collection of vec-tor fields, each representing a single smooth model in aset of modelsP. An example of such a system is a ve-hicle whose wheels can potentially skid. The system’sgoverning dynamics will vary when the wheels slip ordo not slip. Such systems are intimately related to mul-tiple model systems such as studied in [18]. However,we should emphasize that the “switching” isnot like theswitching phenomena found in [8], [25], [12], [44], oras typically studied in the hybrid control systems litera-ture (e.g., [38], [5]). In these studies, the switching phe-nomena is part of a control strategy to be implemented inthe controller. In our case, the switching is induced byenvironmental factors, such as variations in the contactstate between rigid bodies. Since the phenomena which

govern the switching behavior may not be precisely char-acterized, we make no assumptions about the nature ofthe switching functions, except that they are measurable.A long term goal of our work is to develop systematicmethods for analyzing control systems with the type ofhybrid (and therefore nonsmooth) structure seen in Defi-nition IV.1.

To distinguish between the overall control systemand the smooth control systems that comprise it, we de-fine theindividual control systemsto be the smooth con-trol systems making up the multiple model system, com-prising ofq = g1u1 + · · ·+gkuk · · ·+gnun for gk = gαi

for someαi. A system will be termed amultiple modelaffinesystem if it has the formq = f0(q) + f1(q)u1 +f2(q)u2 + · · · + fm(q)um, where the vector fieldf0(q)(or “drift term”) is also selected from a set of analyticvector fieldsgσ0 .

V. OVERVIEW OF THE POWER DISSIPATION

METHODOLOGY

This section reviews the basic concept behind thePower Dissipation Method (PDM), which we will for-malize in Section VII. Letq again denote a system con-figuration. The relative motions between moving objectsat a point contact can be written in the formω(q)q. Ifω(q)q = 0, then the contact point is not slipping, while ifω(q)q 6= 0, thenω(q)q describes the contact point’s slip-ping velocity. Thepower dissipation functionmeasuresthe object’s total frictional energy dissipation due to con-tact slippage.

Definition V.1: Consider a mechanical system,S(which consists of a single rigid body or a set of rigidbodies) that maintainsn frictional contacts, where someor all of the contacts may be slipping. TheDissipationorFriction Functionalfor n-contact states that are governedby Coulomb friction is defined to be

D =n∑

i=1

µiNi | ωi(q)q | (3)

whereωi(q)q describes the relative slipping velocity,µi

is the Coulomb friction coefficient, andFNi is the normal

force at theith contact.The form of this function reflects the Coulomb fric-

tion model, but it can easily be extended to different fric-tion models (see [37]) by replacing the linear termµiNi

with a more general state-dependent function,hi(q). Ev-ery slipping contact dissipates energy. Based on this ob-servation, Alexander and Maddocks proposed the follow-ing axiom:

Power Dissipation Principle: A system’s motion atany given instant is the one that minimizesD (Eq. 3)with respect toq.


The power dissipation methodis built upon this axiom.That is, the first order equations of motion generated bythe system are precisely the ones that minimize the dissi-pation.

Remark V.1:Some insight into the relationship be-tween the motions predicted by the PDM and those givenby the Lagrangian approach can be seen in the follow-ing example. Consider a particle constrained to moveon a surface, with friction between the particle and thesurface. Lagrangian analysis suggests that there are twopossible contact states–one slipping and one not slipping.The PDM predicts that the particle will not slip. Hence,it misses some of the contact states predicted by the La-grangian framework. However, the non-slip motions thatit does predict are consistent with a Lagrangian analysis.

For overconstrained systems with control inputs, thePDM leads to more interesting and useful results. When aconfigurationq can be decomposed into two componentsq = (s, r) (where we refer tos as the group variable andr as the shape variable), Eq. (3) implies that the PDM willpredict s given r. In most cases of interest, the variabler corresponds to the control inputs, while the variablesscorresponds the system motion of interest.

VI. EXAMPLE : A TWO-WHEELED BICYCLE

Consider the planar bicycle (Fig. 1(a)) which is con-strained to move along a line. We will revisit this exampleusing the PDM formalism, but for now we treat it in theLagrangian framework. Letq = [x, φ1, φ2]T , whereφ1 isthe front wheel angle,φ2 is the rear wheel angle, andx de-notes the bicycle’s translation along the line. The down-ward normal force on each wheel depend upon the bicy-cle’s weight distribution, which is assumed to be known.Assume that each wheel is actuated, with torquesτ1 andτ2, and that each wheels may possibly slip.

Using Eq. (1) and solving for the Lagrange multipli-ers, there are four different governing equations of motion(see Table I), each corresponding to a different type ofcontact state. The analysis based on Lagrangian mechan-ics suggests that there arefour possible contact states, cor-responding to Eq. (A) where neither wheel slips, Eq. (B)where the front wheel slips, Eq. (C) where the rear wheelslips, and Eq. (D) where both wheels slip.

When theith wheel slips, the tangential reactionforce at theith contact point is governed by the Coulomb

friction law: FTi = − x−Rφi

‖x−Rφi‖µiF

Ni , whereµi is the

Coulomb friction coefficient, andFNi is the normal force

bearing down upon theith wheel contact. When theith

wheel does not slip, the tangential reaction force is givenby the Lagrange multiplierλi. The Coulomb frictionmodel implies that the boundary between slipping andnonslipping states occurs at some value of the Lagrangemultiplier, denoted byλnom. Whenλi > λnom

i , the ith

q =

R2J+mR2

12J+mR2

12J+mR2

τ1 +

R2J+mR2

12J+mR2

12J+mR2

τ2 (A)

q =

F R

1J+mR2

−RF R1

JRF R

1J+mR2

+

01J0

τ1 +

RJ+mR2

01

J+mR2

τ2 (B)

q =

F R

2J+mR2

RF R2

J+mR2

−RF R2

J

+

RJ+mR2

1J+mR2

0

τ1 +

001J

τ2 (C)

q =

F R

1 +F R2

m−F R

1 RJ

−F R2 RJ

+

01J0

τ1 +

001J

τ2 (D)

TABLE ITHE LAGRANGIAN DYNAMICS OF THE PLANAR BICYCLE IN THE

FOUR POSSIBLE CONTACT STATES. J IS A WHEEL’ S MOMENT OF

INERTIA ABOUT ITS ROTATIONAL AXIS , m IS TOTAL BICYCLE

MASS, AND R IS THE WHEEL RADIUS.

contact slips. Consequently, theλ space is divided intoregions corresponding to different contact slipping states.The problem of contact state determination arises fromthe inherently complicated dependency ofλ on the cur-rent state. For the planar bicycle model, the Lagrangemultipliers assume the following values when model (A)holds:

λ1 =J(τ1 − τ2)−R2mτ1

R(R2m+ 2J)λ2 =

J(τ2 − τ1)−R2mτ2R(R2m+ 2J)

.

Under the Coulomb friction model, the critical valueof λ for this example takes the valueλnom = µiF

Ni .

However, depending on the friction modelλnom will takedifferent values. This fact implies that the boundary ofthese regions is both terrain dependent and sensitive tothe details of the friction model. One of the purposes ofthis paper is to provide a modeling foundation for controlstrategies that are not sensitive to the friction model, suchas those we employ in [34].

Alexander and Maddocks showed thatD is convexas a function ofq; therefore its local minima are globalminima [3]. Note that the minimum ofD must occurat a nondifferentiable point ofD, since the function ismonotone everywhere else. By direct comparison of thetwo nondifferentiable states, which correspond to one ofthe wheels not slipping, the minimum is associated withwhichever wheel is associated with a lower value ofµN .Consequently, the zero level set of the function

Ψ(q) = µ1N1 − µ2N2


determines the contact state of the bicycle. This determi-nation is nonunique whenµ1N1 = µ2N2. (This is alsotrue for the Lagrangian system.) This model has only twostates, making it simpler to analyze than the Lagrangianmodel. Additionally, the governing equations take thesimplified form:

x = Rui (4)

wherei indexes the wheel not slipping and theui are ve-locity inputs.

To compare the PDM method to Lagrangian analy-sis, consider the bicycle example with torque inputs onboth the front wheelW1 and the back wheelW2. ThePDM predicts two different contact states correspondingto either the front or rear wheel slipping. In comparison,Lagrangian analysis predicts four possible contact states.Eqs. (A) and (D) in Table I both imply that the inertialterms dominate the system’s dynamics, thereby violatingthe “quasi-static” assumption. Eq. (D) implies that the bi-cycle is skidding out of control. The physical conditionscorresponding to Eq. (A) are unlikely to be found in anactual system, as they imply that both contacts must bedriven atexactlycompatible speeds. Moreover, this con-tact state will be predicted by the PDM so long as the twowheels are driven at exactly the same speed—we will seelater these conditions can be interpreted as a special de-generate case. This leaves the second two contact statesrepresented by Eqs. (B) and (C), which are the same asthose found in Eq. (4) using the power dissipation model.This is an indication of how the quasi-static assumptionhelps to simplify our problem, while yielding results thatare consistent with the contact state analysis of the La-grangian. With the additional analyses introduced below,we can investigate the relationship between the motionspredicted by the Lagrangian method and the PDM in com-parable contact states.

VII. C HARACTERISTICS OF THEPDM

In this section we formalize the Power DissipationMethod and show that the PDM generically gives rise tomultiple model driftless affinesystems, as described inIV.1.

Before proceeding, let us recall a few facts thatwere already established by Alexander and Maddocks [3].They showed that the dissipation function of Eq. 3 is con-vex, so that its local minima are also its global minima,should they exist. They also show that if such a minimumexists, it must exist at a point of nondifferentiability ofDdue to the piecewise continuity ofD.

Let Ω = ω1, · · · , ωm and denote theconstraint1-forms. Furthermore, letQ = q1, q2, · · · , qr consistof the

(n

n − m

)velocities that have the property thatqk is a

kinematic solution to a non-overconstrained subsetΩ′ ⊂

Ω consisting ofn−m constraints, i.e.,

Ω′qk =

ωk1

...ωkm−n

qk = 0.

That is, at all points inQ the derivatives ofD are non-smooth. From Alexander and Maddocks, we have only afinite number of points to check in order to find the min-ima of D. It is straightforward to show that these min-ima mustat leastoccur at points inQ. See, for instance,Clarke [11]. ReorderQ so thatD(q1) ≤ D(q2) ≤ · · · ≤D(qr). AlthoughQ is associated with at least one of theminima achieved byD, it does not necessarily containall of them. In fact, if more than one element ofQ is aminimum, then every element of the convex hull of theseminima are also minima. Hence, if there is more than onesolution, there are an infinite number of solutions.

Proposition VII.1: If q1 and q2 both minimize thedissipation functional found in Definition V.1, then sodoescoq1, q2.

Proof: AssumeD(q1) = D(q2) = a andδ ∈ [0, 1].Then

D(q) (δq1 + (1− δ)q2) =n∑

i=1

µiFNi |ωi (δq1 + (1− δ)q2)|

≤ δn∑

i=1

µiFNi |ωi (q1)|+ (1− δ)

n∑i=1

µiFNi |ωi (q2)| = a

Assume thatD is strictly less thanD(q1) somewhere incoq1, q2. Then∃ δ′ such thatD(δ′q1+(1−δ′)q2) is at aminimum by an extension of Rolle’s Theorem for the realline [15]. Thenq′ = δ′q+(1−δ′)q is at a point whereD isnonsmooth in all its directional derivatives [3] (becauseDis monotone elsewhere). This implies thatq′ ∈ Q and thatD(q′) < D(q1), thus violating our assumption thatD(q1)is a minimum ofD. ThereforeD(q) (δq1 + (1− δ)q2) =a ∀δ ∈ [0, 1]. The proof for higher numbers ofqi havingequal dissipation is by induction on this argument.

This result formalizes the intuition that if the powerdissipated is equal for two velocitiesqi, then all possi-ble trajectories whose velocity lies in the convex hull ofthe qi will satisfy the minimum also. That is, in the non-generic case whenD does not have a unique minimum,we can still bound the object’s motion. (We will see laterthat these solutions correspond exactly to kinematic solu-tions of the Lagrangian dynamics.) Nowcoqi, i ∈ J isa set of points on whichD is nondifferentiable, just notin all directions. It therefore still meets the criterion to bea minimum [3]. Let us consider the extent to which thefunctionD having a unique minimum is generic. We de-note the function space of the coefficient of friction byΞ,the function space of normal forces byN .


Proposition VII.2: AssumeD : (U ,Ξ,N , TQ) →R is of the form in Definition V.1 and that theµ is mea-surable inx andt. Then the dissipation functionalD hasa unique minimum almost always (i.e., except on a set ofmeasure zero1 relative to the space(U ,Ξ,N , TQ))

Proof: Case 1: If q1 is a unique minimum inQ,then it is the unique global minimum since Alexander andMaddocks showed that the minimum must occur inQ.Case 2: If ∃ q1 and q2 such that both are minima, thenby Proposition VII.1, we know thatcoq1, q2 also mini-mizes theD. However, this situation can only occur whenthe parameters(ui, Nj , µk) ∈ U × N × Ξ are chosen tosatisfy the constraintD(q1) = · · · = D(qn). This impliesthat the constraint is only satisfied on a set of measure 0in the spaceU ×N × Ξ.

That is, the PDM will almost always lead to a uniqueset of governing equations. The reader should note thatthe proof of Proposition VII.2 is only useful if we have al-ready foundQ, and moreover for a high number of statesit may be computationally expensive to find the mini-mum ofQ.2 Also note that in the non-overconstrainedcase ofn − m constraints, the dissipation method leadsto the classical kinematic solution in the sense of theAppendix. Proposition VII.2 allows us to now state whatwe mean by the dissipation functional leading genericallyto an MMDA system. A direct consequence of Proposi-tion VII.2 is the following Corollary.

Corollary VII.3: The multivalued mapF : TQ →TQ implicitly defined byD(q) = min(D) is single val-ued almost everywhere.

Corollary VII.3 implies that we can generically ex-pect the power dissipation method to lead to a uniqueand well defined set of first-order governing equations—it will almost never lead to an indeterminate system. Thismakes rigorous the comment made in [3] referring to thephysical expectation of continually switching back andforth between the dominance of one wheel or another,rather than staying in an indeterminate state. See [13]for a discussion of implicitly defined multivalued maps.Corollary VII.3 additionally establishes a relationship be-tween solutions that minimizeD and MMDA systems.

1Intuitively, sets of measure0 can be as sparse as disjoint points inQ or as replete as a submanifold ofQ. For example, consider a vehiclemoving on smooth terrain. In its ambient Euclidean space, a vehicle isalways constrained to a set of measure 0, yet that set is precisely wherethe interesting dynamics occur. On the other hand, sets of measure 0can represent arbitrary algebraic relationships between parameters andthe state space. Unless there is some reason to believe that these rela-tionships are necessarily satisfied, we can feel physically motivated inasserting they will not occur in practice. This is the case that we areconsidering, and therefore we feel that the ensuing results do imply thegenericity we assert. Nevertheless, whether or not these sets are impor-tant in the analysis is aphysical assumption, not a mathematical result.For a reference on measure theory, see [2].

2This problem, thus stated, bears more than a passing resemblance tothe simplex method found in LP theory and techniques from that theorycan be applied to the problem of finding the minimum of the functionDin the presence of high numbers of contact states.

Moreover, we will see that the contact states predicted bythe PDM are(U ,U) reductions of a class of mechanicalcontrol systems onTQ.

Corollary VII.3 also implies that multiple modelsystems are a natural result of frictional interactions. Con-sequently, multiple model modeling and control tech-niques should be developed for systems involving fric-tional contact. In Section IX we will explore more for-mally the relationship between solutions to the PDM andsolutions to the Lagrangian dynamics.

VIII. K INEMATIC REDUCIBILITY FOR MULTIPLE

MODEL SYSTEMS

This section introduces the formal tools and resultsrequired to relate solutions arising from the power dis-sipation method to solutions arising from the full La-grangian analysis. A rigorous understanding of thePDM’s properties and its relationship to conventional La-grangian mechanical analysis has heretofore been miss-ing. We structure our analysis of this issue in two steps. Inthe previous section we developed a more formal mathe-matical framework for the PDM. In particular, we showedthat the PDM leads generically to multiple model sys-tems. This section introduces kinematic reducibility the-ory for multiple model systems. We then use our multi-model reduction theory to formally study the relationshipbetween the properties of the PDM solutions and those ofthe associated Lagrangian models (in Section IX.2).

A. Review of Kinematic Reducibility for Smooth Systems

We briefly review the relevant notions of kinematicreduction here, without going into details of the under-lying formalism. For some of these details, refer to theAppendix and to [23]. First we start with what we meanby a solution to a control system. In the following,Qis the configuration space andTQ is its tangent bundle.Moreover, ifXi are kinematic vector fields andYjare dynamic vector fields (see the Appendix for notationaldetails), we let thedistributionsDkin andDdyn be de-fined byDkin = spanXi andDdyn = spanYj.

Definition VIII.1: Let Σs be a smooth control sys-tem q = f(q, u) onQ and letu ∈ U ⊆ Rm. A (U , T )-solutionto Σs is a pair(c, u), whereu : [0, T ] → U andc : [0, T ] → Q satisfyc′(t) = f(c(t), u(t)).We now can define what it means for a mechanical systemof the form in Eq. (20) to be(U ,U) reducible to Eq. (21).Let τQ

τQ : TQ → Q(vq, q) → q

denote the tangent bundle projection.Definition VIII.2: Let∇ be an affine connection on

Q (see the Appendix), and letU andU be two families


of control functions. The system in Eq. (20) is(U ,U

)-

reducibleto the system in Eq. (21) (also in the Appendix)if the following two conditions hold:

i ) for each(U , T )-solution (η, u) of the dynamicEq. (20) with initial conditionsη(0) in the distri-butionDkin, there exists a

(U , T

)-solution(γ, u)

of the kinematic Eq. (21) with the property thatγ = τQ η;

ii ) for each(U , T

)-solution(γ, u) of the kinematic

Eq. (21), there exists a(U , T )-solution (η, u) ofthe dynamic Eq. (20) with the property thatη(t) =γ′(t) for almost everyt ∈ [0, T ].

Condition i) says that for every solution of a dynamic sys-tem there must exist a kinematic solution that is the pro-jection of the dynamic system. In the case of a vehicle,this corresponds to requiring that for everytrajectory ofthe vehicle there exists a correspondingpath that can beobtained from kinematic considerations alone. Conditionii) says that every kinematic solution must be the integralof a dynamic solution. For a vehicle, this means that theremust exist a dynamic solution for every feasible kinematicpath. We should point out here that this is related to theclasses of admissible inputs. Because kinematic inputsmust be essentially integrals of dynamic inputs, they mustbe absolutely continuous if the dynamic inputs are mea-surable. Otherwise, infinite forces would be required (see[23]).

Let χ∞(D) denote thoseC∞ vector fields takingvalues in a distributionD. The following theorem statesthe local test for Eq. (20) to be(U ,U) reducible toEq. (21).

Theorem VIII.1—Lewis [23]:Let ∇ be an affineconnection, and letY1, . . . , Ym andX1, . . . , Xm be vec-tor fields on a manifoldQ. The control system in Eq. (20)is

(U ,U

)reducibleto a system of the form in Eq. (21) if

and only if the following two conditions hold:i ) spanRX1(q), . . . , Xm(q) =spanRY1(q), . . . , Ym(q) for each q ∈ Q(in particular,m = m)

ii ) 〈X : Y 〉 ∈ χ∞(Ddyn) for every X,Y ∈χ∞(Ddyn) where〈·, ·〉 is the symmetric product ofvector fields, defined in the Appendix.

This theorem says that if the input distributions of both thekinematic system and the dynamic system are the sameand the dynamic system is closed under symmetric prod-ucts, then the system is kinematic.

B. Main Result on Reducibility of Multiple Model Sys-tems

We now consider the problem of whether or nota dynamic multiple model system is kinematically re-ducible to an MMDA system. Lemma VIII.2 states thatif switches in system dynamics are separated by a small

amount of time (making the switching signal piecewisecontinuous), the resulting solution is also kinematicallyreducible.

Lemma VIII.2: Let Σ be a multiple model systemwhose switching signalσ is piecewise constant. Then,Σis (U ,U) reducible iff the individual model componentsΣσi,··· ,σj

are all(U ,U) reducible.Proof: Sinceσ is piecewise constant,σ switches a

countable number of times. Therefore, let the times whenσ changes its value be denotedt1, t2, · · · , for i in someindex setI. Then on the intervals(ti, ti+1), Σ is (U ,U)reducible, making it(U ,U) reducible almost always.3 Ittherefore satisfies the requirements of Definition VIII.2.

We will use this lemma to prove Theorem VIII.4,which says that solutions to the differential inclusion de-fined by multiple model systems are kinematically re-ducible if and only if the individual models are kinemat-ically reducible. Before proving that this is true, we willneed the following result from Filippov [14].

Theorem VIII.3—Filippov [14]:Let f : Q × R →TQ be a compact, set-valued map and letΦi be a se-quence of solutions to the differential inclusion

q ∈ f(t, q) (5)

such that limi→∞

Φi → Φ. Then Φ is also a solution to

Eq. (5).Note that solutions to the differential inclusionf are ingeneral not unique, meaning that there is often an infinitefamily of solutions. This theorem says that for a compactdifferential inclusion, a converging sequence of solutionsconverges to a solution. Theorem VIII.3 will be used sev-eral times in the proof of Theorem VIII.4. Roughly speak-ing, piecewise continuous(U ,U) reducible solutions ofthe multiple model mechanical system can be used as ap-proximations to flows of elements inf , wheref assumesthe form of the right half side of Eq. (6). Theorem VIII.3can then be used to show that their kinematic counterpartsonTQ must also converge to an element of the differen-tial inclusion defined onTQ. This brings us to our mainresult.

Theorem VIII.4:A multiple model systemΣ is(U ,U) reducible iff the individual dynamical modelsΣσi,··· ,σj are all(U ,U) reducible.

Proof: First note that it is obviously necessary thatall the individual models be(U ,U)-reducible in order forthe resulting multiple model system to be reducible. Oth-erwise, a valid solution to a multiple model system is thesmooth, non-reducible solution of one of the models inthe set of models. To show sufficiency, we must showthat when the individual models are(U ,U) reducible,

3That is, it is reducible everywhere except for a set of measure 0.


the MMDA system satisfies partsi) and ii) of Defini-tion VIII.2. We show this in two steps. The first stepconstructs kinematic solutions given dynamic ones, andthe second step constructs dynamic solutions given kine-matic ones.

(i) A multiple model mechanical system has the form (seethe Appendix for notation)

Gl∇c′(t)c′(t) ∈ uα lYα(c(t)) (6)

where l ∈ Λ ⊂ N is the index for a given model,Gl

is the metric appropriate to that model,Gl∇ is the affineconnection associated with the metricGl, andlYα is thevector field representing the force input corresponding touα of the lth model of the multiple model system. Incoordinates, Eq. (6) is equivalent to

qi + GlΓijkq

j qk = uα lY iα. (7)

Set lYi = −GlΓijkq

j qk + uα lY iα andYi = colYi :

l ∈ Λ, with co·, · denoting the convex hull. In [14] itwas shown that solutions to a discontinuous system coin-cide with solutions of a differential inclusion of the con-vex hull of the discontinuous system. Applying this toour systems of interest, we see that solutions to a multiplemodel system coincide with solutions to the differentialinclusionqi ∈ Yi, or in vector notation:

q ∈ Y. (8)

Eq. (8) is a second order system onQ that we can easilyrewrite as a first order system onTQ (see [23] for de-tails of this procedure). Then, for a given solutionΦ(t)of Eq. (8) rewritten as a first order system, we know thatddtΦ ∈ Y. Therefore, we can choose a selection (an ele-ment) ofY, denoteds(Y) ∈ Y, such thatΦs(Y) locallyapproximates the flowΦ. BecauseY is convex, we canrewrite a selection ofY as

s(Y) = δ11Y + δ2

2Y + · · ·+ δmmY (9)

for anyδj such thatδj > 0 andm∑j

δj = 1. Let us denote

the composition of a flowΦ with itself n times byΦn.That is,Φn(q) = ΦΦ· · ·ΦΦ(q). In [28], it was shownthat we can choose the following map to approximate (inthe sense of pointwise convergence to a set) the flow ofthe selections(Y):

Φt,ndyn(q)

def=

(Φδ1

1Y tn Φδ2

2Y tn · · · Φδm

mY tn

)n

(q)(10)

Each of the component flowsΦδmiY t

n contributing to theflow Φt,n

dyn(q) consists of a flow along a(U ,U) reducible

mechanical system. Moreover,Φt,ndyn(q) is a solution of

Eq. (8) onTQ which is absolutely continuous for everyn. This is due to the fact that we assume that the switch-ing is measurable and the forces are measurable and thatthe Lebesgue integral of measurable signals is absolutelycontinuous. Lastly, it converges to the flow of the se-lection s(Y) asn → ∞. That is, by applying Theo-rem VIII.3 to the Taylor expansion ofΦt,n

dyn, we locallyget

limn→∞

Φt,ndyn = Φs(Y).

By assumption, we know that each segmentΦδiiY t

n

of Φt,ndyn is (U ,U)-reducible. Therefore, for every choice

of n, Φt,ndyn is (U ,U)-reducible by Lemma VIII.2. These

results then yield us, for eachn, a corresponding map onQ:

Φt,nkin(q)

def= τQ Φt,n

dyn(q) =(Φδ1

1X tn Φδ2

2X tn · · · Φδm

mX tn

)n

(q)(11)

where eachΦδiiX t

n is the flow of equations that are(U ,U)-reductions (as in Eq. (21)) from equations thatgenerate the flowΦδi

iY tn . Moreover, from Theorem

VIII.3 we know that limn→∞

Φt,nkin exists and that its limit

is a solution toq ∈ X (12)

whereX = colX|l ∈ L and thelX come from thereduced equations in Eq. (21). Therefore, parti) of Defi-nition VIII.2 is satisfied.

(ii) The analysis of this second condition uses the sameessential steps as above, but begins with the solution tothe kinematic equations and works towards a dynamicsolution. Starting with the solutions from Eq. (21), weknow that for an individual model with indexl we haveqi = ua lXi

a, or in vector form:

q = ua lXa. (13)

Therefore, this MMDA system can be associated withgoverning equations having the form of Eq. (12). Again,for any given solutionΦ of Eq. (12) we haved

dtΦ ∈ X,so we can choose a selections(X) such thatΦs(X) lo-cally approximates the flow for that solution. As before,we construct a sequence of solutions converging toΦs(X).By construction, there exists aΦt,n

kin whose limit isΦs(X).From Def VIII.2 we know we must show there exists

anη solution with

d

dtΦs(X) = η.

By our construction, we know that

limn→∞

Φt,nkin = Φs(X)(q0, t) .


From part(i) above, for everyn andΦt,nkin there exists a

correspondingΦt,ndyn such thatΦt,n

kin(q) = τQ Φt,ndyn(q).

In the limit,lim

n→∞Φt,n

dyn = Φs(Y),

for some selection of the differential inclusions(Y).Consequently,Φs(Y) is a solution to Eq. (8), again byTheorem VIII.3. Taking the derivative of both sides, weget (after repeated application of the chain rule)

d

dtΦs(X) =

d

dtlim

n→∞Φt,n

kin = limn→∞

d

dtΦt,n

kin

= limn→∞

Φt,ndyn = Φs(Y)

so partii) is satisfied. This ends the proof.Notice that the proof of Theorem VIII.4 relied heav-

ily on specifically constructing a solution with the de-sired properties based onknownsolutions to the individ-ual models comprising the multiple model system. Thisresult shows that determining the kinematic properties ofthe individual models in a multiple model system is suf-ficient for determining the kinematic properties of the en-tire system.

IX. T HE PDM AND (U ,U) REDUCIBILITY

This section addresses the relationship between themodels produced by the power dissipation methodologyand the kinematically reducible states of a generic me-chanical system. An informal restatement of this is thequestion: does the PDM produce equations of motion thatare kinematic reductions of Euler-Lagrange equations?First, we derive a result that will be shortly used to showthe relationship between PDM solutions and solutions ofmechanical, second order, systems.

Proposition IX.1: Given a configuration manifoldQand a set of constraintsωi(q) which span the cotangentspaceT ∗q Q, then the input distributionDkin(q) minimiz-ing D(q) will always satisfyDkin(q) = Null(Ωsat)(q)whereΩsat(q) is the collection ofwi(q) which satisfywi(q)q = 0 for q ∈ Dkin.

Proof: Suppose that this was not the case. Thenthere would existv 6= 0 which minimizesD such thatif ωi

s are the constraints which are satisfied, thenv ∈Nullωi

s andv /∈ Dkin. This implies that for the choiceof uk = 0 ∀k, v still minimizesD. However, becausetheωi spanT ∗Q, 0 is the unique minimizer since D isconvex inq. This contradicts the assumption thatv 6= 0and is a minimizer ofD.

This result roughly corresponds to the intuition thatthe minimum dissipation in any unactuated direction isto not move at all in that direction. We should commentthat this can still lead to a solution of no motion in thegroup variabless–if the unactuated constraints dominatethe motion, then the actuators will all slip.

Next we consider the case where we are given a met-ricG for some mechanical system and a set of constraintsdescribed by one-formsωj. What are sufficient con-ditions for the resulting system to be(U ,U) reducible?Lemma IX.2 gives one sufficient condition which is in-variant with respect to the metricG, and is a simple corol-lary to the work found in [24].

Lemma IX.2:Given a “constraint distribution”Dcon ⊆ TQ which annihilates the constraintsωjand an input distributionDdyn, if Ddyn = Dcon themechanical system described by∇q q = uY is (U ,U)reducible.

Proof: Denote by∇ the connection and by∇the constrained connection defined by the Lagrange-dAlembert principle (see Lewis [23] for details of thisconstruction). We know that

∇XY ∈ Dcon ∀ Y ∈ Dcon and X ∈ T (M),

which implies

∇XY +∇Y X ∈ Dcon ∀X,Y ∈ Dcon.

This in turn implies by Theorem VIII.1 that∇q q = uY is(U ,U) reducible.

Therefore,(U ,U) reducibility of a multiple modelmechanical system is guaranteedregardless of the met-ric G when the constraint distribution is covered by theinput distribution. Moreover, we already know that thepower dissipation model only admits solutions where thisis true. This allows us to interpret the use of the powerdissipation method. The power dissipation method is away of choosing a more tractablesubsetof contact statesfrom the full Lagrangian contact mechanics. In otherwords, when we make the “quasistatic” assumption, weare merely restricting our attention to(U ,U) reduciblesystems. Moreover, when the reaction forces due to fric-tion do not lie inDkin, then those contact states are not(U ,U) reducible. However, we should be very clear thatthis only shows that the power dissipation method cap-tures(U ,U) reducible states whenDcon = Dkin. Thatis, the correspondence only goes one direction: all PDMcontact states are kinematic states, but not all kinematicstates can be predicted by the PDM. There are exam-ples of mechanical systems which are(U ,U) reducibleby virtue of properties of the metricG. For examples ofsuch systems, see Lewis [23].

In summary, we have shown is the following.Theorem IX.3:Given a configuration manifoldQ

with tangent spaceTQ and constraints represented byone-formsωi, then all solutions to the PDM are(U ,U)reductions of solutions to the Euler-Lagrange equationsonTQ constrained by a subset ofωi.

We should also remark on the relationship betweenTheorem VIII.1 (reduction for smooth systems) and The-orem VIII.4 (reduction for multiple model systems). In


the smooth case,(U ,U) reducibility is equivalent togeodesic invariance (for details, see Lewis [23]). How-ever, in the nonsmooth case there is no well defined no-tion of geodesic invariance because the metric changesover time. Nevertheless, we were able to extend the no-tion of (U ,U) reducibility relatively easily. Therefore, theconcept of(U ,U) reducibility is in some sense more gen-eral than that of geodesic invariance.

X. EXAMPLES

To illustrate how the results presented in this paperare useful, and point towards more general applicationsof theories developed here, we now revisit the examplesfrom Section II. First, we come back to the bicycle ex-ample to illustrate all of the theory details. We study thebicycle example in detail as illustration, and then quicklysummarize several applications in other related work. Forinstance, we show how this analysis helps to establishcontrollability characteristics for the Mars rover familyof vehicles and stability analysis for distributed manipu-lation problems. We end this section with a brief discus-sion of how the method presented here can be applied tograsping and locomotion.

A. Bicycle

Now, we return to the bicycle example of Section IIin detail. Assume that the bicycle is constrained to moveon a line. Using the mechanics formulation as describedin the Appendix, the configuration space isx, φ1, φ2 ∈R×S2, and the Riemannian metric describing the kineticenergy is

G = (m+ 2J)dx⊗ dx+ Jdφ1 ⊗ dφ1 + Jdφ2 ⊗ φ2.

The two non-rolling constraints are

x−Rφ1 = 0 x−Rφ2 = 0

and the constraint covectors can be written as

ω1 = dx−Rdφ1 ω2 = dx−Rdφ2

As inputs, we have

F 1 = dφ1 F 2 = dφ2.

Now, for each combination of slipping and no slippingof the wheels, we have a set of equations to solve for.Therefore, we have four sets of equations to solve. More-over, because the Christoffel symbolsΓi

jk are all identi-cally zero for this problem, the equations depend entirelyon the input forces and external forces due to friction.

1) No slipping: When both wheels do not slip, thesystem must satisfyφ1 = φ2. This, in turn, implies thatthe constraint distribution is 1-dimensional, spanned by

R∂

∂x+

∂

∂φ1+

∂

∂φ2.

Moreover, one can readily compute that the orthogonalcomplement ofD is

span

− J

mR

∂

∂x+

∂

∂φ2,− J

mR

∂

∂x+

∂

∂φ1

.

The associated input vector fields are

Y1 = Y2 =1

2J +mR2

(R∂

∂x+

∂

∂φ1+

∂

∂φ2

)and the equations of motion are therefore:

q = Y1u1 + Y2u

2.

It is easy to see that〈Y1, Y2〉 = 0, so this is a kinematicsystem (that is, it is reducible to Eq. 4).

2) One wheel slipping:In the case where one wheelslips, we may assume without loss of generality that theslipping wheel is wheel number 1. In this case, the con-straint distribution is

span

R∂

∂x+

∂

∂φ1,∂

∂φ2

.

Moreover, one can readily compute that the orthogonalcomplement ofD is

− J

mR

∂

∂x+

∂

∂φ1.

To compute the reaction force due to the other wheel slip-ping, note that such a reaction force can be consideredan external force, and can therefore be added to the righthand side of Eq. (20) with the associated control assum-ing constant unity valueua ≡ 1. The associated inputvector fields and external force vector fields are

Y1 = 12J+mR2

(R ∂

∂x + ∂∂φ1

+ ∂∂φ2

)Y2 = 1

J∂

∂φ2

E = R2F R2

J+mR2∂∂x + RF R

2J+mR2

∂∂φ1

− RF R2

J∂

∂φ2

and the equations of motion are therefore:

q = Y1u1 + Y2u

2 + E.

To determine whether this system is kinematically re-ducible or not, we first note that〈Y1, Y2〉 is again identi-cally zero. Moreover, note that although Theorem VIII.1does not directly address the case of external forces, wecan by direct inspection of Definition VIII.2 see that if


E /∈ spanYi then the system cannot in general be re-ducible. However, ifE ∈ spanYi and theYi satisfythe conditions for reducibility, then the system is automat-ically reducible because the external forces are “covered”by the inputs. Therefore, we need only check thatE liesin the span ofY1 andY2. Indeed,E ∈ spanY1, Y2 forthis example. Therefore, this system is kinematically re-ducible. Note that this property does not depend on theparticular description of the reaction force, and is more-over invariant with respect to the reaction forces’ differ-entiability.

3) Both wheels slipping: When both wheels slip,there are no constraints to enforce. In this case, the con-straint distribution is identically zero and the orthogonalcomplement is trivially the entire tangent space. More-over, we can compute the reaction force due to the wheelsslipping to bew1(FR

1 ) andw2(FR2 ). The associated input

vector fields and external vector fields are

Y1 = 1J

∂∂φ1

Y2 = 1J

∂∂φ2

E = F R1 +F R

1m

∂∂x −

RF R1

J2∂

∂φ1− RF R

2J2

∂∂φ2

and the equations of motion are therefore:

q = Y1u1 + Y2u

2 + E.

In this case, it is clear thatE /∈ spanY1, Y2. There-fore this system (not surprisingly) is not kinematically re-ducible, at least for genericFR.

B. Simplified Mars Rover

θ

ψψ

θ

x

y

Fig. 2. Simplified Rocky 7. a) Is a cartoon of a six wheeled rover, andb) is a cartoon of a simplification of the rover.

Next we revisit the example of Fig. 1(b), whosegeometry we simplify here for the sake of discussion.This simplification has three wheels, with all three wheelsdriven. This model can be interpreted as a simplificationof the Mars rover Rocky 7 vehicle, also seen in Fig. 1. Thethree wheeled vehicle seen in the cartoon has a config-uration space consisting of(x, y, θ, ψ, φ1, φ2, φ3). This

system has six nonholonomic constraints (one associatedwith each wheel having both a no roll constraint and a nosideways slip constraint). Therefore, there are26 = 64possible models governing the dynamics of the vehicle.For this reason, we do not relate all the calculations forthis vehicle. However, it is easy to show, using a sym-bolic mathematics package such asMathematica, that thissystem also has a subset of kinematic solutions, and thatthese solutions correspond to the the solutions to the PDM

for this system. There only exist

(63

)=20 kinematic solu-

tions for this system. Such a correspondence is importantbecause the power dissipation method is very straight for-ward to solve and these solutions can be used for bothcontrollability analysis and for purposes of motion plan-ning (we have carried out this analysis in [32], [33]).

In [32], [33] we showed that this system’s control-lability properties can be analyzed using a set-valued ex-tension of the Lie bracket (the prerequisite calculation forunderstanding controllability using the classical Lie Al-gebra Rank Condition (LARC)) that arises naturally inMMDA analysis. Controllability is important for systemslike the Rocky 7 primarily because many motion planningalgorithms for vehicles are based on controllability prop-erties. For instance, Rapidly Exploring Random Trees(RRT) have been used with much success to develop mo-tion planning strategies. However, the computational in-tensity of these calculations is formidable, and recently[10] showed that significant advantage can be taken by re-ducing mechanical systems to kinematic ones when usingRRTs for motion planning. Work is currently underwayto extend RRTs to the multiple model systems of this pa-per. See [28] for a preliminary motion planning that isbased on the MMDA structure found here.

We should comment on the relationship betweenkinematic reducibility results and controllability resultswhich can be obtained for multiple model systems [32],[33]. One of the intuitive aspects of Theorem VIII.4 isprecisely that it is sufficient for each model to be(U ,U)reducible in order to guarantee that the multiple modelmechanical system is(U ,U) reducible. That is, piece-wise (U ,U) reducibility is enough to guarantee(U ,U)reducibility across discontinuities. However, in the caseof controllability, this no longer holds. An MMDA sys-tem can switch among individually controllable systemsin such a way as to destroy controllability [33]. Thus,controllability of each model in an MMDA is not suffi-cient for overall controllability.

The fact that there is such a high number of modelsfor the Rocky 7 suggests the need for a reduction theoryfor multiple model systems. Indeed, for a six-wheeledsystem like the actual Rocky 7, there are212 = 4096possible models governing its dynamics, a completely un-manageable number. For the three wheeled vehicle in the


cartoon, 20 kinematic models is also perhaps an unrea-sonably large number of models to analyze. In [33] wedid an adhoc reduction of this model which turned it intoa two model multiple model system (although it can beshown that no additional reduction is possible). Combin-ing kinematic reduction with this multiple model reduc-tion reduced the number of models from4096 to2. There-fore, formally utilizing reductions (both discrete and con-tinuous) to reduce the dimensionality of the problem willbe very useful, both for motion planning and estimationpurposes. This will be a focus of future research.

C. Distributed Manipulation with Changing Contacts

2π

4π

43π

23π

47π

45π

I

IIIII

IV

V

VI VII

VIII

π 0

Fig. 3. Photograph and cartoon of 4 cell distributed manipulator.

Figure 3(a) shows a photograph of a particular con-figuration of a distributed manipulation experiment atCaltech pictured in Fig. 1(c) which has been used pre-viously to test algorithms for distributed manipulation[34].4 In the photograph we see four driving wheelswhose rims are oriented towards the origin. Each actu-ator is a one degree of freedom actuator. We use a pieceof plexiglass (for purposes of visualization) on top of thefour wheels to represent a manipulated object. The whiteline seen in the photograph indicates the outline of theplexiglass. The goal is to control the center of mass to the

4Video of these experiments can be found at the websitehttp://robotics.caltech.edu/∼murphey.

origin in R2 with a desired orientation ofθ = 0. To dothis, we obtain feedback of the plexiglass’ configurationby affixing a piece of paper with a black triangle (alsoseen in the photo) whose right angle corner coincideswith the plexiglass’ center of mass. Using this, we ob-tain the position and orientation of the plexiglass throughvisual feedback. Figure 3(b) is a cartoon of the experi-ment, where the four arrows correspond to actuators andthe regions denoted byI-VIII and0-7π

4 will be importantin our subsequent description of the equations of motiondescribed by the PDM.

Note that this system thus described is overactuatedbecause there are four inputs and only three outputs. As-sume the coefficient of friction is the same for all fourdriving actuators. In this case we can show that the modelswitches as the center of mass moves across the array. Infact, under these assumptions, the actuator wheel nearestto the center of mass will have both its “rolling” constraintand its “sideways” slip constraint satisfied. The actuatorwheel second closest to the center of mass will have oneof its two constraints satisfied. In the case of the wheelsshown in the figure, it will be the rolling constraint. Fordetails on this analysis, see [31]. Denote the actuator in-put associated with the closest actuator byui and the ac-tuator input associated with the second closest actuator byuj . Then these considerations lead to first order govern-ing equations of motion of the form: x

y

θ

= g1ui + g2uj (14)

where

g1 ∈

−yi

(xj−xi) sin(θj)+(yi−yj) cos(θj)xi

(xj−xi) sin(θj)+(yi−yj) cos(θj)uj

(xi−xj) sin(θj)+(yj−yi) cos(θj)

(15)

g2 ∈

sin(θj)((xi−xj) cos(θi)+yi sin(θi))+cos(θi) cos(θj)yj

(xj−xi) sin(θj)+(yi−yj) cos(θj)− cos(θi) cos(θj)xi−sin(θi)(xj sin(θj)−(yi−yj) cos(θj))

(xj−xi) sin(θj)+(yi−yj) cos(θj)− cos(θi−θj)

(xi−xj) sin(θj)+(yj−yi) cos(θj)

(16)

In these equationsxi, yi, andθi refer to the planar coor-dinates and orientation of theith actuator. The set-valuednotation of (15) and (16) refers to the fact that at a transi-tion between actuatorsi andj being the two closest actu-ators to actuatorsk andl being the closest the kinematicsare discontinuous. Therefore, at these points we must al-low multi-valued differentials in order to guarantee exis-tence of solutions to the differential equation in (14). See[27] for more details. It should be noted that here the in-dex notation should be thought of as mapping(i, j) pairsto equations of motion in some neighborhood (not nec-essarily small) around theith andjth actuator. In each


regionI − V III the kinematics are smooth, but when atrajectory crosses a boundary0-7π

4 , there is a discontinu-ity in the kinematics. It is possible to obtain point stabi-lization to (x, y, θ) = (0, 0, 0) from any initial conditionusing discontinuous control laws based on the kinemat-ics and knowing the current model (see [27] for details ofthis control design). Moreover, this stability is provablyexponential. However, there are many questions relevantto this system which remain unanswered. In particular,we are currently developing algorithms which do not re-quire any knowledge of the slipping state, and instead usean online estimation process based on hierarchical controllike that found in [4], [19], [20], [18].

D. Relationship to Grasping and Locomotion

We briefly give our vision of how the preceding ideascan be related to both grasping and locomotion. Tra-ditionally, analysis of grasping and locomotion has as-sumed clean interactions between the robot and its en-vironment. Moreover, kinematic analysis has proven tobe a very computationally and theoretically useful venuefor understanding many issues in both areas. However,in real robotic systems, interactions in contact are oftennot clean, and we expect slipping to take place. Consider,for example, the hand shown in Fig. 1. As the hand ma-nipulates the ball, its fingers will slip against the surface.However, we generally expect such motions to not inter-fere with the stability of the motion. The analysis pre-sented in this paper provides a forum for robustness anal-ysis as well as development of algorithms that explicitlyrequire slipping.

XI. SOME FINAL REMARKS

In this paper we derived conditions that are both nec-essary and sufficient for a multiple model system to bekinematically reducible. Such an understanding of a sys-tem’s kinematic motions is important for the purposes oftasking and motion planning. The structure we describehere is put to advantage in [34] in an application to dis-tributed manipulation and in [33] where we analyze thecontrollability properties of an example like that found inFig. 1. Moreover, it has future potential for greatly sim-plifying friction compensation problems in robotics. Thenotion of kinematic reducibility we presented can be re-lated to the Power Dissipation Method, a method for de-termining the quasistatic equations of motion for an over-constrained system (see [3], [39]). We have been able toshow that the solutions to the power dissipation methodcorrespond to kinematic solutions of multiple model sys-tems.

We do not claim that the PDM is a better model thanthe full Lagrangian setup, only that it is more tractable. Itproduces first order equations of motion that are amenable

to analysis. Moreover, the fact that it allows us to com-pute explicit controllers that work on a real experimentis an indication of its validity [34]. Nevertheless, thereare certainly important systems that must be treated in thefull Lagrangian mechanical framework, since even in theexample of the planar bike there are important dynamicstates not accounted for in the PDM. This determinationwill in general have to be made by the control designer.

Lastly, this work leaves several open questions to beanswered. First of all, in the definition presented in thispaper the dissipation functional is only applicable to a fi-nite number of contacts. However, in many pushing prob-lems the frictional interaction occurs at the interface be-tween two continuous media. The example of the Marsrover in Section X-B makes it clear that reduction the-ory (beyond kinematic reduction theory presented here)needs to be formally explored for multiple model systems.Lastly, there is the question of external forces. Our use ofkinematic reducibility in the example avoids the problemsof differentiation of friction forces because the manifoldstructure provides all the information we need. However,this cannot be expected in general, and there is a clearneed to extend the work in [23] to cases that generic reac-tion forces entering the equations of motion.

APPENDIX

We assume the reader is familiar with the basic no-tation and formalism of differential geometry and nonlin-ear controllability theory. See [35], [40], [1], [7], [43] formore details.

The notion of(U ,U)-reducibility formalizes whatis meant by kinematic reducibility. For mechanical sys-tems, we consider inputsu : [0, T ] → Rm that are essen-tially bounded and Lebesgue integrable. In Lewis [23], itwas assumed that inputs are absolutely continuous func-tions, since piecewise continuity implies that instanta-neous changes in system velocity are possible. In thepresence of inertial effects, such changes can only occurwhen infinite forces are allowed. We keep this assumptionon the inputs. However, here state transitions are beingapproximatedwith piecewise continuous signals. This isa common approximation in many areas of physical mod-eling [42]–, such as impacting bodies. Therefore, we onlyrequire that absolute continuity hold locally rather thanglobally.

Definition .1: f : [a, b] → Rm is absolutely contin-uousif for eachε > 0 ∃ δ > 0 such that for every finitecollection(ti, t

′

i)1≤i≤N of non-overlapping intervals in[a, b] with the property that

N∑i=1

|t′

i − ti| < δ we haveN∑

i=1

‖f(t′

i)− f(ti)‖ < ε


This definition implies thatDf exists almost everywhere.Like Lewis [23], we restrict our attention to systems

that can be modeled assimple mechanical systemsin apiecewise sense. In simple mechanical systems, the La-grangian takes the formL = K.E. − V . Assume thatQ is ann-dimensional configuration manifold, andG isa Riemannian metric onQ defining the kinetic energy.Since many of the applications of interest are systemswith no potential energy, let us simplify to the case whereL = K.E. (i.e., V = 0). Denote byvq elements in thetangent space ofQ atq, TqQ. With zero potential energy,the system Lagrangian takes the formL = 1

2g(vq, vq).Definition .2: TheChristoffel symbolsfor the Levi-

Civita connectiong∇ (associated with the metricG) are

Γijk =

12Gil

(∂Gjl

∂qk+∂Gkl

∂qj− ∂Gjk

∂ql

)(17)

where summation over repeated indices is implied usedunless otherwise stated, and upper indices indicate the in-verse.

Definition .3: In coordinates, thecovariant deriva-tiveof Y with respect toX is

G∇XY =(∂Y i

∂qjXj + Γi

jkXjY k

)∂

∂qj(18)

Definition .4: The symmetric productbetween twovector fieldsX andY is defined to be

〈X : Y 〉 =G ∇XY +G ∇Y X (19)

With these definitions in mind, we can quickly summarizeappropriate notions of dynamic and kinematic mechanicalsystems. Given a metricG on the manifoldQ and inputsua, it is possible to show that the Euler-Lagrange dynam-ical equations can be written in the form:

G∇c′(t)c′(t) = ua(t)Ya(c(t)) (20)

wheret→ c(t) is a path onQ andc′(t) = ddtc(t). On the

other hand, given input velocitiesuα, kinematicequationscan be written in the form:

q(t) = uα(t)Xα(q(t)) (21)

Theorem VIII.1 relates Eq (20) to Eq (21). As noted inLewis [23], the symmetric product plays a similar role inestablishing

(U ,U

)reducibility to the Lie bracket in es-

tablishing integrability. Some other things to note aboutkinematic reducibility include the following. First, allfully actuated systems are automatically kinematically re-ducible because their dynamic input vector fields are al-ways closed under symmetric products. For instance, theforward kinematics of a robotic manipulator are kine-matic whether moving in air (where the kinematic ap-proximation is obvious) or in a viscous fluid of some

sort. Also, kinematic reducibility is not the same thing asthe “quasistatic” assumption commonly made in robotics.This is because kinematic reducibility only requires thatthere be a complete correspondence between dynamicmotions and kinematic motions, whereas “quasistatic” as-sumptions, when formalized at all, typically require thatthe system be moving slowly in some sense. As notedin [39], the quasistatic case can only be equated to New-ton’s laws when the friction is Coulombic, but we notethat here kinematic motions are independent of frictionmodel. This fact seems to have reasonably deep implica-tions for friction compensation, and will be the topic offuture study.

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The Power Dissipation Method and Kinematic Reducibility of