Issues in Controllability and Motion Planning forOverconstrained Wheeled Vehicles

Todd Murphey, Joel BurdickEngineering and Applied Science, California Institute of Technology

Mail Code 104-44, Pasadena, CA 91125 USAmurphey,jwb @robotics.caltech.edu

Keywords: Overconstrained wheeled vehicles, switchedsystems, controllability, motion planning.

Abstract

Conventional nonholonomic motion planning and controltheories do not directly apply to “overconstrained vehicles,”such as the Sojourner vehicle of the Mars Pathfinder mission.This paper investigates some basic issues that are necessaryto build a motion planning and control framework for thispotentially important class of mobile robots. A power dis-sipation approach is used to model the governing equationsof overconstrained vehicles that move quasi-statically. Theseequations are shown to be switched hybrid systems. Stan-dard notions, such as the Lie bracket, are extended to theseswitched systems. We then develop a controllability test forsuch systems. We explore motion planning primitives in thecontext of simplified examples.

1 Introduction and Motivation

Most mobile robots use wheels since they provide one ofthe simplest means for mobility. Wheels impose nonholo-nomic constraints on a vehicle’s motion, and thus the subjectof control and motion planning for nonholonomic wheeledvehicles has been widely pursued [1, 2, 3, 4]. Most wheeledrobots operate in relatively benign man-made environments.Legged robots may be used for rough, unstructured terrains[5].

In order to operate in moderately rough terrains without re-sorting to the inherent complexity of legged system design,“overconstrained” wheeled vehicle designs have been pro-posed. The most famous example is the Sojourner robot de-ployed during the Mars Pathfinder mission. Fig. 1 shows the“Rocky 7,” a prototype for future Mars rover vehicles whosesuspension and wheel kinematics are essentially identical tothe Sojourner. The Rocky 7 employs six wheels, with bothfront wheels independently steered and all six wheels inde-pendently driven, making it an eight input system. The rearwheels on each side are coupled through a “bogey” linkagemechanism that helps the vehicle negotiate obstacles that areup to 1.5 times the wheels’ diameter. Below we shall showthat standard nonholonomic motion planning and control the-ories can not be applied to this vehicle.

Figure 1: Photo of Rocky 7 Mars Rover Prototype

To motivate the issues that are considered in this paper,consider a highly simplified model of the Rocky 7 vehicle(Fig. 2(b)). In this stripped down model, hereafter referredto as the Simplified Rocky 7 (or SR7), the vehicle operateson flat terrain. To realize the model of Fig. 2, each pairof Rocky 7 wheels is conceptually “collapsed” into a sin-gle wheel, as is done in conventional models of the classi-cal kinematic car (Fig. 2(a)). Furthermore, we assume thatonly the front wheel is actuated. While highly simplified,this model captures many of the essential features and chal-lenges of overconstrained wheeled vehicles. Obviously, thefact that Rocky 7 operates in non-planar terrain and has ad-ditional wheel actuation will pose further complexities.

The motion of every planar body can be characterized ateach instant by its Instantaneous Center of Rotation (ICR). Inthe classical kinematic car model (Fig. 2(a)), the assumptionthat the wheels do not slip defines an instantaneous center ofrotation at the intersection of the lines that are collinear withthe two wheel axes. Note that because of the overconstraintdue to the presence of an additional wheel, kinematic issuesalone can not be used to determine the ICR of the SR7 vehiclein Fig. 2(b). Standard nonholonomic motion planning andcontrol (NMPC) techniques implicitly assume that the ICRis uniquely defined by non-slip wheel constraints that candescribed by purely kinematical relationships. Furthermore,the overconstraint implies that unless the vehicle is stoppedor moving straight ahead, one or more wheels are alwaysslipping—i.e, the no-slip assumption that is at the basis ofNMPC techniques is violated.

One might argue that the “extra” wheels in the systemsof Fig.s 1 and 2(b) can be practically ignored. After all,18-wheeled trucks have similar overconstrained geometries.

r

l

ICR

Figure 2: (a) kinematic car; (b) simplified Rocky 7.

However, we seek dextrous maneuvering of such robots farbeyond that which is required for 18-wheelers. For example,future Mars rover mission scenarios call for a manipulatorarm with only 2 or 3 degrees of freedom (see Fig. 1 for anexample) to collect rock samples and position sensors. Arbi-trary displacements of the arm’s end-effector will require thevehicle to make non-trivial local sideways motions so as tocompensate for the arm’s kinematic deficiency.

As represented by Mars exploration opportunities, over-constrained vehicles are a potentially important class ofrobots. We seek to develop theories and algorithms thatparallel those for nonholonomic mobile robots. While themechanics of overconstrained vehicles have previously beenconsidered [6], no systematic control and motion planningtheory exists. This paper makes some first steps in this di-rection. We first consider how to model such systems for thepurpose of motion planning. A power dissipation model sug-gested by Alexander and Maddocks [7] is used to develop thegoverning equations. We formalize the techniques suggestedin their paper and show that the resulting equations have thestructure of a switched hybrid system. We next show howto extend notions such as the Lie bracket to the switchedcase. Based on these extensions, we develop a small-timelocal controllability test for such systems. This test answersthe practical question: “can the overconstrained vehicle lo-cally move in any direction?” Controllability tests for smoothnonholonomic systems are well known [8, 9], and analogoustests have been developed for some other types of hybrid sys-tems [10]. To our knowledge, this is the first controllabilitytest for such systems. In the closest related work, Liberzonand Morse [11] have considered the issue of asymptotic sta-bilizability of a class of switched systems. Finally, in thecontext of the model of Fig. 2(b), we consider motion plan-ning primitives for overconstrained mobile robots.

2 Mathematical Preliminaries

We assume the reader is familiar with the basic notationand formalism of differential geometry and nonlinear con-trollability theory. We remind the reader of the followingdefinitions and classical theorems so that it may be clearwhere we are starting from. We assume that is an di-mensional configuration space, that is the associateddimensional tangent bundle, and that , the space of inputs,is dimensional. We denote the function space of the coef-ficient of friction by , the function space of wheel normalforces by , and the function space of switching signals by

.The mechanics of classical nonholonomic systems can be

put into the form of a drift free affine system:

(1)

where is the system’s state space, arethe controls, and are termed the control vectorsfields. Controllability tests for such systems are based on thefollowing definitions.

Definition 1 The Lie Bracket between two vector fieldsand is defined to be

(2)

Definition 2 The distribution of a set of vector fields is

(3)

where the span is defined over the set of real valued functionson . A distribution is involutive if it is closed under theLie Bracket operation. The involutive closure of a distribu-tion is the smallest involutive distribution containing ,and is denoted by .

Definition 3 A drift free control system of the form Eq. 1 iscontrollable if for any there exists a and

such that and .

Definition 4 Given an open set , let bethe set of states such that steers thesystem from to and satisfiesfor . Moreover, define:

.The main controllability test for systems of the form Eq. (1)is Chow’s theorem.

Theorem 1 (Chow) The control system (1) is small time lo-cally controllable at if .

As modeled using the power dissipation method of Section3.2, the motions of overconstrained wheeled vehicles aregoverned by switched systems, a class of hybrid dynami-cal systems. One of the purposes of this paper is to extendChow’s theorem to switched systems.

Definition 5 (Switched Systems) A system is a switchedsystem if it can be written as

where for any and

with an index set and f measurable in .

With this definition, theorems from classical nonsmoothanalysis ensure existence and uniqueness of solutions for sys-tems of this type. These issues, which are beyond the scopeof this paper, are adequately covered in Deimling [12]. Forsystems of our type, one can generally expect existence, butcannot always expect uniqueness.

Definition 6 A system is a switched driftless affine system(SDA) if it can be expressed in the form

where for any and , , with an indexset and measurable in and smooth in for all.

This is a switched system as in Definition 5. Since the com-position of measurable functions is again measurable, onecan equivalently consider to be a function of andsay that .

Definition 7 The multiplicity of a switched vector is definedas the number of elements in , and the multiplicity of aswitched driftless affine system is defined to be the maximumnumber of elements of the for all .

For the purposes of clarity and practicality we focus onswitched driftless affine systems of multiplicity two. How-ever, most of our results hold for higher multiplicity SDAsystems.

3 Models for Motion Planning

We seek models for overconstrained systems that capturethe essential physics of the problem, and that are tractableand amenable to control and motion planning analysis. Thissection briefly reviews Lagrangian modeling of such sys-tems, and then considers in more detail a power-dissipationapproach that is appropriate for quasi-statically moving ve-hicles, such as the Rocky 7 in Fig. 1. We assume that thevehicle’s wheel contact with the ground is governed by theCoulomb friction law.

3.1 Lagrangian models.

Constrained mechanical systems can be modeled usingconventional Lagrangian mechanics via the use of Lagrangemultipliers. Let denote the Lagrangian (kinetic mi-nus potential energy) of the vehicle system. If the wheelis not slipping, then this constraint on the vehicle’s motioncan be expressed as . If the wheel is slip-ping, then the Coulomb law governs the reaction force at thatwheel: , where , , and are respec-tively the Coulomb friction coefficient, normal force to theground, and slipping velocity of the wheel at the contact.Hence, the system’s equations of motion are described by:

(4)

where is the slipping set, the are undetermined La-grange multipliers, and are the generalized applied forces.That is, if the wheel contact is slipping. If the

wheel is not slipping, corresponds to the reactionforce necessary to maintain the no-slip constraint at thewheel. There are two primary practical problems with theLagrangian approach. First, one must solve for the Lagrangemultipliers—a tedious task that often leads to complex equa-tions. Second, an additional (and often complicated) analysisis necessary to determine which wheels are slipping at anygiven instant.

NN 2 1

2 1

12

Figure 3: Planar BikeAs an example, consider the planar bike (Fig. 3). The

downward normal force on each wheel will depend on thebike’s weight distribution. We assume that each wheel is ac-tuated, with torques and , and that each of the wheelsmay slip (depending upon the ground reaction force). UsingEq. (4) and solving for the Lagrange multipliers, the equa-tions of motion are

where

where is a wheel’s moment of inertia about its rotationalaxis, is total bike mass, and is the wheel radius. This isa multiplicity 4 switched system. If is the reaction forcefor each wheel, the Coulomb friction model implies that theboundary between slipping and nonslipping states occurs atsome value of , thereby implying that the spaceis divided into regions of different slipping states. Generally,for an -wheeled vehicle, the slipping regions are separatedby hyperplanes which bound a hypercube. The problem ofstate determination arises from the inherently complicateddependency of on the current state. In the case of the planarbicycle, we compute that

and get a symmetric result for . Moreover, the criticaltakes the value , where and are the

Coulomb friction coefficient and normal force at contact .This fact implies that the boundary of these regions is ter-rain dependent. I.e., the hypercube delineation of switchingregions is a purely local phenomena.

3.2 Power Dissipation Models.Many overconstrained vehicles, such as the Rocky 7

whose speed is measured in 10’s of centimeters per sec-ond, move slowly enough that their motion can be consideredquasi-static. Hence, it makes sense to develop a modeling ap-proach that is suited to this realistic situation. Recall that atleast one wheel always slips during overconstrained vehiclemotion, thereby dissipating energy. The power dissipationmethod assumes that the system’s motion at any given instantis the one that instantaneously minimizes power dissipationdue to wheel slippage. This method is adapted from Alexan-der and Maddocks [7]. The power dissipation function mea-sures the vehicle’s total dissipation due to wheel slippage.

Definition 8 The Dissipation or Friction Functional for an-wheeled vehicle is defined to be

where , with the normal force.

Alexander and Maddocks showed that is convex; there-fore its local minima are global minima [7]. Let’s revisit theplanar bicycle example. Note that the minimum of mustoccur at a nondifferentiable point of , since the functionis monotone everywhere else. By direct comparison of thetwo nondifferentiable states, which correspond to one wheelnot slipping or the other wheel not slipping, the minimumis associated with whichever has a lower value. Conse-quently, the zerol level set of the functiondetermines which state the bicycle lies in. This determina-tion becomes nonunique when . This model onlyhas two states, making it much simpler to analyze than theLagrangian model. Additionally, the dynamics take the sim-plified SDA system form: , where is the wheelwhich isn’t slipping. Note that the Lagrangian approach pro-duces four possible dynamic states, whereas the power dissi-pation approach yields only two states. However, the statesin the Lagrangian model are: a) no wheels slipping, b) bothwheels slipping, c) the back wheel slipping, and d) the frontwheel slipping. The first two possibilities both imply thatthe inertial terms dominate the system’s dynamics, whichgoes against our quasistatic assumption. This leaves the sec-ond two states, which are the same as what we found in thepower dissipation model. This is an indication of how thequasistatic assumption, which we believe to be valid in thecase of the rover, helps to simplify our problem.

3.3 The Power Dissipation Modeling ApproachLeads to Switched Hybrid Systems.

Ideally the dissipation function defined in Definition 8would always have a unique minimum. Unfortunately, as

shown by the planar bike’s indeterminacy at , aunique minimum can not be expected. This section showsthat the power dissipation approach generically leads toswitched hybrid systems, which generically exhibit uniqueminimum in . Alexander and Maddocks [7] show that thedissipation model is convex, so local minima are global min-ima, should they exist. They also show that if they exist, theyexist at a point of nondifferentiability of . However, theremay be other points at which the minimum is obtained. Let

and where is thekinematic solution to a nonoverconstrained subsetconsisting of constraints. i.e.

...

This means that consists of points for which nodirectional derivatives of exist and that we therefore onlyhave a finite number of points to check in order to find theminima. It is straightforward to show that these minimamust at least occur at points in . Now, reorder so that

. Although has at least oneof the minima achieved by , it does not necessarily containall of them. In fact, if more than one element of is a min-imum, then the convex hull of these minima also achieve it.Unfortunately, the condition is only necessary for aminima, but the next proposition proves that in the case ofthe function it is also sufficient.

Proposition 1 If and both minimize the dissipationfunctional found in Definition 8, then so does .

Proof: Assume and . Then

Now assume that is strictly less than somewhere in. Then such that is at

a minimum by an extension of Rolle’s Theorem for the realline. Then is at a point where isnonsmooth in all its directional derivatives [7] (becauseis monotone elsewhere). This implies that and that

, thus violating our assumption thatis a minimum of . Therefore

The proof for higher numbers of having equaldissipation is by induction on this argument.

Now is a set of points on which is nondif-ferentiable, just not in all directions. It therefore still meetsthe criterion to be a minimum [7]. We now consider the ex-tent to which the function having a unique minimum isgeneric.

Proposition 2 Assume is of theform 8 and that the is measurable in and . Then the dis-

sipation functional has a unique minimum almost always(i.e. except on a set of measure 0)

Proof:Case 1: If is a unique minimum , then it is the uniqueglobal minimum since Alexander and Maddocks showed thatthe minimum must occur in by our definition of .Case 2: If and such that both are minima, then byProposition 1, we know that also minimizes the .However, this only occurs whensatisfy the constraint . This impliesthat the constraint is only satisfied on a set of measure 0 inthe space

The reader should note that the proof of Proposition 2 isonly useful if we have already found , and moreover fora high number of states it may be very difficult to find theminimum of . This problem, thus stated, bears more thana passing resemblance to the simplex method found in LPtheory and techniques from that theory can be applied tothe problem of finding the minimum of the function inthe presence of high numbers of contact states. Also, in thenonoverconstrained case of constraints, the dissipationmethod leads to the classical kinematic solution. Proposition2 allows us to now state what we mean by the dissipationfunctional leading generically to a switched system.

Corollary 3 The multivalued map implicitlydefined by is single valued almost every-where.

Corollary 3 implies that we can generically expect thepower dissipation method to lead to a uniquely defined set ofkinematics. In particular, it implies that the dissipation mod-eling approach will genericly give a well defined set of kine-matics, and that it will almost never lead to an indeterminatesystem. This makes rigorous the comment made in Alexan-der and Maddocks referring to the physical expectation ofcontinually switching back and forth between the dominanceof one wheel or another, rather than staying in an indetermi-nate state [7]. See Ref. [12] for a discussion of implicitlydefined multivalued maps.

3.4 Dissipation model for the simplified Rocky 7

Using the dissipation approach, we first show using [7]that the minimum of must occur when either the middle orback wheel slips. If the configuration of the simplified Rocky7 is and the controls and are associatedwith the drive velocity and steering velocity respectively, wefind that the equations that govern the motion of the simpli-fied Rocky 7 are:

Our system is an SDA system as in Definition 6, with(i.e., only two possible dynamic states). The

function which determines the current state is:

When , ; when , . Thisformula implies that unless the is know for all , thecurrent vehicle state is unknown. This observation suggeststhe practical use of wheel slip sensors.

4 Switched Controllability

We now address the issue of controllability, i.e, can theoverconstrained vehicle locally move in any direction? Thereare two important cases to distinguish in the controllability ofoverconstrained vehicles: when the switching is controlledand when the switching is not under the vehicle’s control.The case of controlled switching requires a modest extensionof controllability theory, while the other case requires newdefinitions.

4.1 Controlled Switching

Let denote the involutive closure of the control vectorfield distribution associated with switching state . When

is full rank, controllability is immediately realized, asone can (with the assumption of complete control over theswitching process) switch to the controllable state . Con-ventional results for smooth systems then apply to this state.However, if none of the are full rank, then controllabilitymay still exist, but is not obvious. To motivate this situation,consider the example in Fig 4. This fixed wheel kinematic

Mechanical Manipulator

Passive Wheel Driven Wheel Passive Wheelat 45 degrees

Figure 4: The Fixed Wheel Kinematic Car

car (FWKC) has three wheels, of which only the middle isdriven. None of the wheels are steerable: the back one re-mains straight, and the front remains at a constant angle of

. We include the mechanical arm above the body as anexample of a mechanism that can control switching. As thearm moves forward and backward, it can shift the center ofmass sufficiently to switch the vehicle into a new dynamicstate—i.e., the arm position determines which wheel is slip-ping. Each dynamic state by itself is uncontrollable, thoughwe shall see that this system is indeed controllable. Whilethis example generally has no practical value, it is illustra-tive of the idea, and may possibly represent the vehicle in a

singular configuration of its suspension or a state of steeringactuator failure. Based on the power dissipation approach,the governing dynamics of this vehicle are

Recall that the classical Lie bracket between two differen-tiable vector fields, which does not have any meaning in thisswitched system context, is equivalent to:

where represents the flow along for time . The totalflow can be seen schematically in Fig. 5.

gf

fg

f

gnet motionnonzero

Figure 5: Flows associated with a continuous Lie bracketmotion.

The interpretation of the Lie bracket as a flow makesthe following extension to the switched case almost trivial.Rather than forming the Lie Bracket between two separatesmooth control input vector fields, we form a Lie Bracket ofa control input vector field where .We do this by setting for andotherwise, while for and ,otherwise. This produces the flow seen in Figure 5. This sim-ple controlled switched Lie bracket(CSLB) can therefore beused to control the mechanism. This leads us to the followingcorollary of Chow’s Theorem.

Corollary 4 Consider an SDA system of the form Defini-tion 6, where the switching can be controlled directly. Let

denote the distribution of the control vector fields as-sociated with state . Let denote the involutive clo-sure of . The system is locally controllable if

for all .

The proof of this corollary is similar to a standard proof ofChow’s Theorem (see Ref. [8] for example) with the modifi-cation that the flows are produced by a switching vector field.For this reason we omit details of the proof. Applying thisresult to the FWKC, we find the Lie bracket of the two vectorfields to be , which leads to a full span ofthe three-dimensional vector space. Therefore the FWKC iscontrollable. Simulations in Section 5 bear out this result.

4.2 Uncontrolled Switching

To simplify the discussion, we will continue to considerswitched vector fields where there are only two possible vec-tors associated with any given input; i.e., . Thefollowing definition encapsulates the notion that switchingcan happen arbitrarily quickly.

Definition 9 The switched vector is defined to be

where .

This definition allows for switching to occur even as. It also avoids the complications that would otherwise arise

from discontinuous functions in a derivative-based defini-tion. Additionally, it does reduce to a classical vector when

, which implies that , the empty set. Note thatis generally a set valued object. In fact, we can show

the following:

Proposition 5 The switched vector of definition 9 isthe convex hull of the component vectors, i.e. .

Proof: Let denote the convex hull formed bywhere . We first show that if , then. We then show that if , . To demonstrate

the first part, let us show that is a switchingsequence that produces . Taking the limit

I.e., , thereby show-

ing that . To show the second part note thatimplies that

for some choice of such that . This in turnimplies that

Rearrange terms to find

By assumption, . Thus

where . This implies that . For multiplic-ity the argument proceeds by induction.

An interesting implication of the above calculation is thatany switching of the above form is locally equivalent to asmooth vector field . Physically what this means is thatwe are switching much quickly than we are flowing, therebyensuring that at every time we are approximating . In par-ticular, if is the number of switches that occur in time , theflow associated with switching vector field is arbitrarily ap-proximated by a smooth vector field as . It shouldbe noted that this is an extremely strong assumption, partic-ularly in how we use it later in Prop. 7. Moreover, the abovecalculation implies that the limit in Definition 9 exists forany switching sequence. We would also like to know that thefunction seen as a function of time is indeedmeasurable in time. To see this, we recall the following the-orem from measure theory. (see [13] volume 1, page 433 fora good reference)

Theorem 2 (Luzin) Let M be a measurable subset of .Then, the function

is measurable iff it is continuous up to small sets. i.e. foreach such that

is continuous and .

It is easy to see that satisfies the requirements of The-orem 2. Take with fi-nite, and assume that the switching occurs at times

. Because is a finite or possibly count-ably infinite collection of points, (see [14]).Moreover, is continuous.Therefore, by Theorem 2, is measurable.

Having defined a switched vector field, we can address theissue of whether the set valued dynamics has a Lie Algebraicstructure. While there is a bracket, it is not quite the mostnatural operator one might hope for. Nevertheless, we cannow motivate the definition of a Switched Lie bracket. First,we define some notation.

Definition 10 Let the setflow be the set of all flowsgenerated by starting from the set of initial conditions andflowing along the switched vector for time .

Using Definition 10 one can define a the Switched LieBracket which encapsulates the possibility of uncontrolledlocal switching. However, the most convenient mathemati-cal formulation of this problem (in Definition 11) is not themost useful, as we will see. Nevertheless, it is tempting to in-corporate switching into a bracket operation in the followingway.

nonzeronet motion

Figure 6: Schematic of the Switched Lie bracket motion.

Definition 11 The Two Sided Uncontrolled Switched Liebracket (2USLB) is defined as:

where and are switched vectors.

Note that because and are set valued, is thereforealso set valued. We can consider to be the set of allpossible infinitesimal flows associated with any measurableswitching sequence. This concept is illustrated in Fig. 6.Note that in this definition, first order terms necessarily arisewhen the infinitesimal flows are generated. This is becausethe “average” flow along the first half of the bracket may notbe the same as the negative “average” flow along the secondhalf. A straightforward calculation shows that both first andsecond order terms arise in the composition of flows foundin Definition 11, which means that the limit does not typi-cally exist. To address this situation, we introduce a slightlymore narrow bracket, the One Sided Bracket, in order to en-sure that only second order terms are included. To see whyDefinition 11 is not quite the desired definition, consider thefollowing case. Let be of multiplicity 1 and let be ofmultiplicity 2. Then, using Proposition 5 we can allowto vary over the normal Lie Bracket. Using the fact that thatby Prop. 5 ,

This calculation suggests that even if is in , this bracketmay not in general be of interest due to the first term com-posed of terms associated with the initial vector fields. More-over, we would like a definition which will not use deriva-tives of . Lastly, as mentioned previously, this definitionoften has first order terms associated with it, so it does notalways exist. We are interested in determining the closure

of such flows, and are therefore only interested in second or-der terms. With this motivation, we define the One SidedBracket, which effectively symmetrizes the above definitionby dropping first order terms.

Definition 12 The One Sided Uncontrolled Switched Liebracket (1USLB) is defined as:

and where and are switched vectors.

The first half of the composed flow in Definition 12 is thesame as that of Definition 11, but the latter half of Defini-tion 12 guarantees symmetry of the flow, thereby ensuringthat only second order terms arise in the limit. Using this def-inition and Prop. 5, one can show that forms a cone ofthe individual Lie bracket vectors. To make the proof moreclear, we only consider the case relevant to us, where has noswitching and only has two possible switched states. Theresult is true for switched vectors of arbitrary multiplicity.

Proposition 6 If is a multiplicity 1 switched vector andis a multiplicity 2 switched vectors formed by ,

forms the convex hull of vectors arising from the lie brackets.

Proof: Using Def. 12 and Prop. 5 we see that

with . We form the algebraic Lie bracketwhile holding constant.

The problem of controllability can be approached with thisnotion of switched vector Lie brackets. An extension of thedefinitions in 2 is needed.

Definition 13 Let

and additionally define

where , is the number ofinput vectors, and the are the brackets under theoperation.

Proposition 7 Let be the total number of switches that oc-cur in time . Then, taking the limit , the controlsystem of Definition 6 is small time locally controllable at

iff .

Proof: Here we follow the essence of the proof of Chow’stheorem found in [8]. To show necessity, assume that theswitched vectors satisfying the above property are .If , then such that where isthe classical distribution obtained by holding the constant.Then by the classical Chow’s theorem the system is not lo-cally controllable. To show sufficiency, first note that a vector

associated with a given switching sequence is smoothin the limit as defined in Definition 6. Additionally recall thatlocal controllability is equivalent to be-ing nonempty. Now proceed by recursion. Associated withany switching sequence up to time is where

. For small, is asmooth surface (of dimension 1) because is a smooth vec-tor field (because ). Now assume that is a

dimensional manifold. We can do this because is theminimum rank distribution, which implies that there does notexist any switching sequence that can reduce the dimensionof . If , then there exists and suchthat (i.e. ) Theexistence of such a is guaranteed by two facts: 1) that ifsuch a one did not exist, then either for any insome open set , which would implyor , and 2) the definition of implies thatthe at least of the do not intersect, thereby not allowingany switching sequence to force a drop in rank. Statement 1)contradicts the assumption that .Now for small,

is a dimensional manifold. More-over, since can be made arbitrarily small, we can assume

Now if , is an dimensional man-ifold, which implies that is nonempty.

Intuitively, Proposition 7 says that if we have some setof nominally independent control directions, there existsa switching sequence which can make two control direc-tions the same if and only if their associated cones intersect.Therefore, worst case scenario switching can make the sys-tem locally uncontrollable if and only if the associated conesintersect. One advantage of this approach is that it does leadto such a geometrically simple interpretation of the controlla-bility condition. At the same time, we were forced to use theassumption of taking the limit of in order to be ableto make the flow produce a smooth manifold. This certainlylimits the applicability of this controllability test, but we nev-ertheless feel that this is a good first step towards understand-ing the essential elements of the controllability question forswitched systems. This does not imply that the system is notcontrollable, only that it is not locally controllable. However,in the case of the rover, we are interested in local controllabil-ity so that fine maneuvering can be achieved. An equivalentstatement of Proposition 7 can be found in Corollary 8, wherean algebraic, as opposed to a geometric, condition for neces-

sary and sufficient conditions on controllability is found.

Corollary 8 Let , be as in Proposition 7. Writing assuch that

and , then the control system is locally control-lable iff

For the SR7, this amounts to assuring that the determinantof a modest matrix is always nonzero, thereby reducing thequestion of controllability to an algebraic inequality. For theSR7, controllability is ensured so long as , whereis the front wheel angle.

5 Motion Primitives

This section describes some simple motion planning tech-niques for overconstrained wheeled vehicles. As stated, clas-sical controllability results and nonholonomic motion plan-ning ideas can easily be extended to the case of controlledswitching. The case of uncontrolled switching is more com-plicated, as seen in the simulation of Fig. 7. In this sim-ulation, the SR7 drives with constant steering wheel speedand angle. With no switching, the vehicle’s center woulddescribe a constant radius circular motion. However, in thissimulation the vehicle passes over terrain regions with differ-ent friction coefficients, thereby causing switches in the dy-namics. The path’s non-circular geometry clearly indicatesthat switching can introduce considerable error. This simu-lation shows why standard nonholonomic techniques are notdirectly applicable. This unpredictable switching behaviormay make the open loop motion planning problem seem in-surmountable, and suggests that feedback is an appropriatestrategy in this case. However, some heuristic techniques areavailable.

0 5 10 15 207.5

5

2.5

0

2.5

5

7.5

10

Region A

Region B

1

Figure 7: SR7 with Constant Wheel Angle and Speed

In nonholonomic systems, motion in the linearly uncon-trollably directions is created by periodic inputs (or “Liebracket motions”). This observation guided our extension ofthe classical Lie bracket to the switched case. With this ex-tension, it appears true that many (though probably not all)of the open loop local motion planning concepts from theclassical nonholonomic literature can be adapted to the over-constrained case.

5.1 Controlled Switching.

The fixed wheel kinematic car example illustrates this sim-ple case. The FWKC is controllable when switching is takeninto account. Hence, arbitrary motions will generally require

induced switching. Motion in the linearly uncontrollable di-rection is obtained by creating a flow along , inducing aswitch (by “waving” the manipulator arm) so that the systemflows along , inducing another switch to flow along(by arm waving and wheel rotation reversal), and switchingagain to induce flow along , all for time .

In general, one can adapt the approach suggested in Ref.[2] to this situation, with discrete switches replacing the roleof some sinusoidal inputs. In summary, the algorithm is: a)turning until the desired value of is achieved b) move for-ward until (i.e. the car isparallel with the desired goal) c) compute the time neededfor the car to pass through the desired point d) stop at the de-sired goal point ( will have already returned to its originalvalue). This simple algorithm was used to drive the FWKCfrom the origin to the point (2,2) in Fig. 8.

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

XRegion 1

X

Region 2

Region 3

1

Figure 8: X and Y variables of the FWKC

5.2 Uncontrolled Switching.

In the uncontrolled switching case, the primary difficultyshifts from control of an otherwise underactuated mecha-nism to the errors associated with switching. For motionplanning purposes, one must use some form of feedback.The algorithmic approach is similar to the controlled switch-ing case. While we wish to use the Lie bracket motion toarrive at our final state, the actual Lie bracket is

. Fortunately, there is a natural choiceof . If we know something about the environment’s proper-ties, is chosen to be the percentage of time that the vehicleis expected to be in state 1: corresponds to equalpossibility of being in one state or another. We heuristicallychoose to base our Lie bracket motion on the vector field

. Error is then introduced due toswitching, and we iterate.

Traditionally, the use of sinusoids has arisen as the so-lution to optimal control problem that asks how to move achained form system in its Lie bracket direction while min-imizing [8, 9]. We can likewise introduce anoptimal control framework where the cost function is chosento minimize a weighted cost combining power and error dueto switching. To achieve this, choose

(5)

instead of as in Refs. [8, 9]. Here is the anglebetween and defined by the euclidian metric. I.e.,

(6)

Equation 6 is unfortunately too complicated to yield a nicesolution. However, to third order in , making

(7)

a natural choice of cost function. Interestingly, this cost func-tion is the same as in Refs. [8, 9], except that is nowassociated with a new metric. This choice of cost functionencapsulates the fact that the more one is willing to keep thewheels pointing close to straight forward (and therefore iswilling to move increasingly slowly in the Lie bracket di-rection) the more one can reduce error. In the simulation inFig. 9, this method converges in only two iterations to thegoal state of .

1 2 3 4 5 6

1

2

3

4

5

Region ARegion B

1

Figure 9: X and Y Coordinates for SR7

6 Conclusions

Standard nonholonomic motion planning and control the-ories can not be directly applied to the potentially impor-tant class of overconstrained wheeled vehicles. This paperoutlined some initial steps towards a motion planning andcontrol theory for such vehicles. We extended Chow’s theo-rem to both the controlled and uncontrolled switching cases.While we demonstrated useful primitives for open-loop mo-tion planning, a formal and general motion planning schemeis still forthcoming. The uncertainty associated with the un-controlled switching case clearly suggests the investigationof feedback schemes to stabilize such systems. Finally, weexpect that further investigation of the likely strong relation-ship between the simplex method in LP theory and the nu-merical techniques required to solve for the minimum of thedissipation functional will be fruitful.

Acknowledgements: This work was partially supportedby the National Science Foundation.

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