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IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, NO. 3, JUNE 2006 491

Simplified Motion Control of a Two-AxleCompliant Framed Wheeled Mobile Robot

Mark A. Minor, Member, IEEE, Brian W. Albiston, and Corey L. Schwensen

Abstract—Kinematic models and motion control algorithms fora two-axle compliant frame mobile robot are examined. Generalkinematics describing the compliantly coupled nonholonomickinematics are derived using velocity constraints that minimizetraction forces and consider foreshortening of the frame. Giventhe complexity of these equations, the steering ratio is definedto describe the relative heading angles of the front and rear axles.Simplified kinematic models are developed based upon (TypesI, II, and III) and the reference point used to guide the robot.Physical limitations and performance metrics (lateral mobility andmaneuverability per unit of traction force) are derived to evaluatethe models. Six groups of simulations and 24 experimental testsconsisting of 120 trials evaluate the performance of the algorithmson carpet, sand, and sand with rocks. Results indicate that Type I(curvature-based steering) provides superior maneuverability andregulation accuracy, whereas Type II provides excellent lateralmobility at the cost of high traction forces, reduced accuracy, andpotential singularities. Both models offer significant reductions incomplexity for simplified control using standard curvature-basedunicycle control algorithms. These results support expectationsderived from performance metrics and physical limitations. Ex-perimental results also demonstrate the efficacy of the robot toadapt to and maneuver over extremely rugged rocky terrain.

Index Terms—Compliant, kinematics, mobile robot, motioncontrol.

I. INTRODUCTION

KINEMATIC models and motion-control algorithms fora two-axle compliant framed wheeled modular mo-

bile robot (CFMMR), Fig. 1, are the subject of this paper. TheCFMMR concept is unique in two ways. First, it uses a novel yetsimple structure to provide suspension and highly controllablesteering capability without adding any additional hardware tothe system. This is accomplished by using compliant frameelements to couple rigid differentially steered axles. In thispaper, a partially compliant frame provides roll and yaw degreesof freedom (DOFs) between the axles. Relative roll providessuspension capability in order to accommodate uneven terrain,and yaw allows the axles to independently change heading for

Manuscript received January 25, 2005; revised October 14, 2005. This paperwas recommended for publication by Associate Editor K. Yoshida and EditorH. Arai upon evaluation of the reviewers’ comments. This work was supportedin part by the National Science Foundation under Grant IIS-0308056, and inpart by the University of Utah.

M. A. Minor is with the Department of Mechanical Engineering, Universityof Utah, Salt Lake City, UT 84112 USA (e-mail: [email protected]).

B. W. Albiston was with the Department of Mechanical Engineering,University of Utah, Salt Lake City, UT 84112 USA. He is now with SagetechCorporation, Hood River, OR 97031 USA (e-mail: [email protected]).

C. L. Schwensen was with the Department of Mechanical Engineering, Uni-versity of Utah, Salt Lake City, UT 84112 USA. He is now with Rosetta Inphar-matics, Seattle, WA 98109 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TRO.2006.875503

Fig. 1. CFMMR two-axle scout experimental configuration.

steering. Steering and maneuvering of the system are thus ac-complished via coordinated control of the differentially steeredaxles. Since each axle can be steered independently, the systemprovides the capability to control the shape of its frame, andthus enhance maneuverability in confined environments.

A second unique aspect of the CFMMR is its predispositionfor modular mobile robotics. Reconfigurable modular roboticsystems have been of keen interest to researchers during the lastdecade, due to their improved ability to overcome obstacles andperform more tasks using a single hardware platform. Towardsthis goal, researchers have investigated homogenous roboticmodules for reconfigurable manipulation [1], [2], mobility [3],[4], or combinations thereof [5]–[7]. Homogeneity is arguedto reduce maintenance, offer increased robustness throughredundancy, provide compact and ordered storage, and increaseadaptability [3], [8]. The CFMMR allows these concepts to beextended to wheeled mobile robots. Several modular configu-rations are shown in Fig. 2, which includes two-axle scouts, afour-axle train, and a four-axle moving platform. The scout issuited to reconnaissance and exploration, the train is tailoredto transporting payloads extended distances, and the platformis adapted to moving large objects. The utility of the systemis greatest for resource limited applications, such as space ex-ploration or military operations. Potential civilian applicationsinclude farming, forestry, and mining. This paper focuses inparticular on the two-axle scout configuration.

A third unique aspect of the CFMMR is the simplicity ofits mechanical design. At the most fundamental level, the axlemodules are basic differentially steered mobile robots; they arerigid structures providing an interface for the frame that sup-ports two independently controlled wheels. Other than the wheeldrive systems, there are no moving parts in the axle modules.The compliant frame then provides flexible coupling betweenthe axles to allow them to steer independently and conform to

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492 IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, NO. 3, JUNE 2006

Fig. 2. Modular configurations. (a) Two-axle scouts. (b) Four-axle train.(c) Four-axle array/platform.

terrain variations. This alleviates the need for complicated link-ages and associated hardware typical of steering and suspen-sion systems. The compliant frame thus significantly reducescomplexity and cost of the mechanical structure. Since the onlymoving parts of the CFMMR are the wheel drive systems, veryfew components are subject to wear. Those that do wear are sim-ilar throughout the entire system, which simplifies design lifeand maintenance issues. Thus, the simplicity of the CFMMRdesign allows straightforward reduction of mechanical failureprobability.

While the CFMMR provides new capabilities in steering, mo-bility, and reconfiguration, it also introduces new challenges thatmust be addressed in motion control [9], [10], dynamic control[11], sensor instrumentation, and data fusion [12]. While eachof these is important to the implementation of the system, thispaper addresses issues related to controlling the motion of therobot. Frame compliance allows a wide range of steering algo-rithms to be applied, but it also complicates the kinematic modelappreciably and provides significant challenges to motion-con-trol algorithms. To simplify the motion-control task, steeringconstraints are established in this research by the ratio of thefront and rear axle headings. These constraints can provide sig-nificant simplification of the kinematics and allow existing uni-cycle motion-control algorithms to be applied. Hence, we firstdevelop general kinematic models of the system, and then illus-trate the effect of these simplifications when considering per-formance metrics such as mobility, maneuverability, traction re-quirements, and control simplicity.

The structure of the paper follows. The CFMMR will first becompared with other similar robots in Section II. General kine-matics are derived in Section III, and three simplified models are

proposed in Section IV. Performance metrics and limiting fac-tors are discussed in Section V. Motion-control algorithms aredescribed in Section VI, which are applied in simulation andexperiment to evaluate model performance in Section VII. Con-cluding remarks are provided in Section VIII.

II. BACKGROUND

A limited number of compliant vehicles can be found in theliterature, and none possess a similar highly compliant framewhose deflection is controlled by coordinated actuation of thewheels. The earliest found reference is a system proposed forplanetary exploration using compliant members to provide rolland pitch DOFs for suspension of the axles [13]. This conceptwas later extended [14] to the frame of a vehicle composed ofhelical spring(s) with hydraulic cylinders used to control deflec-tion. In each of these cases, compliance was introduced for ac-commodating terrain. The CFMMR uses passive compliance ina similar spirit to provide independent suspension and advancedsteering control between the axles without additional hardwareor actuators. This vastly reduces the number of components re-quired to construct a system, reduces probability of componentfailure, and allows aspects of modularity to be exploited.

More recent research has introduced compliance for accom-modating measurement error and preventing wheel slip from oc-curring between independently controlled axle units on a servicerobot [15]. This robot is similar in spirit to the CFMMR, in thatit allows relative axle yaw, but this is provided by rotary jointsconnected to the ends of a frame with limited prismatic com-pliance. Other flexible robots use actuated articulated joints toprovide relative motion between axles, as in the case of the Mar-sokhod rover [16] and other six-wheeled research rovers withhigh relative DOFs [17]. These actuated kinematic structuresprovide more direct control of their shape than the CFMMR,but it is accomplished at the expense of system complexity. Thepoint is that the CFMMR provides similar capability to adaptto terrain, but does not require any additional hardware or me-chanical systems.

Snake-like mobile robots and continuum-type manipulatorsalso bear resemblance to the CFMMR, although their applica-tions are usually oriented towards search and rescue or mate-rial handling, respectively [18]. Each consists of a serial chainof modules. Trunk-like continuum manipulators are similar tothe CFMMR in that the interconnection is achieved by com-pliant beams or springs, but articulation is driven by tendons thatexert bending moments on the modules [19], [20]. Hyper-re-dundant manipulators replace the compliant member with rigidsegments and active articulated joints [21]. This is similar toactive-joint articulated snake-like mobile robots [3], [4] wherethe serial chain interacts with the ground to propel the robot.Many of these have wheels on the modules for reduced frictionor to establish well-defined kinematics. In some cases, the artic-ulated joints are active and the wheels are passive [22], and inother cases, the wheels are active and articulation is either par-tially active [23], [24] or entirely passive [25]. Active wheelsprovide direct control over forward velocity and are better fortraveling over terrain. Active joints allow direct control overrobot shape, such as for climbing over very large obstacles, but

MINOR et al.: SIMPLIFIED MOTION CONTROL OF A TWO-AXLE COMPLIANT FRAMED WHEELED MOBILE ROBOT 493

Fig. 3. Single-axle kinematic model.

they are usually slow due to high torque demands and limitedspace. Thus, passive compliant joints have evolved for naturalterrain adaptation, to reduce impact loads damaging to activejoints, and to facilitate faster travel over rugged terrain [25]. Ofthe snake-like robots, the CFMMR is similar to [25] in termsof rover-type application and architecture, but complex and ex-pensive mechanical joints with higher potential for failure arecreated for [25] to emulate the simple and cost-effective com-pliance of the CFMMR. The CFMMR is also modular and fa-cilitates numerous configurations and applications.

Control of mobile robots and nonholonomic systems has re-ceived a great deal of attention in recent years. For a thoroughsurvey of nonholonomic control techniques, see the review in[26]. Many of these are well-suited to unicycle-type velocityconstraints. While the kinematics of the compliant framed mo-bile robot are much more complex, we will show that they can bedescribed in an equivalent coordinate frame that admits familiarunicycle motion-control algorithms. In particular, we apply con-trollers discussed by Indiveri [27] and Tayebi [28] in conjunc-tion with a dynamic extension to accommodate nonideal initialconditions (ICs) and provide drift-free motion control. This issimilar in spirit to the extension performed by Astolfi [29], inorder to accommodate nonholonomic systems with drift.

III. GENERAL KINEMATIC MODEL

Fundamentally, each axle module in the CFMMR is a dif-ferentially steered unicycle-type mobile robot (Fig. 3). Unlikeunicycle robots that gain stability from additional castor wheelsfore and aft of the axle, the CFMMR uses frame members tocouple and stabilize multiple axles. Compliant coupling pro-vides suspension to the axles and suits the system to uneventerrain, since the axles can deflect to accommodate surface vari-ations. Compliant axle coupling also implies that the kinematicmodel of the system is more complicated, and that axle behaviormust be coordinated.

Consider first the simplest component of the system: thesingle axle. Assuming no slip, the orientation angle andforward velocity are determined by the wheel velocities

(1)

where is the wheel radius, is the angular velocity of theth wheel, and denotes the axle position in the system. The

resulting nonhololomic Cartesian kinematic equations are thus

(2)

where is the angular velocity of the axle about its centerpoint . In the following developments, the control inputs toeach axle will thus be and , which can be used to calculatethe individual wheel velocities in (1). Based upon these inputsand nonholonomic constraints, each axle imposes displacementboundary conditions on the compliant frame member. Thesedisplacements may produce nonnegligible reaction forcesacting on the axles, which draw from available wheel traction.Hence, consideration of frame coupling in the motion-controlalgorithms and kinematic models is critical. To consider thisissue, we first define several steering strategies that simplifythe kinematics of the system, and then we will examine framecoupling in these situations.

A. Steering Configurations

As Fig. 3 indicates, each axle can move in a forward direction,with orientation determined by , and rotate instantaneouslyabout its center point . At any instant, these constraints aresimilar to the pin-slot type of boundary condition described inmechanics [30]. In order to consider axle spacing variations thatresult from changing axle headings, this analysis can be simpli-fied without loss of generality if we consider one end of the robotto be pinned and the other end to be constrained by a pin-slot.This is due to the fact that a straight line can always be drawnbetween points and , which can then be used to transformthe orientation of the robot and the axle deflections relative to

.Examples of the robot post-transformation are shown in Fig. 4

with relative axle deflections and referenced to the hor-izontally oriented axis . These diagrams actually representthree steering configurations selected for their model-reductioncharacteristics and similarity to existing steering systems. Tosimplify evaluation of these configurations, we introduce thesteering ratio , expressed as the ratio of the relative axleheadings and , and angular rates and such that

and (3)

Type I Kinematics are defined such that each axleis steered in equal and opposite directions [Fig. 4(a)]. This con-figuration can most significantly reduce model complexity andtraction forces, simplifies motion control, and provides goodmaneuverability. It can reduce the kinematics of the system tothat of a simple unicycle defined by a curvature-based path, andreadily accepts such motion-control algorithms for full controlof robot posture. This steering arrangement is similar to that ofan articulated vehicle, but no additional joints or actuators arerequired.

Type II Kinematics correspond to equal deflectionsof each axle such that the frame assumes a shape similar to that


Fig. 4. Boundary conditions and kinematic steering scenarios. (a) Type I kine-matic configuration. (b) Type II kinematic configuration. (c) Type III kinematicconfiguration.

of a sinusoid [Fig. 4(b)]. This model provides simplified kine-matics and excellent lateral mobility, but control of orientationis reduced and traction forces are higher. Thus, while this con-figuration is better for lateral mobility, it provides minimal sim-plification of motion control where orientation of the robot isconsidered, and ultimately requires large traction forces.

Type III Kinematics , Fig. 4(c), are similar toAckerman (automobile) steering, since the rear axle is alwaysdirected towards the center of the front axle, and each wheelon the robot approximately travels about a common instanta-neous center of rotation (ICR) located along a line extendedfrom the rear axle axis. This varies from the strict Ackermanmodel, however, since the steering geometry of the CFMMR isproduced by steering the entire front axle, as opposed to usinga complicated linkage to individually steer each front wheel.This model derives some kinematic simplification, but doesnot simplify motion control appreciably where strict control ofrobot posture must be considered. It requires higher tractionforces, and provides a compromise of maneuverability andlateral mobility.

B. Frame Coupling

To minimize compressive and tensile forces on the beamalong the axis that would consume energy and availabletraction for steering, velocity constraints can be imposed onthe axles such that the length of remains consistent withthe frame length under pin-slot boundary conditions. This taskis complicated by the fact that as the axles steer, the distancebetween points and must vary to accommodate the newshape of the frame. This shortening effect can be calculated by

Fig. 5. Frame foreshortening (l = 0:350 m).

first considering the deflected shape of the frame. Modeling theframe as an Euler–Bernoulli beam in order to derive tractablevelocity-constraint expressions, we impose the pin and pin-slotboundary conditions to derive a third-order polynomial de-scribing the lateral frame deflection as a function of theimposed angles and

(4)

where is the undeflected length of frame, and is the positionalong the frame in the direction. This is used to calculate thedecrease in length of defined as foreshortening, , [31]as

(5)

The foreshortened length of is then denoted as

(6)

The effect of foreshortening on the length of is illus-trated in Fig. 5 for Types I–III steering as functions of . Thisfigure compares foreshortening estimates provided by (5) withthose determined experimentally for a flexible frame character-ized by the parameters shown in Table I. Experimental hard-ware consisted of fixtures maintaining and , where one fix-ture used a linear bearing to allow free axial deflection in orderto measure foreshortening. Angular boundary conditions weremeasured by single-turn potentiometers ( 2% linearity) cali-brated by precision fixtures ( accuracy) across the rangeof motion. Axial displacements were measured by a Litton RVTK25-3 linear potentiometer with 4% linearity.

As Fig. 5 indicates, for , (5) predictswithin 5.0, 1.7, and 0.2 mm for Type I, II, and III constraints,respectively. Even though larger errors are incurred with the


TABLE IPROTOTYPE PARAMETERS

Fig. 6. Axial frame forces when displacing from foreshortened length.

Type I constraints, they are such that the frame is placed in slightcompression, which requires significantly less force than whenthe frame is in tension. Errors incurred by (5) with the Type IIIconstraints, in contrast, are such that the theoretical deflectionis too small and the beam is in slight tension.

Axial forces produced by deviation from the actual foreshort-ened length of the frame are shown in Fig. 6. These have beenexperimentally determined using the fixture described above,with axial tension and compression forces applied to the linearbearing by a Chattillon Model DPPH100 load cell. Regressionlines closely matching experimental data are shown. As the dataindicates, the stiffness of the frame is significantly higher whenin tension (positive displacement), as compared with compres-sion. The deadband near zero displacement indicates a slightamount of stiction in the linear bearing. Thus, per Fig. 5, for aheading angle of , errors in the theoretical foreshort-ening calculations are mm, which per Fig. 6are expected to produce axial forces of N,respectively, for the Type I–III constraints. It is observed thata small amount of tension (positive error) produces significantforces, whereas a small amount of compression produces muchsmaller forces. This is attributed to the postbuckled configura-

Fig. 7. General steering kinematics. Frame omitted for clarity.

tion of the frame [11]. The figures further verify that is notnegligible for , and hence, velocity constraints main-taining axle spacing are critical to minimizing traction forces.

To assure that the axles maintain proper spacing, velocityconstraints are established for the axles that satisfy (5). To de-termine the velocity constraint, the change in frame length as afunction of time is expressed as

(7)

which establishes the general velocity constraint

(8)

Combining (7) and (8), the velocity constraint for Axle 2 is ex-pressed in terms of as

(9)Next, this will be expressed entirely as a function of the velocity

and the kinematic state variables.

C. General Kinematics in Polar Coordinates

Fig. 7 is a diagram of the robot in a general configurationwhere and have not yet been specifically constrained orcoupled. We strive to control the robot for purposes of postureregulation (position and orientation), path following, or generaltrajectory tracking by controlling the position of a point on therobot, such as or , and the angle of , which de-scribes robot orientation.

Brockett’s Theorem [32] shows that a smooth time-invariantcontrol law cannot be used to provide globally asymptotic sta-bility to continuous nonholonomic systems. This is easily cir-


cumvented, though, by introduction of a discontinuous polarcoordinate description of the kinematics, which then admit asmooth time-invariant control law [29]. In the polar description(Fig. 7), the location of a point along , chosen to be forsimplification, and the orientation of its velocity trajectory canbe represented by the variables

(10)

which result in the polar kinematic equations

(11)

that become discontinuous when . Progressing towardsa curvature-based controller, consider that the angular velocityof Axle 1 can also be described as a function of path radius orcurvature, and , respectively, and forward velocity

(12)

Imposing velocity constraint (9) in of (11) and using thesteering ratio to eliminate , the rate of change of robotorientation is then

(13)where and are now viewed as inputs to the system. canthen be expressed as a function of robot orientation and thepolar coordinates such that

(14)

to provide

(15)

given . Substituting (15) into (13), solvingfor , and applying (14) such that the kinematic state equationsare purely functions of the polar coordinates with velocity andpath curvature inputs and , we have

(16)

where and is determinedby (6). The actual Axle 1 heading can then be evaluated by

(17)

which prescribes the angular rate to be

(18)

Given (3), we have the angle of Axle 2 described as

(19)

where can be eliminated by substitution of (14) such thatonly the polar coordinates and remain. The result is

(20)

and the angular rate of Axle 2 is determined by differentiationto be a function of the kinematic variables

(21)

where (15) is imposed on (9) in conjunction with (3) such thatthe Axle 2 velocity constraint is

(22)

Hence, the forward and angular velocities of Axle 2 can be ex-pressed purely in terms of the state variables by application of(14) and (16).

Hence, given a motion controller for (16) that prescribesand as control inputs, the velocities for each axle can be de-termined. While (16) provides a generic description of the kine-matics, the task of motion control is not simple, owing largelyto the complexity of in (16).

IV. SIMPLIFIED KINEMATIC MODELS

Simplifications based on selection of the steering ratio, , cansignificantly reduce the complexity of the kinematic model andfacilitate motion control via standard algorithms.

A. Type I Kinematics: Curvature-Based Steering

As indicated in Section III, Type I steering occurs if we con-strain . This provides the most appealing descriptionof the system, since it simplifies the kinematics significantly,eliminating the need to consider as described in the next para-graph, and allows standard unicycle motion planners to providefull motion control. This is derived in part from the fact that

imposes a constant moment across the frame, whichideally takes the shape of an arc segment. Geometric propertiescan then be used to easily describe the postures of the axles.

To facilitate simplification, Point is defined at the mid-point of , Fig. 8. Point is unique, since its motion under


Fig. 8. Type I steering kinematics.

Type I steering is determined to be that of a simple differen-tially steered axle, and the orientation becomes equivalent tothe velocity heading , and ultimately orientation . Thus, theposture of at Point ultimately possesses kinematics de-scribed by (2) where and describe the velocity trajectoryof the robot. This is apparent, since the angle describing thepolar orientation of also describes the robot orientation ofthe segment , due to the symmetry of . This pro-vides tremendous simplification of the motion-control task since

is eliminated, and polar coordinates and describe both theorientation of the robot and the velocity heading. The reducedkinematic model of the robot subject to Type I constraints is

(23)

where the velocity and path curvature of Point , , and ,respectively, are the control inputs.

Towards implementation, note that at any instant, is ap-proximately traveling about an ICR, indicated as Point , witha path curvature and radius

(24)

where represents the relative angle of axles to the seg-ment . While these equations must be solved numericallyto determine for a desired radius of curvature, note the sim-plicity of this task, since length is used rather than , becausethis equation is based on the actual arc length of the frame.

The control inputs and and the polar coordinates canthen be used to determine the desired trajectory of each axle.Based on , (24) determines the relative heading to give theabsolute axle headings

and (25)

which can be represented in terms of the polar coordinates, since. Axle headings and rates are determined by

and (26)

and (27)

where the rate of relative heading change is

(28)

as determined by differentiation of (24). The velocity magnitudecan then be related to that of the axles by

and (29)

where . Substituting (7), (3), and , theaxle velocities are determined by

(30)

Hence, a reduced-order kinematic model is obtained that fullydescribes the system posture and allows easy implementation ofcurvature-based motion controllers. Even greater simplificationcan be achieved if velocity constraints are relaxed to neglectforeshortening for small steering angles, which provides

(31)

as was the case in earlier work [10].

B. Type II Kinematics:

A different simplification of the kinematics can be obtainedif such that to produce the Type II kine-matics, Fig. 9. The difference between the Type I and II modelsis readily apparent, however, after applying the constraintto (13), which results in

(32)

This nonzero component is due to the modified velocity thatAxle 2 must assume relative to Axle 1 in order to compensate forforeshortening. If the velocity constraints are neglected, fore-shortening is an issue and traction forces increase, but .


Fig. 9. Type II steering kinematics.

This nullifies the state variable and results in a reduced modelsimilar to (23), but referenced to .

C. Type III Kinematics:

A third technique for simplifying the kinematic model canbe achieved if we impose the constraint . In this case,

, and hence the rear axle is always directed towards .The kinematic model is simplified appreciably, but not entirelyreduced, since

(33)

Since the state variable is not entirely reduced as in the caseof Type I steering, a simple unicycle-type motion-control algo-rithm will not drive to zero.

V. MOBILITY AND MANEUVERABILITY

In order to evaluate the performance of the steering modes,we develop metrics based upon achievable mobility and maneu-verability. Forces imposed by the frame play a critical role inevaluating these capabilities, since they directly impact wheeltraction forces and achievable steering angles. These capabil-ities are also affected by physical interference of componentsof the robot that may limit steering angles and potentially un-stable configurations of the system. In the process of developingthese factors and performance metrics, they are evaluated rela-tive to the experimental platform characterized by the parame-ters shown in Table I.

A. Limiting Factors

1) Physical Interference: Depending on the proportions ofthe robot, interference can occur in two different scenarios: thewheels on one side of the robot may touch, or one of the wheelsmay contact the frame. Wheel–wheel interference is apt to limitType I steering, which provides the following equation basedupon the geometry shown in Fig. 6:

(34)

which must be solved numerically for . Based upon the robotparameters in Table I, the limit is .

Wheel–frame interference can occur in any of the steeringmodes if the frame is sufficiently long. Since the deflected shapeof the frame and collision point is very difficult to evaluate an-alytically under these circumstances, this boundary is approxi-mated by the angle of the axle when the leading or trailing edgeof the wheel interferes with . This results in the followinglimit:

(35)

which corresponds to .2) Traction Forces: Quasi-static behavior is assumed since

dynamic interactions are not considered. Thus, traction forcesrequired to impose boundary conditions on the frame are eval-uated. Assuming foreshortening has been considered in the ve-locity constraints, boundary condition forces are described bylateral reactions and moments of the frame that resultfrom axle orientations [35]

(36)

(37)

where is the Young’s modulus of elasticity of the frame andis its cross-sectional moment of inertia of the frame about the

bending axis [30]. These forces and moments can then be usedto calculate wheel traction forces

(38)

where and , respectively, represent the moment andreaction forces on a tire on the th axle oriented with respect tothe frame coordinates. The net traction force on a tire on the thaxle ( and representing the left and right wheels, respec-tively) is then the vector sum

(39)

and the maximum traction forces, , on the th axle is

(40)


Fig. 10. Ideal required maximum wheel traction forces, and available tractionbased on typical surfaces [33], [34].

The maximum of these axle forces then determines the max-imum tire force on the robot

(41)

Ultimately, the achievable traction force is limited by theweight of the robot and the wheel slip characteristics [36].Assuming that the robot has an evenly distributed mass , anormal force will be supported by each wheel. Thus, themaximum traction force can be approximated by

(42)

given a coefficient of friction corresponding to the tire–sur-face interaction [33], [34]. Equations (40) and (42) can thenbe solved to determine the ideal maximum steering angleachievable for a steering ratio . Based upon the parameters inTable I, Fig. 10 illustrates as a function of steering ratiofor several relative axle headings. These results indicate that ide-ally, Type I steering requires minimum traction force, followedby Type III, and then Type II.

Based upon published tire–surface interaction, these resultsestimate that all steering modes should function effectively onsandy loam [34], and even snow-covered ice [33]. An impor-tant point to remember, though, is that these traction force es-timates do not include forces introduced by error in the fore-shortening calculations . If we include these forces, it isevident that the maximum traction forces including foreshort-ening error are larger and dependent upon foreshorteningpredictions (Table II). In situations where , the frameis in tension and traction forces may increase appreciably, as inthe case of the Type III steering, and it is evident that the wheelslip will occur on slippery surfaces. In the case of Types I andII steering, though, the traction force with foreshortening error

TABLE IIIDEAL MAXIMUM WHEEL TRACTION FORCE F , EXPECTED

FORESHORTENING ERROR �L , AND EXPECTED MAXIMUM WHEEL

TRACTION FORCES WITH FORESHORTENING ERROR F

is still sufficiently small that the robot should function well onslippery surfaces.

3) Configuration Instability: Instability could occur in theform of tip-over if the axles are colinear. In the quasi-static case,this occurs when axle orientations satisfy

and

As the physical interference limits indicate, tip-over instabilityis not a limiting factor in the case study.

B. Performance Criteria

1) Maneuverability: The capability to manipulate robot ori-entation, and thus maneuver the robot around obstacles, can beevaluated by , which is generally described by (16). Note thatthis expression is a complex function of steering ratio , rel-ative axle heading , and path curvature . In general, isactually determined by the current configuration of the robot asdetermined by the evolution of the kinematics (16).

2) Lateral Mobility: In confined environments, the ability ofthe robot to move laterally without significant maneuvers is im-portant. This mobility can be quantified by the average relativeheading described by

(43)

3) Scaled Performance Metrics (SPM): Since force is re-quired to deflect the frame, wheel traction and energy are neces-sary. Objective evaluation of the performance criteria is accom-plished by examining the magnitude of and per unit ofmaximum traction force required, represented, respectively, by

and as

and (44)

Evaluation of the SPM was performed based upon the param-eters shown in Table I. Larger performance factors are generallydesirable for maximizing the use of available wheel traction, buttraction requirements, shown in Fig. 10 and Table II, and kine-matic simplifications must be considered.

The SPM plot for maneuverability is shown in Fig. 11 for arelative steering angle and several path curvatures

, as a function of steering ratio . For smaller ,the variation between the curves becomes smaller and less illus-trative of the SPM sensitivity to path curvature. Considering the


Fig. 11. Scaled maneuverability performance metric indicates the ability tochange robot orientation per unit of traction force. = 30 is shown.

three case studies, note that is largest for (Type I) andsmallest for (Type II). The actual maximum of the SPMis at about , but the traction forces are actually muchlarger and the kinematic simplifications are not significant.

In the case of , note that Fig. 11 indicatesat . This indicates potential destabilization, since therobot orientation is decreasing, while axle-heading anglesand are increasing, Fig. 9. Thus, increases and be-comes even more negative. If is maintained for a suf-ficient time, and (22) becomes singular. Prior tothis point, though, destabilization can be remedied if isapplied. Similar destabilizing and restabilizing scenarios occurwhen , and or , respectively.

For robot configuration stability, we thus desire to remainbounded and to converge to a steady-state value proportional tothe path curvature. Given , this is achieved by ,and for . Thus, we can guarantee configurationstability if

(45)

where is derived numerically based uponFig. 11 for at . Note thatas approaches zero, but the aforementioned range of isgiven to indicate the nominal range of that is dependentupon . In contrast, we find that is a function of physicallimitations (35) and is determined by

(46)

based on (3) and (35), which results in for. This lower limit actually becomes more negative

for smaller , allowing a greater range of steering ratios, butthis provides a conservative static boundary. Steering outside ofthe range (45) is certainly allowable for short periods, but ob-serving (45) is generally desired. Steering near the boundariesof (45) is possible, as in Type II steering, but this may lead to

Fig. 12. Lateral mobility performance metric indicates the ability to translatethe robot laterally per unit of traction force without complex maneuvers.

stability issues in extreme maneuvers, as we indicate in the eval-uation section of the paper.

The SPM plot for lateral mobility is shown in Fig. 12 forand 30 as a function of steering ratio . Type II

steering provides the best lateral mobility, while Type Iis the worst. actually is larger for , but sta-

bility becomes an issue. When , becomes negative,indicating behavior counterproductive to lateral mobility, whichresults from large rear axle steering maneuvers that are actuallybetter for maneuvering the robot, as described above.

As the SPM indicate, the Type I steering is the bestfor maneuverability, while Type II is the best for lateralmobility. Type III provides a nominal mix of lateral mo-bility and maneuverability, but this kinematic model cannot besimplified to the same degree as the others, and it introduces sig-nificant traction forces in lieu of foreshortening error. The TypeI and II kinematics, though, each provide specific performancespecialties with reduced kinematic models that vastly simplifymotion control to provide control of full posture regulation withsimple unicycle motion controllers.

VI. MOTION-CONTROL ALGORITHM

Using the polar coordinate representation and the simplifiedmodels, a pose-regulation controller is developed based uponthe Lyapunov function

(47)

Indiveri [27] suggests the use of the smooth control law for thevelocity and the curvature

(48)

where the parameters , , and are constant gains. Inserting(48) and (23) into (47), the Lyapunov derivative becomes

(49)


where it may be shown that the origin is the solitary systemequilibrium point; hence by LaSalle’s theorem [37], the states

will be asymptotically driven to the origin [35].A dynamic extension must be performed to implement the

control law on real systems with drift and nonideal ICs. Themodel meets the requirements for a cascade system as definedby Bacciotti’s Theorem 19.2 [38], and therefore the smoothstabilizers

(50)

may be added with the overall system being smoothly stabiliz-able, where the subscript represents the desired values from(48), and the subscript represents the actual values. The stateequations of the extended system with the control inputs (48)inserted now become

(51)

This algorithm works well in simulation, but in experiments,difficulties are encountered in a small neighborhood of theorigin ( 5–7 cm) where drift causes (48) to command steeringangles that the CFMMR cannot achieve. Thus, in a smallneighborhood of the origin, (48) is modified to and

. As a result, final convergence of , , and are notperfect, as indicated in the experimental results.

The dynamic extension also provides the system with theability to track a desired trajectory by replacing the control in-puts with any desired velocity and curvature trajectory gener-ator. Further details of this and other similar proofs can be foundin [10], [27], [29], [35], and [39]. Application to the Type IIand III kinematics with velocity constraints considered is easilyachieved, but is not actively controlled.

A similar controller may be developed for the path followingcase where the path is expressed as a smooth directed functions as suggested by Tayebi [28]. Since path following is not usedto evaluate performance of the robot here, the reader is referredto previous publications to further examine implementation ofpath following with this robot [11], [35].

VII. EVALUATION OF SIMPLIFIED MOTION CONTROL

A. Methods and Procedures

The steering strategies discussed above were tested via sim-ulation and experiments on the CFMMR platform, Fig. 1, atthe University of Utah. The system was controlled via tetherby a dSpaceTM 1103 DSP from the Matlab Simulink envi-ronment via Real Time Workshop and Control Desk. Wheel

odometry was used for feedback with simple servo-type propor-tional-integral-derivative wheel controllers. Manually obtainedfinal posture measurements with an accuracy of 0.1 cm arepresented in Table III to compare experimental results. Videofootage overlaid with trajectories according to odometry is usedto indicate and document system performance, Fig. 16.

Simulation of the motion controllers applied to the CFMMRusing the Types I–III steering ratios were conducted to evaluatetheir efficacy while performing posture regulation. Thoroughexperimental evaluation of posture regulation was conductedto evaluate the ability of each of the steering modes to ac-curately regulate the final posture of the robot, given theirinherent traction requirements and predispositions for con-trolling orientation, . In particular, Type I steering relative tothe reference frame versus was considered. An IC (a)

m m was selectedfor these tests via simulation since it produced large steeringangles that would be prone to traction loss in experiments.Since this IC actually caused the Type II steering to drive therobot to an unstable configuration in simulation, an IC (b)of m m was also used to evaluate TypeII steering under more nominal conditions. In all cases, thedesired final posture was at the origin with the robot alignedwith the -axis, which corresponds to .

Carpet and sand surfaces were used to evaluate the affects ofwheel slip on final posture error resulting from steering-modetraction requirements. The carpet was a dense short nap thatprovided high traction. Sand tests were conducted with a 1-cmthick layer of sand spread evenly over a plastic sheet to simulatelow traction. The sand was leveled between each trial, and wassufficiently thick that the robot did not dig through unless signif-icant obstacles were present. The ability of the frame foreshort-ening velocity constraints to reduce traction forces and wheelslip was evaluated on sand for each of the steering modes. Thesteering modes were also evaluated on sand with scattered rocks(1–2 cm thick, flat hard sandstone nominally 4 cm 4 cm spaced

10 cm) to evaluate the performance of the robot traversing ob-stacles in a low traction environment. Rocks were 13%–27%of the wheel radius in order to provide nontrivial obstacles.During each trial, 12 rocks were traversed by the tires, andthese were nominally distributed amongst the wheels and axles.For each test, five trials were performed to estimate repeata-bility. The robot was also evaluated on extremely rugged rockyterrain (undulating rock piles as large as the wheel radius) toevaluate its operability.

Typical challenges near the final posture caused by smallpath-radius steering maneuvers were initially encounteredwhile implementing the motion controllers experimentally[10]. This occurs with all steering modes and is caused by driftand phase lag resulting from the low-level wheel controllersthat become problematic as the motion controller (48) attemptsto compensate aggressively near the final posture. This is easilyresolved by commanding the robot to follow a straight line, asindicated in Section VI, once it is within 7.5 cm of the originuntil the position error converges to zero. At the switchingpoint, the configuration variables and have convergednearly to zero, and as the robot drives asymptotically to zero,there is inevitably a small error remaining in . Thus, error in


TABLE IIIFINAL POSTURE ERROR WITH IC (A) [x(0); y(0); �(0)] = [�1:445 m;�1:221 m; 0 ] AND (B) [�1:445 m;�0:500 m; 0 ]

the configuration variables and typically increases near thefinal posture.

In the particular case of Type I steering by Point , twomethods of eliminating this phenomenon near the origin wereexamined. The first was as described above, where the postureand velocity of Point were determined algebraically fromthe axle data. The second method forced a unicycle kinematicmodel (23) to describe the motion of Point , where the axlevelocities along and perpendicular to were used to deter-mine the forward and angular velocity of the coordinate frameattached to . This method of tracking Point denotes the pointas in the results.

B. Results and Discussion

Table III indicates the final posture error data and respectivestandard deviations after pose regulation. Based on simulationresults (Tests 1, 6, 11, 17, 22, and 27), note that only the TypeI steering (relative to Point or , Tests 1 and 5) is capableof completely regulating the posture of the robot from the IC(a). Type II steering is capable of maintaining if initialposture error is small [Test 27, IC (b)], but not in general givenlarger initial error [Test 17, IC (a)]. Type III is unable to reg-ulate (Test 22), which is also true of Type I steering via

(Test 11). In all simulations, though, position error of the refer-ence point ( , , or ) characterized by and convergescompletely to zero. These results are all predicted based uponprevious analysis.

Experimental evaluation indicates nonzero final posture errorin all tests, Table III. In general, distance error was smallestfor Type I steering (Tests 2–4, 7–8, and 12–14), larger for TypeIII (Tests 23–25), and largest for Type II (Tests 18–20). Giventhat Type I traction forces were expected to be minimum, thefirst trend correlates well with predictions. At first glance, how-ever, it is counterintuitive that the Type II steering produced thelargest error, since Type III was expected to induce the max-imum traction forces for a given steering angle .

The source of the discrepancy is recognized upon examina-tion of steering angles, Fig. 13, and traction forces, Fig. 14, de-rived from simulations. These show that the Type II steeringencounters a discontinuity as approaches 80 , and this pro-duces estimated traction forces peaking at 9 N, which does notinclude foreshortening error that would be appreciable at theseextreme deflections. These angles and forces are significantlylarger (nearly three times) than those of the Type III steering,which has and N. The large steering angleresults from potential Type II steering instability, as described


Fig. 13. Simulation of Axle 1 steering angle per Tests 1, 11, 17, and 22.

Fig. 14. Simulation of maximum traction force per Tests 1, 11, 17, and 22.

in Section V-B. In the case of IC (a) (Tests 18–20), islarge enough sufficiently long such that increases until thesingularity occurs. Path plots indicate this discontinuity as a no-ticeable source of slip that causes a large net final error. Giventhe smaller IC (b) (Tests 27–30) requiring smaller path curva-ture, only increases slightly and then decreases whenin the later part of the trajectory (Test 27). Thus, orientationconverges back to zero, which is also true of IC (b) after the sin-gularity has occurred. Owing to the potential instability of TypeII steering, it thus must be applied with caution. Such effects arenot present in the other steering modes, where the relationshipbetween , , and is stable.

Tests performed on sand versus carpet in general indicate anincrease in that is more characteristic of the expected trac-tion forces. This is quantified by the percent increase in error,denoted as in Table III, with respect to the appropriately in-dicated reference test (Ref. Test). As expected, Type I possessesthe least percent increase (10% using to guide the robot),whereas the percent increase for Type II is larger (88%), andType III is largest (243%). Note that the Type I steering guided

Fig. 15. Simulation of axle spacing per Tests 1, 11, 17, and 22.

by or actually yields a much larger percent increase. This isattributed to the fact that the unicycle kinematics of the coordi-nate frame attached to are entirely dependent on maintainingexact ratios of , , , and , which is difficult to achievegiven the servo-type wheel controllers implemented here. Thus,while Type I steering via or provides simplified kinematicsby eliminating , the challenge is synchronously controllingand for each axle.

Tests performed on sand/rock indicate similarly expectedtrends of increased error . Tests guided by indicate thebest performance for Type I (Test 15) and the worst for TypeIII (Test 26). Tests guided by Point and (Tests 5 and 10,respectively) indicate nearly identical error that is much largerthan when guided by (Test 15). This again follows fromdifficulties controlling the exact ratios of and , which areexacerbated by the rocky terrain.

Deviations in final orientation from expected results, ex-pressed as in Table III with respect to the indicated Ref.Test, also agree well with performance predictions based upontraction forces. Namely, was smallest for Type I steeringand largest for Type III steering.

Also note that guiding the robot with the forced unicycle kine-matics produced the smallest final error on carpet and sand(Tests 6 and 7, respectively). While these results are outwardlyvery promising, the subtlety is that the posture of estimated bythe axle postures and posture estimated by the forced unicyclekinematics differed appreciably in their estimates. For example,the posture estimates attained from the axles were nominally10.5 and 18.5 cm larger on carpet (Test 7) and sand (Test 8), re-spectively, than those derived from the forced unicycle model.As such, usage of the forced unicycle kinematic model has po-tential to vary appreciably from the posture of the robot. Mod-eling this slip and more accurately accounting for the kinematicsof Point is the subject of future work.

The effectiveness of the velocity constraints to account forframe foreshortening is amply demonstrated on the sand surfacefor each of the steering modes. Foreshortening on the order of5–7 mm was typical, Fig. 15. In Tests 4, 14, 20, and 25, the beamforeshortening velocity constraints were disabled, and the re-sulting error was compared with the error observed during the


Fig. 16. Robot paths during posture regulation on carpet for Type I (a)–(c) (Test 2). Type II (d)–(f) (Test 18). Type III (g)–(i) (Test 23) steering kinematics.Sand/rock for Type I (j)–(l) (Test 15). [x(0); y(0); �(0)] = [�1:445 m;�1:221 m; 0 ].

initial sand tests. As the results indicate, disabling beam fore-shortening velocity constraints increased error between 24%and 36% for IC (a) on sand. Tests on the sand/rock surface withType I steering demonstrated that disabling the beam velocityconstraints increased error by only 15%, which is attributed tothe larger error already characteristic of the rock field. It canthus be concluded that the frame velocity constraints are quiteeffective for decreasing traction forces and wheel slip.

The robot path during posture regulation (Fig. 16), final pos-ture error (Table III), and predicted traction forces (Table II), in-dicate effectiveness of the SPM to estimate robot performancefor different steering ratios . As Fig. 16(a)–(c) indicates, Type Isteering clearly requires the minimum space and is most ef-fective for maneuvering the robot along a curvature-based path

with minimum final error. In contrast, the Type II kinematicsrequire more space to maneuver, but they are best adapted totranslating the robot laterally small distances (e.g., lateral ma-neuverability), although the motion controller (48) is not opti-mized for this. Type II traction forces are higher, though, and asa result, final posture is more prone to error. Type III kinematicsprovide a compromise between Type I and II characteristics.

All things considered, it is concluded that the Type I steeringguided by or provide the most simplified control of robotposture, but that Type I kinematics guided via provide themost consistent behavior given the odometry measurementsand servo-type wheel controllers implemented here. Overall,the SPM provide a good first estimate of these behaviors, but itis necessary to consider variations in traction forces resulting


Fig. 17. Robot traversing extremely rugged undulating rocky terrain.

from foreshortening errors and the influence of larger relativesteering angles , and particular steering ratios that cause therobot to approach its limitations.

Evaluation of the Type I kinematics on extremely rocky ter-rain demonstrates the capability of the robot to conform to ter-rain variations, Fig. 17. In frame (b), opposite wheels can beobserved (Right Front, RF, and Left Rear, LR) ascending ob-stacles simultaneously. In frame (c), LR continues its ascentand RF descends. In (d), the Left Front (LF) wheel ascends andLR descends. In (e), both LF and LR descend. Finally, RF as-cends in (f) and descends in (g). As this sequence of imagesillustrates, the CFMMR is very adept to conforming to ruggedterrain that would leave traditional rigid mobile robots teeteringon two wheels. While this terrain is more severe than that usedin the sand/rock tests (Tests 10, 15, 21, and 26), it does illustratetypical compliant frame undulations observed in those trials.

VIII. CONCLUSIONS

General kinematics for a two-axle compliant frame mobilerobot have been presented and characterized by steering ratio

. Based upon three special cases of , simplified kinematicmodels have been derived and evaluated. Of these models, onlyType I steering guided by Point provides full control overposture regulation. Types I and III steering guided by pro-vide full position control with limited control over orientation

. Type II steering provides control over position, but orienta-tion became unstable in simulation, given ICs requiring ag-gressive maneuvers. Experimental evaluation corroborates thatthe Type I steering requires minimum traction, provides themost accurate posture regulation, and provides maximum ma-neuverability. The ability of the robot to adapt to rugged ter-rain and maneuver was confirmed by posture regulation on sandwith scattered rocks and the ability to traverse extremely ruggedrocky terrain was also demonstrated. Future work is on mod-ular configurations, dynamic control laws, and improved sensorinstrumentation.

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Mark A. Minor (S’99–M’00) received the B.S.degree from the University of Michigan, Ann Arbor,and the M.S. and Ph.D. degrees from MichiganState University, East Lansing, all in mechanicalengineering.

He is currently an Assistant Professor with the De-partment of Mechanical Engineering, University ofUtah, Salt Lake City, where he has been a facultymember since 2000. Research interests include mo-tion control of mobile robots, terrain-adaptive mobilerobot locomotion, sensing strategies and fusion struc-

tures, and nonlinear robust control of distributed parameter systems.

Brian W. Albiston received the B.S. degree fromUtah State University, Logan, in 1999, and the M.S.degree in 2003 from the University of Utah, Salt LakeCity, both in mechanical engineering.

He was with the Williams Companies, Salt LakeCity, UT, from 1999 to 2002. He was with OrbitalSciences Corporation, Chandler, AZ, from 2003 to2005, and is currently with Sagetech Corporation,Hood River, OR.

Corey L. Schwensen received the B.S. degree fromRose-Hulman Institute of Technology, Terre Haute,IN, in 1995, and received the M.S. degree in 2001from the University of Utah, Salt Lake City, both inmechanical engineering.

He is currently a Senior Engineer with Rosetta In-pharmatics, a research division of Merck & Co., Inc.,Seattle, WA.

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