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164IEEE TRANSACTIONS ON ROBOTICS, VOL. 25, NO. 1, FEBRUARY 200940 C. Galletti and P. Fanghella, “Single-loop kinematotropic mechanisms,”Mech. Mach. Theory, vol. 36, no. 3, pp. 437450, 2009.41 P. Fanghella, C. Galletti, and E. Giannotti, “Parallel robots that changetheir group of motion,”in Advances in Robot Kinematics. TheNetherlands: Springer, 2006, pp. 4956.42 S. Refaat, J. M. Herv e, S. Nahavandi, and H. Trinh, “Two-modeover-constrained three-dofs rotational-translational linear-motor-basedparallel-kinematics mechanism for machine tool applications,” Robot-ica, vol. 25, no. 4, pp. 461466, 2007.43 X. W. Kong, C. M. Gosselin, and P. L. Richard, “Type synthesis of par-allel mechanisms with multiple operation modes,” ASME J. Mech. Des.,vol. 129, no. 6, pp. 595601, 2007.44 D. Zlatanov, I. A. Bonev, and C. M Gosselin, “Constraint singularities ofparallel mechanisms,” in Proc. IEEE Int. Conf. Robot. Autom., Washing-ton, D.C., 2002, pp. 496502.45 D. Zlatanov, I. A. Bonev, and C. M Gosselin, “Constraint singularities asc-space singularities,” in Advances in Robot Kinematics, J. Lenarcic andF. Thomas, Eds.Dordrecht, The Netherlands: Kluwer, 2002, pp. 183192.46 K. H. Hunt, “Constant-velocity shaft couplings: A general theory,” J.Eng. Ind., vol. 95B, no. 2, pp. 455464, 1973.Swedish Wheeled Omnidirectional Mobile Robots:Kinematics Analysis and ControlGiovanni IndiveriAbstractSwedish wheeled robots have received growing attention overthe last few years. Their kinematic models have interesting properties interms of mobility and possible singularities. This paper addresses the is-sue of kinematic modeling, singularity analysis, and motion control for ageneric vehicle equipped with N Swedish wheels.Index TermsMobile robot kinematics, mobile robots, motion control,wheeled robots.I. INTRODUCTIONIn the last few years, Swedish wheeled omnidirectional mobilerobots have received growing attention among the mobile roboticsresearch community. A Swedish wheel differs from a common wheelin the fact that rollers are mounted on its perimeter (see Fig. 1). If allthe rollers are parallel to each other and misaligned with respect tothe wheel hub axis, they will provide an extra degree of mobility withrespect to a traditional perfectly rolling wheel.The wheels depicted in Fig. 1 are often called mecanum or Swedishwheels: one of their design parameters is the angle between therollers rolling direction g and the wheel hub axis direction h. Typicalvalues are = 45and = 0, as shown in Fig. 1 (left and right cases,respectively). Note that the degenerate case = 90has no practicalinterestasitwouldallowthesamemobilityoftraditionalwheels.IntheManuscript received June 27, 2008; revised October 6, 2008 and November19, 2008. First published January 21, 2009; current version published February4, 2009. This paper was recommended for publication by Associate Editor F.Lamiraux and Editor W. K. Chung upon evaluation of the reviewers comments.G. Indiveri is with the Dipartimento Ingegneria Innovazione, University ofSalento, 73100 Lecce, Italy (e-mail: giovanni.indiveriunile.it).Color versions of one or more of the figures in this paper are available onlineat .Digital Object Identifier 10.1109/TRO.2008.2010360Fig. 1.Mecanum wheel: = 45(left) and = 0(right).literature, wheels with = 45are more often called mecanum wheelswhereas the ones with = 0are generally called Swedish wheels. Inthe following, the term Swedish wheel will denote the general case forsome fixed . As reported in 1, the mecanum wheel was invented in1973 by Bengt Ilon, an engineer working for the Swedish companyMecanum AB. Since then, this wheel design has attracted the attentionof the mobile robotics research community. The interest in such kindof wheels is related to the possibility of developing omnidirectionalrobots in the sense of 2, i.e., robots that “have a full mobility inthe plane meaning thereby that they can move at each instant in anydirectionwithoutanyreorientation”2.Theneedtoreorientthewheelsor not prior to implementing any desired linear velocity is related tothe presence or not of nonholonomic constraints 3, 4. Since theearly study of Agull o et al. 5, the kinematics analysis of Swedish ormecanum wheel robots has been addressed in several papers 612.All these except the study of Agull o et al. 5 and Saha et al. 9consider either three or four Swedish wheeled robots for some valueof . One of the objectives of this paper is to derive, in the mostgeneral setting of N Swedish wheels with fixed (but arbitrary) rollerwheelangle,necessaryandsufficientgeometricalconditionsonrelativewheel arrangement that guarantee: 1) the absence of singularities and2) the possibility of decoupling commanded linear and angular robotvelocities. This kind of information can be most valuable to guidethe design of the robot a priori. The results obtained addressing theN-wheel case, besides realizing at once a unified analysis of the mostcommon three- and four-wheel designs, allow to easily explore, forexample, possible six-wheel designs that have a large interest in thefield of outdoor and rough terrain applications 14. As a side result ofthe proposed analysis, all possible kinematics singularities occurringasafunctionoftherollerwheelangleandwheelarrangementcanbeidentified. Moreover, building on well-known methods, the trajectorytracking and pose regulation problems are solved for these systemstaking explicitly into account actuator velocity saturation.The paper is organized as follows: the general kinematics model foran N Swedish wheeled vehicle is derived in Section II. The guidancetrajectorytrackingandposeregulationcontrolproblemsinthepresenceofactuatorvelocitysaturationareaddressedinSectionIII.Experimentalresults are reported in Section IV. At last, some concluding remarksare addressed in Section V.II. KINEMATICSMODELWith reference to Fig. 2, for the sake of introducing the notation,a three-wheel omnidrive mobile robot is considered. All wheel mainaxes, i.e., hub axes, are assumed to always lie parallel to the fixedground plane P having unit vector k P. Each of the N Swedishwheels is indexed from 1 to N: for the hth wheel, the roller in contactwith the ground plane P is depicted as an ellipse with main axes ori-ented along the unit vectors nh: ?nh? = 1 and uh: ?uh? = 1.1552-3098/$25.00 2009 IEEEAuthorized licensed use limited to: Nanchang University. Downloaded on January 12, 2010 at 20:15 from IEEE Xplore. Restrictions apply. IEEE TRANSACTIONS ON ROBOTICS, VOL. 25, NO. 1, FEBRUARY 2009165Fig. 2.Three-wheel omnidrive robot: geometrical model.The unit vector nhis aligned with the rollers rolling axis on the mainwheel perimeter and uh:= nh k indicates the instantaneous tan-gent velocity direction of the roller associated with its rotation aroundnh. All wheels are assumed to be identical and have the same radius. The position of the hth wheel in the body-fixed frame is denotedby bh. The unit vector of each wheel hub axis, i.e., the unit vector ofthe wheels main rotation axis, is denoted by nh: ?nh? = 1. At lastfor each wheel, the unit vector uh:= nh k is defined indicating theinstantaneous tangent velocity direction of the wheel as a consequenceof its rotation around nh. Note that with reference to Fig. 1, the unitvectors uhand nhwould correspond to g and h, respectively. In thegiven hypothesis, all the introduced vectors except k are parallel to theground plane P.Given the components of any two 3-D vectors a and b on a commonorthonormal frame, their vector product will be computed using theskew-symmetric matrix S() such thata b = S(a)b.(1)Calling vcthe linear velocity of the robots center (indicated as pointc in Fig. 2) and k its angular velocity vector, the velocity vector vhof the center of each omnidirectional wheel hub will be given byvh= vc+ k bh,h = 1,2,3,.,N.(2)Considering the generic hth wheel and dropping for the time being theindex h for the sake of notational clarity, in the case of perfect rolling,the velocity v = vhgiven by (2) will be physically realized by theroller rotation around nand the wheel rotation around n. Namely,assuming that nand n are not aligned, i.e., ?= (2 + 1)90, where is an integerv = u+ uimplyingnTv = ?nTu?nTv = ?nTu?and consequentlyv =nTvnTuu+nTvnTuu.(3)Note that the rollers rotation around ngiving rise to the first termon the right-hand side of (3) is completely passive, whereas the wheelrotation around n giving rise to the second term on the right-hand sideof (3) is assumed to be actively produced by a motor. Calling nh qhtheangular speed associated with the hth motor in the body-fixed frame,in case of perfect rolling, the mapping between the “joint” speed q andthe corresponding velocity v of the hub of any given wheel will begiven bynTuTnv = q(4)where the contributionnTvuTnuTuof v in direction of u has been explicitly assumed not to contribute to q, as in the given hypothesis of perfect rolling, it is fully generated bythe passive rotation of the roller. Substituting (2) back into (4), giventhat uThnh= cos for any h, one gets1cosnThvc+1cosnThS(bh)k = qh.(5)Byprojectingallthevectorsin(5)onacommonbody-fixedframewithits third axis equal to k P, the term nThS(bh)k results innThS(bh)k = nxhbyh nyhbxh= bThuh.(6)Summarizing, (5) can be interpreted as a generic component of theinverse differential kinematics equation in matrix formM?vc?= q cos(7)whereM = nx1ny1bT1u1nx2ny2bT2u2.nxNnyNbTNuN IRN 3(8)and q IRN 1is the vector of joint velocities. Equations (7) and (8)represent the general kinematics model 9 of a Swedish wheeled ve-hicle with N wheels. Contrary to most of the other models presentedin the literature that are relative to three or four wheels in fixed config-urations, they allow a fully detailed analysis of the vehicle kinematicsproperties as a function of not only the rollerwheel hub orientation ,but also the relative wheel position. Such an analysis may be extremelyuseful for the mechanical and controls system design of the vehicle.Assuming cos ?= 0 and that M has rank 3, equation (7) can beused to compute joint velocity commands for a desired vehicle speed(vTcd,d)T. In particular, the following lemma holds.Lemma 1: Given a Swedish wheeled mobile robot with N iden-tical wheels of radius satisfying (7), any desired vehicle velocity(vTcd,d)Tcan be implemented by using proper joint velocities qdifand only if all of the following conditions hold:c.1 cos ?= 0;c.2 rank M = 3.In particular, qd=1 cosM?vcdd?.(9)Theviolationofanyofthepreviousconditionsc.1orc.2correspondstokinematicssingularitiesofadifferentkind:theviolationofconditionc.1 would correspond to a total loss of control authority, while theviolation of condition c.2 would correspond to a loss of controllability,as for any choice of the input q, the state derivative (vTc,)Twouldnot be uniquely defined.Matrix M defined in (8) can be decomposed as M = MlMa,where Ml IRN 2and Ma IRN 1such thatMlvc+ Ma = q cos.(10)Authorized licensed use limited to: Nanchang University. Downloaded on January 12, 2010 at 20:15 from IEEE Xplore. Restrictions apply. 166IEEE TRANSACTIONS ON ROBOTICS, VOL. 25, NO. 1, FEBRUARY 2009III. GUIDANCECONTROL OFSWEDISHWHEELEDOMNIDIRECTIONALROBOTSBased on the derived kinematical model, the trajectory tracking andpose regulation motion control problems for a generic robot equippedwithN 3Swedishwheelsaresolved.Theproposedsolution,besidesbeing general in terms of number of wheels, value of , and wheel con-figuration, explicitly accounts for joint velocity saturation. The controlproblem is formulated as a guidance control problem, namely the jointvelocities qhare assumed to be control inputs and the robots linear andangular velocities vcand satisfying (7) are the outputs. Of course,in practice, such an approach requires the implementation of a lowerlevel control loop mapping the joint reference speeds qdin actuatorcommands.A. Low-Level Control DesignAs in standard robot motion control architectures 22, the low-levelcontrol system can be designed either in a decentralized or in a central-ized fashion. In the former case, each actuator is controlled separately,typically with a velocity PID loop (as for the experimental validationreported in this paper); in the latter case, a centralized control solutioncan be derived based on computed torque (i.e., feedback linearization)methods.The decentralized control (or independent joint control) methodis simpler: each component of qdis used as a reference signal forthe corresponding PID actuator velocity loop, and dynamic couplingsamong the actuators are neglected. If such low-level control system(i.e., each actuator velocity servo loop) is fast with respect to the robotdynamic-navigation one, the lag between the desired joint velocitiesand the real joint velocities is negligible: in such a case, as long as theperfectrollingconstraintissatisfied,themappingbetweenthevehiclesvelocity (vTc,)Tand the joint speeds would be approximated by thesystems kinematical model (7) having the desired (or commanded)joint velocities qdon the right-hand side in place of the real jointvelocities q.Centralized control solutions are generally based on the dynamicrobot model 17, Ch. 12, pp. 493502, and the dynamic equationof Swedish wheeled robots with a geometry as the one described inSection II has the following structure:I q + (C() + F) q = (11)with I IRN Nbeing the positive-definite inertia matrix, C() IRN Nthe skew-symmetric Coriolis and centrifugal forces matrix(k is the robots angular velocity), F IRN Nthe diagonal fric-tion matrix, and IRN 1the actuator torques vector. Given thenondiagonal nature of matrices I and C() and the dependency (7)of from q, (11) is nonlinear and coupled. Yet, as commonly donefor robotic manipulators 22, the control input vector can be com-putedbaseduponanonlinearstatefeedbacklinearization(orcomputedtorque) solution, namely = (C() + F) q + I y(12)giving rise (recall that I is full rank) to a linear and decoupled model q = ythat can then be used to design a closed-loop solution for y in order totrack the reference signal qd.Given that the computed torque solution explicitly accounts for thesystems dynamic coupling terms, it is expected to exhibit a bettertracking performance, in particular, for high speed and accelerationreferences. Nevertheless, as also demonstrated by the reported experi-mental results, the independent joint solution appears to be sufficientlyprecise and accurate for tracking simple trajectories at constant speed.Moreover, given that the main objective of this research was to vali-date an actuator velocity management strategy at the guidance level,only the independent joint low-level control solution (i.e., actuator PIDvelocity servo loop) was implemented.In the following, the Swedish wheeled robot will be assumed tohave such kind of low-level independent joint controls system, andthe trajectory tracking and pose regulation problems will be solvedat the guidance level, i.e., considering the commanded joint speeds qdas control inputs and (vTc,)Tas outputs according to the purelykinematicalmodel(10).Inordertoformulatethetrajectorytrackingandpose regulation problems, the following notation will be used: given aninertial (global) frame ?G? = (i,j,k) with k := (i j) P, whereP is the floor plane, a reference (planar) trajectory is a differentiablecurve in Prd(t) = i?rTd(t)i?+ j?rTd(t)j?(13)with curvilinear abscissas(t) :=?tt0?drd()d?d(14)and unit tangent vectortd=drdds.(15)The kinematics trajectory tracking problem consists in finding acontrollawforthesystemsinput qdsuchthatthepositionandheadingtracking errorser(t) := rd(t) rc(t)(16)e(t) := d(t) (t)(17)converge to zero, with rc(t) being the position in ?G? of a referencepoint (e.g., the geometrical center or the center of mass) of the robot,(t) itsheading,andd(t) thedesiredreferenceheading.Notethatfornonholonomicvehicleshavingaunicycleorcar-likekinematicsmodel,the reference heading d(t) is not arbitrary, but needs to coincide withthe heading of the trajectories unit tangent vector td. Conversely, givenany position reference trajectory rd(t), a Swedish wheeled vehicle willbe free to track any arbitrary heading d(t) that does not necessarilyneed to coincide with the heading of td.The pose regulation problem is a special case of the trajectory track-ing one occurring when the position and orientation references areconstant, i.e., when rd= 0 and d= 0.B. Trajectory Tracking Controller DesignIn accordance with the notation previously introduced, consider thesystem in (10) with rc(t) = vcand (t) = being the robots lin-ear and angular velocities. Assuming that conditions c.1 and c.2 ofLemma 1 are satisfied, N 3, and Ma span(Ml) (this can be al-ways guaranteed by a proper design of the robot geometry), then anydesired robot linear velocity vc= (,0)Tand angular velocity kare uniquely mapped to the control input qdas follows: qd= qdl+ qda(18) qdl=1cosMlvc(19) qda=1cosMa.(20)Authorized licensed use limited to: Nanchang University. Downloaded on January 12, 2010 at 20:15 from IEEE Xplore. Restrictions apply. IEEE TRANSACTIONS ON ROBOTICS, VOL. 25, NO. 1, FEBRUARY 2009167Followingastandardapproach,tosolvethetrajectorytrackingproblem,consider the Lyapunov candidate functionV =12eTrKrer+12eTKe(21)with Kr IR22being a symmetric positive definite (Kr 0) matrixand Ka positive constant. The time derivative of V results inV = eTrKr( rd(t) vc) + eTK( d(t) ).(22)If vcand satisfyvc= rd(t) + Kr(rd(t) rc(t)(23) = d(t) + K(d(t) (t)(24)then the time derivative of V will be negative definite, i.e.V = eTrKrKrer (Ke)2 0, the absolute value of the maximumpossible velocity of actuator j, a threshold for ? qd?can be selectedas qmax= minj qmaxj,where j = 1,2,.,N. Whatever the gainsKrand K, depending on d(t), rd(t), er(t), or e(t), the saturationcondition? qd? qmax(29)may always be violated. Note that while the feedforward signals d(t)and rd(t) can eventually always be bounded, the tracking errors initialconditions are not design parameters. Hence, a commanded qdwith anexceeding infinity norm due to odd initial conditions cannot be a prioriexcluded.D. Tracking in Presence of Actuator Velocity SaturationThe presence of actuator velocity saturation has a severe impact onperformance:inparticular, giventheadditive structureof (26),actuatorvelocity saturation can affect the decoupling between commanded an-gular and linear vehicle velocities in spite of the fact that Madoes notbelong to the span of Ml. With reference to (26), suppose, for example,that all the components of qdl(t) are within the actuator limits, but thatas a consequence of adding qda(t), some of the overall commandedjoint speeds qd(t) exceed the actuator limits. In this case, both therobot-commanded linear and angular velocities will be corrupted in anunpredictable fashion. A (known) quick and dirty way out of this prob-lem could be to simply scale down qd(t) such that all its componentsare within acceptable bounds. This, of course, will always guaranteethat the commanded joint speeds are within the proper bounds, butone would need to prove that such a strategy does not jeopardize theasymptotic stability of the tracking error to zero. Moreover, in manyapplications, it may happen that either linear or angular velocity com-mands may have highest priority. In such a case, scaling down theoverall joint commands will have a negative effect of slowing downthe convergence speed of the highest priority task due to the presenceof the lower priority one. In order to cope with this and to guaranteea prioritized execution of the position and heading tracking tasks, thefollowing modification of the proposed control law is suggested: thesum in (26) should be weighted with error- and reference-dependentweightssuchthat:1)theresulting qdcommandhasnormwithintheac-tuator limits; 2) the tasks (position and heading tracking in the presentcase) are executed with a priority-based time order (higher prioritytasks first); and 3) the tracking error converges to zero.Consider the saturation function : IR 0,) IR(x,c) =0,ifx = 01,if0 |x| inamely if a given task is assigned zero capacity, all the lower prioritytasks will also automatically have zero capacity and all their weightsin the sum (33) will be zero. The capacity of task i can be viewedas the residual capacity after the higher priority task i 1 has beencommanded; thus, for example, c2will be zero (and also cj: j 2)if the task 1 input q1is saturating all its capacity c1. In other words,each task will be commanded with a nonnull weight only if the higherpriority tasks have not saturated. The fact that c1should not exceed qmaxis due to the fact that task 1 alone should not saturate the actuatorcapacity qmax; moreover, given that cj+1 cj j 1,n 1, thecondition c1 qmaxguarantees that each term in the sum (33) willhave infinity norm smaller or equal to the threshold qmax. A proof that(33) and (34) also imply? qd?=?n?h=1 qh (? qh?,ch)? qmax(35)is reported in the Appendix. In order to implement the aforementioneddescribed schema in the present trajectory tracking case, assume thatthe reference feedforward linear and angular velocities are sufficientlysmall, namely that1|cos|?Ml rd(t)?12 qmax t(36)1|cos|?Ma d(t)?12 qmax t.(37)These conditions are necessary to guarantee that the tracking task isasymptotically feasible, namely that when the position and headingtracking errors are null, the control effort of the control law (26) iscompatible with the actuator velocity saturation limit, i.e.er= 0,e= 0 =? qd(t)?=1|cos|?Ml rd(t) + Ma d(t)?1|cos|?Ml rd(t)?+ ?Ma d(t)? 0 tthat together with the feasibility condition (36) implies0 12 qmaxc2 qmaxi.e., the tasks 1 and 2 always have nonnull capacity. Moreover, byhypothesis ? q1? 0.5 qmaxsee (36) and c1= qmax, it followsthat q1(? q1?,c1) = q1 t.This fact together with the assumption that Ma span(Ml) impliesthat by substituting (38)(41) in (33) and (34), the vehicle will havelinear and angular velocities given byvc(t) = rd(t) + Krer(t)(? q2?,c2)(42)(t) = d(t)(? q3?,c3) + Ke(t)(? q4?,c4).(43)Note that the term (? q2?,c2) will be zero if and only if ? q2?= 0due to the fact that c2is strictly positive. ConsequentlyV1=12eTrKrer=V1= eTrKr( rd(t) vc(t)= eTrKrKrer(t)(? q2?,c2) 0 t timplying(t)|tt= d(t) + Ke(t)(? q4?,c4)(45)where (? q4?,c4) is zero if and only if ? q4?= 0 due to the factthat c4is strictly positive for t t. ConsequentlyV2=12eTKe=V2(t)?tt= eTK( d(t) (t)= eTK2e(t) (? q4?,c4) 0(46)namely there exists a finite time tafter which the time derivative ofV2is always negative, thus proving convergence to zero of the headingerror e(t). Prior to t, the heading (secondary task) error e(t) is notguaranteed to be decreasing. Note that q1and q2, being in the rangeof Mland normal to Ma, do not contribute toV2.In case the heading should be selected to be the highest priority task,it would be sufficient to select q1,., q4as q1:=1cosMa d(t)(47) q2:=1cosMaKe(48) q3:=1cosMl rd(t)(49) q4:=1cosMlKrer(t)(50)in(33)and(34);Lyapunovstabilityoftheheadingerrorandasymptoticconvergence of the position error could be proven accordingly.As for constant pose control, the trajectory tracking problem isreduced to a pose (position and orientation) regulation one as soonas the reference position and heading trajectories rd(t) and d(t) areconstant.IV. EXPERIMENTALVALIDATIONThe described guidance law for trajectory tracking in the pres-ence of actuator velocity saturation has been experimentally validatedon a three Swedish wheel robot with = 0and = 5 cm 20,21. The robot, named Volksbot, has been designed and built at theFraunhofer Autonomous Intelligent Systems Institute (AiS)1of SanktAugustin, Germany. The vehicles geometry depicted in Fig. 3 issuch that ?bi? = ?bj? = 25 cm for all i,j. This implies that Maspan(Ml).The platform is actuated by three 24 V dc motors of 90 W each with1:8gearratioandisabout8kginweight.Opticalencodersaremounted1Ais has now merged with another Fraunhofer Institute giving rise to the newFraunhofer Institute for Intelligent Analysis and Information Systems (IAIS).Fig. 3.Geometrical model of the Volksbot robot used for experimental vali-dation of the trajectory tracking control law.Fig. 4.Experimental results: driven paths and tracking errors. The highestpriority task is shown to always converge faster in spite of the fact that the lowerpriority ones causes actuator velocity saturation.on each motor shaft to provide feedback for the motor speed regulator.The controls system architecture is rather standard: it has an externalkinematics loop closed on the low-level actuator velocity servo loops.In particular, the low-level motor speed feedback control is realizedby a three-channel digital PID board (the TMC200 designed and builtat AiS) with pulsewidth modulation (PWM) output. The maximumAuthorized licensed use limited to: Nanchang University. Downloaded on January 12, 2010 at 20:15 from IEEE Xplore. Restrictions apply. 170IEEE TRANSACTIONS ON ROBOTICS, VOL. 25, NO. 1, FEBRUARY 2009continuous current provided to each motor by the TMC200 board is8 A with the peak value being 20 A. The TMC200 board is interfacedthrough a serial RS232 link with an onboard laptop PC implementingthe guidance and navigation systems. The navigation system providingthe estimates of the robot poses used to close the guidance control loopis based on the integration of odometry and an omnidirectional visionsystem. This builds on a 30-Hz, 640 480 pixels YUV color FireWirecamera pointing toward a 70-mm-diameter hyperbolic mirror.Inordertovalidatetheperformanceofthedesignedtrajectorytrack-ingcontrollaw,areliablemeasurementoftherobotsposeismandatory.Tothisextent,anexperimentalsetupwasdesignedatAiSwherethepo-sition of the robot was measured by a fixed laser range finder pointingtoward the robot. The robots heading, instead, was evaluated based onthe onboard omnidirectional vision system. The reference position andheading to be tracked and all the collected data were suitably synchro-nized. Examples of the experimental results are reported in Fig. 4: theposition reference was a circular trajectory having radius of 1 m whilethe heading reference was constant (with respect to a fixed frame). Thereported results refer to the cases where either heading (47)(50) orposition (38)(41) tracking was the task with high priority (HP). Thereference trajectory was designed so to satisfy the feasibility condi-tions (36) and (37), and the robots initial conditions were specificallychosen to be large enough to trigger actuator velocity saturation: dueto experimental setup constraints, the actuators were artificially forcedto saturate at 8.7 rad/s via low-level control software (correspond-ing to maximum robot linear and angular velocities of about 0.5 m/sand99.7/srespectively).Asexpected,bothheadingandpositionerrorsarealwaysasymptoticallyconvergingtozero(besidesmeasuringerrorsvisible in the error plots), but the task with HP always converges first.Notethatthepositionerrorsareincreasing(alsowhenpositiontrackinghas highest priority) in the very first few seconds of the experiment asa consequence of the neglected dynamics of the vehicle.V. CONCLUSIONA general kinematics model of an N Swedish wheeled vehicle wasderivedandanalyzed.HavingaddressedtheN Swedishwheelcase,theprovided results allow to unify the analysis of the common three- andfour-wheel designs that are usually performed separately. Moreover,the derived analysis can be immediately and simply applied to explore,for example, eventual six-wheel designs for outdoor and rough terrainapplications or steering (i.e., with variable orientation) wheel designs,asin12,thatallowtheimplementationofcontinuouslyvariabletrans-mission (CVT) systems.Thetrajectorytrackingmotioncontrolprobleminthepresenceofac-tuator velocity saturation has been addressed. Joint velocity saturationisalwayspresentandmayheavilyimpactontheperformanceofmotioncontrolsolutions.Theoriginalityandrelevanceoftheproposedsolutionis related to the handling of the actuator velocity saturation constraints.The designed solution guarantees that each task joint velocity com-mand and the overall commanded joint velocity have upper-boundedinfinity norm compatible with each actuator limit. This is achieved bydynamically allocating control effort to the tasks according to theirpriority. Contrary to standard-task-based controllers, the presence ofvelocity saturations related to lower level tasks does not jeopardize theexecution of higher priority ones. Global convergence of the trackingerror has been theoretically proven and experimentally validated.Future research will be devoted to the extension of the proposedsaturation management technique in case of dynamic-based low-levelcontrol and to the generalization of the solution in case of multipletasks in redundant robotic systems.ACKNOWLEDGMENTThe author is deeply grateful to P. G. Pl oger and J. Paulus of theBonn Rhein Sieg University of Applied Science of Sankt Augusting,Germany for useful discussions and for having developed the setup forthe experimental validation of the proposed control law. The authoralso thanks the anonymous reviewers for their comments.APPENDIXPROOF OFSTATEMENT35Proof: For the sake of notation compactness, define qjdef= ? qj? qj?,cj?.(51)Then, summing the last n 1 equations (34) for j = 2,3,.,n, thefollowing holds:n?j=2cj= c1+n?l=2cl cnn1?k=1 qk=c1= cn+n1?k=1 qkn?k=1 qk(52)because cn qn. 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Pl oger, “Task based kinematical robotcontrol in the presence of actuator saturation and its application to tra-jectory tracking for an omniwheeled mobile robot,” in Proc. IEEE Int.Conf. Robot. Autom. (ICRA 2007), Rome, Italy, Apr. 1014, pp. 26112616.21 G. Indiveri, J. Paulus, and P. G. Pl oger, “Motion control of Swedishwheeledmobilerobotsinthepresenceofactuatorsaturation,”inRoboCup2006: Robot Soccer World Cup X, Lecture Notes in Computer Science,vol. 4437, Berlin, Heidelberg: Springer, 2007, pp. 3546.22 L. Sciavicco and B. Siciliano, Modelling and Control of Robot Manipula-tors, 2nd ed.Berlin, Germany: Springer-Verlag, 2000.Compliant Terrain Adaptation for Biped HumanoidsWithout Measuring Ground Surface and Contact ForcesSang-Ho Hyon, Member, IEEEAbstractThis paper reports the applicability of our passivity-basedcontact force control framework for biped humanoids. We experimentallydemonstrate its adaptation to unknown rough terrain. Adaptation to un-even ground is achieved by optimally distributed antigravitational forcesapplied to preset contact points in a feedforward manner, even withoutexplici