Swedish Wheeled Omnidirectional Mobile Robots Kinematics Analysis and Control.pdf

164IEEE TRANSACTIONS ON ROBOTICS, VOL. 25, NO. 1, FEBRUARY 2009 40 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 change their group of motion,”in Advances in Robot Kinematics. The Netherlands: Springer, 2006, pp. 4956. 42 S. Refaat, J. M. Herv e, S. Nahavandi, and H. Trinh, “Two-mode over-constrained three-dofs rotational-translational linear-motor-based parallel-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 of parallel 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 as c-space singularities,” in Advances in Robot Kinematics, J. Lenarcic and F. Thomas, Eds.Dordrecht, The Netherlands: Kluwer, 2002, pp. 183 192. 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 Control Giovanni Indiveri AbstractSwedish wheeled robots have received growing attention over the last few years. Their kinematic models have interesting properties in terms of mobility and possible singularities. This paper addresses the is- sue of kinematic modeling, singularity analysis, and motion control for a generic vehicle equipped with N Swedish wheels. Index TermsMobile robot kinematics, mobile robots, motion control, wheeled robots. I. INTRODUCTION In the last few years, Swedish wheeled omnidirectional mobile robots have received growing attention among the mobile robotics research community. A Swedish wheel differs from a common wheel in the fact that rollers are mounted on its perimeter (see Fig. 1). If all the rollers are parallel to each other and misaligned with respect to the wheel hub axis, they will provide an extra degree of mobility with respect to a traditional perfectly rolling wheel. The wheels depicted in Fig. 1 are often called mecanum or Swedish wheels: one of their design parameters is the angle between the rollers rolling direction g and the wheel hub axis direction h. Typical values are = 45and = 0, as shown in Fig. 1 (left and right cases, respectively). Note that the degenerate case = 90has no practical interestasitwouldallowthesamemobilityoftraditionalwheels.Inthe Manuscript received June 27, 2008; revised October 6, 2008 and November 19, 2008. First published January 21, 2009; current version published February 4, 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 of Salento, 73100 Lecce, Italy (e-mail: giovanni.indiveriunile.it). Color versions of one or more of the fi gures in this paper are available online at . Digital Object Identifi er 10.1109/TRO.2008.2010360 Fig. 1.Mecanum wheel: = 45(left) and = 0(right). literature, wheels with = 45are more often called mecanum wheels whereas the ones with = 0are generally called Swedish wheels. In the following, the term Swedish wheel will denote the general case for some fi xed . As reported in 1, the mecanum wheel was invented in 1973 by Bengt Ilon, an engineer working for the Swedish company Mecanum AB. Since then, this wheel design has attracted the attention of the mobile robotics research community. The interest in such kind of wheels is related to the possibility of developing omnidirectional robots in the sense of 2, i.e., robots that “have a full mobility in the plane meaning thereby that they can move at each instant in any directionwithoutanyreorientation”2.Theneedtoreorientthewheels or not prior to implementing any desired linear velocity is related to the presence or not of nonholonomic constraints 3, 4. Since the early study of Agull o et al. 5, the kinematics analysis of Swedish or mecanum wheel robots has been addressed in several papers 612. All these except the study of Agull o et al. 5 and Saha et al. 9 consider either three or four Swedish wheeled robots for some value of . One of the objectives of this paper is to derive, in the most general setting of N Swedish wheels with fi xed (but arbitrary) roller wheelangle,necessaryandsuffi cientgeometricalconditionsonrelative wheel arrangement that guarantee: 1) the absence of singularities and 2) the possibility of decoupling commanded linear and angular robot velocities. This kind of information can be most valuable to guide the design of the robot a priori. The results obtained addressing the N-wheel case, besides realizing at once a unifi ed analysis of the most common three- and four-wheel designs, allow to easily explore, for example, possible six-wheel designs that have a large interest in the fi eld of outdoor and rough terrain applications 14. As a side result of the proposed analysis, all possible kinematics singularities occurring asafunctionoftherollerwheelangleandwheelarrangementcanbe identifi ed. Moreover, building on well-known methods, the trajectory tracking and pose regulation problems are solved for these systems taking explicitly into account actuator velocity saturation. The paper is organized as follows: the general kinematics model for an N Swedish wheeled vehicle is derived in Section II. The guidance trajectorytrackingandposeregulationcontrolproblemsinthepresence ofactuatorvelocitysaturationareaddressedinSectionIII.Experimental results are reported in Section IV. At last, some concluding remarks are addressed in Section V. II. KINEMATICSMODEL With reference to Fig. 2, for the sake of introducing the notation, a three-wheel omnidrive mobile robot is considered. All wheel main axes, i.e., hub axes, are assumed to always lie parallel to the fi xed ground plane P having unit vector k P. Each of the N Swedish wheels is indexed from 1 to N: for the hth wheel, the roller in contact with 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 IEEE 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 2009165 Fig. 2.Three-wheel omnidrive robot: geometrical model. The unit vector nhis aligned with the rollers rolling axis on the main wheel perimeter and uh:= nh k indicates the instantaneous tan- gent velocity direction of the roller associated with its rotation around nh. All wheels are assumed to be identical and have the same radius . The position of the hth wheel in the body-fi xed frame is denoted by bh. The unit vector of each wheel hub axis, i.e., the unit vector of the wheels main rotation axis, is denoted by nh: ?nh? = 1. At last for each wheel, the unit vector uh:= nh k is defi ned indicating the instantaneous tangent velocity direction of the wheel as a consequence of its rotation around nh. Note that with reference to Fig. 1, the unit vectors uhand nhwould correspond to g and h, respectively. In the given hypothesis, all the introduced vectors except k are parallel to the ground plane P. Given the components of any two 3-D vectors a and b on a common orthonormal frame, their vector product will be computed using the skew-symmetric matrix S() such that a b = S(a)b.(1) Calling vcthe linear velocity of the robots center (indicated as point c in Fig. 2) and k its angular velocity vector, the velocity vector vh of the center of each omnidirectional wheel hub will be given by vh= vc+ k bh,h = 1,2,3,.,N.(2) Considering the generic hth wheel and dropping for the time being the index h for the sake of notational clarity, in the case of perfect rolling, the velocity v = vhgiven by (2) will be physically realized by the roller rotation around nand the wheel rotation around n. Namely, assuming that nand n are not aligned, i.e., ?= (2 + 1)90, where is an integer v = u+ u implying nTv = ?nT u ? nT v = ?nT u ? and consequently v = nTv nTu u+ nT v nT u u.(3) Note that the rollers rotation around n giving rise to the fi rst term on the right-hand side of (3) is completely passive, whereas the wheel rotation around n giving rise to the second term on the right-hand side of (3) is assumed to be actively produced by a motor. Calling nh qhthe angular speed associated with the hth motor in the body-fi xed frame, in case of perfect rolling, the mapping between the “joint” speed q and the corresponding velocity v of the hub of any given wheel will be given by nT uTn v = q(4) where the contribution nTv uT n uT u of 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 by the passive rotation of the roller. Substituting (2) back into (4), given that uT hnh = cos for any h, one gets 1 cos nT h vc+ 1 cos nT hS(bh)k = qh. (5) Byprojectingallthevectorsin(5)onacommonbody-fi xedframewith its third axis equal to k P, the term nT hS(bh)k results in nT hS(bh)k = nxhbyh nyhbxh = bT h uh.(6) Summarizing, (5) can be interpreted as a generic component of the inverse differential kinematics equation in matrix form M ? vc ? = q cos(7) where M = nx1ny1bT 1 u1 nx2ny2bT 2 u2 . . . . . . . . . nxNnyNbT N uN 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 presented in the literature that are relative to three or four wheels in fi xed confi g- urations, they allow a fully detailed analysis of the vehicle kinematics properties as a function of not only the rollerwheel hub orientation , but also the relative wheel position. Such an analysis may be extremely useful for the mechanical and controls system design of the vehicle. Assuming cos ?= 0 and that M has rank 3, equation (7) can be used to compute joint velocity commands for a desired vehicle speed (vT cd,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 (vT cd,d) T can be implemented by using proper joint velocities qdif and only if all of the following conditions hold: c.1 cos ?= 0; c.2 rank M = 3. In particular, q d= 1 cos M ? vcd d ? .(9) Theviolationofanyofthepreviousconditionsc.1orc.2corresponds tokinematicssingularitiesofadifferentkind:theviolationofcondition c.1 would correspond to a total loss of control authority, while the violation of condition c.2 would correspond to a loss of controllability, as for any choice of the input q, the state derivative (vT c,) T would not be uniquely defi ned. Matrix M defi ned in (8) can be decomposed as M = MlMa, where Ml IRN 2and Ma IRN 1such that Mlvc+ 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 2009 III. GUIDANCECONTROL OFSWEDISHWHEELED OMNIDIRECTIONALROBOTS Based on the derived kinematical model, the trajectory tracking and pose regulation motion control problems for a generic robot equipped withN 3Swedishwheelsaresolved.Theproposedsolution,besides being general in terms of number of wheels, value of , and wheel con- fi guration, explicitly accounts for joint velocity saturation. The control problem is formulated as a guidance control problem, namely the joint velocities qhare assumed to be control inputs and the robots linear and angular velocities vcand satisfying (7) are the outputs. Of course, in practice, such an approach requires the implementation of a lower level control loop mapping the joint reference speeds qdin actuator commands. A. Low-Level Control Design As in standard robot motion control architectures 22, the low-level control 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 validation reported in this paper); in the latter case, a centralized control solution can be derived based on computed torque (i.e., feedback linearization) methods. The decentralized control (or independent joint control) method is simpler: each component of qdis used as a reference signal for the corresponding PID actuator velocity loop, and dynamic couplings among the actuators are neglected. If such low-level control system (i.e., each actuator velocity servo loop) is fast with respect to the robot dynamic-navigation one, the lag between the desired joint velocities and the real joint velocities is negligible: in such a case, as long as the perfectrollingconstraintissatisfi ed,themappingbetweenthevehicles velocity (vT c,) T and the joint speeds would be approximated by the systems kinematical model (7) having the desired (or commanded) joint velocities qdon the right-hand side in place of the real joint velocities q. Centralized control solutions are generally based on the dynamic robot model 17, Ch. 12, pp. 493502, and the dynamic equation of Swedish wheeled robots with a geometry as the one described in Section II has the following structure: I q + (C() + F) q = (11) with I IRN N being the positive-defi nite 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 the nondiagonal nature of matrices I and C() and the dependency (7) of from q, (11) is nonlinear and coupled. Yet, as commonly done for robotic manipulators 22, the control input vector can be com- putedbaseduponanonlinearstatefeedbacklinearization(orcomputed torque) solution, namely = (C() + F) q + I y(12) giving rise (recall that I is full rank) to a linear and decoupled model q = y that can then be used to design a closed-loop solution for y in order to track the reference signal qd. Given that the computed torque solution explicitly accounts for the systems dynamic coupling terms, it is expected to exhibit a better tracking performance, in particular, for high speed and acceleration references. Nevertheless, as also demonstrated by the reported experi- mental results, the independent joint solution appears to be suffi ciently precise 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 PID velocity servo loop) was implemented. In the following, the Swedish wheeled robot will be assumed to have such kind of low-level independent joint controls system, and the trajectory tracking and pose regulation problems will be solved at the guidance level, i.e., considering the commanded joint speeds q das control inputs and (vTc,)Tas outputs according to the purely kinematicalmodel(10).Inordertoformulatethetrajectorytrackingand pose regulation problems, the following notation will be used: given an inertial (global) frame ?G? = (i,j,k) with k := (i j) P, where P is the fl oor plane, a reference (planar) trajectory is a differentiable curve in P rd(t) = i ?rT d(t)i ? + j ?rT d(t)j ? (13) with curvilinear abscissa s(t) := ? t t0 ? ? ? ? drd() d ? ? ? ?d (14) and unit tangent vector td= drd ds .(15) The kinematics trajectory tracking problem consists in fi nding a controllawforthesystemsinput qdsuchthatthepositionandheading tracking errors er(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 reference point (e.g., the geometrical center or the center of mass) of the robot, (t) itsheading,andd(t) thedesiredreferenceheading.Notethatfor nonholonomicvehicleshavingaunicycleorcar-likekinematicsmodel, the reference heading d(t) is not arbitrary, but needs to coincide with the heading of the trajectories unit tangent vector td. Conversely, given any position reference trajectory rd(t), a Swedish wheeled vehicle will be free to track any arbitrary heading d(t) that does not necessarily need 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 are constant, i.e., when rd= 0 and d= 0. B. Trajectory Tracking Controller Design In accordance with the notation previously introduced, consider the system in (10) with rc(t) = vcand (t) = being the robots lin- ear and angular velocities. Assuming that conditions c.1 and c.2 of Lemma 1 are satisfi ed, N 3, and Ma span(Ml) (this can be al- ways guaranteed by a proper design of the robot geometry), then any desired robot linear velocity vc= (,0)Tand angular velocity k are uniquely mapped to the control input qdas follows: q d= qdl+ qda (18) q dl= 1 cos Mlvc(19) q da= 1 cos Ma.(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 2009167 Followingastandardapproach