Control of a Class of Underactuated Mechanical Systems Using Sliding Modes.pdf

IEEE TRANSACTIONS ON ROBOTICS, VOL. 25, NO. 2, APRIL 2009459 Control of a Class of Underactuated Mechanical Systems Using Sliding Modes V. Sankaranarayanan, Member, IEEE, and Arun D. Mahindrakar, Member, IEEE AbstractIn this paper, we present a sliding mode control algorithm to robustly stabilize a class of underactuated mechanical systems that are not linearly controllable and violate Brocketts necessary condition for smooth asymptotic stabilization of the equilibrium, with parametric uncertainties. In defi ning the class of systems, a few simplifying assumptions are made on the structure of the dynamics; in particular, the damping forces are assumed to be linear in velocities. We fi rst propose a switching surface design for this class of systems, and subsequently, a switched algorithm to reach this surface in fi nite time using conventional and higher order sliding mode controllers. The stability of the closed-loop system is investigated with an undefi ned relative degree of the sliding functions. The controller gains are designed such that the controller stabilizes the actual system with parametric uncertainty. The proposed control algorithm is applied to two benchmark problems: a mobile robot and an underactuated underwater vehicle. Simulation results are presented to validate the proposed scheme. Index TermsMobile robot, nonholonomic, sliding mode, underactu- ated underwater vehicle (UUV). I. INTRODUCTION Control of underactuated mechanical systems, systems with fewer number of control inputs than the degrees of freedom, has received much attention over the past few years. This is because of the theo- retical challenges as well as practical applicability. Among the host of control methods proposed for the individual applications, some gen- eral methodologies have also been proposed to stabilize these sys- tems 13. These methodologies use passivity techniques, energy shaping, and discontinuous control methods. A challenging problem is to stabilize a particular class of systems that are not linearly con- trollable and disobey Brocketts necessary condition 4. While there exists piecewise analytic or time-periodic continuous feedback laws that can asymptotically stabilize the equilibrium, the problem of robust and bounded controller design are two important issues that are still open in this fi eld. In this paper, we consider the robust stabilization of this particular class of systems. In particular, we present the design of a sliding mode controller for the aforementioned class of systems. Sliding mode is one of the robust controller design methods and has been successfully applied to underactuated and nonholonomic sys- tems 58. Stabilization of several underactuated systems under this particular class have been solved using sliding mode control tech- niques (see, for example, the nonholonomic integrator and its extended version 9). However, to our knowledge, there is no constructive pro- Manuscript received March 11, 2008; revised July 1, 2008 and September 13, 2008. First published February 6, 2009; current version published April 3, 2009. This paper was recommended for publication by Associate Editor E. Papadopoulos and Editor H. Arai upon evaluation of the reviewers com- ments. The work of A. D. Mahindrakar was supported by the Depart- ment of Science and Technology, India, under the research Grant ELE/07- 08/153/DSTX/ARUD. V. Sankaranarayanan is with the Department of Electrical and Electronics Engineering, National Institute of Technology, Tiruchirapalli-620015, India (e-mail: ). A. D. Mahindrakar is with the Department of Electrical Engineering, Indian Institute of Technology Madras, Chennai-600036, India (e-mail: arun_dmiitm.ac.in). Color versions of one or more of the fi gures in this paper are available online at . Digital Object Identifi er 10.1109/TRO.2008.2012338 cedure to design a switching surface and a sliding mode controller for this class of systems. First, we present an outline to design the switching surface by estab- lishing a connection between the regular form 10 and the defi nition of the switching surface. Three cases are considered to design the switching surface. Subsequently, the sliding mode controller design is presented to reach the switching surface in fi nite time using both conventional and higher order sliding mode controllers. In addition, a switched algorithm is also proposed to reach the switching surface if the relative degree of the sliding functions are not well defi ned. Finally, we presentthe designof the controller gains torender the systemstable with parametric uncertainties. The rest of the paper is organized as follows. A classifi cation of underactuated mechanical systems in the context of sliding mode is presented in Section II. In Section III, a design outline is pre- sented for the generation, fi nite-time reachability with well-defi ned and undefi ned relative degrees of the sliding functions. In Sections IV and V, the proposed methodology is applied to two benchmark exam- ples: a mobile robot and the underactuated underwater vehicle (UUV), respectively. Simulation results are presented in Section VI, followed by conclusions in Section VII. We begin with the defi nition of the class of underactuated mechanical systems considered in this paper. II. CLASS OFUNDERACTUATEDMECHANICALSYSTEMS The dynamics of many mechanical systems can be expressed as D(q) q + C(q, q) = Fu(1) where q = (q1,.,qp ) is the parametrization of the confi guration space Q, D(q) IRppis the inertia matrix, C(q, q) IRpconsists of damping, Coriolis, stiffness, etc., u IRmis the vector of external inputs,andF IRpmisthecorrespondinginputmatrix.Weconsider a class of underactuated mechanical systems that is characterized as follows. 1) The inertia matrix D is diagonal and constant. 2) The vector C in (1) depends only on q, and further, the damping forces are linear in velocities. 3) m 0. This leads to an equivalent dynamics x1= Kx1, which is exponentially stable. But thischoiceofl(x1)resultsinaswitchingsurfacethatistheintersection of(n m)slidingsurfaces,andmoreover,ifthecorrespondingsliding functionsarelinearlyindependent,thentheycannotbereachedinfi nite time using m control inputs. Case 2: To overcome the conservative procedure outlined in case 1, we present a Lyapunov-function-based design of l(x1). Consider the following candidate Lyapunov function V : IRnm IR defi ned as V (x1) = (1/2)x? 1x1, and further construct l(x1) such that V (x1) 0 on O, and further, O is the intersection of m sliding surfaces. Case 3: This case pertains to the design of l(x1) wherein the equiv- alent dynamics is not Lyapunov stable with respect to V (x1). In such a case, construct l(x1) such that the equivalent dynamics possesses a weak notion of stability such as convergence to the origin. B. Finite-Time Reachability of the Switching Surface Oncetheswitchingsurfaceisdesigned,thecontroltaskisreducedto reaching the switching surface in fi nite time. We next defi ne fi nite-time reachability 13 of the switching surface. Defi nition 3.2: The switching surface O is said to be fi nite-time reachable if for any x(0) N IRn, there exists T 0,) and an admissible control u : 0,T IRmwith the property that x(T) O. We assume that the design procedure outlined in the earlier sec- tion yields m functions Sj(x) such that O = .m j=1(Sj(x) = 0). The dynamicsobtainedafterdifferentiatingeachSj(forsimplicity,wesup- press the dependence on x), rjnumber of times, along the trajectories of (5) yields S(r1 ) 1 S(r2 ) 2 . . . S(rm ) m = R(x) + Q(x)u(7) where R(x)?= Lr1 f S1Lr2 f S2 Lrm f Sm?and Q(x) = Lg1Lr1 1 f S1.LgmLr1 1 f S1 Lg1Lr2 1 f S2.LgmLr2 1 f S2 . . . . . . . . . Lg1Lrm 1 f Sm.LgmLrm 1 f Sm . By the defi nition of well-defi ned vector relative degree 14, Q(x) is invertible at xe, and thus, for all x in the neighbourhood of xe, we have u = Q1(x)P(x) R(x).(8) The choice of P(x) IRmdepends on the relative degree of the sliding functions. For example, if the vector relative degree is 1,2, then a few possible choices of P(x) = P1(x) P2(x)?are P1(x) = ! K1sign(S1) S 1 P2(x) = sign(S2)|S2|a sign(S2)|S2|b sign(S2)|S2|1/3 sign % (S2, S2) current version published April 3, 2009. This paper was recommended for pub- lication by Associate Editor Z. W. Luo and Editor F. Park upon evaluation of the reviewers comments. H. Kino is with the Department of Intelligent Mechanical Engineering, Fac- ultyofEngineering,FukuokaInstituteofTechnology,Fukuoka811-0295,Japan (e-mail: kinofi t.ac.jp). T. Yahiro is with the Department of Production Engineering, Izumo Murata Manufacturing Company, Ltd., Shimane 699-0696, Japan (e-mail: yahiro_). S. Taniguchi is with the Research and Dvelopment Center, Home Appliances Manufactuing Business Unit, Panasonic Electric Works Company, Ltd., Osaka 571-8686, Japan (e-mail: taniguchi.shoheipanasonic-denko.co.jp). K. Tahara is with the Organization for the Promotion of Advanced Research, Kyushu University, Fukuoka 819-0395, Japan (e-mail: ). Color versions of one or more of the fi gures in this paper are available online at . Digital Object Identifi er 10.1109/TRO.2009.2013495 Fig. 1.CR parallel-wire-driven system using seven wires for 6-DOF motion. the gravitational force. A CR type requires redundant actuation so that atleastn + 1wiresarenecessarytocontrolthen-DOFmotionbecause wires can generate only tension. Consequently, an internal force exists among wires. On the other hand, an IR type has advantages of fewer actuators and larger workspace than a CR type. Oscillation problems, however, are apt to arise in IR systems because of the incomplete restriction. The CR systems tend to be used for applications requiring high-speed motion or high stiffness 13, 5, 7, 11. On the other hand, IR systems are better suited for the applications requiring a large workspace or high payload 1217. Generally, point-to-point (PTP) position control for the parallel- wire-driven systems must add a balancing internal force term vto actuator input to avoid slackening of the wires 1, 2, 5, 18 = FB+ g+ v(1) where FBis a feedback term based on information from some dis- placement sensor and gdenotes the gravitational compensation. The internal force vcan be renewed corresponding to a current position in real time; alternatively, it can be feedforward input balancing at a desired position. In previous studies related to position control for CR parallel-wire- driven systems, almost all discussion has been centered upon the feed- back term FB. To date, few studies of the effect of the internal force term vhave been reported. Because it is canceled out during op- eration, it theoretically does not generate driving force. Kawamura et al. 5 pointed out that the internal force from redundant actuation can reduce vibration by making good use of nonlinear elasticity of a CR system. Li et al. 19 clarifi ed the effect of internal force on the structural stiffness of an antagonized rotational drive mechanism,