Guldner_Utkin-SlidingMode-CartLikeRobot - Proceedings of...

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Proceedings of the 33rd Conference on Decision and Control Lake Buena Vista, FL - December 1994 FA-12 10110 Stabilization of Non-Holonomic Mobile Robots using Lyapunov Functions for Navigation and Sliding Mode Control .Jiirgon Giildiier Vadim I. Ct,kin DLR, Inst. for Robotics urd Sgsteni Dgnumics Postfuch 11 16, 0-82230 Wessling, Germany Abstract Mohile robots with noii-holonomic kinematics have three degrees of freedom for planar motion, but only there are two control inputs available. The stabilization problem for such robots is known not to be solvable via smooth time-invariant feedback. We propose to utilize a Lyapunov function to prescribe a set of desired trajectories to navigate the robot to a specified configuration. Ideal tracking of the prescribed traject.ories is achieved by exploiting the invariance property ancl the order reduction property of sliding mode control. The mobile robot is shown to be exponentially stahilizable for a class of quadratic Lyapunov functions. 1 Introduction The problem of steering a mobile robot with non- lioloiioniic kinematics to a specified configuration (.ro, yo, do) has recently enjoyed quit,e some at,tent,ion in the robot,ics community. The challenge of t.he prob- leni is reflcct,c!d in t.he fact that, a mobile robot in the plane possesses t,liree degrees of freedom of motion, wliicli have be cont,rolled by only two control inputs and under non-holonomic motion constraii1t.s. Several researchers have shown, based on Brocketk's Theo- rem [l], that, such a system is open-loop controllable, but, not, stabilizable by pure smooth t,ime-invariant feedback, see for example [2-41. To overcome the above difficulties, several di- rections of research have evolved. Among the pro- posed solutions are time-varying feedback [5], result- ing in oscillatory trajectories [6]; nonlinear continuous state feedback depending on an exogeneous time vari- able [7] and yielding asymptotic stabilizat,ion about (soly0,@o) at. a rate of l/t; and piecewise continuous control resulting in exponential st,ability [SI. Further- more, there have been a number of open-loop strate- gies seeking a bounded sequence of control inputs, t,he existence of such sequences having been established in [9]. We propose to utilize sliding mode control t,o solve the stabilization problem, i.e. to employ a discon- tinuous control strategy. The first step is to design a Lyapunov navigat,ion function such that its gradi- ent prescribes a suitable set of trajectories leading to the goal configuration in the robot workspace in the desired manner. Exact tracking of the t.rajecto- ries is guarant.eed via sliding mode cont,rol and hence the robot. is successfully navigated to t,he goal con- figuration. In a case study with a class of quadrat,ic 0-7803-1 968-0/94$4.0001994 IEEE Lyapunov functions, exponential stabi1it.y is achieved along parabolic trajectories. Simulations are utilized to illustrate the performance of the proposed control algorithm.
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This note was uploaded on 02/07/2011 for the course MECH 646 taught by Professor Danielnassrallah during the Fall '10 term at American University of Beirut.

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Guldner_Utkin-SlidingMode-CartLikeRobot - Proceedings of...

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