Motion Control of a Nonholonomic System Based on the Lyapunov Control Method

摘要

AnewdesignmethodofafeedbackcontrollerfornonholonomicsystemsbasedonLyapunovcontrolispresented. In Lyapunov control, the control input is obtained by multiplying the gradient vector of the Lyapunov function by a tensor. The main contribution of our method is that this tensor is composed of two components, one of which is a negative deenite symmetric tensor and the other of which is an asymmetric one. As a result, the goal point in the state space of the controlled system becomes the only globally stable equilibrium point, and exponential convergence tothe goal point can beachieved. The proposed method isapplied toa two-wheeled mobile robot, and the effectiveness is conermed by numerical simulations. HEmotioncontrolofanonholonomicsystemhasbeenfocused on by many researchers. The motion control of a two-wheeled mobile robot is a typical terrestrial application, whereas typical ap- plications in space include the attitude control of a rigid spacecraft with two reaction wheels and the motion control of a planar space manipulator composed of three links. In nonholonomic systems, there exist constraints that are not integrable, for example, a no-slip condition for the wheels of the wheeled vehicle, the law of conser- vation of angular momentum for a space robot, and soon. Although this class of nonlinear systems are not controllable locally, they may be controlled globally by exploiting the constraints. They can not be controlled with the method of linear control theory, however, because they are not exactly linearizable. 1 Moreover, they can not be stabilized to an equilibrium point by any smooth state feedback control, even if they are controllable globally. 2 Therefore, it is dife cult to design a feedback controller that sta- bilizes a nonholonomic system to an equilibrium point, and the re- search on this problem has been extensive.The controllers thathave been proposed thus far are classie ed as time-varying controllers 3i6 and discontinuous time-invariant controllers. 7i9 Time-varying con- trollers were originated by Samson. 3 Pomet proposed a method of designing this type of controller by using a time-varying Lyapunov function. 4 Theircontrollersaresmoothtime-varyingcontrollers,but the ratesofconvergence arenot exponential.Sordalen and Egeland 5 and M'Closkey and Murray 6 proposed nonsmooth time-varying controllers that provide exponential rates of convergence. On the other hand, discontinuous time-invariant feedback con- trollers have also been proposed, such as the class of controllers based on the idea of sliding mode control, proposed by Khennouf and Canudas de Wit. 7 They designed the controller by combining a linear feedback law and a discontinuous feedback law. The discon- tinuous feedback law stabilizes the invariant manifolds formed by the linear feedback law. 7 Lafferriere and Sontag proposed a method of designing a controller by using a piecewise smooth Lyapunov function. 10 Astole proposed a method of designing a controller by transforming an original system through a nonsmooth coordinate transformation and designing a smooth, time-invariant controller for the transformed system. 8;9 The controllers designed in Refs. 7 and 9 provide exponential rates of convergence.