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Fuzzy adaptive EKF motion control for non-holonomic and underactuated cars with parametric and non-parametric uncertainties F M Raimondi and M Melluso Abstract: A new fuzzy adaptive motion control system including on-line extended Kalman’s filter (EKF) for wheeled underactuated cars with non-holonomic constraints on the motion is presented The presence of parametric uncertainties in the kinematics and in the dynamics is treated using suitable differential adaptation laws We merge adaptive control with fuzzy inference system By using fuzzy system, the parameters of the kinematical controller are functions of the lateral, longitudinal and orientation errors of the motion In this way we have a robust control system where the dynamics of the motion errors is with lower time response than the adaptive control without fuzzy Also Lyapunov’s stability of the motion errors is proved based on the properties of the fuzzy maps If data from incremental encoders are employed for the feedback directly, sensor noises can damage the performance of the motion control in terms of the motion errors and of the parametric adaptation These noises are aleatory and denote a kind of non-parametric uncertainties which perturb the nominal model of the car Therefore an EKF is inserted in the adaptive control system to compensate for the above non-parametric uncertainties The control algorithm efficiency is confirmed through simulation tests in Matlab environment Introduction In recent years much attention has been focused upon the control of non-holonomic mechanical systems [1] A mobile wheeled car actuated by a differential drive is usually studied as a typical non-holonomic system, where non-holonomic constraints arise under the no-slip constraints Our approach is about motion control of nonholonomic wheeled cars Due to non-holonomic motion, the cars are underactuated In fact there are three generalized coordinates i.e lateral position, longitudinal position and car orientation to be controlled, whereas there are two control inputs only, that is steering and longitudinal inputs About the motion control, older research effort use only kinematical controllers [2] The main idea is to define velocity control inputs which stabilize the closed loop system However, if one considers kinematical controller only, then one assumes that there exists perfect velocity tracking, that is the control signals affect the car velocities instantaneously and this is not true More recently control researchers have targeted the problem of motion control of underactuated wheeled cars using a backstepping approach [3, 4] which allows many of the steering system commands to be converted to torques, taking into account dynamic parameter (mass, inertia, friction) The fundamental problems of the backstepping control are the parameters uncertainties, i.e the unknown dynamical parameters of the car, that is mass, inertia, position of the mass centre and the partialy known kinematical parameters, that is ray of # The Institution of Engineering and Technology 2007 doi:10.1049/iet-cta:20060459 Paper first received 25th October 2006 and in revised form 22nd January 2007 The authors are with the Dipartimento di Ingegneria dell’Automazione e dei Sistemi, University of Palermo, Italy E-mail: maurizio.melluso@dias.unipa.it IET Control Theory Appl., 2007, 1, (5), pp 1311 –1321 the wheels, distance from the reference point of the motion to the mass centre In this sense adaptive control schemes have been presented [5– 7] In [5, 7] adaptive laws of the dynamical parameters only have been developed, whereas in [6] adaptive laws both the dynamical and the kinematical parameters have been presented However in all the works above the parameters of the kinematic control laws are constant for every value of the motion errors, so that they must be chosen suitably to guarantee a good dynamics of the motion errors; also the problem of on-line localization, that is an optimal estimation of the car’s position, has not been treated Really, at each sampling instant the position of the car is estimated on the basis of the encoders’ increment along the sampling interval A drawback of this method is that the errors of each measure caused by the encoder are summed up Many researchers have solved the problem of localization, by an off-line sensors data fusion based on the use of extended Kalman’s filters (EKF) [8 – 10] An interesting approach has been developed in [11], where a conventional PID control strategy with a Kalman-based active observer controller has been used to solve a problem of path following for non-holonomic cars To apply EKF it is necessary for a discrete non-linear system model which describes the state transition relationship during a sampling interval A discrete model for non-holonomic cars has been proposed [10] where the position coordinates of the car have been expressed with respect to a ground reference About fuzzy inference mechanism for non-holonomic and underactuated cars, fuzzy controllers have been presented [12, 13], where the stability of the motion is not always assured In [14] a dynamical fuzzy control law with Lyapunov’s stability proof has been proposed, whereas in [15] a kinematical control law, where the parameters have been obtained using a suitably fuzzy inference system, has been shown and the asymptotical stability of the 1311 motion error has been proved However, by using the approach of [15], because of the noises of the encoders positioned in right and left wheels of the car, the feedback signals of the control system (i.e the actual position and orientation of the car provided by the encoders) are noised The aleatory noises above are responsible for nonparametric uncertainties which perturb the model of the car The non-parametric uncertainties above can corrupt the performance of the fuzzy control system in terms of dynamics of the motion errors and in terms of the dynamical parametric adaptation In order to continue this research line, a fuzzy adaptive motion control system with on-line EKF for non-holonomic and underactuated cars is presented The following contributions are given: Merging of adaptive kinematic and dynamic controllers with a fuzzy inference system where stability and convergence analysis is built on Lyapunov’s theory, based on the properties of the fuzzy maps This assures robustness and lower time response than the adaptive control without fuzzy [6] A sampled chained form model for non-holonomic car with non parametric uncertainties In [10] the position coordinates of the car are expressed in a ground reference, whereas in our work the coordinates are expressed in a body fixed reference In this way the noises of the proprioceptive sensors (i.e encoders positioned in right and left wheels of the car) are naturally expressed in a frame attached to the car body A methodology for solving the on-line sensors data fusion problem through EKF An EKF has to be introduced in the fuzzy adaptive control system above to fuse data from multiple proprioceptive sensors (i.e encoders, vector compass and sensor position) and to estimate the filtered feedback signals, that is the actual position of the car by on-line recursive predictions and corrections The EKF compensates the non-parametric uncertainties effects (i.e discontinuities in the dynamics of the motion errors and of the parametric adaptation) By using our fuzzy solution, the constant parameters of the conventional kinematic control laws [3, 6] are obviated In fact, in our approach, the parameters are nonlinear functions of the motion errors and this assures faster convergence and more robustness than the conventional adaptive controller [6] Based on the input – output properties of the fuzzy map, the asymptotical stability of the motion errors has been proved in Section 4.2 Section 4.3 adds an adaptive kinematical control with stability proof to solve the problem of partialy known kinematical parameters In Section 5, based on the adaptive backstepping approach [6], a dynamical extension is presented The EKF estimates the filtered state from the noised outputs provided from more odometric sensors and assures a good localization The filter above requires to derive a linear discrete time stochastic state space representation of the car model and of the measure process About the state space representation, we introduce a sampled form of the kinematic ‘chained form’ model [16] where the inputs are provided by the data of the encoders About the measure equations, we consider data provided from proprioceptive sensors of position and orientation Time continuous models for wheeled cars Fig Constrained wheeled car with references dynamic model in generalized coordinates is given in [3, 5, 6] _ q_ ¼ E(q)t À AT (q)l M(q)q ỵ C(q,q) (1) where M(q) [
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