Tracking control of a mobile robot based on Taylor Formula

−− In this work, a strategy to calculate the control actions for a mobile robot following a pre established trajectory is proposed. For this purpose, the Taylor series development of controlled variables is used and the control action is calculated to make the system follow the reference trajectory. As main result, an easy form to implement the strategy is obtained. Simulation and experimental results are presented, showing the advantages of the control strategy proposed. Keywords−− Control system design, discrete systems, models, tracking, trajectory control, numerical methods. I. INTRODUCCIÓN Trajectory tracking is one of the main problems of control systems for which, in a search for a small tracking error, various controllers have been proposed in the literature, for example, Kim (2003) proposes a receding horizon tracking control for time-varying linear systems with constraints both on the control signal and on the tracking error, based on the minimization of a functional for finite-time costs. Besides, Linear Matrix Inequalities (LMI) are used in order to synthesize the controller. In Chem et al. (1995) a controller is proposed only for linear, non-varying systems which are exponentially stable and of nonminimum phase, which needs a set of input/output data. In Fujimoto et al. (2001), a perfect-tracking control based on multirate feedforward control for linear systems is presented. Specifically, the design is made for a SISO system which, nevertheless, can be extended for a MIMO system. In Young-Hoon (2000), the case in which the dynamic equations of the tracking error are described by a set of non-linear, time-varying, periodic differential equations, is considered. In Shuli (2005) a mobile robot controller that is based on the error model of Kanayama (1990) is proposed. As a result, instead of just one controller, two are hended, which are used depending on whether the angular velocity is null or not. The work of Shuli (2005) shows only simulation results. In Scaglia et al. (2005) a control based on numeric methods is shown, where the mobile robot linear speed depends on the current and desired orientation. In Scaglia et al. (2006a) the integral trapezoidal method is used to compute the control signal. The design is based on the model of Kanayama (1990) and here, conditions are established over the tracking error so that the equations system can always have solution. In (Scaglia G., et al., 2006b) the controller is computed solving the normal equations. In (Scaglia G., et al., 2007) the system evolution is approximated by means of a linear interpolation method, this allows a better system precision. The strategies previously mentioned can be applied to other systems, as can seen in Rosales et al. (2006), where the controller design for the RTAC system (also called TORA, a classic nonlinear problem, which is interesting due to the interaction between the traslational and rotational movements) is proposed in order that the system tends to the origin from any initial position. The proposed design is based on a system approximation using numerical methods and on the control signal calculus solving a linear equations system. Simulation results are presented; these results show tha advantages of the proposed control strategy which is obtained by using a simple design procedure. In this work, a strategy based on Taylor’s development of a variable to control is presented, in which the control signal is calculated modifying some terms of this mathematic development to obtain that the system output follows the reference signal. The main advantage of this methodology is that depending on the approximation by Taylor formula considered, different control signal expressions are obtained. Here, the proposed methodology is applied to a multivariable non-linear system (considering the problem of tracking trajectory of a mobile robot). Experimental and simulation results for this case show the advantages of the proposed methodology. The main contribution of this article is that the proposed methodology is based Scaglia Gustavo Mut Vicente Rosales Andres Universidad Nacional de San Juan(U.N.S.J.),Argentina upon easily understandable concepts and there is no need for complex calculations to get the control si gnal. The paper is organized as follows: Section 2 states th problem and presents the methodology proposed to design the controller. Section 3 describes the appl ication of the strategy to a typical robotic system describ ed y a multi-variable nonlinear system as the mobile robot model is. Conclusions and suggested future work are detailed in Section 4. II. STATEMENT OF THE PROBLEM A non-linear kinematic model for a mobile robot will be used (see Fig. 1), which is represented by Campion (1996), . cos sin x V y V W θ θ θ  =  =    =  ɺ ɺ (1) where, V is the linear velocity of the mobile robot, W is the angular velocity of the mobile robot, ( , ) x y is the Cartesian position, θ is the orientation of the mobile robot, { } R is the inertial frame and { } c R is the frame attached to the robot. The values of ( ) ( ) ( ) ( ) , , , x t y t t V t θ and ( ) W t at discrete time t = nTo, where To is the sampling period, and { } 0,1,2,3, n∈ ⋯ will be denoted as , , , n n n n x y V θ y n W . Then, the aim is to find the values of n V and n W so that the mobile robot may follow a pre-established traje ctory ( ) ( ), ( ) xd t yd t . Fig. 1 Geometric description of the mobile robot. The trajectory followed by the mobile robot is described by parametric equations as shown in Eq.(2 ), ( ) ( ) ( ) x x t y y t t =   =   = θ θ (2) If the function ( ) y y t = can be derived until 1 m+ order, with neighborhood of t = nTo included, so the following Taylor’s Formula is valid: 2 3 ( ) ( ) ( ) ( ) ( )( ) ( ) 2! ( ) ( ) ( ) ( ) ( ) 3! m m t nTo y t y nTo y nTo t nTo y nTo t nTo t nTo y nTo y nTo Rm t m − = + − + + − − + + + + ɺ ɺɺ