STABILIZATION OF A UNICYCLE-TYPE MOBILE ROBOT USING HIGHER ORDER SLIDING MODE CONTROL J.P. Barbot * , M. Djema¨ ı * , T. Floquet ** and W. Perruquetti ** * Equipe Commande des Syst` emes (ECS), ENSEA, 6 Avenue du Ponceau, 95014 Cergy Cedex, France e-mail: (barbot,djemai)@ensea.fr ** LAIL UMR CNRS 8021, Ecole Centrale de Lille, BP 48, 59651 Villeneuve-d’Ascq, France e-mail: (Thierry.Floquet,Wilfrid.Perruquetti)@ec-lille.fr Keywords: Higher order sliding modes, nonholonomic sys- tems, robust practical stabilization. Abstract This paper deals with the problem of the practical stabiliza- tion of a unicycle-type mobile robot. The control strategy is divided into three steps and switches between different slid- ing mode controllers: a new third order sliding mode control with smooth manifolds that provides a practical stabilization and other sliding mode controls that perform finite time con- vergence (first order sliding mode and twisting algorithm). A simulation illustrates the results on the studied mobile robot. 1 Introduction One of the motivations for tackling the stabilization (or track- ing) of nonholonomic systems is the large number of applica- tions, such as mobile robots. Obstacles to the stabilization of nonholonomic systems are the uncontrollability of their linear approximation and the fact that the Brockett’s necessary con- dition to the existence of a smooth time-invariant state feed- back is not satisfied [3]. To overcome those difficulties, vari- ous methods have been investigated: homogeneous and time- varying feedbacks [18, 19], sinusoidal and polynomial controls [15], piecewise controls [10, 14], flatness [8] or backstepping approaches [11]. In the present paper, it is aimed to design a control law for a unicycle-type mobile robot which: • is a good compromise between performance and robust- ness, • solves the disturbance rejection problem for some bounded matching perturbations, • takes into account the actuator dynamics, • leads to a practical stabilization: the system is stabilized in a ball containing the origin whose radius may be chosen as small as desired. This objective will be achieved by switching between different sliding mode control laws. To this end, some smooth higher order sliding mode controllers will be introduced. 2 Problem statement In this paper, we particularly focus on nonholonomic systems whose trajectories can be written as the solutions of the driftless system: ˙ x = g 1 (x)u 1 + g 2 (x)u 2 + p(x) (1) where p(x) is a perturbation vector field (assumed to be smooth enough and thus bounded over some compact set). u 1 ,u 2 are the control inputs and the g ′ j s are smooth vector fields on R 3 that are linearly independent for all x ∈ R 3 . For instance, this is the case for the unicycle-type robot, which behavior can be described by the following system (see [4] for details):    ˙ x = cos(θ) u 1 + p 1 (x) ˙ y = sin(θ) u 1 + p 2 (x) ˙ θ = u 2 + p 3 (x) , (2) where x and y are the coordinates of the center gravity of the robot, θ is the orientation of the car with respect to the x-axis, p 1 (x), p 2 (x) and p 3 (x) are some additive perturbations and u 1 and u 2 refer respectively to the applied linear and the angular velocities (see Fig. 1). Figure 1: Unicycle robot kinematic Using the smooth state change of coordinates and input trans- formations given in [16] (that allow to transform some classes of nonholonomic systems in the so-called one chained form), it has been shown in [9] that the system (1) can be written into the perturbed one-chained form    ˙ z 1 = v 1 + p 1 (z) ˙ z 2 = v 2 + p 2 (z) ˙ z 3 = z 2 (v 1 + p 1 (z)) (3)