Sliding Mode and Adaptive Control for an Underactuated Process E. Lucet * , Y. Liu * , N. Mechbal * and M. Vergé * * ENSAM/LMSP, Paris, France Abstract— In this paper we design two controllers: a sliding mode controller and a new designed adaptive controller for an under actuated process with important dry friction. The two controllers are compared. Simulations and experiments are performed to evaluate the efficiency of the designed controllers. I. INTRODUCTION Many kinds of mechanical systems used in the industrial world are difficult to control because of the non linear model or because of the presence of the dry friction. Many friction models [1, 7] try to represent the complexity of physic phenomena. However, friction is still one of the great unknowns in mechanical systems. Besides, dry friction model is not linear [1, 2] and control of such system is quite difficult. In this work, two controllers are designed. First a sliding mode controller with varying parameter is presented. In this controller, dry friction model is not needed. Second, a multi loop adaptive controller using dry friction model is developed. Both controllers are simulated, implemented and compared. A test bed (called MACHA) was build to analyze and to test several controllers for such a mechanical system. It consists of a cart, which is moving by the gravity on a guide rail along a sloping beam. The slope of this beam is controlled by two brushless motors with their own servo drivers (Parvex Inc.), which drive a belt mechanism mounted on the right hand side of the system. Here this system is configured with one input and one output, but the important phenomenon is the dry friction of the cart because the slope of the beam is very weak. This paper is organized as follows. In the second part, the plant is described and the dynamical model is given. In the third part, a non-linear sliding mode controller is designed, following the method presented in [4]. In the fourth part, an adaptive controller designed for two outputs, following the theory of [4]. A new controller is designed and is brought into application. In the fifth part simulations are presented and in the sixth part, experimental results are obtained by applying these methods to the actual process. In the conclusion, the differences between two controllers are observed. II. DYNAMICAL MODEL A non-linear dynamic model of the 2-link machine (see fig. 1) is established using notations of fig. 2. The linear position of the cart on the beam is the underactuated joint and the angular position ) t ( x ) t ( θ of the beam is the actuated joint as it is shown in figure 2 where plan X0, Y0 is vertical. Here the bell elasticity is neglected and the motor is controlled thanks to torque reference so that the input is proportional to F 2 . Two magnetostrictive sensors are used to give cart position and B position, thus beam slope ) t ( θ is known. Figure 1. Process MACHA Figure 2. Notations TABLE 1. Some values Description Value Distance between the cart centre of mass G5 and the beam. h 0.10 m Distance between A and B. d 1.44 m Acceleration of gravity. g 9.81m/s² Overall length of the tilted beam. L 2.20 m Distance between point A and the beam centre of mass: G3. L1 0.72 m Mass of the beam. M3 18.00 kg Mass of the cart. M5 2.86 kg Inertia of the beam. J 29.04 kg.m² B F2 θ G 3 G 5 L1 L x h A d Y0 X0 Y0 3URFHHGLQJV RI WKH WK 0HGLWHUUDQHDQ &RQIHUHQFH RQ &RQWURO  $XWRPDWLRQ -XO\     $WKHQV  *UHHFH 7