High Precision Point-To-Point Maneuvering of an Experimental Structure Subject To Friction Via Adaptive Impulsive Control. Rajaey Kased and Tarunraj Singh kased@eng.buffalo.edu tsingh@eng.buffalo.edu State University of New York at Buffalo, Buffalo, NY 14260 Abstract—Two adaptive impulsive control techniques de- signed to eliminate steady state error for a rigid body system subject to friction, and undergoing a point-to-point maneuver are implemented. Pulse width and pulse amplitude pulse width modulations are explored. It is shown that the results of the pulse amplitude modulation never generates limit cycles and has lower steady-state error than the pulse width modulation. I. I NTRODUCTION The significance of friction to the control community is in its effects on positioning systems and velocity tracking operation. Positioning apparatuses include telescopes, anten- nas, machine tools, disk drives and robot arm positioning. Velocity control is also relevant in machine tool, disk drive and robot arm industrial applications which require the accurate tracking of a pre-determined trajectory. The effect of friction becomes magnified in the low velocity region near the reference position. The majority of work done on control of frictional systems is on rigid body systems. Yang and Tomizuka [1] exploited a simple relationship between a pulse input width and the displacement of the rigid body. This utilizes the fact that the rigid body subject to a pulse input never changes the direction of the velocity and thus the Coulomb friction acts like a bias to the input. This scheme, known as Pulse Width Control (PWC), is presented in an adaptive control setting where an estimate of the friction is provided. Wijdeven and Singh [2], modified the PWC approach to increase accuracy in actual discrete implementation of the input. Their technique modulates the pulse height to compensate for a rounded up pulse width and is called Pulse Amplitude Pulse Width Control (PAPWC). Additional schemes developed for rigid body systems include internal-model following error control [3], PID and state feedback linearization control [4] and variable structure control in order to try to handle qualitatively different fric- tion regimes [5], [6]. Nonlinear PID control has also been developed to overcome the stick-slip behavior of friction [7]. In this paper two techniques, the Adaptive Pulse Width Control (PWC) and the Adaptive Pulse Amplitude Pulse Width Control (PAPWC) proposed by [1] and [2] respec- tively, are presented for the rigid body system, subject to stiction and Coulomb friction. These techniques are imple- mented experimentally on the setup presented in section III- A. Experiments and simulations are presented for qualitative comparison only. This is due to the fact that the actual fric- tion in the system is unknown and an assumed actual value must be used for the simulation. Generally, the assumed value used for the simulation is close to the final estimation of friction obtained during the corresponding experiment run. This accounts for some quantitative comparison obtained from the simulation. II. MATHEMATICAL FORMULATION The equation of motion of the rigid body in consideration subject to a positive pulse input, is given as: ¨ θ =          1 J 1 ( u H − f c − c 1 ˙ θ ) if ˙ θ =0 0 if ˙ θ =0& |u H | <f s 1 J 1 ( u H − sgn(u H )f s ) if ˙ θ =0& |u H | >f s (1) The input to the system u H can be expressed as: u H = f pm H(t − T 1 ) − f pm H(t − T 2 ), where f pm is the pulse height magnitude and H(t − T ) is the Heavy-side function which is equal to one for time greater than T . The Coulomb friction only appears as a bias force since the direction of the rigid body will never change during a single pulse. This eliminates the sgn  ˙ θ  term in the con- ventional Coulomb friction model permitting linear analysis. The total distance travelled by the rigid body due to u H can be found by solving equation (1). Ignoring damping, the distance travelled is given as [1]: θ(t end )= ± f pm (f pm − f c ) t 2 p 2J 1 f c (2) Equation (2) is the basis of the development of the control schemes presented in this paper. A. Pulse Width Control Formulation (PWC) The essential idea of the Pulse Width Control (PWC), is to provide a single pulse to the system near the reference point, such that the total energy (from inertia, damping, control, and friction) is zero at the reference point. This exploits the ability of the friction force to slow down the system with no added control effort (coasting).