Research: Robotics

Lyapunov-Based Control of a Robot and Mass-Spring System Undergoing An Impact Collision


Abstract

The control of a dynamic system with impact conditions is an interesting problem with practical importance. One difficulty in controlling systems subject to impact collisions is that the equations of motion are different when the system status changes suddenly from a non-contact state to a contact state. In this paper a nonlinear controller is designed to regulate the states of two dynamic systems that collide. The academic example of a planar robot colliding with an unactuated mass-spring system is used to represent a broader class of such systems. The control objective is to command a robot to collide with an unactuated mass-spring system and regulate the spring-mass to a desired compressed state. Lyapunov-based methods are used to develop a continuous controller that yields global asymptotic regulation of the spring-mass and robot links. Unlike some other results in literature, the developed continuous force controller does not depend on sensing the impact, measuring the impact force, or the measurement of other acceleration terms.

Experimental Results

The testbed depicted below was developed for experimental demonstration of the proposed controller. The testbed is composed of a mass-spring system and a two-link robot. The body of the mass-spring system includes a U-shaped aluminum plate (item (8) in the figure) mounted on an undercarriage with porous carbon air bearings which enables the undercarriage to glide on an air cushion over a glass covered aluminum rail. A steel core spring (item (1) in the figure) connects the undercarriage to an aluminum frame, and a linear variable displacement transducer (LVDT) (item (2) in the figure) is used to measure the position of the undercarriage assembly. The impact surface consists of an aluminum plate connected to the undercarriage assembly through a stiff spring mechanism (item (7) in the figure). A Precision capacitance probe (item (3) in the figure) is used to measure the deflection of the hard spring system. The two-link robot (items (4-6) in the figure) is made of two aluminum links, mounted on 240.0 Nm (base link) and 20.0 Nm (second link) direct-drive switched reluctance motors. The motors are controlled through power electronics operating in torque control mode. The motor resolvers provide rotor position measurements with a resolution of 614400 pulses/revolution. A Pentium 2.8 GHz PC operating under QNX hosts the control algorithm, which was implemented via Qmotor 3.0, a graphical user-interface, to facilitate real-time graphing, data logging, and the ability to adjust control gains without recompiling the program. Data acquisition and control implementation were performed at a frequency of 2.0 kHz using the ServoToGo I/O board.



The tip of the second link of the robot was initially 217 mm from the desired setpoint and 187 mm from impact along the X1-axis (see above figure). Therefore, once the the initial impact occurs, the robot is required to depress the spring (item (1) in the above figure) to move the mass 30 mm along the X1-axis.

The mass-spring and robot errors are shown in the following plot. The peak steady-state position error of the robot tip along the X1 and X2-axes (plot (a) and plot (b)) are 9.6 µm and 92 µm, respectively. The peak steady-state position error (plot (c)) of the spring-mass is 7.7 µm. The 92 µm is due to the lack of the ability of the model to capture friction and slipping effects on the contact surface. In this experiment, a significant friction effect is present along the X2-axis between the robot tip and the contact surface due to a large normal spring force that is applied along the X1-axis. Future efforts will target adding additional dynamic effects during the impact between the contact surfaces.

A movie of the experiment is provided below.



Publications

K. Dupree, W. E. Dixon, G. Q. Hu, and C. Liang, “Lyapunov-Based Control of a Robot and Mass-Spring System Undergoing An Impact Collision,” IEEE American Control Conference, Minneapolis, Minnesota, 2006, pp. 3241-3246.