Equilibria, Saturation and Basin Selection in Fixed-Gain Sliding-Mode Contact Control

Abstract

A robot presses on an object whose stiffness it knows only approximately, and its actuator can push only so hard. A fixed-gain sliding-mode controller is usually trusted to cope with such an error as long as its switching gain exceeds the error force. We show that this trust has a sharp limit. With a finite actuator, a second, wrong resting place appears as soon as the stiffness error reaches the ratio of the switching gain to the actuator limit, long before the controller's own margin is used up. Which resting place the robot reaches depends on how fast it enters contact. We give closed-form conditions for when the wrong resting place exists, an exact dividing entry speed for linear contact and closed-form two-sided bounds for nonlinear contact, and the largest hit a robot at rest can absorb. A single fixed controller can serve a whole class of objects if it assumes the stiffness of the softest. A robot that approaches under the same controller always enters safely, so the danger lies in impacts and disturbances. When the exponent of the contact law is also wrong, the direction of the error, not its size, decides failure. The entry-speed bracket covers the sign and boundary-layer laws; the power-rate law is outside it. Simulations of the full saturated loop agree with every formula; convergence to each resting place is shown numerically, not proved. These closed forms were not found reported for this controller and plant.