Semi-globally Exponential Trajectory Tracking for a Class of Spherical Robots

TL;DR

This paper presents a geometric PID control method for spherical robots that achieves semi-global exponential trajectory tracking with robustness to uncertainties and disturbances, under specific actuation assumptions.

Contribution

It introduces a controller design for a class of spherical robots that guarantees robust, semi-global exponential tracking despite uncertainties and disturbances.

Findings

The controller ensures asymptotic trajectory tracking of the sphere's geometric center.

Robustness to bounded, constant uncertainties and disturbances is achieved.

The control scheme guarantees bounded actuator velocities and stability.

Abstract

A spherical robot consists of an externally spherical rigid body rolling on a two-dimensional surface, actuated by an auxiliary mechanism. For a class of actuation mechanisms, we derive a controller for the geometric center of the sphere to asymptotically track any sufficiently smooth reference trajectory, with robustness to bounded, constant uncertainties in the inertial properties of the sphere and actuation mechanism, and to constant disturbance forces including, for example, from constant inclination of the rolling surface. The sphere and actuator are modeled as distinct systems, coupled by reaction forces. It is assumed that the actuator can provide three independent control torques, and that the actuator center of mass remains at a constant distance from the geometric center of the sphere. We show that a necessary and sufficient condition for such a controller to exist is that for any constant disturbance torque acting on the sphere there is a constant input such that the sphere and the actuator mechanism has a stable relative equilibrium. A geometric PID controller guarantees robust, semi-global, locally exponential stability for the position tracking error of the geometric center of the sphere, while ensuring that actuator velocities are bounded.