Mathematical models of binary spherical-motion encoders

This paper presents several algorithms that solve the problem of determining the orientation of a freely rotating ball that is partially enclosed in a housing. The ball is painted in two colors (black and white) and the housing has a number of sensors that detect these colors. The question which we attempt to answer is: knowing how the ball is painted, knowing the location of the sensors, and given a complete set of sensor measurements, how does one determine the orientation of the ball to within an acceptable error threshold? The algorithms we present to solve this problem are based on methods and terminology from geometric control theory. Essentially, we generate dynamical systems that evolve on the group SO(3). These dynamical systems are constructed so as to attract the computed orientation of the ball to the actual one being detected by the sensors. Solving this spherical decoding problem is important in applications where the spherical motion must be detected. One such application is the feedback control of spherical motors.