Time-optimal transfer in the plate-ball problem

This paper attempts to provide a tutorial mode presentation of a time-optimal transfer problem in the geometric framework. The control objective is to change both the orientation and position of a ball sandwiched between two plates in minimum time. The lower plate is fixed and the upper plate can be actuated in two lateral directions. In the geometric framework, the problem of time-optimal reorientation and translation translates to a state transfer on the Lie group - ℝ² × SO(3). The energy-optimal control version of the plate-ball system was initially proposed and solved by Jurdjevic in [1]. Subsequent to this, attempts have been made on the time-optimal version, but in the authors´ opinion, complete insight into the problem is still missing. The paper here focusses on the reorientation part alone with the hope that this analysis extends easily to the part including translation.