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On Time-Optimal Control of Elastic Joints under Input Constraints

We highlight the equivalence between the motion of an elastic joint and the two-body problem in classical mechanics. Based on this observation, a change of coordinates is introduced that reduces the two-body problem to a pair of decoupled one-body problems. This allows to treat the rest-to-rest motion problem with bounded actuator torque in an elegant geometric fashion. Instead of dealing directly with the fourth-order dynamics, we consider two equivalent masses whose motions have to be synchronized in separate phase spaces. Based on this idea, we derive a complete synthesis method for time-optimal rest-to-rest motions of this elastic system. The solution is a bang-bang control policy with one or three switches. We also introduce the concept of natural motions, when the minimum-time solution for the elastic and the rigid system is the same. The closed-form solutions obtained with our purely geometric approach verify the standard optimality conditions.