The Minimum Energy Movement for a Spring Muscle Model

There are many ways of programming an actuator or effector for movement between the same two points. In the interest of efficiency it is sometimes desirable to program that trajectory which requires the least amount of energy. This paper considers the minimum energy movement for a spring-like actuator abstracted from muscle mechanics and energetics. It is proved that for this actuator a bang-coast-bang actuation pattern minimizes the energy expenditure. For some parameter values this pattern is modified by a singular arc at the first switching point. A surprising limitation on the duration of coast is demonstrated. Some relaxations of the restrictions underlying the spring model are shown to preserve the bang-coast-bang solution