The number of saturated actuators and constraint forces during time-optimal movement of a general robotic system

The authors formulate the time-optimal control problem for general robotic systems and show that the required maximum (or minimum) value of the path acceleration is the solution of a linear programming problem. The fact that such a solution is an extreme point of the set of feasible solutions makes it possible to determine the minimum number of actuators and internal forces that must be saturated during the time-optimal movement. Specifically, it is proved that, if the dynamics of a general robot system are defined by n coordinates, m differential constraint equations, and p actuators, then some combination of at least L=m+p+1-n of the actuators and internal constraint forces is saturated during a time-optimal movement of the system along a prescribed path. The result applies to general class of dynamic systems with both holonomic and non-holonomic constraints