Time-Optimal Regulation of a Chain of Integrators with Saturated Input and Internal Variables: An Application to Trajectory Planning

The design of a planner for time-optimal trajectories with constraints on velocity, acceleration, jerk, . . . , is translated into a regulation problem for a chain of integrators with saturations not only in the input but also in all the internal (state) variables. Then the problem is solved by designing a regulator, based on the variable structure control, able to steer the state vector to the origin in minimum time, being compliant with all the constraints. For this purpose, a modular structure with a cascade of controllers, each one devoted to the regulation to the origin of a specific component of the state vector, is demonstrated to be effective and ideally suitable to cope with systems of any order. Analytical examples are provided for filters of first, second and third order.