Energy-optimal trajectory planning for robot manipulators with holonomic constraints

In this paper, we study two different trajectory planning problems for robot manipulators. In the first case, the end-effector of the robot is constrained to move along a prescribed path in the workspace, whereas in the second case, the trajectory of the end-effector has to be determined in the presence of obstacles. Constraints of this type are called holonomic constraints. Both problems have been solved as optimal control problems. Given the dynamic model of the robot manipulator, the initial state of the system, some specifications about the final state and a set of holonomic constraints, one has to find the trajectory and the actuator torques that minimize the energy consumption during the motion. The presence of holonomic constraints makes the optimal control problem particularly difficult to solve.Our method involves a numerical resolution of a reformulation of the constrained optimal control problem into an unconstrained calculus of variations problem in which the state space constraints and the dynamic equations, also regarded as constraints, are treated by means of special derivative multipliers. We solve the resulting calculus of variations problem using a numerical approach based on the Euler–Lagrange necessary condition in the integral form in which time is discretized and admissible variations for each variable are approximated using a linear combination of piecewise continuous basis functions of time. The use of the Euler–Lagrange necessary condition in integral form avoids the need for numerical corner conditions and the necessity of patching together solutions between corners. In this way, a general method for the solution of constrained optimal control problems is obtained in which holonomic constraints can be easily treated. Numerical results of the application of this method to trajectory planning of planar horizontal robot manipulators with two revolute joints are reported.