Minimum effort motions for open chain manipulators with task-dependent end-effector constraints

In this article we examine the solution of minimum effort optimal control problems for open-chain manipulators. A local solution to the optimal control problem is determined by a constrained parameter optimization over a set of B-spline basis functions. We demonstrate that the parameter optimization formulation of the problem is numerically ill-conditioned and that it is therefore essential to include analytic, or exact, gradients of the objective function and the constraints in order to guarantee a solution. A main contribution of our approach is an explicit expression for these gradients for general serial chains. Our formulation relies on the use of matrix exponentials for the manipulator kinematics, dynamics, and task constraints. Several examples are presented that demonstrate the power and flexibility of our approach