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. An approximate 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 recursive expression for these gradients is developed for general serial chains. Constraints on end-effector motions are taken into account using the logarithm of the spatial displacement. 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.