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.

In this paper we compute approximate minimum effort motions for open chains by solving a finite dimensional minimization problem using B-splines for the path representation. We begin by deriving the gradient of the recursive Newton-Euler dynamics formulation with respect to the parameters of a curve defining the motion, where the kinematics are expressed via the product of matrix exponentials. We then use this gradient to solve joint torque minimization problems for spatial open chains. The advantage of our approach is that the gradients are computed for each local frame recursively, therefore sharing the same efficiency advantages as the original recursive algorithm, and also can be computed for any spatial open chain. The optimization approach is applied to several open chains where minimum effort paths are found.

The equation AX=XB on the Euclidean group arises in the problem of calibrating wrist-mounted robotic sensors. In this article the authors derive, using methods of Lie theory, a closed-form exact solution that can be visualized geometrically, and a closed-form least squares solution when A and B are measured in the presence of noise.>