A simplified criterion for repeatability and its application in constraint path planning problems

Compared to typical industrial manipulators, redundant manipulators offer more dexterity and flexibility because of their ability to reach points in their workspace in many different ways, e.g. they can perform self-motions which do not change the position of the TCP. As a consequence a given path in workspace x(t), t₀<t<t₁ can be tracked by different paths q(t) in jointspace. When some additional objective function h(q) for avoidance of collisions or singularities has to be optimized simultaneously there exists one path q_opt(t) which is optimal with respect to h(q). In contrast to other work, the search for q_opt(t) is not done in jointspace directly, but in a space of lower dimension. Therefore, a map q(t,s) is constructed numerically in such a way that f(q(t,s))=x(t), i.e. varying s results in self-motions. Then, the optimal path q(t) can be searched in the (t,s) space whose dimension is less than the dimension of the jointspace and therefore the complexity of the search is reduced significantly. Furthermore, a novel and simplified criterion for checking repeatability is introduced which is strongly related to the proposed algorithm