The redundancy resolution schemes based on the optimization of an integral performance index are investigated from the topological point of view. The topological notions of self-motion manifold, C-path-homotopy and extended aspect are clarified in relation to the limitations of the necessary conditions of optimality provided by calculus of variations. On one hand, they do not guarantee the achievement of the optimal solution, and on the other hand, they translate into a two-point boundary value problem (TPBVP), whose resolution, under certain circumstances, may not lead to a feasible solution at all. In response to the limitations of calculus of variations, a dynamic-programming-inspired formalism is developed, which is based on the discretization of the state space and on its representation in the form of multiple grids. Building upon the topological analysis, effective algorithms are designed that are able to find the optimal solution in any condition, across all C-path homotopy classes and self-motion manifolds, with no limitation due to the passage through singularities. Moreover, if the grids are representative of the manipulator’s extended aspects, the topological notion of the transitional point can be used to reduce the computational complexity of the optimal redundancy resolution algorithm. The results are demonstrated on a canonical 4R planar robot in two different scenarios.
Optimal motion planning along prescribed paths can be solved with several techniques, but most of them do not take into account the wrenches exerted by the end-effector when in contact with the environment. When a dynamic model of the environment is not available, no consolidated methodology exists to consider the effect of the interaction. Regardless of the specific performance index to optimize, this article proposes a strategy to include external wrenches in the optimal planning algorithm, considering the task specifications. This procedure is instantiated for minimum-time trajectories and validated on a real robot performing an interaction task under admittance control. The results prove that the inclusion of end-effector wrenches affect the planned trajectory, in fact modifying the manipulator's dynamic capability.
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