Prioritized Inverse Kinematics: Generalization

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This letter proposes a generalization of the prioritized inverse kinematics (PIK) problem as the multi-objective optimization with the lexicographical ordering. We specify three properties for a vector-valued objective function to be proper for the PIK problem, so that the set of all PIK solutions can be generated from the set of all proper objective functions. The dependence property requires that higher priority tasks do not constrain lower priority tasks unnecessarily, the uniqueness property demands that there exists one and only one PIK solution given a proper objective function, and the representation property asks that if there are two distinct references that are realizable by a mechanism, then the PIK solutions for those references should differ. We also include the preconditioning of the velocity mapping functions in our generalization that can be used to handle the numerical imbalance in the orthogonalization that raises a difficulty in choosing damping functions and degenerates the performance of lower priority tasks. We justify our generalization by showing that it discards trivial solutions such as a constant function that is not intended and contains several PIK solutions including two successful PIK solutions, so-called Nakamura's and (weighted) Chiaverini's solutions, with and without damping. We compare those PIK solutions by a simulation with a seven-degree-of-freedom manipulator, KUKA-LWR.