We are proposing a general framework that incorporates multiple task definitions in the prioritized inverse kinematics problem. First, a mathematical description of multiple task definitions is constructed that provides an efficient way to show unprioritized or prioritized accumulations of tasks. Then, smooth transitions between all task definitions are studied, so a method, called task transition control, is developed that interpolates joint trajectories using barycentric coordinates and linear dynamical systems to overcome difficulties of interpolating task trajectories in the conventional methods. Consequently, smooth, arbitrary, and consecutive task transitions are achieved in a simple, direct, and general manner and also boundedness of joint trajectories is assured regardless of singularity. Lastly, the idea is tested by two kinematic simulations: obstacle avoidance with the KUKA LWR and task scheduling of a humanoid robot.
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 degrees of freedom manipulator, KUKA-LWR.
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