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Today's complex robotic designs comprise in some cases a large number ofdegrees of freedom, enabling for multi-objective task resolution (e.g.,humanoid robots or aerial manipulators). This paper tackles the stabilityproblem of a hierarchical losed-loop inverse kinematics algorithm for suchhighly redundant robots. We present a method to guarantee system stability byperforming an online tuning of the closedloop control gains. We define asemi-definite programming problem (SDP) with these gains as decision variablesand a discrete-time Lyapunov stability condition as a linear matrix inequality,constraining the SDP optimization problem and guaranteeing the stability of theprioritized tasks. To the best of authors' knowledge, this work represents thefirst mathematical development of an SDP formulation that introduces stabilityconditions for a multi-objective closed-loop inverse kinematic problem forhighly redundant robots. The validity of the proposed approach is demonstratedthrough simulation case studies, including didactic examples and a Matlabtoolbox for the benefit of the community.

Multi-task closed-loop inverse kinematics stability through semidefinite programming