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