On the hybrid stability of the collocated virtual holonomic constraints based walking design
This paper presents proof of the hybrid stability of the pre-designed walking like trajectory and its feedback tracking controller for the so-called four-link. The four-link is a planar mechanical chain having four degrees of freedom and three actuators placed between its links. In such a way it resembles a pair of legs with knees. The proof of hybrid stability is based on computing the appropriate Poincare map linear approximation and showing numerically that its eigenvalues are inside the unit disk in the complex plane. Unlike the frequent approach in robotic walking showing the stable path following, nature of our designed trajectory enables to prove its tracking including the time dependence. The tested trajectory and the feedback controller were obtained via combination of the design for the so-called Acrobot and suitable selected collocated holonomic constraints enforced by feedback imposed in knees actuators. This approach was published before but it is briefly repeated here for the sake of completeness. Finally, the simulations showing the hybrid stability of 150 steps walking of the four-link with lengths and masses configurations corresponding to an existing laboratory model are presented.
CYBERNETICS AND PHYSICS, Vol. 6, No. 2, 2017, 47-56.