Analytical design of the Acrobot exponential tracking with application to its walking

This paper aims to further improve previously developed design for the Acrobot walking based on the partial exact feedback linearization of order 3. Namely, such an exact system transformation leads to an almost linear system where error dynamics along trajectory to be tracked is a 4 dimensional linear time varying system having 3 time varying entries only, the remaining entries are either zero or equal to one. Previously, the exponentially stable tracking was obtained by solving quadratic stability of a linear system with polytopic uncertainty applying LMI methods to solve this problem numerically. Here, the new approach is presented allowing to design the tracking feedback and to prove the corresponding stability completely analytically. Moreover, this approach gives even better results than the LMI based one in the sense of the convergence speed. The key idea of the novel approach presented here is that it manages to use part of the information about the mentioned time dependent entries, thereby reducing overly conservativeness of the previous LMI based design. Numerical simulations of the Acrobot walking based on the above mentioned new analytical design are demonstrated as well.