Conference Paper (international conference)
serial: Proceedings of the 2011 IEEE International Symposium on Computer-Aided Control System Design (CACSD)
action: CACSD 2011, (Denver Colorado, US, 28.09.2011-30.09.2011)
project(s): GAP103/10/0628, GA ČR, LA09026, GA MŠk
keywords: stability, linear system, tracking
This paper aims to the further improve of the 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. Unlike previous approaches treating time varying entries as uncertainties with various extent of conservatism, the present paper takes into the account an information about these time varying functions including their derivatives up to order 4. Using that, the time varying state and the feedback transformation enable to design a fundamental matrix of the error dynamics in an explicit form and pre-designed stability properties. In particular, product of that fundamental matrix at the end of the single support walking phase by the impact map Jacobian enables directly prove stability of the hybrid cyclic walking like trajectory by computing certain 4x4 matrix and determining numerically whether its eigenvalues lie within the unit circle. This combination of analytical and numerical computations provides the justification of the exponential stability of the walking-like trajectory tracking. Moreover, it is supported by numerical simulations showing practically unlimited number of steps of the Acrobot ”walking”.