Finite time tracking of a fully actuated biped robot with pre-specified settling time: A second order sliding mode synthesis

Abstract

Abstract-A second order sliding mode controller is utilised to track reference trajectories for all the joints of a fully actuated biped robot. The existing tuning rules for the 'twisting' controller are used to guarantee a priori attainment of a prescribed settling time between two successive impacts. The joint torque is modeled as the control input. Smoothing of the discontinuous controller is carried out by introducing a high gain linear controller inside a boundary layer defined by an arbitrarily small region around the origin thereby avoiding numerical errors in the simulations. The overall accuracy of motion control is dictated by the size of this layer leading to practical stability of the closed-loop system. The main contribution of the paper is to provide straightforward and realizable engineering guidelines for the reference trajectory generation and for the tuning of a robust finite time controller for achieving stable gait of a biped in the presence of disturbances in both continuous and impact phases. Numerical simulations of a biped robot are shown to support the theoretical results.

Modeling and Finite-Time Walking Control of a Biped Robot with Feet

“…In this section, we present numerical results from the solution of the bipedal walking model. In particular, we show the solutions for hybrid system (16) under the linearizing control law u defined in Theorem 10. We evaluate three different approaches to achieve finite-time convergence as stated in Theorem 11 from the auxiliary control k. The evaluated approaches are the discontinuous and continuous twisting controls of Sections 4.2.1 and 4.2.2, respectively, and the continuous integral twisting control of Section 4.2.3.…”

“…In this section, we present the design of a control law of the form u = (x) ∈ R 6 in order to yield a stable solution of system (16). At the same time, the solution must correspond to stable walking and must be consistent with hypotheses (HW1), .…”

“…Proposition 13. For the continuous part of hybrid system (16), assume that the control input u and the auxiliary control v are defined in (43) and (54), respectively. Then, for each x 0 ∈ X there exists 0 < * < ∞ such that the solution (x 0 ) of (3) satisfies * (x 0 ) ∈ S.…”

“…Thus, an adequate tuning of control gains must be carried out to ensure that the settling time of the variable 5 is lower than the settling times of 2 , 3 , 4 , and 6 , and in turn, the settling times of these variables are lower than the settling time of the horizontal position of the swing foot 1 . To achieve these conditions, we use the method proposed in [16,17] as an initial approximation. However, other important aspects must be also considered in the tuning process, for instance, reducing overshooting and oscillation of the angular positions and verifying the condition for the CoP (Proposition 4).…”

“…Second-order sliding modes can be achieved by using a discontinuous [13] or a continuous [14] control law. Both approaches have been proposed for the walking control of biped robots [16,17]; however, a comparison and robustness evaluation of both approaches have not been carried out.…”