Discrete finite-time robust fault-tolerant high-order sliding mode control of uncertain quadruped robot: an experimental assessment

Abstract

This study addresses a novel discrete finite-time fault compensation method for a quadruped robot leg when the parameter uncertainties and the actuator faults affect the robot dynamics. The occurrence times, shapes, and pattern of the faults are completely unknown. The actuator faults and dynamic uncertainties are reconstructed precisely in finite-time based on discrete-time super twisting estimator. The lumped fault estimator has simple structure. Defining a hybrid high-order sliding surface, a constructive fault-tolerant tracking controller is achieved. The proposed controllers guarantee robustness against uncertainties associated with the robotic manipulator and all type of actuator faults. It is guaranteed that all signals in the closed-loop system are finite-time stable. Finally, experiments are performed on one leg of a quadruped robot to evaluate the effectiveness and feasibility of the proposed control scheme.

Appendix: details of leg dynamics

The structure and details of matrixes $M$, $C$, and $G$ presented in Sect. 2 are as:

$$M(q)=\left[\begin{array}{ccc}{M}_{11}& {M}_{12}& {M}_{13}\\ {M}_{21}& {M}_{22}& {M}_{23}\\ {M}_{31}& {M}_{32}& {M}_{33}\end{array}\right], \, C(q,\dot{q})=\left[\begin{array}{ccc}{C}_{11}& {C}_{12}& {C}_{13}\\ {C}_{21}& {C}_{22}& {C}_{23}\\ {C}_{31}& {C}_{32}& 0\end{array}\right], \, G(q)=\left[\begin{array}{l}{G}_{1}\\ {G}_{2}\\ {G}_{3}\end{array}\right]$$

where:${M}_{11}=\left({m}_{2}+{m}_{3}\right){d}^{2}+\left({m}_{2}/4+{m}_{3}\right){L}_{2}^{2}\mathrm{cos}\left({q}_{2}^{2}\right)+{m}_{2}{d}^{2}\mathrm{sin}\left({q}_{2}^{2}\right)-{m}_{2}d{L}_{2}\mathrm{sin}\left(2{q}_{2}\right)/2+{m}_{3}{L}_{3}\mathrm{cos}\left({q}_{2}+{q}_{3}\right)\left({L}_{3}/4+{L}_{2}\mathrm{cos}\left({q}_{2}\right)\right)+{n}_{r1}^{2}{I}_{m}$, ${M}_{12}=d\left({m}_{3}{L}_{3}\mathrm{sin}\left({q}_{2}+{q}_{3}\right)+{m}_{2}{L}_{2}\mathrm{sin}\left({q}_{2}\right)\right)+2{m}_{3}{L}_{2}\mathrm{sin}\left({q}_{2}\right)+2d{m}_{3}\mathrm{cos}\left({q}_{2}\right)/2$, ${M}_{13}=d{m}_{3}{L}_{3}\mathrm{sin}\left({q}_{2}+{q}_{3}\right)/2$, ${M}_{21}={M}_{12}$, ${M}_{22}={m}_{2}{L}_{2}^{2}/4+{m}_{3}({L}_{2}^{2}+{L}_{3}^{2}/4)+{m}_{2}{d}^{2}+{m}_{3}{L}_{2}{L}_{3}\mathrm{cos}\left({q}_{3}\right)++{n}_{r2}^{2}{I}_{m}$, ${M}_{23}={m}_{3}{L}_{3}\left({L}_{3}+2{L}_{2}\mathrm{cos}\left({q}_{3}\right)\right)/4+{n}_{r3}^{2}{I}_{m}/2$, ${M}_{31}={M}_{13}$, ${M}_{32}={M}_{23}$, ${M}_{33}={m}_{3}{L}_{3}^{2}/4+{n}_{r3}^{2}{I}_{m}$,${C}_{11}={m}_{2}d\left(d\mathrm{sin}\left(2{q}_{2}\right)-{L}_{2}\mathrm{cos}\left(2{q}_{2}\right)\right){\dot{q}}_{2}/2-{m}_{3}{L}_{3}^{2}\mathrm{sin}\left(2\left({q}_{2}+{q}_{3}\right)\right)({\dot{q}}_{2}+{\dot{q}}_{3})/8-({m}_{2}+4{m}_{3}){L}_{2}^{2}\mathrm{sin}\left(2{q}_{2}\right){\dot{q}}_{2}/8-{m}_{3}{L}_{2}{L}_{3}\mathrm{sin}\left({q}_{3}\right){\dot{q}}_{3}/4-{m}_{3}{L}_{2}{L}_{3}\mathrm{sin}\left(2{q}_{2}+{q}_{3}\right)(2{\dot{q}}_{2}+{\dot{q}}_{3})/4$,${C}_{12}={m}_{2}\left(\mathrm{sin}\left(2{q}_{2}\right)\left({d}^{2}-{L}_{2}^{2}/4\right)-d{L}_{2}\mathrm{cos}\left(2{q}_{2}\right)\right){\dot{q}}_{1}/2-{m}_{3}{L}_{3}\left({L}_{3}\mathrm{sin}\left(2\left({q}_{2}+{q}_{3}\right)\right)/8+{L}_{2}\mathrm{sin}\left(2{q}_{2}+{q}_{3}\right)/2\right){\dot{q}}_{1}-{m}_{3}{L}_{2}^{2}\mathrm{sin}\left(2{q}_{2}\right){\dot{q}}_{1}/2-d\left(d{m}_{2}\mathrm{sin}\left({q}_{2}\right)-{L}_{2}\mathrm{cos}\left({q}_{2}\right)({m}_{2}/2+{m}_{3})\right){\dot{q}}_{2}+d{m}_{3}{L}_{3}\mathrm{cos}\left({q}_{2}+{q}_{3}\right)\left({\dot{q}}_{2}+{\dot{q}}_{3}\right)/2$,${C}_{13}=d{m}_{3}{L}_{3}\mathrm{cos}\left({q}_{2}+{q}_{3}\right){\dot{q}}_{2}/2-{m}_{3}{L}_{3}\mathrm{sin}\left({q}_{2}+{q}_{3}\right)\left({L}_{3}\mathrm{cos}\left({q}_{2}+{q}_{3}\right)+{L}_{2}\mathrm{cos}\left({q}_{2}\right)\right){\dot{q}}_{1}/2+d{m}_{3}{L}_{3}\mathrm{cos}\left({q}_{2}+{q}_{3}\right){\dot{q}}_{3}/2$, ${C}_{21}={L}_{2}^{2}\mathrm{sin}\left(2{q}_{2}\right)\left({m}_{2}/4+{m}_{3}\right){\dot{q}}_{1}+\left({m}_{3}{L}_{3}^{2}\mathrm{sin}\left(2\left({q}_{2}+{q}_{3}\right)\right)/4-{m}_{2}{d}^{2}\mathrm{sin}\left(2{q}_{2}\right)\right){\dot{q}}_{2}+{L}_{2}\left({m}_{3}{L}_{3}\mathrm{sin}\left(2{q}_{2}+{q}_{3}\right)+d{m}_{2}\mathrm{cos}\left(2{q}_{2}\right)/2\right){\dot{q}}_{3}$, ${C}_{22}=-{m}_{3}{L}_{2}{L}_{3}\mathrm{sin}\left({q}_{3}\right){\dot{q}}_{3}/2$, ${C}_{23}=-{m}_{3}{L}_{2}{L}_{3}\mathrm{sin}\left({q}_{3}\right)({\dot{q}}_{2}+{\dot{q}}_{3})/2$, ${C}_{31}={m}_{3}{L}_{3}\mathrm{sin}\left({q}_{2}+{q}_{3}\right)\left({L}_{3}\mathrm{cos}\left({q}_{2}+{q}_{3}\right)+2{L}_{2}\mathrm{cos}\left({q}_{2}\right)\right){\dot{q}}_{1}/4$, ${C}_{32}={m}_{3}{L}_{2}{L}_{3}\mathrm{sin}\left({q}_{3}\right){\dot{q}}_{2}/2$,${G}_{1}=g{m}_{2}{L}_{2}\mathrm{sin}\left({q}_{1}\right)\mathrm{cos}\left({q}_{2}\right)/2-d\mathrm{sin}\left({q}_{2}\right)+d\mathrm{cos}\left({q}_{1}\right)+{m}_{3}{L}_{3}\mathrm{sin}\left({q}_{1}\right)\mathrm{cos}\left({q}_{2}+{q}_{3}\right)/2+{L}_{2}\mathrm{cos}\left({q}_{2}\right)+d\mathrm{cos}\left({q}_{1}\right)$, ${G}_{2}=g\mathrm{cos}\left({q}_{1}\right)\left({m}_{3}{L}_{3}\mathrm{sin}\left({q}_{2}+{q}_{3}\right)+{m}_{2}{L}_{2}\mathrm{sin}\left({q}_{2}\right)+2{m}_{3}{L}_{2}\mathrm{sin}\left({q}_{2}\right)+2d{m}_{2}\mathrm{cos}\left({q}_{2}\right)\right)/2$, ${G}_{3}=g{m}_{3}{L}_{3}\mathrm{sin}\left({q}_{2}+{q}_{3}\right)\mathrm{cos}\left({q}_{1}\right)/2$.