Strict Lyapunov functions for finite-time control of robot manipulators

Abstract

Motivated by the energy shaping framework and the properties of homogeneous systems, we introduce a methodology to derive strict Lyapunov Functions (LFs) for a class of global finite-time (FT) controllers for robot manipulators. As usual in the energy shaping methodology, these controllers are described by the gradient of the controller potential energy plus the gradient of (nonlinear) energy dissipation-like functions. Sufficient conditions on the controller potential energy and energy dissipation-like functions are provided in order to obtain, in a straightforward manner, strict LFs ensuring global FT stability at the desired equilibrium. As an important practical outcome, we illustrate the proposed methodology by constructing strict LFs for some particular FT controllers. The construction of such strict LFs allows us to solve the open problem of FT trajectory tracking of robot manipulators with continuous control laws to track any bounded trajectory.

Keywords:

• Robot manipulators
• Lyapunov stability
• Finite-time control
• Finite-time convergence
• Energy shaping

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Since in the tracking case, in general, the resulting closed-loop is non-autonomous (due to the time dependence of the desired trajectory), the proofs of uniform global asymptotic stability and uniform finite-time stability are not correctly addressed in Su (2009). Our work includes, as an example, the controller proposed in Su (2009) with a formal proof.

2 Notice that ${\mathrm{\nabla }}_{\mathbf{q}}\mathcal{U}(\tilde{\mathbf{q}}+{\mathbf{q}}_d)={\mathrm{\nabla }}_{\tilde{\mathbf{q}}}\mathcal{U}(\tilde{\mathbf{q}}+{\mathbf{q}}_d)$.

3 In this regulation setting, the gradient of ${\mathrm{\nabla }}_{\dot{\mathbf{q}}}\mathcal{F}(\dot{\mathbf{q}})$ is the well-known damping injection term.

4 We apply the inequalities $\Vert \dot{\mathbf{q}}{\Vert }^{p_{\mathcal{F}}}\le \sum _{i=1}^n|{\dot{q}}_i{|}^{p_{\mathcal{F}}}$ and $\Vert \lceil \dot{\mathbf{q}}{\rfloor }^{p_{\mathcal{F}}}\Vert \le n^{\frac{1-p_{\mathcal{F}}}{2}}\Vert \dot{\mathbf{q}}{\Vert }^{p_{\mathcal{F}}}$, $\mathrm{\forall }{p}_{\mathcal{F}}\in (0,1]$. These inequalities have been obtained from the fact that $(\sum _{i=1}^n{x}_i{)}^{p_{\mathcal{F}}}\le \sum _{i=1}^n{x}_i^{p_{\mathcal{F}}}\le n^{1-p_{\mathcal{F}}}(\sum _{i=1}^n{x}_i{)}^{p_{\mathcal{F}}}$, for all $\mathbf{x}\in {\mathbb{R}}^n$ such that $x_1,\dots ,x_n\in {\mathbb{R}}_{\ge 0}$ (Bernstein, 2009).

5 The case when ${\beta }_{\mathcal{F}2}=0$, ${\kappa }_8({\beta }_j^{\prime })={\kappa }_5({\beta }_j^{\prime })$ corresponds to the case where in (6(6) $\mathcal{F}(\dot{\mathbf{q}}):=\frac{1}{p_{\mathcal{F}}+1}{\dot{\mathbf{q}}}^{\mathrm{\top }}\mathbf{D}\lceil \dot{\mathbf{q}}{\rfloor }^{p_{\mathcal{F}}}+\frac{1}{2}{\dot{\mathbf{q}}}^{\mathrm{\top }}{\mathbf{D}}_L\dot{\mathbf{q}},$(6) ) does not appear the quadratic term.

Additional information

Funding

This work has been partially supported by CONACyT basic scientific research projects CB-241171 and CB-282807; the SEP-PRODEP Apoyo a la incorporación NPTC project 511-6/18-9169 UDG-PTC-1400; the PAPIIT-UNAM project IN110719; and the Fondo de Colaboración II-FI UNAM project IISGBAS-100-2015.