Robust control of reaction wheel bicycle robot via adaptive integral terminal sliding mode

Abstract

In this paper, the modelling of a reaction wheel bicycle robot (RWBR) is identified from a second-order mathematical model which is similar to an inverted pendulum, and an adaptive integral terminal sliding mode (AITSM) control scheme is developed for balancing purpose of the RWBR. The proposed AITSM control scheme can not only stabilize the bicycle robot and reject external disturbances generated by uncertainties and unmodelled dynamics, but also eliminate the need of the required bound information in the control law via the designed adaptive laws. The experimental results verify the excellent performance of the proposed control scheme in terms of strong robustness, fast error convergence in comparison with other control schemes.

Appendices

Appendix 1

Given the following first-order nonlinear differential inequality

$$ \dot{V}\left( x \right) + {\Gamma }V^{\mu } \left( x \right) \le 0 $$

(30)

where $V\left( x \right):R^{n} \to R$ represents a continuous positive definite function with respect to the state $x \in R^{n}$, ${\Gamma } > 0$, $0 < \mu < 1$, then the origin is a globally finite-time stable equilibrium for (30) with its convergence time being

$$ t_{\sigma } \le \frac{{V^{1 - \mu } \left( {x_{0} } \right)}}{{{\Gamma }\left( {1 - \mu } \right)}} $$

(31)

The derivation is referred to [33] and references therein.

Appendix 2

When the sliding motion $s = 0$ is established, for the system $\dot{e} = - \beta e - \alpha \smallint e^{{q_{1} /q_{2} }} {\text{ d}}t$, by choosing the Lyapunov function $V = 0.5e^{2}$, the derivative of $V$ can be given as:

$$ \begin{aligned} \dot{V} & & = e\dot{e} \\ & = e\left( { - \beta e - \alpha \smallint e^{{q_{1} /q_{2} }} {\text{ d}}t} \right) \\ & = - \beta e^{2} - \frac{{\alpha q_{2} }}{{\left( {q_{1} + q_{2} } \right)}}e^{{q_{1} /q_{2} + 2}} \\ & = - \beta \left| e \right|^{2} - \frac{{\alpha q_{2} }}{{\left( {q_{1} + q_{2} } \right)}}\left| e \right| \cdot \left| e \right|^{{q_{1} /q_{2} + 1}} {\text{sign}}\left( e \right) \\ & = - \left| e \right|\left( {\beta \left| e \right| + \frac{{\alpha q_{2} }}{{\left( {q_{1} + q_{2} } \right)}}\left| e \right|^{{q_{1} /q_{2} + 1}} {\text{sign}}\left( e \right)} \right) \\ & \le - \xi \left| e \right| \\ \end{aligned} $$

(32)

where $\xi = \beta \left| e \right| - \frac{{\alpha q_{2} }}{{\left( {q_{1} + q_{2} } \right)}}\left| e \right|^{{q_{1} /q_{2} + 1}}$ with following condition being satisfied when $e \ne 0$:

$$ \frac{{\beta \left( {q_{1} + q_{2} } \right)}}{{\alpha q_{2} }} \ge \left| e \right|^{{q_{1} /q_{2} }} $$

(33)

By properly choosing the values for $\alpha$, $\beta$, $q_{1}$, $q_{2}$ such that Eq. (32) can be always fulfilled, then according to Appendix 1, $e$ converges to origin with finite time

$$ t_{s} \le \frac{{q_{2} \left| {e_{0} } \right|}}{{\xi \left( {q_{2} - q_{1} } \right)}} $$

(34)