Finite-time attitude tracking control for a rigid spacecraft using time-varying terminal sliding mode techniques

Abstract

This paper investigates the finite-time attitude tracking control for a rigid spacecraft in the presence of inertia uncertainties and external disturbances. Two novel time-varying terminal sliding mode control algorithms are derived for attitude tracking control system. The proposed two control algorithms not only eliminate the reaching phase of the conventional sliding mode control but also guarantee the tracking errors converge to zero in finite time. Moreover, the singularity problem can be avoided. Simulation results are provided to demonstrate the effectiveness of the proposed design methods.

Keywords:

• attitude tracking
• finite-time control
• time-varying terminal sliding mode (TVTSM)

Acknowledgement

The authors would like to thank the anonymous reviewers for their constructive comments and valuable suggestions which greatly improved the presentation of this article.

Disclosure statement

No potential conflict of interest was reported by the authors.

Appendix

Proof of Theorem 3.3: From Equation (32(32) $$\left\{\begin{array}{c}{x}_{1i}\left(t\right)=\mathrm{sgn}(x_{1i}\left(0\right)+C_i)[{|x_{1i}\left(0\right)+C_i|}^{1-\frac{q}{p}}\hfill \\ -\left(1-\frac{q}{p}\right){\alpha }_{i}t{]}^{\frac{1}{1-\frac{q}{p}}}-\frac{C_i}{T^2}{t}^2+\frac{2C_i}{T}t-C_i,t\le T^{*}\hfill \\ x_{1i}\left(t\right)=-\frac{C_i}{T^2}{t}^2+\frac{2C_i}{T}t-C_i,T^{*}<t\le T\hfill \end{array}\right.$$(32) ), we have (48) $$\begin{array}{ccc}& & {\int }_0^{\infty }\left|x_{1i}\left(t\right)\right|\mathrm{d}t\hfill \\ & & =\mathrm{sgn}\left(x_{1i}\left(0\right)\right){\int }_0^{\infty }{x}_{1i}\left(t\right)\mathrm{d}t\hfill \\ & & =\mathrm{sgn}\left(x_{1i}\left(0\right)\right)\left[{\int }_0^{T^{*}}{x}_{1i}\left(t\right)\mathrm{d}t+{\int }_{T^{*}}^T{x}_{1i}\left(t\right)\mathrm{d}t\right]\hfill \\ & & =\mathrm{sgn}\left(x_{1i}\left(0\right)\right)\left[\frac{T{(x_{1i}\left(0\right)+C_i)}^2}{2(2-\frac{q}{p})C_i}-\frac{1}{3}{C}_{i}T\right]\hfill \end{array}$$(48) Denote $\frac{x_{1i}\left(0\right)}{C_i}=\lambda$, we have (49) $$\begin{array}{ccc}\hfill J& =& max\left\{{\int }_0^{\infty }\left|x_{1i}\left(t\right)\right|\mathrm{d}t\right\}\hfill \\ & =& min\{\mathrm{sgn}\left(x_{1i}\left(0\right)\right)T{x}_{1i}\left(0\right)\hfill \\ & & \times \left(\frac{1}{2(2-\frac{q}{p})}\frac{{\lambda }^2+2\lambda +1}{\lambda }-\frac{1}{3\lambda }\right)\}\hfill \\ & =& max\left\{|x_{1i}\left(0\right)|T\left[\frac{1}{2\left(2-\frac{q}{p}\right)}\frac{{\lambda }^2+2\lambda +1}{\lambda }-\frac{1}{3\lambda }\right]\right\}\hfill \\ & =& max\left\{\right|x_{1i}\left(0\right)\left|\right\}T[\frac{1}{\left(2-\frac{q}{p}\right)}+\frac{1}{2\left(2-\frac{q}{p}\right)}\lambda \hfill \\ & & +\frac{\frac{2q}{p}-1}{6\left(2-\frac{q}{p}\right)}\frac{1}{\lambda }]\hfill \end{array}$$(49) Thus, we can choose $\lambda =1-\frac{2q}{p}$ such that $\frac{1}{2(2-\frac{q}{p})}\lambda +\frac{\frac{2q}{p}-1}{6(2-\frac{q}{p})}\frac{1}{\lambda }$ achieves minimum, i.e. $C_i=\frac{x_{1i}\left(0\right)}{1-\frac{2q}{p}}$.

Additional information

Funding

This work was supported by the National Basic Research Program of China (973 Program) [grant number 2012CB821200], [grant number 2012CB821201]; the NSFC [grant number 61134005], [grant number 61221061], [grant number 61327807].