Trajectory tracking of a class of under-actuated thrust-propelled vehicle with uncertainties and unknown disturbances

Abstract

This paper deals with the problem of designing a controller for a thrust-propelled vehicle which steers the vehicle to track a 3D spatial path, while effective compensation for both time-varying disturbances and uncertainties is achieved as well. Taking advantage of extraction algorithm, we separate the design for the translational and rotational dynamics. A back-stepping-based controller and a sliding mode controller are, respectively, designed for the translational and rotational dynamics in succession. The stability of the control framework is established through Lyapunov analysis. A numerical simulation is also included in the paper to render the effectiveness of the proposed control scheme.

Appendices

Appendix

A Extraction algorithm

Here we introduce the extraction algorithm for obtaining $Q_d$ and T form the intermediate control $F=(F_1, F_2, F_3)^T$ given in (4).

$$\begin{aligned}&T=\frac{1}{\hat{\theta }} ||F-g\hat{z}||,\end{aligned}$$

(63)

$$\begin{aligned}&\eta _{d}=\sqrt{\frac{1}{2}+\frac{g-F_{3}}{2||F-g\hat{z}||}},\,\, q_{d}=\frac{1}{2 ||F-g\hat{z}|| \eta _{d}}\begin{pmatrix} F_{2} \\ -F_{1} \\ 0 \end{pmatrix}.\nonumber \\ \end{aligned}$$

(64)

As it is clear from (64), this extraction is well defined if

$$\begin{aligned} F\ne g\hat{z}. \end{aligned}$$

(65)

The desired angular velocity $\omega _{d}$ and its derivative $\dot{\omega }_{d}$ can also be obtained by the following expressions

$$\begin{aligned} \omega _d= & {} \varXi (F)\dot{F}, \end{aligned}$$

(66)

$$\begin{aligned} \dot{\omega }_d= & {} \dot{\varXi }(F, \dot{F})\dot{F}+\varXi (F)\ddot{F}, \end{aligned}$$

(67)

with

$$\begin{aligned} \varXi (F)=\frac{1}{\ell _1^2 \ell _2}\begin{pmatrix} -F_{1}F_{2} &{} -F^{2^2}+\ell _1\ell _2 &{} -F_{2}\ell _2 \\ F^{1^2}-\ell _1\ell _2 &{} F_{1}F_{2} &{} -F_{1}\ell _2\\ F_{2}\ell _1 &{} -F_{1}\ell _1 &{} 0 \end{pmatrix}, \end{aligned}$$

(68)

where $\ell _1=||F-g\hat{z}||,\quad \ell _2=\ell _1+(g-\mu _{3})$ and $\dot{\varXi }(F, \dot{F})$ is the time derivative of $\varXi (F)$ and the subscript i is omitted for notational simplicity. The proof can be found in [2].

B Analysis of boundedness of $\omega _d$ and $\dot{\omega }_d$

From (66) to (67), boundedness of $\omega _d$ and $\dot{\omega }_d$ can be guaranteed if F, $\dot{F}$, $\ddot{F}$ are bounded. Regarding the structure of F defined in (9)–(10), it is obvious that F is bounded and $\dot{F}$ and $\ddot{F}$ are bounded if, respectively, w and its derivative are bounded. From (29), we have

$$\begin{aligned} w=-k_3s_2-\varPhi _2+\ddot{v}_d+k_1^2\tilde{v}, \end{aligned}$$

and boundedness of w can be easily concluded by Assumption 2 and boundedness of $s_2$ and $\tilde{v}$ which are provided by the discussion in Sect. 4. The derivative of w is obtained by

$$\begin{aligned} \dot{w}=-k_3\dot{s}_2-\dot{\varPhi }_2+v^{(3)}_d+k_1^2\dot{\tilde{v}}. \end{aligned}$$

Viewing (3) and Assumption 2, the last two terms in the above equation are bounded based on boundedness of $s_1$, F, $\tilde{F}$, $\tilde{{\bar{\theta }}}$ which is concluded from the discussion in Sect. 4. Based on (9)–(10) and (15), we have

$$\begin{aligned} \dot{s}_2=\dot{f}(u)-\dot{f_d}=w+k_2 \tanh (s_1)+\varPhi _1-\dot{v}_d+k_1\tilde{v}, \end{aligned}$$

which is also bounded. It now just remains to prove that $\dot{\varPhi }_2$ is bounded. From (30), we can obtain

$$\begin{aligned} \dot{\varPhi }_2= & {} u_{m2}\frac{\dot{s}_2^T\left( ||s_2||^2+(\kappa _2\sigma _2)^2\right) -s_2^T\left( s_2^T \dot{s}_2+\kappa _2\dot{\kappa }_2\sigma _2 \right) }{\left( ||s_2||^2+(\kappa _2\sigma _2)^2\right) ^{3/2}},\nonumber \\ \end{aligned}$$

(69)

which is also bounded since $\dot{s}_2$, $s_2$, $\kappa _2$ are bounded, $\dot{\kappa }_2$ is bounded from (31), and the fact that $\kappa _2$ is kept away from zero by suitable selection of the gain $\lambda _2$ and the initial value $\kappa _2(0)$ as explained in Sect. 4.