Adaptive sliding-mode control for spacecraft autonomous rendezvous with elliptical orbits and thruster faults

Abstract

This article studies the problem of spacecraft autonomous rendezvous control for elliptical target orbits with external disturbance and thruster faults. The Clohessy–Wiltshire equation of elliptical target orbits without linearization is employed to describe the relative dynamic model as a linear time-varying system, such that the model errors in traditional linearization modeling methods are avoided. By on-line estimation of thruster faults and external disturbance, an adaptive sliding-mode control scheme is designed to eliminate the effects of the thruster faults and external disturbances, such that the reachability of the proposed integral sliding surface is ensured. Finally, a numerical example is established to illustrate the effectiveness of the proposed control approach.

Introduction

Spacecraft autonomous rendezvous has long been a hot research topic in the areas of manned spaceflight technology,^1–3 where Clohessy–Wiltshire (C-W) equation has been one of the most popular modeling strategies to investigate the control problems of spacecraft autonomous rendezvous. It is well known that spacecraft orbit control system refers to a class of typical mechanical systems. In fact, the classical C-W equation was first presented by Clohessy and Wiltshire ⁴ in 1960, which has been widely applied to address the problem of linear relative motion between two neighboring spacecrafts.^5–7 Generally speaking, in a C-W equation, the target orbit patterns are generally classified as circular and elliptical orbits, where elliptical orbits are more practical and realistic in engineering applications. However, most literatures on spacecraft autonomous rendezvous are only concerned with circular target orbits, while the investigation on elliptical orbits has received little research attention. In fact, the difficulties of this research topic mainly lies in the following two folds: (1) it is difficult to develop appropriate methodology to completed spacecraft autonomous rendezvous due to the time-varying nature of system parameters in elliptical orbits and (2) most of the obtained designed methods in the existing literature are based on linearized dynamical model, which, however, introduced model error and further results in possible control system performance degradation. However, if we do not perform the linearization for the C-W equation, the resulting system refers to a linear time-varying system, which is difficult to make analysis and synthesis work. ⁸

Thruster faults are frequently encountered in practical aerospace engineering, such as nozzle leakage and nozzle blocking.^9–13 Thruster faults may render to instability, poor performance of systems, and even missions failed.^14–16 Therefore, in spacecraft control systems design, thruster faults should be considered together in order to guarantee the performances of control systems. In the past few years, a few results have been reported in the literature on robust fault-tolerant control for spacecraft autonomous rendezvous.^17–21

On another research forefront, it is well known that sliding-mode control is an effective robust control approach for uncertain systems, which possesses various valuable features such as insensitiveness against parameter variations and external disturbance, and fast response.^22–26 In the past decades, sliding-mode control strategy has been used to solve a wide range of practical engineering problems arising in networked control systems and hypersonic vehicle systems.^27–30 In Li and Zheng, ³¹ a robust adaptive second-order sliding-mode control scheme with finite reaching time is proposed for a class of uncertain non-linear systems. In Shi et al., ²⁵ a new sliding-surface function design method is proposed for a class of linear continuous-time systems with stochastic jumps. The developed method can be used to the sliding-mode control of discrete-time stochastic systems. In Chen and Geng, ³² a novel high-order sliding-mode control approach has been developed for spacecraft rendezvous and docking. Recently, the sliding-mode controller design problems have been extensively investigated for fault-tolerant control of spacecraft autonomous rendezvous in He et al. ³³ In Pukdeboon and Kumam, ³⁴ Xu et al., ³⁵ Li et al., and ^36,37 Wang et al., ³⁸ the sliding-mode control-design problems are considered involved with the adaptive mechanism, which can deal with energy-unbounded disturbance effectively. Adaptive control is an effective way to account for system uncertainties or unknown parameter variations.^24,39–42 During the last few decades, a few adaptive robust control methods have been developed for spacecraft attitude control. ⁷ Therefore, in this article, an adaptive sliding-mode control strategy will be designed to solve the problem of disturbance rejection control with thruster faults for spacecraft autonomous rendezvous.

It should be pointed out that, however, in almost all the aforementioned works on thruster faults of spacecraft autonomous rendezvous, the thruster faults are simply modeled by scaling factors. Apparently, the realistic thruster faults are more complex than such a simple mathematical model. Hence, it is necessary to establish new thruster fault modes for spacecraft autonomous rendezvous. Motivated by the above observations, in this article, we study the adaptive sliding-mode control problem of spacecraft autonomous rendezvous with elliptical orbits and thruster faults, where the non-linearization C-W equation of elliptical target orbits is employed. First, an on-line adaptive estimation scheme is employed to the thruster faults and external disturbance. Second, an adaptive control strategy is developed to deal with the unknown external disturbance and reject the effects of the thruster faults effectively. The proposed adaptive sliding-mode control scheme cab guarantee the reachability of the integral sliding surface despite thruster faults. Finally, the usefulness and effectiveness of the proposed design method is illustrated via a practical example.

Problem formulation

Suppose that target’s orbit is an elliptical orbit and the distance between two spacecrafts is quite small compared with the orbit radius. We define the relative coordinate system as follows: the orbital coordinate frame is a right-hand Cartesian coordinate with the origin attached to the target spacecraft center of mass, and the x-axis is aligned with the vector from the earth center to the target center of mass; the y-axis is along the target orbit circumference; and the z-axis completes an orthogonal right-handed frame. The relative dynamic model between the two spacecrafts can be described as

$$\{\begin{array}{l}\stackrel{··}{x}-2\omega \stackrel{.}{y}-\stackrel{.}{\omega }y-{\omega }^{2}x=2\frac{\mu }{r^3}x+\frac{1}{m}(F_x+w_x)\\ \stackrel{··}{y}+2\omega \stackrel{.}{x}+\stackrel{.}{\omega }x-{\omega }^{2}y=-\frac{\mu }{r^3}y+\frac{1}{m}(F_y+w_y)\\ \stackrel{··}{z}=-\frac{\mu }{r^3}z+\frac{1}{m}(F_z+w_z)\end{array}$$

(1)

where $x$ , $y$ , and $z$ are the components of the relative position in corresponding axes; $m$ is the mass of the chaser; $F_i(i=x,y,z)$ is the relative motion dynamic; $w_i(i=x,y,z)$ is the external disturbance, in which $w_i$ satisfies ${\underset{¯}{w}}_i\le w_i\le {\overline{w}}_i$ , where ${\underset{¯}{w}}_i>0$ , ${\overline{w}}_i>0$ , and ${\underset{¯}{w}}_i$ and ${\overline{w}}_i$ are unknown constants representing lower and upper bounds of the external disturbance $w_i$ , respectively; and $\mu$ is the constant of earth gravitation. In elliptical target orbits, $r$ is the geocentric distance of the target spacecraft which can be described as

$$r=\frac{a·(1-e^2)}{(1+e·\cos \theta )}$$

where a is the semi-major axis, $e\in [0,1)$ is the eccentricity, and $\theta$ is the true anomaly, which has time-varying characteristics and can be represented as

$$\theta =2·\arctan \left(\sqrt{\frac{1+e}{1-e}}\tan \frac{E}{2}\right)$$

where $E$ is the eccentric anomaly. The angle velocity of the target $\omega$ and the angle acceleration of the target $\stackrel{·}{\omega }$ can be described by

$$\omega =\sqrt{\frac{\mu }{r^3}·(1+e\cos \theta )}\stackrel{.}{\omega }=-\frac{2\mu }{r^3}·e\sin \theta$$

Then, equation (1) can be rewritten as

$$\{\begin{array}{c}{\stackrel{.}{q}}_1(t)=q_2(t)\\ {\stackrel{.}{q}}_2(t)=A_1{q}_1(t)+A_2{q}_2(t)+\frac{1}{m}B(u(t)+w(t))\end{array}$$

(2)

where

$$\begin{array}{l}{q}_1(t)={\left[\begin{array}{ccc}x& y& z\end{array}\right]}^T{q}_2(t)={\left[\begin{array}{ccc}\stackrel{.}{x}& \stackrel{.}{y}& \stackrel{.}{z}\end{array}\right]}^T\\ u(t)={\left[\begin{array}{ccc}{F}_x& F_y& F_z\end{array}\right]}^{T}w(t)={\left[\begin{array}{ccc}{w}_x& w_y& w_z\end{array}\right]}^T\\ A_1=\left[\begin{array}{ccc}{\omega }^2+2\frac{\mu }{r^3}& \stackrel{.}{\omega }& 0\\ -\stackrel{.}{\omega }& {\omega }^2-\frac{\mu }{r^3}& 0\\ 0& 0& -\frac{\mu }{r^3}\end{array}\right]\\ A_2=\left[\begin{array}{ccc}0& 2\omega & 0\\ -2\omega & 0& 0\\ 0& 0& 0\end{array}\right],B=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\end{array}$$

During spacecraft autonomous rendezvous process, the thruster faults are usually inevitable. Hence, in this article, it is assumed that the thruster faults may happen and the mathematical description of the thruster failures can be described by

$$u_{ik}^F(t)={\rho }_i^k{u}_i(t)+{\varphi }_i^k{\sigma }_i(t)$$

(3)

where $i=x,y,z$ is along i-axes thruster; $k=1,2,\dots ,\varrho$ is the i-axes thruster in the case of the kth failure; $\varrho =5$ is the total number of thruster fault modes, and the thruster faults modes are generally classified as opening failure, closing failure, nozzle leakage, and nozzle blocking; ${\rho }_i^k$ is the control effectiveness factor of the i-axes thruster; ${\varphi }_i^k$ is the unknown constant defined as ${\varphi }_i^k=0$ or ${\varphi }_i^k=1$ ; and ${\sigma }_i$ is the unknown time-varying bounded fault along i-axes thruster. In the practical application, the constraints of ${\rho }_i^k$ can be described by

$$0<{\underset{\_}{\rho }}_i^k\le {\rho }_i^k\le {\overline{\rho }}_i^k\le 1$$

where ${\underset{¯}{\rho }}_i^k$ and ${\overline{\rho }}_i^k$ are the known constants representing the lower and upper bounds of the fault parameter ${\rho }_i^k$ . Based on equation (3), the thruster fault model is shown in Table 1.

Table 1.

Description of the thruster fault model.

kth thruster faults model	${\underset{¯}{\rho }}_i^k$	${\overline{\rho }}_i^k$	${\varphi }_i^k$
1. Normal operation	1	1	0
2. Opening failure	0	0	0
3. Closing failure	$>0$	$\le 1$	1
4. Nozzle leakage	$>0$	$\le 1$	1
5. Nozzle blocking	$>0$	$<1$	0

Then, we define the following sets

$$\begin{array}{l}{\rho }^k\triangleq diag\{{\rho }_1^k,{\rho }_2^k,\dots ,{\rho }_i^k\}\\ {\varphi }^k\triangleq diag\{{\varphi }_1^k,{\varphi }_2^k,\dots ,{\varphi }_i^k\}\\ {△}_{{\rho }^k}=\{{\rho }^k:{\rho }^k=diag\{{\rho }_1^k,{\rho }_2^k,\dots ,{\rho }_i^k\}\\ {\rho }_i^k\in {[}_i^k,{\overline{\rho }}_i^k]\}\\ N_{{\rho }^k}=\{{\rho }^k:{\rho }^k=diag\{{\rho }_1^k,{\rho }_2^k,\dots ,{\rho }_i^k\}\\ {\rho }_i^k={\underset{\_}{\rho }}_i^{k}or{\rho }_i^k={\overline{\rho }}_i^k\}\\ u^F(t)\triangleq {[u_{ik}^F(t),\dots ,u_{ik}^F(t)]}^T={\rho }^{k}u(t)+{\varphi }^k\sigma (t)\end{array}$$

In order to facilitate the subsequent research, we consider the closing failure and nozzle leakage of the thruster. The description of the thruster faults (equation (3)) can be described by

$$u^F(t)=\rho u(t)+\sigma (t)$$

(4)

where $\rho =\mathrm{diag}\{{\rho }_1,\dots ,{\rho }_i\}\in \mathrm{diag}\{{\rho }^1,\dots ,{\rho }^5\}\in {\Delta }_{{\rho }^k}$ , $\sigma (t)$ satisfies ${\underset{¯}{\sigma }}_i\le {\sigma }_i(t)\le {\overline{\sigma }}_i$ where ${\underset{¯}{\sigma }}_i>0$ and ${\overline{\sigma }}_i>0$ are unknown fault parameters to be estimated. Thus, substituting equation (4) into equation (6) yields

$$\{\begin{array}{c}{\stackrel{.}{q}}_1(t)=q_2(t)\\ {\stackrel{.}{q}}_2(t)=A_1{q}_1(t)+A_2{q}_2(t)+\frac{1}{m}B(\rho u(t)+\sigma (t)+w(t))\end{array}$$

(5)

Given the desired states $(q_{1d},q_{2d})$ , where ${\stackrel{·}{q}}_{1d}=q_{2d},q_{1d}=[x_d,y_d,z_d{]}^T$ , then the error-state vector of the system (equation (5)) is define as

$$e(t)=e_1(t)=q_1(t)-q_{1d},\stackrel{.}{e}(t)=e_2(t)=q_2(t)-q_{2d}$$

(6)

and $\stackrel{·}{e}(0)$ is assumed to satisfy $\stackrel{·}{e}(0)=e_2(0)=0$ . The superscript “^” is used to represent the estimates of variables; the disturbance estimations and thruster fault estimations are described as

$$\begin{array}{l}\widehat{\overline{w}}(t)={[{\widehat{\overline{w}}}_x,{\widehat{\overline{w}}}_y,{\widehat{\overline{w}}}_z]}^T\\ \underset{\_}{\widehat{w}}={[{\underset{\_}{\widehat{w}}}_x,{\underset{\_}{\widehat{w}}}_y,{\underset{\_}{\widehat{w}}}_z]}^T\\ \widehat{\overline{\sigma }}(t)={[{\widehat{\overline{\sigma }}}_x(t),{\widehat{\overline{\sigma }}}_y(t),{\widehat{\overline{\sigma }}}_z(t)]}^T,\\ \underset{\_}{\widehat{\sigma }}(t)={[{\underset{\_}{\widehat{\sigma }}}_x(t),{\underset{\_}{\widehat{\sigma }}}_y(t),{\underset{\_}{\widehat{\sigma }}}_z(t)]}^T\end{array}$$

Sliding-mode control and adaptive control have strong robustness for the unknown parameters and nonlinearity. So in this work, our aim is to design an adaptive sliding-mode control to solve the control problem of the dynamic system (equation (5)), wherein the integral sliding-surface function $s(t)$ is chosen as

$$s(t)=\stackrel{.}{e}(t)+k_{p}e(t)+k_I{\displaystyle \underset{0}{\overset{t}{\int }}e(\tau )d\tau }$$

(7)

where $k_p>0$ and $k_I>0$ both are design parameters. The symbolic variables are defined as

$${\beta }_i(t)=\{\begin{array}{cc}0& s_i(t)>0\\ 1& s_i(t)\le 0\end{array}$$

(8)

Define two new variable matrices as

$$\begin{array}{l}D(t)=\widehat{\overline{w}}(t)+\beta (t)(\underset{\_}{\widehat{w}}(t)-\widehat{\overline{w}}(t))\\ Q(t)=\widehat{\overline{\sigma }}(t)+\beta (t)(\underset{\_}{\widehat{\sigma }}(t)-\widehat{\overline{\sigma }}(t))\end{array}$$

The derivative with respect to the time of the sliding-surface function is calculated as

$$\begin{array}{l}\stackrel{.}{s}(t)=\stackrel{··}{e}(t)+k_p\stackrel{.}{e}(t)+k_{I}e(t)\\ =A_1{q}_1(t)+A_2{q}_2(t)+\frac{1}{m}B(\rho u(t)+\sigma (t)\\ +w(t))-{\stackrel{.}{q}}_{2d}+k_p{q}_2(t)-k_p{q}_{2d}\\ +k_I{q}_1(t)-k_I{q}_{1d}\end{array}$$

(9)

For $\stackrel{·}{s}(t)=0$ , the equivalent control law of equation (9) can be expressed as

$$\begin{array}{l}{u}_{eq}=-{\rho }^{-1}m{B}^{-1}(A_1{q}_1(t)+A_2{q}_2(t)-{\stackrel{.}{q}}_{2d})\\ -{\rho }^{-1}(\sigma (t)+w(t))\\ -{\rho }^{-1}m{B}^{-1}(k_p{q}_2(t)-k_p{q}_{2d})\\ -{\rho }^{-1}m{B}^{-1}(k_I{q}_1(t)-k_I{q}_{1d})\end{array}$$

(10)

By substituting equation (10) into the control system (equation (5)), we obtain

$$\{\begin{array}{c}{\stackrel{.}{q}}_1(t)=q_2(t)\\ {\stackrel{.}{q}}_2(t)={\stackrel{.}{q}}_{2d}-k_p{q}_2(t)+k_p{q}_{2d}-k_I{q}_1(t)+k_I{q}_{1d}\end{array}$$

(11)

Equation (11) can be rewritten as

$$\stackrel{.}{\xi }(t)=\tilde{A}\xi (t)+\tilde{B}$$

(12)

where

$$\begin{array}{l}\xi (t)={[q_1(t),q_2(t)]}^T\tilde{A}=\left[\begin{array}{cc}0& 1\\ -k_p& -k_I\end{array}\right]\\ \tilde{B}=\left[\begin{array}{c}0\\ {\stackrel{.}{q}}_{2d}+k_p{q}_{2d}+k_I{q}_{1d}\end{array}\right]\end{array}$$

The following theorem presents the sufficient condition for the asymptotic stability for sliding dynamic system (equation (12)).

Theorem 1

Considering the spacecraft relative dynamic control system (equation (5)), if there exists symmetric positive definite matrix $P$ such that the following condition holds

$$\left[\begin{array}{cc}{\tilde{A}}^{T}P+P\tilde{A}& {\tilde{B}}^{T}P\\ \ast & 0\end{array}\right]<0$$

(13)

then, the system (equation (5)) is asymptotically stable on the sliding surface $s(t)=0$ .

Proof

Let the Lyapunov function candidate for the system is chosen as

$$V_c(t)={\xi }^T(t)P\xi (t)$$

(14)

It is easy to prove Theorem 1, and we omit the detailed proof for space reason.

In order to ensure the reachability of the sliding surface $s(t)$ in equation (7), the adaptive sliding-mode controller will be constructed. Designing this control law and illustrating this control scheme can ensure that the system (equation (5)) is asymptotically stable, and we require the following theorem.

Theorem 2

Considering the relative dynamic system (equation (5)), the adaptive sliding-mode control law is given by

$$\begin{array}{l}u(t)=u_1(t)+u_2(t)+u_3(t)+u_4(t)\hfill \\ u_1(t)=-m{\widehat{\rho }}^{-1}{B}^{-1}(A_1{q}_1(t)+A_2{q}_2(t)\hfill \\ -{\stackrel{.}{q}}_{2d}+k_p{e}_2(t)+k_I{e}_1(t))\hfill \\ u_2(t)=-0.5m{\widehat{\rho }}^{-1}\mathrm{sgn}(\mathrm{s}(\mathrm{t}))\hfill \\ u_3(t)=-{\widehat{\rho }}^{-1}(t)D(t)\hfill \\ u_4(t)=-{\widehat{\rho }}^{-1}(t)Q(t)\hfill \end{array}$$

(15)

where updating laws of the adaptive variable ${\hat{\rho }}_i(t)$ is designed as

$$\begin{array}{l}{\stackrel{.}{\widehat{\rho }}}_i(t)=Pro{j}_{[mi{n}_k\{{\rho }_i^k\},ma{x}_k\{{\rho }_i^k\}]}\{L_i(t)\}\hfill \\ =\{\begin{array}{l}0{\widehat{\rho }}_i(t)=mi{n}_k\left\{{\rho }_i^k\right\},and{L}_i\le 0\hfill \\ 0{\widehat{\rho }}_i(t)=ma{x}_k\left\{{\rho }_i^k\right\},and{L}_i\ge 0\hfill \\ L_i(t)otherwise\hfill \end{array}\hfill \\ L_i(t)=-[{\widehat{\rho }}_i^{-1}(t)s_i(t)(A_1{q}_1(t)+A_2{q}_2(t)-{\stackrel{.}{q}}_{2d}\hfill \\ +k_p{e}_2(t)+k_I{e}_1(t))\{i\}+\frac{1}{m}{\widehat{\rho }}_i^{-1}(t)s_i(t)D_i(t)\hfill \\ +\frac{1}{m}{\widehat{\rho }}_i^{-1}(t)s_i(t)Q_i(t)+0.5{\widehat{\rho }}_i^{-1}(t)|s_i(t)|]l_i\hfill \end{array}$$

(16)

where $[*]\{i\}$ is the ith row element of the matrix [*], and the updating laws of the parameter estimation variables ${\hat{\overline{w}}}_i$ , ${\widehat{\underset{¯}{w}}}_i$ , ${\hat{\overline{\sigma }}}_i(t)$ , ${\widehat{\underset{¯}{\sigma }}}_i(t)$ are designed as

$$\begin{array}{l}{\stackrel{.}{\widehat{\overline{w}}}}_i={\chi }_i(1-{\beta }_i(t))s_i(t),{\stackrel{.}{\widehat{\underset{\_}{w}}}}_i={\chi }_i{\beta }_i(t)s_i(t)\\ {\stackrel{.}{\widehat{\overline{\sigma }}}}_i(t)={\eta }_i(1-{\beta }_i(t))s_i(t),{\stackrel{.}{\widehat{\underset{\_}{\sigma }}}}_i(t)={\eta }_i{\beta }_i(t)s_i(t)\end{array}$$

(17)

where ${\chi }_i$ and ${\eta }_i$ are the parameters to be designed, and the convergence rate of the estimation variables is to be determined. Then, the relative dynamic system (equation (5)) is asymptotically stable.

Proof

Considering the system (equation (5)) of spacecraft autonomous rendezvous in elliptical target orbits, the Lyapunov function is chosen as

$$\begin{array}{l}V(t)=V_1(t)+V_2(t)+V_3(t)+V_4(t)\hfill \\ V_1(t)=\frac{1}{2}{s}^T(t)s(t)\hfill \\ V_2(t)=\frac{1}{2m{\chi }_i}{\displaystyle \sum _{i=1}^3({\tilde{\overline{w}}}_i^2+{\widetilde{\underset{\_}{w}}}_i^2)}\hfill \\ V_3(t)=\frac{1}{2m{\eta }_i}{\displaystyle \sum _{i=1}^3({\tilde{\overline{\sigma }}}_i^2(t)+{\widetilde{\underset{\_}{\sigma }}}_i^2(t))}\hfill \\ V_4(t)=\frac{1}{2}{\displaystyle \sum _{i=1}^3\frac{{\tilde{\rho }}_i^2(t)}{l_i}}\hfill \end{array}$$

(18)

The derivative of the function $V(t)$ is

$$\begin{array}{l}\stackrel{.}{V}(t)={\stackrel{.}{V}}_1(t)+{\stackrel{.}{V}}_2(t)+{\stackrel{.}{V}}_3(t)+{\stackrel{.}{V}}_4(t)\\ =s^T(t)\stackrel{.}{s}(t)+{\stackrel{.}{V}}_2(t)+{\stackrel{.}{V}}_3(t)+{\stackrel{.}{V}}_4(t)\\ =s^T(t)[{\stackrel{.}{q}}_2(t)-{\stackrel{.}{q}}_{2d}(t)+k_p{e}_2(t)+k_I{e}_1(t)]\\ +{\stackrel{.}{V}}_2(t)+{\stackrel{.}{V}}_3(t)+{\stackrel{.}{V}}_4(t)\\ =s^T(t)[A_1{q}_1(t)+A_2{q}_2(t)+\frac{1}{m}B(\rho u(t))\\ +\sigma (t)+w(t))-{\stackrel{.}{q}}_{2d}(t)+k_p{e}_2(t)+k_I{e}_1(t)]\\ +{\stackrel{.}{V}}_2(t)+{\stackrel{.}{V}}_3(t)+{\stackrel{.}{V}}_4(t)\\ =s^T(t)[A_1{q}_1(t)+A_2{q}_2(t)+\frac{1}{m}B(\rho (u_1(t)+u_2(t)\\ +u_3(t)+u_4(t))+\sigma (t)+w(t))-{\stackrel{.}{q}}_{2d}(t)\\ +k_p{e}_2(t)+k_I{e}_1(t)]+{\stackrel{.}{V}}_2(t)+{\stackrel{.}{V}}_3(t)+{\stackrel{.}{V}}_4(t)\\ =\underset{term1)}{\underbrace{s^T(t)[A_1{q}_1(t)+A_2{q}_2(t)+\frac{1}{m}B\rho u_1(t)-{\stackrel{.}{q}}_{2d}(t)}}\\ \underset{term1)}{\underbrace{+k_p{e}_2(t)+k_I{e}_1(t)]}}\\ +\underset{term2)}{\underbrace{\frac{1}{m}{s}^T(t)B\rho u_3(t)+\frac{1}{m}{s}^T(t)Bw(t)+{\stackrel{.}{V}}_2(t)}}\\ +\underset{term3)}{\underbrace{\frac{1}{m}{s}^T(t)B\rho u_4(t)+\frac{1}{m}{s}^T(t)B\sigma (t)+{\stackrel{.}{V}}_3(t)}}\\ +\frac{1}{m}{s}^T(t)B\rho u_2(t)+{\stackrel{.}{V}}_4(t)\end{array}$$

(19)

The adaptive parameter estimation error is defined as $\tilde{\rho }(t)=\hat{\rho }(t)-\rho$ . By Substituting this into term (1) of equation (19), we obtain

$$\begin{array}{l}{s}^T(t)[A_1{q}_1(t)+A_2{q}_2(t)+\frac{1}{m}B\rho u_1(t)-{\stackrel{.}{q}}_{2d}(t)\hfill \\ +k_p{e}_2(t)+k_I{e}_1(t)]\hfill \\ =s^T(t)[A_1{q}_1(t)+A_2{q}_2(t)-{\stackrel{.}{q}}_{2d}(t)+k_p{e}_2(t)+k_I{e}_1(t)\hfill \\ -\frac{1}{m}B\rho \cdot {\widehat{\rho }}^{-1}(t)m{B}^{-1}(A_1{q}_1(t)+A_2{q}_2(t)-{\stackrel{.}{q}}_{2d}\hfill \\ +k_p{e}_2(t)+k_I{e}_1(t))]\hfill \\ =s^T(t)[\widehat{\rho }(t){\widehat{\rho }}^{-1}(t)(A_1{q}_1(t)+A_2{q}_2(t)-{\stackrel{.}{q}}_{2d}(t)+k_p{e}_2(t)\hfill \\ +k_I{e}_1(t))-\rho {\widehat{\rho }}^{-1}(t)(A_1{q}_1(t)+A_2{q}_2(t)\hfill \\ -{\stackrel{.}{q}}_{2d}+k_p{e}_2(t)+k_I{e}_1(t))]\hfill \\ =s^T(t)\tilde{\rho }(t){\widehat{\rho }}^{-1}(t)[A_1{q}_1(t)+A_2{q}_2(t)-{\stackrel{.}{q}}_{2d}(t)\hfill \\ +k_p{e}_2(t)+k_I{e}_1(t)]\hfill \\ ={\displaystyle \sum _{i=1}^3{\tilde{\rho }}_i(t){\widehat{\rho }}_i^{-1}(t)s_i(t)[A_1{q}_1(t)+A_2{q}_2(t)-{\stackrel{.}{q}}_{2d}(t)}\hfill \\ +k_p{e}_2(t)+k_I{e}_1(t)]\{i\}\hfill \end{array}$$

(20)

where $[*]\{i\}$ is the ith row element of the matrix [*]. Term (2) of equation (19) is calculated as

$$\begin{array}{l}\frac{1}{m}{s}^T(t)B\rho u_3(t)+\frac{1}{m}{s}^T(t)Bw(t)+{\stackrel{.}{V}}_2\\ \le \frac{1}{m}{s}^T(t)B\rho u_3(t)+\frac{1}{m}{s}^T(t)B[\overline{w}(t)+\beta (t)(\underset{\_}{w}(t)-\overline{w}(t)]\\ +\frac{1}{m{\chi }_i}{\displaystyle \sum }_{i=1}^3[{\underset{¯}{\tilde{\overline{w}}}}_i{\underset{¯}{\stackrel{.}{\stackrel{~}{\overline{w}}}}}_i+{\widetilde{\underset{\_}{w}}}_i{\stackrel{.}{\stackrel{~}{\underset{\_}{w}}}}_i]\\ \le -\frac{1}{m}{s}^T(t)B\rho {\widehat{\rho }}^{-1}D(t)\\ +\frac{1}{m}{s}^T(t)B[\overline{w}(t)+\beta (t)(\underset{\_}{\overset{^}{w}}(t)-\overline{w}(t)]\\ +\frac{1}{m}{\displaystyle \sum }_{i=1}^3\left[\frac{{\tilde{\overline{w}}}_i{\stackrel{.}{\stackrel{~}{\overline{w}}}}_i}{{\chi }_i}+\frac{{\widetilde{\underset{\_}{w}}}_i{\stackrel{.}{\stackrel{~}{\underset{\_}{w}}}}_i}{{\chi }_i}\right]\end{array}$$

(21)

In equation (21), the following equalities hold

$$\begin{array}{l}-\frac{1}{m}{s}^T(t)B\rho {\widehat{\rho }}^{-1}(t)D(t)\\ =-\frac{1}{m}{s}^T(t)B\rho {\widehat{\rho }}^{-1}(t)[\widehat{\overline{w}}(t)+\beta (t)(\widehat{\underset{\_}{w}}(t)-\widehat{\overline{w}}(t))]\\ =-\frac{1}{m}{\displaystyle \sum _{i=1}^3{s}_i}(t){\rho }_i{\widehat{\rho }}_i^{-1}(t)[{\widehat{\overline{w}}}_i+{\beta }_i(t)({\widehat{\underset{\_}{w}}}_i-{\widehat{\overline{w}}}_i)]\\ =-\frac{1}{m}{\displaystyle \sum _{i=1}^3{s}_i}(t)[{\widehat{\overline{w}}}_i+{\beta }_i(t)({\widehat{\underset{\_}{w}}}_i-{\widehat{\overline{w}}}_i)]\\ +\frac{1}{m}{\displaystyle \sum _{i=1}^3{s}_i}(t){\tilde{\rho }}_i(t){\widehat{\rho }}_i^{-1}(t)[{\widehat{\overline{w}}}_i+{\beta }_i(t)({\widehat{\underset{\_}{w}}}_i-{\widehat{\overline{w}}}_i)]\end{array}$$

(22)

Because ${\stackrel{·}{\widetilde{\overline{\underset{¯}{w}}}}}_i={\stackrel{·}{\hat{\overline{w}}}}_i$ and ${\underset{¯}{\stackrel{·}{\stackrel{~}{w}}}}_i={\underset{¯}{\stackrel{·}{\hat{w}}}}_i$ , the $\frac{1}{m}\sum _{i=1}^3[\frac{{\tilde{\overline{w}}}_i{\stackrel{·}{\stackrel{~}{\overline{w}}}}_i}{{\chi }_i}+\frac{{\widetilde{\underset{¯}{w}}}_i{\stackrel{·}{\stackrel{~}{\underset{¯}{w}}}}_i}{{\chi }_i}]$ of equation (21) is described as

$$\begin{array}{l}\frac{1}{m}{\displaystyle \sum }_{i=1}^3\left[\frac{{\tilde{\overline{w}}}_i{\stackrel{.}{\stackrel{~}{\overline{w}}}}_i}{{\chi }_i}+\frac{{\widetilde{\underset{\_}{w}}}_i{\stackrel{.}{\stackrel{~}{\underset{\_}{w}}}}_i}{{\chi }_i}\right]\\ =\frac{1}{m}{\displaystyle \sum }_{i=1}^3\left[\frac{{\tilde{\overline{w}}}_i{\stackrel{.}{\widehat{\overline{w}}}}_i}{{\chi }_i}+\frac{{\widetilde{\underset{\_}{w}}}_i{\stackrel{.}{\widehat{\underset{\_}{w}}}}_i}{{\chi }_i}\right]\\ =\frac{1}{m}{\displaystyle \sum }_{i=1}^3\left[\frac{{\tilde{\overline{w}}}_i{\chi }_i(1-{\beta }_i(t))s_i(t)}{{\chi }_i}+\frac{{\widetilde{\underset{\_}{w}}}_i{\chi }_i{\beta }_i(t)s_i(t)}{{\chi }_i}\right]\\ =\frac{1}{m}{\displaystyle \sum }_{i=1}^3[{\tilde{\overline{w}}}_i+{\beta }_i({\widetilde{\underset{\_}{w}}}_i-{\tilde{\overline{w}}}_i)]s_i(t)\\ =\frac{1}{m}{\displaystyle \sum }_{i=1}^3\{[{\widehat{\overline{w}}}_i-{\overline{w}}_i]+{\beta }_i[({\widehat{\underset{\_}{w}}}_i-{\underset{\_}{w}}_i)-({\widehat{\overline{w}}}_i-{\overline{w}}_i]\}{s}_i(t)\\ =\frac{1}{m}{\displaystyle \sum }_{i=1}^3[{\widehat{\overline{w}}}_i+{\beta }_i({\widehat{\underset{\_}{w}}}_i-{\widehat{\overline{w}}}_i)]s_i(t)\\ -\frac{1}{m}{\displaystyle \sum }_{i=1}^3[{\overline{w}}_i+{\beta }_i({\underset{\_}{w}}_i-{\overline{w}}_i)]s_i(t)\end{array}$$

(23)

Substituting equations (22) and (23) into equation (21), it can be rewritten as

$$\begin{array}{l}\frac{1}{m}{s}^T(t)B\rho u_3(t)+\frac{1}{m}{s}^T(t)Bw(t)+{\stackrel{.}{V}}_2\\ \le -\frac{1}{m}{\displaystyle \sum _{i=1}^3{s}_i}(t)[{\widehat{\overline{w}}}_i+{\beta }_i(t)({\widehat{\underset{\_}{w}}}_i-{\widehat{\overline{w}}}_i)]\\ +\frac{1}{m}{\displaystyle \sum _{i=1}^3{s}_i}(t){\tilde{\rho }}_i(t){\widehat{\rho }}_i^{-1}(t)[{\widehat{\overline{w}}}_i+{\beta }_i(t)({\widehat{\underset{\_}{w}}}_i-{\widehat{\overline{w}}}_i)]\\ +\frac{1}{m}{\displaystyle \sum }_{i=1}^3[{\widehat{\overline{w}}}_i+{\beta }_i({\widehat{\underset{\_}{w}}}_i-{\widehat{\overline{w}}}_i)]s_i(t)\end{array}$$

(24)

$$\begin{array}{l}-\frac{1}{m}{\displaystyle \sum }_{i=1}^3[{\overline{w}}_i+{\beta }_i({\underset{\_}{w}}_i-{\overline{w}}_i)]s_i(t)\\ +\frac{1}{m}{\displaystyle \sum _{i=1}^3{s}_i}(t)[{\overline{w}}_i+{\beta }_i(t)({\underset{\_}{w}}_i-{\overline{w}}_i]\\ \le \frac{1}{m}{\displaystyle \sum _{i=1}^3{s}_i}(t){\tilde{\rho }}_i(t){\widehat{\rho }}_i^{-1}(t)[{\widehat{\overline{w}}}_i+{\beta }_i(t)({\widehat{\underset{\_}{w}}}_i-{\widehat{\overline{w}}}_i)]\end{array}$$

Considering the term (3) of equation (19), the inequality, which uses the similar method of dealing with $w(t)$ is obtained as

$$\begin{array}{l}\frac{1}{m}{s}^T(t)B\rho u_4(t)+\frac{1}{m}{s}^T(t)B\sigma (t)+{\stackrel{.}{V}}_3\\ \le \frac{1}{m}{\displaystyle \sum _{i=1}^3{s}_i}(t){\tilde{\rho }}_i(t){\widehat{\rho }}_i^{-1}(t)[{\widehat{\overline{\sigma }}}_i(t)+{\beta }_i(t)({\widehat{\underset{\_}{\sigma }}}_i(t)-{\widehat{\overline{\sigma }}}_i(t))]\end{array}$$

(25)

Considering the following term of the equation (19) is calculated as

$$\begin{array}{l}\frac{1}{m}{s}^T(t)\rho B{u}_2(t)=-0.5\frac{1}{m}{s}^T(t)\rho Bm{\widehat{\rho }}^{-1}\mathrm{sgn}(\mathrm{s}(\mathrm{t}))\\ =-0.5{\displaystyle \sum _{i=1}^3|}{s}_i(t)|+0.5{\displaystyle \sum _{i=1}^3{\tilde{\rho }}_i}(t){\widehat{\rho }}_i^{-1}(t)|s_i(t)|\end{array}$$

(26)

By substituting equations (20) and (24)–(26) into equation (19), we obtain

$$\begin{array}{l}\frac{dV}{dt}\le {\displaystyle \sum _{i=1}^3{\tilde{\rho }}_i}(t){\widehat{\rho }}_i^{-1}(t)s_i(t)[A_1{q}_1(t)+A_2{q}_2(t)-{\stackrel{.}{q}}_{2d}(t)\\ +k\_pe\_2\left(t\right)+k\_Ie\_1\left(t\right)]\left\{i\right\}\\ +\frac{1}{m}{\displaystyle \sum _{i=1}^3{s}_i}(t){\tilde{\rho }}_i(t){\widehat{\rho }}_i^{-1}(t)[{\widehat{\overline{w}}}_i+{\beta }_i(t)({\widehat{\underset{\_}{w}}}_i-{\widehat{\overline{w}}}_i)]\\ +\frac{1}{m}{\displaystyle \sum _{i=1}^3{s}_i}(t){\tilde{\rho }}_i(t){\widehat{\rho }}_i^{-1}(t)[{\widehat{\overline{\sigma }}}_i(t)+{\beta }_i(t)({\widehat{\underset{\_}{\sigma }}}_i(t)\\ -{\widehat{\overline{\sigma }}}_i(t))]+0.5{\displaystyle \sum _{i=1}^3{\tilde{\rho }}_i}(t){\widehat{\rho }}_i^{-1}(t)|s_i(t)|\\ -0.5{\displaystyle \sum _{i=1}^3|}{s}_i(t)|+{\displaystyle \sum _{i=1}^3\frac{{\tilde{\rho }}_i(t){\stackrel{.}{\stackrel{~}{\rho }}}_i(t)}{l_i}}\end{array}$$

(27)

By substituting the updating laws (equation (16)) into equation (27), we obtain

$$\frac{dV}{dt}\le -0.5{\displaystyle \sum _{i=1}^3|}{s}_i(t)|+{\displaystyle \sum _{i=1}^3\left\{-{\tilde{\rho }}_i(t)\frac{L_i(t)}{l_i}+\frac{{\tilde{\rho }}_i(t){\stackrel{.}{\stackrel{~}{\rho }}}_i(t)}{l_i}\right\}}$$

(28)

where

$$\sum _{i=1}^3\{-{\tilde{\rho }}_i(t)\frac{L_i(t)}{l_i}+\frac{{\tilde{\rho }}_i(t){\stackrel{·}{\stackrel{~}{\rho }}}_i(t)}{l_i}\}\le 0$$

Then, the following inequality is obtained

$$\frac{\mathrm{dV}}{\mathrm{dt}}\le -0.5\sum _{i=1}^3|s_i(t)|$$

(29)

Thus, the inequality $\frac{\mathrm{dV}}{\mathrm{dt}}<0$ holds for $\forall s_i(t)\ne 0$ , which illustrate the reachability of the sliding surface.

Illustrative example

In this section, an example is presented to illustrate the effectiveness of the adaptive sliding-mode control strategy proposed in this article. For simplicity, we assume that the initial state $q(0)=[800600500000{]}^T$ and the expected state $q_d=[000000{]}^T$ . The elliptical orbit parameters of the target are shown in Table 2. The upper and lower bounds of external disturbance and thruster faults are $\underset{¯}{w}=[0.650.60.6{]}^T$ , $\overline{w}=[0.650.60.6{]}^T$ and ∼ $\underset{¯}{\sigma }=[0.650.60.6{]}^T$ , $\overline{\sigma }=[0.650.60.6{]}^T$ , respectively. The initial estimates of the adaptive device are $\hat{\overline{w}}(0)=[0.650.60.6{]}^T$ , $\widehat{\underset{¯}{w}}(0)=[-0.2-0.4-0.4{]}^T$ , $\hat{\overline{\sigma }}(0)=[0.050.050.05{]}^T$ , and $\hat{\overline{\sigma }}(0)=[0.050.050.05{]}^T$ .The coefficients of the adaptive updating laws are chosen as $l_x=l_y=l_z=0.001$ , ${\chi }_x={\chi }_y={\chi }_z=0.01$ , and ${\eta }_x={\eta }_y={\eta }_z=0.01$ . The sliding-mode control parameters are $k_I=0.0005$ and $k_p=0.0005$ . To demonstrate the fault-tolerant performance of the adaptive sliding-mode control method with thruster faults, it respectively carries out two simulations of different working conditions with different $\rho$ .

1. Working condition 1. The upper and lower bounds of $\rho$ are ${\overline{\rho }}_{\mathrm{xx}}=0.8$ , ${\underset{¯}{\rho }}_{\mathrm{xx}}=0.5$ , ${\overline{\rho }}_{\mathrm{yx}}=0.5$ , ${\underset{¯}{\rho }}_{\mathrm{yx}}=0.4$ , ${\overline{\rho }}_{\mathrm{zx}}=0.85$ , ${\underset{¯}{\rho }}_{\mathrm{zx}}=0.25$ , ${\overline{\rho }}_{\mathrm{xy}}=0.7$ , ${\underset{¯}{\rho }}_{\mathrm{xy}}=0.5$ , ${\overline{\rho }}_{\mathrm{yy}}=0.5$ , ${\underset{¯}{\rho }}_{\mathrm{yy}}=0.3$ , ${\overline{\rho }}_{\mathrm{zy}}=0.9$ , and ${\underset{¯}{\rho }}_{\mathrm{zy}}=0.7$ . The fault parameters of $\rho$ are ${\rho }_x=0.6$ , ${\rho }_y=0.4$ , and ${\rho }_z=0.4$ ; the initial values of $\hat{\rho }$ are ${\hat{\rho }}_x(0)={\hat{\rho }}_y(0)={\hat{\rho }}_z(0)=1$ .

2. Working condition 2. The upper and lower bounds of $\rho$ are ${\overline{\rho }}_{\mathrm{xx}}=0.8$ , ${\underset{¯}{\rho }}_{\mathrm{xx}}=0.7$ , ${\overline{\rho }}_{\mathrm{yx}}=0.8$ , ${\underset{¯}{\rho }}_{\mathrm{yx}}=0.2$ , ${\overline{\rho }}_{\mathrm{zx}}=0.75$ , ${\underset{¯}{\rho }}_{\mathrm{zx}}=0.25$ , ${\overline{\rho }}_{\mathrm{xy}}=0.75$ , ${\underset{¯}{\rho }}_{\mathrm{xy}}=0.6$ , ${\overline{\rho }}_{\mathrm{yy}}=0.7$ , ${\underset{¯}{\rho }}_{\mathrm{yy}}=0.5$ , ${\overline{\rho }}_{\mathrm{zy}}=0.4$ , and ${\underset{¯}{\rho }}_{\mathrm{zy}}=0.25$ . The fault parameters of $\rho$ are ${\rho }_x=0.7$ , ${\rho }_y=0.5$ , and ${\rho }_z=0.3$ ; the initial values of $\hat{\rho }$ are ${\hat{\rho }}_x(0)={\hat{\rho }}_y(0)={\hat{\rho }}_z(0)=1$ .

Table 2.

Orbit parameters of the target spacecraft.

Parameter name	Value
Semi-major (km)	350
Eccentricity	0.005
Inclination (∘)	45
Ascending node (∘)	90
Argument of perigee (∘)	90
True anomaly (∘)	0
Mass (kg)	00

Remark 1

Remark notes that the estimated values are needed to construct adaptive controller in our controller design procedure. According to the correlative analysis in Ioannou and Sun, ⁴³ whether they can converge to their true values or not is not necessary.

The simulation results are given in Figures 1–12. Among them, Figures 1–6 show the simulation results for working condition 1, and the relative position of the two spacecrafts is given in Figure 1. The relative velocity of the two spacecrafts is given in Figure 2, the sliding mode is given in Figure 3, and the fault parameters of $\rho$ and its estimated values are given in Figures 4–6. The simulation figures of working condition 2 are shown in Figures 7–12. It is seen that the closed-loop system is asymptotically stable and the adaptive estimators have good performance.

Figure 1.

Trajectories of position (Condition 1).

Figure 2.

Trajectories of velocity (Condition 1).

Figure 3.

Trajectories of sliding variable (Condition 1).

Figure 4.

Fault parameter ${\rho }_x$ and its estimated value (Condition 1).

Figure 5.

Fault parameter ${\rho }_y$ and its estimated value (Condition 1).

Figure 6.

Fault parameter ${\rho }_z$ and its estimated value (Condition 1).

Figure 7.

Trajectories of position (Condition 2).

Figure 8.

Trajectories of velocity (Condition 2).

Figure 9.

Trajectories of sliding variable (Condition 2).

Figure 10.

Fault parameter ${\rho }_x$ and its estimated value (Condition 2).

Figure 11.

Fault parameter ${\rho }_y$ and its estimated value (Condition 2).

Figure 12.

Fault parameter ${\rho }_z$ and its estimated value (Condition 2).

In order to make comparison, a proportional–derivative (PD) control law is also applied to the spacecraft relative dynamic system with and without thruster faults, and the PD control strategy is designed as

$$u_c=-k_{\varpi }e(t)-k_{\upsilon }\stackrel{·}{e}(t)$$

(30)

where $k_{\varpi }$ and $k_{\upsilon }$ are the control gains, and we design $k_{\varpi }=1$ and $k_{\upsilon }=1$ .

Figure 13 depicts the relative positions along x-, y-, and z-axis of the controller (equation (30)) without thruster faults. It is seen clearly that the spacecraft relative position stabilization could be achieved in 6000 s. In the case of thruster faults, we choose the same value of fault parameter $\rho$ as working condition 2. Figure 14 depicts the relative positions along x-, y-, and z-axis of the controller with thruster faults. From Figures 13 and 14, we can see that the PD controller (equation (30)) will significantly degrade the control performance with thruster faults. As we can see in Figures 7 and 14, the designed adaptive sliding-control strategy in this article renders to faster convergence performance and more ideal failure-tolerant performance than PD controller (equation (30)).

Figure 13.

Trajectories of position of controller (equation (30)) without faults.

Figure 14.

Trajectories of position of controller (equation (30)) with faults.

Conclusion

In this article, we have studied the stabilization problem with external disturbance and thruster faults which uses the unhandled C-W equation of the elliptical target orbits for the spacecraft autonomous rendezvous. Considering the external disturbance and thruster faults, an adaptive sliding-mode control scheme has been designed such that the resulting closed-loop system is asymptotically stable and also achieves spacecraft autonomous rendezvous. A simulation example has been provided to show the effectiveness of the proposed design method.