Attitude-constrained reorientation control for spacecraft based on extended state observer

Abstract

This article investigates the spacecraft attitude reorientation control problem in the presence of attitude constraint, actuator saturation, parametric uncertainty, and external disturbance. First, a nonlinear tracking law based on a strictly convex potential function is proposed to generate the virtual control angular velocity which has only one global minimum. Then, utilizing the auxiliary system governed by the difference between the upper bound of actuator torque and the untreated command torque, a novel backstepping controller is presented, which is able to satisfy the constraint of actuator saturation and guarantee the stability of control system. In addition, an extended state observer with the uniformly ultimately bounded estimation error and finite-time stability is put forward to realize the real-time compensation of the compound disturbance consisting of parametric uncertainty and external disturbance. Therefore, it enhances the robustness and improves the accuracy of the extended state observer–based backstepping controller. Finally, simulation results validate the effectiveness and reliability of the proposed schemes.

Introduction

During the large angle reorientation maneuver of spacecraft, there exist many constraints caused by flight tasks, on-board sensors, practical actuators, and payloads. For example, optical devices cannot be exposed to strong light sources, and the solar array needs to catch the sunlight. Thus, the attitude should be maintained in the admissible zone which consists of keep-out and keep-in zones. In recent years, considerable research attentions have been attracted to attitude planning and control with constrained attitude zones, and the literature is divided into two research directions.

One direction is planing-and-tracking. The analytic solution of the optimal control problem of the geometric algorithm has been proposed by Biggs and Colley. ¹ However, the local minima need to be avoided by careful parameter selection. For some special types of constraints, the constraint monitor method has been proposed by Singh et al. ² and applied to the Cassini spacecraft. Some image search algorithms have been introduced for attitude maneuver control. Such as the A* path planning method^3,4 the distortion-minimizing attitude parameterizations^5,6 and the random search tree method ⁷ with a smaller computational complexity. Based on the previous literature, ⁸ the pseudo-spectral method ⁹ has been used to generate the paths without any constraint violations between local nodes, which has achieved a better performance. Melton^10,11 has proposed a hybrid algorithm consisting of particle swarm optimization, differential evolution, and bacterial foraging optimization methods to determine the initial guess, which plays a important role in the convergence property of the pseudo-spectral method. The particle swarm algorithm ¹² has also been introduced in the maneuver path planning. However, more practical constraints need to be considered.

The other direction is embedding attitude constraints in the controller. Kim and colleagues^8,13 have proposed a semi-definite planning method, which has a drawback of large computational complexity. Inspired by Kim and Mesbahi, ⁸ Cui et al. ¹⁴ have applied the receding-horizon-control to the design of controller. They obtained a better performance of obstacle avoidance due to its multistep prediction technology. The non-convex constraint set has been modified as a convex one by Tam and Lightsey. ¹⁵ On this basis, a navigation system has been carried out. However, its performance index depends on the selection of the weighting factors, which is a major shortcoming. Sun and Dai ¹⁶ have presented the quadratically constrained quadratic programming for the original optimization problem which can drive solution converge to the optimum.

Considering the computational complexity, the potential-function-based controller is smaller than the above algorithms. The reorientation maneuver task in the presence of attitude constraint, which is caused by optical equipment’s requirement of avoidance to bright celestial bodies, and described by the Euler angle, the quaternion, and the Rodrigues parameters,^17–19 has been studied. Avanzini et al. ²⁰ have proposed a potential-function-based backstepping controller. Furthermore, the existence conditions for the repulsive potential function (RPF) have been presented by Guo et al., ²¹ and the external disturbance has been considered by Zheng et al. ²² To achieve a better performance of convergence time and accuracy, Mengali and Quarta ²³ have designed a variable gain controller. To deal with the local minima in potential functions, Lee and Mesbahi ²⁴ have demonstrated that a logarithmic barrier potential function with respect to attitude quaternion is strictly convex such that it has only one global minimum.

For the practical spacecraft missions, there exist some other constraints. To deal with the actuator saturation and partial loss of control effectiveness, an uniformly ultimately bounded controller has been presented by Xiao et al. ²⁵ The external disturbance and parametric uncertainty have been attenuated, and the actuator saturation has been guaranteed by a sliding-mode-observer-based adaptive controller proposed by Xiao and Yin. ²⁶ To improve the convergence property, Xiao et al. ²⁷ have carried out an observer-based estimation law to completely eliminate the unwanted uncertain dynamics. Thus, the attitude tracking system is asymptotically stable with the controller consisting of the estimation-law-based proportional–derivative-type controller.

Thereupon, this article applies a strictly convex potential function as the navigation function to guarantee that the attitude reaches the target in absence of any constraint violation. Moreover, taking advantage of the auxiliary system governed by the difference between untreated command torque and the upper bound of control torque, the proposed novel backstepping controller is able to be stable with unknown actuator saturation and obtain better universality and reliability. In traditional methods, the external disturbance and parametric uncertainty need to be stabilized by the inherent characteristic of the controller, which causes the loss of control accuracy. Therefore, this article presents an extended state observer (ESO) to realize real-time compensation of the compound disturbance consisting of external disturbance and parametric uncertainty, which is able to enhance the robustness of the control system.

The rest of this article is organized as follows. In section “Problem formulation,” the spacecraft modeling, the description of the attitude-constrained zones, the potential function, the lemmas, and problem statement are presented. Section “Attitude controller design” gives the detailed derivations of attitude controllers. The ESO is designed in section “ESO-based controller design.” To prove the effectiveness and reliability of the proposed controllers, some numerical simulations are carried out in section “Simulation results and discussions” and followed by the conclusion in section “Conclusion.”

Problem formulation

Attitude kinematics and dynamics

Spacecraft attitude kinematics and dynamics equations with actuator saturation and compound disturbance can be written as follows

$$\stackrel{·}{\mathit{q}}=\frac{1}{2}\mathit{q}\otimes \bar{\mathbf{\omega }}=\frac{1}{2}\left[\begin{array}{c}-q_0\hfill \\ \multicolumn{1}{c}{q_0{\mathit{I}}_{3\times 3}+{\mathit{q}}_v^{\times }}\end{array}\right]\mathbf{\omega }$$

(1)

$$\mathit{J}\stackrel{·}{\mathbf{\omega }}=-\mathbf{\omega }\mathbf{\times }\mathbf{J}\mathbf{\omega }\mathbf{+}\mathrm{sat}\mathbf{(}\mathit{u}\mathbf{)}\mathbf{+}{\mathit{d}}_{\mathbf{t}}$$

(2)

where $\mathit{q}=[q_0,{\mathit{q}}_v^{\mathrm{T}}{]}^{\mathrm{T}}=[q_0,q_1,q_2,q_3{]}^{\mathrm{T}}$ is the attitude quaternion of spacecraft body coordinate system $f_b$ relative to inertia coordinate system $f_I$ ; ⊗ represents the quaternion multiplication; ω denotes the angular velocity of $f_b$ relative to $f_I$ ; $\bar{\mathit{a}}=[0,{\mathit{a}}^{\mathrm{T}}{]}^{\mathrm{T}}$ denotes the quaternion form of the vector $\mathit{a}$ ; ${\mathit{I}}_{a\times a}$ represents the $a\times a$ identity matrix; $\mathit{J}$ denotes the nominal inertia matrix; $\Delta \mathit{J}$ is the additive error inertia matrix; $\mathit{u}$ is the control torque of spacecraft; $\mathrm{sat}(\mathit{u})$ is the actuator torque which can be written as

$$\mathrm{sat}(u_i)=\{\begin{array}{cc}\mathrm{sign}(u_i)u_{\max }\hfill & |u_i|>u_{\max }\hfill \\ \multicolumn{1}{c}{u_i}& |u_i|\le u_{\max }\hfill \end{array}i=1,2,3$$

(3)

where $u_{\max }$ is the upper bound of the actuator torque; ${\mathit{d}}_t$ denotes the combination of parametric uncertainty $\Delta \mathit{J}$ and external stochastic disturbance $\mathit{d}$ , that is, ${\mathit{d}}_t=\mathit{d}-\Delta \mathit{J}\stackrel{·}{\mathbf{\omega }}-\mathbf{\omega }\mathbf{\times }\Delta \mathit{J}\mathbf{\omega }$ ; the skew-symmetric matrix ${\mathit{a}}^{\times }$ represents the cross-product operation, and

$${\mathit{a}}^{\times }=\left[\begin{array}{ccc}0\hfill & -a_3\hfill & a_2\hfill \\ \multicolumn{1}{c}{a_3}& 0\hfill & -a_1\hfill \\ \multicolumn{1}{c}{-a_2}& a_1\hfill & 0\hfill \end{array}\right]$$

(4)

Assumption 1

The external stochastic disturbance $\mathit{d}$ is continuous and satisfies

$$\parallel \mathit{d}\parallel <{\delta }_1,\parallel \stackrel{·}{\mathit{d}}\parallel \le {\delta }_2$$

(5)

where $\parallel \mathit{a}\parallel$ is short for $\parallel \mathit{a}{\parallel }_2$ , which denotes the two-norm of $\mathit{a}$ ; ${\delta }_1$ and ${\delta }_2$ are positive constants.

The command attitude quaternion which depends on the attitude maneuver task is denoted by ${\mathit{q}}_c$ ; then the error quaternion can be written as

$$\mathit{e}={\mathit{q}}_c^{*}\otimes \mathit{q}$$

(6)

where $\mathit{e}=[e_0,{\mathit{e}}_v^{\mathrm{T}}{]}^{\mathrm{T}}=[e_0,e_1,e_2,e_3{]}^{\mathrm{T}}$ ; ${\mathit{a}}^{*}$ represents the conjugate quaternion of $\mathit{a}$ .

Assumption 2

In the attitude reorientation task investigated by this article, ${\stackrel{·}{\mathit{q}}}_c=0$ .

Attitude-constrained zones description

Suppose the unit vector $\mathit{y}\in {\mathbb{R}}^3$ represents optical axis direction of spacecraft instrument, and the unit vector $\mathit{x}\in {\mathbb{R}}^3$ denotes the direction vector from spacecraft to a certain celestial object. If an instrument needs avoid the bright celestial object, it should be maintained that the angle between $\mathit{y}$ and $\mathit{x}$ is strictly greater than $\theta$ , as illustrated in Figure 1.

Figure 1.

Optical axis $\mathit{y}$ and celestial object direction $\mathit{x}$ in body coordinate system $f_b$ .

Assumption 3

In the attitude reorientation task researched by this article, the direction vector $\mathit{x}$ of any certain celestial object is fixed in $f_I$ , that is, $\stackrel{·}{\mathit{x}}=0$ .

Thereupon, we have

$${\mathit{x}}^I·{\mathit{y}}^I<\cos \theta$$

(7)

where ${\mathit{x}}^I$ and ${\mathit{y}}^I$ denote $\mathit{x}$ and $\mathit{y}$ represented in $f_I$ , respectively. And

$$\begin{array}{c}{\mathit{y}}^I=\mathit{q}\otimes {\mathit{y}}^B\otimes {\mathit{q}}^{*}\hfill \\ \multicolumn{1}{c}{={\mathit{y}}^B-2({\mathit{q}}_v^{\mathrm{T}}{\mathit{q}}_v){\mathit{y}}^B+2({\mathit{q}}_v^{\mathrm{T}}{\mathit{y}}^B){\mathit{q}}_v+2q_0({\mathit{q}}_v\times {\mathit{y}}^B)}\end{array}$$

(8)

where ${\mathit{y}}^B$ represents $\mathit{y}$ in $f_b$ . Substituting equation (8) into equation (7) yields

$$2{\mathit{q}}_v^{\mathrm{T}}{\mathit{yq}}_v^{\mathrm{T}}x-{\mathit{q}}_v^{\mathrm{T}}{\mathit{q}}_v{\mathit{x}}^{\mathrm{T}}\mathit{y}+q_0^2{\mathit{x}}^{\mathrm{T}}\mathit{y}+2q_0{\mathit{q}}_v^{\mathrm{T}}(\mathit{y}\times \mathit{x})<\cos \theta$$

(9)

where $\mathit{x}$ and $\mathit{y}$ are shortened notations of ${\mathit{x}}^I$ and ${\mathit{y}}^B$ , respectively. After some algebraic manipulation, it is obtained that

$${\mathit{q}}^{\mathrm{T}}\left[\begin{array}{cc}d\hfill & {\mathit{b}}^{\mathrm{T}}\hfill \\ \multicolumn{1}{c}{\mathit{b}}& \mathit{A}\hfill \end{array}\right]\mathit{q}<0$$

(10)

where $d={\mathit{x}}^{\mathrm{T}}\mathit{y}-\cos \theta$ , $\mathit{b}=\mathit{y}\times \mathit{x}$ , $\mathit{A}=\mathit{x}{\mathit{y}}^{\mathrm{T}}+{\mathit{yx}}^{\mathrm{T}}-({\mathit{x}}^{\mathrm{T}}\mathit{y}+\cos \theta ){\mathit{I}}_3$ . Based on equation (10), the definitions of three attitude zones are given as follows:

1. Keep-out zone ${\mathit{Q}}_{\mathrm{KO}}$

Consider the attitude $\mathit{q}$ which make the angle between the optical axis direction ${\mathit{y}}_j$ of the jth spacecraft instrument and the direction vector ${\mathit{x}}_i$ from spacecraft to the ith certain celestial object strictly be greater than ${\theta }_i^j$ . Then the keep-out zone ${\mathit{Q}}_{{\mathrm{KO}}_i^j}$ of jth instrument and ith celestial object can be defined as

$${\mathit{Q}}_{{\mathrm{KO}}_i^j}=\{\mathit{q}|{\mathit{q}}^{\mathrm{T}}{\mathit{M}}_i^j({\theta }_i^j)\mathit{q}<0\}$$

(11)

where

$${\mathit{M}}_i^j({\theta }_i^j)=\left[\begin{array}{cc}{d}_i^j\hfill & {\mathit{b}}_i^{{\mathit{j}}^{\mathrm{T}}}\hfill \\ \multicolumn{1}{c}{{\mathit{b}}_i^j}& {\mathit{A}}_i^j\hfill \end{array}\right],i=1,2,\dots ,n,j=1,2,\dots ,m$$

(12)

• 2. Keep-in zone ${\mathbf{Q}}_{\mathrm{KI}}$

Being similar to the definition of keep-out zone, keep-in zone represents the attitudes which are guaranteed that the angle between $\mathit{y}$ and $\mathit{x}$ is strictly smaller than ${\theta }_{\mathrm{KI}}$ , that is

$${\mathit{Q}}_{\mathrm{KI}}=\{\mathit{q}|{\mathit{q}}^{\mathrm{T}}{\mathit{M}}_{\mathrm{KI}}({\theta }_{\mathrm{KI}})\mathit{q}>0\}$$

(13)

where ${\mathit{M}}_{\mathrm{KI}}$ is defined analogously to equation (12).

• 3. Admissible zone ${\mathit{Q}}_A$

When the satellite carries different kinds of instruments, and some instruments need to be kept in ${\mathit{Q}}_{\mathrm{KO}}$ , some others need to be kept in ${\mathit{Q}}_{\mathrm{KI}}$ , we define a new attitude set, called admissible zone ${\mathit{Q}}_A$ . It can be given by

$${\mathit{Q}}_A={\mathit{Q}}_{\mathrm{KO}}\cap {\mathit{Q}}_{\mathrm{KI}}$$

(14)

During the reorientation maneuvering, the attitude quaternion $\mathit{q}$ should be maintained in ${\mathit{Q}}_A$ , that is, $\mathit{q}(t)\in {\mathit{Q}}_A,\forall t\ge 0$ . Considering three attitude zones defined above, we will construct the potential function in the next subsection.

Potential function design

The attractive potential function $V_a:{\mathbb{S}}^3\to {ℝ}_{+}$ is designed by the distance between $\mathit{e}$ and ${\mathit{q}}_I$ , where ${\mathbb{S}}^3$ and ${\mathbb{R}}_{+}$ represent the unit quaternion set and nonnegative real set, respectively; ${\mathit{q}}_I=[1,0,0,0{]}^{\mathrm{T}}$ is the identity unit quaternion. Moreover, $V_a$ moves to the global minimum when $\mathit{q}$ coincides with ${\mathit{q}}_c$ . Let

$$V_a=\parallel \mathit{e}-\mathrm{sign}(e_0(t_0)){\mathit{q}}_I{\parallel }^2=\parallel {\mathit{q}}_c-\mathrm{sign}(e_0(t_0))\mathit{q}{\parallel }^2$$

(15)

where $e_0(t_0)$ is the initial value of $e_0$ ; $\mathrm{sign}(·)$ means the signum function.

Remark 1

Utilizing the term $\mathrm{sign}(e_0(t_0))$ , the unwinding phenomenon of the quaternion kinematics can be remedied by choosing a certain direction at the initial state.

Assumption 4

The initial attitude quaternion ${\mathit{q}}_{\mathrm{init}}$ satisfies that ${\mathit{q}}_{\mathrm{init}}\in {\mathit{Q}}_A$ .

The RPF $V_r:S^3\to {\mathbb{R}}_{+}$ should increase when $\mathit{q}$ is close to the edge of ${\mathit{Q}}_A$ . Thus, it can be denoted as

$$V_r=-k_1\sum _{j=1}^m\sum _{i=1}^n\log \frac{-{\mathit{q}}^{\mathrm{T}}{\mathit{M}}_i^j\mathit{q}}{2}-k_2\log \frac{{\mathit{q}}^{\mathrm{T}}{\mathit{M}}_{\mathrm{KI}}\mathit{q}}{2}$$

(16)

Evidently, $V_r\to +\infty$ when $\mathit{q}$ approaches the edge of ${\mathit{Q}}_A$ . Therefore, $\mathit{q}$ will be maintained in the admissible attitude zone ${\mathit{Q}}_A$ . In order to unify the form of two terms in equation (16), using some algebraic manipulation, equation (16) can be modified as

$$\begin{array}{c}{V}_r=-k_1\sum _{j=1}^m\sum _{i=1}^n\log \frac{{\mathit{q}}^{\mathrm{T}}{\hat{\mathit{M}}}_i^j\mathit{q}-{\beta }_1}{2}\hfill \\ \multicolumn{1}{c}{-k_2\log \frac{{\mathit{q}}^{\mathrm{T}}{\hat{\mathit{M}}}_{\mathrm{KI}}\mathit{q}-{\beta }_2}{2}}\end{array}$$

(17)

where ${\beta }_1>{\lambda }_{max}({\widehat{M}}_i^j)$ ; ${\widehat{M}}_i^j=-M_i^j+{\beta }_1{I}_{3\times 3}$ ; ${\beta }_2>-{\lambda }_{min}({\widehat{M}}_{KI})$ ; ${\hat{\mathit{M}}}_{\mathrm{KI}}={\mathit{M}}_I+{\beta }_2{\mathit{I}}_{3\times 3}$ ; ${\lambda }_{\min }(\mathit{a})$ and ${\lambda }_{\max }(\mathit{a})$ are the minimum and maximum eigenvalues of $\mathit{a}$ .

Until now, we can construct a logarithmic barrier potential function

$$V_p=V_a·V_r$$

(18)

Lemmas

Lemma 1

The potential function $V_p(\mathit{q})$ defined by equation (18) is smooth and strictly convex for all $\mathit{q}\in {\mathit{Q}}_A$ and admits a global minimum at ${\mathit{q}}_c\in {\mathit{Q}}_A$ . ²⁴

Lemma 2

If a continuous positive definite function $V(x):{\mathbb{R}}^n\to \mathbb{R}$ satisfies the following differential inequality ²⁸

$$\stackrel{·}{V}(\mathit{x})+\vartheta V^{\varpi }(\mathit{x})\le 0,\mathit{x}\in \mathbf{\Omega }\mathbf{,}\mathit{x}\mathbf{\ne }0$$

(19)

where $\vartheta >0$ and $\varpi \in (0,1)$ ; $\mathbf{\Omega }\mathbf{\in }{\mathbf{R}}^{\mathbf{n}}$ is a neighborhood of the equilibrium, $V(x)$ can converge to the equilibrium in finite time

$$t_s=\frac{V^{1-\varpi }(\mathit{x}(t_0))}{\vartheta (1-\varpi )}$$

(20)

Problem statement

In this article, spacecraft reorientation task in the presence of attitude constraint is addressed. Under Assumptions 1–3, the control objective herein is to design a controller to drive the system state $(\mathit{q},\mathbf{\omega }\mathbf{)}$ to converge to $({\mathit{q}}_c,0)$ asymptotically, considering the parametric uncertainties, external disturbances, and actuator saturation. Meanwhile, the attitude $\mathit{q}$ should be kept in the admissible zone ${\mathit{Q}}_A$ .

Attitude controller design

Design of backstepping controller based on nonlinear tracking law

During the design of the controller in this section, the actuator saturation and compound disturbance are not considered for the time being.

The time derivative of $V_p$ is obtained as follows

$${\stackrel{·}{V}}_p=\nabla V_p^{\mathrm{T}}(\frac{1}{2}\mathit{q}\otimes \bar{\mathbf{\omega }})={\bar{\mathbf{\omega }}}^{\mathrm{T}}(\frac{1}{2}{\mathit{q}}^{*}\otimes \nabla V_p)$$

(21)

In order to reduce the system potential function $V_p$ , to ensure $\mathit{q}\in {\mathit{Q}}_A$ during attitude maneuver path planning, and to improve tracking performance, a nonlinear virtual control angular velocity is designed as

$${\bar{\mathbf{\omega }}}_{c1}=-\alpha \arctan (\beta {\mathit{q}}^{*}\otimes \nabla V_p)$$

(22)

$$\begin{array}{c}{\stackrel{·}{\stackrel{̄}{\omega }}}_{c1}={\mathit{C}}_1({\stackrel{·}{\mathit{q}}}^{*}\otimes \nabla V_p+{\mathit{q}}^{*}\otimes \left({\nabla }^2{V}_p\stackrel{·}{\mathit{q}}\right))\hfill \\ \multicolumn{1}{c}{={\mathit{C}}_1({(\frac{1}{2}\mathit{q}\otimes \bar{\mathbf{\omega }})}^{*}\otimes \nabla V_p+{\mathit{q}}^{*}\otimes \left({\nabla }^2{V}_p(\frac{1}{2}\mathit{q}\otimes \bar{\mathbf{\omega }})\right))}\end{array}$$

(23)

where $\alpha$ and $\beta$ are positive constants; $\alpha$ can be assigned according to the admissible maximum angular velocity ${\omega }_{\max }$ ; ${\mathit{C}}_1=\alpha \beta [{\mathit{I}}_{4\times 4}+{\beta }^2\mathrm{diag}({\mathit{q}}^{*}\otimes \nabla V_p{)}^2{]}^{-1}$

$$\begin{array}{c}\nabla V_p=\sum _{j=1}^m\sum _{i=1}^n(2k_1{\mathit{q}}_c\log \frac{{\mathit{q}}^{\mathrm{T}}{\hat{\mathit{M}}}_i^{j}q-{\beta }_1}{2}\hfill \\ \multicolumn{1}{c}{-\frac{2k_1(2-2{\mathit{q}}_c^{\mathrm{T}}\mathit{q})}{{\mathit{q}}^{\mathrm{T}}{\hat{\mathit{M}}}_i^{j}q-{\beta }_1}{\hat{\mathit{M}}}_i^j\mathit{q})}\\ \multicolumn{1}{c}{+2k_2{\mathit{q}}_c\log \frac{{\mathit{q}}^{\mathrm{T}}{\hat{\mathit{M}}}_{\mathrm{KI}}\mathit{q}-{\beta }_2}{2}}\\ \multicolumn{1}{c}{-\frac{2k_2(2-2{\mathit{q}}_c^{\mathrm{T}}\mathit{q})}{{\mathit{q}}^{\mathrm{T}}{\hat{\mathit{M}}}_{\mathrm{KI}}\mathit{q}-{\beta }_2}{\hat{\mathit{M}}}_{\mathrm{KI}}\mathit{q}}\end{array}$$

(24)

$$\begin{array}{c}{\nabla }^2{V}_p=\sum _{j=1}^m\sum _{i=1}^n(\frac{8k_1}{{\mathit{q}}^{\mathrm{T}}{\hat{\mathit{M}}}_i^j\mathit{q}-{\beta }_1}{\mathit{q}}_c{({\hat{\mathit{M}}}_i^j\mathit{q})}^{\mathrm{T}}\hfill \\ \multicolumn{1}{c}{+\frac{4k_1(2-2{\mathit{q}}_c^{\mathrm{T}}\mathit{q})}{{({\mathit{q}}^{\mathrm{T}}{\hat{\mathit{M}}}_i^j\mathit{q}-{\beta }_1)}^2}\left({\hat{\mathit{M}}}_i^j\mathit{q}\right){\left({\hat{\mathit{M}}}_i^j\mathit{q}\right)}^{\mathrm{T}}}\\ \multicolumn{1}{c}{-\frac{2k_1(2-2{\mathit{q}}_c^{\mathrm{T}}\mathit{q})}{{\mathit{q}}^{\mathrm{T}}{\hat{\mathit{M}}}_i^j\mathit{q}-{\beta }_1}{\hat{\mathit{M}}}_i^j)}\\ \multicolumn{1}{c}{+\frac{8k_2}{{\mathit{q}}^{\mathrm{T}}{\hat{\mathit{M}}}_{\mathrm{KI}}\mathit{q}-{\beta }_2}{\mathit{q}}_c{({\hat{\mathit{M}}}_{\mathrm{KI}}\mathit{q})}^{\mathrm{T}}}\\ \multicolumn{1}{c}{+\frac{4k_2(2-2{\mathit{q}}_c^{\mathrm{T}}\mathit{q})}{{({\mathit{q}}^{\mathrm{T}}{\hat{\mathit{M}}}_{\mathrm{KI}}\mathit{q}-{\beta }_2)}^2}\left({\hat{\mathit{M}}}_{\mathrm{KI}}\mathit{q}\right){\left({\hat{\mathit{M}}}_{\mathrm{KI}}\mathit{q}\right)}^{\mathrm{T}}}\\ \multicolumn{1}{c}{-\frac{2k_2(2-2{\mathit{q}}_c^{\mathrm{T}}\mathit{q})}{{\mathit{q}}^{\mathrm{T}}{\hat{\mathit{M}}}_{\mathrm{KI}}\mathit{q}-{\beta }_2}{\hat{\mathit{M}}}_{\mathrm{KI}}}\end{array}$$

(25)

Thereupon

$${\stackrel{·}{V}}_p=\bar{\omega }{c1}^{\mathrm{T}}(\frac{1}{2}{\mathit{q}}^{*}\otimes \nabla V_p)\le 0$$

(26)

The state equations that need to be processed can be written as

$$\{\begin{array}{c}{\stackrel{·}{V}}_p={\bar{\omega }}^{\mathrm{T}}(\frac{1}{2}{\mathit{q}}^{*}\otimes \nabla V_p)\hfill \\ \multicolumn{1}{c}{J\stackrel{·}{\omega }=-{\mathbf{\omega }}^{\mathbf{\times }}J\mathbf{\omega }\mathbf{+}\mathit{u}}\end{array}$$

(27)

Introduce new variables

$$\{\begin{array}{c}{z}_1=V_p\hfill \\ \multicolumn{1}{c}{{\mathit{z}}_2=\mathbf{\omega }\mathbf{-}\mathbf{\omega }\mathbf{c}1}\end{array}$$

(28)

Theorem 1

Consider the system plant in equation (27) with the new variables in equation (28). The equilibrium $(\mathit{q}={\mathit{q}}_c,\mathbf{\omega }\mathbf{=}0\mathbf{)}$ is globally asymptotically stable under the following backstepping controller

$$\begin{array}{c}{\mathit{u}}_1={\mathbf{\omega }}^{\mathbf{\times }}J\mathbf{\omega }\mathbf{+}\mathbf{J}{\stackrel{·}{\mathbf{\omega }}}_{\mathbf{c}1}\mathbf{-}{\mathbf{s}}_1\mathit{J}{\mathit{z}}_2\hfill \\ \multicolumn{1}{c}{-s_2^{-1}\mathit{J}\mathrm{vec}[\frac{1}{2}{\mathit{q}}^{*}\otimes \nabla V_p]}\end{array}$$

(29)

where $s_1>0$ and $s_2>0$ ; $\mathrm{vec}[\mathit{a}]$ means the vector part of the quaternion $\mathit{a}$ .

Proof

Since $V_p>0$ , the candidate Lyapunov function can be constructed as

$$V_1=V_p+\frac{1}{2}{s}_2{\mathit{z}}_2^{\mathrm{T}}{\mathit{z}}_2$$

(30)

where $s_2>0$ .

Taking the time derivative of $V_1$ , and substituting equation (27) into it, yields

$$\begin{array}{c}{\stackrel{·}{V}}_1=\nabla V_p^{\mathrm{T}}(\frac{1}{2}\mathit{q}\otimes \bar{\omega })+s_2{\mathit{z}}_2^{\mathrm{T}}(\stackrel{·}{\omega }-{\stackrel{·}{\omega }}_{c1})\hfill \\ \multicolumn{1}{c}{=\nabla V_p^{\mathrm{T}}(\frac{1}{2}\mathit{q}\otimes {\bar{\mathit{z}}}_2)+\nabla V_p^{\mathrm{T}}(\frac{1}{2}\mathit{q}\otimes {\bar{\omega }}_{c1})}\\ \multicolumn{1}{c}{+s_2{\mathit{z}}_2^{\mathrm{T}}(-{\mathit{J}}^{-1}{\mathbf{\omega }}^{\mathbf{\times }}J\mathbf{\omega }\mathbf{+}{\mathit{J}}^{\mathbf{-}1}\mathit{u}\mathbf{-}{\stackrel{·}{\mathbf{\omega }}}_{\mathbf{c}1})}\\ \multicolumn{1}{c}{={\mathit{z}}_2^{\mathrm{T}}\mathrm{vec}[\frac{1}{2}{\mathit{q}}^{*}\otimes \nabla V_p]+{\bar{\omega }}_{c1}(\frac{1}{2}{\mathit{q}}^{*}\otimes \nabla V_p)}\\ \multicolumn{1}{c}{+s_2{\mathit{z}}_2^{\mathrm{T}}(-{\mathit{J}}^{-1}{\mathbf{\omega }}^{\mathbf{\times }}J\mathbf{\omega }\mathbf{+}{\mathit{J}}^{\mathbf{-}1}\mathit{u}\mathbf{-}{\stackrel{·}{\mathbf{\omega }}}_{\mathbf{c}1})}\end{array}$$

(31)

Then, substituting controller (equation (29)) into above equation, it is obtained that

$$\begin{array}{c}{\stackrel{·}{V}}_1=-\frac{1}{2}\alpha \arctan (\beta {\mathit{q}}^{*}\otimes \nabla V_p)({\mathit{q}}^{*}\otimes \nabla V_p)\hfill \\ \multicolumn{1}{c}{-s_1{s}_2{\mathit{z}}_2^{\mathrm{T}}{\mathit{z}}_2\le 0}\end{array}$$

(32)

Letting ${\stackrel{·}{V}}_1\equiv 0$ , we have $\mathbf{\omega }\mathbf{=}0$ and $\nabla V_p=0$ . With Lemma 1, $V_p$ is a strictly convex function with respect to $\mathit{q}$ , which implies that $\nabla V_p=0$ if and only if $\mathit{q}={\mathit{q}}_c$ . Therefore, by LaSalles invariance principle, $(\mathit{q}={\mathit{q}}_c,\mathbf{\omega }\mathbf{=}0\mathbf{)}$ is asymptotically stable. In addition, when $\mathbf{q}$ tends to the edge of ${\mathit{Q}}_A$ or $\mathbf{\omega }\mathbf{\to }\mathbf{\infty }$ , there exists $V_t\to \infty$ . Hence, $(\mathit{q}={\mathit{q}}_c,\mathbf{\omega }\mathbf{=}0\mathbf{)}$ is globally asymptotically stable. Thus, the proof is completed.

Remark 2

Traditional methods to construct artificial potential function cannot guarantee the strictly convex property, which is easier to lead to local minima in generating virtual angular velocity command. When the system is trapped into local minima, it is necessary to design an anti-loacal-minima control scheme. Taking advantage of the strictly convex potential function, the globally asymptotically stable controller proposed in this article has a better application value.

Remark 3

Introducing the existence conditions for the RPF helps avoid unnecessary detour in the attitude maneuver process. ²¹ However, because of the complex admissible attitude set ${\mathit{Q}}_A$ which consists of three keep-out zones and one keep-in zone in this article, the existence conditions may be changeable which implies that the control scheme may be chattering. As a consequence, the existence conditions are ignored in this article.

Design of backstepping controller with actuator saturation

In practical engineering applications, the actuators cannot provide infinite energy. If they cannot execute the controller command completely, the spacecraft may be unstable. Consequently, it is necessary to address the actuator saturation constraint.

State equations under actuator saturation are

$$\{\begin{array}{c}{\stackrel{·}{V}}_p={\bar{\omega }}^{\mathrm{T}}(\frac{1}{2}{\mathit{q}}^{*}\otimes \nabla V_p)\hfill \\ \multicolumn{1}{c}{\mathit{J}\stackrel{·}{\omega }=-{\mathbf{\omega }}^{\mathbf{\times }}\mathit{J}\mathbf{\omega }\mathbf{+}\mathrm{sat}\mathbf{(}\mathit{u}\mathbf{)}}\end{array}$$

(33)

We construct an auxiliary system

$$\stackrel{·}{\mathbf{\xi }}=-\mathit{G}\mathbf{\xi }+{\mathit{J}}^{-1}\Delta \mathit{u}$$

(34)

where $\mathbf{\xi }$ represents the auxiliary variable with respect to $\mathbf{\omega }$ ; $\mathit{G}$ is a positive definite matrix to be determined; $\Delta \mathit{u}=\mathrm{sat}(\mathit{u})-\mathit{u}$ .

Introduce new variables

$$\{\begin{array}{c}{z}_3=V_p\hfill \\ \multicolumn{1}{c}{{\mathit{z}}_4=\mathbf{\omega }\mathbf{-}\mathbf{\omega }\mathbf{c}1}\end{array}$$

(35)

Theorem 2

Considering the system governed by equation (33) with extended states (35), and the auxiliary system (34), there exist controller gains $s_3>0$ and $s_4>0$ such that the equilibrium $(\mathit{q}={\mathit{q}}_c,\mathbf{\omega }\mathbf{=}0\mathbf{)}$ is globally asymptotically stable under the following backstepping controller

$$\begin{array}{c}{\mathit{u}}_2=\mathbf{\omega }\mathbf{\times }\mathit{J}\mathbf{\omega }\mathbf{+}\mathit{J}{\stackrel{·}{\mathbf{\omega }}}_{\mathbf{c}1}\mathbf{-}\mathit{JG}\mathbf{\xi }\mathbf{-}{\mathbf{s}}_3\mathit{J}\mathbf{(}{\mathit{z}}_4\mathbf{-}\mathbf{\xi }\mathbf{)}\hfill \\ \multicolumn{1}{c}{-s_4^{-1}\mathit{J}\frac{{\mathit{z}}_4-\mathbf{\xi }}{\parallel {\mathit{z}}_4-\mathbf{\xi }{\parallel }^2}{\mathit{z}}_4^{\mathrm{T}}\mathrm{vec}[\frac{1}{2}{\mathit{q}}^{*}\otimes \nabla V_p]}\end{array}$$

(36)

Proof

Since $V_p>0$ , the extended Lyapunov function can be constructed as

$$V_2=V_p+\frac{1}{2}{s}_4{({\mathit{z}}_4-\mathbf{\xi })}^{\mathrm{T}}({\mathit{z}}_4-\mathbf{\xi })$$

(37)

where $s_4$ is a positive constant to be determined.

The time derivative of $V_2$ is given by

$$\begin{array}{c}{\stackrel{·}{V}}_2=\nabla V_p^{\mathrm{T}}(\frac{1}{2}\mathit{q}\otimes \bar{\mathbf{\omega }})+s_4{({\mathit{z}}_4-\mathbf{\xi })}^{\mathrm{T}}({\stackrel{·}{\mathit{z}}}_4-\stackrel{·}{\mathbf{\xi }})\hfill \\ \multicolumn{1}{c}{=\nabla V_p^{\mathrm{T}}(\frac{1}{2}\mathit{q}\otimes {\bar{\mathit{z}}}_4)+\nabla V_p^{\mathrm{T}}(\frac{1}{2}\mathit{q}\otimes {\bar{\mathbf{\omega }}}_{c1})}\\ \multicolumn{1}{c}{+s_4{({\mathit{z}}_4-\mathbf{\xi })}^{\mathrm{T}}(-{\mathit{J}}^{-1}\mathbf{\omega }\mathbf{\times }\mathit{J}\mathbf{\omega }\mathbf{+}{\mathit{J}}^{\mathbf{-}1}\mathrm{sat}\mathbf{(}\mathit{u}\mathbf{)}}\\ \multicolumn{1}{c}{-{\stackrel{·}{\mathbf{\omega }}}_{c1}+\mathit{G}\mathbf{\xi }\mathbf{-}{\mathit{J}}^{\mathbf{-}1}\mathbf{(}\mathrm{sat}\mathbf{\left(}\mathit{u}\mathbf{\right)}\mathbf{-}\mathit{u}\mathbf{)})}\\ \multicolumn{1}{c}{={\mathit{z}}_4^{\mathrm{T}}\mathrm{vec}[\frac{1}{2}{\mathit{q}}^{*}\otimes \nabla V_p]+{\bar{\omega }}_{c1}(\frac{1}{2}{\mathit{q}}^{*}\otimes \nabla V_p)}\\ \multicolumn{1}{c}{+s_4{({\mathit{z}}_4-\mathbf{\xi })}^{\mathrm{T}}(-{\mathit{J}}^{-1}\mathbf{\omega }\mathbf{\times }\mathit{J}\mathbf{\omega }}\\ \multicolumn{1}{c}{-{\stackrel{·}{\mathbf{\omega }}}_{c1}+\mathit{G}\mathbf{\xi }\mathbf{+}{\mathit{J}}^{\mathbf{-}1}\mathit{u})}\end{array}$$

(38)

Then, substituting controller (equation (36)) into the equation above, it follows that

$$\begin{array}{c}{\stackrel{·}{V}}_2=-\frac{1}{2}\alpha \arctan (\beta {\mathit{q}}^{*}\otimes \nabla V_p)({\mathit{q}}^{*}\otimes \nabla V_p)\hfill \\ \multicolumn{1}{c}{-s_3{s}_4{({\mathit{z}}_4-\mathbf{\xi })}^{\mathrm{T}}({\mathit{z}}_4-\mathbf{\xi })\le 0}\end{array}$$

(39)

It should be noted that if the actuators are always in saturation, the state $(z_3,{\mathit{z}}_4-\mathbf{\xi })$ will hardly be able to convergence to the origin. ²⁹ When the actuators are unsaturated, $\Delta \mathit{u}=0$ . Since $\mathit{G}$ is a positive definite matrix, the auxiliary system (equation (34)) will asymptotically converge to the origin. Thus, we have $\mathbf{\omega }\mathbf{=}0$ and $\nabla V_p=0$ when ${\stackrel{·}{V}}_2\equiv 0$ . Evidently, the global asymptotical stability proof of the equilibrium $(\mathit{q}={\mathit{q}}_c,\mathbf{\omega }\mathbf{=}0\mathbf{)}$ is analogous to that in Theorem 1. Thus, the proof is completed.

ESO-based controller design

The above backstepping controller has not considered the compound disturbance, which consists of external disturbance and parametric uncertainties. Although the controller is asymptotically stable, the robustness is weak. If the compound disturbance is excessively large, the system will be hard to be stabilized. For orbiting spacecraft, the estimation and compensation of the compound disturbance is difficult due to the complex sources, including gravitational perturbation, aerodynamic torque, and other environmental or non-environmental torques. In this section, the ESO will be proposed to enhance the robustness of the system.

The system can be rewritten as

$$\stackrel{·}{\mathbf{\omega }}=\mathit{f}\left(\mathbf{\omega }\right)+\mathit{g}·\mathrm{sat}\left(\mathit{u}\right)+\tilde{\mathit{d}}$$

(40)

where $\mathit{f}(\mathbf{\omega }\mathbf{)}\mathbf{=}\mathbf{-}{\mathit{J}}^{\mathbf{-}1}\mathbf{\omega }\mathbf{\times }\mathit{J}\mathbf{\omega }$ , $\mathit{g}={\mathit{J}}^{-1}$ , $\tilde{\mathit{d}}={\mathit{J}}^{-1}{\mathit{d}}_t$ . Introducing the ESO technique, ³⁰ let ${\mathit{x}}_1=\mathbf{\omega }$ , ${\mathit{x}}_2=\tilde{\mathit{d}}$ , then we have

$$\{\begin{array}{c}{\stackrel{·}{\mathit{x}}}_1=\mathit{f}\left(\mathbf{\omega }\right)+\mathit{g}·\mathrm{sat}\left(\mathit{u}\right)+{\mathit{x}}_2\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\mathit{x}}}_2=\mathit{h}(t)}\end{array}$$

(41)

where $\mathit{h}(t)$ is the time derivative of ${\tilde{\mathit{d}}}_t$ . According to the designed controller, and the assumption that $\stackrel{··}{\mathbf{\omega }}$ is bounded, $\mathit{h}(t)$ is bounded, that is, $\parallel \mathit{h}(t)\parallel <\delta$ .

Based on the finite-time observer in Zou, ³¹ the ESO is carried out as

$$\begin{array}{c}{\stackrel{·}{\mathit{z}}}_5=\mathit{f}\left(\mathbf{\omega }\right)+\mathit{g}·\mathrm{sat}\left(\mathit{u}\right)+{\mathit{z}}_6-{\varrho }_1\mathrm{si}{\mathrm{g}}^{\kappa }\left({\mathit{e}}_1\right)\hfill \\ {\stackrel{·}{\mathit{z}}}_6=-{\varrho }_2\mathrm{si}{\mathrm{g}}^{2\kappa -1}\left({\mathit{e}}_1\right)\hfill \end{array}$$

(42)

where ${\mathit{z}}_5$ and ${\mathit{z}}_6$ are the estimation of $\mathbf{\omega }$ and $\tilde{\mathit{d}}$ , respectively; ${\mathit{e}}_1={\mathit{z}}_5-{\mathit{x}}_1$ ; $\kappa \in (0.5,1)$ ; ${\varrho }_{1i},{\varrho }_{2i}>0,i=1,2,3$ .

The observation error ${\mathit{e}}_{\mathrm{eso}}=[{\mathit{e}}_1^{\mathrm{T}},{\mathit{e}}_2^{\mathrm{T}}{]}^{\mathrm{T}}$ of ESO (equation (42)) designed for system (equation (41)) can be expressed as

$$\{\begin{array}{c}{\mathit{e}}_1={\mathit{z}}_5-{\mathit{x}}_1\hfill \\ \multicolumn{1}{c}{{\mathit{e}}_2={\mathit{z}}_6-{\mathit{x}}_2}\end{array}$$

(43)

In addition, the dynamics of the observation error ${\mathbf{e}}_{\mathrm{eso}}$ can be described as

$$\{\begin{array}{c}{\stackrel{·}{\mathit{e}}}_1=-{\varrho }_1\mathrm{si}{\mathrm{g}}^{\kappa }\left({\mathit{e}}_1\right)+{\mathit{e}}_2\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\mathit{e}}}_2=-{\varrho }_2\mathrm{si}{\mathrm{g}}^{2\kappa -1}\left({\mathit{e}}_1\right)-\mathit{h}(t)}\end{array}$$

(44)

Theorem 3

For the uncertain system (equation (41)) and the ESO (equation (42)), set the initial estimation ${\mathit{z}}_5(t_0)$ and ${\mathit{z}}_6(t_0)$ with appropriate ${\varrho }_{1i},{\varrho }_{2i}>0,i=1,2,3$ . If external stochastic disturbance $\mathit{d}$ meets Assumption 1, and $\stackrel{··}{\mathbf{\omega }}$ is bounded, the observation error ${\mathit{e}}_{\mathrm{eso}}$ described in equation (44) will converge to a small neighborhood of origin in finite time.

Proof

The candidate Lyapunov function is selected as

$$V_3={\mathbf{\varsigma }}^{\mathrm{T}}\mathit{P}\mathbf{\varsigma }$$

(45)

where $\mathbf{\varsigma }\mathbf{=}\mathbf{[}\mathrm{si}{\mathrm{g}}^{\mathbf{\kappa }}\mathbf{(}{\mathit{e}}_{1\mathbf{i}}\mathbf{)}\mathbf{,}{\mathit{e}}_{2\mathbf{i}}{\mathbf{]}}^{\mathrm{T}}$ and

$$\mathit{P}=\frac{1}{2}\left[\begin{array}{cc}\frac{2{\varrho }_{2i}}{\kappa }+{\varrho }_{1i}^2\hfill & -{\varrho }_{1i}\hfill \\ \multicolumn{1}{c}{-{\varrho }_{1i}}& 2\hfill \end{array}\right]$$

(46)

Thus, $V_3$ is continuous but is not differentiable at ${\mathit{e}}_{1i}=0$ . $V_3$ is a strong Lyapunov function, that is, $V_3$ is positive definite and radially unbounded, and

$${\lambda }_{\min }\left(\mathit{P}\right)\parallel \mathbf{\varsigma }{\mathbf{\parallel }}^2\mathbf{\le }{\mathbf{V}}_3\mathbf{\le }{\mathbf{\lambda }}_{\mathbf{max}}\mathbf{\left(}\mathit{P}\mathbf{\right)}\mathbf{\parallel }\mathbf{\varsigma }{\mathbf{\parallel }}^2$$

(47)

where $\parallel \mathbf{\varsigma }{\mathbf{\parallel }}^2\mathbf{=}\mathbf{|}{\mathit{e}}_{1\mathbf{i}}{\mathbf{|}}^{2\mathbf{\kappa }}\mathbf{+}{\mathit{e}}_{2\mathbf{i}}^2$ .

The time derivative of $\mathbf{\varsigma }$ is achieved as

$$\begin{array}{c}\stackrel{·}{\mathbf{\varsigma }}=\left[\begin{array}{c}\kappa |{\mathit{e}}_{1i}{|}^{\kappa -1}(-{\varrho }_{1i}\mathrm{si}{\mathrm{g}}^{\kappa }({\mathit{e}}_{1i})+{\mathit{e}}_{2i})\hfill \\ \multicolumn{1}{c}{-{\varrho }_{2i}\mathrm{si}{\mathrm{g}}^{2\kappa -1}({\mathit{e}}_{1i})-{\mathit{h}}_{\mathit{i}}(t)}\end{array}\right]\hfill \\ \multicolumn{1}{c}{=\mathit{A}|{\mathit{e}}_{1i}{|}^{\kappa -1}\varsigma +\mathit{B}{\mathit{h}}_i(t)}\end{array}$$

(48)

where

$$\mathit{A}=\left[\begin{array}{cc}-{\varrho }_{1i}\kappa \hfill & \kappa \hfill \\ \multicolumn{1}{c}{-{\varrho }_{2i}}& 0\hfill \end{array}\right]$$

and $\mathit{B}=[0,-1{]}^{\mathrm{T}}$ .

The time derivative of $V_3$ is

$$\begin{array}{c}{\stackrel{·}{V}}_3={\stackrel{·}{\mathbf{\varsigma }}}^{\mathrm{T}}\mathit{P}\mathbf{\varsigma }\mathbf{+}\mathbf{\varsigma }\mathrm{T}\mathit{P}\stackrel{·}{\mathbf{\varsigma }}\hfill \\ \multicolumn{1}{c}{={(\mathit{A}|{\mathit{e}}_{1i}{|}^{\kappa -1}\mathbf{\varsigma }\mathbf{+}\mathit{B}{\mathit{h}}_{\mathbf{i}}\mathbf{(}\mathbf{t}\mathbf{)})}^{\mathrm{T}}\mathit{P}\mathbf{\varsigma }}\\ \multicolumn{1}{c}{+\mathbf{\varsigma }\mathrm{T}\mathit{P}(\mathit{A}|{\mathit{e}}_{1i}{|}^{\kappa -1}\mathbf{\varsigma }\mathbf{+}\mathit{B}{\mathit{h}}_{\mathbf{i}}\mathbf{(}\mathbf{t}\mathbf{)})}\\ \multicolumn{1}{c}{=|{\mathit{e}}_{1i}{|}^{\kappa -1}\mathbf{\varsigma }\mathrm{T}({\mathit{A}}^{\mathrm{T}}\mathit{P}+\mathit{PA})\varsigma +2{\mathit{h}}_i(t)\mathit{R}\mathbf{\varsigma }}\end{array}$$

(49)

where $\mathit{R}={\mathit{B}}^{\mathrm{T}}\mathit{P}=[{\varrho }_{1i}/2,-1]$ . Since $\mathit{A}$ is a Hurwitz matrix, there exists a positive definite matrix $\mathit{Q}$ satisfying

$${\mathit{A}}^{\mathrm{T}}\mathit{P}+\mathit{PA}=-\mathit{Q}$$

(50)

where

$$\mathit{Q}=\frac{{\varrho }_{1i}}{2}\left[\begin{array}{cc}2{\varrho }_{2i}+2{\varrho }_{1i}^2\kappa \hfill & -2{\varrho }_{1i}\kappa \hfill \\ \multicolumn{1}{c}{-2{\varrho }_{1i}\kappa }& 2\kappa \hfill \end{array}\right]$$

(51)

Therefore

$$\begin{array}{c}{\stackrel{·}{V}}_3=-|{\mathit{e}}_{1i}{|}^{\kappa -1}\mathbf{\varsigma }\mathrm{T}\mathit{Q}\mathbf{\varsigma }\mathbf{+}2{\mathit{h}}_{\mathbf{i}}\mathbf{(}\mathbf{t}\mathbf{)}\mathit{R}\mathbf{\varsigma }\hfill \\ \multicolumn{1}{c}{\le -|{\mathit{e}}_{1i}{|}^{\kappa -1}{\lambda }_{\min }(\mathit{Q})\parallel \mathbf{\varsigma }{\mathbf{\parallel }}^2\mathbf{+}2{\mathbf{\lambda }}_{\mathbf{max}}\mathbf{(}{\mathit{J}}^{\mathbf{-}1}\mathbf{)}\mathbf{\delta }\mathbf{\parallel }\mathit{R}\mathbf{\parallel }\mathbf{\parallel }\mathbf{\varsigma }\mathbf{\parallel }}\end{array}$$

(52)

Considering $|{\mathit{e}}_{1i}{|}^{\kappa -1}\ge \parallel \mathbf{\varsigma }{\mathbf{\parallel }}^{\frac{\mathbf{\kappa }\mathbf{-}1}{\mathbf{\kappa }}}$ , it follows that

$$\begin{array}{c}{\stackrel{·}{V}}_3\le -{\lambda }_{\min }(\mathit{Q})\parallel \mathbf{\varsigma }{\mathbf{\parallel }}^{\frac{3\mathbf{\kappa }\mathbf{-}1}{\mathbf{\kappa }}}\mathbf{+}2{\mathbf{\lambda }}_{\mathbf{max}}\mathbf{(}{\mathit{J}}^{\mathbf{-}1}\mathbf{)}\mathbf{\delta }\mathbf{\parallel }\mathit{R}\mathbf{\parallel }\mathbf{\parallel }\mathbf{\varsigma }\mathbf{\parallel }\hfill \\ \multicolumn{1}{c}{=-({\lambda }_{\min }(\mathit{Q})\parallel \mathbf{\varsigma }{\mathbf{\parallel }}^{\frac{2\mathbf{\kappa }\mathbf{-}1}{\mathbf{\kappa }}}\mathbf{-}2{\mathbf{\lambda }}_{\mathbf{max}}\mathbf{(}{\mathit{J}}^{\mathbf{-}1}\mathbf{)}\mathbf{\delta }\mathbf{\parallel }\mathit{R}\mathbf{\parallel })\parallel \mathbf{\varsigma }\mathbf{\parallel }}\end{array}$$

(53)

When

$$\parallel \mathbf{\varsigma }{\mathbf{\parallel }}^{\frac{2\mathbf{\kappa }\mathbf{-}1}{\mathbf{\kappa }}}>\frac{2{\mathbf{\lambda }}_{\mathbf{max}}\mathbf{(}{\mathit{J}}^{\mathbf{-}1}\mathbf{)}\mathbf{\delta }\mathbf{\parallel }\mathit{R}\mathbf{\parallel }}{{\mathbf{\lambda }}_{\mathbf{min}}\mathbf{(}\mathit{Q}\mathbf{)}}$$

(54)

there exists a constant $\vartheta$ satisfying

$${\lambda }_{\min }(\mathit{Q})\parallel \mathbf{\varsigma }{\mathbf{\parallel }}^{\frac{2\mathbf{\kappa }\mathbf{-}1}{\mathbf{\kappa }}}\mathbf{-}2{\mathbf{\lambda }}_{\mathbf{max}}\mathbf{(}{\mathit{J}}^{\mathbf{-}1}\mathbf{)}\mathbf{\delta }\mathbf{\parallel }\mathit{R}\mathbf{\parallel }>\mathbf{\vartheta }>0$$

(55)

Since

$$\parallel \mathbf{\varsigma }\mathbf{\parallel }\mathbf{\ge }\frac{{\mathbf{V}}_3^{\frac{1}{2}}}{{\mathbf{\lambda }}_{\mathbf{max}}^{\frac{1}{2}}\mathbf{(}\mathit{P}\mathbf{)}}$$

(56)

it yields

$${\stackrel{·}{V}}_3<-\vartheta \parallel \mathbf{\varsigma }\mathbf{\parallel }\mathbf{\le }\mathbf{-}\frac{\mathbf{\vartheta }}{{\mathbf{\lambda }}_{\mathbf{max}}^{\frac{1}{2}}\mathbf{(}\mathit{P}\mathbf{)}}{\mathbf{V}}_3^{\frac{1}{2}}$$

(57)

Then, by Lemma 2, ${\mathbf{e}}_{\mathrm{eso}}$ converges to the region

$$\{{\mathit{e}}_{\mathrm{eso}}|\parallel \mathbf{\varsigma }\mathbf{\parallel }\mathbf{\le }{\mathbf{\left(}\frac{2{\mathbf{\lambda }}_{\mathbf{max}}\mathbf{(}{\mathit{J}}^{\mathbf{-}1}\mathbf{)}\mathbf{\delta }\mathbf{\parallel }\mathit{R}\mathbf{\parallel }}{{\mathbf{\lambda }}_{\mathbf{min}}\mathbf{(}\mathit{Q}\mathbf{)}}\mathbf{\right)}}^{\frac{\mathbf{\kappa }}{2\mathbf{\kappa }\mathbf{-}1}}\}$$

(58)

in finite time. Thus, the proof is completed.

Consequently, the ESO-based controllers can be modified based on controller (equations (29) and (36)) as

$$\begin{array}{c}{\mathit{u}}_1=\mathbf{\omega }\mathbf{\times }\mathit{J}\mathbf{\omega }\mathbf{+}\mathit{J}{\stackrel{·}{\mathbf{\omega }}}_{\mathbf{c}1}\mathbf{-}{\mathbf{s}}_1\mathit{J}{\mathit{z}}_2\hfill \\ \multicolumn{1}{c}{-s_2^{-1}\mathit{J}\mathrm{vec}[\frac{1}{2}{\mathit{q}}^{*}\otimes \nabla V_p]-\mathit{J}{\mathit{z}}_6}\end{array}$$

(59)

$$\begin{array}{c}{\mathit{u}}_2=\mathbf{\omega }\mathbf{\times }\mathit{J}\mathbf{\omega }\mathbf{+}\mathit{J}{\stackrel{·}{\mathbf{\omega }}}_{\mathbf{c}1}\mathbf{-}\mathit{JG}\mathbf{\xi }\mathbf{-}{\mathbf{s}}_3\mathit{J}\mathbf{(}{\mathit{z}}_4\mathbf{-}\mathbf{\xi }\mathbf{)}\hfill \\ \multicolumn{1}{c}{-s_4^{-1}\mathit{J}\frac{{\mathit{z}}_4-\mathbf{\xi }}{\parallel {\mathit{z}}_4-\mathbf{\xi }{\parallel }^2}{\mathit{z}}_4^{\mathrm{T}}\mathrm{vec}[\frac{1}{2}{\mathit{q}}^{*}\otimes \nabla V_p]}\\ \multicolumn{1}{c}{-\mathit{J}{\mathit{z}}_6}\end{array}$$

(60)

The system block diagram for the ESO-based controller in this article is shown in Figure 2.

Figure 2.

System block diagram for the ESO-based controller.

Theorem 4

Under Assumption 1, the ESO-based controllers (equations (59) and (60)) are uniformly bounded.

Proof

1. Proof of controller (equation (59)):

Considering equation (40) without actuator saturation, equation (31) can be modified as

$$\begin{array}{c}{\stackrel{·}{V}}_1={\mathit{z}}_2^{\mathrm{T}}\mathrm{vec}[\frac{1}{2}{\mathit{q}}^{*}\otimes \nabla V_p]+{\bar{\mathbf{\omega }}}_{c1}(\frac{1}{2}{\mathit{q}}^{*}\otimes \nabla V_p)\hfill \\ \multicolumn{1}{c}{+s_2{\mathit{z}}_2^{\mathrm{T}}(-{\mathit{J}}^{-1}\mathbf{\omega }\mathbf{\times }\mathit{J}\mathbf{\omega }\mathbf{+}{\mathit{J}}^{\mathbf{-}1}\mathit{u}\mathbf{+}\tilde{\mathit{d}}\mathbf{-}{\stackrel{·}{\mathbf{\omega }}}_{\mathbf{c}1})}\end{array}$$

(61)

Then, substituting controller (equation (59)) into above equation, it will be

$$\begin{array}{c}{\stackrel{·}{V}}_1=-\frac{1}{2}\alpha \arctan (\beta {\mathit{q}}^{*}\otimes \nabla V_p)({\mathit{q}}^{*}\otimes \nabla V_p)\hfill \\ \multicolumn{1}{c}{-s_1{s}_2{\mathit{z}}_2^{\mathrm{T}}{\mathit{z}}_2-s_2{\mathit{z}}_2^{\mathrm{T}}{\mathit{e}}_2}\\ \multicolumn{1}{c}{\le -\frac{1}{2}\alpha \arctan (\beta {\mathit{q}}^{*}\otimes \nabla V_p)({\mathit{q}}^{*}\otimes \nabla V_p)}\\ \multicolumn{1}{c}{-s_1{s}_2\parallel {\mathit{z}}_2{\parallel }^2+s_2\parallel {\mathit{z}}_2\parallel \parallel {\mathit{e}}_2\parallel }\end{array}$$

(62)

According to Theorem 3, there exists a positive constant ${\vartheta }_1$ satisfying $\parallel {\mathbf{e}}_2\parallel \le {\vartheta }_1$ . So, it can be obtained that

$${\stackrel{·}{V}}_1\le 0,\forall \parallel {\mathit{z}}_2\parallel \ge \frac{{\vartheta }_1}{s_1}$$

(63)

Thus, controller (equation (59)) is uniformly bounded.

• 2. Proof of controller (equation (60)):

Due to Theorem 2, the auxiliary system $\mathbf{\xi }$ will asymptotically converge to the origin, which is an useful property to the proof process. And the process is analogously to the above one, which is omitted here.

Thus, the proof is completed.

Simulation results and discussions

In this section, to illustrate the performance of proposed ESO-based controllers, numerical simulations have been carried out for the attitude reorientation maneuver mission in the presence of attitude constraint, parametric uncertainty, external disturbance, and actuator saturation. The simulation environment is given in Table 1, including parameters of spacecraft, maneuver target, and descriptions of keep-in and keep-out zones.

Table 1.

The simulation parameters.

	Parameter value
Spacecraft	$\mathit{J}=\mathrm{diag}(80,60,70)\mathrm{kg}{\mathrm{m}}^2$
	$\Delta \mathit{J}=\mathrm{diag}(8,-6,7)\mathrm{kg}{\mathrm{m}}^2$
	${\mathit{u}}_{\max }=0.5\mathrm{N}\mathrm{m}$
Target	${\mathit{q}}_{\mathrm{init}}=[0.2600,0.7140,0.6370,-0.1300{]}^{\mathrm{T}}$
	${\mathit{q}}_c=[-0.8401,-0.2300,0.0080,0.4911{]}^{\mathrm{T}}$
Keep-out zones	${\mathit{y}}_1=[0,0,1{]}^{\mathrm{T}}$
	${\mathit{x}}_1=[0,-1,0{]}^{\mathrm{T}}$ , ${\theta }_1^1={40}^{\circ }$
	${\mathit{x}}_2=[0,0.8194,0.5733{]}^{\mathrm{T}}$ , ${\theta }_2^1={40}^{\circ }$
	${\mathit{x}}_3=[-0.1221,-0.1391,-0.9827{]}^{\mathrm{T}}$
	${\theta }_3^1={20}^{\circ }$
Keep-in zone	${\mathit{y}}_2=[0,1,0{]}^{\mathrm{T}}$
	${\mathit{x}}_4=[0.8138,-0.5485,0.1922{]}^{\mathrm{T}}$
	${\theta }_{\mathrm{KI}}={60}^{\circ }$

It is assumed that the maximum angular velocity of spacecraft is $\mathbf{\omega }\mathbf{max}=2^{\circ }/\mathrm{s}$ . The slowly varying bounded external disturbance is supposed to be

$$\mathit{d}=\left[\begin{array}{c}4-5\sin (2\pi t/600)\hfill \\ \multicolumn{1}{c}{4+5\sin (2\pi t/600)}\\ \multicolumn{1}{c}{4-5\sin (2\pi t/600)}\end{array}\right]\times {10}^{-3}\mathrm{N}\mathrm{m}$$

(64)

In order to achieve the control object, the parameters of controllers and ESO are chosen as Table 2.

Table 2.

Controller parameters.

	Parameter value
Potential function	$k_1=0.05$ , $k_2=0.05$
Controller	$\alpha =0.02$ , $\beta =200$
	$s_1=1$ , $s_2=1$
	$s_3=1$ , $s_4=1$ , $\mathit{G}=100{\mathrm{J}}^{-1}$
ESO	${\varrho }_1=[1,1,1{]}^{\mathrm{T}}$ , ${\varrho }_2=[1,1,1{]}^{\mathrm{T}}$ , $\kappa =2/3$

ESO: extended state observer.

To illustrate the performance and effectiveness, numerical simulations are realized through MATLAB/Simulink and divided into three cases. All three cases include parametric uncertainty and external disturbance. The differences are shown as follows:

• Case 1: Ignore the actuator saturation, that is, utilize the controller (equation (59)).

• Case 2: Take into account the constraint of actuator saturation, that is, utilize the controller (equation (60)).

• Case 3: Simulate the backstepping controller proposed by Lee and Mesbahi ²⁴ as a comparison.

Figure 3 shows the time response of Case 1, including the zyx-ordered error Euler angles, the control torque $\mathit{u}$ , and the compound disturbance ${\mathit{d}}_t$ together with its observation error ${\mathit{e}}_{\mathrm{dt}}$ .

Figure 3.

Simulation results of Case 1.

It can be seen from Figure 3 that the error Euler angles fall into the tolerance within $33\mathrm{s}$ and be less than ${10}^{-5}\circ$ . The observer error ${\mathit{e}}_{\mathrm{dt}}$ converges to a small neighborhood of origin in $7\mathrm{s}$ , which illustrating the good performance of the proposed ESO for estimating the compound disturbance. Figure 4 illustrates the movement trajectories (from point “∘” to “×”) of instruments ${\mathit{y}}_1$ and ${\mathit{y}}_2$ presented in celestial coordinate system, which demonstrates that the attitude $\mathit{q}\in {\mathit{Q}}_A$ is satisfied for all times. Although the control object is achieved, the peak torque in the maneuver mission is almost $125\mathrm{N}\mathrm{m}$ , which is too large to be realized in engineering applications. Considering the physical constrain of actuators, the untreated command torque cannot be fully executed. Consequently, without the stability guarantee, the control object will possibly be failed, including being unable to converge to target and violating the constraint of the admissible zone.

Figure 4.

Trajectories of ${\mathit{y}}_1$ and ${\mathit{y}}_2$ in 2D projection of Case 1.

In order to eliminate this drawback of not being able to deal with actuator saturation, the controller (equation (60)) has been proposed with the same controller parameters. The numerical simulation results are shown in Figure 5, including the zyx-ordered error Euler angles, spacecraft angular velocity $\mathbf{\omega }$ , the control torque $\mathit{u}$ , the auxiliary system state $\mathbf{\xi }$ , and the compound disturbance ${\mathit{d}}_t$ together with its observation error ${\mathit{e}}_{\mathrm{dt}}$ .

Figure 5.

Simulation results of Case 2.

As can be seen from Figure 5, the controller (equation (60)) is able to drive the spacecraft attitude to the target without any constraint violation, which is also illustrated in Figure 6. It is clearly that the command torque maintains in the limitation during the whole maneuver process. Compared with Case 1, the convergence time is extended to $130\mathrm{s}$ due to the limitation of actuator saturation.

Figure 6.

Trajectories of ${\mathit{y}}_1$ and ${\mathit{y}}_2$ in 2D projection of Case 2.

What is more, by introducing the auxiliary system $\mathbf{\xi }$ , without any changes of controller parameters, the controller ensures the system stability while satisfying the actuator saturation constraint. Thus, the controller (equation (60)) is more effective and reliable. The auxiliary system is driven by $\Delta \mathit{u}$ , which is the difference of the saturated torque $\mathrm{sat}(\mathit{u})$ and untreated command torque $\mathit{u}$ . Looking into the time responses of $\mathit{u}$ and $\mathbf{\xi }$ , it can be found that if the command torque is saturated, that is, $\Delta \mathit{u}\ne 0$ , $\mathbf{\xi }$ will no longer be zero. On the contrary, $\mathbf{\xi }$ will asymptotically converges to the origin with $\Delta \mathit{u}=0$ while making a contribution to the command torque $\mathit{u}$ . In other words, the excess part of $\mathit{u}$ due to the actuator saturation constraint will be stored through the auxiliary system. Moreover, as a highlight, it will be released when $\mathit{u}$ is unsaturated. The method proposed in this article does not rely on experience and trail and has better universality and reliability.

Analyzing the time responses of the observation error ${\mathit{e}}_{\mathrm{dt}}$ in Figures 3 and 5, it can be seen that both of them fall into an admissible tolerance with a short period of time. Hence, the ESO employed in controller (equations (59) and (60)) can compensate the compound disturbance ${\mathit{d}}_t$ effectively, which is beneficial for robustness and accuracy of attitude maneuver controller.

To illustrate the effective of the proposed controllers in this article, some comparative simulation using the backstepping controller proposed by Lee and Mesbahi ²⁴ has also been carried out, as shown in Figures 7 and 8. Comparing Figures 5 and 7, the controller proposed in this article has a smaller convergence time and higher attitude accuracy, that is, it is more effective than the traditional algorithm.

Figure 7.

Simulation results of Case 3.

Figure 8.

Trajectories of ${\mathit{y}}_1$ and ${\mathit{y}}_2$ in 2D projection of Case 3.

Conclusion

The spacecraft attitude reorientation task in the presence of attitude constraint, parametric uncertainty, external disturbance, and actuator saturation was investigated. A novel feedback controller was designed based on the potential function, the nonlinear tracking law, and the ESO. First, a strictly convex logarithmic barrier potential function was designed to ensure that the tracking law has one and only one minimum, namely global minimum. To guarantee the stability of the control system under actuator saturation, an auxiliary system was introduced in the backstepping controller. Then, to achieve real-time compensation of the compound disturbance which consists of parametric uncertainty and external disturbance, the ESO was proposed. Therefore, a better robustness of controller was obtained. The stability of all proposed algorithms was proved by Lyapunov method. Finally, the numerical simulations with fine performance illustrated the reliability and effectiveness of the proposed controller.

However, time or fuel optimization problem is not considered in this article, so the follow-up work needs to deal with the optimization problem in the presence of attitude constraint.