Reaction Wheel Installation Deviation Compensation for Overactuated Spacecraft with Finite-Time Attitude Control

Abstract

A novel attitude tracking control scheme is presented for overactuated spacecraft to address the attitude stabilization problem in presence of reaction wheel installation deviation, external disturbance and uncertain mass of moment inertia. An adaptive sliding mode control technique is proposed to track the uncertainty. A Lyapunov-based analysis shows that the compensation control law can guarantee that the desired attitude trajectories are followed in finite-time. The key feature of the proposed control strategy is that it globally asymptotically stabilizes the system, even in the presence of reaction wheel installation deviation, external disturbances, and uncertain mass of moment inertia. The attitude track performance using the proposed finite-time compensation control is evaluated through a numerical example.

1. Introduction

In present, nearly all of the highly accurate slewing maneuvers necessitate the use of nonlinear differential equations for the kinematics and dynamics during the control system design [1]. However, the attitude tracking problem is further complicated by the external disturbance and uncertain mass of moment inertia. To address these issues, there have been several important developments in the design of feedback control laws for spacecraft maneuvering. A number of control design approaches using adaptive control [2, 3], sliding mode control [4–7], 𝐻_∞ [8, 9], optimal control [10–13], and data driven control [14–16] have been proposed. However, few of them focus on the reaction wheel installation deviation that are of great theoretical and practical interest. In fact, the installation deviation is a widespread phenomenon, such as the actuator misalignment which is limited by the installation technique or generated by materials deforms the vehicle violent vibration during the launching process. In the area of actuator misalignment compensation, there currently exist few unified frameworks for the design of simple control structures.

Several solutions to actuator installation deviation have been presented in the literature [17–20]. In [17], the authors presented a general adaptive tracking attitude controller design framework for a spacecraft subject to the actuator installation minor angle deviation. In [18], an adaptive attitude tracking method is proposed to compensate the actuator misalignment of nearly 15 degree. And in [19], a novel algorithm is employed precisely to estimate the information, such as installation angle of wheel and CMG alignments. And then the controller design can be on for the estimation information. Moreover, another recent paper in [20] proposed an adaptive control approach for satellite formation flying, in which backstepping technique is used to synthesize a controller to handle thrust magnitude error and misalignment. However, the torque is different between thruster and reaction wheel, one is literal and the other is time-variable. That is to say that this control strategy is not suitable for reaction wheel installation deviation compensation for overactuated spacecraft attitude control.

Treating the uncertain mass of moment inertia caused by the reaction wheel misalignment is another impossibly avoided problem. In practice, in order to ensure the reliability of on-orbit spacecraft operation, especially under high altitude sever external environment, overactuatation is widely employed to guarantee the control system reliability service. And finite-time is meanwhile necessary for time critical missions. As a result, more and more investigations also have focused on attitude control design with finite-time convergence. In [21–23], the finite-time control technique was applied to design an attitude controller. Feng et al. [24] proposed a terminal sliding mode controller to solve the singular problem for a second-order nonlinear dynamic system. A terminal sliding mode and the Chebyshev neural network were used in [25] to guarantee that the attitude manoeuvre was accomplished in finite time, even in the presence of an unknown inertia matrix, external disturbances, and control input constraints. Furthermore, two robust sliding mode controllers were proposed in [26] to realize attitude tracking in finite-time. Similar finite-time fault tolerant controllers for spacecrafts were investigated in [27–29].

This work focuses on developing a control scheme to perform attitude compensation for an overactuated spacecraft with reaction wheel installation deviation, external disturbances, and uncertain inertia parameters. More specifically, the attitude tracking error is required to be zero in finite time. The proposed approach is illustrated in Figure 1. The compensation control module is added to the output of the nominal controller to compensate for the reaction wheel misalignment, disturbances, and uncertain moment of inertia. The proposed scheme solves a difficult problem of reliable and high accuracy attitude tracking control in finite time that rejects external disturbances and, at the same time, compensates for actuator misalignment and system uncertainties so that the control objective is met.

Figure 1 

Structure of the attitude compensation controller.

The remainder of this paper is organized as follows. In Section 2, we summarize the mathematical model for the rigid spacecraft attitude and control problem. A compensation control solution with the misalignment, disturbance, and mass moment of inertia is presented in Section 3. Simulation results are presented in Section 4. Some conclusions are given in Section 5.

2. Mathematical Model and Problem Formulation

The notation adopted throughout this paper is introduced as follows. The symbol ‖⋅‖ denotes the standard Euclidean norm or its induced norm; the symbol ‖⋅‖_∞ denotes the infinite norm of a vector or matrix. For any given matrix 𝐀∈𝐑^𝑝×𝑞 with full row rank, 𝐀^† denotes its pseudoinverse.

2.1. Dynamic Model of Rigid Spacecraft

Consider a rigid space system described by the following attitude kinematics and dynamics equations [30]: ̇1𝐪=2𝐪^×+𝑞₀𝐈₃𝑞̇𝝎,₀1=−2𝐪^𝑇𝐉̇𝝎,𝝎+𝝎^×𝐉𝝎=𝐮+𝐝, (1)(2)(3)

where 𝝎∈𝐑³ is the angular velocity of a body-fixed reference frame expressed in the body-fixed reference frame, 𝐉∈𝐑^3×3 (positive and definite) is the total inertia matrix of the spacecraft, 𝐮=𝑢₁𝑢₂𝑢₃^𝑇∈𝐑³ denotes the combined control torque produced by the actuators, and 𝐝(𝑡)=𝑑₁𝑑₂𝑑₃^𝑇∈𝐑³ denotes the external disturbance torque from the environment, which is assumed to be unknown but bounded; 𝑞₀, 𝐪 are the scalar and vector components of the unit quaternion, respectively, with 𝐪=𝑞₁𝑞₂𝑞₃^𝑇∈𝐑³, satisfying the constraint 𝑞²₀+𝐪^𝑇𝐪=1; 𝐈₃ represents the identity matrix with proper dimensions, and for ∀𝐚=𝑎₁𝑎₂𝑎₃^𝑇, 𝐚^× denotes a skew-symmetric matrix, more precisely, 𝐚^×=0−𝑎₃𝑎₂𝑎₃0−𝑎₁−𝑎₂𝑎₁0. (4)

2.2. Reaction Wheel Configuration with Installation Deviation

For orbiting spacecraft, loosely speaking, they have more than three reaction wheels aligned with the spacecraft body axes. However, in practice, the configuration of actuators will never be perfect; that is, to say, whether due to finite manufacturing tolerances or warping of the spacecraft structure during launch, some alignment errors can always exist. Thus, in this section, the reaction wheels misalignment is taken into consideration; the faulty dynamics can be described by 𝐉𝝎=−𝝎^×𝐉𝝎+(𝐃+Δ𝐃)𝝉+𝐝, (5)

where 𝐃∈𝐑^3×𝑁 denotes the actuator distribution matrix, Δ𝐃 denotes the actuator distribution matrix induced by misalignment, and 𝝉=[𝜏₁,…,𝜏_𝑁]^𝑇∈𝐑^𝑁 denotes the actual output torque of the 𝑁 reaction wheels.

Due to the rotation of the payload or the existence of the flywheel installation deviation, the moment of inertia 𝐉 is uncertain but positive definite symmetric matrices and record 𝐉=𝐉₀+Δ𝐉, where 𝐉₀ denotes the nominal rotational inertia and Δ𝐉 denotes the uncertain rotational inertia. Here set 0<‖Δ𝐉‖≤‖𝐉‖≤𝐽_max<∞ and 𝐽_max is a positive constant.

2.3. Attitude Tracking Model

Assume that the desired attitude to be followed is described with a desired frame 𝒯 with respect to ℐ. It is specified by the desired unit quaternion 𝐐_𝐝=(𝑞_𝑑0,𝐪^𝑇_𝐝)∈𝐑×𝐑³. The desired angular velocity is denoted by 𝝎_𝐝∈𝐑³. Let the error quaternion 𝐐_𝐞=(𝑒₀,𝐞^𝑇)∈𝐑×𝐑³ denote the attitude between ℬ and 𝒯, and let 𝝎_𝐞∈𝐑³ represent the corresponding error angular velocity. One has 𝝎_𝐞=𝝎−𝐑𝝎_𝐝, (6)

where 𝐑∈𝐑^3×3 denote the corresponding rotation matrix that brings 𝒯 onto ℬ, and 𝐑=𝑒²₀−𝐞^𝑇𝐞𝐈₃+2𝐞𝐞^𝑇−2𝑒₀𝐞^×. (7)

With (1)–(5), the attitude tracking error dynamics is given by: 𝐉𝝎̇_𝐞+𝝎̇_𝐞+𝐑𝝎_𝐝^×𝐉𝝎_𝐞+𝐑𝝎_𝐝−𝐉𝝎^×_𝐞𝐑𝝎_𝐝−𝐑𝝎_𝐝=()𝐄(𝑡)𝝉(𝑡)(𝑡),𝑒̇𝐃+Δ𝐃+𝐝₀1=−2𝐞^𝑇𝝎_𝐞,̇1𝐞=2𝐞^×+𝑒₀𝐈₃𝝎_𝐞. (8)(9)(10)

2.4. Control Objective

The control objective of this work can be stated as considering the uncertain attitude tracking system (8)–(10) and design a control law to guarantee that the attitude tracking error converges to zero in finite-time, even in the presence of actuator misalignment, uncertain inertia matrix, and external disturbance 𝐝(𝑡).

We present now the main results of this study.

3. Finite-Time Attitude Compensation Control

For the proposed control approach shown in Figure 1, the nominal control power and the compensation control effort are presented in this section. First, a finite-time sliding mode surface is proposed. Then, based on the finite-time sliding mode surface, a compensation controller is synthesized and added to the nominal controller to guarantee the global asymptotic stability of the resulting closed-loop attitude tracking system with finite-time convergence.

3.1. Finite-Time Sliding Mode Surface Design

We first introduce some lemmas which will be utilized in the subsequent control development and analysis.

Lemma 1 (see [31]). If 𝑝∈(0,1), then the following inequality holds for any vector 𝐱=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇∈𝐑^𝑛: ^𝑛_𝑖=1𝑥_𝑖^1+𝑝≥^𝑛_𝑖=1𝑥_𝑖^2(1+𝑝)/2. (11)

Lemma 2 (see [32]). Suppose that 𝐕(𝐱) is a 𝐶¹ smooth positive-definite function such that 𝐕̇(𝐱)+𝜆₁𝐕(𝐱)+𝜆₂𝐕^𝛽(𝐱)≤0, (12)
where 𝜆₁∈𝐑⁺, 𝜆₂∈𝐑⁺, 𝛽∈𝐑⁺ and 0<𝛽<1. Then for any initial value 𝑥(0)=𝑥₀, it follows that 𝐕(𝐱(𝑡))=0 for all the time 𝑡≥𝑡_𝐹1, 𝑡_𝐹1≤1𝜆₁𝜆1−𝛽ln¹𝐕^1−𝛽𝑥₀+𝜆₂𝜆₂. (13)
To this end, in this work, a sliding mode surface is introduced as 𝐬=𝝎_𝐞+𝜇₁𝐞+𝜇₂sgn(𝐞)^𝑟, (14)
where 𝐬=𝑠₁𝑠₂𝑠₃^𝑇∈𝐑³, 𝜇₁>0, 𝜇₂>0, and 0<𝛾<1 are the design parameters and sgn(𝐞) is the sign function defined by sgn(𝐞)^𝑟=𝑒₁^𝑟sgn𝑒₁,𝑒₂^𝑟sgn𝑒₂,𝑒₃^𝑟sgn𝑒₃^𝑇. (15)

Theorem 3. If an controller 𝐮(𝑡) is appropriately designed to let the states reach the sliding surface 𝐬, then it has 𝐞(𝑡)≡0, 𝑒₀(𝑡)≡1, and 𝝎_𝐞(𝑡)≡0 for all the 𝑡≥𝑡_𝐹1.

Proof. From the sliding mode theory [33], it is known that once the state trajectories of the attitude tracking system reach the sliding surface, that is, 𝐬=𝟎, it follows that 𝝎_𝐞=−𝜇₁𝐞−𝜇₂sgn(𝐞)^𝑟. (16)
Consider a candidate Lyapunov function as 𝑉₁=1−𝑒₀²+𝐞^𝑇𝐞. (17)
Because the inequality ∑³_𝑖=0|𝑒_𝑖|^𝑟+1≥(∑³_𝑖=0|𝑒_𝑖|²)^(𝑟+1)/2 holds for 0<𝑟<1, it is obtained from (15) that 𝑉̇₁=−2𝑒₀̇=𝐞^𝑇𝝎_𝐞=−𝜇₁𝐞^𝑇𝐞−𝜇₂³_𝑖=1𝑒_𝑖^𝑟+1 (18)
which implies that 𝑉̇₁=0 if and only if 𝐞=0. Thus, 𝑉₁ is really a Lyapunov function such that the signal 𝐞 will converge to zero and, accordingly, 𝑒₀ tends to ±1 as 𝑡→∞ by using the constraint in (18). Note that the equilibrium point (𝑒₀,𝐞)=(−1,𝟎) is not a stable equilibrium point [34]. Then, by Lemma 1, we obtain 𝑉̇₁≤−𝜇₁𝐞^𝑇𝐞−𝜇₂𝐞^𝑇𝐞^(𝑟+1)/2. (19)
Because 𝐞=(−1,0)^𝑇 is not the stable equilibrium point, the signal 𝐞 will converge to zero. Thus, lim_𝑡→∞𝑒₀(𝑡)=1 can be obtained from the constraint 𝐞^𝑇𝐞+𝑒²₀=1. There exists a finite time 𝑡≥0 such that 𝑒₀(𝑡)>0 for 𝑡≥𝑡. Then, for 𝑡≥𝑡, one has 1−𝑒₀²=2𝑒²₀+𝐞^𝑇𝐞−2𝑒₀=2𝑒₀𝑒₀−1+𝐞^𝑇𝐞−2≤𝐞^𝑇𝐞. (20)
Then, 𝑉₁≤2𝐞^𝑇𝐞. (21)
Using (21), (19) can be further bounded by 𝑉̇₁≤−12𝜇₁𝑉₁−12^(𝑟+1)/2𝜇₂𝑉₁^(𝑟+1)/2. (22)
Using 0.5<(𝑟+1)/2<1 and Lemma 2, one has 𝑉₁(𝑡)≡0 for all 𝑡≥𝑡_𝐹1. According to definition of 𝑉₁(𝑡) in (17), 𝑒₀(𝑡)≡1, 𝐞(𝑡)≡0, and 𝝎_𝐞(𝑡)≡0 for all 𝑡≥𝑡_𝐹1 are concluded. Thereby, the proof is completed here.

3.2. Attitude Compensation Controller Design

Considering the reaction wheel installation deviation and external disturbance, it is obtained from the sliding surface (14) that 𝐉𝐬̇=𝐃𝝉+Δ𝐃𝝉+𝐋−𝛽𝐪_𝑒−0.5𝐉̇𝐬. (23)

Because 𝐉 is unknown but bounded, then 𝐉̇=0 is established. Then, 𝐋 can be represented to be bounded by ‖𝐋‖≤𝛼₀+𝛼₁‖𝝎‖+𝛼₂‖𝝎‖², where 𝜶_𝑖, 𝑖=1,2,3 are positive constants [35].

In order to facilitate analysis and proof, firstly, define 𝜅=𝜆_min(𝐃𝐃^𝑇), 3‖Δ𝐃‖_∞‖𝐃^†‖_∞=𝜀<1, and 𝐃^† is the pseudoinverse of 𝐃. Now, we are ready to present the main result in Theorem 4.

Theorem 4. Considering the uncertainty attitude tracking dynamics described by (5) with actuator misalignment Δ𝐃 and external disturbance torque 𝐝(𝑡), design an attitude compensation control law as 𝝉=𝝉_nom(𝑡)+𝝉_𝑎𝑑𝑝(𝑡)+𝝉_𝑚𝑖𝑠(𝑡), (24)
where 𝝉_nom(𝑡)=−𝑘₁𝛽𝐃^𝑇𝐪_𝑒𝐬‖𝐬‖𝐃−𝐾^𝑇𝐬‖𝐬‖²,𝝉_𝑎𝑑𝑝(𝑡)=𝐃^𝑇−𝑘₃−𝑘₄‖𝝎‖−𝑘₅‖𝝎‖²𝐬‖𝐬‖,𝝉_𝑚𝑖𝑠(𝑡)𝜋=−¹−1𝜑(𝑡)𝐃^†𝐬‖𝐬‖_∞, (25)(26)(27)
where 𝐾∈𝐑⁺ is control parameter and 𝑘₁ is carefully chosen such that 𝑘₁𝜅−1>0; 𝜋₁ is the estimate of 𝜋₁=1/(1−𝛿); 𝑘₃ is the estimate of 𝑘₃=𝛼₀/𝜅; 𝑘₄ is the estimate of 𝑘₄=𝛼₁/𝜅; 𝑘₅ is the estimate of 𝑘₅=𝛼₂/𝜅. Moreover, 𝜋̇₁, 𝑘_𝑖, 𝑖=3,4,5 are adaptively updated by ̇𝑘₂=(‖𝐬^𝑇‖/ℓ₄)𝑘_𝑖, ̇𝑘₃=‖𝝎‖‖𝐬‖/ℓ₅, ̇𝑘₄=‖𝝎‖²‖𝐬‖/ℓ₆, and 𝜋̇₁=𝑙₁𝜑(𝑡)‖𝐬‖_∞, respectively. Then the system states reach the sliding mode surface 𝐬(𝑡)=0 in finite-time for any initial state 𝐐(0) and 𝝎(0).

Proof. When 𝐬≠𝟎, consider a candidate Lyapunov function: 𝑉₂=12𝐬^𝑇()𝐉𝐬+1−𝜀2𝑙₁𝜋²₁+12𝑙₃𝑘²₃+12𝑙₄𝑘²₄+12𝑙₅𝑘²₅, (28)
where 𝜋₁=𝜋₁−𝜋₁, 𝑘_𝑖=𝑘_𝑖−𝑘_𝑖, 𝑖=3,4,5, and 𝑙₁, 𝑙₃, 𝑙₄, 𝑙₅ are the positive constants.
Calculating the time-derivative of 𝑉₂, it yields 𝑉̇₂≤𝐬^𝑇𝐉𝐬̇−(1−𝜀)𝑙₁𝜋𝜋̇−𝑙₃𝑘₃̇𝑘₃−𝑙₄𝑘₄̇𝑘₄−𝑙₅̇𝑘₃=𝐬𝐃𝝉_nom−𝛽𝐪_𝑒+𝐬𝐃𝝉_adp+𝐋+𝐬𝐃𝝉_mis−()+Δ𝐃𝝉1−𝜀𝑙₁𝜋𝜋̇−𝑙₃𝑘₃̇𝑘₃−𝑙₄𝑘₄̇𝑘₄−𝑙₅̇𝑘₃. (29)
With (25), it follows that 𝐬𝐃𝝉_nom−𝛽𝐪_𝑒≤𝐬𝐃𝝉_nom+𝛽𝐪_𝑒‖𝐬‖≤−𝜅𝑘₁𝛽‖𝐬‖𝐪_𝑒+𝛽𝐪_𝑒𝐬‖𝑘‖−𝐾𝜅=−₁𝜅𝛽−𝛽𝐪_𝑒−𝜅𝐾≤−𝜅𝐾. (30)
In the same way, with (26), the following equality is yielded: 𝐬𝐃𝝉_adp+𝐋≤𝐬𝐃𝝉_adp+‖𝐬𝐋‖𝑘‖‖≤−𝜅₃−𝛼₀𝜅‖𝐬‖−𝜅𝑘₄−𝛼₁𝜅‖𝐬𝝎‖𝑘‖‖−𝜅₅−𝛼₂𝜅‖𝐬‖‖𝝎‖²=𝜅𝑘₃‖𝐬‖+𝜅𝑘₄‖𝐬‖‖𝝎‖+𝜅𝑘₅‖𝐬‖‖𝝎‖². (31)
Define 𝜑(𝑡)=‖𝐃^†(𝝉_nom+𝝉_adp)‖_∞. If the choice of the control gains is such that 3‖Δ𝐃‖_∞‖𝐃⁻¹‖_∞=𝜀<1, using the inequality 𝐱𝐲≤3‖𝐱‖_∞‖𝐲‖_∞>0 for any vector 𝐱,𝐲∈𝐑^𝑛 and applying (29) lead to 𝐬𝐃𝝉_mis𝝉+Δ𝐃𝝉=𝐬Δ𝐃_nom+𝝉_adp+𝝉_mis+𝐬𝐃𝝉_mis≤3‖Δ𝐃‖_∞𝐃^†_∞𝜑(𝑡)‖𝐬‖_∞+𝐬Δ𝐃𝝉_mis+𝐬𝐃𝝉_mis≤𝜀𝜑(𝑡)‖𝐬‖_∞−𝜋₁−1𝜑(𝑡)‖𝐬‖_∞−𝜋+𝐬Δ𝐃¹−1𝜑(𝑡)𝐃^†𝐬‖𝐬‖_∞=(1−𝜀)𝜋₁𝜑(𝑡)‖𝐬‖_∞. (32)
Substituting (30)–(32) into (29), consequently, it follows that 𝑉̇≤−𝜅𝐾. (33)
And then, 𝑉̇^𝑡₀₂𝜇𝑑𝜇≤−𝜅𝐾^𝑡₀𝑉𝑑𝜇;thatis,₂(𝑡)−𝑉₂(0)≤−𝜅𝐾𝑡. (34)
Due to 𝑉₂(𝑡)≥0, solving (34) leads to 𝑉₂(𝑡)≡0, for 𝑡≥𝑡_𝐹2, 𝑡_𝐹2𝑉≤−²(𝑡)−𝑉₂(0).𝜅𝐾 (35)
Then, it can be concluded that the system states reach the surface 𝐬(𝑡)=𝟎 in finite time. Thereby, the proof is completed here.

Remark 5. The controller (24) includes three parts: 𝝉_adp(𝑡) is used to compensate for system uncertainty caused by the external disturbance and moment inertia, 𝝉_mis(𝑡) is used to accommodate actuator misalignment, and 𝝉_nom(𝑡) is the nominal control.

Theorem 6. Consider the attitude tracking system given by (5), (9), and (10). If the control scheme (24) is implemented, then the attitude tracking maneuver can be accomplished in a finite time 𝑡_𝐹=𝑡_𝐹1+𝑡_𝐹2; that is, 𝝎_𝐞(𝑡)≡0 and 𝐞(𝑡)≡0 are guaranteed for all the time 𝑡≥𝑡_𝐹.

Proof. It is obtained from Theorem 4 that all the states of the attitude tracking system reach the sliding mode surface 𝐬(𝑡)=0 in finite-time 𝑡_𝐹2 and maintain the motion state on the slide mode surface. Furthermore, from Theorem 3, it is obtained that once the system state reaches the slide mode surface (14) the system state can reach the equilibrium point (𝑒₀,𝐞)=(1,𝟎) in finite time 𝑡_𝐹1. Therefore, for any initial state 𝐐(0) and 𝝎(0), the desired attitude trajectory can be followed in a finite time 𝑡_𝐹; that is, 𝐞(𝑡)≡0, 𝑒₀(𝑡)≡1, and 𝝎_𝐞(𝑡)≡0 are achieved for all the time 𝑡≥𝑡_𝐹. Thereby, the proof is completed here.

4. Numerical Simulation Results

4.1. Reaction Wheel Configuration

To demonstrate the effectiveness and performance of the proposed compensation control scheme, numerical simulations have been carried out using the rigid spacecraft system (3) and (6) in conjunction with the developed compensation control law (24). The spacecraft is activated by four reaction wheels with a limited control torque 𝑢_max=0.1 N·m. The configuration of those four actuators is shown in Figure 2. 𝜶_𝑖=35.26° and 𝜷_𝑖=45° are the nominal alignment angles, 𝑖=1,2,3,4. Δ𝜶_𝑖 and Δ𝜷_𝑖 are the misalignment angles.

Figure 2 

Configuration of four reaction wheels.

With the configuration shown in Figure 2, the relation between the actual output torque of reaction wheel and the total torque acting on the spacecraft is to be calculated as 𝐮(𝑡)=𝜏₁cos𝛼₁+Δ𝛼₁sin𝛽₁+Δ𝛽₁−sin𝛼₁+Δ𝛼₁cos𝛼₁+Δ𝛼₁cos𝛽₁+Δ𝛽₁+𝜏₂−cos𝛼₂+Δ𝛼₂cos𝛽₂+Δ𝛽₂−sin𝛼₂+Δ𝛼₂cos𝛼₂+Δ𝛼₂sin𝛽₂+Δ𝛽₂+𝜏₃−cos𝛼₃+Δ𝛼₃sin𝛽₃+Δ𝛽₃−sin𝛼₃+Δ𝛼₃−cos𝛼₃+Δ𝛼₃cos𝛽₃+Δ𝛽₃+𝜏₄cos𝛼₄+Δ𝛼₄cos𝛽₄+Δ𝛽₄−sin𝛼₄+Δ𝛼₄−cos𝛼₄+Δ𝛼₄sin𝛽₄+Δ𝛽₄. (36)

Although the misalignment angles exist due to finite-manufacture technique and vehicle vibration, those angles Δ𝜶_𝑖, Δ𝜷_𝑖  (𝑖=1,2,3,4) are small values. They can be approximated by cosΔ𝜶_𝑖≈1,sinΔ𝜶_𝑖≈Δ𝜶_𝑖,cosΔ𝜷_𝑖≈1,sinΔ𝜷_𝑖≈Δ𝜷_𝑖,sinΔ𝜶_𝑖sinΔ𝜷_𝑖().≈0𝑖=1,2,3,4 (37)

Hence, (36) can be re written as 𝐮(𝑡)=𝐃𝝉(𝑡)+Δ𝐃𝝉(𝑡), (38)

where 𝐃 and Δ𝐃=(Δ𝐃₁,Δ𝐃₂,Δ𝐃₃,Δ𝐃₄) are calculated as 𝐃=cos𝛼₁sin𝛽₁−cos𝛼₂sin𝛽₂−cos𝛼₃sin𝛽₃cos𝛼₄sin𝛽₄−sin𝛼₁−sin𝛼₂−sin𝛼₃−sin𝛼₄cos𝛼₁cos𝛽₁cos𝛼₂cos𝛽₂−cos𝛼₃cos𝛽₃−cos𝛼₄cos𝛽₄=√33−√33−√33√33−√33−√33−√33−√33√33√33−√33−√33,Δ𝐃₁=Δ𝛽₁cos𝛼₁cos𝛽₁−Δ𝛼₁sin𝛼₁sin𝛽₁−Δ𝛼₁cos𝛼₁−Δ𝛽₁cos𝛼₁sin𝛽₁−Δ𝛽₁sin𝛼₁cos𝛽₁Δ𝐃₂=−Δ𝛽₂cos𝛼₂cos𝛽₂+Δ𝛼₂sin𝛼₂sin𝛽₂−Δ𝛼₂cos𝛼₂−Δ𝛽₂cos𝛼₂sin𝛽₂−Δ𝛽₂sin𝛼₂cos𝛽₂Δ𝐃₃=−Δ𝛽₃cos𝛼₃cos𝛽₃+Δ𝛼₃sin𝛼₃sin𝛽₃−Δ𝛼₃cos𝛼₃Δ𝛽₃cos𝛼₃sin𝛽₃+Δ𝛽₃sin𝛼₃cos𝛽₃Δ𝐃₄=Δ𝛽₄cos𝛼₄cos𝛽₄−Δ𝛼₄sin𝛼₄sin𝛽₄−Δ𝛼₄cos𝛼₄Δ𝛽₄cos𝛼₄sin𝛽₄+Δ𝛼₄sin𝛼₄cos𝛽₄. (39)

Remark 7. Theorem 4 gives out the sufficient condition 3‖Δ𝐃‖_∞‖𝐃^†‖_∞=𝜀<1 of efficacious processes on reaction wheel installation deviation Δ𝐃 for guaranteeing the attitude controller (24). Particulary, according to the definition Δ𝐃 and matrix norm, then ‖Δ𝐃‖_∞≤max_𝑖=1,2,3𝜃_𝑖, where 𝜃₁=⁴_𝑖=1Δ𝜶_𝑖sin𝜶_𝑖sin𝜷_𝑖+⁴_𝑖=1Δ𝜷_𝑖cos𝜶_𝑖cos𝜷_𝑖=√66⁴_𝑖=1Δ𝜶_𝑖+√33⁴_𝑖=1Δ𝜷_𝑖,𝜃₂=⁴_𝑖=1Δ𝜶_𝑖cos𝜶_𝑖=√63⁴_𝑖=1Δ𝜶_𝑖,𝜃₃=⁴_𝑖=1Δ𝜶_𝑖sin𝜶_𝑖cos𝜷_𝑖+⁴_𝑖=1Δ𝜷_𝑖cos𝜶_𝑖sin𝜷_𝑖=√66⁴_𝑖=1Δ𝜶_𝑖+√33⁴_𝑖=1Δ𝜷_𝑖. (40)
On the other hand, inequality 3‖Δ𝐃‖_∞‖𝐃^†‖_∞=𝜀<1 means that ‖Δ𝐃‖_∞=𝜀/(3‖𝐃^†‖_∞)<1/(3‖𝐃^†‖_∞). Therefore, the establishment conditions max_𝑖=1,2,3,4𝜃_𝑖<1/3‖𝐃^†‖_∞=0.2566 rad of Theorem 4 from (40) can be obtained, that is to say, the installation deviation angle of any two reaction wheels is not larger than 0.2566 rad. Relying on this, this largest installation deviation angle, that is max_𝑖=1,2,3,4𝜃_𝑖 = 14.7021°, of the reaction wheel installation structure is considered in this paper.

4.2. Simulation Results

The nominal inertia matrix of the considered spacecraft is specified by [36] 𝐉₀=353−1.53282−1.5230kg⋅m²,Δ𝐉=1+𝑒^−0.1𝑡+2𝜗(𝑡−10)−4𝜗(𝑡−20)diag(3,2,1) (41)(42)

which incorporated into the model, where 𝜗(⋅) is defined as 𝜗(𝑡≥0)=1 and 𝜗(𝑡<0)=0. External disturbance 𝐝(𝑡) is chosen as [35] 𝐝(𝑡)=‖𝝎‖²+0.05(sin0.8𝑡,cos0.5𝑡,cos0.3𝑡)^𝑇. (43)

The reaction wheel misalignment angle Δ𝜶_𝑖  (𝑖=1,2,3,4) can be selected randomly between −4.5°~+4.5°, and Δ𝜷_𝑖  (𝑖=1,2,3,4) can be selected randomly between −5.5°~+5.5°.

In this simulation, spacecraft initial parameter is set as follows: initial angular velocity 𝝎_𝑖(0)=0.1−0.1−0.05^∘/𝑠, 𝑖=1,2,3; initial attitude quaternary 𝐪_𝑖(0)=0.181−0.2870.792−0.524^𝑇, 𝑖=0,1,2,3; the corresponding initial roll angle, pitch angle, and yaw angle are set 0.2°, −0.4° and −0.3° respectively. The control gains are selected by the following list: 𝛽=0.7, 𝐾=0.05, 𝜅=1.05, 𝑘_𝑖=1.5  (𝑖=1,3,4,5), as 𝑙_𝑖=1.5  (𝑖=1,3,4,5); moreover, the initial of the adaptive update laws are 𝜋̇₁(0)=1.25, ̇𝑘₃(0)=0.68, ̇𝑘₄(0)=0.42, ̇𝑘₅(0)=0.22.

To demonstrate the effectiveness of the proposed misalignment compensation and disturbance rejection scheme, a spacecraft is numerically simulated using the proposed control compensation strategy (24).

We see in Figures 3–8 the controller managed to stabilize the origin equilibrium point in 30 seconds with great pointing accuracy. Indeed, since the knowledge of spacecraft inertia parameters was not required and an implicit integral item was incorporated in the control law design, external disturbance effect on the attitude control performance can be compensated efficiently, and also great robustness to system uncertainties, such as misalignment, can be guaranteed.

Figure 3 

Time responses of angular velocity.

Figure 4 

Time response of satellite attitude angle.

Figure 5 

Time response of control torque.

Figure 6 

Time response of control torque for reaction wheels.

Figure 7 

Time responses of quaternion.

Figure 8 

Time response of sliding mode surface.

We can see in Figures 3 and 4 the time responses of angle velocity and attitude angle; the proposed control scheme surely realized the high precision stable control in the presence of external disturbance, uncertain moment of inertia, and reaction wheel misalignment, and the pointing accuracy is superior to 0.01°; the attitude stable precision is superior to 0.001°/s. Meanwhile, from Figures 5 and 6, we can see that for compensating the misalignment and other uncertainties, the designed control command of control redundancy configuration for 4 reaction wheels 𝝉 is allocated to the three-actual-output torque 𝐮, and then the purpose of compensation, attitude high precision control is realized. In addition, the finite-time control validity is shown in Figure 8. And from Figure 8, we can see that the spacecraft attitude control system status has realized the tracking control at 𝑡_𝐹=30.5. Thereinto, the spacecraft attitude has arrived at slide mode surface 𝐬 at 𝑡_𝐹1=25.9; afterwards, the statuses converge to equilibrium point at 𝑡_𝐹2=5.4 under the normal control 𝝉_nom(𝑡). The same validity of finite-time attitude compensation control strategy proposed in this paper can be further proved from the time response of the quaternion as shown in Figure 7.

From the above illustrated simulation results, it is shown that the proposed scheme can accomplish the attitude stabilization in finite-time in presence of time-varying external disturbances, uncertain inertial parameters, and even reaction wheel installation deviation.

5. Conclusions and Future Works

Considering the spacecraft issues about reaction misalignment, external disturbances, and parameters uncertainty, in this paper, a finite-time adaptive attitude compensation control has been proposed. A quantitative installation deviation angle analysis has been done and given out the value range of the reaction wheel misalignment angle. In the end the system stability and engineering practical value have been discussed from the perspective of theory and engineering. Numerical simulation of this novel control strategy was also presented to confirm the advantages and improvements over existing controllers. The case of actuator misalignment mentioned in Section 4 had only discussed for four reaction wheel configuration, but this compensation control scheme is suitable for more than that reaction wheel number. Moreover, the actuator faults have not been considered. The latter case should be as one of subjects for future research. Meanwhile, the method optimal control approach combined robust control [37, 38] also can be applied in this field.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The present work was supported partially by the National Natural Science Foundation of China (Project no. 61304149) and Natural Science Foundation of Liaoning, China (Project no. 2013020044). The authors highly appreciate the above financial supports.