1 Introduction

With the development of space science and technology, the solar array has been an important component of the spacecraft since it can provide necessary power for the whole spacecraft system. The solar arrays are folded in the carrier during the spacecraft launch and ascent. When the carrier enters its orbit, the solar arrays are going to extend to their working position. In this process, the attitude of the spacecraft is changed due to the strong dynamic coupling between the base spacecraft and the hinged solar arrays. In order to ensure that the spacecraft locates in the designed position, it is necessary to study the control regularity during the deployment process of the spacecraft solar arrays.

In the past decades, domestic and foreign researchers have done a lot of researches on the dynamics and control of the deployment of the spacecraft solar array system. Wallrapp and Wiedemann^[1] simulated the deployment process of a flexible spacecraft solar array system using the multibody program SIMPACK. Kuang et al.^[2] investigated the nonlinear dynamics and chaotic motions of a satellite with flexible solar arrays under the effect of gravity-gradient torques. Kwak et al.^[3] studied the dynamic response of a satellite with rigid solar arrays equipped with strain energy hinges, and the simulation results were similar to those of the ground deployment experiments. Zhang and Zhou^[4] and Zhang^[5] reported the recursive Lagrangian dynamic modeling and simulation of flexible-link flexible-joint robot systems. The optimal control problem of the deployment of the spacecraft solar arrays was transformed into a nonholonomic motion planning problem, which was solved by numerical optimization algorithms^[6-8]. Robust and adaptive controllers for the attitude control and vibration suppression of a flexible space manipulator system were developed^[9-12]. Jiang and Li^[13-14] presented a robust H_∞ output feedback control scheme to suppress the vibration of a smart composite solar array structure in the presence of external disturbances. Zhang et al.^[15] investigated the effects of joint clearance on the attitude motion of a spacecraft installed with rigid solar arrays using a modified Coulomb friction model. A novel computational approach for modeling and analysis of a spacecraft with symmetric flexible solar arrays within the framework of global analytical modes was proposed^[16-19]. Li et al.^[20-23] investigated the deployment dynamics of a spacecraft solar array system with joint friction and developed a fuzzy proportional derivative(PD) controller to eliminate the drift of the base spacecraft.

Recently, a pseudospectral method has been widely used to solve the optimal control problems in aerospace engineering^[24-27]. It is a kind of direct optimization method, and transforms the optimal control problem into a nonlinear programming problem by using global orthogonal polynomials to approximate both the states and controls at a set of collocation points. According to the use of different types of collocation points, pseudospectral methods can be divided into the Legendre pseudospectral method(LPM), the Gauss pseudospectral method(GPM), the Radau pseudospectral method(RPM), and the Chebyshev pseudospectral method(CPM). The LPM is adopted in this paper. Fahroo and Ross^[28] proved the equivalence between the costates of the optimal control problem and the Karush-Kuhn-Tucker (KKT) multipliers of the nonlinear programming problem discretized with the LPM. The accuracy of the results obtained with the LPM is almost the same as that of the indirect methods, which makes up a shortage of direct optimization methods. However, the LPM is essentially an open-loop control method which is sensitive to system parametric uncertainties and external disturbances. Therefore, researchers developed several feedback control methods to track the optimal reference trajectory obtained with the LPM and guarantee the system's robustness against the uncertainties and disturbances^[29-33]. Among them, a closed-loop state feedback control method based on the indirect pseudospectral method is adopted in this paper.

Motivated by the aforementioned discussion, a control strategy combining feedforward control and feedback control is presented for the optimal deployment of spacecraft solar array system with the initial state uncertainty. The rest of this paper is organized as follows. In Section 2, the dynamic equation of the spacecraft solar array system is established under the assumption that the initial linear momentum and angular momentum of the system are zero. In Section 3, the dissipation energy of each revolute joint is selected as the performance index of the system. The LPM is used to transform the attitude maneuver problem into a nonlinear programming problem. Then, the sequential quadratic programming algorithm is used to solve the nonlinear programming problem and generate the optimal reference trajectory of the system. In Section 4, the dynamic equation is linearized along the reference trajectory in the presence of initial state errors. The trajectory tracking problem is converted to a two-point boundary value problem based on Pontryagin's minimum principle. The LPM is used to discretize the two-point boundary value problem and transform it into a set of linear algebraic equations which can be easily calculated. Then, the closed-loop state feedback control law is designed based on the resulting optimal feedback control. Numerical simulations are performed in Section 5. The main conclusions of this work are summarized in Section 6.

2 Dynamics of spacecraft solar array system

The spacecraft with deployable solar arrays adopted in this paper is shown in Fig. 1. The spacecraft solar array system consists of a base spacecraft B₀ and n solar arrays B_i(i=1, 2, …, n). B_i(i=0, 1, …, n) is assumed to be a rigid body. O_ci(i=0, 1, …, n) denotes the center of mass of B_i. O_c denotes the center of mass of the whole system. O_i(i=1, 2, …, n) denotes the rotational center of the revolute joint between B_i and B_i-1. O₀ coincides with O_c0. m_i and I_i(i=0, 1, …, n) are the mass and moment of inertia of the rigid body B_i, respectively. During the deployment process of the spacecraft solar arrays, the base spacecraft B₀ is free-floating in order to save fuel and enhance the reliability of the whole system.

The spacecraft solar array system is typically a rootless multibody system with an open-chain structure. Set an inertial reference frame OXYZ. Let O_i x_iy_iz_i(i=0, 1, …, n) be the body-fixed frame of B_i, in which the axis O_ix_i is along O_iO_i+1 with e_i as the base vector. Assume that each body makes planar movement in a plane spanned by x_i(i=0, 1, …, n). θ₀ denotes the attitude angle of the base spacecraft B₀ with respect to the reference frame. θ_i(i=1, 2, …, n) denotes the relative rotational angle between the adjacent rigid bodies B_i-1 and B_i. Let r_i(i=0, 1, …, n) be the position vector of the center of mass of B_i with respect to the origin of the reference frame and r_c be the position vector of the center of mass of the whole system with respect to the origin of the reference frame, respectively. According to the geometric relationship and the definition of the center of mass of the whole system, we have

(1)

where C_ij(i=0, 1, …, n and j=0, 1, …, n) is the combination of the system inertia parameters. Differentiating Eq.(1) with respect to time yields

(2)

Neglecting the weak gravity gradient, the spacecraft solar array system exhibits nonholonomic behaviour due to the linear and angular momentum conservation of the system. Without loss of generality, it is assumed that the initial linear momentum and angular momentum of the system are zero. The relationship of the linear momentum conservation of the system is given as

(3)

The relationship of the angular momentum conservation of the system is given as

(4)

where ω_i(i=0, 1, …, n) denotes the angular velocity of the rigid bodies B_i. By substituting Eqs.(1) and(2) into Eqs.(3) and(4), Eq.(4) can be deduced into the following form:

(5)

where q=[θ₁, θ₂, …, θ_n]^T∈ Rⁿ, and P=[P₁, P₂, …, P_n]. P_i(i=1, 2, …, n) is the combination of the sine and cosine functions of the system generalized coordinates θ₀, θ₁, …, θ_n. The specific expressions of P_i can be found in Refs. [6] and [7].

Note that the angular momentum conservation is nonintegrable in Eq.(5), which indicates that the spacecraft with solar arrays is a system with nonintegrable velocity constraints or nonholonomic constraints. Any relative motions of the components in the spacecraft solar array system may lead to a perturbation on the attitude of the base spacecraft.

3 Feedforward control method 3.1 Optimal control problem

The generalized coordinates of the spacecraft solar array system x=[θ₀, θ₁, …, θ_n]^T∈ R^m are selected as the state variables of the system. The relative angular velocities (i=1, 2, …, n) between the adjacent bodies are selected as the control inputs of the system, denoted as . Thus, the spacecraft solar array system is stated as a nonlinear control system with m state variables and n control inputs, in this problem, m=n+1. From the dynamic equation of the system(5), the state equation of the system is expressed as

(6)

where E_n denotes an n× n identity matrix.

According to the minimum energy control principle, the dissipation energy of each revolute joint is selected as the performance index of the system,

(7)

Equality boundary conditions of the system contain the initial and terminal conditions of the states and control inputs,

(8)

Inequality path constraints of the system contain the limits of the joint angles and the control inputs,

(9)

where u_m denotes the maximum control input of the system, and u_m>0.

Thus, the minimum energy deployment process of the spacecraft solar arrays is considered as an optimal control problem, i.e., find the optimal control inputs u(t) to steer the system from the initial state x₀ to the desired position x_f, which minimizes the performance index(7), subject to the state equation(6), boundary conditions(8), and path constraints(9).

3.2 Optimal reference trajectory generation

The LPM is used to transform the aforementioned optimal control problem into a nonlinear programming problem. First, the time interval of the optimal control problem t∈ [t₀, t_f is converted to τ ∈ [-1, 1] through the affine transformation,

(10)

Then, the optimal control problem can be rewritten into the following general form:

(11)

where B(·) and C(·) represent the boundary conditions and path constraints of the system, respectively. Equation(11) is referred to as a continuous optimal control problem in the Bolza form.

Let P_K(τ) be the Legendre polynomial of K degrees over the interval [-1, 1]. The Legendre-Gauss-Lobatto(LGL) points are defined as τ₀=-1, τ_K=1, and τ_i(i=1, 2, …, K-1), i.e., the zeros of , where is the derivative of the Legendre polynomial P_K(τ). The states and controls are approximated by a basis of Lagrange polynomials L_i(τ)(i=0, 1, …, K) at the LGL points, denoted as

(12)

(13)

where L_i(τ) is defined by

(14)

Differentiating Eq.(12) with respect to time yields

(15)

where is expressed by a differential approximation matrix D∈ R^(K+1)×(K+1). The elements of D can be offline calculated as follows^[28]:

(16)

By substituting Eq.(15) into Eq.(6), the state equation of the system is transformed into algebraic constraints,

(17)

where X_k denotes X(τ_k), and U_k denotes U(τ_k) for simplicity. Then, the performance index is approximated by the Gauss-Lobatto quadrature rule,

(18)

where μ_k(k=0, 1, …, K) is the Gauss-Lobatto weight, denoted as

(19)

Likewise, the boundary conditions and path constraints are approximated at the LGL points as

(20)

(21)

Thus, the optimal control problem is transformed into a nonlinear programming problem, i.e., find the optimal control inputs U which minimizes the performance index(18), subject to the state equation constraints(17), boundary conditions(20), and path constraints(21). Then, the nonlinear programming problem is solved by the SNOPT, an MATLAB toolbox which is based on the sequential quadratic programming algorithm and suits for the large-scale constrained nonlinear optimization^[34]. The optimal trajectory for the deployment of spacecraft solar array system is generated and then can be used as the reference trajectory in the next section.

Note that the solution obtained with the LPM is discretized. If fewer LGL points are selected, the results obtained are of poor accuracy. If more LGL points are selected, the number of design variables will be relatively large. When the initial guess values are selected improperly, the problem will not converge to the feasible solution or fall in local minimum. Therefore, an optimization framework is used to obtain the optimum solution from a feasible solution. First, fewer LGL points K₁ are used to solve the nonlinear programming problem and generate the feasible solution of the system. Then, the values of states and controls at more LGL points can be obtained by interpolating the feasible solution. Using the values obtained by interpolation as the initial guess values, the nonlinear programming problem can be easily solved at more LGL points K₂(K₂>K₁) to generate the optimum solution of the system. This approach can accelerate the convergence rate of the program effectively.

4 Feedback control method 4.1 Two-point boundary value problem

The optimal reference trajectory obtained with the LPM is only the nominal trajectory that small perturbations on the initial states may have a large effect on the deployment process of the spacecraft solar arrays. Therefore, a closed-loop state feedback control scheme based on the indirect pseudospectral method is developed in this section to track the reference trajectory obtained with the LPM and modify the initial state errors of the system.

The feedback control scheme is based on the trajectory errors with respect to the reference trajectory. Thus, the tracking control problem is considered as a trajectory state regulation problem. By using the Taylor series expansion and neglecting the higher-order terms, the linearized dynamics of Eq.(6) is conducted as the following linear time-varying system:

(22)

where δx(t)=x-x^* and δu(t)=u-u^* denote the deviations between the actual and reference values of the states and controls, respectively. δx₀ denotes the initial state error of the system. A(t) and B(t) are the Jacobian matrices, denoted as

(23)

The trajectory state regulation problem is stated as an optimal control problem, i.e., find the required control inputs δu(t) and corresponding state variables δx(t) which satisfy Eq.(22) and minimize the following performance index:

(24)

where P_f, Q(t)∈ R^{m× m} and R(t)∈ R^{n× n} are symmetric positive definite weight matrices.

Based on Pontryagin's minimum principle, the following Hamiltonian function is introduced:

(25)

where δλ(t) denotes the costate vector of the system. According to the calculus of variations, the costate equation of the system is written as

(26)

and from the first-order necessary optimality condition , the optimal control equation of the system is obtained,

(27)

where the boundary conditions are

(28)

Equations(22), (26), (27), and(28) construct the following two-point boundary value problem:

(29)

4.2 Indirect pseudospectral method

The traditional numerical method, such as the tool function LQR in MATLAB, is used to solve the strong backward integration problem of the matrix Riccati differential equation(29), which not only consumes a large amount of calculation time, but also may lead to numerical instability. Thus, the indirect pseudospectral method is used in this paper to obtain the optimal feedback control of the linear control system(22).

The LPM is used to discretize the two-point boundary value problem (29) and transform it into a set of linear algebraic equations which can be easily calculated. Through the affine transformation(10), the time interval of the two-point boundary value problem t∈ [t₀, t_f] is converted to τ ∈ [-1, 1],

(30)

Similarly, as in Subsection 3.2, the states δx and costates δλ are approximated by a basis of Lagrange polynomials L_i(τ)(i=0, 1, …, K) at the LGL points,

(31)

(32)

Differentiating Eqs.(31) and(32) with respect to time yields

(33)

(34)

where the differential approximation matrix D is defined in Eq.(14). Substituting Eqs.(33) and(34) into Eq.(30), the two-point boundary value problem is discretized and transformed into the following algebraic equations:

(35)

(36)

where δx_k denotes δx(τ_k), δλ_k denotes δλ(τ_k), A_k denotes A(τ_k), B_k denotes B(τ_k), R_k denotes R(τ_k), and Q_k denotes Q(τ_k) for simplicity. Define X=[δx₀^T, δx₁^T, …, δx_K^T]^T, Λ=[δλ₀^T, δλ₁^T, …, δλ_K^T]^T, and Z=[X^T, Λ^T]^T. Then, Eqs.(35) and(36) can be rewritten as

(37)

(38)

where F, G, M, and N are m(K+1)× m(K+1) matrices whose ijth blocks are m× m matrices with the following form^[30]:

(39)

(40)

(41)

(42)

where E_m denotes the m× m identity matrix, and 0_m denotes the m× m zero matrix, respectively. In order to solve Eqs.(37) and(38) subject to the boundary conditions(28) of the two-point boundary value problem, we rewrite Eqs.(28), (37), and(38) into the following form:

(43)

where S₁=[0_m, …, 0_m, P_f]∈ R^{m× m(K+1)}, S₂=[0_m, …, 0_m, -E_m]∈ R^{m× m(K+1)}, and there are K zero matrices in S₁ and S₂. Here, we divide V into two portions V=[V₀, V_e], then, Eq.(43) is of the form

(44)

where Z_e=[δx₁^T, …, δx_K^T, δλ₀^T, δλ₁^T, …, δλ_K^T]^T∈ R^{m(2K+1)× 1}. In this way, V₀ and V_e are m(2K+3)× m and m(2K+3)× m(2K+1) block matrices of V, respectively. From Eq.(44), we further have

(45)

where W=-V_e\V₀∈ R(^2mK+m)×m. The operator "\" represents the least squares solution in MATLAB. Then, we have

(46)

where W_x∈ R^{m(K+1)× m}and W_λ∈ R^{m(K+1)× m} are two portions of the matrix [E_m, W]^T such that

(47)

(48)

Substituting Eq.(48) into Eq.(27), we obtain the optimal feedback control law,

(49)

where W_λk∈ R^{m× m} is the kth block of the matrix W_λ.

Therefore, if the reference trajectory and the initial state errors of the system are known, the states, costates, and controls at the LGL points can be solved from Eqs.(47)-(49). Then, the values of variables at instants of time between the LGL points can be obtained by interpolation. Regarding the deviations between the current states and the reference states as the new initial state errors, Eq.(49) constitutes a closed-loop state feedback control law for the continuous trajectory tracking for the deployment process of the spacecraft solar arrays. The parameters R(τ_k), B(τ_k), and W_λk in Eq.(49) are determined by the system parameters of the spacecraft solar array system and can be obtained offline before the implementation of the optimal feedback control. Furthermore, Eq.(49) has no integral terms and can be quickly calculated. Therefore, the proposed feedback control method does not require any integration calculations and can be used to track the optimal reference trajectory in real time.

The steps of the feedback control scheme are briefly listed as follows. First, we obtain the optimal reference trajectory x^*(τ) and the reference control inputs u^*(τ) based on the feedforward method in Section 3. Then, we calculate the feedback control law δu(τ) based on Eq.(49) and the initial state differences δx₀(τ)=x₀(τ)- x₀^*(τ). Then, the new control inputs u(τ)=u^*(τ)+δu(τ) are implemented on the spacecraft solar array system and generate the corresponding states x(τ). Then, the deviations between the new states and the reference states δx(τ)=x(τ)-x^*(τ) are considered as the new initial state errors. This yields a map x^*↦δx^*↦δu^*↦u^*, and this process is repeated until the terminal position is reached.

5 Numerical simulations

Numerical simulations are performed in this section to illustrate the control strategy presented previously. Consider a spacecraft solar array system consisting of four rigid bodies, where B₀ is the base spacecraft, B₁ is the connecting array, and B₂ and B₃ are the internal and external solar arrays, respectively. The inertia and geometric parameters of the spacecraft solar array system used in this paper are listed in Table 1.

In the simulation, the solar arrays are required to completely extend from the folded condition while the attitude of the base spacecraft remains unchanged in the initial and terminal positions. The initial and terminal configurations of the system are chosen as

However, the following initial state uncertainty is considered in the actual system:

In this paper, we define the maximum positive and negative initial state errors as the test Cases A and B, respectively.

First, the feedforward control method in Section 3 is used to generate the optimal reference trajectory and the reference control inputs of the spacecraft solar array system. Set the simulation time t=5 s. In the simulation, 5 LGL points are used to calculate the feasible solution of the system, while the initial guess values of the feasible solution are created by the RAND function in MATLAB. Then, by interpolating the feasible solution at more LGL points, 60 LGL points are used to generate the optimum solution of the system.

The simulation results for the feedforward control are shown in Figs. 2 and 3. Figure 2 shows the optimal reference trajectory of the base spacecraft θ₀ and the solar arrays θ₁, θ₂, and θ₃. The motion curves are smooth and stable, which indicates that there is no detour behavior during the deployment process of the spacecraft solar arrays. Figure 3 shows the reference control inputs of the solar arrays. The control inputs are zero at initial and terminal positions and satisfy the path constraints of the system. It can be clearly seen that, when the joint speeds of the solar arrays , and are selected as the control inputs, the deployment process of the spacecraft solar array system is controllable. The base spacecraft and the hinged solar arrays are moved to the required positions from their initial states in the fixed time, with the performance index reaching its optimality and all the constraints satisfied.

Then, the feedback control method in Section 4 is used to track the reference trajectory and modify the initial state errors in the test Cases A and B. In the simulation, the weight matrices P_f, Q, and R used in Eq.(24) are determined by Bryson's rule^[35].

The simulation results for the feedback control are shown in Table 2 and Fig. 4. Figure 4 shows the trajectory tracking of the base spacecraft and the hinged solar arrays in the presence of initial state uncertainty, where "reference" denotes the nominal result under the open-loop control without considering the initial state errors, and "Case A-closed" and "Case B-closed" denote the results under the closed-loop feedback control subject to the initial state errors. From Fig. 4, the deviations between the two actual cases and the reference trajectory are obvious at the initial stage. After the feedback control is implemented to the system, the states of the reference and the two test cases asymptotically converge within 5 s. The terminal state deviations of the system are listed in Table 2. The magnitude of the terminal state deviations is around 10^-4 rad, which implies that the closed-loop feedback control system has a good robustness against the initial state errors. Thus, the closed-loop feedback control method can guarantee the system to track the planned reference trajectory efficiently in the presence of initial state uncertainty.

It must be noted that the proposed tracking control method is an optimal state feedback control method. Compared with the output feedback control methods using the joint torques as the control inputs such as the proportional integral and derivative(PID) control and sliding mode control, the proposed control method requires the dynamic and kinematic information of the spacecraft solar array system in each time period during the tracking process. Thus, the control inputs, namely, the joint speeds of the system will be smoother than those of the output feedback control methods. However, the proposed tracking control method may cost more time to track the desired trajectory than the output feedback control methods with the same initial condition.

6 Conclusions

The control strategy combining feedforward control and feedback control is presented for the optimal deployment of a spacecraft solar array system with the initial state uncertainty. In the design of feedforward control, the LPM is used to transform the optimal control problem into a nonlinear programming problem. Then, the sequential quadratic programming algorithm is used to solve the nonlinear programming problem and offline generate the optimal reference trajectory of the system. In the design of feedback control, the dynamic equation is linearized along the reference trajectory obtained with the LPM. The tracking control problem is converted to the two-point boundary value problem based on Pontryagin's minimum principle. Then, the LPM is used to discretize the two-point boundary value problem and transform it into a set of linear algebraic equations which can be easily calculated. This process does not require any integration calculations and has good performance in real time. Numerical simulations are carried out for two test cases in the presence of initial state errors. The results show that the base spacecraft and the hinged solar arrays can achieve the required terminal configuration from their initial states with all the constraints satisfied. Hopefully, the control strategy proposed in this paper can also be used for the optimal control and trajectory tracking of other space multibody systems with nonholonomic constraints.