Agile Attitude Maneuvers with Active Vibration-suppression for Flexible-spacecraft

This paper addresses the agile attitude stabilization maneuver control of ﬂexible-spacecraft using control moment gyros (CMGs) in the absence of modal information. Here, piezoelectric actuators are employed to actively suppress the vibration of ﬂexible appendages. Both the attitude dynamics and the proposed robust controller are globally developed on the Special Orthogonal Group SO(3), avoiding ambiguities and singularities associated with other attitude representations. More speciﬁcally, an observer is ﬁrst designed to estimate the modal information of vibration. A robust control law is developed by synthesizing a proportional derivative (PD) controller, an adaptive sliding mode controller, and an active vibration-suppression controller, which use the information of the estimated structural modes. The stability of the closed-loop system is proved using Lyapunov stability theory. Finally, numerical examples are performed to show the effectiveness of the proposed method.

information of appendage vibrations is hard to measure for on-orbit spacecraft, making vibration-suppression a difficult task for an attitude control system. Furthermore, the spacecraft is subject to external disturbances, which will degrade control performance and unavoidably lead to additional vibration of the flexible appendages [8]. These challenges compound the control difficulties when one strives for high performance agile attitude maneuver using CMGs.
Attitude stabilization and tracking problems for flexible-spacecraft have been studied extensively [9]. For instance, proportional derivative (PD) control [10] and PD plus feedforward compensation control [6] were proposed to investigate the attitude tracking control problem. The attitude stabilization problem was addressed in [11] and [12], where the requirement on the angular velocity was removed. In addition, other control methods, such as backstepping control [13], sliding mode control (SMC) [14], PD plus SMC [15] and H ∞ control [16], are also utilized in the area of flexible-spacecraft attitude control. However, these methods do not attempt to suppress the vibration itself but provide robustness to the disturbance caused by it. This approach means that the control is over-engineered, requiring unnecessarily large torques. For example, a sign function of the sliding surface was used in [15] to deal with the upper bound of the disturbance caused by vibrations. Though SMC can guarantee the robustness of the system, the controller is conservative and suffers from chattering.
To realize vibration-suppression, Fracchia et al. [17] developed a low-jerk attitude path planning method based on an analytical smoothing of a bang-off-bang maneuver, Hu et al. [18] proposed a path planning method to generate a smooth attitude maneuver trajectory, and in [19], an integral SMC with input shaping method was proposed for a slewing flexible spacecraft equipped with on-off thrusters. These methods can reduce the vibration of flexible structures during attitude maneuvers, however, path planning and input shaping cannot guarantee the convergence of modal coordinates and will deteriorate the steady-state performance of the attitude control system [8]. Therefore, active vibration control must be incorporated in the loop of the attitude control system [20]. Based on an observer to estimate the appendage vibrations, the concept of using piezoelectric actuators was proposed in [21] for active vibration-suppression of the flexible modes, where the advantages of piezoelectric materials include small size, low mass, and high bandwidth control capability [22]. However, the proposed control algorithm in [21], based on PD control, can only guarantee the bounded stability of the perturbed system.
In addition, the previous work on control design for flexible-spacecraft were all based on coordinate representations, such as the modified Rodriguez parameters (MRPs) or quaternions. As stated in [23], the unwinding problem exists in quaternions, and MRPs and other local coordinate representations suffer from singularities. Fortunately, pioneering studies on the attitude control have been conducted recently based on rotation matrices, such as [24], [25], [26], which provide a global and unique attitude representation on the Special Orthogonal Group SO(3).
Since SO(3) is a non-Euclidean manifold the classical methods developed for nonlinear control systems defined on Euclidean space cannot be applied. An alternative way to deal with this problem is to construct a smooth positive definite configuration error function on SO(3), and then the attitude controller is designed based on the error function [23], [27], [28]. This method has also been applied for multiple flexible-spacecraft attitude control [29]. However, actuator dynamics and vibration-suppression were not taken into account. This paper extends the state-of-the-art in control law design on SO(3) to flexible-spacecraft attitude control March 25, 2021 DRAFT using CMGs and active vibration-suppression using piezoelectric actuators. The controllers are designed without knowledge of the modal coordinates, and without using a co-ordinate representation. Based on the developed model, an observer is designed to estimate the modal information of the flexible appendages, and then an adaptive PD plus SMC (PD+) coupled with an active vibration-suppression controller is proposed. The control algorithm uses the observer outputs, where a modified singular direction avoidance (SDA) steering law based on [30] is employed to generate the attitude control signal. The designed controllers guarantee the uniformly asymptotic stability of the perturbed attitude stabilization and vibration-suppression system.
The remainder of this paper is organized as follows. Sec. II develops the mathematical model of the flexiblespacecraft attitude stabilization system and formulates the control objective of this paper. The main results of this paper including the designs of the appendage vibration observer and the controllers, and the stability analysis of

A. Flexible-Spacecraft Dynamics Based on Rotation Matrix
In this paper, we investigate the rotational motion of a fully-actuated flexible-spacecraft, where the rigid part is controlled by a cluster of angular momentum exchange actuators, i.e., SGCMGs, and vibration-suppression is realized using piezoelectric actuators. In addition, without loss of generality, the spacecraft is assumed to be equipped with two flexible appendages, such as solar arrays, antennas, or other flexible structures. Firstly, a standard Earth centered inertial frame F I is introduced to be the reference frame, whose origin is located in the spacecraft mass center. The body-fixed frame F B is defined with its origin located in the mass center of the spacecraft, and its three axes coincide with the spacecraft's principal body axes. Then, the spacecraft's configuration, namely its attitude, is the orientation from F B to F I , which can be described by the set of 3 × 3 rotation matrices R ∈ SO(3). The Special Orthogonal Group SO(3) is a Lie group with a determinant of one, and can be described by SO(3) = R ∈ R 3×3 : R T R = I 3×3 , det(R) = 1 , where I 3×3 is an 3 × 3 identity matrix [31]. Then, the attitude kinematics and dynamics of a flexible-spacecraft can be described as [4], [5], [23] where Ω ∈ R 3×1 denotes the angular velocity of the spacecraft expressed in F B , J ∈ R 3×3 is the total inertia matrix of the flexible-spacecraft, Ξ ∈ R N ×3 is the coupling coefficient matrix between the flexible structures and the rigid body, η ∈ R N ×1 is the modal coordinate vector reflecting the elastic deflection, N is the number of the vibration modes considered in the dynamics, h i is the angular momentum of each actuator, n is the number March 25, 2021 DRAFT of the actuators, u c ∈ R 3×1 and d ∈ R 3×1 are the control torque and external disturbance respectively, C = diag([2ξ 1 ω 1 , 2ξ 2 ω 2 , · · · , 2ξ N ω N ]) ∈ R N ×N represents the modal damping matrix, K = diag([ω 2 1 , ω 2 2 , · · · , ω 2 N ]) ∈ R N ×N represents the stiffness matrix, where ξ j and ω j (j = 1, 2, · · · , N ) denote the damping ratios and the natural frequencies respectively, u p ∈ R M ×1 is the M-dimensional piezoelectric input, and Ξ p ∈ R N ×M is the corresponding coupling matrix.
The symbol (·) × is defined as follows: For any vectors x ∈ R 3×1 and y ∈ R 3×1 , the mapping (·) × : R 3×1 → ςo(3) can convert x into a 3 × 3 skew-symmetric matrix, such that x × y = x × y. ςo(3) is the Lie algebra of SO(3), and can be described by ςo(3) = {H ∈ R 3×3 |H T = −H}. The inverse mapping of (·) × is defined as In addition, some useful properties are summarized as follows [26], [28] where tr(·) is the trace of a matrix. In this paper, four (n=4) SGCMGs configured on the x-y plane of the spacecraft constructing a pyramid-type is considered, as depicted in Fig. 1.

B. Actuator Dynamics
(2) can be described by where h 0i is the magnitude of angular momentum of each CMG and without loss of generality, they are equal to h 0 . β is the skew angle, (δ i , i = 1, 2, 3, 4) is the ith gimbal angle, c(x) = cos(x), and s(x) = sin(x). The dynamics of the SGCMGs can be obtained by [2], [5] u c = −ḣ = Aδ where δ = [δ 1 δ 2 δ 3 δ 4 ] T denotes the gimbal angle vector, and A is the Jacobian matrix with the following The control signal allocation problem can be described by the following constraint The solution of this minimization problem is [2] where A † = A T (AA T ) −1 , which is referred to as the pseudoinverse steering law. If rank(A) < 3 for certain sets of δ, the system has been caught in a singular state. A singular state occurs when all the momentum vectors h i of the SGCMGs system have either maximal or minimal projections in the direction of u c . One of the principal difficulties in the widely application of SGCMGs to attitude control is the geometric singularity problem. Therefore, a singular-free steering law is required.
The inputs of the piezoelectric actuators are the vibration displacements of the flexible appendages. However, the modal information of appendage vibrations is hard to measure for on-orbit spacecraft. Thus, the inputs of u p will be the estimation of the modal information, which can be expressed by the following equation whereη,η are the estimation of η and its derivative.

C. Relative Attitude Dynamics of a Spacecraft
The attitude stabilization control problem investigated in this paper requires the development of a control law, such that the spacecraft can rotate to a desired smooth attitude command R d ∈ SO(3) where Ω d is the desired angular velocity. Then we can get the attitude error R e = R T d R ∈ SO(3). An appropriate smooth positive configuration error function Ψ : SO(3) × SO(3) → R is required to measure the attitude error R e , and then the derivative of Ψ(R, R d ) is used to obtain an attitude error vector and a velocity error vector in the tangent bundle of SO(3). Following these, the attitude controller can be designed with these error vectors via Lyapunov analysis. In this paper, the so-called modified trace function Ψ(R, R d ) defined in [23], [28] is adopted where G = diag([g 1 , g 2 , g 3 ]), g i > 0, i = 1, 2, 3. Taking the derivative of Ψ, we have where e R ∈ R 3×1 is the attitude tracking error, and e Ω ∈ R 3×1 is the angular velocity tracking error. Since the attitude stabilization problem is considered in this paper, the desired anglular velocity is equal to zero, which means e Ω = Ω, and Ω d = 0.
, taking the derivatives of e R and e Ω , substituting Eqs. (2) and (3) into the results, and using the relationships in Eq. (4), we can obtain the relative dynamics of the flexible-spacecraft attitude stabilization system studied in this papeṙ It is obvious that A ϑ is Hurwitz. For the developed relative attitude dynamics, the following properties are given (the proof is available in [28]) 2) The critical points of where e 1 , e 2 , e 3 are the unit where , e R denotes the Euclidean norm, and ψ is a positive 4) The matrix E(R, R d ) in the relative attitude kinematics (16) is bounded by In addition, the following assumptions hold March 25, 2021 DRAFT Assumption 1: The disturbance d that the spacecraft suffers is unknown but bounded, and satisfies d ≤d, whered is an unknown constant.
Assumption 2: Among the feedback states used in the control law design, only the modal information η and its first and second order derivativesη,η cannot be measured by the spacecraft.

Remark 1:
The disturbance in the space environment is due to solar radiation pressure, gravity gradient torque, and aerodynamic torque, which is bounded [4]. Thus, Assumption 1 is satisfied in practice. By introducing the attitude error function Ψ, we develop the model in a coordinate free way [23]. In addition, the piezoelectric input u p in Eq. (18) is applied to realize active vibration-suppression. Since it also appears in Eq. (17) and has an influence on the convergence of e Ω , the designs of u p and u c should be considered together in the stability analysis of the overall closed-loop system.

D. Control Objective
The attitude stabilization problem of a flexible-spacecraft is studied in this paper, and the control objective herein can be stated as follows. Based on the relative attitude dynamics of the flexible-spacecraft described by Eqs. (16)-(18), by designing u c and u p properly, we aim to control the spacecraft to rotate to any given desired attitude R d , and suppress the flexible vibration η simultaneously. Meanwhile, the above reasonable assumptions hold, and the controller u c should be robust against external disturbance d. Moreover, u p is designed based on the modal information, however, η and its derivatives cannot be measured. Therefore, an observer should be designed to estimate the modal information. These goals can be described mathematically as lim t→∞ e R = e Ω = 0, lim t→∞ η =η = 0, lim t→∞ (η −η) = 0, and lim t→∞ (η −η) = 0, whereη andη are the observer outputs.

III. GEOMETRIC STABILIZATION CONTROL AND ACTIVE VIBRATION-SUPPRESSION ON SO(3)
In this section, based on the relative attitude dynamics described by Eqs. (16)- (18), we solve the flexible-spacecraft attitude stabilization problem, and realize vibration-suppression at the same time. Since the modal information is not available in the controller design, the first step is to design an observer to estimate η andη. This is the key point to guarantee the fine performance of the designed control system. Then an adaptive PD plus SMC and an active vibration-suppression controller u p are proposed by using the observer outputs. In addition, the stability of the closed-loop system resulted by u c and u p is proved. Furthermore, since SGCMGs are employed to provide control torque, the method to distribute control signals to actuators without singularity will be discussed. First, the following necessary lemmas are given to provide theoretical support for the stability analysis of the designed controller.  [32]) For a differentiable function V (t) and its

A. Controller Design and Stability Analysis
An observer is required to estimate the modal information as a stepping-stone in the design of u c and u p . The piezoelectric input u p is designed by employing a feedback method with where K 1 , K 2 ∈ R 1×N are constant matrices.θ is the output of the observer, and the estimation error is denoted byθ = ϑ −θ. Substituting the expression of u p into Eq. (18), we havė where Then, the observer is designed with the following forṁ where P ∈ R 2N ×2N is a symmetric positive definite matrix, satisfying A T ϑ P + P A ϑ < 0 and A T ϑ1 P + P A ϑ1 < 0, , and S is an auxiliary error vector defined as where c is a positive constant. It can be observed that S has the same structure as a sliding mode surface, by which we propose the following adaptive controller where K p , K d , k 1 , k 2 , γ, and ǫ are all positive constants, and k 1 >d.
To analyze the stability of the closed-loop system yielded by the controller (27) Using the property of Ψ in (19), we have By defining ξ = [ e R ; e Ω ; θ ] ∈ R 3×1 , Eq. (31) can be written as where λ max and λ min represent the maximum and minimum eigenvalues of a matrix, respectively. From Lemma 1, we have that if c satisfies then Q 1 > 0. Thus, it is obvious that V is positive and radially unbounded.
Taking the derivative of V with respect to time along the system trajectories leads tȯ Substituting the adaptive control law (27) and using (26), yielḋ Substituting the observer (25) and u p into (36), and after some algebraic manipulations, one can obtaiṅ From (20), we have E(R, R d ) ≤ √ 2 2 tr(G). Furthermore, A ϑ1 + B = A ϑ . Then (37) can be changed tȯ where W > 0 is the solution of A T ϑ P + P A ϑ = −W . Since k 1 >d is guaranteed in the controller, therefore, we have the following inequalitẏ This inequality can be rewritten asV where From Lemma 1, we have that if c satisfies then Q 2 is positive. As a result, we haveV ≤ 0. V > 0, andV ≤ 0 mean that V has a lower bound and it is nonincreasing. Therefore, V,V ∈ L ∞ .
Integrating both sides of (40), one can obtain which can be further expressed as Hence, ξ ∈ L 2 , and ξ ∈ L ∞ . According to Lemma 2, we have e R , e Ω ,θ → 0 as t → ∞.

March 25, 2021 DRAFT
If the attitude maneuver has been completed, which means e Ω = Ω =θ = 0, then Eq. (24) becomeṡ Then, another candidate Lyapunov function is considered Taking the time derivative of V 1 , and substituting Eq. (46) into the result, yielḋ Then using similar stability analysis process as Eqs. (43) and (44) and Lemma 2, we have lim t→∞ ϑ = 0. In summary, the above analysis can be concluded by the following theorem  2) The controller can guarantee the almost global asymptotic stability of the closed-loop system, i.e., R → R d and Ω = 0 as t → ∞.

Remark 2:
The proposed controller (27) includes an adaptive term − k 1 S S +ǫk 2 2 with updated law (29). This part is applied to deal with the system disturbance. In addition, to eliminate the influence of the flexible deformation on attitude motion, two terms Ω × Zθ and −L 1θ are added into the controller, where η andη are replaced by the observer estimated information. In [15], sign function is used to realize similar control objective with an assumption that the modal information has an upper bound, which will cause chattering phenomenon. Though tanh function is employed to replace sign function, it implies deterioration in control accuracy. Therefore, we can conclude that the control law of this paper is less conservative. Moreover, for the adaptive term, there is a problem that if k 2 goes to zero before the sliding surface S, then it will turn into −k 1 sgn(S). As a result, the chattering phenomenon cannot be avoided. Considering (29), we have γ ǫk 1 k 2 S S +ǫk 2 2 ≤ γǫk 1 k 2 , and then the inequalityk 2 ≥ −γǫk 1 k 2 holds. Integrating both sides of this inequality, we can obtain k 2 (t) ≥ k 0 exp(−γǫk 1 t), where k 2 (t = 0) = k 0 > 0.
We can conclude that k 2 (t) > 0 holds all the time, and only tends to zero as t → ∞. Consequently, it can be guaranteed that k 2 converges to some positive nonzero values in simulation by selecting γ and ǫ suitably.

Remark 3:
As stated in the works of Liu [4] and Chen [26], spillover is an important problem in the vibrationsuppression of flexible-spacecraft. In this paper, the proposed controller u c and the active vibration-suppression March 25, 2021 DRAFT control u p can achieve the control objective with the first N modes. By selecting N large enough, the effect of residual modes on the overall system performance can be neglected, which means the spillover effects can be ignored with the controller proposed in this paper. In addition, from the above stability analysis, u p can realize active vibration-suppression if A ϑ1 is Hurwitz, and the values of K 1 and K 2 can be obtained by choosing P and W properly.

Remark 4:
In Theorem 1, we declare that the closed-loop system yielded by u c and u p is almost globally asymptotically stable. Ψ = 0 means the attitude of the spacecraft has been rotated to the ideal attitude by the designed controller u c . However, it should be noted that Ψ = 0 has three critical points, which means that when e R = 0, R can converge to R d or the other three critical points. If we show that these three undesired equilibria are unstable and the attraction regions to them have zero measure, then we can conclude that for almost all initial attitude errors, the spacecraft attitude can converge to the desired attitude R d . Therefore, the statement in Theorem 1 holds. At the first critical point If e Ω , θ , and k 2 /γ are sufficiently small, then V 2 > 0 can be guaranteed, Moreover, in the domain U ,V 2 = −V > 0. Therefore, the critical point R 1 is unstable and the instabilities of the other two critical points can be proved by the similar way. Furthermore, to guarantee the global asymptotic stability of the closed-loop system, if the spacecraft attitude moves to the neighborhoods of these undesired points, the hybrid controller such as in [33] could be adopted.

B. Steering Law Design
In this paper, to reduce the impact of singularity, the SDA steering law in [30] is considered. A singularity measure κ = det(AA T ) can be defined to describe the singularity degree [5], and κ = 0 means the system encounters singular state. Modifications have been made to Eq. (9) to generate the so-called singularity-robust (SR) steering laws such as [2], [5] to avoid the singularity problem. However, it has been proved in [30] that the SDA steering law has less torque error and the rotational motion is much smoother than the SR law. Therefore, a modified SDA law will be adopted in this paper to generate the required control torques.
First, the Jacobian matrix A can be decomposed into the product of three special matrices by the singular value decomposition (SVD) method where U a is an 3 × 3 unitary matrix, V a is an 4 × 4 unitary matrix, and S a is an 3 × 4 diagonal matrix and the last column of S a is zero. By discarding the zero column of S a and V a , we have March 25, 2021 DRAFT where V t and S t are the truncated matrices.
For S t , its singular values are positive and ordered such that When a singularity occurs, the rank of A is 2, and S t [3, 3] = 0. Preventing S t [3,3] going to zero will make sure that the singularity is avoided. Therefore, the steering law is designed as followṡ where  Table I. It should be noted that the diagonal values of J , namely J 0 = diag ([350, 270, 190])kg · m 2 , will be used to compute the control signal in order to show the robustness of the proposed controller against model uncertainty. In addition, four natural modes of the flexible appendages are considered in the simulation, and the main parameters of the flexible part are selected from [4], [6], [21] as follows. The coupling matrices Ξ and Ξ p are   In this section, we consider two cases. In Case 1, the spacecraft attitude is governed by the proposed controller (27) and the modal information is estimated by the observer (25), and the parameters are: K p = 20, K d = 100, is employed for the purpose of comparison to show the advantages of the controller proposed in this paper. Though the controller in [28] is applied to a rigid body, assuming that the modal information is available and the spacecraft operates in an environment without disturbance, the controller can also guarantee the stability of the closed-loop flexible-spacecraft attitude stabilization system with the following candidate Lyapunov function The controller parameters are consistent with each other in these two cases. It can be observed that the modal coordinates can be estimated by the proposed observer (25) with high accuracy.
The time responses of the modal coordinates ϑ are shown in Fig. 6, which illustrate that the vibration is suppressed effectively by the proposed active vibration-suppression controller u p . The plots of the control torques are given in Fig. 7, which are generated by the SGCMGs. The corresponding gimbal angles and gimbal rates are presented in Figs. 8 and 9, from which we can see that the limit on the gimbal rate in Table I is satisfied. Moreover, the time responses of the singularity measure are given in Fig. 10, which imply that the system avoids being caught in a singular state during the attitude maneuver. In addition, time responses of the adaptive gain k 2 are given in Fig. 11. It can be seen that k 2 decreases monotonically to nonzero values as illustrated in Remark 2. From the above simulation results, we can conclude that the tested results of Case 1 mean the control objective of this paper has been achieved with fine performance.   Table I is satisfied and the system also avoids being caught in a singular state during the attitude maneuver. However, comparing the control accuracy of these two cases with each other, we can find that not only the control accuracy of the attitude stabilization maneuver has been improved, but also the vibration-suppression performance is better than the traditional PD-like controller in (55).
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