AttitudeManeuvering andVibrationReducingControl of Flexible Spacecraft Subject to Actuator Saturation and Misalignment

A robust adaptive constrained control scheme is proposed for flexible spacecraft attitude maneuver and vibration suppression, in which multiple constraints are simultaneously considered, such as uncertain inertia parameters, external disturbance, unmeasured elastic vibration, actuator saturation, and even actuator misalignment. More specifically, a novel path planning scheme based on quintic polynomial transition is firstly developed to realize smooth acceleration variate and therefore decrease the vibration of flexible appendages. Secondly, an elastic modal estimator is employed to estimate the unmeasured variables, such as the modal position and velocity. *irdly, an adaptive updating technique is used to spare the extra knowledge of system parameters and the bound of the external disturbance. In addition, an auxiliary design system is constructed to address the actuator saturation problem, and a compensation term is synthesized and integrated into the controller to handle the actuator misalignment. Finally, overall system stabilization is proved within the framework of Lyapunov theory, and numerical simulation results are presented to illustrate the effectiveness of the proposed scheme.


Introduction
Flexible spacecraft with large flexible structures is usually expected to achieve high pointing and fast attitude maneuvering in future space missions.However, the attitude maneuvering operation will introduce certain levels of vibration to flexible appendages due to the rigid-flexible coupling effect, which will deteriorate its pointing performance.For flexible spacecrafts, the governing differential equations for attitude kinematics and dynamics are strongly nonlinear in nature.e attitude maneuvering problem is further complicated by the uncertainty of spacecraft inertia parameters due to onboard payload motion and fuel consumption.Furthermore, it is also affected by various external disturbances that influence the mission objectives significantly.Additionally, the actuator misalignment during installation and actuator saturation increases the complexity and difficulty further.
All these factors in a realistic environment cause a considerable difficulty in the design of attitude control system for meeting high-precision pointing requirement and desired control performance during the attitude maneuver process, especially when all these issues are treated simultaneously.
Over the last few decades, considerable works have been found for vibration suppression and attitude control system design.A nonlinear state feedback attitude control law [1] in combination with path planning is proposed.Specifically, the planned attitude maneuver path is smooth, and thus, appendages' vibration excited by attitude maneuver can be attenuated greatly.Specially, variable structure control (VSC) is known as an efficient way to deal with system uncertainty and external disturbance and has been applied to attitude control problem of flexible spacecrafts in [2,3].
However, these design methods require the information on the bounds on the uncertainties or disturbances for the computation of control gains.Recently, to overcome the drawbacks of each method, a combination of these techniques with adaptation mechanism to tune the controller gains are also studied for flexible spacecrafts with parameter uncertainties and disturbances [4][5][6].A new adaptive system [4] for rotational maneuver and vibration suppression of an orbiting spacecraft with flexible appendages has been designed.en adaptive output regulation of the closed-loop system is accomplished in spite of parameter uncertainties and disturbances.A new variable structure control approach [5] has been proposed for attitude control and vibration suppression of flexible spacecrafts during attitude maneuvering, and the adaptive version of the proposed controller is achieved through releasing the limitation of knowing the bounds of the uncertainties and perturbations in advance.
Relevant drawback of these control strategies is either the extra necessity to measure the modal variables or to treat the rigid-flexible coupling effect as an additional disturbance acting on a rigid structure.With regard to the latter situation, an extended state observer [7,8] is designed to estimate and thus to attenuate the total disturbance in finite time, including an external disturbance torque and the coupling effect.As a result, the prior knowledge of the total disturbance is not required.
Unfortunately, in some cases, the availability of the measured modal variables is an unrealistic hypothesis due to the impracticability of using appropriate sensors or the economical requisite.For removing this disadvantage, reconstructing the unmeasured modal position and velocity by means of appropriate dynamics is an alternative way.A class of nonlinear controllers incorporating modal state estimator [9][10][11][12][13] has been derived for spacecraft with flexible appendages.It does not ask for measures of the modal variables, but only uses the parameters describing the attitude and the spacecraft angular velocity.e controller derived then uses estimates of the modal variables and its rate to avoid direct measurement.
However, a typical feature in all of the mentioned attitude-control schemes and methods is that the control device is assumed to be able to produce big enough control torque without taking actuator saturation into account.Extensive results pertaining to spacecraft attitude control systems containing actuator saturation nonlinearities have been presented in [14][15][16][17][18][19][20][21][22].A robust variable structure controller in [14,15] has been skillfully designed to control the spacecraft attitude under input saturation.However, these control schemes lose the generality to nonlinear flexible spacecraft system.A modified adaptive backstepping attitude controller [17,18] is developed considering external disturbances and input saturation.But the way to include the effects of the flexible dynamics in the lumped disturbance for the rigid dynamics deprives the controller of a direct compensation of the dynamic terms caused by the flexibility.
A typical feature in the previous approaches is that they do not consider actuator misalignments.However, whether due to finite manufacturing tolerances or warping of the spacecraft structure during launch, some actuator alignment error will definitely exist in practice.is problem may cause mission performance to degrade and thus pose significant risk to the successful operation of the spacecraft.erefore, it is desirable to design a control scheme to handle the actuator misalignments.Unfortunately, there has been insufficient research on attitude control in the presence of actuator misalignments.An adaptive control law [23] is developed to accomplish attitude maneuver in the presence of relatively small gimbals' alignment error of variable speed control moment gyros.
More specially, an adaptive control approach [24] was proposed for satellite formation flying.
e backstepping technique is used to synthesize the controller, and the thrust misalignment is successfully handled.In another related work, a nonlinear model reference adaptive control scheme [25] is tested in the presence of misalignment errors up to fifteen degrees.Although an extended Kalman filter is used in another approach to develop methods for on-orbit actuator alignment calibration, uncertain inertia properties have not been taken into account.
With a view to handle the above challenges and potential problems simultaneously, a new constrained robust adaptive control scheme is proposed in this paper.e main contributions are summarized as follows: (1) A novel path planning scheme based on quantic polynomial transition is applied to smooth acceleration variate and therefore decrease the vibration of flexible appendages.( 2) is paper investigates the feasibility of attitude maneuver and vibration suppression in the presence of uncertain inertia parameters, external disturbances, unmeasured elastic vibrations, actuator saturation, and even actuator misalignment simultaneously, which has not been previously examined.
As compared with the controller in [16] which only considers attitude stabilization, that is, rest-to-rest maneuver of a flexible spacecraft, the control scheme in the present paper can be applied to handle attitude tracking control problem.e controller in [6] investigates the attitude tracking problem for flexible spacecrafts, while it does not take the actuator saturation and actuator misalignment into consideration explicitly.(3) e proposed controller is designed without requiring prior knowledge of the measurement of vibration variables, of the parameter uncertainties, and the upper bound of external disturbance.Robust control terms are synthesized to compensate all uncertainties including parametric uncertainties, unmeasured elastic vibration, external disturbances, actuator saturation, and actuator misalignment.
e rest of this paper is organized as follows: Section 2 states flexible spacecraft modelling with actuator misalignment and control problem formulation.A constrained robust adaptive-control scheme is proposed in Section 3. Numerical simulation results are presented in Section 4 to demonstrate the effectiveness and superiority of the proposed control scheme.Finally, conclusion is given in Section 5.

2
Shock and Vibration

Kinematics and Dynamics Equation.
is section briefly reviews the quaternion mathematical description of the attitude motion of a flexible spacecraft.
e attitude kinematic equation of the spacecraft can be written in terms of unit quaternion for global representation without singularities as follows [26]: where q � [q 0 , q T v ] T represents the attitude quaternion, and I is an identity matrix of the dimension specified by its subscript.
e dynamic model of a flexible spacecraft is governed by the following differential equations: where ω is the spacecraft angular velocity in the body-fixed frame, ω × is the skew-symmetric matrix of ω, η is the modal coordinate vector of the flexible appendages, the matrix J denotes the total inertia of spacecraft moment, δ is the coupling matrix between the flexible appendages and the rigid spacecraft, u is the control torque, d is the external disturbance torque vector, and C and K denote the damping and stiffness matrices, respectively, and they are defined as where N is the number of elastic modes considered, ω ni is the natural frequency, and ξ i is the corresponding damping ratio.
For simplicity of the development, let us first introduce the following variable ψ � _ η + δω representing the total angular velocity expressed in modal variables, so the dynamics of the flexible spacecraft from (2) to (3) can be further expressed as where J m � J − δ T δ, with δ T δ as the contribution of the flexible parts to the total inertia matrix.

Attitude Tracking Model.
Let q d � [q d0 , q T dv ] T be the unit quaternion representing the desired attitude, ω d be the desired angular velocity.en, the quaternion error q e and angular velocity error ω e are given by q e � q −1 d ∘ q � q 0 q d0 + q T v q dv q d0 q v − q 0 q dv + S q v q dv  , ) where q −1 d is the inverse of the desired attitude unit quaternion, the symbol ∘ is the operator for quaternion multiplication; with ω r � R(q e )ω d , we have where R(q e ) is the transformation matrix from the desired coordinate frame to the body coordinate frame with R q e  � q 2 e0 − q T ev q ev  I 3 + 2q ev q T ev − 2q e0 S q ev .
From ( 1) to (10), tracking error dynamic equation can be obtained as follows: where A and B are given by

Actuator Uncertainty Model.
In practical aerospace engineering, redundant actuators are fixed to improve the reliability of the attitude control system.e considered spacecraft is controlled by using four reaction wheels.e configuration structure of four actuators in [27,28] is adopted.
ree reaction wheels are fixed orthogonally aligned with the axes of the body-fixed frame.e fourth, redundant, actuator is mounted skewed at α 4 and β 4 .
With this configuration, the total control torque u can be calculated as where τ � [τ 1 , τ 2 , τ 3 , τ 4 ] T denotes the nominal torque generated by the four reaction wheels, and D is the reaction wheel configuration matrix, representing the influence of each wheel on the angular acceleration of the spacecraft.However, in practice, the knowledge of orthogonal configuration of actuator will never be perfect.Due to finitemanufacturing tolerances or warping of the spacecraft structure during launch, actuator misalignments may exist.

Shock and Vibration
e reaction wheel mounted on the X-axis is offset from the nominal direction by constant angles, Δα 1 and Δβ 1 .e reaction wheels mounted on Y-axis and Z-axis are assumed to be tilted away from their nominal directions by Δα 2 , Δβ 2 , Δα 3 , and Δβ 3 , while the redundant reaction wheel is tilted from its nominal direction by Δα 4 and Δβ 4 .
en, the real control torque acting on spacecrafts with misalignment is expressed as e definition of the misalignment angles suggests that Δα i , Δβ i , i ∈ 1, 2, 3, 4 { } are small angles.Hence, the following relationships are used to approximate (16): en for the considered actuator configuration, the configuration matrix D can be represented as where D 0 denotes the nominal configuration matrix, and ΔD denotes the actuator misalignment.ey can be written as In view of ( 15) and ( 18), tracking error dynamics in (12) can be rewritten as To facilitate control system design, the following assumptions and lemmas are presented and will be used in the subsequent developments.

Assumption 1.
e components of external disturbance d in ( 21) are assumed to be bounded by a set of unknown bounded constants, that is Assumption 2. Due to physical limitations on the reaction wheels considered, the maximum output torque of each actuator output torque has the same limit value (τ, � τ), that is [29] where τ and � τ represent the known saturation levels of reaction wheels.

Assumption 3. e uncertainty ΔD due to misalignment is an unknown but bounded matrix satisfying
where Δ m is a known positive constant.
Remark 1.According to (20), every element of actuator misalignment matrix ΔD is a function of misalignment angle errors which are small in practice, then the Frobenius norm of ΔD is bounded by a known quantity Δ m .It is easy to find a feasible Δ m in practice.

Control Problem Formulation.
Given any initial attitude and angular velocity, the control objective can be stated as considering the flexible spacecraft attitude system described by ( 1)-( 3), design a torque command τ such that the following goals are met in the presence of uncertain inertia parameters, external disturbance, unmeasured elastic vibration, actuator saturation (23), and even actuator misalignment: (1) e attitude orientation and angular velocity tracking errors are driven to zero, or a small set containing the origin.(2) e vibration induced by the maneuver rotation should be attenuated as soon as possible in the presence of parametric uncertainties, external disturbance, actuator saturation, and even actuator misalignment.

Robust Adaptive Constrained Controller Design
e proposed attitude control scheme for the flexible spacecraft is shown in Figure 1, in which the controller is composed of an auxiliary design system to compensate the effect of actuator saturation and modal estimator to estimate 4 Shock and Vibration unmeasured modal position and velocity, as well as path planning, which will be given in the following subsections.
3.1.Path Planning.From ( 2) and ( 3), it is known that attitude maneuvering would excite appendages' vibration [8], which is closely related to the attitude angular acceleration.A sudden change in attitude signal, especially in the form of step signal, may cause a serious appendages' vibration.Here, in order to achieve high performance of attitude control and attenuate the residual vibration, a novel path planning scheme based on quintic polynomial transition is developed, which is motivated by the previous study of robot trajectory planning in joint space [32].With regard to the limit of the angular velocity and the angular acceleration, the maneuvering path is designed as follows: e angular acceleration signal of the path is planned as a quintic polynomial curve.
en, parameters a 5 , a 4 , a 3 , a 2 , a 1 , and a 0 should be selected appropriately to satisfy the following equality and inequality constraints.
where _ φ max and € φ max are the permitted maximum angular velocity and acceleration, respectively.
Using (27) yields Angular velocity and acceleration with regard to ( 26) and ( 29) are obtained as With the notation τ t/t f and â5 a 5 t 5 f , (30) can be rewritten as For notational convenience, we de ne f(τ) as It can be easily veri ed that

Controller Design.
To remove the hypothesis of the measurability of the modal position and velocity, an elastic modal estimator to supply their estimates is constructed as follows: where η, ψ are the estimates of modal variables, and e η � η − η, e ψ � ψ − ψ are their estimation errors.From ( 13) and (37), the response of e η , e ψ can be algebraically arranged as Since matrix A is a Hurwitz matrix, the estimation errors e η , e ψ will converge to zero asymptotically.
Considering the actuator saturation in (23), the actual control input τ � [τ 1 , τ 2 , τ 3 , τ 4 ] T in ( 21) can be further defined as where τ c � [τ c1 , τ c2 , τ c3 , τ c4 ] T is the desired control input that will be designed in the subsequent developments.en, ( 21) can be rewritten as Step 1.We start with ( 11) and ( 13) by considering ω e as the virtual control variable.Define the tracking error as where α c is the output of the following first-order filter utilized to approximate the derivative of the virtual control _ α.
e auxiliary signal ξ in (41) is designed to compensate the effect of actuator saturation and can be produced by the following system: where K ξ is a positive matrix to be chosen, Δτ denotes the difference between the applied control and the designed control input e first Lyapunov candidate function is chosen as where the positive definite matrices K 1 and K 2 are partitioned as Using ( 11), (37), ( 38) and (41), the time derivative of ( 45) is given by 6 Shock and Vibration ( Define the virtual control α as Based on what is mentioned above, (47) be rewritten as Step 2. Take the derivate of z left-multiplied by inertia matrix with respect to (40), then we have Although the inertia matrix is unknown for the system design, it can be observed that the inertia parameters appear linearly in (51).To isolate these parameters, a linear operator Let θ T m � [J m,11 , J m,22 , J m,33 , J m,12 , J m,13 , J m,23 ], it follows that J m a � L(a)θ m and then (51) can be rewritten as ).  θ m denotes the estimate error of θ m and is defined as  θ m � θm − θ m .e design procedure can be summarized in the following theorem.

Theorem 1. Consider the flexible spacecraft system involving uncertain inertia parameters, external disturbance, unmeasured elastic vibration, actuator saturation, and even actuator misalignment. If the control law is designed by
where tanh(z) � diag(tanh(z j /ε dj )), the adaptive control is selected as where F 2 � F 1 − L( _ ξ), the projection operator Proj(•) is defined in [33] to avoid the parameter drift problem, then the attitude orientation and angular velocity tracking errors are uniformly ultimately bounded.
Proof.Consider the composite Lyapunov function V 2 as Substituting the aforementioned control torque (54), taking the derivative of the above Lyapunov function with regard to (50) and (53), it follows In view of the conclusion in [34], we have ‖z α ‖ ≤ μ, then Shock and Vibration e last two terms in (57) can be expanded as According to Lemma 1, we have where en, substituting the aforementioned adaptive law and considering the property of projection operator as well as the following inequality gives Assumption 2 dictates that ‖sat(τ c )‖ ≤ 2τ m , τ m is the permitted maximum control torque.Combining Assumption 2 and Lemma 1 yields to From the above development, (57) can be further derived as If parameters K 1 , K 2 , and K 3 are selected such that then _ V 2 can be further upper bounded by where ε t is given by Hence, V 2 is bounded.It follows the definition of V 2 that the quaternion tracking error q ev , angular velocity control error z, modal variables estimates [η T , ψT ] T , and parameter estimation error  θ m are all bounded.erefore, the controlled closed-loop system is uniformly ultimately bounded.
In addition, following the definition of V 2 also yields max q T ev q ev , such that quaternion tracking error q ev and angular velocity control error z, respectively, converge to the following compact sets: According to (41), the angular velocity tracking error ω e is also uniformly ultimately bounded.
e convergence domain Ω 1 and Ω 2 can be adjusted by an explicit choice of design parameters.
e rst four elastic modes have been taken into account for the implemented spacecraft model resulting from the modal analysis of the structure, with natural frequency and damping presented in Table 1.
e maximum angular velocity and acceleration of the exible spacecraft are v max 0.0343 rad/s and a max 0.0286 rad/s 2 , respectively.Herein, we consider a three-axis large angle maneuver.e initial attitude in Euler angles is 0 °30 °45 ° T , and the corresponding initial attitude in quaternion is q T (0) [0.8924, −0.0990, 0.2391, 0.3696].
To examine the robustness to external disturbance, simulation is done corresponding to the following periodic disturbance torque d(t) 0.03 cos(0.01t)+ 0.01 0.015 sin(0.02t)+ 0.03 cos(0.025t)0.03 sin(0.01t)+ 0.01 For the purpose of comparison, two di erent sets of simulation are conducted to demonstrate the e ectiveness of the proposed control approach as follows: Case 1. Attitude control without considering the actuator saturation and even actuator misalignment.
e robust adaptive control (RAC) law and corresponding adaptive law are derived in steps identical to those employed in Section 3.2 as Case 2. Attitude control using constrained robust adaptive control (CRAC) law (54) and adaptive law (55), taking actuator saturation and even actuator misalignment into account explicitly.

Shock and Vibration
In the following simulations, the control and adaptation gains are provided in Table 2. e maximum output torque of reaction wheel is assumed to be 10 Nm.For the overactuated reaction wheel with con guration misalignments, the normal assembling angles are α 4 35.26  e planned maneuvering path is illustrated in Figures 2-4.For the comparison, the conventional path planning method based on bang-coast-bang (BCB) is also performed at the same simulation conditions.It is illustrated that the maneuver path utilizing planning scheme based on quantic polynomial transition is much smoother than the one by applying conventional method.Smooth angular acceleration variate will introduce the least level of vibration to exible appendages.
With the application of the proposed CRAC scheme with path planning in Case 2, the simulation results are summarized in Figures 5-21.Since the attitude variables in terms of Euler angles are easier to understand than quaternion, the attitude tracking error in Euler angles denoted as e φ , e θ , and e ϕ are also displayed in Figures 5 and 6.

Shock and Vibration
It can been seen from Figures 5, 7, and 8 that it takes about 100 s to drive the tracking errors of attitude Euler angles, attitude quaternion, and angular velocity to their stable points.rough enlarged version of attitude tracking errors illustrated in Figure 6, it can be observed that steady errors of Euler angles are maintained within a range of less than 0.0006 rad. Figure 8 shows the response of angular velocity tracking error, and the steady errors are no more than 2.9e − 6 rad/s.us, the high-precision attitude control performance can be achieved despite the presence of parameters uncertainties, external disturbance, actuator saturation, and even actuator misalignments.
e responses of estimated inertial parameters corresponding to update law of (55) and (78) are illustrated in Figures 9-12.It is clear that the convergence of these estimated parameters can be achieved, but not to the true values.
at is because sufficient frequency components in the tracking error states are not guaranteed.In other words, the persistent excitation (PE) condition is not satisfied.
e behavior of the modal displacements and their estimates are given in Figure 13.It is noted that all the elastic vibrations and their rates approach zero at time 70 s.It can be observed that not only the vibrations excited by the attitude maneuver are effectively suppressed but also the modal displacements can be well estimated by the modal observer whose performance is explicitly demonstrated in Figure 14. e steady observation errors of modal observer in (37) are tabulated in Table 3.Furthermore, the time responses of total required control torques and reaction wheel output torques are shown in  respectively.e output torque of each reaction wheel is within the saturation limitation, that is, ± 10 Nm, which is because the proposed control scheme addresses the control saturation constraints explicitly.
For comparison, the system is also controlled by RAC scheme in Case 1.When it is applied under the same initial conditions and control parameters, the control performance is also illustrated in Figures 5-21 where it is denoted as "RAC."e steady errors of Euler angles depicted in Figure 6 are 0.005 rad.As demonstrated in Figure 8, it can be observed that severe oscillations of the angular velocity tracking error occur in transient phase and its steady values are no more than 4.1e − 5 rad/s, which are larger than those of the proposed CRAC scheme in Case 2. us, although the attitude rotational maneuver using RAC scheme is successfully performed, the control performances are much poorer than those using the proposed CRAC scheme owing to not considering the actuator misalignment.
Further, as is shown in Figures 15-21, the saturation phenomenon of the reaction wheel torques using RAC scheme is not addressed.us, it can be concluded that desired performance of the system can be achieved by the proposed control scheme Shock and Vibration 13 subject to multiple undesired constraints including uncertain inertia parameters, external disturbance, unmeasured elastic vibration, actuator saturation, and even actuator misalignment.

Conclusion
Taking uncertain inertia parameters, external disturbance, unmeasured elastic vibration, actuator saturation, and even actuator misalignment into account simultaneously, this paper investigates the problem of attitude maneuver and vibration suppression of exible spacecrafts.Firstly, a new way of path planning, based on quintic polynomial transition, is designed to suppress the exible appendages' vibration e ectively.en, by utilizing the inherent physical properties of exible appendages, a modal observer is constructed to supply elastic modal estimates.Plus, an adaptive law is derived so that the requirements to know the prior knowledge of parameter uncertainties and the upper bound of external disturbance is eliminated.In addition, an auxiliary design system is introduced to compensate the actuator saturation e ect, and a compensation term is

□ 4 .
Numerical SimulationNumerical simulations have been carried out in this section to illustrate and verify the effectiveness of the proposed control scheme.e spacecraft is characterized by a nominal main body inertia matrix

Figure 7 :Figure 8 :
Figure 7: Time response of attitude tracking error in quaternion.

Figure 9 :
Figure 9: Time response of estimated parameters of inertia in Case 1.

Figure 12 :Figure 11 :
Figure 12: Time response of estimated parameters of the product of inertia in Case 2.

Figure 10 :
Figure 10: Time response of estimated parameters of the product of inertia in Case 1.

Table 1 :
Parameters of the exible dynamics.

Table 2 :
Design parameters for the di erent controllers.

Table 3 :
Steady observation errors of modal observer.