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Experiments study on attitude coupling control method for flexible spacecraft
Accepted Manuscript Experiments study on attitude coupling control method for flexible spacecraft Jie Wang, Dongxu Li PII:
S0094-5765(17)31463-7
DOI:
10.1016/j.actaastro.2018.03.023
Reference:
AA 6765
To appear in:
Acta Astronautica
Received Date: 16 October 2017 Revised Date:
20 February 2018
Accepted Date: 12 March 2018
Please cite this article as: J. Wang, D. Li, Experiments study on attitude coupling control method for flexible spacecraft, Acta Astronautica (2018), doi: 10.1016/j.actaastro.2018.03.023. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Experiments Study on Attitude Coupling Control Method for Flexible Spacecraft Wang Jie*, Li Dongxu College of Aerospace Science and Engineering, National University of Defense Technology
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NO.47 Yanwachi Street, Changsha 410073
*Corresponding author. E-mail: [email protected]
Abstract
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High pointing accuracy and stabilization are significant for spacecrafts to carry out Earth observing, laser communication and space exploration missions. However, when the a spacecraft undergoes large angle maneuver, the excited elastic oscillation
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of flexible appendages, for instance, solar wing and onboard antenna, would downgrade the performance of the spacecraft platform. This paper proposes a coupling control method, which synthesizes the adaptive sliding mode controller and the positive position feedback (PPF) controller, to control the attitude and suppress the elastic vibration simultaneously. Because of its prominent performance in for
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attitude tracking and stabilization, the proposed method is capable of slewing the flexible spacecraft with a large angle. Also, the method is robust to parametric uncertainties of the spacecraft model. Numerical simulations are carried out with a hub-plate system which undergoes a single-axis attitude maneuver. An attitude
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control testbed of for the flexible spacecraft is established and experiments are conducted to validate the coupling control method. Both the numerical and
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experimental results demonstrate that the method discussed above can effectively decrease the stabilization time and improve the attitude accuracy of the flexible spacecraft.
Keywords: Flexible Spacecraft; Sliding mode control; Positive position feedback;
Attitude Coupling Control; Experiment
1 Introduction Early spacecrafts is were usually idealized as a rigid bodies for attitude control while the flexible parts is were ignored because of its their small mass or inertia [1-4]. With the application of large scale solar wings and antennas on modern spacecrafts, the influences of flexibilities on characteristics and attitude of the whole system 1 / 25
ACCEPTED MANUSCRIPT cannot be ignored. The slew maneuver of the platform of the spacecraft will excite the elastic oscillations of flexible appendages, which will downgrade the pointing accuracy of the platform [5-7]. Therefore, flexible appendages must be considered while designing the attitude control method. Besides, spacecrafts often contain propellant tanks with large quantities of liquid fuel, whose slosh excited by the main
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bus would also influence the spacecraft’s attitude [8-11]. In addition, the vibrations of the flexible appendages and fuel slosh will change the physical properties of the spacecraft [8, 12]. And the fuel consumption over time would introduce further uncertainty on the satellite spacecraft model description. Hence, the design of the
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attitude stabilization theories is crucial issues for flexible spacecrafts with parametric uncertainties.
In the field of attitude control for flexible spacecrafts, a mass of control methods
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have been proposed in last decades, for example, H∞ control method [13]、sliding mode controller [14], model predictive control [15], fuzzy control [16], adaptive fuzzy control [17], neural network controller [18], etc. For further improving the performances of the attitude control methods, the vibration controllers for flexible appendages has have been introduced and the coupling control method has been developed. The basic idea of the coupling control method is to combine the
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stabilization theory with the a vibration controller. The attitude of the platform and the elastic vibrations of the flexible appendages are controlled simultaneously. In general, the frequency bands of the attitude control subsystem and the
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vibration control subsystem are non-overlapping or the coupling is weak. Under this circumstance, the closed loops for the two subsystems can be designed separately. And the designed controllers can comply with the requirements. As the scale of the
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flexible appendages increases and the natural frequencies decrease, severe overlap occurs in the frequency bands of the two subsystems. One way to conceive of the controller is to decouple the two subsystems before the controller design. Quinnt and Meirovitch [19] proposed a first-order perturbation approach to separate the equations of motion into a set of equations governing rigid-body slewing of the spacecraft and a set of time-varying linear equations governing small elastic motions. Then a maneuver force distribution was developed which excites the least amount of elastic deformation of the flexible parts of the spacecraft. Kakad [20] used four Euler parameters to express the motion of a flexible spacecraft and the maneuver problem was reduced to be a system of uncoupled equations. Azadi et al. [21] divided the 2 / 25
ACCEPTED MANUSCRIPT system dynamics into two fast and slow subsystems using singular perturbation theory. To date the applied attitude control subsystem is mainly based on PD, LQR, sliding mode control and back-stepping control theories, etc. The flexible control subsystem is based on passive or active vibration control method and the actuators are
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mainly piezoelectric materials. Sales et al. [22] constructed a passive control system for flexible appendages using piezoelectric transducers. The results demonstrated that the plate vibrations levels and coupling between the flexible and rigid body motions were significantly reduced during the spacecraft maneuver. Silverberg and Baruh [23]
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used a natural control to suppress the vibration so that the control of the elastic motion does not distort the maneuver. Then the maneuver of the platform can be designed and performed independently. Then Baruh [24] applied this method to lightweight
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multibody structures, and designed a vibration controller based on piezoelectric film actuators. But the decouping procedure depends on the accuracy of the system’s dynamic model. And the error and uncertainties of the model were ignored. Meirovitch [25] designed an both open loop and closed loop controllers, in which the former one was to cancel the environmental disturbance and the later one was utilized to control the spacecraft and flexible appendage based on the optimal control theory.
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Gennaro [26] studied the active suppression of appendages during attitude tracking or large-angle maneuver. A PD controller was designed for attitude with thruster and fly wheel while the piezo-electrical materials were used to suppress the
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vibration of a flexible beam. Grewal and Modi [27] proposed a two-level strategy using the LQG/LTR approach and the strategy achieved good attitude control and vibration suppression behavior. Zhu et al. [28] studied a robust hybrid control design
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method which combines the backstepping control law with the strain rate feedback control method. Azadi et al. [29] applied an adaptive-robust control scheme to three axes maneuver of a flexible satellite, And the control method also suppressed the elastic vibration. Cui and Xu [30] designed the attitude control method via θ-D method and adopted the positive position feedback (PPF) for vibration control. Shahravi and Azimi [31] compared the performance of the collocated and non-collocated piezoceramic patches acting as sensors and actuators during attitude maneuver. Azadi et al. [21] established the dynamic model for a flexible spacecraft under three axes slewing maneuver. And they proposed an adaptive-robust control scheme for attitude control and a Lyapunov based controller for the vibration 3 / 25
ACCEPTED MANUSCRIPT suppression of the flexible structure. Some literature combines the input shaping method for attitude maneuver of and the active vibration control for flexible appendages. SONG and AGRAWAL [32] presented an approach which is built on the input shaping method and the pulse-width pulse-frequency (PWPF) modulator to reduce elastic vibrations during the attitude
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control. The closed loop sub-system of attitude feedback control employs a PD controller while the active vibration suppression sub-system uses the PPF control strategy. Experiments was were conducted on a system which consists of a central hub and a L-shape flexible appendage. Based on Song’s work, in order to solve the
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problem of model uncertainties, Hu and Ma [33, 34] proposed an approach based on PPF to vibration reduction during maneuver by using the theory of variable structure control [35]. The attitude controller consisted of linear and noncontinuous feedback
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items, which ensure the sliding manifold existing and global reachable. Hu and Ma [36] programmed the maneuver trajectory based on the component synthesis vibration suppression (CSVS) method with the pulse-width pulse-frequency (PWPF) modulation. And the PPF control technique using piezoelectric materials, acted on the flexible parts. The CSVS method and the input shaping method share the similar principle to reduce the residual vibration. Then Hu [37] used the backstepping method
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and modal velocity feedback method to attitude control and elastic vibration. In general, depending on the idea of input shaping method, the behavior of attitude actuators is programmed, for instance, the on-off of thrusters or the output torques of
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flywheels. Na et al. [38] proposed another way to reduce vibration through controlling the rotation at the root of the solar array using zero-placement input shaping technique.
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In former research, the attitude control subsystem and the vibration control
subsystem were mostly designed independently. And the stabilization of the whole system was not considered. Few literatures conducted experiments to validate the controller. This paper synthesizes the adaptive sliding mode controller and the PPF controller to stabilize flexible spacecrafts which undergo large-angle maneuvers. This paper is organized as follows. The physical and mathematical models of the flexible spacecraft are given in the second section. The coupling control method is presented in the third section. Numerical simulation results are presented in section 4. Finally, in section 5, experiments are conducted in order to demonstrate the efficacy of the designed controller. Section 6 provides concluding remarks. 4 / 25
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2 Physical and Mathematical Models Figure 1 presents a typical flexible spacecraft, which consists of a central core and several flexible appendages, for instance, solar arrays and antennas. The flexible spacecraft experiences rigid motions in six freedoms, including translations and
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rotations. On the other hand, the maneuver of the spacecraft or other environmental
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forces would excite elastic oscillations of flexible appendages.
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Figure 1 Scheme of the flexible spacecraft
In order to derive the governing equations for the whole system, the following assumptions are made. (1) The central core is typically modeled as a rigid body which is free to undergo unrestricted rotations relative to inertial space. (2) The appendages
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are modeled as flexible bodies that can be discretized by the finite element method (FEM). Moreover, they only have rigid rotations relative to the central core while
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translations of the hinge point are restricted. (3) The elastic deformations of the appendages is are assumed small and yield to the linear elasticity theory. Therefore, the center of mass of the whole system remains fixed in the body frame of reference. According to the Lagrange’s equations in terms of quasi-coordinates, the
mathematical model of the flexible spacecraft is established, and the governing equations can be expressed as
J ( t ) ωɺ + D ( t )ηɺɺ + ω × ( J ω + D ( t )ηɺ ) = τ D ( t ) ωɺ + ηɺɺ + 2ξΛηɺ + Λ2η = 0 T
(1)
where J is the inertia matrix of the spacecraft, which is the sum of the inertia matrix of the central body and that of the flexible appendage; ω is the angular velocity vector 5 / 25
ACCEPTED MANUSCRIPT of the central body with respect to the inertial frame; τ=[τߠ, τ߮, τΨ]T denotes the control torque applied to the spacecraft around three axes produced by the actuators such as thruster, reaction wheels, and CMG; D is the rigid-flex rotational coupling coefficient matrix; η (1×N, N is the truncation number) is the modal coordinate of the flexible appendage; ξ =diag[ξ1, ξ2, … , ξN] is the damping ratio matrix of the
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appendage, where ξi is the ith damping ratio; Λ=diag[Λ1, Λ2, … , ΛN] is the natural frequency matrix of the cantilevered appendage, where Λi is the ith natural frequency. In general, when the flexible appendage is fixed to the main body of the spacecraft, the inertia matrix J and the coupling coefficient matrix D remain constant
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with respect to time. However, when the appendage rotates relative to the main body, the inertia of the appendage about the body frame changes continuously so that the inertia matrix of the whole system is time-variable. Also, the rigid-flex coupling
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coefficient matrix varies with respect to time because of the time-varying position between the main body and the appendage.
3 Coupling Control Strategy
The coupling control system is composed of two subsystems, the attitude
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compensator subsystem and the vibration control subsystem, as shown in Figure 2. The coupling controller receives signals of the elastic vibration and attitude. And then
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the controller calculates the output signals to vibration and attitude actuators.
Figure 2 Scheme of the coupling control strategy
3.1 Positive position feedback control The elastic vibration of a the flexible structure appendage is suppressed by the forces generated by bonded piezoelectric patches. Then the governing equation of the elastic vibration has the form D ( t ) ωɺ + ηɺɺ + 2ξΛηɺ + Λ2η = BaV T
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(2)
ACCEPTED MANUSCRIPT where V (m×1，m is the number of piezoelectric patches) denotes the control voltage， Ba is the distribution matrix of actuators. In order to damp the elastic oscillation of the flexible appendage, the PPF controller, which is robust to parametric uncertainties, is introduced. The PPF controller consists of a second-order controller. The second-order system is forced by
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the position response which is then feedback to give the force input to the structure. The governing equation of the second-order controller is given by
εɺɺ + 2ξ c Λcεɺ + Λ2cε = Λ2c BsTη
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and
(3)
V = G Λ2ε (4) Where G is the feedback gain matrix of PPF，ߝ is the compensator state，ξc is damp of
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the compensator, Λc is the compensator stiffness matrix, and Bs is the distribution matrix of sensors.
The global dynamic equation with PPF control law is given by:
0 ηɺ Λ2 ηɺɺ 2ξΛ + + εɺɺ 0 2ξ c Λc εɺ − Λ2c BsT
T − BaG Λ2 η D ( t ) ωɺ = Λ2c ε 0
(5)
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In order to effectively damp out a structural mode, obviously the case of active damping is required. In former literature, the stiffness matrix Λc is consistent with the natural frequencies of the target structure.
Λ2 − BaGBs > 0
(6)
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satisfied
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The closed-loop system in Eq.(5) is stable if only the following equation is
The proof can be seen in literature[39].
3.2
Adaptive sliding mode control
Then the nonlinear model for the spacecraft motion including the PPF controller
is summarized by J ( t ) ωɺ + D ( t )ηɺɺ + ω × ( J ω + D ( t )ηɺ ) = τ D ( t ) ωɺ + ηɺɺ + 2ξΛηɺ + Λ2η = BaG Λ2ε T
εɺɺ + 2ξ c Λcεɺ + Λ2cε = Λ2c BsTη Define 7 / 25
(7)
ACCEPTED MANUSCRIPT η q= ε
(8)
Inserting q into Eq.(7)，and we obtain
J ( t ) ωɺ + Dq ( t ) qɺɺ + ω × ( J ω + Dq ( t ) qɺ ) = τ
(9)
Dq ( t ) ωɺ + qɺɺ + 2ξ q Λq qɺ + Κ qq = 0
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T
where
Λ 0 Λq = 0 Λc
ξ 0 ξq = 0 ξ c − BaG Λ2 Λ2c
D q = D ( t )
(10)
0
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Λ2 Κq = 2 T − Λc Bs
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In order to avoid singularity of attitude configuration, the Modified Rodrigues Parameterization (MRP) is introduced to describe the attitude of the spacecraft [40, 41]
σ = a tan (φ / 4 )
(11)
where σ=(σ1, σ2, σ3)T∈ℝ3, a and ϕ are the principal axis and angles of rotation from Euler’s theorem. The rotational rate kinematics relating the angular velocity to the
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MRP rates are given by [42]
σɺ = G ( σ ) ω,
σ ( 0) = σ0
(13)
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1 1 + σT σ G ( σ ) = I + σɶ + σσT − I 2 2
(12)
( ɶ⋅ )
is defined as
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The matrix
0 σɶ = σ 3 −σ 2
−σ 3 0
σ1
σ2 −σ 1
(14)
0
And the matrix G(σ) satisfies the following two identities 1 + σT σ T σT G ( σ ) ω = σ ω 4 2
1 + σT σ GT ( σ ) G ( σ ) = I 4 Then inverse of the matrix G(σ) can be derived by direct calculation as 8 / 25
(15)
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4 T G (σ) = G (σ) T 1+ σ σ −1
3.2.1
(16)
Sliding manifold
σe (t ) = σ (t ) − σd (t )
ωe ( t ) = ω ( t ) − ωd ( t )
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The errors MRP and angular velocities are defined as (17)
where σd is the desired reference trajectory, and ωd is the desired reference angular
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velocity.
The state trajectories move onto a sliding manifold (s=0), where s is given by
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ˆ ( t ) + λG −1 ( σ ) σ e ( t ) s ( t ) = ω ( t ) − ω
(18)
ˆ ( t ) is defined related where λ is a diagonal matrix with positive elements, and the ω to σd as
ˆ ( t ) = G −1 ( σ ) σɺ d ( t ) ω
(19)
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Lemma 3.1. After the states of the system arrive the sliding manifold in Eq.(18),
the attitude tracking error σe → 0 and ωe → 0 as t → ∞. Proof. When the states of the system arrive the sliding manifold, s satisfies
ˆ ( t ) + λ G −1 ( σ ) σ e ( t ) = 0 s ( t ) = ω ( t ) − ω
(20)
or
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Multiplying by G(σ) and inserting Eqs.(12) and (19) result in σɺ ( t ) − σɺ d ( t ) + λ σ e ( t ) = 0
(21)
σɺ e + λ σ e ( t ) = 0
(22)
Choose the Lyapunov function candidate as Vσe =
1 T σe σe 2
(23)
Differentiating Eq. (23) and it follows that Vɺσe satisfies
Vɺσe = σ Te σɺ e = −λ σTe σ e ≤ 0
(24)
If Vσe ≠ 0 then Vɺσe < 0 , and the attitude tracking error σe will decrease. If 9 / 25
ACCEPTED MANUSCRIPT ˆ ( t ) =0, and Vɺσe =0 then σe=0 and from Eqs.(20) and (19) also ω ( t ) − ω ˆ ( t ) − ωd ( t ) =0. So the angular velocities error ωe =0. ω From Eq.(22), the MRP error satisfies − λ ( t − th )
(25)
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σ e ( t ) = σ e ( th ) e
We can conclude that after the states of the system arrive the sliding manifold, the convergence rate of the MRP error σe depends on the coefficient λ. Control law
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3.2.2
After the design of the sliding surface, the control law is established to ensure the
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sliding manifold to be reachable.
Construct a Lyapunov function for the closed-loop system 1 T s Js 2 The inertial matrix J is symmetric and positive. Hence, Vs satisfies Vs =
Vs ≥ 0
(26)
(27)
Vɺs = sT Jsɺ
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The derivative of the Lyapunov function is
ɺˆ ( t ) + λ G ɺ −1 ( σ ) σ ( t ) + λ G −1 ( σ ) σɺ ( t ) ɺ (t ) − ω = sT J ω e e ɺˆ ( t ) + λ JG ɺ −1 ( σ ) σ ( t ) + λ JG −1 ( σ ) σɺ ( t ) + ... = s T [τ − ω × J ω − Jω e e
(28)
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... − ω × D ( t )ηɺ − D ( t )ηɺɺ]
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The torques on the platform of the spacecraft induced by the flexible appendages are taken as disturbance τ d = τ d ,1 , τ d ,2 , τ d ,3 = −ω × D ( t )ηɺ − D ( t )ηɺɺ T
(29)
In a physical spacecraft, the platform’s rotation would excite vibrations of lower
modes for the flexible appendages. And the amplitudes for the multi-mode vibrations are generally limited. Moreover, the excited elastic oscillations of the flexible appendages would decrease with respect to time with structural damping or active control forces. Hence, both the deformation and velocity of the flexible appendages are assumed to be limited. So the disturbance induced by the appendages is limited and there exists 10 / 25
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( i = 1, 2,3)
τ d ,i ≤ τˆd ,i
(30)
where |∙| denotes the absolute value of a scalar.
( i = 1, 2,3)
Define the control law as τ d ,i ≤ τˆd ,i
where τ ′ = [ τ1′ τ 2′ τ3′ ]
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τi′ = − ki ⋅ sgn ( si ) ,
( i = 1, 2, 3)
(31)
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ɺˆ ( t ) − λ JG ɺ −1 ( σ ) σ ( t ) − λ JG −1 ( σ ) σɺ ( t ) + τ ′ τ = ω × J ω + Jω e e
(32)
Vɺs = sT ( τ ′ + τ d )
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Sgn(·) denotes the sign function. Inserting Eq.(31) into (28) induces
(33)
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3 τ s = -∑ ki si 1 − d ,i ⋅ i ki si i =1
ki is selected to satisfy the inequalities ki≥ τˆd ,i (i=1,2,3). Hence, Vɺs ≤ 0 . If s≠0, Vs > 0 and Vɺs < 0 . Then the state of the system is moved to the sliding manifold. Therefore, the control law globally asymptotically stabilizes the error-based
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spacecraft system at the origin.
The control law in Eq.(31) is noncontinuous and would induce chattering because of the existence of the function sgn(∙). In order to eliminate chattering, the saturation function sat(si, ϵ) is introduced to instead the sign function si > ε si ≤ ε
(34)
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sgn ( si ) , sat ( si , ε ) = s i , ε
After the saturation function is included, the state of the system would move in
the sliding surface boundary layer |si|≤ϵ instead of to be strictly restricted on the sliding manifold s=0. Hence, the control torque is continuous and the chattering is lowered. 3.2.3
Control law with parametrical uncertainties In this paper, we consider the parametric uncertainties due to the spacecraft
moment of inertia uncertainty and the coupling coefficient matrix uncertainty between
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ACCEPTED MANUSCRIPT the rigid-body attitude and the flexible structure appendage. Suppose that the normalized inertial matrix of the system is Jnom. The difference between the actual and the normalized value is defined as ∆J = J − J nom
(35)
Then the control law is defined as
Eq.(28) becomes Vɺs = sT ( τ ′ + τ d + δ )
(36)
(37)
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3 τ + δ i si = -∑ ki si 1 − d ,i ⋅ ki si i =1
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ɺˆ ( t ) − λ J G ɺ −1 ( σ ) σ ( t ) − λ J G −1 ( σ ) σɺ ( t ) + τ ′ τ = ω × J nomω + J nom ω nom e nom e
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where
ɺˆ ( t ) − λ∆JG ɺ −1 ( σ ) σ ( t ) − λ∆JG −1 ( σ ) σɺ ( t ) δ = ω × ∆J ω + ∆Jω e e
(38)
Also, the uncertainty of the spacecraft moment of inertia is usually limited. Therefore, in the above equation, independent variables of δ are all limited. So there exists δˆi (i=1,2,3) such that the following inequality satisfies
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δ i ≤ δˆi
( i = 1, 2,3)
(39)
In order to ensure the states of the system move to the sliding manifold, ki is
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selected to satisfy ki≥ τˆd ,i + δˆi (i=1,2,3).
4 Numerical Simulations
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4.1 Hub-plate system Figure 3 shows the adopted model for simulation, which consists of a main bus
and a flexible plate bonded with PZT layers. The flexible plate is fixed on the hub through a yoke. The moment of inertia of the system is selected as 10 kg·m2. The body coordinate of the hub is denoted as oxyz and the body coordinate for the plate is defined as osxsyszs. The hub can rotate about the y axis so that only the flexual modes in ys direction of the plate are coupled with the hub’s attitude. Therefore, the first four flexual modes in ys direction for the plate and the hub-plate system are obtained, as given in Table 1. 12 / 25
ACCEPTED MANUSCRIPT Compared to natural frequencies of the plate, the corresponding natural frequencies of the hub-plate system are higher, especially for lower modes. The table also gives the coupling coefficient between the flexible modes and the attitude. It can be noted that with the decrease of the modal index, the coupling coefficient between the flexible mode and the attitude becomes larger, which indicates that the influence on the
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angular velocity becomes more great greater.
Figure 3 Schematic drawing of the Hub-Plate system Table 1 Natural frequencies (Hz) and coupling ratio of the Plate and Hub-Plate
Plate
Hub-Plate
Coupling Coefficient
1
0.520
0.645
1.87
2
3.26
3.33
-0.537
3
9.17
9.23
-0.268
4
18.1
18.1
-0.176
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Modal index
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Table 2 Typical properties of piezoelectric materials
Parameters
Young’s Modulus (GPa)
30.336
Poisson ratio
0.31
Density (g/cm3)
5.44
d33(pC/N)
460
d31(pC/N)
-210
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Items
PZT layers of dimensions 85×14×0.3mm are bonded on the fixed end of the plate to act as actuators. Properties of PZT are given in the Table 2. The stiffness and the mass of the piezoelectric layers are incorporated into the model and the structural damping is ignored in this case. The input voltage of actuator was restricted to the range of [-1000, 1000]V.
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4.2 Simulation results The hub-plate system undergoes a 60 deg angle rest to rest slew, using the coupling control method. And the results are compared to the case that only the adaptive sliding mode control is utilized. Parameters of the PPF controller are set as
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Λc=Λ, ξc=0.4. Parameters of the adaptive sliding mode controller are set as λ=1.0, k=1.0, ϵ =0.001. For simplicity, we only consider the first four flexible modes of the plate. And the modal damping ratios are all considered to be 0.01.
Results of the simulation are presented in Figure 4~Figure 9. Figure 4 shows the amplitudes of generalized coordinates of the first four modes with and without PPF
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compensator. Figure 5 curves the time response of the free end of the plate in the body coordinate. Among the four modal coordinates, it is obvious that the first one is
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the dominate one by comparing Figure 4 and Figure 5. The tip deflection of the plate converges to zero with respect to time. It is clear that the controller including PPF achieves a better performance. Figure 6 shows the time response of the input voltage of the piezoelectric patch. In the maneuver process, the input voltage of actuator varies in the range of [-959, 926]V, within the limits of endurance. 0.04
0
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-0.02
20 time (s)
0 -2 sliding mode control coupling control
-4
40
1
x 10
-5
2
-1
-3
5
x 10
10
-6
-2 -4
sliding mode control coupling control
0
5
0 η4
η3
0
-2
0
time (s)
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0
-4
η2
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η1
0.02
-0.04
2
sliding mode control coupling control
x 10
-6
10
time (s)
sliding mode control coupling control
0
5
10
time (s)
Figure 4 Time history of the generalized coordinates of the first four modes 14 / 25
ACCEPTED MANUSCRIPT Figure 4 also shows that a slight vibration amplitude is obtained even for the case in which the active vibration controller is not applied. It should be pointed out that this decay is not only due to the structural damping of the plate, but also the compensator of the attitude controller. Even the structural damping is assumed to be zero, the elastic vibration of the plate will converge to zero finally under the attitude
20
500
-20 -40
-500
sliding mode control coupling control
0
10
20 time (s)
30
-1000
40
Figure 5 The tip deflection response of the plate
0
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0
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1000 Voltate (V)
40
0
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deflection (mm)
controller.
20
40
time (s)
Figure 6 Input voltage of the actuator
Figure 7~Figure 9 present the time histories of the items related to the hub. It can be concluded that the vibration control of the plate does not alter the attitude
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maneuver trajectory. Trends for control torques, angular velocities, sliding manifold and Euler angles are consistent. The angular velocity is initially perturbed by the excited oscillation of the flexible plate. Then in the coupling control case, the
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oscillation is damped rapidly under the PPF controller. As the decrease of the vibration, the amplitude of jitter of the angular velocity decreases such that the control
1
0.2
0.5
0.1
torque (Nm)
torque (Nm)
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torque is smoother.
0
-0.5 -1
sliding mode control coupling control
0
10
20 time (s)
30
0 -0.1 -0.2 30
40
(a) complete time history
sliding mode control coupling control
35 time (s) (b) zoomed portion
Figure 7 Control torque comparison of a rest to rest slew 15 / 25
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ACCEPTED MANUSCRIPT As shown in Figure 8 and Figure 9, after time t>12s, the angle of the hub is approximated to the targeted value and the angular velocity of the hub is about zero. At time t=35.9s, the amplitude of the angular velocity under the coupling control case is reduced as 1.0×10-4 deg/s while the amplitude is 5.5×10-3 deg/s under the sliding control case.
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After about 24.9s, the angle error with the targeted value is less than 1.0×10-3 deg. However, at the moment the angle experiences severe oscillation under the sliding mode controller, whose amplitude is about 5.2×10-3 deg while the amplitude is less than 5.0×10-4 deg under the coupling control case. It is evident to see that the
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coupling control reduces the stabilization time and improves the pointing accuracy of the spacecraft.
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5 Experiments Validation In this section, experimental researches are carried out and the results are compared with simulations.
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5.1 Attitude control testbed for flexible spacecraft Figure 10 presents the photograph of the attitude control testbed of for flexible spacecraft, which consists of a hub and a fixed aluminum rectangular plate. The hub represents the platform of a flexible spacecraft. And an optical fibre gyro and a
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reaction wheel are mounted on the hub. The gyro senses the angular velocity of the hub and the wheel supplies control moment to the system. The maximum control moment that the wheel can provide is ±0.1 Nm. The plate is designed to simulate the
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behavior of flexible appendages, such as solar wings. The plate is bonded with a strain gauge and collocated PZT patches. They serve as sensors and actuators to sense or control the plate. The hub-plate system is mounted on a single axis air bearing table, which is used to provide similar conditions of low friction and the weightlessness of outer space. The testbed can perform single axis rotation to test attitude control
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algorithms and vibration control methods.
Figure 10 Picture of the Attitude Control Testbed of Flexible Spacecraft
The testbed is a hardware in loop simulation platform, which is capable of validating coupling control strategies. Meanwhile, this system can be utilized to evaluate performances of vibration control methods for appendages and maneuver trajectories for flexible spacecraft. The testbed is realized using a dSpace simulation computer to provide real-time calculation and multi-channel A/D, D/A connector
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wheel through serial port. The control voltage values are transmitted to an amplifier through the D/A panel and then drive the PZT patches. To eliminate the influence of noise and high modes, low pass filters are utilized to smooth the outputs of sensors.
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5.2 Analysis of the coupling control strategy
In this experiment, the testbed is commanded to perform a 60 deg slew, with respect to the initial attitude. For comparative purposes, two different cases are
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conducted: (1) slew using the adaptive sliding mode control method, (2) slew using the coupling control strategy, i.e. the adaptive sliding mode control for attitude and the PPF control for elastic vibration. Parameters of the PPF controller are same as that of the simulation. Parameters of the adaptive sliding mode controller are set as λ=0.2, k=0.1, ϵ =0.05.
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Figure 11~Figure 16 presents the experimental results. Figure 11 presents the torque applied to the hub. Though the time histories are close to each other, the zoomed portion illustrated in Figure 11(b) shows that there is indeed an influence of the plate elastic motion on the control torque. With only the adaptive sliding mode
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control is applied, there exists jitter for the control torque. It is clear that, with the PPF control method, the excited vibration of the plate is suppressed rapidly, as shown in Figure 13. Such influence, indicated by oscillations, is reduced by the PPF controller.
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Similarly, Figure 12 (a) depicts the time histories of the angular velocities of the hub. The corresponding zoomed portion is depicted in Figure 12 (b). One can see once again the influence of the flexibility of the plate on the rigid-motion responses. This reflects advantages of the coupling control strategy. Figure 14~Figure 16 show the results in terms of attitude behavior. Also, it can be seen that the goals established for attitude control are satisfactorily achieved, namely, a steady 60 deg position is reached in nearly 40s of control action.
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Figure 15 The sliding manifold
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three simulation cases with λ=0.2, λ=0.1 and λ=0.05 are compared. And the results are shown in Figure 17~Figure 20. When λ is increased, the time when states of the system move in the sliding surface boundary layer is delayed. Meanwhile, states of the system converge toward the targeted value with a faster speed. 1
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Figure 21~Figure 26 compare the experimental results and the simulation results. We can see from these figures that, it shows a good agreement between simulation and experimental results. Apparently, the mathematic model of the hub-plate system is similar to the actual system. From Figure 24 it is possible to conclude that the experimental maneuver trajectory of the hub is closely followed with the numerical
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The problem of maneuvering a flexible spacecraft has been tackled in this work. A coupling control strategy is designed, which synthesizes an adaptive sliding mode controller and a PPF controller. Also, the method is robust to parametric uncertainties
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of the spacecraft model. Numerical simulations have provided satisfactory results in terms of vibration control and rigid body motion. Simulations and experiments show that, the coupling control strategy reduces the stabilization time and improves the pointing accuracy of the platform. Hence, the method is significant for the flexible spacecraft to accomplish attitude maneuver missions.
References [1] H. Gui, L. Jin, S. Xu, Attitude maneuver control of a two-wheeled spacecraft with bounded wheel speeds, Acta Astronautica, 88 (2013) 98-107. [2] D. Verbin, V.J. Lappas, Rapid rotational maneuvering of rigid satellites with 22 / 25
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hybrid actuators configuration, Journal of Guidance, Control, and Dynamics, 36 (2013) 532-547. [3] D. Verbin, V.J. Lappas, Rapid rotational maneuvering of rigid satellites with reaction wheels, Journal of Guidance, Control, and Dynamics, 36 (2013) 1538-1544. [4] X. Huang, Y. Yan, Y. Zhou, Dynamics and control of spacecraft hovering using the geomagnetic Lorentz force, Advances in Space Research, 53 (2014) 518-531. [5] R.L. Farrenkopf, Optimal Open-Loop Maneuver Profiles for Flexible Spacecraft, Journal of Guidance & Control, -1 (2015) 272-280. [6] L. Meirovitch, M.K. Kwak, Dynamics and control of spacecraft with retargeting flexible antennas, Journal of Guidance, Control, and Dynamics, 13 (1990) 241-248. [7] T. Loquen, H. de Plinval, C. Cumer, D. Alazard, Attitude control of satellites with flexible appendages: a structured H∞ control design, in: AIAA Guidance, Navigation, and Control Conference, Minneapolis, USA, 2012. [8] A.G. de Souza, L.C. de Souza, Satellite attitude control system design taking into account the fuel slosh and flexible dynamics, Mathematical Problems in Engineering, 2014 (2014). [9] P. Gasbarri, M. Sabatini, A. Pisculli, Dynamic modelling and stability parametric analysis of a flexible spacecraft with fuel slosh, Acta Astronautica, 127 (2016) 141-159. [10] A.M. Khoshnood, O. Kavianipour, VIBRATION SUPPRESSION OF FUEL SLOSHING USING SUBBAND ADAPTIVE FILTERING (RESEARCH NOTE), International Journal of Engineering-Transactions A: Basics, 28 (2015) 1507-1514. [11] A.G.d. Souza, L.C.G.d. Souza, Design of Satellite Attitude Control System Considering the Interaction between Fuel Slosh and Flexible Dynamics during the System Parameters Estimation, Applied Mechanics and Materials, 706 (2015) 14-24. [12] P. Gasbarri, R. Monti, C. De Angelis, M. Sabatini, Effects of uncertainties and flexible dynamic contributions on the control of a spacecraft full-coupled model, Acta Astronautica, 94 (2014) 515-526. [13] T. Loquen, H.d. Plinval, C. Cumer, D. Alazard, Attitude control of satellites with flexible appendages: a structured Hinf control design, in: AIAA Guidance, Navigation, and Control Conference, Minneapolis, Minnesota, 2012, pp. AIAA 2012-4845. [14] H. Bang, C.-K. Ha, J.H. Kim, Flexible spacecraft attitude maneuver by application of sliding mode control, Acta Astronautica, 57 (2005) 841-850. [15] H. Bang, C.-S. Oh, Predictive control for the attitude maneuver of a flexible spacecraft, Aerospace science and technology, 8 (2004) 443-452. [16] R.G. Knapp, Fuzzy Based Attitude Controller for Flexible Spacecraft with On/Off Thrusters, in: Aeronautical and Astronautical Engineering, Massachusetts Institute of Technology, Cambridge, MA, 1991. [17] C. Dong, L. Xu, Y. Chen, Q. Wang, Networked flexible spacecraft attitude 23 / 25
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maneuver based on adaptive fuzzy sliding mode control, Acta Astronautica, 65 (2009) 1561-1570. [18] M.R.D. Nayeri, A. Alasty, K. Daneshjou, Neural optimal control of flexible spacecraft slew maneuver, Acta Astronautica, 55 (2004) 817-827. [19] R.D. Quinnt, L. Meirovitch, EQUATIONS FOR THE VIBRATION OF A SLEWING FLEXIBLE SPACECRAFT, AIAA 86-0906, (1986). [20] Y.P. Kakad, DYNAMICS AND CONTROL OF SLEW MANEWER OF LARGE FLEXIBLE SPACECRAFT, AIAA 86-47472, (1986) 629-634. [21] M. Azadi, M. Eghtesad, S. Fazelzadeh, E. Azadi, Dynamics and control of a smart flexible satellite moving in an orbit, Multibody System Dynamics, (2015) 1-23. [22] T. Sales, D. Rade, L. De Souza, Passive vibration control of flexible spacecraft using shunted piezoelectric transducers, Aerospace Science and Technology, 29 (2013) 403-412. [23] L. Silverberg, H. Baruh, Simultaneous maneuver and vibration suppression of flexible spacecraft, Applied Mathematical Modelling, 12 (1988) 546-555. [24] H. Baruh, Control of the elastic motion of lightweight structures, in: AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit, 42 nd, Seattle, WA, 2001. [25] L. Meirovitch, M.K. Kwak, Dynamics and Control of Spacecraft with Retargeting Flexible Antennas, J. GUIDANCE, 13 (1990) 241-248. [26] S.D. Gennaro, Active vibration suppression in flexible spacecraft attitude tracking, Journal of Guidance, Control, and Dynamics, 21 (1998) 400-408. [27] A. Grewal, V. Modi, Robust attitude and vibration control of the space station, Acta Astronautica, 38 (1996) 139-160. [28] L. Zhu, Y. Liu, D. Wang, Q. Hu, Backstepping-based attitude maneuver control and active vibration reduction of flexible spacecraft, in: Control and Decision Conference, 2008. CCDC 2008. Chinese, IEEE, 2008, pp. 887-891. [29] M. Azadi, S. Fazelzadeh, M. Eghtesad, E. Azadi, Vibration suppression and adaptive-robust control of a smart flexible satellite with three axes maneuvering, Acta Astronautica, 69 (2011) 307-322. [30] C. Meiyu, X. Shijie, Optimal attitude control of flexible spacecraft with minimum vibration, in: AIAA Guidance, Navigation and Control Conference, AIAA-2010-8201, AIAA Toronto, 2010. [31] M. Shahravi, M. Azimi, A comparative study for collocated and non-collocated sensor/actuator placement in vibration control of a maneuvering flexible satellite, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, (2014) 0954406214544182. [32] G. Song, B.N. Agrawal, Vibration suppression of flexible spacecraft during attitude control, Acta Astronautica, 49 (2001) 73-83. [33] Q. Hu, G. Ma, Active Vibration Suppression in Flexible Spacecraft Attitude Maneuver Using Variable Structure Control and Input Shaping Technique, in: AIAA Guidance, Navigation, and Control Conference and Exhibit, 2005, pp. 15-18. 24 / 25
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[34] Q. Hu, Robust adaptive sliding mode attitude control and vibration damping of flexible spacecraft subject to unknown disturbance and uncertainty, Transactions of the Institute of Measurement and Control, 34 (2012) 436-447. [35] Q. Hu, G. Ma, Variable structure control and active vibration suppression of flexible spacecraft during attitude maneuver, Aerospace Science and Technology, 9 (2005) 307-317. [36] Q. Hu, G. Ma, Vibration suppression of flexible spacecraft during attitude maneuvers, Journal of guidance, control, and dynamics, 28 (2005) 377-380. [37] Q. Hu, B. Xiao, Robust adaptive backstepping attitude stabilization and vibration reduction of flexible spacecraft subject to actuator saturation, Journal of Vibration and Control, 17 (2011) 1657-1671. [38] S. Na, G.-a. Tang, L.-f. Chen, Vibration reduction of flexible solar array during orbital maneuver, Aircraft Engineering and Aerospace Technology: An International Journal, 86 (2014) 155-164. [39] J.L. Fanson, T.K. Caughey, Positive Position Feedback Control for Large Space Structures, AIAA JOURNAL, 28 (1990) 717-724. [40] Y.-P. CHEN, S.-C. LO, Sliding-mode controller design for spacecraft attitude tracking maneuvers, IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS, 29 (1993) 1328-1333. [41] M.R. Binette, C.J. Damaren, L. Pavel, Nonlinear H∞ Attitude Control Using Modified Rodrigues Parameters, Journal of Guidance Control & Dynamics, 37 (2014) 2017-2021. [42] P. Tsiotras, Stabilization and optimality results for the attitude control problem, Journal of Guidance, Control and Dynamics, 19 (1996) 1-9.
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ACCEPTED MANUSCRIPT 1. A coupling control method synthesizes sliding mode control and PPF 2. Attitude control testbed of flexible spacecraft is established
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3. The method can decrease the stabilization time and improve attitude accuracy
S0094-5765(17)31463-7
DOI:
10.1016/j.actaastro.2018.03.023
Reference:
AA 6765
To appear in:
Acta Astronautica
Received Date: 16 October 2017 Revised Date:
20 February 2018
Accepted Date: 12 March 2018
Please cite this article as: J. Wang, D. Li, Experiments study on attitude coupling control method for flexible spacecraft, Acta Astronautica (2018), doi: 10.1016/j.actaastro.2018.03.023. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Experiments Study on Attitude Coupling Control Method for Flexible Spacecraft Wang Jie*, Li Dongxu College of Aerospace Science and Engineering, National University of Defense Technology
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NO.47 Yanwachi Street, Changsha 410073
*Corresponding author. E-mail: [email protected]
Abstract
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High pointing accuracy and stabilization are significant for spacecrafts to carry out Earth observing, laser communication and space exploration missions. However, when the a spacecraft undergoes large angle maneuver, the excited elastic oscillation
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of flexible appendages, for instance, solar wing and onboard antenna, would downgrade the performance of the spacecraft platform. This paper proposes a coupling control method, which synthesizes the adaptive sliding mode controller and the positive position feedback (PPF) controller, to control the attitude and suppress the elastic vibration simultaneously. Because of its prominent performance in for
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attitude tracking and stabilization, the proposed method is capable of slewing the flexible spacecraft with a large angle. Also, the method is robust to parametric uncertainties of the spacecraft model. Numerical simulations are carried out with a hub-plate system which undergoes a single-axis attitude maneuver. An attitude
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control testbed of for the flexible spacecraft is established and experiments are conducted to validate the coupling control method. Both the numerical and
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experimental results demonstrate that the method discussed above can effectively decrease the stabilization time and improve the attitude accuracy of the flexible spacecraft.
Keywords: Flexible Spacecraft; Sliding mode control; Positive position feedback;
Attitude Coupling Control; Experiment
1 Introduction Early spacecrafts is were usually idealized as a rigid bodies for attitude control while the flexible parts is were ignored because of its their small mass or inertia [1-4]. With the application of large scale solar wings and antennas on modern spacecrafts, the influences of flexibilities on characteristics and attitude of the whole system 1 / 25
ACCEPTED MANUSCRIPT cannot be ignored. The slew maneuver of the platform of the spacecraft will excite the elastic oscillations of flexible appendages, which will downgrade the pointing accuracy of the platform [5-7]. Therefore, flexible appendages must be considered while designing the attitude control method. Besides, spacecrafts often contain propellant tanks with large quantities of liquid fuel, whose slosh excited by the main
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bus would also influence the spacecraft’s attitude [8-11]. In addition, the vibrations of the flexible appendages and fuel slosh will change the physical properties of the spacecraft [8, 12]. And the fuel consumption over time would introduce further uncertainty on the satellite spacecraft model description. Hence, the design of the
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attitude stabilization theories is crucial issues for flexible spacecrafts with parametric uncertainties.
In the field of attitude control for flexible spacecrafts, a mass of control methods
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have been proposed in last decades, for example, H∞ control method [13]、sliding mode controller [14], model predictive control [15], fuzzy control [16], adaptive fuzzy control [17], neural network controller [18], etc. For further improving the performances of the attitude control methods, the vibration controllers for flexible appendages has have been introduced and the coupling control method has been developed. The basic idea of the coupling control method is to combine the
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stabilization theory with the a vibration controller. The attitude of the platform and the elastic vibrations of the flexible appendages are controlled simultaneously. In general, the frequency bands of the attitude control subsystem and the
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vibration control subsystem are non-overlapping or the coupling is weak. Under this circumstance, the closed loops for the two subsystems can be designed separately. And the designed controllers can comply with the requirements. As the scale of the
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flexible appendages increases and the natural frequencies decrease, severe overlap occurs in the frequency bands of the two subsystems. One way to conceive of the controller is to decouple the two subsystems before the controller design. Quinnt and Meirovitch [19] proposed a first-order perturbation approach to separate the equations of motion into a set of equations governing rigid-body slewing of the spacecraft and a set of time-varying linear equations governing small elastic motions. Then a maneuver force distribution was developed which excites the least amount of elastic deformation of the flexible parts of the spacecraft. Kakad [20] used four Euler parameters to express the motion of a flexible spacecraft and the maneuver problem was reduced to be a system of uncoupled equations. Azadi et al. [21] divided the 2 / 25
ACCEPTED MANUSCRIPT system dynamics into two fast and slow subsystems using singular perturbation theory. To date the applied attitude control subsystem is mainly based on PD, LQR, sliding mode control and back-stepping control theories, etc. The flexible control subsystem is based on passive or active vibration control method and the actuators are
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mainly piezoelectric materials. Sales et al. [22] constructed a passive control system for flexible appendages using piezoelectric transducers. The results demonstrated that the plate vibrations levels and coupling between the flexible and rigid body motions were significantly reduced during the spacecraft maneuver. Silverberg and Baruh [23]
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used a natural control to suppress the vibration so that the control of the elastic motion does not distort the maneuver. Then the maneuver of the platform can be designed and performed independently. Then Baruh [24] applied this method to lightweight
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multibody structures, and designed a vibration controller based on piezoelectric film actuators. But the decouping procedure depends on the accuracy of the system’s dynamic model. And the error and uncertainties of the model were ignored. Meirovitch [25] designed an both open loop and closed loop controllers, in which the former one was to cancel the environmental disturbance and the later one was utilized to control the spacecraft and flexible appendage based on the optimal control theory.
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Gennaro [26] studied the active suppression of appendages during attitude tracking or large-angle maneuver. A PD controller was designed for attitude with thruster and fly wheel while the piezo-electrical materials were used to suppress the
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vibration of a flexible beam. Grewal and Modi [27] proposed a two-level strategy using the LQG/LTR approach and the strategy achieved good attitude control and vibration suppression behavior. Zhu et al. [28] studied a robust hybrid control design
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method which combines the backstepping control law with the strain rate feedback control method. Azadi et al. [29] applied an adaptive-robust control scheme to three axes maneuver of a flexible satellite, And the control method also suppressed the elastic vibration. Cui and Xu [30] designed the attitude control method via θ-D method and adopted the positive position feedback (PPF) for vibration control. Shahravi and Azimi [31] compared the performance of the collocated and non-collocated piezoceramic patches acting as sensors and actuators during attitude maneuver. Azadi et al. [21] established the dynamic model for a flexible spacecraft under three axes slewing maneuver. And they proposed an adaptive-robust control scheme for attitude control and a Lyapunov based controller for the vibration 3 / 25
ACCEPTED MANUSCRIPT suppression of the flexible structure. Some literature combines the input shaping method for attitude maneuver of and the active vibration control for flexible appendages. SONG and AGRAWAL [32] presented an approach which is built on the input shaping method and the pulse-width pulse-frequency (PWPF) modulator to reduce elastic vibrations during the attitude
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control. The closed loop sub-system of attitude feedback control employs a PD controller while the active vibration suppression sub-system uses the PPF control strategy. Experiments was were conducted on a system which consists of a central hub and a L-shape flexible appendage. Based on Song’s work, in order to solve the
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problem of model uncertainties, Hu and Ma [33, 34] proposed an approach based on PPF to vibration reduction during maneuver by using the theory of variable structure control [35]. The attitude controller consisted of linear and noncontinuous feedback
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items, which ensure the sliding manifold existing and global reachable. Hu and Ma [36] programmed the maneuver trajectory based on the component synthesis vibration suppression (CSVS) method with the pulse-width pulse-frequency (PWPF) modulation. And the PPF control technique using piezoelectric materials, acted on the flexible parts. The CSVS method and the input shaping method share the similar principle to reduce the residual vibration. Then Hu [37] used the backstepping method
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and modal velocity feedback method to attitude control and elastic vibration. In general, depending on the idea of input shaping method, the behavior of attitude actuators is programmed, for instance, the on-off of thrusters or the output torques of
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flywheels. Na et al. [38] proposed another way to reduce vibration through controlling the rotation at the root of the solar array using zero-placement input shaping technique.
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In former research, the attitude control subsystem and the vibration control
subsystem were mostly designed independently. And the stabilization of the whole system was not considered. Few literatures conducted experiments to validate the controller. This paper synthesizes the adaptive sliding mode controller and the PPF controller to stabilize flexible spacecrafts which undergo large-angle maneuvers. This paper is organized as follows. The physical and mathematical models of the flexible spacecraft are given in the second section. The coupling control method is presented in the third section. Numerical simulation results are presented in section 4. Finally, in section 5, experiments are conducted in order to demonstrate the efficacy of the designed controller. Section 6 provides concluding remarks. 4 / 25
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2 Physical and Mathematical Models Figure 1 presents a typical flexible spacecraft, which consists of a central core and several flexible appendages, for instance, solar arrays and antennas. The flexible spacecraft experiences rigid motions in six freedoms, including translations and
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rotations. On the other hand, the maneuver of the spacecraft or other environmental
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forces would excite elastic oscillations of flexible appendages.
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Figure 1 Scheme of the flexible spacecraft
In order to derive the governing equations for the whole system, the following assumptions are made. (1) The central core is typically modeled as a rigid body which is free to undergo unrestricted rotations relative to inertial space. (2) The appendages
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are modeled as flexible bodies that can be discretized by the finite element method (FEM). Moreover, they only have rigid rotations relative to the central core while
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translations of the hinge point are restricted. (3) The elastic deformations of the appendages is are assumed small and yield to the linear elasticity theory. Therefore, the center of mass of the whole system remains fixed in the body frame of reference. According to the Lagrange’s equations in terms of quasi-coordinates, the
mathematical model of the flexible spacecraft is established, and the governing equations can be expressed as
J ( t ) ωɺ + D ( t )ηɺɺ + ω × ( J ω + D ( t )ηɺ ) = τ D ( t ) ωɺ + ηɺɺ + 2ξΛηɺ + Λ2η = 0 T
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where J is the inertia matrix of the spacecraft, which is the sum of the inertia matrix of the central body and that of the flexible appendage; ω is the angular velocity vector 5 / 25
ACCEPTED MANUSCRIPT of the central body with respect to the inertial frame; τ=[τߠ, τ߮, τΨ]T denotes the control torque applied to the spacecraft around three axes produced by the actuators such as thruster, reaction wheels, and CMG; D is the rigid-flex rotational coupling coefficient matrix; η (1×N, N is the truncation number) is the modal coordinate of the flexible appendage; ξ =diag[ξ1, ξ2, … , ξN] is the damping ratio matrix of the
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appendage, where ξi is the ith damping ratio; Λ=diag[Λ1, Λ2, … , ΛN] is the natural frequency matrix of the cantilevered appendage, where Λi is the ith natural frequency. In general, when the flexible appendage is fixed to the main body of the spacecraft, the inertia matrix J and the coupling coefficient matrix D remain constant
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with respect to time. However, when the appendage rotates relative to the main body, the inertia of the appendage about the body frame changes continuously so that the inertia matrix of the whole system is time-variable. Also, the rigid-flex coupling
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coefficient matrix varies with respect to time because of the time-varying position between the main body and the appendage.
3 Coupling Control Strategy
The coupling control system is composed of two subsystems, the attitude
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compensator subsystem and the vibration control subsystem, as shown in Figure 2. The coupling controller receives signals of the elastic vibration and attitude. And then
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the controller calculates the output signals to vibration and attitude actuators.
Figure 2 Scheme of the coupling control strategy
3.1 Positive position feedback control The elastic vibration of a the flexible structure appendage is suppressed by the forces generated by bonded piezoelectric patches. Then the governing equation of the elastic vibration has the form D ( t ) ωɺ + ηɺɺ + 2ξΛηɺ + Λ2η = BaV T
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(2)
ACCEPTED MANUSCRIPT where V (m×1，m is the number of piezoelectric patches) denotes the control voltage， Ba is the distribution matrix of actuators. In order to damp the elastic oscillation of the flexible appendage, the PPF controller, which is robust to parametric uncertainties, is introduced. The PPF controller consists of a second-order controller. The second-order system is forced by
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the position response which is then feedback to give the force input to the structure. The governing equation of the second-order controller is given by
εɺɺ + 2ξ c Λcεɺ + Λ2cε = Λ2c BsTη
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and
(3)
V = G Λ2ε (4) Where G is the feedback gain matrix of PPF，ߝ is the compensator state，ξc is damp of
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the compensator, Λc is the compensator stiffness matrix, and Bs is the distribution matrix of sensors.
The global dynamic equation with PPF control law is given by:
0 ηɺ Λ2 ηɺɺ 2ξΛ + + εɺɺ 0 2ξ c Λc εɺ − Λ2c BsT
T − BaG Λ2 η D ( t ) ωɺ = Λ2c ε 0
(5)
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In order to effectively damp out a structural mode, obviously the case of active damping is required. In former literature, the stiffness matrix Λc is consistent with the natural frequencies of the target structure.
Λ2 − BaGBs > 0
(6)
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satisfied
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The closed-loop system in Eq.(5) is stable if only the following equation is
The proof can be seen in literature[39].
3.2
Adaptive sliding mode control
Then the nonlinear model for the spacecraft motion including the PPF controller
is summarized by J ( t ) ωɺ + D ( t )ηɺɺ + ω × ( J ω + D ( t )ηɺ ) = τ D ( t ) ωɺ + ηɺɺ + 2ξΛηɺ + Λ2η = BaG Λ2ε T
εɺɺ + 2ξ c Λcεɺ + Λ2cε = Λ2c BsTη Define 7 / 25
(7)
ACCEPTED MANUSCRIPT η q= ε
(8)
Inserting q into Eq.(7)，and we obtain
J ( t ) ωɺ + Dq ( t ) qɺɺ + ω × ( J ω + Dq ( t ) qɺ ) = τ
(9)
Dq ( t ) ωɺ + qɺɺ + 2ξ q Λq qɺ + Κ qq = 0
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T
where
Λ 0 Λq = 0 Λc
ξ 0 ξq = 0 ξ c − BaG Λ2 Λ2c
D q = D ( t )
(10)
0
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Λ2 Κq = 2 T − Λc Bs
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In order to avoid singularity of attitude configuration, the Modified Rodrigues Parameterization (MRP) is introduced to describe the attitude of the spacecraft [40, 41]
σ = a tan (φ / 4 )
(11)
where σ=(σ1, σ2, σ3)T∈ℝ3, a and ϕ are the principal axis and angles of rotation from Euler’s theorem. The rotational rate kinematics relating the angular velocity to the
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MRP rates are given by [42]
σɺ = G ( σ ) ω,
σ ( 0) = σ0
(13)
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1 1 + σT σ G ( σ ) = I + σɶ + σσT − I 2 2
(12)
( ɶ⋅ )
is defined as
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The matrix
0 σɶ = σ 3 −σ 2
−σ 3 0
σ1
σ2 −σ 1
(14)
0
And the matrix G(σ) satisfies the following two identities 1 + σT σ T σT G ( σ ) ω = σ ω 4 2
1 + σT σ GT ( σ ) G ( σ ) = I 4 Then inverse of the matrix G(σ) can be derived by direct calculation as 8 / 25
(15)
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4 T G (σ) = G (σ) T 1+ σ σ −1
3.2.1
(16)
Sliding manifold
σe (t ) = σ (t ) − σd (t )
ωe ( t ) = ω ( t ) − ωd ( t )
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The errors MRP and angular velocities are defined as (17)
where σd is the desired reference trajectory, and ωd is the desired reference angular
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velocity.
The state trajectories move onto a sliding manifold (s=0), where s is given by
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ˆ ( t ) + λG −1 ( σ ) σ e ( t ) s ( t ) = ω ( t ) − ω
(18)
ˆ ( t ) is defined related where λ is a diagonal matrix with positive elements, and the ω to σd as
ˆ ( t ) = G −1 ( σ ) σɺ d ( t ) ω
(19)
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Lemma 3.1. After the states of the system arrive the sliding manifold in Eq.(18),
the attitude tracking error σe → 0 and ωe → 0 as t → ∞. Proof. When the states of the system arrive the sliding manifold, s satisfies
ˆ ( t ) + λ G −1 ( σ ) σ e ( t ) = 0 s ( t ) = ω ( t ) − ω
(20)
or
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EP
Multiplying by G(σ) and inserting Eqs.(12) and (19) result in σɺ ( t ) − σɺ d ( t ) + λ σ e ( t ) = 0
(21)
σɺ e + λ σ e ( t ) = 0
(22)
Choose the Lyapunov function candidate as Vσe =
1 T σe σe 2
(23)
Differentiating Eq. (23) and it follows that Vɺσe satisfies
Vɺσe = σ Te σɺ e = −λ σTe σ e ≤ 0
(24)
If Vσe ≠ 0 then Vɺσe < 0 , and the attitude tracking error σe will decrease. If 9 / 25
ACCEPTED MANUSCRIPT ˆ ( t ) =0, and Vɺσe =0 then σe=0 and from Eqs.(20) and (19) also ω ( t ) − ω ˆ ( t ) − ωd ( t ) =0. So the angular velocities error ωe =0. ω From Eq.(22), the MRP error satisfies − λ ( t − th )
(25)
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σ e ( t ) = σ e ( th ) e
We can conclude that after the states of the system arrive the sliding manifold, the convergence rate of the MRP error σe depends on the coefficient λ. Control law
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3.2.2
After the design of the sliding surface, the control law is established to ensure the
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sliding manifold to be reachable.
Construct a Lyapunov function for the closed-loop system 1 T s Js 2 The inertial matrix J is symmetric and positive. Hence, Vs satisfies Vs =
Vs ≥ 0
(26)
(27)
Vɺs = sT Jsɺ
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The derivative of the Lyapunov function is
ɺˆ ( t ) + λ G ɺ −1 ( σ ) σ ( t ) + λ G −1 ( σ ) σɺ ( t ) ɺ (t ) − ω = sT J ω e e ɺˆ ( t ) + λ JG ɺ −1 ( σ ) σ ( t ) + λ JG −1 ( σ ) σɺ ( t ) + ... = s T [τ − ω × J ω − Jω e e
(28)
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... − ω × D ( t )ηɺ − D ( t )ηɺɺ]
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The torques on the platform of the spacecraft induced by the flexible appendages are taken as disturbance τ d = τ d ,1 , τ d ,2 , τ d ,3 = −ω × D ( t )ηɺ − D ( t )ηɺɺ T
(29)
In a physical spacecraft, the platform’s rotation would excite vibrations of lower
modes for the flexible appendages. And the amplitudes for the multi-mode vibrations are generally limited. Moreover, the excited elastic oscillations of the flexible appendages would decrease with respect to time with structural damping or active control forces. Hence, both the deformation and velocity of the flexible appendages are assumed to be limited. So the disturbance induced by the appendages is limited and there exists 10 / 25
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( i = 1, 2,3)
τ d ,i ≤ τˆd ,i
(30)
where |∙| denotes the absolute value of a scalar.
( i = 1, 2,3)
Define the control law as τ d ,i ≤ τˆd ,i
where τ ′ = [ τ1′ τ 2′ τ3′ ]
T
τi′ = − ki ⋅ sgn ( si ) ,
( i = 1, 2, 3)
(31)
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ɺˆ ( t ) − λ JG ɺ −1 ( σ ) σ ( t ) − λ JG −1 ( σ ) σɺ ( t ) + τ ′ τ = ω × J ω + Jω e e
(32)
Vɺs = sT ( τ ′ + τ d )
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Sgn(·) denotes the sign function. Inserting Eq.(31) into (28) induces
(33)
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3 τ s = -∑ ki si 1 − d ,i ⋅ i ki si i =1
ki is selected to satisfy the inequalities ki≥ τˆd ,i (i=1,2,3). Hence, Vɺs ≤ 0 . If s≠0, Vs > 0 and Vɺs < 0 . Then the state of the system is moved to the sliding manifold. Therefore, the control law globally asymptotically stabilizes the error-based
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spacecraft system at the origin.
The control law in Eq.(31) is noncontinuous and would induce chattering because of the existence of the function sgn(∙). In order to eliminate chattering, the saturation function sat(si, ϵ) is introduced to instead the sign function si > ε si ≤ ε
(34)
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sgn ( si ) , sat ( si , ε ) = s i , ε
After the saturation function is included, the state of the system would move in
the sliding surface boundary layer |si|≤ϵ instead of to be strictly restricted on the sliding manifold s=0. Hence, the control torque is continuous and the chattering is lowered. 3.2.3
Control law with parametrical uncertainties In this paper, we consider the parametric uncertainties due to the spacecraft
moment of inertia uncertainty and the coupling coefficient matrix uncertainty between
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ACCEPTED MANUSCRIPT the rigid-body attitude and the flexible structure appendage. Suppose that the normalized inertial matrix of the system is Jnom. The difference between the actual and the normalized value is defined as ∆J = J − J nom
(35)
Then the control law is defined as
Eq.(28) becomes Vɺs = sT ( τ ′ + τ d + δ )
(36)
(37)
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3 τ + δ i si = -∑ ki si 1 − d ,i ⋅ ki si i =1
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ɺˆ ( t ) − λ J G ɺ −1 ( σ ) σ ( t ) − λ J G −1 ( σ ) σɺ ( t ) + τ ′ τ = ω × J nomω + J nom ω nom e nom e
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where
ɺˆ ( t ) − λ∆JG ɺ −1 ( σ ) σ ( t ) − λ∆JG −1 ( σ ) σɺ ( t ) δ = ω × ∆J ω + ∆Jω e e
(38)
Also, the uncertainty of the spacecraft moment of inertia is usually limited. Therefore, in the above equation, independent variables of δ are all limited. So there exists δˆi (i=1,2,3) such that the following inequality satisfies
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δ i ≤ δˆi
( i = 1, 2,3)
(39)
In order to ensure the states of the system move to the sliding manifold, ki is
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selected to satisfy ki≥ τˆd ,i + δˆi (i=1,2,3).
4 Numerical Simulations
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4.1 Hub-plate system Figure 3 shows the adopted model for simulation, which consists of a main bus
and a flexible plate bonded with PZT layers. The flexible plate is fixed on the hub through a yoke. The moment of inertia of the system is selected as 10 kg·m2. The body coordinate of the hub is denoted as oxyz and the body coordinate for the plate is defined as osxsyszs. The hub can rotate about the y axis so that only the flexual modes in ys direction of the plate are coupled with the hub’s attitude. Therefore, the first four flexual modes in ys direction for the plate and the hub-plate system are obtained, as given in Table 1. 12 / 25
ACCEPTED MANUSCRIPT Compared to natural frequencies of the plate, the corresponding natural frequencies of the hub-plate system are higher, especially for lower modes. The table also gives the coupling coefficient between the flexible modes and the attitude. It can be noted that with the decrease of the modal index, the coupling coefficient between the flexible mode and the attitude becomes larger, which indicates that the influence on the
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angular velocity becomes more great greater.
Figure 3 Schematic drawing of the Hub-Plate system Table 1 Natural frequencies (Hz) and coupling ratio of the Plate and Hub-Plate
Plate
Hub-Plate
Coupling Coefficient
1
0.520
0.645
1.87
2
3.26
3.33
-0.537
3
9.17
9.23
-0.268
4
18.1
18.1
-0.176
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Modal index
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Table 2 Typical properties of piezoelectric materials
Parameters
Young’s Modulus (GPa)
30.336
Poisson ratio
0.31
Density (g/cm3)
5.44
d33(pC/N)
460
d31(pC/N)
-210
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Items
PZT layers of dimensions 85×14×0.3mm are bonded on the fixed end of the plate to act as actuators. Properties of PZT are given in the Table 2. The stiffness and the mass of the piezoelectric layers are incorporated into the model and the structural damping is ignored in this case. The input voltage of actuator was restricted to the range of [-1000, 1000]V.
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4.2 Simulation results The hub-plate system undergoes a 60 deg angle rest to rest slew, using the coupling control method. And the results are compared to the case that only the adaptive sliding mode control is utilized. Parameters of the PPF controller are set as
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Λc=Λ, ξc=0.4. Parameters of the adaptive sliding mode controller are set as λ=1.0, k=1.0, ϵ =0.001. For simplicity, we only consider the first four flexible modes of the plate. And the modal damping ratios are all considered to be 0.01.
Results of the simulation are presented in Figure 4~Figure 9. Figure 4 shows the amplitudes of generalized coordinates of the first four modes with and without PPF
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compensator. Figure 5 curves the time response of the free end of the plate in the body coordinate. Among the four modal coordinates, it is obvious that the first one is
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the dominate one by comparing Figure 4 and Figure 5. The tip deflection of the plate converges to zero with respect to time. It is clear that the controller including PPF achieves a better performance. Figure 6 shows the time response of the input voltage of the piezoelectric patch. In the maneuver process, the input voltage of actuator varies in the range of [-959, 926]V, within the limits of endurance. 0.04
0
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-0.02
20 time (s)
0 -2 sliding mode control coupling control
-4
40
1
x 10
-5
2
-1
-3
5
x 10
10
-6
-2 -4
sliding mode control coupling control
0
5
0 η4
η3
0
-2
0
time (s)
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0
-4
η2
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η1
0.02
-0.04
2
sliding mode control coupling control
x 10
-6
10
time (s)
sliding mode control coupling control
0
5
10
time (s)
Figure 4 Time history of the generalized coordinates of the first four modes 14 / 25
ACCEPTED MANUSCRIPT Figure 4 also shows that a slight vibration amplitude is obtained even for the case in which the active vibration controller is not applied. It should be pointed out that this decay is not only due to the structural damping of the plate, but also the compensator of the attitude controller. Even the structural damping is assumed to be zero, the elastic vibration of the plate will converge to zero finally under the attitude
20
500
-20 -40
-500
sliding mode control coupling control
0
10
20 time (s)
30
-1000
40
Figure 5 The tip deflection response of the plate
0
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0
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1000 Voltate (V)
40
0
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deflection (mm)
controller.
20
40
time (s)
Figure 6 Input voltage of the actuator
Figure 7~Figure 9 present the time histories of the items related to the hub. It can be concluded that the vibration control of the plate does not alter the attitude
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maneuver trajectory. Trends for control torques, angular velocities, sliding manifold and Euler angles are consistent. The angular velocity is initially perturbed by the excited oscillation of the flexible plate. Then in the coupling control case, the
EP
oscillation is damped rapidly under the PPF controller. As the decrease of the vibration, the amplitude of jitter of the angular velocity decreases such that the control
1
0.2
0.5
0.1
torque (Nm)
torque (Nm)
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torque is smoother.
0
-0.5 -1
sliding mode control coupling control
0
10
20 time (s)
30
0 -0.1 -0.2 30
40
(a) complete time history
sliding mode control coupling control
35 time (s) (b) zoomed portion
Figure 7 Control torque comparison of a rest to rest slew 15 / 25
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ACCEPTED MANUSCRIPT As shown in Figure 8 and Figure 9, after time t>12s, the angle of the hub is approximated to the targeted value and the angular velocity of the hub is about zero. At time t=35.9s, the amplitude of the angular velocity under the coupling control case is reduced as 1.0×10-4 deg/s while the amplitude is 5.5×10-3 deg/s under the sliding control case.
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After about 24.9s, the angle error with the targeted value is less than 1.0×10-3 deg. However, at the moment the angle experiences severe oscillation under the sliding mode controller, whose amplitude is about 5.2×10-3 deg while the amplitude is less than 5.0×10-4 deg under the coupling control case. It is evident to see that the
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coupling control reduces the stabilization time and improves the pointing accuracy of the spacecraft.
0.01
-2 -4 -6
sliding mode control coupling control
0
10
20 time (s)
30
0
-0.005
-0.01 30
40
35
40
time (s)
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-8
sliding mode control coupling control
0.005
ωy (deg/s)
ωy (deg/s)
0
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2
(a) complete time history
(b) zoomed portion
Figure 8 Angular velocity of the Hub
sliding mode control coupling control
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40 20
0
-20
0
10
20 time (s)
x 10
-3
sliding mode control coupling control
2 angle (deg)
60 angle (deg)
4
EP
80
0 -2 -4
30
-6 20
40
25
30
35
time (s)
(a) complete time history
(b) zoomed portion
Figure 9 Euler angle of the Hub
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5 Experiments Validation In this section, experimental researches are carried out and the results are compared with simulations.
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5.1 Attitude control testbed for flexible spacecraft Figure 10 presents the photograph of the attitude control testbed of for flexible spacecraft, which consists of a hub and a fixed aluminum rectangular plate. The hub represents the platform of a flexible spacecraft. And an optical fibre gyro and a
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reaction wheel are mounted on the hub. The gyro senses the angular velocity of the hub and the wheel supplies control moment to the system. The maximum control moment that the wheel can provide is ±0.1 Nm. The plate is designed to simulate the
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behavior of flexible appendages, such as solar wings. The plate is bonded with a strain gauge and collocated PZT patches. They serve as sensors and actuators to sense or control the plate. The hub-plate system is mounted on a single axis air bearing table, which is used to provide similar conditions of low friction and the weightlessness of outer space. The testbed can perform single axis rotation to test attitude control
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algorithms and vibration control methods.
Figure 10 Picture of the Attitude Control Testbed of Flexible Spacecraft
The testbed is a hardware in loop simulation platform, which is capable of validating coupling control strategies. Meanwhile, this system can be utilized to evaluate performances of vibration control methods for appendages and maneuver trajectories for flexible spacecraft. The testbed is realized using a dSpace simulation computer to provide real-time calculation and multi-channel A/D, D/A connector
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ACCEPTED MANUSCRIPT panels for data acquisition and control output. When conducting experiments, analog signals generated by the gyro, strain gauge and PZT patches are fed back to the simulation computer through the A/D panel. The reaction torque value for the wheel and the control voltage values for PZT patches are determined by the coupling control strategy. The torque value is converted to data flow and the data is transmitted to the
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wheel through serial port. The control voltage values are transmitted to an amplifier through the D/A panel and then drive the PZT patches. To eliminate the influence of noise and high modes, low pass filters are utilized to smooth the outputs of sensors.
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5.2 Analysis of the coupling control strategy
In this experiment, the testbed is commanded to perform a 60 deg slew, with respect to the initial attitude. For comparative purposes, two different cases are
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conducted: (1) slew using the adaptive sliding mode control method, (2) slew using the coupling control strategy, i.e. the adaptive sliding mode control for attitude and the PPF control for elastic vibration. Parameters of the PPF controller are same as that of the simulation. Parameters of the adaptive sliding mode controller are set as λ=0.2, k=0.1, ϵ =0.05.
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Figure 11~Figure 16 presents the experimental results. Figure 11 presents the torque applied to the hub. Though the time histories are close to each other, the zoomed portion illustrated in Figure 11(b) shows that there is indeed an influence of the plate elastic motion on the control torque. With only the adaptive sliding mode
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control is applied, there exists jitter for the control torque. It is clear that, with the PPF control method, the excited vibration of the plate is suppressed rapidly, as shown in Figure 13. Such influence, indicated by oscillations, is reduced by the PPF controller.
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Similarly, Figure 12 (a) depicts the time histories of the angular velocities of the hub. The corresponding zoomed portion is depicted in Figure 12 (b). One can see once again the influence of the flexibility of the plate on the rigid-motion responses. This reflects advantages of the coupling control strategy. Figure 14~Figure 16 show the results in terms of attitude behavior. Also, it can be seen that the goals established for attitude control are satisfactorily achieved, namely, a steady 60 deg position is reached in nearly 40s of control action.
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0.05 0 -0.05
-0.004 -0.006 -0.008
0
10
20 time (s)
30
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-0.1
sliding mode control coupling control
-0.002 torque (Nm)
torque (Nm)
0.1
-0.01 10
40
12
14
time (s)
(a) complete time history
(b) zoomed portion
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Figure 11 Control torque comparison of a rest to rest slew 1
-2.2 sliding mode control coupling control
ωy (deg/s)
-1 -2 -3
0
10
20 time (s)
30
-2.3
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ωy (deg/s)
0
sliding mode control coupling control
-2.4
-2.5 10
40
12
13
14
15
time (s)
(b) zoomed portion
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(a) complete time history
11
Figure 12 Angular velocity of the Hub
6
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0
σ
2
0
10
20 time (s)
0.2 0.1
-2 -4
sliding mode control coupling control
0.3
EP
4 microstrain
0.4
sliding mode control coupling control
30
0
40
0
10
20
30
40
time (s)
Figure 13 Strain on the root of plate
Figure 14 Comparison of MRP
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sliding mode control coupling control
angle (deg)
0.2 s
0.15 0.1 0.05
40 20 0
0
10
20
30
40
0
10
time (s)
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0
sliding mode control coupling control
20
30
40
time (s)
Figure 16 Euler angle of the Hub
5.3 Parameter analysis of the controller
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Figure 15 The sliding manifold
In order to analyze the effect of the controller parameter on transient responses,
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three simulation cases with λ=0.2, λ=0.1 and λ=0.05 are compared. And the results are shown in Figure 17~Figure 20. When λ is increased, the time when states of the system move in the sliding surface boundary layer is delayed. Meanwhile, states of the system converge toward the targeted value with a faster speed. 1
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-0.05
λ=0.2 λ=0.1 λ=0.05
-0.1 0
10
20
30
ωy (deg/s)
0
0
EP
torque (Nm )
0.05
-1 -2 -3 -4
40
time (s)
λ=0.2 λ=0.1 λ=0.05
0
10
20
30
40
time (s)
Figure 18 Angular velocity comparison
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Figure 17 Control torque comparison with different parameters
with different parameters
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s
0.15
angle (deg )
λ=0.2 λ=0.1 λ=0.05
0.1 0.05 0
0
10
20
30
60 40 20 0
40
λ=0.2 λ=0.1 λ=0.05
0
10
time (s)
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0.2
20
30
40
time (s)
Figure 19 The sliding manifold
Figure 20 Euler angle comparison with different parameters
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comparison with different parameters
5.4 Comparison with simulation results
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Figure 21~Figure 26 compare the experimental results and the simulation results. We can see from these figures that, it shows a good agreement between simulation and experimental results. Apparently, the mathematic model of the hub-plate system is similar to the actual system. From Figure 24 it is possible to conclude that the experimental maneuver trajectory of the hub is closely followed with the numerical
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result. Similarly, unwanted elastic oscillations presented clearly in the flexible plate. However, the oscillation in simulation decays faster than that in experiment. The mathematic model needs further modification to accommodate the experiment. 0.25
0.1
simulation experiment
s
0.15
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0.1
torque (Nm)
EP
0.2
0.05
0
0
10
20 time (s)
30
0.05 0 -0.05 -0.1
40
simulation experiment
0
10
20
30
time (s)
Figure 21 Plot of sliding surface
Figure 22 Control torques
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0 ωy (deg/s)
60
simulation experiment
-1
simulation experiment
40 20
-2 0
10
20
30
40
0
10
time (s)
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0.4 simulation experiment
σ
2
0.3
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microstrain
30
40
Figure 24 Euler angle of the Hub
6
0
simulation experiment
0.2 0.1
-2 -4
20
time (s)
Figure 23 Angular velocities of the Hub
4
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0 -3
0
10
20 time (s)
30
0
40
0
6 Conclusion
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Figure 25 Strain on the root of plate
10
20
30
40
time (s)
Figure 26 Comparison of MRP
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The problem of maneuvering a flexible spacecraft has been tackled in this work. A coupling control strategy is designed, which synthesizes an adaptive sliding mode controller and a PPF controller. Also, the method is robust to parametric uncertainties
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of the spacecraft model. Numerical simulations have provided satisfactory results in terms of vibration control and rigid body motion. Simulations and experiments show that, the coupling control strategy reduces the stabilization time and improves the pointing accuracy of the platform. Hence, the method is significant for the flexible spacecraft to accomplish attitude maneuver missions.
References [1] H. Gui, L. Jin, S. Xu, Attitude maneuver control of a two-wheeled spacecraft with bounded wheel speeds, Acta Astronautica, 88 (2013) 98-107. [2] D. Verbin, V.J. Lappas, Rapid rotational maneuvering of rigid satellites with 22 / 25
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ACCEPTED MANUSCRIPT 1. A coupling control method synthesizes sliding mode control and PPF 2. Attitude control testbed of flexible spacecraft is established
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3. The method can decrease the stabilization time and improve attitude accuracy