Research on solar sail attitude maneuvering method based on adaptive sliding mode control

A solar sail is a spacecraft propulsion system that uses the pressure of light to push a spacecraft. It consists of a lightweight support structure that deploys a large area of flexible film. During attitude maneuvering in space, the solar sail will excite vibrations in the flexible support structure and film, which can affect the accuracy of the attitude control system. To address this issue, this article first establishes a rigid-flexible coupled dynamic model of the solar sail that includes attitude motion and flexible structure elastic vibrations. Then, based on an adaptive sliding mode control method, a solar sail attitude maneuvering method is proposed. Finally, a numerical simulation is conducted using a hexagonal-shaped solar sail as the object. The numerical simulation results show that the attitude maneuvering method has the performance of attitude tracking and attitude stabilization, can meet the requirements of large-angle attitude maneuvering for solar sail spacecraft, and has strong robustness to model parameter uncertainty.


Introduction
The solar sail accelerates the spacecraft by momentum obtained by reflecting solar photons projected onto the sail surface.Due to the very weak solar pressure generated per unit area, solar sail spacecraft tend to have larger areas and lighter masses.However, the lightweight, ultra-thin construction of the solar sail spacecraft presents new dynamics and poses additional problems.When a solar sail spacecraft performs orbital or attitude maneuvers in space, it tends to excite nonlinear deformation of the flexible structure, which in turn affects the rigid body motion of the spacecraft.Initially, scholars used the rigid-body model of the solar sail for attitude control system and control strategies using traditional methods such as PID.With the continuous increase in functional demands of spacecraft, the requirements for its attitude control accuracy are also getting higher.It is unable to satisfy the needs of practical missions via traditional control theories.As a result, the control system design used the flexible body model and modern control methods, such as sliding mode control and robust control.Wie [1] used the nonlinear PID control logic for the attitude control of the sailcraft.Actuators for attitude control are two-axis gimbaled boom or vanes [2].However, the model of the solar sail is a rigid model.Based on the shape variation of booms, Zhang et al. [3] proposed a novel attitude control method, and PD control law was applied to realize the attitude control of three-axis.Eldad et al. [4] established a minimum-time control method of a solar sail with model uncertainty so that the sails were able to perform large angle rotation in a short period of time.Gong et al. [5] proposed an attitude maneuver strategy for the solar sail with axisymmetric spinning.For the control of a flexible 40×40m solar sail, Rotunno et al. [6] used the H ∞ , QFT, and input shaping method and the synthesis of four controller types.Firuzi and Gong [7] used the geometrically nonlinear finite element method to study the attitude dynamics and control of a flexible solar sail considering the solar radiation pressure, gravity gradient torque and so on.Furthermore, a robust optimal control approach based on the servomechanism linear quadratic regulator was applied.Although several studies have been carried out in the literature on attitude maneuvers and attitude stabilization of solar sails, in-orbit flight tests of LightSail-2 have shown that the actual control results are significantly different from the design values [8].Therefore, the high precision attitude stabilization of solar sails is still one of the difficulties of research.On the other hand, the adaptive sliding mode control method has been widely used for attitude control of spacecraft and has achieved good results.Mostafa et al. [9] broadened the previous applications of variable structure control systems theory to flexible spacecraft's maneuvering.Hu and Ma [10] proposed a robust control algorithm that considered control input nonlinearities.Lo and Chen [11] derived a sliding-mode control (SMC) algorithm, in which the Lyapunov function was utilized to avoid the inverse operation of the inertia matrix.Giri [12] presented a fast finite-time attitude control method using sliding mode strategy.With the bandwidth constraints and unknown disturbances considered, Amrr et al. [13] developed an ET-based adaptive SMC strategy to solve the attitude stabilization problem of spacecraft.From the above brief review, it is found that the adaptive sliding mode control is effective, and has strong controllability for external interference and internal parameter uncertainty.However, at present, there are few studies on the attitude maneuvering of large-angle of solar sail using sliding mode control.This paper investigates the effectiveness of adaptive sliding mode control methods on rigidflexible coupled models of solar sails.The structure of this paper is as follows: Section 2 presents the physical configuration of a solar-sail spacecraft; Section 3 presents the rigid-flexible coupling dynamics model of a solar-sail spacecraft; Section 4 proposes a method based on adaptive sliding mode attitude control; Section 5 conducts numerical simulation research; and Section 6 presents conclusions.

Problem description
As shown in Figure 1, the solar sail in orbital configuration is a hexagonal shape with a side length of several meters or even tens of meters.The main components of the solar sail include the central body, support booms, and sail surface.The central body is located at the center of the hexagon, with six booms fixed to it.The sail surface consists of six trapezoidal thin films, each tensioned by the central body and booms.At the free ends (i.e., tips) and roots (near the central body) of the trapezoids, the booms connect with the film through a sail-to-structure interface [14].The material of the film is aluminized polyimide on the front side (i.e., the sun-facing side).The support booms are lenticular tubes, made of laminating carbon fiber/epoxy prepreg material.Moreover, to ensure that the film remains as flat as possible to maximize the light pressure generated by the sunlight, there is a certain prestress inside the film, which generates a certain pressure on the support booms.However, the lightweight and ultra-thin structural properties bring severe problems to the solar The solar sail film and support booms are so flexible that nonnegligible elastic vibration would be excited when the spacecraft performs attitude maneuvers.And the excited vibration will in turn cause great disturbance to the attitude of the spacecraft.Therefore, it is very important to develop robust attitude control methods to ensure the normal operation of the spacecraft.

Rigid-flexible coupling model of solar sail
The solar sail is simplified to a system consisting of a central rigid body, flexible films, and booms.As a result, a hybrid coordinate system is used to describe the movement of the central rigid body and flexible appendage.The Lagrangian method is employed to derive the mathematical model of the system.Firstly, the following coordinate systems need to be defined, as shown in Figure 1.Firstly, the following coordinate systems need to be defined, see.The coordinate system OXYZ is an inertial coordinate system.And the body coordinate system of the spacecraft is represented as oxyz.The coordinate system o e x e y e z e is an element coordinate system used to describe the elastic deformation relative to the central body [15].Lagrange's equations in terms of quasi-coordinates are shown below [16] 0 Flexible films and booms are discretized using the finite element method.And then Lagrange functions of the system are constructed and the rigid-flexible coupling equation of the solar sail system is derived as where m is the spacecraft's mass; J is the spacecraft's inertia matrix; V is the spacecraft's velocity vector; ω is the spacecraft's angular velocity vector; F v and τ at the right end of the equation represent the force and moment vector acting on the center body, F r is the external force on the flexible attachment, and are expressed in the body coordinate system.S denotes the coupling matrix of the translation and rotation.C and D are the translation and rotation coefficient matrices for rigid-flexible coupling; η(1×N, N is the truncation number) is the mode coordinate of the flexible appendage; ξ=diag[ξ 1 , ξ 2 , …, ξ N ] and Λ=diag[Λ 1 , Λ 2 , …, Λ N ] are the damping ratio matrix and natural frequency matrix for a cantilevered appendage respectively, where ξ i and Λ i are the ith damping ratio and frequency [17].The motion equations of the spacecraft consist of six ordinary differential equations and partial differential equations describing elastic deformation of the connecting appendage.The six ordinary differential equations describe rotation of the rigid body.

Sliding mode control method
In this section, an adaptive sliding mode control method for attitude control of solar sail spacecraft is proposed.
The attitude of the spacecraft is described by the modified Rodrigues parameterization ˆtan 4 where σ = (σ 1 , σ 2 , σ 3 ) T ∈ℝ 3 , ê and  are the principal axis and angles of rotation.The kinematics of the rotational speed associated with the angular velocity related to the MRP rate is given as follows Inverse the matrix G(σ) and we can derive Let σ d denotes the desired trajectory, and ω d denotes the desired angular velocity.The errors of MRP and angular velocities can be expressed as The time-varying surface s(t) is defined by the errors of MRP and angular velocities In the above equation, λ is a diagonal matrix with positive elements, and ( ) ˆt ω is defined as ( ) ( ) ( ) (11) After the state moves onto the sliding manifold (s=0), the state would slide along the surface towards σ d and ω d .To ensure the sliding manifold is reachable, a feedback control law should be developed.Choose a Lyapunov function for the closed-loop system as follows 1 2 T s V = s Js (12) Since the inertial matrix J is a symmetric array and positive definite, V s satisfies The elastic vibrations of the flexible booms and films induce torques acting on the central body, which can be taken as a disturbance ,, In a physical spacecraft, the vibrations of the flexible structures are assumed to be limited.As a result, the disturbance can be assumed to be limited and there is ( ) ,, ˆ1, 2,3 where |•| indicates the absolute value of a scalar.The control law is constructed as where sgn(•) is the sign function.Let (17) inserts into (14), and the following formula is induced By choosing k i large enough, and k i should satisfy the inequalities k i ≥ , ˆdi τ (i=1,2,3).Then As a result, the state of the system moves to the sliding manifold.
Therefore, the control law makes the global asymptotic stability of the spacecraft system with an origin error.The feedback control law in (17) may cause chattering, that is the states are oscillated within a neighborhood of the switching surface.Hence, the sign function is instead by the saturation function sat(s i , ϵ) to eliminate chattering, ( )

Numerical simulation
To verify the effectiveness of the adaptive sliding mode control method, numerical simulations are carried out.A solar sail performing a large angle attitude maneuver is taken as the example.Firstly, parameters of the solar sail are given, and then the simulation results are presented.The parameters of the solar sail are shown in Table 1.The response curves of the variables related to the central body attitude are shown in Figure 4 and Figure 5, respectively.As can be seen in Figure 4, the spacecraft reached the preset desired attitude after about 300s.The pitch angle showed a small overshoot during the maneuver.In Figure 5, there is some deviation in the initial state attitude angular velocity of the spacecraft, which finally converges to 0 after the process of acceleration and then deceleration.damping consideration, the vibration of the support boom decays with time.Figure 7 shows the reaction torque generated on the central body by the vibrations of the support boom, with a maximum disturbance torque of 1.3×10 -4 Nm on the y-axis.

Conclusion
In this paper, the research on the attitude control method of flexible solar sail spacecraft is carried out, and the attitude maneuvering method of the spacecraft is proposed based on the adaptive sliding mode control method.The numerical simulation results using hexagonal solar sails show that the adaptive sliding mode control method can meet the attitude maneuvering and attitude stabilization of the flexible solar sail spacecraft and achieve high precision attitude stabilization under model parameter uncertainty.

Figure 1 .
Figure 1.Geometry of a hexagonal solar sail.The material of the film is aluminized polyimide on the front side (i.e., the sun-facing side).The support booms are lenticular tubes, made of laminating carbon fiber/epoxy prepreg material.Moreover, to ensure that the film remains as flat as possible to maximize the light pressure generated by the sunlight, there is a certain prestress inside the film, which generates a certain pressure on the support booms.However, the lightweight and ultra-thin structural properties bring severe problems to the solar . The controller parameters are λ = 0.05, k = 0.02, ϵ = 0.002.The control torque and MRP parameter curves of the spacecraft are shown in Figure 2 and Figure 3.The maximum three-axis control torque during the attitude maneuver is 0.026 Nm.In Figure3, the dashed line indicates the expected value of the MRP parameter, and the solid line is the simulated value of the MRP parameter over time.About 300 seconds from the start of the maneuver, the control moment reaches 0 while the MRP parameters reach their target values.

Figure 4 .
Figure 4. Angular of the central body.Figure5.Angular velocity response.Figure6demonstrates the displacement response of the end of the film support boom due to elastic deformation during the maneuver, i.e., the displacement response of the end relative to the center point in the x, y, and z-axis directions of the main body.As can be seen from the figure, the attitude maneuvering of the solar sail excites the out-of-plane vibrations of the support boom.The tip displacement response is mainly determined by the corresponding first-order mode.Meanwhile, due to

Figure 5 .
Figure 4. Angular of the central body.Figure5.Angular velocity response.Figure6demonstrates the displacement response of the end of the film support boom due to elastic deformation during the maneuver, i.e., the displacement response of the end relative to the center point in the x, y, and z-axis directions of the main body.As can be seen from the figure, the attitude maneuvering of the solar sail excites the out-of-plane vibrations of the support boom.The tip displacement response is mainly determined by the corresponding first-order mode.Meanwhile, due to

Table 1 .
Solar Sail Parameters

Table 2 .
Initial And Target Attitude for Solar Sail