A novel unloading strategy for solar sails using reflectivity control device

For near-Earth spacecraft, many environmental forces are utilized to unload reaction wheels. However, effects like magnetic force or atmospheric friction hardly exist in deep-space environments where solar sails operate. Thus, this paper proposes an unloading method that uses reflectivity control devices to generate Solar Radiation Pressure (SRP) torque to conduct the unloading process of solar sails operating in deep space. Firstly, the basic principles of the Reflectivity Control Device (RCD), satellite kinetic model, and Pulse Width Pulse Frequency (PWPF) modulator are given. Secondly, based on the torque envelope obtained by traversing all switch combinations of RCD, a novel strategy using a PWPF modulator is specially designed to achieve efficient unloading. In the end, we carry out numerical simulations and analyze the influence of different parameters, through which the validity of the proposed torque resolution scheme is demonstrated.


Introduction
Compared to traditional spacecraft, solar sails are subjected to additional solar radiation pressure.Utilizing this characteristic, it is possible to achieve orbital movements with solar sails that are impossible for conventional spacecraft, such as heliocentric suspension orbits and equivalent Kepler orbits [1].Solar radiation pressure is closely related to the attitude angle of the solar sail.Therefore, to ensure the long-term stable operation of solar sails in the predetermined orbit, it is necessary to develop a solar sail attitude control method that does not require propellants.In previous studies, many scholars have proposed various schemes by using the optical pressure effect to achieve attitude control, such as Diedrich [2] who first proposed the method of using control rods to generate torque; Wie [3] who proposed an attitude control scheme combining a roll axis stabilizer and four moving slide masses; Mettler E [4], Lawrence [5] and others who have also conducted in-depth research on various feasible methods by using corner sails to achieve active and passive attitude control.In 2010, JAXA's IKAROS spacecraft first verified the feasibility of attitude control by using the Reflectivity Control Device (RCD) [6].By switching the liquid crystal element on and off, the change in spin axis towards the sun was successfully observed [7].This provides a new idea for the execution mechanism of solar sail control.Because this type of scheme does not require any moving parts, it greatly reduces the structural complexity and total mass of the solar sail spacecraft, attracting scholars to further research the use of RCD for attitude control.Andreas and his colleagues [8] achieved attitude stabilization by using a 4x4 partitioned RCD sail to generate control torque that closely matched the controller output.Mashtakov et al. [9] investigated the relative position and attitude control problems of RCD in a dual-satellite formation.
IOP Publishing doi:10.1088/1742-6596/2764/1/012066 2 In addition to using the RCD optical pressure torque for direct attitude control, it can be also applied to momentum wheel unloading.Andrea and James [10] developed a control scheme for momentum management of conventional satellites by using RCD, and the feasibility of this technology was proved through numerical simulations in Earth and lunar orbits.Ji et al. [11] investigated the unloading problem and proposed an unloading method based on the angular deviation to determine the RCD state.The control strategies proposed above all require a sorting and selection process to decide the RCD states, which relies on adequate memory size and computing resources.
For solar sails using RCD, this paper proposes a novel unloading strategy that requires very little storage space and computational requirements.The remainder of the paper is organized as follows.Section 2 describes the preliminaries involving RCD, solar radiation pressure, attitude control model, and PWPF modulator.Section 3 proposes the unloading strategy in detail.Section 4 presents the results of simulations to illustrate the effectiveness of the designed strategy.Finally, Section 5 concludes the paper.

RCD variable reflectivity principle
RCD is a multi-layer material component with electrochromic properties.A commonly used RCD component consists of an emitter layer, a polyimide film layer, a transparent indium-tin-oxide (ITO) layer, and a polymer dispersion liquid crystal (PDLC) layer [12].The microstructure of the liquid crystal will scatter the incident light, which will be absorbed and dissipated in multiple reflections.As shown in Figure 1, when an electric field is applied, the micro-liquid crystals dispersed in the solid organic polymer matrix will be transformed from a disordered arrangement to an oriented arrangement, resulting in a significant enhancement of light transmission and a near specular reflection effect.This feature can be utilized to control the optical pressure for attitude control.This material has been produced in the form of thin films and is initially used on exterior walls, indoor partitions, etc.

Solar radiation pressure model
Momentum exchange occurs when solar photons impact the surface of the solar sail.This effect is reflected in the propulsive force of the sunlight on the sail surface.Solar photons appear in three behaviors after hitting the sail surface: specular reflection, diffuse reflection, and absorption.The photon reflection coefficient, diffuse reflection coefficient, and absorption coefficient are defined to represent the proportion of each part.These three coefficients naturally meet Equation (1).
(1) Using the solar radiation pressure (SRP) model introduced in [13], the SRP acting on the sail surface is expressed in Equation (2).
Where represents the SRP coefficient at 1 AU distance from the sun; 1AU a   and R is the distance between the spacecraft and the sun; S is defined as the effective reflection area; Moreover, n and t n is the normal vector and tangent vector of the sail reflector, and s is the light IOP Publishing doi:10.1088/1742-6596/2764/1/0120663 vector.Further decomposing the solar radiation pressure into the normal and tangential directions of the sail, as shown in Figure 2, Equation (3) to Equation ( 5) can be obtained.
Where ,   s n is used to represent the angle between the sunlight vector and the sail normal vector, which is called the solar angle.Applying the SRP model to every single RCD and combining the geometric and optical parameters of the RCD, the integrating moment generated by n pieces of RCD can be calculated as: shows the switching states of every RCD, RCD S is the valid area of one RCD, and i p indicates the relative position of the ith RCD to the sail surface shape center.The optical coefficients depend on the switching state of the corresponding RCD as shown in Table 1.If the term including d  is considered relatively small, Equation ( 6) can be further simplified to a form where only the specular reflection coefficient s  is considered.

Attitude control model
To help describe the motion state of a solar sail, it is necessary to define a set of spatial coordinate systems in advance. Using the inertial coordinate system s s s s o x y z to describe the inertia state of the sail.


Using the ontology coordinate system b b b ox y z to describe the ontology attitude of the sail.


Using the orbital coordinate system oxyz to describe the orbital motion of the sail.The specific relative position relationship of the coordinate systems can be referred to in Figure 2 and Figure 3.Then, based on the rigid body assumption, the motion state of the spacecraft is described as [14]: Where represent the attitude quaternion and the angular velocity in the ontology coordinate system relative to the heliocentric inertial coordinate system, respectively; v  q is the antisymmetric matrix of v q , and 3 I is the third order identity matrix.If c u is defined as the momentum wheel control torque, RCD T as the torque generated by RCD, and d  as the disturbance torque, the kinetic equation can be written as: Where  is the angular velocity of momentum wheels; J and  J symbolize the angular momentum of the solar sail body and momentum wheels.
To achieve rapid attitude stabilization and disturbance elimination, the momentum wheel's output torque utilizes a tracking control rate [11] designed through the Lyapunov method, i.e.

(
) ( ) (10) Where  and  are positive tunable control parameters.Quaternion error e q and angular velocity error e  can be calculated in Equations ( 11) and ( 12) from the corresponding expected state d q and d  .Considering that the transition process is short compared with the orbital period, the influence of orbital angular velocity can be ignored.e e ev ev ev ev e ev q q      C q q q I q q q stands for the transformation matrix from the desired coordinate system to the ontology coordinate system.The scalar part and vector part of the quaternion are written as 0 e q and ev q .The structure of the control system is shown in Figure 4.

Pulse width pulse frequency modulator
The Pulse Width Pulse Frequency (PWPF) modulator [15] automatically modulates the frequency and width of the pulse, converting continuous control quantities into switching control quantities, so the attitude control can be carried out according to the quasi-linear control law.The structure of this modulator is shown in Figure 5.A first-order inertial link is connected in series in front of a Schmidt trigger to form a negative feedback loop.In the figure, u is the continuous control command generated by the previous controller; m K and m T are the amplification factor and time constant of the inertial link; V is the output of error e through the inertial link.The Schmidt trigger can be defined by three parameters, which are the switching threshold values on U and off U , as well as pulse amplitude m u .The final output of the modulator is either m u , m u  or 0.
Next, five parameters are listed to describe the characteristics of the PWPF regulator [16].
(1) Pulse width (2) Gap width (3) Pulse period (4) Duty cycle When the input value of the Schmidt trigger satisfies on V U  , zero output will be given.Only when , there can be pulse output.At this point, the corresponding pulse width is the minimum, and is defined as the dead zone of the PWPF modulator.In Equation ( 13), when  , we have: From Equation ( 18), it can be found that DC is approximated as a linear function of x , i.e., a linear function of the input quantity u .When u is in the linear zone, we assume u is a constant in one pulse period, then the average torque av u generated by the modulator over one pulse period can be calculated.

Unloading strategy using RCD
Once the exact layout of RCD has been determined, the reachable moment points are calculated through Equation ( 6) and together form the set E. Here, we assume each of the eight RCDs can be controlled independently.
The same moment point might correspond to multiple RCD combinations.The outermost moment points form the reachable envelope of the SRP moment.The control torque given by Lyapunov theory consists of three independent components.However, in the region between the outer octagon envelope and the chosen square envelope, the components of the moment in the y and z axes are coupled, which means that the RCD torque of these two axes cannot be selected independently.Thus, in order to simplify the RCD state-solving process, the RCD moments are restricted to a square envelope shown in Figure 6, where the torque changes in one axis do not affect the other.To achieve a fast unloading process, the ideal unloading torque is defined in the opposite direction of the angular momentum of the wheels as exp , where k is a positive tunable control parameter.To investigate the stability of unloading, a Lyapunov function is defined as: Where  J is the angular momentum of the reaction wheels.0 q , and ( ) T are defined, then: Since 0   is satisfied when t   , we will have 0 V   for every  , which indicates that  will finally cover to zero when t   .The maximum RCD torque in the y and z directions is equal to m u , and then the ideal unloading torque can be mapped into the square envelope through Equation (24).It is worth noting that since RCD torque can only produce control torque in the y and z directions, the component in the x direction is constantly equal to zero.
It can be found that av u u  and item  need to be minimized to follow the expected torque as precise as possible.Thus, in order to achieve faster unloading, on U should be sufficiently small, while m K and on off U U  should be large enough.Normally, on U and off U are determined and cannot be easily changed.It is more convenient to change the amplification factor m K .It should be noted that m K cannot be infinitely large, as being too large not only makes it physically difficult to achieve but also amplifies the noise in the system.Considering that RCD cannot switch between the two states immediately, indicating that the maximum number of switches per second is limited.In this research, it is assumed that the minimum time interval between two state switches is 0.1 s.After the determination of m K , time constant m T needs to be reselected to satisfy the restrictions of on T and off T .Table 4 gives a group of feasible parameters of the PWPF session.With Equation (13) and Equation ( 14), we have: It is checked that the selected PWPF parameters are reasonable and satisfy the requirements of actual devices.

4.2.Numerical simulation
Firstly, the proposed attitude control algorithm is verified.Since the attitude settling process occurs over a much shorter period compared to the orbital period, the effect of orbital angular velocity is negligible.For the desired sun-oriented attitude, the control target is set as . As shown in Figure 7, the angle and angular velocity of the solar sail have both converged within 20 seconds.When the desired attitude maneuver process is completed, the three-axis angular velocity of the momentum wheel will stabilize to [125.9; 165.8; 244.6] rad s.Secondly, Group 1 parameters in Table 4 are used to simulate the unloading procedure.In Figure 8, the momentum wheel rotation speed finally covers [-0.01, 0.01] rad/s in 3231 s.So far, the simulation process has been completed, demonstrating the feasibility and effectiveness of the proposed unloading strategy.In Figure 9    T is set to barely meet the requirement of minimum switching time.When m K reaches about 100, the effect of further increasing m K on reducing the unloading time becomes no longer significant.Figure 11 shows the impact of k .When 3 k  , the time required rapidly decreases from over 3000 s to 2835 s.Further increments of k will not result in any additional reduction of the unloading time.Overall, by selecting Group 2 parameters in Table 4, the unloading procedure ends in 2835 s, equivalent to a 12.3% increase in efficiency.

Conclusions
This paper proposes a novel solar sail unloading strategy for a solar sail using a reflectivity control device.Firstly, an RCD state solution algorithm is proposed.We apply a PWPF session to help convert the desired control signal to specific discrete values that can be produced by RCDs.The PWPF link is simple in structure and requires little storage space and computing power.In summary, the strategy proposed in this paper can help the solar sail achieve fast and reliable unloading during long-term navigation and has a certain reference value for the further promotion and application of solar sail spacecraft.

Figure 2 .
Figure 2. Decomposition of solar pressure acting on RCD.

Figure 3 .
Figure 3.The heliocentric orbit of the solar sail.Applying the SRP model to every single RCD and combining the geometric and optical parameters of the RCD, the integrating moment generated by n pieces of RCD can be calculated as:
T   .This indicates that the output of the PWPF modulator will change from periodic pulses to constant value m u .Here, s u is marked as the width of the saturation region of the PWPF modulator.defined, and then brought into Equation (16), we can get:

1 x 1 DC
 , and x is a linear function of u ; when  , and the modulator is in saturation; when d u u  , 0 DC  , and the modulator is in the dead zone.Taylor expansion is performed around 0.5 x 

Figure 6 .
Figure 6.Outer octagon envelope and inner square envelope.The maximum RCD torque in the y and z directions is equal to m u , and then the ideal unloading torque can be mapped into the square envelope through Equation (24).It is worth noting that since RCD torque can only produce control torque in the y and z directions, the component in the x direction is constantly equal to zero.

Figure 7 .
Figure 7. Attitude stabilization process of the solar sail (in order of momentum wheel speed, spacecraft quaternion, and angular velocity).
, the RCDs keep switching in five different state combinations, and the time interval between two states meets the requirement of ,

Figure 9 .
Figure 9. RCD output torque.Thirdly, we attempt to optimize different parameters to further enhance the efficiency of unloading.Figure 10 displays the unloading time when

Figure 10
Figure 9. RCD output torque.Thirdly, we attempt to optimize different parameters to further enhance the efficiency of unloading.Figure 10 displays the unloading time when [0.1; 200] m K  and mT is set to barely meet the requirement of minimum switching time.When m K reaches about 100, the effect of further increasing m K on reducing the unloading time becomes no longer significant.Figure11shows the impact of k .When

Figure 10 .
Figure 10.Unloading time with different k ( 100, 84 m m K T   ).Figure 11.Unloading time with different m

Figure 11 . 3 k
Unloading time with different m K (  ).
the modulation process by the PWPF regulator, the resulting pulse signal is approximately equivalent to the desired input signal u .