Switch Programming of Reflectivity Control Devices for the Coupled Dynamics of a Solar Sail
Tianjian Hu a,b[1], Shengping Gong a, Junshan Mu a,c, Junfeng Li a, Tianshu Wang a, Weiping Qian b
a School of Aerospace Engineering, Tsinghua University, Beijing 100084, China
b Beijing Institute of Tracking and Telecommunication Technology, Beijing 100094, China
c China Satellite Maritime Tracking and Control Department, Jiangyin 214431, China
Abstract:
As demonstrated in the Interplanetary Kite-craft Accelerated by Radiation Of the Sun (IKAROS), reflectivity control devices (RCDs) are switched on or off independently with each other, which has nevertheless been ignored by many previous researches. This paper emphasizes the discrete property of RCDs, and aims to obtain an appropriate switch law of RCDs for a rigid spinning solar sail. First, the coupled attitude-orbit dynamics is derived from the basic solar force and torque model into an underdetermined linear system with a binary set constraint. Subsequently, the coupled dynamics is reformulated into a constrained quadratic programming and a basic gradient projection method is designed to search for the optimal solution. Finally, a circular sail flying in the Venus rendezvous mission demonstrates the model and method numerically, which illustrates approximately 103km terminal position error and 0.5 m/s terminal velocity error as 80 independent RCDs are switched on or off appropriately.
Keywords: Solar sail; Coupled attitude-orbit dynamics; Reflectivity control devices; Gradient projection method
I. Introduction
Due to its capability of generating propulsion without fuel consumption, the solar sail has been envisioned to enable or enhance a wide range of mission applications throughout the solar system (Macdonald and McInnes, 2011). Some previous literature (Gong, S. and Li, J., 2014; Zeng, X. et al., 2014) mainly focused on the orbit design of a solar sail and assumed that the attitude can change instantaneously. However, the propulsion exerted by solar radiation pressure (SRP) is related to the angle between the sunlight and the normal vector of the sail surface, so the orbit dynamics and attitude dynamics are not completely independent of each other (Gong, S., et al. 2009; Zhang, J. and Wang, T., 2013). Thus, neither the attitude maneuver process nor the attitude-orbit coupled effects can be neglected.
Starting from the above idea, Gong, S., et al. (2009) developed the coupled attitude-orbit dynamics for displaced solar orbits and discussed the stability of its motion. Zhang, J. and Wang, T., (2013) considered the flexibility of the sail membrane and derived a coupled dynamical equation for a flexible displaced solar orbiter. Nevertheless, neither of these authors discussed the physical realization of the desired trajectories in both orbit and attitude. An interesting implementation of reflectivity control devices (RCDs) on the attitude adjustment of a solar sail was demonstrated by the Japanese sail “interplanetary kitecraft accelerated by radiation of the sun” (IKAROS) (Tsuda, Y. et al, 2011; 2011; 2013). RCD is a thin film attached to the surface of the sail that can switch its reflectivity between the specular and diffusive states versus electricity power being switched on or off. Different reflectivity states can result in different SRP. By adjusting the total amount and distribution of switched-on RCDs, the propulsion and torque acting on the sail can be controlled to follow the desired orbit and attitude. Compared to other methods for attitude control utilizing the gimbaled control boom (Wie, B., 2004), sail control vanes (Wie, B., 2004), or sliding masses (Scholz, C., 2011), the application of RCDs is obviously propellantless, much simpler in structure and has lower cost (Tsuda, Y. et al, 2011; 2011).
Such benefits have prompted increasing research interest in the utilization of RCDs in solar sail missions. Mu et al. (2013; 2015) extended the coupled attitude-orbit dynamics to formation flying for a GeoSail mission. The reflectivity modulation ratio and its distribution on the sail are used as continuous control variables, while RCDs consist of finite panels so that the achievable magnitudes of solar propulsion and torque are actually discrete rather than continuous. This point was noticed by Aliasi et al. (2013) in their exploration of the active stabilization of L1-type artificial equilibrium points (AEPs) for a solar sail using RCDs. They considered the discretization and saturation effect and successfully generated AEPs by switching on or off the RCDs, which are symmetrically distributed on the sail.
This paper naturally extends the application of finite discrete RCDs to the coupled attitude-orbit control of a spinning rigid solar sail in a Venus rendezvous mission. The paper aims to determine the switch law of RCDs, i.e., which part and when the devices should be switched on to follow the desired orbit and attitude. The coupled attitude-orbit dynamical equation is derived from the SRP force model of RCDs implemented in IKAROS. A type of gradient projection algorithm is successfully developed to efficiently calculate the optimal switch sequences of RCDs, which is verified in the application to a Venus rendezvous mission.
This paper are outlined as follows. In Section II, the model of coupled attitude-orbit dynamics of spinning solar sail using RCDs is derived. Section III rearranges the derived linear form into an underdetermined linear system (ULS) with a binary set constraint (BSC), converts the formulated multi-objective least squares estimation (MLSE) and l1-l2 hybrid optimization into a classical quadratic programming (QP), and develops a type of basic gradient projection (BGP) method. As numerical verifications, switch regulations of RCDs are programmed applying previously proposed methods to accomplish the Venus rendezvous mission in Section IV. Section V draws conclusions of this paper.
II. Coupled Attitude-orbit Dynamics Model of Spinning Solar Sail Utilizing Reflectivity Control Devices
A. Configuration and Coordinates of Solar Sail
As depicted in Fig. 1, the solar sail is assumed to be an ultra-thin (zero thickness) rigid sheet with arbitrary contour shape and a total mass ms. To maintain the stability and deployment, the sail is spinning along its normal axis at a constant angular rate Ω. On the surface of the sail, RCDs are fixed with rpi as the vector from the mass center of the sail to the i-th unit.
A series of right-handed orthogonal reference coordinates are established. The J2000 ecliptic coordinates are chosen as the inertial coordinates oIxIyIzI to describe the orbital motion and the SRP force exerted on the sail. The sunlight pointing coordinates oExEyEzE are used to describe the relative attitude of the sail with respect to the sunlight. The origin oE is located at the sail’s mass center, the zE axis points from the Sun’s center to the sail’s center of mass, the yE axis is parallel to the orbital normal of the sail and the xE axis is located in the orbital plane and forms a right-handed coordinate system with yE and zE axes. Both the spin-free coordinates oFxFyFzF and the body-fixed coordinates oBxByBzB are established to describe the attitude motion and the SRP torque of the sail. The origins oF and oB are located at the same point as oE. The zF and zB axes are directed along the normal of the sail surface away from the Sun. Both the xF, xB and yF, yB axes are located in the sail plane; meanwhile, the xB axis rotates with the sail, and the xF axis does not follow the spinning motion. Assume that at the very beginning of the motion, the coordinates oFxFyFzF and oBxByBzB coincide with each other.
Fig. 1 Configuration (left) and coordinates (right) of solar sail.
Define the sun angle between the zF axis and the zE axis as α. The angle between the projection of the zF axis in the xEoEyE plane and the xE axis is denoted as δ. As the sail is spinning along the normal vector of the sheet surface at constant angular speed Ω, the spin angle ξ can be easily obtained as
1)
where t denotes the spinning time. The intersection of the ecliptic plane and the orbital plane is defined as the node line oINI, and the angles between xI and oINI, and between oINI and zE are denoted as ψ and φ, respectively. In addition, the angle θ is the one between the yE and zI axes. The transition from oIxIyIzI to oExEyEzE and from oExEyEzE to oBxByBzB follows the 3-1-2 order with the angle sequence (ψ, π/2+θ, π/2+φ) and the 3-2-3 order with the angle sequence (δ, α, ξ), respectively. Denote cosα and sinα as cα and sα, respectively. The transition matrices from oIxIyIzI to oExEyEzE and from oExEyEzE to oBxByBzB can be represented as
2)
and
3)
, respectively. The transition matrix from oIxIyIzI to oBxByBzB can be derived as RIB =REBRIE. Angles ψ, θ and φ describe the orbital motion of the sail, and angles δ, α and ξ represent the attitude motion of the sail.
B. Force and Torque Models Utilizing RCDs
The surface of the sail is covered with k devices (k ∈ N*) of RCDs in total. Each device can be switched between different reflection modes by powering on and off. Number the k devices from yB axis clockwise as i = 1, 2, …, k, with Δsi as the surface area of each device, and the SRP force Fi (McInnes, C., 1999; Rios-Reyes, L. and Scheeres, D., 2005) exerted on the i-th device can be written as
4)
in which P represents the SRP on the sail, which can be calculated (Rios-Reyes, L. and Scheeres, D., 2005) as
5)
where I0 represents the frequency integrated specific intensity, c represents the speed of light, Rs represents the sun’s radius and r represents the distance from the sun; nR is the unit sun vector parallel to the zE axis; n is the unit normal vector of the sail parallel to the zF axis, and ρa, ρd and ρs represent the coefficients of absorption, diffuse and specular reflections, respectively. In order to simplify the theoretical analysis, coefficients take values (Mu, J., et al, 2015) as ρa_on = 0, ρd_on = 0, ρs_on = 1, ρa_off = 0, ρd_off = 1 and ρs_off = 0, which is similar to the SRP model implemented in the IKAROS mission (Tsuda, Y. et al, 2011). Thus, Eq. (4) can be rewritten as
6)
in which coefficients ρdi ρsi = 0 and ρdi + ρsi = 1. If ρdi is substituted by 1-ρsi, ρsi takes a value from the following binary set
. 7)
Thus, Eq. (6) can be rewritten as
. 8)
Denote vector ρs = [ρs1, ρs2, …, ρsk]T and parameter Ps = P cosα. The total SRP force FΣ exerted on the sail can be obtained as
9)
where ρsi ∈ {0, 1} and the vector S ∈ R1×k is defined as
. 10)
Define the matrix ΓF and the vector bF as
11)
and
12)
, respectively. The total SRP force can be rearranged as the following linear form
. 13)
The torque Ti due to the SRP acting on the i-th device can be obtained as
14)
and the total torque of RCDs TΣ can be obtained as
. 15)
Denote the skew symmetric matrix of a vector χ as χ×, and Eq. (15) can be rewritten as the following linear form
16)
in which the matrix [Δsirpi]3×k = [Δs1rp1, Δs2rp2, …, Δskrpk].
C. Coupled attitude-orbit dynamics
The orbit dynamics can be expressed as the following perturbed equation in oIxIyIzI
17)
where r is the position vector from the Sun’s center oI to the center of mass of the solar sail oE and as is the SRP acceleration exerted on the sail as the dominant term of nonspherical geopotential J2 of the Sun is neglected. Since the performance of the J2 perturbation is much weaker (approximately 3 times less in order of magnitude) than SRP (Roxburgh, I., 2001), the neglect of J2 perturbation is reasonable. Derived from Eqs. (13) and (17), the orbit dynamics can be expressed in oIxIyIzI as
18)
where the superscript to the left of the symbol Iχ means that the matrix or vector χ is expressed in oIxIyIzI, and eF denotes the error between the expected solar force and the practical solar force generated by RCDs. The values of IΓF and IbF can be calculated as
19)
and
20)
where matrices RIE and REB can be obtained as Eqs. (2) and (3), respectively. Eq. (20) indicates that the orbital motion of the sail is coupled with the attitude motion due to the time-varying direction of vector n in the inertial coordinate.
The angular velocity of the sail ωBI can be obtained as
21)
in which the angular velocity of oBxByBzB to oExEyEzE and oExEyEzE to oIxIyIzI can be expressed, respectively, in oBxByBzB as
22)
and
. 23)
Matrices Rx, Ry and Rz are the rotating matrices along the x-, y- and z- axes, respectively. The superscript to the left of the symbol Bω means that the vector ω is expressed in oBxByBzB. Denote the angular momentum of the sail as H = J BωBI, where the inertial moment matrix J = diag(Jx, Jy, Jz) is expressed in oBxByBzB and Jx, Jy and Jz and are the inertia moments about the xB, yB and zB axes, respectively. The attitude dynamics can be expressed using the following equation of the theorem of angular momentum