# Three-Axes Attitude Control of Solar Sail Based on Shape Variation of Booms

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Description

^{−3}and 1 < p ≤ 2, respectively. In this case, the deformation is minimal relative to the size of the sail. In the maximum deformation case, the length of the projection of the 30 m boom in the Y-axis is 29.982 m and its maximum curvature is only 0.002, the effect of deformation on the areal density of the film is ignored in this manuscript.

_{0}stands for one astronomical unit, P(r

_{0}) is the pressure of sunlight at one astronomical unit, P(r

_{0}) = 4.5 × 10

^{−6}N/m

^{2}, r is the distance from the sail to the sun, P(r) represents the solar radiation pressure acting on the solar sail, A

_{off}is the area of the electrochromic device which the state is turned off. A is the sum of the area of the open electrochromic device and the surface area of the uncovered electrochromic device. α is the angle between the normal of the solar sail

**n**and

**n**

_{s}.

**n**

_{s}is the unit vector of the sunlight direction in the body-coordinate frame, and it has the following expression:

_{s}Y

_{s}Z

_{s}]

^{T}is the coordinate of the sunlight unit vector in the heliocentric ecliptic inertial reference frame, φ, θ and ψ are the Euler angles which describe the attitude orientation of the body-coordinate system with respect to the heliocentric ecliptic inertial reference frame: first rotation around x-axis at an angle φ, then rotation around y-axis at an angle θ, finally rotation around z-axis at an angle ψ.

_{0}. L

_{0}can continuously change in the range of (0, Lmax). Therefore, the torque calculation of the OAD sail surface can be considered to consist of the ADGK (the part of the complete absorption of the sunlight) and the OGK (the part of the complete specular reflection of the sunlight). Both ADGK and OGK are composed of many rectangular elements parallel to BD. The solar radiation pressure on each element is a set of parallel forces of equal magnitude, so it can be regarded as the resultant force at the center of the rectangular element. The normal of each element can be calculated by

^{T}and [0 ${y}_{\mathrm{f}}$$a{y}_{{}_{\mathrm{f}}}^{p}$]

^{T}, respectively. y

_{k}is the value related to L

_{0}, a, p and length of boom L. y

_{f}is the value related to a, p and L.

#### 2.1. Torque Generated by OGK Part

^{T}. L(y) can be denoted as the function of the y coordinate of W, which can be written as

_{k}), we can obtain the force and torque relative to the center of mass of the OGK part.

#### 2.2. Torque Generated by ADGK Part

^{T}, the vector from the center of mass of the solar sail to the pressure center of the element is as follows

_{k}), we can obtain the force and torque of the DEKG part relative to the center of mass.

^{T}, the vector from the center of mass of the solar sail to the pressure center of the element is as follows

_{k}, y

_{f}), we can obtain the force and torque of the AEK part relative to the center of mass.

_{1}and exponent p

_{1}characterize the deformation of boom OA, while deformation of boom OB is measured by deformation coefficient a

_{2}and exponent p

_{2}, deformation of boom OC is measured by deformation coefficient a

_{3}and exponent p

_{3}, deformation of boom OD is measured by deformation coefficient a

_{4}and exponent p

_{4}. The variables characterizing the area of the electrochromic device in the closed state on the OAD, OAB, OCB, and OCD portion can be defined as L

_{1}, L

_{2}, L

_{3}and L

_{4}.

## 3. Torque and Force Analysis

_{0}characterizing the area of an electrochromic device in a closed state, the distance from the sail to the sun r, and the unit vector of the sunlight direction in the body-coordinate frame [n

_{sx}n

_{sy}n

_{sz}]

^{T}. According to the coordinate system transition matrix from the heliocentric ecliptic inertial reference frame to the body-coordinate system, [n

_{sx}n

_{sy}n

_{sz}]

^{T}are related to the three Euler angles φ, θ, ψ and the coordinate of the sunlight unit vector in the heliocentric ecliptic inertial reference frame [X

_{s}Y

_{s}Z

_{s}]

^{T}. In order to analyze the solar radiation pressure force and torque, the time history of the attitude angles, the distance from the sail to the sun and the coordinate of the sunlight unit vector in the heliocentric ecliptic inertial reference frame [X

_{s}Y

_{s}Z

_{s}]

^{T}, in the rendezvous mission with Asteroid 2000 SG344 in literature [58], are used as relevant parameters. According to the position and velocity and the direction of the spin axis of the solar sail, the attitude angles φ and θ, describing the body-coordinate system with respect to the inertial reference frame, can be calculated. Since the orbital driving force of the solar sail in the plane state is not a function of the spin angle ψ, the spin angle is not included in the literature [58]. Unless otherwise specified, the relevant parameters used in this section are shown in Table 1. For the sake of distinction, in the following figures, the solid line is used to indicate the deformation of the booms in the x-axis direction, the dashed line is used to indicate the deformation of the booms in the y-axis direction, and different colors are used to indicate different electrochromic device control.

#### 3.1. The Influence of Spin Angle ψ on Force and Torque

_{1}, a

_{2}, a

_{3}, a

_{4}) that characterize the deformation of the booms, and the variables (L

_{1}, L

_{2}, L

_{3}, L

_{4}) that characterize the area of the electrochromic devices in a closed state, are used as the control variables. This section studies the influence of the spin angle on the force and torque of the deformed solar sail and provides a reference for selecting the control target of the spin angle. The effects of the spin angle on the force and torque, under different deformation and variable reflectivity device states, are studied. As a comparison, the data in literature [58] is used as reference variables, and the changing trends under different Euler Angles φ and θ are studied.

_{x}and F

_{y}show the same trend under different deformation conditions, indicating that the deformation of the booms in different directions will not change the projection of the force on the x-y plane of the body coordinate system for any spin angle, but the deformation of the booms in different directions makes the trend of the force F

_{z}in the z-axis direction different. In general, the deformation of booms in different directions has little effect on the solar radiation pressure force.

_{x}, M

_{y}and M

_{z}with the value of ψ. This will affect the attitude control process of the solar sail.

#### 3.2. The Influence of The State of The Electrochromic Devices on The Force and Torque

_{1}= L

_{4}= 0.5 m, L

_{2}= L

_{3}= 0 m), the amplitude of the curve is larger, and the period is longer. This change is mainly caused by the change of F

_{z}. Obviously, the distribution of the state of the electrochromic device has a greater impact on the force than the deformation of the booms in different directions.

_{2}= a

_{4}= 1 × 10

^{−3}. In the legend, ”L

_{1},L

_{4}” indicate that the values of L

_{1}and L

_{4}increase from 0 to 1 at the same time, and the values of L

_{2}and L

_{3}are always zero. The rest of the illustrations are similar.

_{0,}which characterizes the area of the electrochromic device in the closed state. For the distribution of electrochromic devices ”L

_{1},L

_{2}”, ”L

_{1},L

_{3}”, ”L

_{3},L

_{4}” and ”L

_{2},L

_{4}”, the trend of force has the same slope, but for the distribution of electrochromic devices ”L

_{1},L

_{4}” and ”L

_{2},L

_{3}”, the slope of the lines are different. The reason for this is that for the deformation of the booms in the x-axis direction (OB and OD), the angle between the sunlight and the film or electrochromic devices on both sides of the y-axis is not the same. However, due to the small deformation of the solar sail, the change in the slope is also small.

_{1},L

_{2}” and “L

_{3},L

_{4}” are both symmetrical to the x-axis, the trend of the force and torque are the same. The state distributions of “L

_{1},L

_{3}” and “L

_{2},L

_{4}” are both symmetrical to the original point, the trend of the force and torque are also the same, but the trend of the torque in the state distributions of “L

_{1},L

_{2}” is not the same as the trend of the torque in the state distributions of “L

_{1},L

_{3}”, although the trend of the force in the two states is the same.

_{0}does not show a monotonous decrease law similar to the force. Especially for the state distributions of “L

_{1},L

_{4}”, the magnitude of the torque with L

_{0}first decreases and then increases. This is because, in this case, the torque around the y-axis has undergone a transition from negative to positive, and its magnitude first decreases and then increases.

_{0}. As shown in the Figure 7, when the state distribution of “L

_{1},L

_{2}”, “L

_{1},L

_{3}”, “L

_{3},L

_{4}”, “ L

_{2},L

_{4}” and “L

_{2},L

_{3}” is adopted, only the negative torque around the y-axis can be generated, and except for “L

_{2},L

_{3}”, the torque does not change in the rest of the state distribution. For the state distribution of “L

_{1},L

_{4}”, although the positive torque can be generated, no torque around x-axis is generated, and the torque around z-axis and the positive torque around the y-axis are also very small. In summary, in order to realize the three-axis attitude control of the solar sail, it is necessary to control the deformation of the solar sail and L

_{0}at the same time.

#### 3.3. The Influence of Deformation on The Force and Torque

_{1}= 1 × 10

^{−3}” means that the boom OA always keeps an upward bending deformation, and the deformation coefficient of the boom OC continuously changes; “a

_{1}= 0” means that the boom OA is not deformed at all; “a

_{1}= −1 × 10

^{−3}” means that the boom OA always keeps bending downward.

_{0}(characterizing the area of electrochromic device in the closed state) and a (characterizing the deformation of the solar sail), thus achieving the three-axes attitude control of the solar sail.

_{1}= L

_{4}= 1 m, L

_{2}= L

_{3}= 0 m.

_{0}.

## 4. Attitude Maneuver

#### 4.1. Attitude Dynamics and Kinematics

**I**

^{b}and

**ω**

^{b}is the projection of the inertia matrix and the angular velocity vector of the solar sail in the body-coordinate system, respectively.

**M**

^{b}is the projection of the solar radiation pressure torque vector. Their scalar forms are as follows

**r**and

**V**are the position and velocity vectors.

#### 4.2. Control Law

**e**represents the error between the actual value and the target value of the solar sail attitude angle, Δ

**ω**represents the error between the actual value and the target value of the solar sail angular velocity. U represents the maximum torque that can be obtained in the control process, which is related to the attitude angles (φ, θ, ψ) and control variables L

_{0}, a and p. sat means a saturation controller. The torque cannot exceed the maximum control force in the process of attitude adjustment.

_{P}is beneficial to reduce the system steady-state error and improve the response speed, but it will also cause oscillation. Increasing K

_{D}is beneficial to reduce overshoot and improve the stability of the control system. By adjusting K

_{P}and K

_{D}to appropriate values, the desired attitude control process can be obtained.

#### 4.3. Numerical Examples

_{1}, L

_{2}, L

_{3}, L

_{4}, a

_{1}, a

_{3}, p

_{1}, p

_{3}) that can control the torque of the solar sail, the torque can reach the required accuracy only with six control variables. In order to simplify the calculation, make p

_{1}= p

_{3}= 2, the range of other control variables is limited to: 0 ≤ (L

_{1}, L

_{2}, L

_{3}, L

_{4}) ≤ 1 m, −1 × 10

^{−3}≤ (a

_{1}, a

_{3}) ≤ 1 × 10

^{−3}. The attitude change process of the solar sail is plotted, as shown in Figure 11.

^{−6}due to the coupling effect. Although this is a very small value, the uncontrolled spin angle will seriously deviate from the expected value with the accumulation of time in long-term missions where spin angles are required.

**M**

_{x}around the x-axis and the torque

**M**

_{z}around the z-axis are very small, and the angular velocities

**ω**

_{x}and

**ω**

_{z}are also both small. In contrast, due to the larger adjustment range of Euler Angle θ, the required control torque

**M**

_{y}is also larger, and

**ω**

_{y}also changes significantly.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 5.**The trend of force with ψ in the different distribution of the electrochromic device state.

**Table 1.**The relevant parameters in Section 1.

Parameter | Value |
---|---|

The Length of Boom L (m) | 30 |

The sun-sail distance at the initial time r (m) | 1.508 × 10^{11} |

The sunlight unit vector at the initial time [X_{s} Y_{s} Z_{s}]^{T} | [−0.789 −0.614 −1.253 × 10^{−5}]^{T} |

The attitude of solar sail {φ, θ, ψ} (rad) | {1.569046, −0.565996, 0} |

Booms OA and OC Are Deformed | Booms OB and OD Are Deformed |
---|---|

$\begin{array}{l}{a}_{1}={a}_{3}=1\times {10}^{-3}\\ {a}_{2}={a}_{4}=0\\ {L}_{1}={L}_{4}=0.5\begin{array}{c}\mathrm{m}\end{array}\\ {L}_{2}={L}_{3}=0\begin{array}{c}\mathrm{m}\end{array}\end{array}$ | $\begin{array}{l}{a}_{2}={a}_{4}=1\times {10}^{-3}\\ {a}_{1}={a}_{3}=0\\ {L}_{1}={L}_{2}=0.5\mathrm{m}\\ {L}_{3}={L}_{4}=0\mathrm{m}\end{array}$ |

The Initial Time: MJD 59721.2196 | The Target Time: MJD 59722.2082 | ||
---|---|---|---|

The sun-sail distance at the initial time (m) | The sunlight unit vector at the initial time | Initial attitude (rad) and angular velocity (rad/s) | Target attitude (rad) and angular velocity (rad/s) |

$r=1.508\times {10}^{11}$ | $\left[\begin{array}{c}{X}_{\mathrm{s}}\\ {Y}_{\mathrm{s}}\\ {Z}_{\mathrm{s}}\end{array}\right]=\left[\begin{array}{c}-0.789\\ -0.614\\ -1.253\times {10}^{-5}\end{array}\right]$ | $\left[\begin{array}{c}\phi \\ \theta \\ \psi \\ {\omega}_{\mathrm{x}}\\ {\omega}_{\mathrm{y}}\\ {\omega}_{\mathrm{z}}\end{array}\right]=\left[\begin{array}{c}1.569046\\ -0.565996\\ 0\\ 0\\ 0\\ 0\end{array}\right]$ | $\left[\begin{array}{c}\phi \\ \theta \\ \psi \\ {\omega}_{\mathrm{x}}\\ {\omega}_{\mathrm{y}}\\ {\omega}_{\mathrm{z}}\end{array}\right]=\left[\begin{array}{c}1.569058\\ -0.552697\\ 0\\ 0\\ 0\\ 0\end{array}\right]$ |

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## Share and Cite

**MDPI and ACS Style**

Zhang, F.; Gong, S.; Baoyin, H.
Three-Axes Attitude Control of Solar Sail Based on Shape Variation of Booms. *Aerospace* **2021**, *8*, 198.
https://doi.org/10.3390/aerospace8080198

**AMA Style**

Zhang F, Gong S, Baoyin H.
Three-Axes Attitude Control of Solar Sail Based on Shape Variation of Booms. *Aerospace*. 2021; 8(8):198.
https://doi.org/10.3390/aerospace8080198

**Chicago/Turabian Style**

Zhang, Feng, Shengping Gong, and Hexi Baoyin.
2021. "Three-Axes Attitude Control of Solar Sail Based on Shape Variation of Booms" *Aerospace* 8, no. 8: 198.
https://doi.org/10.3390/aerospace8080198