# Automated Orbital Transfer and Hovering Control Using Artificial Potential

Authors: Xun Wang; Zhaokui Wang; Yulin Zhang
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

On-orbit servicing for GEO targets has attracted great attention due to particular significance. In this study, the GEO nearby flying is considered, which denotes that the servicing spacecraft approaches GEO targets within tens of meters. Based on the analysis of differential spherical Earth gravity,J2 perturbation, third-body perturbation, and solar radiation pressure (SRP) between two spacecraft, a relative dynamics of GEO nearby flying is built, in which the differential SRP has great influence on relative motion. Therein, considering the differential SRP, an analytical solution to relative dynamics is obtained, which is an extension to CW equation’s solution. Based on the derived relative dynamics, a novel control method utilizing feedback compensation and artificial potential is developed for spacecraft orbital transfer and hovering at the desired position. During orbital transfer, a bounded space is constrained with maximum and minimum relative distance between two spacecraft. An improved repulsive potential based on Gauss function is designed to drive the servicer off the bound and guarantee the potential value converge to zero at the desired position. Meanwhile, a novel attractive potential about relative distance that drives the servicer to the desired position and to hover with high accuracy against perturbations is designed. Besides, a velocity potential with variable gain is designed to ensure that the servicer achieve the desired position without overshoot. The stability of the system is proved with Lyapunov theory, and the feasibility of the proposed control law is verified through numerical simulations with obvious advantages.


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## Body

## 1. Introduction

GEO satellites, in general, are high-value assets because of the crucial role they play in global communications. This highlights the importance of on-orbit servicing (OOS) and spacecraft situational awareness (SSA) in GEO. A number of missions, such as SMART-Orbital Life Extension Vehicle (SMART-OLEV) [1], Deutscle Orbital Servicing Mission (DEOS) [2], Phoenix Program [3], and Robotic Servicing of Geosynchronous Satellites Program (RSGS) [4], have been conducted or proposed during previous decade, which demonstrates practical demands as well as technological readiness for OOS in GEO. A typical rendezvous and docking (RVD) process can be divided into several stages, including phasing, close-range rendezvous, final approaching, and docking [5–7]. In the final approaching stage, the servicing spacecraft approaches the target within hundreds of meters. This study particularly studies orbital transfer and hovering control for GEO nearby flying which denotes that the servicing spacecraft approaches GEO targets within tens of meters, in order to conduct high-resolution observation or adjust the relative position for OOS. It will be demonstrated that, compared with LEO missions, additional considerations are required for GEO nearby flying.The simplest model of relative motion between two spacecraft is CW equations [8] with assumptions that the orbit is circular with no orbital perturbations and the distance between spacecraft is negligible with respect to the distance from the reference spacecraft to the Earth. An extension to elliptic Keplerian orbits is the Lawden or TH equations, yet still assuming no orbital perturbations [9, 10]. Other mentionable approaches for modeling relative motion are orbit element differences [11, 12]. It is worth reporting that Sullivan offers a comprehensive survey and assessment of spacecraft relative motion dynamics [13]. However, as for GEO nearby flying, the differential spherical Earth gravity between two spacecraft is approximately 10-7m/s2 magnitude and a tiny thrust force or even the perturbations will affect the relative motion between two spacecraft. In response, a relative dynamics of GEO nearby flying is built with consideration of the differential perturbations, and the low-thrust maneuver is focused in this study. Then, the analytical solution considering the solar radiation pressure (SRP) is obtained, which indicates the drifting pattern of relative position between two spacecraft without control inputs. Thus, during orbital transfer, bounded space for relative distance is constrained to make sure of collision avoidance with the target through lower limit and relative distance with little divergence through upper limit, which is conducive to high-resolution observation and high-precision measurement. After transferring to the desired position, hovering control to achieve a stable relative position with high accuracy is studied.A large number of guidance and control methods for orbital transfer have been researched in RVD missions [14, 15]. Considering the collision avoidance with the target, optimal control based guidance methods have been widely used [16–18]. In [16], the Gauss pseudospectral method was employed to solve the optimal rendezvous trajectory within collision avoidance constraints. In [18], by inverting the dynamics model of approaching and parameterizing the trajectory of a chaser spacecraft via high-order polynomials, the polynomial coefficients were optimized through nonlinear programming. Model predictive control (MPC) methods used for docking with spacecraft and avoiding collision were discussed in detail [19–21]. In [22], a method to improve the computational efficiency of MPC was applied to rendezvous maneuvering using low-thrust propulsion. However, it is still a challenge for above two methods to ensure the real-time implementation. Therein, artificial potential function (APF) is a promising method, which has analytical form of control law and low computational complexity. Its clear physical meaning makes it suitable to describe the motion constraints, as well. To date, APF, combining with other control methods, such as fuzzy control [23], sliding mode control [24], and LQR control [25], has been widely used in spacecraft RVD [26], formation flying [27], and on-orbit assembly [28]. Recently, the APF with orbital element acting as variables for satellite cluster is focused. Reference [29] proposed a set of self-organizing control rules described by artificial potential functions using relative orbital elements for the reconfiguration, uniform distribution, and collision avoidance operations of satellite clusters. Although this is also an effective method for the control of relative states between spacecraft, the relative position and velocity are more intuitive than relative orbital elements, which can be measured by the on-board spacecraft instruments, such as camera and LIDAR system. Reference [30] has given a review of spacecraft pose determination techniques for close-proximity operations with cameras and LIDAR systems. Reference [31] focused on the close-proximity estimation of relative motions between spacecraft using stereovision, and the relative position and relative velocity can be acquired with high precision. In this study, a novel control method based on feedback compensation and APF is developed, and the relative position and velocity are adopted as the variables for the APF. Based on the nearby flying dynamics, feedback compensation control is adopted to make the dynamics a double-integrator model with additional noises, which is conducive to potential design and performance of the orbital transfer. When spacecraft conducts orbital transfer, Gauss function is always adopted to ensure collision avoidance. In order to avert additional impact on the desired position from the Gauss function, an improved potential by introducing the quadratic function about relative position has been designed [32]. Nonetheless, the introduced quadratic potential has great influence on the Gauss repulsive potential when the spacecraft is far away from the desired position. Motivated by this, a novel potential based on Gauss function is designed to improve the performance of the repulsive potential.Spacecraft hovering was firstly studied aiming at asteroids [33, 34]. Until recently, more attentions were paid to hovering around artificial spacecraft for OOS missions to which the accuracy of relative position is important. Affected by the orbital perturbations, open-loop control methods cannot achieve good performance for hovering problem [35]. As a result, closed-loop control methods have been widely investigated. Lin [36] derived a closed-loop control law to enable the active spacecraft hovering with a target in circular orbits along the radial direction. In [37], Yan developed a control law to achieve hovering at any desired relative position. A multiobjective and reliable output feedback control algorithm for spacecraft orbital transfer and hovering was investigated in [38]. In this study, both the orbital transfer and hovering are fulfilled by the proposed feedback compensation and APF control method. Although the quadratic potential about the relative position and velocity, discussed in [23], can drive the spacecraft to the desired position, its robustness against perturbations is limited. What’s more, its control input highly depends on the relative distance between the spacecraft and the desired position. When the spacecraft is far away from the desired position, the larger force produced by the quadratic potential makes it difficult to support the bounded orbital transfer. To ensure the smooth orbital transfer and achieve high-accuracy hovering, a position potential inspired by tangent function and a velocity potential with variable gain are designed. Finally, the feasibility of the proposed methods is evaluated by numerical simulations.The rest of this study is organized as follows: Section2 presents a relative dynamics of GEO nearby flying; Section 3 states the developed control law based on feedback compensation and artificial potential for orbital transfer and hovering, and the design of potentials is discussed in detail. Section 4 gives the numerical simulation results and illustrates the performance of the proposed methods. Conclusions are drawn in Section 5.


## 2. Equations of Relative Motion

In this section, the differential spherical Earth gravity and differential perturbations between two spacecraft, such asJ2 perturbation, third-body perturbation, and SRP, are discussed in detail. Then, relative dynamics with consideration of perturbations for GEO nearby flying is set forth. This model is described in order to be specific in further developments present in this study.


### 2.1. Relative Dynamics Model

The relevant coordinate systems are (1) the Inertial Frame fixed in the center of Earth withX-axis pointing toward the Vernal equinox,Z-axis aligned with the rotation axis andY-axis completing the right-handed coordinate system and (2) the Local-Vertical–Local-Horizontal (LVLH) Frame centered on the center of mass of the target withx-axis pointing radially outward from its orbit,y-axis perpendicular tox-axis along its direction of motion, andz-axis completing the right-handed coordinate system, as shown in Figure 1.Figure 1
Geometrical relationship and coordinate system.The motion of spacecraft in Earth orbit is governed by the vector differential equation.(1)d2Rtdt2=-μRtRt3+at(2)d2Rsdt2=-μRsRs3+as+uwhere Rt and Rs are the target and servicer’s position with respect to the center of Earth, respectively, μ is the geocentric gravitational constant, u is the servicer’s control acceleration, and at and as are the perturbation forces per unit mass (accelerations) of target and servicer, respectively. For GEO spacecraft, the J2 perturbation aJ2, third-body perturbation asun and amoon, and SRP perturbation aSRP are mainly considered. It is assumed that the plane O-XY and the plane o-xy are coplanar, and f is the orbital angle.The relative position between servicer and target is defined asr=Rs-Rt. Assuming that Δap=as-at is the differential perturbation, combining with (1) and (2) yields(3)d2rdt2=-μRsRs3-RtRt3+Δap+uDifferential Spherical Earth Gravity. The orbital eccentricity of GEO target is near zero. Considering the nearby flying, the differential spherical Earth gravity can be represented as(4)-μRsRs3-RtRt3≈n2r-er+3eRteRt·erwhere n=μ/Rt3 is the orbital mean angular velocity of the target, and er and eRt are the unit vectors of r and Rt, respectively. It is noted that n2≈5.3×10-9 and -er+3eRt(eRt·er)∈[1,2] are satisfied, where · denotes the Euclidean norm of a vector. Therein, the differential spherical Earth gravity of GEO nearby flying is approximately 10-7m/s2 magnitude. According to (4), the numerical propagation of differential spherical Earth gravity in two absolute orbits (servicer and target) is shown in Figure 2.Figure 2
Magnitude of differential spherical Earth gravity.Differential J 2 Perturbation. The most important nonspherical Earth gravity perturbation to be considered is the J2 perturbation, which can be represented as(5)aJ2=grad⁡-J2μ2R3Re23sin2φ-1where R is the distance of a spacecraft with respect to the center of Earth, Re is the Earth radius, and φ is the geocentric latitude. The differential J2 perturbation can be represented as(6)ΔaJ2=asJ2-atJ2Similarly, according to (5) and (6), the differential J2 perturbation is shown in Figure 3. As observed in Figure 3, the differential J2 perturbation of GEO nearby flying is approximately 10-11m/s2 magnitude.Figure 3
Magnitude of differential J2 perturbation.Differential Third-Body Perturbation. The perturbation due to the third body can be represented as(7)asun=μsunRsun-RRsun-R3-RsunRsun3(8)amoon=μmoonRmoon-RRmoon-R3-RmoonRmoon3where μsun and μmoon are the Sun and Moon’s gravitational constants, respectively, and Rsun and Rmoon are the position of Sun and Moon with respect to the center of Earth, respectively. The differential third-body perturbation can be represented as(9)Δathird-body=μsunrR⊙3-er+3eR⊙eR⊙·er+μmoonrR⊗3-er+3eR⊗eR⊗·erwhere R⊙ and R⊗ are the position of target with respect to the center of Sun and Moon, respectively, and eR⊙ and eR⊗ are their unit vectors, respectively. Similarly, according to (9), the differential third-body perturbation is shown in Figure 4. As observed in Figure 4, the differential third-body perturbation of GEO nearby flying is approximately 10-11m/s2 magnitude.Figure 4
Magnitude of differential third-body perturbation.Differential SRP Perturbation. Generally, the simplest model used to describe the SRP perturbation is(10)as=-PCRAmR⊙R⊙3AU2where P≈4.56×10-6Nm-2 is the constant of the SRP, CR accounts for the mean reflectivity of the surface,A is the spacecraft cross-sectional area assumed constant,m is the mass of the spacecraft, AU is the Astronomical Unit, and the norm of R⊙/R⊙3·AU2 approximates 1. SRP model in (10) has been widely used in analyzing the effect of SRP on spacecraft absolute orbit and relative motion between two spacecraft [12, 39, 40]. It is noted that model of differential SRP perturbation in (10) is derived with assumption that the normal vector of spacecraft’s surface points in the direction of the Sun [41]. Strictly speaking, the noise should also be considered, and then the SRP perturbation can be rewritten as(11)aSRP=-PCRAmR⊙R⊙3AU2+anwhere an is a bounded noise produced by above assumption. It is considered that an is a weak term with respect to as, which is modeled as(12)an=I3×3-2×diag⁡δ1,δ2,δ3×κ·aswhere δ is a random number between 0 and 1, and κ is a scale coefficient indicating the amount of noise. The modeled an is used in numerical simulations to evaluate the performance of the analytical solutions and the robustness of the designed control law, which is discussed in detail later.Given the large distance to the Sun as compared with the distance between the spacecraft, it suffices to assume that the distance to the Sun and the unit direction-to-the-Sun vector are the same for both spacecraft [12]. In this way, the differential SRP perturbation can be represented as(13)ΔaSRP=-PR⊙R⊙3AU2DR+Δanwhere DR is the differential coefficient which can be represented as DR=CRsAs/ms-CRtAt/mt. The magnitude of the differential SRP perturbation mainly depends on the product of P and DR. As for two orbital spacecraft, the DR may reach 0.02 [39], and then the differential SRP perturbation may have similar order of magnitude with the differential spherical Earth gravity, which denotes that relative dynamics of GEO nearby flying cannot be simplified as CW equations with neglecting the perturbations. Additionally, under the following assumptions:(1) by retaining only first-order terms in the Taylor expansion ofR⊙/R⊙3, R⊙/R⊙3 can be approximated as Rsun/Rsun3,(2) for a short duration (a few days), the inertial direction of the Sun is fixed (actually, it changes in about 1 deg per day) [39],(3) the Earth shadowing effects are not taken into account for GEO targets (actually, the GEO is shadow-free most of the year); the differential SRP perturbation in (13) can be rewritten as(14)ΔaSRP=Δas+Δanwhere Δas=(Δaxs,Δays,Δazs) is modeled as a constant resolved in Inertial Frame according to above assumptions, which can be represented as(15)Δas=-PRsunRsun3AU2DRReconsidering (3), (4), (6), (9), and (14), the relative dynamics of two spacecraft resolved in Inertial Frame can be written as(16)d2rdt2=n2r-er+3eRteRt·er+u+Δas+Δawhere Δa=Δan+ΔaJ2+Δathird-body. According to above analysis, Δa is assumed as a bounded and weak noise. Assuming that the orbital inclination of GEO spacecraft is zero, the relative dynamics resolved in target’s LVLH Frame for GEO nearby flying can be written as (17)x¨=2ny˙+3n2x+Δaxscos⁡f0+nt+Δayssin⁡f0+nt+Δax+uxy¨=-2nx˙-Δaxssin⁡f0+nt+Δayscos⁡f0+nt+Δay+uyz¨=-n2z+Δazs+Δaz+uzwhere x=(x,y,z) and v=(x˙,y˙,z˙) are the relative position and velocity in the target’s LVLH Frame, f0 is the initial orbital angle of the target, Δa=(Δax,Δay,Δaz) is the perturbation noise in the target’s LVLH Frame, and u=(ux,uy,uz) is the control acceleration of the servicer in the target’s LVLH Frame.


### 2.2. Solutions to Equation

The relative dynamics of GEO nearby flying takes the differential perturbations into account, which is an extension to CW equations. Furthermore, in order to analyze the impact of SRP on relative motion, the analytical solutions to relative dynamics are also obtained. Without loss of generality, the initial orbital angle of the targetf0 is set to zero. Without considering the control input and the perturbation noises, analytical solutions to relative dynamics in (16) can be derived by a superposition of the homogenous solution to the CW equations and the particular solution due to the modeled differential SRP in (15).(18)x=3axs4n2cos⁡nt+3ays4n2sin⁡nt+3axs2ntsin⁡nt-3ays2ntcos⁡nt+C1cos⁡nt+C2sin⁡nt+2nC4x˙=3axs4nsin⁡nt-3ays4ncos⁡nt+3axs2tcos⁡nt+3ays2tsin⁡nt-nC1sin⁡nt+nC2cos⁡nty=-7axs2n2sin⁡nt+7ays2n2cos⁡nt+3axsntcos⁡nt+3aysntsin⁡nt-2C1sin⁡nt+2C2cos⁡nt-3C4t+C3y˙=-axs2ncos⁡nt-ays2nsin⁡nt-3axstsin⁡nt+3aystcos⁡nt-2nC1cos⁡nt-2nC2sin⁡nt-3C4z=C5cos⁡nt+C6sin⁡nt+azsn2z˙=-nC5sin⁡nt+nC6cos⁡ntwhere C1, C2, C3, C4, C5, and C6 are the integral constants, which can be represented as(19)C1=-3x0-2ny˙0+5axs4n2C2=1nx˙0+3ays4n2C3=y0-2nx˙0-5aysn2C4=2nx0+y˙0-axsnC5=z0-azsn2C6=1nz˙0where (x0,y0,z0) and (x˙0,y˙0,z˙0) are the initial relative position and initial relative velocity, respectively. From (17) and (18), it is noted that the out-of-plane motion is also decoupled from the in-plane motion. What’s more, the out-of-plane motion is a simple harmonic. The in-plane motion can be represented as(20)x-2C4/nΔa2+y+2axssin⁡nt/n2-2ayscos⁡nt/n2+3C4t-C32Δa2=1where(21)Δa=3axs4n2+C1-3ays2nt2+3ays4n2+C2+3axs2nt2.As can be seen from (19), the in-plane motion is like a cycloidal motion with relative distance diverging. Different from analytical solutions to CW equations, there are no proper initial relative states to ensure a stable relative motion with no divergence. Absolutely, when neglecting the SRP perturbation, the analytical solutions to relative dynamics of GEO nearby flying reduce to the analytical solutions to CW equations.What’s more, a numerical simulation in a high-fidelity perturbation environment, complete with a 60 degree of gravity field, third-body Sun and Moon forces, and SRP, is conducted to evaluate the feasibility of the derived analytical solutions. It is noted that the atmospheric drag in GEO is negligible. The positions of Soon and Moon are obtained referring to [41]. The parameters of servicer and target are shown in Table 1.Table 1
Parameters of servicer and target.
Orbital parameters Parameters related to SRP servicer h s = 35786000  m, es=0, m s = 100  k g, CRs=1.4, As/ms=0.01 f s = 4.9 × 1 0 - 5  d e g, is=0deg target h t = 35786000  m, et=0, m t = 1000  k g, CRt=1.5, At/mt=0.02 f t = 0  d e g, it=0degAccordingly, the differential coefficient of SRP isDR=-1.6×10-2, and the scale coefficient κ in (12) is set to 0. The trajectories of relative position generated by numerical simulation and predicted through solutions to CW equations and relative dynamics of GEO nearby flying are shown in Figure 5, respectively.Figure 5
Comparison and analysis of the analytical solutions.As observed in Figure5, the obtained solution has the ability to predict the relative motion of GEO nearby flying with high accuracy, compared with solutions to CW equation. The main contribution to the prediction error is the simplified differential SRP model described in (15) with the assumptions (1)—(3). Actually, the effects of differential J2 and third-body perturbation on short-period relative motion are in the millimeter magnitude.Additionally, the influence of SRP noise on prediction error is also analyzed with different scale coefficientκ described in (12). The prediction error ep is defined as(22)ep=x^-x2+y^-y2+z^-z2where ∗^ denotes the prediction position through solutions to CW equations or relative dynamics of GEO nearby flying. The prediction error is shown in Figure 6.Figure 6
Prediction errors with different scale coefficient.As observed in Figure6, with the increasing of SRP noise, the prediction accuracy decreases. However, the obtained solution has the ability to predict the relative motion with better performance than CW equation’s solution. What’s more, according to the parameters design of servicer and target, it is noted that the configuration of two spacecraft is a natural hovering configuration when neglecting perturbations, discussed in [35, 38]. However, due to the longer orbital period of GEO than LEO, the differential SRP perturbation will result in a great divergence of relative distance within one orbital period. It is concluded that when two spacecraft keep GEO nearby flying, the differential SRP perturbation has great influence on relative motion. Meanwhile, a tiny control input will also have great influence on relative motion. Therein, the low-thrust maneuver is focused. In order to ensure the relative distance with little divergence, an upper bound is constrained for the process of orbital transfer, which is discussed in detail later.


## 2.1. Relative Dynamics Model

The relevant coordinate systems are (1) the Inertial Frame fixed in the center of Earth withX-axis pointing toward the Vernal equinox,Z-axis aligned with the rotation axis andY-axis completing the right-handed coordinate system and (2) the Local-Vertical–Local-Horizontal (LVLH) Frame centered on the center of mass of the target withx-axis pointing radially outward from its orbit,y-axis perpendicular tox-axis along its direction of motion, andz-axis completing the right-handed coordinate system, as shown in Figure 1.Figure 1
Geometrical relationship and coordinate system.The motion of spacecraft in Earth orbit is governed by the vector differential equation.(1)d2Rtdt2=-μRtRt3+at(2)d2Rsdt2=-μRsRs3+as+uwhere Rt and Rs are the target and servicer’s position with respect to the center of Earth, respectively, μ is the geocentric gravitational constant, u is the servicer’s control acceleration, and at and as are the perturbation forces per unit mass (accelerations) of target and servicer, respectively. For GEO spacecraft, the J2 perturbation aJ2, third-body perturbation asun and amoon, and SRP perturbation aSRP are mainly considered. It is assumed that the plane O-XY and the plane o-xy are coplanar, and f is the orbital angle.The relative position between servicer and target is defined asr=Rs-Rt. Assuming that Δap=as-at is the differential perturbation, combining with (1) and (2) yields(3)d2rdt2=-μRsRs3-RtRt3+Δap+uDifferential Spherical Earth Gravity. The orbital eccentricity of GEO target is near zero. Considering the nearby flying, the differential spherical Earth gravity can be represented as(4)-μRsRs3-RtRt3≈n2r-er+3eRteRt·erwhere n=μ/Rt3 is the orbital mean angular velocity of the target, and er and eRt are the unit vectors of r and Rt, respectively. It is noted that n2≈5.3×10-9 and -er+3eRt(eRt·er)∈[1,2] are satisfied, where · denotes the Euclidean norm of a vector. Therein, the differential spherical Earth gravity of GEO nearby flying is approximately 10-7m/s2 magnitude. According to (4), the numerical propagation of differential spherical Earth gravity in two absolute orbits (servicer and target) is shown in Figure 2.Figure 2
Magnitude of differential spherical Earth gravity.Differential J 2 Perturbation. The most important nonspherical Earth gravity perturbation to be considered is the J2 perturbation, which can be represented as(5)aJ2=grad⁡-J2μ2R3Re23sin2φ-1where R is the distance of a spacecraft with respect to the center of Earth, Re is the Earth radius, and φ is the geocentric latitude. The differential J2 perturbation can be represented as(6)ΔaJ2=asJ2-atJ2Similarly, according to (5) and (6), the differential J2 perturbation is shown in Figure 3. As observed in Figure 3, the differential J2 perturbation of GEO nearby flying is approximately 10-11m/s2 magnitude.Figure 3
Magnitude of differential J2 perturbation.Differential Third-Body Perturbation. The perturbation due to the third body can be represented as(7)asun=μsunRsun-RRsun-R3-RsunRsun3(8)amoon=μmoonRmoon-RRmoon-R3-RmoonRmoon3where μsun and μmoon are the Sun and Moon’s gravitational constants, respectively, and Rsun and Rmoon are the position of Sun and Moon with respect to the center of Earth, respectively. The differential third-body perturbation can be represented as(9)Δathird-body=μsunrR⊙3-er+3eR⊙eR⊙·er+μmoonrR⊗3-er+3eR⊗eR⊗·erwhere R⊙ and R⊗ are the position of target with respect to the center of Sun and Moon, respectively, and eR⊙ and eR⊗ are their unit vectors, respectively. Similarly, according to (9), the differential third-body perturbation is shown in Figure 4. As observed in Figure 4, the differential third-body perturbation of GEO nearby flying is approximately 10-11m/s2 magnitude.Figure 4
Magnitude of differential third-body perturbation.Differential SRP Perturbation. Generally, the simplest model used to describe the SRP perturbation is(10)as=-PCRAmR⊙R⊙3AU2where P≈4.56×10-6Nm-2 is the constant of the SRP, CR accounts for the mean reflectivity of the surface,A is the spacecraft cross-sectional area assumed constant,m is the mass of the spacecraft, AU is the Astronomical Unit, and the norm of R⊙/R⊙3·AU2 approximates 1. SRP model in (10) has been widely used in analyzing the effect of SRP on spacecraft absolute orbit and relative motion between two spacecraft [12, 39, 40]. It is noted that model of differential SRP perturbation in (10) is derived with assumption that the normal vector of spacecraft’s surface points in the direction of the Sun [41]. Strictly speaking, the noise should also be considered, and then the SRP perturbation can be rewritten as(11)aSRP=-PCRAmR⊙R⊙3AU2+anwhere an is a bounded noise produced by above assumption. It is considered that an is a weak term with respect to as, which is modeled as(12)an=I3×3-2×diag⁡δ1,δ2,δ3×κ·aswhere δ is a random number between 0 and 1, and κ is a scale coefficient indicating the amount of noise. The modeled an is used in numerical simulations to evaluate the performance of the analytical solutions and the robustness of the designed control law, which is discussed in detail later.Given the large distance to the Sun as compared with the distance between the spacecraft, it suffices to assume that the distance to the Sun and the unit direction-to-the-Sun vector are the same for both spacecraft [12]. In this way, the differential SRP perturbation can be represented as(13)ΔaSRP=-PR⊙R⊙3AU2DR+Δanwhere DR is the differential coefficient which can be represented as DR=CRsAs/ms-CRtAt/mt. The magnitude of the differential SRP perturbation mainly depends on the product of P and DR. As for two orbital spacecraft, the DR may reach 0.02 [39], and then the differential SRP perturbation may have similar order of magnitude with the differential spherical Earth gravity, which denotes that relative dynamics of GEO nearby flying cannot be simplified as CW equations with neglecting the perturbations. Additionally, under the following assumptions:(1) by retaining only first-order terms in the Taylor expansion ofR⊙/R⊙3, R⊙/R⊙3 can be approximated as Rsun/Rsun3,(2) for a short duration (a few days), the inertial direction of the Sun is fixed (actually, it changes in about 1 deg per day) [39],(3) the Earth shadowing effects are not taken into account for GEO targets (actually, the GEO is shadow-free most of the year); the differential SRP perturbation in (13) can be rewritten as(14)ΔaSRP=Δas+Δanwhere Δas=(Δaxs,Δays,Δazs) is modeled as a constant resolved in Inertial Frame according to above assumptions, which can be represented as(15)Δas=-PRsunRsun3AU2DRReconsidering (3), (4), (6), (9), and (14), the relative dynamics of two spacecraft resolved in Inertial Frame can be written as(16)d2rdt2=n2r-er+3eRteRt·er+u+Δas+Δawhere Δa=Δan+ΔaJ2+Δathird-body. According to above analysis, Δa is assumed as a bounded and weak noise. Assuming that the orbital inclination of GEO spacecraft is zero, the relative dynamics resolved in target’s LVLH Frame for GEO nearby flying can be written as (17)x¨=2ny˙+3n2x+Δaxscos⁡f0+nt+Δayssin⁡f0+nt+Δax+uxy¨=-2nx˙-Δaxssin⁡f0+nt+Δayscos⁡f0+nt+Δay+uyz¨=-n2z+Δazs+Δaz+uzwhere x=(x,y,z) and v=(x˙,y˙,z˙) are the relative position and velocity in the target’s LVLH Frame, f0 is the initial orbital angle of the target, Δa=(Δax,Δay,Δaz) is the perturbation noise in the target’s LVLH Frame, and u=(ux,uy,uz) is the control acceleration of the servicer in the target’s LVLH Frame.


## 2.2. Solutions to Equation

The relative dynamics of GEO nearby flying takes the differential perturbations into account, which is an extension to CW equations. Furthermore, in order to analyze the impact of SRP on relative motion, the analytical solutions to relative dynamics are also obtained. Without loss of generality, the initial orbital angle of the targetf0 is set to zero. Without considering the control input and the perturbation noises, analytical solutions to relative dynamics in (16) can be derived by a superposition of the homogenous solution to the CW equations and the particular solution due to the modeled differential SRP in (15).(18)x=3axs4n2cos⁡nt+3ays4n2sin⁡nt+3axs2ntsin⁡nt-3ays2ntcos⁡nt+C1cos⁡nt+C2sin⁡nt+2nC4x˙=3axs4nsin⁡nt-3ays4ncos⁡nt+3axs2tcos⁡nt+3ays2tsin⁡nt-nC1sin⁡nt+nC2cos⁡nty=-7axs2n2sin⁡nt+7ays2n2cos⁡nt+3axsntcos⁡nt+3aysntsin⁡nt-2C1sin⁡nt+2C2cos⁡nt-3C4t+C3y˙=-axs2ncos⁡nt-ays2nsin⁡nt-3axstsin⁡nt+3aystcos⁡nt-2nC1cos⁡nt-2nC2sin⁡nt-3C4z=C5cos⁡nt+C6sin⁡nt+azsn2z˙=-nC5sin⁡nt+nC6cos⁡ntwhere C1, C2, C3, C4, C5, and C6 are the integral constants, which can be represented as(19)C1=-3x0-2ny˙0+5axs4n2C2=1nx˙0+3ays4n2C3=y0-2nx˙0-5aysn2C4=2nx0+y˙0-axsnC5=z0-azsn2C6=1nz˙0where (x0,y0,z0) and (x˙0,y˙0,z˙0) are the initial relative position and initial relative velocity, respectively. From (17) and (18), it is noted that the out-of-plane motion is also decoupled from the in-plane motion. What’s more, the out-of-plane motion is a simple harmonic. The in-plane motion can be represented as(20)x-2C4/nΔa2+y+2axssin⁡nt/n2-2ayscos⁡nt/n2+3C4t-C32Δa2=1where(21)Δa=3axs4n2+C1-3ays2nt2+3ays4n2+C2+3axs2nt2.As can be seen from (19), the in-plane motion is like a cycloidal motion with relative distance diverging. Different from analytical solutions to CW equations, there are no proper initial relative states to ensure a stable relative motion with no divergence. Absolutely, when neglecting the SRP perturbation, the analytical solutions to relative dynamics of GEO nearby flying reduce to the analytical solutions to CW equations.What’s more, a numerical simulation in a high-fidelity perturbation environment, complete with a 60 degree of gravity field, third-body Sun and Moon forces, and SRP, is conducted to evaluate the feasibility of the derived analytical solutions. It is noted that the atmospheric drag in GEO is negligible. The positions of Soon and Moon are obtained referring to [41]. The parameters of servicer and target are shown in Table 1.Table 1
Parameters of servicer and target.
Orbital parameters Parameters related to SRP servicer h s = 35786000  m, es=0, m s = 100  k g, CRs=1.4, As/ms=0.01 f s = 4.9 × 1 0 - 5  d e g, is=0deg target h t = 35786000  m, et=0, m t = 1000  k g, CRt=1.5, At/mt=0.02 f t = 0  d e g, it=0degAccordingly, the differential coefficient of SRP isDR=-1.6×10-2, and the scale coefficient κ in (12) is set to 0. The trajectories of relative position generated by numerical simulation and predicted through solutions to CW equations and relative dynamics of GEO nearby flying are shown in Figure 5, respectively.Figure 5
Comparison and analysis of the analytical solutions.As observed in Figure5, the obtained solution has the ability to predict the relative motion of GEO nearby flying with high accuracy, compared with solutions to CW equation. The main contribution to the prediction error is the simplified differential SRP model described in (15) with the assumptions (1)—(3). Actually, the effects of differential J2 and third-body perturbation on short-period relative motion are in the millimeter magnitude.Additionally, the influence of SRP noise on prediction error is also analyzed with different scale coefficientκ described in (12). The prediction error ep is defined as(22)ep=x^-x2+y^-y2+z^-z2where ∗^ denotes the prediction position through solutions to CW equations or relative dynamics of GEO nearby flying. The prediction error is shown in Figure 6.Figure 6
Prediction errors with different scale coefficient.As observed in Figure6, with the increasing of SRP noise, the prediction accuracy decreases. However, the obtained solution has the ability to predict the relative motion with better performance than CW equation’s solution. What’s more, according to the parameters design of servicer and target, it is noted that the configuration of two spacecraft is a natural hovering configuration when neglecting perturbations, discussed in [35, 38]. However, due to the longer orbital period of GEO than LEO, the differential SRP perturbation will result in a great divergence of relative distance within one orbital period. It is concluded that when two spacecraft keep GEO nearby flying, the differential SRP perturbation has great influence on relative motion. Meanwhile, a tiny control input will also have great influence on relative motion. Therein, the low-thrust maneuver is focused. In order to ensure the relative distance with little divergence, an upper bound is constrained for the process of orbital transfer, which is discussed in detail later.


## 3. Control Strategy Design

In this section, a novel control strategy based on feedback compensation and artificial potential, developed for servicer’s orbital transfer and hovering, is discussed. During orbital transfer, an upper limit of relative distance, denoted asrmax, is constrained to ensure the relative distance with little divergence. What’s more, a lower limit of relative distance, denoted as rmin, is also constrained to make sure of the collision avoidance with the target. After orbital transfer, the servicer is controlled to hover at the desired position with high precision. The scheme of control objective is shown in Figure 7.Figure 7
Scheme of control objective.


### 3.1. Design of Repulsive Potential

The definition of repulsive potential is associated with the distance between the spacecraft and bounds. Usually, the repulsive potential becomes large when approaching the bounds and becomes small or near zero when moving away from bounds. The control accelerations, obtained through the negative gradient of the repulsive potential, will drive the spacecraft away from the bounds. In this study, Gaussian function is applied to represent the repulsive potential(23)Rdi=λr1exp⁡-12λr22di2i=1,2where d1=rmax-x2+y2+z2 is the distance between the servicer and upper bound, and d2=x2+y2+z2-rmin is the distance between the servicer and lower bound. The parameters λr1>0 and λr2>0 determine the amplitude and influence range of the repulsive potential, respectively. However, the repulsive potential (23) do not converge to zero at the desired position, which will deflect the balance point and reduce the control precision. Accordingly, the repulsive potential is innovatively modified as (24)Rd0,di=1-exp⁡-12λr02d02·λr1exp-12λr22di2i=1,2where d0=(x-xd)2+(y-yd)2+(z-zd)2 is the distance between the servicer and the desired position. It is noted that the repulsive potential (24) will converge to zero when reaching at the desired position. Contrasted with improvement of repulsive potential by introducing a quadratic function about the relative position between the spacecraft and desired point, discussed in [23, 32], the introduced term 1-exp⁡-1/2λr02d02∈0,1 has little influence on the amplitude and influence range of Gauss repulsive potential (23).In order to illustrate the design principle of repulsive potential (24), a similar potential in one-dimensional space is constructed as follows(25)y=1-exp⁡-12λr02x-x02·λr1exp⁡-12λr22x-x12x∈-10,10Analogously,x=x0 is the desired point, and x=x1 is the bound. Parameters in (25) are set to λr1=6×10-6, λr0=λr2=2, x0=5, and x1=0. The distribution of potential (25) is shown in Figure 8. Apparently, when x=x0 is satisfied, the potential converges to zero, which is a global minimum. When approaching the bound x1, the potential becomes larger. What’s more, when far away from the bound, the potential only has weak effect.Figure 8
Repulsive potential in one-dimensional space.


### 3.2. Design of Attractive Potential

The quadratic function and hyperbolic function have been widely used for the design of attractive potential. In this study, inspired by tangent function, a novel attractive potential about relative distance is designed as follows(26)Rd0=λp1d0arctan⁡λp2d0-12λp2In⁡1+λp22d02Similarly, in order to illustrate the design principle of attractive potential (26), a potential in one-dimensional space is constructed as follows(27)y=λp1xarctan⁡λp2x-12λp2In⁡1+λp2x2x∈-10,10The control acceleration, obtained through the negative gradient ofy, can be represented as(28)f=-∂y∂x=-λp1arctan⁡λp2xx∈-10,10The parameterλp1>0 determines the amplitude of acceleration and λp2>0 is a slope coefficient. Parameters in (27) and (28) are set to λp1=2.5×10-7 and λp2=100. The distributions of potential (27) and acceleration are shown in Figure 9.Figure 9
Attractive potential (a) and corresponding acceleration (b).
(a) (b)As observed in Figure9, the desired position x=0 is the global minimum in potential. When the object is not at the desired position, a reverse force with respect to its position will drive it to the desired position. It is noted that the amplitude of the acceleration changes little when the object is far away from the desired position, which is similar to bang-bang control law. Compared with attractive potential with quadratic function, this designed attractive potential can reduce the effect of relative distance on control acceleration. What’s more, when the object has a small offset with the desired position, a larger force will act on the object, which will improve the system’s robustness against the disturbing force.To ensure the smooth orbital transfer, an attractive potential about relative velocity with variable gain is designed, which will control the relative velocity between the servicer and desired position to converge to zero.(29)Rv=12λvvTvwhere λv>0 is the velocity damping coefficient defined as(30)λv=ελv0+1-ελv1ε=11+exp⁡-ςd0-awhere λv0>0 and λv1>0 are constants determining the upper and lower limits of the velocity damping coefficient, ε is a variable related to the relative distance, ς>0 is a coefficient, and a>0 is a threshold. The design scheme of the variable gain λv is shown in Figure 10 with λv0=1×10-3, λv1=5×10-2, ς=5, and a=1.Figure 10
Design scheme of velocity damping coefficient.


### 3.3. Control Law and Stability Proof

The system potential is defined as the summation of all above designed potentials, which can be represented as(31)R=Rd0+Rv+Rd0,d1+Rd0,d2Reconsidering the relative dynamics of GEO nearby flying, the shorter form of (16) can be written as(32)v˙=AX+gt+u+Δawhere X=(x,y,z,x˙,y˙,z˙)T is the state vector, and matrix A and function matrix g can be represented as(33)A=3n20002n0000-2n0000-n2000g=Δaxscos⁡f0+nt+Δayssin⁡f0+nt-Δaxssin⁡f0+nt+Δayscos⁡f0+ntΔazsThe control law is designed as follows(34)u=uinv+up+uv+urep=-AX-gt-∇xRd0-∇vRv-∇xRd0,d1-∇xRd0,d2where uinv=-AX-g(t) is the feedback compensation part, up=-∇xR(d0) is the attractive acceleration about relative position, uv=-∇vR(v) is the attractive acceleration about relative velocity, and urep=-∇xR(d0,d1)-∇xR(d0,d2) is the repulsive acceleration. The detailed description of up, uv, and urep can be represented as(35)up=-λp1arctan⁡λp2d0x-xdd0(36)uv=-λvv(37)urep=-λr1exp⁡-d122λr22d1λr22xx+λr1exp⁡-d022λr02exp⁡-d122λr22d1λr22xx-x-xdλr02+λr1exp⁡-d222λr22d2λr22xx-λr1exp⁡-d222λr22exp⁡-d022λr02d2λr22xx+x-xdλr02Theorem 1.
Given the relative dynamics of GEO nearby flying in (32) and neglecting the perturbation noise Δa, if the control law is considered as (34), the servicer will transfer to the desired position and the error of hovering will converge to zero.Proof.
Choosing a Lyapunov function candidate as(38)Ψ=Rd0+12vTv+Rd0,d1+Rd0,d2>0
the time derivative ofΨ is (39)Ψ˙=∇xRd0x˙+vTv˙+∇xRd0,d1x˙+∇xRd0,d2x˙
Substituting control law equation (34) and relative dynamics equation (32) into (39) yields(40)Ψ˙=-λvvTv≤0
It is noted thatΨ˙ is negative semidefinite, and if Ψ˙≡0 is satisfied, v≡0 and v˙≡0 are obtained. Substituting control law equation (34) into relative dynamics equation (32) (neglecting the noise), x=xd is obtained. According to the LaSalle theory, the system is asymptotically stable. This completes the Proof.Remark 2.
The APF method always suffers from local minimum problem, especially when applied to nonlinear dynamics system. In this study, the feedback compensation control is applied to make the relative dynamics a double-integrator model with additional noises. The designed system potential guarantees that the desired position is at the global minimum as discussed in the design of attractive potential. Once the desired position is at the global minimum, Theorem1 will guarantee that the servicer with any initial relative states will converge to its global minimum. And the designed attractive potential about relative distance in (26) will improve the system’s robustness against the disturbing force.Remark 3.
After the feedback compensation being applied, the negative gradient of repulsive potential and attractive potential, respectively, represent the repulsive acceleration and the attractive acceleration applied on the servicer. As discussed in (24), (26), and (29), both the potentials and their related control accelerations are bounded. However, parameters λr, λp, and λv have great influence on the size of repulsive acceleration and the attractive acceleration. Therein, in order to make sure the servicer do not cross the upper and limit bounds, the parameters λr, λp, and λv should be appropriately designed.


## 3.1. Design of Repulsive Potential

The definition of repulsive potential is associated with the distance between the spacecraft and bounds. Usually, the repulsive potential becomes large when approaching the bounds and becomes small or near zero when moving away from bounds. The control accelerations, obtained through the negative gradient of the repulsive potential, will drive the spacecraft away from the bounds. In this study, Gaussian function is applied to represent the repulsive potential(23)Rdi=λr1exp⁡-12λr22di2i=1,2where d1=rmax-x2+y2+z2 is the distance between the servicer and upper bound, and d2=x2+y2+z2-rmin is the distance between the servicer and lower bound. The parameters λr1>0 and λr2>0 determine the amplitude and influence range of the repulsive potential, respectively. However, the repulsive potential (23) do not converge to zero at the desired position, which will deflect the balance point and reduce the control precision. Accordingly, the repulsive potential is innovatively modified as (24)Rd0,di=1-exp⁡-12λr02d02·λr1exp-12λr22di2i=1,2where d0=(x-xd)2+(y-yd)2+(z-zd)2 is the distance between the servicer and the desired position. It is noted that the repulsive potential (24) will converge to zero when reaching at the desired position. Contrasted with improvement of repulsive potential by introducing a quadratic function about the relative position between the spacecraft and desired point, discussed in [23, 32], the introduced term 1-exp⁡-1/2λr02d02∈0,1 has little influence on the amplitude and influence range of Gauss repulsive potential (23).In order to illustrate the design principle of repulsive potential (24), a similar potential in one-dimensional space is constructed as follows(25)y=1-exp⁡-12λr02x-x02·λr1exp⁡-12λr22x-x12x∈-10,10Analogously,x=x0 is the desired point, and x=x1 is the bound. Parameters in (25) are set to λr1=6×10-6, λr0=λr2=2, x0=5, and x1=0. The distribution of potential (25) is shown in Figure 8. Apparently, when x=x0 is satisfied, the potential converges to zero, which is a global minimum. When approaching the bound x1, the potential becomes larger. What’s more, when far away from the bound, the potential only has weak effect.Figure 8
Repulsive potential in one-dimensional space.


## 3.2. Design of Attractive Potential

The quadratic function and hyperbolic function have been widely used for the design of attractive potential. In this study, inspired by tangent function, a novel attractive potential about relative distance is designed as follows(26)Rd0=λp1d0arctan⁡λp2d0-12λp2In⁡1+λp22d02Similarly, in order to illustrate the design principle of attractive potential (26), a potential in one-dimensional space is constructed as follows(27)y=λp1xarctan⁡λp2x-12λp2In⁡1+λp2x2x∈-10,10The control acceleration, obtained through the negative gradient ofy, can be represented as(28)f=-∂y∂x=-λp1arctan⁡λp2xx∈-10,10The parameterλp1>0 determines the amplitude of acceleration and λp2>0 is a slope coefficient. Parameters in (27) and (28) are set to λp1=2.5×10-7 and λp2=100. The distributions of potential (27) and acceleration are shown in Figure 9.Figure 9
Attractive potential (a) and corresponding acceleration (b).
(a) (b)As observed in Figure9, the desired position x=0 is the global minimum in potential. When the object is not at the desired position, a reverse force with respect to its position will drive it to the desired position. It is noted that the amplitude of the acceleration changes little when the object is far away from the desired position, which is similar to bang-bang control law. Compared with attractive potential with quadratic function, this designed attractive potential can reduce the effect of relative distance on control acceleration. What’s more, when the object has a small offset with the desired position, a larger force will act on the object, which will improve the system’s robustness against the disturbing force.To ensure the smooth orbital transfer, an attractive potential about relative velocity with variable gain is designed, which will control the relative velocity between the servicer and desired position to converge to zero.(29)Rv=12λvvTvwhere λv>0 is the velocity damping coefficient defined as(30)λv=ελv0+1-ελv1ε=11+exp⁡-ςd0-awhere λv0>0 and λv1>0 are constants determining the upper and lower limits of the velocity damping coefficient, ε is a variable related to the relative distance, ς>0 is a coefficient, and a>0 is a threshold. The design scheme of the variable gain λv is shown in Figure 10 with λv0=1×10-3, λv1=5×10-2, ς=5, and a=1.Figure 10
Design scheme of velocity damping coefficient.


## 3.3. Control Law and Stability Proof

The system potential is defined as the summation of all above designed potentials, which can be represented as(31)R=Rd0+Rv+Rd0,d1+Rd0,d2Reconsidering the relative dynamics of GEO nearby flying, the shorter form of (16) can be written as(32)v˙=AX+gt+u+Δawhere X=(x,y,z,x˙,y˙,z˙)T is the state vector, and matrix A and function matrix g can be represented as(33)A=3n20002n0000-2n0000-n2000g=Δaxscos⁡f0+nt+Δayssin⁡f0+nt-Δaxssin⁡f0+nt+Δayscos⁡f0+ntΔazsThe control law is designed as follows(34)u=uinv+up+uv+urep=-AX-gt-∇xRd0-∇vRv-∇xRd0,d1-∇xRd0,d2where uinv=-AX-g(t) is the feedback compensation part, up=-∇xR(d0) is the attractive acceleration about relative position, uv=-∇vR(v) is the attractive acceleration about relative velocity, and urep=-∇xR(d0,d1)-∇xR(d0,d2) is the repulsive acceleration. The detailed description of up, uv, and urep can be represented as(35)up=-λp1arctan⁡λp2d0x-xdd0(36)uv=-λvv(37)urep=-λr1exp⁡-d122λr22d1λr22xx+λr1exp⁡-d022λr02exp⁡-d122λr22d1λr22xx-x-xdλr02+λr1exp⁡-d222λr22d2λr22xx-λr1exp⁡-d222λr22exp⁡-d022λr02d2λr22xx+x-xdλr02Theorem 1.
Given the relative dynamics of GEO nearby flying in (32) and neglecting the perturbation noise Δa, if the control law is considered as (34), the servicer will transfer to the desired position and the error of hovering will converge to zero.Proof.
Choosing a Lyapunov function candidate as(38)Ψ=Rd0+12vTv+Rd0,d1+Rd0,d2>0
the time derivative ofΨ is (39)Ψ˙=∇xRd0x˙+vTv˙+∇xRd0,d1x˙+∇xRd0,d2x˙
Substituting control law equation (34) and relative dynamics equation (32) into (39) yields(40)Ψ˙=-λvvTv≤0
It is noted thatΨ˙ is negative semidefinite, and if Ψ˙≡0 is satisfied, v≡0 and v˙≡0 are obtained. Substituting control law equation (34) into relative dynamics equation (32) (neglecting the noise), x=xd is obtained. According to the LaSalle theory, the system is asymptotically stable. This completes the Proof.Remark 2.
The APF method always suffers from local minimum problem, especially when applied to nonlinear dynamics system. In this study, the feedback compensation control is applied to make the relative dynamics a double-integrator model with additional noises. The designed system potential guarantees that the desired position is at the global minimum as discussed in the design of attractive potential. Once the desired position is at the global minimum, Theorem1 will guarantee that the servicer with any initial relative states will converge to its global minimum. And the designed attractive potential about relative distance in (26) will improve the system’s robustness against the disturbing force.Remark 3.
After the feedback compensation being applied, the negative gradient of repulsive potential and attractive potential, respectively, represent the repulsive acceleration and the attractive acceleration applied on the servicer. As discussed in (24), (26), and (29), both the potentials and their related control accelerations are bounded. However, parameters λr, λp, and λv have great influence on the size of repulsive acceleration and the attractive acceleration. Therein, in order to make sure the servicer do not cross the upper and limit bounds, the parameters λr, λp, and λv should be appropriately designed.


## 4. Numerical Simulation

To demonstrate the effectiveness of the proposed control law, numerical simulations realized in MATLAB are conducted in this section. High-fidelity perturbation environment discussed in Section2 is also adopted. Three cases of numerical simulations are discussed. All of the three cases refer to the same parameters of the servicer and target, as shown in Table 2. In the first case, the proposed control law in (34) is conducted. In order to evaluate the performance of improved design for repulsive potential in (24), repulsive potential in (23) is instead applied in the second case. In order to evaluate the performance of improved design for attractive potential, the quadratic attractive potential is instead applied in the third case.Table 2
Parameters of servicer and target for testing the control law.
Orbital parameters Parameters related to SRP servicer h s = 35785970  m, es=0, m s = 100  k g, CRs=1.4, As/ms=0.01 f s = 4.9 × 1 0 - 5  d e g, is=0deg target h t = 35786000  m, et=0, m t = 1000  k g, CRt=1.5, At/mt=0.02 f t = 0  d e g, it=0degAccordingly, the initial relative conditions arex0=(-30,35.73,0)m and v0=(0,3.3×10-3,0)m/s. The desired hovering position is xd=(30,-15.73,30)m. What’s more, a bounded space is constrained for servicer’s orbital transfer with rmin=30m and rmax=50m. The differential coefficient of SRP is DR=-1.6×10-2 and the scale coefficient κ is set to 0.1. This study focuses on the low-thrust maneuver, and the control acceleration is limited to 10-6m/s2 magnitude.


### 4.1. Case A

The proposed control law in (34) is conducted with the designed repulsive potential in (24) and attractive potential in (26), (29), and (30). The parameters of repulsive potential in (24) are set to λr0=2, λr1=6×10-6, and λr2=2. The parameters of attractive potential in (26) are set to λp1=2.5×10-7 and λp2=100. The parameters of attractive potential in (29) and (30) are set to λv0=2×10-4, λv1=5×10-3, ς=1, and a=0.1. The simulation results are shown in Figures 11–16.Figure 11
History of relative position in Case A.Figure 12
History of relative velocity in Case A.Figure 13
History of control input in Case A.Figure 14
History of repulsive control input in Case A.Figure 15
History of relative distance in Case A.Figure 16
History of transfer trajectory in Case A.In Figures15 and 16, it is observed that the servicer can maneuver to the desired position from the initial position within the bounded space. The histories of relative position and velocity are shown in Figures 11 and 12; it is observed that the time for orbital transfer is about 0.7 orbital period and the hovering precision is 10-4m magnitude. It is concluded that the designed control law is robust to noises. In Figure 13, it is observed that the upper limit of control input is approximately 10-6m/s2 magnitude. When the servicer approximates the target about 0.1 m, the designed attractive potential in (29) and (30) helps reduce the trajectory overshoot. Using the designed control law for orbital transfer and hovering within two orbital periods in Case A, the ΔV required is approximately 0.11 m/s. In Figure 14, the ur1=(urx1,ury1,urz1)T and ur2=(urx2,ury2,urz2)T are control accelerations produced by repulsive potential in (24), which are defined as follows(41)ur1=-λr1exp⁡-d122λr22d1λr22xxur2=λr1exp⁡-d022λr02exp⁡-d122λr22d1λr22xx-x-xdλr02where ur1 is produced by the Gauss repulsive potential, and ur2 is another part produced by the improved design for repulsive potential in order to eliminate the deflection from the desired position. It is observed that when hovering at the desired position, the force produced by the improved repulsive potential is zero.


### 4.2. Case B

As a comparison, control law with Gauss repulsive potential in (23) is also simulated and analyzed. The parameters of the control law are set the same with Case A. The simulation results are shown in Figures 17–20.Figure 17
History of relative distance in Case B.Figure 18
History of relative position in Case B.Figure 19
History of repulsive control input in Case B.Figure 20
History of position error in Case B.In Figures17 and 18, it is observed that the servicer can also maneuver to the desired position within the bounded space, and the history of transfer trajectory is similar to Case A. However, the hovering error is about 0.1 m, which is caused by the repulsive potential as shown in Figure 19. When the servicer arrives at the desired position, the Gauss repulsive potential still has an effect on the servicer, and then the deflection happens.


### 4.3. Case C

In order to illustrate the improved design for attractive potential, control law with the quadratic attractive potential is simulated and analyzed as a comparison. The quadratic attractive potential is designed as follows(42)Rx=12Kpx-xdTx-xdRv=12KvvTvThe repulsive potential in (24) is also adopted, and the corresponding control law is designed as follows(43)u=-AX-gt-∇xRx-∇vRv-∇xRd0,d1-∇xRd0,d2The parameters of repulsive potential are set the same as Case A. The parameters of attractive potential in (42) are set to Kp=5×10-9 and Kv=5×10-5. The simulation results are shown in Figures 21–24.Figure 21
History of relative distance in Case C.Figure 22
History of control input in Case C.Figure 23
History of relative position in Case C.Figure 24
History of position error in Case C.In Figures21 and 23, it is observed that the servicer can also maneuver to the desired position within the bounded space. As observed in Figure 22, the magnitude of control acceleration meets the requirements. Owing to the fact that the attractive control acceleration highly depends on the relative distance between the servicer and desired position, the parameter Kp should be set littler to ensure the bounded orbital transfer and the parameter Kv should be set larger to damp the relative velocity. The time for orbital transfer is longer than in Case A. What’s more, the servicer oscillates between the upper and lower bounds during orbital transfer, and the hovering precision is about 0.2 m.


### 4.4. Discussions

Through above simulations, it is noted that the proposed control law based on feedback compensation and artificial potential is feasible for orbital transfer and hovering. However, the artificial potential has great influence on the transfer trajectory and the final hovering precision. In Case A, the designed novel attractive and repulsive potential are adopted. As shown in Figures11–16, the transfer trajectory of the servicer relative to the target is smooth, and the hovering precision is 10-4m magnitude, which is acceptable for the spacecraft proximity missions.In Case B, the attractive potential is the same as Case A. However, the Gauss function with no improvement is applied as the repulsive potential. Owing to the fact that the Gauss function still has effect on the servicer when arriving at the desired position, the hovering position is about 0.1 m, which is apparently worse than in Case A. The simulation results illustrate that the improvement design for the repulsive potential is effective.In Case C, the repulsive potential is the same as Case A. However, the quadratic attractive potential is applied. Owing to the fact that the attractive control acceleration obtained from the quadratic attractive potential is influenced by the relative distance between the servicer and desired position, the smoothness of the transfer trajectory is difficult to guarantee. As can be seen from Figures21–24, the transfer trajectory has larger oscillations, and the hovering precision is about 0.2 m. It is evidently shown that the designed novel attractive potential has better performance than the quadratic attractive potential when conducting orbital transfer and hovering control.


## 4.1. Case A

The proposed control law in (34) is conducted with the designed repulsive potential in (24) and attractive potential in (26), (29), and (30). The parameters of repulsive potential in (24) are set to λr0=2, λr1=6×10-6, and λr2=2. The parameters of attractive potential in (26) are set to λp1=2.5×10-7 and λp2=100. The parameters of attractive potential in (29) and (30) are set to λv0=2×10-4, λv1=5×10-3, ς=1, and a=0.1. The simulation results are shown in Figures 11–16.Figure 11
History of relative position in Case A.Figure 12
History of relative velocity in Case A.Figure 13
History of control input in Case A.Figure 14
History of repulsive control input in Case A.Figure 15
History of relative distance in Case A.Figure 16
History of transfer trajectory in Case A.In Figures15 and 16, it is observed that the servicer can maneuver to the desired position from the initial position within the bounded space. The histories of relative position and velocity are shown in Figures 11 and 12; it is observed that the time for orbital transfer is about 0.7 orbital period and the hovering precision is 10-4m magnitude. It is concluded that the designed control law is robust to noises. In Figure 13, it is observed that the upper limit of control input is approximately 10-6m/s2 magnitude. When the servicer approximates the target about 0.1 m, the designed attractive potential in (29) and (30) helps reduce the trajectory overshoot. Using the designed control law for orbital transfer and hovering within two orbital periods in Case A, the ΔV required is approximately 0.11 m/s. In Figure 14, the ur1=(urx1,ury1,urz1)T and ur2=(urx2,ury2,urz2)T are control accelerations produced by repulsive potential in (24), which are defined as follows(41)ur1=-λr1exp⁡-d122λr22d1λr22xxur2=λr1exp⁡-d022λr02exp⁡-d122λr22d1λr22xx-x-xdλr02where ur1 is produced by the Gauss repulsive potential, and ur2 is another part produced by the improved design for repulsive potential in order to eliminate the deflection from the desired position. It is observed that when hovering at the desired position, the force produced by the improved repulsive potential is zero.


## 4.2. Case B

As a comparison, control law with Gauss repulsive potential in (23) is also simulated and analyzed. The parameters of the control law are set the same with Case A. The simulation results are shown in Figures 17–20.Figure 17
History of relative distance in Case B.Figure 18
History of relative position in Case B.Figure 19
History of repulsive control input in Case B.Figure 20
History of position error in Case B.In Figures17 and 18, it is observed that the servicer can also maneuver to the desired position within the bounded space, and the history of transfer trajectory is similar to Case A. However, the hovering error is about 0.1 m, which is caused by the repulsive potential as shown in Figure 19. When the servicer arrives at the desired position, the Gauss repulsive potential still has an effect on the servicer, and then the deflection happens.


## 4.3. Case C

In order to illustrate the improved design for attractive potential, control law with the quadratic attractive potential is simulated and analyzed as a comparison. The quadratic attractive potential is designed as follows(42)Rx=12Kpx-xdTx-xdRv=12KvvTvThe repulsive potential in (24) is also adopted, and the corresponding control law is designed as follows(43)u=-AX-gt-∇xRx-∇vRv-∇xRd0,d1-∇xRd0,d2The parameters of repulsive potential are set the same as Case A. The parameters of attractive potential in (42) are set to Kp=5×10-9 and Kv=5×10-5. The simulation results are shown in Figures 21–24.Figure 21
History of relative distance in Case C.Figure 22
History of control input in Case C.Figure 23
History of relative position in Case C.Figure 24
History of position error in Case C.In Figures21 and 23, it is observed that the servicer can also maneuver to the desired position within the bounded space. As observed in Figure 22, the magnitude of control acceleration meets the requirements. Owing to the fact that the attractive control acceleration highly depends on the relative distance between the servicer and desired position, the parameter Kp should be set littler to ensure the bounded orbital transfer and the parameter Kv should be set larger to damp the relative velocity. The time for orbital transfer is longer than in Case A. What’s more, the servicer oscillates between the upper and lower bounds during orbital transfer, and the hovering precision is about 0.2 m.


## 4.4. Discussions

Through above simulations, it is noted that the proposed control law based on feedback compensation and artificial potential is feasible for orbital transfer and hovering. However, the artificial potential has great influence on the transfer trajectory and the final hovering precision. In Case A, the designed novel attractive and repulsive potential are adopted. As shown in Figures11–16, the transfer trajectory of the servicer relative to the target is smooth, and the hovering precision is 10-4m magnitude, which is acceptable for the spacecraft proximity missions.In Case B, the attractive potential is the same as Case A. However, the Gauss function with no improvement is applied as the repulsive potential. Owing to the fact that the Gauss function still has effect on the servicer when arriving at the desired position, the hovering position is about 0.1 m, which is apparently worse than in Case A. The simulation results illustrate that the improvement design for the repulsive potential is effective.In Case C, the repulsive potential is the same as Case A. However, the quadratic attractive potential is applied. Owing to the fact that the attractive control acceleration obtained from the quadratic attractive potential is influenced by the relative distance between the servicer and desired position, the smoothness of the transfer trajectory is difficult to guarantee. As can be seen from Figures21–24, the transfer trajectory has larger oscillations, and the hovering precision is about 0.2 m. It is evidently shown that the designed novel attractive potential has better performance than the quadratic attractive potential when conducting orbital transfer and hovering control.


## 5. Conclusion

In this study, the GEO nearby flying is focused, which denotes that the servicing spacecraft approaches GEO targets within tens of meters. For instance, when the relative distance is between 10 m and 100 m, the differential spherical Earth gravity is approximately10-8~10-7m/s2 magnitude, and the differential J2 and third-body perturbation is approximately 10-11m/s2 magnitude, which are weak terms. However, the differential SRP perturbation may have similar order of magnitude with the differential spherical Earth gravity due to the differential coefficient of SRP. Therein, a relative dynamics of GEO nearby flying has been built with considering the perturbations. Furthermore, considering the differential SRP, an analytical solution to relative dynamics is obtained, which is an extension to CW equation’s solution. A feedback compensation and APF control law has been developed for bounded orbital transfer and high-precision hovering. The advantages of the improved design for repulsive potential and attractive potential have been discussed in detail. The stability of the system with the proposed control law is proved with Lyapunov theory. Numerical results illustrate the effectiveness and robustness of the proposed control law. In addition, owing to the fact that differential SRP has great influence on GEO nearby flying, the maneuver control with only using SRP should be focused in future, which is meaningful to long-term OOS for GEO targets.



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*Source: 6186283-2019-03-24.xml*