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, 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.
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 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: Section 2 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 as 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 with X-axis pointing toward the Vernal equinox, Z-axis aligned with the rotation axis and Y-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 with x-axis pointing radially outward from its orbit, y-axis perpendicular to x-axis along its direction of motion, and z-axis completing the right-handed coordinate system, as shown in Figure 1.

The motion of spacecraft in Earth orbit is governed by the vector differential equation.where and are the target and servicer’s position with respect to the center of Earth, respectively, is the geocentric gravitational constant, is the servicer’s control acceleration, and and are the perturbation forces per unit mass (accelerations) of target and servicer, respectively. For GEO spacecraft, the perturbation , third-body perturbation and , and SRP perturbation are mainly considered. It is assumed that the plane and the plane are coplanar, and is the orbital angle.
The relative position between servicer and target is defined as . Assuming that is the differential perturbation, combining with (1) and (2) yields
Differential Spherical Earth Gravity. The orbital eccentricity of GEO target is near zero. Considering the nearby flying, the differential spherical Earth gravity can be represented aswhere is the orbital mean angular velocity of the target, and and are the unit vectors of and , respectively. It is noted that and are satisfied, where denotes the Euclidean norm of a vector. Therein, the differential spherical Earth gravity of GEO nearby flying is approximately magnitude. According to (4), the numerical propagation of differential spherical Earth gravity in two absolute orbits (servicer and target) is shown in Figure 2.

Differential Perturbation. The most important nonspherical Earth gravity perturbation to be considered is the perturbation, which can be represented aswhere is the distance of a spacecraft with respect to the center of Earth, is the Earth radius, and is the geocentric latitude. The differential J2 perturbation can be represented as
Similarly, according to (5) and (6), the differential perturbation is shown in Figure 3. As observed in Figure 3, the differential perturbation of GEO nearby flying is approximately magnitude.

Differential Third-Body Perturbation. The perturbation due to the third body can be represented aswhere and are the Sun and Moon’s gravitational constants, respectively, and and are the position of Sun and Moon with respect to the center of Earth, respectively. The differential third-body perturbation can be represented aswhere and are the position of target with respect to the center of Sun and Moon, respectively, and and 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 magnitude.

Differential SRP Perturbation. Generally, the simplest model used to describe the SRP perturbation iswhere is the constant of the SRP, accounts for the mean reflectivity of the surface, A is the spacecraft cross-sectional area assumed constant, m is the mass of the spacecraft, is the Astronomical Unit, and the norm of 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 aswhere is a bounded noise produced by above assumption. It is considered that is a weak term with respect to , which is modeled aswhere is a random number between 0 and 1, and is a scale coefficient indicating the amount of noise. The modeled 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 aswhere is the differential coefficient which can be represented as . The magnitude of the differential SRP perturbation mainly depends on the product of and As for two orbital spacecraft, the 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 of , can be approximated as ,
(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 aswhere is modeled as a constant resolved in Inertial Frame according to above assumptions, which can be represented as
Reconsidering (3), (4), (6), (9), and (14), the relative dynamics of two spacecraft resolved in Inertial Frame can be written aswhere . According to above analysis, 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 where and are the relative position and velocity in the target’s LVLH Frame, is the initial orbital angle of the target, is the perturbation noise in the target’s LVLH Frame, and 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 target 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).where , , , , , and are the integral constants, which can be represented aswhere and 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 aswhereAs 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.
Accordingly, the differential coefficient of SRP is , and the scale coefficient 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.

As observed in Figure 5, 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 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 coefficient described in (12). The prediction error is defined aswhere denotes the prediction position through solutions to CW equations or relative dynamics of GEO nearby flying. The prediction error is shown in Figure 6.

As observed in Figure 6, 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 as , is constrained to ensure the relative distance with little divergence. What’s more, a lower limit of relative distance, denoted as , 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.

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 potentialwhere is the distance between the servicer and upper bound, and is the distance between the servicer and lower bound. The parameters and 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 where 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 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
Analogously, is the desired point, and is the bound. Parameters in (25) are set to , , , and . The distribution of potential (25) is shown in Figure 8. Apparently, when is satisfied, the potential converges to zero, which is a global minimum. When approaching the bound , the potential becomes larger. What’s more, when far away from the bound, the potential only has weak effect.

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
Similarly, in order to illustrate the design principle of attractive potential (26), a potential in one-dimensional space is constructed as follows
The control acceleration, obtained through the negative gradient of , can be represented as
The parameter determines the amplitude of acceleration and is a slope coefficient. Parameters in (27) and (28) are set to and . The distributions of potential (27) and acceleration are shown in Figure 9.

(a)

(b)
As observed in Figure 9, the desired position 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.where is the velocity damping coefficient defined aswhere and are constants determining the upper and lower limits of the velocity damping coefficient, is a variable related to the relative distance, is a coefficient, and is a threshold. The design scheme of the variable gain is shown in Figure 10 with , , , and .

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
Reconsidering the relative dynamics of GEO nearby flying, the shorter form of (16) can be written aswhere is the state vector, and matrix and function matrix can be represented as
The control law is designed as followswhere is the feedback compensation part, is the attractive acceleration about relative position, is the attractive acceleration about relative velocity, and is the repulsive acceleration. The detailed description of , , and can be represented as
Theorem 1. Given the relative dynamics of GEO nearby flying in (32) and neglecting the perturbation noise , 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 asthe time derivative of is Substituting control law equation (34) and relative dynamics equation (32) into (39) yieldsIt is noted that is negative semidefinite, and if is satisfied, and are obtained. Substituting control law equation (34) into relative dynamics equation (32) (neglecting the noise), 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, Theorem 1 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 , , and 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 , , and 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 Section 2 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.
Accordingly, the initial relative conditions are and . The desired hovering position is . What’s more, a bounded space is constrained for servicer’s orbital transfer with and . The differential coefficient of SRP is and the scale coefficient is set to 0.1. This study focuses on the low-thrust maneuver, and the control acceleration is limited to 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 , , and . The parameters of attractive potential in (26) are set to and . The parameters of attractive potential in (29) and (30) are set to , , , and . The simulation results are shown in Figures 11–16.






In Figures 15 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 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 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 required is approximately 0.11 m/s. In Figure 14, the and are control accelerations produced by repulsive potential in (24), which are defined as followswhere is produced by the Gauss repulsive potential, and 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.




In Figures 17 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
The repulsive potential in (24) is also adopted, and the corresponding control law is designed as follows
The parameters of repulsive potential are set the same as Case A. The parameters of attractive potential in (42) are set to and . The simulation results are shown in Figures 21–24.




In Figures 21 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 should be set littler to ensure the bounded orbital transfer and the parameter 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 Figures 11–16, the transfer trajectory of the servicer relative to the target is smooth, and the hovering precision is 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 Figures 21–24, 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 approximately magnitude, and the differential and third-body perturbation is approximately 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.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This paper was sponsored by the National Natural Science Foundation of China 11572168.