Fuel-Optimal In-Track Satellite Formation Trajectory with J₂ Perturbation Using Pontryagin Neural Networks

Abstract

Satellite formation flying faces significant challenges in maintaining its desired configurations due to various orbital perturbations, particularly in low-Earth-orbit environments. This paper presents a novel approach to generating fuel-optimal reference trajectories for in-track satellite formations by incorporating both the Earth’s oblateness ($J_2$ perturbation) and the inherent nonlinearity of the two-body problem. The resulting indirect optimal control problem is solved using Pontryagin Neural Networks (PoNNs). The proposed method transforms the conventional two-point boundary value problem into a mathematical programming problem, enabling the efficient computation of optimal trajectories. The effectiveness of our approach is validated through extensive numerical simulations at different inclinations of the chief satellite (0–90°) and cross-track separation distances (1–400 km), demonstrating significant reductions in annual fuel consumption compared to conventional approaches. The feasibility of these optimal trajectories is verified through closed-loop simulations using a PD controller, confirming their practical applicability in realistic mission scenarios. This research contributes to enhancing the long-term sustainability of satellite formation flying missions by optimizing fuel efficiency while maintaining precise formations.

1. Introduction

Formation flying satellites represent an advanced space technology paradigm wherein multiple satellites maintain precise positions relative to each other while operating as a coordinated system. This approach enables complex missions beyond the capabilities of single satellites, including synthetic aperture radar interferometry, distributed sensing, and large-scale space-based interferometry. The versatility of formation flying has proven valuable across a diverse range of applications, from Earth observation and space science to practical implementations in satellite navigation enhancement, ionospheric studies, and disaster monitoring [1].

The emergence of small satellite platforms, particularly CubeSats, has transformed the landscape of formation flying missions. Their modular architecture and cost-effective development have created new opportunities for flexible mission design. While multiple small satellites can provide enhanced mission reliability and improved observation capabilities compared to traditional single-satellite approaches, they face a critical challenge: limited fuel capacity. This constraint makes fuel-efficient control strategies essential for maintaining precise relative positions during long-term operations.

Significant amounts of research have addressed station-keeping techniques aimed at minimizing fuel consumption while maintaining stable spacecraft formations. These studies have extensively explored various methodologies, including robust control frameworks, adaptive trajectory planning, and collision avoidance strategies, to improve the reliability and efficiency of formation flying missions [2,3,4,5,6,7,8]. However, current research exhibits two notable limitations. First, most studies concentrate on formations within 50 km separation distances, leaving substantial uncertainty about control strategies for larger-scale formations. Second, tracking references for optimal control typically rely on approximated models, which undermine the effectiveness of fuel optimization under real operational conditions. While recent work has explored nonlinear dynamic solutions for formation flying references, these approaches have not adequately addressed $J_2$ perturbation effects, which dominate satellite dynamics in low Earth orbit [9,10].

Our research addresses these limitations by implementing optimal control techniques based on Xu and Wang’s high-fidelity $J_2$ nonlinear relative dynamics model [11]. Their model provides a precise representation of Earth’s gravitational field asymmetry without approximations, enabling the accurate prediction of long-term $J_2$ perturbation effects in low-Earth-orbit formation flying. To solve the optimal control problem, we employ a Physics-Informed Neural Network (PoNN) approach, which is specifically designed to handle indirect optimal control problems through Pontryagin’s Minimum Principle (PMP) [12]. The PoNN framework incorporates the Theory of Functional Connections (TFC) for efficient function interpolation [13] and utilizes the Extreme Learning Machine (ELM) methodology to simplify neural network training through Newton–Raphson methods [14]. This approach has demonstrated robust convergence in various aerospace applications, including celestial landing and orbital transfer problems [12,15,16,17,18].

This study focuses on the in-track formation flying adopted by leading satellite formation-flying companies such as HawkEye 360. By analyzing the Two-Line Element (TLE) data from these companies’ satellites, it was confirmed that they maintain an in-track formation. Based on this observation, this study addresses the problem of generating fuel-optimal reference trajectories for this type of formation. In-track formation refers to a configuration in which satellites fly in the same orbital plane while maintaining fixed separations in the along-track direction. Due to its relatively simple structure, this formation is easier to manage in terms of orbital maintenance and control and is widely adopted in actual satellite operations. This paper proposes a trajectory generation method that produces fuel-optimal reference trajectories while considering the orbital dynamics and constraints specific to in-track formations. Although the proposed method is not limited to in-track formations and can be generalized to other configurations, in-track formation is selected as the primary case study due to its practical relevance and ease of validation.

Our research examines formation flying dynamics across a broad range of parameters, analyzing cross-track separation distances from 1 km to 400 km and inclinations from 0° to 90°. Through a systematic comparison with traditional methods such as the Clohessy–Wiltshire equations, we evaluate the importance of incorporating $J_2$ perturbation in nonlinear dynamics models and establish quantitative criteria for optimal orbital parameter selection for long-term missions. This comprehensive analysis provides practical insights for the design of future formation flying missions, particularly regarding the relationships between formation size, orbital inclination, and fuel consumption optimization.

This paper is composed of three main chapters. Section 2 introduces a high-fidelity nonlinear relative dynamics model that incorporates $J_2$ perturbation to describe the motion of the chief and deputy satellites and explains the differences between this and conventional linear models. In Section 3, using this dynamics model, an optimal control method is proposed to generate fuel-optimal reference trajectories using the Pontryagin Neural Network (PoNN). Then, Section 4 verifies the performance of the proposed fuel-optimal trajectories through simulations and quantitatively analyzes the annual fuel consumption of various formation sizes and orbital inclinations.

2. Coordinate System of Formation Flying Satellites

2.1. Coordinate System of Formation Flying

Formation satellites consist of a chief satellite and a deputy satellite. Generally, the motion of the deputy satellite is described within the LVLH (Local Vertical Local Horizontal) coordinate frame, with the chief satellite at the origin. The basis vectors of the ECI (Earth-Centered Inertial) coordinate system are ($\widehat{X}$, $\widehat{Y}$, $\widehat{Z}$). The LVLH coordinate frame, as shown in Figure 1, is defined by the unit vectors ($\widehat{x}$, $\widehat{y}$, $\widehat{z}$), such that

$$\widehat{x}={\displaystyle \frac{\overrightarrow{\overline{r}}}{|\overrightarrow{\overline{r}}|}},\widehat{z}={\displaystyle \frac{\overrightarrow{h}}{|\overrightarrow{h}|}},\widehat{y}=\widehat{z}\times \widehat{x}$$

(1)

where $\overrightarrow{\overline{r}}$ is the position vector of the chief satellite and $\overrightarrow{h}$ is the orbital angular momentum vector, which is expressed as the cross product of the position and velocity vectors of the chief satellite.

2.2. High-Fidelity $J_2$ Nonlinear Relative Dynamics

In this paper, the relative motion equations of the chief satellite proposed by Xu and Wang (2008), which consider orbital eccentricity and $J_2$ perturbation, are used [11]. These equations form an accurate model that incorporates nonlinearity, eccentricity, and $J_2$ effects and are derived using Lagrangian mechanics without any simplifying approximations. Unlike the traditional Clohessy–Wiltshire equations, this model does not assume a circular orbit or neglect perturbations, and it is formulated independently of the right ascension of the ascending node using only a small number of physical parameters.

$$\begin{array}{cc}\hfill \ddot{x}& =2\dot{y}{\omega }_z-x(n^2-{\omega }_z^2)+y{\alpha }_z-z{\omega }_x{\omega }_z-(\zeta -\overline{\zeta })sin\left(i\right)sin\left(\theta \right)-r(n^2-{\overline{n}}^2)+u_x\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill \ddot{y}& =-2\dot{x}{\omega }_z+2\dot{z}{\omega }_x-x{\alpha }_z-y(n^2-{\omega }_z^2-{\omega }_x^2)+z{\alpha }_x-(\zeta -\overline{\zeta })sin\left(i\right)cos\left(\theta \right)+u_y\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill \ddot{z}& =-2\dot{y}{\omega }_x-x{\omega }_x{\omega }_z-y{\alpha }_x-z(n^2-{\omega }_x^2)-(\zeta -\overline{\zeta })cos\left(i\right)+u_z\hfill \end{array}$$

(4)

where x, y, and z represent the components of the vector $\overrightarrow{r}$. The control accelerations, which are denoted by $u_x$, $u_y$, and $u_z$, are defined such that i represents the inclination and $\theta$ denotes the argument of the latitude of the chief satellite. Additionally, (${\omega }_x$, ${\omega }_z$) and (${\alpha }_x$, ${\alpha }_z$) represent the angular velocity and angular acceleration of the LVLH coordinate frame, respectively, as shown in Equations (5)–(15).

$$\begin{array}{cc}\hfill {\omega }_x& =-{\displaystyle \frac{k_{J2}sin\left(2i\right)sin\left(\theta \right)}{h{\overline{r}}^3}}\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill {\omega }_z& ={\displaystyle \frac{h}{{\overline{r}}^2}}\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill {\alpha }_x& =-{\displaystyle \frac{k_{J2}sin\left(2i\right)cos\left(\theta \right)}{{\overline{r}}^5}}+{\displaystyle \frac{3v_r{k}_{J2}sin\left(2i\right)sin\left(\theta \right)}{{\overline{r}}^{4}h}}-{\displaystyle \frac{8k_{J2}^2{sin}^3\left(i\right)cos\left(i\right){sin}^2\left(\theta \right)cos\left(\theta \right)}{{\overline{r}}^6{h}^2}}\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {\alpha }_z& =-{\displaystyle \frac{2h{v}_r}{{\overline{r}}^3}}-{\displaystyle \frac{k_{J2}{sin}^2\left(i\right)sin\left(2\theta \right)}{{\overline{r}}^5}}\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill {\overline{n}}^2& ={\displaystyle \frac{\mu }{{\overline{r}}^3}}+{\displaystyle \frac{k_{J2}}{{\overline{r}}^5}}-{\displaystyle \frac{5k_{J2}{sin}^2\left(i\right){sin}^2\left(\theta \right)}{{\overline{r}}^5}}\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill n^2& ={\displaystyle \frac{\mu }{r^3}}+{\displaystyle \frac{k_{J2}}{r^5}}-{\displaystyle \frac{5k_{J2}{r}_Z^2}{r^7}}\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill \overline{\zeta }& ={\displaystyle \frac{2k_{J2}sin\left(i\right)sin\left(\theta \right)}{{\overline{r}}^4}}\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill \zeta & ={\displaystyle \frac{2k_{J2}{r}_Z}{r^5}}\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill k_{J2}& ={\displaystyle \frac{3J_2\mu R_e^2}{2}}\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill r& =\sqrt{{(\overline{r}+x)}^2+y^2+z^2}\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill r_Z& =(\overline{r}+x)sin\left(i\right)sin\left(\theta \right)+ysin\left(i\right)cos\left(\theta \right)+zcos\left(i\right)\hfill \end{array}$$

(15)

where $\mu$ is the gravitational parameter of the Earth, $J_2$ is the second zonal harmonic coefficient of the Earth, and $R_e$ is the Earth’s equatorial radius.

The equation of motion for the chief satellite is given in Equations (16)–(20).

$$\begin{array}{cc}\hfill \dot{\overline{r}}& =v_r\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill \dot{v_r}& =-{\displaystyle \frac{\mu }{{\overline{r}}^2}}+{\displaystyle \frac{h^2}{{\overline{r}}^3}}-{\displaystyle \frac{k_{J2}\left(1-3{sin}^2\left(i\right){sin}^2\left(\theta \right)\right)}{{\overline{r}}^4}}\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill \dot{h}& =-{\displaystyle \frac{k_{J2}{sin}^2\left(i\right)sin\left(2\theta \right)}{{\overline{r}}^3}}\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill \dot{\theta }& ={\displaystyle \frac{h}{{\overline{r}}^2}}+{\displaystyle \frac{2k_{J2}{cos}^2\left(i\right){sin}^2\left(\theta \right)}{h{\overline{r}}^3}}\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill \dot{i}& =-{\displaystyle \frac{k_{J2}sin\left(2i\right)sin\left(2\theta \right)}{2h{\overline{r}}^3}}\hfill \end{array}$$

(20)

where $\overline{r}$, $v_r$, h, $\theta$, and i denote the radial distance, the radial speed, the orbital angular momentum, the mean argument of latitude, and the inclination, respectively, of the chief satellite.

3. Fuel-Optimal In-Track Formation Using Pontryagin Neural Networks

In-track formation is a configuration derived from the analytical solution of the CW (Clohessy–Wiltshire) equations [19]. It oscillates in the cross-track direction in LVLH coordinates, as shown in Figure 2. The in-track formation, which is based on the CW equation, is expressed by Equation (21).

$$\begin{array}{cc}\hfill x\left(t\right)& =0,\hfill \\ \hfill y\left(t\right)& =0,\hfill \\ \hfill z\left(t\right)& =Lcos\left({\omega }_{\mathrm{orbit}}t\right)\hfill \end{array}$$

(21)

where L is the maximum cross-track separation distance between the chief and deputy satellite and ${\omega }_{\mathrm{orbit}}$ is the circular orbital angular velocity of the chief satellite, which is defined as ${\omega }_{\mathrm{orbit}}=\sqrt{\mu /|\overrightarrow{\overline{r}}{|}^3}$.

An in-track formation based on the CW equations faces limitations in its practical implementation because it relies on linearized orbital dynamics. As shown in Figure 3a, directly applying the initial conditions derived from the CW equations causes the deputy satellite to drift in the $-y$ direction. While adjusting the altitude-related initial offset $x_0$ can eliminate this drift under the two-body assumption, once $J_2$ perturbations are taken into account, the formation geometry still becomes twisted, as illustrated in Figure 3b.

These results indicate that the classical CW equations are incomplete due to linearization and the omission of $J_2$ perturbations. While the CW equations offer simplicity for control logic, relying on them as a guidance trajectory for continuous-thrust control is likely to lead to substantial fuel consumption.

This study proposes a new in-track formation strategy that considers both the nonlinearity of orbital dynamics and $J_2$ perturbation effects. To overcome the limitations of conventional approaches based on the CW equations, we aim to develop a formation flying strategy that satisfies the following key requirements:

• The deputy satellite, relative to the chief satellite, shall perform a periodic oscillatory motion in the cross-track direction, similar to that in a formation based on the CW equations.
• Both the chief and deputy satellites shall incorporate nonlinear orbital effects and $J_2$ perturbations.
• If fuel consumption is necessary for formation maintenance, it must be minimized.

To satisfy these requirements, this study presents a solution that employs the high-fidelity nonlinear relative dynamics model proposed by Xu and Wang [11] and an optimal control method, aiming to achieve a fuel-minimized in-track formation. Specifically, we transform the optimal control problem into a two-point boundary value problem (TPBVP) via Pontryagin’s Minimum Principle (PMP) and solve it using a Pontryagin Neural Network (PoNN).

Fuel-Optimal In-Track Formation Through High-Fidelity $J_2$ Nonlinear Dynamics and Pontryagin Neural Network

The optimal control problem of minimizing the deputy satellite’s fuel consumption is formulated as shown in Equation (22).

$$minJ={\displaystyle \frac{1}{2}}{\int }_{t_0}^{t_f}{u}^{T}udt$$

(22)

To achieve in-track formation, the deputy satellite’s relative position and velocity must form a periodic function, meaning its initial and final states must be identical. Therefore, as shown in Equation (23), the deputy satellite’s initial and final position and velocity have been set accordingly. The chief and deputy satellites have different orbital inclinations, causing an angular rate difference that can induce drift in the along-track direction. To compensate for this drift, the altitude-related initial offset $x_0$ for the deputy satellite is not explicitly defined. The constraints are as follows:

$$\dot{\xi }=f\left(\xi \right),$$

$$x\left(t_0\right)=x\left(t_f\right),y\left(t_0\right)=y\left(t_f\right)=0,z\left(t_0\right)=z\left(t_f\right)=L$$

$$v_x\left(t_0\right)=v_x\left(t_f\right),v_y\left(t_0\right)=v_y\left(t_f\right),v_z\left(t_0\right)=v_z\left(t_f\right)$$

(23)

$$\overline{r}\left(t_0\right)=\overline{r}\left(t_f\right)={\overline{r}}_0,v_r\left(t_0\right)=0,h\left(t_0\right)=h_0$$

$$\theta \left(t_0\right)={\theta }_0,i\left(t_0\right)=i_0$$

where $\xi$ represents the state of the satellite formation flying system, which is defined as

$$\xi ={[x,y,z,v_x,v_y,v_z,\overline{r},v_r,h,\theta ,i]}^T$$

(24)

and u is the control input vector of the deputy satellite, which is defined as

$$u={[u_x,u_y,u_z]}^T$$

(25)

The Hamiltonian for this optimal control problem is given as follows [20]:

$$H={\displaystyle \frac{1}{2}}{u}^{T}u+{\lambda }^{T}f\left(\xi \right)$$

(26)

where $\lambda$ is the costate vector, which is defined as

$$\lambda ={[{\lambda }_x,{\lambda }_y,{\lambda }_z,{\lambda }_{v_x},{\lambda }_{v_y},{\lambda }_{v_z},{\lambda }_{\overline{r}},{\lambda }_{v_r},{\lambda }_h,{\lambda }_{\theta },{\lambda }_i]}^T$$

(27)

According to the Pontryagin Minimum/Maximum Principle (PMP), the optimal control input must satisfy the optimality condition, which is given as follows:

$${\displaystyle \frac{\partial H}{\partial u}}=0$$

(28)

The optimal control input determined by the optimality condition is determined using the costate associated with the deputy satellite’s velocity, as follows:

$$u=-{\lambda }_v=-\left[\begin{array}{c}{\lambda }_{v_x}\\ {\lambda }_{v_y}\\ {\lambda }_{v_z}\end{array}\right]$$

(29)

The optimal control input derived from Equation (29) is substituted into Equation (26). The substituted Hamiltonian is given by Equation (30):

$$\overline{H}={\displaystyle \frac{1}{2}}{\lambda }_v^T{\lambda }_v+{\lambda }^T\overline{f}\left(\xi \right)$$

(30)

where $\overline{f}$ is the dynamics into which Equation (29) is substituted.The necessary condition for obtaining the optimal control input based on the Pontryagin Minimum Principle (PMP) is given as follows:

$$\dot{\xi }-{\displaystyle \frac{\partial \overline{H}}{\partial \lambda }}=0$$

(31)

$$\dot{\lambda }+{\displaystyle \frac{\partial \overline{H}}{\partial \xi }}=0.$$

(32)

The partial differentiation of the necessary first-order conditions for the state and costate is very complicated. Therefore, it was derived using the `Symbolic Math Toolbox’ in MATLAB R2024a.

Additionally, the transversality condition, which specifies the optimal requirements for the undefined initial and final states, as well as the final time, is given as follows:

$${\lambda }_0=-{\displaystyle \frac{\partial \mathcal{J}}{\partial {\xi }_0}},{\lambda }_f={\displaystyle \frac{\partial \mathcal{J}}{\partial {\xi }_f}}$$

(33)

$$H_f=-{\displaystyle \frac{\partial \mathcal{J}}{\partial t_f}}$$

(34)

The result of the transversality conditions is expressed as follows:

$${\lambda }_{x_0}={\lambda }_{x_f}={\lambda }_{v_{x_0}}={\lambda }_{v_{x_f}}={\lambda }_{v_{y_0}}={\lambda }_{v_{y_f}}={\lambda }_{v_{z_0}}={\lambda }_{v_{z_f}}=0$$

(35)

$${\lambda }_{v_{rf}}={\lambda }_{h_f}={\lambda }_{{\theta }_f}={\lambda }_{i_f}=0$$

(36)

$${\overline{H}}_f=0$$

(37)

Through the PMP, the optimal control problem is transformed into a two-point boundary value problem (TPBVP), which comprises ordinary differential equations (ODEs) from the necessary conditions and the boundary conditions (BCs) defined by the initial and final states and the transversality condition.

In this study, a PoNN (Pontryagin Neural Network) is employed to solve the TPBVP [12]. Within the PoNN framework, the state and costate are expressed as parameterized functions of time. They are represented as linear combinations of an artificial neural network that takes the parameterized time as input, and the Theory of Functional Connections (TFC) is utilized to enforce boundary values. The state and costate represented through the neural network and TFC are given as follows:

$${\widehat{\psi }}_j=C_j^T\sigma (W\tau +B)+{\alpha }_j+{\beta }_j\tau$$

(38)

where ${\widehat{\psi }}_j$ is the j-th element of $\widehat{\psi }$, which is a concatenation of the state vector and the costate vector ($\widehat{\psi }=[\widehat{\xi };\widehat{\lambda }]$). $C_j$ is the coefficient of the linear combination, $\sigma$ is the activation function, W and B denote the fixed weights and bias of the neural network, and $\tau$ is the independent variable defined to account for the unspecified final time $t_f$, as described in Equation (40), where c is the mapping coefficient.

$$\tau ={\tau }_0+c(t-t_0)$$

(39)

Additionally, ${\alpha }_j$ and ${\beta }_j\tau$ are terms derived through TFC that enforce the boundary conditions. They are applied differently depending on the type of boundary condition present. The ${\alpha }_j$ and ${\beta }_j$ for each type of boundary condition are as follows:

• BC Type 1: When only initial or final conditions are given.

  $${\alpha }_j=y_{0,j}-C_j^T\sigma (W{\tau }_0+B)$$

  (40)

  $${\beta }_j=0$$

  (41)
• BC Type 2: When the initial condition is given and the final condition is the same as the initial condition.

  $${\alpha }_j={\displaystyle \frac{{\tau }_f{y}_{0,j}-{\tau }_0{y}_{f,j}-{\tau }_f{C}_j^T\sigma (W{\tau }_0+B)+{\tau }_0{C}_j^T\sigma (W{\tau }_f+B)}{{\tau }_0-{\tau }_f}}$$

  (42)

  $${\beta }_j={\displaystyle \frac{y_{0,j}-y_{f,j}-C_j^T\sigma (W{\tau }_0+B)+C_j^T\sigma (W{\tau }_f+B)}{{\tau }_0-{\tau }_f}}$$

  (43)
• BC Type 3: When neither the initial nor final conditions are given, but their values are the same.

  $${\alpha }_j=0$$

  (44)

  $${\beta }_j={\displaystyle \frac{C_j^T\sigma (W{\tau }_f+B)-C_j^T\sigma (W{\tau }_0+B)}{{\tau }_0-{\tau }_f}}$$

  (45)
• BC Type 4: When neither the initial nor final conditions are given.

  $${\alpha }_j=0$$

  (46)

  $${\beta }_j=0$$

  (47)

The derivative of $\widehat{\psi }$ with respect to time can be expressed as follows, using the chain rule:

$${\displaystyle \frac{d\widehat{\psi }}{dt}}={\displaystyle \frac{d\widehat{\psi }}{d\tau }}{\displaystyle \frac{d\tau }{dt}}=c{\displaystyle \frac{d\widehat{\psi }}{d\tau }}$$

(48)

The expressions for the state and costate used in this study, as well as their time derivatives, are detailed in Appendix A.

The necessary conditions at a specific time ${\tau }_p$, which are derived from the state and costate expressed through the neural network and TFC, are given as follows:

$$L_p(\Gamma )={\left[{\left(\dot{\xi }-{\displaystyle \frac{\partial \overline{H}}{\partial \lambda }}\right)}^T,{\left(\dot{\lambda }+{\displaystyle \frac{\partial \overline{H}}{\partial \xi }}\right)}^T\right]}_{\tau ={\tau }_p}$$

(49)

where ${\tau }_p$ is the collocation point, such that

$${\tau }_p={\tau }_0+p·{\displaystyle \frac{{\tau }_f-{\tau }_0}{N_{\tau }}},\mathrm{for}p=0,1,2,\cdots ,N_{\tau },{\tau }_{N_{\tau }}={\tau }_f$$

(50)

where ${\tau }_0=-1$, ${\tau }_f=1$, and $\Gamma$ is the learnable parameter defined in Equation (52).

$$\begin{array}{cc}\hfill \Gamma =& \left[C_x^T,C_y^T,C_z^T,C_{v_x}^T,C_{v_y}^T,C_{v_z}^T,C_{\overline{r}}^T,C_{v_r}^T,C_h^T,C_{\theta }^T,C_i^T,\right.\hfill \\ \hfill & \left.C_{{\lambda }_x}^T,C_{{\lambda }_y}^T,C_{{\lambda }_z}^T,C_{{\lambda }_{v_x}}^T,C_{{\lambda }_{v_y}}^T,C_{{\lambda }_{v_z}}^T,C_{{\lambda }_{\overline{r}}}^T,C_{{\lambda }_{v_r}}^T,C_{{\lambda }_h}^T,C_{{\lambda }_{\theta }}^T,C_{{\lambda }_i}^T,c\right]\hfill \end{array}$$

(51)

The training loss vector, formed by concatenating the necessary conditions at all collocation points with the transversality condition at the final time, is as follows:

$$L(\Gamma )={\left[L_1,L_2,...,L_{N_{\tau }},{\overline{H}}_f\right]}^T$$

(52)

The solution to the optimal control problem is obtained by computing the linear combination coefficients and the mapping coefficients through the training of the loss vector. The learnable parameter $\Gamma$ is optimized using the Newton–Raphson method, which is conducted as follows [21]:

$${\Gamma }_{q+1}={\Gamma }_q+\Delta {\Gamma }_q$$

(53)

$$\Delta {\Gamma }_q=-{\left(J{\left({\Gamma }_q\right)}^{T}J\left({\Gamma }_q\right)\right)}^{-1}J{\left({\Gamma }_q\right)}^{T}L\left({\Gamma }_q\right)$$

(54)

where J is the Jacobian matrix of L with respect to $\Gamma$. The Newton–Raphson method is repeated until the 2-norm value of L becomes smaller than the threshold:

$$\Vert L\left({\Gamma }_q\right){\Vert }_2<ϵ$$

(55)

The architecture of the PoNN is shown in Figure 4.

The objective of this study is to compute a relatively fuel-optimal trajectory for formation flying spacecraft. The proposed formation flying scenario is designed under the assumption that continuous thrust is available, and under this condition, an ideal trajectory that minimizes fuel consumption exists. While continuous thrust may affect satellite systems in real operational environments, this study focuses on fuel optimization to derive its trajectories. Based on an ideally regulated thrust model, the proposed trajectory offers a theoretical lower bound on fuel consumption and can serve as a reference baseline for future practical applications.

4. Simulations

4.1. Simulation Conditions

Using the LVLH coordinate system, the deputy satellite’s initial position is set such that x is a free variable, $y=0$, and z ranges from 1 km to 400 km for the simulation. Its initial velocity is also designated as a free variable. The chief satellite is positioned at an altitude of 500 km, with an initial radial velocity of 0 and an initial angular momentum of $\sqrt{{\overline{r}}_0\mu }$. The initial argument of the latitude is set to 0, and the orbital inclination is varied from $0^{\circ }$ to ${90}^{\circ }$ in the simulation.

In our PoNN framework, the neural network is configured with a single hidden layer. Although multiple hidden layers could be used, increasing the number of hidden layers complicates the automatic differentiation process, consequently extending the computational time required to solve the TPBVP. Therefore, consistent with other PoNN studies, we selected a single hidden layer structure to achieve more efficient computations [12]. The sine function was adopted as the activation function based on existing research on implicit neural representation. Such studies have demonstrated that periodic activation functions effectively map input coordinates (in this study, time) to outputs (state and costate) [22]. Initial weights and biases were randomly initialized within the range of $[-2\pi ,2\pi ]$, guided by convergence experiments. After progressively expanding the sampling range from $[-1,1]$ to $[-4\pi ,4\pi ]$ and comparing convergence behaviors, the $[-2\pi ,2\pi ]$ interval was found to yield the optimal convergence. For the initial values of $\Gamma$, the coefficients associated with the deputy satellite $(C_x,C_y,C_z,C_{v_x},C_{v_y},C_{v_z})$ were set to zero, while those related to the chief satellite $(C_{\overline{r}},C_{v_r},C_h,C_{\theta },C_i)$ were determined to satisfy the numerically propagated initial conditions. Additionally, the initial mapping coefficient was based on the orbital period of a circular orbit at an altitude of 500 km.

To evaluate the performance of the proposed fuel-optimal trajectory, a control simulation was conducted in which a PD controller tracked the reference trajectory. The configuration of the PD controller is described as follows [23]:

$$\begin{array}{cc}\hfill u_{\mathrm{con}}& =Ke\hfill \\ \hfill K& =\left[\begin{array}{cccccc}{k}_p& 0& 0& k_d& 0& 0\\ 0& k_p& 0& 0& k_d& 0\\ 0& 0& k_p& 0& 0& k_d\end{array}\right]\hfill \end{array}$$

(56)

where $k_p$ is the proportional gain, set to 100; $k_d$ is the derivative gain, set to 100; and e is the error defined as the difference between the deputy satellite’s reference position and velocity and its actual position and velocity.

The simulation loop used to verify the performance of the proposed fuel-optimal trajectory is shown in Figure 5. In this simulation, the orbits of both the chief satellite and the deputy satellite are propagated in the ECI coordinate system.

The orbital equations of the chief and deputy satellites in the ECI coordinate system are given as follows, and their orbit propagation is performed using the Runge–Kutta 45 numerical integration method [24]:

$${\ddot{\overrightarrow{R}}}_c=-\nabla V_c+{\overrightarrow{a}}_{c,drag}$$

(57)

$${\ddot{\overrightarrow{R}}}_d=-\nabla V_d+{\overrightarrow{a}}_{d,drag}+\overrightarrow{U}$$

(58)

$$V_c={\displaystyle \frac{\mu }{R_c}}\left[1+\sum _{n=2}^N{\left({\displaystyle \frac{R_e}{R_c}}\right)}^n\sum _{m=0}^n\left(C_{nm}cos\left(m{\lambda }_c\right)+S_{nm}sin\left(m{\lambda }_c\right)\right)P_{nm}(sin{\varphi }_c)\right]$$

(59)

$$V_d={\displaystyle \frac{\mu }{R_d}}\left[1+\sum _{n=2}^N{\left({\displaystyle \frac{R_e}{R_d}}\right)}^n\sum _{m=0}^n\left(C_{nm}cos\left(m{\lambda }_d\right)+S_{nm}sin\left(m{\lambda }_d\right)\right)P_{nm}(sin{\varphi }_d)\right]$$

(60)

$${\overrightarrow{a}}_{c,drag}=-{\displaystyle \frac{1}{2}}{C}_D{\displaystyle \frac{A}{m}}\rho ∥{\overrightarrow{v}}_{c,rel}∥{\overrightarrow{v}}_{c,rel}$$

(61)

$${\overrightarrow{a}}_{d,drag}=-{\displaystyle \frac{1}{2}}{C}_D{\displaystyle \frac{A}{m}}\rho ∥{\overrightarrow{v}}_{d,rel}∥{\overrightarrow{v}}_{d,rel}$$

(62)

${\overrightarrow{R}}_c$ represents the position of the chief satellite in the ECI coordinate system, with $R_c=∥{\overrightarrow{R}}_c∥$. Likewise, ${\overrightarrow{R}}_d$ represents the position of the deputy satellite, with $R_d=∥{\overrightarrow{R}}_d∥$. The parameters ${\lambda }_c$ and ${\lambda }_d$ represent the longitudes of the chief and deputy satellites, respectively, while ${\varphi }_c$ and ${\varphi }_d$ denote their corresponding latitudes. These quantities are used to evaluate the spherical harmonic gravity potential at the positions of each satellite within the Earth-centered–Earth-fixed (ECEF) reference frame. The relative velocities ${\overrightarrow{v}}_{c,\mathrm{rel}}$ and ${\overrightarrow{v}}_{d,\mathrm{rel}}$ denote the velocities of the chief and deputy satellites with respect to the rotating atmosphere and are obtained by subtracting the velocity of the atmosphere (due to the Earth’s rotation) from their inertial velocities. These vectors are used in the computation of atmospheric drag acceleration. $\overrightarrow{U}$ denotes the control accelerations of the deputy satellite in the ECI coordinate system. The deputy satellite’s control acceleration is determined using a PD controller. The Earth’s equatorial radius, gravitational parameter, and spherical harmonic coefficients adopted were based on the EGM2008 gravity model. The simulation time step is $\Delta t=1\mathrm{s}$.

4.2. Feasibility Verification of Optimal Control Problem

In this chapter, we discuss the feasibility of 1 km and 400 km fuel-optimal formations, using a chief satellite with a 45° inclination. To examine the feasibility of the proposed fuel-optimal control approach for formation-flying satellites, we gradually increased the number of neurons $N_{\sigma }$ and the number of collocation points $N_{\tau }$, analyzing the convergence characteristics of the cost and final time.

• Case 1: L = 1 km

As shown in

Table 1. Performance summary for $L=1\mathrm{km}$ based on the number of artificial neural networks and number of collocation points used.

PoNN					Shooting
$N_{\sigma }$	$N_{\tau }$	$\mathcal{J}$	$t_f$ (min.)	Cal. Time	$N_s$	$\mathcal{J}$	$t_f$ (min.)	Cal. Time
8	8	$5.1178\times {10}^{-12}$	94.7431	54.13	$1\times {10}^4$	$1.0257\times {10}^{-10}$	94.2352	168.15
10	10	$5.6256\times {10}^{-12}$	94.6528	58.20	$2\times {10}^4$	$1.0123\times {10}^{-16}$	94.2347	273.57
12	12	$1.4589\times {10}^{-14}$	94.2230	69.03	$3\times {10}^4$	$1.0090\times {10}^{-16}$	94.2343	400.88
15	15	$1.0173\times {10}^{-16}$	94.2345	75.62	$4\times {10}^4$	$0.9967\times {10}^{-16}$	94.2340	537.52
20	20	$0.9967\times {10}^{-16}$	94.2339	84.24	$5\times {10}^4$	$0.9951\times {10}^{-16}$	94.2341	710.24

, in the case of an amplitude of 1 km (L = 1 km), the cost converged to approximately $1\times {10}^{-16}$ when around 15 neurons and 15 collocation points were used, while the final time converged to approximately 94.23 min.

With $N_{\sigma }$ and $N_{\tau }$ set to 15, the optimal reference trajectory shows negligible displacement and velocity in both its x and y components. The z component exhibits a sinusoidal pattern consistent with the CW equations, and the control inputs remain near zero, as shown in Figure 6. A comparison between the PoNN method and the Shooting method reveals that while the Shooting method shows a high convergence rate when it has many propagation nodes ($N_s$), it requires a significantly longer computation time. In contrast, the PoNN method was found to generate an optimal trajectory quickly, in about 80 s.

• Case 2: L = 400 km

In the case of an maximum separation distance of 400 km, as shown in

Table 2. Performance summary for $L=400\mathrm{km}$ based on the number of artificial neural networks and number of collocation points used.

PoNN					Shooting
$N_{\sigma }$	$N_{\tau }$	$\mathcal{J}$	$t_f$ (min.)	Cal. Time	$N_s$	$\mathcal{J}$	$t_f$ (min.)	Cal. Time
8	8	$6.2249\times {10}^{-5}$	94.8754	39.98	$1\times {10}^4$	$2.9980\times {10}^{-10}$	94.2359	96.74
10	10	$1.4381\times {10}^{-8}$	94.3217	42.88	$2\times {10}^4$	$2.7665\times {10}^{-10}$	94.2356	178.34
12	12	$1.6117\times {10}^{-9}$	94.2405	44.81	$3\times {10}^4$	$2.7263\times {10}^{-10}$	94.2357	244.93
15	15	$2.8903\times {10}^{-10}$	94.2382	52.49	$4\times {10}^4$	$2.7171\times {10}^{-10}$	94.2353	298.90
20	20	$2.8433\times {10}^{-10}$	94.2364	61.67	$5\times {10}^4$	$2.7158\times {10}^{-10}$	94.2364	401.67

, the cost converged to approximately $2.8\times {10}^{-10}$ with 15 neurons and 15 collocation points, while the final time converged to 94.23 min.

When $N_{\sigma }$ and $N_{\tau }$ are set to 15, the displacement and velocity in the z direction of the optimal reference trajectory exhibit a sinusoidal pattern, as shown in Figure 7. Unlike the 1 km formation case, significant components appear in both the x and y directions, and the control inputs are primarily concentrated in the z direction. For a formation separation of 400 km, a comparison between the PoNN method and the Shooting method shows that similar to the 1 km case, the Shooting method exhibits a high convergence rate when using many propagation nodes, but it requires a significantly longer computation time.

This study demonstrates the stable convergence of the PoNN-based, fuel-optimal trajectory generation method for in-track formation satellites under various amplitude conditions, showing that it successfully derives periodic solutions that minimize fuel consumption. A notable finding is that as the formation amplitude increases, the dynamic impact on the altitude (x) and along-track (y) directions becomes increasingly significant. This demonstrates the necessity of control strategies that consider not only the in-track direction but also fully integrated three-dimensional motion when used for large-scale formation satellites.

4.3. Performance Review of Optimal Reference Trajectory for In-Track Formation

In this section, we discuss the annual performance of the proposed fuel-optimal reference trajectory. First, we compare the performance of the CW equation-based reference and the fuel-optimal reference by analyzing their annual $\Delta V$ requirements and fuel consumption for formation distances ranging from 1 km to 50 km, with chief satellite inclinations of 0°, 45°, and 90°. The results of this comparison are shown in Figure 8, where the usage mass ratio $\eta$ is defined as $\eta =1-e^{-{\displaystyle \frac{\Delta V}{I_{sp}{g}_0}}}$, $I_{sp}=300$ s is the specific impulse, and $g_0=9.80665$ m/s² is standard gravitational acceleration.

Before conducting the annual fuel consumption analysis, we verified whether the PD controller could reliably track the reference trajectory. To this end, we compared the tracking errors in both the position and velocity for formation separations of 1 km and 400 km under a chief satellite inclination of 45° (Figure 9). The results confirmed that the PD controller exhibited stable behavior, with tracking errors remaining close to zero, thereby ensuring accurate reference-following throughout the simulation.

As shown in Figure 10, while the conventional CW equation-based method requires a significantly high annual $\Delta V$, the proposed optimization method achieves remarkably low $\Delta V$ requirements. The fuel mass ratio calculated based on $\Delta V$ shows that the CW method’s fuel consumption increases dramatically with the formation distance, reaching up to 100%, whereas the proposed method maintains extremely low fuel consumption, nearly 0%, when using continuous control thrust for formation maintenance.

Next, we broaden our analysis to examine the impact of $J_2$ perturbation across a wider range of orbital parameters. This investigation provides a comprehensive assessment of the annual $\Delta V$ requirements for orbital inclinations from 0° to 90° and chief–deputy separation distances from 0 km to 400 km, enabling a systematic comparison of fuel consumption characteristics under different conditions. Figure 11 illustrates both the computed annual $\Delta V$ requirements and the corresponding fuel mass ratio over a representative 5-year operational period for formation-flying satellites.

From our analysis of the relationship between orbital inclination and $Z_0$ (formation size), we observed distinct patterns in fuel consumption, which have been illustrated in the two contour plots above. The left plot presents the annual $\Delta V$ (m/s), and the right plot shows the fuel mass ratio (%) over a 5-year period.

The annual $\Delta V$ consumption and fuel mass ratio vary significantly with orbital inclination and L. At low orbital inclinations (10–${30}^{\circ }$) and high inclinations (70–${90}^{\circ }$), fuel consumption remains relatively modest as L increases. However, near a ${50}^{\circ }$ inclination, fuel consumption escalates dramatically with an increasing L. As L approaches 400 km, the annual $\Delta V$ requirement spikes to approximately 500 m/s, translating to a substantial fuel mass ratio of about 8% over a typical 5-year operational period.

These findings highlight the critical role of orbital inclination in the design of large satellite formation flight missions. The data clearly show that fuel consumption depends on both formation size and inclination, indicating that the careful selection of inclination is essential for minimizing operational costs and extending mission lifetimes. In particular, our results show a significant increase in $\Delta V$ requirements near the 50° inclination, which can be attributed to the inclination-dependent nature of the $J_2$ perturbation. The $J_2$ effect causes a gradual rotation of the orbital planes (nodal precession), and at mid-range inclinations even small differences in orbital parameters (e.g., altitude or inclination) result in noticeable variations in the precession rate. This leads to a divergence in the orbital planes of the satellites over time, a phenomenon known as differential nodal drift. In larger formations, where the initial separation between satellites is greater, this divergence becomes more pronounced, requiring more fuel to maintain the intended formation geometry. This explains the observed increase in $\Delta V$ in our simulations around 50°, as this demonstrates how the impact of $J_2$ perturbations is amplified when combined with formation size and inclination.

Another notable feature in the contour plots is the red line indicating the points of maximum fuel consumption. This line represents the orbital inclination at which maximum fuel consumption occurs for each L value. Its near-horizontal position around ${50}^{\circ }$ indicates that the maximum fuel consumption does not vary markedly with L. In particular, even as L grows from 50 km to 400 km, the maximum fuel consumption point stays between ${48}^{\circ }$ and ${52}^{\circ }$. This suggests that this inclination region demands the most fuel for orbit maintenance, regardless of L. Consequently, the line serves as a crucial indicator of which inclination range to avoid during mission design.

5. Conclusions

This study presents a method for generating fuel-optimal reference trajectories for satellite formation flying by combining Pontryagin Neural Networks (PoNNs) with a high-fidelity $J_2$ nonlinear relative dynamics model. The main conclusions of our work are as follows below.

First, the PoNN-based trajectory generation method demonstrates both stability and reliability. Using a model with 15 neurons and 15 collocation points, stable convergence was achieved across a range of formation sizes, from 1 km to 400 km, with the final time consistently converging to approximately 94.23 min. This indicates that the proposed method can produce stable periodic orbits while minimizing fuel consumption under various operating conditions, which is expected to significantly improve the reliability of trajectory planning in actual missions.

Second, this study clearly identifies that changes in motion characteristics are based on formation size. In small formations (1 km), motion is predominantly along the z axis, with negligible x and y components. However, in larger formations (400 km), significant motion components appear in both the altitude (x) and along-track (y) directions, indicating that the nonlinearity of three-dimensional motion intensifies in larger-scale formations and requires more complex control strategies. These findings underscore the need for different control strategies based on formation size, with the integration of three-dimensional motion being particularly necessary in large formations.

Third, the importance of inclination angle selection was clearly demonstrated in this study, and it is closely related to the complex orbital variations caused by $J_2$ perturbation. $J_2$ perturbation arises from the Earth’s slightly oblate shape, which leads to an imbalance in its gravitational field and causes continuous changes in orbital elements such as RAAN, AOP, and mean anomaly. For a single satellite, these changes may not pose significant issues. However, in formation flying, where there are multiple satellites, each satellite is affected differently by the $J_2$ perturbation, causing their relative positions and velocities to gradually distort over time. This distortion becomes more severe as the distance between satellites increases, requiring more fuel for correction maneuvers. According to the results of this study, when the formation size is large, fuel consumption reaches its maximum at around an inclination angle of 50 degrees, whereas in smaller formations, this peak shifts toward 60 degrees. This indicates that the effects of the $J_2$ perturbation depend on both the inclination angle and the size of the formation, and that these two factors interact in a complex way. Ultimately, when the inclination has a symmetric value like 0 or 90 degrees, the differential effect of the $J_2$ perturbation between satellites is minimal, leading to lower fuel consumption. In contrast, in the range between 45 and 60 degrees, the perturbation becomes stronger and more unbalanced, resulting in a sharp increase in fuel usage. These findings suggest that inclination should not be selected as a fixed or arbitrary value, but rather optimized strategically by considering both the size and configuration of the satellite formation during mission planning.

Lastly, this study emphasizes the significance of nonlinear dynamics models that account for $J_2$ perturbation. While the conventional Clohessy–Wiltshire equations may be sufficient for smaller formations, for those exceeding 400 km, neglecting $J_2$ perturbation can lead to orbital deviation and increased fuel consumption. The proposed method effectively addresses these issues, enabling more realistic and efficient trajectory generation, which is expected to substantially reduce fuel consumption and extend the operational lifetime of large formations.

Furthermore, to evaluate the feasibility of the proposed trajectory from a design perspective, we considered a scenario involving a formation-flying microsatellite with an approximate mass of 15 kg [25] that was equipped with an electric thruster capable of providing a maximum thrust of 1 mN–5 mN [26]. This corresponds to a maximum achievable control acceleration in the range of $0.06$ mm/s²–$0.3333$ mm/s², which comfortably exceeds the required control accelerations observed in Figure 6 ($0.0000025$ mm/s²) and Figure 7 ($0.02$ mm/s²). Although environmental disturbances such as atmospheric drag (up to ${10}^{-4}$ mm/s²) and higher-order gravity effects (approximately $0.0981$ mm/s²) exist in low Earth orbit, the effective relative disturbance acting on the formation is much smaller, as it is determined by the difference between the disturbances of the deputy and the chief satellites [27]. Therefore, these disturbances also remain within the controllable range of currently available electric propulsion systems. Accordingly, the proposed station-keeping and trajectory control strategy can be realistically implemented with current technology, supporting the practical applicability of this approach in real-world formation-flying missions.

However, several limitations remain. In actual space environments, various dynamical factors, including subtle perturbations and satellite-to-satellite interactions, can influence fuel consumption and orbital dynamics. Although this study primarily employs a $J_2$-based model, future work must expand the model by incorporating these additional elements and develop corresponding control strategies.

In this study, the reference track for the formation-flying satellites was generated using the PoNN framework. To verify the validity of the proposed trajectory, we employed a simple PD controller as a baseline method due to its simplicity and ease of implementation. We expect that if a more advanced control law were applied (e.g., model predictive control or nonlinear controllers), it would result in an improved tracking performance and potentially lower $\Delta V$ consumption. However, since our focus is on evaluating the quality of the reference trajectory generated by the PoNN, the PD controller was deemed sufficient for demonstrating its effectiveness in a straightforward and interpretable manner.

Moreover, further research is needed to validate the practicality and feasibility of using this method for real missions, including verifying theoretical models and simulations in actual flight conditions and developing systems capable of real-time fuel-consumption optimization. Nevertheless, this study provides critical foundations for the design of satellite formation flying missions and offers practical guidelines for optimizing fuel efficiency in long-term formation missions. The proposed methodology is expected to be applicable to future missions involving large formations of satellites.