Passivity-Based Control with Disturbance Observer of Electromagnetic Formation Flight Spacecraft in the Port-Hamiltonian Framework

Abstract

Satellite formation flying technology currently represents a focal point in space mission research. Traditional spacecraft payload performance and lifespan are often constrained by propellant limitations. Electromagnetic Formation Flying (EMFF), a propellant-free formation flying technique, has garnered widespread attention. Its inherent strong nonlinearity and coupling present challenges for high-precision control within EMFF. This paper presents the relative motion dynamics of a two-satellite EMFF in the port-Hamiltonian framework and constructs an accurate nonlinear model of the dynamics. Utilizing the concept of Interconnection and Damping Assignment and nonlinear disturbance observer, a composite disturbance-rejection passivity-based controller is designed, offering a method for controlling the magnetic dipole strength of formation satellites. Finally, numerical simulations are conducted to demonstrate the viability of the proposed dynamics model and control strategy.

1. Introduction

As technological progress continues, satellite capabilities are progressively evolving to address the escalating requirements of space missions [1,2]. Notably, the advent of satellite formation flying technology has inaugurated novel opportunities for deploying small satellites [3]. This technology, known as Satellite Formation Flying (SFF), involves the precise coordination of multiple satellites operating in a defined arrangement. In contrast to conventional large solitary satellites, formation flying presents substantial benefits including cost reduction, heightened redundancy, enhanced upgrade potential, and improved performance. These qualities define SFF as a multifaceted instrument with broad applications in fields like terrestrial monitoring, celestial research, and ventures into deep space, positioning it as an attractive pathway for future space missions.

However, the upkeep of exact formation configurations in SFF demands regular maneuvering controls. A large number of conventional space missions reliant on SFF employ propellant-dependent control actuators for these maneuvers, resulting in significant propellant depletion. Such a constraint not only curtails the overall mission duration but also poses a risk of plume contamination to the optical instruments aboard the satellites.

Electromagnetic Formation Flight (EMFF), recognized as a propellant-free approach to formation flying, has garnered considerable attention [4,5]. In EMFF, every satellite comes equipped with electromagnetic coils that act as magnetic dipoles. The mutual interaction of these dipoles produces electromagnetic forces and torques, facilitating the control of satellite relative motion. Advantages of EMFF include non-contact, continuously adjustable inter-satellite electromagnetic forces, effective over distances ranging from several tens to hundreds of meters [6]. However, the challenges in dynamic modeling and controller design for EMFF are heightened by the nonlinearity and interaction of electromagnetic forces/torques, uncertainties in electromagnetic modeling, satellite inertia moments, the effects of Earth’s magnetic field on electromagnetic satellites, and the exacting demands for precise control required by space missions with satellite formations.

Initial research into the complexities and foundational aspects of EMFF was undertaken by the MIT Space Systems Laboratory and the University of Tokyo. Elias explored an EMFF system comprising two spacecraft, each equipped with three orthogonal electromagnets and reaction wheels [7]. They elucidated the nonlinear dynamics of the EMFF system, delineated its linear dynamics, and devised a linear controller. Cai’s research delved into the nonlinear translational dynamics of EMFF reconfiguration, anchored in in/out-of-plane angles and relative distance [8]. Utilizing analytical mechanics theory, they formulated a nonlinear translational dynamics model and proposed a linear feedback control approach based on optimal control theory and the Gauss pseudospectral method. Ahsun examined the deployment of adaptive control laws within linear dynamics, proposing a control allocation strategy for angular momentum management [9]. Later studies predominantly focused on decoupling and frequency multiplexing applications. Huang suggested combining frequency multiplexing with an adaptive sliding mode controller, informed by learning algorithms, to manage the relative positioning of satellites [10]. Zhang employed sinusoidal and steady electromagnetic fields to regulate the relative positioning and to mitigate the accumulation of angular momentum caused by the Earth’s magnetic field [11]. Youngquist innovated a new frequency multiplexing technique, facilitating simultaneous control of both force and torque [12].

Obviously, prevailing research in the realm of EMFF is predominantly centered on control strategies derived from linear error dynamics, with a notable scarcity of studies directly implementing nonlinear controllers founded on nonlinear dynamics.

A variety of modeling strategies are crucial for handling the relative translational dynamics in EMFF, particularly because of the internal force nature attributed to electromagnetic forces. In fact, Elias’s work has substantiated that, for the modeling of complex systems, analyzing the system from the perspectives of energy and constraints offers substantial advantages [7]. Additionally, given the system’s strong coupling and nonlinearity, the complexity of the model and the control challenges notably escalate when formations include more than two satellites. Consequently, researchers typically commence with studies on electromagnetic force control in two-satellite systems.

In this research, a unique input-state-output port-Hamiltonian (pH) system technique is utilized to construct an accurate nonlinear dynamic model for a two-satellite EMFF and to directly develop a corresponding control law. The pH system, an extension of Hamiltonian systems, accommodates inputs and dissipation, referred to as ’ports.’ This methodology facilitates the modeling and analysis of intricate physical systems and their interconnections, thereby harmonizing the processes of modeling and control [13]. Owing to its reliance on energy methods, the pH dynamic model effectively preserves nonlinearities without enforcing specific distance constraints between satellites, enhancing model accuracy.

External disturbances, uncertainty in the inertia matrices, and coupled nonlinear elements represent the uncertain factors in the satellite attitude and orbit control systems [14]. Moreover, it is crucial to address uncertainties in the electromagnetic modeling and actuator saturation within EMFF studies. Attaining high-precision formation control performance amidst these complexities presents a significant challenge. Various scholars have delved into control methodologies for EMFF, exploring adaptive control [9,15], sliding mode control [8,16], linear quadratic regulation (LQR) control [17,18], and robust suboptimal control [19].

In the realm of passivity-based control, research on disturbance rejection controllers is relatively sparse. Juhoon introduced an inner-loop controller for robotic systems, generating compensation signals by estimating aggregated disturbances defined by object uncertainty and external perturbations, thus achieving tracking error control and enhancing the robustness of passive controllers [20]. Joel demonstrated new findings on reducing mismatched disturbances in pH systems via control by interconnection (CbI) [21]. Controllers were designed to suppress constant disturbances originating from non-passive outputs. In the port-Hamiltonian framework, the disturbance suppression problem for permanent magnet synchronous motors was addressed by incorporating integral actions and changing coordinates. Li investigated the issue of grid imbalance in Vienna rectifiers, proposing a passive control approach with inherent disturbance rejection capabilities [22]. Alazard designed a PD control law for rotating flexible spacecraft, achieving disturbance rejection in a port-Hamiltonian approach [23]. However, there is as yet no specific design for disturbance rejection control in EMFF systems in the port-Hamiltonian framework.

This article derives the Lagrangian for a two-spacecraft EMFF within the port- Hamiltonian structure, utilizing Euler-Lagrange equations. Leveraging these equations, we introduce the dynamics of EMFF. Inspired by the IDA-PBC paradigm, we have designed a controller specifically for the dynamics of a two-satellite EMFF. Based on the nonlinear disturbance observer technique, the robust composite control strategy is designed. Simulation tests are conducted on a given Circular Orbit reference trajectory. The contributions of this study are outlined below:

• This study presents the EMFF dynamics for two satellites within the pH framework, preserving the system’s inherent nonlinear characteristics;
• Guided by the tIDA-PBC concept, a controller tailored to the dynamics of two-satellite EMFF is designed;
• Utilizing the nonlinear disturbance observer technique, a robust composite control strategy is formulated.

The structure of this paper is outlined as follows: Section 2 presents the dynamic model of the two-satellite electromagnetic formation within the port-Hamiltonian framework. Section 3 proposes the magnetic dipole composite passivity-based controller, aimed at achieving trajectory tracking and disturbance rejection. Then, numerical simulation results are discussed in Section 4. Lastly, Section 5 provides valuable conclusions derived from the study.

2. Relative Motion Dynamics Model

2.1. Port-Hamiltonian System

A port-Hamiltonian system is composed of a Dirac structure, which illustrates the interconnections of the system’s energy components, a resistive structure, which embodies the damping effects, and a Hamiltonian function, which dictates the dynamic behaviors of these energy components [24]. The configuration of input-state-output for port-Hamiltonian systems is specified as follows.

$$\dot{\mathit{x}}=\left(\mathit{J}\left(\mathit{x}\right)-\mathit{R}\left(\mathit{x}\right)\right)\nabla H\left(\mathit{x}\right)+\mathit{g}\left(\mathit{x}\right)\mathit{u},\mathit{x}\in {\mathbb{R}}^n,\mathit{u}\in {\mathbb{R}}^m$$

(1)

$$\mathit{y}={\mathit{g}}^{\mathrm{T}}\left(\mathit{x}\right)\nabla H\left(\mathit{x}\right),\mathit{y}\in {\mathbb{R}}^m$$

(2)

where $\mathit{x},\mathit{y},\mathit{u}$ and $\mathit{g}\left(\mathit{x}\right)$ are the state, output, control signal, and input matrix of the system, respectively. $\mathit{J}\left(\mathit{x}\right)$ is a skew-symmetric matrix (i.e., $\mathit{J}\left(\mathit{x}\right)=-{\mathit{J}}^{\mathrm{T}}\left(\mathit{x}\right)$), and $\mathit{R}\left(\mathit{x}\right)$ is a semi-positive definite matrix for $\mathit{x}$. In Equation (1), the matrices $\mathit{J}\left(\mathit{x}\right)$ and $\mathit{R}\left(\mathit{x}\right)$ are identified as the interconnection matrix and the damping matrix, respectively. The functions denoted by $H\left(\mathit{x}\right)$ are Hamiltonian functions, and $\nabla H\left(\mathit{x}\right)$ represents their partial derivative vectors.

Remark: By differentiating the Hamiltonian function $H\left(\mathit{x}\right)$, we can obtain

$$\dot{H}=-{\nabla }_{\mathit{x}}^{\mathrm{T}}H\left(\mathit{x}\right)\mathit{R}\left(\mathit{x}\right)\nabla H\left(\mathit{x}\right)+{\mathit{u}}^{\mathrm{T}}\mathit{y}⩽{\mathit{u}}^{\mathrm{T}}\mathit{y}$$

(3)

2.2. Coordinate System Definition

The principal coordinate systems for formation flight dynamics involve the Earth Centered Inertial (ECI) frame and the Hill frame as shown in Figure 1. In the ECI frame, the X-axis follows the center of the earth with the vernal equinox, the Z-axis is normal to the equator plane and points to the geographic north pole, and the Y-axis is determined by the right-hand rule. In the Hill frame, the center is at the formation mass center. The x-axis is along the orbital radial direction, the y-axis is along the velocity vector of the mass center, and the z-axis is along the orbital angular momentum vector of the mass center.

Electromagnetic formation flight is characterized by the orchestration of multiple satellites, each subject to electromagnetic forces produced by the collective ensemble of the formation. In our exploration of relative kinematics, we utilized two electromagnetic satellite, designated SatA and SatB, to derive the preliminary equations of relative motion in the port-Hamiltonian framework. The formation center of mass is invariant to electromagnetic interactions, as they do not modify the total momentum of the system. In the scenario where no other forces act on the formation, the center of mass would follow a trajectory consistent with Keplerian motion. It is critical to acknowledge, especially in low Earth orbit scenarios, the significance of the interplay between the magnetic dipoles and Earth’s geomagnetic field, which introduces complexities in attitude stabilization and momentum regulation. Substantial efforts have been invested in the study of this issue, with numerous approaches being proposed for the control of attitude and the management of angular momentum [4]. Our study focuses on the translational reconfiguration of the EMFF, involving a control system with three degrees of freedom (3-DOF), primarily under the assumption of ideal attitude dynamics.

2.3. Formation Dynamics Modeling

As illustrated in Figure 1, the system under consideration consists of two satellites: SatA and SatB. The formation’s mass center’s coordinates in ECI coordinates are denoted as ${\mathit{Q}}_0={\left[X_0,Y_0,Z_0\right]}^{\mathrm{T}}$. The projection of this vector in the Hill frame is expressed as ${\mathit{q}}_0={\left[q_0,0,0\right]}^{\mathrm{T}}$. The coordinates of SatA are defined as ${\mathit{Q}}_a={\left[X_a,Y_a,Z_a\right]}^{\mathrm{T}}$. In the context of formation flight, particular attention is paid to the relative positions of the satellites within the Hill coordinate frame. The vector describing the position of SatA relative to the formation’s mass center is denoted by ${\mathit{q}}_a={\left[x_a,y_a,z_a\right]}^{\mathrm{T}}$. Likewise, the position vector for SatB is denoted as ${\mathit{q}}_b={\left[x_b,y_b,z_b\right]}^{\mathrm{T}}$. The dynamics pertaining to these relative positions will be explored further in this discussion.

According to the defined parameters of the formation mass center, the relative position vectors ${\mathit{q}}_a$ and ${\mathit{q}}_b$ comply with the following relationship:

$$m_a{\mathit{q}}_a+m_b{\mathit{q}}_b=\mathbf{0}$$

(4)

In the formula, $m_a$ and $m_b$ represent the masses of satellites A and B, respectively. Thus, we can define $k=\frac{m_a}{m_b}$.

Assuming that electromagnetic formation flying is propellant-free, the mass of the satellites can be treated as a constant. Consequently, it follows that

$$m_a{\dot{\mathit{q}}}_a+m_b{\dot{\mathit{q}}}_b=\mathbf{0}$$

(5)

Consider SatA, whose kinetic energy $\mathcal{T}$ can be expressed as

$$\mathcal{T}\left({\dot{\mathit{Q}}}_a\right)=\frac{1}{2}{m}_a{∥{\dot{\mathit{Q}}}_a∥}^2$$

(6)

The potential energy of SatA encompasses two components: gravitational potential energy ${\mathcal{U}}_g$ and electromagnetic potential energy ${\mathcal{U}}_e$. The gravitational potential energy, subject to $J_2$ perturbations, is formulatable in the following manner.

$${\mathcal{U}}_g\left({\mathit{Q}}_a\right)=-\frac{\mu m_a}{R_a}\left[1-\frac{J_2}{2}\frac{R_e^2}{R_a^2}\left(3\frac{Z_a^2}{R_a^2}-1\right)\right]$$

(7)

where $R_a$ is the distance between SatA and the Earth’s center, $\mu$, $R_e$, and $J_2$ are the gravitational constant, the Earth’s radius, and the second zonal gravitational coefficient, respectively.

By conceptualizing the electromagnetic coils on each satellite as a maneuverable magnetic dipole, the magnetic field attributed to SatB can be delineated as

$${\mathit{B}}_b\left(\mathsf{\rho }\right)=\frac{{\mu }_0}{4\pi }\left(\frac{3{\mathsf{\mu }}_b^{\mathrm{T}}\mathsf{\rho }}{{∥\mathsf{\rho }∥}^5}\mathsf{\rho }-\frac{{\mathsf{\mu }}_b}{{∥\mathsf{\rho }∥}^3}\right)$$

(8)

where ${\mathsf{\mu }}_b$ represents the magnetic dipole vector on SatB, $\mathsf{\rho }$ is the vector from SatB to SatA, and ${\mu }_0=4\pi \times {10}^{-7}\mathrm{N}/{\mathrm{A}}^2$ is the permeability of free space.

The electromagnetic potential energy of the formation is given by [7]

$${\mathcal{U}}_e=-{\mathsf{\mu }}_a^{\mathrm{T}}{\mathit{B}}_b\left(\mathsf{\rho }\right)$$

(9)

Subsequently, the Lagrangian function is formulable as follows:

$$\mathcal{L}=\mathcal{T}-{\mathcal{U}}_g-{\mathcal{U}}_e$$

(10)

To articulate the dynamics of spacecraft within the Hill frame, it is imperative to execute specific coordinate transformations between the Hill and ECI frames. The transformation from the Hill frame to the ECI frame is stipulated by the subsequent equation.

$$\mathit{T}=\left[\begin{array}{ccc}{c}_{\Omega }{c}_{\theta }-s_{\Omega }{s}_{\theta }{c}_i& -c_{\Omega }{s}_{\theta }-s_{\Omega }{c}_{\theta }{c}_i& s_{\Omega }{s}_i\\ s_{\Omega }{c}_{\theta }+c_{\Omega }{s}_{\theta }{c}_i& -s_{\Omega }{s}_{\theta }+c_{\Omega }{c}_{\theta }{c}_i& -c_{\Omega }{s}_i\\ s_{\theta }{s}_i& c_{\theta }{s}_i& c_i\end{array}\right]=\left[\begin{array}{c}{\mathit{T}}_1\\ {\mathit{T}}_2\\ {\mathit{T}}_3\end{array}\right]$$

(11)

where $s_o=sin\left(o\right),c_o=cos\left(o\right)$, $s_o=sin\left(o\right),c_o=cos\left(o\right)$, $\Omega$ is the right ascension of the satellite ascending node, $\theta$ is the argument of latitude, and i is the inclination of the formation mass center.

Given that $\mathit{T}$ represents the orthogonal matrix, hence ${\mathit{T}}^{\mathrm{T}}\mathit{T}={\mathit{I}}_3$, differentiating both sides of this equation yields ${\dot{\mathit{T}}}^{\mathrm{T}}\mathit{T}+{\mathit{T}}^{\mathrm{T}}\dot{\mathit{T}}=\mathbf{0}$, confirming its skew-symmetric matrix property. Thus, we designate the skew-symmetric matrix function as $\mathit{S}={\dot{\mathit{T}}}^{\mathrm{T}}\mathit{T}$.

Subsequently, the relationship between the ECI and Hill frames can be articulated.

$${\mathit{Q}}_a=\mathit{T}\left({\mathit{q}}_0+{\mathit{q}}_a\right)$$

(12)

As a result, the kinetic energy of the spacecraft within the Hill frame can be delineated as follows:

$$\mathcal{T}\left({\mathit{q}}_a,{\dot{\mathit{q}}}_a\right)=\frac{1}{2}{m}_a{∥{\dot{\mathit{Q}}}_a∥}^2=\frac{1}{2}{m}_a{∥\left({\dot{\mathit{q}}}_0+{\dot{\mathit{q}}}_a\right)+\mathit{S}\left({\mathit{q}}_0+{\mathit{q}}_a\right)∥}^2$$

(13)

The gravitational potential energy of the spacecraft can be described as

$${\mathcal{U}}_g\left({\mathit{q}}_a\right)=-\frac{\mu m_a}{R_a}\left[1-\frac{J_2}{2}\frac{R_e^2}{R_a^2}\left(3\frac{{\left({\mathit{T}}_3\left({\mathit{q}}_0+{\mathit{q}}_a\right)\right)}^2}{R_a^2}-1\right)\right]$$

(14)

Putting Equations (5), (13), (14) into Equation (10), the Lagrangian equation can be written as

$$\begin{array}{cc}\hfill \mathcal{L}\left({\mathit{q}}_a,{\dot{\mathit{q}}}_a\right)& =\mathcal{T}-{\mathcal{U}}_g-{\mathcal{U}}_e\hfill \\ \hfill & =\frac{1}{2}{m}_a{∥\left({\dot{\mathit{q}}}_0+{\dot{\mathit{q}}}_a\right)+\mathit{S}\left({\mathit{q}}_0+{\mathit{q}}_a\right)∥}^2\hfill \\ \hfill & +\frac{\mu m_a}{R_a}\left[1-\frac{J_2}{2}\frac{R_e^2}{R_a^2}\left(3\frac{{\left({\mathit{T}}_3\left({\mathit{q}}_0+{\mathit{q}}_a\right)\right)}^2}{R_a^2}-1\right)\right]\hfill \\ \hfill & +\frac{{\mu }_0}{4\pi }{\mathsf{\mu }}_a^{\mathrm{T}}\left(\frac{3{\mathsf{\mu }}_b^{\mathrm{T}}\left(1+k\right){\mathit{q}}_a}{{∥\left(1+k\right){\mathit{q}}_a∥}^5}\left(1+k\right){\mathit{q}}_a-\frac{{\mathsf{\mu }}_b}{{∥\left(1+k\right){\mathit{q}}_a∥}^3}\right)\hfill \end{array}$$

(15)

Considering the focus on the motion of the mass center and follower spacecraft within the Hill frame, ${\mathit{q}}_a$ is selected as the generalized coordinates. Following the Legendre transformation, the corresponding generalized momentum is derived.

$${\mathit{p}}_a=\frac{\partial \mathcal{L}}{\partial {\mathit{q}}_a}=m_a\left[{\dot{\mathit{q}}}_0+{\dot{\mathit{q}}}_a+\mathit{S}\left({\dot{\mathit{q}}}_0+{\dot{\mathit{q}}}_a\right)\right]$$

(16)

Consequently, the dynamics of relative motion are determined by the Euler–Lagrange equation.

$$\begin{array}{cc}\hfill & \frac{\mathrm{d}}{\mathrm{dt}}\left(\frac{\partial \mathcal{L}}{\partial {\dot{\mathit{q}}}_a}\right)-\frac{\partial \mathcal{L}}{\partial {\mathit{q}}_a}=\mathit{u}\hfill \\ \hfill & \Rightarrow m_a\left[{\ddot{\mathit{q}}}_0+{\ddot{\mathit{q}}}_a+\dot{\mathit{S}}\left({\mathit{q}}_0+{\mathit{q}}_a\right)+\mathit{S}\left({\dot{\mathit{q}}}_0+{\dot{\mathit{q}}}_a\right)\right]\hfill \\ \hfill & -m_a{\mathit{S}}^{\mathrm{T}}\left[\left({\dot{\mathit{q}}}_0+{\dot{\mathit{q}}}_a\right)-\mathit{S}\left({\mathit{q}}_0+{\mathit{q}}_a\right)\right]+{\nabla }_{{\mathit{q}}_a}{\mathcal{U}}_g+{\nabla }_{{\mathit{q}}_a}{\mathcal{U}}_e=\mathit{u}\hfill \end{array}$$

(17)

Rectifying the equation yields

$$\begin{array}{cc}\hfill \left[\begin{array}{c}{\dot{\mathit{q}}}_a\\ {\dot{\mathit{p}}}_a\end{array}\right]& =\left[\begin{array}{c}\frac{1}{m_a}{\mathit{p}}_a-{\dot{\mathit{q}}}_0-\mathit{S}\left({\mathit{q}}_0+{\mathit{q}}_a\right)\\ m_a{\mathit{S}}^{\mathrm{T}}\left[\left({\dot{\mathit{q}}}_0+{\dot{\mathit{q}}}_a\right)-\mathit{S}\left({\mathit{q}}_0+{\mathit{q}}_a\right)\right]-{\nabla }_{{\mathit{q}}_a}{\mathcal{U}}_g-{\nabla }_{{\mathit{q}}_a}{\mathcal{U}}_e\end{array}\right]+\left[\begin{array}{c}{\mathbf{0}}_3\\ {\mathit{I}}_3\end{array}\right]\mathit{u}\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill \left[\begin{array}{c}{\dot{\mathit{q}}}_a\\ {\dot{\mathit{p}}}_a\end{array}\right]& =\left[\begin{array}{cc}{\mathbf{0}}_3& {\mathit{I}}_3\\ -{\mathit{I}}_3& {\mathbf{0}}_3\end{array}\right]\left[\begin{array}{c}\frac{\partial H}{\partial {\mathit{q}}_a}\\ \frac{\partial H}{\partial {\mathit{p}}_a}\end{array}\right]+\left[\begin{array}{c}{\mathbf{0}}_3\\ {\mathit{I}}_3\end{array}\right]\mathit{u}\hfill \\ \hfill & =\left(\mathit{J}-\mathit{R}\right)\nabla H\left(\mathit{x}\right)+\mathit{g}\mathit{u}\hfill \end{array}$$

(19)

where $\mathit{J}=\left[\begin{array}{cc}{\mathbf{0}}_3& {\mathit{I}}_3\\ -{\mathit{I}}_3& {\mathbf{0}}_3\end{array}\right]$, $\mathit{R}={\mathbf{0}}_{6\times 6}$, $\mathit{g}=\left[\begin{array}{c}{\mathbf{0}}_3\\ {\mathit{I}}_3\end{array}\right]$. It should be noted that, unless stated otherwise, $\mathit{x}$ in the text represents ${\left[{\mathit{q}}_a,{\mathit{p}}_a\right]}^{\mathrm{T}}$.

Equation (19) represents the dynamics of EMFF spacecraft relative motion within the port-Hamiltonian framework, where the Hamiltonian function is deduced from the Lagrangian equations.

$$\begin{array}{cc}\hfill H\left(\mathit{x}\right)& ={\mathit{p}}_a^{\mathrm{T}}{\dot{\mathit{q}}}_a-\mathcal{L}\hfill \\ \hfill & =\frac{1}{2m_a}{\mathit{p}}_a^{\mathrm{T}}{\mathit{p}}_a-{\mathit{p}}_a^{\mathrm{T}}{\dot{\mathit{q}}}_0-{\mathit{p}}_a^{\mathrm{T}}\mathit{S}\left({\mathit{q}}_0+{\mathit{q}}_a\right)+{\mathcal{U}}_g+{\mathcal{U}}_e\hfill \end{array}$$

(20)

Notably, certain factors may negatively impact the reconfiguration design, including errors in modeling far-field electromagnetic forces and external environmental disturbances. While these factors might be overlooked during the nominal design of reconfiguration trajectories, they are crucial considerations in the design of the formation controller.

3. Design of the Control System

This chapter delves into the control issues of EMFF port-Hamiltonian systems under non-zero disturbances. Utilizing damping injection techniques and Nonlinear Disturbance Observer (NDOB) technology, a composite disturbance rejection passivity-based controller is designed as ${\mathit{u}}_{DRPBC}={\mathit{u}}_{FB}+{\mathit{u}}_{DO}$, where ${\mathit{u}}_{FB}$ is a baseline feedback controller in the absence of disturbances and ${\mathit{u}}_{DO}$ is a disturbance compensation controller with $\mathit{d}\left(t\right)\ne 0$.

3.1. Baseline Feedback Controller Design

Assuming that the function H, within system (1), exhibits a strict minimum at the point ${\dot{\mathit{x}}}^{*}$ (where ${\dot{\mathit{x}}}^{*}$ represents an equilibrium point), it follows that $H\left(\mathit{x}\right)$ may be locally regarded as a Lyapunov function. This, in turn, leads to the stability of the equilibrium point ${\dot{\mathit{x}}}^{*}$ with $\mathit{u}=\mathbf{0}$. In this regard, $\dot{H}⩽0$ due to the positive semi-definiteness of $\mathit{R}$. Nevertheless, if the targeted equilibrium point does not coincide with the minimum of the Hamiltonian function, it becomes necessary to suitably shape the function through the control signal $\mathit{u}$. Beyond adjusting the Hamiltonian function, it may also be requisite to alter the interconnection and damping matrices. This approach is encapsulated in the method known as Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC), which is elucidated in the subsequent theorem [25].

Theorem 1.

(IDA-PBC) Suppose that there exist $H_d:{\mathbb{R}}^n\to \mathbb{R},{\mathit{J}}_d\left(\mathit{x}\right)=-{\mathit{J}}_d^T\left(\mathit{x}\right)$, ${\mathit{R}}_d\left(\mathit{x}\right)={\mathit{R}}_d^T\left(\mathit{x}\right)\succcurlyeq 0$ satisfying the following equation for system (1).

$$\begin{array}{c}\hfill {\mathit{g}}^{\perp }\left(\mathit{x}\right)\left(\left(\mathit{J}\right(\mathit{x})-\mathit{R}(\mathit{x}\left)\right)\nabla H\left(\mathit{x}\right)\right)={\mathit{g}}^{\perp }\left(\mathit{x}\right)\left(({\mathit{J}}_d\left(\mathit{x}\right)-{\mathit{R}}_d\left(\mathit{x}\right))\nabla H_d\left(\mathit{x}\right)\right)\end{array}$$

(21)

where ${\mathit{g}}^{\perp }$ is the full-rank left annihilator of $\mathit{g}$, ≽ represents the positive semi-definite. If ${\mathit{x}}^{*}$ is a minimum of $H_d$, then there exists a controller

$$\mathit{u}={\left({\mathit{g}}^{\mathrm{T}}\left(\mathit{x}\right)\mathit{g}\left(\mathit{x}\right)\right)}^{-1}{\mathit{g}}^{\mathrm{T}}\left(\mathit{x}\right)\left(({\mathit{J}}_d\left(\mathit{x}\right)-{\mathit{R}}_d\left(\mathit{x}\right))\nabla H_d\left(\mathit{x}\right)-(\mathit{J}\left(\mathit{x}\right)-\mathit{R}\left(\mathit{x}\right))\nabla H\left(\mathit{x}\right)\right)$$

(22)

that is capable of locally stabilizing system (1), rendering the closed-loop system as follows.

$${\dot{\mathit{x}}}_d=\left({\mathit{J}}_d\left(\mathit{x}\right)-{\mathit{R}}_d\left(\mathit{x}\right)\right)\nabla H_d\left(\mathit{x}\right)$$

(23)

In a recent study, the IDA-PBC method has been expanded to encompass a tracking design utilizing contraction analysis, as detailed by Yaghmaei and Yazdanpanah [26]. The key findings are summarized as follows.

Theorem 2.

(tIDA-PBC) Consider the following system

$${\dot{\mathit{x}}}_d=\left({\mathit{J}}_d\left(\mathit{x}\right)-{\mathit{R}}_d\left(\mathit{x}\right)\right)\nabla H_d(\mathit{x},{\mathit{x}}^{*}\left(t\right))$$

(24)

$H_d$ is a desired Hamiltonian function conforming to the subsequent conditions.

$$\begin{array}{cc}\hfill \left({\mathit{J}}_d\left(\mathit{x}\right)-{\mathit{R}}_d\left(\mathit{x}\right)\right)& \nabla H_d(\mathit{x},{\mathit{x}}^{*}\left(t\right)){|}_{{\mathit{x}}^{*}\left(t\right)}={\dot{\mathit{x}}}^{*}\left(t\right),\hfill \\ \hfill {\alpha }_1\mathit{I}\prec {\nabla }^2{H}_d& \left(\mathit{x},{\mathit{x}}^{*}\left(t\right)\right)\prec {\alpha }_2\mathit{I},\forall \mathit{x}\in D_T\hfill \end{array}$$

(25)

${\alpha }_1$, ${\alpha }_2$ are positive constants and ${\alpha }_1<{\alpha }_2$. $D_T$ is an open subset of ${\mathbb{R}}^n$ which for all $t⩾0$ constitutes a sphere containing ${\mathit{x}}^{*}\left(t\right)$ and possessing a constant radius T centered around ${\mathit{x}}^{*}\left(t\right)$. Assuming that all eigenvalues of ${\mathit{A}}_d\triangleq {\mathit{J}}_d-{\mathit{R}}_d$ possess strictly negative real parts and that the ensuing matrix exhibits no eigenvalues on the imaginary axis for a given positive constant ϵ,

$$\mathit{N}=\left[\begin{array}{cc}{\mathit{A}}_d& \eta {\mathit{A}}_d{\mathit{A}}_d^{\mathrm{T}}\\ -\left(\eta +ϵ\right)\mathit{I}& -{\mathit{A}}_d^{\mathrm{T}}\end{array}\right]$$

(26)

Then, the controller

$$\mathit{u}={\left({\mathit{g}}^{\mathrm{T}}\left(\mathit{x}\right)\mathit{g}\left(\mathit{x}\right)\right)}^{-1}{\mathit{g}}^{\mathrm{T}}\left(\mathit{x}\right)\left(({\mathit{J}}_d\left(\mathit{x}\right)-{\mathit{R}}_d\left(\mathit{x}\right))\nabla H_d(\mathit{x},{\mathit{x}}^{*}\left(t\right))-(\mathit{J}\left(\mathit{x}\right)-\mathit{R}\left(\mathit{x}\right))\nabla H\left(\mathit{x}\right)\right)$$

(27)

makes the system (1) an exponential tracker for ${\mathit{x}}^{*}\left(t\right)$.

Then, for the EMFF trajectory tracking controller, the designed closed-loop system is as follows

$$\left[\begin{array}{c}{\dot{\mathit{q}}}_a\\ {\dot{\mathit{p}}}_a\end{array}\right]=\left({\mathit{J}}_d-{\mathit{R}}_d\right)\nabla H_d\left(\mathit{x},{\mathit{x}}^{*}\left(t\right)\right)$$

(28)

The Hamiltonian function form for the closed-loop system is as follows

$$H_d\left(\mathit{x},{\mathit{x}}^{*}\left(t\right)\right)=\frac{1}{2m_a}{\mathit{p}}_a^{\mathrm{T}}{\mathit{p}}_a-{\mathit{p}}_a^{\mathrm{T}}{\dot{\mathit{q}}}_0-{\mathit{p}}_a^{\mathrm{T}}\mathit{S}\left({\mathit{q}}_0+{\mathit{q}}_a\right)+\frac{1}{2}{({\mathit{q}}_a-{\mathit{L}}_j)}^{\mathrm{T}}{\mathit{K}}_j({\mathit{q}}_a-{\mathit{L}}_j)$$

(29)

where $\begin{array}{ccc}& {\mathit{J}}_d=\left[\begin{array}{cc}{\mathbf{0}}_3& {\mathit{I}}_3\\ -{\mathit{I}}_3& {\mathbf{0}}_3\end{array}\right],\hfill & \hfill {\mathit{R}}_d=\left[\begin{array}{cc}{\mathbf{0}}_3& {\mathbf{0}}_3\\ {\mathbf{0}}_3& \tilde{\mathit{R}}\end{array}\right]\end{array}$, ${\mathit{K}}_j$, and ${\mathit{L}}_j$ must satisfy the conditions of Equation (25) for a desired feasible trajectory ${\mathit{x}}^{*}\left(t\right)={\left[{\mathit{q}}_a^{*}\left(t\right),{\mathit{p}}_a^{*}\left(t\right)\right]}^{\mathrm{T}}$. A simple calculation using Equation (25) results in the following:

$${\mathit{L}}_j={\mathit{q}}_a^{*}\left(t\right)-{\mathit{K}}_j^{-1}\left(-\mathit{S}{\mathit{p}}_a^{*}\left(t\right)-\left({\dot{\mathit{p}}}_a^{*}\left(t\right)+\tilde{\mathit{R}}{\dot{\mathit{q}}}_a^{*}\left(t\right)\right)\right)$$

(30)

Then, the baseline feedback controller ${\mathit{u}}_{FB}$ is achieved as

$${\mathit{u}}_{FB}={\left({\mathit{g}}^{\mathrm{T}}\mathit{g}\right)}^{-1}{\mathit{g}}^{\mathrm{T}}\left(\left({\mathit{J}}_d-{\mathit{R}}_d\right)\nabla H_d\left(\mathit{x},{\mathit{x}}^{*}\left(t\right)\right)-\left(\mathit{J}-\mathit{R}\right)\nabla H\left(\mathit{x}\right)\right)$$

(31)

3.2. Nonlinear Disturbance Observer Design

As detailed previously, the design of the controller necessitates considerations for electromagnetic force modeling errors and external disturbances. Given the minimal relative separation between satellites, the overall external disturbance impacting the formation can be considered as bounded forces [27]. Moreover, the modeling error for electromagnetic forces stays below 10% when the separation distance significantly surpasses the radius of the electromagnetic coils, and the intensity of the electromagnetic force is notably minor [8]. Consequently, it is justifiable to treat the effects of both external disturbances and electromagnetic force modeling errors as equivalent bounded total disturbances, denoted as $\mathit{w}$. Subsequently, this allows us to establish a necessary assumption for the discussions within this paper [28].

Assumption 1.

The disturbance $\mathit{d}\left(t\right)={(d_1,d_2,d_3)}^{\mathrm{T}}$ and their derivatives $\dot{\mathit{d}}\left(t\right)$ are bounded, and the condition that as $t\to \infty$, ${\dot{d}}_i\left(t\right)=0$ is satisfied, $i=1,2,3$.

When considering system (1) with disturbances and incorporating the baseline feedback control law (31), we obtain the revised expression for system (1) as follows:

$${\dot{\mathit{x}}}_d=\left({\mathit{J}}_d-{\mathit{R}}_d\right)\nabla H_d\left(\mathit{x},{\mathit{x}}^{*}\left(t\right)\right)+\mathit{g}{\mathit{u}}_{DO}+\mathit{g}\mathit{d}\left(t\right)$$

(32)

Due to the presence of disturbance $\mathit{d}\left(t\right)$ in system (32), which necessitate estimation and compensation for effective system control, a Nonlinear Disturbance Observer (NDOB), inspired by [28], is structured as follows

$$\left\{\begin{array}{c}\begin{array}{cc}\hfill \dot{\mathit{M}}& =-\mathit{l}\left(\mathit{x}\right)\mathit{gM}-\mathit{l}\left(\mathit{x}\right)\left[\mathit{gw}\left(\mathit{x}\right)+\left({\mathit{J}}_d-{\mathit{R}}_d\right)\nabla H_d\left(\mathit{x},{\mathit{x}}^{*}\left(t\right)\right)+{\mathit{gu}}_{DO}\right]\hfill \\ \hfill \widehat{\mathit{d}}& =\mathit{M}+\mathit{w}\left(\mathit{x}\right)\hfill \end{array}\end{array}\right.$$

(33)

where $\widehat{\mathit{d}}$ represents the estimates of disturbances $\mathit{d}\left(t\right)$; the NDOB gain $\mathit{l}\left(\mathit{x}\right)$ is defined as follows:

$$\mathit{l}\left(\mathit{x}\right)=\frac{\partial \mathit{w}\left(\mathit{x}\right)}{\partial \mathit{x}}$$

(34)

where the vector-valued function $\mathit{w}\left(\mathit{x}\right)$ is designed as $\mathit{w}\left(\mathit{x}\right)={\left[{\mathbf{0}}_3,{\mathit{K}}_w{\mathit{p}}_a\right]}^{\mathrm{T}}$, ${\mathit{K}}_w$ is the parameter of the function, and $\mathit{M}$ represents the state of the disturbance observer.

Define the disturbance estimation error as ${\mathit{e}}_d=\widehat{\mathit{d}}-\mathit{d}$, leading to the formulation of the disturbance estimation error system.

$${\dot{\mathit{e}}}_d=-\mathit{l}\left(\mathit{x}\right)\mathit{g}{\mathit{e}}_d-\dot{\mathit{d}}$$

(35)

Substituting an error term into system (32), we obtain the following system

$${\dot{\mathit{x}}}_d=\left({\mathit{J}}_d-{\mathit{R}}_d\right)\nabla H_d(\mathit{x},{\mathit{x}}^{*}\left(t\right))+\mathit{g}{\mathit{u}}_{DO}+\mathit{g}\widehat{\mathit{d}}-\mathit{g}{\mathit{e}}_d$$

(36)

In system (36), the disturbance compensation controller ${\mathit{u}}_{do}$ is intended to counterbalance the estimated disturbances $\widehat{\mathit{d}}$, conforming to the equation specified below

$${\mathit{u}}_{DO}=-\widehat{\mathit{d}}$$

(37)

Thus, the disturbance rejection composite controller is designed as

$${\mathit{u}}_{DRPBC}={\mathit{u}}_{FB}+{\mathit{u}}_{DO}$$

(38)

However, for EMFF satellites, the modulation of control force is accomplished through the adjustment of their magnetic dipole strength. It is coupled with the formation’s position and the satellite’s magnetic field strength. Therefore, the control force is not directly exerted on the satellite; rather, it is based on the current magnetic field strength, altering the satellite’s own magnetic dipole strength to align with the required magnitude of the control force. The final practical form of the EMFF controller should be as follows:

$$\begin{array}{cc}\hfill & -{\nabla }_{{\mathit{q}}_a}\mathit{B}{\mu }_a^{*}-\left(-{\nabla }_{{\mathit{q}}_a}{\mathcal{U}}_e\right)=\left({\mathit{g}}^{\mathrm{T}}\mathit{g}\right)\mathit{u}\hfill \\ \hfill & \Rightarrow -{\nabla }_{{\mathit{q}}_a}\mathit{B}{\mu }_a^{*}={\mathit{g}}^{\mathrm{T}}\left(\left({\mathit{J}}_d-{\mathit{R}}_d\right)\nabla H_d\left(\mathit{x},{\mathit{x}}^{*}\left(t\right)\right)-\left(\mathit{J}-\mathit{R}\right)\nabla H\left(\mathit{x}\right)\right)-{\nabla }_{{\mathit{q}}_a}{\mathcal{U}}_e-\widehat{\mathit{d}}\hfill \\ \hfill & \Rightarrow {\mu }_a^{*}=-{\nabla }_{{\mathit{q}}_a}^{-1}\mathit{B}\left({\mathit{g}}^{\mathrm{T}}\left(\left({\mathit{J}}_d-{\mathit{R}}_d\right)\nabla H_d\left(\mathit{x},{\mathit{x}}^{*}\left(t\right)\right)-\left(\mathit{J}-\mathit{R}\right)\nabla H\left(\mathit{x}\right)\right)-{\nabla }_{{\mathit{q}}_a}{\mathcal{U}}_e-\widehat{\mathit{d}}\right)\hfill \end{array}$$

(39)

${\mu }_a^{*}$ is the ideal magnetic dipole strength in the current state.

4. Numerical Simulation

4.1. Simulation of Controller Performance

In this section, we introduce a numerical simulation for reconfiguring a two-satellite electromagnetic formation. This simulation assesses the performance of our nonlinear relative motion dynamics model and the integrated control strategy. This example is motivated by the use of Satellite Formation Flying (SFF) in Synthetic Aperture Radar (SAR) missions, which require non-Keplerian orbits and continuous control to maintain the desired formation configuration [29]. We assume that the center of mass of the formation follows a circular orbit at an altitude of 500 km. The two satellites, which are identically configured, are designed to orbit the formation’s center of mass with a rotation period of 2 h and a radius of 15 m. The mass of the satellite is 50 kg, and the magnetic dipole on SatB is selected with a constant value of $3\times {10}^6\mathrm{A}·{\mathrm{m}}^2$ point to SatA. The super-boundary magnitude of the magnetic dipole is $5\times {10}^5\mathrm{A}·{\mathrm{m}}^2$. The total external disturbance acceleration acting on the formation system is taken as ${\left[6,-5-4\right]}^{\mathrm{T}}\times {10}^{-6}\mathrm{m}/{\mathrm{s}}^2$ [30]. The initial relative state error is chosen as

$${\mathit{e}}_r={\left[-0.5,0.2,0.6\right]}^{\mathrm{T}}\mathrm{m},{\dot{\mathit{e}}}_r={\left[0.06,0.01,-0.04\right]}^{\mathrm{T}}\mathrm{m}/\mathrm{s}$$

The mass center orbit parameters and initial conditions are shown in

Table 1. Parameters of mass center orbits.

Item	Values
Radius $q_0$ [km]	6878.137
Eccentricity e	0
Inclination i [deg]	60
Argument of perigee $\omega$ [deg]	90
Right ascension of ascending node $\Omega$ [deg]	0
True anomaly $\upsilon$ [deg]	0

and

Table 2. Initial condition of simulations.

Item	Values
${\mathit{q}}_a\left(0\right)$ [m]	${\left[\begin{array}{ccc}12.7950& 1.4490& 7.6934\end{array}\right]}^{\mathrm{T}}$
${\dot{\mathit{q}}}_a\left(0\right)\left[\mathrm{m}/\mathrm{s}\right]$	${\left[\begin{array}{ccc}0.002& 0.0117& -0.0055\end{array}\right]}^{\mathrm{T}}$
${\mathit{q}}_b\left(0\right)\left[\mathrm{m}\right]$	${\left[\begin{array}{ccc}-12.7950& -1.4490& -7.6934\end{array}\right]}^{\mathrm{T}}$
${\dot{\mathit{q}}}_b\left(0\right)\left[\mathrm{m}/\mathrm{s}\right]$	${\left[\begin{array}{ccc}0.002& 0.0117& -0.0055\end{array}\right]}^{\mathrm{T}}$

. The subscripts a and b represent SatA and B.

The controller parameters are set as

$$\tilde{\mathit{R}}=\left[\begin{array}{ccc}6.42& 0& 0\\ 0& 6.25& 0\\ 0& 0.0& 6\end{array}\right],{\mathit{K}}_j=\left[\begin{array}{ccc}0.128& 0& 0\\ 0& 0.12& 0\\ 0& 0& 0.12\end{array}\right],{\mathit{K}}_w=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$$

For comparison, some simulation results from [30] are cited in the result figures with their control strategy ${\mathit{u}}_{ADRC}$. Figure 2 illustrates the ideal relative trajectory and the trajectories of SatA/B under the control of ${\mathit{u}}_{DRPBC}$, respectively. Subsequently, Figure 3 presents the variation in relative error between the actual trajectory and the relative orbit, while Figure 4 depicts the variation of the magnetic dipole on SatA. As observed in the figures, employing the DRPBC method results in the trajectory error reducing to below 1 mm within 150 s, with the eventual error consistently maintained below ${10}^{-4}$ m. The strength of the magnetic dipoles remains within the super-boundary limits. It is evident that the DRPBC method exhibits significant advantages in terms of error convergence speed and control accuracy.

4.2. Monte Carlo Analysis

To rigorously assess the performance of the proposed DRPBC method against defined performance thresholds, we conducted a standard Monte Carlo simulation. This analysis serves dual purposes: first, to ascertain the robustness of our controller against initial positional uncertainties; second, to confirm that the method consistently adheres to the predetermined performance boundaries. Initially, we select random values for the uncertain parameters ${\mathit{q}}_a$ and ${\mathit{q}}_b$ within $\pm 1.5\%$ of their nominal values. For each pair of ${\mathit{q}}_a$ and ${\mathit{q}}_b$, the simulation is executed over 400 s to determine the norms of the steady-state relative position errors [31].

We categorize the relative position error data into eight evenly spaced bins ranging from the minimum to the maximum observed values. As illustrated in Figure 5, the bin spanning from $2.880\times {10}^{-4}$ to $2.885\times {10}^{-4}$ contains the highest frequency of values, with a central value of $2.883\times {10}^{-4}$. The cumulative probability distribution allows us to estimate that there is approximately an $80\%$ likelihood that the relative orbit error does not exceed $2.885\times {10}^{-4}$.

Table 3. Monte Carlo percentage difference.

Item	Center Value	Min	Max	Difference
Relative position error [$\times {10}^{-4}$ m]	2.883	2.864	2.898	1.19%

presents the percentage difference between the maximum and minimum values relative to the center value, indicating that this difference is smaller than the variation in parameters, thereby demonstrating the robustness of the proposed controller to initial positional uncertainties.

Figure 6 illustrates the distribution of relative position errors resulting from the randomly selected uncertain parameters, which are compactly aligned into distinct bands. In terms of relative position errors, despite the presence of parameter uncertainty, the steady-state errors align with the magnitude of those observed under precise initial conditions and consistently fall beneath the thresholds required for engineering applications. This substantiates the efficacy of the proposed method in maintaining control performance within defined limits.

5. Conclusions

This research incorporates the EMFF system into the port-Hamiltonian framework, facilitating the analysis of satellite EMFF relative motion dynamics. By adopting ${\mathit{q}}_a$ as generalized coordinates within the Lagrangian framework, we successfully formulate the equations governing the non-linear relative motion dynamics under the port-Hamiltonian structure, along with the system’s Hamiltonian function. For satellites under disturbance, a disturbance-reaction passivity-based controller is designed, combining a feedback control law based on tIDA-PBC with an error observer. This controller achieves trajectory tracking by adjusting the strength of the satellite’s magnetic dipole, thereby modifying the electromagnetic forces exerted on it. Numerical simulations validate the effectiveness of the designed controller on a predefined circular formation reference trajectory. The results demonstrate that this method effectively controls the relative motion of EMFF spacecraft, exhibiting superiority in both convergence speed and control accuracy. Given the applicability of EMFF in the port-Hamiltonian framework and the potential for future research, this paper holds considerable practical significance.