Artificial potential field-based method for multi-spacecraft loose formation control

This study presents a control strategy based on artificial potential fields to maintain loose formation flying of satellites without a fixed formation, ensuring collision avoidance and stability under various conditions. The approach employs artificial potential fields for obstacle evasion and maintaining relative distances between satellites. Comprehensive simulations validate this method, with Monte Carlo techniques confirming control stability despite variations in initial conditions. The strategy effectively manages the integration of external satellites, maintaining optimal formation even as operational demands increase.


Introduction
The evolution of space technology has accentuated the significance of satellite formation flying (SFF) in complex missions, surpassing the capabilities of standalone satellites in tasks like high-resolution earth monitoring, deep-space missions, and space infrastructure development.Despite its advantages, SFF poses operational challenges, primarily ensuring inter-satellite safe distances and effective collision avoidance maneuvers.
Recent studies have explored dynamic interplay and collision evasion in SFF.Notable works include those by Ran D, Chen X, and others, analyzing relative satellite motion within formations [1], and Lu P, Jiang Q, and others, investigating control strategies for geometric stability in adverse space conditions [2].Wu B, Xu G, and others examined the implications of J2 perturbations on control strategies [3].Nonetheless, critical gaps remain, particularly regarding collision avoidance under extreme environmental constraints.
This paper proposes using artificial potential field methods for addressing pivotal SFF challenges, including follower-leader dynamics, obstacle evasion, and propellant efficiency.Prior research by Wang C, Chen D, and others highlighted the effectiveness of potential fields in debris rendezvous and obstacle avoidance in intricate space scenarios [4], while other studies confirmed the method's applicability in aerospace dynamics [5] [6].Issues related to potential local minima in artificial potential fields have been discussed extensively in literature [7] [8].
In response to traditional methodologies' shortcomings, this study introduces an innovative loose formation control algorithm, leveraging artificial potential fields.The algorithm accounts for relative satellite velocities and proximities, invoking control forces to prevent imminent collisions.It also encompasses an integrative control framework for assimilating new members into the formation, utilizing a unique drag potential field to ensure seamless, safe incorporation.

Relative motion dynamics
To effectively study and control the relative motion between satellites [9], the Local-Vertical-Local-Horizontal (LVLH) coordinate system is extensively adopted.As shown in figure 1, the coordinate system is affixed to the reference satellite, with its origin at the centroid of the reference satellite, offering a lucid and intuitive framework for describing inter-satellite relative motion [10]., and correspondingly, the relative velocity vector is [ , , ] .
The dynamics within the LVLH frame are influenced by gravitational forces and orbital perturbations.The radius Ri from Earth's center to satellite i is computed as The leader satellite follows an elliptical orbit with semi-major axis ac and eccentricity ec , the gravitational parameter is denoted by μ.The mean motion nc is given by   , respectively.The dynamics of satellite i are then governed by the equations [11]: Where mi is the mass of satellite i , represents external disturbances.This model provides a rigorous framework for relative motion analysis and control in satellite formation flight.

Control targets
In satellite formation flight missions, the structural hierarchy consists primarily of lead and follower satellites.The primary satellite, pivotal for the formation, adheres to a pre-defined trajectory, negating the need for supplementary control.Other lead satellites sustain strategic distances from the primary satellite, forming a specific configuration essential for mission objectives, while all follower satellites maneuver around the primary satellite, ensuring collision-free operation.This research is dedicated to developing an optimized control strategy for leader-follower satellite formations, with a special emphasis on maintaining the integrity of a loose formation.Within the lead satellite's LVLH frame, N trailing satellites operate.The strategy prioritizes the creation of a satellite cluster surrounding the leader, avoiding strict configurations to reduce fuel consumption significantly.It is vital to ensure that the relative distances among follower satellites remain within the communication radius, D, of the primary satellite, while sustaining a safe margin beyond the collision radius, d.Additionally, the integration of extraneous satellites into the formation necessitates velocity modulation, particularly deceleration for those exhibiting excessive relative velocities.
To achieve the objectives outlined above, this study will conduct an exhaustive analysis of various factors, including relative coordinates between satellites, velocity considerations, and external disturbances, culminating in a sophisticated control framework.

Collision avoidance control based on artificial potential field method
The artificial potential field technique conceptualizes a virtual potential field enveloping the satellites.This field propels movement from regions of high to low potential, instrumental in circumventing obstacles [12].As illustrated in Figure 2, the two-dimensional field promotes autonomous migration towards areas of low potential, which are the desired regions for maintaining a loose formation.The presence of convex zones around obstacles facilitates autonomous collision evasion.This method combines both attractive and repulsive potentials.The control force on follower satellites correlates with the potential field's gradient, computed relative to position, yielding a force vector toward decreasing potential.The aggregate force on follower satellite i in operational space is: Where upi is the chief's repulsive force, uxi the attractive force, uli the repulsion from formation's lead satellites, ufi the inter-follower repulsion, and uzi the deceleration control.This approach offers an efficacious paradigm for navigational control and collision evasion in satellite formations.

Control strategy considering relative positioning
To prevent collisions between satellites and ensure they remain within the communication range of the leading satellite, this communication range is divided into four crucial zones: Collision, Collision Risk, Free Flight, and Escape Risk.As shown in figure 3, Repulsive forces are operative within the Collision Risk zone, with attractive forces prevailing in the Escape Risk zone.Additionally, the Free Flight zone necessitates collision avoidance maneuvers relative to other lead satellites, identified by the vectors lm.
The repulsive potential from the primary satellite, attractive potential, and the repulsive potential from other leading satellites in the formation are denoted as ϕpi, ϕxi and ϕli respectively.These potentials adhere to a uniform structure, expressed as follows: Table 1.Parameters for potential functions and control forces.
Here, control gains Ki1, Ki2 and Kim regulate force intensity.The ai and aim elements, derived from the inter-satellite communication matrices Al and Am respectively,while constant C is derivable from boundary conditions.The parameters σi1,σi2 and σim facilitate a gradual increase from zero for both repulsive and attractive forces, mitigating over-control and abrupt control transitions.

Collision avoidance strategy considering relative velocity
In satellite formations, significant relative velocity between follower satellites can induce excessive propellant use under conventional artificial potential field controls.To mitigate collision risks, a deflection-enabled repulsive potential function is essential.The relative velocity between satellites i and j is articulated as vij ,computed by: The repulsive force within the potential field, ufi , is defined thus: / / / and 0 and 0 Where Kij denotes the constant dictating the intensity of the control potential field.The deflection force, ufij ⊥ , is oriented perpendicularly to the repulsive force ufij , co-planar with ufij and vij, forming an acute angle with vij.The magnitude and direction of the deflection force are collaboratively determined by the relative position and velocity between satellites, proficiently navigating satellite i to evade impending collisions with other follower satellites.
As mission requirements escalate, the existing satellite constellation may prove insufficient, necessitating the integration of additional external satellites.Inadequate deceleration protocols could preclude these satellites from maintaining positions within the leader's communicative reach.Moreover, if follower satellites neglect to decelerate, their trajectories would oscillate substantially between repulsive and attractive zones.This erratic motion not only precipitates excessive fuel consumption but, in certain scenarios, may also propel followers beyond the leader's communication perimeter.Consequently, this study proposes a smoothly transitional drag potential field, articulated as follows: Here, Kzi is the control gain correlating to uzi.

Numerical simulation
This section evaluates the proposed collision avoidance control strategies in multi-satellite loose formation flight through comprehensive numerical simulations.The simulations are grounded in accurately replicated orbital dynamics and account for various initial conditions.By simulating satellites' three-dimensional trajectories, observing relative distance alterations, and assessing the and and control strategies' impact on velocity and acceleration, we ascertain the strategies' competence in maintaining formation stability and preventing potential collisions.

Simulation results
The simulation initially unveils the three-dimensional motion trajectories of the follower satellites under the proposed control strategy.All satellites are capable of navigating their trajectories safely without collisions, as depicted in Figure 4.Even with the integration of an external entity, Satellite 3, into the satellite formation, the control efficacy remains intact, ensuring all satellites operate within the anticipated parameters, as illustrated in Figure 5. Subsequently, the focus shifts to the temporal evolution of the distances between the satellites and their respective leaders.Figure 6 unequivocally illustrates that, notwithstanding orbital perturbations and inherent dynamic intricacies, the control methodology effectively sustains each satellite within a designated safe and stable distance bracket.
Additionally, relative inter-satellite distances play a pivotal role in ensuring the safety of formation flight.As depicted in Figure 7, the implemented control strategy effectively sustains each inter-satellite distance beyond the prescribed safety margins.

Monte Carlo simulation
This methodology facilitated the execution of numerous experimental iterations, each featuring initial conditions randomly sampled from a Gaussian distribution.During the simulations, critical determinants of the outcomes encompassed the satellites' initial coordinates, the initial statuses and velocities of dynamic impediments, and the fixed positions of static obstructions.The investigation proceeded with a random assortment of 500 distinct initial setups, applying the articulated control paradigm throughout the simulations.Subsequent statistical analyses of these iterations revealed a performance distribution of the control approach, depicted in Figure 8.The entirety of the results conformed to a Gaussian distribution, indicating consistent behavior.Notably, in every simulation, there was a 100% likelihood that the follower satellites maintained a minimum distance from the primary satellite exceeding the established safety margin of 200 meters.Furthermore, the likelihood stood at 98.6% that they preserved a safety-compliant proximity relative to other follower satellites.These statistics affirm the efficacy and stability of the control methodology introduced.

Conclusions
The simulations conducted underscore the effectiveness of the proposed collision avoidance control strategy.The outcomes indicate that the strategy excels in maintaining safe distances, adjusting velocities, and managing complex dynamics.However, there are certain limitations and avenues for potential refinement.For instance, the current strategy does not account for communication delays and collaborative mechanisms among satellites.

Figure 1 .
Figure 1.Dynamics Environment of Satellite Formation System.

Figure 2 .
Figure 2. Artificial Potential Field.Figure 3. Zone division near the reference satellite

Figure 3 .
Figure 2. Artificial Potential Field.Figure 3. Zone division near the reference satellite states of the follower satellites.3646 -3066] T [3.4 -5.3 4.4] T To demonstrate the applicability of the control methods for intricate space missions, we posit that Satellite 1 and Satellite 2 are already within the vicinity of the lead satellite.Satellite 3, originating from beyond the communication range, is slated to assimilate into the formation.The initial conditions of the follower satellites, including both position and velocity vectors, are cataloged in Table

Figure 8 .
Figure 8. Closest distances to reference satellite and other followers distribution