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Suboptimal Formation Reconfiguration of Satellites Under Input Directional Constraints

Yasuhiro Yoshimura

arXiv:2609.27841v1eess.SY

TL;DR

Satellite reconfiguration must respect thrust-direction constraints arising in missions such as observation and servicing. The paper combines an energy-optimal controller with additional inputs to handle constraints in inertial and leader-fixed frames. Simulations reached the target orbit under both constraints, while added inputs increased energy consumption in the leader-fixed case.

  • Problem

    Formation reconfiguration must account for thrust-direction limits imposed by mission needs such as astronomical observation and plume avoidance.

    Source: §1 · p. 2, §2.2 · p. 7

  • Method

    The method superimposes constraint-satisfying inputs on an energy-optimal controller and compensates for their trajectory effects with a virtual target orbit.

    Source: §3 · p. 7

  • Results

    Simulations reached the target orbit under both frame constraints; in the leader-fixed case, added inputs increased energy consumption 56% over the nominal controller.

    Source: §4 · pp. 17, 19

  • Takeaways & Limitations

    The controller's singularity-free form permits analytical constraint handling, while the virtual target orbit preserves successful reconfiguration despite added inputs.

    Source: §3 · p. 7, §3.1 · p. 12, §5 · p. 21

    For the leader-fixed constraint, the controller is limited near δ̄ ≈ 90°; the paper describes δ̄ ≤ 60° as more reasonable in practice.

    Source: §4 · p. 19

Abstract

from arXiv · show

Proximity operations of satellites such as formation flying and on-orbit servicing offer more advanced missions than missions achieved by a single satellite. In a practical situation of formation flying, thrust directions for keeping and controlling a relative orbit is limited, e.g., for astronomical observation and plume impingement avoidance. The aim of this paper is to provide an energy efficient control method for a formation reconfiguration under input directional constraints with respect to both an inertial and a leader-fixed frames. The proposed controller is designed consisting of two parts: 1) guaranteeing a formation reconfiguration to a desirable formation on the basis of an energy optimal controller and 2) satisfying the input directional constraints by superimposing additional inputs. The analytical form of the control input shows that the input direction forms an ellipse in the leader-fixed frame when a particular boundary condition is satisfied, which is exploited as a nominal controller to take into account the input directional constraints. Due to the singular avoidance of the nominal controller, the additional inputs can be analytically obtained. The effect on the follower trajectory due to the additional inputs is compensated by setting a virtual target orbit, and thus the successful formation reconfiguration is still guaranteed. Some numerical simulation results verify the effectiveness of the proposed method and compare the energy efficiency.

1. Introduction

Satellite formation reconfiguration must account for mission-specific thrust-direction limits while preserving energy efficiency. This paper combines an energy-optimal basis controller with additional inputs to address constraints in inertial and leader-fixed frames.

1. Introduction

  • Astronomical observation and servicing impose thrust-direction limits in inertial and target-relative frames, respectively.The introduction also cites millimeter-scale relative positioning accuracy for PROBA-3 coronagraph observation.Source: §1 · p. 2
  • Earlier constrained-control work addressed along-track thrusts and target-relative constraints, while an inertial-frame method caused thrust-direction oscillation.The cited inertial-frame approach also enabled simultaneous position and attitude control with two thrusters.Source: §1 · pp. 2–3
  • Energy-optimal reconfiguration methods had been developed, but the introduction notes that they gave little attention to input-direction constraints.Source: §1 · p. 3
  • The proposed method combines an energy-optimal nominal controller with additional inputs for inertial- and leader-fixed constraints, avoiding oscillation around a target direction.The added inputs mean the control is not strictly energy optimal.Source: §1 · p. 3

2. Problem Formulation

The problem formulation models a follower's relative motion around a leader in a leader-fixed frame and specifies thrust-direction constraints in inertial and leader-relative frames.

2.1. Hill–Clohessy–Wiltshire equations

  • The leader is assumed to orbit circularly, while the nearby follower is controlled to maintain a desired relative orbit; disturbances such as Earth oblateness and solar radiation pressure are omitted.Source: §2.1 · p. 4
  • The leader-fixed frame uses x for the radial direction, z for orbital angular momentum, and y to complete a right-handed coordinate system.Source: §2.1 · p. 4, Figure 1 · p. 5
  • The model uses in-plane Hill–Clohessy–Wiltshire motion; cross-track motion is decoupled and simple harmonic, and the input can generate arbitrary accelerations.Source: §2.1 · p. 4
  • The state transition matrix relates states across time without inputs, with ct and st defined as cosine and sine of the orbital phase difference.Source: §2.1 · p. 5
  • The initial formation assumes the follower flies around the leader with d = b = 0.For b = 0, the relative orbit is elliptical with |x(t)| / |y(t)| = 0.5 and center (d, 0).Source: §2.1 · p. 6

2.2. Input directional constraints

  • The formulation considers in-plane thruster forces, which can be generated by distributing thrust magnitudes among multiple thrusters.Source: §2.2 · p. 6
  • For the inertial-frame constraint, the desired direction is the positive X-axis, and its thrust-direction angle δ_i is kept at zero throughout the maneuver.Source: §2.2 · p. 7
  • For the leader-relative constraint, the thrust must not point toward the leader, so δ_l must remain outside the forbidden range |δ_l| ≤ δ̄.The angle δ_l is measured between the follower position vector and acceleration vector.Source: §2.2 · p. 7

3. Control Method

The method combines an energy-optimal controller for reaching a desired relative orbit with additional inputs that enforce directional constraints. It compensates for the added inputs’ effects by using a virtual target orbit.

3.1. Energy optimal controller

  • The energy-optimal controller derives Fourier-series inputs from boundary states and terminal time without the singularity that previously barred setting tf=2πN/Ω.That terminal-time choice simplifies later calculations for satisfying directional constraints.Source: §3.1 · pp. 10–12

3.2. Additional input for the constraint with respect to the inertial frame

  • For an inertial-frame desired direction, nominal input forms an ellipse when λ1 = 0; an additional input matches the desired direction’s rotation rate, and reconfiguration uses a virtual target orbit.Source: §3.2 · pp. 12, 13, 15

3.3. Additional inputs for the constraint with respect to the leader-fixed frame

  • For leader-fixed constraints, the method corrects the 90-degree phase mismatch between the follower trajectory and nominal input by adding a y-axis input that changes the ellipse ratio to 2, then compensates the position variation with a virtual target.The elliptically varying input direction is used to keep thrust within the admissible region as the forbidden direction varies.Source: §3.3 · pp. 15–16

4. Numerical simulation results

Simulations reconfigure the follower to the target orbit under both inertial-frame and leader-fixed-frame input-direction constraints. The added inputs satisfy the constraints, with higher costs than the corresponding nominal controllers.

4. Numerical simulation results

  • 1.78 × 10−5 m2/s3 versus 1.60 × 10−5 m2/s3; the inertial-frame case increases energy consumption by 11.2%.The follower reaches the target orbit while the input angle remains at 0 deg throughout the maneuver.Source: §4 · p. 17, Figure 3 · p. 18, Figure 4 · p. 18
  • The inertial-frame simulation uses tf = 2π/Ω; the leader-fixed-frame simulation uses tf = 6π/Ω and δ̄ = 30 deg.The leader is modeled in a circular orbit with Ω = 6.31 × 10−4 rad/s, corresponding to a 10000 km semi-major axis.Source: §4 · p. 17
  • The leader-fixed-frame simulation keeps δl outside the forbidden region throughout reconfiguration, but the controller's stated limitation is δ̄ ≈ 90 deg.The passage says δ̄ ≤ 60 deg would be reasonable in practical plume-avoidance situations.Source: §4 · p. 19, Figure 8 · p. 21
  • 8.34×10−6 m2/s3 versus 5.34 × 10−6 m2/s3; the leader-fixed case increases energy consumption by 56%.The actual input direction becomes elliptical with |ux| / |uy| = 2, and the follower reaches the target orbit.Source: §4 · pp. 17, 19, Figure 6 · p. 20, Figure 7 · p. 20
  • Longer maneuver time is one proposed way to reduce the added energy cost in the leader-fixed-frame case.The second simulation uses a longer terminated time than the first and has a smaller cost function.Source: §4 · p. 19

5. Conclusions

The method combines a nonsingular energy-optimal controller with additional inputs that satisfy directional constraints. Simulations verify reconfiguration under inertial and leader-fixed constraints, with virtual target orbits compensating for added-input effects on the follower trajectory.

5. Conclusions

  • The controller combines an energy-optimal reconfiguration law with additional inputs that enforce input-direction constraints.The Fourier-series-based optimal controller avoids singularities, enabling analytical constraint conditions and virtual target orbits to compensate for added-input effects.Source: §5 · p. 21
  • The additional inputs avoid thrust-direction oscillation around the desired direction, and simulations verify both inertial-frame and leader-fixed-frame cases.For the leader-fixed case, phase difference is compensated with additional inputs.Source: §5 · p. 21
  • Formation control in an elliptic orbit and control under disturbances are identified as future extensions.Source: §5 · p. 21

Appendix A. Fourier Coefficients

The appendix supplements the derived optimal controller with definitions and equations, including matrix A components and a simplification under a specified termination time.

Appendix A. Fourier Coefficients

  • The appendix summarizes definitions of variables in Eq. (24).Source: Appendix A · p. 21
  • It gives analytical expressions for components of matrix A, including A12, A13, A14, A22, A23, A24, and A33.The listed component expressions are provided across the appendix equations.Source: Appendix A · p. 23
  • Setting tf = 2Nπ/Ω simplifies the analytical form of matrix A; the Lagrange multipliers are calculated from Eqs. (21) and (38).Source: Appendix A · p. 24
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