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Decentralized Power-Optimal Coordination for Spacecraft Swarms Using Time-Varying Magnetorquer Actuation

Yuta Takahashi, Shin-ichiro Sakai

arXiv:2610.02118v1eess.SYcs.MAcs.SY

TL;DR

Large spacecraft swarms need formation control despite limited onboard resources and magnetic interactions that can degrade tracking. This paper proposes decentralized power-optimal graph, frequency-grouping, and control design; its closed-loop results report precise formation holding, while bounded-error approximate integration supports long-horizon reconfiguration.

  • Problem

    Magnetic interactions couple nearby spacecraft and can degrade trajectory tracking, while whole-swarm grouping at every control update is undesirable with limited onboard computing.

    Source: §I · p. 1

  • Method

    A decentralized framework jointly designs the interaction graph and frequency grouping, allocates dipoles by group, and uses momentum-preserving control.

    Source: §I · pp. 1–2

  • Results

    In the closed-loop reconfiguration, formation error held to a 0.61 mm median under a 64 nN median commanded force, and mission-average coil power was 74 mW per spacecraft.

    Source: §3 · p. 9

  • Takeaways & Limitations

    The framework’s approximate integration, with a proven error bound, carries the coordination method to a long-horizon orbital reconfiguration.

    Source: §I · p. 2, §VI · p. 10

Abstract

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This paper presents a decentralized power-optimal coordination framework for magnetically actuated spacecraft swarms. Swarms that form large space structures overcome the aperture limit set by the launch vehicle and hold their shape on solar-generated power alone. Magnetic actuation is propellant-free and generated by a magnetorquer, which is commonly used for attitude control. However, every spacecraft interacts with every other within range, and its effect depends on the actuation power and a carrier frequency. We therefore design a decentralized power-optimal framework to jointly derive the interaction graph, frequency grouping, and controller gains. Our decentralized controller preserves angular momentum, which is a nonholonomic constraint. Then, this framework for connected groups whose memberships overlap across carriers guarantees that the relative position errors, the absolute attitude errors, and the imbalance of the reaction-wheel momenta converge to the desired states under the decentralized power-optimal allocation. A closed-loop simulation of a thousand spacecraft with the complete alternating-current interaction confirms the framework. A fast approximate integration with a proven error bound extends the framework to a long-horizon orbital reconfiguration held with high precision.

I. Introduction

The introduction motivates decentralized magnetic coordination for large spacecraft swarms and proposes a framework for power-optimal graph selection, frequency grouping, and control. The preliminaries describe the magnetic actuation model and a momentum-preserving command construction.

I. Introduction

  • Magnetic interactions couple nearby spacecraft, can degrade trajectory tracking, and make whole-swarm regrouping at every update undesirable under limited onboard computing.The open problem is a simple grouping that secures power optimality while bounding interactions between groups.Source: §I · p. 1
  • The framework selects the interaction graph and frequency grouping per reference frame, while dipole allocation and each group’s control use only member states.Its contributions include momentum-preserving convergence, tuning that bounds unintended interactions and loop delay, and approximate integration with a proven error bound.Source: §I · pp. 1–2

II. Preliminaries: Magnetic Swarm Control Model

  • Magnetorquer dipoles are sinusoidally modulated, and first-order averaging yields electromagnetic force and torque for the swarm model.The supplied model passage introduces dipoles as μ_j(t) = s_j sin(ωt) + c_j cos(ωt).Source: §II · p. 2
  • Magnetic interactions are internal, so pairwise actuation preserves the swarm’s linear and angular momentum.For multiple groups, groups sharing a carrier are indexed separately, and groups on different frequencies decouple in first-order averaged dynamics.Source: §II · p. 2
  • The kinematics control lemma maps commands through the null space of the momentum constraints to preserve linear and angular momentum while retaining commandability.The constraint matrix represents momentum conservation, and the command dimension is 6n + 3m − 6.Source: §II · p. 2
  • The optimal dipole allocation is certified by a dual-cost bound, while the inverse baseline fixes one endpoint amplitude and uses the best of K random initializations.The implemented allocation power lies between the dual lower bound and γ_W times that bound.Source: §II · p. 2

III. Problem Formulation

The formulation represents swarm guidance and coordination with a network-based controller, then treats time-averaging error and cross-group actuation as disturbances. It also constrains consensus gains through implementation delay.

III. Problem Formulation

  • The formulation accommodates general guidance and trajectory-generation laws, using a network-based PD controller as a representative example.The swarm state stacks positions, attitude parameters, and angular velocities, while an undirected graph defines neighborhoods and its Laplacian.Source: §III · p. 2

A. Cross-Group Disturbances and Loop Delay

  • Multi-frequency dipoles and wheel torques realize the swarm command, while time-averaging error enters the dynamics as a disturbance.The complete interaction includes cross-group terms at twice the channel frequency.Source: A. Cross-Group Disturbances and Loop… · p. 2
  • The time-averaging and cross-group errors have zero mean over a period, and their magnitudes are bounded using geometry and power.The passage states a geometry-and-power bound for the oscillatory interaction and proves a counterpart for cross-group disturbance.Source: A. Cross-Group Disturbances and Loop… · p. 3
  • The cross-group disturbance bound uses a root-sum-square envelope of magnetic interaction norms over cross-group spacecraft pairs.The envelope takes the supremum of the interaction-matrix norm over distances at least d.Source: A. Cross-Group Disturbances and Loop… · p. 3
  • Pairwise disturbance terms are bounded using spacecraft power allocations and the distance envelope, then summed across the two groups.The derivation applies the triangle and Cauchy–Schwarz inequalities, then uses the allocation objective and dual-cost bound.Source: A. Cross-Group Disturbances and Loop… · p. 3
  • Summing bounds over group pairs yields the aggregate cross-group disturbance bound.The supplied passage identifies this as the result obtained by summing the group-pair terms.Source: A. Cross-Group Disturbances and Loop… · p. 3
  • Holding actuation amplitudes over an update interval contributes a zero-order-hold delay of T_u/2 to the consensus loop.The classical delay condition bounds the velocity-feedback rate using the largest Laplacian eigenvalue and loop delay.Source: A. Cross-Group Disturbances and Loop… · p. 3
  • The loop delay combines communication latency and hold delay, while the number of commensurate carriers is bounded by the carrier and divergence-delay limits.The passage gives N_f < (τ_div − τ_comm)ω_max/π as the condition for remaining below measured divergence delay.Source: A. Cross-Group Disturbances and Loop… · p. 3

B. Decentralized Power-Optimal Grouping

The grouping design jointly chooses the commanded interaction graph and a connected partition into frequency groups to optimize power-related criteria.

B. Decentralized Power-Optimal Grouping

  • The design jointly selects the commanded interaction graph and its partition into frequency groups.The admissible partitions contain connected groups of at most n_max spacecraft, and N_f(Z) counts the distinct carriers required per neighborhood.Source: B. Decentralized Power-Optimal Grouping · p. 3

IV. Decentralized Coordination Framework

The framework constructs coordination in stages: it selects a delay-admissible graph, groups spacecraft for magnetic actuation, and derives group commands. Its graph criteria balance actuation power, time-averaging error, and consensus time.

IV. Decentralized Coordination Framework

  • The framework solves the design in three stages: neighbor selection, magnetic interaction grouping, and swarm kinematics control.The stages select a delay-admissible graph, partition nodes into frequency groups, and derive force-and-torque commands; the evaluation uses a 1000-spacecraft, 60-second reconfiguration.Source: §IV · p. 3

A. Stage 1: Controller Neighbor Selection for Nodes

  • Controller neighbor selection reduces actuation power, time-averaging error, and consensus time through three graph criteria.The first criterion admits short edges because magnetic coupling decays with the fourth power of distance.Source: A. Stage 1: Controller Neighbor… · p. 3, §1 · p. 3

1. Criterion 1: Fourth-Power Distance Law

  • Pairing each spacecraft with its 30th-nearest rather than nearest neighbor raises median distance from 0.65 to 1.39 m and mission energy 7.3 times.The dipole model is reported accurate to a few percent in the median at the flight scale considered.Source: §1 · p. 3

2. Criterion 2: Delay-and-Hold Divergence Threshold

  • The second criterion admits a k-nearest-neighbor graph only when its worst measured divergence delay exceeds the implementation loop delay.At common consensus stiffness, the delayed-feedback mode depends on λ_N/λ_2; the passage also contrasts graph-selection scaling of k^4.7 with k^1.9.Source: §2 · p. 3

3. Criterion 3: Information Graph for Consensus

  • The third criterion uses long edges in an information graph to shorten consensus time without assigning those edges magnetic force.On a connected commanded graph, the desired forces are realized by a minimum-norm edge flow solved once per update interval.Source: §3 · pp. 3–4

B. Stage 2: Magnetic Interaction Grouping for Edges

The grouping stage limits magnetic interaction groups to reduce power and carrier count, then uses effective-distance costs and constrained agglomeration to choose groups. Its repartition procedure terminates at a locally stationary partition with a stated power bound.

B. Stage 2: Magnetic Interaction Grouping for Edges

1. Effective-Distance Score and Agglomeration

  • Effective distance equates the power of the reference wrench at that distance with the demanded wrench’s power at the actual separation.Edge power grows with the fourth power of distance and linearly with demanded wrench; alignment changes power by at most a factor of two.Source: §1 · p. 4
  • The algorithm initializes with a minimum-weight perfect matching, then merges connected neighboring groups only when repartitioning decreases the dual cost.Candidate repartitions have group sizes from 2 to n and diameter at most ρ̄; the group score uses the minimum effective edge distance and maximum within-group separation.Source: §1 · p. 5
  • Algorithm 1 terminates after finitely many cost-decreasing repartitions at a locally stationary partition Z_final, with J(Z_final) ≤ J(Z_match).The implemented optimal dipole allocation also satisfies W_ODA(Z_final) ≤ γ_W J(Z_final) at every time.Source: §1 · p. 5

2. Offline Evaluation of the Grouping Stage

The offline evaluation tests demand-direction sensitivity, within-group routing, and the power effects of groups larger than pairs. Grouping at n=4 can exceed pair-based allocation substantially, but the gain is concentrated where neighbors are close and aligned.

2. Offline Evaluation of the Grouping Stage

  • 1.16–1.25×: the grouped power ratio varies within this range as alignment weight wa increases until the grouping result stops changing.The reported comparison is the power ratio over wa in Fig. 4(a).Source: §2 · p. 5, Figure 4 · p. 5
  • 0.07 to 0.98: demand alignment ρg rises from the transient to aligned demands, when routing follows lines of sight and direction changes power by at most 1.05.The evaluation measures alignment within groups; the stated maximum direction effect is 2 − ρg² = 1.05.Source: §2 · p. 5
  • 1.39 in the median and 15.9 at the maximum: grouping at n=4 exceeds the ODA pairing on mission samples, while best-of-K inverse remains below ODA pairing in the median even at K=10^5.The authors limit group size by diameter because gains are large only where neighbors are close and aligned; larger groups reduce group count but raise the cross-group bound Φ_AC.Source: §2 · p. 5, Table I · p. 6
  • The octahedron gains less than the 5-clique despite having more edges, indicating that grouping gains depend on edge directions, not edge count.The comparison also includes isolated cliques of three to five spacecraft under random commands.Source: §2 · pp. 5–6

C. Stage 3: Swarm Kinematics Controller for Groups

The group controller extends momentum conservation to the full swarm and drives position, attitude, and wheel-momentum errors toward their targets. Its convergence and error bounds depend on connected group structure, Lipschitz amplitudes, carrier frequency, and cross-group disturbance.

C. Stage 3: Swarm Kinematics Controller for Groups

  • The full-swarm momentum constraint is reduced through embedded group subspaces, leaving 6(Ng −1) directions that carry relative group momenta under disjoint partitions.The group embeddings lie in the null space of the full momentum constraint.Source: C. Stage 3: Swarm Kinematics… · pp. 6–7
  • Each group’s error dynamics use stacked position and MRP errors with proportional and velocity feedback, while wheel exchange acts as a consensus step that does not increase the Lyapunov function.The velocity feedback is nonpositive in the Lyapunov analysis, and connected wheel coupling yields momentum consensus.Source: C. Stage 3: Swarm Kinematics… · p. 6
  • With a connected commanded graph and disjoint group partitions on each carrier, positions and attitudes converge to reference and wheel momenta to their uniform share.Theorem 4 assumes each group follows the specified controller.Source: C. Stage 3: Swarm Kinematics… · p. 6
  • With Lipschitz-continuous allocated amplitudes, state errors remain bounded, shrink with carrier frequency ω0, and grow with cross-group disturbance.For the AC-driven system, first-order averaging gives an O(1/ω0) finite-time discrepancy, and input-to-state stability extends the bound to all time.Source: C. Stage 3: Swarm Kinematics… · pp. 6–7

D. Calculation Mitigation with Frozen Approximation

The frozen-geometry approximation fixes interaction matrices and amplitudes over update intervals of whole common periods. For a connected graph, the proposition bounds its steady-state position and attitude errors relative to the actual trajectory.

D. Calculation Mitigation with Frozen Approximation

  • The multiple carriers require small integration steps and make direct integration computationally expensive, motivating the approximation.Source: D. Calculation Mitigation with Frozen… · p. 7
  • For a connected graph, freezing each interaction matrix at the interval start and holding amplitudes over whole common periods bounds steady-state position and attitude errors.The bound depends on relative displacement, edge force and torque bounds, and the maximum number of magnetic partners.Source: D. Calculation Mitigation with Frozen… · p. 7
  • The approximation’s residual comes from interaction-geometry motion: force errors on an edge cancel between its spacecraft, but torque errors do not.The force error scales with the d^-4 interaction law, while the torque error scales with d^-3.Source: D. Calculation Mitigation with Frozen… · p. 7

V. Application to Spacecraft Swarm Reconfiguration

The reconfiguration simulations assess framework validation and MIG group-size tuning. They show lower power for larger groups and compare the decentralized allocation with a best-of-K inverse baseline.

A. Tower Reconfiguration for Framework Validation

1. MIG-Tuning of Group-Size Limit

  • At n=7, dipole power is 40% of n=2 and peak carriers fall from ∼3400 to ∼840; position error falls from 0.47 to 0.09 times median spacing.The median spacing is 0.38 m; tracking improves monotonically as n increases.Source: §1 · pp. 7–8, Figure 6 · p. 8

V. Application to Spacecraft Swarm Reconfiguration

  • The Tokyo Tower terminal formation is the visual target, while Fig. 6 compares closed-loop trajectories with the averaged model during MIG tuning.Source: Figure 6 · p. 8

1. MIG-Tuning of Group-Size Limit

  • ODA lowers total power against best-of-K inverse allocation (K=1000) in all 57 reconfiguration frames at every tested group-size limit.Source: §1 · p. 7, Table II · p. 8

2. CNS-Tuning of Neighbor Count

The tuning results expose trade-offs among neighbor count, controller gains, tracking accuracy, carrier count, and power. With tuned settings, the tower reconfiguration tracks closely while using less power than inverse allocation.

2. CNS-Tuning of Neighbor Count

  • Across 601 tower geometries, the worst divergence delay rises from 12.5 ms at k=8 to 42.5 ms at k=24, so sparser graphs tolerate shorter holds.At k=16, the worst divergence delay is 27.5 ms.Source: §2 · p. 8
  • At k=8, the terminal k-NN graph has λ2=0.0092; adding 150 information edges raises it to 0.27, while larger k trades power and carriers for improved tracking.The power follows the fourth power of mean effective distance within 9%; worst tracking error at 120 s is 4.3 m at k=16 versus 8.3 m at k=4.Source: §2 · p. 8

3. SKC-Tuning of Controller Gains

  • The tuned k=8 loop settles to 6 mm tracking error by 120 s, whereas the same gains diverge on k=4; the measured divergence boundary follows the hold-delay gain bound.Tuning uses a 0.21 s hold and gains 4–50 times those of the sweeps; the measured boundary is (kv/ms)λNτ0≈2.7–2.8.Source: §3 · pp. 8–9

4. Tower Reconfiguration with Tuned Settings

  • With tuned settings, the median trajectory follows the averaged model within 0.40 mm, and the wheel exchange reaches a uniform momentum distribution.The magnetic forces realize commanded forces with a median relative error of 4 × 10^-9.Source: §4 · p. 9
  • Inverse allocation carries 2.3 times the disturbances and 2.4 times the power; the duality-gap ratio γW is 1.0 at n=2 and 1.11–1.43 for n∈[3,7].The reported maximum γW is below 2.2.Source: §4 · p. 9, Table III · p. 9

B. Orbital Reconfiguration with Frozen Approximation

The paper applies its coordination framework to a 64-spacecraft orbital reconfiguration and evaluates a frozen-geometry approximation. The simulation reports sub-millimeter median formation error and close agreement with carrier-resolving integration over 24 orbits.

B. Orbital Reconfiguration with Frozen Approximation

  • The reconfiguration uses 64 spacecraft on a 500 km circular orbit, moving from a staggered 4×3×3 formation to an 8×2 grid with 0.3 m pitch.The 24-orbit mission includes one initial-formation orbit, 20 transfer orbits, and three final-grid orbits.Source: B. Orbital Reconfiguration with Frozen… · p. 9, §1 · p. 9

1. Accuracy of the Frozen Approximation

  • Across 2,261 holds, frozen and carrier-resolving loops differ by 0.59 mm at the median and 2.0 mm at most; attitude parameters agree to 2 × 10^-5.The position bound is below 81 mm on this trajectory and 18 mm at the median hold.Source: §1 · p. 9

2. Coordination Settings for MIG, CNS, and SKC

  • Twenty-six information edges raise λ2 from 0.23 to 1.03 initially and from 0.28 to 1.15 at the median hold, widening the gain window from 1.5× to 5.7×.The selected gains are kp=1.6 × 10^-5 N/m and kv=1.0 × 10^-3 N s/m.Source: §2 · p. 9

3. Numerical Results

  • The 64-spacecraft simulation holds median formation error to 0.61 mm, with 74 mW average coil power per spacecraft and a 3.4 W peak for the worst spacecraft.The worst attitude falls below 1° within 15 minutes of the 20-minute slew reference.Source: §3 · p. 9, Figure 8 · p. 10

VI. Conclusions

  • The framework combines momentum-preserving coordination, grouping that reduces carrier count and admits higher consensus gains, and bounded-error approximate integration for long-horizon reconfiguration.The power-minimizing allocation keeps time-averaging error small enough to hold the formation.Source: §VI · p. 10
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