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Decentralized Power-Optimal Coordination for Spacecraft Swarms Using Time-Varying Magnetorquer Actuation
Yuta Takahashi, Shin-ichiro Sakai
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 93%
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Summary
This paper proposes a decentralized power-optimal coordination framework for spacecraft swarms using time-varying magnetorquer actuation. The framework addresses the challenge of forming large space structures by jointly deriving the interaction graph, frequency grouping, and controller gains to minimize power consumption while maintaining stability. It ensures the preservation of angular momentum (a nonholonomic constraint) and guarantees convergence of position, attitude, and reaction-wheel momentum errors. The method includes a Controller Neighbor Selection stage to define the interaction graph based on distance, delay, and consensus criteria, and a Magnetic Interaction Grouping stage to partition the swarm into frequency groups. A closed-loop simulation with 1,000 spacecraft validates the approach, demonstrating its scalability and precision in long-horizon orbital reconfiguration.
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Yuta Takahashi → affiliatedwith → Institute of Science Tokyo
confidence 98% · Yuta Takahashi is with the Department of Mechanical Engineering, Institute of Science Tokyo, Tokyo, Japan
Shin-ichiro Sakai → affiliatedwith → Japan Aerospace Exploration Agency
confidence 98% · Shin-ichiro Sakai is with the Department of Spacecraft Engineering, Japan Aerospace Exploration Agency, Kanagawa, Japan
Decentralized Power-Optimal Coordination Framework → appliedto → Spacecraft Swarm
confidence 95% · This paper presents a decentralized power-optimal coordination framework for magnetically actuated spacecraft swarms.
Decentralized Power-Optimal Coordination Framework → utilizes → Magnetorquer
confidence 95% · This paper presents a decentralized power-optimal coordination framework for magnetically actuated spacecraft swarms... generated by a magnetorquer
Decentralized Power-Optimal Coordination Framework → validateson → Tokyo-Tower formation
confidence 85% · We quantify all three stages on a 1000-spacecraft reconfiguration that transits from a dense trunk to the sparse Tokyo-Tower formation
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Abstract
Abstract: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.
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- Source: https://arxiv.org/abs/2610.02118v1
- Canonical: https://arxiv.org/abs/2610.02118v1
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Decentralized Power-Optimal Coordination for Spacecraft Swarms Using Time-Varying Magnetorquer Actuation Yuta Takahashi Affiliation: Department of Mechanical Engineering, Institute of Science Tokyo, Tokyo, Japan Shin-ichiro Sakai Affiliation: Department of Spacecraft Engineering, Japan Aerospace Exploration Agency, Kanagawa, Japan Abstract 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. †volume: X†issue: X†month: X†paper-type: 1234567†year: 2026†doi: TAES.2026.Doi Number†received: Manuscript received —, 2026; revised —, —; accepted —, —. This work was not supported by any organization.†address: Yuta Takahashi is with the Department of Mechanical Engineering, Institute of Science Tokyo, Tokyo, Japan (e-mail: stateofyuta@gmail.com). Shin-ichiro Sakai is with the Department of Spacecraft Engineering, Japan Aerospace Exploration Agency, Kanagawa, Japan (e-mail: sakai@isas.jaxa.jp). keywords Spacecraft swarms, spacecraft formation flying, cooperative control, graph partitioning. Fig. 1: Magnetically actuated swarm reconfiguration and system diagram of grouping and controller design. I Introduction Decentralized coordination of magnetically actuated multi-agent systems achieves scalable, propellant-free formation control for the resource constraints of space systems and for high-precision tracking of the target trajectory. Forming large space structures from spacecraft swarms is among the most challenging applications of autonomous multi-agent systems: distributed apertures overcome the aperture limit set by the launch vehicle [1, 2] and hold their shape on solar-generated power alone. The actuation has matured as electromagnetic formation flight (EMFF), which uses the fields of spacecraft-mounted magnetorquers (MTQs) for relative position and attitude control [3]. Because the actuator is underactuated, alternating-current (AC) modulation and time-division control realize six-degree-of-freedom (6-DoF) force and torque control [4, 5, 3]. However, the magnetic interaction couples every agent with every other within range; the unintended interactions degrade the tracking of the reference trajectory, and the force decays rapidly with separation and is anisotropic. Since the on-board computing resources of a space system are limited, an optimization that searches the grouping of the entire swarm at every control update is undesirable. Previous studies established a control law that respects the linear- and nonholonomic angular-momentum conservation [3] and graph-based coordination for small formations [6]. A simple grouping that secures power optimality and bounds the interactions between groups within the on-board computation is an open problem. To this end, this paper proposes a decentralized power-optimal control framework for magnetorquer-driven spacecraft swarms. A coordinator selects the interaction graph and frequency grouping once per reference frame, whereas dipole allocation and each group’s control act only on its members’ states. Our contributions are threefold. First, the averaged commands of connected overlapping groups preserve the swarm momentum and converge, with the reaction-wheel momenta reaching a uniform distribution, which we verify in a closed-loop simulation with the complete magnetic interaction. Second, a tuning method selects the interaction graph and the frequency grouping through which the decentralized control acts. The same partition bounds the unintended interactions between spacecraft and keeps the loop delay within the requirement, and the power-minimizing allocation therefore also suppresses the time-averaging error. Third, an approximate integration that freezes the interaction geometry over each update interval carries the framework to a long-horizon orbital reconfiguration with a proven error bound. Together, these results give a decentralized coordination framework whose power optimality and convergence are certified at swarm scale. The overall structure is as follows. Section I introduces the magnetic swarm control model, and Section I formulates the design problem under disturbance using time-averaged error and delay constraints. Section IV constructs the decentralized coordination framework including graph selection, edge grouping, and the swarm kinematics controller. Section V demonstrates the proposed framework in spacecraft swarm reconfigurations along with numerically friendly integration methods with a proven error bound. Section VI concludes the paper. I Preliminaries: Magnetic Swarm Control Model We first consider one group of n spacecraft, each carrying a three-axis MTQ, of which m spacecraft carry three-axis reaction wheels (RWs), 1≤m≤n1≤ m≤ n. Spacecraft j applies the dipole j(t)≜jsin(ωt)+jcos(ωt)∈ℝ3 μ_j(t) s_j (ω t)+ c_j (ω t) ^3. First-order averaging over one period gives the electromagnetic force and torque j←ka≈¯j←k u_j← k^a≈ u_j← k [7, 8] where ¯j←k≜[j←ka⊤j←ka⊤]=μ08πQj←k(k⊗j+k⊗j), u_j← k bmatrix f_j← k^a \\ τ_j← k^a bmatrix= _08πQ_j← k ( s_k s_j+ c_k c_j ), (1) and Qj←kQ_j← k is the matrix function of dj←k=‖j←k‖d_j← k=\| r_j← k\| and the line-of-sight (LOS) matrix CA/Lj←kC^A/L_j← k (see [7]): Qj←k Q_j← k =[CA/Lj←kΨfCA/Lj←kΨτ](CLj←k/A⊗CLj←k/A), = bmatrixC^A/L_j← k _f\\ C^A/L_j← k _τ bmatrix(C^L_j← k/A C^L_j← k/A), (2) Ψf _f =1dj←k4[−600030003030300000003000300], = 1d_j← k^4 bmatrix-6&0&0&0&3&0&0&0&3\\ 0&3&0&3&0&0&0&0&0\\ 0&0&3&0&0&0&3&0&0 bmatrix, Ψτ _τ =1dj←k3[0000010−100020001000−20−100000]. = 1d_j← k^3 bmatrix0&0&0&0&0&1&0&-1&0\\ 0&0&2&0&0&0&1&0&0\\ 0&-2&0&-1&0&0&0&0&0 bmatrix. Since magnetic interactions are internal, ¯j←k u_j← k preserves the linear momentum P and the angular momentum L. Lemma 1 (Kinematics Control [3]). We define the linear- and angular-momentum conservation as A(n,m,)=[⊤I,⊤I]⊤A(n,m; q) ζ=[^I P ,^I L ] where A(n,m,)∈ℝ6×(6n+3m)A(n,m; q) ^6×(6n+3m) and =[˙i⊤,b⊤,b⊤]⊤ ζ=[ r^i , ω^b , h^b ] . Then, arbitrary command c=B(n,m)−1M(n,m)S(n,m)c u_c=B(n,m)^-1M(n,m)S(n,m) u_c, c∈ℝ6n+3m−6 u_c ^6n+3m-6, preserves angular momenta and is commandable where S(n,m)S(n,m) is the null space of A(n,m)A(n,m) and M(n,m)M(n,m) and B(n,m)B(n,m) are the mass–inertia and input matrix. For multiple groups comprising N≫nN n spacecraft, let the leader set lV_l index the groups, p g_p consisting of the spacecraft that share the carrier ωp _p; the kkth dipole is [7] k(t)=∑p∈l[k[p]sin(ωpt)+k[p]cos(ωpt)]. μ_k(t)= _p _l [ s_k[p] ( _pt)+ c_k[p] ( _pt) ]. (3) Groups on different frequencies decouple in the first-order averaged dynamics, and this realizes swarm control [3, 7]. The minimum-power allocation problem is W⋆() W ( g) ≜min,(⊤+⊤)/2 _ s_ g, c_ g ( s_ g s_ g+ c_ g c_ g )/2 (4) s.t.cj .t. u_cj =μ08π∑k∈gpQj←k(k⊗j+k⊗j), = _08π _k∈ g_pQ_j← k ( s_k s_j+ c_k c_j ), with cj u_cj the command input. Its Lagrange dual Jdual()J_dual( g) is a semidefinite program in the multipliers of the n wrench constraints [7]. The power WODA()W_ODA( g) of the implemented optimal dipole allocation (ODA) then satisfies Jdual≤W⋆≤WODA≤γWJdualJ_dual≤ W ≤ W_ODA≤ _WJ_dual, where γW≥1 _W≥ 1 is the duality-gap ratio, the certificate ratio with WODA≤γWJdualW_ODA≤ _WJ_dual on every allocation. The inverse baseline [7, 8] fixes one endpoint amplitude, inverts Qj←kQ_j← k for the other, and keeps the best of K random initializations (best-of-K inverse). I Problem Formulation For a given feasible reference trajectory Pr(t)P_r(t), this work accommodates general guidance and trajectory-generation laws, and we adopt a network-based PD controller as a representative example: F(t) F(t) =−kpL(P−Pr)−kvL(P˙−P˙r), =-k_pL(P-P_r)-k_vL( P- P_r), (5) T(t) T(t) =−kσLΣ−kωLΩ, =-k_σL -k_ωL , where P, Σ , and Ω stack the positions, attitude parameters, and angular velocities of the swarm. We define an undirected graph as =(,ℰ)G=(V,E), N=||N=|V|, that has neighborhoods j=k∣(j,k)∈ℰN_j=\k (j,k) \, adjacency A~ A, and Laplacian L=Diag(A~)−A~L=Diag( A 1)- A with eigenvalues 0=λ1(L)≤⋯≤λN(L)0= _1(L)≤·s≤ _N(L) [9]. I-A Cross-Group Disturbances and Loop Delay The swarm reconfiguration command, such as (5), is realized by the multi-frequency dipoles (3) and wheel torques, and leaves the time-averaging error of (1) as a disturbance. For the dipole (3) and channels p,qp,q, the complete interaction is j←k(t)=¯j←k+∑p∈lj←k2ωp(t)+∑p,q∈lp≠qj←kpq(t), u_j← k(t)= u_j← k+ _p _l d_j← k^2 _p(t)+ _ subarraycp,q _l\\ p≠ q subarray d_j← k^pq(t), (6) with the intra-group disturbance j←k2ωp d_j← k^2 _p at twice the channel frequency [7, 8] and the cross-group disturbance j←kpq d_j← k^pq at Δωpq=ωp−ωq _pq= _p- _q and Σωpq=ωp+ωq _pq= _p+ _q, j←kpq(t)=μ08πQj←k[jkpqcos(Δωpqt)+jkpqsin(Δωpqt)+jkpqcos(Σωpqt)+jkpqsin(Σωpqt)],jkpq=k[p]⊗j[q]+k[p]⊗j[q],jkpq=k[p]⊗j[q]−k[p]⊗j[q],jkpq=k[p]⊗j[q]−k[p]⊗j[q],jkpq=k[p]⊗j[q]+k[p]⊗j[q]. gathered d_j← k^pq(t)= _08πQ_j← k [ A_jk^pq ( _pqt)+ B_jk^pq ( _pqt)\\ + C_jk^pq ( _pqt)+ D_jk^pq ( _pqt) ],\\ A_jk^pq= c_k[p] c_j[q]+ s_k[p] s_j[q],\ \ B_jk^pq= s_k[p] c_j[q]- c_k[p] s_j[q],\\ C_jk^pq= c_k[p] c_j[q]- s_k[p] s_j[q],\ \ D_jk^pq= s_k[p] c_j[q]+ c_k[p] s_j[q]. gathered (7) Both errors have zero mean over a period [3] and supt‖j←k2ωp(t)‖ _t\| d_j← k^2 _p(t)\| is bounded by a geometry and a power [7], and we prove the counterpart for j←kpq(t) d_j← k^pq(t). Lemma 2. Consider a grouping Z=[1,Ng]Z= g_[1,N_g] in which group ℓ g_ carries ωℓ _ and has the dual cost Jℓ=Jdual(ℓ)J_ =J_dual( g_ ). Then, the cross-group disturbance that q g_q exerts on ℓ g_ , ℓ←qAC=∑j∈ℓsupt‖∑k∈qj←kℓq(t)‖2D_ ← q^AC= _j∈ g_ _t \| _k∈ g_q d_j← k q(t) \|_2, summed as ℓ↔qAC=ℓ←qAC+q←ℓACD_ q^AC=D_ ← q^AC+D_q← ^AC, obeys ∑ℓ<qℓ↔qAC≤μ02πγWΦAC(Z)≜μ02πγW∑ℓ<qΞℓq(Z)(Jℓ+Jq), _ <qD_ q^AC≤ _02π _W _AC(Z) _02π _W _ <q _ q(Z)(J_ +J_q), (8) where Ξℓq(Z)≜[∑j∈ℓ∑k∈qγQ2(djk)]1/2 _ q(Z) [ _j∈ g_ _k∈ g_q _Q^2(d_jk)]^1/2 is the root-sum-square of the distance envelope γQ(d)≜sup‖Q()‖2∣‖2≥d _Q(d) \\|Q( r)\|_2 \| r\|_2≥ d\ over the cross-group pairs. Proof. With pj[ℓ]=‖j[ℓ]‖2+‖j[ℓ]‖2p_j[ ]=\| s_j[ ]\|^2+\| c_j[ ]\|^2, the amplitudes jkℓq,…,jkℓq A_jk q,…, D_jk q of (7) have norm at most pj[ℓ]pk[q] p_j[ ]p_k[q], and each pairwise term is bounded by μ04πγQ(djk)pj[ℓ]pk[q] _04π _Q(d_jk) p_j[ ]p_k[q]. Summing over the two groups by the triangle inequality and applying the Cauchy–Schwarz inequality gives ℓ←qAC≤μ04πΞℓq(∑j∈ℓpj[ℓ]∑k∈qpk[q])1/2=μ02πΞℓqWℓWq,D_ ← q^AC≤ _04π _ q ( _j∈ g_ p_j[ ] _k∈ g_qp_k[q] )^1/2= _02π _ q W_ W_q, where ∑j∈ℓpj[ℓ]=2Wℓ _j∈ g_ p_j[ ]=2W_ with Wℓ=WODA(ℓ)W_ =W_ODA( g_ ) from the objective of (4). The arithmetic-geometric mean inequality and Wℓ≤γWJℓW_ ≤ _WJ_ replace 2WℓWq2 W_ W_q by γW(Jℓ+Jq) _W(J_ +J_q), and summing over all group pairs yields (8). ∎ The multi-frequency actuation holds the amplitudes over an update interval Tu=NTT_u=N_TT, which acts on the consensus loop as a zero-order hold, that is, a pure delay of Tu/2T_u/2. The classical delay condition of consensus [10, 9] bounds the velocity feedback rate kλ=kv/msk_λ=k_v/m_s by kλλNτ<π/2k_λ _Nτ<π/2 under the loop delay τ such that kλλ2<(π/2τ)λ2/λN.k_λ _2<(π/2τ) _2/ _N. (9) The loop delay τ is the sum of the communication latency τcomm _comm and the hold delay, which is π/ω0π/ _0 at NT=1N_T=1. Because NfN_f commensurate carriers must fit below the highest admissible carrier ωmax _ , the base frequency obeys ω0≤ωmax/Nf _0≤ _ /N_f, and the loop stays below its measured divergence delay τdivτ^div only if Nf<(τdiv−τcomm)ωmax/πN_f<(τ^div- _comm)\, _ /π. I-B Decentralized Power-Optimal Grouping The design freedom is therefore the commanded interaction graph A~ A on the given node set V and its partition Z into frequency groups, for the power-optimal grouping (A~⋆,Z⋆)=argminA~,Z∈nmax(A~)∑∈ZW⋆() ( A ,Z )= arg~min _ A,\ Z _n_ ( A)\ _ g∈ ZW ( g) (10) s.t.Nf(Z)<(τdiv(A~)−τcomm)ωmax/π, .t. N_f(Z)< (τ^div( A)- _comm )\, _ /π, where Nf(Z)N_f(Z) is the number of distinct carriers per neighborhood that the partition requires, nmax(A~)Z_n_ ( A) collects the partitions of A~ A into connected groups of at most nmaxn_ spacecraft. IV Decentralized Coordination Framework We solve the design problem (10) in three stages: 1. Controller neighbor selection (CNS) selects the delay-admissible nearest-neighbor graph 2. Magnetic interaction grouping (MIG) partitions its nodes into frequency groups along CNS’s edges 3. Swarm kinematics controller (SKC) derives the force-and-torque commands for MIG’s groups We quantify all three stages on a 10001000-spacecraft reconfiguration that transits from a dense trunk to the sparse Tokyo-Tower formation in Tend=60T_end=60 s (Fig. 1). IV-A Stage 1: Controller Neighbor Selection for Nodes CNS fixes the commanded graph A~ A through three criteria to reduce the actuation power, the time-averaging error, and the consensus time. IV-A1 Criterion 1: Fourth-Power Distance Law The first criterion admits short edges. The dipole model Qj←kQ_j← k of (2) decays as djk−4d_jk^-4, and a longer edge therefore demands larger dipoles and a larger actuation power W⋆W in (4). On the reference mission, pairing each spacecraft with its 3030th-nearest neighbor rather than its nearest raises the median paired distance from 0.650.65 to 1.391.39 m and the mission energy 7.37.3 times in Fig. 2(a). Note that this model is accurate to a few percent in the median at the flight scale of Fig. 2(b), where RcoilR_coil is the coil radius and d/2Rcoil≈3d/2R_coil≈ 3 at the median paired distance for Rcoil=0.1R_coil=0.1 m. IV-A2 Criterion 2: Delay-and-Hold Divergence Threshold The second criterion floors the number of neighbors under the loop delay τ of (9). At a consensus stiffness kpλ2k_p _2 common to the candidate graphs, the delayed feedback of the fastest mode scales with λN/λ2 _N/ _2, which falls as the graph densifies. We measure the divergence delay τdivτ^div of the linear part of (14), the consensus loop ¨=−kpLk(t−τ)−kvLk˙(t−τ) x=-k_pL_k x(t-τ)-k_vL_k x(t-τ) on the k-nearest-neighbor (k-N) graph with Laplacian LkL_k at that common stiffness, and take its worst value over the mission geometries. A k-N graph is admissible as A~ A if this worst τdivτ^div exceeds the loop delay of the implementation. Optimizing edge selection does not scale because orienting the Delaunay-triangulation graph toward the leaders via a linear program grows as k4.7k^4.7 in the neighbor count, versus k1.9k^1.9 for the neighborhood-greedy construction of the k-N graph. (a) (b) (c) Fig. 2: Mission energy over the neighbor distance, finite-coil correction with a minimum ratio of 0.800.80 in force and 0.670.67 in torque, and delay margin. IV-A3 Criterion 3: Information Graph for Consensus The third criterion shortens the consensus time with long information edges. The consensus rate of (5) is set by λ2 _2 of the graph that exchanges the states, and long edges raise λ2 _2, whereas realizing them magnetically is prohibitive because the interaction of (2) decays as d−4d^-4. We therefore form (5) on the information graph, the commanded graph A~ A with long edges added, and these edges carry no magnetic force. Any set of long edges serves this purpose, and a fixed set is used throughout. The per-spacecraft forces F of (5) sum to zero over the swarm, and on a connected A~ A with incidence matrix E they are realized exactly by the minimum-norm edge flow =E+ f=E^+ F with E=E f= F, solved once per update interval. IV-B Stage 2: Magnetic Interaction Grouping for Edges MIG partitions the swarm into frequency groups of at most nmaxn_ to reduce the power and the number of carriers. 1: Input N,N,A~,nmax,ρ¯,R⋆ r_N, u_N, A,n_ , ρ,R , and ω0 _0 such that each reference frame spans an integer number of common periods T; set the force cutoff and same-color separation to the effective radius R⋆R . Form the minimum-weight no-singleton pairing using deff,jkd_eff,jk 2: for n=3,…,nmaxn=3,…,n_ do 3: repeat 4: Form disjoint neighborhoods of nearby groups with at most 2n2n spacecraft 5: for each neighborhood S in parallel do 6: Enumerate connected partitions π satisfying 2≤||≤n2≤| g|≤ n and ρ()≤ρ¯ρ( g)≤ ρ 7: Minimize the far-field dual cost ∑∈πJdual() _ g∈πJ_dual( g) and commit only a decrease 8: end for 9: until no neighborhood reduces the cost 10: end for 11: return the partition Z Algorithm 1 Agglomerative grouping. IV-B1 Effective-Distance Score and Agglomeration We score a candidate edge by its effective distance deff,jkd_eff,jk. The power of an edge grows with the fourth power of the distance and linearly with the demanded wrench in (2). We therefore define the effective distance as the distance at which the reference wrench urefu_ref costs the same power as the demanded wrench at djkd_jk, deff,jk=djk‖~jk‖1/4uref1/4,~jk=[jk/γf,jkjk/(djkγτ,jk)]d_eff,jk=d_jk \| u_jk\|^1/4u_ref^1/4,\ u_jk= bmatrix f_jk/ _f,jk\\ τ_jk/(d_jk _τ,jk) bmatrix (11) where γf,jk=∥djk4Qj←k[:3,:]∥2 _f,jk=\|d_jk^4Q_j← k[1\!:\!3,:]\|_2 and γτ,jk=∥djk3Qj←k[:6,:]∥2 _τ,jk=\|d_jk^3Q_j← k[4\!:\!6,:]\|_2 are the gains of the force and torque blocks, and the torque term has the extra djkd_jk of its d−3d^-3 law. The alignment changes the power by at most a factor of two. With ρ the cosine of the angle between the demanded force and the line of sight, the power of a pair is P/P(ρ=1)=2−ρ2≤2P/P(ρ=1)=2-ρ^2≤ 2. This factor becomes at most 21/4≈1.192^1/4≈ 1.19 in deffd_eff and 21/32^1/3 on torque. The exact alignment is evaluated in the dual cost at every merge. We also use this score for groups, since a group routes its demand along the lines of sight and does not cancel it. For existing groups, we define deff(p,q)=mindeff,jk∣j∈p,k∈q,(j,k)∈ℰ(A~)d_eff( g_p, g_q)= \d_eff,jk j∈ g_p,\,k∈ g_q,\,(j,k) ( A)\ and the group diameter ρ()=maxj,k∈djkρ( g)= _j,k∈ gd_jk. Algorithm 1 grows the groups with this score. We first form the minimum-weight perfect matching ZmatchZ_match on deff,jkd_eff,jk by the blossom algorithm [11] in O(N3)O(N^3) time. We then merge neighboring groups only when the dual cost J(Z)=∑∈ZJdual()J(Z)= _ g∈ ZJ_dual( g) decreases. The candidate set of a group g is the connected repartitions of g and its neighboring groups in A~ A into groups of size 22 to n and diameter at most ρ¯ ρ. A partition is locally stationary when no candidate decreases J. Proposition 1. Consider Algorithm 1 that commits only the repartitions that decrease J. Then the algorithm terminates after finitely many repartitions at a locally stationary partition ZfinalZ_final with J(Zfinal)≤J(Zmatch)J(Z_final)≤ J(Z_match). Moreover, the ODA satisfies WODA(Zfinal)≤γWJ(Zfinal)W_ODA(Z_final)≤ _W\,J(Z_final) at every time. Proof. Every candidate set is finite, and every committed repartition strictly decreases J≥0J≥ 0. Only finitely many repartitions are committed, which gives the first claim. Summing the certificate WODA()≤γWJdual()W_ODA( g)≤ _WJ_dual( g) of Section 1 over the groups of ZfinalZ_final yields the second claim. ∎ IV-B2 Offline Evaluation of the Grouping Stage (a) (b) Fig. 3: Grouping mechanisms: (a) routing and the alignment penalty; (b) ρg _g per group size. (a) (b) Fig. 4: (a) Power ratio over the alignment weight waw_a. (b) Mission ratio W(n=2)/W(Method)W(n=2)/W(Method). We evaluate the grouping offline on the reference trajectory, independently of the closed loop. We check three points, namely the effect of the demand direction on the chosen edges, the routing of the demands within a group, and the power saved by groups larger than a pair. For the first point, we increase the alignment weight waw_a until the result no longer changes, and the grouped power ratio moves only within 1.161.16–1.25×1.25× in Fig. 4. Note that 57%57\% of the competing candidate pairs are within the 21/42^1/4 band of each other, and the demand direction therefore rarely changes which pair is chosen. For the second point, we measure the demand alignment ρg _g, which rises from 0.070.07 in the transient to 0.980.98 once the demands align in Fig. 3(b). The direction can then change the power by at most 2−ρg2=1.052- _g^2=1.05, and the demand is routed along the lines of sight. For the third point, we compare groups with pairs on the mission samples in Table I, where the best-of-K inverse stays below the ODA pairing in the median even at K=105K=10^5, whereas grouping at n=4n=4 exceeds the pairing by 1.391.39 in the median and by 15.915.9 at the maximum. The gain is therefore large only where the neighbors are close and aligned, and we limit the size of a group by its diameter ρ¯ ρ in Algorithm 1. Note that larger groups reduce the number of groups but raise the cross-group bound ΦAC _AC of (8), whose djk−8d_jk^-8 kernel is dominated by the closest cross-group pair, and we therefore limit the group size by nmaxn_ . Remark 1. We also compare isolated cliques of three to five spacecraft and an octahedron under random commands, where the ratio grows with the clique size, whereas the octahedron gains less than the 5-clique despite more edges. The gain therefore depends on the directions of the edges of a spacecraft, not on their number. TABLE I: Power ratios J(Z(2))/J(Z(n))J(Z(2))/J(Z(n)) on isolated cliques and W(Z(2))/W(method)W(Z(2))/W(method). Clique Med. Max Mission Med. Max 3-clique 1.751.75 2.92.9 Inv. K=102K=10^2 0.790.79 0.940.94 4-clique 2.422.42 5.15.1 Inv. K=105K=10^5 0.930.93 0.940.94 5-clique 4.024.02 9.09.0 ODA n=2n=2 1.001.00 1.001.00 Octahedron 3.193.19 6.36.3 Grp. n=4n=4 1.391.39 15.915.9 IV-C Stage 3: Swarm Kinematics Controller for Groups The third stage constructs a decentralized controller that converges to target states while conserving angular momentum. We first generalize the momentum constraint [3] to the full state of N spacecraft and NwN_w wheels: (N,Nw,)=[m1E3m1[1i]×⊤mNE3mN[Ni]×⊤(CI/B1I¯1)⊤(CI/BNI¯N)⊤CI/Bj1⊤CI/BjNw⊤]⊤=[nini],A(N,N_w; q) ζ= [ smallmatrixm_1E_3&m_1[^i r_1]_× \\ & \\ m_NE_3&m_N[^i r_N]_× \\ 0&(C^I/B_1 I_1) \\ & \\ 0&(C^I/B_N I_N) \\ 0&C^I/B_j_1 \\ & \\ 0&C^I/B_j_N_w smallmatrix ]^\! \! ζ= bmatrix P_n^i\\ L_n^i bmatrix,\\ where [⋅]×[·]_× is the skew operator and E3E_3 the identity. We define the local constraint with n members and m wheels: A(n,m,)=[m1E3m1[1i]×⊤mnE3mn[ni]×⊤(CI/B1I¯1)⊤(CI/BnI¯n)⊤CI/Bj1⊤CI/Bjm⊤]⊤=[NiNi].A(n,m; q) ζ= [ smallmatrixm_1E_3&m_1[^i r_1]_× \\ & \\ m_nE_3&m_n[^i r_n]_× \\ 0&(C^I/B_1 I_1) \\ & \\ 0&(C^I/B_n I_n) \\ 0&C^I/B_j_1 \\ & \\ 0&C^I/B_j_m smallmatrix ]^\! \! ζ= bmatrix P_N^i\\ L_N^i bmatrix. (12) Each local group computes S(nℓ,mℓ)S(n_ ,m_ ) from A(n,m,)A(n,m; q) and applies the admissible input of Lemma 1 [3]: ℓ u_ =B(nℓ,mℓ)−1M(nℓ,mℓ)S(nℓ,mℓ)ℓ, =B(n_ ,m_ )^-1M(n_ ,m_ )S(n_ ,m_ ) u_ , (13) ℓ u_ =˙r,ℓ+M¯ℓ−1(C¯ℓr,ℓ−K1,ℓδs,ℓ−K2,ℓδℓ), = v_r, + M_ ^-1 ( C_ v_r, -K_1, \,δ q_s, -K_2, \,δ v_ ), where δs,ℓ,δℓδ q_s, ,δ v_ are the tracking errors, K1,ℓ,K2,ℓ≻0K_1, ,K_2, 0, and M¯ℓ=S⊤MS M_ =S MS and C¯ℓ=S⊤(MS˙+CS) C_ =S (M S+CS) are evaluated at (nℓ,mℓ)(n_ ,m_ ) with C the Coriolis matrix of [3]. Lemma 3. Consider the member embeddings Kℓ∈ℝ(6N+3Nw)×(6nℓ+3mℓ)K_ ^(6N+3N_w)×(6n_ +3m_ ), the selection matrices with orthonormal columns that place the coordinates of the members of ℓ g_ at their positions in the swarm coordinates, stacked into =[K1S(n1,m1)⋯KNgS(nNg,mNg)]S=[K_1S(n_1,m_1)\ ·s\ K_N_gS(n_N_g,m_N_g)]. Then, (N,Nw,)=A(N,N_w; q)S= 0, and for a disjoint partition S has 6N+3Nw−6Ng6N+3N_w-6N_g columns and leaves 6(Ng−1)6(N_g-1) directions of ker(N,Nw,) (N,N_w; q) unspanned. Proof. The embedding gives (N,Nw,)Kℓ=A(nℓ,mℓ,)A(N,N_w; q)K_ =A(n_ ,m_ ; q), and applying it block by block, (N,Nw,) (N,N_w; q)S =[(N,Nw;)K1S(n1,m1)⋯] = [\,A(N,N_w; q)K_1S(n_1,m_1)\ ·s\, ] =[A(n1,m1;)S(n1,m1)⋯]=, = [\,A(n_1,m_1; q)S(n_1,m_1)\ ·s\, ]= 0, since A(nℓ,mℓ,)S(nℓ,mℓ)=A(n_ ,m_ ; q)S(n_ ,m_ )= 0. For a disjoint partition Kℓ⊤Kq=K_ K_q= 0 for ℓ≠q ≠ q, and S has full column rank, 6(Ng−1)6(N_g-1) below dimker(N,Nw,)=6N+3Nw−6 (N,N_w; q)=6N+3N_w-6. ∎ Lemma 3 derives the averaged dynamics of the swarm with the constraint force of the momentum conservation: M˙+C=B∑ℓKℓℓ−(N,Nw,)⊤,M ζ+C ζ=BΣ _ K_ u_ -A(N,N_w; q) η, where M, C, and B are block diagonal over the spacecraft and wheels. On the zero-momentum leaf of every group, = ζ=S v with v stacking the reduced velocities ℓ v_ . Multiplying the dynamics from the left by ⊤S removes the constraint force, since ⊤⊤=S A = 0, and the block-diagonal structure with Kℓ⊤Kq=0K_ K_q=0 for ℓ≠q ≠ q decouples the result over the groups, M¯ℓ˙ℓ+C¯ℓℓ=M¯ℓℓ M_ v_ + C_ v_ = M_ u_ . Substituting (13) gives the error dynamics of each group, M¯ℓδ˙ℓ+C¯ℓδℓ=−K1,ℓδs,ℓ−K2,ℓδℓ, M_ δ v_ + C_ δ v_ =-K_1, δ q_s, -K_2, δ v_ , (14) where δs,ℓδ q_s, stacks the position errors δjδ r_j and the MRP errors δjδ σ_j of the members and K1,ℓ=diag(kpE,kσE,)K_1, =diag(k_pE,k_σE, 0). Theorem 4. Assume that the commanded graph is connected, that the groups of each carrier form a disjoint partition, and that each group follows (14). Then all positions and attitudes converge to the reference, and the wheel momenta to the uniform share limt→∞ji=(i−di)/Nw,j=1,…,Nw. _t→∞ h_j^i=( L^i- L_d^i)/N_w, j=1,…,N_w. (15) Moreover, if the allocated amplitudes are Lipschitz continuous in the state, the state errors remain bounded, shrinking with the carrier frequency ω0 _0 and growing with the cross-group disturbance in (8). Proof. Memberships overlap across carriers through (3) and superpose, and the averaged interaction (1) is additive over carriers. Then, a spacecraft in several groups receives the sum of their commands, its wheels absorb the sum of their shares, and the sums below run over spacecraft. Let V=∑ℓVℓV= _ V_ with Vℓ=δℓ⊤M¯ℓδℓ2+kp2∑j∈ℓ‖δj‖2+2kσ∑j∈ℓln(1+‖δj‖2)V_ = δ v_ M_ δ v_ 2+ k_p2 _j∈ g_ \|δ r_j\|^2+2k_σ _j∈ g_ (1+\|δ σ_j\|^2) [3]. Along (14), the skew symmetry of M¯˙ℓ−2C¯ℓ M_ -2 C_ and the attitude kinematics cancel the K1,ℓK_1, terms, and between exchanges V˙=−∑ℓδℓ⊤K2,ℓδℓ≤0 V=- _ δ v_ K_2, δ v_ ≤ 0. The wheel exchange is the implicit consensus step (E+kξTuℒh⊗E3)i+=i(E+k_ξT_u\,L_h E_3)\, h^i+= h^i with the Laplacian of the wheel-coupling graph ℒhL_h. For any kξTu>0k_ξT_u>0, this does not increase V and it becomes the consensus flow ˙i=−kξ(ℒh⊗E3)i h^i=-k_ξ(L_h E_3) h^i, for which dt‖i‖22=−kξi⊤(ℒh⊗E3)i≤0 ddt\, \| h^i\|^22=-k_ξ\, h^i (L_h E_3) h^i≤ 0. Then, the connected graph makes the largest invariant set the zero-velocity-error set with position and wheel-momentum consensus. LaSalle’s theorem [12] gives convergence, and the conserved total momentum fixes the wheel momenta at (15). The 6(Ng−1)6(N_g-1) directions of Lemma 3 that a disjoint partition leaves unspanned carry the relative momenta of the groups. For the AC-driven swarm, the interaction (6) is smooth on separated configurations and the Lipschitz amplitudes give a unique solution. With all carriers integer multiples of ω0 _0, the closed loop is T-periodic with (1) as its period average, and first-order averaging [13] keeps the true trajectories within O(1/ω0)O(1/ _0) of the averaged ones over finite time. Since the equilibrium of (14) is locally exponentially stable on the level sets of the conserved momenta, a strict converse Lyapunov function exists, and input-to-state stability against the cross-group residual bounded by (8) extends the error bound to all time. The ODA solves a sequence of convex programs with a unique minimizer on separated configurations, whose solution is Lipschitz in the command and the geometry.∎ IV-D Calculation Mitigation with Frozen Approximation The multiple carriers in (6) require small steps and are computationally expensive, motivating the approximation. Proposition 2. Consider every interaction matrix Qj←kQ_j← k of (2) fixed at the start of each interval and the closed loop integrated with the amplitudes fixed over update intervals TuT_u of whole common periods. Then, on a connected graph, its steady-state position and attitude errors with the actual trajectory are bounded by ‖δP‖≤4kmaxqmaxF¯kpλ2,‖δj‖≤3qmaxT¯jkσ,\|δ P\|≤ 4k_ q_ Fk_p _2, \|δ σ_j\|≤ 3q_ T_jk_σ, (16) where qmax=maxeveTu/deq_ = _ev_eT_u/d_e is the largest relative displacement of an edge, vev_e bounds the relative speed over the edge length ded_e, F¯ F bounds the force of every magnetic edge, T¯j=∑e∋j‖j,e‖ T_j= _e j\| τ_j,e\| is the torque of the magnetic edges of spacecraft j, and kmaxk_ is the largest number of magnetic partners of a spacecraft, which exceeds 2k2k when more than k spacecraft select the same neighbor. Proof. The disturbances j←k2ωp d_j← k^2 _p and j←kpq d_j← k^pq of (6) have zero mean over whole common periods, and the only remaining error comes from the motion of Qj←kQ_j← k, at most veTuv_eT_u on an edge e=(j,k)e=(j,k). For the force, the d−4d^-4 law gives ‖∂Q/∂‖≤4‖Q‖/d\|∂ Q/∂ r\|≤ 4\|Q\|/d and bounds the error of the edge by ‖δe‖≤4qe‖e‖\|δ f_e\|≤ 4q_e\| F_e\| with qe=veTu/deq_e=v_eT_u/d_e. This error acts as +δe+δ f_e on spacecraft j and −δe-δ f_e on spacecraft k, and the per-spacecraft errors δδ F therefore sum to zero. Summing at most kmaxk_ incident edges gives ‖δj‖≤4kmaxqmaxF¯\|δ F_j\|≤ 4k_ q_ F. In steady state the position errors of (5) satisfy kpLδP=δk_pL\,δ P=δ F, the zero sum places δδ F orthogonal to the kernel of L on a connected graph, and the smallest nonzero eigenvalue λ2 _2 of L gives the first bound. Likewise, the d−3d^-3 law bounds the torque error of edge e on spacecraft j by 3qe‖j,e‖3q_e\| τ_j,e\|. These errors are not equal and opposite on the two members, and they act on each body directly with ‖δj‖≤3qmaxT¯j\|δ T_j\|≤ 3q_ T_j, and the attitude command settles at kσδj=δjk_σδ σ_j=δ T_j.∎ V Application to Spacecraft Swarm Reconfiguration This section verifies the proposed framework on two illustrative examples. Note that the simulation integrates the full dynamics by fourth-order Runge–Kutta and re-forms the groups at every frame. The amplitudes are updated every common period T=2π/ω0=0.53T=2π/ _0=0.53 s of the base carrier, and the step Δt=T/(Nsubmaxpνp) t=T/(N_sub _p _p) resolves the highest carrier with NsubN_sub sub-steps. V-A Tower Reconfiguration for Framework Validation Fig. 5 shows the terminal formation of the 10001000 spacecraft. Each spacecraft has ms=20m_s=20 kg and an isotropic inertia Js=20J_s=20 kg m2, and the dipoles are uncapped, an idealization that ignores the actuator limits of a flight unit. The loop delay is the hold delay τ0/2 _0/2 alone with τcomm=0 _comm=0, the highest carrier is the 15211521st harmonic of ω0 _0 at 7.27.2 kHz, and Nsub=20N_sub=20. (a) (b) Fig. 5: The terminal formation representing the Tokyo Tower (Tokyo Digital Twin Project, C-BY). V-A1 MIG-Tuning of Group-Size Limit We sweep the group-size limit n from two to seven under the fixed commanded graph in Fig. 7, in which every spacecraft commands its eight nearest neighbors. The controller gains of (5) are set to (kp,kv,kσ,kω)=(0.1,0.4,1.2,1.2)(k_p,k_v,k_σ,k_ω)=(0.1,0.4,1.2,1.2) at the 0.530.53 s hold. The dipole power decreases monotonically with n and n=7n=7 operates at 40%40\% of the power of n=2n=2 in Fig. 7(b). MIG partitions that graph, and the peak number of carriers in Fig. 7(f) falls from ∼3400 3400 at n=2n=2 to ∼840 840 at n=7n=7. Against the best-of-K inverse allocation with K=1000K=1000, the ODA lowers the total power in all 5757 frames of the reconfiguration at every n (Table I). The time-averaging error, the root-mean-square AC disturbance force of (6), is not minimized and falls less uniformly over the 5.7×1045.7× 10^4 spacecraft-frames, since the intra-group bound of [7] ties the disturbance to the power without ordering two allocations of the same pair. The tracking improves monotonically with n, and the position error falls from 0.470.47 to 0.090.09 times the median spacing of 0.380.38 m and from 2.62.6 to 0.60.6 for the worst spacecraft in Fig. 7(c). TABLE I: Gain over the best-of-K inverse allocation (K=1000K=1000) in the closed-loop sweep, total power per frame and time-averaging error per spacecraft. n=2n=2 n=3n=3 n=4n=4 Power gain (median) 1.15×1.15× 1.34×1.34× 1.58×1.58× Power gain (range) 1.051.05–1.94×1.94× 1.231.23–2.29×2.29× 1.391.39–2.78×2.78× Error gain (median) 1.04×1.04× 1.28×1.28× 1.54×1.54× Error gain (10–90%) 0.400.40–2.50×2.50× 0.580.58–2.94×2.94× 0.760.76–3.67×3.67× Lower error 54%54\% 73%73\% 83%83\% The wheel momenta converge toward (15) and the power falls by a further factor of two, as the loop only holds the formation in Fig. 7(e). (a) (b) (c) (d) (e) (f) Fig. 6: Reconfiguration for MIG-tuning in Section V-A: closed loop (solid) against the averaged model (dotted). (g) (h) (i) (j) (k) (l) Fig. 7: Reconfiguration for CNS- and SKC-tuning in Section V-A, with the conventions of Fig. 7. V-A2 CNS-Tuning of Neighbor Count Criterion 2 of Section IV-A floors the neighbor count first. Over the 601601 geometries of the tower reconfiguration, at the consensus stiffness calibrated so that the worst geometry of k=20k=20 diverges at 3535 ms, the worst τdivτ^div rises monotonically with k, from 12.512.5 ms at k=8k=8 and 27.527.5 ms at k=16k=16 to 42.542.5 ms at k=24k=24 (Fig. 2(c)); a sparser graph tolerates only a shorter hold at the same consensus rate. With the grouping fixed at n=4n=4, we sweep the symmetrized k-N graph from k=4k=4 to k=16k=16 in Fig. 7. A larger k lengthens the mean effective distance ⟨deff⟩ d_eff of (11), and the power follows its fourth power within 9%9\% in Fig. 7(b). A larger k also raises λ2 _2 at the cost of this power and of more carriers. The worst tracking error among all spacecraft at t=60t=60 s is 1.71.7 m at k=16k=16, whereas it is 2.32.3 m at k=4k=4, and by t=120t=120 s the two differ twofold, 4.34.3 against 8.38.3 m. Even so, the consensus is slow at every k, and the k-N graph gives λ2=0.0092 _2=0.0092 at k=8k=8 in the terminal formation. We therefore add 150150 information edges of Criterion 3, 3%3\% of the edge list, drawn once uniformly at random among the pairs more than 88 m apart in the terminal formation, which raise λ2 _2 to 0.270.27. V-A3 SKC-Tuning of Controller Gains We finally tune the gains of (5) under the grouping fixed at n=4n=4 and the graph fixed at k=8k=8. Since MIG of n=4n=4 cuts the peak carrier count from ∼3400 3400 to 13271327, (9) admits a 2.6×2.6× shorter hold, from 0.530.53 s to 0.210.21 s, and higher gains. The gains are tuned on the linearized averaged model from an LQR initialization to minimize the error within the mission time, and the selected set (kp,kv,kσ,kω)=(5.0,4.0,56,5.0)(k_p,k_v,k_σ,k_ω)=(5.0,4.0,56,5.0) is 44–5050 times the gains of the sweeps. Against the mean target spacing d¯(t) d(t), every lane of Sections V-A-1 and V-A-2 peaks at 1.21.2–5.0d¯5.0\, d, whereas the tuned loop peaks at 0.24d¯0.24\, d and its tracking error settles to 66 m by 120120 s. The same gains diverge on the sparser k=4k=4 graph. In the averaged model the worst-spacecraft error reaches 3.93.9 m at t=60t=60 s and 8989 m at t=120t=120 s, against 0.020.02 m at t=60t=60 s on k=8k=8. Remark 2. The gain limit scales inversely with the amplitude hold. In the averaged model with k=8k=8, the position loop diverges at kv=6.4k_v=6.4, 1616, and 2626 (kp=kv/4k_p=k_v/4) for holds τ0=0.53 _0=0.53, 0.210.21, and 0.130.13 s, respectively. With the hold delay τ=τ0/2τ= _0/2, the bound (9) reads (kv/ms)λNτ0<π(k_v/m_s) _N _0<π, and the measured divergence boundary (kv/ms)λNτ0≈2.7(k_v/m_s) _N _0≈ 2.7–2.82.8 follows it at all three holds. V-A4 Tower Reconfiguration with Tuned Settings Finally, we run the tower reconfiguration with the tuned settings (n=4n=4, k=8k=8 with the 150150 information edges, 0.210.21 s hold), shown in orange in Fig. 7. The magnetic interaction forces realize the commanded forces to a median relative error of 4×10−94× 10^-9, and the resulting trajectory follows the averaged model (1) within 0.400.40 m for the median spacecraft over the run while the worst tracking error of the two loops differs by less than 1%1\%. The wheel exchange achieves uniform distribution of (15), which validates Theorem 4 (Fig. 7(e)). We compare this result with the inverse allocation for K≤K≤ 10, shown in pink in Fig. 7 under the same settings. The inverse allocation carries 2.32.3 times the disturbances j←k2ωp d_j← k^2 _p and j←kpq d_j← k^pq of (6) and 2.42.4 times the power. Over the sweep n∈[2,7]n∈[2,7], the duality-gap ratio γW _W of Proposition 1 is 1.01.0 at n=2n=2 [2] and 1.111.11–1.431.43 for n∈[3,7]n∈[3,7] with max<2.2 <2.2 (Table I), in agreement with [7]. TABLE I: Empirical certificate γW=WODA/Jdual _W=W_ODA/J_dual. n=2n=2 n=3n=3 n=4n=4 n=5n=5 n=6n=6 n=7n=7 Count 130598130598 9748397483 6037860378 3785737857 2296622966 1329113291 Mean 1.0001.000 1.1101.110 1.2181.218 1.3151.315 1.3821.382 1.4311.431 Std 0.0000.000 0.1280.128 0.1900.190 0.2280.228 0.2370.237 0.2390.239 Max 1.0001.000 1.5331.533 1.8121.812 2.0062.006 2.0872.087 2.1332.133 Remark 3. The d−4d^-4 decay decouples distant groups, and the cross-group force and torque disturbances of (8) on the densest frame evaluated within R⋆=5R =5 m reach 0.9980.998 and 0.9940.994 of their values at 88 m, whereas R⋆=3R =3 m leaves 0.9940.994 and 0.9580.958. We drop interactions beyond R⋆=5R =5 m. Carriers are distinct within R⋆R and reused beyond it. V-B Orbital Reconfiguration with Frozen Approximation We next consider spacecraft swarm reconfiguration on an Earth orbit for the 6464 spacecraft of 1.331.33 kg, each with three orthogonal coils of 200200 turns, 20Ω20~ , and 55 cm radius, so that a dipole amplitude μ costs Rμ2/(2N2A2)=4.1Rμ^2/(2N^2A^2)=4.1 W per (A m2)2. These move from a staggered 434^3 cube onto an 828^2 grid of 0.30.3 m pitch on a circular orbit at 500500 km altitude and inclination 30∘30 with the linearized dynamics [6]. Each spacecraft starts with 5∘5 in attitude error and at rest in angular rate. The amplitudes are fixed during Tu=59.8T_u=59.8 s, with the carriers on multiples of 4.754.75 Hz. V-B1 Accuracy of the Frozen Approximation Our orbital reconfiguration takes 2424 orbits, including the initial formation for one orbit, transfers over 2020 orbits, and the final grid for three orbits. Since one orbit TorbT_orb takes 94.794.7 min, neighbors move relative to each other by qe≈2πTu/Torb≈7%q_e≈ 2π T_u/T_orb≈ 7\% of their distance ded_e per 59.859.8 s hold, with 19%19\% at most over the magnetic edges. This keeps the position bound of (16) below 8181 m on this trajectory and 1818 m at the median hold, with kmax=15k_ =15 at k=6k=6, the largest edge force of the minimum-norm routing F¯=56 F=56 nN at the median hold, and λ2=1.15 _2=1.15. The measured difference between the frozen and carrier-resolving loops is 0.590.59 m at the median hold and 2.02.0 m at most over all 22612261 holds, with the attitudes agreeing to 2×10−52× 10^-5 in the attitude parameter. V-B2 Coordination Settings for MIG, CNS, and SKC For MIG and CNS, we set n=4n=4 and k=6k=6 with 2626 routed information edges, which raise λ2 _2 from 0.230.23 to 1.031.03 at the initial formation and from 0.280.28 to 1.151.15 at the median hold. For SKC, an offset along the nadir direction meets a negative stiffness forbitf_orbit per unit mass and drifts along the orbital motion [6], and the gains of (5) must therefore exceed forbitf_orbit on the slowest mode and respect the hold delay on the stiffest, which gives the window forbitmsλ2<kp<msλNTu2 f_orbit\,m_s _2<k_p< m_s _NT_u^2. The 2626 information edges widen this window from 1.5×1.5× to 5.7×5.7×. To satisfy these constraints, we select kp=1.6×10−5k_p=1.6× 10^-5 N/m and kv=1.0×10−3k_v=1.0× 10^-3 N s/m inside 4.8×10−6<kp<2.8×10−54.8× 10^-6<k_p<2.8× 10^-5 N/m, which gives (kv/ms)λNTu=0.6(k_v/m_s) _NT_u=0.6. Likewise, the hold delay bounds the attitude gains by kσ<Js/Tu2k_σ<J_s/T_u^2 and kω<Js/Tuk_ω<J_s/T_u for the spacecraft inertia JsJ_s, and we set kσ=0.6Js/Tu2=3.7×10−7k_σ=0.6J_s/T_u^2=3.7× 10^-7 N m and kω=0.6Js/Tu=2.2×10−5k_ω=0.6J_s/T_u=2.2× 10^-5 N m s for Js=2.2×10−3J_s=2.2× 10^-3 kg m2. V-B3 Numerical Results The calculation result in Fig. 8 holds the formation to 0.610.61 m in the median under a commanded force of 6464 nN in the median and 1.11.1 μ at most. The attitude of the worst spacecraft falls below 1∘1 within 1515 min of the 2020 min slew reference, and after the slew the swarm median settles at 6×10−56× 10^-5 degrees with the worst spacecraft below 0.005∘0.005 on every hold (Fig. 8(e)). The wheels absorb the angular momentum that the reconfiguration releases and converge to the uniform share of (15), ‖−d‖/Nw=0.628\| L- L_d\|/N_w=0.628 mNms. At the end of the mission the median wheel holds this share and the farthest one 0.6360.636 mNms, 1.3%1.3\% apart (Fig. 8(f)). The coil power of Fig. 8(d) is 7474 mW per spacecraft averaged over the mission and 3.43.4 W at the peak of the worst spacecraft. (a) (b) (c) (d) (e) (f) Fig. 8: Orbital reconfiguration of 6464 spacecraft (y along-track, z normal, viewed from the radial direction). VI Conclusions This paper developed a decentralized power-optimal coordination framework for magnetically actuated spacecraft swarms. The designed commands of overlapping groups converge while preserving momentum. Grouping reduces the number of carriers and admits higher consensus gains. The power-minimizing allocation keeps the time-averaging error small enough to hold the formation. The approximate integration with a bounded error carries the framework to a long-horizon orbital reconfiguration. Acknowledgment The first author would like to thank Prof. Hiraku Sakamoto at the Institute of Science Tokyo for technical discussions. The authors acknowledge the Tokyo Digital Twin Project for providing the open 3D building point-cloud data used for visualization in this paper. The authors used Claude to improve the English text. References [1] S. Shim, Y. Takahashi, N. Usami, and S.-i. Sakai, “System design framework for distributed space antennas using electromagnetic formation flight,” 2026, manuscript under review. [2] Y. Takahashi, S. Shim, H. Sakamoto, and S.-i. Sakai, “Power-efficiency and scalability analysis of magnetically-actuated satellite swarms via convex optimization,” arXiv preprint arXiv:2605.06286, 2026. [3] Y. Takahashi, H. Sakamoto, and S.-i. Sakai, “Kinematics control of electromagnetic formation flight using angular-momentum conservation constraint,” Journal of Guidance, Control, and Dynamics, vol. 45, no. 2, p. 280–295, 2022. [4] R. C. Youngquist, M. A. Nurge, and S. O. Starr, “Alternating magnetic field forces for satellite formation flying,” Acta Astronautica, vol. 84, p. 197–205, 2013. [5] Z. Abbasi, J. B. Hoagg, and T. M. Seigler, “Decentralized electromagnetic formation flight using alternating magnetic field forces,” IEEE Transactions on Control Systems Technology, vol. 30, no. 6, p. 2480–2489, 2022. [6] Y. Takahashi, S. Shim, and S.-i. Sakai, “Distance-based relative orbital transition for palm-sized satellite swarm with guaranteed escape-avoidance,” in AIAA Scitech 2025 Forum, 2025, p. 2068. [7] Y. Takahashi and S.-i. Sakai, “Neural power‐optimal magnetorquer solution for multi‐agent formation and attitude control,” IEEE Robotics and Automation Letters, 2026. [8] Y. Takahashi, H. Tajima, and S.-i. Sakai, “Certified coil geometry learning for short-range magnetic actuation and spacecraft docking application,” IEEE Robotics and Automation Letters, 2026. [9] M. Mesbahi and M. Egerstedt, Graph Theoretic Methods in Multiagent Networks. Princeton University Press, 2010. [10] R. Olfati-Saber and R. M. Murray, “Consensus problems in networks of agents with switching topology and time-delays,” IEEE Transactions on Automatic Control, vol. 49, no. 9, p. 1520–1533, 2004. [11] V. Kolmogorov, “Blossom v: a new implementation of a minimum cost perfect matching algorithm,” Mathematical Programming Computation, vol. 1, no. 1, p. 43–67, 2009. [12] H. K. Khalil, Nonlinear Systems. Prentice Hall, 2002, vol. 3. [13] J. A. Sanders, F. Verhulst, and J. A. Murdock, Averaging methods in nonlinear dynamical systems. Springer, 2007, vol. 59. Biography Yuta Takahashi received his B.S. and M.S. degrees in mechanical engineering (ME) in 2019 and 2021 (early graduations), respectively, from the Tokyo Institute of Technology, Japan. He was the recipient of the Best Student Awards 2017 and 2018 (school class rank 1 of 7/149) and 2019–2021 (school class rank 1/140) in ME at Tokyo Tech. He was a Visiting Student Researcher at the California Institute of Technology in 2022 and 2023. He is currently pursuing a Ph.D. degree in ME at the Institute of Science Tokyo, Japan, under the JAXA/ISAS research program. He has also been working at Astroscale as a spacecraft GNC engineer since 2026, designing GNC algorithms for commercial rendezvous-and-docking missions. His research fields include multi-agent and learning-based nonlinear control theory, with primary applications in distributed space and under-actuated systems. He led the avionics team for the space demonstration mission of the membrane-array antenna device “5G/HELIOS” from 2019 to 2022, with launches in 2023 and 2026. He collaborated with Interstellar Technologies Inc., providing the spacecraft swarm guidance and control algorithm from 2023 to 2026. Mr. Takahashi was the recipient of the Best Paper Finalist Award of the GNC Graduate Student Paper Competition at the AIAA SciTech Forum 2025, Best Paper Nominee of the 2nd Space Robotics Workshop at IEEE SMC-IT/SCC 2025, First Place Award at the MTT-Sat Challenge by the IEEE MTT-S in 2023, Miura prize from The Japan Society of Mechanical Engineers in 2021 (1 of 221/31000), Best Group Presentation Award of the AOTULE Conference, which he joined for his B.S. work in 2019, and two conference awards for collaborative research on the hardware implementation of his machine-learning algorithms. Biography Shin-ichiro Sakai is a professor at the Institute of Space and Astronautical Science / Japan Aerospace Exploration Agency (JAXA). He received his Ph.D. in Electrical Engineering from the University of Tokyo in 2000. He designed attitude control systems such as REIMEI, ASTRO-G, HISAKI, and ARASE. He managed JAXA’s “Smart Lander for Investigating Moon” for a pinpoint moon landing.