Paper deep dive
Power-Efficiency and Scalability Analysis of Magnetically-Actuated Satellite Swarms via Convex Optimization
Yuta Takahashi, Seang Shim, Hiraku Sakamoto, Shin-ichiro Sakai
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 90%
Last extracted: 7/8/2026, 10:47:48 AM
Summary
This paper introduces a convex-optimization-based evaluation framework for analyzing the power efficiency and scalability of magnetically actuated satellite swarms designed as spaceborne distributed apertures. It addresses the challenge of maintaining formation control under unstable orbital dynamics (e.g., J2 perturbations) by utilizing magnetorquer-generated magnetic interactions for propellant-free position control. The authors transform inherently nonconvex power-consumption constraints into a tractable convex optimization problem, proving global optimality for two-satellite interactions and extending the approach to a decentralized architecture. The analysis demonstrates that increasing the number of satellites improves formation-keeping power efficiency, positioning magnetically actuated swarms as a scalable, power-efficient alternative to conventional few-satellite electromagnetic formation-flight systems.
Entities (7)
Relation Signals (6)
Magnetorquers → generate → Magnetic Field Interactions
confidence 94% · magnetic interactions generated by satellite-mounted magnetorquers (MTQs) provide a promising solution.
Convex Optimization → transforms → Nonconvex Power-Consumption Constraints
confidence 92% · the nonlinearities of the electromagnetic force and torque model lead to a nonconvex power-consumption constraint, making system-level configuration analysis difficult. To address this issue, we develop a convex optimization-based framework
Satellite Swarms → improve → Formation-Keeping Power Efficiency
confidence 90% · The resulting analysis shows that increasing the number of satellites can improve formation-keeping power efficiency.
J2 Gravity Effect → cause → Unstable Orbital Dynamics
confidence 88% · relative motion is continuously affected by unstable orbital dynamics and environmental perturbations, including the J2 effect
Dipole Allocation → optimizedvia → Convex Optimization
confidence 87% · First, we show the global optimality of the power allocation problem for two-satellite formation keeping, even though the original dipole allocation problem is nonconvex.
Electromagnetic Formation Flight → extends → Magnetic Actuation Principle
confidence 85% · electromagnetic formation flight (EMFF) extends this magnetic actuation principle to relative-position control among multiple satellites
Cypher Suggestions (0)
No Cypher suggestions yet.
Abstract
Abstract:This correspondence presents a convex-optimization-based evaluation framework of satellite-swarm-based apertures maintained by magnetic-field interactions. Spaceborne distributed apertures are composed of multiple satellites and are attractive for scientific and commercial missions because their scalability enables high-gain, narrow-beam, and large-aperture capabilities beyond the launch-size limitations. A key challenge is that the long-term maintenance of such virtual structures requires consistent formation control amid unstable orbital dynamics, and magnetic interactions generated by satellite-mounted magnetorquers offer a desirable propellant-free position-control strategy. However, the nonlinearities of the electromagnetic force and torque model lead to a nonconvex power-consumption constraint, making system-level configuration analysis difficult. To address this issue, we develop a convex optimization-based framework to analyze the power consumption of large magnetically actuated satellite swarms. The resulting analysis shows that increasing the number of satellites can improve formation-keeping power efficiency. This indicates that magnetically actuated swarm architectures provide a power-efficient alternative to the conventional few-satellite electromagnetic formation-flight concept for constructing large-scale space systems.
Tags
Links
- Source: https://arxiv.org/abs/2605.06286v1
- Canonical: https://arxiv.org/abs/2605.06286v1
Trouble viewing inline? Open PDF directly →
Full Text
50,500 characters extracted from source content.
Expand or collapse full text
Power-Efficiency and Scalability Analysis of Magnetically-Actuated Satellite Swarms via Convex Optimization Yuta Takahashi1, Seang Shim2, Hiraku Sakamoto1, Shin-Ichiro Sakai3 1 Mechanical Engineering, Institute of Science Tokyo, Meguro-ku, Tokyo 152-8550, Japan2 Department of Space and Astronautical Science, The Graduate University for Advanced Studies, Sagamihara, Kanagawa 252-5210, Japan3Spacecraft Engineering, Institute of Space and Astronautical Science, Sagamihara, Kanagawa 252-5210, JapanCorr. author: Yuta Takahashi, stateofyuta@gmail.com Abstract This correspondence presents a convex-optimization-based evaluation framework of satellite-swarm-based apertures maintained by magnetic-field interactions. Spaceborne distributed apertures are composed of multiple satellites and are attractive for scientific and commercial missions because their scalability enables high-gain, narrow-beam, and large-aperture capabilities beyond the launch-size limitations. A key challenge is that the long-term maintenance of such virtual structures requires consistent formation control amid unstable orbital dynamics, and magnetic interactions generated by satellite-mounted magnetorquers offer a desirable propellant-free position-control strategy. However, the nonlinearities of the electromagnetic force and torque model lead to a nonconvex power-consumption constraint, making system-level configuration analysis difficult. To address this issue, we develop a convex optimization-based framework to analyze the power consumption of large magnetically actuated satellite swarms. The resulting analysis shows that increasing the number of satellites can improve formation-keeping power efficiency. This indicates that magnetically actuated swarm architectures provide a power-efficient alternative to the conventional few-satellite electromagnetic formation-flight concept for constructing large-scale space systems. I INTRODUCTION Spaceborne distributed apertures are a promising architecture for future large-scale space systems. By synthesizing an aperture from multiple satellites, they can provide high-gain, narrow-beam, and large-aperture capabilities beyond the launch-size limitations of monolithic satellites. Such capabilities are important for scientific and commercial missions, including deep-space communication, direct-connectivity services with small ground or user terminals in Fig 1a, and resilient communication in cellular frequency bands [1, 2]. The membrane structure system also offers a large surface area, small volumes, and reliable deployment along with software-based alignment calibration [3] in Fig 1b. While the scalability of membrane structures depends on material advances, distributed apertures rely on state-estimation sensors that advance more rapidly than materials. In conventional monolithic architectures, increasing antenna size directly increases structural mass, deployment complexity, and payload volume requirements. Distributed architectures mitigate these limitations by assigning the aperture function to multiple spacecraft, while distributing payload, thermal, and structural requirements among multiple satellites [4]. One of the main difficulties in realizing such virtual space structures is the long-term maintenance of the formation, and the lifetime of a distributed aperture is limited by fuel consumption. In low Earth orbit, the relative motion is continuously affected by unstable orbital dynamics and environmental perturbations, including the J2J_2 effect [5]. The communication performance of a distributed aperture depends directly on the accuracy of relative positions, and deviations from the desired formation can degrade the sidelobe level, effective aperture, and overall link performance. Therefore, a distributed aperture requires a formation-keeping strategy that can preserve the desired geometry over long mission durations. (a) Space Rigid-panel array. (b) Space membrane array [6]. Figure 1: Monolithic space antenna arrays (Conceptual illustration of the BlueWalker3 © AST SpaceMobile and HELIOS-R [6]). Figure 2: Our conceptual illustration of spaceborne distributed apertures example using satellite swarms. A propellant-free actuation strategy is desirable for this purpose, and magnetic interactions generated by satellite-mounted magnetorquers (MTQs) provide a promising solution. MTQs are widely used as attitude actuators for Earth-orbiting satellites, and electromagnetic formation flight (EMFF) extends this magnetic actuation principle to relative-position control among multiple satellites [7, 8]. Microgravity demonstrations have validated the basic feasibility of magnetic interaction swarm control [9, 10]. Recent studies have also examined MTQ-based formation control for distributed space antenna concepts under unstable orbital dynamics [11, 2]. The central challenge in power evaluation for magnetically actuated swarm-keeping stems from the nonconvex nature of the magnetic interaction model. The generated force and torque are strongly coupled through the relative geometry and the dipole commands of multiple satellites. The resulting input map is bilinear, and the power required to realize a prescribed force–torque command cannot be directly inferred from independent force and torque limits. Moreover, the achievable magnetic input depends on the coil geometry and the nominal inter-satellite distance. Therefore, the formation-keeping problem in a user-defined orbit must be evaluated together with the system design variables, rather than treated as a separate control problem. This difficulty becomes more pronounced as the number of satellites increases, because the magnetic interaction network grows rapidly, and each satellite can be affected by multiple neighboring interactions. Consequently, estimating the peak and total formation-keeping power via direct nonconvex numerical optimization is computationally challenging and obscures the architecture’s scalability trend. This correspondence addresses the issue by developing a convex-optimization-based framework to evaluate the power consumption of large magnetically actuated satellite swarms. First, we show the global optimality of the power allocation problem for two-satellite formation keeping, even though the original dipole allocation problem is nonconvex. Second, we combine this global optimal dipole allocation with a decentralized formation-keeping architecture. Consequently, the proposed method transforms a nonconvex system-level evaluation problem into a tractable convex-optimization-based analysis framework. The remainder of this correspondence is organized as follows. Section I summarizes the orbital motion model and magnetic interaction model used in the analysis. Section I formulates the grid-structured distributed aperture and the disturbance model for user-defined stable relative orbits. Then, we prove the global optimality of dipole allocation of the two satellite systems. Section IV derives the decentralized bucket-brigade model and presents the convex-optimization-based power-consumption analysis to show that increasing the number of satellites can improve formation-keeping power efficiency under the considered architecture. Finally, Section VI concludes this correspondence. I Preliminaries I-A Magnetically Actuated Swarm Control Model This subsection introduces the magnetic field interaction model. We define the magnetic moment μ and the resistance of a single-axis coil RcoilR_coil as =πNtacoil2ccoil,Rcoil=2acoilNtpc/rcoil2 μ=π N_ta_coil^2c_coil n, R_coil=2a_coilN_tp_c/r_coil^2 (1) where NtN_t is the number of coil turns, acoila_coil is the coil radius, ccoilc_coil is the coil current, n is a vector normal to the coil plane, pcp_c is the wire resistivity, and rcoilr_coil is the wire radius. We assume the following sinusoidal magnetic moment for each agent [9]: j(t)≈¯jsin(ωjt+j)=jsin(ωjt)+jcos(ωjt), μ_j(t)≈ μ_j ( _jt+ θ_j)= s_j ( _jt)+ c_j ( _jt), (2) where the amplitudes of the cosine and sine components are j∈ℝ3 s_j ^3 and j∈ℝ3 c_j ^3, respectively, and j∈ℝ3 θ_j ^3 are phases. The first-order time-averaged input u¯j←k∈ℝ6 u_j← k ^6 exerted on the j-th agent by the k-th agent is [9] u¯j←k u_j← k ≜[fj←kavgτj←kavg]=∫Tμ04πQj←k(μkb(u)⊗μjb(u))duT bmatrixf^avg_j← k\\ τ^avg_j← k bmatrix= _T _04πQ_j← k(μ^b_k(u) μ^b_j(u)) duT (3) ≈12μ04πQj←k(skb⊗sjb+ckb⊗cjb)ifωj=ωk ≈ 12 _04πQ_j← k (s^b_k s^b_j+c^b_k c^b_j )\ if\ _j= _k where μ0=4π×10−7 _0=4π× 10^-7 T⋅·m/A and a position vector from k-th coil to j-th one rj←kr_j← k yields Qj←k∈ℝ6×9Q_j← k ^6× 9 as [9] Qj←k=(I2⊗CA/Lj←k)[ΨfΨτ](CLj←k/A⊗CLj←k/A), Q_j← k=(I_2 C^A/L_j← k) bmatrix _f\\ _τ bmatrix(C^L_j← k/A C^L_j← k/A), Ψf=1‖rj←k‖4[−600030003030300000003000300]Ψτ=1‖rj←k‖3[0000010−100020001000−20−100000]. \ aligned & _f= 1\|r_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\\ & _τ= 1\|r_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 aligned .. The line-of-sight frame ℒj←k\LOS_j← k\ [9] is attached to the k-th agent and oriented toward the j-th agent and the associated coordinate transformation matrix CO/Lj←k∈ℝ3×3C^O/Lj← k ^3× 3 is CA/Lj←k=(rj←ka,fja×rj←ka)C^A/L_j← k=C(r_j← k^a,\ f^a_j× r^a_j← k) where (v,aw)a=[x≜va∥v∥a,y≜v×awa∥v×aw∥a,x×y].C(v^a,w^a)= bmatrix e_x v^a\|v^a\|, e_y v^a× w^a\|v^a× w^a\|, e_x× e_y bmatrix. (4) I-B Passively Stable Orbits for Satellite Swarm We introduce the passively stable trajectories under J2J_2 Earth gravity. Let rjk=rj−rk=[x;y;z]r_jk=r_j-r_k=[x;y;z] be the relative position from the k-th satellite to the j-th one. The dynamics of the linearized relative motion in the local vertical, local horizontal (LVLH) frame are [11, 12] x¯¨−2ωxyy¯˙−3ωxy2x¯−4ωxy2c−2/sJ2(2x¯+y¯˙ωxy)=c+(ux+dx) x-2 _xy y-3 _xy^2 x- 4 _xy^2c_-^2/s_J_2 (2 x+ y _xy )=c_+(u_x+d_x) (5) y¯¨+2ωxyx¯˙=c−(uy+dy) y+2 _xy x=c_-(u_y+d_y) z¨+ωz2z=2lωzcos(ωzt+θz)+(uz+dz) z+ _z^2z=2l _z ( _zt+ _z)+(u_z+d_z) x¯=c+x,y¯=c−y,ωxy=c−ωo,ωz=ωzref+f1(δΩ˙avg)≈ωzref,rzsinθz=z,lsinθz+ωzrzcosθz=z˙, \ aligned & x=c_+x, y=c_-y, ω_xy=c_- _o,\\ & _z= _zref+f_1(δ _avg)≈ _zref,\ r_z _z=z,\\ &l _z+ _zr_z _z= z, aligned . where l(δΩ˙avg)=−rrefsinijsinikf2(δΩ˙avgjk)≈0l(δ _avg)=-r_ref i_j i_kf_2(δ _avgjk)≈ 0 since we can naturally assume that the satellites have identical i. The analytical solution of (5) is [11, 12]: [x(t)y(t)z(t)]=[ro0]+[rxysin(ωxyt+θxy)/c+2rxycos(ωxyt+θxy)/c−(rz+lt)sin(ωzt+θz)], bmatrixx(t)\\ y(t)\\ z(t) bmatrix= bmatrixr_o\\ 0 bmatrix+ bmatrixr_xy ( _xyt+ _xy)/c_+\\ 2r_xy ( _xyt+ _xy)/c_-\\ (r_z+lt) (ω_zt+ _z) bmatrix, (6) ro=[2C1C4−ϵ2C1t]⊤,ϵ2=3+5sJ2c+c−ωxy, r_o= bmatrix2C_1&C_4- _2C_1t bmatrix ,\ _2= 3+5s_J_2c_+c_- _xy, where ro(0,t)∈ℝ2r_o(0,t) ^2 is the center position of the relative orbit. The orbital indices calculated at t=0t=0 are [11, 12] C1=c+/c−2(2x¯+y¯˙/ωxy),C4=(y¯−2x¯˙/ωxy)/c−rxy2=C22+C32,θxy=tan−1(C3,C2)C2=(y¯−c−C4)/2,C3=x¯−2c+C1rz2=C62+C52,θz=tan−1(C6,C5)C5=z˙/ωz,C6=z. \ aligned &C_1=c_+/c_-^2(2 x+ y/ω_xy),\ C_4=( y-2 x/ω_xy)/c_-\\ &r_xy^2=C_2^2+C_3^2,\ _xy= ^-1(C_3,C_2)\\ &C_2=( y-c_-C_4)/2,\ C_3= x-2c_+C_1\\ &r_z^2=C_6^2+C_5^2,\ _z= ^-1(C_6,C_5)\\ &C_5= z/ _z,\ C_6=z. aligned . (7) Enforcing ωz=ωxy _z= _xy and θz=θxy+tan−1(2tanΘz−xy) _z= _xy+ ^-1(2 _z-xy) derives the desired stable trajectories pd(t)p_d(t) [11, 12] pd(t)=[(1/c+)rxydsin(ωxyt+θxy)(1/c−)2rxydcos(ωxyt+θxy)rxydtanΘPcos(Θz−xy)cos(θz−θxy)sin(ωxyt+θzd(Θz−xy,t))].p_d(t)= bmatrix(1/c_+)r_xyd ( _xyt+ _xy)\\ (1/c_-)2r_xyd ( _xyt+ _xy)\\ r_xyd _P ( _z-xy) ( _z- _xy) (ω_xyt+ _zd( _z-xy,t)) bmatrix. (8) Note that ωzd=ωxy _zd= _xy is realized via active control, or equivalently, the mismatch acts as the disturbance dfz=(ωxy2−ωz2)zd_fz=( _xy^2- _z^2)z on pdp_d [11, 12]: dfz d_fz =rzd(ωxy2sin(ωxyt+θz)−ωz2sin(ωzt+θz)). =r_zd( _xy^2 ( _xyt+ _z)- _z^2 ( _zt+ _z)). (9) I Problem Formulation: Perturbed Grid Aperture This section formulates our problem by introducing a simplified model and states the computational problems in power estimation. We model our satellite swarm as a square formation with equally spaced satellites, as illustrated in Fig. 2. Each grid line includes a linear formation consisting of 2n+12n+1 satellites shown in Fig. 3, where the central satellite is indexed as 0, and the satellites at either edge are indexed as n and −n-n, respectively. We define the vector from the (−n)(-n)th satellite to the nnth satellite as Rl(τ)=rl(τ)p^(τ),τ∈[0,2π/ωxy).R_l(τ)=r_l(τ) p(τ), τ∈[0,2π/ _xy). (10) The total number of satellites is given by Nall=Nl2=(2n+1)2,N_all=N_l^2=(2n+1)^2, (11) where NlN_l denotes the number of elements along the array. The total system mass m¯sys=Nallmsat m_sys=N_allm_sat, inter-distance, and array side length are user-defined constants: m¯sys=const.,dsat=const.,rl=(2n+1)dsat. m_sys=const.,\ d_sat=const.,\ r_l=(2n+1)d_sat. (12) We assume that the on-orbit environmental disturbance force fdf_d is given by fd=msatKorb(t)p.f_d=m_sat\ K_orb(t)\ p. (13) where time-varying coefficient matrix Korb(t)∈ℝ3×3K_orb(t) ^3× 3 and relative position vector p∈ℝ3p ^3. Note that we neglect the environmental disturbance forces and torques except (13) because they are generally independent of the distance p from the center and bounded. Example 1. Consider the satellites on stable relative orbit pd(t)p_d(t) in (8). The averaging error disturbance under J2J_2 gravity is fd(t)=msatKJ2(Pref,i,θ)pf_d(t)=m_sat\ K_J_2\ (P_ref,i,θ)p where fd(t) f_d(t) =msat(∇2UJ2−∫02π∇2UJ2dθ2π)+msatdfz(t) =m_sat (∇^2U_J_2- _0^2π∇^2U_J_2 dθ2π )+m_satd_fz(t) =msat(K(Pref,i,θ)+[00000000−(ωz2−ωxy2)])p, =m_sat (K(P_ref,i,θ)+ bmatrix0&0&0\\ 0&0&0\\ 0&0&-( _z^2- _xy^2) bmatrix )p, dfz(t)d_fz(t) in (9) and K(Pref,i(t),θ(t))∈ℝ3×3K(P_ref,i(t),θ(t)) ^3× 3 is [5] K(Pref,i,θ)=kJ22Pref5[12si2c2θ4si2s2θ4s2isθ4si2s2θ−7si2c2θ−s2icθ4s2isθ−s2icθ−5si2c2θ].K(P_ref,i,θ)= k_J_22P_ref^5 bmatrix12s^2_ic_2θ&4s^2_is_2θ&4s_2is_θ\\ 4s^2_is_2θ&-7s^2_ic_2θ&-s_2ic_θ\\ 4s_2is_θ&-s_2ic_θ&-5s^2_ic_2θ bmatrix. To evaluate the required powers, one straightforward approach is to use numerical evaluations, but this becomes computationally burdensome as the number of satellites NallN_all increases. As shown in subsection I-B, the relative orbital dynamics can be approximated as the linear time-invariant system x˙=Ax(t)+d(t) x=Ax(t)+d(t) with a time-varying external input d(t)d(t). Our goal is to evaluate x(T)x(T) for Nall≫1N_all 1 and its analytical solution under x(0)=0x(0)=0 is x(T)=∫0TeA(T−τ)d(τ)dτ=∑k=1K[∫tktk+1eA(T−τ)dτ]dkx(T)= _0^Te^A(T-τ)d(τ)dτ= _k=1^K [ _t_k^t_k+1e^A(T-τ)dτ ]d_k where d(t)=dkd(t)=d_k for t∈[tk,tk+1)t∈[t_k,t_k+1). This method incurs a total computational cost of (4Nall2K)O(4N_all^2K), which is not scalable. Alternatively, we can reduce the total cost using the eigendecomposition A=VΛV−1A=V V^-1 and the fact that eA(T−τ)=VeΛ(T−τ)V−1e^A(T-τ)=Ve (T-τ)V^-1. This allows for the use of parallel computing, and the analytical solution is x(T)=∫0TVeΛ(T−τ)V−1d(τ)dτ.x(T)= _0^TVe (T-τ)V^-1d(τ)dτ. Although its total cost is improved as (Nall3)+(Nall2K/P)O(N_all^3)+O(N_all^2K/P) with the parallelism P and eigendecomposition cost (Nall3)O(N_all^3), this still does not provide a scalable evaluation method. Therefore, a scalable and uniquely determined evaluation method is required for system-level design of large magnetically actuated satellite swarms. IV Magnetically-Actuated Swarm Keeping Evaluation This section derives the framework to estimate the satellite’s maximum power and the system’s total power for formation keeping. These indices are the key factors in scalability for magnetically actuated swarms, constrained by power and thermal budgets, rather than by time-integrated consumption. Figure 3: Grid-structured approximation for distributed space system design. Linear formation of 2n+12n+1 satellites with coil actuators. IV-A Optimal Decentralized Formation-Keeping Model We derive the decentralized control model for a magnetically-actuated swarm system. Our goal of the formation-keeping is to cancel out the disturbance force fdjf_dj acting on each satellite by the control input fcjf_cj, i.e., fdj=fcjforj=1,…,Nall.f_dj=f_cj j=1,…,N_all. (14) As introduced in subsection I-A, magnetic field interaction between different frequencies does not interact with each other in the first-order averaged dynamics. To represent a set of satellites driven by the same frequency, we define an arbitrary collection of satellite groups as ⊆1,…,Nall:||≥2G \ g \1,…,N_all\:| g|≥ 2\ where each group ∈ g can contain an arbitrary number of satellites. Then, the time-averaged dynamics admit nonunique feasible realizations of (14) as fdj=∑∈:j∈fcj()forj=1,…,Nall.f_dj= _ g :\ j∈ gf_cj^( g) j=1,…,N_all. The simplest approach is to assign a single frequency to all satellites, i.e., =1,…,Nall g=\1,…,N_all\. However, finding a globally optimal solution is computationally expensive, as the optimal dipole allocation problem is an NP-hard program [9], and relying on locally optimal solutions may compromise the reliability of our analysis. Then, we show that the globally optimal solution in the two-satellite case can be obtained by convex optimization. We consider the optimal dipole allocation for two satellites, which is a non-convex optimization with six equality constraints and belongs to the QCQP class: min Jp=‖m‖2/2=‖[sj;sk;cj;ck]‖2/2 J_p=\|m\|^2/2=\|[s_j;s_k;c_j;c_k]\|^2/2 (15) s.t. .t. Qj←k(sk⊗sj+ck⊗cj)=(8π/μ0)uj←klos Q_j← k (s_k s_j+c_k c_j )=(8π/ _0)u_j← k^los We derive its Lagrange dual problem in (15) as maxJd=−λ⊤uj←klosμ0/(8π)s.t.Pλ=[E3RλRλ⊤E3]⪰0max\ J_d= -λ u^los_j← k _0/(8π) .t. P_λ= bmatrixE_3&R_λ\\ R_λ &E_3 bmatrix 0 (16) where Lagrange multiplier vector λ∈ℝ6λ∈R^6, Rλ∈ℝ3×3R_λ ^3× 3 satisfies vec(Rλ)=Qj←k⊤λvec(R_λ)=Q_j← k λ. Despite QCQP, including our problem, not being convex, strong duality holds for QCQP with one quadratic inequality constraint provided Slater’s condition holds [13]. We show that the problem in (15) holds strong duality, i.e., no duality gap exists between the primal and dual problems, and the proof is in Appendix VI-A. Lemma 1. Consider the relative position rj←kr_j← k and the command input uj←klos=[fj←klos;τj←klos]∈ℝ6u_j← k^los=[f_j← k^los; _j← k^los] ^6 from kkth agent to jjth one. Let λ∗∈ℝ6λ^* ^6 be defined as the optimal Lagrange multiplier vector. Then, a global-optimum solution of the non-convex optimization is [sk∗,ck∗]=−Rλ∗⊤[sj∗,cj∗][s_k^*,c_k^*]=-R_λ^* [s_j^*,c_j^*] and [sj∗,cj∗][s_j^*,c_j^*] satisfies sj∗=μ¯jcosθj,cj∗=μ¯jsinθj s_j^*= μ_j _j, c_j^*= μ_j _j μ¯j=[123],|θL(1)−θL(2)|=cos−1(4/12)|θL(3)−θL(1)|=cos−1(5/31)|θL(2)−θL(3)|=cos−1(6/23) μ_j= bmatrix L_1\\ L_2\\ L_3 bmatrix, \ aligned &| _L(1)- _L(2)|= ^-1( L_4/ L_1 L_2)\\ &| _L(3)- _L(1)|= ^-1( L_5/ L_3 L_1)\\ &| _L(2)- _L(3)|= ^-1( L_6/ L_2 L_3)\\ aligned . where j L_j is derived by vec(j←ki)=Qj←k(i,:)⊤vec(Q_j← k^i)=Q_j← k(i,:) and −tr[Rλ∗j←ki⊤[145∗26∗3]]=8πμ0uj←k(i)los,i∈[1,6]-tr [R_λ^*Q_j← k^i bmatrix L_1& L_4& L_5\\ *& L_2& L_6\\ *&*& L_3\\ bmatrix ]= 8π _0u_j← k(i)^los,\ i∈[1,6] (17) Remark 1. This global optimal result is useful for nonlinear controller design for a magnetically actuated robot swarm under the strong duality assumption [14]. To suppress unintended coupling among nonadjacent satellites in close proximity, we assume the use of multiple frequency allocations to confine electromagnetic interactions to neighboring satellites. Assumption 1. Control pairs are defined by assigning distinct AC angular frequencies ωfk _fk for k∈[−n,n]k∈[-n,\ n] to adjacent satellite groups, as illustrated in Fig. 3 (Please refer the detailed selection of angular frequencies [9].). We introduce a power index WpowerW_power [9] by Lemma 1 Wpower≜Rcoil∫Tccoil2(τ)dτ/T=(Rcoil/γμ/c2)JdW_power R_coil _Tc_coil^2(τ)dτ/T=(R_coil/ _μ/c^2)J_d (18) where the coil resistance RcoilR_coil in (1), γμ/c _μ/c [m2] is the coil design ratio to convert ccoilc_coil into μ in (1), and the lower bound JdJ_d in (16) into the WpowerW_power. The definition of μ and RcoilR_coil in (1) derives by the results of Lagrange dual problem in (16) γμ/c≜πNtacoil2 _μ/c π N_ta_coil^2 and Rcoilγμ/c2=2pc/rcoil2π2Ntacoil3 R_coil _μ/c^2= 2p_c/r_coil^2π^2N_ta_coil^3. IV-B Recursive Disturbance Elimination Model We estimate the maximum electric power required to eliminate the orbital disturbances in (13) via magnetic-field interactions. We use a simplified analytical model introduced in the previous subsection. We derive jjth error disturbance introduced in subsection I fd(j)=msatKorb(t)(jdsatp^),‖p^‖2=1f_d(j)=m_satK_orb(t)\ (jd_sat p), \| p\|_2=1 (19) where dsatp^d_sat p is the constant spacing. We define the equilibrium conditions between adjacent satellites. Definition 1 (“Bucket-Brigade” Model). Equilibrium conditions of linear formation under Assumption 1 are f(j)←(j−1)+fd(j)+f(j)←(j+1) f_(j)←(j-1)+f_d(j)+f_(j)←(j+1) =0 =0 (20) τ(j)←(j−1)+τd(j)+τ(j)←(j+1) _(j)←(j-1)+ _d(j)+ _(j)←(j+1) =0 =0 where we use fd(j)f_d(j) in (19) and τd(j)=0 _d(j)=0. Then, we uniquely derive the required feedforward input for a recursive environmental disturbance elimination, and the proof is deferred to Appendix VI-B. Lemma 2. The required magnetic force and torque for the recursive disturbance elimination in Definition 1 are u(j−2)←(j−1)≜χsys(m¯sys,n)L(n,j)U^(rl,t) u_(j-2)←(j-1) \ _sys( m_sys,n)L(n,j) U(r_l,t) (21) χsys(m¯sys,n)≜m¯sysn(n+1)6(2n+1)3∈ℝL(n,j)≜[(n−j+2)(n+j−1)n(n+1)I3O3O3(n−j+2)(n−j+3)(2n+j−1)n(n+1)(2n+1)I3]U^(rl,t)≜[3Korb(t)Rl(t)Rl(t)×Korb(t)Rl(t)]∈ℝ6 aligned & _sys( m_sys,n) m_sys n(n+1)6(2n+1)^3 \\ &L_(n,j) bmatrix (n-j+2)(n+j-1)n(n+1)I_3&O_3\\ O_3& (n-j+2)(n-j+3)(2n+j-1)n(n+1)(2n+1)I_3 bmatrix\\ & U(r_l,t) bmatrix3K_orb(t)R_l(t)\\ R_l(t)× K_orb(t)R_l(t) bmatrix ^6 aligned where j∈[2,n+1]j∈[2,\ n+1], L(n,j)∈ℝ6×6L(n,\ j) ^6× 6, and orbital disturbance coefficient matrix Korb(t)K_orb(t) in (13). IV-C Convex-Optimization-based Power Estimation We finally derive a power consumption for formation keeping. We derive the power distribution trend required by the overall distributed space systems. Theorem 3. Consider the grid-structured satellites on the user-defined plane with m¯sys=(2n+1)2msat m_sys=(2n+1)^2m_sat and rl=(2n+1)dsatr_l=(2n+1)d_sat. Then, the upper-bound of maximum power consumption W¯ W [W] along all satellites is W¯≜χsys(m¯sys,n)supt∈[0,Torb)[w(rl,n,2,t)∗=w(rl,t)∗] W _sys( m_sys,n) _t∈ [0,\ T_orb ) [w^*_(r_l,n,2,t)=w^*_(r_l,t) ] (22) where w∗w^* is given by a convex optimization for j∈[2,n+1]j_∈[2,n+1]: w(rl,n,j,t)∗≜2Rcoilγμ/c2maxλ−∈ℝ6λ+∈ℝ6∑x=−,+λx⊤L(n,j)U^(rl,tx)−μ0/(8π) w^*_(r_l,n,j,t) 2R_coil _μ/c^2 _ subarrayc _- ^6\\ _+ ^6 subarray _x=\-,+\ _x L_(n,\ j) U_ (r_l,t_x )- _0/(8π) (23) s.t.[E3Rλ+Rλ+⊤E3]⪰0,[E3Rλ−Rλ−⊤E3]⪰0. \ \ s.t.\ \ aligned & bmatrixE_3&R_ _+\\ R_ _+ &E_3 bmatrix 0,\ bmatrixE_3&R_ _-\\ R_ _- &E_3 bmatrix 0 aligned .. Proof. The power consumption between (j−2)(j-2)th and (j−1)(j-1)th satellites for j∈[2,n+1]j∈[2,\ n+1] under the decentralized control of Assumption 1 is derived the convex optimizaiton: maxλ∈ℝ6Jd=λ⊤u(j−2)←(j−1)(t)−μ0/(8π)s.t.[E3RλRλ⊤E3]⪰0 _λ ^6\ J_d= λ u_(j-2)←(j-1)(t)- _0/(8π)\ s.t. bmatrixE_3&R_λ\\ R_λ &E_3 bmatrix 0 (24) where r(j−2)←(j−1)(t)=−dsatp^(t)r_(j-2)←(j-1)(t)=-d_sat p(t) derives RλR_λ vec(Rλ(t))=Q(j−2)←(j−1)⊤(−dsatp^(t))λ.vec(R_λ(t))=Q _(j-2)←(j-1)(-d_sat p(t))λ. Each satellite belongs to two linear formations under the decentralized control scheme and another linear formation exists as the π/2π/2-phase-shifted linear formation. We define these two phase-shifted times t−,+∈ℝt_-,+ in (25): ∀t,t−≜t,t+≜t+Torb/4,Torb≜2π/ωxy∀ t, t_- t, t_+ t+T_orb/4, T_orb 2π/ _xy (25) Since the linear formation spans satellites from −n-n to n, we account for the symmetric satellite pairs from −n-n to 0 by multiplying (24) by a factor of two. Therefore, the total power between (j−2)(j-2)th and (j−1)(j-1)th satellite pair and its symmetric (−j+2)(-j+2)th and (−j+1)(-j+1)th satellite pair for j∈[2,n+1]j∈[2,\ n+1] is calculated as χsys(m¯sys,n)w(rl,n,j,t)∗ _sys( m_sys,n)w^*_(r_l,n,j,t) [W] through the convex optimization in (23). The coefficients in (21) show that the maximum control force and torque is given j=2j=2 and work on the center satellite u(j−2)←(j−1)=χsys(m¯sys,n)U^u_(j-2)←(j-1)= _sys( m_sys,n) U. Thus, the problem in (24) derives W¯ W along all satellites in (22). ∎ (a) Averaged squared dipole moment M(rl,n)M(r_l,n) for formation keeping. (b) Surface area ratio of the distributed system to the monolithic system. Figure 4: The averaged total power consumption per total system mass msysm_sys for formation keeping during TorbT_orb and the ratio γSsat(Nl) _S_sat(N_l) in (27) of total surface Ssat(n)S_sat(n) with the monolithic structure one. V Discussion This section presents the underlying trends and trade-off between the power consumption and the number of satellites for magnetic formation-keeping. V-A Peak Power and Number of Satellites Tradeoff We investigate the trend in peak power for magnetic formation keeping. The results in Theorem 3 show that, as the total system mass m¯sys m_sys increases, the upper bound of power consumption grows linearly regardless of the array size rlr_l or the number of satellites NallN_all, which is evident from W∮W_ in (26) and W¯ W in (22). Moreover, since w(rl,n,n,t)∗w^*_(r_l,n,n,t) is independent of n if j=nj=n, i.e., w(rl,n,n,t)∗=w(rl,t)∗w^*_(r_l,n,n,t)=w^*_(r_l,t), the result in (22) indicates that W¯ W is linear with χsys(m¯sys,n) _sys( m_sys,n) along with the constant supt∈[0,Torb)w(rl,t)∗ _t∈ [0,\ T_orb )w^*_(r_l,t) for given formation length rlr_l. Since χsys(m¯sys,n)→0 _sys( m_sys,n)→ 0 as n→∞n→∞, W¯ W converges to zero and the formation-keeping constraints imposed on the center satellite become less restrictive. This indicates the advantage of a distributed architecture based on magnetic-field interactions. V-B Minimization Trend of Total Power Consumption The total power budget W∮W_ in (26) is evaluated numerically since an analytical expression is not available. Corollary 1. Consider the grid-structured satellites on the user-defined plane with m¯sys=(2n+1)2msat m_sys=(2n+1)^2m_sat and rl=(2n+1)dsatr_l=(2n+1)d_sat. The upper-bound of its averaged total power consumption W∮(m¯sys,rl,n)W_ ( m_sys,r_l,n) for formation keeping during one orbit TorbT_orb is W∮≜χsys(m¯sys,n)∫0Torb(2n+1)∑j=2n+1w(rl,n,j,t)∗dtTorbW_ _sys\ ( m_sys,n) _0^T_orb(2n+1) _j=2^n+1\ w^*_(r_l,n,j,t)\ dtT_orb (26) where w(rl,n,j,t)∗w^*_(r_l,n,j,t) for j∈[2,n+1]j_∈[2,n+1] is the optimal solution of the convex optimization in (23). Proof. The total power between (j−2)(j-2)th and (j−1)(j-1)th satellite pair (and its symmetric (−j+2)(-j+2)th and (−j+1)(-j+1)th satellite pair) for the orthogonal linear formation is calculated as χsys(m¯sys,n)w(rl,n,j,t)∗ _sys( m_sys,n)w^*_(r_l,n,j,t) [W] through the convex optimization in (23). Then, (26) sums the power distribution trend required by the overall distributed system. ∎ We define the averaged squared dipole moment M(rl,n)M(r_l,n) as the averaged total power consumption, normalized by the system mass m¯sys m_sys and scaled by Rcoil/γμ/c2R_coil/ _μ/c^2 M(rl,n)≜W∮(m¯sys,rl,n)/(m¯sys(Rcoil/γμ/c2))M(r_l,n) W_ ( m_sys,r_l,n)/( m_sys(R_coil/ _μ/c^2)) and Figure 4a shows its values for an altitude of 500500 km, an inclination of 45∘45 , θ0=0 _0=0, and a passively stable orbital plane of (θP,θZmXY)=(30∘,0∘)( _P, _ZmXY)=(30 ,0 ). Although the total power scales with (2n+1)2(2n+1)^2, the formation-keeping power decreases as the number of satellites increases; similar trends were observed for other parameter settings. This suggests that, for magnetic-field-interaction-based formation keeping, increasing the number of satellites while holding total mass and array size fixed is energetically favorable because shorter inter-satellite distances improve magnetic actuation efficiency. V-C Maximization Trend of Solar Panel Area Moreover, the increased number of satellites increases the available electrical power and communication performance. For a given constant satellite volume VsysV_sys, we can derive satellite size 2asat2a_sat and total surface area of overall satellites Ssat≜Nl2×6(2asat)2S_sat N_l^2× 6(2a_sat)^2 as 2asat≜(VsysNl2)1/3⇒Ssat(Nl)=6Nl2Nl43Vsys23=6Nl23Vsys23.2a_sat ( V_sysN_l^2 )^1/3 \ S_sat(N_l)= 6N_l^2N_l 43V_sys 23=6N_l 23V_sys 23. For n, the ratio of total surface Ssat(n)S_sat(n) with the monolithic space structure one, i.e., Ssat(0)=6Vsys2/3S_sat(0)=6V_sys^2/3, is γSsat(Nl)≜Ssat(Nl)/Ssat(0)=Nl23 _S_sat(N_l) S_sat(N_l)/S_sat(0)=N_l 23 (27) and this is illustrated in Fig. 4b. This increase in the total surface area improves both electrical power and radio-frequency performance, including antenna gain, sidelobe level, and effective isotropic radiated power. V-D Limitations and Future Work Excessively increasing the number of satellites would eventually lead to unrealistically small satellite sizes in practice. We can formulate the detailed design problem as a potentially nonconvex optimization [2], yielding feasible design solutions and more refined scaling trends. Moreover, environmental disturbances cannot be perfectly modeled; nevertheless, stabilizing the satellites within a tolerable position error reduces power consumption, rather than relying on ideal feedforward. The worst-case power estimate in (21) is not directly suitable for satellite design. From the disturbance model in (19), the largest disturbance within a formation is supjfd(j)=fdn=msat2Korb(t)Rl(t) _jf_d(j)=f_dn= m_sat2K_orb(t)R_l(t) at the satellite located farthest from the formation center. In contrast, the largest feedforward force in (21) increases approximately in proportion to n supjf(j−2)←(j−1)=f0←1=n(n+1)(2n+1)supjfd(j) _jf_(j-2)←(j-1)=f_0← 1= n(n+1)(2n+1) _jf_d(j) Note that the total system power decreases as the number of satellites increases, as shown in subsection V-B, and such an excessive force is canceled out among different groups, i.e., f0←1+f0←−1=0f_0← 1+f_0←-1=0. However, each group must independently generate such large intermediate forces in our decentralized control model. For a homogeneous satellite swarm, the design should be based on the worst-case satellite, leading to overly conservative component specifications. For detailed satellite configuration design, a different evaluation metric is required that better reflects the actual per-satellite power distribution. VI CONCLUSION This correspondence presented a convex-optimization-based framework for evaluating the formation-keeping power of magnetically actuated distributed apertures. We showed that the nonconvex two-satellite dipole-allocation problem admits a uniquely characterizable global optimum through a convex formulation, and incorporated this result into a decentralized bucket-brigade model for large satellite swarms. The resulting analysis indicated that increasing the number of satellites can improve formation-keeping power efficiency under the considered grid-structured architecture. These results support the use of magnetically actuated swarms as a power-efficient approach for constructing large-scale space systems. APPENDIX VI-A Proof of Lemma 1 First, we show that the non-convex primal problem and its associated Lagrange dual problem have zero duality gap for the considered feasible command input. Equivalently, there exists an optimal Lagrange multiplier λ∗λ^* such that the corresponding Lagrangian provides a tight global lower bound for the primal objective. Using Roth’s column lemma, λ⊤Qj←k(sk⊗sj+ck⊗cj)λ Q_j← k(s_k s_j+c_k c_j) is converted into tr[Rλ⊤(sjsk⊤+cjck⊤)]tr[R_λ (s_js_k +c_jc_k )] where RλR_λ satisfies vec(Rλ)=Qj←k⊤λvec(R_λ)=Q_j← k λ. Thus, the Lagrangian is written as L(m,λ)=12m⊤(I2⊗Pλ)m−8πμ0λ⊤uj←klosL(m,λ)= 12m (I_2 P_λ)m- 8π _0λ u^los_j← k where Pλ=[I3,Rλ;Rλ⊤;I3]P_λ=[I_3,R_λ;R_λ ;I_3]. For the dual optimal multiplier λ∗λ^*, we have Pλ∗⪰0P_λ^* 0. Therefore, L(m,λ∗)L(m,λ^*) gives a global lower bound of the primal objective. Hence, any feasible point satisfying the stationarity condition with λ∗λ^* attains this lower bound and is globally optimal. The first-order optimality conditions are ∂L/∂λ=g(m)=0∂ L/∂λ=g(m)=0 and ∂L/∂m=(I2⊗Pλ)m=0∂ L/∂ m=(I_2 P_λ)m=0. The second equality yields Pλ[sj;sk]=Pλ[cj;ck]=0P_λ[s_j;s_k]=P_λ[c_j;c_k]=0. Thus, [sj,cj]+Rλ[sk,ck]=0,[sk,ck]+Rλ⊤[sj,cj]=0.[s_j,c_j]+R_λ[s_k,c_k]=0,\ [s_k,c_k]+R_λ [s_j,c_j]=0. In particular, at the optimal multiplier λ∗λ^*, [sk∗,ck∗]=−Rλ∗⊤[sj∗,cj∗][s_k^*,c_k^*]=-R_λ^* [s_j^*,c_j^*]. This relationship also yields [sj∗,cj∗]=Rλ∗Rλ∗⊤[sj∗,cj∗],[sk∗,ck∗]=Rλ∗⊤Rλ∗[sk∗,ck∗],[s_j^*,c_j^*]=R_λ^*R_λ^* [s_j^*,c_j^*],\ [s_k^*,c_k^*]=R_λ^* R_λ^*[s_k^*,c_k^*], and therefore ‖sj∗‖=‖Rλ∗sk∗‖=‖sk∗‖,‖cj∗‖=‖Rλ∗ck∗‖=‖ck∗‖\|s_j^*\|=\|R_λ^*s_k^*\|=\|s_k^*\|,\ \|c_j^*\|=\|R_λ^*c_k^*\|=\|c_k^*\|. Consequently, global-optimum solutions satisfy ‖sk∗‖2+‖ck∗‖2=‖sj∗‖2+‖cj∗‖2=const\|s_k^*\|^2+\|c_k^*\|^2=\|s_j^*\|^2+\|c_j^*\|^2=const. Next, introduce the lifted variables i L_i by Gj∗≜sj∗sj∗⊤+cj∗cj∗⊤≜[145∗26∗3].G_j^* s_j^*s_j^* +c_j^*c_j^* bmatrix L_1& L_4& L_5\\ *& L_2& L_6\\ *&*& L_3 bmatrix. where satisfies Gj∗⪰0G_j^* 0 and rank(Gj∗)≤2rank(G_j^*)≤ 2. For arbitrary x∈ℝ9x ^9 and X∈ℝ3×3X ^3× 3 satisfying vec(X)=xvec(X)=x, Roth’s column lemma gives x⊤(sk∗⊗sj∗+ck∗⊗cj∗) x (s_k^* s_j^*+c_k^* c_j^*) =tr[X⊤(sj∗sk∗⊤+cj∗ck∗⊤)] =tr [X (s_j^*s_k^* +c_j^*c_k^* ) ] =−tr[Rλ∗X⊤(sj∗sj∗⊤+cj∗cj∗⊤)] =-tr [R_λ^*X (s_j^*s_j^* +c_j^*c_j^* ) ] =−tr[Rλ∗X⊤Gj∗]. =-tr [R_λ^*X G_j^* ]. Replacing x in the above relationship by Qj←k(i,:)⊤∈ℝ9Q_j← k(i,:) ^9 reduces the primal equality constraints to (17). Finally, because Gj∗=sj∗sj∗⊤+cj∗cj∗⊤G_j^*=s_j^*s_j^* +c_j^*c_j^* , there exist amplitudes and phases such that sj∗=μ¯jcosθjs_j^*= μ_j _j and cj∗=μ¯jsinθjc_j^*= μ_j _j in Lemma 1. Together with [sk∗,ck∗]=−Rλ∗⊤[sj∗,cj∗][s_k^*,c_k^*]=-R_λ^* [s_j^*,c_j^*], this gives the global-optimum dipole allocation. VI-B Proof of Lemma 2 The boundary conditions restrict the edge (n)(n)th and (−n)(-n)th satellites to interact only with their neighboring (n−1)(n-1)th and (−n+1)(-n+1)th satellites, respectively. These derive the required electromagnetic force and torque: f(n)←(n−1)+fd(n) f_(n)←(n-1)+f_d(n) ≜0,f(−n)←(−n+1)+fd(−n)≜0 0, f_(-n)←(-n+1)+f_d(-n) 0 τ(n)←(n−1) _(n)←(n-1) =τ(−n)←(−n+1)≜0 = _(-n)←(-n+1) 0 The angular momentum conservation satisfies the interaction between neighboring satellites: τ(j)←(j−1)+τ(j−1)←(j)+dsat2p^×(f(j)←(j−1)−f(j−1)←(j))=0 _(j)←(j-1)+ _(j-1)←(j)+ d_sat2 p×(f_(j)←(j-1)-f_(j-1)←(j))=0 and the summary of the linear and angular momentum conservation between j and kkth neighboring satellites is f(j)←(j−1)+f(j−1)←(j)=0τ(j)←(j−1)+τ(j−1)←(j)+dsatp^×f(j)←(j−1)=0 \ aligned &f_(j)←(j-1)+f_(j-1)←(j)=0\\ & _(j)←(j-1)+ _(j-1)←(j)+d_sat\ p× f_(j)←(j-1)=0\\ aligned . (28) where we use f(k)←(j)=−f(j)←(k)f_(k)←(j)=-f_(j)←(k) for angular momentum conservation. The conservation in (28) derive reaction force and torque in the (n−1)(n-1)th satellites: f(n−1)←(n) f_(n-1)←(n) =−f(n)←(n−1)=fd(n) =-f_(n)←(n-1)=f_d(n) τ(n−1)←(n) _(n-1)←(n) =−τ(n)←(n−1)−dsatp^×f(n)←(n−1) =- _(n)←(n-1)-d_sat\ p× f_(n)←(n-1) =−dsatp^×f(n)←(n−1) =-d_sat\ p× f_(n)←(n-1) The conservation in (28) also derive reaction force and torque in the (n−2)(n-2)th satellites as follows f(n−2)←(n−1)= f_(n-2)←(n-1)= −f(n−1)←(n−2)=fd(n−1)+f(n−1)←(n) -f_(n-1)←(n-2)=f_d(n-1)+f_(n-1)←(n) τ(n−2)←(n−1)= _(n-2)←(n-1)= −τ(n−1)←(n−2)−dsatp^×f(n−1)←(n−2) - _(n-1)←(n-2)-d_sat\ p× f_(n-1)←(n-2) = = dsatp^×(f(n−1)←(n)+f(n−2)←(n−1)) d_sat\ p×(f_(n-1)←(n)+f_(n-2)←(n-1)) where we use equilibrium condition τ(n−1)←(n−2)+τ(n−1)←(n)=0 _(n-1)←(n-2)+ _(n-1)←(n)=0. Generalizing these for the j∈[2,n+1]j_∈[2,n+1]th satellite derives the required electromagnetic force as f(j−2)←(j−1) f_(j-2)←(j-1) =−f(j−1)←(j−2)=fd(j−1)+f(j−1)←(j) =-f_(j-1)←(j-2)=f_d(j-1)+f_(j-1)←(j) (29) =∑k=j−1nfd(k)=∑k=j−1nmsatKorb(t)(kdsatp^) = _k=j-1^nf_d(k)= _k=j-1^nm_satK_orb(t)(kd_sat p) =(n−j+2)(n+j−1)2/(msatdsat)Korb(t)p = (n-j+2)(n+j-1)2/(m_satd_sat)K_orb(t) p and the required electromagnetic torque as τ(j−2)←(j−1)=dsatp^×∑k=j−1nf(k−1)←(k) _(j-2)←(j-1)=d_sat\ p× _k=j-1^nf_(k-1)←(k) (30) = = dsatp^×∑k=j−1n(n−k+1)(n+k)2msatdsatKorb(t)p d_sat\ p× _k=j-1^n (n-k+1)(n+k)2m_satd_satK_orb(t) p = = (n−j+2)(n−j+3)(2n+j−1)6/(msatdsat2)p^×Korb(t)p (n-j+2)(n-j+3)(2n+j-1)6/(m_satd_sat^2)\ p× K_orb(t) p Following the bucket-brigade logic, the disturbances from both sides of the linear array accumulate and are ultimately canceled at the central 0th satellite, i.e., f(0)←(1)+f(0)←(−1)=0,τ(0)←(1)+τ(0)←(−1)=0f_(0)←(1)+f_(0)←(-1)=0, _(0)←(1)+ _(0)←(-1)=0 where fd(0)=0f_d(0)=0. Applying m¯sys=(2n+1)2msat m_sys=(2n+1)^2m_sat and rl=(2n+1)dsatr_l=(2n+1)d_sat into (29) and (30) yields (21). References [1] D. Tuzi, T. Delamotte, and A. Knopp Satellite swarm-based antenna arrays for 6g direct-to-cell connectivity IEEE Access, vol. 11, p. 36 907–36 928, 2023. [2] S. Shim, Y. Takahashi, N. Usami, M. Kubota, and S.-i. Sakai Feasibility study of distributed space antennas using electromagnetic formation flight In 2025 IEEE Aerospace Conference. IEEE, 2025, p. 1–18. [3] D. You et al. A ka-band 16-element deployable active phased array transmitter for satellite communication In 2021 IEEE MTT-S International Microwave Symposium. IEEE, 2021, p. 799–802. [4] F. Y. Hadaegh, S.-J. Chung, and H. M. Manohara On development of 100-gram-class spacecraft for swarm applications IEEE Systems Journal, vol. 10, no. 2, p. 673–684, 2014. [5] S. A. Schweighart and R. J. Sedwick High-fidelity linhigh-fidelity linearized J2J_2 model for satellite formation flight Journal of Guidance, Control, and Dynamics, vol. 25, no. 6, p. 1073–1080, 2002. [6] T. Komaba et al. On-orbit demonstration of a deployable ka-band 16-element active phased-array transmitter In 2026 IEEE International MTT Symposia (IMS). IEEE, June 2026. [7] E. M. C. Kong, D. W. Kwon, S. A. Schweighart, L. M. Elias, R. J. Sedwick, and D. W. Miller Electromagnetic formation flight for multisatellite arrays Journal of Spacecraft and Rockets, vol. 41, no. 4, p. 659–666, 2004. [8] 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. [9] Y. Takahashi and S.-i. Sakai Neural power‐optimal magnetorquer solution for multi‐agent formation and attitude control IEEE Robotics and Automation Letters, 2026. [10] 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. [11] 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. [12] Y. Takahashi and S. Shin-Ichiro Scalable satellite swarm deployment via distance-based orbital transition under j2j_2 perturbation arXiv preprint, 2025. [13] S. P. Boyd and L. Vandenberghe Convex Optimization. Cambridge University Press, 2004. [14] Y. Takahashi, A. Ochi, Y. Tomioka, and S.-I. Sakai Noda-mmh: Certified learning-aided nonlinear control for magnetically-actuated swarm experiment toward on-orbit proof In International Conference on Space Robotics. IEEE, 2025.