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Optimal Design Framework for Distributed Array Using Magnetically-Actuated Satellite Swarm
Seang Shim, Yuta Takahashi, Naoto Usami, Shin-ichiro Sakai
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 89%
Last extracted: 7/8/2026, 7:29:31 PM
Summary
This paper introduces a system-level design framework for electromagnetic formation flight (EMFF)-based distributed space antennas. It links phased-array requirements with satellite-level constraints (mass, power, coil geometry) and incorporates a control index from distributed-control simulations to maximize feasible aperture. Case studies demonstrate that increasing system mass only improves performance within design headroom, while larger inter-satellite spacing (0.60 m) can exceed satellite capacities, rendering designs infeasible despite favorable communication metrics.
Entities (8)
Relation Signals (5)
Electromagnetic Formation Flight (EMFF) → enables → Distributed Space Antenna
confidence 95% · EMFF provides a physically attractive solution to this requirement... enables simultaneous control of relative positions and absolute attitudes
Inter-satellite Spacing → affects → Antenna Performance
confidence 90% · excessive intersatellite spacing induces grating lobes, whereas insufficient spacing increases mutual coupling
0.60 m Spacing → exceeds → Satellite-Level Capacity
confidence 90% · In the 0.60 m large-spacing case, the required coil burden can exceed satellite-level mass, size, and power capacities, making the design infeasible
Control Index → integratesinto → Antenna-Aperture Maximization Problem
confidence 85% · This index is integrated into an antenna-aperture maximization problem with sizing, power, coil, and sidelobe-envelope constraints.
Satellite Mass → limits → Feasible Aperture
confidence 85% · increasing system mass improves footprint reduction or effective isotropic radiated power only while satellite-level design headroom remains.
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Abstract
Abstract:Distributed space antennas using electromagnetic formation flight (EMFF) are a promising architecture for large-aperture, long-life space communication systems. Their feasible aperture, however, is governed by coupled constraints on antenna performance, satellite mass, power generation, coil geometry, and formation-keeping power. This paper proposes a system-level design framework for EMFF-based distributed space antennas. It links phased-array requirements with satellite-level sizing constraints and provides a static grid-based reference for designing feasible apertures under a fixed system mass. Unlike our previous bucket-brigade disturbance-compensation model, the formation-maintenance requirement is incorporated through a control index derived from distributed-control simulations. This index is integrated into an antenna-aperture maximization problem with sizing, power, coil, and sidelobe-envelope constraints. Parametric case studies examine margin magnetic moment, prescribed transmit power, and large inter-satellite spacing. Results show that increasing system mass improves footprint reduction or effective isotropic radiated power only while satellite-level design headroom remains. In direct-to-device cases with 0.15 m spacing, generated-power and coil-geometry constraints dominate the feasible aperture. In the 0.60 m large-spacing case, the required coil burden can exceed satellite-level mass, size, and power capacities, making the design infeasible despite favorable communication performance. The proposed framework enables the design and evaluation of feasible static grid-based EMFF distributed antennas under coupled antenna, satellite, and control constraints.
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- Source: https://arxiv.org/abs/2605.23481v1
- Canonical: https://arxiv.org/abs/2605.23481v1
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Optimal Design Framework for Distributed Array Using Magnetically-Actuated Satellite Swarm Seang Shim, Yuta Takahashi, , Naoto Usami, , and Shin-ichiro Sakai Seang Shim and Yuta Takahashi contributed equally as co-first authors. This work was supported by JST SPRING, Japan Grant Number JPMJSP2104.S. Shim is with The Graduate University for Advanced Studies, Sagamihara, Kanagawa 252-5210, Japan.Y. Takahashi is with Institute of Science Tokyo, Tokyo 152-8550, Japan, and Interstellar Technologies Inc., Hiroo, Hokkaido 089-2113, Japan.N. Usami and S.-i. Sakai are with Japan Aerospace Exploration Agency, Sagamihara, Kanagawa 252-5210, Japan.Corresponding author: Seang Shim (e-mail: shim.seang@ac.jaxa.jp). Abstract Distributed space antennas using electromagnetic formation flight (EMFF) are a promising architecture for large-aperture, long-life space communication systems. Their feasible aperture, however, is governed by coupled constraints on antenna performance, satellite mass, power generation, coil geometry, and formation-keeping power. This paper proposes a system-level design framework for EMFF-based distributed space antennas. It links phased-array requirements with satellite-level sizing constraints and provides a static grid-based reference for designing feasible apertures under a fixed system mass. Unlike our previous bucket-brigade disturbance-compensation model, the formation-maintenance requirement is incorporated through a control index derived from distributed-control simulations. This index is integrated into an antenna-aperture maximization problem with sizing, power, coil, and sidelobe-envelope constraints. Parametric case studies examine margin magnetic moment, prescribed transmit power, and large inter-satellite spacing. Results show that increasing system mass improves footprint reduction or effective isotropic radiated power only while satellite-level design headroom remains. In direct-to-device cases with 0.15-m spacing, generated-power and coil-geometry constraints dominate the feasible aperture. In the 0.60-m large-spacing case, the required coil burden can exceed satellite-level mass, size, and power capacities, making the design infeasible despite favorable communication performance. The proposed framework enables the design and evaluation of feasible static grid-based EMFF distributed antennas under coupled antenna, satellite, and control constraints. I Introduction Figure 1: Overview of design optimization of distributed space antennas for the trade-off between formation control and antenna performance. The magnetic coils in each satellite generate magnetic fields, and their interactions enable long-term formation maintenance. Detailed guidance and control frameworks are provided in our previous studies [1, 2, 3]. Distributed space antennas composed of palm-sized satellites provide a scalable architecture for high-performance communication systems under fuel-free formation maintenance, such as electromagnetic formation flight (EMFF). Previous space antennas typically required narrow beams to suppress interference with other beams [4] and high-gain transmission over long distances, which in turn required larger antenna structures through increased element counts or inter-element spacing [5, 6]. Examples include deep-space antennas [7], ground terminals with tiny receivers [8], and space membrane antennas [9]. Compared with monolithic antennas, distributed architectures alleviate single-point failures, payload-size limitations, and cost escalation while enabling large effective apertures [10, 11, 12, 7, 13, 14, 1, 15]. Their practical realization, however, requires active position control to preserve the formation against orbital disturbances such as the J2J_2 effect [16]. Without such control, antenna performance deteriorates over time: excessive intersatellite spacing induces grating lobes, whereas insufficient spacing increases mutual coupling [17, 5, 6]. Because the propulsion required for formation maintenance directly affects mission lifetime, fuel-free control is of particular interest. EMFF provides a physically attractive solution to this requirement. By exploiting magnetic torquers (MTQs), which are standard actuators on small satellites, EMFF enables simultaneous control of relative positions and absolute attitudes [18, 14, 13, 19, 1]. Electromagnetic formation control using alternating current (AC) has also been studied with learning-based control [20, 3]. Prior studies have established exact interaction models based on the Biot–Savart law [21], learning-based approximations [22], and dipole approximations for far-field applications [18]. In addition, the Earth’s magnetic field can be exploited to regulate the angular momentum of the overall system and mitigate unnecessary interference [14]. Alternative fuel-free approaches, such as differential drag and tethers [23, 24], are less suitable for distributed space antennas because the former is strongly attitude-dependent, whereas the latter lacks scalability and reconfigurability [25]. Existing EMFF studies have largely focused on small numbers of satellites [13], whereas practical distributed space antennas may require substantially larger arrays [8]. The distributed architecture becomes attractive precisely in this large-scale regime, where monolithic systems are constrained by payload, cost, and reliability considerations [12]. In general, a large array size improves communication performance and enables a narrow beam to prevent interference with other communication areas. However, the electromagnetic force decays with the fourth power of the distance. This distance-dependent limit directly limits the inter-satellite distance for a given power consumption. Therefore, it is important to clarify the trade-off between control and antenna performance with respect to distance. Despite these advantages, the system-level potential of EMFF-based distributed antennas remains unclear. A previous study clarified that magnetically actuated satellite swarms can improve power efficiency as the number of satellites increases when constructing large-scale distributed space structures, using a convex-optimization-based formation-keeping power analysis [26]. Although the convex-optimization-based framework is suitable for evaluating the overall system and per-satellite power trends, this framework is unsuitable for detailed component-level satellite design due to conservative worst-case specifications. This motivates a system-level feasibility analysis that jointly accounts for communication metrics, including sidelobe levels and effective isotropic radiated power (EIRP), and the control power required for deployment and long-term formation maintenance [27]. To address this need, this paper presents an extended system-level design methodology for EMFF-based distributed space antennas that meets prescribed antenna performance requirements while enabling practical satellite sizing. This paper builds on our previous work on the feasibility study of EMFF-based distributed space antennas [28]. In that study, the formation-maintenance requirement was derived using a bucket-brigade compensation model for a grid formation. Although this model provided an analytically tractable first estimate of the required magnetic moment and control power, it relied on an idealized compensation structure and did not directly reflect the behavior of distributed formation control. In contrast, this paper estimates the required magnetic moment and control power from numerical simulations of distributed formation control, thereby providing a more realistic basis for system-level sizing. The remainder of this paper is organized as follows. Section I reviews the underlying orbital dynamics, magnetic interaction model, and planar phased-array antenna formulation. Our problem formulation is presented in Section I to define the variables describing the distributed antenna. As a main result, Section IV presents a nonconvex optimization-based framework for designing distributed space antennas, subject to satellite-level mass and power constraints, as well as antenna performance and formation-maintenance constraints for grid-based distributed antennas. Section V presents the numerical case studies based on the integrated optimization framework. Section VI concludes the paper. Throughout the paper, perfect knowledge of all satellite states is assumed in order to focus on trend analysis. I Preliminaries This section summarizes the control of the relative distance from J2J_2 disturbance using EMFF and the derivation of the antenna performance, focusing on the EIRP and beamwidth. I-A Linearized/Averaged Relative Orbital Dynamics We introduce the approximated dynamics of relative orbital motion under J2J_2 disturbances [16]. Let Pj=[rref+xj;yj;zj]P_j=[r_ref+x_j;y_j;z_j] be the position vector of the j-th satellite from the center of the Earth in the local vertical, local horizontal (LVLH) coordinate frame, where rrefr_ref is the constant orbital radius of the reference orbit. We obtain P¨j=∇UJ2 P_j=∇ U_J_2 where ∇UJ2∈ℝ3∇ U_J_2 ^3 is the gravity gradient associated with its orbital dynamics [16]: ∇UJ2(Pj,i,θ)=−[μg‖Pj‖200]−kJ2‖Pj‖4[1−3sin2isin2θsin2isin2θsin2isinθ],∇ U_J_2(P_j,i,θ)=- bmatrix _g\|P_j\|^2\\ 0\\ 0 bmatrix- k_J_2\|P_j\|^4 bmatrix1-3 ^2i ^2θ\\ ^2i 2θ\\ 2i θ bmatrix, (1) where the constant μg _g, the inclination i, the argument of latitude θ, and J2J_2 coefficient kJ2=2.63e10km5/s2k_J_2=2.63e^10~km^5/s^2. A previous study defined a reference orbit to compensate J2J_2 effect: ωref=[00c+ωo]s.t.ωref×ωref×rref=∫02π∇UJ2(θ)dθ2π, _ref= bmatrix0\\ 0\\ c_+ _o bmatrix\ s.t.\ _ref× _ref× r_ref= _0^2π∇ U_J_2(θ) dθ2π, (2) ωo=μgrref3,c±=1±sJ2,sJ2=kJ2(1+3cos2iref )4μgrref2. _o= _gr_ref^3,\ c_±= 1± s_J_2,\ s_J_2= k_J_2(1+3 2i_ref )4 _gr_ref^2. We include an additional compensation term for the separation of the orbital planes owing to the time-varying longitudes of the ascending nodes Ω(t) (t) such that θ(t)=ωzreftθ(t)=ω_zreft where ωzref=c+ω−Ω˙avgjcosij=ωo(c++kJ2cos2iμrref2),ω_zref=c_+ω- _avgj i_j= _o (c_++ k_J_2 ^2iμ r_ref^2 ), (3) where Ω˙(θ)=−2kJ2cosisin2θ/(‖rref3) (θ)=-2k_J_2 i ^2θ/(\|h\|r_ref^3), |Ω˙avg|=|∫02πΩ˙(θ)dθ/2π|≲2e−6cosiref| _avg|=| _0^2π (θ)dθ/2π| 2e^-6 i_ref under ij≈iki_j≈ i_k [1],[29]. Linearization based on the reference orbit is used to derive the relative motion between two satellites. Let rjk=rj−rk=[xjk;yjk;zjk]r_jk=r_j-r_k=[x_jk;y_jk;z_jk] be the relative position from the k-th satellite to the j-th one. The dynamics of the linearized relative motion around the reference orbit in the LVLH frame are [1, 29] 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) (4) 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 the subscript jkjk will be omitted, l(δΩ˙avgjk)=−rrefsinijsinikf2(δΩ˙avgjk)l(δ _avgjk)=-r_ref i_j i_kf_2(δ _avgjk), and the dynamics along the z-axis include a term for compensating for errors in the cross-track motion owing to time averaging. Assuming that the satellites have identical i, the amplitude of the relative orbital motion along the z-axis reaches a constant value, which means that l(δΩ˙avg)≈0l(δ _avg)≈ 0 in (4). I-B Averaged J2J_2 Relative Orbital Parameters We derives the orbital indices, averaged J2J_2 relative orbital parameters, and this denotes desired stable trajectories pd(t)p_d(t). An analytical solution of (4) is [1, 29]: [x(t)y(t)z(t)] bmatrixx(t)\\ y(t)\\ z(t) bmatrix =[ro(0,t)0]+[rxysin(ωxyt+θxy)/c+2rxycos(ωxyt+θxy)/c−(rz+lt)sin(ωzt+θz)], = bmatrixr_o(0,t)\\ 0 bmatrix+ bmatrixr_xy ( _xyt+ _xy)/c_+\\ 2r_xy ( _xyt+ _xy)/c_-\\ (r_z+lt) (ω_zt+ _z) bmatrix, (5) ro(0,t) r_o(0,t) =[2C1C4−ϵ2C1t]⊤,ϵ2=3+5sJ2c+c−ωxy, = bmatrix2C_1&C_4- _2C_1t bmatrix ,\ _2= 3+5s_J_2c_+c_- _xy, where ro(0,t)∈ℝ2r_o(0,t) ^2 represents the center position of the relative orbit at time t, l(δΩ˙avgjk)≈0l(δ _avgjk)≈ 0, where f1,2∈ℝf_1,2 is a function of δΩjk0δ _jk0. The orbital indices calculated at the estimation time (t=0t=0) are specified as [1] 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 . (6) It is clear from (5) that the motions are governed by rxy,zr_xy,z and δθ=θz−θxyδθ= _z- _xy, as defined in (6). A simple calculation yields the following relationship between rxy,zr_xy,z, δθδθ, and the satellite swarm angle (ΘP,Θz−xy)( _P, _z-xy) for the aperture design: rz r_z =rxytanΘPcosΘz−xycos(θz−θxy),θz=θxy+tan−1(2tanΘz−xy). = r_xy _P _z-xy ( _z- _xy), _z= _xy+ ^-1(2 _z-xy). (7) The desired trajectory along the z-axis is selected based on the x–y motion: ωzd=ωxy _zd= _xy and θzd=θxy+tan−1(2tanΘz−xy) _zd= _xy+ ^-1(2 _z-xy). This derives the desired stable trajectories pd(t)p_d(t) 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 the z-axis orbit control to maintain the z-axis orbital frequency ωz _z in ωxy _xy or it works with the disturbance dfz=(ωxy2−ωz2)zd_fz=( _xy^2- _z^2)z on pdp_d [1, 29]: dfz(t) d_fz(t) =rzd(ωxy2sin(ωxyt+θz)−ωz2sin(ωzt+θz)). =r_zd( _xy^2 ( _xyt+ _z)- _z^2 ( _zt+ _z)). (9) I-C Magnetic Interaction Approximation Model This subsection introduces the approximation of 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, (10) 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. A previous study [20] simplified a dipole model approximation [18] as the bilinear polynomial formulation using the line-of-sight frame ℒj←k\LOS_j← k\ (see Definition 1). Then, the input exerted on the j-th agent by the k-th agent is : uj←ka=[fj←kaτj←ka]=μ04πQj←k(rj←k)(μka⊗μja),u_j← k^a= bmatrixf_j← k^a\\ _j← k^a bmatrix= _04πQ_j← k(r_j← k) ( _k^a _j^a ), (11) where μ0 _0 is the permeability of free space, j,k μ_j,k are the magnetic moments of the j,kj,k-th coils, Qj←k∈ℝ6×9Q_j← k ^6× 9 is [20] Qj←k=(I2⊗CA/Lj←k)[Ψf(rj←k)Ψτ(rj←k)](CLj←k/A⊗CLj←k/A),Q_j← k=(I_2 C^A/L_j← k) bmatrix _f(r_j← k)\\ _τ(r_j← k) bmatrix(C^L_j← k/A C^L_j← k/A), and the vector from the k-th coil to the j-th coil rj←kr_j← k yields Ψf=1‖rj←k‖4[−600030003030300000003000300]Ψτ=1‖rj←k‖3[0000010−100020001000−20−100000]. aligned & \ 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 . aligned. Definition 1. The line-of-sight frame ℒj←k\LOS_j← k\ [20] is attached to the k-th agent and oriented toward the j-th agent. It is defined in (12), with the associated coordinate transformation matrix CO/Lj←k∈ℝ3×3C^O/Lj← k ^3× 3 expressed as 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 we define the coordinate transformation matrix given the arbitrary vectors v,aw∈aℝ3v^a,w^a ^3: (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. (12) I-D Optimal Electromagnetic Swarm Control We consider the position and attitude control of multiple satellites using the magnetic interaction model defined in the previous subsection. We assume that the j-th magnetic moment is driven by a sinusoidal wave [20]: j(t)=¯jsin(ωjt+j)=jsin(ωjt)+jcos(ωjt), μ_j(t)= μ_j ( _jt+ θ_j)= s_j ( _jt)+ c_j ( _jt), (13) 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 for the j-th MTQ. The first-order averaged input u¯j←k∈ℝ6 u_j← k ^6 is u¯j←k=12μ04πQj←k(ska⊗sja+cka⊗cja)ifωj=ωk. u_j← k= 12 _04πQ_j← k (s_k^a s_j^a+c_k^a c_j^a )\ if\ _j= _k. (14) We formulate the power-optimal dipole allocation [20] as minJp=‖[sja;ska;cja;cka]‖2/2s.t.μ08πQj←k(ska⊗sja+cka⊗cja)=uj←ka. aligned min &J_p= \|[s_j^a;s_k^a;c_j^a;c_k^a] \|^2/2\\ s.t. & _08πQ_j← k (s_k^a s_j^a+c_k^a c_j^a )=u_j← k^a aligned. (15) and its Lagrange dual problem of the problem in (15) is maxλ∈ℝ6Jd=−λ⊤uj←kaμ0/(8π)s.t.Pλ=[E3RλRλ⊤E3]⪰0, _λ ^6\ J_d= -λ u^a_j← k _0/(8π) .t. P_λ= bmatrixE_3&R_λ\\ R_λ &E_3 bmatrix 0, (16) where the Lagrange multiplier vector λ∈ℝ6λ ^6 and Rλ∈ℝ3×3R_λ ^3× 3 satisfies vec(Rλ)=Qj←k⊤λvec(R_λ)=Q_j← k λ [20]. Thus, the required magnetic dipole power magnitude for control can be derived using (16) [20] because the lower bound is obtained as Jd≤JpJ_d≤ J_p [30] and Jd=JpJ_d=J_p for n=2n=2 [26]. I-E Planar Phased-Array Antenna Formulation This subsection introduces the effective isotropic radiated power (EIRP) for a planar phased-array antenna characterized by uniform element spacing, omnidirectional elements, and identical excitation amplitudes across all elements. The EIRP is the product of antenna gain GTG_T and the transmitting power PTP_T [27], where PTP_T is the total of each element transmitting power PtP_t. Then, we obtain EIRP=PTGT=(PtNlxNly)GTEIRP=P_TG_T=(P_tN_lxN_ly)G_T where NlxN_lx and NlyN_ly are the number of satellites per grid side, respectively. Assuming that the antenna loss is negligible, GTG_T is equal to the maximum directivity D¯ D [5]: D¯≜maxθ,ϕD(θ,ϕ)=maxθ,ϕ4πB(θ,ϕ)∫02π∫0πB(θ,ϕ)sinθdθdϕ, D _θ,φD(θ,φ)= _θ,φ 4π B(θ,φ) _0^2π _0^πB(θ,φ) θ\,d φ, (17) where θ and ϕφ are the elevation and azimuth angles in the polar coordinate system, as shown in Fig. 2a. Here, B(θ,ϕ)B(θ,φ) denotes the dimensionless normalized gain pattern, which is defined from the normalized field array factor A(θ,ϕ)A(θ,φ) as B(θ,ϕ)=|A(θ,ϕ)|2.B(θ,φ)=|A(θ,φ)|^2. (18) For a rectangular planar array with uniform excitation, the normalized field array factor is expressed as the product of two linear-array factors [5]: A(θ,ϕ)=Ax(ux)Ay(uy),Ax(ux)=1Nlxsin(Nlxux)sin(ux),Ay(uy)=1Nlysin(Nlyuy)sin(uy). gatheredA(θ,φ)=A_x(u_x)A_y(u_y),\\ A_x(u_x)= 1N_lx (N_lxu_x ) (u_x ), A_y(u_y)= 1N_ly (N_lyu_y ) (u_y ). gathered (19) Here, uxu_x and uyu_y are defined as ux u_x =kdx(sinθcosϕ−sinθ0cosϕ0), =kd_x ( θ φ- _0 _0 ), (20) uy u_y =kdy(sinθsinϕ−sinθ0sinϕ0), =kd_y ( θ φ- _0 _0 ), where k=π/λk=π/λ, λ is the wavelength, dxd_x and dyd_y are the element spacings along the two array axes, and (θ0,ϕ0)( _0, _0) denotes the main-beam direction. When mutual coupling is neglected, averaging (17) over all directions gives the following approximation [31, 32]: GT=D¯≈Nl2.G_T= D≈ N_l^2. (21) Finally, Nl=Nlx=NlyN_l=N_lx=N_ly derives EIRP as follows: EIRPmax≜PTGT=PtNl4.EIRP_ P_TG_T=P_tN_l^4. (22) I Problem Formulation I-1 Grid Structure Array Model We primarily use a simplified model of a square phased array in Fig. (2) and the elements are equally spaced on a grid. Assumption 1. We consider a distributed space antenna of a square phased array as shown in Fig. 2b, i.e., Nall≜Nl2≜(2n+1)2,N_all N_l^2 (2n+1)^2, (23) where NlN_l is used for antenna context in subsection I-E. Its total system mass m¯sys m_sys, their inter-element distance dsatd_sat, and associated total length rlr_l are user-defined constant values: m¯sys≜const.,dsat≜const.,rl≜(2n+1)dsat, m_sys ., d_sat ., r_l (2n+1)d_sat, Its arbitrary row-wise linear formation illustrated in Fig. 2c havs the distance vector Rl(τ)∈ℝ3R_l(τ) ^3 from the (−n-n)th satellite to the (n)th satellite as Rl(τ)≜rl(τ)p^(τ)∥pd(τ),τ∈[0,2π/ωxy).R_l(τ) r_l(τ) p(τ)\ \ p_d(τ), τ∈[0,2π/ _xy). Example 1. The distance dsatd_sat is set to be no greater than half a wavelength λ to suppress grating lobes—undesired radiation with comparable strength to the main lobe peak. dsat≜const.≤λ/2.d_sat .≤λ/2. (24) Definition 2. The total system mass msysm_sys and its relaxed constraint are defined by the total number of satellites Nall=(2n+1)2N_all=(2n+1)^2 by (23) in Assumption 1: Nallmsat≜m¯sys,|Nallmsat−m¯sys|≤△γsys,N_all\ m_sat m_sys, |N_all\ m_sat- m_sys| ≤ _sys, (25) where m¯sys m_sys is the user-defined total system mass and γsys _sys is the tolerance of the total system mass for computation. (a) Planar-array antenna. (b) Distributed space antenna [1, 29]. (c) Linear formation model of 2n+12n+1 satellites [26]. Figure 2: Grid-structured approximation for distributed space system design. This formation consists of equally spaced linear arrays. I-2 Decentralized Control Model To suppress unintended coupling among nonadjacent satellites in close proximity, we first assume the use of frequency allocation introduced in subsection I-D to confine electromagnetic interactions to neighboring satellites. Assumption 2. 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. 2c (Please refer the detailed selection of angular frequencies [20].). IV Distributed Antenna Design Optimization This section presents a nonconvex optimization-based framework to evaluate the feasibility of distributed space antennas. In practice, however, excessively increasing the number of satellites would lead to unrealistically small satellites due to physical density limits. In the next section, considering these practical constraints, the detailed design problem is formulated as a nonconvex optimization problem, yielding feasible design solutions and more refined scaling trends. IV-A Overview of Design Optimization Problem We provide an overview of the design framework for the distributed space antenna and clarify the constraints. We consider the design parameters X of our framework for a given satellite distance dsatd_sat in (24) of Example 1 and system mass m¯sys m_sys in Assumption 1: X=2acoil, 2asat,qcoil,n,umsl,X= \2a_coil,\ 2a_sat,\ q_coil,\ n,\ u_msl \, (26) where the coil diameter 2acoil2a_coil, satellite size 2asat2a_sat, coil parameter qcoilq_coil, satellite number n, and peak sidelobe variable upslu_psl. Then, the design optimization DSAP_DSA is formulated as follow: X∗=argminX∈χfeasLbw=argmaxX∈χfeasNallsubject to: \ X^*= arg~min _X∈ _feasL_bw= arg~max _X∈ _feasN_all to: (27) (A) Satellite specification constraints(B) Phased-array antenna constraints(C) Magnetic formation-keeping constraints \ aligned &(A) Satellite specification constraints\\ &(B) Phased-array antenna constraints\\ &(C) Magnetic formation-keeping constraints aligned . where the constraints in (A)–(C) are described in Subsections IV-B, IV-C, and IV-D, respectively. The optimal X∗X^* yields other dependent parameters, such as the generation and consumption power, EIRP, and component masses. IV-B Constraints 1/3: Satellite Specification This section defines the design parameters for each satellite and formulates the power consumption and satellite constraints. We define the satellite configuration as follows. Definition 3. The satellite consists of 1) an identical three-axis coil, 2) solar panels, 3) a battery, 4) a satellite body, and 5) a bus that includes antennas, transmitters, and avionics: msat≜m3coil+m4sap+mbat+mstr+mbus,m_sat m_3coil+m_4sap+m_bat+m_str+m_bus, where each term corresponds to the mass of each subsystem. IV-B1 Component Size Constraints This subsection summarizes the constraints on the satellite and coil sizes. Size is a critical parameter that involves trade-offs between mass, power acquisition, and achieving of the magnetic moment. We formulate Assumption 3 considering the obvious physical constraints of satellites. Assumption 3. The geometric relationship among the satellite radius, coil radius, and inter-satellite distance is illustrated in Fig. 2c. The coil radius acoila_coil is smaller than the satellite one asata_sat and 2asat2a_sat is smaller than the distance between the satellite centers dsatd_sat with the coil size margin rmarr_mar acoil≤△asat−rmar,2asat≤△dsat,kFacoil≤△dsat,a_coil ≤a_sat-r_mar, 2a_sat ≤d_sat, k_Fa_coil ≤d_sat, (28) where the final inequality is from the assumption that the distance between the coils is sufficiently large compared with the coil radius with defined coefficient kFk_F. IV-B2 Component Mass Definitions and Assumptions This subsection defines the masses of the satellite components. We implement Assumption 4 based on Definition 3 to define the component masses m3coilm_3coil, m4sapm_4sap, mbatm_bat, mstrm_str, and mbusm_bus as dependent parameters of the other parameters. Assumption 4. The satellite has a cubic shape and an identical three-axis coil, that is, with an equal number of turns NtN_t, wire radius rcoilr_coil, and coil radius acoila_coil. Solar panels are attached to the four sides of the satellite. The bus mass mbusm_bus is treated as a user-defined constant, mbus0m_bus0. The structural mass mstrm_str is defined as a proportion of the total satellite mass. PsapP_sap denotes the power generated during sunlit operation: Psap(asat)≜ksapPW/m2(2asat)2,P_sap(a_sat) k_sap\,P_W/m^2(2a_sat)^2, (29) where PW/m2P_W/m^2 is the power generated per unit area of the solar panel and ksapk_sap is the effective area coefficient. The battery storage requirement is modeled as the solar energy that is harvested over a sunlit charging duration hchargeh_charge and a design coefficient that captures the charging efficiency and usable fraction kbatk_bat using PsapP_sap in (29). Example 2 (Requirement for battery storage from sunlit charging). As a concrete sizing example, consider a baseline case in which the satellite stores 10%10\% of the energy that is harvested during a 1212-h sunlit period. This corresponds to setting kbat=0.1k_bat=0.1 and hcharge=12h_charge=12. Under this setting, the required battery storage is modeled as kbathchargePsap(asat)=1.2Psap(asat),k_bat\,h_charge\,P_sap(a_sat)=1.2\,P_sap(a_sat), (30) where Psap(asat)P_sap(a_sat) is determined by (29). This Assumption 4 defines the mass of the three-axis coil with the coil parameter qcoilq_coil, four solar panels, battery, structure, and bus system: m3coil(acoil,qcoil)≜3(2π2acoil)qcoilρc,qcoil≜Ntrcoil2,msap(asat)≜νsapρsap(2asat)2,mbat(asat)≜ρbatkbathchargePsap(asat),mstr(msat)≜ηstrmsat,mbus≜mbus0=const, \ aligned m_3coil(a_coil,\ q_coil)& 3(2π^2a_coil)q_coil _c,\\ q_coil& N_tr_coil^2,\\ m_sap(a_sat)& _sap _sap(2a_sat)^2,\\ m_bat(a_sat)& _batk_bat\ h_charge\ P_sap(a_sat),\\ m_str(m_sat)& _strm_sat,\\ m_bus& m_bus0=const, aligned . (31) where ρc _c denotes the mass density of the wire, ρsap _sap is the mass per unit area density, νsap _sap is the number of solar panels that are mounted on the satellite, ηstr _str is a user-defined constant, and ρbat _bat is the mass per unit power capacity of the battery. Throughout this study, PbatP_bat is modeled as a function of the satellite volume. The coil parameter qcoilq_coil in (31) allows NtN_t and rcoilr_coil to be adjusted according to the satellite geometry by obtaining the optimal qcoilq_coil. IV-B3 Component Mass Constraints Finally, we calculate the component mass constraints. As indicated in (31), our assumption limits the masses of almost all components as dependent variables of asata_sat, except for the coil mass m3coil(acoil,qcoil)m_3coil(a_coil,q_coil). This m3coilm_3coil should be designed to satisfy the requirements of antenna performance and formation control. Subsequently, we derive the upper bound m¯sat m_sat to avoid impractical coil configurations by assuming that msat∝asat3m_sat a_sat^3. Because the total mass should converge to a finite lower bound, we statistically derive this relationship from the existing satellite design parameters. Assumption 5. m¯sat m_sat is a polynomial function of asata_sat: m¯sat(asat)≜km¯asat3if2asat≥0.1femp(asat)otherwise, m_sat(a_sat) \ aligned &k_ m\ a_sat^3&&if 2a_sat≥ 0.1\\ &f_emp(a_sat)&&otherwise,\ \\ aligned . (32) where km¯k_ m is an arbitrary parameter for which we mainly select km¯=1e3k_ m=1e^3 for the standard CubeSat design, and femp(asat)f_emp(a_sat) denotes an empirically approximated function that is obtained by fitting the previously designed satellite mass, e.g., the parameters in Table 3 and red markers in Fig. 3. This methodology using fempf_emp yields design solutions that are consistent with current technology and is adaptable using updated data from next-generation satellites. Subsequently, we derive the upper and lower bounds msat(asat)m_sat(a_sat) and msat≤△m¯sat(asat) m_sat ≤ m_sat(a_sat) (33) m4sap(asat)+mbat(asat)+mstr(msat)+mbus≤△msat, m_4sap(a_sat)+m_bat(a_sat)+m_str(m_sat)+m_bus ≤m_sat, where m¯sat(asat) m_sat(a_sat) is determined by (32) and the lower bound is derived as m3coil=0m_3coil=0. TABLE I: Previous satellite designs V (cm3) msatm_sat (g) [15] 2×2×0.0752× 2× 0.075 10 [33] 9×9.5×19× 9.5× 1 70 [34] 9×9×19× 9× 1 100 [12] 4×4×4.254× 4× 4.25 95.5 [35, 36] 3.3×3.3×0.53.3× 3.3× 0.5 9.9 Figure 3: Relationship between satellite volume and component mass. IV-C Constraints 2/3: Phased-Array Antenna The antenna performance constraints of the distributed space antenna are formulated in this section. We adopt a simplified antenna model consisting of an Nl×NlN_l× N_l square grid to render the analysis tractable while preserving system-level trends. IV-C1 Tractable Beam Footprint Model We first formulate a computationally tractable beam footprint model by approximating the beam edge with the first null of the main lobe. The footprint boundary is characterized by the beamwidth in the diagonal cut (ϕ=π/4)(φ=π/4), which reduces the two-dimensional beam projection to a uniaxial representation. Assumption 6 (Azimuth restriction for footprint evaluation). The azimuth is fixed at ϕ=ϕ0=π/4φ= _0=π/4, that is, the diagonal cut where ux=uy=u_x=u_y=u, to reduce the dimensionality of the footprint evaluation. Under this restriction, the two-dimensional normalized field array factor of the square grid array is transformed into Adiag(u)=(sin(Nlu)Nlsin(u))2,u=kdsat2(sinθ−sinθ0).A_diag(u)= ( (N_lu )N_l (u ) )^2, u= kd_sat 2 ( θ- _0 ). (34) Here, Adiag(u)A_diag(u) is the direction-restricted normalized field array factor. The corresponding normalized gain pattern is given by |Adiag(u)|2|A_diag(u)|^2, but only the null location of Adiag(u)A_diag(u) is required for the footprint approximation. The beam edge is therefore defined by the first null condition Nlu=πN_lu=π, which yields the null angle θn1 _n1. Assumption 7 (Footprint Definition). The beam footprint is defined by the first null of the main lobe evaluated along the specific azimuth direction defined in Assumption 6. It is assumed that the orbital altitude h is significantly larger than the element spacing dsatd_sat, i.e., h≫dsath d_sat. Using the geometric relationship based on Assumption 7, the footprint diameter DfpD_fp is derived as a function of θn1 _n1 and the orbital altitude h: θn1≜arcsin(2λNldsat+sinθ0),Dfp≈2|θn1−θ0|h. _n1 ( 2λN_ld_sat+ _0 ), D_fp≈ 2| _n1- _0|h. (35) IV-C2 Peak-Sidelobe and Transmitter Power Derivation The local maximizer of Benv(u)B_env(u) within the first sidelobe region defines the peak-sidelobe envelope point upslu_psl: Nlsin(upsl)cos(Nlupsl)−cos(upsl)sin(Nlupsl)≜0,πNl<△|upsl|<△2πNl. \ aligned &N_l (u_psl) (N_lu_psl)- (u_psl) (N_lu_psl) 0,\\ & πN_l <|u_psl| < 2πN_l. aligned . (36) For peak-sidelobe-constrained transmit-power sizing, we use the one-dimensional sidelobe envelope of the separable square array. From the separable field array factor in (19) and the normalized gain definition in (18), the two-dimensional response can be expressed as B2D(ux,uy)=Bx(ux)By(uy),B_2D(u_x,u_y)=B_x(u_x)B_y(u_y), where BxB_x and ByB_y are normalized by the main-beam peak. Since the maximum contribution from the other array axis is unity, i.e., maxuyBy(uy)=1 _u_yB_y(u_y)=1, the largest two-dimensional response for a given ux=u_x=u is bounded by maxuyB2D(u,uy)=Bx(u)maxuyBy(uy)=Bx(u). _u_yB_2D(u,u_y)=B_x(u) _u_yB_y(u_y)=B_x(u). Therefore, the sidelobe-envelope measure used in this study is defined as Benv(u)=Bx(u)=|sin(Nlu)Nlsin(u)|2.B_env(u)=B_x(u)= | (N_lu )N_l (u ) |^2. (37) This quantity is a dimensionless normalized gain pattern used for system-level sizing. It is distinguished from the direction-restricted field array factor Adiag(u)A_diag(u) used for the footprint approximation. The sidelobe EIRP evaluated by the envelope model must be smaller than the received power indicator IRI_R: EIRPmaxBenv(upsl)≤△IR.EIRP_ B_env(u_psl) ≤I_R. (38) The RF transmit power satisfying the equality in (38) is obtained from EIRPmax=PtNl4EIRP_ =P_tN_l^4 and (37) as follows: Pt≜IRNl4Benv(upsl)=IRNl2(sin(upsl)sin(Nlupsl))2.P_t I_RN_l^4B_env(u_psl)= I_RN_l^2 ( (u_psl) (N_lu_psl) )^2. (39) Remark 1. Assuming a direct-to-device (D2D) communication scenario, IRI_R is expressed as an example as follows: IR≜ζPRLf/GR,Lf=(4πhλ)2,I_R ζ P_RL_f/G_R, L_f= ( 4π hλ )^2, (40) where PRP_R is the required received power, GRG_R is the antenna gain, ζ∈(0,1]ζ∈(0,1] is the attenuation factor, and LfL_f is the free-space path loss. Then, the transmit power is Pt≜ζPRLfNl2GR(sin(upsl)sin(Nlupsl))2.P_t ζ P_RL_fN^2_lG_R ( (u_psl) (N_lu_psl) )^2. (41) Remark 2. The variable upslu_psl is introduced to specify the finite-array peak point of the sidelobe envelope used in transmit-power sizing. It is not intended to represent a full two-dimensional radiation-pattern search. IV-D Constraints 3/3: Magnetic Formation-Keeping We describe the power requirements for a distributed space antenna to execute its communication mission while maintaining relative inter-satellite positions against disturbances. IV-D1 Power Budget for Steady-Formation Maintenance A steady-state power-budget constraint for long-term magnetic formation maintenance is formulated in this subsection. Definition 4. The steady-state power consumption includes a control power PcontP_cont, mission power PmisP_mis, bus power PbusP_bus, and margin power PmarP_mar, where PbusP_bus is a constant: Ptot≜Pcont+Pmis+Pbus+Pmar.P_tot P_cont+P_mis+P_bus+P_mar. (42) Assumption 8. As a steady-state condition, the power is supplied only by solar panels, and the time-averaged consumption remains below the generated power: Ptot≤PsapP_tot≤ P_sap. Assumption 9. The required power for each axis is evaluated from the peak coil output. The control power for one axis is defined as P1axis=Rcoilccoil2=2pcμ¯2dir2π2qcoilacoil3,P_1axis=R_coil\,c_coil^2= 2p_c μ_2dir^2π^2q_coila_coil^3, (43) where RcoilR_coil is the coil resistance defined in (10), ccoilc_coil is the coil current, and μ¯2dir μ_2dir is the upper bound of magnetic moment required for bidirectional compensation along one axis. When AC actuation is used to alternate the moment command between two opposite directions along one axis, the synthesized magnetic moment μ2dir(t) _2dir(t) is written as μ2dir(t) _2dir(t) =μ1dircos(ωfnt)+μ1dircos(ωfnt+π2) = _1dir ( _f_nt)+ _1dir ( _f_nt+ π2 ) (44) =2μ1dirsin(ωfnt+π4), = 2\, _1dir ( _f_nt+ π4 ), where ωfn _f_n is the AC frequency assigned to the n-th satellite and the upper bound of μ¯2dir μ_2dir is 2μ1dir 2 _1dir. Accordingly, the total control power at the central satellite is given by Pcont=P2axis=2P1axis.P_cont=P_2axis=2P_1axis. (45) We also model the mission power PmisP_mis as the transmitter power consumption and the margin power PmarP_mar for another purpose, e.g., deployment, are defined as Pmis=Pt/ηtra,Pmar=2pcμ¯mar2π2qcoilacoil3,P_mis=P_t/ _tra, P_mar= 2p_c μ_mar^2π^2q_coila_coil^3, (46) where ηtra _tra is the transmitter efficiency and PtP_t is the per-satellite transmit power and μ¯mar μ_mar denotes the upper bound of margin magnetic moment. IV-D2 Magnetic Moment Estimation by Numerical Integration We numerically calculate the power consumption during steady-state operations for a square-grid formation. We compute the maximum control power at each time step through actual numerical simulations in order to evaluate the largest power consumption within the formation. This avoids the conservatism of the convex optimization-based power analysis in [26], which assumes perfect disturbance cancellation. Although the optimal values Jd∗J_d^* in (16) are equal to the total squared magnetic-dipole requirement over the three axes [26], the problem in (16) does not specify its axis-wise allocation. Then, we adopt the simplified coil-sizing assumption. Assumption 10. A single-axis coil is sized to meet the worst-case axis power demand, and the same coil specification is applied to all three axes. By removing the time-averaging factor and considering the peak requirement, the one-direction magnetic moment μ1d _1d is defined as μ1d2≜Jd∗(n).μ^2_1d J_d^*(n). (47) Instead of strictly stabilizing the formation to an exact target trajectory, we relax the control objective. As an example, this study considers stabilizing the satellites within a tolerable position-error bound sufficient for antenna functionality to improve performance. The drift term in (5) increases the relative distance between satellites over time and negatively affects the connectivity maintenance. Then, we apply previous controller [1, 29] to reduce both the drift and the control input. Its closed-loop system related to this drift term [1, 29]: [˙−2C1˙C4] bmatrix e_-2C_1\\ e_C_4 bmatrix =[−kA2LeOϵ22(I−γkγ2Le)−γkA2Le][−2C1C4]−k0[dydx] = bmatrix- k_A2L_e&O\\ _22 (I- γ k_γ2L_e )&- γ k_A2L_e bmatrix bmatrix e_-2C_1\\ e_C_4 bmatrix-k_0 bmatrix e_d_y\\ e_d_x bmatrix (48) where −2C1=E⊤[−2C1] e_-2C_1=E [-2C_1], C4=E⊤[C4] e_C_4=E [C_4], and dx,y=E⊤[dx,y] e_d_x,y=E [d_x,y]. We construct approximate function Jd∗(n)J_d^*(n) by exhaustively solving (16) for each n in Section V-A. We emphasize that we can replace with different controllers and control objectives. Remark 3. Our goal is to evaluate the steady states of the formation keeping for Nall≫1N_all 1 using scalable computational methods. Since the theoretical bound estimation of the steady state can yield conservative results for Nall≫1N_all 1 [2], we utilize the numerical estimation to derive its exact values. One drawback to using the numerical method is the high computational costs as the number of satellites grows. For example, we consider the linear time-invariant system x˙=Ax(t)+d(t) x=Ax(t)+d(t). The previous study [26] avoids conducting a straightforward numerical integration, such as the fourth-order Runge–Kutta method, and deriving analytical solutions for its analytical solution under x(0)=0x(0)=0: x(T)=∫0TeA(T−τ)d(τ)τ=∑k=1K[∫tktk+1eA(T−τ)τ]dk,x(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), A∈ℝNall×NallA ^N_all× N_all is a stable matrix, and d(t)∈ℝNalld(t) ^N_all is a time-varying external input. Previous study [26] shows its computational costs can be reduced to (4Nall2K)O(4N_all^2K) or (Nall3)+(Nall2K/P)O(N_all^3)+O(N_all^2K/P) with the parallelism P, which takes a high computational cost. In Section V-A, we use Krylov subspace methods [37] to reduce the total costs for Nall≫1N_all 1. This method approximately derives eA(T−τj)d(τj)e^A(T- _j)d( _j) with a cost of (NalllogNall)O(N_all N_all) and the final cost is (NalllogNallK/P)O(N_all N_allK/P), making the computation more tractable than direct numerical integration. IV-E Overall Framework and Feasibility Region Algorithm 1 Distributed space antenna design framework 1: Input: 1) satellite distance dsatd_sat, system mass m¯sys m_sys, and investigation parameters χinv _inv: \e.g., margin magnetic moment μ¯mar μ_mar or transmit power PtP_t \ 2) feasible region χfeas _feas for design parameters \coil diameter 2acoil2a_coil, satellite size 2asat2a_sat, coil parameter qcoilq_coil, satellite number n, and maximum sidelobe variable upslu_psl\ in Table I, 3) multi-start trials NGSN_GS, 4) constants in Table I, 5) user-defined evaluation function (X)J(X) 2: Output: Optimal X∗X^* and other parameters f(X∗)f(X^*): generation and consumption power, EIRP, satellite and component masses msat=m3coil+m4sap+mbat+mstr+mbusm_sat=m_3coil+m_4sap+m_bat+m_str+m_bus 3: for s=1,…,NGSs=1,…,N_GS do 4: Generate random X0(s)∈χfeasX_0^(s)∈ _feas and solve DSAP_DSA 5: end for 6: Select the best solution X∗X^* that maximizes (X)J(X) 7: 8: Distributed array design optimization DSA(dsat,msys)P_DSA(d_sat,m_sys) X∗=argmaxX∈χfeas(X)subject to: \ X^*= arg~max _X∈ _feas (X) to: (A) Satellite specification constraintsacoil≤asat−rmar,kFacoil≤dsat2asat≤dsat,|msysNall−msat|≤γmassm¯sat(asat)−m3coil−m4sap≤msat≤m¯sat(asat)|m¯sys−msys|≤γsysPcont+Pmis+Pbus+Pmar≤Psap(B) Phased-array antenna constraints|Nlsin(upsl)cos(Nlupsl)−cos(upsl)sin(Nlupsl)|≤γpslπNl<|upsl|<2πNl,Pmis=Pt/ηtraGT=D¯≈Nl2,Pt≜IRNl2(sin(upsl)sin(Nlupsl))2(C) Magnetic formation-keeping constraints4P1d−cont=42pcμ1d−cont2π2qcoilacoil3,μ1d−cont2=Jd∗(n) -14.22636pt \ aligned &(A) Satellite specification constraints\\ & a_coil≤ a_sat-r_mar, k_Fa_coil≤ d_sat\\ & 2a_sat≤ d_sat, | m_sysN_all-m_sat |≤ _mass\\ & m_sat(a_sat)-m_3coil-m_4sap≤ m_sat≤ m_sat(a_sat)\\ & | m_sys-m_sys|≤ _sys\\ & P_cont+P_mis+P_bus+P_mar≤ P_sap\\ &(B) Phased-array antenna constraints\\ & |N_l _(u_psl) _(N_lu_psl)- _(u_psl) _(N_lu_psl)|≤ _psl\\ & πN_l<|u_psl|< 2πN_l, P_mis=P_t/ _tra\\ & G_T= D≈ N_l^2, P_t I_RN_l^2 ( (u_psl) (N_lu_psl) )^2\\ &(C) Magnetic formation-keeping constraints\\ & 4P_1d-cont=4 2p_c _1d-cont^2π^2q_coila_coil^3, μ^2_1d-cont=J_d^*(n)\\ aligned . We integrate the objectives and constraints introduced in previous sections into a optimization-based design framework DSAP_DSA in (27) and the overall procedure is summarized in Algorithm 1 with user-defined evaluation function (X)J(X) and the constraints in (A)–(C) are described in Subsections IV-B, IV-C, and IV-D, respectively. To avoid an infeasible start of our nonconvex optimization problem, this subsection also denotes a warm-start strategy to specify the feasibility region χfeas _feas of design parameters X=2acoil, 2asat,qcoil,n,umslX= \2a_coil,\ 2a_sat,\ q_coil,\ n,\ u_msl \ in (26). Here, (⋅)¯ (·) and (⋅)¯ (·) denote the lower and upper bounds, subscript 0 denotes the initial value, and ξi∼(0,1) _i (0,1) are independent samples. The primary geometric variables are drawn uniformly within their bounds and then capped by simple geometric limits: acoil0 a_coil0 =min(dsatkF,a¯coil+ξ2(a¯coil−a¯coil)), = ( d_satk_F,\; a_coil+ _2 ( a_coil- a_coil ) ), (49) asat0 a_sat0 =min(dsat2,a¯sat+ξ1(a¯sat−a¯sat)), = ( d_sat2,\; a_sat+ _1 ( a_sat- a_sat ) ), msys0 m_sys0 =m¯sys+ξ3(m¯sys−m¯sys). = m_sys+ _3 ( m_sys- m_sys ). Then, we compute the remaining mass budget for the coil wiring, denoted m¯coil0 m_coil0, and translate it into q¯coil q_coil using the coil-mass model (Definition 3 and Assumption 4): qcoil q_coil ≤q¯coil0≜m¯coil0/(6π2ρcacoil0). ≤ q_coil0 m_coil0/(6π^2 _c\,a_coil0). (50) qcoil0 q_coil0 =q¯coil+ξ6(min(q¯coil0,q¯coil)−q¯coil). = q_coil+ _6 ( ( q_coil0, q_coil)- q_coil ). Note that we bound qcoilq_coil in (31) qcoil∈[N¯tr¯coil2,N¯tr¯coil2]q_coil∈ [ N_t\, r_coil^2,\; N_t r_coil^2 ] (51) where the N¯t N_t and N¯t N_t define N¯t=1,N¯t=a¯satr¯coil. \ aligned N_t&=1, N_t= a_sat r_coil. aligned . (52) Next, n is sampled using an instance-dependent upper bound obtained from the empirical mass model in Assumption 5. We enforce msys0/(2n0+1)2≥m¯satm_sys0/(2n_0+1)^2≥ m_sat and solving this for n0n_0 yields n0≤n¯≜(−1+msys0/m¯sat)/2,n_0≤ n (-1+ m_sys0/ m_sat )/2, (53) to avoid unrealistically small satellite masses, and we use n0=n¯+ξ4(n¯−n¯).n_0= n+ _4 ( n- n ). (54) The initial value upsl0u_psl0 is randomly generated within a narrowed subset of its theoretical peak-sidelobe neighborhood. By introducing a small positive tolerance γu _u to the interval boundaries, the warm-start point is determined as follows: upsl0=((1−2γu)π2n0+1)ξ5−(2−γu)π2n0+1.u_psl0= ( (1-2 _u)π2n_0+1 ) _5- (2- _u)π2n_0+1. (55) This formulation ensures that upsl0u_psl0 is constrained within the range [(−2+γu)π2n0+1,(−1−γu)π2n0+1][ (-2+ _u)π2n_0+1, (-1- _u)π2n_0+1], providing a well-conditioned starting point for the subsequent local optimization. V Array Design Examples and Discussion This section presents design examples obtained through the proposed framework and highlights the design trends of EMFF-based distributed space antennas. V-A Analysis of Representative Case Studies TABLE I: Calculation constants of design parameters for Algorithm 1: Distributed space antennas design. Satellite Design Parameters Description Constant Value Resistivity of wire pcp_c (10) 1.68e-8Ωm [38] Mass density of wire ρc _c (31) 89608960 kg/m3kg/m^3[39] Mass per unit power capacity of the battery ρbat _bat (31) 0.005 kg/Whkg/Wh Mass per unit area of the solar panel ρsap _sap (31) 0.60.6 kg/m2kg/m^2 Number of solar panels νsap _sap (31) 44 Power generation per unit area of solar panel PW/m2P_W/m^2 (29) 1367×0.3W/m21367× 0.3\ W/m^2 Effective area coefficient ksapk_sap (29) 11 Coefficient of structure mass ηstr _str (31) 0.250.25 Margin of coil size rmarr_mar (28) 0.005m0.005~m Bus power PbusP_bus (42) 0.200W0.200~W Bus mass mbusm_bus (31) 0.200kg0.200~kg Battery storage coefficient kbatk_bat (30) 0.10.1 Satellite/system-mass tolerance γmass/γsys _mass/ _sys 10−2/10−210^-2/10^-2 Peak-sidelobe/upsl0u_psl0 tolerance γpsl/γu _psl/ _u 10−5/0.110^-5/0.1 Orbital and Communication Constants Description Constant Value Orbital height h (35) 500500 kmkm Orbital inclination ioi_o 45π/180rad45π/180~rad Receiving power PRP_R (38) −87.2dBm-87.2dBm[40] Receiving antenna gain GRG_R (38) 0 dBidBi Attenuation rate ζ (38) 0.50.5 Transmitter efficiency ηtra _tra (46) 0.30.3 Direction of main beam θ0 _0 (21), (35) π/6radπ/6~rad Battery charging duration hchargeh_charge (30) 1212 Feasible region χfeas _feas of optimization variable Variable Min Max System mass msysm_sys (1−γsys)m¯sys(1- _sys) m_sys (1+γsys)m¯sys(1+ _sys) m_sys Satellite radius asata_sat 0.015 dsat/2d_sat/2 Coil radius acoila_coil 0.005 dsat/2−rmard_sat/2-r_mar Coil parameter qcoilq_coil N¯tr¯coil2 N_t r_coil^2 N¯tr¯coil2 N_t r_coil^2 Satellite number n 33 n¯ n Peak sidelobe variable upslu_psl −2π/3-2π/3 0 Note: The lower and upper bounds of the coil-wire radius used in the bounds of qcoilq_coil are r¯coil=0.03937×10−3m r_coil=0.03937× 10^-3~m and r¯coil=0.00105m r_coil=0.00105~m, respectively. To demonstrate the applicability of the proposed framework, this subsection considers three design scenarios. For Case 1, the design objective is to achieve a narrow beam by minimizing the footprint. This objective can be written as argminarcsin(2λNldsat+sinθ0)=argmaxNall. arg~min _X ( 2λN_ld_sat+ _0 )= arg~max _XN_all. (56) For Cases 2 and 3, the design objective is to maximize the communication or sensing capability. In these cases, the EIRP maximization problem is likewise reduced to the maximization of NallN_all: argmaxEIRP∝argmax(PtNl2⋅GT)=argmaxNall. arg~max _XEIRP arg~max _X (P_tN_l^2· G_T )= arg~max _XN_all. (57) The numerical settings are shared across the three case studies to ensure a consistent comparison. All design solutions are obtained using Algorithm 1 and MATLAB’s GlobalSearch solver[41]. The constants and feasible design ranges are listed in Table I, and the overall system mass is varied from 500 kg to 6000 kg. The disturbance-compensation requirement enters the EMFF power model through Jd∗(n)J_d^*(n): according to (47), Jd∗(n)J_d^*(n) determines the required single-axis magnetic moment, and this value is then used in the control-power expressions in (46). Therefore, Jd∗(n)J_d^*(n) is estimated in advance as a function of the satellite number for each inter-satellite distance. We numerically integrate (48) for a square-grid formation with fixed spacing dsatmd_satm over multiple values of kAk_A. Based on Fig. 4, we select kA=0.0560k_A=0.0560. For this kAk_A, we derive JdJ_d as a function of the number of satellites from the relationship between the position error and the control input. Figure 4 shows the fitted Jd∗(n)J_d^*(n) curves for dsat=0.15md_sat=0.15~m and dsat=0.60md_sat=0.60~m, and these fitted curves are used in the EMFF power constraint during the optimization. (a) Residual drift vector (b) Maximum control input (c) Power index Jd∗J_d^* at dsat=0.15md_sat=0.15m. (d) Power index Jd∗J_d^* at dsat=0.60md_sat=0.60m. Figure 4: Numerical estimation of the distributed-control requirement used in the EMFF power constraint. Panels (a) and (b) show the residual drift vector and maximum control input obtained from the distributed formation-control simulation for different feedback gains kAk_A. Based on these results, kA=0.0560k_A=0.0560 is selected as the nominal gain used to construct the control requirement. Panels (c) and (d) show the resulting power index Jd∗(n)J_d^*(n) for dsat=0.15md_sat=0.15~m and dsat=0.60md_sat=0.60~m, respectively. The red circles denote the numerical values obtained by solving the magnetic-dipole allocation problem in (16), and the blue curves denote the fourth-order polynomial fits used in the design optimization. V-A1 Case 1: Margin Magnetic Moment Analysis Case 1 evaluates how the margin magnetic moment affects footprint-oriented antenna sizing in a D2D-oriented 1-GHz case with λ=0.30mλ=0.30~m and dsat=0.15md_sat=0.15~m. This spacing corresponds to half a wavelength. The per-element RF transmit power PtP_t is calculated from the peak-sidelobe-envelope-limited model in (41) using the D2D received-power requirement, and μ¯mar μ_mar is treated as the investigation parameter representing the control capability reserved beyond disturbance compensation. The numerical setup, representative design values, optimized trends, and constraint margins are shown in Table 5, Table 5, Fig. 5, and Figs. 8a–8c, respectively. V-A2 Case 2: Transmit-Power Analysis Case 2 evaluates how prescribed transmit power affects EIRP and satellite sizing at λ=0.30mλ=0.30~m and dsat=0.15md_sat=0.15~m. The margin magnetic moment is fixed at μ¯mar=0.25Am2 μ_mar=0.25~Am^2, and PtP_t is treated as the investigation parameter; unlike Case 1, the peak sidelobe level is evaluated as an output rather than used to determine PtP_t inside the optimization. The numerical setup, representative design values, optimized trends, and constraint margins are shown in Table 6, Table 6, Fig. 6, and Figs. 8d–8f, respectively. V-A3 Case 3: Large-Spacing Feasibility Analysis Case 3 evaluates the feasibility limit caused by increasing the inter-satellite spacing to dsat=0.60md_sat=0.60~m with λ=1.20mλ=1.20~m. The margin magnetic moment is fixed at μ¯mar=0.25Am2 μ_mar=0.25~Am^2, and PtP_t is swept as in Case 2 to compare how the larger spacing increases the coil burden required for EMFF formation maintenance. The numerical setup, representative design values, optimized trends, and constraint margins are shown in Table 7, Table 7, Fig. 7, and Figs. 8g–8i, respectively. TABLE I: Numerical setup and input parameters for Case 1. Frequency [GHz] Wavelength λ [m] Inter-satellite distance dsatd_sat [m] Surplus magnetic moment μ¯mar μ_mar [Am2] Transmitting power from an element PtP_t [W] 1 0.30 0.15 [0, 0.25, 0.50, 0.75, 1.0] (41) (Calculated for Pr=−87.2P_r=-87.2 dBm and GR=0G_R=0 dBi.) TABLE IV: Case 1: Breakdown of size, mass, and power with overall system masses from 500 kg to 6000 kg. In terms of power, only the solar panel generates power, while all other components consume power. The bus mass and bus power are set as constants at 200 g and 200 mW, respectively. The SLL denotes the normalized peak sidelobe-envelope level relative to the main-beam peak. μmar _mar (Am2) 0 0.25 0.50 0.75 1.0 Overall system mass m¯sys m_sys (kg) 500 3000 6000 500 3000 6000 500 3000 6000 500 3000 6000 500 3000 6000 Size Satellite 2asat2a_sat (m) 62.7 59.4 61.3 85.0 85.0 85.1 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 Coil 2acoil2a_coil (m) 40.3 34.9 33.0 75.0 75.0 75.0 75.0 75.0 75.0 75.0 75.0 75.0 75.0 75.0 75.0 Coil parameter qcoilq_coil (m2) 0.218 0.397 0.356 1.47 1.47 1.47 4.15 4.15 4.15 9.35 9.33 9.33 16.7 16.6 16.6 Mass (g) Satellite 293 292 292 348 348 348 436 436 436 572 571 571 763 761 761 3-axis coil 2.33 3.68 3.12 29.3 29.3 29.3 82.7 82.6 82.6 186 186 186 332 330 330 Battery 9.67 8.69 9.24 17.8 17.8 17.8 24.6 24.6 24.6 24.6 24.6 24.6 24.6 24.6 24.6 Body 73.2 73.0 73.0 87.0 87.0 87.0 109 109 109 143 143 143 191 190 190 4 Solar panels 9.43 8.47 9.01 17.3 17.4 17.4 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 Power (mW) 4 Solar panels (×103× 10^3) 1.61 1.45 1.54 2.96 2.97 2.97 4.10 4.10 4.10 4.10 4.10 4.10 4.10 4.10 4.10 Disturbance 683 795 1130 17.8 25.3 27.3 7.43 11.0 11.8 4.00 6.14 6.71 2.74 4.38 4.87 Transmitter 3.65 0.102 0.0254 5.13 0.143 0.0358 8.03 0.224 0.0561 13.8 0.385 0.0963 24.5 0.683 0.171 Margin (×103× 10^3) 0 0 0 2.74 2.74 2.74 3.89 3.89 3.89 3.88 3.89 3.89 3.87 3.90 3.90 Performance Main-beam EIRP (dBW) 39.5 39.5 39.5 39.5 39.5 39.5 39.4 39.5 39.5 39.4 39.5 39.5 39.4 39.5 39.5 Maximum antenna gain (dBi) 32.4 40.1 43.1 31.6 39.4 42.4 30.6 38.4 41.4 29.4 37.2 40.3 28.2 36.0 39.0 Peak SLL envelope (dB) -13.2 -13.3 -13.3 -13.2 -13.3 -13.3 -13.2 -13.3 -13.3 -13.2 -13.3 -13.3 -13.2 -13.3 -13.3 Footprint diameter (km) 77.1 31.9 22.6 83.8 34.7 24.6 93.6 38.7 27.5 107 44.3 31.4 123 51.0 36.2 Satellite number Nl×NlN_l× N_l 41×4141× 41 101×101101× 101 143×143143× 143 39×3939× 39 93×9393× 93 131×131131× 131 35×3535× 35 83×8383× 83 117×117117× 117 29×2929× 29 73×7373× 73 103×103103× 103 25×2525× 25 63×6363× 63 89×8989× 89 (a) Antenna diameter (μ¯mar=0.25 μ_mar=0.25 Am2) (b) Antenna diameter (c) Footprint diameter (d) Satellite number (e) Satellite mass (f) Coil mass (g) Satellite size (h) Coil diameter (i) Coil parameter Figure 5: Representative design solutions for Case 1, where the margin magnetic moment is swept at λ=0.30mλ=0.30~m and dsat=0.15md_sat=0.15~m. Panel (a) compares the original formulation with the fixed-PtP_t formulation at μ¯mar=0.25Am2 μ_mar=0.25~Am^2, showing the scatter of local solutions caused by the peak-sidelobe-related variable. Panels (b)–(i) show the optimal design trends for μ¯mar=0 μ_mar=0–1.0Am21.0~Am^2; the zero-margin case approaches the satellite-number upper bound with near-minimum satellite mass, whereas nonzero margin magnetic moment reduces the feasible aperture and shifts the coil adjustment toward qcoilq_coil as the generated-power and satellite-size margins become restrictive. TABLE V: Numerical setup and input parameters for Case 2. Frequency [GHz] Wavelength λ [m] Inter-satellite distance dsatd_sat [m] Surplus magnetic moment μ¯mar μ_mar [Am2] Transmitting power from an element PtP_t [W] 1 0.30 0.15 0.25 [0.1, 0.2, 0.3, 0.4, 0.5] TABLE VI: Case 2: Breakdown of size, mass, and power with overall system masses from 500 kg to 6000 kg. In terms of power, only the solar panel generates power, while all other components consume power. The bus mass and bus power are set as constants at 200 g and 200 mW, respectively. The SLL denotes the normalized peak sidelobe-envelope level relative to the main-beam peak. PtP_t (W) 0.1 0.2 0.3 0.4 0.5 Overall system mass m¯sys m_sys (kg) 500 3000 6000 500 3000 6000 500 3000 6000 500 3000 6000 500 3000 6000 Size Satellite 2asat2a_sat (m) 87.6 87.6 87.6 92.1 92.1 92.2 96.4 96.5 96.5 100.0 100.0 100.0 100.0 100.0 100.0 Coil 2acoil2a_coil (m) 75.0 75.0 75.0 75.0 75.0 75.0 75.0 75.0 75.0 75.0 75.0 75.0 75.0 75.0 75.0 Coil parameter qcoilq_coil (m2) 1.56 1.56 1.56 1.56 1.56 1.56 1.56 1.56 1.56 1.58 1.59 1.59 1.82 1.82 1.83 Mass (g) Satellite 353 353 353 358 358 358 363 363 363 368 369 369 375 375 375 3-axis coil 30.9 31.0 31.0 30.9 31.0 31.0 30.9 31.0 31.0 31.5 31.6 31.6 36.2 36.3 36.3 Battery 18.9 18.9 18.9 20.9 20.9 20.9 22.9 22.9 22.9 24.6 24.6 24.6 24.6 24.6 24.6 Body 88.2 88.2 88.3 89.5 89.5 89.6 90.8 90.8 90.9 92.1 92.2 92.2 93.7 93.7 93.7 4 Solar panels 18.4 18.4 18.4 20.4 20.4 20.4 22.3 22.3 22.3 24.0 24.0 24.0 24.0 24.0 24.0 Power (mW) 4 Solar panels (×103× 10^3) 3.14 3.15 3.15 3.48 3.48 3.48 3.81 3.82 3.82 4.10 4.10 4.10 4.10 4.10 4.10 Disturbance 17.0 24.3 26.1 17.2 24.6 26.4 17.4 24.9 26.8 17.3 24.7 26.6 15.2 21.9 23.5 Transmitter 333 333 333 667 667 667 1000 1000 1000 1330 1330 1330 1670 1670 1670 Margin (×103× 10^3) 2.59 2.59 2.59 2.59 2.59 2.59 2.59 2.59 2.59 2.55 2.54 2.54 2.22 2.21 2.21 Performance Main-beam EIRP (dBW) 53.1 68.7 74.7 56.0 71.6 77.6 57.6 73.2 79.2 58.8 74.3 80.3 59.6 75.1 81.2 Maximum antenna gain (dBi) 31.6 39.3 42.3 31.5 39.3 42.3 31.4 39.2 42.2 31.4 39.1 42.2 31.3 39.1 42.1 Peak SLL envelope (dB) -13.2 -13.3 -13.3 -13.2 -13.3 -13.3 -13.2 -13.3 -13.3 -13.2 -13.3 -13.3 -13.2 -13.3 -13.3 Footprint Diameter (km) 84.4 34.9 24.8 85.0 35.2 24.9 85.6 35.4 25.1 86.2 35.7 25.3 86.9 36.0 25.5 Satellite number Nl×NlN_l× N_l 37×3737× 37 93×9393× 93 131×131131× 131 37×3737× 37 91×9191× 91 131×131131× 131 37×3737× 37 91×9191× 91 129×129129× 129 37×3737× 37 91×9191× 91 129×129129× 129 37×3737× 37 89×8989× 89 127×127127× 127 (a) Antenna diameter (Pt=0.2P_t=0.2 W) (b) Antenna diameter (c) EIRP (d) Satellite number (e) Satellite mass (f) Coil mass (g) Satellite size (h) Coil diameter (i) Coil parameter Figure 6: Representative design solutions for Case 2, where the transmit power is swept at λ=0.30mλ=0.30~m, dsat=0.15md_sat=0.15~m, and μ¯mar=0.25Am2 μ_mar=0.25~Am^2. Panel (a) shows the concentrated local-solution trend for the fixed-PtP_t formulation at Pt=0.2WP_t=0.2~W. Panels (b)–(i) show that increasing PtP_t improves EIRP but changes the optimized aperture only moderately; the main design response is an increase in satellite size for solar-power generation, while the coil mass, coil diameter, and coil parameter remain comparatively stable because the EMFF control requirement is fixed. TABLE VII: Numerical setup and input parameters for Case 3. Frequency [MHz] Wavelength λ [m] Inter-satellite distance dsatd_sat [m] Surplus magnetic moment μ¯mar μ_mar [Am2] Transmitting power from an element PtP_t [W] 250 1.2 0.60 0.25 [0.1, 0.2, 0.3, 0.4, 0.5] TABLE VIII: Case 3: Breakdown of size, mass, and power with overall system masses from 500 kg to 6000 kg. In terms of power, only the solar panel generates power, while all other components consume power. The bus mass and bus power are set as constants at 200 g and 200 mW, respectively. The SLL denotes the normalized peak sidelobe-envelope level relative to the main-beam peak. PtP_t (W) 0.1 0.2 0.3 0.4 0.5 Overall system mass m¯sys m_sys (kg) 500 3000 6000 500 3000 6000 500 3000 6000 500 3000 6000 500 3000 6000 Size Satellite 2asat2a_sat (m) 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 – 100.0 – – Coil 2acoil2a_coil (m) 90.0 90.0 90.0 90.0 90.0 90.0 90.0 90.0 90.0 90.0 90.0 – 90.0 – – Coil parameter qcoilq_coil (m2) 6.57 12.0 14.3 7.49 14.2 17.4 8.71 17.4 22.1 10.4 22.1 – 12.8 – – Mass (g) Satellite 533 702 776 562 773 874 601 872 1020 653 1020 – 728 – – 3-axis coil 157 285 342 179 339 416 208 415 527 248 528 – 305 – – Battery 24.6 24.6 24.6 24.6 24.6 24.6 24.6 24.6 24.6 24.6 24.6 – 24.6 – – Body 133 176 194 141 193 219 150 218 255 163 255 – 182 – – 4 Solar panels 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 – 24.0 – – Power (mW) 4 Solar panels (×103× 10^3) 4.10 4.10 4.10 4.10 4.10 4.10 4.10 4.10 4.10 4.10 4.10 – 4.10 – – Disturbance 3210 3370 3400 2920 3070 3100 2630 2770 2800 2340 2460 – 2050 – – Transmitter 333 333 333 667 667 667 1000 1000 1000 1330 1330 – 1670 – – Margin (×103× 10^3) 0.356 0.195 0.163 0.312 0.165 0.134 0.268 0.134 0.106 0.225 0.106 – 0.183 – – Performance Main-beam EIRP (dBW) 49.5 62.7 67.8 52.1 64.9 69.8 53.3 65.6 70.2 53.8 65.5 – 53.8 – – Maximum antenna gain (dBi) 29.8 36.3 38.9 29.5 35.9 38.4 29.2 35.4 37.7 28.9 34.7 – 28.4 – – Peak SLL envelope (dB) -13.2 -13.3 -13.3 -13.2 -13.3 -13.3 -13.2 -13.3 -13.3 -13.2 -13.3 – -13.2 – – Footprint Diameter (km) 103 49.1 36.6 106 51.4 38.8 109 54.6 41.9 114 59.0 – 120 – – Satellite number Nl×NlN_l× N_l 31×3131× 31 65×6565× 65 89×8989× 89 29×2929× 29 63×6363× 63 83×8383× 83 29×2929× 29 59×5959× 59 77×7777× 77 27×2727× 27 55×5555× 55 – 27×2727× 27 – – (a) Antenna diameter (Pt=0.2P_t=0.2 W) (b) Antenna diameter (c) EIRP (d) Satellite number (e) Satellite mass (f) Coil mass (g) Satellite size (h) Coil diameter (i) Coil parameter Figure 7: Representative design solutions for Case 3, where the transmit power is swept under large inter-satellite spacing at λ=1.20mλ=1.20~m, dsat=0.60md_sat=0.60~m, and μ¯mar=0.25Am2 μ_mar=0.25~Am^2. The larger spacing increases the geometric aperture but makes EMFF control more demanding. Because the satellite-size margin is nearly exhausted, the coil diameter cannot be enlarged further, and the optimizer increases qcoilq_coil to reduce coil-related power; this increases the coil and satellite mass, producing infeasible points when the satellite-level mass, size, and power capacities can no longer accommodate the required coil. V-B Discussion (a) Case 1: Satellite mass lower margin (b) Case 1: Satellite size upper margin. (c) Case 1: Power margin. (d) Case 2: Satellite mass upper margin (e) Case 2: Satellite size upper margin (f) Case 2: Power margin (g) Case 3: Satellite mass upper margin (h) Case 3: Satellite size margin (i) Case 3: Power margin Figure 8: Constraint margins of satellite mass, satellite size, and power in each case. For satellite mass and satellite size, the active margin with respect to either the upper or lower bound is shown. The power-constraint margin is defined as the generated power from the solar panels minus the total consumed power. The margins indicate how the active bottleneck changes from generated power and satellite size in the baseline-spacing cases to reduced power and size margins with a decreasing satellite-mass upper margin in the large-spacing case. V-B1 System-Level Trade-Off The three case studies show that aperture enlargement improves antenna performance only while the satellite-level design retains sufficient headroom for EMFF control. Figure 5 and Fig. 6 show that increasing the overall system mass generally enlarges the optimized aperture in the baseline-spacing cases, leading to footprint reduction in Case 1 and EIRP improvement in Case 2. Figure 7, however, shows that the large-spacing case reaches infeasible design points even when the overall system mass is increased. Figure 8 supports this interpretation by showing that, in Case 3, the satellite-size and power margins are already nearly exhausted while the satellite-mass margin decreases toward the upper-bound limit. Tables IV, VI, and VIII summarize the corresponding representative satellite specifications and antenna-performance values. The following subsections discuss the mechanisms that determine this design headroom: fixed-transmit-power simplification, margin magnetic moment, transmit power, and large inter-satellite spacing. V-B2 Trends in Optimized Design Solutions Case 1 shows that requiring margin magnetic moment shifts the dominant design adjustment from coil diameter to coil parameter. For dsat=0.15md_sat=0.15~m, the coil diameter is limited by the distance-dependent coil-size constraint, as shown in Fig. 5h.Because PcontP_cont and PmarP_mar scale as 1/(qcoilacoil3)1/(q_coila_coil^3), larger values of acoila_coil reduce the required power and are therefore selected when allowed by the constraints; after this limit is reached, the remaining adjustment is made through qcoilq_coil, as shown in Fig. 5i. This increases the coil mass in Fig. 5f and reduces the power and size margins in Figs. 8c and 8b. Case 2 should be interpreted as a sweep of non-coil mission power rather than as a general sensitivity analysis of transmit power. In this case, PtP_t is prescribed and increases the mission-power term, but it does not directly change the margin magnetic moment or the EMFF control requirement. Because this power increase is not large enough to substantially change the active satellite-level constraints, the optimized aperture and coil design remain nearly unchanged. Increasing PtP_t therefore mainly improves EIRP, as shown in Fig. 6c, while the antenna diameter changes only moderately in Fig. 6b. The design response appears mainly as satellite-size adjustment for additional power generation and the corresponding reduction in satellite-size margin, as shown in Figs. 6g and 8e. Case 3 shows that the feasibility limit is governed by the coil burden required for large-spacing EMFF control. Figure 4 shows that Jd∗(n)J_d^*(n) increases more severely for dsat=0.60md_sat=0.60~m than for dsat=0.15md_sat=0.15~m, while Figs. 8h and 8i show that the satellite-size and power margins are nearly exhausted. Since the coil diameter is limited by the satellite-size geometry in Fig. 7h, the optimizer increases qcoilq_coil instead, which increases the coil mass and drives the satellite mass toward its upper bound, as shown in Figs. 7i, 7f, and 8g. The comparison between Figs. 6c and 7c also shows that the coil mass fraction becomes larger in Case 3 than in Case 2, indicating that a larger portion of each satellite must be allocated to EMFF actuation under the large-spacing condition. Thus, the infeasible points arise from satellite-level mass, size, and power limits rather than from insufficient communication performance. V-B3 Convex Relaxation of the Fixed-PtP_t Formulation The fixed-PtP_t formulation explains why the local solutions become concentrated in Case 2. In the original Case 1 formulation, PtP_t is derived through upslu_psl, which introduces a strongly nonlinear peak-sidelobe constraint. When the PtP_t values obtained from Case 1 are fixed and the problem is recalculated, the solutions nearly follow the selected optimal trend of the original formulation, as shown in Fig. 5a. Case 2 has the same simplified structure because PtP_t is prescribed, and Fig. 6a shows a concentrated local-solution trend. This behavior indicates that removing upslu_psl leaves a mostly monotonic active-constraint structure, but it does not prove global convexity of the original formulation. Remark 4. The original problem is not convex because it includes upslu_psl, integer-like satellite-number sizing, empirical satellite-mass bounds, and a numerically fitted Jd∗(Nl)J_d^*(N_l) with negative polynomial coefficients. A relaxed fixed-PtP_t problem can be written as a geometric program if upslu_psl is removed, NlN_l is relaxed as a positive continuous variable, msysm_sys is fixed, msatm_sat is eliminated as a dependent posynomial, the total-mass constraint is relaxed to Nl2msat≤msysN_l^2m_sat≤ m_sys, the satellite-mass bound is replaced by a monomial envelope, and Jd∗(Nl)J_d^*(N_l) is replaced by a positive-coefficient posynomial upper envelope. Under these relaxations, the problem becomes convex after logarithmic variable transformation [30]. This convexified problem is a conservative relaxation, not a proof of convexity of the original formulation. (a) Low-gain antenna element approximating an omnidirectional element (b) Simulated array gain pattern including mutual coupling for Case 1 with μ¯mar=0.25Am2 μ_mar=0.25~Am^2 and m¯sys=3000kg m_sys=3000~kg Figure 9: Electromagnetic-field simulation used to check the effect of mutual coupling on the simplified antenna model. Panel (a) shows the low-gain antenna element used to approximate the omnidirectional element assumed in the system-level array model. Panel (b) shows the radiated gain pattern of the representative Case 1 design with μ¯mar=0.25Am2 μ_mar=0.25~Am^2 and m¯sys=3000kg m_sys=3000~kg, including mutual coupling in the electromagnetic-field simulation. The electromagnetic-field simulation provides a representative check of the no-mutual-coupling antenna model used in the system-level optimization. The proposed framework assumes a simplified array model with idealized antenna elements to keep the satellite-sizing problem tractable. To examine the influence of this assumption, Fig. 9a shows a low-gain antenna element used to approximate the omnidirectional element assumed in the array model, and Fig. 9b shows the simulated gain pattern for the representative Case 1 design with μ¯mar=0.25Am2 μ_mar=0.25~Am^2 and m¯sys=3000kg m_sys=3000~kg. The electromagnetic-field simulation indicates an approximately 2 dB gain reduction relative to the simplified gain model. The antenna gain computed from the no-mutual-coupling approximation in (21) is Gmodel=39.4dBiG_model=39.4~dBi, whereas the electromagnetic-field simulation including the element model and mutual coupling gives GEM=37.4dBiG_EM=37.4~dBi. The difference is therefore ΔG=2.00dB G=2.00~dB. This result indicates that the practical element model and mutual coupling reduce the gain, while the simulated value remains close to the value predicted by the simplified model for this representative design. A mutual-coupling-aware radiation model is still required for final antenna design and for evaluating non-uniform or more compact formations. V-B4 Design Implications and Future Work The proposed framework should be regarded as a design-space analysis framework based on a static grid reference, rather than as a final antenna shape-optimization method. The sizing results in Figs. 5–8 show how the feasible antenna aperture is governed by coupled constraints on satellite mass, size, power, and coil design. Under the assumptions of a uniform square grid, simplified radiation modeling, fixed beam direction, and no mutual-coupling-aware optimization, the framework clarifies how antenna requirements are translated into satellite designs. These results indicate that the static grid-based EMFF antenna is a useful reference configuration for identifying the dominant bottlenecks in distributed space antenna design under a fixed launch mass. Future work should extend this reference configuration to more realistic antenna and formation models. Important directions include non-uniform and time-varying formations, relaxed beam-direction constraints, and radiation models that explicitly account for mutual coupling. Dynamic reconfiguration is also important because it requires trajectory optimization that jointly considers orbital motion, design headroom, antenna performance, and control time. VI Conclusion This paper proposed a system-level design-space analysis framework for distributed space antennas using electromagnetic formation flight. By linking antenna requirements with satellite-level mass, power, and coil-design constraints, the framework provides a static grid-based reference for designing feasible distributed apertures under a fixed system mass. Unlike our previous bucket-brigade model [28], the formation-maintenance requirement was incorporated through a control index derived from distributed-control simulations. The case studies showed that antenna performance improves with system mass only while satellite-level design headroom remains. In the direct-to-device-oriented case with dsat=0.15md_sat=0.15~m, generated-power and coil-geometry constraints mainly govern the feasible aperture, whereas the dsat=0.60md_sat=0.60~m large-spacing case can become infeasible because the required coil burden exceeds satellite-level mass, size, and power capacities. 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