Multi-Layer Walker Delta Constellations

• Multi-layer Walker Delta constellations are parametric orbital configurations that stack LEO, MEO, and GEO layers to ensure robust, scalable NTN coverage.
• They leverage classic Walker parameters (P, S, F, i, h) to design satellite planes and enable efficient inter-layer handovers for improved connectivity.
• The FTA-NTN framework employs Bayesian optimization and adaptive clustering to jointly maximize system throughput and user fairness under realistic simulation conditions.

A multi-layer Walker Delta constellation is a parametric configuration of orbital satellite planes and phasing, extended across multiple altitude regimes to establish robust, scalable non-terrestrial network (NTN) coverage. In such designs, each layer corresponds to a distinct orbital shell—typically Low Earth Orbit (LEO), Medium Earth Orbit (MEO), and Geostationary Earth Orbit (GEO)—each parameterized as a classic Walker Delta constellation distinguished by the tuple (P, S, F, i, h): number of planes (P), satellites per plane (S), “star” phasing parameter (F), inclination (i), and altitude (h). The integration of multiple layers enables synergistic coverage, enhances user connectivity through inter-layer coordination, and supports dynamic handover mechanisms. The FTA-NTN (Fairness and Throughput Assurance in Non-Terrestrial Networks) framework employs these multi-layered Walker Delta constellations in a multi-objective setting, optimizing both aggregate throughput and user-level fairness via Bayesian optimization over the integer space of plane and satellite counts in each layer. The framework demonstrates the joint coverage, user association, and handover features of these constellations and validates their performance constraints using a simulation with realistic user mobility, resource allocation, and radio access models (Trankatwar et al., 27 Jan 2026).

1. Formal Fundamentals of Walker Delta Constellations

A single-layer Walker Delta constellation is defined by the tuple (P, S, F, i, h), classically written as Walker Δ:P/S/F. The key geometric properties are:

• Planes: Evenly spaced in right ascension of the ascending node (RAAN), with RAAN for plane $p$ given by $\Omega_p = 2\pi p / P$, for $p = 0, \ldots, P-1$.
• Satellite Phasing: Satellites in each plane are evenly distributed in mean anomaly with an inter-plane phase offset governed by $F$; for satellite $s$ in plane $p$, the mean anomaly is

$$M_{p,s} = \frac{2\pi s}{S} + \frac{2\pi p F}{P} \quad (\mathrm{mod}\ 2\pi),\quad s = 0, \ldots, S-1.$$

• Orbit and Coverage Geometry: The semi-major axis is $a = R_\oplus + h$, with $R_\oplus$ as mean Earth radius, $h$ orbital altitude, and orbital period

$$T_{orb} = 2\pi \sqrt{\frac{a^3}{\mu}},\quad \mu = 398,600\, \mathrm{km}^3/\mathrm{s}^2.$$

Derived quantities include inter-plane RAAN spacing $\Delta\Omega = 2\pi/P$, intra-plane satellite spacing $\Delta\theta = 2\pi/S$, and chord distances for in-plane and inter-plane satellites: $d_{intra} \approx 2a\sin(\pi/S)$, $d_{inter-plane} \approx 2a\sin(\Delta\Omega/2)$. Ground-track nodal periods account for Earth's rotation:

$$T_{nodal} = \frac{2\pi}{n - \Omega_E},\quad \Omega_E = 7.2921159\times10^{-5}\ \mathrm{rad/s}$$

and longitudinal shift per orbit:

$$\Delta\lambda_{GT} = 360^\circ\left[1 - \frac{\Omega_E T_{orb}}{2\pi}\right].$$

Ground track repeat intervals (revisit) require integer $m, n$ such that $m T_{orb} \approx n T_{Earth-day}$, with $F$ chosen to facilitate desired revisit cycles.

2. Stacking and Coordination in Multi-Layer Architectures

The multi-layer extension operates with $k \in \{L, M, G\}$ for LEO, MEO, and GEO. Each layer $k$ is a Walker Delta at its own $(P_k, S_k, F_k, i_k, h_k)$, resulting in the union of all satellites across layers as

$$S_{total}(t) = \bigcup_{k, p, s} \text{satellite}(k,p,s)@( \varphi_{k,p,s}(t), \lambda_{k,p,s}(t) )$$

Coverage at layer $k$ at time $t$ corresponds to all ground points within elevation threshold,

$$C_k(t) = \{\text{points within }\theta_{elev-min}\text{ central angle to any satellite at } h_k\}$$

and global coverage is $C_{total}(t) = \bigcup_k C_k(t)$.

Inter-layer handovers arise for users in $$H_{k\to\ell}(t) = C_k(t) \cap C_\ell(t) \cap \{ \text{points where } \text{SINR}_\ell > \text{SINR}_k\}$$. The footprint half-angle per layer is

$$\theta_k = \arccos \left( \frac{R_\oplus}{R_\oplus+h_k}\cos\epsilon_{min} \right),$$

with instantaneous area fraction

$$A_k = \frac{2\pi R_\oplus^2 (1-\cos \theta_k)}{4\pi R_\oplus^2}.$$

3. Joint Throughput-Fairness Optimization in FTA-NTN

FTA-NTN frames the multi-layer parameter selection as a multi-objective problem:

• The aggregate system throughput is

$$R_{total} = \sum_{\ell \in \{L, M, G\}} \sum_{i \in U_\ell} B_{i,\ell} \log_2(1 + \text{SINR}_{i,\ell}),$$

where $U_\ell$ are users served at layer $\ell$, $B_{i,\ell}$ denotes per-user bandwidth.

• Jain’s fairness index is

$$\text{JFI} = \frac{ \left( \sum_{\ell,i}R_{i,\ell} \right)^2 }{ n \sum_{\ell,i} R_{i,\ell}^2 },$$

with $n$ the total user count.

The joint multi-objective, subject to resource and association constraints,

$$\text{max} \ \{ R_{total},\ \text{JFI} \} \quad \text{subject to:} \ \# \text{active beams}/\text{satellite} \leq X, \ \# \text{users}/\text{beam} \leq Z, \ A_{ij} \in \{0,1\}.$$

is reduced to weighted-sum scalarization,

$$f(P_k, S_k) = \omega R_{total} + (1 - \omega)\text{JFI},$$

with $\omega \in [0,1]$.

4. Bayesian Optimization of Constellation Parameters

The optimization utilizes a Gaussian Process (GP) surrogate over $(P_L, S_L, P_M, S_M)$ in integer domains $P_k\in[2, 10]$, $S_k\in[2,15]$. The performance objective $f(L)$ is a black box, evaluated per parameterization; the algorithm iteratively selects candidate $L$ vectors via acquisition functions such as Expected Improvement (EI),

$$\text{EI}(L) = \mathbb{E}[ \max(0, f(L) - f_{best} - \xi) ],$$

using a zero-mean GP prior and Matérn kernel. Typically, 50–100 simulation iterations suffice for convergence to optimal parameters.

5. Simulation, Mobility, and User Association Models

The simulation scenario samples 500 users spatially over Canadian land areas, employing the STEPS mobility process:

$$v_i(t+1) = v_i(t) + a_i + \eta,\quad x_i(t+1) = x_i(t) + v_i(t+1)$$

for user velocity $v_i$, stochastic acceleration $a_i$, and random process $\eta$. Simulated over 24 hours in 1-hour epochs and 50 random seeds, the radio link assumes 2.2 GHz S-band, 20 MHz bandwidth, $P_t = 40$ dBm, $G_t = 30$ dBi, $G_r = 0$ dBi, NF = 2 dB.

Beamforming and user association employ adaptive K-Means clustering per layer and epoch:

• At each $t$ and $k$, set $$K^{(k)} = \min( X|S_k|, \lceil|U_{rem}|/Z\rceil, |U_{rem}| )$$ clusters.
• Apply K-Means to user positions, allocate clusters $C_b$ to satellites with available beam slots, and seat top $Z$ users per beam according to computed $\text{SINR}_i$.

6. Optimal Multi-Layer Configuration and Network Performance

FTA-NTN’s Bayesian search converges to:

• LEO: $P_L^* = 9$ planes, $S_L^* = 15$ satellites per plane at $h_{LEO} = 600$ km, $i_{LEO} = 53^\circ$
• MEO: $P_M^* = 7$, $S_M^* = 3$ at $h_{MEO} = 20,200$ km, $i_{MEO} = 56^\circ$
• GEO: 1 plane, 3 satellites at $h_{GEO} = 35,786$ km, $i_{GEO} = 0^\circ$

At this configuration, system throughput is $R_{total} \approx 9.88$ Gbps, average fairness $\text{JFI} \approx 0.42$ for $\omega = 0.5$. In LEO, excessive $S > 15$ or $P > 9$ produces diminishing returns due to beam resources ($X = 15$) saturating fairness. In MEO, $S_M = 3$ (wide footprints) with $P_M = 7$ (distributed coverage) prevents holes and supports fairness. User service statistics per epoch (averaged):

• LEO: 280–400 users covered, 2–5 active satellites, 9–11 beams/satellite
• MEO: 100–210 users, $\approx$2 satellites, 9–12 beams/satellite
• GEO: $<30$ users, 0–1 satellite, 2–3 beams/satellite (never exceeding $X=15$)
• System-wide sum-rate 8.5–11 Gbps, $\text{JFI} \approx 0.40–0.46$

Layer	Planes ($P_k$)	Sats/Plane ($S_k$)	Altitude ($h_k$)
LEO	9	15	600 km
MEO	7	3	20,200 km
GEO	1	3	35,786 km

7. Significance for NTN Design and Future Networks

The FTA-NTN framework demonstrates that multi-layer Walker Delta constellations, optimized via GP-guided search and integrated with adaptive clustering for user association, achieve practical trade-offs between capacity and fairness under realistic service constraints. The resulting constellations reflect both the operating demands (user handover management, regional mobility, coverage continuity) and physical system limits (beams per satellite, per-beam user seats). These principles align with 3GPP NTN evaluation scenarios, corroborating their relevance for prospective satellite network deployments (Trankatwar et al., 27 Jan 2026). A plausible implication is that further scalability and customization—potentially by increasing altitude-layer granularity or adjusting beam/user constraints—could adapt this methodology to broader coverage regions and evolving spectrum allocations.