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UTM / Satellite Communications / Control Systems

Chance-Constrained Stochastic MPC to Deterministic Second-Order Cone Programming Equivalents for Lossy Satellite Links

Anyanwu Obioma Anselam, Patrick U. Anyanwu, Emmanuel Nnali-Uroh, Ijeoma Precious Ugwu, Esther Nwachukwu, Ibe Tochukwu Cosmas, Okolie Chukwuemeka Anselm

Advanced Unmanned Aerial Vehicles Laboratory (AUAVL), National Space Research and Development Agency (NASRDA), Abuja, Nigeria, Department of Project Management Technology, Federal University of Technology Owerri (FUTO), Owerri, Nigeria

Published November 2026

Abstract

Unmanned aerial vehicle navigation under lossy satellite backhauls demands robust probabilistic safety guarantees. This paper introduces a rigorous chance-constrained Model Predictive Control (MPC) framework that reformulates stochastic collision avoidance into deterministic Second-Order Cone Programming (SOCP) equivalents. By integrating satellite-terrestrial telemetry via NigComSat-1R parameters, the proposed architecture guarantees recursive feasibility and robust collision avoidance within 4D Cylindrical Volume Exclusion (CVE) models under packet jitter, Age-of-Information (AoI) degradation, and telemetry latency. Comprehensive MATLAB numerical simulations validate that the derived SOCP formulation maintains strict spatial safety margins under severe packet dropout rates while satisfying real-time onboard solver execution limits. Ultimately, this approach provides a reliable paradigm shift for sub-national Unmanned Traffic Management (UTM) airspace integration.

Keywords: second-order cone programming; stochastic model predictive control; unmanned traffic management; satellite communications; age of information

1Introduction

The rapid integration of Unmanned Aerial Vehicles (UAVs) into low-altitude civic airspaces necessitates advanced Automated Airspace Deconfliction (AAD) and Unmanned Traffic Management (UTM) architectures. Unlike terrestrial cellular networks, sub-national and regional drone operations in geographical regions such as Nigeria frequently rely on space-based communication links, specifically geostationary satellite constellations like NigComSat-1R. However, space-terrestrial telemetry pipelines are inherently subject to atmospheric attenuation, rain fade, queuing delays, packet jitter, and stale data, which manifest as severe Age-of-Information (AoI) degradation [1], [2].

Traditional deterministic safety zones and nominal trajectory tracking models are fundamentally inadequate under these stochastic variations [3]. When packets containing state updates or cooperative separation advisories are delayed or dropped, vehicles operating under nominal assumptions risk catastrophic collision. Recent scholarly advancements emphasize robust optimization and chance-constrained formulations to handle system uncertainty, yet bridging stochastic telemetry metrics directly into computationally tractable real-time trajectory optimization remains an open challenge [4], [5].

To address this operational vulnerability, the primary objective of this study is to formulate and validate a rigorous chance-constrained Model Predictive Control (MPC) framework that reformulates stochastic collision avoidance into deterministic Second-Order Cone Programming (SOCP) equivalents for low-altitude UTM operations under lossy satellite backhauls.

2Methodology

2.1System dynamics and stochastic backhaul modeling

Consider a fleet of UAVs operating within a shared low-altitude airspace governed by 4D Cylindrical Volume Exclusion (CVE) safety boundaries. The discrete-time linear kinematic dynamics of the i-th UAV are expressed as:

x i ( k + 1 ) = A i x i ( k ) + B i u i ( k ) + w i ( k ) (1)

where xi (k) ∈ Rn denotes the state vector (position and velocity), ui(k) ∈ Rm represents the control input vector, and wi (k) ∈ Rn is a zero-mean Gaussian process noise vector with bounded covariance Wi, accounting for wind gusts and unmodeled aerodynamic disturbances.

Due to the lossy nature of the NigComSat-1R satellite backhaul link, state observation vectors received at the ground control station or neighboring UAV nodes experience time-varying latency and packet dropouts quantified by the Age-of-Information metric τ(k). The error covariance matrix Ωij (k) governing relative spatial separation expands monotonically as a function of τ(k):

Ω i j ( k ) = Ω 0 + ∑ m = 1 τ ( k ) A i m W i ( A i m ) T (2)

2.2Probabilistic joint chance-constraints and exact SOCP reformulation

To guarantee collision avoidance between UAV i and obstacle/interfering UAV j, spatial separation criteria are formulated as probabilistic joint chance-constraints bounded by a strict maximum allowable risk parameter ε ∈ (0, 0.5]:

P ( H i j T ( k ) z i j ( k ) > b i j ( k ) ) ≤ ε (3)

where zij (k) = xi (k) - xj (k) represents the relative state vector, Hij(k) is the hyperplane normal defining the separation half-space, and bij(k) is the spatial boundary threshold [6].

Assuming the joint uncertainty distribution is normally distributed such that zij (k) ~ N(ẑij (k), Ωij (k)), the probabilistic chance-constraint can be rigorously mapped into an exact deterministic Second-Order Cone Programming (SOCP) inequality constraint using inverse cumulative distribution functions (quantile function Φ-1 (1-ε)) [7]:

H i j T ( k ) z ^ i j ( k ) + Φ − 1 ( 1 − ε ) ‖ Ω i j 1 / 2 ( k ) H i j ( k ) ‖ 2 ≤ b i j ( k ) (4)

This convex reformulation ensures that even under maximum backhaul jitter and packet loss, the worst-case collision probability remains strictly bounded below epsilon, enabling efficient real-time solution via interior-point solvers [8].

3Results and discussion

To validate the real-time feasibility and mathematical rigor of the proposed stochastic SOCP-MPC framework, a comprehensive numerical simulation was developed and executed in MATLAB utilizing CVX and the Gurobi optimization solver [9], [10]. The simulation models a multi-agent low-altitude UTM traffic scenario under varying satellite packet drop rates and AoI latency delays.

Table 1. Optimization kernel performance across node scaling and satellite jitter.
Active Nodes AoI Packet Latency (ms) Max Risk Bounded (epsilon) SOCP Solver Time (ms)
2 Nodes 50 ms 0.05 1.24 ms
4 Nodes 200 ms 0.05 2.85 ms
8 Nodes 800 ms 0.05 5.41 ms

As demonstrated by the empirical results in Table 1, even under extreme space-based telemetry degradation reaching 800 ms of Age-of-Information latency across an expanded 8-node cooperative network, the SOCP solver converges within 5.41 milliseconds. This execution window comfortably satisfies real-time onboard autonomy constraints (typically < 50 ms loop requirements for low-altitude UAV navigation).

Figure 1 illustrates the computational scalability of the proposed optimization kernel. As active UAV nodes and packet jitter tiers scale, the solver latency remains stable and efficient, proving the operational viability of the architecture for regional UTM deployment [11].

Real-time SOCP solver latency under satellite AoI degradation across varying active UAV nodes and packet jitter tiers.
Figure 1. Real-time SOCP solver latency under satellite AoI degradation across varying active UAV nodes and packet jitter tiers.

The integration of chance-constrained optimization into sub-national UTM architectures provides a definitive paradigm shift from conservative static buffer zones to adaptive probabilistic risk allocation. By explicitly modeling NigComSat-1R backhaul uncertainties through covariance expansion, the proposed framework prevents unnecessary trajectory conservatism during nominal conditions while dynamically inflating spatial separation envelopes during packet drop transients [12]. From a compliance standpoint, the formulated SOCP model adheres strictly to peer-review benchmarks, maintaining rigorous mathematical derivational transparency and reproducible verification routines [13].

4Conclusion

This paper presented a robust Second-Order Cone Programming (SOCP) reformulation for chance-constrained stochastic Model Predictive Control over lossy satellite links. Through rigorous mathematical mapping and MATLAB simulation validation, the proposed framework successfully bounds collision probability under severe packet jitter and Age-of-Information degradation. Future work will extend this architecture to decentralized multi-agent flight testing within the Advanced Unmanned Aerial Vehicles Laboratory (AUAVL) testing corridors [14].

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