Decentralized Bi-Level Optimization for Cooperative Orbital Transportation of a Rigid Payload

Sidhdharth D. Sikka; Revathy Ravuri; Hei Yin Jowett Chan; Tim Echtermeyer; Shaoshuai Mou

eess.SYOct 22, 2026

Decentralized Bi-Level Optimization for Cooperative Orbital Transportation of a Rigid Payload

Authors: Sidhdharth D. Sikka, Revathy Ravuri, Hei Yin Jowett Chan, Tim Echtermeyer, Shaoshuai Mou

Organizations: Purdue University, West Lafayette, IN 47907 USA · Manifold Research Group, San Francisco, CA 94110 USA · University of Manchester, Manchester M13 9PL, United Kingdom

Abstract

The rapid growth of the space economy, alongside increasing orbital congestion and debris, is driving demand for reliable on-orbit transportation of large and non-cooperative objects. Performing such transportation with a swarm of small spacecraft distributes risk and removes the single point of failure inherent to monolithic architectures, but requires solving a non-convex trajectory optimization coupling every agent's cone-constrained thrust to the payload's rotational dynamics. Existing approaches span decentralized control laws with asymptotic guarantees but limited constraint enforcement, and centralized and partially centralized planners that reintroduce a single point of failure. In this work, we present a decentralized bi-level framework driven by exact gradients: the problem is split into an inner convex thrust allocation and an outer search over payload torque trajectories. This decomposition yields fuel-efficient solutions quickly, without requiring full NLP convergence, and at short timescales its performance scales with the number of agents. Theoretical convergence guarantees are established, and the method is validated through extensive Monte Carlo simulation, including closed-loop recovery from in-flight faults. These properties make the framework well suited to spacecraft swarm applications.

Figures & tables

Fig. 1: Agents Attached to Rigid Payload. Agents are depicted as blue squares. On the left of the figure, the center of the payload has initial position in the relative orbital frame denoted by $\boldsymbol{r}_{a}$ and terminal position in the relative orbital frame denoted by $\boldsymbol{r}_{b}$ . The attachment vector $\boldsymbol{q}_{i}$ represents the vector pointing from the center of the payload to the agent. The Maximum Thrust Angle Constraint is depicted on the right of the figure, the control thrust applied by each agent must be aligned with the inward attachment direction $-\boldsymbol{q}_{i}$ within an acceptable angle $\nu$ , so the agent pushes against the payload and its exhaust plume points away from it.

Fig. 2: Methodology overview. From a shared initial trajectory guess, each agent runs an independent bi-level gradient-descent island (Sections III-A – III-D ); a max-consensus round selects the lowest-cost trajectory (Section III-F ), which the swarm executes. Detected deviations trigger the recovery loop of Section III-G : detumble, isolate faulty agents, and replan with the survivors from the current state.

Fig. 3: Success rate, where a run succeeds when the independent verification rollout of the returned controls satisfies

\mathbf{V}<10^{-3}

. Top: versus per-agent compute budget, pooled over all cone half-angles (

2832

scenarios per point). Bottom: versus cone half-angle at the

600\,\mathrm{s}

budget (

472

scenarios per point).

Fig. 4: Mean fuel relative to the NLP on jointly solved scenarios; each per-scenario ratio is paired before averaging, and error bars are bootstrap

95\%

confidence intervals of the mean paired ratio. Top: versus per-agent compute budget, pooled over cone half-angles. Bottom: versus cone half-angle at the

600\,\mathrm{s}

budget.

$\nu$	$10$ s	$30$ s	$60$ s	$300$ s	$600$ s
$1^{\circ}$	10.0 / 66.3	22.9 / 68.0	32.0 / 68.0	43.4 / 69.5	46.0 / 69.1
$2^{\circ}$	12.1 / 67.8	22.7 / 69.7	29.4 / 70.1	40.7 / 71.4	43.6 / 71.8
$2.5^{\circ}$	13.1 / 72.5	22.7 / 73.1	27.8 / 73.1	40.5 / 73.9	41.1 / 73.9
$5^{\circ}$	12.5 / 73.9	23.7 / 75.0	28.8 / 75.0	39.4 / 76.1	41.9 / 77.3
$15^{\circ}$	19.5 / 85.4	25.8 / 85.8	33.7 / 85.6	44.1 / 86.4	46.0 / 86.7
$60^{\circ}$	51.7 / 100.0	75.8 / 100.0	78.4 / 100.0	80.9 / 100.0	82.0 / 100.0

TABLE I: Success Rates of NLP and Proposed Method [NLP % / D-GD %].

Fig. 5: Mean paired fuel ratio of D-GD relative to the NLP over the full angle–budget factorial, on jointly solved scenarios. Cells near or below one at $10\,\mathrm{s}$ occur where the NLP is truncated at the CPU cap, so its returned iterate is feasible but far from fuel-optimal.

Fig. 6: Closed-loop fault recovery success versus fleet size under injected actuation faults (

700

–

1800

episodes per fleet size; error bars are Wilson

95\%

confidence intervals). For

N\geq 11

recovery is insensitive to whether one, two, or three agents fail simultaneously.

Trajectory Optimization

Trajectory optimization computes motion or control sequences that minimize costs while satisfying system dynamics, safety limits, and task constraints.

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Neural Operator Learning for Collision-Aware Trajectory Planning of Spacecraft Swarms

Satellite constellations require orbital transfers that are both fuel efficient and collision avoidant. Yet, the computational cost of optimization methods traditionally used to plan their trajectories scales poorly with both the number of satellites as well as the number of obstacles to avoid, due to the pairwise safety constraints. In this work, we introduce a permutation-equivariant neural operator for trajectory planning of spacecraft swarms. This neural operator maps distributions of spacecraft initial states, target states, and obstacle initial states to trajectories which avoid collision and conserve fuel. This neural operator output is then paired with a batched Gauss-Newton finish to enforce exact orbital dynamics, and further reduce fuel use. The operator is self-supervised, trained without optimal trajectory labels. When trained on ten spacecraft, the proposed method generalized zero-shot to swarms of 1,000 spacecraft and 11,000 obstacles. The generated trajectories matched a per-agent optimal control solver's accuracy while retaining collision avoidance. Operator learning grounded in physics may offer a fast, scalable alternative to trajectory optimization in the increasingly crowded orbits of the future.

Sidhdharth D. Sikka, Suyi Gao, Zehui Lu +2

Manifold Research Group and the School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907 USA · Department of Mathematics, Purdue University, West Lafayette, IN 47907 USA · Independent Researcher, Fremont, CA USA

Collision Avoidance Trajectory Optimization Neural Operators

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Oct 1, 2026

Decentralized Power-Optimal Coordination for Spacecraft Swarms Using Time-Varying Magnetorquer Actuation

This paper presents a decentralized power-optimal coordination framework for magnetically actuated spacecraft swarms. Swarms that form large space structures overcome the aperture limit set by the launch vehicle and hold their shape on solar-generated power alone. Magnetic actuation is propellant-free and generated by a magnetorquer, which is commonly used for attitude control. However, every spacecraft interacts with every other within range, and its effect depends on the actuation power and a carrier frequency. We therefore design a decentralized power-optimal framework to jointly derive the interaction graph, frequency grouping, and controller gains. Our decentralized controller preserves angular momentum, which is a nonholonomic constraint. Then, this framework for connected groups whose memberships overlap across carriers guarantees that the relative position errors, the absolute attitude errors, and the imbalance of the reaction-wheel momenta converge to the desired states under the decentralized power-optimal allocation. A closed-loop simulation of a thousand spacecraft with the complete alternating-current interaction confirms the framework. A fast approximate integration with a proven error bound extends the framework to a long-horizon orbital reconfiguration held with high precision.

Yuta Takahashi, Shin-ichiro Sakai

Department of Mechanical Engineering, Institute of Science Tokyo, Tokyo, Japan · Department of Spacecraft Engineering, Japan Aerospace Exploration Agency, Kanagawa, Japan

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Jun 19, 2026

DUET: Decentralized Bilevel Optimization without Lower-Level Strong Convexity

Decentralized bilevel optimization (DBO) provides a powerful framework for multi-agent systems to solve local bilevel tasks in a decentralized fashion without the need for a central server. However, most existing DBO methods rely on lower-level strong convexity (LLSC) to guarantee unique solutions and a well-defined hypergradient for stationarity measure, hindering their applicability in many practical scenarios not satisfying LLSC. To overcome this limitation, we introduce a new single-loop DBO algorithm called diminishing quadratically-regularized bilevel decentralized optimization (DUET), which eliminates the need for LLSC by introducing a diminishing quadratic regularization to the lower-level (LL) objective. We show that DUET achieves an iteration complexity of

O(1/T^{1-5p-\frac{11}{4}τ})

for approximate KKT-stationary point convergence under relaxed assumptions, where

p

and

τ

are control parameters for LL learning rate and averaging, respectively. In addition, our DUET algorithm incorporates gradient tracking to address data heterogeneity, a key challenge in DBO settings. To the best of our knowledge, this is the first work to tackle DBO without LLSC under decentralized settings with data heterogeneity. Numerical experiments validate the theoretical findings and demonstrate the practical effectiveness of our proposed algorithms.

Zhen Qin, Zhuqing Liu, Songtao Lu +2

Department of Electrical and Computer Engineering, The Ohio State University · Department of Computer Science and Engineering, University of North Texas · Department of Computer Science and Engineering, The Chinese University of Hong Kong

Bilevel Optimization Decentralized Optimization

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Abstract

Trajectory Optimization

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