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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Jul 31, 2026J/K move · Enter open · S save

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

Sidhdharth D. Sikka, Suyi Gao, Zehui Lu, Rongjie Lai+1

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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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.

Abstract

Trajectory Optimization

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