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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

arXiv:2610.31562v1eess.SYcs.MAcs.ROcs.SY

TL;DR

Cooperative orbital transport requires planning with coupled payload rotation and cone-constrained thrust, while existing approaches have limitations in constraint enforcement or decentralization. The paper presents a decentralized bi-level planner using an exact gradient and independent agent restarts. In Monte Carlo tests, it matched or exceeded baseline reliability across cone angles and raised closed-loop fault survivability to 93–96% for fleets of eleven or more agents.

  • Problem

    Decentralized trajectory optimization with hard constraint satisfaction remains open for orbital transportation of a shared payload.

    Source: B. Optimal Control Problem · p. 3

  • Method

    The framework searches over torque trajectories using the exact envelope-theorem gradient of an inner convex thrust-allocation problem, with independent agent restarts and max-consensus.

    Source: §III · p. 4

  • Results

    The method matched or exceeded every baseline’s reliability at every cone angle, succeeding on 78–80% of rollout-verified scenarios against 20–50% for the direct NLP.

    Source: §VI · p. 9

  • Takeaways & Limitations

    Closed-loop fault survivability rose from 54% at four agents to 93–96% at eleven or more agents.

    Source: §VI · p. 9

    The implemented search explores only the principal homotopy class of rotations, so optimality claims apply to its reachable subset of torque trajectories.

    Source: D. Equivalence of the Bi-Level… · p. 4

Abstract

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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.

II. PROBLEM FORMULATION

The formulation represents the rigid payload and mounted agents in a Hill-frame state, with thrust vectors as controls and attitude propagated on SO(3). Its discrete-time dynamics combine relative orbital motion, rigid-body rotation, and the effects of individual thrusts.

A. State, Input, and System Dynamics

  • The payload state contains relative position and velocity, a rotation-vector attitude parameterization, and angular velocity.The rotation matrix is updated on SO(3) using the exponential map to eliminate norm drift.Source: A. State, Input, and System Dynamics · pp. 2–3
  • Each agent is rigidly mounted at a fixed body-frame attachment point on the payload.The full state stacks the payload state with the agents’ attachment-point positions.Source: A. State, Input, and System Dynamics · p. 3
  • The stacked control input is the collection of all agents’ three-dimensional thrust vectors.The input has dimension 3N.Source: A. State, Input, and System Dynamics · p. 3
  • The continuous-time model combines linearized relative orbital dynamics with Euler rotational equations.The orbital dynamics use the time-varying Tschauner–Hempel equations, valid for an eccentric reference orbit.Source: A. State, Input, and System Dynamics · p. 3
  • The input map sends individual thrusts to translational accelerations and body torques.The torque mapping uses the skew-symmetric operator [·]×.Source: A. State, Input, and System Dynamics · p. 3
  • The discrete-time map propagates attitude directly on SO(3), recovering the rotation vector from the rotation matrix logarithm.The inertia tensor is treated as constant because payload mass dominates agent mass.Source: A. State, Input, and System Dynamics · p. 3

B. Optimal Control Problem

The optimal-control problem minimizes total fuel subject to orbital and attitude dynamics, endpoint conditions, and thrust-cone constraints. The bi-level reformulation fixes a torque trajectory for a convex inner thrust-allocation problem, then searches over torque trajectories to minimize that inner cost.

B. Optimal Control Problem

  • The original control problem is non-convex because of nonlinear attitude dynamics and the thrust-cone constraint.Each agent’s thrust lies within a cone centered on its inward attachment direction, so thrust presses against the payload while exhaust points away.Source: B. Optimal Control Problem · p. 3

C. Bi-Level Reformulation

  • Separating net force from net torque makes the inner optimization over thrusts and translational states convex when the torque trajectory is fixed.The fixed torque determines rotational quantities independently of the translational states.Source: C. Bi-Level Reformulation · p. 3
  • The inner problem returns the minimum-fuel thrust sequence consistent with the specified torque trajectory.It enforces translational dynamics and position and velocity boundary conditions.Source: C. Bi-Level Reformulation · p. 3
  • The outer problem searches for the torque trajectory that minimizes the inner problem’s cost.Source: C. Bi-Level Reformulation · p. 3

D. Equivalence of the Bi-Level Reformulation

The bi-level formulation has the same optimal value as the original thrust-sequence problem: feasible solutions map between the formulations while preserving the constraints and cost. The implemented search, however, explores only a reachable subset of admissible torque trajectories.

D. Equivalence of the Bi-Level Reformulation

  • A feasible original thrust sequence induces an admissible torque trajectory whose inner feasible set contains that sequence.This gives the bi-level optimum no greater than the original optimum.Source: D. Equivalence of the Bi-Level… · p. 4
  • Conversely, an admissible torque trajectory paired with an inner-feasible thrust sequence satisfies the original problem’s constraints.The construction preserves rotational and translational boundary conditions, thrust cones, and torque consistency.Source: D. Equivalence of the Bi-Level… · p. 4
  • The bi-level optimal value equals the original problem’s optimal value, and their optimizers correspond through the induced torque trajectory and inner minimizer.Source: D. Equivalence of the Bi-Level… · p. 4
  • The implemented search explores the principal homotopy class of rotations, excluding admissible terminal-attitude paths that reach the goal the long way around.Optimality statements for the implemented method therefore apply to the projector-reachable subset.Source: D. Equivalence of the Bi-Level… · p. 4
  • The overview connects independent agent optimization to swarm-wide selection, execution, and replanning after detected deviations.After deviations, the recovery loop detumbles, isolates faulty agents, and replans with survivors from the current state.Source: Figure 2 · p. 4

III. PROPOSED APPROACH

The approach descends the exact gradient of the inner convex problem’s optimal value over torque trajectories. Independent restart streams run across agents, whose max-consensus selects the best result.

III. PROPOSED APPROACH

  • The outer objective is the optimal value of a convex program whose data vary smoothly with the torque trajectory.Source: §III · p. 4
  • The method uses the envelope-theorem exact gradient of the inner optimal value for projected descent.Source: §III · p. 4
  • Random restarts address multimodality, with N agents running independent streams and max-consensus selecting the best result.Source: §III · p. 4

A. Envelope-Theorem Gradient of the Inner Value

The inner value function is differentiable under standard regularity conditions, allowing the outer search to use an exact envelope-theorem gradient. Its evaluation uses the converged inner solution without differentiating the optimizer’s sensitivity to torque.

A. Envelope-Theorem Gradient of the Inner Value

  • Torque enters the inner constraints directly through allocation equalities and indirectly through attitude-dependent rotation matrices and attachment vectors.The direct channel uses equality multipliers; the indirect channel uses a backward adjoint pass through the attitude recursion.Source: A. Envelope-Theorem Gradient of the… · pp. 4–5
  • Under LICQ and strict complementarity, the inner value function is differentiable and admits an envelope-theorem gradient.The primal-dual solution of the inner problem supplies the point at which the gradient is evaluated.Source: A. Envelope-Theorem Gradient of the… · pp. 4–5
  • The gradient needs only the Lagrangian’s partial derivative at the converged solution, not the sensitivity of the optimizer to torque.The gradient has dimension 3T, and its evaluation cost is independent of the number of agents.Source: A. Envelope-Theorem Gradient of the… · p. 5
  • The inner allocation is implemented as an SOCP with an epigraph fuel objective, exact thrust-cone constraints, and linear translational dynamics.A small quadratic regularizer selects a unique allocation when multiple allocations have identical fuel.Source: A. Envelope-Theorem Gradient of the… · p. 5

B. Feasibility Projection

The outer search projects candidate torque trajectories onto rotationally feasible ones, then uses projected gradient descent and random restarts. Independent decentralized islands explore separate restart streams and retain the best result found.

B. Feasibility Projection

  • A multiple-shooting projection makes each evaluated torque candidate rotationally feasible; failed projections are rejected, with failures increasing as the cone half-angle shrinks.The projection problem is non-convex but small-dimensional and solved quickly.Source: B. Feasibility Projection · p. 5

C. Projected Gradient Descent

  • Projected gradient descent re-projects each trial step, while backtracking evaluates the true objective and scales steps relative to torque magnitude.Relative scaling supports transfer across scenarios with torque magnitudes spanning several orders of magnitude.Source: C. Projected Gradient Descent · p. 5

D. Random Restarts and Decentralized Multistart

  • Because the value function is multimodal, each island runs independent descent-and-restart streams; island 1 starts from the shooting nominal, while others start from random draws.No communication occurs during the search, so the swarm evaluates N restart streams per unit of per-agent wall-clock.Source: D. Random Restarts and Decentralized… · pp. 5–6

E. Convergence of the Random Multistart

Repeated independent restarts converge almost surely to the best cost attainable through the accepted projection and projected descent. With multiple islands, the aggregate search gains an N-fold reduction in elapsed time, while retaining the warm-started single-start baseline.

E. Convergence of the Random Multistart

  • The minimum cost across independent restart draws converges almost surely to L‡, the essential infimum of costs reachable by projected descent.The result follows from independent draws and the positive probability of reaching costs arbitrarily close to L‡.Source: E. Convergence of the Random Multistart · p. 6
  • L‡ is the best cost over reachable stationary points, not unconditionally the global optimum; equality requires positive-measure access to the global minimizer’s basin and successful projection.Cold-draw projection acceptance can decline as the cone half-angle shrinks and can vanish for some scenarios below about 2.5°.Source: E. Convergence of the Random Multistart · p. 6
  • With N independent islands, the first n restarts finish in ⌈n/N⌉ per-agent wall-clock rounds, yielding an N-fold elapsed-time reduction.The warm-started island ensures the swarm does no worse than the deterministic single-start baseline.Source: E. Convergence of the Random Multistart · p. 6

F. Consensus Round

After the search budget expires, agents use max-consensus to select the island with the best cost and recover their thrust sequences from its torque trajectory.

F. Consensus Round

  • Each agent compares the pair (1/L_i, i) and replaces it with the maximum over itself and its neighbors in the line-of-sight graph.Islands whose starts all failed have fitness zero and cannot win.Source: F. Consensus Round · p. 6
  • If the line-of-sight graph is connected, all agents hold the global winner after at most diam(G) synchronous iterations.Connectedness of the graph is assumed throughout.Source: F. Consensus Round · p. 6
  • The winning agent broadcasts its torque trajectory along the consensus tree, and each agent recovers its thrust sequence with one final inner solve..Source: F. Consensus Round · p. 6

G. Closed-Loop Fault Recovery

The paper describes closed-loop recovery after trajectory deviations or communication timeouts, then specifies sampling-based and direct-NLP baselines for comparison. Recovery detumbles the payload, removes unresponsive or misbehaving agents, and replans with the survivors.

G. Closed-Loop Fault Recovery

  • Recovery proceeds through detumbling, identifying and removing unresponsive or misbehaving agents, and replanning from the current state to the original terminal condition.If the survivors cannot span the required wrench space, replanning is infeasible and the episode is scored as a survivability failure.Source: G. Closed-Loop Fault Recovery · p. 6

A. Baseline Methods

  • The baselines share the bi-level structure, inner solver, feasibility projection, and multiple-shooting warm start, isolating the outer search strategy.Source: A. Baseline Methods · pp. 6–7

1. Genetic Algorithm (GA)

  • The genetic algorithm evolves projected torque trajectories using crossover, mutation, elitist steady-state selection, and reciprocal inner cost as fitness.Source: §1 · p. 7

2. Model Predictive Path Integral (C-MPPI / D-MPPI)

  • MPPI perturbs and projects sampled trajectories, then updates its nominal toward the weighted mean of evaluated candidates.Weighting projected candidates keeps the update target among evaluated points; decentralized MPPI uses independent islands followed by consensus.Source: §2 · p. 7
  • Failed MPPI inner solves receive infinite cost and zero weight, while the best-ever projected candidate is retained.Source: §2 · p. 7

3. Direct NLP

  • The direct NLP solves the original problem as one joint optimization over thrust and state trajectories, using the same dynamics and a cold-started multiple-shooting nominal.Source: §3 · p. 7

B. Experimental Design

The experiments randomize orbital, payload, terminal-state, fleet-size, and attachment parameters, then compare solver reliability and fuel under varying cone angles and compute budgets. The proposed decentralized gradient method matches or exceeds baseline reliability across tested angles, while its fuel gap to the NLP narrows at moderate angles and longer budgets.

B. Experimental Design

  • Trials sample mission parameters and terminal states across 472 unique scenarios per angle, with returned controls independently re-simulated and success defined by V < 10−3.The baseline grid spans six methods, six cone half-angles, and budgets from 10 to 600 s; separate fleet-size tests include randomized agent dropouts.Source: B. Experimental Design · p. 7

A. Baseline Comparison

  • D-GD succeeds on 69–87% of scenarios across cone angles, reaching 100% at the widest angle, and matches or exceeds every baseline in every angle cell.At budgets of 30 s and above it has the highest success rate; the NLP succeeds on 41–46% for angles up to 15° and 82% at 60°.Source: A. Baseline Comparison · pp. 7–8, Figure 3 · p. 8
  • D-GD’s mean fuel ratio to the NLP falls from 1.18 at 10 s to 1.12 at 600 s, and it is the closest outer search to the NLP at every budget.The comparison uses jointly solved scenarios and paired per-scenario fuel ratios.Source: A. Baseline Comparison · p. 8, Figure 4 · p. 8
  • For cone angles of at least 15° and budgets of 300 s or more, D-GD is within 5–8% of the converged NLP cost.At 10 s, near-parity ratios of 0.99–1.01 for 5°–15° arise because CPU-truncated NLP iterates are feasible but expensive.Source: A. Baseline Comparison · p. 8, Figure 5 · p. 8

B. Value of Decentralisation

The decentralized method matched or exceeded every baseline’s reliability across tested cone angles and budgets, while recovery reliability increased with fleet size. Independent restart streams maintained fuel quality comparable to centralized search, and the conclusion identifies tight-angle reachability and additional fault settings as future work.

C. Scalability and Fault Survivability

  • Recovery success rose from 54% at N = 4 to 93% at N = 11 and 94–96% at N ∈ {18, 25}; for N ≥ 11 it was insensitive to one, two, or three simultaneous faults.The dominant failure was a wrench-infeasible replan when surviving attachment cones could not span the wrench needed to brake and re-steer the payload.Source: Figure 6 · p. 9, C. Scalability and Fault Survivability · p. 9

B. Value of Decentralisation

  • D-GD’s mean paired fuel ratio matched or bettered C-GD’s at every budget: 1.18 versus 1.20 at 10 s and 1.12 versus 1.14 at 600 s.The two variants stayed within 0.02 once either had explored reachable basins, and D-GD had the higher success rate at every budget.Source: B. Value of Decentralisation · pp. 8–9

VI. CONCLUSION

  • The framework splits the problem into inner convex thrust allocation and outer torque-trajectory search, solved using projected gradient descent on the exact envelope-theorem gradient.Independent restart streams and max-consensus select the swarm-wide best trajectory; almost-sure convergence is established for the projector-reachable set.Source: §VI · p. 9
  • 78–80% of rollout-verified scenarios succeeded, versus 20–50% for direct NLP, across cone half-angles from 1° to 60° and per-agent budgets from 10 to 600 s.The proposed approach also came closest to converged NLP fuel among the outer searches, within 5–8% at moderate cone angles and generous budgets, while sampling baselines paid 34–46%.Source: §VI · p. 9
  • Future work targets tight-cone reachability, online restart adaptation, time-varying communication topologies, and faults during detumbling.The conclusion proposes widening the projection’s reachable set or using cone-angle homotopy.Source: §VI · p. 9
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