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Distributionally Robust Adaptive Iterative Covariance Steering for Small-Body Proximity Operations

Vivek Khatana, Aditya Gahlawat, Naira Hovakimyan, Petros Voulgaris

arXiv:2610.12774v1eess.SYcs.ROmath.FAmath.OCmath.PR

TL;DR

Small-body proximity guidance must manage uncertain gravity, SRP, and stochastic disturbances while avoiding collisions under continuous thrust. The paper combines iCS and L1-DRAC, using a Wasserstein radius to account for residual distribution mismatch in robust CVaR keep-out constraints. Across 1000 paired Monte Carlo trials, L1-DRAC reduced keep-out violations from 32.7% to 0.2% and physical collisions from 16.4% to zero, without actuator saturation.

  • Problem

    Small-body proximity guidance faces uncertain gravity and SRP, and nominal covariance-steering safety guarantees can be lost under persistent dynamics mismatch.

    Source: §I · p. 1

  • Method

    iCS plans a nominal policy under distributionally robust CVaR keep-out constraints, while L1-DRAC compensates for uncertainty and bounds residual distribution mismatch through a Wasserstein radius.

    Source: §I · p. 1, A. Risk-Aware Small-Body Collision… · p. 2, C. L1-DRAC Augmentation · p. 2

  • Results

    32.7% to 0.2% keep-out violation rate and 16.4% to zero physical collision rate across 1000 paired Monte Carlo trials with L1-DRAC; no actuator saturation was observed.

    Source: §V · p. 3

  • Takeaways & Limitations

    With L1-DRAC, lower-tail CVaR remains nonnegative over the controlled horizon despite two adaptive violations within the displayed worst-1% tail.

    Source: §V · p. 4

    Further work is planned to assess sensitivity across additional uncertainty models and small-body scenarios, reduce iCS planning latency, and validate with higher-fidelity and hardware-in-the-loop simulations.

    Source: §VI · pp. 4–5

Abstract

from arXiv · show

Autonomous small-body proximity operations require continuous-thrust guidance under uncertain gravity, solar-radiation pressure (SRP), and stochastic disturbances. We combine iterative covariance steering (iCS) with $L_1$ distributionally robust adaptive control ($L_1$-DRAC) to provide risk-aware collision avoidance despite mismatch between the nominal planning model and the true dynamics. Iterative covariance steering plans a nominal feedback policy satisfying distributionally robust conditional value-at-risk keep-out constraints, while $L_1$-DRAC compensates for dynamics uncertainty and bounds the residual distributional mismatch through a Wasserstein radius used by the planner. In $\mathbf{1000}$ paired Monte Carlo trials with gravity, SRP, and diffusion mismatch, $L_1$-DRAC adaptive control reduces the keep-out violation rate from $\mathbf{32.7\%}$ to $\mathbf{0.2\%}$ and the physical collision rate from $\mathbf{16.4\%}$ to zero, without actuator saturation.

I. INTRODUCTION

The paper combines iterative covariance steering with L1 distributionally robust adaptive control to maintain collision safety under dynamics mismatch. Its stochastic model describes spacecraft position and velocity in a rotating asteroid-centered frame, with uncertainty in drift and diffusion.

I. INTRODUCTION

  • The proposed framework pairs nominal-model iCS with L1-DRAC, which compensates for uncertain dynamics and bounds residual state-distribution mismatch for robust safety constraints.The mismatch bound defines the Wasserstein ambiguity radius used by the planner.Source: §I · p. 1

II. SMALL-BODY STOCHASTIC MODEL

  • The state X stacks three-dimensional position R and velocity V in a rotating asteroid-centered frame, with commanded acceleration U.The dynamics are described using a normalized augmented nonlinear Hill three-body problem model.Source: §II · p. 1
  • The model distinguishes known dynamics from drift and diffusion uncertainties, while the planner uses the nominal model obtained by setting both uncertainties to zero.The stochastic model also includes Brownian motion W_t and an independent Brownian motion.Source: §II · p. 1

A. Risk-Aware Small-Body Collision Avoidance

Collision avoidance uses a keep-out boundary beyond the physical collision radius, with safety expressed through signed clearance and a risk constraint. iCS plans a nominal trajectory and feedback policy, while L1-DRAC addresses model uncertainty and residual distributional mismatch.

A. Risk-Aware Small-Body Collision Avoidance

  • The keep-out radius adds a positive, mission-defined safety buffer to the physical collision radius, which is the sum of asteroid and spacecraft radii.The signed keep-out clearance is defined relative to this boundary.Source: A. Risk-Aware Small-Body Collision… · p. 1
  • Safety requires nonnegative signed clearance, and the formulation imposes a risk constraint together with terminal mean and covariance requirements.The safety-risk tolerance is prescribed, and the control problem minimizes a statistical control cost.Source: A. Risk-Aware Small-Body Collision… · p. 2
  • iCS computes a nominal trajectory and feedback policy satisfying safety and terminal-distribution constraints, while L1-DRAC compensates for model uncertainty and certifies residual distributional mismatch.Both components address the limitation that nominal-model planning alone does not represent uncertain true dynamics.Source: A. Risk-Aware Small-Body Collision… · p. 2

III. DISTRIBUTIONALLY ROBUST ADAPTIVE ICS

Iterative covariance steering repeatedly linearizes nominal nonlinear dynamics around a reference trajectory and solves linear time-varying covariance-steering subproblems. The optimization uses discrete-time dynamics and updates a zero-order-hold control policy at sampled instants.

A. Nominal Iterative Covariance Steering

  • iCS repeatedly linearizes nominal nonlinear dynamics about a reference trajectory and solves linear time-varying covariance-steering subproblems.This extends finite-horizon covariance steering to nonlinear stochastic systems.Source: A. Nominal Iterative Covariance Steering · p. 2
  • The iCS optimization uses sampled discrete-time dynamics under a zero-order-hold input.The sampled nominal state and input are denoted at times t_k.Source: A. Nominal Iterative Covariance Steering · p. 2
  • At each sampled time t_k, the zero-order-hold control policy is updated using an optimization with a statistical control cost.The discrete-time model terms are obtained from the analytic solution under piecewise-constant input.Source: A. Nominal Iterative Covariance Steering · p. 2

B. Wasserstein DR-CVaR Safety

The Wasserstein DR-CVaR safety constraint accounts for differences between the predicted nominal state distribution and the true distribution. It requires robust safety over an ambiguity set whose radius is supplied by L1-DRAC.

B. Wasserstein DR-CVaR Safety

  • Uncertain gravity, SRP, nonlinear propagation, and residual adaptive-control error can make the true state distribution differ from the predicted Gaussian distribution.The nominal Gaussian safety constraint is valid when that prediction accurately represents the true state.Source: B. Wasserstein DR-CVaR Safety · p. 2
  • The ambiguity set contains state distributions within 2-Wasserstein distance rho of the nominal distribution, where rho bounds distribution mismatch.The safety requirement must hold for every distribution in this set.Source: B. Wasserstein DR-CVaR Safety · p. 2
  • A tractable sufficient condition is used for the distributionally robust safety requirement.The nominal safety constraint is made conservative under the ambiguity set.Source: B. Wasserstein DR-CVaR Safety · p. 2
  • L1-DRAC provides the Wasserstein ambiguity-set radius used by the planner.The controller's radius bounds the mismatch between nominal and true state distributions.Source: B. Wasserstein DR-CVaR Safety · p. 2

C. L1-DRAC Augmentation

L1-DRAC adds an adaptive input to the uncertain system to compensate for model uncertainty. Under stated regularity and stability conditions, it bounds the distance between nominal and true state distributions, with the remaining mismatch represented by a Wasserstein radius.

  • L1-DRAC generates an adaptive control input through a predictor, piecewise-constant adaptation law, and low-pass filter.The iCS policy supplies the other control component.Source: C. L1-DRAC Augmentation · p. 2
  • Under its stated regularity and stability conditions, L1-DRAC bounds the Wasserstein distance between nominal and true state distributions.Source: C. L1-DRAC Augmentation · pp. 2–3
  • The Wasserstein radius accounts for residual distributional mismatch in the planner's constraints.Source: C. L1-DRAC Augmentation · p. 3

IV. SIMULATION STUDY

The simulation study considers a representative half-revolution proximity maneuver under normalized ANH3BP dynamics. It specifies the collision and keep-out radii, maneuver horizon, initial and final distributions, and safety-risk tolerance.

  • The study models a representative half-revolution proximity maneuver with normalized ANH3BP dynamics and β = 0.01.Source: §IV · p. 3
  • The physical collision radius is 0.1887 km, and the buffered keep-out radius is 0.1897 km.The safety buffer is 0.001 km; the corresponding normalized radii are 1.02 and 1.0254.Source: §IV · p. 3
  • The normalized maneuver horizon is T = 4, with ε_s = 0.01 and specified initial and final state means and covariances.The reference radius is 1.1, and the iCS time step is Δt = 0.1.Source: §IV · p. 3

A. System Uncertainty

The true system includes asteroid-gravity and SRP uncertainty absent from nominal iCS planning dynamics, along with diffusion mismatch. The specified uncertainty levels are a representative stress test, not a mission-calibrated model for a particular asteroid.

  • True dynamics include uncertainty in the asteroid gravitational parameter and solar-radiation-pressure acceleration, which are absent from nominal iCS planning dynamics.Source: A. System Uncertainty · p. 3
  • The gravity error is a fractional perturbation δμ, and its lower clipping limit keeps the true gravitational scaling, 1 + δμ, positive.Source: A. System Uncertainty · p. 3
  • A constant matched diffusion-uncertainty function Hσ(X) = 0.30I3 is used, and the uncertainty levels are a representative stress test rather than a mission-calibrated asteroid model.Source: A. System Uncertainty · p. 3

B. Control Policy Implementation

The implementation combines the iCS control policy with an L1-DRAC input. The reported iCS solution time concerns the high-level planning update, not the lower-level adaptive-control update.

V. SIMULATION RESULTS

Across paired trials, L1-DRAC sharply reduced keep-out violations and eliminated physical collisions without actuator saturation. It also improved lower-tail clearance and kept trajectory distributions closer to the nominal plan.

V. SIMULATION RESULTS

  • 32.7% to 0.2% keep-out violation rates and 16.4% to zero physical collision rates across 1,000 paired trials with L1-DRAC, without actuator saturation.Both cases used the same nominal iCS policy, including optimized state feedback; only the adaptive augmentation differed.Source: §V · p. 3
  • With adaptation, lower-tail CVaR remained nonnegative over the controlled horizon despite two violations in the worst 1% tail; without it, CVaR was negative over most of the maneuver.The figure shows median clearances, empirical 0th–1st percentile tails, lower-tail CVaR, and the keep-out boundary.Source: Figure 1 · p. 3, §V · p. 4
  • Adaptation kept trajectories concentrated around the planned path and empirical means and 95% KDE-HDRs close to nominal; without it, distributions dispersed toward the keep-out region.Figure 2 compares representative trajectories and empirical distributions under the same nominal policy with adaptation disabled or enabled.Source: §V · p. 4, Figure 2 · p. 4

VI. CONCLUSION

The paper combines nominal iterative covariance steering with adaptive control to address dynamics mismatch in continuous-thrust small-body proximity operations. In paired trials, adaptation reduced keep-out violations and eliminated physical collisions; broader validation remains future work.

VI. CONCLUSION

  • In 1,000 paired Monte Carlo trials, adaptation reduced the keep-out violation rate from 32.7% to 0.2% and eliminated physical collisions.The framework also uses a Wasserstein ambiguity set in its distributionally robust CVaR keep-out constraints.Source: §VI · p. 4
  • The framework uses iCS to plan a safe state distribution under nominal dynamics and L1-DRAC to compensate for matched acceleration uncertainty.A Wasserstein ambiguity set captures distributional mismatch in the DR-CVaR keep-out constraints.Source: §VI · p. 4
  • Future work will assess additional uncertainty models and small-body scenarios, reduce iCS planning latency, and validate the framework with higher-fidelity and hardware-in-the-loop simulations.Source: §VI · pp. 4–5
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