Chance constraints transcription and failure risk estimation for stochastic trajectory optimisation

Roberto Armellin; St\'ephanie Lizy-Destrez; Thomas Caleb

arxiv: 2502.15949 · v3 · submitted 2025-02-21 · 🧮 math.OC

Chance constraints transcription and failure risk estimation for stochastic trajectory optimisation

Thomas Caleb , Roberto Armellin , St\'ephanie Lizy-Destrez This is my paper

Pith reviewed 2026-05-23 02:17 UTC · model grok-4.3

classification 🧮 math.OC

keywords chance constraintsstochastic trajectory optimizationGaussian uncertaintyrisk estimationoptimal controlfailure probabilityconstraint transcription

0 comments

The pith

Two new methods transcribe multi-dimensional Gaussian chance constraints into deterministic forms for stochastic trajectory optimization with reduced conservatism.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops transcription techniques to turn probabilistic chance constraints into solvable deterministic problems for trajectory optimization when uncertainties are present. The spectral radius approach extends prior work to handle arbitrary multi-dimensional constraints. A refined first-order method delivers tighter bounds at linear computational cost. A separate d-th order risk estimator supplies accurate failure probability bounds that stay conservative yet practical even as dimension grows. These tools are demonstrated on an uncertain optimal control problem where the first-order transcription meets the risk target with near-optimal fuel use.

Core claim

The spectral radius and refined first-order transcriptions convert multi-dimensional Gaussian chance constraints into tractable deterministic inequalities for trajectory optimization under uncertainty, while the d-th order risk estimation method supplies conservative yet accurate failure probability estimates in quadratic complexity; the first-order transcription achieves near-optimal fuel consumption while keeping failure risk below target, the spectral radius method adds roughly 0.7 kg fuel from extra conservatism, and the risk estimator avoids the exponential conservatism growth seen in earlier high-dimensional methods.

What carries the argument

The refined first-order transcription method, which approximates the chance constraint boundary using a first-order expansion refined for tightness, together with the spectral radius method that uses the largest eigenvalue of the covariance to bound the probability.

If this is right

The first-order transcription maintains failure risk below target while achieving near-optimal fuel consumption.
The spectral radius transcription handles arbitrary multi-dimensional constraints but adds approximately 0.7 kg fuel and increases computation time by 51 percent.
The d-th order risk estimator remains accurate in high dimensions with quadratic complexity.
Earlier risk estimation methods exhibit exponential growth in conservatism as constraint dimension increases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The linear complexity of the refined first-order method could enable scaling the approach to problems with hundreds of constraints.
The risk estimation technique might be paired with non-Gaussian uncertainty models by replacing the underlying distribution assumptions.
The transcriptions could be applied directly to other domains such as spacecraft rendezvous or autonomous vehicle path planning under sensor noise.

Load-bearing premise

The uncertainties follow multi-dimensional Gaussian distributions and the transcriptions convert the probabilistic constraints into deterministic forms without unmodeled approximation errors that would violate the target failure risk.

What would settle it

A numerical test on the same optimal control problem in which the realized failure rate under the refined first-order transcription exceeds the prescribed risk threshold.

Figures

Figures reproduced from arXiv: 2502.15949 by Roberto Armellin, St\'ephanie Lizy-Destrez, Thomas Caleb.

**Figure 2.** Figure 2: Solution to the energy-optimal Earth-Mars transfer. [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Figure 3: Earth-Mars transfer covariance propagation. Magnified [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

**Figure 4.** Figure 4: Evolution of the conservatism with the dimension. [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

read the original abstract

Stochastic trajectory optimisation under uncertainty requires robust constraint satisfaction through chance constraints. However, existing transcription methods remain limited to scalar constraints or highly specific structures while introducing substantial conservatism. This work presents two general-purpose transcription methods for multi-dimensional Gaussian chance constraints for trajectory optimisation problems under uncertainty. The spectral radius method extends existing methods to arbitrary multi-dimensional constraints with reduced conservatism. The refined first-order method achieves superior tightness with linear complexity. In addition, a d-th order risk estimation methodology provides conservative failure probability estimates with limited conservatism in high dimensions in quadratic complexity. Applied to an optimal control with uncertainties setting, the first-order transcription achieves near-optimal fuel consumption while maintaining the failure risk below the target. The spectral radius method incurs approximately 0.7 kg additional fuel consumption due to excessive conservatism and a 51% increase in computational time due to its cubic complexity. High-dimensional tests show that the proposed risk estimation method provides accurate risk estimates, while previously developed methods exhibit exponential growth in conservatism with respect to constraint dimension.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper introduces two transcriptions for multi-dimensional chance constraints and a quadratic risk estimator that performs well in the reported tests.

read the letter

This paper gives two transcription methods for multi-dimensional Gaussian chance constraints in trajectory optimization and a d-th order risk estimator with quadratic complexity. The spectral radius method extends existing approaches to arbitrary dimensions with reduced conservatism. The refined first-order method is tighter and linear in cost. The risk estimator avoids the exponential conservatism growth that hits older methods in high dimensions. In the reported optimal control example the first-order transcription keeps fuel consumption near optimal while the realized failure risk stays below the target. The spectral radius version uses roughly 0.7 kg more fuel and runs 51 percent slower. High-dimensional tests show the new estimator stays accurate. The work is clear on its assumptions and ties the performance claims to the Gaussian case and the specific tests. The comparisons on fuel, risk, and time are concrete. The main limitation is that everything rests on the tested regimes and the Gaussian distribution. If the uncertainties deviate or the constraints have structures not covered in the experiments, the tightness and risk guarantees could slip. The abstract does not provide detailed error bounds or sensitivity analysis, so those would need checking in review. This is for people working on stochastic trajectory problems in control and aerospace. Readers who need practical ways to handle multi-dimensional chance constraints will find the methods and the scaling results useful. It is a focused piece that engages the prior literature directly. I think it deserves peer review. The contribution is narrow but the methods look implementable and the experiments target the right practical questions.

Referee Report

0 major / 3 minor

Summary. The paper presents two transcription methods for converting multi-dimensional Gaussian chance constraints into deterministic equivalents suitable for trajectory optimization: the spectral radius method (extending prior work with reduced conservatism) and the refined first-order method (with linear complexity and superior tightness). It also introduces a d-th order risk estimation procedure that yields conservative failure probability estimates in quadratic time. These are demonstrated on a stochastic optimal control problem, where the first-order transcription yields near-optimal fuel consumption while satisfying the target failure risk; the spectral radius approach adds ~0.7 kg fuel and 51% runtime due to greater conservatism and cubic scaling. High-dimensional numerical tests indicate that the proposed risk estimator remains accurate with limited conservatism, unlike prior methods whose conservatism grows exponentially with dimension.

Significance. If the derivations and experimental controls hold, the contribution would be significant for stochastic optimal control under Gaussian uncertainty, supplying general-purpose, scalable transcriptions that reduce conservatism relative to existing scalar or structure-specific approaches. The quadratic-complexity risk estimator and explicit fuel/runtime deltas in the optimal-control example provide concrete, falsifiable evidence of practical utility in aerospace or robotics trajectory planning.

minor comments (3)

[Abstract] The abstract reports precise numerical outcomes (0.7 kg fuel penalty, 51% runtime increase) without cross-references to the corresponding tables, figures, or experimental parameters (e.g., uncertainty covariance, constraint dimension, or solver tolerances) that produced them; adding such pointers would improve traceability.
[§3] Notation for the refined first-order transcription and the d-th order estimator should be introduced with explicit definitions of all auxiliary matrices or scalars before their use in the complexity and conservatism claims.
[§5] The high-dimensional risk-estimation tests would benefit from an explicit statement of the maximum dimension tested and the precise definition of 'exponential growth in conservatism' for the baseline methods.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its significance for stochastic optimal control, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation chain is self-contained

full rationale

The abstract frames the contributions as extensions of existing transcription methods for multi-dimensional Gaussian chance constraints, with the spectral radius and refined first-order approaches presented as new general-purpose techniques having stated computational complexities. Performance claims (near-optimal fuel consumption, risk below target, 0.7 kg gap, quadratic vs. cubic scaling) are described as outcomes of numerical experiments in optimal-control settings and high-dimensional tests, not as quantities derived by construction from fitted parameters or prior self-citations. No equations, definitions, or load-bearing steps in the provided text reduce the central results to self-referential inputs, self-citation chains, or renamed known patterns. The d-th order risk estimator is positioned as an independent methodology whose accuracy is validated empirically rather than assumed. This matches the expectation that most papers exhibit no circularity when their claims rest on external benchmarks and stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claims rest on the domain assumption of Gaussian uncertainties and the validity of the proposed approximation orders for chance constraint transcription; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (1)

domain assumption Uncertainties are modeled as multi-dimensional Gaussian random variables
All methods are specified for Gaussian chance constraints, a standard modeling choice in stochastic optimization.

pith-pipeline@v0.9.0 · 5704 in / 1318 out tokens · 45984 ms · 2026-05-23T02:17:01.161640+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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H. Holt, R. Armellin, N. Baresi, Y . Hashida, A. Tur- coni, A. Scorsoglio, and R. Furfaro. Optimal q-laws via reinforcement learning with guaranteed stability. Acta Astronautica, 187:511–528, Oct. 2021. ISSN 0094-5765. doi: 10.1016/j.actaastro.2021.07.010

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