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Risk-Constrained Belief-Space Optimization for Safe Control under Latent Uncertainty

Clinton Enwerem, John S. Baras, Calin Belta

Abstract

Many safety-critical control systems must operate under latent uncertainty that sensors cannot directly resolve at decision time. Such uncertainty, arising from unknown physical properties, exogenous disturbances, or unobserved environment geometry, influences dynamics, task feasibility, and safety margins. Standard methods optimize expected performance and offer limited protection against rare but severe outcomes, while robust formulations treat uncertainty conservatively without exploiting its probabilistic structure. We consider partially observed dynamical systems whose dynamics, costs, and safety constraints depend on a latent parameter maintained as a belief distribution, and propose a risk-sensitive belief-space Model Predictive Path Integral (MPPI) control framework that plans under this belief while enforcing a Conditional Value-at-Risk (CVaR) constraint on a trajectory safety margin over the receding horizon. The resulting controller optimizes a risk-regularized performance objective while explicitly constraining the tail risk of safety violations induced by latent parameter variability. We establish three properties of the resulting risk-constrained controller: (1) the CVaR constraint implies a probabilistic safety guarantee, (2) the controller recovers the risk-neutral optimum as the risk weight in the objective tends to zero, and (3) a union-bound argument extends the per-horizon guarantee to cumulative safety over repeated solves. In physics-based simulations of a vision-guided dexterous stowing task in which a grasped object must be inserted into an occupied slot with pose uncertainty exceeding prescribed lateral clearance requirements, our method achieves 82% success with zero contact violations at high risk aversion, compared to 55% and 50% for a risk-neutral configuration and a chance-constrained baseline, both of which incur nonzero exterior contact forces.

Risk-Constrained Belief-Space Optimization for Safe Control under Latent Uncertainty

Abstract

Many safety-critical control systems must operate under latent uncertainty that sensors cannot directly resolve at decision time. Such uncertainty, arising from unknown physical properties, exogenous disturbances, or unobserved environment geometry, influences dynamics, task feasibility, and safety margins. Standard methods optimize expected performance and offer limited protection against rare but severe outcomes, while robust formulations treat uncertainty conservatively without exploiting its probabilistic structure. We consider partially observed dynamical systems whose dynamics, costs, and safety constraints depend on a latent parameter maintained as a belief distribution, and propose a risk-sensitive belief-space Model Predictive Path Integral (MPPI) control framework that plans under this belief while enforcing a Conditional Value-at-Risk (CVaR) constraint on a trajectory safety margin over the receding horizon. The resulting controller optimizes a risk-regularized performance objective while explicitly constraining the tail risk of safety violations induced by latent parameter variability. We establish three properties of the resulting risk-constrained controller: (1) the CVaR constraint implies a probabilistic safety guarantee, (2) the controller recovers the risk-neutral optimum as the risk weight in the objective tends to zero, and (3) a union-bound argument extends the per-horizon guarantee to cumulative safety over repeated solves. In physics-based simulations of a vision-guided dexterous stowing task in which a grasped object must be inserted into an occupied slot with pose uncertainty exceeding prescribed lateral clearance requirements, our method achieves 82% success with zero contact violations at high risk aversion, compared to 55% and 50% for a risk-neutral configuration and a chance-constrained baseline, both of which incur nonzero exterior contact forces.

Paper Structure

This paper contains 26 sections, 3 theorems, 26 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Risk Measures & Risk-Sensitive Optimization
  4. Model-Predictive Control (MPC) & Chance Constraints
  5. Belief-Space Planning
  6. Safety-Critical Control
  7. Notation and Preliminaries
  8. Problem Formulation
  9. Risk-Sensitive Belief-Space MPPI
  10. MPPI Algorithm
  11. Online Solution of \ref{['prb:rsbel']}
  12. Theoretical Guarantees
  13. Case Study: Vision-Guided Dexterous Stowing
  14. Object Stowing Task and Simulation Environment
  15. Instantiation of the General Framework
  16. ...and 11 more sections

Key Result

Theorem 1

Let $\beta_s \in (0,1)$ and suppose $\mathrm{CVaR}_{\beta_s}\!\bigl(-M_H\bigr) \le 0$. Then $\blacktriangleleft$$\blacktriangleleft$

Figures (4)

  • Figure A1: Running Example. (a). A robotic manipulator inserts a grasped object into an occupied receptacle with uncertain pose. (b). With low risk aversion ($\beta_s=0.50$), the controller allows trajectories that produce catastrophic contact forces. (c). With high risk aversion ($\beta_s=0.95$), the CVaR safety constraint maintains safer insertion margins and avoids contact.
  • Figure E1: Object Stowing Task.$\mathcal{F}_{\mathcal{R}}$, $\mathcal{F}_{\mathcal{H}}$, and $\mathcal{F}_{\mathcal{C}}$ denote the robot base, hand, and camera frames. $\mathcal{B}_{\mathrm{obj}}$ is the object to be transported, $\mathcal{B}_{\mathrm{slot}}$ denotes the receptacle, and $\mathcal{B}_{\mathrm{env}}$ represents previously stowed objects inside the receptacle. The goal region $\mathcal{X}_{\mathrm{goal}}$ specifies admissible placement poses within the receptacle, and all frames are expressed with respect to $\mathcal{F}_{\mathcal{R}}$.
  • Figure F1: Representative Trajectory. Colored bands indicate task phases ( approach and insert). $\varepsilon_{p}$ is the stowing proximity threshold. Both CVaR-MPPI ($\beta_s \ge 0.90$) configurations reach the goal while maintaining positive safety margins, whereas $\beta_s{=}0.50$ and CcMppi ($\delta_H{=}0.05$) fail to complete insertion.
  • Figure F2: CVaR Safety Constraint under Varying $\boldsymbol{\beta_s}$. Each panel shows $\mathrm{CVaR}_{\beta_s}({-}{M_H})$ for one risk configuration; green region indicates constraint satisfaction, red dashed line marks the violation boundary. Higher $\beta_s$ evaluates a narrower worst-case tail (worst 5% vs. 50%), making the constraint harder to satisfy---the trace rides closer to the boundary. Despite this, all three configurations remain feasible throughout approach and insertion, consistent with \ref{['eq:cvarcnst']}. Only $\beta_s \ge 0.90$ eliminates contact (\ref{['tab:results']}).

Theorems & Definitions (17)

  • Definition 1: Value-at-Risk
  • Definition 2: Conditional Value-at-Risk
  • Remark 1: Shorthand Notation
  • Definition 3: Belief and Latent Parameter
  • Definition 4: Particle Belief
  • Definition 5: Safety Function
  • Definition 6: Trajectory Safety Margin
  • Remark 2: Underlying Distribution and Problem Data
  • Remark 3: Monotonicity in $\beta_c$
  • Theorem 1: CVaR Safety Implication
  • ...and 7 more