Belief CVaR-MPPI: Safe Control under Latent Uncertainty

Low Risk Aversion, $\beta_s = 0.50$. At low risk aversion, the object being transported by the robot collides with the blue stowed object, and the robot subsequently fails the insertion task.

High Risk Aversion, $\beta_s = 0.95$. At high risk aversion, the robot avoids collision with the receptacle exterior and successfully places the object on the shelf.

82%

task success
CVaR-MPPI, $\beta_s = 0.95$

0%

exterior-contact rate
CVaR-MPPI, $\beta_s = 0.95$

55%

task success
CVaR-MPPI, $\beta_s = 0.50$

50%

task success
CCMPPI baseline

Abstract

Many safety-critical control systems operate under latent uncertainty that sensors cannot resolve at decision time. Such uncertainty, arising from unknown physical properties, disturbances, or unobserved geometry, affects 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 systems with measured state and an unknown, time-invariant parameter represented by a belief distribution. We propose a risk-sensitive belief-space Model Predictive Path Integral (MPPI) controller that plans under this belief, regularizes performance using Conditional Value-at-Risk (CVaR), and imposes a CVaR constraint on a trajectory safety margin over the horizon. For the exact risk-constrained formulation underlying this controller, we establish three properties: (1) the CVaR constraint implies a probabilistic safety guarantee, (2) the controller recovers the risk-neutral optimum as the objective risk weight tends to zero, and (3) a union-bound argument extends the per-horizon guarantee to cumulative safety over repeated solves. In contact-rich MuJoCo simulations of vision-guided dexterous stowing, where a manipulator inserts a grasped object 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 with $55\%$ and $50\%$ for a risk-neutral configuration and a chance-constrained baseline, both of which incur nonzero exterior contact forces.

Overview & Working Principle

CVaR-MPPI is a risk-sensitive belief-space MPPI controller designed to be robust to latent uncertainty in contact model parameters represented by a particle-based belief distribution. Our controller jointly guarantees probabilistic safety by constraining the adverse tail of a trajectory safety margin. At each receding-horizon solve, CVaR-MPPI evaluates sampled control sequences across belief particles, applies the first control, and updates the belief from the next observation.

Particle Belief

We use a particle belief that captures time-invariant parameters, including receptacle pose and geometry unknown at decision time.

Sampled MPPI Rollouts

Sampled MPPI rollouts propagate candidate control sequences across belief particles and compute trajectory cost and safety margin.

Tail Risk Minimization

The objective penalizes the trajectory-wise CVaR cost, and a separate CVaR constraint bounds the adverse tail of a trajectory-wise safety violation measure.

Feedback Planning

The controller applies the first action, updates its belief, and re-solves the finite-horizon problem using the updated belief.

Theoretical Guarantees

1

CVaR Safety Implication If $\operatorname{CVaR}_{\beta_s}(-M_H) \le 0$, then $\Pr(M_H \ge 0) \ge \beta_s$.

2

Risk-Neutral Limit Under the paper’s compactness and continuity assumptions, every cluster point of the optimizer minimizes expected trajectory cost as $\lambda_r \downarrow 0$.

3

Cumulative Safety If the constraint holds at $T$ successive solves, all predicted horizons are safe with probability at least $1 - T(1 - \beta_s)$.

Simulation Results

Effect of $\beta_s$ on Success Rate

Increasing the safety confidence level from $\beta_s = 0.50$ to $0.95$ raised task success from $55\%$ to $82\%$ and reduced the exterior-contact rate from $27\%$ to $0\%$. CVaR-MPPI at $\beta_s = 0.90$ and CCMPPI both produced zero exterior contact, but CVaR-MPPI achieved $73\%$ success compared with $50\%$ for CCMPPI. Because CCMPPI uses a point estimate and omits both CVaR terms, this comparison evaluates the complete controllers rather than either CVaR component in isolation.

Effect of $\beta_s$. Across $11$ randomized MuJoCo trials per setting, increasing $\beta_s$ from $0.50$ to $0.90$ or $0.95$ eliminated exterior-contact violations while task success rose from $55\%$ to $73\%$ and $82\%$, respectively.

BibTeX

@misc{enwerem2026riskconstrained,
  title         = {Risk-Constrained Belief-Space Optimization for Safe Control under Latent Uncertainty},
  author        = {Enwerem, Clinton and Baras, John S. and Belta, Calin},
  year          = {2026},
  eprint        = {2604.03868},
  archivePrefix = {arXiv},
  primaryClass  = {eess.SY},
  url           = {https://arxiv.org/abs/2604.03868}
}