BC-MPPI: A Probabilistic Constraint Layer for Safe Model-Predictive Path-Integral Control

TL;DR

Model Predictive Path Integral (MPPI) control struggles with hard constraint guarantees in safety-critical robotics. BC-MPPI introduces a Bayesian neural-network surrogate for constraints and integrates it by weighting MPPI rollouts with the joint feasibility probability, effectively biasing sampling toward safe trajectories without explicit rejection. The surrogate provides a predictive mean and uncertainty, enabling probabilistic constraint satisfaction via factors like $\prod_{j} \Pr[c_j(\boldsymbol{\theta}) \le 0]$ and updating weights as $\tilde{\mu}^k = \exp[-(J^k-\rho)/\lambda] \prod_{j} \Pr[c_j(\boldsymbol{\theta}^k) \le 0]$. In MuJoCo quadrotor experiments with static and moving obstacles, BC-MPPI maintained safety margins, achieved target tracking better than a penalty-based baseline, and reduced collisions relative to Classic MPPI, though at the cost of higher computation due to surrogate evaluation. The approach offers a verifiable, single-scalar runtime safety score suitable for integration with verification-and-validation pipelines in certifiable autonomous systems.

Abstract

Model Predictive Path Integral (MPPI) control has recently emerged as a fast, gradient-free alternative to model-predictive control in highly non-linear robotic tasks, yet it offers no hard guarantees on constraint satisfaction. We introduce Bayesian-Constraints MPPI (BC-MPPI), a lightweight safety layer that attaches a probabilistic surrogate to every state and input constraint. At each re-planning step the surrogate returns the probability that a candidate trajectory is feasible; this joint probability scales the weight given to a candidate, automatically down-weighting rollouts likely to collide or exceed limits and pushing the sampling distribution toward the safe subset; no hand-tuned penalty costs or explicit sample rejection required. We train the surrogate from 1000 offline simulations and deploy the controller on a quadrotor in MuJoCo with both static and moving obstacles. Across K in [100,1500] rollouts BC-MPPI preserves safety margins while satisfying the prescribed probability of violation. Because the surrogate is a stand-alone, version-controlled artefact and the runtime safety score is a single scalar, the approach integrates naturally with verification-and-validation pipelines for certifiable autonomous systems.

Figures (6)

• Figure 1: BC-MPPI workflow. A Gaussian sampler perturbs the nominal parameter vector $\bar{\boldsymbol{\theta}}$ and launches $K$ parallel roll-outs; their costs $J^{1},\dots,J^{K}$ feed the usual MPPI weight update, producing importance weights $\omega$. A Bayesian surrogate multiplies these weights by the joint feasibility probability, suppressing unsafe trajectories and yielding the filtered parameter $\boldsymbol{\theta}^{\!*}$. The first input $\mathbf{u}^{\!*}$ of the associated control sequence is applied to the robot, closing the feedback loop with the measured state $\hat{\mathbf{x}}$.
• Figure 2: Snapshot sequence from a complex scenario involving five moving obstacles. Spheres model obstacles with inflated radii.
• Figure 3: Distance of quadrotor from the target
• Figure 4: Average obstacle distance of BC-MPPI vs Classic MPPI in scenarios with stationary and moving obstacles.
• Figure 5: Distance of quadrotor from the target