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Parameter-Robust MPPI for Safe Online Learning of Unknown Parameters

Matti Vahs, Jaeyoun Choi, Niklas Schmid, Jana Tumova, Chuchu Fan

TL;DR

This work addresses the challenge of maintaining safety for robots operating with unknown and time-varying parameters by introducing Parameter-Robust MPPI (PRMPPI). PRMPPI combines a Stein Variational Gradient Descent (SVGD) particle-based belief over unknown parameters with Conformal Prediction (CP) to enforce probabilistic safety, and integrates these into a Model Predictive Path Integral (MPPI) controller that optimizes a nominal trajectory in parallel with a robust backup trajectory. The approach yields a non-Gaussian parameter belief that continuously adapts online, guaranteeing safety over the prediction horizon while improving performance as parameter estimates converge. Demonstrations on simulation benchmarks and hardware quadrotor-payload experiments show higher success rates, reduced tracking error, and more accurate parameter estimates compared to baselines, highlighting its practical impact for safe online learning in uncertain robotic systems.

Abstract

Robots deployed in dynamic environments must remain safe even when key physical parameters are uncertain or change over time. We propose Parameter-Robust Model Predictive Path Integral (PRMPPI) control, a framework that integrates online parameter learning with probabilistic safety constraints. PRMPPI maintains a particle-based belief over parameters via Stein Variational Gradient Descent, evaluates safety constraints using Conformal Prediction, and optimizes both a nominal performance-driven and a safety-focused backup trajectory in parallel. This yields a controller that is cautious at first, improves performance as parameters are learned, and ensures safety throughout. Simulation and hardware experiments demonstrate higher success rates, lower tracking error, and more accurate parameter estimates than baselines.

Parameter-Robust MPPI for Safe Online Learning of Unknown Parameters

TL;DR

This work addresses the challenge of maintaining safety for robots operating with unknown and time-varying parameters by introducing Parameter-Robust MPPI (PRMPPI). PRMPPI combines a Stein Variational Gradient Descent (SVGD) particle-based belief over unknown parameters with Conformal Prediction (CP) to enforce probabilistic safety, and integrates these into a Model Predictive Path Integral (MPPI) controller that optimizes a nominal trajectory in parallel with a robust backup trajectory. The approach yields a non-Gaussian parameter belief that continuously adapts online, guaranteeing safety over the prediction horizon while improving performance as parameter estimates converge. Demonstrations on simulation benchmarks and hardware quadrotor-payload experiments show higher success rates, reduced tracking error, and more accurate parameter estimates compared to baselines, highlighting its practical impact for safe online learning in uncertain robotic systems.

Abstract

Robots deployed in dynamic environments must remain safe even when key physical parameters are uncertain or change over time. We propose Parameter-Robust Model Predictive Path Integral (PRMPPI) control, a framework that integrates online parameter learning with probabilistic safety constraints. PRMPPI maintains a particle-based belief over parameters via Stein Variational Gradient Descent, evaluates safety constraints using Conformal Prediction, and optimizes both a nominal performance-driven and a safety-focused backup trajectory in parallel. This yields a controller that is cautious at first, improves performance as parameters are learned, and ensures safety throughout. Simulation and hardware experiments demonstrate higher success rates, lower tracking error, and more accurate parameter estimates than baselines.
Paper Structure (22 sections, 2 theorems, 21 equations, 4 figures, 3 tables, 1 algorithm)

This paper contains 22 sections, 2 theorems, 21 equations, 4 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries and Problem Formulation
  4. Stein Variational Gradient Descent
  5. Conformal Prediction
  6. Model Predictive Path Integral Control
  7. Problem Setting
  8. Parameter-Robust MPPI
  9. Online Parameter Estimation
  10. Receding Horizon Controller
  11. Uncertainty Quantification in Safety Violations
  12. Control Design
  13. Experiments
  14. Simulation Setup
  15. Evaluation Metrics
  16. ...and 7 more sections

Key Result

Lemma 1

Let $Z, Z^{(1)},\dots, Z^{(k)}$ be k + 1 independent and identically distributed (i.i.d.) real-valued random variables. Let $Z^{(1)},\dots, Z^{(k)}$ be sorted in non-decreasing order and define $Z^{k+1} := \infty$. For $\delta \in (0, 1)$, it holds that where $\bar{Z} = Z^{(r)}$ with $r = \lceil(k+1)(1-\delta)\rceil$ and where $\lceil\cdot\rceil$ is the ceiling function.

Figures (4)

  • Figure 1: Long exposure shot of our hardware experiments of a Crazyflie 2.1 brushless quadrotor with a suspended payload of unknown length. The quadrotor has to track a square trajectory while avoiding collisions and has to improve its belief about its physical parameters from online observations.
  • Figure 2: Illustration of the simulated 2D Quadrotor environment. Left: The circular path to be tracked is shown in black while the traversed path is colored in blue. The nominal trajectory is entering the unsafe region which is shaded in red. Right: A snapshot of the parameter belief in the depicted time step. Particles are denoted by blue circles and the contours of the KDE are shown in red.
  • Figure 3: RMSE and PA over multiple laps in the simulated Quadrotor environment. Our method identifies parameters more quickly resulting in lower RMSEs.
  • Figure 4: Experimental result for both parameter learning settings. (a) Illustration of the traversed $xy$ trajectories of the Crazyflie. Obstacles are shown in red and a colormap indicates the time scale. (b) The parameter beliefs at different time steps. (c) A barplot indicating that learning damping parameters and length jointly can result in more accurate beliefs leading to lower trajectory RMSEs over time. (d) The minimum distance to obstacles over time.

Theorems & Definitions (4)

  • Lemma 1
  • Theorem 1
  • proof
  • Remark 1