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Safe Navigation in Dynamic Environments Using Data-Driven Koopman Operators and Conformal Prediction

Kaier Liang, Guang Yang, Mingyu Cai, Cristian-Ioan Vasile

TL;DR

The paper tackles safe navigation for systems with nonlinear dynamics and state uncertainty by marrying data-driven Koopman lifting with conformal prediction to yield probabilistic safety guarantees within a linear MPC framework. Offline learning of a Koopman-based lifted model paired with a neural encoder enables linear dynamics in the lifted space, while conformal prediction provides quantifiable error bounds that tighten MPC constraints. A reference generator guides the MPC, and slack-based constraint handling preserves feasibility, resulting in a computationally efficient, safe navigation strategy validated by simulations on a unicycle model with dynamic obstacles. This approach offers a practical pathway to robust, real-time safe navigation in dynamic environments with uncertain models.

Abstract

We propose a novel framework for safe navigation in dynamic environments by integrating Koopman operator theory with conformal prediction. Our approach leverages data-driven Koopman approximation to learn nonlinear dynamics and employs conformal prediction to quantify uncertainty, providing statistical guarantees on approximation errors. This uncertainty is effectively incorporated into a Model Predictive Controller (MPC) formulation through constraint tightening, ensuring robust safety guarantees. We implement a layered control architecture with a reference generator providing waypoints for safe navigation. The effectiveness of our methods is validated in simulation.

Safe Navigation in Dynamic Environments Using Data-Driven Koopman Operators and Conformal Prediction

TL;DR

The paper tackles safe navigation for systems with nonlinear dynamics and state uncertainty by marrying data-driven Koopman lifting with conformal prediction to yield probabilistic safety guarantees within a linear MPC framework. Offline learning of a Koopman-based lifted model paired with a neural encoder enables linear dynamics in the lifted space, while conformal prediction provides quantifiable error bounds that tighten MPC constraints. A reference generator guides the MPC, and slack-based constraint handling preserves feasibility, resulting in a computationally efficient, safe navigation strategy validated by simulations on a unicycle model with dynamic obstacles. This approach offers a practical pathway to robust, real-time safe navigation in dynamic environments with uncertain models.

Abstract

We propose a novel framework for safe navigation in dynamic environments by integrating Koopman operator theory with conformal prediction. Our approach leverages data-driven Koopman approximation to learn nonlinear dynamics and employs conformal prediction to quantify uncertainty, providing statistical guarantees on approximation errors. This uncertainty is effectively incorporated into a Model Predictive Controller (MPC) formulation through constraint tightening, ensuring robust safety guarantees. We implement a layered control architecture with a reference generator providing waypoints for safe navigation. The effectiveness of our methods is validated in simulation.

Paper Structure

This paper contains 11 sections, 2 theorems, 17 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Dynamical System
  4. Koopman Operator
  5. Conformal Prediction
  6. Problem Formulation
  7. Solution
  8. Uncertainty quantification for Koopman Operator
  9. Model Predictive Controller
  10. Experiment
  11. Conclusion

Key Result

Proposition 1

Consider a Koopman dynamics in lifted space in eq:lifted_system, for any control $u\in\mathcal{U}$. Let $\{(x_i, \hat{x}_i)\}_{i=1}^{n}$ be a calibration dataset where $x_i$ represents the true state and $\hat{x}_i$ the state estimated using the Koopman model with stochastic sensor measurements. Ass where $Q_{1-\alpha} = \text{Quantile}_{1-\alpha}\{s_1, \ldots, s_n, \infty\}$ is the $(1-\alpha)$-t

Figures (4)

  • Figure 1: Half Space Constraints
  • Figure 2: Navigation with Dynamic Environment: Agent's trajectory is shown in orange line, where it goes through a sequence of designated target points (marked as green dots). Dynamic obstacles are shown as blue shaded regions where the opacity represents their position over time. Darker blue indicates the obstacle's more recent position, while lighter blue shows where it was previously. Agent starts at location $(-2, -2)$ and ends at location $(2, 0)$.
  • Figure 3: Comparison between different confidence levels. Collision time intervals are highlighted. Lower confidence levels lead to increased risk, resulting more collisions. Any lines that dips into the grey area represent collisions.
  • Figure 4: Comparison between the proposed approach with reference generator and baseline using only soft constraints. Agent starts from $(-1, 0)$ and ends at $(0.5,0)$ (a) The trajectory comparison reveals that our agent follows a smooth, efficient path when guided by the reference generator, in contrast to the zigzag movement exhibited by the baseline method. (b) Distance to obstacles during navigation, where the shaded region indicates soft constraint activation without collision. Our method also outperforms the baseline in terms of task completion time.

Theorems & Definitions (4)

  • Proposition 1
  • proof
  • Theorem 1
  • proof