Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fine-Tuned Convex Approximations of Probabilistic Reachable Sets under Data-driven Uncertainties

Pengcheng Wu, Sonia Martinez, Jun Chen

TL;DR

This work develops a data-driven framework to compute convex, $n$-sided polygon approximations of probabilistic reachable sets (PRS) for uncertain discrete-time dynamics with unbounded disturbances. It combines FFT-accelerated kernel density estimation to model unknown uncertainty and obtain a PRS, with a mixed-integer nonlinear programming (MINLP) formulation that yields a tight convex polygon whose interior mass meets a prescribed confidence level. A two-track solution approach is proposed: an exact MINLP optimal solver and a faster MINLP heuristic using weighted sampling of grid points to enable real-time applicability. Case studies demonstrate that the convex approximations are significantly less conservative than bounding boxes, while achieving near-optimal accuracy and substantial computational efficiency, enabling safer real-time motion planning under data-driven uncertainties.

Abstract

This paper proposes a mechanism to fine-tune convex approximations of probabilistic reachable sets (PRS) of uncertain dynamic systems. We consider the case of unbounded uncertainties, for which it may be impossible to find a bounded reachable set of the system. Instead, we turn to find a PRS that bounds system states with high confidence. Our data-driven approach builds on a kernel density estimator (KDE) accelerated by a fast Fourier transform (FFT), which is customized to model the uncertainties and obtain the PRS efficiently. However, the non-convex shape of the PRS can make it impractical for subsequent optimal designs. Motivated by this, we formulate a mixed integer nonlinear programming (MINLP) problem whose solution result is an optimal $n$ sided convex polygon that approximates the PRS. Leveraging this formulation, we propose a heuristic algorithm to find this convex set efficiently while ensuring accuracy. The algorithm is tested on comprehensive case studies that demonstrate its near-optimality, accuracy, efficiency, and robustness. The benefits of this work pave the way for promising applications to safety-critical, real-time motion planning of uncertain dynamic systems.

Fine-Tuned Convex Approximations of Probabilistic Reachable Sets under Data-driven Uncertainties

TL;DR

This work develops a data-driven framework to compute convex, -sided polygon approximations of probabilistic reachable sets (PRS) for uncertain discrete-time dynamics with unbounded disturbances. It combines FFT-accelerated kernel density estimation to model unknown uncertainty and obtain a PRS, with a mixed-integer nonlinear programming (MINLP) formulation that yields a tight convex polygon whose interior mass meets a prescribed confidence level. A two-track solution approach is proposed: an exact MINLP optimal solver and a faster MINLP heuristic using weighted sampling of grid points to enable real-time applicability. Case studies demonstrate that the convex approximations are significantly less conservative than bounding boxes, while achieving near-optimal accuracy and substantial computational efficiency, enabling safer real-time motion planning under data-driven uncertainties.

Abstract

This paper proposes a mechanism to fine-tune convex approximations of probabilistic reachable sets (PRS) of uncertain dynamic systems. We consider the case of unbounded uncertainties, for which it may be impossible to find a bounded reachable set of the system. Instead, we turn to find a PRS that bounds system states with high confidence. Our data-driven approach builds on a kernel density estimator (KDE) accelerated by a fast Fourier transform (FFT), which is customized to model the uncertainties and obtain the PRS efficiently. However, the non-convex shape of the PRS can make it impractical for subsequent optimal designs. Motivated by this, we formulate a mixed integer nonlinear programming (MINLP) problem whose solution result is an optimal sided convex polygon that approximates the PRS. Leveraging this formulation, we propose a heuristic algorithm to find this convex set efficiently while ensuring accuracy. The algorithm is tested on comprehensive case studies that demonstrate its near-optimality, accuracy, efficiency, and robustness. The benefits of this work pave the way for promising applications to safety-critical, real-time motion planning of uncertain dynamic systems.
Paper Structure (29 sections, 37 equations, 18 figures, 3 tables, 3 algorithms)

This paper contains 29 sections, 37 equations, 18 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Contributions and Organization
  4. Problem Statement
  5. PRS Identification
  6. FFT-Based KDE
  7. Online Computation of PRS
  8. MINLP Formulation for PRS Approximation
  9. Problem Description and Objective Function Formulation
  10. Constraints on Continuous Variables
  11. Constraints Relating Continuous and Discrete Variables
  12. Constraints on Discrete Variables
  13. Formulation of MINLP Optimization Framework
  14. Solution Method
  15. MINLP Optimal Algorithm
  16. ...and 14 more sections

Figures (18)

  • Figure 1: Graphical representation of $2$-variate linear binning
  • Figure 2: Evaluation time of Algorithm 1 with respect to the increasing number of data samples
  • Figure 3: Three cases of paths that are not enclosing
  • Figure 4: A graph that is not enclosing
  • Figure 5: Two cases of degenerate paths
  • ...and 13 more figures

Theorems & Definitions (3)

  • proof
  • proof
  • proof