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Uniform Sampling from the Reachable Set Using Optimal Transport

Karthik Elamvazhuthi, Sachin Shivakumar

TL;DR

This paper reframes reachability sampling under bounded controls as an Optimal Transport problem, aiming to push the joint distribution on initial conditions and controls so that the terminal-state distribution matches a uniform target over the reachable set. It introduces a tractable, particle-based approach by relaxing the hard transport constraint with an $L_2$-based penalty and mollifying it to enable kernel-coupled terminal costs, then demonstrates convergence via Gamma-convergence across successive relaxations. Theoretical results show existence of minimizers and convergence of the relaxed and finite-dimensional problems to the original formulation. Numerical experiments on systems with strong attractors, such as the Vander Pol oscillator and a three-arm planar robot, show significantly improved uniformity of reachable-set sampling compared to random-control strategies, underscoring the method’s potential for robust safety verification and planning.

Abstract

Estimating the reachable set of a dynamical system is a fundamental problem in control theory, particularly when control inputs are bounded. Direct simulation using randomly sampled admissible controls often leads to trajectories that cluster near attractors, resulting in poor coverage of the reachable set. To achieve a more uniform distribution of terminal states, we formulate the problem within an Optimal Transport (OT) framework. In this setting, the goal is to steer the system so that the final state distribution, determined by the chosen controls and initial conditions, matches a desired target distribution. Enforcing this condition exactly is not possible since the reachable set is not known. So we introduce an $L_2$-norm based regularization of the terminal distribution that relaxes the constraint while promoting uniform coverage. The resulting formulation can be approximated by a finite-dimensional, particle-based optimal control problem with kernel-coupled terminal cost. We show that this approach converges to the original formulation and demonstrate through numerical examples that it provides significantly more uniform reachable-set sampling than random control strategies.

Uniform Sampling from the Reachable Set Using Optimal Transport

TL;DR

This paper reframes reachability sampling under bounded controls as an Optimal Transport problem, aiming to push the joint distribution on initial conditions and controls so that the terminal-state distribution matches a uniform target over the reachable set. It introduces a tractable, particle-based approach by relaxing the hard transport constraint with an -based penalty and mollifying it to enable kernel-coupled terminal costs, then demonstrates convergence via Gamma-convergence across successive relaxations. Theoretical results show existence of minimizers and convergence of the relaxed and finite-dimensional problems to the original formulation. Numerical experiments on systems with strong attractors, such as the Vander Pol oscillator and a three-arm planar robot, show significantly improved uniformity of reachable-set sampling compared to random-control strategies, underscoring the method’s potential for robust safety verification and planning.

Abstract

Estimating the reachable set of a dynamical system is a fundamental problem in control theory, particularly when control inputs are bounded. Direct simulation using randomly sampled admissible controls often leads to trajectories that cluster near attractors, resulting in poor coverage of the reachable set. To achieve a more uniform distribution of terminal states, we formulate the problem within an Optimal Transport (OT) framework. In this setting, the goal is to steer the system so that the final state distribution, determined by the chosen controls and initial conditions, matches a desired target distribution. Enforcing this condition exactly is not possible since the reachable set is not known. So we introduce an -norm based regularization of the terminal distribution that relaxes the constraint while promoting uniform coverage. The resulting formulation can be approximated by a finite-dimensional, particle-based optimal control problem with kernel-coupled terminal cost. We show that this approach converges to the original formulation and demonstrate through numerical examples that it provides significantly more uniform reachable-set sampling than random control strategies.

Paper Structure

This paper contains 14 sections, 4 theorems, 37 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation and Solution
  3. Main Results
  4. Numerical Examples
  5. Vander Pol Oscillator
  6. Three-arm Planar Robot
  7. Conclusion
  8. More Examples
  9. Duffing Oscillator
  10. Single-arm Robot
  11. Numerical Algorithm
  12. Connection to Classical Optimal Transport
  13. Proof of Continuity of the End-point Map
  14. A Qualitative Comparison

Key Result

Theorem III.1

Given Assumptions asmp1:sublin, asmp2:ctrl and asmp:fullleb, and assuming a weak topology on $\mathcal{U}$, the optimization problem eq:OTsamp2 is feasible. Moreover, an optimal solution exists for each of the optimization problems eq:OTsamp2, eq:OTsamp2ep, eq:OTsamp2epdel, and eq:OTsamp2epdelN.

Figures (8)

  • Figure 1: Final positions (at $t=15$, marked by '$\times$') of $N=100$ particles obeying Eq. \ref{['eq:vanderpol']} starting within $[-1,1]^2$ (marked by '$\bullet$'). States evolve under two control laws: (a) projected Brownian control $u(t)\in[-0.1,0.1]$, and (b), control obtained by solving the optimization problem in \ref{['eq:OTsamp2epdelN']}.
  • Figure 2: Final positions (at $t=20$, marked by '$\times$') of $N=600$ particles obeying Eq. \ref{['eq:3arm']} starting with joint angles and velocities within $[-0.1,0.1]$ (marked by '$\bullet$'). States evolve under two control laws: (a) projected Brownian control $u(t)\in[-0.2,0.2]$, and (b), control obtained by solving the optimization problem in \ref{['eq:OTsamp2epdelN']}.
  • Figure 3: Final positions (at $t=5$, marked by '$\times$') of $N=100$ particles obeying Eq. \ref{['eq:duffing']} starting at $[0,0]$ (marked by '$\bullet$'). States evolve under uniformly distributed stochastic control $u(t)\in[-1,1]$.
  • Figure 4: Final positions (at $t=5$, marked by '$\times$') of $N=100$ particles obeying Eq. \ref{['eq:duffing']} starting at $[0,0]$ (marked by '$\bullet$'). States evolve under the control obtained by solving the optimization problem in \ref{['eq:OTsamp2epdelN']}.
  • Figure 5: Final positions (at $t=15$, marked by '$\times$') of $N=100$ particles obeying Eq. \ref{['eq:nonlinearpendulum']} starting within $[-1,1]^2$ (marked by '$\bullet$'). States evolve under uniformly distributed stochastic control $u(t)\in[-1,1]$.
  • ...and 3 more figures

Theorems & Definitions (10)

  • Definition II.1
  • Theorem III.1
  • proof
  • Remark III.2
  • Proposition III.3
  • proof
  • Proposition III.4: Convergence of minimizers
  • proof
  • Theorem 1.1
  • proof