Reachability Analysis for Black-Box Dynamical Systems

TL;DR

This work proposes a novel reachability method to compute reachable sets and safe controllers for black-box dynamical systems and demonstrates the effectiveness of this approach in accurately obtaining the reachable sets for blackbox dynamical systems.

Abstract

Hamilton-Jacobi (HJ) reachability analysis is a powerful framework for ensuring safety and performance in autonomous systems. However, existing methods typically rely on a white-box dynamics model of the system, limiting their applicability in many practical robotics scenarios where only a black-box model of the system is available. In this work, we propose a novel reachability method to compute reachable sets and safe controllers for black-box dynamical systems. Our approach efficiently approximates the Hamiltonian function using samples from the black-box dynamics. This Hamiltonian is then used to solve the HJ Partial Differential Equation (PDE), providing the reachable set of the system. The proposed method can be applied to general nonlinear systems and can be seamlessly integrated with existing reachability toolboxes for white-box systems to extend their use to black-box systems. Through simulation studies on a black-box slip-wheel car and a quadruped robot, we demonstrate the effectiveness of our approach in accurately obtaining the reachable sets for blackbox dynamical systems.

Figures (4)

• Figure 1: An overview of the proposed method: we use the samples from the black-box dynamics to construct a dataset where each sample consists of state $x_t$, the spatial derivative of value function $\nabla V$, and the corresponding Hamiltonian label $H^*$. This dataset is then used to train a neural network to estimate the Hamiltonian at any given state and $\nabla V$. The trained network can be integrated within existing reachability toolboxes to solve the Hamilton-Jacobi PDE to compute the value function and the BRT for black-box dynamical systems.
• Figure 2: Bicycle Robot: A slice of the value function for different methods at $t = 2$s. The zero level set (BRT) and the failure set are shown in black and green respectively.
• Figure 3: The figures illustrate value function slices (corresponding to initial states $(p_x,p_y,0,12,0,0)$) for the slip-wheel car system. The failure set and sub-$\delta$ level sets (BRTs), with $\epsilon=10^{-2}$ and $10^{-3}$, are also shown.
• Figure 4: (a) An overview of value function prediction: DeepReach takes in the full state $x_{full}$, internally compresses it using an encoder, and feeds the condensed state $x_c$ to its safety value NN. The Hamiltonian estimation NN also takes $x_c$ as input. (b) The BRTs verified with $\epsilon = 10^{-3}$ for different variants. Ham-NN consistently outperforms all other baselines. (c) A snapshot of the Issac Gym environment that is used to learn the BRT and for deploying the safety filter. (d) The safety-filtered trajectories for the quadruped example. The obstacle contours are shown in black dashed lines, and the star represents the goal position. Maroon line segments indicate the activation of the safety controller. Due to the over-conservatism of the MB approach, the robot takes an inefficient detour.

Theorems & Definitions (4)

• Remark 1
• Remark 2
• Remark 3
• Remark 4