In this paper, we present a method for finding approximate Nash equilibria in a broad class of reachability games. These games are often used to formulate both collision avoidance and goal satisfaction. Our method is computationally efficient, running in real-time for scenarios involving multiple players and more than ten state dimensions. The proposed approach forms a family of increasingly exact approximations to the original game. Our results characterize the quality of these approximations and show operation in a receding horizon, minimally-invasive control context. Additionally, as a special case, our method reduces to local gradient-based optimization in the single-player (optimal control) setting, for which a wide variety of efficient algorithms exist.
Game-Theoretic Safe Multi-Agent Motion Planning with Reachability Analysis for Dynamic and Uncertain Environments (Extended Version)
Department of Electrical and Computer Engineering, National University of Singapore（国立新加坡大学电气与计算机工程系）；Department of Control Science and Engineering, Shanghai Institute of Intelligent Science and Technology（上海智能科学与技术研究院控制科学与工程系）
Reachability analysis is at the core of many applications, from neural network verification, to safe trajectory planning of uncertain systems. However, this problem is notoriously challenging, and current approaches tend to be either too restrictive, too slow, too conservative, or approximate and therefore lack guarantees. In this paper, we propose a simple yet effective sampling-based approach to perform reachability analysis for arbitrary dynamical systems. Our key novel idea consists of using random set theory to give a rigorous interpretation of our method, and prove that it returns sets w…
This study focuses on reachability problems in differential games. An improved level set method for computing reachable tubes is proposed in this paper. The reachable tube is described as a sublevel set of a value function, which is the viscosity solution of a Hamilton-Jacobi equation with running cost. We generalize the concept of reachable tubes and propose a new class of reachable tubes, which are referred to as cost-limited one. In particular, a performance index can be specified for the system, and a cost-limited reachable tube is a set of initial states of the system's trajectories that …
We present a method to compute the stochastic reachability safety probabilities for high-dimensional stochastic dynamical systems. Our approach takes advantage of a nonparametric learning technique known as conditional distribution embeddings to model the stochastic kernel using a data-driven approach. By embedding the dynamics and uncertainty within a reproducing kernel Hilbert space, it becomes possible to compute the safety probabilities for stochastic reachability problems as simple matrix operations and inner products. We employ a convergent approximation technique, random Fourier feature…
In this work, we perform safety analysis of linear dynamical systems with uncertainties. Instead of computing a conservative overapproximation of the reachable set, our approach involves computing a statistical approximate reachable set. As a result, the guarantees provided by our method are probabilistic in nature. In this paper, we provide two different techniques to compute statistical approximate reachable set. We have implemented our algorithms in a python based prototype and demonstrate the applicability of our approaches on various case studies. We also provide an empirical comparison b…
This paper aims to synthesize a reachability controller for an unknown dynamical system. We first learn the unknown system using Gaussian processes and the (probabilistic) guarantee on the learned model. Then we use the funnel-based controller synthesis approach using this approximated dynamical system to design the controller for a reachability specification. Finally, the merits of the proposed method are shown using a numerical example.
We study a class of two-player competitive concurrent stochastic games on graphs with reachability objectives. Specifically, player 1 aims to reach a subset $F_1$ of game states, and player 2 aims to reach a subset $F_2$ of game states where $F_2\cap F_1=\emptyset$. Both players aim to satisfy their reachability objectives before their opponent does. Yet, the information players have about the game dynamics is asymmetric: P1 has a (set of) hidden actions unknown to P2 at the beginning of their interaction. In this setup, we investigate P1's strategic planning of action deception that decides w…
Carnegie Mellon University（卡内基梅隆大学）；Bosch Research Pittsburgh（博世研究院匹兹堡）；University of California, San Diego（加州大学圣地亚哥分校）
Racing demands each vehicle to drive at its physical limits, when any safety infraction could lead to catastrophic failure. In this work, we study the problem of safe reinforcement learning (RL) for autonomous racing, using the vehicle's ego-camera view and speed as input. Given the nature of the task, autonomous agents need to be able to 1) identify and avoid unsafe scenarios under the complex vehicle dynamics, and 2) make sub-second decision in a fast-changing environment. To satisfy these criteria, we propose to incorporate Hamilton-Jacobi (HJ) reachability theory, a safety verification met…
We propose a new approach to compute an interval over-approximation of the finite time reachable set for a large class of nonlinear systems. This approach relies on the notions of sensitivity matrices, which are the partial derivatives representing the variations of the system trajectories in response to variations of the initial states. Using interval arithmetics, we first over-approximate the possible values of the second-order sensitivity at the final time of the reachability problem. Then we exploit these bounds and the evaluation of the first-order sensitivity matrices at a few sampled in…
TTT: A Temporal Refinement Heuristic for Tenuously Tractable Discrete Time Reachability Problems
Division of Robotics, Perception and Learning, Intelligent Systems Department, School of Electrical Engineering & Computer Science（机器人、感知与学习 division，智能系统部门，电气工程与计算机科学学院）；Digital Futures at KTH Royal Institute of Technology（KTH皇家理工学院数字未来）
We show that, for any spatially discretized system of reaction-diffusion, the approximate solution given by the explicit Euler time-discretization scheme converges to the exact time-continuous solution, provided that diffusion coefficient be sufficiently large. By "sufficiently large", we mean that the diffusion coefficient value makes the one-sided Lipschitz constant of the reaction-diffusion system negative. We apply this result to solve a finite horizon control problem for a 1D reaction-diffusion example. We also explain how to perform model reduction in order to improve the efficiency of t…
