This paper investigates performance guarantees on coverage-based ergodic exploration methods in environments containing disturbances. Ergodic exploration methods generate trajectories for autonomous robots such that time spent in each area of the exploration space is proportional to the utility of exploring in the area. We find that it is possible to use techniques from reachability analysis to solve for optimal controllers that guarantee ergodic coverage and are robust against disturbances. We formulate ergodic search as a differential game between the controller optimizing for ergodicity and an external disturbance, and we derive the reachability equations for ergodic search using an extended-state Bolza-form transform of the ergodic problem. Contributions include the computation of a continuous value function for the ergodic exploration problem and the derivation of a controller that provides guarantees for coverage under disturbances. Our approach leverages neural-network-based methods to solve the reachability equations; we also construct a robust model-predictive controller for comparison. Simulated and experimental results demonstrate the efficacy of our approach for generating robust ergodic trajectories for search and exploration on a 1D system with an external disturbance force.

翻译：本文研究了存在扰动的环境中基于覆盖率的遍历探索方法的性能保证。遍历探索方法为自主机器人生成轨迹，使得在探索空间各区域停留的时间与在该区域探索的效用成正比。我们发现可以利用可达性分析技术来求解保证遍历覆盖且对扰动具有鲁棒性的最优控制器。我们将遍历搜索建模为优化遍历性的控制器与外部扰动之间的微分博弈，并通过遍历问题的扩展状态Bolza形式变换推导出遍历搜索的可达性方程。主要贡献包括：计算遍历探索问题的连续值函数，以及推导出在扰动下提供覆盖保证的控制器。我们的方法利用基于神经网络的技术求解可达性方程；同时构建了鲁棒模型预测控制器进行比较。仿真和实验结果表明，我们的方法在存在外部扰动力的1D系统中，能够有效生成用于搜索与探索的鲁棒遍历轨迹。