We present a method for feedback motion planning of systems with unknown dynamics which provides probabilistic guarantees on safety, reachability, and goal stability. To find a domain in which a learned control-affine approximation of the true dynamics can be trusted, we estimate the Lipschitz constant of the difference between the true and learned dynamics, and ensure the estimate is valid with a given probability. Provided the system has at least as many controls as states, we also derive existence conditions for a one-step feedback law which can keep the real system within a small bound of a nominal trajectory planned with the learned dynamics. Our method imposes the feedback law existence as a constraint in a sampling-based planner, which returns a feedback policy around a nominal plan ensuring that, if the Lipschitz constant estimate is valid, the true system is safe during plan execution, reaches the goal, and is ultimately invariant in a small set about the goal. We demonstrate our approach by planning using learned models of a 6D quadrotor and a 7DOF Kuka arm. We show that a baseline which plans using the same learned dynamics without considering the error bound or the existence of the feedback law can fail to stabilize around the plan and become unsafe.

中文翻译：

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利用学习动态进行规划：通过 Lipschitz 常数对安全性和可达性进行概率保证


我们提出了一种用于未知动力学系统的反馈运动规划的方法，该方法为安全性、可达性和目标稳定性提供了概率保证。为了找到一个可以信任真实动态的学习控制仿射近似的域，我们估计真实动态和学习动态之间差异的 Lipschitz 常数，并确保估计在给定概率下有效。假设系统至少具有与状态一样多的控制，我们还推导出单步反馈定律的存在条件，该定律可以将真实系统保持在用学习动态规划的标称轨迹的小范围内。我们的方法将反馈律的存在作为基于采样的规划器的约束，该规划器返回围绕名义计划的反馈策略，确保如果 Lipschitz 常数估计有效，则真实系统在计划执行期间是安全的，达到目标，并且在关于目标的小集合中最终是不变的。我们通过使用 6D 四旋翼飞行器和 7DOF Kuka 臂的学习模型进行规划来演示我们的方法。我们表明，在不考虑误差界限或反馈定律的存在的情况下使用相同的学习动态进行计划的基线可能无法在计划周围稳定并变得不安全。