Hybrid dynamical systems, i.e., systems that have both continuous and discrete states, are ubiquitous in engineering but are difficult to work with due to their discontinuous transitions. For example, a robot leg is able to exert very little control effort, while it is in the air compared to when it is on the ground. When the leg hits the ground, the penetrating velocity instantaneously collapses to zero. These instantaneous changes in dynamics and discontinuities (or jumps) in state make standard smooth tools for planning, estimation, control, and learning difficult for hybrid systems. One of the key tools for accounting for these jumps is called the saltation matrix. The saltation matrix is the sensitivity update when a hybrid jump occurs and has been used in a variety of fields, including robotics, power circuits, and computational neuroscience. This article presents an intuitive derivation of the saltation matrix and discusses what it captures, where it has been used in the past, how it is used for linear and quadratic forms, how it is computed for rigid body systems with unilateral constraints, and some of the structural properties of the saltation matrix in these cases.

中文翻译：

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跃迁矩阵：线性化混合动力系统的基本工具


混合动力系统，即同时具有连续状态和离散状态的系统，在工程中普遍存在，但由于其不连续的转变而难以处理。例如，与在地面上相比，机器人腿在空中时能够施加非常小的控制力。当腿落地时，穿透速度瞬间为零。这些动态的瞬时变化和状态的不连续性（或跳跃）使得用于规划、估计、控制和学习的标准平滑工具对于混合系统来说变得困难。解释这些跳跃的关键工具之一称为跳跃矩阵。跳跃矩阵是混合跳跃发生时的灵敏度更新，已被用于多种领域，包括机器人、电力电路和计算神经科学。本文介绍了跳动矩阵的直观推导，并讨论了它捕获的内容、过去的用途、如何将其用于线性和二次形式、如何计算具有单边约束的刚体系统，以及一些这些情况下跃迁矩阵的结构特性。