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 paper 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.

翻译：混合动力系统，即同时包含连续状态和离散状态的系统，在工程领域普遍存在，但由于其不连续的过渡特性，处理起来较为困难。例如，机器人腿部在空中时几乎无法施加控制力，而在地面时则可施加较大控制力。当腿部接触地面时，穿透速度瞬间降至零。这些动力学的瞬时变化及状态的不连续（或跳跃）使得标准的平滑工具（如规划、估计、控制与学习）难以应用于混合系统。解决这些跳跃问题的关键工具之一便是Saltation矩阵。该矩阵是混合跳跃发生时灵敏度的更新方式，已被广泛应用于机器人学、电力电路和计算神经科学等多个领域。本文介绍了Saltation矩阵的直观推导过程，并探讨了其所涵盖的内容、在过去的应用场景、在线性和二次型中的使用方法、针对具有单边约束的刚体系统的计算方法，以及在这些情况下Saltation矩阵的一些结构特性。