Saltation Matrices: The Essential Tool for Linearizing Hybrid Dynamical Systems

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.

I. Introduction

Many interesting problems in engineering can be modeled as hybrid dynamical systems, meaning that they involve both continuous and discrete evolution in state [1], [2], [3], [4]. These systems can be hybrid, e.g., due to physical contact, a result of digital logic circuits or they can be triggered by control—reacting to sensor feedback or switching control modes. Meanwhile, most of the tools that exist for planning, estimation, control, and learning assume continuous (if not smooth) systems. A common strategy to adapt tools that were designed for smooth systems to hybrid systems is to minimize the effect of discontinuities [5], [6], e.g., by slowing down to near zero velocity at the time of an impact event [7]. However, these strategies do not make use of the underlying dynamics of the system and only seek to mitigate them. This may work out for certain fully actuated systems, but many hybrid systems of interest are underactuated and cannot always cancel out the discontinuous dynamics.