# Saltation Matrices: The Essential Tool for Linearizing Hybrid Dynamical Systems

arXiv: https://arxiv.org/abs/2306.06862 · HTML: https://mlsanity.com/paper/2306.06862/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Jun 2023.

## Built on by (most cited first, 1 of 1)

- [Differentiable Parameter Optimization for DAEs with State-Dependent Events](https://mlsanity.com/paper/2605.05395/) (2026)

## Abstract (arXiv)

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.