Saltation Matrices: The Essential Tool for Linearizing Hybrid Dynamical Systems

TL;DR

Saltation matrices provide the key to first-order linearization of hybrid dynamical systems at events by capturing both timing shift and reset effects. The paper presents two derivations of the saltation matrix, demonstrates its use for linear and quadratic form propagations, and applies it to simple and rigid-body contact models to reveal structure and eigenproperties. The survey of applications links saltation matrices to stability analysis (monodromy, Poincaré maps, Floquet theory), covariance propagation (Kalman filtering) and control design (LQR, Riccati updates), highlighting practical use in robotics and estimation. The work enables engineers to incorporate hybrid transitions into planning, estimation, and control with clarified formulas and concrete examples for both toy and complex rigid-body systems.

Abstract

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.

Figures (8)

• Figure 1: Example drop on a slanted surface with initial covariance. The saltation matrix ($\Xi$) correctly estimates the end distribution's covariance where covariance in the direction of the constraint is eliminated. Using the incorrect update, only the Jacobian of the reset map ($\textbf{$\mathrm{D}$}_x R$) leads to retaining belief in the direction of the constraint.
• Figure 2: The saltation matrix has been used in many different fields, including the control of legged robots in tasks such as (a) a quadrupedal backflip paper:kong-hybrid-2022 and (b) robust bipdeal walking tucker2022bipedsaltation; the analysis and control of power circuits such as (c) supervising control of a buck converter Giaouris2008Stability and (d) the bifurcation behavior of DC drives okafor2010analysis; and the modelling of neural activity in the brain such as (e) the stability analysis of a Wilson-Cowan neural mass modelcoombes2018networks and (f) the modelling of synaptic filter behaviorlai2018analysis.
• Figure 3: An example 2 mode hybrid system where the domains are shown in black circles $D$, the dynamics are shown with gray arrows $F$, the guard for the current domain is shown in red dotted$g$, and the reset from the current mode to the next mode is shown in dashed lines$R$.
• Figure 4: Linearizations made about the nominal trajectory shown in black where a perturbation is shown in green and the perturbed trajectory is shown in blue. At $a)$ describes $\vec{v} = F_{\mathrm{I}}^- \delta t + \delta x(t^-)$. At $b)$$\delta x (\widetilde{t}^+)$ is $\textbf{$\mathrm{D}$}_x R^- \vec{v} - F_{\mathrm{J}}^+ \delta t$. At $c)$ the guard condition is $0 = \textbf{$\mathrm{D}$}_x g^-(\delta x (t^-) + F_{\mathrm{I}}^- \delta t)$. Here $\delta t$ is positive (late transition) and for the purposes of this figure it is assumed that the system is autonomous, so the $\textbf{$\mathrm{D}$}_t g$ and $\textbf{$\mathrm{D}$}_tR$ terms drop out.
• Figure 5: Constant flow hybrid system with identity reset map. The Jacobian of the reset map $\textbf{$\mathrm{D}$}_xR$ predicts no variational changes whereas using the saltation matrix $\Xi$ predicts the correct variational changes.

Theorems & Definitions (2)

• Definition 1
• Definition 2