  Many state estimation and control algorithms require knowledge of how
probability distributions propagate through dynamical systems. However, despite
hybrid dynamical systems becoming increasingly important in many fields, there
has been little work on utilizing the knowledge of how probability
distributions map through hybrid transitions. Here, we make use of a
propagation law that employs the saltation matrix (a first-order update to the
sensitivity equation) to create the Salted Kalman Filter (SKF), a natural
extension of the Kalman Filter and Extended Kalman Filter to hybrid dynamical
systems. Away from hybrid events, the SKF is a standard Kalman filter. When a
hybrid event occurs, the saltation matrix plays an analogous role as that of
the system dynamics, subsequently inducing a discrete modification to both the
prediction and update steps. The SKF outperforms a naive variational update -
the Jacobian of the reset map - by having a reduced mean squared error in state
estimation, especially immediately after a hybrid transition event. Compared a
hybrid particle filter, the particle filter outperforms the SKF in mean squared
error only when a large number of particles are used, likely due to a more
accurate accounting of the split distribution near a hybrid transition.
