# Equivariant Systems Theory and Observer Design for Second Order Kinematic Systems on Matrix Lie Groups

arXiv: https://arxiv.org/abs/2105.04797 · HTML: https://mlsanity.com/paper/2105.04797/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 May 2021.

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

- [Equivariant Filter for High Performance Image Tracking using an Event Camera](https://mlsanity.com/paper/2607.09103/) (2026)

## Abstract (arXiv)

This paper presents the equivariant systems theory and observer design for second order kinematic systems on matrix Lie groups. The state of a second order kinematic system on a matrix Lie group is naturally posed on the tangent bundle of the group with the inputs lying in the tangent of the tangent bundle known as the double tangent bundle. We provide a simple parameterization of both the tangent bundle state space and the input space (the fiber space of the double tangent bundle) and then introduce a semi-direct product group and group actions onto both the state and input spaces. We show that with the proposed group actions the second order kinematics are equivariant. An equivariant lift of the kinematics onto the symmetry group is defined and used to design a nonlinear observer on the lifted state space using nonlinear constructive design techniques. A simple hovercraft simulation verifies the performance of our observer.