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.
Constructive Observer Design for Visual Simultaneous Localisation and Mapping
Australian Centre for Robotic Vision（澳大利亚机器人视觉中心）；Australian National University（澳大利亚国立大学）；I3S (University Côte d’Azur, CNRS, Sophia Antipolis)（I3S（蔚蓝海岸大学，法国国家科学研究中心，索菲亚安提波利斯））；Insitut Universitaire de France（法国大学学院）
Systems Theory and Robotics Group（系统理论与机器人组）；Australian Centre for Robotic Vision（澳大利亚机器人视觉中心）；Australian National University（澳大利亚国立大学）；I3S (University Côte d’Azur, CNRS, Sophia Antipolis)（I3S（蔚蓝海岸大学、法国国家科学研究中心、索菲亚安提波利斯））；Insitut Universitaire de France（法兰西学院）
Task-Space Consensus of Networked Robotic Systems: Separation and Manipulability
Beijing Institute of Control Engineering（北京控制工程研究所）；Science and Technology on Space Intelligent Control Laboratory, Beijing Institute of Control Engineering（空间智能控制重点实验室，北京控制工程研究所）
A General Nonlinear Observer Design for Inertial Navigation Systems with Almost Global Stability Guarantees
Capgemini Engineering（凯捷工程）；University of Quebec in Outaouais（渥太华大学）；I3S-UniCA-CNRS, University Cote d’Azur（尼斯大学）；Institut Universitaire de France（法国高等研究院）
Equivariant Symmetries for Inertial Navigation Systems
Control of Networked Systems Group, University of Klagenfurt, Austria（网络化系统控制组，克雷格弗尔特大学）；Robotics Lab, Australian Centre for Robotic Vision, Australian National University, Australia（机器人实验室，澳大利亚机器人视觉中心，澳大利亚国立大学）
