The field of robotics and control is witnessing significant developments in the use of matrix Lie groups for localization, trajectory optimization, and control of rigid bodies and multi-agent systems. Researchers are exploring innovative approaches to address the challenges of consistent localization, pattern formation, and trajectory optimization in complex systems. A key direction is the integration of geometric and stochastic methods to improve the efficiency, reliability, and scalability of control systems. Notably, Riemannian optimization frameworks and intrinsic optimal control approaches are being applied to achieve correct-by-construction control strategies that respect the topological structure of the rotation group. Noteworthy papers include:

• Fault-Tolerant Multi-Modal Localization of Multi-Robots on Matrix Lie Groups, which proposes a novel framework for fault-tolerant localization of multi-robot systems.
• Riemannian Direct Trajectory Optimization of Rigid Bodies on Matrix Lie Groups, which introduces a Riemannian optimization framework for direct trajectory optimization of rigid bodies.
• Geometric Fault-Tolerant Neural Network Tracking Control of Unknown Systems on Matrix Lie Groups, which presents a geometric neural network-based tracking controller for systems evolving on matrix Lie groups.