Differential Geometry in Robotics and Motion Planning

Abstract

When it comes to robotics and motion planning, where geometric structures are naturally used to simulate the configuration and movement of robots, differential geometry offers a strong mathematical foundation for analysis and problem solutions. Differential geometry allows for the exact characterisation of kinematics, dynamics, and constraints in joint space and task space by describing robot configurations as points on differentiable manifolds. how manifolds, tangent spaces, Riemannian metrics, and Lie groups—all notions from differential geometry—are used to design control and mobility plans for robots. Generating trajectories, optimizing motion pathways, and controlling nonholonomic systems like wheeled robots are some of the applications. In particular, we focus on the ways in which differential geometry equips algorithms for motion planning with means of dealing with topological restrictions, geodesics, and curvature. The capacity of geometric approaches to integrate local motion control with global planning is illustrated by case studies in robotic arms, mobile robots, and multi-agent systems. This work demonstrates the foundational role of differential geometry by connecting theoretical mathematics with real-world robotics applications.