Robot Optimal Trajectory Planning Based on Geodesics

Geometric characteristics of geodesics in the Riemannian surface are used to make robotic optimal trajectory planning in this paper. Distance length and kinetic energy are regarded as Riemannian metrics respectively, and the Riemannian surfaces are determined by the corresponding metrics, and they represent the robotics kinematics and dynamics respectively. The geodesies on the Riemannian surface are calculated and are regarded as the optimal trajectory. Geodesic is the necessary condition of the shortest length between two points on the Riemannian surface and the covariant derivative of the geodesic's tangent vector is zero. When to implement optimal trajectory planning with arc length as the Riemannian metric, geodesic makes the shortest length between two points. The end-effector's velocity is invariant along the geodesic and the acceleration is zero. So the motion is very smooth. When system's kinetic energy as the Riemannian metric, the geodesic between two points on the kinetic surface makes the kinetic energy remain invariant. Computer calculation and simulation verify that the method based on geodesic is good at trajectory planning especially when the trajectory is linear or certain index should be minimized.