A novel manipulator trajectory planning approach using geodesic is proposed in this paper. Geodesic is the necessary condition of the shortest length between two points on the Riemannian surface in which the covariant derivative of the geodesic’s tangent vector is zero. The geometric characteristic of geodesic is discussed and used to implement trajectory planning of the manipulator. First, the Riemannian metric is constructed according to the planning task, e.g. to establish a distance metric by arc length of the trajectory to achieve shortest path. Once the Riemannian metric is obtained, the corresponding Riemannian surface is solely determined. Then the geodesic equations on this surface can be attained. For the given initial conditions of the trajectory, the geodesic equations can be solved and the results are the optimal trajectory of the manipulator in the joint space for the given metric. The planned trajectories in the joint space can also be mapped into the workspace. To demonstrate the effectiveness of the proposed approach, simulation experiments are conducted using some typical manipulators, e.g. spatial 3R, 3-PSS parallel and planar 3R manipulators.
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