Robot Manipulability Research Based On Differential Geometry

Some problems about robot manipulability are researched based on modern differential geometry. The main frame of the dissertation is to map the robot kinematics and dynamics properties into geometry surface by moving frame method and Riemannian metrics, and the invariants of the surface, such as volume element, Riemannian curvature and geodesics, are studied then the robot manipulability and optimal trajectory planning are studied based on these geometry invariants.At first, moving frame is established on the end of each linkage by moving frame methods; According to the geometry meanings of moving frame and its relative quantities, a new robot forward kinematics formula is given. And the moving frame on the end-effector is got by this new recursive formula. All the pose of moving frame on end-effector composes of a manifold surface. The concept of volume element is defined on this geometric surface using external differentiation and moving frame method. The volume element is an invariant and is regarded as the manipulability index. Volume element shows the end-effector's general motion ability of translation and rotation. The bigger the volume element is, the better the robot general manipulability is. The optimal pose is determined because of volume element is a function related to joint angle. Then a synthesis index is obtained by integrating the volume element. When the volume element becomes zero, the robot is in the singular pose. The traditional method about singularity is to let the determinant of Jacobi matrix be zero. Volume element has a very simple formulation and calculation when defined by external differentiation. The moving frame method and external differentiation are combined in our work and solve the problem that external differentiation is difficult to be applied to robot manipulability.A new method of optimal trajectory planning is introduced in this paper. The square of arc length and system kinetics are regarded as Riemannian metrics respectively, then the robot kinematics and dynamics are mapped onto different Riemannian surfaces. The geodesic on this surface is invariant and it is the necessary condition of the shortest distance between two points on this surface. The geodesic on the kinematics surface corresponds to the optimal distance trajectory; The geodesic on the dynamic surface corresponds to the optimal kinetics trajectory. Through solving differential equations of geodesics on these surfaces, the optimal kinematics and dynamics trajectories are obtained. This geodesic based method use the couple relations between joints to get the optimal trajectory directly. At last, several trajectory planning examples of 2-dof and 3-dof are given and the simulation results verified the trajectory planning method based on geodesies.In a word, robot kinematics and dynamics are mapped onto the corresponding geometric surfaces by moving frame method and Riemannian metrics. Combined with the geometric properties of the invariants on the surface, such as geodesies, volume element and Riemannian curvature, external differentiation is applied to the manipulability analysis. An optimal trajectory planning method is introduced based on geodesies. The arc length of robot trajectory and the kinetic energy of the robotsystem are represented by the arc length of geodesics and the optimal trajectory is attained directly by solving geodesic differential equation.