Robot manipulability includes the geometry, kinematics and dynamics, trajectory planning and the mode of optima] control of framework, which is the base of the design and control of robot. Many specialists attach importance to it. In this dissertation, using techniques and notation which come from the theory of modern differential manifold, Riemannian geometry, Lie groups and Lie algebras, we have researched the some problems about robot manipulability and trajectory planning under the support by National Natural Science Foundation of China (No.59975013).In the first place, natural frames are established on the end of linkages; According to the transformation between the new natural frames, a new robot forward kinematics and back kinematics formulas are got, which is proved to be correct through the kinematics formulas of the XKH5140 and PUMA560. It describes the movement and configuration of system and avoids the singularity of robot, In the next place, on the base of the natural frame vector field, a mapping is established among the natural frame, Lie groups and joints of robot, which has the characteristic of robot and explicit signification of geometry and kinematics. The velocity and the acceleration have been studied by using the left invariant and right invariant of motion and covariant derivative. The recursive formulations and the Jacobian matrix of the joint space and the operational space have been gained and the singularity of robot has been analyzed through the Jacobian matrix in the meaning time.In succession, based on the recursive Newton-Euler equations and Lagrange equations, the general force and the general velocity are defined, and we utilize the theory of Lie groups and Lie algebras to derive the back-dynamics equations and Lagrange equations of robot in the joint space and operational space. The method makes it easy to compute and reduces the computational complexity. In the meaning time, the real-time control of robot comes true. Through using the inertial matrix, we analyse the manipulation of robot by the Riemannian curvature.In the end, the problem of robot trajectory planning is investigated by the linearization method and Riemannian metric. On the one hand, the optimal control system is linear programming by the Talyor equation and the polynomial. The nonlinear control system is transformed into the linear one. Several trajectory planning examples of 2-DOF and 3-DOF robot are given and the simulative results verified the trajectory planning and the couple among differential joints. On the other hand, the kinetics and dynamics of the system are regarded as Riemannian metrics, so the robot kinematics and dynamics are mapped into different Riemannian surfaces. The geodesic on the surfaces is invariant and it is necessary condition of the shortest distance between two points on the surface. The geodesic on the dynamic surface corresponds to the optimal kinetics trajectory. Through solving differential equations of geodesic on these surfaces, the optimal kinematics and dynamics trajectories are obtained. Through the couple relations between joints of the geodesic, the optimal trajectory is got directly. At last, several trajectory planning examples are given and the simulation results verified the trajectory planning method successfully based on geodesic.In a word, robot kinematics, robot dynamics and robot trajectory planning are studied respectively by differential geometry, Riemannian geometry and Lie groups and Lie algebras. Some problems about robot manipulability are solved effectively. |