基于微分流形的机器人机构操作性能的研究

The approach of the modern differential geometry is applicated to research the robotic mechanism. The Frenet frame of robotic mechanism with plane or spatial 2(RR or RP), 3(RRR, RRP) even 4-6 degree of freedom are set up and the mathematical models of the geometry, kinematics and dynamics are derived. And then, the translation volume, the rotation volume, the dexterity and the geometric properties of the workspace surface and the singularity surfaceas well as the normal curvature and the Gauss curvature etc, are investigated, which lead to the analysis approach for robotic mechanism with difference combination of the joints is obtained. Meanwhile, the arc length of track and the kinetic energy of system are taken as Riemannian metric and the corresponding surface is obtained under Riemannian metrics. As a result, the optimal path of the robotic mechanism can be converted into the properties researching of geodesic on the Riemannian surface by solving a group of the differential equations under the differential Riemannian metric. For convinience to application, the group of differential equations may be simplified and be partly linear, the mathematical model of optimal control with single and two degree of freedom are given. Therefore, the geometry significant can be intuitively expressed by animation and experiment, and a new approach to measure the optimal control of robotic mechanism will be established.

应用现代微分几何学的思想与方法研究机器人机构多参数图形的不变量及其性质，在微分流形上讨论机器人机构的几何学、运动学与动力学性质以及运动规划问题，研究黎曼度量方式与评价机器人机构操作及运动规划路径的性能指标，探索合理利用多变量系统的耦合效应的途径，建立机器人机机构运动微分几何学理论及操作臂操作性能评价的新理论与新方法。