为解决三关节空间机械臂能耗最小路径规划问题, 给出了空间机械臂能耗函数, 将其视为各关节角函数的泛函, 从而将能耗最小路径规划问题转化成了固定边界的泛函求极值问题。利用变分法求出了能耗最小路径的微分方程组。利用上三角矩阵逆矩阵的表示式, 将微分方程组转化成以标准状态方程的形式表示的微分方程组。与微分几何的方法相比, 这种方法避免了逆矩阵的运算, 使转化过程的运算更加简单。针对一组三关节机械臂参数, 利用 Matlab进行了仿真, 求出了以标准状态方程的形式表示的微分方程组初值问题的解。利用随机搜索方法通过初值问题的解得到了边值问题的解, 实现了能耗最小路径规划, 并证实了变分法的可行性。

In order to solve the problem of space manipulator path planning with minimum energy consumption, the energy function of space manipulator is derived first, and then the energy function is regarded as the functional of 3 joint angle functions. The problem of space manipulator path planning with minimum energy consumption is translated into a problem of the functional extreme with fixed boundary. A system of equations for expressing the path with minimum energy consumption is derived from resolving the problem of the functional extreme by using variational method. By means of the inverse matrix expression of the upper triangular matrix, the system of equations is translated into a new one consisted of normal state equations. The method can avoid complex inverse matrix computing, and make the computation of the transition process simpler when compared with differential geometry method. For a group of parameters of the 3R manipulator, the initial value problem of the new system of equations is solved by using Matlab simulation method. The solution of the boundary value problem of the new system of equations, representing the path with minimum energy consumption, is obtained from the solutions of initial value problem of the same system of equations by using random search method, and the feasibility of variational method for solving this kind of problem is also validated.