中文摘要：

研究了在两固定端点之间运动的三连杆平面欠驱动机械臂的最优运动规划问题．该机械臂有两个受驱动的移动关节，而第三个关节是一个既无驱动器又无制动器的被动关节．施加在自由运动连杆上的动力学约束是一个二阶非完整约束．通过坐标和输入变换，系统的状态和输入变量被表示为子状态变量的高阶导数形式；通过这些新的子状态变量，将一个受约束的动力学优化问题变成一个无约束的优化问题．利用变分法原理，推导出满足优化问题的必要条件．给出了PPR型平面欠驱动机械臂最优运动规划的数值算例．

英文摘要：

An optimal motion planning scheme for a three-link planar under-actuated manipulator moving between two stationary states was considered. The manipulator has two prismatic joints that are actuated, while the third joint of the manipulator is a revolute passive joint which has neither an actuator nor a holding brake. The dynamic constraint on the free link is shown to be second-order nonholonomic. The states and inputs of the system were expressed in terms of higher order derivatives of a subset of states by using coordinate and input transformations. The explicit expressions for the states and inputs were used to change a constrained dynamic optimization problem into an unconstrained one. The necessary conditions for optimality were derived by calculus of variations. The scheme was applied to a planar PPB under-actuated manipulator and numerical resuits were reported.