本文使用「直接定位與非線性規劃」(DCNLP) 方法來尋找機器手臂最佳化操作及軌跡。一旦「兩端點邊界值問題TPBVP」被轉成「參數最佳化問題」，DCNLP 便可著手尋找最佳化數值解。本文使用Lagrange-Euler 公式模擬一具三連桿機器手臂，並假此手臂來闡明求解過程。文中的機器手臂須以最節能的最佳化操作、來接住一顆掉落中的球，接球所需之時間f t 須由最佳化技術決定，在手臂工作空間、同時存有一顆路徑障礙球，由於f t 及障礙球的存在，更加深了問題深度。

This study applies the Direct Collocation and Nonlinear Programming (DCNLP) method to lay out the solution for manipulator arm path planning problem. It also demonstrates how DCNLP can efficiently handle the two-point boundary-value problem (TPBVP) by converting a TPBVP into a parameter optimization problem. A three-link manipulator arm, modeled by Lagrange-Euler equation, is used as a vehicle to illustrate the procedures. The manipulator is required to intercept a falling target and a least-energy maneuver sequence control is anticipated. The final time is unspecified and an obstacle sphere is placed in the workspace.