This study is aimed at finding the entire points that a manipulator can launch an object onto by an optimal motion. These points are called throw-able workspace, which are located outside the reachable workspace of the robot. From an optimization point of view, some throwing parameters can be found to decrease motion cost. In this paper, by using this concept, the best combination of throwing and trajectory planning is attempted. The proposed method consists of two basic ideas: first, defining the optimal throwing problem as the optimal control problem (OCP) and solving it using the indirect solution approach based on the fundamental lemma of calculus of variations. To achieve the best release angle and speed, the throwing equation of motion is applied as a moving-end boundary condition (BC). Second, based on the obtained optimal throwing, an algorithm is presented to calculate the maximum throw-able workspace. The simulation results demonstrate the effectiveness of the proposed framework for both single link and spatial two-degree-of-freedom throwing robots.

这项研究旨在找出机械手可以通过最佳运动将物体发射到其上的所有点。这些点称为可抛出工作区，它们位于机器人可访问的工作区之外。从优化的角度来看，可以找到一些投掷参数来降低运动成本。在本文中，通过使用这种概念，尝试了投掷和弹道规划的最佳组合。所提出的方法包括两个基本思想：首先，将最优投掷问题定义为最优控制问题（OCP），并基于变分演算的基本引理，使用间接解法对其进行求解。为了获得最佳的释放角度和速度，将运动的投掷方程式用作移动端边界条件（BC）。其次，根据获得的最佳投掷，提出了一种算法来计算最大可抛出工作空间。仿真结果证明了所提出的框架对于单链接和空间两自由度投掷机器人的有效性。