# A Sampling-Based Unfixed Orientation Search Method for Dual Manipulator Cooperative Manufacturing
## 1 Introduction
### Relative Work  
A 6-DOF industrial manipulator implies a type of serial chain robot with six revolving joints, and its structure is much more complex than that of a mobile robot. Since a single manipulator can be modeled as a particle in its own configuration space (C-space), the particle-based path-planning methods [5] can be used for single manipulator path-planning directly. Limited progress has been made for the non-collision path planning of multiple cooperating manipulators in C-space. Obstacles were modeled as simple geometries by [PERSON] et al. [6], which will then be projected into the C-space of a manipulator. This method is not effective for multiple manipulators due to their complex structures. Following on from this, [PERSON] et al. [7] divided the C-space into multiple layers based on specific dimensions and implemented a rapidly created C-space grid map on each layer. However, because of the massive amount of elements in the maps, the whole algorithm tends to be time-consuming. [PERSON] et al. [8] planned collision-free paths for 2D horizontally articulated dual-arm robots by dividing the C-space of each robot into multiple blocks, marking the obstacle space and free space at different time intervals, and mapping the search method to seek out an optimal path in free space. The method was too computationally expensive for real-time applications, and it is difficult to project a 6-DOF manipulator body into the C-space of another 6-DOF manipulator. [PERSON] et al. [9] proposed a general manipulation planner for a dual-arm industrial manipulator, which combined the C-space of the arms and obstacle space into one overall search map. The method mainly focused on each arm's trajectory arrangement and movement order to allow a target object to pass from starting point to endpoint. However, the possibility of collision between the two working arms was not considered. Hence, how to explore a simple, effective, and reliable execution space path-planning algorithm has become one of the more and more enthusiastic focuses in recent decades. An online collision-free trajectory generation algorithm for dual-arm robots was established by [PERSON] et al. [10] by setting up a Virtual Road Map (VRM) in the execution space and refreshing the map with a new collision-free path. However, the execution time is extended due to difficulties in setting up the VRM, and raising the method to multiple robots is challenging. [PERSON] et al. [11] divided the working area into different zones and set up obstacle models within each zone. A master-slave method was then used to choose a free path; however, dynamic environments cannot be accommodated since the obstacle must be static. [PERSON] et al. [12] combined the C-space with execution space to build a \"motion space\". They used the Lazy Weighted A* method to plan a non-collision path in an environment containing N-manipulators. Still, only the path planning for one robot is allowed at a time, and the planning process is performed offline. While the ARA* method was also tested to search for the optimal path in the workspace (execution space) of a manipulator, the study focused only on dual arms performing the same task. It did not consider the coordination of different tasks. Additional methods have been proposed for the non-collision path planning of multiple manipulators. [PERSON] et al. [13] used the A* method to plan a collision-free path in C-space without considering the motion of other robots, and a Path Modification Sequence (PMS) method was then proposed to arrange the moving sequence of each robot, resulting in a much longer execution time. Another method proposed by [PERSON] et al. [14] used a collision map to detect the collisions between two robots to avoid deadlocks, which can occur if one robot becomes an obstacle to another. Unfortunately, the method simply delays the movement of the robot to avoid collision. [PERSON] et al. [15] suggested an approach based on a variation of the Probabilistic Road Map, called the Probabilistic Road Map with Obstacles (PRMwO).