## 2 Background Concept and Problem Modeling  
### Path Planning for Robot Manipulator and Configuration Space  
In the path planning problem for a robot manipulator, the configuration is defined as a position representation of the robot in the workspace. In other words, the configuration space \(\mathbf{Q}\) (also called joint space in a robot manipulator) is a set of all possible joint angles of the robot manipulator [3; 21; 22]. To be specific, \(\mathbf{Q}\) is defined as a subset of \(n\)-dimensional vector space \(\mathbb{R}^{n}\), where \(n\) is the number of the joints of the robot manipulator. Because of this reason, the position of the robot can be represented as point \(\bar{q}\in\mathbb{R}^{n}\) that has \(n\) angle values \((\bar{q}_{1},...,\bar{q}_{n})\), where each value indicates the joint angle of the robot. The set \(\mathbf{Q}\) consists of two subsets: the first subset is the collision-free space \(\mathbf{Q}_{\text{\tiny{int}}}\) that is comprised of all possible configurations, and in which the robot does not collide with any obstacles or itself. The second subset is called the collision space \(\mathbf{Q}_{\text{\tiny{int}}\pm\text{\tiny{int}}}\) which is the complement of \(\mathbf{Q}_{\text{\tiny{int}}}\) in \(\mathbf{Q}\) and the subset of \(\mathbf{Q}\). In \(\mathbf{Q}_{\text{\tiny{int}}}\), the robot arm manipulator collides with obstacles or itself [21; 23]. The robot manipulator does not collide with any obstacles or itself if its joint angles belong to \(\mathbf{Q}_{\text{\tiny{int}}}\). However, if there is a collision between the robot manipulator and obstacles or itself, the values of the joint angles must be in \(\mathbf{Q}_{\text{\tiny{int}}}\).  
In the sampling-based method, the discrete representation of \(\mathbf{Q}_{\text{\tiny{int}}}\) is obtained by random sampling in \(\mathbf{Q}_{\text{\tiny{int}}}\). The valid path is defined by a connected line between the sampling points. Such sampled points and paths can establish a graph. Because of this construction, the nodes in the graph describe possible configurations of the robot manipulators. In addition, the collision-free paths between any two nodes are represented by connected edges in the graph. For the notation of the path planning problem, let \(\bar{q}_{t}\in\mathbf{Q}\in\mathbb{R}^{n}\) mean the values of the joint angles of the manipulator at the \(t\)th iteration, and \(T\) describe the maximum number of iterations in the algorithm. Thus, the algorithm needs to find the path within \(T\) iterations for a given starting configuration \(\bar{q}_{\text{\tiny{int}}}\in\mathbf{Q}_{\text{\tiny{int}}}\) and goal configuration \(\bar{q}_{\text{\tiny{goal}}}\in\mathbf{Q}_{\text{\tiny{int}}}\).  
Based on this concept, when \(\bar{q}_{\text{\tiny{int}}}\in\mathbf{Q}_{\text{\tiny{int}}}\) and \(\bar{q}_{\text{\tiny{goal}}}\in\mathbf{Q}_{\text{\tiny{int}}}\) are acquired, the algorithm computes a valid continuous and shortest path linking \(\bar{q}_{\text{\tiny{int}}}\) and \(\bar{q}_{\text{\tiny{goal}}}\). The path is defined as a sequence of the state \(\bar{q}_{t}\) such that \(\bar{q}_{0}=\bar{q}_{\text{\tiny{int}}}\), \(\ \cdots\), \(\bar{q}_{T_{1}}=\bar{q}_{\text{\tiny{goal}}}\) where \(T_{1}\leq T\). In addition, the sequence \(\{\bar{q}_{0},\cdots,\bar{q}_{T_{1}}\}\) and the line segment connecting any two neighbors state \(\bar{q}_{t}\) and \(\bar{q}_{t+1}\) from \(\{\bar{q}_{0},\cdots,\bar{q}_{T_{1}}\}\) have to belong to \(\mathbf{Q}_{\text{\tiny{int}}}\).