## 2 The NC-RRT Method
### The Node Control Mechanism  
When the CSA-RRT algorithm selects the extended nodes, the nodes in the tree should be traversed to find the node with the smallest distance from the sampling configuration point in the C-space, but this traversal is usually redundant and time consuming. Under the premise of maintaining the performance of the CSA method, a node control mechanism is introduced in this part to further reduce the extension of invalid nodes and extract the boundary nodes (or near-boundary nodes), thereby further improving the speed of the algorithm and enhancing the adaptability of the environment. On the basis of this mechanism, the \"local trap\" phenomenon in the process of random tree expansion is proposed. We update and record the expansion state of each node when the random tree expands in the C-space. The value of the node control factor is changed according to whether the \"local trap\" phenomenon occurs. Then, the selection of the extension node is adjusted, such that the tree is expanded by the boundary nodes only or by nodes close to the boundary in most cases. The specific process is as follows. We represent the extended state value of the node as \(\delta\) and redefine the following concept that is different from the past: the path that each leaf node of the tree traces back to the initial node \(q_{start}\) is called a branch. Then, the change strategy of the node state value is as follows. Its \(\delta\) value is set to zero each time a new node is successfully added to the tree. Then, starting from the parent node and following the reverse path, one is added to the state value of each node in the branch where this new node is located until it traces back to the initial node. As shown in Figure 2, any node \(q_{n}\) in the process of random tree exploration is regarded as an example. Assuming its \(\delta\) value is recorded as \(m\) at this time, if a child node \(q_{n+1}\) is generated with \(q_{n}\) as the parent node in a certain iteration, then the value of \(\delta\) for \(q_{n+1}\) is set to zero; then, starting from the parent node (i.e., \(q_{n}\)) of \(q_{n+1}\) and following the reverse path, one is added to the value of \(\delta\) for all nodes in the branch where \(q_{n+1}\) is located until it traces back to the \(q_{start}\) node. At this time, the \(\delta\) value of \(q_{n}\) becomes \(m+1\). If the new node \(q_{n+2}\) generated in a subsequent iteration is still a child node of \(q_{n}\), the above process is repeated, and the \(\delta\) value of \(q_{n}\) becomes \(m+2\). However, because the \(q_{n+1}\) node is not in the same branch as the \(q_{n+2}\) node, the \(\delta\) value of the \(q_{n+1}\) node remains unchanged in this iteration. According to the state value of each node, a node control factor \(control\) can be introduced to guide the selection of the expanded node. For each extension, only the node that has a state value less than \(control\) and lies closest to the sampling node is selected as the extension node. To reduce the invalid exploration in the C-space sufficiently, we set \(control\) to one by default. However, the introduction of this control factor creates a problem. Because only the \(\delta\) value of the leaf nodes in the random tree is zero and the value of \(control\) remains one, it means that the last node in the tree will always be used as  
Figure 1: Performance comparison between Basic-RRT and CSA-RRT in the same environment. (**a**) Performance of Basic-RRT. (**b**) Performance of CSA-RRT. the extension node. Combined with the proposed CSA sampling method, this technique will drive the tree to reach the target configuration extremely rapidly if there are no obstacles in the environment. As shown in Figure 3a, we can see that all nodes in the tree exist as \"valid nodes\" in the final query path. However, once there are obstacles in the environment, the tree easily falls into the \"local trap\" state, as shown in Figure 3b.