## 3 Materials and Methods
### Multi-Scale Positioning for Waypoint Navigation
\begin{array}{ll}0&SUCCESS\\ \sigma&FAILURE\end{array} \right. \tag{12}\]

\[COST_{\text{collision}}=\left\{ \begin{array}{ll}\tau&HIT\\ 0&otherwise\end{array} \right. \tag{13}\]

where \(ACC_{gait-cost}\) is the number of points after fitting and the number of planned waypoints without HDPP, and the \(COST_{planning}\) and \(COST_{collision}\), respectively, represents the planning loss and collision loss. The \(\sigma\) and \(\tau\) can be modified according to the application scenario, which is set to 10 and 1000 in our experiment. However, while these indicators have an outstanding effect on optimization results, they are not standard. In order to intuitively analyze the experimental data, we transform and combine these three aspects into clearer data representation, and also add an index for time cost (the name of the indicator is qualitatively determined by the function of performance quantification, which use superscript \"*\" to distinguish):

\[OPT^{*}=\sum\limits_{i=1}^{n}ACC_{gait-cost}/\sum\limits_{i=1}^{n}\alpha \tag{14}\]

\[ACC^{*}=(\sum\limits_{i=1}^{n}\sigma-COST_{planning}+\sum\limits_{i=1}^{n}\tau- COST_{collision})/(\sum\limits_{i=1}^{n}\sigma+\sum\limits_{i=1}^{n}\tau) \tag{15}\]

\[TS^{*}(Time-Score)=\sum\limits_{i=1}^{n}t_{i}(HDPP)/T \tag{16}\]

where \(n\) means the number of rounds tested and \(i\) is the current iteration number, \(T\) means the average \(n\) rounds planning time of the baseline algorithms; \(OPT^{*}\)(waypoint optimization) represents the efficiency of remove the redundancy based on waypoint importance, also the efficiency of gait updating; \(ACC^{*}\) quantify the condition for collision and whether path planning is successful, while \(TS^{*}\) is used to evaluate the real-time performance of HDPP. Obviously, to achieve a high level of planning, we need to minimize the number of waypoints and reduce the number of re-planning gait under the premise of ensuring that the path is successfully planned without collision. Due to the different operating environments of the robots, we use \(OPT^{*}\), \(ACC^{*}\), and \(TS^{*}\) in the results to represent the average score of each image. ### Results of HDPP

**Fast and efficient waypoint planning. ** We conducted experiments on multi-degree-of-freedom robots (for example, 2-DOF Mobile robot, 3-DOF and 6-DOF Robotic arm) to verify HDPP's effectiveness in eliminating redundant computational path points to improve overall performance. The results of HDPP are shown in Table 1. Our HDPP can completely achieve anti-collision path planning, but only use other algorithms with less than 60% waypoints. If the waypoint coefficient is increased from 1 to 100, the number of waypoints can be reduced to 40% or less of other algorithms, and the anti-collision plan can still reach more than 90%. Compared with the traditional method based on search, it reduces the amount of data calculation by 68.73%, and achieves the same better and faster exercise effect. These results show that our approach can improve the efficiency of robot motion. **Comparisons with Baselines. ** In order to conduct a comprehensive and reliable evaluation of HDPP, we conducted a comparative experiment. Table 2 shows the improvement of our algorithm in redundant waypoint screening and security, and also shows that our strategy can dynamically find and divide very well compared with search or sampling-based methods. Category area for waypoint optimization. We noticed that some recent work based on reinforcement learning are also dedicated to reducing the waypoints in robot path planning. They used clever strategies and achieved good results in robot navigation. These works have the same focus and task as ours but put the improvement of motion planning speed in the first place, instead of paying too