## 3 Materials and Methods
### Multi-Scale Positioning for Waypoint Navigation
We hope that this method can be used to compare the impact of introducing a priori information on preliminary path planning. * Cuckoo Search Algorithm [43]. A parallel compact cuckoo search algorithm for three-dimensional path planning. The algorithm uses an improved cuckoo search based on compact parallel technology to find the target in a complex three-dimensional environment. Cuckoo search proposes a new parallel communication strategy that saves the memory of the robot. We train the encoder by treating the parallel search module as a black box function and optimizing it through the black box, while keeping other settings. We transformed the drone in the original application scenario of the algorithm into a multi-joint robot, which is consistent with the HDPP operating environment. And check the plan in a high-dimensional environment. This algorithm is often used in UAVs (unmanned aerial vehicle), underwater robots and other equipment, and has reference significance for high-dimensional space planning. Traditional comparisons. In order to conduct a comprehensive comparative evaluation of HDPP, we selected the most classical methods and variants of the three traditional algorithms (search-based, sampling-based and reinforcement learning) to participate in the experiment:

* Search-based [16, 21, 22, 23]. We mainly choose Dijkstra's Algorithm, A* and D* lite for experiment. These algorithms are representative of traditional path planning based on search strategy, which will find the optimal path by visiting and calculating map nodes and costs. However, the difference is that the search strategies of these three algorithms are constantly refined, and their performance has a progressive relationship. * Sampling-based [17, 24, 25, 26, 27, 28]. RRT and its improved algorithm--RRT\"--are regarded as representative of sample-based path planning. For this strategy, the optimal path point was found by random sampling and added into the planned path tree. RRT\" is improved on the basis of RRT, mainly by reselecting the parent node and rewiring operations. * Reinforcement learning [29, 30, 31, 32, 33]. Classical Monte Carlo algorithm and DQN are transferred to the experiment to explore the effect of reinforcement learning ideas in high-dimensional space, especially for multi-degree-of-freedom objects. Since our data set is dynamic and variable, and time thresholds are designed to emphasize practical applications, underfitting or local optimality can be expected with this strategy of calculating returns by receiving continuous environmental feedback. Two of the most typical algorithms are introduced to test this judgment. ### Implementation Details

We tested ABB's six-axis robotic arm and four-legged trolley in the experiment. Among them, the experimental environment of ABB's six-axis robotic arm is natural light, and the background floor has a single color. There are 10 rectangular paper boxes of different sizes and different colors as the target. During the operation of the robotic arm, there will be random interference of black squares. Our HDPP performs path planning for the robot to run from any position to the target carton to grab this segment of the travel process. For the simulation of the quadruped car, the obstacle avoidance is carried out based on the complex environment of the laboratory. For the parameter setting of our method, we set the maximum number of iterations of the high-dimensional manipulator \(I=5\) K, the quadruped car's \(I=2\) K, the frequency of patch refinement and gradient optimization \(Nk=100\), and the threshold Mk for reducing redundant waypoints = 10, the patch size is 50 \(\times\) 50 (the resize of the receptive field is 1024 \(\times\) 1024), and the upper limit of waypoints is \(N=20\). In order to fully evaluate our algorithm, we have introduced the following indicators:

\[ACC_{gait-\text{cos}t}=\left\{ \begin{array}{ll}\alpha\sum\limits_{r=1}^{N}P_{ nouw}/P_{original}&Poin_{now}<Poin_{original}\\ 0&otherwise\end{array} \right. \tag{11}\]

\[COST_{\text{planning}}=\left\{