# Rapid Planning of an Assembly Path by Reusing the Prior Path
## 1 Introduction  
Digital assembly is one of the key technologies in the design and manufacturing of complex products [1; 2; 3; 4]. Some technologies, such as virtual reality (VR), augmented reality (AR), digital prototyping (DP), and digital twin (DT) assembly, are used to realize the design, verification, and optimization of an assembly in a virtual environment by virtual reality fusion and interactive control and to improve the assembly quality and efficiency of complex products [5; 6; 7; 8; 9; 10; 11; 12; 13]. Assembly planning (AP) is one of the most important bases and premises of the assembly process and has some main subproblems: assembly sequence planning (ASP), assembly motion planning (AMP), assembly resource planning (ARP), assembly path planning (APP), etc. These problems have proven to be either non-deterministic polynomial hard (NP-hard) or non-deterministic polynomial complete (NP-complete), and many studies have been conducted to solve them efficiently [14; 15; 16; 17; 18; 19]. The methods proposed for ASP include a scatter search optimization algorithm [15], ant colony optimization algorithm [20], case-based reasoning algorithm [21], improved particle swarm optimization method [22], etc. The methods proposed for AMP include the rapidly exploring random trees method [23], rapidly exploring deterministic tree method [24], curve-matching method [25], etc. The methods proposed for ARP include a digital twin method [26], geometric reasoning method [27], modularity and reconfiguration method [28], intelligent collaboration method [29], etc. The methods proposed for APP include a particle swarm optimization approach [30], potential field method [31], centroidal Voronoi tessellation algorithm [32], exemplary K-nearest neighbor interpolation approach [33], etc. The APP of a complex product takes much time due to the large number of parts; thus, the optimization of APP will improve assembly efficiency [33]. In addition to the above-mentioned APP methods, there have also been some studies on robot path planning [34; 35; 36], vehicle path planning [37; 38; 39], UAV path planning [40; 41; 42], etc. For most of the APP methods mentioned above, it takes a large amount of repeated computation and leads to low planning efficiency and high costs if generating the assembly path through all parts. In these studies, rapidly exploring random tree (RRT) and probabilistic roadmap planners (PRM) are two typical random sampling algorithms in path planning. The PRM algorithm constructs the road map in the space by random sampling and seeks a feasible solution based on graph searching [43; 44]. However, the PRM algorithm consumes high costs and is not suitable for a dynamic environment [45]. The RRT algorithm is an effective method which explores free space by expanding a tree structure with the leaf nodes generated randomly and uniformly. When a leaf node reaches the target area, a collision-free path is obtained by backtracking from the leaf node to the parent node in sequence [23; 46]. Although the RRT algorithm is an efficient algorithm in path planning, the randomness of the sampling points makes it difficult to obtain the optimal path even if the number of sampling points is sufficient. Therefore, some algorithms have been proposed to improve the RRT algorithm in terms of random point generation, nearest point selection, interference detection, and node connection [47; 48; 49]. One of them is the rapidly exploring random tree star (RRT *) algorithm, which makes the path cost lower by improving the nearest point selection strategy [50; 51]. However, the RRT * algorithm cannot obtain a low-cost assembly path in a short time, or even a valid assembly path when the number of sampling points is small. In fact, the assembly paths of many parts of a complex product are coincident or partially coincident, resulting in a large number of repeated calculations in path planning.