# A Random Sampling-Based Method via Gaussian Process for Motion Planning in Dynamic Environments
## 1 Introduction  
An autonomous robot system usually needs to consider mechanical design [1], visual perception [2], kinematics [3], dynamics [4], motion planning and control [5], etc. Motion planning is an essential tool for autonomous robots, as it enables the description and execution of motion as high-level goals rather than lower-level primitives [6]. It is a fundamental task for motion planning to provide a collision-free path guiding a robot from an initial node to a goal region in an environment full of obstacles [7]. For example, spray-painting robots based on vision guidance have to follow reasonable paths given by motion planning for spraying [8]. Sometimes robots face more difficult situations, such as performing tasks in dynamic environments, which involves dynamic motion planning problems. For solving these problems, various methods have been developed, of which graph-based and sampling-based methods are the most effective and popular. For low-dimensional motion planning, such as path planning of smart vehicles, graph-based methods (e.g., A* [9] and Dijkstra [10] methods) always rapidly provide high-quality feasible paths. The feasible paths refer to collision-free paths and the high-quality paths are the low-cost paths. However, motion planning is a PSPACE-hard problem [11], with complexity growing with the number of a robot's degrees of freedom (DoF). The graph-based methods are no longer applicable due to the excessive computational resources they require in 6 DoF, 7 DoF, and even higher DoF cases. Sampling-based motion planning methods, which randomly sample valid robot configurations and form graphs of valid motions to find collision-free paths, can avoid the explicitconstruction of planning scenarios by random sampling and collision detection. Because of high-planning efficiency, they are suitable for solving both low- and high-dimensional motion planning problems [12; 13]. In a dynamic environment, dynamic obstacles sometimes make part of the global path invalid, so the invalid partial path needs to be local re-planned online [14]. Sampling-based methods with anytime technology, such as informed anytime fast marching tree (IAFMT*) [15] and informed rapidly exploring random trees (informed RRT*) [16], find multiple feasible paths in the allowed time. The * indicates that the methods are asymptotically optimal (AO) methods. However, these methods only solve static motion planning problems because the algorithms will not return solutions until all of the allowed time runs out. Methods based on probabilistic roadmaps, such as probabilistic roadmaps (PRM) [17] and its variants [18; 19], usually build probabilistic roadmaps offline and plan feasible paths online. If environments change, the probabilistic roadmaps have to be rebuilt offline, so these methods are unsuitable for solving dynamic motion planning problems. Offline motion planning refers to robot paths or trajectory planning when movement ceases. Methods based on random trees without anytime technology, such as rapidly exploring random trees (RRT) [20] and its variants [21; 22], can be directly run online and can be used for motion planning in dynamic environments. However, there is strong randomness for the paths planned by these methods, so the path quality is not stable. Compared with these methods, some asymptotically optimal methods, such as fast marching trees (FMT*) [23] and variants [24; 25], can give high-quality paths in online local re-planning; however, their planning efficiency needs to be further improved. A summary of sampling-based methods is presented in Table 1.  
Most sampling-based methods construct probabilistic roadmaps or random trees by sampling many random nodes in configuration-spaces (C-spaces) to find feasible paths. In order to obtain high-quality paths, even optimal paths, the probabilistic roadmaps or random trees have to be further expanded by adding random sampling nodes. For global motion planning that allows offline planning, these sampling-based methods perform well.