# Research on the Path Planning Algorithm of a Manipulator Based on GMM/GMR-MPRM
## 1 Introduction  
Conventional industrial robots work in a safety fence, and their work trajectory is usually preset through a program. In recent years, with the concept of human-robot collaboration, robots have gradually stepped out of the safety fence, which has made the working environment of robots more complicated and unpredictable, and offline programming would not be able to meet such work requirements. LfD can quickly transfer human skills to robots through human-computer interaction and teaching methods, to realize the rapid programming of robots [1]. However, the teaching scene and the recurring scene are usually not completely consistent while repeating the task. At present, most path planning algorithms are aimed at the point-to-point path planning problem, that is, when encountering obstacles, the information of the whole workspace is obtained first, and then, a feasible path from the starting point to the target point is planned. However, the path planning problem of task recurrence in the process of LfD is not a point-to-point path planning problem, but a problem of tracking a tasktrajectory while avoiding obstacles. Meanwhile, not only industrial robots, but also other platforms, such as parallel robot platforms, require path planning. [PERSON] presented the mechanical design of a humanoid robot named LARMbot 2. The torso of LARMbot 2 can be described as a 4 SPS-(3E) parallel mechanism with four degrees of freedom, and the legs of LARMbot 2 are structured by a 3 UPR lower-mobility parallel mechanism. The parallel architectures on which the robot is based allowed it to lift a considerable payload [2]. [PERSON] presented the results of HeritageBot. The novel aspect of the HeritageBot robot is that it was designed with a modular structure. The HeritageBot robot can be recognized in the modular design by combining a legged parallel manipulator locomotor with a drone with a short flight capability, which allows the robot to have greater potential in cultural heritage services [3]. The dynamic control of both LARMbot 2 and the HeritageBot robot requires path planning. The path planning algorithm usually models the free configuration space through random sampling and collision detection and then generates an effective collision-free path in the sampled free configuration space. The most famous sampling-based planning algorithms are PRM [4] and RRT [5], which use roadmaps and tree structures to model the free space, respectively [6]. In addition, in order to extend the capabilities of these algorithms, many variants have been proposed, such as LazyPRM [7], EST [8], SBL [9], RRT-Connect [10], NC-RR [11], and KPIEC [12]. Due to the inherent characteristics of random sampling, these algorithms have some common problems, such as the low efficiency of narrow channel search and the optimal probability of the searched path [13]. In recent years, researchers have proposed many sampling planning algorithms based on optimization, such as TRRT [14], RRT* [15], and BIT* [16]. Although some optimization goals or constraints can be achieved gradually, such as minimizing the length of the path, the computational cost of these optimization planners is relatively expensive. Aiming at the problem that the constrained optimal feedback controllers require extensive calculation, [PERSON] presented two locally optimal feedback controllers that can be efficiently computed online in real time. The controllers use an iterative algorithm to numerically compute the constrained optimal feed-forward control input, so the computation of the inputs can be performed in real time [17]. However, this controller can better mitigate external perturbations and imperfect model information. In addition, the optimization objective of some problems may be difficult to describe as a specific mathematical function. For example, the motion generated by a humanoid robot is an optimization objective that is difficult to describe by mathematical function [18].