Complexity

Volume 2018, Article ID 9104720, 23 pages

https://doi.org/10.1155/2018/9104720

## Maximizing the Coverage of Roadmap Graph for Optimal Motion Planning

^{1}Robotics R&D Group, Korea Institute of Industrial Technology (KITECH), Ansan, Republic of Korea^{2}School of Electrical Engineering, Korea University, Seoul, Republic of Korea

Correspondence should be addressed to Tae-Woong Yoon; rk.ca.aerok@ywt

Received 26 April 2018; Revised 15 September 2018; Accepted 4 October 2018; Published 8 November 2018

Academic Editor: Zhiwei Gao

Copyright © 2018 Jae-Han Park and Tae-Woong Yoon. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Automated motion-planning technologies for industrial robots are critical for their application to Industry 4.0. Various sampling-based methods have been studied to generate the collision-free motion of articulated industrial robots. Such sampling-based methods provide efficient solutions to complex planning problems, but their limitations hinder the attainment of optimal results. This paper considers a method to obtain the optimal results in the roadmap algorithm that is representative of the sampling-based method. We define the coverage of a graph as a performance index of its optimality as constructed by a sampling-based algorithm and propose an optimization algorithm that can maximize graph coverage in the configuration space. The proposed method was applied to the model of an industrial robot, and the results of the simulation confirm that the roadmap graph obtained by the proposed algorithm can generate results of satisfactory quality in path-finding tests under various conditions.

#### 1. Introduction

Automatic motion planning for robots is an important technology for next-generation manufacturing systems, such as Industry 4.0. Sampling-based algorithms have been widely used in robot motion-planning problems because they are efficient at providing reasonable solutions [1–4]. Sampling-based algorithms determine feasible paths for the robot’s motion using information from a graph that consists of randomly sampled nodes and connected edges in the given configuration space. Such randomized approaches have a strong advantage in terms of quickly providing solutions to complex problems, such as in a high-dimensional configuration space. However, sampling-based methods guarantee probabilistic completeness, and thus they require a large number of sampling nodes to find paths for the motion of the robot in arbitrary cases [2]. The* probabilistic roadmap (PRM)* [3] and* rapidly exploring random trees (RRT)* [5] are representative of such sampling-based motion-planning algorithms.

The PRM method is composed of learning and query phases. In the query phase, collision-free paths for the motion of the robot are obtained by using information from a graph known as the roadmap [2, 3]. The learning phase is the process of constructing a graph that represents the morphology of the given configuration space. In this phase, it is important to construct the graph appropriately to obtain the shortest path for the collision-free motion of the robot. Developments have recently been reported on the optimality of sampling-based algorithms [6–9]. For example, the PRM algorithm provides criteria for connecting edges that can guarantee asymptotic optimality in terms of the probability of finding path.

An important factor to consider when constructing a graph for optimal motion planning is the node sampling strategy used. Sampling-based algorithms were originally designed with a random sampling of uniform distribution [2, 10]. However, a few drawbacks have been discovered in these uniform sampling strategies, such as the narrow passage problem and failure to capture the true connectivity of spaces of arbitrary shapes [9–11]. Thus, various sampling schemes, such as the Voronoi diagram [11], Gaussian random sampling [12], the bridge test [13], and visibility PRM [14] algorithms, have been proposed to overcome drawbacks in the uniform sampling method. Nevertheless, the effect of sampling strategies on optimal motion planning remains unclear. The node sampling strategy that fully captures the shape of the free configuration space remains an important problem in optimal motion planning [9, 10]. In the query phase of the roadmap method, the shortest path is obtained as the optimal path using graph information. Thus, constructing an appropriate graph is closely related to the problem of optimal motion planning.

Another area highly relevant to our work here is research on optimization techniques. In recent years, with growing interest in machine learning and Artificial Intelligence, techniques of distributed optimization that can handle large amounts of data and high-dimensional variables have emerged as an important issue. In [15, 16], distributed optimization techniques were considered for multi-agent systems in complex networks. Secure and resilient techniques [17–19] have also been developed to guarantee the stability of distributed algorithms in adversarial environments. As highly relevant to our work here, coverage maximization was studied using methods of distributed optimization in applications of sensor networks [20–22].

This paper proposes an algorithm to optimize the roadmap graph that can cover arbitrary morphologies of the free configuration space to maximize coverage. Our previous study [23] considered an adaptation algorithm for geometric graphs in an intuitive manner. However, in this study, we reanalyze that adaptive graph algorithm and present its results in terms of the maximization of graph coverage. We found that the optimization algorithm updates the positions of the nodes along the direction of expanding the region of covering until it reaches equilibrium, where expansion and contraction factors are balanced. The results of a simulation to test the proposed optimization algorithm show that it can construct a suitable graph to describe arbitrary configuration spaces, and the resulting graph can generate paths of shorter length for various test queries.

#### 2. Problem Statement

The* configuration space * is the space of all possible configurations of the system, where* n* is the number of degrees of freedom (DOF) of the system and is* n*-dimensional Euclidean space [1]. In the formulation of the configuration space, the robot can be abstracted as a point .* Free configuration* is defined as one where the robot does not intersect with any obstacle in the workspace. The set of all free configurations is defined as the* free configuration space *, and it can be determined by checking for collisions between obstacles and the robot in the workspace. Every workspace obstacle is mapped as an obstacle in the configuration space [2]. Figure 1 shows the mapping relations between the workspace and the configuration space for a two-link articulated robot.