Journal of Robotics

Volume 2018, Article ID 4676720, 9 pages

https://doi.org/10.1155/2018/4676720

## Filtered Medial Surface Based Approach for 3D Collision-Free Path Planning Problem

^{1}LARC Laboratory, University Frères Mentouri Constantine 1, Constantine, Algeria^{2}X-Lim Institute, UMR CNRS 7252, University of Limoges, Limoges, France

Correspondence should be addressed to Karima Benzaid; rf.oohay@diazneb_irak

Received 25 September 2017; Revised 14 February 2018; Accepted 1 March 2018; Published 3 June 2018

Academic Editor: Farrokh Janabi-Sharifi

Copyright © 2018 Karima Benzaid et al. 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

This paper introduces an original 3D path planning approach for Unmanned Aerial Vehicle (UAV) applications. More specifically, the core idea is to generate a smooth and collision-free path with respect to the vehicle dimension. Given a 3D grid representation of the environment, the Generalized Voronoi Graph (GVG) is first approximated using a filtered medial surface (FMS) algorithm on the corresponding navigable space. Based on an efficient pruning criterion, the produced FMS excludes GVG portions corresponding to narrow passages unfitting safe UAV navigation constraints, and thus it defines a set of guaranteed safe trajectories within the environment. Given a set of starting and destination coordinates, an adapted* A*-star algorithm is then applied to compute the shortest path on the FMS. Finally, an optimization process ensures the smoothness of the final path by fitting a set of 3D Bézier curves to the initial path. For a comparative study, the* A*-star algorithm is applied directly on the input environment representation and relevant comparative criteria are defined to assert the proposed approach using simulation results.

#### 1. Introduction

This paper deals with the issue of 3D path planning for UAVs in a known environment with stationary obstacles. The problem is to define an efficient method for finding the safest path from a starting point to a target, taking into account the dimension of the UAV.

To deal with this problem, many methods are proposed in the literature. Sampling based algorithms [1], graph based search algorithms [2, 3], mathematic model based algorithms [4, 5], and bioinspired algorithms [6–8] are the fundamental families of path planning algorithms [9]. Most of the path planning solutions were originally developed for 2D problems and then extended to the third dimension, thus increasing the complexity of the approach and its computational cost.

Among the most known sampling based algorithms, we can cite the Rapidly Exploring Random Tree (RRT) [10] and the Probabilistic Road-Maps (PRM) [11] algorithms. These methods are initially designed for 2D path planning problems based on the sampling of the configuration space. The drawback of this kind of algorithms is that they return nonoptimal solutions [12]. To improve the quality of the solution, many variants of these sampling based algorithms have been proposed [12–16]. For aerial vehicles path planning problem, some adapted versions [17–20] are applied. Few methods based on Voronoi diagram have been used to deal with the problem of 3D path planning [21].

The* A*-star is the most popular graph based search algorithm. Developed by Hart et al. [22], it is based on Dijkstra’s algorithm [23] and has been widely implemented in robotics and video games. The conventional* A*-star is used to find the shortest (possible) path connecting edges of 8 neighbors in 2D grids and 26 neighbors in 3D grids. However, this algorithm cannot find the path in any neighbor connection angle. This implies a suboptimal solution represented as a path with continuous heading changes. With the aim of overcoming this limit and/or improving results in terms of time computation, memory requirement, and path length, some variants of the* A*-star algorithm have been developed [24–26]. For 3D applications, we can cite [2, 3, 27, 28]. Even though the solution was improved, the planned path is not unruffled enough and needs more refinements to make it smooth for aerial vehicles tracking.

In this paper, we propose a novel approach based on the skeletonization of the navigable space along with an adapted* A*-star path planner for aerial vehicles. The environment is represented as a discrete 3D occupancy grid supposedly given (its construction is out of scope here). Based on a pruning parameter, the skeletonization algorithm is able to discard noisy skeleton branches and thus exclude narrow passages with respect to the UAV dimensions. This constraint is of major importance but is rarely considered in the literature.

The key idea is to extract the* filtered medial surface* of the grid representation according to the dimension of the aerial vehicle. In the next step, the use of the* A*-star algorithm aims to find the shortest path on the FMS. Since the returned path is not smooth due to the continuous heading changes of the* A*-star path, a smoothing method is finally applied.

This may appear similar to the idea adopted in [29]. However, in this cited paper, the authors propose a 2D solution for a 3D path planning problem. The method is based on a 2D skeleton represented as a thin line. The 3rd dimension is not considered at any time.

The main contributions can be summarized in three essential points:(i)We introduce a 3D skeletonization algorithm for direct applications in aerial robotics context based on a 2D shape representation algorithm.(ii)In order to eliminate oscillations and tight turns in the generated path, we propose a new safety corridor based smoothing approach using Bézier curves.(iii)We perform a comparative study of the proposed approach by applying the* A*-star algorithm directly on an inflated occupancy grid and defining a set of quantitative criteria.

The paper is organized as follows. In Section 2, we present the two steps constituting the proposed solution. The first is the extraction of the FMS on the navigable space, and the second is the computation of the shortest path on the FMS. In Section 3, we introduce a new Bézier curve based method to smooth the initial planned path. In Section 4, our solution is compared to the* A*-star solution applied on the 3D grid representation. We finally conclude the paper and introduce the future work.

#### 2. Path Planning on the Environments GVG

This section is dedicated to the problem of accurate and real-time path planning on the Generalized Voronoi Graph (GVG), i.e., the set of points in the environment at equal distance from their two closest obstacles. Given a 3D occupancy grid delimiting the navigable space from the obstacles , it is approximated here by the medial surface of , filtered with respect to the robot dimension.

##### 2.1. Skeletonization and Filtering of the Navigable Space

Let be a subset of and be its complement. Initially defined by Blum [30], the real medial axis of (also referred to as skeleton) is the set of points at equal distance from their two closest neighbors in . As pointed out in [31], this definition is conceptually equivalent to the GVG of an environment and is thus used here as an accurate approximation. In the specific case of 3D shapes, it generates a set of surfaces and is thus also referred to as medial surface.

For each shape point , let denote the set of its closest points on , referred to as projection of . The corresponding Euclidean distance is denoted as . Accordingly, the skeleton can then be defined by

To efficiently approximate and filter the environment GVG, two important concerns have to be considered. First, the navigable space is expressed in this work in a discrete domain (the occupancy grid ). Applying (1) directly on would fail, since, in most cases, even when a given cell actually belongs to the environment GVG.

Therefore, the definition of the discrete medial axis has to be slightly modified. Although several solutions were proposed over the years, this work is based upon the Integer Medial Axis (IMA) [32]. Let denote the reduced projection of , corresponding to the first element in with respect to a lexical ordering. Given a digital shape and its complement , the discrete skeleton of can then be expressed aswhere is the set of direct neighbors of and is the midpoint of line segment . The first condition measures the (Euclidean) distance between the projections of and to remove skeleton points related to aliasing, while the second condition is used to produce a skeleton as thin as possible, since it only selects the best approximation to the real skeleton. Note that it is theoretically proven [32] that this definition efficiently approximates the medial axis of the corresponding shape in while removing skeleton points induced by discretization.

The second major concern is the question of boundary noise: Skeletons are known to be very sensitive to small shape deformations and tend to produce spurious branches. The IMA efficiently approximates Blum’s real medial axis but is not designed to filter noisy branches. This issue is addressed here using the delta medial axis (DMA) [33], a real-time pruning strategy able to identify and discard spurious skeleton branches in real time. The algorithm was initially designed for 2D shapes but can straightforwardly be extended to higher dimensions.

The core concepts of the algorithm are illustrated in Figure 1.