Mathematical Problems in Engineering

Volume 2015, Article ID 690310, 14 pages

http://dx.doi.org/10.1155/2015/690310

## Representation of 3D Environment Map Using B-Spline Surface with Two Mutually Perpendicular LRFs

^{1}Department of Mechatronics Engineering, Hanyang University, Ansan, Gyeonggi-do 426-791, Republic of Korea^{2}Department of Robot Engineering, Hanyang University, Ansan, Gyeonggi-do 426-791, Republic of Korea

Received 19 November 2014; Revised 27 March 2015; Accepted 2 April 2015

Academic Editor: Yongsheng Ou

Copyright © 2015 Rui-Jun Yan 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 proposes a map representation method of three-dimensional (3D) environment by using B-spline surfaces, which are first used to describe large environment in 3D map construction research. Initially, a 3D point cloud map is constructed based on extracted line segments with two mutually perpendicular 2D laser range finders (LRFs). Then two types of accumulated data sets are separated from the point cloud map according to different types of robot movements, continuous translation and continuous rotation. To express the environment more accurately, B-spline surface with covariance matrix is proposed to be extracted from each data set. Due to the random movements, there must be overlap between extracted B-spline surfaces. However, merging of two overlapping B-spline surfaces with different distribution directions of their control points is a complex problem, which is not well addressed by far. In our proposed method, each surface is divided into overlap and nonoverlap. Then generated sample points with propagated uncertainties from one overlap and their projection points located on the other overlap are merged using the product of Gaussian probability density functions. Based on this merged data set, a new surface is extracted to represent the environment instead of the two overlaps. Finally, proposed methods are validated by using the experimental result of an accurate representation of an indoor environment with B-spline surfaces.

#### 1. Introduction

Two-dimensional (2D) features-based simultaneous localization and mapping (SLAM) is the problem of correcting a robot position and building an environment map by using the extracted features in unknown environment. In the past decade, researchers have investigated many issues in 2D SLAM such as feature characterization [1–3], data association [4–6], and loop closing [7–9]. Even though much work has increased the accuracy of constructed 2D environment map, only the 2D geometrical parameters of the objects in three-dimensional (3D) environment have been obtained.

Recently, several SLAM works have constructed a 3D point cloud map of a real environment to show the geometrical shape of the real objects [10, 11]. Based on constructed 3D point cloud map, navigation [12] and path planning [13] research have been done in a 3D environment. To build the 3D point cloud map, a 3D sensor is necessary to obtain the raw sensor data. Most of the 3D LRF system is composed of a 2D LRF and a mechanism system, such as a vertical rotating system [14], a pitch motion system [15], and a spring-mounted system [16]. The obtained 3D raw sensor data from these sensors should be organized to represent the environment map. The iterative closest point (ICP) algorithm [17] is the most well-known method for registration of 3D shapes described either geometrically or with point clouds. Extended ICP algorithms [18, 19] have been used to represent outdoor terrain maps. However, since the environment is represented with scanned sensor data, large storage space is needed in the experimental process.

To represent the environment well, the most commonly used feature is a plane, which has been considered to be extracted from the point cloud map in current research about 3D map construction. There are many plane extraction methods [20, 21], in which the planes have been chosen as landmarks to build an environment map. Nevertheless, a plane is not a good choice to represent 3D map in consideration of the diversity of real objects.

In this paper, two mutually perpendicular 2D LRFs are used to build the 3D point cloud map. To correct the position of a mobile robot, line segments are extracted from the sensor data obtained from the horizontal LRF. Improved extended Kalman filter (IEKF) SLAM algorithm is applied to update the position of robot by using matched feature pair. Based on the accurate position of robot, point cloud map is constructed by using the sensor data obtained from the vertical LRF, shown in Figure 1.