Optimizing Scan Homogeneity for Building Full-3D Lidars based on Rotating a Multi-Beam Velodyne Rangefinder* Anthony Mandow 1 , Jes´ us Morales 1 , Jose Antonio Gomez-Ruiz 1 and Alfonso J. Garc´ ıa-Cerezo 1 Abstract— Multi-beam lidar (MBL) scanners are compact, light, and accessible 3D sensors with high data rates, but they offer limited vertical resolution and field of view (FOV). Some recent robotics research has profited from the addition of a degree-of-freedom (DOF) to an MBL to build rotating multi- beam lidars (RMBL) that can achieve high-resolution scans with full spherical FOV. In a previous work, we offered a methodology to analyze the complex 3D scan measurement distributions produced by RMBLs with a rolling DOF and no pitching. In this paper, we investigate the effect of introducing constant pitch angles in the construction of the RMBLs with the purpose of finding a kinematic configuration that optimizes scan homogeneity with a spherical FOV. To this end, we propose a scalar index of 3D sensor homogeneity that is based on the spherical formulation of Ripley’s K function. The optimization is performed for the widely used Puck (VLP-16) and HDL-32 sensors by Velodyne. I. INTRODUCTION In the last few years, multi-beam lidar (MBL) rangefind- ers, like those developed by Velodyne, are becoming increas- ingly applied in robotic vehicles, such as in urban scene understanding [1], emergency response robots [12], off-road ground vehicles [8], and mobile manipulators [2]. These sensors can be considered as a hybrid between 2D and 3D scanners, as they consist on a spinning structure that holds a number of independent laser transceivers to scan planes with different elevation angles within a fixed vertical field of view (FOV). With the decreasing cost of MBL sensors [14], rotating multi-beam lidars (RMBL) built by adding a degree of freedom (DOF) to a commercial MBL may arguably become a common solution to obtain affordable rapid full-3D high resolution scans in the close future. This idea is supported by recent examples reported in the literature. The first so- lutions incorporated Velodyne HDL-64E lidars on tilting (or nodding) mechanisms for robotic mapping vehicles in mines [7] and tunnels [3]. Lighter MBL devices, like the Velodyne Puck, have favored more compact tilting systems [8][5]. Furthermore, a complete spherical FOV can be achieved by mounting the MBL on a continuous rotation mechanism [6][2]. In contrast to rotating single-beam 2D lidars, whose non- homogeneous measurement distributions can be character- ized by constant vertical and horizontal angular resolutions *This work was partially funded by the Spanish project DPI2015-65186- R. The publication has received support from Universidad de M´ alaga, Campus de Excelencia Andaluc´ ıa Tech. 1 Robotics and Mechatronics Lab, Andaluc´ ıa Tech, Universidad de M´ alaga, 29071 M´ alaga, Spain {amandow, jesus.morales, janto, ajgarcia}@uma.es [16], overlapping beams in RMBLs provoke much more complex scan patterns. Understanding these patterns is cru- cial for effectively building and applying customized 3D sensors. In a recent work [5], we proposed a methodology based on a spherical formulation of Ripley’s K function [10] to analyze the distribution of 3D range measurements projected on a hollow sphere. This methodology was applied to an RMBL configuration where the vertical axis of the constituent MBL is perpendicular to the rotation axis (i.e., zero pitch angle), as in [7][3][8][2]. However, as indicated by Neumann et al. [6], rotating a Velodyne Puck with a constant pitch inclination can improve measurement density distribution. In this paper, we investigate the effect of pitch angles in RMBL measurement distributions in order to find a kinematic configuration that optimizes scan homogeneity and with a full-sphere FOV. With this purpose, the originality of this work with respect to [5] is threefold: (i) we consider a more general build configuration where the Velodyne sensor is not perpendicular to the axis of the rolling DOF but can have a constant pitch angle; (ii) we define a new scalar index to assess the homogeneity of 3D sensor data distribution based on the spherical formulation of Ripley’s K function; and (iii) we optimize the pitch angle that maximizes the homogeneity for RMBLs based on the Puck VLP-16 and the HDL-32. The rest of the paper is organized as follows. Section II de- fines a general RMBL that includes a pitch angle. Section III reviews the application of the spherical extension of Ripley’s K function to analyze 3D scan measurement distribution and proposes a scalar homogeneity index. Section IV analyzes the effect of the pitch angle and discusses optimal configurations for rotating the Puck and the HDL-32 rangefinders. Finally, Section V offers the conclusions. II. GENERAL RMBL DEFINITION Let us define the local frame X v Y v Z v of an MBL with its origin in the optical center, its Y v axis in the forward direction and Z v pointing upwards. This is illustrated in Fig. 1 for the case of the Puck sensor. An MBL scans points in spherical coordinates (R, ω, α), which correspond to Cartesian coordinates (x v ,y v ,z v ) for each measured point: x v = R cos(ω) sin(α), (1) y v = R cos(ω) cos(α), (2) z v = R sin(ω). (3) The local frame XYZ of the RMBL resulting from the addition of a rotating mechanism to the MBL is illustrated in This is the accepted version of an article published at the 2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2018). Please find the published version at IEEEXplore.