Research Article Kinematics of Spherical Robots Rolling over 3D Terrains Saeed Moazami , 1 Hassan Zargarzadeh , 1 and Srinivas Palanki 2 1 Department of Electrical Engineering, Lamar University, Beaumont, TX, USA 2 Department of Chemical Engineering, Lamar University, Beaumont, TX, USA Correspondence should be addressed to Hassan Zargarzadeh; hzargarzadeh@lamar.edu Received 24 July 2019; Accepted 9 October 2019; Published 16 December 2019 Academic Editor: Dimitri Volchenkov Copyright © 2019 Saeed Moazami et al. is 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. Although the kinematics and dynamics of spherical robots (SRs) on flat horizontal and inclined 2D surfaces are thoroughly investigated, their rolling behavior on generic 3D terrains has remained unexplored. is paper derives the kinematics equations of the most common SR configurations rolling over 3D surfaces. First, the kinematics equations for a geometrical sphere rolling over a 3D surface are derived along with the characterization of the modeling method. Next, a brief review of current mechanical configurations of SRs is presented as well as a novel classification for SRs based on their kinematics. en, considering the mechanical constraints of each category, the kinematics equations for each group of SRs are derived. Afterward, a path-tracking method is utilized for a desired 3D trajectory. Finally, simulations are carried out to validate the developed models and the effectiveness of the proposed control scheme. 1. Introduction Spherical robots are a class of mobile robots that are gen- erally recognized by their ball-shaped shell and internal driving components that provide torques required for their rolling motion. From the ball-shaped exterior, SRs inherit multiple advantages over other types of mobile robots, in which skidding, tipping over, falling, or friction with the surface makes them vulnerable or inefficient [1, 2]. Despite all the unique features, the complicated nonlinear behavior of SRs has remained as a hurdle to fully comprehend their dynamics, motion kinematics, and unveil their maneuver- ability capabilities. Although the earliest efforts to analytically capture the kinematics and dynamics of rolling geometrical balls on mathematical surfaces back to more than a century ago, e.g., in E. Routh and S. A. Chaplygin’s works, this topic is still an open discussion that is being investigated in more recent papers such as [3–5]. Numerous research studies have been conducted on the mathematical modeling of kinematics and dynamics of SRs with a variety of mechanical configurations [6, 7]. Among them, a widespread assumption is that SRs roll over an ideally flat horizontal plane. In a number of available works, where SRs are studied on nonhorizontal surfaces, simplifying assumptions are made. For instance, in [2, 8], two SR designs are investigated that climb obstacles, as- suming the condition to be static. From a different view, rolling of SRs is studied where the desired path is assumed to be a straight line with constant slope or a single step obstacle [9–11] and a 2D curved path with variable slope [12] re- spectively. In [13], the authors have investigated the dy- namics of Martian tumbleweed rovers while this special type of SR rolls in its heading direction and the turning action is not considered for them. In fact, while several pieces of research have been done on 3D kinematics of other types of mobile robots such as legged [14] and wheeled robots [15, 16], to the best of the authors’ knowledge, the general problem of kinematics of SRs rolling on 3D terrains has not been investigated in the literature. e motivation to address this problem is that, while many applications of the SRs are on flat surfaces such as indoor [17] and paved roads [18], for a variety of applications such as agriculture [19], surveillance [20], environmental monitoring [21], and even planetary explorations [22], they would get exposed to uneven terrains. In this work, prior to deriving the kinematics of SRs on 3D terrains, a general method for modeling a geometrical Hindawi Complexity Volume 2019, Article ID 7543969, 14 pages https://doi.org/10.1155/2019/7543969