International Journal of Dynamics and Control https://doi.org/10.1007/s40435-018-0441-z Analytical modeling of a 3-D snake robot based on sidewinding locomotion Mohsen Malayjerdi 1 · Alireza Akbarzadeh 1 Received: 21 January 2018 / Revised: 11 May 2018 / Accepted: 15 May 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract In this paper, we restrict our attention to sidewinding locomotion and present detailed kinematics and dynamics of a 3-D multi-link snake robot. To obtain kinematics of three-dimensional snake-like robot modeling, first, a virtual structure with an additional six degrees of freedom is attached to the tail of the robot. Denavit–Hartenberg method is next employed to derive the kinematics relationships. A spring and damper model is used to realistically model contacts between ground and the robot. Gibbs–Appell’s method is next utilized to obtain the 3-D robot dynamics. To validate the dynamics equations, SimMechanic software is used. Finally, a 3-D snake robot, referred to as FUM-Snake 5, is constructed and utilized to experimentally show the sidewinding locomotion. The theoretical derived equation in this study can also be used to generate both other 2-D and 3-D snake robot locomotions. Keywords Dynamic analysis · Gibbs–Appell · 3-D snake-like robot · Friction and ground model 1 Introduction Snake robots offer potential in assisting in areas such as fire- fighting, rescue missions and maintenance due to their high maneuverability and ability to move through tight spaces. These robots are able to bend and adapt to the form of the terrain on which they move. The most famous gaits used by snakes are lateral undulatory (serpentine), concertina, sidewinding and rectilinear locomotion [1–10]. Snake-like robots were introduced by Shigeo Hirose [1]. Since Hirose initial study, many snake robots are designed. Existing snake- like robot designs have different physical configurations and purpose. They mostly attempt to mimic locomotion of real snakes; however, some use non snake-like gaits [2–4]. Hasanzadeh and Akbarzadeh [2] presented a novel gait, for- ward head serpentine (FHS), for a two-dimensional snake B Alireza Akbarzadeh ali_akbarzadeh_t@yahoo.com Mohsen Malayjerdi Mohsen.ciw@gmail.com 1 Center of Excellence on Soft Computing and Intelligent Information Processing, (SCIIP) Mechanical Engineering Department, Ferdowsi University of Mashhad, Mashhad, Iran robot. In their work, they used Lagrange’s method to obtain dynamic equation. Kalani and Akbarzadeh [3] restricted their attention to worm-like, vertical traveling wave loco- motion and presented detailed kinematics and dynamics of a planar multi-link snake robot. They employed Lagrange’s method to obtain the robot dynamics. Furthermore, authors in [4] used Newton’s method to obtain dynamic equations of traveling wave locomotion. Saito et al. [5] constructed a snake robot without wheels. This robot has great potential to adapt to various environments at the cost of increased power consumption. They obtained total equations of motion for a multi-link snake robot traveling with serpentine locomo- tion. They also showed that the unsymmetrical body curve increases the robot’s performance. Ma [6] formulated the kinematics and the dynamics of a 2-D snake-like robot in closed form. The derived robot dynamics were used to ana- lyze the 2-D creeping locomotion [6–8]. Ma et al. [9] also considered formulation of the kinematics and dynamics of a three-dimensional (3-D) snake robot and analyzed creep- ing locomotion. They investigated the motion efficiency of a sinus-lifting motion in comparison with normal creeping locomotion. The 3-D dynamics of a snake robot during loco- motion across flat surfaces is considered in [9,11–14]. A mathematical model of a 3-D snake robot with 2-d.o.f. revo- lute joints using the Newton–Euler’s algorithm is presented by Liljebäck et al. [11,12]. 123