Singular Configurations Analyses of the Modifiable Theo Jansen-like Mechanism by Focusing on the Jacobian Determinant - A Finding Limitations to Exceed Normal Joint Range of Motion Kazuma Komoda and Hiroaki Wagatsuma Abstract— Adaptive locomotion is an important topic for the design of mobile robots, and smooth leg movement is of interest for investigating an indicator of how much bio-inspired robot represents the essential mechanism in animal walking. Using the multibody dynamics approach, we have investigated a potential extension of the Theo Jansen mechanism, an eleven-bar linkage that generates the locomotion pattern of multi-legged animals. Our extension highlighted a flexibility of the mechanism’s trajectory, but an unclear limitation beyond normal joint range of motion was found. In this study, we examined the theoretical limitation of joint range of motion by singular configuration analysis. Multibody dynamics provide the Jacobian matrix to represent whole kinetics with constraints from close linkages and enables a singularity from the Jacobian determinant to be found. The method of finding a singularity in the mechanism may help to understand spatio-temporal properties of the control parameter, thus generating various stable motions toward mobility in an uneven ground. I. INTRODUCTION Bio-inspired robots are tools for understanding locomotion mechanisms in animals. A simple solution for building flexi- ble leg systems is serial mechanisms, such as Collins et al. [1] provided as a open-loop mechanism. It reproduces a bipedal locomotion as a passive walking devices. The passive walker restricts degrees of freedom (DOF) in individual joints to limited ranges, for providing the locomotion function. Sim- ilarly, Niiyama et al. [2] proposed a musculoskeletal robot with air springs in joints with appropriate tendernesses. The simple serial mechanism may provide an equivalent locomo- tion functionally but the system does not always represent the underlying mechanism of target musculoskeletal systems in animal, because the system relies on the control problem of actuators in joints. In the viewpoint of the structure of muscles surrounding the bone and pulling in parallel and competitively, parallel link mechanisms can be considered as an alternative model of musculoskeletal systems. The link system represents the biological constraint system in the form of coupling between pulling and pushing forces. Recently, a parallel link mechanism is used by a walking assistance device for Kazuma Komoda is with Department of Brain Science and Engineering, Graduate School of Life Science and Systems Engineering, Kyushu Institute of Technology, 2-4 Hibikino, Wakamatsu-Ku, Kitakyushu 808-0196, JAPAN komoda-kazuma@edu.brain.kyutech.ac.jp Hiroaki Wagatsuma is mainly with Graduate School of Life Science and Systems Engineering, Kyushu Institute of Technology, 2-4 Hibikino, Wakamatsu-Ku, Kitakyushu 808-0196, JAPAN and also with RIKEN Brain Science Institute, as visiting scientist, 2-1 Hirosawa, Wako-shi, Saitama, 351-0198 Japan waga@brain.kyutech.ac.jp rehabilitation [3]. In the engineering viewpoint, the linkage requires a minimum number of actuators, for example an cyclic motion in the beginning like a steam engine. As a design method, the linkage is not only an analytical tool of biological locomotions but also a low cost mechanism with less energy consumptions. However, in the linkage, the design of target trajectories remains unsolved. There is a few achievements of walking mechanisms by using the linkage, such as the Chebyshev [4], the Klann linkage [5] and Theo Jansen mechanism [6]. In succession to advantages of parallel link mechanisms for walking mechanism, Giesbrecht et al. [7] investigated an optimized orbit of legs from the viewpoint of a minimum torque. Ingram [8] used the center of gravity for analyzing the walking motion. These approaches depend on basic motion, and there is no deep discussion or mathematical analysis of changing DOF and extension motions. We have investigated a way to extend the given trajectory of motion based on the Theo Jansen mechanism in Fig. 1, which transits from the walking to some other motions, such as climbing [9], skipping, jumping, and so on [10]. Although the original Theo Jansen mechanism has two static joints, we modified the mechanism by changing a static joint to the movable joint in a systematic way, providing flexible trajectories in Fig. 2. The further problems is a safety assurance to avoid breaking of the linkage and locking the movement. The present paper proposed a way to detect the limitation to exceed the normal range of motion, by using determinant analysis. In mathematical analysis, Jacobian matrix of the Theo Jansen-like mechanism is obtained through multi-body dynamics approach (MBD). II. MATHEMATICAL MODEL AND EXTENDED LINK MECHANISM A. Multibody Dynamics for Theo Jansen-like mechanism Multibody dynamics (MBD) is the numerical analytical approach that calculates kinematics and dynamics of rigid and deformable objects in the system [11]. This approach can perform numerical calculations: position, velocity, ac- celeration, and force. Walking robots are often constructed by rigid body linkages and artificial muscle actuators. In these cases, MBD gives the mathematical models to the mechanical and structural systems systematically by using kinematic constraints and the Jacobian matrix. In this paper, we describe a mathematical model on the Theo Jansen-like mechanism by MBD in Fig. 3. Let the 2014 IEEE/ASME International Conference on Advanced Intelligent Mechatronics (AIM) Besançon, France, July 8-11, 2014 978-1-4799-5736-1/14/$31.00 ©2014 IEEE 76