Journal of Automation and Control, 2016, Vol. 4, No. 2, 10-14 Available online at http://pubs.sciepub.com/automation/4/2/1 ©Science and Education Publishing DOI:10.12691/automation-4-2-1 Base Space of Nonholonomic System Tomáš Lipták * , Michal Kelemen, Alexander Gmiterko, Ivan Virgala, Ľubica Miková Department of Mechatronics, Faculty of Mechanical Engineering, Technical University of Košice, Košice, Slovakia *Corresponding author: tomas.liptak@tuke.sk Abstract The article deals with the issue of use of geometric mechanics tools at modelling nonholonomic systems. The introductory part of article contains theory of geometric mechanics that we use at creating mathematical model of nonholonomic locomotion system with undulatory movement. Further it contains the determination of reconstruction equation for three-link snake-like robot where we consider ideal source of velocity. The relation between changes of shape and position variables was expressed using the local connection. After determination of controllability of kinematic snake, in last part we created reduced base dynamic equations in case when base variables do not represent ideal source of velocity. Keywords: reconstruction equation, connection, reduced base dynamic equation, reduced Lagrangian, snake-like robot Cite This Article: Tomáš Lipták, Michal Kelemen, Alexander Gmiterko, Ivan Virgala, and Ľubica Miková, “Base Space of Nonholonomic System.” Journal of Automation and Control, vol. 4, no. 2 (2016): 10-14. doi: 10.12691/automation-4-2-1. 1. Introduction Classical treatment of multibody system dynamics usually leads to long and complicated equations of motion in case when these equations are expressed in scalar form. These equations are often not suitable for the analysis and design of control. Recent development in geometric mechanics provides a powerful tool for the formulation of equations of motion and understanding of the important features of their dynamics. These results can be summarized in the following points: 1. dynamics of multibody system is independent of certain base configurations, which are defined using group variables - this feature is called as symmetry, 2. for mechanical system with symmetries is possible to obtain simplified equations of motion by removing these invariances - this process is called as reduction, 3. the separation of group and base variables has many advantages at formulation of equations, investigation of dynamic properties and at analysis and design of control. An important consequence of symmetries is the existence of conserved quantities, e.g. momentum or energy, according to Noether’s theorem. These conserved quantities, described by algebraic equations, are useful to study dynamics and control problems for multibody systems. On the one hand we can use of these quantities to obtain a smaller number of equations, thus simplifying the equations of motion and on the other hand, these quantities impose constraints on system states. This is crucial to recognize in controllability analysis. Imitation of snake-like locomotion in robotics is an important area of research with respect to high stability and good terrainability compared with wheeled and legged robots. Significant is also their robustness to mechanical failure due to high redundancy. Motion patterns of biological snakes therefore serve as a source of inspiration for snake-like locomotion. To be able to analyze these motion patterns, suitable mathematical models are need. In general, based on observing animal locomotion in nature such as walking of horses or swimming of fishes is possible to find out that the locomotion arises due to change of body shape and its interaction with the environment (Figure 1). This fact is the basis for creating models of robot locomotion. Figure 1. Principle of locomotion In this article we deals with modelling and analysis of flat surface locomotion with sideslip constraints, where the principle of modeling is based on the assumption, that snake body cannot perform a lateral movement, it follows that, it is necessary to establish to motion equations of snake-like robot nonholonomic constraints first kind. In real model of robot we obtain these nonholonomic constraints using wheels. This fact at biological snake is explained by anisotropy of skin friction and surface irregularity on which the snake moves [1]. For creating model of locomotion we inspired by works [2,3,4,5], in which authors tried to research basic mathematical structures that are common for all locomotion systems.