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Research Article International Journal of Advanced Robotic Systems September-October 2016: 1–9 A geometric approach to modeling of ª The Author(s) 2016 DOI: 10.1177/1729881416663714 four- and five-link planar snake-like robot arx.sagepub.com Toma´sˇ Lipta´k, Ivan Virgala, Peter Frankovsky´, Patrik Sˇ arga, Alexander Gmiterko, and Lenka Balocˇkova´ Abstract The article deals with the issue of use of geometric mechanics tools in modelling nonholonomic systems. The introductory part of the article contains fiber bundle theory that we use at creating mathematical model of nonholonomic locomotion system with undulatory movement. Further the determination of general mathematical model for n-link snake-like robot is presented, where we used nonholonomic constraints. The relation between changes of shape and position variables was expressed using the local connection that was used to analyze and control system movement by vector fields. The effect of links number of snake-like robot on its mathematical model was investigated. The last part of this article consists of detailed description of modeling reconstruction equation for four- and five-link snake-like robot. Keywords Snake-like robot, fiber bundle, nonholonomic constraint, reconstruction equation Date received: 9 May 2016; accepted: 20 July 2016 Topic: Special Issue - Manipulators and Mobile Robots Topic Editor: Tomas Brezina Introduction According to the study by Liljeba¨ck et al.,1 the research in area of snake-like robot locomotion modeling is focused Different methods of locomotion were studied separately, and on the following areas: most approaches in the analysis process and design relied on the specific characteristics of given locomotion system. 1. biomechanical studies of biological snakes; However, despite their differences, most robots share com- 2. modeling and analysis of flat surface locomotion mon features. Most of the robots today use wheels and legs, with sideslip constraints; but there are other classes of robots such as snake-like robots. 3. modeling and analysis of flat surface locomotion Imitation of snake-like locomotion in robotics is an without sideslip constraints; important area of research with respect to high stability and 4. modeling and analysis of robotic fish and eel-like good terrainability compared with wheeled and legged mechanisms; and robots. Significant is also their robustness to mechanical 5. modeling and analysis of locomotion in environ- failure due to high redundancy. Motion patterns of biological ments with obstacles. snakes therefore serve as a source of inspiration for snake- like locomotion. To be able to analyze and control these motion patterns, suitable mathematical models are need. In general, based on the observation of animal locomotion in Technical University of Kosˇice, Kosˇice, Slovakia nature such as walking of horses or swimming of fishes, it is possible to find out that the locomotion occurs due to change Corresponding author: Toma´sˇ Lipta´k, Technical University of Kosˇice, Letn 9, Kosˇice 04200, of body shape and its interaction with the environment. This Slovakia. fact is the basis for creating models of robot locomotion. Email: [email protected] Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License (http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage). 2 International Journal of Advanced Robotic Systems Our article deals with area no. 2 in which the principle of modeling is based on the assumption that snake body can- not perform a lateral movement, it follows that, it is nec- essary to substitute nonholonomic constraints first kind to motion equations of snake-like robot. In real model of robot, we obtain given nonholonomic constraints using wheels. In biological snake, this fact is explained by aniso- tropy of skin friction and surface irregularity on which the snake moves.2 Unfortunately, introducing nonholonomic constraints to conventional methods such as Newton—Euler equations or Lagrange equations is quite problematic. In the last 30 years, in the context of rapid developments of differential geometry and global analysis and their application to Figure 1. The scheme of using of kinematics and dynamics at mechanics, this topic has been intensively studied and new control the locomotion system. mathematical models to deal with nonholonomic systems had been invented.3 SE (2) for the movement in plane and this group is typically For creating model of locomotion we inspired by works represented as a homogenous matrix: SE(2). by Shapere and Wilczek,4 Murray and Sastry,5 Kelly and 2 3 cos Àsin x Murray,6 and Ostrowski,7 in which authors tried to 4 sin cos y 5: research basic mathematical structures that are common 001 for all locomotion systems. These research studies have led to the general principle of locomotion that says if We are able to express the total body movement based certain variables in locomotion system change periodi- on the relation between shape variables and external con- cally, movement of the whole locomotion system occurs. straint forces that arise through the interactions of animal The mathematical object, which describes effectively this with its environment. One type of movement that works on phenomenon, is called connection, which is object used in the mentioned principle is called undulatory locomotion. theoretical physics. A key result of this approach is recon- Although this is a relatively primitive and simple move- struction equation for nonholonomic systems, which ment, it can be significantly robust. Not surprising, that is gives to correlation local velocity of locomotion system used across the entire biological spectrum from moving and velocity of changes its internal shape for broad group bacteria to reptiles. of locomotion systems. The classical treatment of multi- Accordingtodefinitionofundulatory locomotion is the body systems dynamics provides length and complicated process that generates displacements of locomotion system equations of motion and these equations are often not via a coupling of internal deformations to an interaction suitable for control analysis and design. Geometric between the robot and its environment. As from this definition mechanics offers a powerful tool for formulating equa- implies, generally undulatory locomotion requires some form tions of motion and understanding important properties of interaction with environment, which we will model as a of their dynamics. constraint. These constraints can have several forms, such as: Although now foundations of the theory of nonholonomic locomotion systems are quite well established, there is still a viscous friction; lack of solved models of locomotion systems appearing in rolling of wheels without slipping; and robotics. In this article, we use mathematical framework interaction of surface with a viscous liquid or air. connection and established techniques from geometric mechanics to derive the reconstruction equation for four- Although constraint forces do not perform mechanical and five-link planar snake-like robot moving on planar work, that is, they do not supply the energy to system, they surface. allow to convert energy generated by shape variables to the kinetic energy of group variables. With the examples of undulatory locomotion, we can The fiber bundle theory include worms, snakes, amoeba, and fish. In this article, we The research studies concerning application of unconven- restrict interactions between the robot and its environment on tional locomotion of technical systems were always nonholonomic kinematic constraints. This constraint allows us inspired by nature. We can state that animal locomotion to model large group of systems and it leads to good structures.8 in homogeneous environment is based on periodic change As we already know animal locomotion is produced by in their internal shape, by which displacement occurs in the shape change of body and its reaction with the environ- their body. This displacement in their body can be mathe- ment. A gait decides about the locomotion, which can be matically determined by element of Lie group, for example, described by kinematics, as it is shown in Figure 1. Lipta´ketal. 3 where i is the force function. When we have k constraints, we can write their as set k vector-valued equations i j !jðqÞq_ ¼ 0; for i ¼ 1; ... ; k (3) This group of constraints includes the most frequently investigated nonholonomic constraints. The constraints can be incorporated into the dynamics using Lagrange multi- pliers. That is, the equation (2) is modified by adding the constraint of force with unknown multiplier d @L @L i À þ j! À i ¼ 0(4) dt @q_i @qi j Figure 2. Modelling process based on fiber bundle theory. Q is Although we use Lagrange multipliers to eliminate the the bundle manifold, G is the structure group, M is the base constraints for obtaining the dynamic description of sys- manifold and (1, 2, ...) is the joint angles. tems, it was found out that fiber bundle theory is quite useful. In the presence of symmetries, we can rewrite the equations of motion into the following form According to definition, the gait is a continuous closed À1 curve in the base space M, that is9: gÀ1g_ ¼AðrÞr_ þ I~ ðrÞp (5) 2 : R + M; t 7! r; such; that ðtÞ¼ðt þ Þ; t; R: 1 T T 1 T

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