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]
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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 sin 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 þ j! 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 1 curve in the base space M, that is9: g 1g_ ¼ AðrÞr_ þ I~ ðrÞp (5)
2 : R + M; t 7! r; such; that ðtÞ¼ ðt þ Þ; t; R: 1 T T 1 T p_ ¼ r_ r_r_ðrÞr_ þ p pr_ðrÞr_ þ p ppðrÞp (6) (1) 2 2 Thus the gait is periodic and so after each period of time M~r€þ r_T C~ðrÞr_ þ N~ðr; r_; pÞ¼TðrÞ (7) , the system returns to the same point in the base space and The reconstruction equation (5) and the momentum in each cycle has the same shape. ’ is the smooth map of equation (6) represent set of l first-order differential equa- real interval on a manifold M.Wewillassumethatallbase tions, where l is dimension of the fiber space G.The variables are fully actuated, that is, we can independently reduced base dynamic equation (7) is set of m second- control each of the base configuration variables and also order differential equations, where m is dimension of the we assume that the base variables represent the ideal base space M. Thus, we reduce original set of n (m l) source of velocity. ¼ þ second-order dynamic equation of motion obtained from The effect of environmental constraint force on locomo- equation (4) to set of 2l first-order differential equations tion can be described with dynamics. If we have given gait for and m second-order differential equations.7,12 locomotion system and the environmental constraint force, Individual equations (5), (6), and (7) are shown using then we can calculate the torques of actuators through the block diagram in Figure 3. inverse dynamics and we can control these torques to adjust the side constraint force based on the inverse dynamics. Using fiber bundle theory, we obtain reduced dynamic Snake-like robot models equations. The modeling process based on fiber bundle In the following sections, we will show how to build a 10 theory is shown in Figure 2. mathematical model of n-link kinematic snake. Then we The principle fiber bundle Q represents the configura- will discuss the case of four- and five-link snake and we tion space of locomotion system. Given that the locomotion will consider the situation when the base variables repre- group is SE and is also a Lie group, it responds to the sent the ideal source of velocity. structure group G of the principle bundle. Then we have enough nonholonomic constraints that can be used to con- N-Link kinematic snake struct the connection of the fiber bundle which responds to the kinematics of locomotion system. Using the redundant The undulatory locomotion of the n-link kinematic snake nonholonomic constraints, we can reduce the dynamics will be provided by modifying the angle size a1, ... , an-1, from the shape space to the gait space and finally we get where the wheel axes are rigidly held perpendicular to the optimal torques by minimizing the constraint force.10,11 the links (Figure 4). The position and orientation of kine- When we work with mechanical systems, we will assume matic snake is represented in the plane by the three fiber the existence of Lagrangian function L(q, :q) on the tangent variables (x, y, ) SE (2), and its configuration space is bundle TQ. In the absence of constraints, the dynamic equa- Q ¼ G M ¼ SEð2Þ ðn 1ÞS (8) tions can be derived from the Lagrange equations The n-link kinematic snake has three degrees of freedom d @L @L i ¼ 0(2)givenbyvariables(x, y, )andn 1 shape variables dt @q_i @qi a1, ..., an 1, while we have only two independent 4 International Journal of Advanced Robotic Systems
Figure 3. Representation of reduced base dynamic equation, reconstruction equation, momentum equation and their relation expressed using block diagram.
Figure 4. The simplified scheme of n-link kinematic snake.
L L x_ sinð a Þ y_cosð a Þþ _ð1 þ cosa Þ a_ ¼ 0 2 2 2 2 2 2 (11) Figure 5. Scheme representing mathematical model of principally kinematic systems. x_ sinð a2 a3Þ y_cosð a2 a3Þ L _ L þ ð1 þ cosða2 þ a3Þþ2cosa3Þ a_ 2ð1 þ 2cosa3Þ variables a1, a2, and a3, ..., an 1, we obtain by combina- 2 2 tion of variables a1 and a2. This means, that n-link kine- matic snake belongs to under-actuated nonholonomic L a_ 3 ¼ 0 ð12Þ mechanical systems first order. 2
This n-link kinematic snake belongs to the category of prin- x_ sinð a2 a3 a4Þ y_cosð a2 a3 a4Þ cipally kinematic systems, where to build the reconstruction L _ þ 1 þ cosða2 þ a3 þ a4Þþ2cosða3 þ a4Þþ2cosa4 equation fully suffice nonholonomic constraints (Figure 5). For 2 L this type of snake, we have n nonholonomic constraints and a_ 2 1 þ 2cosða3 þ a4Þþ2cosa4 thus we get n equations, while the fourth to the nth equation 2 L L represents only the dependence between shape variables. a_ 3ð1 þ 2cosa4Þ a_ 4 ¼ 0 ð13Þ 2 2 For creating the mathematical model–reconstruction x_ sinð a ... a Þ y_cosð a ... a Þ equation, we will use the approach where we work with non- 2 i 1 2 i 1 iP 1 iP 1 iP 1 holonomic constraints expressed in world coordinates (Fig- L _ þ 1 þ 2 cos aj þ cos aj ure 6). For n-link kinematic snake, we have n nonholonomic 2 k¼3 k 2 constraints and we will express constraints for the first five iP 2 iP 1 iP 1 L links and for ith link with respect to coordinates x and y a_ t 1 þ 2 cos ap 2 t¼2 tþ1 p¼tþ1 L _ L x_ sinð þ a1Þ y_cosð þ a1Þ ð1 þ cosa1Þ a_ 1 ¼ 0 L 2 2 a_ i 1 ¼ 0 ð14Þ 2 (9) Equation (8) represents a general notation of nonholo- x_ sin y_cos ¼ 0 (10) nomic constraint for ith link and we have been able to Lipta´ketal. 5
Figure 6. Nonholonomic constraints in world variables. determine this shape of constraint after determining five Four- and five-link kinematic snake constraints. In this part, we will show the creation of reconstruction Now we rewrite nonholonomic constraints to Pfaffian equation for four- and five-link kinematic snake. form with the use of substitution, where by using r , we will 1 The undulatory locomotion of the four-link kine- mark the first two shape variables a1 and a2 and using r2 10 matic snake will be provided by modifying the angle other shape variables a3, ..., an 1 : size a1, a2,anda3 (Figure 7). The position and orienta- tion of kinematic snake is represented in the plane by A B 0 r_ 0 the three fiber variables (x, y, ) SE and its config- þ 1 1 ¼ (15) A~ B~1 B~2 r_2 0 uration space is
Q ¼ G M ¼ SEð2Þ S S S (19) Then we state the general form of the reconstruction equation using local coordinates The four-link kinematic snake has three degrees of "#freedom given by variables (x, y, )andthree shape vari- 1 A B 1 ables a1, a2,anda3, while we have only two independent ¼ 1 1 r_1 (16) r_ ~ ~ 1 ~ ~ variables a1, a2,anda3, we obtain by combination of 2 B2 AA B1 B2 B1 variables a1 and a2. This means, that the four-link If we want to obtain the reconstruction equation expressed snake-like robot belongs to under-actuated nonholonomic using world coordinates, we will use the following relation mechanical systems first order. 2 3 2 32 3 x The four-link kinematic snake belongs to the cos sin 0 g_x y category of principally kinematic systems,whereto 4 5 ¼ 4 sin cos 0 54 g_ 5 (17) y build the reconstruction equation fully suffice non- 001g_ holonomic constraints. For this type of snake, we have using which we transform the body velocity to the world four nonholonomic constraints—equations (9) to (12) velocity (Figure 8). "#Nonholonomic constraints, equations (3) to (6), we 1 I3 0 A B express in Pfaffian form g 1g_ þ r_ ¼ 1 r_ (18) 0 I 2 ~ 1 ~ 1 ~ 1 ~ 1 m B2 AA B1 B2 B1
2 3 L L 6 sinð þ a1Þ cosð þ a1Þ ð1 þ cosa1Þ 0072 3 6 2 2 7 2 3 6 7 x_ 6 76 7 0 6 sin cos 0 00076 y_ 7 6 0 7 6 L L 76 _ 7 6 7 6 sinð a2Þ cosð a2Þ ð1 þ cosa2Þ 0 0 76 7 4 5 ¼ 6 2 2 76 7 0 6 74 a_1 5 0 6 L 7 a_2 6 ð1 þ cosða2 þ a3Þ L L 7 4 sinð a a Þ cosð a a Þ 2 0 ð1 þ 2cosa Þ 5 a_3 2 3 2 3 2 3 2 þ2cosa3Þ ð20Þ 6 International Journal of Advanced Robotic Systems
Figure 7. The simplified scheme of four-link kinematic snake.
Figure 8. Nonholonomic constraints in world variables.
After multiplying the previous equation and separating group from shape variables, we obtain the reconstruction equation in the form
2 3 L L 2 3 6 cos ðcosa2 þ 1Þ cos ðcosa1 þ 1Þ 7 6 2 2 7 x_ 6 7 6 7 1 6 7 6 y_ 7 6 L L 7 a_ 1 4 _ 5 ¼ 6 sin ðcosa2 þ 1Þ sin ðcosa1 þ 1Þ 7 D 6 2 2 7 a_ 2 a_ 4 5 3 sina2 sina1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflsina2 sina3 þ sinða2 a3Þ sinða1 a2Þþ sinða2 a3Þþfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}sina2 sinða1 a2 þ a3Þ
A ð21Þ where D ¼ sina1 sina2 þ sinða1 a2Þ and A represents connection. Now we express reconstruction equation expressed using local velocities
2 3 2 3 x L L 6 ðcosa2 þ 1Þ ðcosa1 þ 1Þ 7 6 y 7 6 2 2 7 6 7 1 6 7 a_ 1 4 5 ¼ 6 007 D 4 5 a_ 2 sina2 sina1 a_ 3 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflsina2 sina3 þ sinða2 a3Þ sinða1 a2Þþ sinða2 a3Þþfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}sina2 sinða1 a2 þ a3Þ
A ð22Þ Lipta´ketal. 7
Figure 9. Serpentine movement of four-link kinematic snake with shown trajectory of the mass center.
As mentioned earlier, the last equation expresses only Q ¼ G M ¼ SEð2Þ S S S S (24) dependence shape variable a3 on a1 and a2. The serpentine movement of four-kinematic snake This five-link kinematic snake has three degrees of freedom simulated in program SolidWorks2015 during one period givenbyvariables(x, y, )andfour-shape variables a1, a2, a3, (2 s) is shown in Figure 9, where shape variables change and a4, while we have only two independent variables a1, a2, according to equation and a3, a4, we obtain by combination of variables a1 and a .Thismeansthatthefive-link snake-like robot belongs to a ¼ 40 sin t; a ¼ 40 sinðt þ 25Þ (23) 2 1 2 under-actuated nonholonomic mechanical systems first order. and dependence of these shape variables represents gait. The five-link kinematic snake belongs to the category of The travelled distance of snake at serpentine movement principally kinematic systems, where to build the recon- during one period is 27.23 mm. struction equation fully suffice nonholonomic constraints. The undulatory locomotion of the five-link kinematic For this type of snake, we have five nonholonomic con- snake will be provided by modifying the angle size a1, straints—equations (9) to (13) (Figure 11). a2, a3, and a4 (Figure 10). The position and orientation of After modifying nonholonomic constraints and separat- kinematic snake is represented in the plane by the three ing group from shape variables, we obtain the reconstruc- fiber variables (x, y, ) SE and its configuration space is tion equation in the form
2 3 L L 6 cos ðcosa2 þ 1Þ cos ðcosa1 þ 1Þ 7 2 3 6 2 2 7 x_ 6 7 6 7 6 L L 7 6 y_ 7 6 sin ðcosa2 þ 1Þ sin ðcosa1 þ 1Þ 7 1 6 2 2 7 a_ 1 6 _ 7 ¼ 6 7 6 7 6 7 a_ 4 5 D 6 sina2 sina1 7 2 a_ 3 6 7 6 sina2 sina3 þ sinða2 a3Þ sinða1 a2Þþ sinða2 a3Þþ sina2 sinða1 a2 þ a3Þ 7 a_ 4 4 5 sinða2 a3Þþ sinða3 a4Þ sinða1 a2 þ a3 a4Þ sinða2 a3Þ þ sina3 sinða2 a3 þ a4Þþsinða1 a2 þ a3Þ sinða2 a3 þ a4Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} A ð25Þ where D ¼ sina1 sina2 þ sinða1 a2Þ, or using local velocities: 8 International Journal of Advanced Robotic Systems
Figure 10. The simplified scheme of five-link kinematic snake.
Figure 11. Nonholonomic constraints in world variables.
Figure 12. Serpentine movement of five-link kinematic snake with shown trajectory of the mass center.
2 3 L L 6 cos ðcosa2 þ 1Þ cos ðcosa1 þ 1Þ 7 2 3 6 2 2 7 x 6 7 6 y 7 6 L L 7 6 7 6 sin ðcosa2 þ 1Þ sin ðcosa1 þ 1Þ 7 1 6 2 2 7 a_ 1 6 7 ¼ 6 7 6 7 6 7 a_ 4 5 D 6 sina2 sina1 7 2 :a3 6 7 6 sina2 sina3 þ sinða2 a3Þ sinða1 a2Þþ sinða2 a3Þþ sina2 sinða1 a2 þ a3Þ 7 :a4 4 5 sinða2 a3Þþ sinða3 a4Þ sinða1 a2 þ a3 a4Þ sinða2 a3Þ þ sina3 sinða2 a3 þ a4Þþsinða1 a2 þ a3Þ sinða2 a3 þ a4Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} A ð26Þ Lipta´ketal. 9
The serpentine movement of five-kinematic snake simu- Funding lated in program SolidWorks2015 during one period (2 s) The author(s) received no financial support for the research, is shown in Figure 12, where shape variables change authorship, and/or publication of this article. according to equation (23). The travelled distance of snake at serpentine movement References during one period is 25.65 mm. 1. Liljeba¨ck P, Pettersen K Y, Stavdahl O, et al. A review on As we could notice, upon increasing the number of modelling, implementation, and control of snake robots. links, we always get the same structure of reconstruction Robot Auton Syst 2012; 60: 29–40. equation. The addition of links leads to decrease in tra- 2. Hu D, Nirody J, Scott T, et al. The mechanics of slithering veled distance during one period, because we have only locomotion. Proc Natl Acad Sci 2009; 106: 10081–10085. two control variables—shape variables and other shape 3. Stolarova´ M.Examples of nonholonomic systems. In: 10th variables only follow control variables and do not perform Czecho-Slovak Student’s Research Competition in Mathe- any work. matics and Informatics, Kosˇice, Slovakia, May 2009. 4. Shapere A and Wilczek F. Geometry of self-propulsion at low reynolds number. J Fluid Mech 1989; 198: 557–585. Conclusion 5. Murray R and Sastry S. Nonholonomic motion planning: steering using sinusoids. IEEE Trans Autom Control 1993; The aim of this article was to introduce the use the geo- 38(5): 700–715. metric mechanics at modeling nonholonomic locomotion 6. Kelly S and Murray RM. Geometric phases and robotic loco- systems. At first we discussed about modeling process motion. J Robot Syst 1995; 12(6): 417–431. based on the fiber bundle theory that we used to create 7. Ostrowski J. The mechanics and Control of Undulatory mathematical model of locomotion systems. We deduced Robotic Locomotion. Thesis, California Institute of Technol- mathematical model that is called as reconstruction equa- ogy Pasadena, USA, 1995. tion in general form for n-link snake-like robot. The 8. Ostrowski J, Burdick J, Lewis A, et al. The mechanics of detailed process of creating reconstruction equation was undulatory locomotion: the mixed kinematic and dynamic presented on the four- and five-link snake. The main advan- case. In: Robotics and Automation, 1995. Proceedings., tage of forming these created reconstruction equations is 1995 IEEE International Conference on, 21–27 May 1995, existence of connection. Individual components of connec- pp. 1945–1951. IEEE. tion we can show using vector fields and then we can 9. Shammas E. Generalized Motion Planning for Underactu- analyse the shape of vector fields to the shape of gait by ated Mechanical Systems. Thesis, Carnegie Mellon Univer- which we obtain the ability to predict how designed loco- sity Pittsburgh, Pennsylvania. motion system will move in the plane. 10. Guo X, Ma S, Li B, et al. Modeling and optimal torque control of a snake-like robot based on the fiber bundle theory. Acknowledgements Sci China Informat Sci 2015; 58: 032205:1–032205:13. 11. Turygin YV, Zubkova JV and Maga D. Investigation of The work has been accomplished under the research projects No. kinematic error in transfer mechanisms of mechatronic sys- VEGA 1/0872/16 ‘‘Research of synthetic and biological inspired tem. In: 15th International Conference on Mechatronics, locomotion of mechatronic systems in rugged terrain’’ and the grant project Tatra bank: DidacticBot—‘‘Innovation of educa- MECHATRONIKA 2012, Prague; Czech Republic, 5 tional process of robotics and mechatronics.’’ December 2012 through 7 December 2012, ISBN: 978- 800104987-7. 12. Mamrykin OV, Kuznetsov AP and Yakimovich BA. Declaration of conflicting interests Simulation model of the new articles mastering process in The author declared no potential conflicts of interest with respect machine-building. Avtomatizatsiya i Sovremennye Tekhnolo- to the research, authorship, and/or publication of this article. gii, Izdatel’stvo Mashinostroenie 2004; 5: 27–33.