Considerations over the Mechanisms that ProvideSnake-like Locomotion

Iulian Iordachita

University of Craiova, Faculty of Mechanics,I.C.Bratianu 165, 1100 Craiova (ROMANIA)

E-mail: iord_ii@mecanica.ucv.ro

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Abstract: In this paper we approach the structure, the kinematics and two experimental models ofdragging locomotion mechanisms. The structure of these mechanisms must be in concordance whit thestudied biomechanisms structure and the specific character of reproduce locomotion. The mechanismsthat can provide dragging locomotion can be considered as an assembly made of the following basestructural modules: dragging kinematic chain (DKC) based on a substratum through a support device(SD), driving motor (DM) and the motion transmission and transforming mechanism (MTTM). Tosolve the kinematic analysis of these mechanisms we implement two computational algorithms. Toelaborate these algorithms we have considered the structure of these mechanisms and the specificcharacter of snakes’ locomotion. For exemplification, we present the kinematic analysis for onemechanism that mimics snakes locomotion. This mechanism includes one dragging kinematic chain(DKC) that is built with dragging elementary chain (DEC) type Rz. We also present the results of theexperimental research regarding mechanisms for dragging locomotion. This research had the purpose tovalidate the principle of dragging locomotion like a snake and the theoretical calculation principles. Twoexperimental mechanisms for dragging locomotion similar to snake were conceived and realized.Satisfactory rapprochement between the theoretical and experimental results was established as a resultof this research

1. INTRODUCTION

e biomechanisms, which provide dragging loco-tion, could be included in the category of biologicctures type vertebral column. Such a structure is

med by serial connection of some reduce lengthments through elastic entities which allow relativelyuced angular displacements. The constitutive ele-nts can be rigid, in this case being named vertebrae the case of vertebrates) or flexible, the worm’s case,ng named segments.

e functions of these structures are:patial sustaining and guidance – vertebral column intebrates’ case;atial guidance and sustaining – giraffe’ s neck;atial guidance – insects’ antennas, scorpion’s tail;

ragging locomotion – snakes, worms, etc.

HE STRUCTURE OF BIOMECHANISMS FORDRAGGING DISPLACEMENTS

e structure of such biomechanism presents a largeiety due to the huge number of species that use thise of locomotion [1]. For this reason we will limity to the apodeu’ study, whose natural locomotion is

dragging. From all apodeus we have chosen, in thispaper, snakes due to their performances in dragginglocomotion.

The snakes do not present a real locomotion system.Body specific movements provide their displacement.As concerning the skeleton (fig. 1) [9], only theendoskeleton is present, presenting a deep positioninside the body. Only the axial skeleton can beremarked: skull, vertebral column and ribs.

Fig. 1

The vertebral column is formed from a large number ofvertebrae (160-400). The amount of bending which cantake place between any two adjacent vertebrae is about25 deg from side to side and 25-31 deg in the verticalplane, some 13 deg of ventral and 12-18 deg of dorsalflexion being possible [2]. The muscles of the trunk andtail, which do all the locomotion work, are remarkablycomplicated.

3. STRUCTURE OF THE MECHANISMS THATPROVIDE DRAGGING LOCOMOTION

The mechanisms that provide dragging locomotion canbe considered as an assembly made of the followingbase structural modules [5], [6]: dragging kinematicchain (DKC) based on a substratum through a supportdevice (SD), driving motor (DM) and the motiontransmission and transforming mechanism (MTTM).

The function of DKC is to provide the ensemble of theliberty of movements, imposed by dragging locomotion.DKC is a fundamental kinematic chain, made ofrotation and/or translation couples. The problem is thatwe have to find also in the case of DKC someelementary identical structures of which repeating toresult the entire DKC. These elementary structures willbe named dragging elementary chains (DEC). 63 typesof DEC-s have been created. In [5] we present theobtained alternatives. For exemplification we presentsome DEC with three liberty degrees (table 1) and withfour liberty degrees (table 2).

The driving motor destination is, of course, that ofgenerating movement. The destination of MTTM is thatof transmitting the movement produced by the motor tothe driving kinematic joint. The role of the supportingdevice (SD) is to provide the contact between themechanism and the surface of support. For a snake, thiscontact is provided by the scales situated on its ventralsides and for the earthworms by the tegument of thesegments [1]. From the technical point of view, the SDcan be replaced by wheels, caterpillars, skis or sets ofneedles. These elements need not be motors, thepropulsion force being generated by the specificmovements of the DKC. In this case unidirectionalwheels and caterpillars and passive needles can be used.

4. THE KINEMATIC ANALYSIS OFMECHANISMS THAT PROVIDE SNAKE-LIKE

LOCOMOTION

One considers in Figure 2 a mechanism that mimicssnake's locomotion. This mechanism includes onedragging kinematic chain (DKC) that is built with onedragging elementary chain (DEC) with indefinite type.One knows the geometry of links and the kinematicparameter's variation in driving joints. As a result, oneknows the order of connection with the base for themechanism links. In Figure 2 one considers the n+1links articulate to the base in contact point Dn+1, on rightside in advance direction.

The computational algorithm for the kinematic analysisof this mechanism it is presented in Figure 3 [4].

Fig. 3

As one shows, this algorithm can be use to thekinematic analysis of any snake-like locomotionmechanism. An alternative for this algorithm ispresented in Figure 4. This last algorithm is less exactlythan the previous, for speeds and accelerations calculusbecause of the approximations of interpolation. At sametime, the algorithms of Figure 4 present the advantageof one improved calculus speed in comparison with thefirst algorithm. In the case of the mechanisms thatmimic snake's locomotion, which include a largenumber of elements, the calculus speed is decisive.

Further on, for exemplification, we present thekinematic analysis for one mechanism that mimicssnakes locomotion.

One considers the mechanism presented in Figure 5.This planar mechanism includes six identical elementsjointed through five rotating joints. In accordance withthe systematization proposed in [5], this mechanism

Fig.2 (LET=DEC)

includes one dragging kinematic chain (DKC) that isbuilt with dragging elementary chain (DEC) type Rz.

Fig. 4

The extreme points of elements, that are placed inmechanism median line, were noted A, B, ..., G and theextreme points placed in laterally, was noted S2, D2, ...,S6, D6.

To solve the kinematic analysis of this mechanism weuse an algorithm similar to the last presented algorithm.

One knows the initial positions for the mechanismpoints A, B, ..., S6, D6 in relation with the fixedreference frame, the elements lengthsAB=BC=...=FG=0,175m, BS2 = BD2 = ... = FD6 = l2=0,110 m, the angles values a12=-9°, a23=-3°, a34=3°,a45=9°, a56=15°.

For the positional analysis of this mechanism we use therelations presented in [5], [3]. The positions of theframes attached to each element, in relation with thefixed reference frame are taken from matrices A, B, ...,D6. To understand the calculus algorithm we present inFigure 6 the matrices scheme that ensures the passingamong the frames attached to the mechanism elements

According to the Figure 6, the passing between framesB and A is ensured with matrix ABA, the passingbetween frames B and C is ensured with matrix A12,a.s.o.

.

The matrices AjD, j=2, 3, 4, 5, 6 have the expression:

where:- ajd - the rotating angle between the frame attached tothe element j in extreme point placed in laterally, rightside and the frame attached to the same element inmedian point.

The matrices AjS, j=2, 3, 4, 5, 6 have the expression:

Fig.5Fig.6

[ ]

⋅+⋅−⋅−⋅

=

10000cos(ajd)sin(ajd)cos(ajd)l2sin(ajd)l10sin(ajd)cos(ajd)sin(ajd)l2cos(ajd)l10001

AjD

(1)

[ ]

⋅−⋅−⋅+⋅

=

10000)cos()sin()cos(2)sin(10)sin()cos()sin(2)cos(10001

ajsajsajslajslajsajsajslajsl

AjS

(2)

where:- ajs - the rotating angle between the frame attached tothe element j in extreme point laterally placed, left sideand the frame attached to the same element in medianpoint.

For start position, the angles values are known. For theother positions the values must be calculated. In Figure5 we consider the segment S6D6 perpendicular onsegment FG.

For the positional analysis of this mechanism, we mustknow the kinematic parameters variation in drivingjoints. In this paper, we consider a parabolicallyvariation of the parameters in driving joints a12, a23, ...,a56 (see Figure 7). On the basis of Figure 7 result theorder for jointed to base of the mechanism elements.

Based on the calculus scheme in Figure 6 and usingMATLAB 4.0 software package, we implemented asoftware program. With this program we calculate thetrajectories and the positions of anyone mechanisms'points.

For initial position, presented in Figure 5, with theprevious initial values and for diagrams presented inFigure 7, we obtain:- in Figure 8 the trajectories of the points A, B, ...D6, forone working cycle;- the trajectories of the points A (Fig.9), B (Fig.10), C(Fig.11), D, E, F and G, for three working cycles;- the trajectories of the points S3 (Fig.12), S4 (Fig.13)and S5, for three working cycles;- the trajectories of the points D2 (Fig.14), D3 (Fig.15)and D5, for three working cycles.

The numerical values of the mechanism point’spositions were registered and used further on like inputdata in another program. With this program, based onpolynomial interpolation, we calculate and plot thevelocities and accelerations diagrams. Some of theobtained results are presented in Figure 16.

Fig.9 Point A trajectory

Fig.7 The angles variation in driving joints

Fig.10 Point B trajectory

Fig.11 Point C trajectory

Fig.8 The points trajectories

5. EXPERIMENTAL MODELS OFMECHANISMS THAT ENSURE DRAGGING

LOCOMOTION

To validate the theoretical principles previously stated,were conceived and realized two experimental modelsthat ensure dragging locomotion similar to snake [7].

First experimental model is presented in Figure 17. Thetwo component elements can perform relativeoscillation with amplitude of 30o. The mechanism isactuated by means of an electrical motor for the windscreen cleaner of the OLTCIT automobile. To obtain theoscillation angle of 30o, lengths of equalizer and leverwere modified.

Fig. 17

Fig.12 Point S3 trajectory

Fig.13 Point S4 trajectory

Fig.14 Point D2 trajectory

Fig.15 Point D3 trajectory

Fig.16 The abscissa, speed vx and acceleration axvariation of point G

Based on the first experimental model was conceivedand realized the second one, of which layout ispresented in Figure 18.

Fig. 18

As it can be observed, locomotion structure is made ofsix modules, five of them identical, and one extreme,shorter. The five identical modules have mounted onthem the operation motors. The structure has a length,with collinear elements, of 1080 mm and a weight of9.350 kg.

On the first and on the last element were mountedcounter-weights that increase the pressing force onextreme "legs". It was attempted that pressing force onearth "leg" to be approximately the same to realize equalfriction forces with the soil. Motion was made with 12V DC motors from the windscreen cleaner of DACIAautomobile. Details regarding construction andoperation of this mechanism are presented in [5] and[7].

To realize the locomotion of structure there is necessarythat between the elements of the structure to be anoscillation movement. Achievement of this movementof oscillation was made by inversion of the rotatingdirection of motors, inverting the sense of feedingcurrent at the extreme position of angular stroke.

6. PROGRAM OF EXPERIMENTAL RESEARCH

To study working of the mechanism that ensure snakelike locomotion several experiments were carried out[8]. These experiments consisted in displacement of theexperimental model on different type of plane surfaces:-surface with textile support (fig. 18);-synthetic carpet (fig. 19);-wood plank (fig. 20);-melamine surface;-mosaic surface;

To visualize the marks made by mechanism, maize andcorn flour were spread on the displacement surface.Although, to make necessary measurements, on thesupport surfaces was traced a square network. As a

result of these experiments, the first finding was thatthrough the oscillation movements of elements, onereported to the other, structure can move ahead. Thisfinding represents, in fact, validation of the proposedlocomotion principle.

During locomotion, median axis of mechanism is acurve (in fact a fractured line that may be approximatedto a curve) of which curve radius depends of theoscillation angles between the elements and of thefriction forces between the elements and the support.

Influence of the friction forces over the behaviour ofmechanism during locomotion is decisive. As a result ofthe experiments it was establishes that on slipperysurfaces (melamine, mosaic) the amplitude of themovements of elements is high and the mechanism isnot advancing, in these case friction forces being small.In case of rugged surfaces (carpet, wood) when frictionforces are high, amplitude is small and structure isadvancing slowly. On the surfaces with textile support,when friction forces are smaller than in the beforementioned case, the mechanism is displacing withhigher speed, movements of the elements being moreample.

All the experiments were shoot. To shoot, a videocamera type PANASONIC VHS-C, PAL system with25 frames/sec. was used. The images recorded onmagnetic tape were studied on a color monitor though avideo player having the possibility to reproduce imagesframe by frame. Same of the images were processed oncomputer trough a Movie Machine Pro. processor. Onthe obtained images, using the square network made onthe support surface, were measured the angles betweenthe elements and the coordinates of the points of the

Fig.19

Fig.20

mechanism. Then, the experimentally obtained valueswere compared with the values theoretically obtained.

7. INTERPRETATION OF THE OBTAINEDRESULTS

To compare the experimental results with theoreticalresults it was used a set of images (for example seefigure 21) in which successive positions of themechanism on a plane surface whit textile support arepresented.

Shooting was made with a speed of 25 frames/sec.Videotape was rewinded frame by frame and eachfourth frame was held. Thus, time between the heldedimages is 4/25 seconds. On the images obtained in thisway, with a protractor, were measured the anglesbetween the axis of the elements. Based on these valuesby mean of a program realized in MATLABprogramming media, variation diagrams of the angles ofleading joints were traced, diagrams presented in Figure22.

Although, reported to the presented systems of axis,were measured by mean of a rule the coordinates ofpoints A, B, ..., G and, tacking into account the scale ofimages (approximately 1/166 [m]/[mm]) values wereobtained.

To establish the theoretical values of the coordinates ofthe points alone mentioned a calculation programpresented in was used.

As input data the real lengths of elements were used:AB=0.178m, BC=0.174m, CD=0.176m, DE=0.174m,EF=0.174m, FG=0.175m, BD2=...=FS6=l2=0.1m.

Based on the diagrams of figure 22 and following to theanalysis of the set of images, order of lateral points ofelements considered to base was established: D5, S4,(on two intervals) S6, D2, D4, S5, D3, D6, S2. Initialcoordinates of point D5 were measured on the firstimage xD5=0.726m, yD5=-0.098m.

Following the analysis of the computer using the abovementioned data, resulted the values of the coordinates ofpoints A, B, ..., G varying with time.

Based on these values several diagrams were obtained(see figures 23, ...,27) which are comparativelypresented variation in time of coordinates of points A,B, ..., G, experimentally respective theoreticallyobtained.

Analyzing the values of the diagrams presented in alonefigures, some conclusions are arising:-during a cycle, theoretical values of the abscissa ofpoints A, B, ..., G increase faster than experimentalvalues;-during a cycle, maximum deviation of theoreticalvalues of abscissa of points A, B, ..., G compared to theexperimental values have the values 1.58% for xG,1.37% for xF, 2.31% for xE, 2.64 for xD, 4.89% for xC,6.33% for xB, 21.8% for xA;

Fig.21

Fig.22

Fig.23

Fig.24

-increase of the value of maximal deviation of abscissaof point G (point of the front of the mechanism) towardspoint A (point of the back) may be on the account of themeasuring errors of these lengths. These errors are dueto the fact that, during shooting, video camera wasplaced over the front element of the mechanism;-in case of variation of ordinates of analyzed points,errors of theoretical values compared to the

experimental ones, are much higher than in case ofabscissa. However it is to emphasize the fact that in caseof yG, yE, yD an even yC diagrams, theoretical curveshave the looking like those of experimental curves;-large differences between theoretical and experimentaldiagrams, in case yF, yB on yA must be due to thefriction forces between the elements of the mechanismand substratum. These forces, of which variations arenot known, determine changing of points of mechanismconsidered to base.

Based on the obtained results, was made a livelinessprogram in TURBO PASCAL, that allows visualizationof the movements of mechanism on the display. Thus,for the theoretical values of the coordinates of themechanism resulted the positions of the medium axis offigure 28.

As it might be found, displacement of theoreticalmechanism is a like displacement of the experimentalone.

Some explanations must be given regarding thedeviations between the theoretical and experimentalvalues of the coordinates of the mechanism:- at the beginning for measurements were used imagesof mechanism shoot at lapse of time of 5/25 sec.Computer program adapted at the data obtained basedon these images led to theoretical values much differentthan the experimental one;- to continue, study of the film was tacked back andwere selected successive images at lapse of time of 4/25sec. This time the differences were smaller but carriageof the theoretical and experimental curves were verydifferent;- finally were used images at a larger scale (with almostdouble resolution) such way that experimental

Fig.25

Fig.26

Fig.27

0 sec

0,16sec

0.32sec

0,48sec

Fig.28

measurements were more accurate, obtaining the abovementioned results.

8. CONCLUSIONS

The mechanisms presented in this paper reproduce themacroscopic structure of dragging locomotionbiomechanisms. These mechanisms are able to providesnakes-like locomotion.

The aim of this research was to validate the principle ofdragging locomotion similar to snakes and thetheoretical principles of kinematics calculus of analyzedmechanisms.

As we shown, to reach the proposed aims, wereconceived and realized two experimental models ofmechanisms that ensure dragging locomotion similar tosnakes.

By mean of these experimental models some aspectsrelated to the kinematics and structure of mechanismsable to displace similar to snakes were clarified.

All experiments were shoot with a video camera.Consequently to the analysis of film were selected aserial of images that were processed on computer andfinally, reproduced on printer. On the images obtainedin this way were made measurements related with theposition of the points of the mechanism and of theangles of elements. For the same sequence ofdisplacement of mechanism using a calculation programpresented before adapted to the concrete condition ofthe analyzed mechanism were determined the positionsof the same points. Finally experimentally obtainedvalues were compared with the values obtainedtheoretically.

As a result of these comparisons it was established asatisfactory approach of the theoretical. Experimentalresults could be more precise by fitting the mechanismwith some active position transducers and by using acommand system led by a computer.

In spite of the mentioned minuses, we consider that theresults of the experimental research validated the resultsof the theoretical research presented by this work.

REFERENCES

[1] Alexander, R. Mcn., Locomotion of Animals.Chapman & Hall, New York, 1982;

[2] Bellaris, A., The Life of Reptiles. vol.I, Weidenfeldand Nicolson, London, 1969;

[3] Iordachita, I., Aspects Regarding the Kinematics ofthe Dragging Locomotion Mechanisms. The SeventhIFToMM International Symposium on Linkages andComputer Aided Design Methods - Theory and Practiceof Mechanisms - Bucharest, Romania, 1997, vol. III. pp.281-284;

[4] Iordachita, I., Considerations over the Kinematics ofMechanisms that Provide Snake-like Locomotion.XXIIth Yugoslav Congress of Theoretical and AppliedMechanics, YUCTAM′97, Vrnjacka Banja, Yugoslavia,1997;

[5] Iordachita, I., Contributions to the Study ofMechanism for Dragging Locomotion, Based upon theStudy of Locomotion in Animal World. Ph.D.Dissertation. University of Craiova. 1996;

[6] Iordachita, I., Contributions Regarding Structure ofMechanisms that Ensure Dragging Locomotion.Proceedings of the 9th International Workshop onRobotics in Alpe-Adria-Danube Region, RAAD’2000,University of Maribor; Slovenia, 2000, pp. 35-40;

[7] Iordachita, I., Dragging Locomotion Mechanism -Experimental Model. Proceedings of the 8th

International Workshop on Robotics in Alpe-Adria-Danube Region, RAAD’99, TU Munchen, DE, 1999;

[8] Iordachita, I., Experimental Research RegardingMechanisms for Dragging Locomotion. The SeventhIFToMM International Symposium on Linkages andComputer Aided Design Methods - Theory and Practiceof Mechanisms - Bucharest, Romania, 1997, vol. III. pp.273-280;

[9] Storer, T., Usinger, R. Elements of Zoology. Mc.Graw-Hill Book Company. Inc., New York, 1955;

Table 1 (partial)

No Type Kinematic opportunity oflocomotion

Kinematic diagram

1 RxRyRzTx locomotion like snakesand like earthworms

2 RxRyRzTy locomotion like snakesand like earthworms

3 RxRyRzTz locomotion like snakesand like earthworms

Table 2 (partial)

No Type Kinematic opportunity oflocomotion

Kinematic diagram

1 RxRyRzTxTy locomotion like snakesand like earthworms

2 RxRyRzTxTz locomotion like snakesand like earthworms