Lu Zhenli(盧振利),Ma Zhipeng,Marko Pencic'*,Maja Cavic*,Borovac Branislav*,Roumiana Ilieva,Bojan Nemec*,Marjan Mernik

(*School of Electrical Engineering and Automation,Changshu Institute of Technology,Changshu 215500,P.R.China)

(**School of Information and Control Engineering,China University of Mining and Technology,Xuzhou 221116,P.R.China)

(***Faculty of Technical Sciences,University of Novi Sad,Novi Sad 621000,Serbia)

(****Faculty of Management,Technical University of Sofia,Sofia 1000,Bulgaria)

(*****Department of Automatics,Biocybernetics and Robotics,Jo?ef Stefan Institute,Ljubljana 1000,Slovenia)

(******Faculty of Electrical Engineering and Computer Science,University of Maribor,Maribor 462000,Slovenia)

Abstract

Key words:V-REP,spiral winding swimming(SWS),motion control,slender legless creature(SLC)

0 Introduction

Spiral motion is a kind of constant motion form and phenomenon in nature.For example,in summer,a whirlwind in the field is a very intuitive spiral motion.The spiral motion in the magnetic field is more vibrating and oscillating.These oscillation and wave phenomena are often related to different physical motion functions.It is an abstract drawing that draws wave curve,spiral curve and various circulation lines with mathematical models of different parameters.

Spiral motion can be a form of dynamic motion for creature in liquid environments.Based on the spiral motion of slender legless creature(SLC)in a liquid environment,some simulation experiments are carried out to study the mechanism of spiral motion and to analyze its movement performance.This work can provide the platform and technical reserves for the study of spiral motion of slender legless bionic snakes and other creatures in water.

1 Related work

Most researches are related to spiral motion performance analysis of slender legless biological creatures in liquid,such as the analyses of the bionic mechanism from aspects,bacteria and sea snakes.

Most bacteria rely on spiral motion to move in the environment.Ref.[1]studied the motion of the caulobacter through tracking technology.The results show that its body seems to move along an invisible spiral tube track,which provides the motive force for limbless microorganisms to move around.Based on the mathematical model of‘resistance theory’,the researchers demonstrated that the thrust of different bacteria can lead to different movements.For bacteria,spiral motion is an effective form of motion,and it is found that the speed of forward motion is faster than backward motion.

As shown in Fig.1,some bacteria(e.g.spirobacteria)can show unique spiral(body spiral)movements.In addition to crawling or creep buckling on the solid surface,these organisms can also adjust the axial fibers around the object to achieve their functions.The chemical composition of axonemes is very similar to flagella,but they are wrapped in spirochetes rather than in vitro.

Fig.1 Helix bacilli

Spiral movement can help sea snakes and sea otters to swim up and down in the ocean,as shown in Fig.2.As for the bionic underwater snake-like robot,the spiral motion is considered to be a new motion for a wide range of application fields.A series of studies on the spiral climbing of snake-shaped robots for bridge cable climbing are conducted in recent years[2-5],which is for demonstrating the practicality of spiral motion in cable testing.The 3D underwater motion control of a snake-like robot is developed in simulation[6-7],and the three-dimensional motion gait of snake-shaped robots,the S-shaped tumbling and spiral tumbling of a real snake-like robot in water[8]are systematically studied.More researchers studied the mechanism design of snake-shaped robots,as well as their different applications[9-11].

Fig.2 Sea snakes swimming

2 Development of SLC model in V-REP

2.1 SLC model

V-REP is adopted as simulation platform to study the helical motion of SLC model.In V-REP simulation platform,cylinder is selected as the main spiral motion model of module.The type of joint is revolute with the dynamic properties of the force/torque model,so as to program to drive to control the joint.Sphere is selected as a carrier of orthogonal joint,as shown in Fig.3,where the vertical joints and horizontal joints are independent movement.

Fig.3 Orthogonal joint

SLC model is developed with 9 modules for the study of spiral motion simulation,as shown in Fig.4.In order to easily distinguish the motion direction,the outer edge of the head module is set to green,and another side is defined as tail.The dynamics parameters of the SLC model are shown in Table 1.

Fig.4 SLCmodel

Table 1 Main parameters in the model

2.2 Define the water dynamic in V-REP

The spiral winding swimming(SWS)motion of the SLCmodel is carried out under the condition of still water,where the environment factors,such as water waves,water velocity,are not considered.In this work,mainly two kinds of force are control the simulation model to realize the SWSin the water.One group force is its own gravity and the buoyancy of water on the model,the other group is generated by SWSof SLC model where the water will produce interaction force between viscous drag[12-13].

The following forces will be analyzed for SWS of SLC model in still water.

(1)The gravityGiof each link in SLC model

where,miis the mass of thei-th module,irefers to the module number,gis the gravitational acceleration.

(2)The buoyancyFiof each link in SLC model

There is no actual surface in the simulation environment,so it is needed to call the SimAddForce function of the software to add a simulation model for the module,so that the reverse direction of gravity is similar to the buoyancy of the actual water environment.Considering that the buoyancy cannot be calculated directly according to the submerged volume,here in each module(perpendicular to the horizontal plane),the approximate position of the centroid ofZaxis is relative to the horizontal plane(Z=0)plus the buoyancyFi.

In this system of the still water,ifZis less than-50,it can be determined by the approximate buoyancy.

(3)The simulation model of the viscous dragf

When the simulation model moves on the water,each module will be affected by the viscous resistance of the water.The viscous resistance model inX,Y,Zcoordinate system is analyzed.The resistance is related to the velocity of the model.In the simulation,the VREP resistance coefficient is set to a negative value,and the component resistances on theX,YandZaxes are

where,vix,viyandvizare the three velocity components of the module speed in the direction of axisx,y,z;scis viscous resistance coefficient which is determined by the fluid state and the surface roughness of the rigid body;miis the mass for each module;cidepends on the height of mass center coordinates relative to the horizontal plane for each module.

3 Control function for SWS motion

In order to achieve the spiral motion control,the Serpenoid curve function proposed to synthesize snake shaped is adopted to generate rhythm of the rotation angle of each joint[14-16].

This function after extension can be used to control spiral motion of SLC model as shown in Fig.5.

Serpentine curve curvature equation is shown as

where,αis the initial amplitude curveting radian,bis the proportionality constant,sis the arc length of the serpentine curves.

where,θ(s)is the angle in the tangent direction along the serpentine curves arc lengths.

Fig.5 Serpentine curve

According to Eq.(5),the joint angleφf(shuō)or connecting adjacent joints of the SLC model is shown as

where 2lis the length of the connecting rod of a single model.

由上表可以看出，在工作滿意度的各個(gè)維度中，同事關(guān)系和工作性質(zhì)這兩個(gè)維度的實(shí)際平均值與理論平均值相差最大，既差值相差最大為4.6分和3.2分，表明在各個(gè)維度中，對(duì)同事關(guān)系和工作性質(zhì)這兩個(gè)維度的工作滿意度相對(duì)較高；在福利報(bào)酬、職業(yè)認(rèn)同感、領(lǐng)導(dǎo)管理、進(jìn)修晉升四個(gè)維度的差值分別為-2.8分，-6.09分，-3.7分，-3.1分，表明在這四個(gè)維度上體育教師的工作滿意度較低，其中職業(yè)認(rèn)同感和領(lǐng)導(dǎo)管理差值較大，說(shuō)明在這兩個(gè)維度上教師滿意度極低；在其他兩個(gè)維度中體育教師對(duì)學(xué)生品質(zhì)、工作環(huán)境的差值分別為0.2分和1.2分，表明實(shí)際平均值與理論平均值差異不大，處于基本滿意狀態(tài)。

According to Eq.(6)to control SLC model to generate the spiral motion,the discretization of the adjacent joints of the formula is shown as

where,iis joint number of the SLC model,αis starting angle,knis s-wave number in the body,Sis displacement of the simulation model along the axis direction,Lis the total length of model,Nis the module number,rad[i]is the correspondingi-th joint rotation angle in radian.The corresponding simulation model of each connecting rod module at a certain moment in the process of movement state is shown in Fig.6.

Fig.6 Connecting rod swing state

In Fig.6,n1,n2,…,n9are respectively simulation model connecting rod module;θ1,θ2,…,θ8are respectively the angle between the adjacent link module from head to tail;the serpentine curve is discrete by linkage massmiwith fixed length 2l.

SWSmotion of SLC model in still water is realized by the rotation of the orthogonal connection joint mechanism,which is a kind of three-dimensional movement gait coupled by vertical and horizontal rotation.Its implementation requires a combination of vertical joints and horizontal joints,which can be obtained from Eq.(7)with the following control function:

where,φv[i]is the vertical deflection angle of thei-th joint,φh[i]is the horizontal deflection angle ofi-th joint,and the phase difference of curve function is π/2,αvandαhare starting angles of serpentine curves respectively applied to vertical joints and horizontal joints.

4 Performance analysis of SWS motion

4.1 Performance analysis of spiral shape

In terms of screw motion control function,the different combinations of initial angleαvandαhdetermine the configuration of initial state simulation model of screw motion and the size of screw radius.

The parameterknin the control function determines the number of S waves in the SLC model,and also determines the number of spirals in the model.The change speed of parametersdetermines the speed of spirals.αvandαhare set as the same values,the experiments are carried out under different screw configurations,as shown in Fig.7.

Fig.7 αv andαh under different radius of a spiral

Whenαvandαhare set as different values,two kinds of situations are discussed.Whenαv>αh,SLC model will generate the spiral winding along the horizontal(X)direction.When the value of angle ofαhis set very small,the simulation model will generate a spiral winding motion alone the horizontal(X)direction.Whenαv<αh,the simulation model will generate the spiral winding along the vertical(Y)direction,the movement is also called the spiral dive up/down swimming.The experiments under differentknspiral coil number are shown in Fig.8.

Fig.8 Experimental results under different kn

4.2 Performance analysis of starting angle

In the spiral winding swimming process,according to the recorded torque information of the vertical and horizontal joints of SLC model,the simulation results are obtained,as shown in Fig.10 and Fig.11.

In the experiment of spiral winding swimming,according to the torque diagram in Fig.10 and Fig.11,in the vertical direction of the dynamic model,joint 4 and joint 5 need larger torque compared with other joints;in the horizontal direction,joint 4 and joint 5 need smaller torque compared with other joints.In the simulation model of spiral configuration,module 4 and module 5 are in two reverse spiral arc joints.In order to maintain the spiral configuration,the torque changes of joint 4 and joint 5 are different from other joints in the vertical and horizontal directions.

Fig.9 SWSmotion(αv=π/2,αh=π/2)

Fig.10 Torque of the vertical joints

Fig.11 Torque of the horizontal joints

During the experiment,the head module center of mass of the dynamic model moves along theXandYdirections in the process of spiral winding,as shown in Fig.12,basically in a straight line direction;meanwhile,the three-dimensional tracks of the model in theX,YandZdirections are shown in Fig.13.

Fig.12 2D trajectory of SWS

Fig.13 3D trajectory of SWS

It can be seen that the trajectory of SLC model under the driving excitation is a spiral roll with lateral resistance.Chooseαv=αh,give different starting angle of helix to carry out the experiment,and the relationship between the spiral winding speed and the starting angle of helix can be obtained,as shown in Fig.14.

The experimental results show that the smaller the initial helix angle is,the smaller the helix radius is,and the lower the helix velocity is.When the initial angle of the spiral is very small gradually,the simulation model can still slowly complete the small radius circular spiral action and move forward.From the point of view of energy consumption,the larger the start angle of the screw controlling the excitation input,the larger the spatial circumferential radius of the screw structure of the SLC model,which will also lead to the greater the joint torque change and the increase of energy consumption.

Fig.14 αv=αh for different spiral starting angles

In the simulation model of spiral winding,whenαv=αh,it is like a circle helical screw configuration.Whenαv≠αh,it is like oval ring spiral screw configuration.If we consider the application in water or on land for similar robots,for this kind of configuration,the simulation model of the helical winding can realize more given direction motion in water,such as spiral rise,and spiral dive in water.This kind of movement gait is not only unstable,but also difficult to control to switch from other movement gaits,such as serpentine locomotion on land and other applications[17-20].The control method of spiral winding is more suitable for the application in water.

Whenαv=π/2,αh=π/8,spiral winding process is shown in Fig.15.Its movement form embodies mainly the swimming along the vertical direction of spiral dive and buoyancy.

Fig.15 Spiral winding process(αv=π/2,αh=π/8)

The experimental results of spiral winding,whereαv=π/8,αh=π/2,are shown in Fig.16,which is the forward movement of spiral winding along the horizontal direction.

According to the movement experiment results of the model given different initial angle input of spiral winding,it can be seen that due to the lack of sufficient water support force,when the vertical and horizontal initial angle of SWS is smaller,the movement effect of SWS is poorer,the efficiency is lower,and the rolling is carried out in situ.The simulation model can realize arbitrary coiling of space without considering the phenomenon of rollover.Its main disadvantages are uncontrollable direction and low efficiency.

Fig.16 Spiral winding process(αv=π/8,αh=π/2)

5 conclusions

Inspired by the spiral motions of different living creature and natural tornado phenomenon,a SWS motion of SLC in water is developed.Firstly,through the mechanism analysis,the spiral motion can be an effective motion of SLC in water.Secondly,the spiral helical radius and the body turns the morphology of spiral motion are analyzed with a math function output controlled the dynamic model of the legless creature.Thirdly,under different bending angle of spiral math function,the performance of SWS motion of the dynamic model is also analyzed.Finally,through changing the value of bending angle of the spiral function,the dynamic model of SLC can realize SWS together with the up and down function.