Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 45, 1, pp. 179-187, Warsaw 2007 WORM-LIKE LOCOMOTION AS A PROBLEM OF NONLINEAR DYNAMICS Klaus Zimmermann Igor Zeidis Technische Universität Ilmenau, Faculty of Mechanical Engineering, Germany e-mail: igor.zeidis@tu-ilmenau.de Non-pedal forms of locomotion show their advantages in inspection tech- niques or applications to the medical technology for diagnostic systems andminimally invasive surgery.Observing the locomotion ofworms, one recognizes conversion of (mostly periodic) internal and internally dri- venmotions into a change in the external position (undulatory locomo- tion). This paper presents some theoretical and practical investigations of worm-like motion systems that have the earthworm as a live pro- totype. The locomotion of worms can be described by introduction of nonlinear non-symmetric frictional forces. In the first part of the paper these systems are modelled in form of straight chains of interconnected mass points. The ground contact can be described by non-symmetric dry friction. The investigations are concentrated onmotion in a tube or channel, and onmotion on a horizontal plane as well. In both cases, the body is modeled as a viscous Newtonian fluid. The effect of the mass flow through a cross section on the disturbance andmaterial parameters (viscosity, dimensions) is discussed. The paper presents first prototypes of technically implemented artificial worms. Key words: worm-like motion, modelling, oscillations, non–symmetric friction, peristaltic motion 1. Introduction Biologically inspired locomotion systems are currently dominated by walking machines, i.e., systems performing pedal locomotion. Non-pedal forms of lo- comotion show their advantages in inspection techniques or applications to the medical technology for diagnostic systems and minimally invasive surge- ry. Observing the locomotion of worms, one recognizes conversion of (mostly periodic) internal and internally driven motions into a change in the external 180 K. Zimmermann, I. Zeidis position (undulatory locomotion). For realization of this type of locomotion, non-symmetry in external friction forces acting on the system is needed. The motion of a chain of mass points placed on a rough straight line and connected consecutively by equal linear viscoelastic elements under action of a non-symmetricCoulombdry frictional force and related by unilateral differen- tial constraints was described by Steigenberger et al. (2003), Zimmermann et al. (2001, 2002), Steigenberger (1999). In the case of three masses, an expres- sion for velocity of ”slow” motion is obtained (Zimmermann et al., 2002). A limiting case of non-symmetric friction, when motion is possible only in one direction (realized by means of scales that prevent backward displacement of points of the ground contact) was proposed by Miller (1998) in connection with a realistic computer animation of worms. A thorough discussion of such systems, where the pointmasses can also be equippedwithmassless steerable runnersdescribed via knife-edge conditions,was given in Steigenberger (1999). 2. Motion under action of nonlinear friction and periodic external forces We consider themotion of n=2k+1mass points in a common straight line. We suppose that all of them except the middle one are equipped with scales contacting the ground. The middle mass is subjected to a harmonic external force −MΦ0 sinΩt (Fig.1). The equations of motion are ẍ1+ω 2(x1−x2)+F(ẋ1)= 0 . . . ẍk+ω 2(2xk −xk−1−xk+1)+F(ẋk)= 0 ẍk+1+ω 2 0(2xk+1−xk−xk+2)+Φ0 sinΩt=0 (2.1) ẍk+2+ω 2(2xk+2−xk+1−xk+3)+F(ẋk+2)= 0 . . . ẍ2k+1+ω 2(x2k+1−x2k)+F(ẋ2k+1)= 0 where ω2 = c m ω20 = c M Φ0 > 0 The function F(ẋs) describes non-symmetric dry friction, i.e., the frictional force is taken tobedifferent inmagnitudedependingon thedirection ofmotion Worm-like locomotion as a problem of nonlinear dynamics 181 of the body. F(ẋs) may be specified as follows F(ẋ)=        F+ for ẋ > 0 F0 for ẋ=0 −F− for ẋ < 0 (2.2) where F− > F+ ­ 0 are fixed, whereas F0 may assume any value in the interval (−F−,F+). Fig. 1. Following the method of direct separation of motion (Zimmermann et al., 2001), we seek for a solution xi to system (2.1) in the form (i=1, . . . ,2k+1) xi(t)=Xi(t)+Ψi(t,Ωt) (2.3) Here Xi denotes the ”slow” component ofmotion xi, whereas Ψi is the ”fast” one. Ψi is assumed to be a smooth periodic function of the ”fast” time τ =Ωt (Ω≫ω) having the average value zero 〈Ψi(t,τ)〉= 1 2π 2π ∫ 0 Ψi(t,τ) dτ =0 We assume that the amplitude of the external force is Φ0 ≫ max(F−,F+), then the equations for ”slow” motion take the final form Ẍ1+ω 2(X1−X2)+V ∗ 1 (Ẋ1)+ 1 2 (F+−F−)= 0 . . . Ẍk+ω 2(2Xk −Xk−1−Xk+1)+V ∗ k (Ẋk)+ 1 2 (F+−F−)= 0 Ẍk+1+ω 2 0(2Xk+1−Xk−Xk+2)= 0 (2.4) Ẍk+2+ω 2(2Xk+2−Xk+1−Xk+3)+V ∗ k (Ẋk+2)+ 1 2 (F+−F−)= 0 . . . Ẍ2k+1+ω 2(X2k+1−X2k)+V ∗ 1 (Ẋ2k+1)+ 1 2 (F+−F−)= 0 182 K. Zimmermann, I. Zeidis where V ∗i (Ẋ)=                  1 2 (F−+F+) for Ẋ >ΘiΩ 1 π (F−+F+)arcsin Ẋ ΘiΩ for |Ẋ| ¬ΘiΩ − 1 2 (F−+F+) for Ẋ <−ΘiΩ (2.5) and Θi =Θi(Φ0,Ω,ω,ω0), i=1, . . . ,k. There is a steadymotionwith constant velocities Ẋ∗1 = . . .= Ẋ ∗ 2k+1 = Ẋ ∗, where Ẋ∗ is determined from the equation k ∑ i=1 V ∗i (Ẋ ∗)= k(F−−F+) (2.6) By means of (2.5), it yields an equation for the velocity of the steady ”slow” motion. Equation (2.6) should be solved numerically and, what is different from the case with three masses where two of which have contact with the ground (Zimmermann et al., 2002), the steady motion does not always exist. Thus in the considered case the ”slow” motion of the chain as a single whole is not always possible. If F+ = F−, then Ẋ ∗ = 0 which follows from (2.6). That means that in the case of symmetrical friction the system ”in average” does notmove at all. On the other hand, if F− >F+, there can be a solution Ẋ ∗ > 0, which entails a unidirectional average motion under non-symmetric friction. Fig. 2. For the case of eleven masses, displacement curves vs. time for masses 1(11), 2(10), 6, are shown in Fig.2a, and for masses 4(8), 5(7), 6, in Fig.2b Worm-like locomotion as a problem of nonlinear dynamics 183 (mass 3(9) is not shown). The result follows from the numerical solution of the exact equations of motion (2.1). The following values of parameters have been taken ω=ω0 =1 Ω=10 Φ=10 F+ =0 F− = 1 2 The numerical solution to equation (2.6) for eleven masses gives Ẋ∗ =0.22. 3. Peristaltic motion We consider themotion of a deformable continuum (index 0, ”worm”) within a second medium (index 1), where the extension of the latter is constrained by straight walls. The deformable body whose motion is to be described is bounded by an impermeable skin. Both media are modeled as incompressible linear-viscous media with different viscosities. The transport of the matter (mediumwith index 0) is based on awave-like disturbance of the skin (Fig.3). This peristaltic motion is the subject of present investigation. The amplitude of thedisturbedsurface is assumed tobe small in comparison to the transversal size of the body. The equations of motion for bothmedia and the constraints at the wave-like boundary were given in Zeidis and Zimmermann (2000). For motion within a channel (plane deformation) or within a tube (rotationally symmetric case), Cartesian coordinates (x,y) or cylindrical coordinates (x,r) are used, respectively. TheReynolds numbers of bothmedia and the frequency ofwave-like disturbances are assumed tobe small. Thismeans that the inertial forces are small compared to theviscous forces. Following these considerations, the flow of an incompressible viscous fluid is described by the equations gradp=µ∆v divv=0 (3.1) where v is the velocity vector, p is the pressure and µ – the shear viscosity. Fig. 3. 184 K. Zimmermann, I. Zeidis For both rotational symmetry (motion in a tube) and plane deformation (motion in a channel), a two-dimensional problemoccurs, and from continuity equation (3.1) the existence of a stream function Ψ follows. In the case of rotational symmetry, the stream function Ψ(x,r,t) satisfies a differential equation ( ∂2 ∂r2 − 1 r ∂ ∂r + ∂2 ∂x2 )( ∂2 ∂r2 − 1 r ∂ ∂r + ∂2 ∂x2 ) Ψs =0 s=0,1 (3.2) whereas in the case of plane deformation the stream function Ψ(x,y,t) must be determined from ∂4Ψs ∂y4 +2 ∂4Ψs ∂y2∂x2 + ∂4Ψs ∂x4 =0 s=0,1 (3.3) Let us introduce dimensionless variables (quantitieswith stars have dimen- sions) m= µ1 µ0 δ= h0 h t= t∗ U h Ψ = Ψ∗ Uh (u,v) = (u∗,v∗) U (x,y,r)= (x∗,y∗,r∗) h with U as a characteristic velocity for motion of the boundary surface. The following boundary conditions can be formulated. At the fixed walls of the tube (channel) there holds: u1 = v1 = 0. On the boundary surface described by the equation η(x,t) = δ+ εcos[ω(x− ct)], a kinematic condi- tion and continuity of normal tensions and of the velocities are required. The conditions of symmetry relative to the axis OX must be added as well. Applying perturbation theory, the interesting functions are represented as power serieswith respect to the small parameter ε, e.g., the stream function Ψ Ψs =Ψs0 +εΨ s 1 +ε 2Ψs2 + . . .= ∞ ∑ k=0 εkΨsk s=0,1 (3.4) To see the efficiency of peristaltic motion, the flow q is determined, i.e., the amount of the matter which flows through a given cross-section per time unit (Zeidis and Zimmermann, 2000) q(x,t)= η(x,t) ∫ 0 u0(y) dy For a flow Q within a period λ=2π/(ωc), there holds Q= λ ∫ 0 q(x,t) dt Worm-like locomotion as a problem of nonlinear dynamics 185 For calculation of Q, terms up to the second order in ε are taken into account Q=Q0+εQ1+ε 2Q2 (3.5) Here Q0 = Q1 = 0. How Q depends on ε (which is the amplitude of the disturbed surface divided by the extension of surroundingmedium1) is shown in Fig.4 for different values of m and δ and the parameters ω=0.5, c=1. Fig. 4. 4. Conclusions The locomotion of worms can be described by introduction of nonlinear non- symmetric frictional forces or by imposing unilateral differential constrains which providemotion only in one direction. The ”slow”motion of the system acted upon by ”small” non-symmetric dry friction and the ”fast” external force are equivalent to motion under viscous friction and a constant external force. The mass flow in the case of a viscous fluid is proportional to the square of amplitude of the surface disturbance. Based on the model-aided analysis of biological motion systems, technical systems with functionally defined rigid and compliant structures are develo- ped. Someworm prototypes applying the principles outlined above have been constructed and proved positive. Future considerations should focus onpossibilities of realizingwave-like de- formations with a given amplitude and frequency of a surface using ferrofluids (Zimmermann et al., 2004). 186 K. Zimmermann, I. Zeidis References 1. Blekhmann I.I., 2000, Vibrational Mechanics: Nonlinear Dynamic Effects, General Approach, Applications,World Scientific 2. Miller G., 1998, Themotion dynamics of snakes andworms,Computer Gra- phics, 22, 4, 169-173 3. Steigenberger J., 1999,On a Class of Biomorphic Motion Systems, Faculty of Mathematics and Natural Sciences, Ilmenau Technical University, Preprint M12/99 4. Steigenberger J., Zeidis I., Zimmermann K., 2003, Model of worm-like motion as a basis of creation of new microrobots, International Advanced Ro- botics Program. Proceeding ofWorkshop onMicro Robots,MicroMachines and Micro Systems, Moscow, 89-86 5. Zeidis I., Zimmermann K., 2000, Ein mathematisches Modell für die peri- staltische Bewegung als Grundlage für das Design wurmartiger Mikroroboter, Technische Mechanik, 20, 1, 73-80 6. Zimmermann K., Zeidis I., Naletova V.A., Turkov V.A., 2004, Waves on the surface of a magnetic fluid layer in a travelling magnetic field, Journal of Magnetism and Magnetic Materials, 268, 227-231 7. Zimmermann K., Zeidis I., Steigenberger J., 2002, Mathematical model ofworm-likemotion systemswithfinite and infinitedegreeof freedom,Romansy 14. Theory and Practice of Robots and Manipulators, 507-515 8. Zimmermann K., Zeidis I., Steigenberger J., Huang Jianjun, 2001,An approach to the modelling of worm-like motion systems with finite degree of freedom–first steps in technical realization,Proceeding of the 4th International Conference of Climbing and Walking Robots, Karlsruhe, 561-568 Ruch robaczkowy jako zagadnienie dynamiki nieliniowej Streszczenie Znanaw naturze forma transportu i przemieszczania się z miejsca namiejsce bez użycia odnóży, tzw. forma niekrocząca, kryje w sobie ogromny potencjał aplikacyjny w technikach badawczych oraz medycynie, zwłaszcza w diagnostyce i nieinwazyjnej chirurgii.Wobserwacjipełzania robakówstwierdza siękonwersję ruchuwewnętrznego, lubwewnętrznie indukowanego (głównie periodycznego)wmierzalne przemieszczenie zewnętrzne (tzw. ruch falujący).Wpracy przedstawiono badania teoretyczne i uwagi praktyczne na temat sztucznych układów przemieszczających się ruchem robaczko- wym, których żywym prototypem może być zwykła dżdżownica. Taki rodzaj trans- portu opisano poprzez wprowadzenie do modelu matematycznego nieliniowej i nie- symetrycznej charakterystyki tarcia.W pierwszej części pracy układy zamodelowano Worm-like locomotion as a problem of nonlinear dynamics 187 prostymi łańcuchami punktówmaterialnych. Kontakt z podłożem odwzorowano nie- symetryczną siłą tarcia suchego.Analizę skoncentrowanona pełzaniuw rurze i kanale oraz ruchu po poziomej powierzchni.Wewszystkich przypadkach poruszające się cia- ło zamodelowano jako ciecz Newtonowska. Przedyskutowano efekt przepływu masy przez przekrój poprzecznyna zakłóceniaw ruchu i zmianę parametrówciała (lepkość, wymiary). Zaprezentowano również pierwsze prototypy sztucznych robakówprzezna- czonych do zadań technicznych. Manuscript received August 3, 2006; accepted for print August 18, 2006