Mathematical model of inchworm locomotion

International Journal of Non-Linear Mechanics 76 (2015) 56–63

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Mathematical model of inchworm locomotion Raymond H. Plaut n Department of Civil and Environmental Engineering, Virginia Tech, Blacksburg, VA 24061, USA

art ic l e i nf o

a b s t r a c t

Article history: Received 4 April 2015 Received in revised form 22 May 2015 Accepted 23 May 2015 Available online 3 June 2015

Inchworms are caterpillars. Their locomotion, involving arching of much of the central portion of their body length, has not been studied as extensively as the peristaltic locomotion of worms or the crawling locomotion of many other caterpillars. A mathematical model is developed to describe the shapes and bending strains of typical inchworm motions. The inchworm is assumed to travel in a straight line on a rigid horizontal substrate. Two basic types of cycles are considered. In Case I, the inchworm body arches and then reverses that motion in becoming flat again. In Case II, the body arches, then cantilevers upward, and then falls down to a flat shape. A continuum model based on an elastica is adopted. The results may be useful in the development of soft robots exhibiting an inchworm mode of motion. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Inchworm Locomotion Arching Elastica

1. Introduction Inchworm locomotion involves alternate releasing and grasping of the front and rear sets of legs, and arching (sometimes called looping or inching) incorporating much of the animal's length (https://www. youtube.com/watch?v=cyasgr9mn3s). The mechanics of the locomotion of an inchworm has received relatively little attention compared to that of similar animals [1,2]. Some inchworm configurations from [3] are reproduced in Fig. 1. The arching shape of an inchworm resembles the post-buckled shape of a continuous flexible beam with fixed (clamped) ends. A beam with such large displacements is often analyzed as an elastica, in which the beam is elastic and the bending moment is assumed to be proportional to the curvature [4–7]. In the present paper, this concept is utilized to develop a mathematical model of the shapes and bending strains of an inchworm. Inchworms are not worms, but caterpillars [8]. Worms typically move using peristalsis, with axial (longitudinal) waves propagating along their bodies and utilization of frictional resistance with the substrate [9,10]. Caterpillars contain a number of sets of short legs: true (thoracic) segmented legs in the head section, and prolegs (unsegmented appendages) in the central and tail sections [11–14]. Caterpillar bodies are segmented, with intersegmental membranes connecting the segments [2]. Many caterpillars exhibit crawling motions that involve arching of a small part of the body, with the arched portion moving in a wave toward the head [8,11,15–23]. Inchworms do not contain some of the central prolegs that exist in

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http://dx.doi.org/10.1016/j.ijnonlinmec.2015.05.007 0020-7462/& 2015 Elsevier Ltd. All rights reserved.

crawling caterpillars, and inchworm locomotion includes arching that involves much of the length of the body (Fig. 1b). Inchworms are the larvae of moths of the family Geometridae and are sometimes called loopers, spanworms, or measuring worms [11,21]. A cycle of motion begins as seen in Fig. 1(a), with the inchworm on the substrate. During the first phase (Phase 1) of the cycle, most of the central portion exhibits arching, as sketched in Fig. 1(b), with the head section remaining stationary (anchored) and the tail section moving toward the head. Two basic types of motion can occur during the second phase (Phase 2). For both, the tail section remains stationary. In Case I, the sequence of Phase 1 is reversed until the arching is eliminated. In Case II, the head moves away from the substrate till the central and head sections are straight or almost straight, as shown in Fig. 1(c), and then the raised (cantilevered) portion moves back toward the substrate until the end of the cycle when the configuration in Fig. 1(a) is attained again. These two parts of Phase 2 will be designated Phases 2A and 2B, respectively. Inchworm locomotion tends to be several times faster than crawling motion [11,21]. One potential application of the results of the present paper is in the burgeoning field of “soft robotics” [24–33]. In 2013, the International Workshop on Soft Robotics and Morphological Computation was held in Switzerland. In 2014, workshops on Soft Robots and on Soft Medical Robots were conducted during the IEEE International Conference on Robotics and Automation (ICRA) in Hong Kong, a workshop on Soft Robotics took place at the Robotics Science and Systems Conference (RSS) in Berkeley, soft robots were on display at the International Conference on Intelligent Robots and Systems (IROS 2014) in Chicago, and a new journal entitled Soft Robotics was created. Flexible inchworm-type robots were described in [14,34]. Some papers use terms like “inchworm-like robots”, “inchworm robot”,

R.H. Plaut / International Journal of Non-Linear Mechanics 76 (2015) 56–63

or “inchworm motion” in their titles, but involve robots that crawl and do not exhibit any or much arching [35–40]. Some robots exhibit arching behavior but are not soft, often containing rigid segments connected by joints [18,41–47]. Further papers describing robots and other devices inspired by inchworm locomotion include [48–52]. Some robots resembling earthworms or caterpillars are used for purposes such as endoscopies, colonoscopies, and inspection of pipes [37,39,53–57]. A discussion of the effect of friction on earthworm-type robots is presented in [58]. Shape memory alloy (SMA) wires are sometimes employed to actuate the locomotion of robots imitating inchworms [2,14,19,27,34,59]. It is possible to predict the bending strains required for such actuations with the use of the type of analysis presented here. The mathematical model will be formulated in Section 2. Results for Cases I and II will be described in Sections 3 and 4, respectively. Bending strains will be discussed in Section 5, followed by concluding remarks in Section 6.

2. Formulation The arched shape of an inchworm on top of a rigid horizontal surface resembles that of the upward buckling of a thin, flexible, horizontal beam with clamped ends and subjected to end shortening (i.e., the ends are moved toward each other). Here the inchworm model is based on a uniform inextensible elastica, for which the material is linearly elastic and the bending moment is proportional to the curvature. The lateral view of the undeformed model is shown in Fig. 2. The weight of the inchworm is neglected [21], so the orientation of the substrate and the direction of arching relative to the substrate are irrelevant. Dynamic effects (inertia, energy dissipation) are also neglected [27]. Configurations in a vertical plane are considered, with the inchworm assumed to move along a straight line. The inchworm's depth (height when flat) is denoted D, which is the diameter if the cross section is circular. The inchworm's length is the sum of the lengths Lh of the head section, L of the central section, and Lt of the tail section. The head and tail sections are assumed to remain straight and to have a continuous slope at their connections with the central section, so that the analysis will only involve the central section. An arched shape of the centerline of

Fig. 3. Arched shape of centerline of central section of model.

this section is depicted in Fig. 3, with zero slopes at its ends. At arc length S, the inchworm has horizontal position X(S), vertical position Y(S), and rotation θ(S). The horizontal base length is B, with B ¼L when the inchworm is lying on the substrate. The analysis is conducted in terms of the following nondimensional quantities: S X Y B D s¼ ; x¼ ; y¼ ; b¼ ; d¼ ; L L L L L Lt Lh ML P o L2 ℓt ¼ ; ℓh ¼ ; m ¼ ; po ¼ : EI L L EI

ð1Þ

In Eq. (1), M is the bending moment (positive if counter-clockwise on a positive face), EI is the bending stiffness, and P0 is the internal force in the horizontal direction (positive if compressive). Due to symmetry and vertical equilibrium, no vertical reaction forces will exist at the ends of the central section, and hence there will be no vertical component of internal force. From geometry dx ¼ cos θ; ds

dy ¼ sin θ: ds

ð2Þ

For an inextensible elastica, the constitutive law and equilibrium provide the equations [5] dθ ¼ m; ds

dm ¼  po sin θ: ds

ð3Þ

The boundary conditions at s ¼0 are x ¼y¼θ¼ 0, and at s¼1 they are x ¼b and y ¼θ¼0. If a uniform inextensible elastica on a rigid horizontal foundation has fixed ends and the ends are displaced toward each other, the beam buckles upward into a symmetrical shape. These buckled shapes were computed here using the subroutines NDSolve and FindRoot in Mathematica [5]. For the nondimensional base length b¼0.32, the resulting rotation θ(s) is plotted as the solid curve in Fig. 4. Rotations were computed for a number of base lengths between b¼0.3 and b¼1. The shapes θ(s) are similar to a sinusoid, and an approximation for the amplitude f(b) was obtained using the subroutine Non linearModelFit. The resulting approximation is θðsÞ ¼ f ðbÞ sin ð2πsÞ;

Fig. 1. Schematic of inchworm configurations, from [3] with permission: (a) initial flat configuration; (b) arched configuration; and (c) cantilevered configuration.

57

0rsr1

ð4Þ

with 2

3

f ðbÞ ¼ 2:22  0:281b 4:533b þ 6:385b 4

 3:494b

if

0:3 r br 0:912

ð5Þ

0:912 o b r1:

ð6Þ

and

Fig. 2. Lateral view of model of inchworm in initial configuration.

f ðbÞ ¼ 2:02

pffiffiffiffiffiffiffiffiffiffiffi 1b

if

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R.H. Plaut / International Journal of Non-Linear Mechanics 76 (2015) 56–63

Eqs. (5) and (6) are plotted in Fig. 5 over the range 0.3 rbr1. The approximation given by Eqs. (4)–(6), based on the elastica analysis, is utilized in the rest of this paper. For b ¼0.32, it furnishes the dashed curve in Fig. 4, with a maximum difference of 2.5%; this difference between the approximation and the result from the elastica analysis decreases as b increases.

3. Case I The shape of the centerline of the central section of the inchworm is obtained by solving Eqs. (2) and (3) numerically, and then plotting y versus x. Some resulting shapes for Phase 1 are presented in Fig. 6. The horizontal axis is the initial flat shape. The dot-dashed shape corresponds to a nondimensional base length b¼ 0.9, the solid shape to b¼ 0.7, the dashed shape to b¼0.5, and the dotted shape to b¼0.32. For a given nondimensional depth d of the cross section, there is a minimum allowable value of nondimensional base length b, denoted bmin, for which the front and back halves of the arched inchworm touch each other (i.e., self-contact occurs). Computations lead to the curve in Fig. 7, which can be approximated by the cubic equation 2

3

bmin ¼  0:0542 þ 0:1688d þ 1:9226d  2:0616d

if

Fig. 6. Nondimensional shapes of centerline of central section during Phase 1 (Cases I and II) for b ¼1 (horizontal x axis), b ¼0.9 (dot-dashed), b ¼0.7 (solid), b¼ 0.5 (dashed), and b ¼0.32 (dotted).

0 r d r 0:19: ð7Þ

Consider a particular example having d¼0.13. Then Eq. (7) yields bmin ¼0.32. Assume that Phase 1 ends when b¼bmin. Also assume for this numerical example that the nondimensional head and tail lengths, respectively, are ℓh ¼ 0:26 and ℓt ¼ 0:26, so that the nondimensional total length is 1.52. During Phase 1 for Cases I and II, based on the present analysis, this inchworm (including head and tail sections) exhibits the shapes depicted in Fig. 8(a–e), respectively, for b¼1, 0.9, 0.7, 0.5, and 0.32 (when self-contact occurs). The maximum height of the centerline is denoted ymax, and the horizontal distance that the midpoint of the central portion (where

Fig. 7. Minimum value of nondimensional base length b of central section as function of nondimensional cross-sectional height d.

ymax occurs) moves relative to its initial flat position in Phase 1 is denoted as the marker xm. Therefore, in Phase 1 xm ¼

ð1  bÞ : 2

ð8Þ

For the example, xm ¼0.34 at the end of Phase 1, and an approximation for ymax during Phase 1 is given by the formula 0:4 0:5 ymax ¼  10:79x0:3 m þ 64:421xm  142:864xm 0:7 þ 142:388x0:6 m  52:931xm

Fig. 4. Rotation θ of arched shape of centerline of central section as a function of nondimensional arc length s for nondimensional base length b ¼ 0.32. Solid curve is based on an elastica analysis; dashed curve is the approximation employed in this paper.

ð9Þ

which is plotted as a function of xm in the left half of Fig. 9. At xm ¼0.34, Eq. (9) yields ymax ¼0.394. During Phase 2 for Case I, the tail section is stationary and the rest of the inchworm moves to the right in the reverse sense of Figs. 6 and 8. Starting with b¼ bmin, the base length increases until b¼1 (when the inchworm is flat again). The base length b is replaced by 2bmin  b in Eq. (8), and xm is replaced by 1  bmin  xm in Eq. (9). Hence, for the example, b is replaced by 0.68  b in Eq. (8) and xm is replaced by 0.68  xm in Eq. (9). The marker xm increases from 0.34 to 0.68 in Phase 2, and the maximum height is given approximately by Eq. (9) with xm replaced by 0.68  xm, as plotted in the right half of Fig. 9. During a cycle, based on this analysis, the inchworm moves forward a nondimensional distance 1 bmin. In the numerical example, this is 0.68, corresponding to a distance 0.68 times the length L of the central part and 0.45 times the total length Lh þ LþLt. This is true for both Cases I and II.

4. Case II

Fig. 5. Function f in Eqs. (5) and (6) as a function of b for the range 0.3 r br 1.

Phase 1 is the same for Case II as for Case I, and is illustrated in Figs. 6 and 8. Phase 2 for Case II is divided into two parts. In Phase 2A, the head of the inchworm moves upward and forward until it reaches its maximum height. In this paper the upper part is assumed to be straight at the end of Phase 2A, as shown in Fig. 1(c). In Phase 2B, the

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59

Fig. 10. Nondimensional shape of inchworm in Phase 2A of Case II with rotation β of chord connecting ends of arched part of central section.

ends having an angle β with the horizontal, as shown by a dashed line in Fig. 10. In comparison to the analysis for Phase 1, the base length along this line is denoted b2, the axes are x2 and y2, the rotation from the chord is θ2, and the total arc length is 1 r1β (i.e., these quantities correspond to b, x, y, θ, and 1, respectively, in Phase 1). The head section has the angle β with the horizontal. For the circular portion, the rotation from the horizontal is denoted θ with 0 rθr β, and the x and y coordinates are given by x ¼ r 1 sin θ;

y ¼ r 1 ð1  cos θÞ:

ð10Þ

For the rest of the central part, x2 and y2 are computed similarly as in Phase 1 with the appropriate correspondence, and the coordinates x and y are then obtained from Fig. 8. Configurations of example inchworm (having d ¼ 0.13 and ℓh ¼ ℓt ¼ 0:26) during Phase 1 (Cases I and II): (a) b ¼ 1; (b) b ¼0.9; (c) b¼ 0.7; (d) b ¼0.5; and (e) b ¼0.32.

x ¼ r 1 sin β þ x2 cos β–y2 sin β y ¼ r 1 ð1– cos βÞ þ x2 sin β þ y2 cos β:

ð11Þ

During Phase 2A, the angle β is increased until it reaches a specified value βmax. The inclined base length b2 is given by the formula  b2 ¼ bmin þ

Fig. 9. Maximum nondimensional height ymax of example inchworm as a function of marker xm for Case I.

central and head sections move downward until the inchworm is horizontal on the substrate. For Phase 2A, the shape of the centerline for the total inchworm is assumed to have the form sketched in Fig. 10 in nondimensional terms. The origin for the x and y axes is at the beginning of the central section, which has a unit length. The initial upraised portion of this section (adjacent to the tail section) is assumed to be circular with radius of curvature r1 and subtended angle β, so that the arc length for this portion is r1β. The rest of the central section is assumed to be arch-shaped as in Phase 1 but with the line (chord) connecting its

 1  r 1 βmax bmin β : βmax

ð12Þ

For the numerical example, with d¼ 0.13, bmin ¼0.32, ℓh ¼ 0:26, and ℓt ¼ 0:26, it is assumed here that r1 ¼0.1 and βmax ¼ 1 rad. A sequence of centerline shapes for the central part of the inchworm is presented in Fig. 11 for Phase 2A (rising from the rightmost shape in Fig. 6), with β ¼0.25 (lowest shape), 0.5 (dashed), 0.7, 0.9 (dot-dashed), and 1.0. In Phase 2B, the central and head sections descend to the substrate. It is assumed that the initial part of the central section is circular as before, and the remaining part is straight. Shapes of the centerline of the central section for the example are depicted in Fig. 12 for β¼ 1 (as also seen in Fig. 11), 0.75, 0.5, 0.25, and 0 (the horizontal axis). For Phase 2A of the example, shapes of the total inchworm (including head and tail sections) corresponding to Fig. 11 are illustrated in Fig. 13(a–e) for β¼0.25, 0.5, 0.7, 0.9, and 1. For Phase 2B, corresponding to Fig. 12, they are shown in Fig. 14(a–d) for β¼ 0.75, 0.5, 0.25, and 0.

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Fig. 11. Nondimensional shapes of centerline of central section during Phase 2A of Case II for β ¼0.25 (solid), β ¼ 0.5 (dashed), β¼ 0.7 (solid), β ¼0.9 (dot-dashed), and β ¼1 (solid).

Fig. 12. Nondimensional shapes of centerline of central section during Phase 2B of Case II for β ¼1, 0.75, 0.5, 0.25, and 0 (horizontal axis).

For arbitrary values of bmin and βmax, the marker xm in Phase 2A is defined by   ð1 bmin Þ 2βmax þ β : ð13Þ xm ¼ 4βmax It begins at xm ¼(1  bmin)/2 when β¼0, and ends at xm ¼ 3 (1 bmin)/4 when β¼ βmax. For Phase 2A of the numerical example, xm ¼0.34 þ 0.17β and it increases from 0.34 to 0.51 as β increases from 0 to 1. For Phase 2B, the marker is chosen to be given by

xm ¼

  ð1 bmin Þ 4βmax  β : 4βmax

ð14Þ

Fig. 13. Configurations of example inchworm during Phase 2A of Case II: (a) β¼ 0.25; (b) β ¼0.5; (c) β ¼ 0.7; (d) β ¼0.9; and (e) β ¼ 1.

It increases from 3(1  bmin)/4 when β¼ βmax to 1 bmin when β¼0. For Phase 2B of the example, xm ¼0.68–0.17β, which increases from 0.51 to 0.68 as β decreases from 1 to 0. The quantity ymax is defined here as the maximum height of the centerline in the central section, so its computation does not include the head section. At the beginning of Phase 2A, ymax is associated with

R.H. Plaut / International Journal of Non-Linear Mechanics 76 (2015) 56–63

61

Fig. 15. Maximum nondimensional height ymax of central section of example inchworm as a function of marker xm for Case II.

Fig. 16. Function g(s) defined in Eq. (20).

5. Bending strain

Fig. 14. Configurations of example inchworm during Phase 2B of Case II: (a) β ¼0.75; (b) β ¼ 0.5; (c) β¼ 0.25; and (d) β¼ 0.

an internal point on the arched section. After a certain inclination β is attained, the maximum height is located at the top end of the central section. For the numerical example, the transition occurs at β¼0.76 with xm ¼ 0.47, so that in Fig. 11 ymax occurs at an internal point for β¼0.25, 0.5, and 0.7, and at the top for β¼0.9 and 1. For Phase 2B, ymax is at the top of the central section, as seen in Fig. 12 for the example, and is given in terms of β as ymax ¼ r 1 ð1  cos βÞ þ ð1  r 1 βÞ sin β:

ð15Þ

Fig. 15 depicts ymax versus xm for Case II for the numerical example. In Phase 1, with 0rxm r0.34, the plot is the same as in Fig. 9 for Case I. For Phase 2, it is noted that the location of ymax is not the location of xm. The marker xm has been chosen to represent the progress of the inchworm during a cycle, and to be steadily increasing (as is time). For the numerical example (bmin ¼ 0.32, βmax ¼1), approximations are derived. For Phase 1, Eq. (9) is applicable. Polynomial approximations for the three portions of Case 2B are now given. For 0.34 oxm r0.46 in Phase 2A ymax ¼  0:364 þ 5:754xm  15:8x2m þ 15:95x3m :

ð16Þ

The peak value of ymax is 0.803 and occurs at xm ¼0.51.

dθ : ds

ð19Þ

For Phase 1 of both cases, and Phase 2 of Case I, with θ(s) given by Eq. (4), the bending strain for 0 oso 1 (the central section) is given by εbending ¼ 2π z f ðbÞ g ðsÞ

where

g ðsÞ ¼  cos θðsÞ:

ð20Þ

The function f(b) was defined in Eqs. (5) and (6). The function g (s) is plotted in Fig. 16. For Phase 2A of Case II, in the circular section 0 os or1β εbending ¼ 

z : r1

In the arched section of Phase 2A   2πzf ðb2 Þ s2 g εbending ¼ ð1  r 1 βÞ 1  r 1 β

ð21Þ

ð22Þ

ð17Þ 6. Concluding remarks

Finally, for 0.51 oxm r0.68 in Phase 2B ymax ¼  5:1015 þ 32:443xm 53:619x2m þ 24:913x3m :

εbending ¼  z

where b2 is defined in Eq. (12), g is defined in Eq. (20), and the arc length s2 runs from 1 to 1  r1β.

For 0.46 oxm r0.51 in Phase 2A ymax ¼  2:319 þ 6:119xm :

Hydrostatic pressure causes axial and circumferential (hoop) strains in the inchworm [60,61]. The strains in muscles of a crawling caterpillar were studied in [62], for example. During inchworm locomotion, arching is associated with additional bending strains εbending. In the present analysis, the bending strain at a distance z perpendicular to the centerline (positive if upward, with – d/2 o zod/2) is obtained from the formula

ð18Þ

Inchworm locomotion is very interesting and can involve relatively high speeds. During one cycle, first the head section is anchored while the tail section moves forward by a distance approximately equal to

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half the body length, with the central section arching significantly. Then the tail section is anchored and the central and head sections move forward, involving either (i) a reduction of the arch height or (ii) cantilevering from the tail section, away from the substrate, and then dropping onto the substrate. Other types of caterpillars have been investigated extensively. The present paper has developed a mathematical model of inchworm locomotion, focusing on shapes and bending strains. The model was based on post-buckling of an inextensible elastica. Numerical results were obtained for a particular set of nondimensional parameters. It is hoped that this work will help in the understanding of inchworm locomotion, and that it will be useful in the design of soft robots mimicking inchworms. For example, if shape memory alloy (SMA) wires are used in a soft robot for actuation, the analysis can estimate the bending strains that would be needed to attain a particular arched configuration. The work also provides a foundation for a potential inverse approach to determine muscular activity associated with the locomotion of inchworms.

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