Peristaltic Wave Locomotion and Shape Morphing with a Millipede Inspired System

Abstract

We present the mechanical model of a bio-inspired deformable system, modeled as a Timoshenko beam, which is coupled to a substrate by a system of distributed elements. The locomotion action is inspired by the coordinated motion of coupling elements that mimic the legs of millipedes and centipedes, whose leg-to-ground contact can be described as a peristaltic displacement wave. The multi-legged structure is crucial in providing redundancy and robustness in the interaction with unstructured environments and terrains. A Lagrangian approach is used to derive the governing equations of the system that couple locomotion and shape morphing. Features and limitations of the model are illustrated with numerical simulations.

Appendices

Appendix 1: Nonzero Entries of Mass Matrix and Christoffel Symbols Matrix

Nonzero entries of the mass matrix $\mathbf {M}$ in (37):

$$\begin{aligned} M_{11}&= M_{22} = 1 \end{aligned}$$

(68a)

$$\begin{aligned} M_{14}&= M_{25} = M_{41} = M_{52} = \epsilon \end{aligned}$$

(68b)

$$\begin{aligned} M_{44}&= M_{55} = \epsilon ^2 \end{aligned}$$

(68c)

$$\begin{aligned} M_{66}&= \alpha _2 \epsilon ^2 \end{aligned}$$

(68d)

$$\begin{aligned} M_{36}&= M_{63} = \alpha _2 \epsilon \end{aligned}$$

(68e)

$$\begin{aligned} M_{13}&= M_{31} =-(d_2 + w^\star ) \end{aligned}$$

(68f)

$$\begin{aligned} M_{34}&= M_{43} = -\epsilon (d_2 + w^\star ) \end{aligned}$$

(68g)

$$\begin{aligned} M_{23}&= M_{32} = d_1 + u^\star + X_1 - \delta \end{aligned}$$

(68h)

$$\begin{aligned} M_{35}&= M_{53} = \epsilon (d_1 + u^\star + X_1 - \delta ) \end{aligned}$$

(68i)

$$\begin{aligned} M_{33}&= \alpha _2(1+\psi ^\star )^2 + (d_1 + u^\star + X_1 - \delta )^2 + (d_2 + w^\star )^2 \end{aligned}$$

(68j)

Nonzero entries $C_{ij}$ in (44)

$$\begin{aligned}&C_{13} = C_{31} = \partial _t d_2 + \frac{1+\epsilon }{2} \partial _\tau w^\star + \partial _t \theta (d_1 + u^\star +X_1 - \delta ) \end{aligned}$$

(69a)

$$\begin{aligned}&C_{43} = C_{34} = \frac{1+\epsilon }{2} \partial _t d_2 + \epsilon \partial _\tau w^\star + \partial _t \theta (d_1 + u^\star +X_1 - \delta ) \end{aligned}$$

(69b)

$$\begin{aligned}&C_{23} = C_{32} = -\partial _t d_1 -\frac{1+\epsilon }{2} \partial _\tau u^\star + \partial _t \theta (d_2 + w^\star ) \end{aligned}$$

(69c)

$$\begin{aligned}&C_{53} = C_{35}= - \frac{1+\epsilon }{2} \partial _t d_1 - \epsilon \partial _\tau u^\star + \partial _t \theta (d_2 + w^\star ) \nonumber \\&C_{33} = -(\partial _t d_1 + \partial _\tau u^\star )(d_1 + u^\star +X_1 - \delta ) \end{aligned}$$

(69d)

$$\begin{aligned}&\quad \qquad - (\partial _t d_2 + \partial _\tau w^\star )(d_2 + w^\star ) - \alpha _2 \psi ^\star \partial _\tau \psi ^\star \end{aligned}$$

(69e)

$$\begin{aligned}&C_{36} = C_{63} = \alpha _2 \psi ^\star \partial _t \theta \end{aligned}$$

(69f)

Appendix 2: Basis Functions in Galerkin Projection

The set of basis functions for the deformation fields of the beam is obtained by solving the following homogeneous system for the nondimensionalized Timoshenko beam with free ends boundary conditions and a distributed system of supporting springs with stiffness per unit length $\kappa$

$$\begin{aligned} \ddot{u} - \alpha _1 u''&= 0 \end{aligned}$$

(70a)

$$\begin{aligned} \ddot{w} - (w'- \psi )' + \alpha _3 w&= 0 \end{aligned}$$

(70b)

$$\begin{aligned} \alpha _2 \ddot{\psi } - \alpha _1 \alpha _2 \psi '' - (w' - \psi )&= 0 \end{aligned}$$

(70c)

$$\begin{aligned} u'(0,t) = u'(1,t)&=0, \end{aligned}$$

(70d)

$$\begin{aligned} w'(0,t) - \psi (0,t) = w'(1,t) - \psi (1,t)&=0 \end{aligned}$$

(70e)

$$\begin{aligned} \psi '(0,t) = \psi '(1,t)&=0 \end{aligned}$$

(70f)

where $\alpha _1$, $\alpha _2$, and $\alpha _3$ are defined in (36), with $\alpha _3$ being a nondimensional measure of the stiffness of the supporting elastic coupling elements (modeling the legs) with respect to the shear stiffness of the body. The solution is obtained by the usual separation of variables $u(X_1,\tau ) = \bar{u}(X_1) \exp (\mathrm {I} \omega \tau )$, $w(X_1,\tau ) = \bar{w}(X_1) \exp (\mathrm {I} \omega \tau )$, and $\psi (X_1,\tau ) = \bar{\psi }(X_1) \exp (\mathrm {I} \omega \tau )$, where $\omega >0$ is the angular frequency, and $\mathrm {I}$ is the imaginary unit. Since (70a) is uncoupled, the solution for $\bar{u}$ is easily obtained as

$$\begin{aligned} \bar{u}_i(X_1) = B \cos \left( i \pi X_1 \right) \end{aligned}$$

(71)

To obtain the natural frequencies and associated eigenfunctions for w and $\psi$, we follow the approach in van Rensburg and van der Merwe (2006), which is based on the solution of a vector eigenvalues problem for the system of two coupled second-order differential equations for the transverse displacement and for the rotation of the cross section. In our case, the two equations are (70b) and (70c). This allows to enforce boundary conditions for free ends in a direct way. A different approach based on the derivation of one-fourth order governing equation obtained by combining (70b) and (70c) is presented in the original work of Timoshenko (1974). However, in this case the application of boundary conditions requires special attention (Majkut 2009). The vector eigenvalues problem has different general solutions depending on the choice of material parameters (van Rensburg and van der Merwe 2006). The general solution of the eigenvalues problem is sought by considering the vector evaluated function $\exp (\lambda x) \left( \begin{array}{cc} \bar{W}&\bar{\varPsi } \end{array} \right) ^\mathsf {T}$, where $\bar{W}$ and $\bar{\varPsi }$ are constants. Such function is a solution for some positive constant $\lambda$ if and only if

$$\begin{aligned} \left( \begin{array}{cc} \lambda ^2 + \beta _1 &{} -\lambda \\ \beta _3 \lambda &{} \lambda ^2 + \beta _2 \end{array} \right) \left( \begin{array}{c} \bar{W} \\ \bar{\varPsi } \end{array} \right) = \left( \begin{array}{c} 0 \\ 0 \end{array} \right) \end{aligned}$$

(72)

with nondimensional parameters $\beta _i$ defined by

$$\begin{aligned} \beta _1 = \omega ^2 - \alpha _3, \quad \beta _2 = \frac{1}{\alpha _1} \left( \omega ^2 - \frac{1}{\alpha _2} \right) , \quad \beta _3 = \frac{1}{\alpha _1 \alpha _2} \end{aligned}$$

(73)

The roots $\lambda ^2$ of the characteristic polynomial $\lambda ^4 + (\beta _1 + \beta _2 + \beta _3) \lambda ^2 + \beta _1 \beta _2 = 0$ are

$$\begin{aligned} \lambda ^2_{1,2}&= -\frac{1}{2} \left( \beta _1 + \beta _2 + \beta _3 \right) \left( 1 \pm \sqrt{\varDelta } \right) \end{aligned}$$

(74)

$$\begin{aligned} \varDelta&= 1 - \frac{4 \beta _1 \beta _2}{\left( \beta _1 + \beta _2 + \beta _3 \right) ^2 } \end{aligned}$$

(75)

In order for $\lambda ^2$ to be real, it must be $\varDelta > 0$, which is satisfied for $\beta _1 \beta _2 < \gamma ^{2}/4$, where $\gamma = \beta _1 + \beta _2 + \beta _3$. The special case $\lambda ^2 = 0$ occurs when $\varDelta = 1$, that is $\beta _1 \beta _2 = 0$ or

$$\begin{aligned} \beta _1&= 0 \Rightarrow \omega ^2 = \alpha _3, \end{aligned}$$

(76)

$$\begin{aligned} \beta _2&= 0 \Rightarrow \omega ^2 = \alpha _2 \text { or } \alpha _1 = 0 \end{aligned}$$

(77)

This corresponds to rigid body motions of the system (Majkut 2009). Here we consider the system made of a material with shear modulus ten times smaller than the Young’s modulus, and we consider the overall shear stiffness $k G A \ell ^2$ to be one hundred times larger than the bending stiffness YI. Moreover, we set the supporting springs to be softer than the beam (with ratio 1 / 2). Therefore, parameters $\alpha _i$ are assigned to the numerical values in Table 1.

Table 1 Nondimensional parameters of the Timoshenko beam

Table 2 First seven roots of the characteristic equation for the linear modes

Let $\omega _\varDelta$ be the root of $\varDelta = 0$ and $\omega _\gamma$ be the root of $\gamma = 0$; with the choice of parameters listed above, we have $\omega _\varDelta = 0.707$ and $\omega _\gamma = 0.674$. Moreover, $\omega _\lambda = 0.707$ is the root of $\beta _1 \beta _2 = 0$. The condition $\varDelta >0$ implies $\omega > \omega _\varDelta$; therefore, it must be $\gamma > 0$ since this is the case for $\gamma (\omega>\omega _\gamma )>0$, and in this case $\omega _\gamma < \omega _\varDelta$. For $\varDelta > 0$ and $\gamma > 0$, we have $\lambda _1 = \pm \mathrm {I} \nu _1$ with $\nu _1^2 = \frac{\gamma }{2} \left( \sqrt{\varDelta } + 1 \right)$; $\varDelta >0$ also implies $\beta _1 \beta _2 < 0$, since this is the case for $\beta _1 \beta _2$ evaluated at $\omega > \omega _\lambda$, and $\omega _\lambda = \omega _\varDelta$. Therefore, $\sqrt{\varDelta }>1$ with $\lambda _2 = \pm \nu _2$ and $\nu _2^2 = \frac{\gamma }{2} \left( \sqrt{\varDelta } - 1 \right)$. Here $\varDelta > 0$ dictates $\omega >\omega _{\varDelta }$. The general solution is therefore given by

$$\begin{aligned} \varPhi (x)&= C_1 \left( \begin{array}{c} \sin \nu _1 X_1 \\ - \frac{\beta _1 - \nu _1^2}{\nu _1} \cos \nu _1 X_1 \end{array} \right) + C_2 \left( \begin{array}{c} \cos \nu _1 X_1 \\ \frac{\beta _1 - \nu _1^2}{\nu _1} \sin \nu _1 X_1 \end{array} \right) \nonumber \\&+ C_3 \left( \begin{array}{c} \sinh \nu _2 X_1 \\ \frac{\beta _1 + \nu _2^2}{\nu _2} \cosh \nu _2 X_1 \end{array} \right) + C_4 \left( \begin{array}{c} \cosh \nu _2 X_1\\ \frac{\beta _1 + \nu _2^2}{\nu _2} \sinh \nu _2 X_1 \end{array} \right) \end{aligned}$$

(78)

By imposing the free end boundary conditions at $X_1=0$ and $X_1=1$, we obtain the linear algebraic relations involving $C_3$ and $C_4$ and coefficient matrix. The nontrivial solutions of the system are obtained by investigating the condition for rank deficiency of the coefficients matrix, which translates into the following condition for the determinant

$$\begin{aligned} - \cos \nu _1 \cosh \nu _2 + \frac{\nu _1 (\nu _1^2 - \beta _1)}{\nu _2 ( \nu _2^2 + \beta _1)} \sin \nu _1 \sinh \nu _2 + 1 = 0 \end{aligned}$$

(79)

All parameters in the characteristic equation depend on $\omega$ and on the material and geometric parameters of the system. Therefore, once the material and the geometry are defined the characteristic equation is a nonlinear function of $\omega$ only. The first seven roots of the characteristic equation that determine the corresponding modes are given in Table 2. The mode shapes are normalized with respect to the maximum amplitude for $x \in (0,1)$ (boundaries not included). The corresponding dimensional values of the frequencies are obtained from the ones in Table 2 as $\omega /\bar{t} = \frac{\omega }{\ell } \sqrt{\frac{k G}{\varrho }}$.