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Edited by: Matteo Cianchetti, Scuola Sant'Anna di Studi Avanzati, Italy

Reviewed by: Shinya Aoi, Kyoto University, Japan; Barbara Mazzolai, Fondazione Istituto Italiano di Technologia, Italy

This article was submitted to Soft Robotics, a section of the journal Frontiers in Robotics and AI

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The study of self-propelled locomotors exploiting friction-induced traction as a result of body shape changes, is gaining attention because of the variety of physical systems which take advantage of such a locomotion strategy. One motivation is the desire to understand biological phenomena, such as cell migration on or within solid substrates, matrices, and tissues (Alberts et al.,

In particular, robotic locomotion research has recently considered crawling and burrowing animals (e.g., earthworms, snakes, and caterpillars), whence an increasing number of research projects on bio-inspired metameric (soft) robots (Menciassi et al.,

In the field of robotics, peristalsis has been mostly mimicked by

This paper aims to provide a deeper understanding of harmonic oscillations and peristalsis as result of an optimization problem rather than an a priori hypothesis. Indeed, we prove that - in the regime of small deformations - peristalsis is a symmetry property of the solution to an optimization problem. Symmetry of the solution comes from symmetry properties of operators in the equations governing the optimization problem, which are, in turn, the signature of geometric symmetries of the physical system.

The rest of the paper is organized in three main sections: material and methods, results and discussion. The first one, inspired by nightcrawlers' retractable

Consider a 1D crawler moving along a straight line and assume its reference configuration is the segment

cf. Figure

where _{1}(_{1}, _{2}(_{2},

in order to guarantee the monotonicity of χ(·,

Kinematics of a continuous 1D crawler: reference

In what follows a prime will denote the derivative with respect to

We define the displacement

so that

where we have implicitly defined the strain

in terms of which condition (1) reads

Finally, notice that, since the material (or Lagrangian) velocity is

the spatial (or Eulerian) velocity is given by

Throughout this section we deal with the motility problem, namely, given a history of strain ϵ(_{1}(

The force at the interface between substrate and crawler is modelled through a force-velocity relationship. In particular, we write the density per unit current length of the tangential component of the friction force at time

Several models for the resistance forces are conceivable such as a

Newtonian model, i.e., a linear viscous law

where μ>0 is a friction (or viscosity) coefficient;

or a more general “

for _{p}(ϵ) around ϵ = 0 for different values of

Function _{p}(ϵ) governing the friction law (4) for selected values of parameter

In what follows, we will use the

The total friction is obtained by integrating the force per unit current length on the whole current domain, i.e.,

where

Then, by neglecting inertia, the force balance yields

The square bracket multiplying ẋ_{1}(_{e}(_{1}(

which, in the case of zero external forces, is independent of μ.

Notice that, once the initial position _{1}(0) the strain ϵ(_{e}(_{1}(

one has

and hence the right hand side of (5) is known once ϵ(

We now move to a discrete model, directly inspired by studies on annelid worms (Quillin,

We model the crawler's body as made up of

for _{0}(

whence the definition of strain

Kinematics of a discrete 1D crawler consisting of

Furthermore, we assume that each segment can be contracted or expanded according to a constant stretch so that the overall strain results to be a piecewise constant function of

Consequently its time-derivative,

for _{n−1}, _{n}]. Note that in this new framework the monotonicity condition (2) reads

which is the only constraint for an admissible history of strains, the datum of our motility problem.

Analogously to the previous case, the force balance yields

where, in view of (9),

and, in view of (10),

Solving for ẋ_{0}(

which can be rewritten in the following vectorial form

where

Equation (11) fully describes the dynamics once _{0}(0) and ϵ(

We can rewrite everything in terms of a displacement relative to _{0}(

in order to describe the displacement in a coordinate system which is “co-moving” with the left end _{0}(

In the discrete framework, the relative displacement turns out to be a piecewise affine function of _{n−1}, _{n}],

Setting

we have

where _{0}≡0. Equivalently

where

In this section we address the problem of maximizing the net displacement Δ_{0} among periodic shape changes ϵ(

We now describe the optimization problems with quadratic energy in the non-linear case first, and then in the small-deformation regime, for which general results can be established. We assume _{e}≡0.

We assume that the shape function ^{2} function defined from ℝ to ℝ^{N}.

In addition, we require

where 𝔸 and 𝔹 are symmetric and positive definite

The general (non-linear) optimization problem is

which is an isoperimetric problem (e.g., Van Brunt, _{n}. The corresponding Euler-Lagrange equations lead to a second order non-linear system of ODEs, i.e., for

where

We can focus on the

and, integrating by parts,

whence

In particular, it can be proved that the (skew-symmetric Toeplitz) matrix skw(_{ϵ}(

(see Appendix

Therefore, in the regime of small deformations, problem (18) can be replaced by the following linear problem

The corresponding Euler-Lagrange equations

where

lead to the following system of second order linear ODEs

In general, a solution to (21) might be difficult to determine due to the complexity of finding a common diagonalization of 𝔸 and 𝔹. However, following the procedure adopted by Wiezel et al. (

for

where α∈ℂ\{0} is a constant such that

for 𝔸 symmetric and positive definite and 𝔹 = 0, a solution of (20) with

where α∈ℂ\{0} is a constant such that

Both (22) and (23) have the form

i.e., they are circles in the plane (ℜ(

we get, for

i.e., the optimal gait depends only on the 2_{ϑn}n}, {_{ϱn}n}, ϑ_{a} and ϱ_{a}. Admittedly, since α is a constant with fixed modulus and free argument, we can always assume

thus reducing the number of parameters to 2

The problem for 𝔸 = 0 and 𝔹 = 𝕀_{N} is _{N} and 𝔹 = 0 (provided that unitary time frequency of

then it is a solution also to

and vice versa.

In general the two problems, 𝔸 = 0 with 𝔹 symmetric positive definite and 𝔹 = 0 with 𝔸 symmetric positive definite, are not equivalent. In fact, constraining the norm induced by one operator does not determine the norm induced by the other one, but only provides a bound. Indeed, denoting by λ_{min}(·) and λ_{max}(·) the minimum and maximum eigenvalue respectively, observe that, for

and, analogously,

In this section we discuss two examples of contraction waves to illustrate the behavior of the

For simplicity, in the following examples we neglect external forces, i.e., _{e}≡0.

Consider a smooth traveling contraction wave by prescribing the strain along the body of the crawler as

or equivalently, in terms of the stretch,

where ϵ_{0} is the wave amplitude,

and, by differentiating with respect to time,

Finally, in view of (5),

whence

The Newtonian case is recovered by setting

Figure

Plot of _{1}(_{0} = 0.6,

Consider the square contraction wave

or equivalently, in terms of the stretch,

where _{~L} denotes the “modulo _{~L} stands for

By integrating the stretch over space, we get

whence

Finally, in view of (5),

where

and

Defining

where {·} and ⌊·⌋ denote the fractional part and floor function respectively. The Newtonian case can be obtained as particular case by setting

Plot of _{1}(t) for a square contraction wave (27) for selected values of parameter

In the discrete framework, peristalsis is the result of phase coordination among the harmonic contractions of body segments, i.e., it has the form

where

In this section we work out explicitly the problem of maximizing the displacement for a particular case from which peristalsis emerges, modulo an

Let us define an energy functional ^{2}(ℝ, ℝ^{N}) → ℝ as

where

i.e., the energy cost is the time integral over a period of a dissipation rate which is sum of two terms: ^{2}-norm of the controls suitably weighted to time the input direction.

Some calculations [see section 1 in Appendix

where 𝔻(^{N×N} for any ^{N}, and

where 𝕀_{N} is the

where 𝔾(_{N}.

The non-linear optimization problem associated with energy functional (28) is

The Euler-Lagrange equations lead to a second order non-linear system of ODEs, i.e., for

where

In the regime of small deformations we can expand the terms of problem (29) at the leading orders about

and the energy functional by

where 𝔾: = 𝔾(

[see section 2 in Appendix

The centrosymmetry of 𝔾 and the skew-centrosymmetry of 𝕍 imply a

the moduli of components of

phase differences between adjacent segments are symmetric about the center, i.e.,

so that the

Plot of arguments and moduli of ϵ_{n} for

Equation (25) shows that the optimal gait requires a precise “phase coordination” of locomotion patterns among the segments, which is a common observation in Biology for several kinds of animals.

Numerical simulations show that the optimal solution turns out to be a discrete approximation of a traveling wave. In particular,

the moduli of _{n} for

so that each segment undergoes a harmonic deformation with a certain initial phase;

phase differences between adjacent segments turn out to be almost constant, i.e., for a suitable φ_{0}

holds true for

Therefore, in view of (32) and (33), the solution is a discrete approximation of a continuous traveling wave, i.e.,

where

whence the canonical form of traveling wave which describes peristalsis (cf. Figure

where

Plot of piecewise constant optimal strain

The symmetric structure of the optimal gait (in the small-deformation regime) arises from underlying physical symmetries which clearly stand out in the properties of the matrices 𝔾 and 𝕍. In particular, an “edge-effect” is apparent: the 1D crawler is symmetric about its geometric center and segments near the edges behave differently with respect to adjacent segments, but in the same way as their centrosymmetric counterparts.

As expected, this edge-effect vanishes when considering an “infinite” (periodic) 1D crawler because, due to the shift-invariance symmetry, each segment behaves as a “geometric center.”

To show this claim consider a 1D crawler made up of infinitely many segments and assume that it is a periodic structure of which each module consists of

Kinematics of a discrete infinite 1D crawler consisting of identical segments of reference length

At any time t, we already defined the relative displacement

The hypothesis of periodicity leads to

From (35) we obtain that the friction force is periodic and we can consider the force balance in a single module. Therefore, condition (35) reads

and, in the discrete framework (9), this leads to

The optimal control problem becomes

and it can be proved [see section 3 in Appendix

for some _{1}∈ℂ\{0}, cf. Figure

- each component of _{1}||;

- each component can be obtained from the previous one by a rotation of

where

Complex components of the vector

whence the exact harmonic peristalsis.

Notice that problem (38) can be written in terms of relative displacements _{n} through the periodic version of transformation (15), i.e.,

where

In particular, we get

where

are

Considering the general (i.e., non-periodic) problem in terms of relative displacements yields

where

are “

where 𝔼_{V} and 𝔼_{G} are null a part from the last column and the last row, i.e.,

and

where ^{3}(3−^{3}(9−4

In the “periodic case” we can study the wavenumber (that is the number of waves travelling along the body of the crawler) of the optimal gait in relation to the number of metameres

and we let ^{2} for

for

for

Wavenumber of optimal gaits as a function of

This behavior is qualitatively unaffected by the type of friction model which is adopted (i.e., by the choice of the parameter

To put our study in perspective, we consider the discrete framework and we compare our results with the ones presented by Fang et al. (_{s} among harmonic shape functions having the form (in our notation)

where _{n} is the actuation phase for the

Since the average steady-state velocity is given by

the optimization problem reads

and in the small-deformation regime it can be replaced by

Denote the actuation phase differences between adjacent segments by

From observations of numerical simulations, Fang et al. (

In fact these properties can be rigorously proved under the assumption that problem (45) admits a unique solution in [0, 2π)^{N} (see Appendix

Furthermore, property (47) can be proved also for (46), assuming it admits a unique solution in [0, 2π)^{N}. To this aim, denote the unique solution to (46) by

and consider the shape change

where

Notice that for

and hence, by exploiting the fact that ^{T}𝕍𝕂 = −𝕍),

Thus

which leads to (47).

Problem (46) constrains the ^{2}-norm of the time-derivatives, i.e., for strains having the form (43) we get

regardless of ^{2} periodic strains whose time derivative fulfills the same constraint, i.e.,

Since problem (21) reduces to (48) when 𝔸 = 0 and 𝔹 = 𝕀_{N}, a solution to (48) must be of the form

where _{en)n} is a unit eigenvector associated with the maximum-modulus eigenvalue of _{a} is a constant. Notice that the

Average velocities _{s}, (44) obtained by the solution ^{⋆} to (48) (yellow bars) for different numbers of segments: ^{−10},

Our analysis confirms the effectiveness of mimicking peristalsis in bio-inspired robots, at least in the small-deformation regime. This bio-inspired actuation strategy has been implemented on a trial-and-error basis many times in the robotics literature and, more recently, also proposed as optimal (in some suitably defined sense, and in some suitably defined class of actuation strategies). Our main result is a mathematically rigorous proof that, in the small deformation regime, actuation by peristaltic waves is an optimal control strategy emerging naturally from the geometric symmetry of the system, namely, the invariance under shifts along the body axis. This is true exactly in the periodic case, and approximately true in the case of finite length, modulo edge-effects. Our results is of theoretical nature. Nevertheless, we believe that it has important consequences for applications. For example, it helps us not to fall into the naive temptation to expect that peristaltic waves are always an optimal actuation strategy just because they are observed in nature, but to exercise critical judgment whenever the hypotheses on the geometric symmetry that are “responsible” for the optimality result in our case (invariance under shift of a homogeneous one-dimensional system) are false.

Actuation by phase coordination, optimal actuation by identical phase difference, and the connections between this and traveling waves have been already discussed in the literature (e.g., Fang et al.,

Further work will be needed to test the effectiveness of peristaltic waves as a locomotion strategies if large deformations are allowed. In addition, future work will explore the issue of how peristalsis is actually enforced in biological systems. Of particular interest is the dichotomy between the paradigm of actuations via a Central Pattern Generator (CPG), as opposed to local sensory and feedback mechanisms. The CPG paradigm is apparent in several different organisms (Marder and Bucher,

AD and FA conceived research. DA executed research and performed numerical simulation. AD supervised research. FA contributed expertise on circulant matrices. All authors analyzed the data and wrote the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. The handling Editor declared a shared affiliation, though no other collaboration, with one of the authors, AD.

This article was written while DA and AD where visiting Division C of the Engineering Department of the University of Cambridge. We thank Prof. Vikram Deshpande and Corpus Christi College for their warm hospitality.

The Supplementary Material for this article can be found online at: