Journal of the Mechanics and Physics of Solids Active filaments I: Curvature and torsion generation

In many filamentary structures, such as hydrostatic arms, roots, and stems, the active or growing part of the material depends on contractile or elongating fibers. Through their activation by muscular contraction or growth, these fibers will generate internal stresses that are partially relieved by the filament acquiring intrinsic torsion and curvature. This process is fundamental in morphogenesis but also in plant tropism, nematic solid activation, and muscular motion of filamentary organs such as elephant trunks and octopus arms. Here, we provide a general theory that links the activation of arbitrary fibers at the microscale to the generation of curvature and torsion at the macroscale. This theory is obtained by dimensional reduction from the full anelastic description of three-dimensional bodies to morphoelastic Kirchhoff rods. Hence, it links the geometry and material properties of embedded fibers to the shape and stiffness of the rod. The theory is applied to fibers that are wound helically around a central core in tapered and untapered filaments.


Introduction
Filaments are soft slender mechanical structures that are roughly defined by having one dimension much larger than the typical scale of their cross section. Around us, filaments are the strings, ropes, strands, wires, cables, and cords that we encounter in our daily lives. In mechanics, they include beams, strings, strips, ribbons, and rods (Antman, 2005). Passive filaments have fixed material and geometric properties such as length, girth, intrinsic shape, and rigidity. The central problem in the mechanics of these passive objects is to obtain their shapes for given boundary conditions, body forces, and external loads. Due to their particular aspect ratio, the shape of such objects can be captured from their central axis and modeled using a combination of differential geometry of curves and physical balance laws for forces and moments, leading ultimately to the Kirchhoff equations of rod theory (Kirchhoff, 1859;Dill, 1992;Goriely and Tabor, 2000). These equations have been successfully applied to structures as varied in size and functions as DNA, proteins (Benham, 1979;Hoffman et al., 2003;Beveridge et al., 2004;Neukirch et al., 2008a,b), polymers and liquid crystals (Sheley and Ueda, 2000), whips and lassos (McMillen and Goriely, 2002;Brun et al., 2014), and bridge cables (Costello, 1990;Yokota et al., 2001).
In the engineering world, it includes some soft robotic arms (Calisti et al., 2011;Hannan and Walker, 2003;Laschi et al., 2012;Paley and Wereley, 2021;Walker et al., 2005;Jones et al., 2021), liquid crystal elastomers (Goriely et al., 2022) and actuators (de Payrebrune and O'Reilly, 2016;Sano et al., 2021). Internal remodeling can be generated by external fields as in the case of magnetic actuation, by growth during morphogenesis, or by muscular contraction. In most situations, the internal changes can be modeled by the relative change of internal geometry of a volume element along a principal direction. Since the activation is done in a single distinguished direction, we refer to such a volume element as a fiber, and the activation direction as the fiber direction, as shown in Fig. 1. There are now two main questions: What is the intrinsic shape of the filament, for a given activation field, in the absence of body forces and boundary loads? And, what is the shape of the filament when loaded given this intrinsic shape? Here, we will focus on the first question to obtain the active filament formula linking fiber activation to intrinsic curvature and torsion.
Our starting point is the general theory developed in Moulton et al. (2020a) for the problem of determining the curvature, extension, and torsion of a filament for an arbitrary anelastic field. Accordingly, we treat the filament as a morphoelastic solid. This is a continuum that can grow, remodel, support stresses and can be subject to large deformations (Goldstein and Goriely, 2006;Goriely and Moulton, 2010;. We use the theory of morphoelasticity, which represents deformations due to elasticity, growth, and remodeling through a multiplicative decomposition of the deformation gradient into elastic and growth tensors . In the present work, the decomposition is naturally extended to include the activation process, such as mass-preserving contraction or elongation, in the growth tensor. The problem is then to obtain from the specification of a growth tensor representing fiber activation for the full three dimensional problem, the corresponding reduced one-dimensional morphoelastic rod as defined in Lessinnes et al. (2017), Moulton et al. (2012).

General set-up
We briefly recall the basic assumptions from Moulton et al. (2020a). We assume that the filament is a three-dimensional tubular body that is allowed to grow and whose shape varies slowly along its axis. The initial structure is stress-free and growth or anelastic activation is defined at every point as a local change of a volume element. Following the procedure in Moulton et al. (2020a), we define a growth tensor, characterizing at each point the local change of shape of a volume element by the addition, removal, or redistribution of mass. We assume that this filamentary structure deforms into another tubular structure defined by variations along a deformed centerline. The slenderness of this structure introduces naturally a small parameter in the problem that can be used to asymptotically expand the energy of the system. This energy can then be minimized, and the stresses and strains within the section can be obtained explicitly, leaving an energy that can be identified with the energy of a rod. This type of dimensional reduction is related to a large body of work in rational mechanics focused on obtaining systematically reduced models from threedimensional elasticity (Cimetière et al., 1988;Sanchez-Hubert and Palencia, 1999) and anelasticity (Mora and Müller, 2003;Audoly and Lestringant, 2021;Kohn and O'Brien, 2018;Cicalese et al., 2017;Bauer et al., 2019;Kupferman and Solomon, 2014). The emphasis here is not in justifying the reduction but using it to explore the possible shapes that can be created through simple and universal filamentary structures.

The growth tensor for activation
We consider an initial elastic tubular configuration  0 ⊂ R 3 with material points ( , , ) ∈  0 that can be decomposed as the product [0, ] ×  of a segment of the -axis between 0 and and a family of cross-sections  whose centroids are on the -axis and oriented with the condition where is the Young's modulus. The typical length scale of each section is ( ), corresponding to a typical or averaged radius, and each cross section is a slowly varying function of the arc length , so that, on short scales, the tubular structure is cylindrical. We consider the deformation ( ) ∶  0 →  from the initial configuration  0 to the current configuration  and model the growth or activation through a tensor so that where the gradient is taken with respect to the coordinates and is an elastic tensor describing the elastic stretches related to a strain-energy density function = ( ). The tensor is given with a strictly positive determinant. In the case of fiber activation, the determinant is governed by the Poisson effect of the activating element. In the general description of growth mechanics, the determinant is larger than 1 for growth, and smaller than 1 for shrinkage. For the rest of the paper, we refer to any process with ≠ as activation. Activation is naturally expressed as a map from the cylindrical coordinates ( , , ) of the reference configuration to the cylindrical coordinates ( , , ) of the current configuration where ( , , ) and ( , , ) are the usual unit cylindrical basis vectors in the current and reference configuration, respectively, and = ( , , ). We restrict our analysis to growth tensors that are small deviations with respect to identity, where the deviation is measured with respect to the small parameter : (4)

Fig. 2.
We consider a tubular structure in the reference configuration (left) and its deformation in the current configuration (right). The deformed configuration is fully parameterized by the centerline ( ) and the deformation of each cross section.
A key assumption of our theory is that we restrict our attention to a particular family of possible deformations, mapping a straight tubular structure to a filament in space  with centerline ( ) as shown in Fig. 2. This centerline is the image of a segment of the -axis defining the centerline of the initial configuration. From this centerline, we define a local director basis ( 1 ( ), 2 ( ), 3 ( )) where ′ ( ) = 3 , is the axial extension, and ( ) ′ denotes derivatives with respect to the material coordinate . From the director basis, we define the Darboux curvature vector = 1 1 + 2 2 + 3 3 . This vector describes the evolution of the director basis along the filament, satisfying The mapping ∶  0 →  is then written where the reactive strains correspond to deformations of the sections and are to be determined; these satisfy (0, 0, ) = 0 so that the -axis maps to the centerline ( ). The particular form (6) expresses the deformation of a tubular body in terms of its centerline and director basis. We note that, since the section has a typical radius , the variable is an order 1 quantity. Taking = 1 + , the deformation gradient = ⊗ , ∈ {1, 2, 3}, ∈ { , , } is then given by Next, we consider a continuum with a distinguished direction that we call a fiber. Whereas we utilize a model for a material containing fibers that can be activated, it is worth noting that, in our model, the material does not actually contain physical fibers within a matrix. Rather, we assume that the density of fibers is large enough so that they can be represented, locally, by a vector field , as shown in Fig. 3. In cylindrical coordinates, this arbitrary fiber direction is described by two angles and : Helical fibers are tangent to a cylinder centered around the axis and are therefore prescribed by = 0 and ∈ [− ∕2, ∕2] with limiting cases of a hoop fiber at = + ∕2 and an axial fiber at = 0. A right-handed helical fiber is given by 0 < < ∕2 and a left-handed helical fiber is specified by − ∕2 < < 0. Sectional fibers lie in the cross-section and are characterized by = ∕2, with radial fibers given by = ∕2 and, in the limit = 0, we recover hoop fibers, as before.
Since we are only considering elongation or contraction along this fiber, must be an eigenvector of the tensor with eigenvalue . Similarly, the perpendicular vector ⊥ = cos − sin and ′ ⊥ = × ⊥ are also eigenvectors of : We consider two cases: growth and activation (see Fig. 1c). In the growth case, there is a change of length of the fiber = 1+ ( , ) that is not accompanied by a change in the transverse direction and ⊥ = ′ ⊥ = 1. For active fibers, the extension (contraction) of the fiber generates a contraction (extension) in the transverse directions, with any change in volume due to the action of described by the Poisson's ratio. Therefore, for small , we have ⊥ = ′ ⊥ = 1 − ( , ) and = 1 + ( , ), where is the Poisson's ratio. We see that both cases can be combined by taking either = 0 for growth and ∈ [0, 1∕2] for an active fiber (in the case = 0, there is no distinction between an active fiber and growth as the material extends without lateral contraction). We emphasize that , and may be functions of , , and, possibly, slowly varying functions of . Combining both cases, the growth tensor in cylindrical coordinates is (1+ ) sin 2 sin cos (1+ ) sin cos sin (1+ ) sin 2 sin cos −(1+ ) sin 2 sin 2 − (1+ ) sin cos cos (1+ ) sin cos sin (1+ ) sin cos cos

The energy density
We assume that the growing material is a compressible hyperelastic material with strain-energy density = ( ). The total energy of the system is which can be written in terms of and as The problem is then to minimize this energy for a given over the set of allowable deformations gradients considered above. In cylindrical coordinates, the energy functional can be written We proceed by expanding the inner variables , with the auxiliary energy density taking the form where . The Euler-Lagrange equations then become with the appropriate natural boundary conditions. To lowest order, the solution of the Euler-Lagrange equations is given by B. Kaczmarski et al. To order ( ), the Euler-Lagrange equations are identically satisfied. To order ( 2 ), the solution for the reactive strains is ( 1 sin 2 − 2 cos 2 + 2 cos (1) 2 = 4( + ) ( 1 cos 2 + 2 sin 2 − 2 sin (1) where 1,2 are functions that only enter in the so-called reactive part of the energy and are not needed to compute the moduli and intrinsic curvature. The function ( , ) is the classic warping function and is a solution of the Neumann problem for the Laplace equation: where is the unit outward normal vector to the cross section boundary .
The function ( , ) is a new function that we call the torsion function. It is given by the solution of the Poisson equation with a null Neumann condition: (2 ) cos + cos 2 sin − sin cos sin + sin cos cos + sin cos sin )] With the solution for the reactive strains, the energy takes the form Since the strain-energy density is isotropic and its contribution in the expression for includes at most quadratic terms in the strains, we can use without loss of generality the quadratic approximation of : where = − and , are the Lamé parameters. After integration, up to order ( 4 ), we find Here we have used = ( − 1)∕ 0 where 0 is the typical scale of the cross section. We recover the classic extensional, bending, and torsional stiffness coefficients of rod theory where = (3 + 2 )∕( + ) is the Young's modulus which can be taken as a function of position. If and are constant, we recover where  is area of the cross section whereas 1,2 are its second moments of area, and is a parameter that depends only on the cross-sectional shape and the warping function. In addition, we define From these quantities, we extract the intrinsic extension and curvatures: We refer to the last three sets of definitions for , and̂as the active filament formulas as they describe in a fundamental way how curvatures are related to internal stresses induced by growth or activation. These formulas are named in honor of the fundamental helical spring formulas obtained by Thomson and Tait that relate the curvatures to external stresses (Thomson and Tait, 1867;.

Particular case of a circular cross-section
The expressions above simplify greatly if we consider a circular cross-section of radius 0 . In that case, the stiffnesses have been tabulated for different types of inclusions and the appropriate formulas can be found in textbooks for different profiles of . For a uniform material with no variations of , we have: More interestingly, the warping function is identically zero and, since is periodic in , it will not contribute. We then obtain an explicit expression in terms of the given functions = ( , ), = ( , ), = ( , ), and the material parameters and :
From these expressions, we compute the intrinsic stretch and curvatures: These intrinsic curvatures have the familiar form̂= (̂̂sin̂,̂̂coŝ,̂̂+̂′) (Goriely, 2017, p. 103) from which we obtain the intrinsic Frenet curvature and torsion: Note that we must include the factor̂here to take into account the change of length in the rod due to activation. A few comments are in order: • In the case of incompressible activation ( = 1∕2), the factor (1 − + (1 + ) cos 2 ) that appears in (38)-(40) is 1 = 1 + 3 cos 2 . It vanishes at the magic angle  given by ≈ ±54.73 degrees. Therefore, we can design a system with fiber angles close to that particular angle so that 1 = 2 = 0. For this system there is no curvature induced by the activation of the fibers (Goriely and Tabor, 2013). The absence of extension is well known in the theory of McKibben actuators. The absence of curvature at that angle seems to be new and unexpected. It will be further analyzed in the next section. • The contribution to the curvature from activation is specified by the amplitude of the first Fourier components = √ 2 1 + 2 1 . This amplitude can be controlled by a function that can be either continuous or discrete.
• The activation of a ring sector with 1 ≠ 0 leads in general to an intrinsic helical centerline since both curvature and torsion are constant. • Right-handed helical fibers can lead to a left-handed helical shape if − 0 is sufficiently large, which requires average contraction of the fiber on the ring. Any extension of the fibers will lead to a shape with the same handedness. • We emphasize again that the contribution of any activation function ( ) only enters through its first three Fourier coefficients.
Hence, on a ring, there are at most three independent degrees of freedom dictating the curvatures.

Ring solution with piecewise constant and uniform activation
We further constrain the system by assuming that activation is uniform along and piecewise constant along , as shown in Fig. 4, and take 2 = 0 . Let [ ; 0 , ] be the real 2 -periodic function that is zero everywhere except in the interval [ 0 − ∕2, 0 + ∕2] where it is equal to one. Then, our activation function has equally spaced helical activators in the ring: where ≤ 2 ∕ to avoid overlap. The corresponding Fourier coefficients are: We observe that there are only three independent variables for activation: 0 , 1 , and 1 . Therefore, one does not need more than three independent activators, i.e., the choice = 3 is sufficient in terms of their relative effect, and their angular extent can be adjusted to increase the magnitude of the response. The variable 0 is a phase that is used to adjust the initial position of the activator. It can be set to zero without loss of generality by assuming that one of the filament ends can be rotated arbitrarily. Therefore, for the rest of our analysis, we set 0 = 0.
In order to understand the possible material and activation controls, we consider two extreme cases.

Curvature without torsion
We first look at the possibility of creating intrinsic curvature with helical fibers ( 2 ≠ 0) without intrinsic torsion. From the relation (57), we see that it requires a combination of material properties and activation. Indeed, 3 has the same sign as tan 2 . Hence, zero torsion can only occur if 0 is negative, which implies contraction on average, and requires In addition, we need = √ 2 1 + 2 1 to be non-vanishing, which implies that all the cannot be equal to each other. We show in

Twist without curvature
In the absence of curvature, there can be no torsion. Thus, the complementary problem to the problem of curvature without torsion is twisting a rod in the absence of curvature. There are two ways that can be used to remove curvature from the system.
First, by symmetry, we have that 1 = 1 = 0 if = for all . In this case, the system does not develop any curvature but only twist.

Multiple rings
We can design a filamentary structure with multiple rings as shown in Fig. 4. In this case, each new ring contributes to the intrinsic curvatures additively. Denoting by a superscript ( ) ( ) a quantity attached to the th ring from 1 to , we simply have: where the expressions for ( ) 0 , ( ) 1 , ( ) 3 are obtained by replacing 1 and 2 by the internal and external radii of the th ring ( ) 1 and ( ) 2 , respectively. For these solutions, unless the helical angles follow the same rule (42) in all rings, the curvature and torsion are not constant anymore and the activation induces therefore non-helical solutions that can be controlled through the activation parameters. The space of solutions becomes quite rich even with two rings, as demonstrated in Fig. 4. However, a difficulty arises here in that the space of potential configurations cannot be easily quantified, since the actual position of the rod's centerline in space, ( ), is difficult to describe as a function of activation.

Tapered filaments
Many filamentary structures are tapered. It is therefore of interest to understand the advantages or differences that these structures present with respect to a simpler cylindrical profile. The tapering is characterized by a function = ( ) with (0) = 1 such that the external and internal radii are given by 2 ( ) = 2 (0) ( ), and 1 ( ) = 1 (0) ( ), respectively. At a point , the graph of the function 2 ( ) makes an angle 2 ( ) with the -axis as shown in Fig. 7, while = ( , ) is the general fiber tapering angle which varies in both and , such that 2 ( ) = ( 2 , ). We want to define local angles for the fiber on the surface that have the same interpretation as in the case of untapered filaments. To do so, we introduce the local angles̃and̃, and a rotation matrix A fiber defined by a vector = siñsiñ+ siñcos̃+ cos̃,̃,̃∈ ] − ∕2, + ∕2] is mapped to = ⋅̃(69) = (siñsiñcos − cos̃sin ) + siñcos̃+ (siñsiñsin + cos̃cos ) .
These angles now have the same interpretation as the ones given in Fig. 3. For instance, an axial fiber in the tapered case lies in a plane that contains the -axis and is characterized bỹ=̃= 0, and so on. Using these new angles, we can compute the two angular functions that enter the active filament formulas: cos 2 = sin 2̃siñsin 2 − sin 2̃( sin 2̃c os 2 + cos 2̃) + cos 2̃c os 2 , cos sin 2 = 2 siñcos̃(siñsiñsin + cos̃cos ) . (71)

Curvature and torsion in tapered filaments
In the case of the ring solution, we utilize a uniform twist relationship for the tapered fiber field akin to (42), such that tañ= tañ2 wherẽ=̃( ), = ( ) are introduced for notational brevity. For such a construction of a tapered fiber field, the activation function can be written as ( , ) = ( −̃), where the angular shift functioñis given bỹ and 0 ∈ [ 1 (0), 2 (0)] is the radius at which a given activated fiber originates at = 0. Under the assumption of a slow tapering profile ( ), the variation of̃with respect to 0 is negligible. Thus, we can perform the substitution 0 ← 2 (0) to obtain an approximate form̃2( ) =̃( 2 (0), ) of the angular shift, which depends only on . Then, substituting ( , ) = ( −̃2) into (38)-(41) yields where 0 , 1 = 2 , 3 are functions of̃,̃, , 1 , 2 , , and all other quantities are defined as before. In the case of tangentially helical fibers (̃= 0), reduce to where ( ) = arctan The intrinsic extension and curvatures are then obtained via (36). Utilizing this result, we consider the intrinsic curvature,̂, for axially tapered fibers (̃2 = 0,̃= 0), and torsion,̂, for helically tapered fibers (̃2 ≠ 0,̃= 0). These quantities are computed for three different functional forms of the tapering profile ( ), as shown in Fig. 8. Consistently with intuition, both curvature and torsion are monotonically increasing functions of for all considered tapering profiles, as manifested by the spiraling shapes of the deformed filaments. Interestingly, for each of the three cases, the normalized curvature and torsion functions are approximately equal for outside of the close neighborhood of = 0; hence the collapse of̂∕̂m ax and̂∕̂m ax into one curve in each of the three plots. Further, the shapes of the normalized̂andĉ urves are not trivial despite the monotonicity of ( ) and its derivatives, as exemplified by the inflection point in the curvature and torsion functions for the logarithmic profile. Such a complexity arises primarily since we account for the activation of a tapered fiber field embedded in a tapered domain geometry. Such an approach is more biologically relevant because a non-tapered field contained in a tapered domain would result in premature termination of fibers inside the filament.

Activation in non-tapered and tapered filaments
We now compile the main results to inform the comparison of configurations resulting from activation in non-tapered and tapered filaments. In particular, as shown in Fig. 9, we consider a multi-ring geometry in both cases. The first ring contains axial fibers (̃( 1) =̃( 1) = 0), and the second ring consists of helical fibers (̃( 2) ≠ 0,̃( 2) = 0), with ( ) 2 = 0 and ( ) 2 > 0 in the non-tapered and tapered scenarios, respectively. In such a setup, the first ring enables direct curvature control, while the activation of the second ring promotes torsion. Generally, these effects are not decoupled when both rings are activated simultaneously, so special care needs to be taken in designing the activation patterns for a desired configuration to be attained.
The same activation distributions, (1) ( ), (2) ( ) (Fig. 9e), are prescribed in the corresponding rings in both geometries, for the comparison to be meaningful. Moreover, the initial ring thicknesses ( ) 2 (0) − ( ) 1 (0) are set to be the same in both cases, so that the activation magnitudes are comparable as well. In order to ensure that the outer boundary of the first ring and the inner boundary of the second ring coincide at all , (1) 2 is chosen such that (1) 2 (0) = (2) 1 (0) and (1) = (2) at all , for some (2) 2 . For simplicity, the tapering profile ( ) is chosen to be linear, but the same analysis can be readily applied for an arbitrary form of ( ).
Activation of the tapered filament (Fig. 9g) results in a notably more elaborate configuration, as compared to the deformed shape of the non-tapered geometry (Fig. 9f). The variation in the magnitude of curvature and torsion is considerably larger in the tapered case, as expected based on the prior analysis in Fig. 8, but it is now confirmed for the multi-ring scenario. Further, the tapered configuration is more compact in space due to the same activation magnitudes being applied in rings of decreasing thickness, which induces significant curvature and torsion especially towards the end of the filament.

Conclusion
Active filaments are one-dimensional structures that remodel internally. Here, we have assumed that the lone source of this remodeling process is a fiber activation field given at any point in the material, specifying extension or contraction in a given direction. The case of activation by growth or muscular contraction are both taken into account through the specification of a Poisson-like ratio for the extending fiber. This fiber activation is not the most general case of activation as, in principle, a full anelastic tensor could be specified at each point. Yet, from a modeling point of view, these fiber-driven active structures are ubiquitous and universal. Within this framework, we derived the so-called active filament formulas linking fiber stretch and orientation to the intrinsic curvatures generated by the remodeling process.
We further restricted our attention to the case of filaments with circular cross sections and ring solutions for rope-like structures. In these materials, the orientation of the fiber is slaved to the orientation of the active material. As the active material coils around the central core, we assume that the activation takes place in the tangential direction to the coil. A mathematically pleasing model for these structures is given by ring solutions with well-defined sectors of activation, each specified by a single angle, hence restricting the number of parameters. In this case, further analytical progress leads to an explicit form of the curvatures in terms of the Fourier decomposition of the activation functions. The sets of possible shapes of these structures, even in the simple case of a single active ring with a finite number of activation sectors, are remarkably rich. In particular, we showed that a chiral helically-wrapped filament can be tuned to create an achiral intrinsic shape with curvature but no torsion, or a structure with twist but without curvature. It is straightforward to consider these interesting limiting cases, since most of the derived expressions have an analytic form.
From an engineering perspective, the case of curvature without torsion corresponds to the typical application in metallic bilayer actuators. The possibility that one can also achieve twisting without curvature is more surprising, and it constitutes a type of actuation that could be of potential interest to the soft-robotics community.
Our analysis is limited to filaments that are initially straight. If the initial curvature is relatively small with respect to the length, we expect that the intrinsic curvature will be the sum of the initial curvature and the curvature induced by activation. However, for larger curvature, the analysis should be done properly, e.g., starting with a ring or helical initial configuration.
We generalized the ring solutions to the case of tapered filaments, another ubiquitous feature of the natural world. We showed that for a given tapering function, intrinsic curvatures can still be obtained, albeit at the expense of increasingly more complicated analytical formulas. Tapering provides yet another opportunity for control, especially in its ability to create large curvature at the small end of the filament. One would naturally expect that the intrinsic curvature increases as the radius decreases and, once the internal balance of forces is computed, this is mostly what we observe. Yet, the change in radius has another important effect, as it changes the stiffness of the structure. The typical scaling for the bending stiffness, inherited from the second moment of area, is 4 . Hence, the tapering not only affects the curvature, but also allows to connect a stiff region (large radii), that needs to support the weight of the filament, to a soft region (small radii) dedicated to fine manipulations. This is exactly what is observed in the massive elephant trunk in the animal kingdom.
The framework presented here provides a general analytical formulation for the problem of active filaments. It can now be adapted for specific challenges in physics and engineering -particularly, solving inverse problems found in robotics, where a given geometrical property or a final shape of a soft-robotic filament is sought, and the activation functions need to be determined. It can also be used to understand typical and universal design of natural manipulators, such as the elephant trunk.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.