Optimizing energetics of lateral undulatory locomotion: unveiling morphological adaptations in different environments

Abstract

Ongoing efforts seek to unravel theories that can make simple, quantitative and reasonably accurate predictions of the morphological adaptive changes that arise with the size variation. Yet, relatively scant attention has been directed towards lateral undulatory locomotion. In the current study, we explore: (i) the constraints imposed by the variation of length and mass in viscous and dry friction environments on the cost of transport (COT) of lateral undulatory locomotion and (ii) the role of the body, environment and input oscillations in such an intricate interplay. In a dry friction environment, minimum COT correlates with stiffer and longer bodies, higher frictional anisotropy and angular amplitudes greater than approximately 10^o. Conversely, a viscous environment favours flexible long bodies, higher frictional anisotropy and angular amplitudes lower than approximately 30^o. In both environments, optimizing mass and maintaining low angular frequencies minimizes COT. Our conclusions are applicable only in the low-Reynolds-number regime, and it is essential to consider the interdependence of parameters when applying the generalized results. Our findings highlight musculoskeletal and biomechanical adaptations that animals may use to mitigate the consequences of size variation and to meet the energetic demands of lateral undulatory locomotion. These insights enhance foundational biomechanics knowledge while offering practical applications in robotics and ecology.

1. Introduction

The quest to identify how mass and size vary with the energetic cost of transport (COT)—a measure of the mass-specific energy required to move an object a unit of distance—has a long history [1]. Research in this area has provided valuable insights into preferred morphology, physiology and biomechanics of locomotion for different animal groups and locomotory modes [2–13].

Investigations into locomotion efficiency and COT have elucidated evolutionary transitions and trade-offs between different performance measures, which, in turn, can inform the design of adaptive and energy-efficient robotic solutions [14]. Among the various modes of locomotion, undulatory locomotion is pervasive across terrains and body sizes in limbless animals [15,16]. However, the study of the energetics of undulatory locomotion remained relatively unexplored. Most studies have focused on comparing the energetics of different gaits of undulatory locomotion with legged locomotion. For example, in [17], peristaltic locomotion (figure 1a) of larval blowflies was observed to be eight to ten times more energetically expensive than legged locomotion of similarly sized organisms. In [19], the metabolic COT of gypsy moth caterpillar was found to be 4.5 times higher than that of similar vertebrates and arthropods. The resistance generated by internal fluid pressure and increased contact of the body with the ground increased the COT compared with limbed animals. In Coluber constrictor, the COT of lateral undulatory locomotion (figure 1b) was equal to the legged animal of similar size [20]. Extrapolated data from limbed lizards suggest that COT for sidewinding would be comparable to that of similarly sized lizards [21]. Further comparisons among different modes of undulatory locomotion show that the COT of Crotalus cerastes, which performs sidewinding (figure 1c), is lower than the COT of Coluber constrictor performing either lateral undulatory or concertina locomotion (figure 1d) [21].

Figure 1.

Different modes of limbless locomotion. (a) Peristatic locomotion is when the body moves by the contraction and relaxation of the body segments. (b) Lateral undulatory locomotion is in the form of sinusoidal body movements. (c) Sidewinding is when the body moves sinusoidally and lifts itself off the ground, creating discontinuous tracks. (d) Concertina locomotion involves alternately straightening and bending sections of the body. (e) Polychaete-like locomotion with body waves aligning with the direction of motion [18].

Although the efficiency of undulatory locomotion has been investigated to determine optimal gaits [22,23], the COT of lateral undulatory locomotion has never been investigated as a function of the properties of the body and the surrounding environment. Motivated by the existing research gap in the study of COT of limbless animals, we investigate how COT varies for lateral undulatory locomotion with body and environmental properties. The contribution of our work is twofold. First, we present the first attempt to elucidate the variation of COT in limbless lateral undulatory locomotion as a function of body properties through mathematical modelling. Second, we examine the influence of environmental factors on COT, addressing a significant gap in existing research where limited work has quantified the impact of environmental factors on COT [24].

We specifically focus on studying lateral undulatory locomotion among various locomotion strategies of limbless animals due to its significance within this group and its existence in organisms with a variety of body shapes and sizes [25–27]. We do not address a specific organism, but we model an archetypical organism that exhibits undulatory locomotion. Our mathematical model is tested on a prototype [28] on land, as well as on biological data for a corn snake [29] and for Caenorhabditis Elegans [30], demonstrating its efficacy in capturing the normal models of lateral undulatory locomotion across environments.

2. Mathematical modelling

Our model is based on the resistive force theory (RFT), first proposed by Gray for low-Reynolds-number lateral undulatory locomotion [31]. Its main postulates are as follows: (i) the body of an organism can be regarded as the combination of elements; (ii) the magnitude of the thrust depends on internally generated bending couples and the component of friction force acting normally to the body; (iii) the phase of each element varies along the length; and (iv) disturbances in the surroundings remain in the vicinity of the body, and viscous forces dominate inertial forces [32]. The effectiveness of resistance force theory was tested in granular media [33,34], in viscous media [35,36] and on land [28,37]. RFT has also been applied in conjunction with geometric mechanics to unveil morphological adaptations in long and cylindrical animals [38,39].

To understand how propulsion can be achieved in lateral undulatory locomotion, we consider a small segment $\mathrm{\delta }s$ from an undulating body (figure 2). The forces acting normal and tangential to the element are ${\displaystyle \delta N}$ and ${\displaystyle \delta \mathrm{T}}$, respectively. The element is inclined to the $x$-axis by an angle ${\displaystyle \theta }$. The velocity of the element in the $x$, ${\displaystyle v_x}$, and $y$, ${\displaystyle v_y}$, directions can be decomposed into velocities in the normal, ${\displaystyle v_n}$, and tangential, ${\displaystyle v_t}$, directions as shown in figure 2. The reaction forces on this element in the normal and tangential directions are ${\displaystyle \delta N=-C_n{v}_n\delta s}$ and ${\displaystyle \delta T=-C_t{v}_t\delta s}$, respectively. These reaction forces can be written as

Figure 2.

Schematic illustrates the forces acting on an element $\delta s$, with velocity components along the $x$ and $y$ directions represented by $v_x$ and $v_y$, respectively. The net propulsive force is ${\displaystyle \delta N\mathrm{s}\mathrm{i}\mathrm{n}\left(\theta \right)-\delta T\mathrm{c}\mathrm{o}\mathrm{s}\left(\theta \right)}$.

$${\displaystyle \delta N=C_n\left(v_y\mathrm{c}\mathrm{o}\mathrm{s}\left(\theta \right)-v_x\mathrm{s}\mathrm{i}\mathrm{n}\left(\theta \right)\right)\delta \mathrm{s}}$$ (2.1)

and

$${\displaystyle \delta T=C_t\left(v_y\sin \left(\theta \right)+v_x\mathrm{c}\mathrm{o}\mathrm{s}\left(\theta \right)\right)\delta \mathrm{s}.}$$ (2.2)

Equations (2.1) and (2.2) are obtained by substituting the components of $v_x$ and $v_y$ in the directions of $v_n$ and $v_t$. ${\displaystyle C_t}$ is the coefficient of viscous resistance in the tangential direction and ${\displaystyle C_n}$ is the coefficient of viscous resistance in the normal direction. The resultant forward thrust is ${\displaystyle \delta F=\delta N\mathrm{s}\mathrm{i}\mathrm{n}\left(\theta \right)-\delta T\mathrm{c}\mathrm{o}\mathrm{s}\left(\theta \right)}$ (figure 2). Substituting the values of ${\displaystyle \delta N}$ and ${\displaystyle \delta T}$ in the resultant thrust and after simplification, we obtain

$${\displaystyle \delta F=\left(\frac{\left(C_n-C_t\right)v_y\mathrm{t}\mathrm{a}\mathrm{n}\left(\theta \right)-v_x\left(C_t+C_n{\tan }^2\left(\theta \right)\right)}{1+{\tan }^2\left(\theta \right)}\right)\delta \mathrm{s}.}$$ (2.3)

To attain a positive thrust in the $x$ direction, equation (2.3) should be greater than 0, which implies ${\displaystyle C_n>C_t}$. Thus, in order to attain a positive thrust, the normal frictional coefficient should be greater than the tangential frictional coefficient.

2.1. Conceptual model of the body

We conceptualize the organism as a discrete system of rigid links, with internal damping and stiffness at the joints (figure 3). The total length, ${\displaystyle l_{\mathrm{t}\mathrm{o}\mathrm{t}}}$, of the body is divided into equal length links, ${\displaystyle l_i=l_{\mathrm{t}\mathrm{o}\mathrm{t}}/N}$. Here, $l_i$ is the length of the ith link and $N$ is the total number of links. Three-link models have been studied extensively in the context of lateral undulatory locomotion and are deemed to be sufficient to approximate the basic gaits [40–45]. Based on these efforts and other prior research [29,46–49] we use three links throughout the study. The body is heaved sinusoidally at one end, and the rest of the body follows passively. This establishes the input to the system as the angle of the one end link, represented by ${\theta }_N\left(t\right)$ (figure 3).

Figure 3.

Schematic of a three-link body (N = 3) where sinusoidal oscillations are applied to one end, ${\theta }_3\left(t\right)$, and the links are joined together by viscoelastic joints with rotational stiffness constant $k_1$ and $k_2$ and internal damping constant $b_1$ and $b_2$. $u_1$ and $u_2$ are the x and y coordinates of the end point. $n_i$ and $t_i$ represent the normal and tangential directions, respectively.

The equations of motion are derived using an Euler–Lagrange formulation,

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\mathrm{\partial }L}{\mathrm{\partial }{\overrightarrow{\dot{q}}}_h}\right)-\frac{\mathrm{\partial }L}{\mathrm{\partial }{\overrightarrow{q}}_h}=Q_h-\frac{\mathrm{\partial }R}{\mathrm{\partial }{\overrightarrow{\dot{q}}}_h},}$$ (2.4)

where $L$ is the Lagrangian, which involves the kinetic and potential energies of the system [29], ${\overrightarrow{q}}_h$ is the set of generalized coordinates that define the state of the body,

$${\displaystyle {\overrightarrow{q}}_h\left(t\right)=\left[\begin{array}{c}{u}_1\left(t\right)\\ u_2\left(t\right)\\ {\theta }_1\left(t\right)\\ ⋮\\ {\theta }_{N-1}\left(t\right)\end{array}\right].}$$ (2.5)

${\displaystyle {\mathit{Q}}_h}$ is the interaction with the environment (the generalized force of friction) and $R$ is the internal damping term,

$${\displaystyle {\displaystyle R\left({\overrightarrow{\dot{q}}}_h,t\right)=\frac{1}{2}\sum _{i=1}^{N-1}{b}_i{\left({\dot{\theta }}_{i+1}-\dot{{\theta }_i}\right)}^2.}}$$ (2.6)

Here, $b_i$ is the internal damping constant of the ith joint, simulating energy dissipation as the body deforms.

2.2. Modelling the interaction of the body with the environment

To conduct a comparative analysis of organism performance across various environments, we categorize generalized forces of friction in two forms: dry friction and viscous.

2.2.1. Locomotion in a dry friction environment

Locomotion in a dry friction environment is modelled using Coulomb’s law of friction. The generalized force of friction is represented by ${\displaystyle Q_{h,\mathrm{d}\mathrm{r}\mathrm{y}}}$,

$${\displaystyle {\displaystyle Q_{h,\mathrm{D}\mathrm{r}\mathrm{y}}=\sum _{i=1}^N{\overrightarrow{F}}_{i,\mathrm{D}\mathrm{r}\mathrm{y}}\left({\overrightarrow{\dot{q}}}_h,{\overrightarrow{q}}_h,t\right)\frac{\mathrm{\partial }{\overrightarrow{r}}_{l_i/2}}{\mathrm{\partial }{\overrightarrow{q}}_h},}}$$ (2.7)

where ${\displaystyle {\overrightarrow{r}}_{l_i/2}}$ is the position vector of the midpoint of the ith link, corresponding to the contact points between the body and the ground. Thus, the number of contact points is equal to the number of links. In a dry friction environment, the entire body of the animal does not make continuous contact with the ground [37,49]. This makes the model relevant to actual locomotion, where body–ground interactions occur only at specific contact points. The force of friction acting on each link is the sum of its normal and tangential components,

$${\displaystyle {\overrightarrow{F}}_{i,\mathrm{D}\mathrm{r}\mathrm{y}}=-g{\mu }_t{m}_i\mathrm{s}\mathrm{g}\mathrm{n}({\overrightarrow{\dot{r}}}_i.{\hat{t}}_i){\hat{t}}_i-g{\mu }_n{m}_i\mathrm{s}\mathrm{g}\mathrm{n}({\overrightarrow{\dot{r}}}_i.{\hat{n}}_i){\hat{n}}_i.}$$ (2.8)

We note that the total friction force comprises the normal and tangential components, due to frictional anisotropy. The unit vectors in the tangential and normal directions are ${\widehat{t}}_i$ and ${\widehat{n}}_i$, respectively (see figure 3). Frictional anisotropy provides the driving force to lateral undulatory locomotion if the normal coefficient of frictional resistance (${\displaystyle {\mu }_n}$) is greater than the tangential coefficient of frictional resistance (${\displaystyle {\mu }_t}$). Here, sgn() is the sign function [28,29]; $g$ is the gravitational acceleration; $m_i$ is the mass of the ith link and ${\displaystyle {\overrightarrow{\dot{r}}}_i}$ is a time derivative of the position vector for the ith link. Electronic supplementary material, video S1 shows undulatory locomotion of a three-link organism in a dry friction environment.

2.2.2. Locomotion in a viscous medium

Locomotion in a viscous medium is characterized by viscosity. Generalized force of friction in a viscous medium is represented as

$${\displaystyle {\displaystyle Q_{h,\mathrm{V}\mathrm{i}\mathrm{s}\mathrm{c}}=\sum _{i=1}^N{\int }_0^{l_i}{\overrightarrow{F}}_{i,\mathrm{V}\mathrm{i}\mathrm{s}\mathrm{c}}\left(\overrightarrow{\dot{q}},\overrightarrow{q},t,s_i\right).\frac{\mathrm{\partial }{\overrightarrow{r}}_i}{\mathrm{\partial }{\overrightarrow{q}}_h}d{s}_i.}}$$ (2.9)

For further details on the modelling of locomotion in a viscous medium (equation (2.9)) see [30]. The force of friction acting on each link in a viscous medium is defined as

$${\displaystyle {\overrightarrow{F}}_{i,\mathrm{V}\mathrm{i}\mathrm{s}\mathrm{c}}=-C_t({\overrightarrow{\dot{r}}}_i.{\hat{t}}_i){\hat{t}}_i-C_n({\overrightarrow{\dot{r}}}_i.{\hat{n}}_i){\hat{n}}_i.}$$ (2.10)

Different notations for the coefficients of resistance are used to differentiate locomotion in dry friction and viscous environments. Electronic supplementary material, video S2 demonstrates lateral undulatory locomotion of a three-link organism in a viscous environment.

2.2.3. Calculating the cost of transport

We assume that during lateral undulatory locomotion, energy dissipation is due to friction and internal damping. COT is estimated by summing up the energy dissipated according to each unknown generalized coordinate during a given time interval. The energy dissipated for a generalized coordinate is calculated by integrating the power dissipated, which is obtained by multiplying generalized velocity with non-conservative generalized force (right-hand side of equation (2.4)). Therefore, the total energy dissipation during the time interval $t_0$ to $t_1$ is calculated as follows:

$${\displaystyle {\displaystyle {\mathcal{E}}_{\mathrm{d}}=-\sum _{h=1}^{N+1}{\int }_{t_0}^{t_1}\left[\left(Q_h-\frac{\mathrm{\partial }R}{\mathrm{\partial }{\dot{q}}_h}\right){\dot{q}}_h\right]\text{d}t.}}$$ (2.11)

Here, generalized non-conservative force can be dry friction or viscous, depending on the type of environment, as defined in equations (2.7) and (2.9). The time interval is kept constant throughout the parametric study. For further details of the energy calculation, see [29,50]. We define COT in joules per kilogram per distance travelled in metres (J (kg m)⁻¹), by dividing the energy dissipation (${\mathcal{E}}_{\mathrm{d}}$) by the total distance covered by the body and its mass. The work done by the actuation during the time interval can be calculated as follows [51]:

$${\displaystyle \mathrm{W}={\int }_{t_0}^{t_1}{\lambda }_{{\theta }_N}{\dot{\theta }}_N\text{d}t,}$$ (2.12)

where ${\lambda }_{{\theta }_N}$ is defined as

$${{\mathrm{\lambda }}_{\mathrm{\theta }}}_N=\frac{\text{d}}{\text{d}t}\left(\frac{\text{∂}L}{\text{∂}{\dot{\mathrm{\theta }}}_N}\right)-\frac{\text{∂}L}{\text{∂}{\mathrm{\theta }}_N}-Q_{{\mathrm{\theta }}_N}+\frac{\text{∂}R}{\text{∂}{\dot{\mathrm{\theta }}}_N} .$$ (2.13)

The relationship between the work done by the actuation and the dissipated energy is given as

$${\displaystyle \mathrm{\Delta }U=\mathrm{W}-{\mathcal{E}}_{\mathrm{d}}.}$$ (2.14)

In equation (2.14), $\mathrm{\Delta }U$ represents the variation in the elastic and kinetic energy, which solely depends on the initial and final states of the system. $\mathrm{\Delta }U$ can be directly calculated as

$${\displaystyle {\displaystyle \mathrm{\Delta }U=\left(\sum _{h=1}^{N+2}\frac{\mathrm{\partial }L}{\mathrm{\partial }{\dot{q}}_h}{\dot{q}}_h-L\right){|}_{t_0}^{t_1}.}}$$ (2.15)

At the end of the solicitation cycle in the steady state, $\mathrm{\Delta }U$ approaches zero [51]. Hence, defining COT either based on the work done by the input or the dissipation energy is equivalent.

In living organisms, a part of the total available energy supports basic life functions, even when they are at rest [52]. When we have access to the information about the total available energy, efficiency can be defined as the fraction of energy translated into locomotion relative to the total available energy [53,54]. In our case, the work done by the input is equal to the dissipated energy. We defined efficiency as the ratio between the energy required to pull a straightened body along its axis at a constant speed and the total energy dissipated by the undulatory locomotion. We adopted this definition, also known as quasi-propulsive efficiency, due to its relevance to low-Reynolds-number swimming [55,56]

$${\displaystyle e=\frac{{\text{COT}}_{\mathrm{m}\mathrm{i}\mathrm{n}}}{\text{COT}},}$$ (2.16)

where ${\text{COT}}_{\mathrm{m}\mathrm{i}\mathrm{n}}=\frac{{\mathcal{E}}_{\mathrm{m}\mathrm{i}\mathrm{n},\mathrm{d}}}{m_{\mathrm{t}\mathrm{o}\mathrm{t}}D}$, with $D$ being the horizontal distance being covered in time $\mathrm{\Delta }t$, and ${\displaystyle m_{\mathrm{t}\mathrm{o}\mathrm{t}}}$ the mass of the body. In our model of undulatory locomotion, the minimum energy configuration is obtained under the following conditions, $u_2\left(t\right)=0$, ${\mathrm{\theta }}_1\left(t\right)=0$, ${\mathrm{\theta }}_2\left(t\right)=0$ and ${\mathrm{\theta }}_3\left(t\right)=0$. The minimum dissipated energy, ${\mathcal{E}}_{\mathrm{m}\mathrm{i}\mathrm{n},\mathrm{d}}$, is

$${\displaystyle {\mathcal{E}}_{\mathrm{m}\mathrm{i}\mathrm{n},{\mathrm{d}}_{\mathrm{V}\mathrm{i}\mathrm{s}\mathrm{c}}}=C_t{v}^2{l}_{\mathrm{t}\mathrm{o}\mathrm{t}}\mathrm{\Delta }t}$$ (2.17)

for viscous medium and

$${\displaystyle {\mathcal{E}}_{\mathrm{m}\mathrm{i}\mathrm{n},{\mathrm{d}}_{\mathrm{D}\mathrm{r}\mathrm{y}}}=m_{\mathrm{t}\mathrm{o}\mathrm{t}}\mathrm{g}{\mu }_{t}v\mathrm{\Delta }t,}$$ (2.18)

for a dry friction environment. Here, $\mathrm{\Delta }t=t_1-t_0$ and $v$ is the velocity calculated by dividing the horizontal distance covered by the body during time $\mathrm{\Delta }t$. Equation (2.16) shows that COT is inversely related to the efficiency of locomotion. Note that further simplifications of the minimum COT led to

$${\displaystyle {\text{COT}}_{\mathrm{m}\mathrm{i}\mathrm{n},\mathrm{V}\mathrm{i}\mathrm{s}\mathrm{c}}=\frac{C_{t}v{l}_{\mathrm{t}\mathrm{o}\mathrm{t}}}{m_{\mathrm{t}\mathrm{o}\mathrm{t}}}}$$ (2.19)

and

$${\displaystyle {\text{COT}}_{\mathrm{m}\mathrm{i}\mathrm{n},\mathrm{D}\mathrm{r}\mathrm{y}}=\mathrm{g}{\mu }_t.}$$ (2.20)

3. Validation of power calculations

We performed a comparative analysis of the locomotion power predicted by our model and the values reported in the literature by selecting test organisms that exhibit lateral undulatory locomotion in the respective environments; corn snake (Pantherophis guttatus) for dry friction environment and Caenorhabditis elegans for viscous medium.

3.1. Validation in a dry friction environment

To calculate COT of a corn snake, the body mass is taken as 0.73 kg, the length as 1.313 m and the rotational stiffness $k_i$ as 0.01196 N m [57]. The normal coefficient of frictional resistance ${\displaystyle \left({\mu }_n\right)}$ and the angular amplitude of undulation are chosen as 0.2 and 7π/37 rad (34.05°), respectively [37]. The frequency of input is fixed at 1.5 Hz [58]. The frictional coefficient ratio is fixed at 4 and internal damping constant set at 0.15 N m s [29]. By utilizing the above-mentioned parameters, we estimate that the locomotion utilizes 3.17 W of power and achieves a speed of 2.7 cm s⁻¹ which corresponds to the speed of the snake [37]. Power calculated using the contractive force of muscles for various snakes’ body profiles and winding angles ranges from 0.914 to 1.862 W [59], which is of the same order of magnitude as our predictions.

3.2. Validation in a viscous medium

We chose C. elegans as our test organism for the validation in a viscous medium because it is extensively studied in low-Reynolds-number regimes [35,60–62]. We sourced body properties for our simulations from literature on C. elegans, the mass is calculated using the formula ${\displaystyle l_{C. Elegans}D/1.6\times {10}^6}$ μg [63], where $l_{C. Elegans}$ is the length of the organism (1.15 mm [64]) and ${\displaystyle D}$ is the diameter (80 μm [65]). Rotational stiffness is calculated using, $EI/l_{C.Elegans}$ [30], where $E$ is the Young’s modulus (table 1) sourced from various references to account for differences in magnitude due to different measurement methods used. $I$ is the second moment of area as approximated by respective sources [60–62,66–68] (table 1). Simulations are performed with the frequency and angular amplitude of the input set to 4π rad s⁻¹ [61] and a = 11π/100 rad (${\displaystyle {19.8}^{\circ }}$) [26]. Internal damping ($b_i=4.34\times {10}^{-13}$ N m s) is calculated from the coefficient of internal viscosity as mentioned in [62]. Power is calculated using a frictional coefficient ratio of 4.1 [35].

Table 1.

List of Young’s modulus used to estimate the rotational stiffness for joints in the simulations and corresponding values of obtained speed and power in picowatts.

order	Young’s modulus ($E$)	estimated rotational stiffness (N m)	speed (mm s−1)	predicted power per cycle (pW)
a.	140 ± 20 kPa [66]	$2.45\times {10}^{-10}$	0.198	72.63
b.	10 MPa [67]	$2.45\times {10}^{-9}$	0.17	583.26
c.	110 ± 30 kPa [68]	$1.92\times {10}^{-10}$	0.1943	58.16
d.	13 MPa [62]	$2.27\times {10}^{-8}$	0.024	713.12
e.	0.62 ± 0.05 MPa [61]	$5.96\times {10}^{-13}$	0.0024	11.86
f.	3.77 ± 0.62 kPa [60]	$4.42\times {10}^{-13}$	0.00166	11.86

Table 1 lists the predicted power dissipation per cycle of locomotion and corresponding speeds of C. elegans. Our predictions for the power dissipation per cycle of C. elegans’ locomotion in the viscous medium are an order of magnitude greater than 3−6 pW reported in [35], corresponding to the speed of 0.35 mm s⁻¹. This difference is expected, as the reported value in [35] represents a lower limit of power dissipation, given the inherent assumptions and limitations of the estimation conducted therein. We believe that the higher values reported in our work are consistent with a more comprehensive model of energy dissipation.

4. Results

Here, we report how COT of lateral undulatory locomotion varies with body mass and length, and how body properties, frictional anisotropy, angular amplitude and frequency influence these variations.

As size increases, the validity of RFT can be challenged because of the dominance of inertial effects and increase in Reynolds numbers [27]. However, RFT has been successfully validated for physical models of lengths 86 [69], 48 [70] and 28.7 cm [28], and for organisms of sizes ranging from micrometre [36] up to 44.1 cm [71]. Hence, considering the limitation of RFT and evidence from the literature, we selected 1 m as the upper limit for our investigation. This choice aligns with the suggested transitional length of slow swimmers, above which the fluid–structure interaction becomes salient [72]. Similarly, we selected 12 kg as the upper limit for mass, based on the allometric relationship between the mass and length of undulatory animals, L = 0.44M^0.33 (L is the length of the organism in meters, M is the mass in kilograms and 0.44 is expressed in m kg^−0.33) [72], for a length of 1 m. During the analysis, we kept the mass constant and varied the length—and vice versa—to isolate the effects of each variable. This approach helps us set lower (0.05 m for the length and 0.1 kg for the mass) and upper (1 m for the length and 12 kg for the mass) bounds for the length and mass, for stability.

4.1. The effect of body properties on the variation of cost of transport

In a dry friction environment, COT is low for all body lengths when internal damping is sufficiently low (figure 4a). However, as the internal damping increases, the body length and stiffness become important factors to minimize COT. Therefore, for elevated internal damping, longer and stiffer bodies offer an optimal combination to minimize COT; contrarily, shorter bodies necessitate minimum internal damping. In contrast, shorter bodies with internal damping and stiffness exhibit low COT in a viscous medium (figure 4b); however, their speed is also low (electronic supplementary material, figure S1). The optimal configuration is a longer and flexible body with less internal damping (figure 4b and electronic supplementary material, figure S1).

Figure 4.

The effect of body properties on the variation of COT with length and mass. Results are obtained by setting angular amplitude to π/6 rad (30°), angular frequency of the input to 2π rad s⁻¹, frictional coefficient ratio to 42.8, normal coefficients of frictional and viscous resistance to 3 and tangential coefficients of frictional and viscous resistance to 0.07. Legends represent the combinations of k1, k2, k3, k4 with b1, b2, b3 and b4. k1 is 0.5 N m rad⁻¹, k2 is 1 N m rad⁻¹, k3 is 4 N m rad⁻¹, k4 is 8 N m rad⁻¹, b1 is 0 N m s rad⁻¹, b2 is 0.1 N m s rad⁻¹, b3 is 0.5 N m s rad⁻¹ and b4 is 1 N m s rad⁻¹. (a,b) The total mass of the body is 0.35 kg (c, d), the total length of the body is 0.5 m. (a) An appropriate combination of length and stiffness with low internal damping can optimize COT in a dry friction environment. (b) In a viscous medium, reduced internal damping and flexibility are favoured for an optimal COT. (c) In a dry friction environment, an optimal COT can be achieved through the combination of mass and stiffness at a reduced internal damping. In the regions of discontinuity, a transition between polychaete-like and lateral undulatory locomotion occurs. (d) In a viscous medium, flexibility and low internal damping are desirable, where COT can be optimized through mass selection.

In a dry friction environment, low stiffness and high damping yield polychaete-like locomotion (figure 1e) with high COT (figure 4c). Polychaete-like locomotion is associated with a negative COT that represents the reversal of the direction of locomotion. For low internal damping and high stiffness, the optimum mass can minimize COT (figure 4c and electronic supplementary material, figure S1). Similarly, in a viscous medium, flexible bodies with low internal damping and optimum mass minimize COT (figure 4d and electronic supplementary material, figure S1).

We present a summary of the results in table 2.

Table 2.

Summary of the results to obtain minimal COT by varying body properties in different environments.

environment	internal damping	stiffness	length/mass
dry friction environment	zero	optimum	all/optimum
	non-zero	high	long/optimum
viscous medium	low	low	long/optimum

In terms of energy efficiency, both environments favour longer and flexible bodies with minimum internal damping (figure 5a,b). The efficiency of lighter bodies is higher than heavier bodies in a viscous medium (figure 5d). Polychaete-like locomotion is marked by both inefficiency and a high COT.

Figure 5.

Parameters and legends are the same as figure 4. (a,b) Percentage efficiency in both environments is high for flexible and longer bodies with minimum internal damping. (c) In a dry friction environment, flexibility and minimum internal damping are preferred for efficient locomotion. (d) In a viscous medium, light bodies are more energy efficient with flexibility and minimum internal damping.

4.2. Effect of frictional anisotropy on the variation of cost of transport

In a dry friction environment, the COT of a shorter body reduces by increasing the frictional anisotropy (figure 6a). In environments with lower frictional anisotropy, longer body lengths are preferable.

Figure 6.

Plots showing the trends of COT with length and mass, and how frictional anisotropy affects it. Results are obtained by setting angular amplitude to π/6 rad (30°), angular frequency to 2π rad s^-1, tangential coefficients of frictional and viscous resistance to 0.07 and then normal coefficients of frictional and viscous resistance are varied. Stiffness is set to 1 N m rad⁻¹ and damping to 0.1 N m s rad⁻¹. (a,b) The total mass is 0.35 kg. (c,d) The total length is 0.5 m. (a) In a dry friction environment, either longer lengths are preferred in a low frictional anisotropic environment or shorter lengths in a high frictional anisotropic environment. (b) In a viscous medium, longer bodies in high frictional anisotropic environments are more efficient. (c) In a dry friction environment, an optimal value of frictional anisotropy can be found for a lighter body. However, for a heavier body frictional anisotropy greater than 10 is favourable. (d) In a viscous medium, as the frictional anisotropy increases range of masses where COT is minimum also increases.

In a viscous medium, cost-effective locomotion becomes feasible when a longer body slithers in an environment with high frictional anisotropy (figure 6b and electronic supplementary material, figure S2).

In a dry friction environment, lighter bodies have low COT when frictional anisotropy is greater than 10 (figure 6c) and are more sensitive to frictional anisotropy, unlike heavier bodies. Therefore, frictional anisotropy plays a significant role in optimizing COT for a lighter body.

Similarly, in a viscous medium, we can see the regions where COT is minimal, and we can exploit them by selecting appropriate mass according to frictional anisotropy (figure 6d). Furthermore, as frictional anisotropy increases, the range of optimum masses widens (figure 6d, electronic supplementary material, figure S2).

Table 3 presents the summary of the results to optimize COT based on frictional anisotropy.

Table 3.

Summary of the results to obtain minimal COT based on frictional anisotropy.

environment	frictional coefficient ratio	length/mass
dry friction environment	high	short/light
viscous medium	high/low	long/light

Similarly, for efficient locomotion in a dry friction environment, short bodies require high frictional anisotropy, while low frictional anisotropic environment requires longer bodies (figure 7a). In a viscous medium, efficiency increases by increasing frictional anisotropy (figure 7b). In a dry friction environment, higher frictional anisotropy improves the robustness of the system by allowing a broader range of masses to achieve high efficiency (figure 7c). In a low frictional anisotropic environment, increasing the body mass is crucial to enhance efficiency. However, in a viscous medium, light bodies are more energy efficient (figure 7d).

Figure 7.

Simulation parameters and legends are the same as in figure 6. (a) In a dry friction environment, efficient locomotion requires optimum length. (b) In a viscous medium, efficiency increases with frictional anisotropy. (c) In a dry friction environment, higher anisotropy enables a broader range of masses to achieve high efficiency, whereas low anisotropy requires heavier bodies. (d) In a viscous medium, lighter bodies are more energy efficient.

4.3. The influence of input parameters on the variation of cost of transport

COT decreases with increasing length and remains constant with further increase in length, for frequency π/2 rad s⁻¹ and angular amplitudes from 10^o to 40^o (figure 8a). However, speed increases when the angular amplitude is high (electronic supplementary material, figure S3). This trend reflects a preference for angular amplitudes greater than 10^o for lateral undulatory locomotion in a dry friction environment at intermediate to longer bodies. Conversely, lower angular amplitudes do not prefer higher frequencies (figure 8a). Furthermore, the optimum length is required at higher angular frequencies to minimize COT. Contrarily, viscous media prefers the combination of longer length, lower angular amplitude and frequency to achieve optimum COT (figure 8b).

Figure 8.

The influence of angular amplitude and frequency on the trends of COT with length and mass. Results are obtained by setting the frictional coefficient ratio to 42.8, normal coefficients of frictional and viscous resistance to 3 and tangential coefficients of frictional and viscous resistance to 0.07, damping constant to 0.1 N m s rad⁻¹ and stiffness to 1 N m rad⁻¹. Legends represent the combinations of angular amplitude (a1, a2, a3, a4) and frequency (ω1, ω2, ω3, ω4). a1 is 10^ο, a2 is 20^ο, a3 is 30^ο, a4 is 40^ο, ω1 is π/2, ω2 is π, ω3 is 4π and ω4 is 8π rad s⁻¹. (a,b) The total mass of the body is 0.35 kg. (c,d) The total length of the body is 0.5 m. (a) COT in a dry friction environment can be reduced by an optimal combination of frequency, length and angular amplitude. (b) In a viscous medium, a lower angular amplitude and a longer length can produce efficient locomotion (low COT). (c) In a dry friction environment, higher frequencies are not desirable for heavy bodies, and lighter bodies with higher angular amplitude are preferred (electronic supplementary material, figure S3). (d) In a viscous medium, low to intermediate mass is favoured for all tested ranges of angular frequencies and angular amplitudes.

In a dry friction environment, lighter bodies with high angular amplitudes have optimum COT for any frequency (figure 8c and electronic supplementary material, figure S3). However, a range of masses shows low COT at low frequency.

Similarly, in a viscous medium, minimal COT can be achieved by an optimal combination of mass and angular amplitude, with mass in the lower to intermediate range (figure 8d). As frequency increases, COT increases (figure 8d).

Table 4 shows optimum combinations of angular frequency and angular amplitude to minimize COT.

Table 4.

Summary of the key results for achieving minimal COT through a combination of angular amplitude and frequency.

environment	angular amplitude	frequency	length/mass
dry friction environment	high	low	intermediate/light
	high	high	optimum/light
viscous medium	low	low	long/light

In both environments, efficient locomotion requires lower frequencies (figure 9a–d). In a dry friction environment, intermediate to higher angular amplitudes with optimum length, and in a viscous medium, intermediate angular amplitude with longer lengths yields efficient locomotion (figure 9a,b).

Figure 9.

Legends and parameters are the same as figure 8. (a–c) Lower frequencies and intermediate angular amplitudes yield efficient locomotion. (a) An optimum body length is required in a dry friction environment. (b) Longer lengths are preferred in a viscous medium. (c) In a dry friction environment, light bodies are less efficient. (d) In a viscous medium, lighter bodies remain more efficient.

5. Discussion and conclusions

We investigated how the COT and efficiency of an organism vary with its size and how it constrains the ability of the organism to undertake lateral undulatory locomotion in dry friction and viscous environments.

In a dry friction environment, a body that is both flexible and long is efficient from an energetic standpoint but possesses high COT. Evolutionary simulations showed that stiffer bodies evolve more effective gaits on land [73]. This observation aligns with the minimization of COT compared with the maximization of efficiency in a dry friction environment.

Contrarily, in a viscous medium, a long body should be flexible to optimize efficiency and COT. Evolutionary simulations of a soft robot also showed the significance of flexibility in a viscous medium [73]. Our findings are only applicable to viscous swimmers where inertial effects are negligible. However, when considering fluid–structure interaction for inertial swimmers, swimming efficiency decreases by increasing flexibility [74]. Contrarily, research involving high-inertia sinusoidal oscillating foils revealed that flexibility improves thrust [75], suggesting that varying swimming regimes can yield different results. This evidence highlights the nuanced relationship between flexibility, inertial effects and swimming efficiency. To deal with such regimes, animals probably modulate their body stiffness to enhance their manoeuverability [76,77] and adjust their performance by varying the stiffness along their body [28,78].

A short body needs high frictional anisotropy and minimal internal damping to minimize COT and increase efficiency in a dry friction environment. Moderate lengths allow optimization of frictional anisotropy to minimize COT in a dry friction environment. In a viscous medium, longer lengths demand heightened frictional anisotropy to minimize COT, posing a challenge for aquatic organisms to modulate frictional anisotropy relative to terrestrial organisms. For example, snakes on land can adjust the angle of their scales [57] and the number of contact points [49] to alter the frictional anisotropy. However, these mechanisms are not available for organisms in viscous environments. In a hydrodynamic environment, where inertial effects dominate and transfer significant momentum to the surrounding fluid, animals exploit other mechanisms to increase efficiency, such as vortices [79,80] and passive energy recapture [81].

In a viscous medium, COT is directly proportional to the length. After a critical length or a transition period, COT becomes inversely proportional to the length. This trend is qualitatively consistent with the predictions of [72], where COT is directly proportional to the length below 0.4 ± 0.2 m, and above 1.3 ± 0.6 m, COT is inversely proportional to the length. Authors categorized slow (below 0.4 ± 0.2 m) and fast (above 1.3 ± 0.6 m) swimmers from the data of 1200 animals and designated 0.4−1.3 m as a transition zone. We observe that the transition zone for low-Reynolds-number swimmers is narrower, ranging from 0.1 to 0.5 m.

Low frequency and intermediate angular amplitudes are necessary in both environments, with optimal body lengths in a dry friction environment and longer lengths in a viscous medium to optimize COT and efficiency. Bhalla et al. have found that efficient forward thrust is generated by lower deformation modes, while higher frequencies hinder forward locomotion by producing higher modes of deformation [82]. Our findings refine this hypothesis by stating that higher frequencies can be employed in a viscous medium for lower angular amplitudes and in a dry friction environment for higher angular amplitudes to minimize COT. Moreover, our observations in a viscous medium also support lower frequency for longer bodies, as observed in [72].

Mass dependence highlights the importance of optimizing mass in both environments to minimize COT. Optimizing mass is required for lighter bodies, whereas heavier bodies require greater frictional anisotropy to minimize COT. Variation of COT with mass shows that in a dry friction environment, light bodies can move with a wide range of angular amplitudes and frequencies, but intermediate to heavy bodies require lower angular frequencies for a wide range of angular amplitudes. Viscous medium prefers the optimal combination of mass and angular amplitude with low angular frequencies.

Researchers typically study polychaete-like locomotion by adjusting frictional anisotropy to less than 1 [18,29,30]. However, we observed this locomotion mode even when frictional anisotropy exceeds 1. This behaviour is more prominent in a dry friction environment. In a viscous medium, polychaete-like locomotion only occurs for narrow combinations of parameters, where slight changes can change the entire mode of locomotion (see figure 4d). This phenomenon indicates a strong dependence of polychaete-like locomotion on body properties when frictional anisotropy is greater than 1. In contrast, in a dry friction environment, high damping for a flexible body or high frequency for a heavy body triggers polychaete-like locomotion.

In the future, we aim to enhance the robustness of our results in a viscous medium by incorporating the effects of added mass, fluid moments and current effects [83,84]. This will broaden the applicability of our findings beyond viscous swimming scenarios, where friction dominates inertia, to situations where increasing length leads to higher Reynolds numbers and locomotion shifts towards inertia dominance [27]. Furthermore, the conclusions are derived by varying interdependent parameters, so applying these findings requires careful selection of parameters while accounting for their interdependence.

Our investigation into the variation of COT for lateral undulatory locomotion in dry friction and viscous environments yielded valuable insights into the complex interplay of various factors influencing the efficiency and COT of locomotion. Optimization based on COT or efficiency can give different sets of optimum parameters also supported by [85]. These findings elucidate the selective adaptations of long and slender limbless animals, providing insights applicable to animal locomotion and the design of robotic systems.

Ethics

This work did not require ethical approval from a human subject or animal welfare committee.

Data accessibility

Supplementary material is available online [86].

Declaration of AI use

We have not used AI-assisted technologies in creating this article.

Authors’ contributions

B.Y.: formal analysis, investigation, methodology, software, validation, visualization, writing—original draft; M.P.: conceptualization, formal analysis, supervision, writing—review and editing; N.M.P.: conceptualization, formal analysis, methodology, supervision, writing—review and editing.

All authors gave final approval for publication and agreed to be held accountable for the work performed therein.

Conflict of interest declaration

We declare we have no competing interests.

Funding

N.M.P. is supported by the Italian Ministry of Education, University and Research (MIUR) under the PRIN grant AMPHYBIA: Advanced Metamaterials from PHYsics and BIomechanics of Axolotls.