Mechanics and optimization of undulatory locomotion in different environments, tuning geometry, stiffness, damping and frictional anisotropy

Abstract

One of the oldest yet most common modalities of locomotion known among limbless animals is undulatory, also recognized for its stability compared to legged locomotion. Multiple forms of active mechanisms, e.g. active gait control, and passive mechanisms, e.g. body morphology and material properties, have adapted to different environments. The current research explores the passive role of body stiffness and internal losses in meeting terrain requirements. Furthermore, it addresses the influence of the environment on the resultant gait and how the interplay between various environments and body properties can lead to different speeds. We modelled undulatory locomotion in a dry friction environment where frictional anisotropy determines propulsion. We found that the body stiffness, the moment of inertia, the dry frictional coefficient ratio between normal and tangential frictional constants, and the internal damping of the body play an essential role in optimizing speed and animal adaptability to external conditions. Furthermore, we demonstrate that various known gaits like swimming, crawling and polychaete-like locomotion are achieved as a result of the interaction between body and environment parameters. Moreover, we validated the model by retrieving a corn snake's speed using data from the literature. This study demonstrates that the dependence between morphology, body material properties and environment can be exploited to design long-segmented robots to perform in specialized situations.

1. Introduction

In nature, locomotion is an inherent necessity of some living organisms to forage for food, escape danger or mate. Undulatory locomotion is widespread among typically legless, long-thin-bodied organisms, as it is the optimal solution for these living beings [1]. Exhibition of this movement is found in aquatic and terrestrial animals. Furthermore, subarenaceous animals like sandfish exploit the aptness of lateral undulations for moving in sub-surfaces without using their limbs [2].

The lateral travel of sinusoidal waves along the length of a body characterizes undulatory locomotion. Gray [3,4] first formulated that, in undulatory locomotion, a body interacts with the environment so that the force of propulsion is generated due to frictional anisotropy (direction-dependent friction). However, several other factors synergistically play an important role in generating efficient undulatory locomotion in adaptation to environmental constraints.

From a morphological perspective, it is supported by experiments that scales on the ventral side of the body assist in generating a frictional anisotropy [5–7]. Besides providing traction, scales also enhance control and help in manoeuvrability by allowing the animals to lift their bodies. The animal can control each scale with a muscle [8,9] and enhance manoeuvrability by controlling contact points and frictional anisotropy. Results drawn by comparing the performance of a shovel-nosed snake and a sandfish lizard show that long cylindrical bodies also favour undulatory locomotion [10]. In addition, sandfish experiments have shown that the head shape is responsible for lift and drag forces required for better manoeuvrability during their underground undulatory locomotion [11]. A tapered head produces 18% less drag and 20% higher speed than a uniform body [12].

Furthermore, limbless animals can also enhance their performance by modulating different gaits. For example, among sidewinding (figure 1a) and lateral undulation (figure 1b), lateral undulation is hypothesized to be more energy efficient at higher speeds, whereas sidewinding at moderate speeds [13]. Moreover, the cost per cycle of sidewinding is lower than that of concertina (figure 1d) and lateral undulatory locomotion, as investigated by Secor et al. [14]. Undulatory locomotion can further be categorized into swimming and crawling-like modes (figure 1e,f). Here, we define swimming-like locomotion where amplitude along the body length increases from head to tail. By contrast, in crawling-like locomotion, the amplitude decreases from head to tail (figure 1e,f) for any environment without discriminating by the number of contact points. Note that the contact points are where friction is active and are not necessarily present in each link, as we are assuming here for simplicity. In swimming-like locomotion, the speed of the organism is observed to be faster than in crawling [15]. Another kind of undulatory locomotion in some organisms is polychaete-like, in which the direction of the travelling wave is the same as the displacement direction. The difference between lateral undulatory and polychaete-like locomotion is illustrated in figure 1b,c. In polychaete-like locomotion, organisms are not totally limbless but have special attachments along their body length called parapodia (figure 2a), and because of their interaction with the environment, these attachments might have a crucial role in defining locomotion [17].

Figure 1.

Different modes of undulatory locomotion as observed in snakes (a,b,d,f) and generally long cylindrical limbless animals (b,c,e,f). (a) In sidewinding, static contact points with the ground result in disconnected tracks oblique to the direction of motion. (b) In lateral undulatory locomotion, the direction of the travelling wave is opposite to the direction of motion, and the dry frictional coefficient ratio (μ = μn/μt) between normal (n) and tangential (t) direction is greater than 1. We have divided the undulatory locomotion further into concertina (d), swimming-like (e) and crawling-like (f) motion. (c) In polychaete-like locomotion, the direction of the travelling body wave is along the same direction as motion, and the dry frictional coefficient ratio is less than 1. (d) Concertina locomotion. Half of the body has lateral bendings in this mode, and the other half makes a rectilinear motion. Forward motion is achieved by the sequential transfer of these two modes along the body length. This mode of locomotion is observed in narrow tunnels with insufficient space for lateral bending. (e) In swimming, the amplitude is shown to be increasing from anterior to posterior. (f) In crawling, the amplitude is shown to be decreasing from anterior to posterior.

Figure 2.

(a) Polychaete-like locomotion exhibited by Alitta virens. Photo courtesy of Alexander Semenov [16]. (b) Swimming-like locomotion of a corn snake (Pantherophis guttatus) on plastic. Picture courtesy of David Hu, Georgia Tech. (c) Crawling-like locomotion of a corn snake on cloth. Picture courtesy of David Hu, Georgia Tech.

Both active and passive mechanisms have evolved to act effectively in a specific environment and adapt to changes in environmental properties. In the case of fluidic environments, a change in the fluid viscosity triggers a switch between swimming- and crawling-like behaviours. When viscosity increases, the undulation wavelength and frequency in C. elegans decrease [18] with the increase of the drag force [19,20]. The impact of a change in environment on the resultant gait is also shown in [21] using mathematical modelling. A factor in changing the body waveform from swimming to crawling is a muscle activity change, as observed in C. elegans [22] and lungfish [23,24]. For lungfish, at higher velocities [23] and also in higher viscosity environments [25], more muscles are recruited to produce thrust. In C. elegans, the propagation of ventral muscle activity towards the posterior is observed to be faster in swimming than in crawling [22]. Besides neural control, factors like changes in body properties and environment are also involved in the transition from swimming to crawling [26,27]. In dry environments, snakes can modulate swimming-like, and crawling-like locomotion by passive mechanics adapting to frictional forces exercised on the body. In a high-frictional environment, crawling-like behaviour is the dominant mode in snakes (figure 2b), while they move with swimming-like behaviour on smooth substrates (figure 2c). Physical constraints provided by the environment also affect motion performance. Parashar et al. [28] validated this hypothesis by making worms move in waved channels of varying amplitudes. It was observed that worms moved faster and more steadily when the channel matched the natural body wave characteristics.

As discussed above, physical factors are involved in optimizing the performance of undulatory locomotion. To attain terrain adaptability and energy compensation, it is crucial to change the way of exploiting the interaction between the body and the substrate [5–9], modulating various gaits [13,14] and having efficient physical features [10]. There are some studies of the mathematical quantification of these physical parameters using computational fluid dynamics [29] and nonlinear resistive penalty models [30]. These studies revealed that the stiffness of the body could be used to optimize undulatory kinematics. To our knowledge, the mathematical quantification of different physical performance parameters like frictional anisotropy, stiffness, internal losses and moment of inertia in a combined form is not yet studied. Here, we model the undulatory locomotion and relate different design and tribological parameters to optimize and quantify motion. We will look at the interconnection between the moment of inertia, joint stiffness, internal damping and frictional anisotropy expressed as the ratio between normal and tangential frictional coefficients, and we will show different modes of locomotion induced by the interplay between these physical quantities. Many organisms show passive dynamics, as observed in snakes when facing unplanned collisions [31]. We also investigate ‘morphological intelligence’, defined as the use of passive properties of the body to perform various tasks [32]. The current study explores this morphological intelligence by analysing the relative contribution of listed physical properties in showcasing different undulatory locomotion gaits. The proposed model is validated using experimental data of a corn snake reported in [5,8].

2. Mathematical modelling

A modular body of a limbless animal is assumed to be composed of N finite number of links, joined together by N −1 visco-elastic springs, as shown in figure 3. The damping factor of a rotational dashpot is represented by b_i, and the stiffness constant of a rotational spring is denoted by k_i. Here, i designates the joint number between the ith and (i + 1)th link. We assume that each spring and dashpot have the same spring constant (k) and damping factor (b), respectively. The length of the link is represented by l_i while l_tot is the total length of system. Here, we are assuming that the total length is divided into equal-length segments such that l_i = l_tot/N, i ∈ [1, N]. u_x and u_y are x and y coordinates of the starting points of the first link with respect to a global coordinate system. s_i ∈ [0, l_i] represents points along the length of the ith link, and θ_i is the link angle which is measured anticlockwise.

Figure 3.

Schematic of the discretized system.

Frictional forces are generated along normal and tangential directions (as shown in figure 3) because of the orthotropic friction between links and ground. The force of friction is opposite to the direction of velocity and is given by equations (2.1) and (2.2):

$${\mathit{T}}_i=-g{\mu }_t{m}_i\hspace{0.17em}\mathrm{sgn}(v_{t,i}){\hat{t}}_i$$ 2.1

and

$${\mathit{N}}_i=-g{\mu }_n{m}_i\hspace{0.17em}\mathrm{sgn}(v_{n,i}){\hat{n}}_i.$$ 2.2

Here i ∈ [1, N], and g is the gravitational acceleration, μ_t and μ_n are dry frictional constants along tangential and normal directions, respectively (figure 3), m_i is the mass of the ith link, which we are considering the same for every link so m_i = m, and sgn is the sign function. ${\hat{t}}_i$ and ${\hat{n}}_i$ are unit vectors in the tangential and normal directions (see electronic supplementary material, information) which translate forces of friction to a local coordinate system. Tangential and normal velocities are given by ${\mathit{v}}_{t,i}={\dot{\mathit{r}}}_i.{\hat{t}}_i$ and ${\overrightarrow{v}}_{n,i}={\overrightarrow{\dot{r}}}_i.{\hat{n}}_i$, respectively, where r_i is the position vector, and the dot represents the time derivative. We replaced the sgn function according to equation (2.3) for the sake of continuity in the numerical model:

$$\mathrm{sgn}(\hspace{0.17em}f(x))\simeq \left[{\displaystyle \frac{\hspace{0.17em}f(x)}{\sqrt{\hspace{0.17em}f\hspace{0.17em}{(x)}^2+\epsilon }}}\right].$$ 2.3

Here, the smaller the value of ε, the more accurate the approximation would be. The Euler–Lagrange equation of the motions can be obtained following the Lagrangian approach, accordingly:

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

where q_h is the set of the unknown generalized coordinate system, L is the Lagrangian of the system, Q_h,Dry represents losses because of friction and R represents energy loss because of internal damping [33]. Further description of the model is provided in electronic supplementary material, information.

2.1. Non-dimensionalization

The non-dimensional form is achieved through the following substitutions: $u_x=l_{\mathrm{tot}}{\tilde{u}}_x$, $u_y=l_{\mathrm{tot}}{\tilde{u}}_y$, $t=\tau \tilde{t}$, μ = μ_n/μ_t, $s_i=l_i{\tilde{s}}_i$. Here ${\tilde{u}}_x$ and ${\tilde{u}}_y$ are dimensionless x and y coordinates of the starting point of the links system, τ is the period of oscillations, and $\tilde{t}$ is the dimensionless time. ${\tilde{s}}_i$ is a dimensionless number that indicates the dimensionless length of the ith link such that ${\tilde{s}}_i\in [0,1]$. Replacing dimensional quantities with their non-dimensional equivalents, as defined above, produces

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}\tilde{t}}\left({\displaystyle \frac{\mathrm{\partial }\tilde{L}}{\mathrm{\partial }{\tilde{\dot{q}}}_h}}\right)+{\displaystyle \frac{\mathrm{\partial }\tilde{L}}{\mathrm{\partial }{\tilde{q}}_h}-{\alpha }_{\mathrm{Dry}}{\tilde{Q}}_{h,\mathrm{Dry}}+\beta {\displaystyle \frac{\mathrm{\partial }\tilde{R}}{\mathrm{\partial }{\tilde{\dot{q}}}_h}=0.}}}$$ 2.5

Here $\tilde{L}$ is the Lagrangian function in the non-dimensionalized form defined by (2.6), ${\tilde{Q}}_{h,\mathrm{Dry}}$ is the dimensionless dissipation due to friction, and $\tilde{R}$ is the dimensionless internal viscous losses. α_Dry and β are dimensionless quantities directly related to the normal frictional constant and internal damping, respectively, and are defined by (2.7) and (2.8):

$$\begin{array}{rl}\tilde{L}({\tilde{\dot{q}}}_h,{\tilde{q}}_h,\tilde{t})& =\sum _{i=1}^N{\widetilde{\dot{\mathit{r}}}}_{i,1/2}^T\cdot {\widetilde{\dot{\mathit{r}}}}_{i,1/2}+A\sum _{i=1}^N{\tilde{\dot{\theta }}}_i^2\\ & -\mathrm{B}\sum _{i=1}^{N-1}{({\theta }_{i+1}-{\theta }_i)}^2,\end{array}$$ 2.6

$${\alpha }_{\mathrm{Dry}}={\displaystyle \frac{2g{\tau }^2{\mu }_n}{l_{\mathrm{tot}}}}$$ 2.7

$$\mathrm{and}\beta =N{\displaystyle \frac{b\tau }{m_{\mathrm{tot}}{l}_{\mathrm{tot}}^2}.}$$ 2.8

In equation (2.6), ${\widetilde{\dot{\mathit{r}}}}_{i,1/2}$ is the dimensionless position vector of the centre point of the ith link (${\tilde{s}}_i=1/2$). Constants A and B are directly related to the moment of inertia and joint stiffness, respectively, and are defined according to (2.9) and (2.10):

$$A=N{\displaystyle \frac{I}{m_{\mathrm{tot}}{l}_{\mathrm{tot}}^2}}$$ 2.9

and

$$B=N{\displaystyle \frac{{\tau }^{2}k}{m_{\mathrm{tot}}{l}_{\mathrm{tot}}^2}.}$$ 2.10

Here, I is the moment of inertia considered the same for every link and m_tot is the total mass, such that m_i = m_tot/N is the mass of the ith link. Harmonic excitation in the system is introduced by soliciting the end link angle sinusoidally, as given by equation (2.11):

$${\theta }_N(\tilde{t})=a\sin [2\pi \tilde{t}],$$ 2.11

where a is the angular amplitude (see electronic supplementary material, information, for further details).

The total energy (${\tilde{E}}_{\mathrm{tot}}$) of the system is evaluated as the sum of the energy dissipated by viscous forces $({\tilde{E}}_{\mathrm{out}})$ and work done by constraint forces $({\tilde{E}}_{\mathrm{in}})$. See electronic supplementary material, information, for detailed definitions.

2.2. Validation settings

We selected corn snakes as our test organism and used data retrieved from the literature to validate the model. When snakes move, it is known that they do not have the entire body in contact with the ground [5]. The concertina is the most complex within lateral undulatory locomotions, requiring the highest number of contact points. Nevertheless, Marvi & Hu [9] noticed that snakes make less than seven contact points during concertina locomotion in open spaces and used three links in their theoretical modelling. By convention, for lateral undulatory locomotion, three contact points are considered [3,34,35]. In our model, every link is in contact with the ground; consequently, for the validation test, we discretized the snake body into three links to follow the convention, and we considered the geometry of a thin rod. To visualize a more realistic motion, in all other simulations, we used a higher number of links (N = 5) (the Results section also reports an analysis of the effects of the number of links). The mass and length of the corn snake are used as reported in [8] (mass = 0.73 [kg], length = 1.313 [m]). We used the torsional stiffness of the scale of a corn snake to be k_i = 0.01196 [Nm] as calculated by Marvi et al. [8]. Normal frictional coefficient (μ_n) is taken as 0.2, as experimentally observed for corn snakes in an unconscious state [5]. We set the angular amplitude of undulation (a) as 7π/37 [rad], which is in the range as specified in [5], and the frequency is fixed at 1.5 [Hz] [36]. We estimated the damping and tangential coefficient of friction with a snake velocity of 2.7 [cm s⁻¹] [5], and then results are compared with the biological model.

3. Results

For the validation of the model, experimental data are substituted to find out the optimum value of the internal damping constant (b) and tangential frictional coefficient (μ_t) corresponding to the speed of 0.027 [m s⁻¹] [5], which are found to be 0.15 ± 0.025 [Nm s rad⁻¹] and 0.05 ± 0.01, respectively (figure 4). We noticed that our estimated dry frictional coefficient ratio is 4 at 0.027 [m s⁻¹], which is of the same order of magnitude as the experimental value equal to 1.82 [5].

Figure 4.

Damping and tangential frictional constants are varied with intervals of 0.025 [Nm s rad⁻¹] and 0.01, respectively. A solution corresponding to the average velocity of 0.027 [m s⁻¹] is selected.

We then characterized undulatory locomotion using dimensionless constants based on environment and body properties. Normal frictional coefficient, internal damping, moment of inertia, stiffness and the ratio between normal and tangential friction components are directly correlated with constants α_Dry, β, A, B and μ, respectively. Using these constants, we can characterize and optimize various modes of undulatory locomotion.

Figure 5a shows the interdependence between stiffness (B) and internal damping (β) factors to achieve the desired speed. Here, we can optimize speed based on stiffness and internal damping factors by fixing the remaining factors. Optimal values of stiffness and internal damping are given in figure 5a in a scenario providing a dry frictional coefficient ratio of μ = 50. In this scenario, the maximum speed is reached with a stiffness factor B = 175 and an internal damping β = 10.4. In the same scenario, by setting the optimal B and β found, we can explore the inertial factor (A) and dry frictional factor (α_Dry) domain and find the combination providing the maximum displacement, as observed in figure 5b. From figure 5, we concluded that undulatory locomotion speed strongly depends on frictional, internal damping, inertial and stiffness factors. This interdependence can be exploited in physical situations to achieve a desired speed. For example, the deployment of a robot in a specific environment can be made more efficient based on its stiffness and geometry optimization. The dry frictional coefficient ratio (μ) and dry frictional factor (α_Dry) characterize the environment–body interaction in our formalized constants. Deciding factors of the environment affecting ground–body interaction include surface roughness, texture and material properties. This information can be used to design features and ventral asperities of the body to reach the desired performance in a specific environment. Similarly, based on the properties of the body, the definition of an ideal environment is also possible, where the body can reach its maximum performance. In this case, the performance can be tuned by acting on the characteristics of the environment, if possible.

Figure 5.

The interdependence of stiffness, internal damping and inertial and dry frictional factors needed to achieve a desired dimensionless speed for lateral undulatory locomotion. Simulations are run at an angular amplitude of a = 8π/125 [rad], dry frictional coefficient ratio μ = 50 and the number of links N = 5. The speed shown is the dimensionless speed of the system's centre of mass. The multiplicative factor to convert dimensionless speed into dimensional form is l_tot/τ. (a) Speed optimization based on stiffness and internal damping factors. The maximum speed value is achieved at 175 ± 5 and 10.4 ± 0.8 values of B and β, respectively. Inertial and dry frictional factors are kept constant at 0.5 and 20, respectively. (b) Speed optimization is achieved from inertial and dry frictional factors. Obtained optimum values of inertial and dry frictional factors are 0.46 ± 0.02 and 22 ± 1, respectively. Stiffness and internal damping factors are kept constant at 175 and 10.4, respectively.

Further characterization of the motion revealed the transition from crawling-like to swimming-like behaviour. We noticed that crawling-like locomotion is triggered when the dry frictional factor is greater than the stiffness factor (figure 6a). While with swimming-like locomotion, the stiffness factor is greater than the dry frictional factor (figure 6b). Animations of these behaviours are provided in the electronic supplementary material, videos S1 and S2.

Figure 6.

Simulations are run at angular amplitude = 8π/125 [rad], damping factor = 1, inertial factor = 0.5, number of links N = 5 and dry frictional coefficient ratio = 50. The multiplicative factor to convert dimensionless displacement and amplitude into dimensional form is l_tot. (a) In crawling-like locomotion, it can be observed that the amplitude of a head is larger than that of a tail. The dry frictional factor is 50 and the stiffness factor is 40, i.e. α_Dry > B. (b) Depiction of swimming-like locomotion, where the amplitude of the head is smaller than that of the tail. The stiffness factor is set at 175 and the dry frictional factor at 40, i.e. B > α_Dry.

Looking at the energy and the speed, they are directly proportional to each other (figure 7a,b). Stiffness, inertial and internal damping factors depend on the number of links except for the dry frictional factor (equations (2.7)–(2.10)). Increasing links increases stiffness members in the system; consequently, the system's stiffness factor and potential energy increase. Likewise, the internal damping factor, on increasing links, increases internal damping points in the system; hence, internal losses also increase. However, the dry frictional factor and the dry frictional coefficient ratio remained the same, but increasing links also increased contact points which have an overall increasing effect on frictional losses (Q_h,Dry). That is why it is observed that higher number of links perform better in a low-frictional environment: a higher number of contact points are required to provide propulsion (figure 7a). On the contrary, a system with fewer contact points, thus fewer links, requires higher frictional anisotropy. This phenomenon is also reported in [13], where it was found that the lateral undulatory gait with fewer contact points required higher frictional anisotropy to perform better. Hence, an increasing number of links by keeping the dry frictional factor constant increases speed until the domination of frictional and internal losses because of the rise in contact points and internal damping factors, respectively (figure 7b). In addition, the inertial factor is also inversely proportional to the number of links keeping total mass and length constant (see electronic supplementary material, information).

Figure 7.

Effect of the number of links and the trend of energy. Simulations are run as A = 0.25/N², B = 5 N, β = N/5, a = 8π/125 [rad], and μ = 50. The multiplicative factor to convert dimensionless total energy into respective dimensions is 2Nτ²/m_totl²_tot. (a) The impact of an increasing number of links on speed and energy in relation to the dry frictional factor. More links are shown to be advantageous in a low-frictional anisotropic environment. Contrarily, fewer links perform better in a high-frictional environment. (b) When all factors are kept constant, as stated earlier, and the dry frictional factor is kept constant at 5, we can observe the optimum number of links in that environment.

When the dry frictional coefficient ratio (μ) is reduced to less than 1, polychaete-like locomotion is observed (figure 1c). On the other hand, when the dry frictional coefficient ratio is greater than 1, the direction of the travelling body wave is opposite to the direction of motion, leading to lateral undulatory locomotion (figure 1b). Results showed that polychaete-like locomotion could also be optimized based on frictional, stiffness, internal damping and inertial factors (figure 8a,b). The comparison of the influence of the dry frictional coefficient ratio on lateral undulatory and polychaete-like locomotion showed opposite results. In the case of undulatory locomotion, increasing the dry frictional coefficient ratio increases the displacement. Eventually, it reaches the saturation point where further increases do not have much effect (figure 8c). Whereas, in the case of polychaete-like locomotion, motion can be optimized based on the dry frictional coefficient ratio (figure 8d).

Figure 8.

(a) Optimization of polychaete-like locomotion based on stiffness and internal damping factors. Simulation is run at angular amplitude (a) = 8π/125 [rad], inertial factor (A) = 0.5, dry frictional coefficient ratio (μ) = 0.01, dry frictional factor (α_Dry) = 5, and number of links (N) = 5. Maximum value of dimensionless speed is achieved at internal damping factor (β) = 0.6 ± 0.2 and stiffness factor (B) = 208 ± 8. (b) Optimization of polychaete-like locomotion based on inertial and dry frictional factors, which come out to be 0.196 ± 0.014 and 1.32 ± 0.12, respectively, where stiffness factor (B) = 100, dry frictional coefficient ratio (μ) = 0.01, damping factor (β) = 0.6 and angular amplitude (a) = 8π/125 [rad]. (c) The trend of dimensionless speed with increasing value of the dry frictional coefficient ratio in the case of lateral undulatory locomotion. (d) Relationship between dimensionless speed and dry frictional coefficient ratio in the case of polychaete-like locomotion.

4. Discussion

We proposed a non-dimensional model to analyse the role of physical quantities, such as stiffness, moment of inertia, frictional anisotropy and internal damping, in undulatory locomotion. We found that the desired locomotion is achieved with proper tuning of these physical quantities. Furthermore, we observed that the optimum set of derived constants at maximum displacement has a greater stiffness factor than dry frictional factor (figure 5a,b). When the body stiffness increases, the energy storage capacity increases, which enhances the elastic potential energy of the body to overcome frictional energy losses. In this scenario, we observed that body wave amplitude increases from anterior to posterior, as shown in figure 6b (electronic supplementary material, video S1), gaining a swimming-like behaviour. When dissipation forces are dominant, body wave amplitude decreases from anterior to posterior, shown in figure 6a (electronic supplementary material, video S2), leading to crawling-like behaviour. The speed in the former case is higher than in the latter. The pattern of increasing amplitude from anterior to posterior is observed among organisms swimming in water or on land [22,37]. Muscles in living beings serve the purpose of energy storage elements like springs [38]. It has been observed that increasing muscle engagement increases speed, which means muscle activity increases, and more muscles get recruited to achieve high speed [23,25]. For example, a study on the activity of two different kinds of muscles, called red and white, in eels showed that only red muscles are used at slower speeds. In comparison, at higher speeds, white muscles are additionally recruited. In addition, the stiffness role has also been tested in C. elegans, and it has been observed that mutants with deteriorated muscles and cuticles moved slower [39]. Frictional anisotropy cannot be neglected in undulatory locomotion. In fact, it is predominantly a friction-based motion where frictional forces overcome inertial forces [20,40,41]. The challenge of adapting to different terrains can be addressed better if the effects of stiffness and friction are synergized together, which is supported by studies on animals. For example, in nematodes, when their environment changes, they shift between different modes of locomotion and speed. During swimming in water, the body's waveform characteristics differ significantly from crawling on harder media, like agar, in terms of amplitude, wavelength and speed [18,22,42,43]. Recently, the influences of the shape and orientation of scales have been explored for lateral undulatory locomotion [44–46]. However, according to the authors' knowledge, the characterization of body frictional and stiffness properties has never been done before. We hypothesized that the passive adaptability of the body according to the dynamic scenarios can be exploited for energy efficiency, simplicity and weight reduction. Indeed, these qualities are desirable for micro-robots (e.g. that are needed to release in the human body) or in situations where onboard electronic components should be avoided, for example, when operating in a high-radiation environment or when recovery of a device is not feasible.

Polychaete-like locomotion, in which body waves travel in the same direction as the direction of movement, is achieved by setting a dry frictional coefficient ratio smaller than one. In reality, polychaetes have numerous parapodia along their body (figure 2a), making tangential frictional coefficient larger than normal frictional coefficient [17]. Previously, it has also been realized in robots by setting the normal frictional constant less than the tangential coefficient [47,48]. We observed that it is not an efficient mode of locomotion compared to normal undulatory locomotion. Simulations suggest the significance of parapodia in animals for polychaete-like locomotion. Without parapodia, locomotion is slow and does not closely mimic natural movements; however, we can successfully initiate polychaete-like locomotion by setting the appropriate dry frictional coefficient ratio and can investigate the role of passive properties of the environment and the body (electronic supplementary material, video S3).

In the validation part, the tangential frictional and damping constants are varied, as shown in figure 4. We found that values of μ_t = 0.05 ± 0.01 and b = 0.15 ± 0.025 [Nm s rad⁻¹] correspond to a snake speed of 2.7 [cm s⁻¹] as mentioned in [5]. Our theoretically estimated dry frictional coefficient ratio (μ) is 4, higher than the coefficient ratio of 1.82 recoded from milk-snakes [5]. This discrepancy could be due to the model simplification that cannot capture the real model complexity at this stage. In fact, it has been observed that snakes lift their bodies during locomotion in such a way that only the points of zero curvature are in contact with the ground [5]. In this way, it has been observed that their efficiency increases. However, in our modelling, we compromise the body's shape continuity by using three links to compensate for comparable contact points. In addition, mechanical properties differ along the length of snake bodies [49,50], and their bodies’ cross-section area and mass distribution also change. Also, the reported frictional constants could be underestimated in part because it has been observed that the frictional constant of a conscious snake is higher than in an unconscious one. During locomotion, snakes can change the orientation of their scales as muscles control scales, while in their unconscious state, this is impossible [51]. Furthermore, they can increase the force on contact points with the ground during the movement by increasing the push provided by muscles. By these considerations, our estimation might be more realistic than the value found experimentally.

5. Conclusion

The effects of physical quantities, such as stiffness, moment of inertia, frictional anisotropy and internal damping, are studied here for undulatory locomotion. It has been established that the exploitation of body and environment features in situated physical systems can promote the passive locomotion of the body in its environment, gaining advantages in terms of energy consumption, adaptability, lightweight and minimal use of electronic components. It is hypothesized that properties of the body can be optimized to get desired speed depending upon the environment, which can assist in developing soft specialized segmented robots in the deemed surrounding, and vice versa. The proposed theoretical model successfully captured different modes of locomotion, like swimming, crawling and polychaete-like movements. It is found that during crawling, dissipation dominates stiffness, while in swimming, stiffness dominates dissipation. Despite model simplifications, feeding our model with experimental data has produced results in good agreement with the experimentally calculated dry frictional coefficient ratio of a corn snake, considering an underestimation due to the unconsciousness of the snake in the experimental tests. This study can be applied to developing better adaptable bioinspired robots for search and rescue, exploration and medical applications. Further validation of this hypothesis on a physical system is our future goal.

Data accessibility

The data are provided in the electronic supplementary material [52].

Authors' contributions

B.Y.: data curation, formal analysis, investigation, methodology, software, validation, visualization and writing—original draft; A.R.: conceptualization, formal analysis, methodology, software and writing—review and editing; E.D.D.: formal analysis, project administration, resources, supervision and writing—review and editing; A.M.: formal analysis, project administration, resources and writing—review and editing; B.M.: conceptualization, formal analysis, resources, supervision and writing—review and editing; N.M.P.: conceptualization, formal analysis, methodology, supervision and 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 MIUR, Italy under the PRIN-20177TTP3S.