Sea stars have slower crawling and faster bouncing gaits. Both speed and oscillation amplitude increase during the transition from crawling to oscillating. In the bouncy gait, oscillating vertical velocities precede oscillating horizontal velocities by 90 deg, as reflected by clockwise circular hodographs. Potential energy precedes horizontal kinetic energy by 9.6 deg and so they are nearly in phase. These phase relationships resemble terrestrial running gaits, except that podia are always on the ground. Kinetic and potential energy scale with body mass as Mb   1.1, with the change in kinetic energy consistently two orders of magnitude less, indicating that efficient exchange is not feasible. Frequency of the bouncy gait scales with Mb−0.14, which is similar to continuously running vertebrates and indicates that gravitational forces are important. This scaling differs from the Hill model, in which scaling of muscle forces determine frequency. We propose a simple torque-stabilized inverted pendulum (TS-IP) model to conceptualize the dynamics of this gait. The TS-IP model incorporates mathematics equivalent to an angular spring, but implemented by a nearly constant upward force generated by the podia in each step. That upward force is just larger than the force required to sustain the underwater weight of the sea star. Even though the bouncy gait is the rapid gait for these sea stars, the pace of movement is still very slow. In fact, the observed Froude numbers (10−2 to 10−3) are much lower than those typical of vertebrate locomotion and are as low or lower than those reported for slow-walking fruit flies, which are the lowest values for pedestrian Froude numbers of which we are aware.

A gait is a pattern of motion characteristic of a range of speeds. Gaits typically change abruptly to a different gait at specific transition speeds (Alexander, 2003). The preferred or typical gait used at a certain speed uses less energy than alternative gaits that could be used at that speed (Hoyt and Taylor, 1981; Alexander, 1995; Minetti and Alexander, 1997; Saibene and Minetti, 2003; Snaterse et al., 2011; Andrada et al., 2020). Although gaits are rarely studied in underwater legged locomotors, there is a large literature on the kinematics of gaits in terrestrial legged locomotors and several well-established paradigms that are often used to frame kinematic, biomechanic and energetic arguments about terrestrial legged locomotion. While some terrestrial legged locomotors such as horses have several gait patterns such as walk, trot, canter and gallop (Nauwelaerts et al., 2015), terrestrial gaits have often been viewed through a dual kinematic characterization and contrast of walking versus running gaits.

In terms of models, an inverted pendulum model has been applied to walking gaits and a bouncing, spring-mass model has been applied to running gaits (reviewed in Dickinson et al., 2000), both of which are characterized by oscillations in the center of mass. More recently, the spring-loaded inverted pendulum (SLIP) model has been used to unify walking and running gaits under one model in contrast to the inverted pendulum model, which can only be applied to walking (Andrada et al., 2020). The need to incorporate hydrodynamics into spring-mass models for underwater legged locomotion was identified by Martinez et al. (1998), which motivated the modification of the SLIP model for crabs locomoting underwater (underwater spring-loaded inverted pendulum: USLIP) (Calisti and Laschi, 2017; Chellapurath et al., 2020).

In terms of speeds, among terrestrial legged locomotors, walking is a slower gait and running is a faster gait. Although it is quite possible to run slowly or to walk quickly, the fastest speeds cannot be achieved with a walking gait. The difference in speed between gaits can be considerable. For example, compared with walking, long distance runners approximately double their speed and sprinters approximately quadruple their speed. Speeds associated with novel gaits have also been considered in underwater legged locomotors. Some octopi accelerate into a bipedal gait (Huffard et al., 2005), and a fast underwater punting gait has been identified in crabs (Martinez et al., 1998), hippopotami (Coughlin and Fish, 2009) and crocodiles (Farlow et al., 2018). Brittle stars move rapidly by using their flexible arms to locomote using rowing and reverse rowing among other gaits (Arshavskii et al., 1976; Astley, 2012).

We focus here on sea stars, whose mode of locomotion has broadly been described as a crawl (Jennings, 1907; Cole, 1913; Brusca and Brusca, 2003; Montgomery and Palmer, 2012; Montgomery, 2014). In addition to the crawling gait, a second, faster oscillatory gait has been recently identified in the sea star Protoreaster nodosus (Ellers et al., 2014) and subsequently in Asterias forbesi and Luidia clathrata (Ellers et al., 2018; Johnson et al., 2019). We have also observed this bouncy gait in Asterias rubens, Asterias amurensis, Pisaster giganteus and Acanthaster planci. Sea stars locomote using numerous pressurized cylindrical tube feet, or podia, and the way in which those podia are used changes from the crawling to the bouncy gait. Heydari et al. (2020) created a model of podial function that has coordination among podia as an emergent property as speed increases from the crawling to the bouncing gait. Because different gaits can be used at some overlapping speeds, speed alone cannot be used to distinguish between gaits; it has to be considered in conjunction with other variables. In general, gaits can be considered as specific kinematic patterns and, in this paper, we describe several kinematic parameters for the sea star's gaits.

One kinematic feature of gaits is the frequency of the repeated motion. For terrestrial locomotors ranging in size from mice to horses, stride frequency f scales with mass Mb as (Heglund et al., 1974), which is consistent with the hypothesis that gait frequency is controlled by the physical dimensions of a pendulum driving the oscillatory motion, as described by Alexander and Jayes (1983). In that hypothesis, gravitational forces are important in driving the motion, and the kinematics and dynamics can be described using the Froude number, which is the ratio of inertial to gravitational forces; gait transitions occur at specific Froude numbers. In contrast, oscillatory frequency of the limbs of a wide range of swimming seabirds, cetaceans and pinnipeds scaled with mass as (Sato et al., 2007, 2010), supporting the hypothesis first suggested by Hill (1950), whereby muscle characteristics explain this scaling relationship.

In addition, phase relationships in energy and velocities are frequently used to characterize gaits. The phase relationship of kinetic energy and potential energy determines the potential for exchange between these two energies and is one way of distinguishing between walking and running gaits. In a perfect pendulum, the change in kinetic energy and potential energy are exactly out of phase and of equal magnitude such that all the energy can be perfectly recovered in each swing, with no energy loss. Similarly to a pendulum, kinetic energy and potential energy fluctuate out of phase and are of similar magnitude during the terrestrial walking gait; however, the kinetic energy–potential energy phase relationship and energy recovery is imperfect. In human walking, for example, Cavagna and Legramandi (2020) found a maximum energy recovery of 66%. Among the highest energy recoveries reported are, surprisingly, for waddling penguins (80%); Griffin and Kram (2000) found that the penguin's waddling gait achieved this remarkable efficiency by causing the kinetic energy and potential energy to fluctuate more completely out of phase and to be of more similar magnitude. In contrast, in stealthy walking in cats, low energy recovery between kinetic energy and potential energy was a consequence of a less efficient phase relationship between these two energies (Bishop et al., 2008). In running, these energies fluctuate in phase; for example, Cavagna et al. (1964) reported a phase shift between kinetic and potential energy of, at most, about 10 deg. The phase relationship between vertical and horizontal velocities, shown on plots called hodographs, can also be used to characterize gaits. In particular, counter-clockwise hodographs are characteristic of walking, whereas clockwise hodographs are characteristic of running (Usherwood, 2010). Experimental hodographs have not previously been examined for underwater legged animal locomoters, although Calisti and Laschi (2017) have produced theoretical limit cycles, based on the USLIP model, whose projections are hodographs.

In this paper, we kinematically characterize the crawling and oscillatory gaits of the sea star Asterias forbesi. Some kinematic variables we consider are: speed, frequency, kinetic and potential energy changes, phases between vertical and horizontal velocities (i.e. hodographs), and phases between kinetic and potential energy. We measure changes with mass of the kinematic variables: frequency, speed, vertical position, and kinetic and potential energy. We further measure changes with mass of the morphological variables: density, underwater weight and podial leg length. To obtain these kinematics, we tracked synchronized side and bottom videos of 54 individuals of the sea star A. forbesi, ranging in mass over three orders of magnitude. We calculated Froude numbers associated with the sea star oscillatory gait and we evaluate whether gravitational forces, as in Alexander and Jayes (1983) or muscle forces, as in Hill (1950), drive the scaling of frequency with mass. Finally, we developed a version of a simple inverted pendulum model that is consistent with the observed kinematics of the sea star oscillatory gait and provides some insight into the dynamics of the gaits.

List of symbols and abbreviations

     
  • A

    total cross-sectional area of all actively lifting podia

  •  
  • az

    vertical acceleration

  •  
  • (az)j,s

    observed vertical acceleration, sth sea star, jth run

  •  
  • DFT

    discrete Fourier transform

  •  
  • Ep

    potential energy

  •  
  • mean potential energy

  •  
  • ΔEp

    change in potential energy

  •  
  • Ep,amp

    potential energy amplitude

  •  
  • Ekh

    horizontal kinetic energy

  •  
  • mean horizontal kinetic energy

  •  
  • ΔEkh

    change in horizontal kinetic energy

  •  
  • Ekh,amp

    horizontal kinetic energy amplitude

  •  
  • Ekz

    vertical kinetic energy

  •  
  • Ek,total

    total kinetic energy

  •  
  • ΔEk,total

    change in total kinetic energy

  •  
  • Ek,amp,total

    total kinetic energy amplitude

  •  
  • f

    frequency

  •  
  • horizontal force due to experimental observed axy

  •  
  • mean for a sea star s

  •  
  • vertical force due to experimental observed az

  •  
  • mean for a sea star s

  •  
  • Fz

    resultant vertical force ()

  •  
  • underwater vertical force (gravity minus bouyancy)

  •  
  • mean underwater vertical force for a sea star s

  •  
  • upward podial forces

  •  
  • force tangent to the arc of a pendulum

  •  
  • Fr

    Froude number

  •  
  • Frz

    underwater Fr, using vertical accelerations

  •  
  • Frxy

    underwater Fr, using horizontal accelerations

  •  
  • g

    acceleration due to gravity

  •  
  • ge

    effective gravity

  •  
  • J

    number of runs

  •  
  • l

    length

  •  
  • Mb

    body mass of animal

  •  
  • MPB

    mass of pendulum bob

  •  
  • Mf

    mass of displaced fluid

  •  
  • (Mb)s

    mass of the sth sea star

  •  
  • ΔM

    sea star mass minus mass of displaced water

  •  
  • P

    podial hydrostatic pressure

  •  
  • ΔP

    difference in pressure across the ampullar wall

  •  
  • R

    radius of curvature of the ampullar wall

  •  
  • ΔR

    ampullar wall thickness

  •  
  • rms

    root mean square

  •  
  • s

    sea star number

  •  
  • S

    total number of sea stars

  •  
  • SLIP

    spring-loaded inverted pendulum model

  •  
  • TS-IP

    torque stabilized inverted pendulum model

  •  
  • t

    time

  •  
  • U

    speed

  •  
  • USLIP

    underwater spring-loaded inverted pendulum model

  •  
  • vx, vy

    velocity in the x and y directions, from bottom videos

  •  
  • vz,side

    velocity in the z direction, from side videos

  •  
  • vx,side

    velocity in the x direction, from side videos

  •  
  • vxy

    speed in the horizontal plane

  •  
  • vxyz

    magnitude of the 3D velocity vector

  •  
  • x, y, z

    axes in the right-handed coordinate system

  •  
  • X

    independent variable

  •  
  • Y

    dependent variable

  •  
  • mean z displacement over an entire video

  •  
  • Δz

    vertical location relative to a reference height

  •  
  • θ

    angle of pendulum rod to vertical

  •  
  • κ

    coefficient of added mass

  •  
  • λ

    step length

  •  
  • ρ

    sea star density

  •  
  • ρPB

    pendulum bob density

  •  
  • ρf

    fluid density

  •  
  • σ

    podial muscle stress

  •  
  • τ

    period in air

  •  
  • τu

    period of an underwater pendulum

  •  
  • υ

    sea star volume

  •  
  • φj

    phase shifts for each of j runs for a given sea star

  •  
  • phase shift mean for all the runs of a given sea star

  •  
  • ω

    angular frequency

Animal collection and maintenance

Adult Asterias forbesi (Desor 1848) were hand-collected in rocky intertidal and shallow subtidal areas adjacent to Basin Point, Harpswell, ME (43.74°N, −70.04°W) and the Rockland Breakwater, Rockland, ME (44.11°N, −69.08°W). Sea stars were maintained in recirculating seawater aquaria at 12–13°C and 30–33 ppt salinity and were fed weekly with mussels and barnacles. Sea stars were acclimated in these aquaria for at least 2 weeks prior to locomotion trials.

Locomotion trials

A subset of 54 sea stars were individually filmed in runways with recirculating sea water maintained at 12–13°C and 30–33 ppt. Sea stars were filmed over 6 weeks during the summer of 2015 and 7 weeks during the summer of 2017.

Two runways were used, with cross sections: (1) 15 cm wide by 10 cm deep and (2) 36 cm wide by 15 cm deep. Locomoting sea stars were filmed at 30 fps (resolution 1920×1080 pixels) with two Nikon 5200 SLR cameras equipped with SanDisk Extreme Pro95 MB s−1 SDXC I V90 C10 U3 memory cards. The lines of sight of the cameras were normal to the bottom and side of the runway, and the lines of sight intersected with the side camera line of sight being 1 cm above the bottom surface and the bottom camera line of sight being centrally located in the bottom of the tank. The bottom camera was equipped with a Nikon DX AF-S Nikkor 10–24 mm 1:3.5–4.5G ED lens, the side camera was equipped with a Nikon DX AF-S Nikkor 18–55 mm 1:3.5–5.6G VR lens. Cameras were synchronized using wireless WR-T10 and WR-R10 controllers.

Camera synchronization was checked by filming the sudden blocking of a laser light beam by a falling object. This blockage resulted in the disappearance of a laser spot in both the side and bottom videos; the difference (side–bottom) of the frame number in which the disappearance of that light spot was detected in the two views was determined; difference in total frames in the video of each view was also calculated. The average difference in frame in which the laser spot disappeared was −0.67±2.5 (mean±s.d.) frames and the average difference in total frames was −0.66±2.2 frames (n=100 side and bottom pairs). In addition, we checked the synchronization of the cameras during the filming of the sea stars by comparing the total frames in paired side and bottom videos for all videos used in this study; the average difference in total frames for these video pairs was −0.14±1.9 frames (n=128 side and bottom pairs).

As a final synchrony check on the sea star videos, we used unfiltered x-position data from the side videos and unfiltered x-position data from the bottom videos to calculate x-velocity from the two views (see Fig. 1 for the coordinate system). We used cross-correlation between those two x-velocities to determine phase shifts between the x-position as seen from the side camera and the x-position as seen from the bottom camera. Mean phase shift among all films (n=128) was 0.021±0.059 radians (1.2±3.4 deg; mean±s.d.), with the side videos preceding the bottom videos on average by that amount. A sign test (P<0.001) shows that side camera tended to start before the bottom camera to a statistically detectable extent. On average, this imposes a 1 deg phase shift and limits the ability to detect phase shifts in kinematic variables to phase shifts greater than 1 deg.

Fig. 1.

The coordinate system is right handed, and oriented with the positive x-axis in the direction of travel.

Fig. 1.

The coordinate system is right handed, and oriented with the positive x-axis in the direction of travel.

We ensured movement from right to left from the point of view of the cameras in the following way. The movement was encouraged by gentle poking with a blunt rod or by inverting the sea stars and allowing them to right themselves. Sea stars can move with any arm or interarm leading and the leading region can change continuously or spontaneously. For a given run, and before starting to film, we noted the leading region of the sea star and gently rotated the moving sea star so that the leading region was oriented toward the left end of the runway from the point of view of the cameras. At the end of each run, two square layout miters (Shinwa, Japan) were placed in the tank to scale the videos.

Kinematic variables

The right-handed coordinate system we used is shown in Fig. 1. Only portions of runs in which the sea stars moved in the positive x-direction without touching the walls of the tank were analyzed.

Sea star movement was tracked using Tracker Video Analysis and Modeling Tool 5.1 (https://physlets.org/tracker/). The mouth and the highest point on the top of the sea star were used as tracking points for the bottom and side videos, respectively.

A total of 128 matched bottom and side videos of 54 sea stars were analyzed (1–6 videos per sea star); the average duration of videos was 39±18 s (mean±s.d.; range: 10–107 s).

Time and position data from Tracker were smoothed using 30-point moving averages as a low-pass filter. The moving average was applied to the time array and to the position array (see supplementary Materials and Methods section 1 for further details). Such averaging results in arrays that are shorter than the originals by 29 points, with equal amounts of information being lost from both ends of the arrays. The time interval between points is nevertheless maintained and is equal to the sampling interval, i.e., the frame interval of the video recording. The smoothed time-array was then renumbered starting at zero.

This moving average filter preserves the motion at the main underlying frequency of the bouncy gait and rejects high frequency noise while maintaining much, or most, of the amplitude of the major low frequency motions of the sea stars (see supplementary Materials and Methods section 2 for further details; Figs S1–S5).

The moving average filter does not introduce any phase distortion into the signal. Therefore, it is possible to calculate phase shifts from the filtered data. Nevertheless, because some kinds of filters do introduce phase distortion, we used unfiltered data and cross-correlation analysis to determine the phase shifts between the horizontal speed and the vertical velocity and between the kinetic and potential energy.

Calculating velocities and accelerations, indeed calculating any higher derivatives from raw data, has the effect of increasing the importance of higher frequencies in the signal (see supplementary Materials and Methods section 3 for further details; Eqns S1–S3, Figs S2D, S4, S5). This problem becomes more severe the higher the derivatives. For this reason, when calculating accelerations, we used a slightly less aggressive (not moving average) low pass filter with a 1 Hz cut off on the position data, implemented in Mathematica (Wolfram) before calculating the accelerations as a function of time for each video.

Finally, velocity squared data, such as kinetic energy, also increases the importance of higher frequencies (see supplementary Materials and Methods section 4 for further details; Eqns S4–S7). Therefore, as a sensitivity analysis, we present in the supplementary Materials and Methods, an alternative fitted scaling equation of the kinetic and potential energy that is produced with 1 Hz low pass filtered position data.

With the exception of acceleration and phase calculations, calculations were done on the smoothed, renumbered time and position signal. In particular, smoothed time and position data were used to calculate position changes and velocity components (vx,vy, vz). Horizontal speed vxy was calculated at each time from the velocity components vx and vy. Vertical velocity vz was calculated from the side videos and vx and vy were calculated from the bottom videos. For purposes of perspective error analysis, vx,side was also calculated from the side videos.

Vertical displacement Δz was calculated as:
formula
(1)
and is the mean over an entire video. As a variable to examine scaling of vertical displacement relative to mass, we also calculated rms(Δz), where rms is the root mean square (RMS). The RMS is a convenient measure of signal size especially when there are a few frequencies all contributing to the overall signal size (see supplementary Materials and Methods section 5 for further details; Eqns S8,S9). The range in Δz can be used as a proxy for podial leg length; for this purpose we calculated .

Assessment of perspective error in side videos

Fractional perspective error in the side videos was estimated as:
formula
(2)

Across 128 matched side and bottom videos and 54 sea stars, the mean fractional perspective error was 0.10±0.0035 (mean±s.e.m.). A plot of vx,side as a function of vx is expected to have a slope of 1 in the absence of perspective error; the observed mean of the slopes is 1.02±0.0078 (R2=0.94±0.0044).

Speed, frequency and gait

Frequency spectra of horizontal plane speed oscillations were estimated using the Fourier function in Mathematica, which performs a discrete Fourier transform (DFT). Frequency resolution equals frame rate divided by the number of frames in a video. Across the 128 video pairs the mean frequency resolution was 0.03 Hz, with a range from 0.1 to 0.01 Hz. We report amplitudes only from frequencies f where 0.1 Hz≤f≤1 Hz. Amplitudes from higher frequencies are negligibly small and amplitudes from lower frequencies represent longer term changes in behavior such as changes in gait. The fundamental frequency was taken to be the frequency with the largest speed amplitude over the time interval analyzed, which was the whole length of each video except when moving windows were analyzed.

Such moving windows were used in determining the relationship between the mean vxy in a time interval and the amplitude of the fundamental frequency for portions of a given video. This is useful for determining changes in the amplitude of the fundamental frequency as gait changes. For this purpose, DFTs and moving averages were performed on 5 s or 45 s windows starting every second. Such analyses from three characteristic videos are shown in the results.

Mass and density

Mass

Mass was obtained for 86 sea stars, including for the subset of 54 used in kinematic analyses. Mass was measured using an Ohaus Adventure digital precision balance Model AX423 (precision: ±0.001 g). Underwater weights were obtained by hanging each sea star from a force beam into a bucket of seawater using a Kyowa force transducer model LVS-20GA, a DPM-911B strain amplifier, micro 1401 Cambridge Electronic Design AtoD converter and Spike II version 7.14 data acquisition software (precision: ±0.001 g). Masses (three to five measurements per sea star) and underwater weights (one to five measurements per sea star) were determined.

A sea star of mass Mb displaces a fluid of mass Mf generating an underwater vertical force due to gravity and buoyancy :
formula
(3)
where ΔMb is the difference in sea star and displaced masses and g is the gravitational acceleration. The ΔMb can be determined experimentally by dividing the underwater weight in Newtons by g.

Density

The definition of density implies that:
formula
(4)
where ρ is sea star density and ρf is fluid density. For our experiment we used a sea water density of 1025 kg m−3, which corresponds to seawater at 13°C and 33 ppt. Combining Eqn 3 and Eqn 4 gives an equation for calculating density from the measured Mb and ΔMb:
formula
(5)
Eqn 5 is equivalent to:
formula
(6)
which gives a ratio for expressing the fraction of mass not supported by buoyancy.
Eqn 6 is an alternative to using an effective gravity ge (Martinez et al., 1998; Chellapurath et al., 2020) where, using Eqns 3 and 6:
formula
(7)

We prefer using Eqn 6 because it makes explicit that g is a constant, whereas ge depends on the assumption of constant density. It has been our experience that the density of benthic animals often scales with mass making the use of an effective gravity potentially misleading.

Energies

Data from the side videos were used to calculate the change in potential energy ΔEp and data from the bottom videos were used to calculated the change in kinetic energy associated with the component of velocity in the horizontal plane ΔEkh relative to respective mean energies in each video.

The change in potential energy was calculated as:
formula
(8)
where the potential energy Ep is:
formula
(9)
Change in horizontal kinetic energy was calculated as:
formula
(10)
where:
formula
(11)
To compare temporal patterns of these energies, they were plotted as functions of time for three characteristic videos in the results. In addition, potential energy amplitude Ep,amp and the kinetic energy amplitude Ekh,amp for each video were calculated as:
formula
(12)
and
formula
(13)

The mean values of Ep,amp and Ekh,amp for each sea star were obtained by averaging values for all videos of a given sea star. These quantities are of interest because they allow the comparison of the amount of energy that could potentially be exchanged between potential and kinetic energy.

To assess the degree to which total energy amplitude Ek,amp,total reflects Ekh,amp, we calculated the ratio where Ek,amp,total was calculated in a manner analogous to Ekh,amp, where total kinetic energy Ek,total is:
formula
(14)
and vxyz is the magnitude of the 3D velocity vector.

Scaling

Allometric relationships were analyzed using double logarithmic plots for the following dependent variables: sea star density, fundamental frequency, mean horizontal speed, ΔMb, rms(Δz), Ep,amp and Ekh,amp. To do this, both the dependent variable and the independent variable (Mb) were log-transformed and linear regression on the log–log transformed data was performed to evaluate the exponential trends observed. Plots of only the first three and last two variables are shown in the results. The density was calculated from the mean Mb and ΔMb using Eqn 5 for each sea star. The frequency was the mean over all videos of each sea star of the fundamental frequencies measured in each video. The speed, rms (Δz), Ep,amp and Ekh,amp were the means over all videos of each sea star in each video. Exponentiating both sides of the equation for the linear relationship between the log-transformed variables gives the coefficient and exponent of the exponential relationship.

In addition, a paired t-test on log-transformed energies was used to test the null hypothesis that Ep,amp and Ekh,amp were equal in a given sea star. Finally, because theory developed in the discussion predicts a scaling exponent of 1.3 for energy amplitudes, we tested the null hypothesis that the slopes of the log-log regressions did not differ from a slope of 1.3 using an extra sum-of-squares F-test in Prism (GraphPad).

Hodographs and phase shifts

Hodographs plotting vz against vxy were produced to observe phase shifts and to compare patterns between crawling and oscillating gaits. In addition, hodograph-like plots were made of ΔEp against ΔEkh to describe the temporal pattern of change in energies. Characteristic patterns from three videos are presented in the results.

For all 128 videos, we estimated the phase shifts on the unfiltered data between ΔEkh and ΔEp, and between vxy and vz by using a cross-correlation analysis written in Mathematica (by O.E.). Means for each sea star were determined using polar statistics. Angular mean was calculated as:
formula
(15)
where is the phase shift mean for all the runs of a given sea star, and ϕj are the phase shifts for each of J runs for each of the energy and velocity pairs for a given sea star. An angle mean similar to Eqn 15 was also used to calculate the mean phase shift for all sea stars from the individual means of each sea star. Depending on underlying distributions in the data either t-tests or non-parametric sign-tests or Mann–Whitney U-tests were used to compare phase shift between vertical and horizontal velocities and between potential and kinetic energy with values of π/2 and zero, respectively and the mean phase shift between energies compared with that between cameras.

Gravitational versus accelerative forces

The mean underwater vertical force due to gravity and bouyancy acting on a specific sea star was calculated from Eqn 3 as:
formula
(16)
where is the average ΔMb for the sth sea star.

The underwater vertical and horizontal forces due to accelerations acting on a specific sea star were determined as follows. Vertical velocities vz and horizontal speeds vxy, were calculated from 1 Hz low pass filtered position data and were then used to calculate vertical accelerations az and horizontal accelerations axy.

The mean of the observed experimental rms(az) for a specific sea star was estimated over several runs as:
formula
(17)
where (az)j,s is the observed vertical acceleration as a function of time for the sth sea star and the jth run.
The mean rms of forces associated with the observed experimental vertical accelerations acting on a specific sea star averaged over several runs were estimated as:
formula
(18)
where is the mean mass for the sth sea star. The mean rms of forces associated with the observed experimental horizontal accelerations acting on a specific sea star were estimated in the appropriate analogous way.
These accelerative forces are compared with gravitational minus bouyancy forces over all sea stars by calculating two kinds of underwater Froude numbers Fr relative to vertical and horizontal accelerations:
formula
(19)
and
formula
(20)
where S is the total number of sea stars in the kinematic analysis.

As a sensitivity analysis on the effects of filtering, in addition to calculating the Froude numbers from 1 Hz low pass filtered position data, we also calculated them from data low pass filtered with 2, 3 and 4 Hz cut offs (see supplementary Materials and Methods section 6 for further details; Table S1).

Analysis software

Data were processed using Mathematica (v. 12.0.0.0) and Julia (v. 1.5). Figures were created using Prism (v. 7.0a) and Julia packages: GRUtils (https://juliapackages.com/p/grutils), CSV (https://juliapackages.com/p/csv) and DataFrames (https://juliapackages.com/p/dataframes). Statistical analyses were done using Prism, Mathematica and SPSS Statistics (v. 26.0.0.0). Mathematica and Julia codes were custom written in house.

Kinematic variables

Sea stars exhibit consistent patterns of motion (Fig. 2). They have two gaits: a slower crawling gait and a faster oscillatory gait in which the sea star oscillates vertically. During the oscillatory gait the vertical displacements and speeds oscillate and the horizontal speeds oscillate around an average speed. Whereas the average vertical speed is close to zero, the average horizontal speed is several mm s−1 and the minimum horizontal speed during a video is typically greater than zero. While transitioning to an oscillatory gait the sea star vertical height increases and develops vertical oscillations that tend to become more regular.

Fig. 2.

The transition from crawling to oscillatory gait in a 24 g sea star (Asterias forbesi). (A) Vertical velocity (vz), shown as a 1 s moving average (black line) and a 3.36 s moving average based on the fundamental period for these data (red line). (B) Vertical displacement (Δz) of the sea star, shown as a 1 s moving average, with the mean vertical displacement indicated by the dotted zero line. (C) Speed in the horizontal plane (vxy), shown as a 1 s moving average (black line) and a 3.36 s moving average (red line), with the mean vxy indicated by the horizontal dotted line. (D) Horizontal displacement shown as a 1 s moving average. The two vertical dashed lines are placed at the times of two peaks in vertical displacement and show that peak horizontal speeds are nearly in phase with peak vertical displacements of the sea star. These data illustrate a typical pattern, where sea stars crawl with increasing Δz and vxy, transitioning to low amplitude and then high amplitude Δz oscillations. The area shaded in grey indicates the cycle that is plotted in Fig. 7A.

Fig. 2.

The transition from crawling to oscillatory gait in a 24 g sea star (Asterias forbesi). (A) Vertical velocity (vz), shown as a 1 s moving average (black line) and a 3.36 s moving average based on the fundamental period for these data (red line). (B) Vertical displacement (Δz) of the sea star, shown as a 1 s moving average, with the mean vertical displacement indicated by the dotted zero line. (C) Speed in the horizontal plane (vxy), shown as a 1 s moving average (black line) and a 3.36 s moving average (red line), with the mean vxy indicated by the horizontal dotted line. (D) Horizontal displacement shown as a 1 s moving average. The two vertical dashed lines are placed at the times of two peaks in vertical displacement and show that peak horizontal speeds are nearly in phase with peak vertical displacements of the sea star. These data illustrate a typical pattern, where sea stars crawl with increasing Δz and vxy, transitioning to low amplitude and then high amplitude Δz oscillations. The area shaded in grey indicates the cycle that is plotted in Fig. 7A.

Speed, frequency and gait

As sea stars transitioned from the crawling to the oscillatory gait, the vertical oscillations increased and the horizontal and vertical speed oscillations increased in amplitude (Figs 2,3). Each sea star has a characteristic frequency that can be determined by doing DFTs over the period of a video; we report those for the vxy oscillations. The frequencies we observed for these sea stars were between 0.14 and 0.40 Hz corresponding to periods between 7.2 and 2.5 s. While the highest period-averaged horizontal speeds are achieved during the oscillatory gait, most of the increase in period-averaged horizontal speed occurs during the crawling gait. There are more gradual increases in period-averaged horizontal speed with increasing oscillation amplitude of the horizontal speed during the oscillatory gait.

Fig. 3.

Examples from videos of three locomoting sea stars. (A–C) vxy as a function of time (left), shown as a 1 s moving average (black line) and longer moving averages based on the fundamental periods determined for each video (red lines; A, 5.9 s; B, 4.4 s; C, 2.8 s) for sea stars of mass 203 g (A), 41 g (B) and 15 g (C). The area shaded in grey indicates the cycle that is plotted in Fig. 7B. DFTs with the vxy oscillation amplitude as a function of frequency (middle), where the vertical red line indicates the fundamental frequency f, which has the largest vxy amplitude. Mean vxy over intervals shown as a function of vxy amplitude over intervals (right); means were calculated for the indicated intervals starting each second. Most of the increase in vxy occurred during low amplitude oscillations of less than 4×10−4 m s−1 (i.e. during crawling); further increases in speed with increasing vxy amplitudes were more gradual.

Fig. 3.

Examples from videos of three locomoting sea stars. (A–C) vxy as a function of time (left), shown as a 1 s moving average (black line) and longer moving averages based on the fundamental periods determined for each video (red lines; A, 5.9 s; B, 4.4 s; C, 2.8 s) for sea stars of mass 203 g (A), 41 g (B) and 15 g (C). The area shaded in grey indicates the cycle that is plotted in Fig. 7B. DFTs with the vxy oscillation amplitude as a function of frequency (middle), where the vertical red line indicates the fundamental frequency f, which has the largest vxy amplitude. Mean vxy over intervals shown as a function of vxy amplitude over intervals (right); means were calculated for the indicated intervals starting each second. Most of the increase in vxy occurred during low amplitude oscillations of less than 4×10−4 m s−1 (i.e. during crawling); further increases in speed with increasing vxy amplitudes were more gradual.

Scaling of density, frequency, speed and vertical displacement

Sea star density was a function of mass (log–log regression: F1,84=37, P<0.0001; Fig. 4A) with heavier sea stars having a lower density. During the oscillatory gait, mass of the sea stars accounted for 53% of the variation in the fundamental frequency of horizontal speed oscillations with heavier sea stars oscillating at lower frequencies (log-log regression: F1,52=58, P<0.0001; Figs 3, 4B, 5). Mass accounted for 14% of the variation in mean speed, with heavier sea stars moving faster (log–log regression: F1,52=8.8, P=0.005; Fig. 4C).

Fig. 4.

Logarithmic plots of sea star density, fundamental frequency of the oscillatory gait and mean horizontal speed. (A) Sea star density, (B) fundamental frequency of the oscillatory gait, and (C) mean horizontal speed as a function of sea star mass. Density and fundamental frequency decreased and mean horizontal speed increased with increasing mass of sea stars. Data points represent the mean values for each sea star. Units in the fitted equations are as on the axes in the figures. In (A) two of the plotted points were excluded from the regression analysis based on Cook's distance. Errors shown are s.e.m.

Fig. 4.

Logarithmic plots of sea star density, fundamental frequency of the oscillatory gait and mean horizontal speed. (A) Sea star density, (B) fundamental frequency of the oscillatory gait, and (C) mean horizontal speed as a function of sea star mass. Density and fundamental frequency decreased and mean horizontal speed increased with increasing mass of sea stars. Data points represent the mean values for each sea star. Units in the fitted equations are as on the axes in the figures. In (A) two of the plotted points were excluded from the regression analysis based on Cook's distance. Errors shown are s.e.m.

Fig. 5.

Change in kinetic energy and potential energy during the oscillatory gait in three individual sea stars. (A–C) Change in KE (ΔEkh, left axes, dashed lines, Eqn 10) and PE (ΔEp, right axes, solid lines, Eqn 8) in sea stars with mass of 203 g (A), 34 g (B) and 19 g (C). Oscillation frequency of energies decreases with increasing mass (see also Figs 3, 4B). Peaks in ΔEkh and ΔEp are almost in phase (see also Fig. 7), with the fastest speeds at the highest point in a sea star's stride. Both ΔEkh and ΔEp increase with increasing size of sea star (see also Fig. 5). The area shaded in grey in B indicates the cycle that is plotted in Fig. 7C. Results shown in C are from Movie 1.

Fig. 5.

Change in kinetic energy and potential energy during the oscillatory gait in three individual sea stars. (A–C) Change in KE (ΔEkh, left axes, dashed lines, Eqn 10) and PE (ΔEp, right axes, solid lines, Eqn 8) in sea stars with mass of 203 g (A), 34 g (B) and 19 g (C). Oscillation frequency of energies decreases with increasing mass (see also Figs 3, 4B). Peaks in ΔEkh and ΔEp are almost in phase (see also Fig. 7), with the fastest speeds at the highest point in a sea star's stride. Both ΔEkh and ΔEp increase with increasing size of sea star (see also Fig. 5). The area shaded in grey in B indicates the cycle that is plotted in Fig. 7C. Results shown in C are from Movie 1.

Horizontal and vertical oscillations in speed were mechanically linked since for 84% of the videos the fundamental frequency was identical when calculated from horizontal versus vertical velocities. In the 16% of cases where they were not identical, they were similar; the mean absolute value of the difference was 0.04 Hz, which was below our ability to resolve differences given the mean frequency resolution of 0.03 Hz.

Eqn 6 indicates that ΔMb is a linear function of Mb if sea star density ρ is constant, but a log–log regression gives:
formula
(21)
with an R2=0.98 (log–log regression: F1,52=3500, P<0.0001), indicating that sea star density changes slightly but systematically with sea star mass, which is also supported by Fig. 4A. Eqn 21 gives a useful approximate rule that ΔMb/Mb≈0.1. However, for the sea stars in Fig. 4A, the range of density from ≈1100–1200 kg m−3 encompasses a range of ge/g (Eqn 7) from 0.054 to 0.14, a difference of a factor more than two. Thus, the force that sea stars need to exert to support their weight underwater is about 10% of what it would be on land, but for more precise calculations or scaling considerations it may be necessary to consider that systematic trend with mass.
Vertical displacement was also a function of sea star mass:
formula
(22)
where units of mass are in g and units of vertical displacement are in m, and with an R2=0.32 (log–log regression: F1,52=25, P<0.0001) indicating that heavier sea stars have slightly greater, but highly variable, changes in oscillatory height.

Energies, hodographs and phases

Oscillations in potential and kinetic energy were nearly in phase in the oscillatory gait and the change in potential energy Ep,amp was much greater than the change in horizontal kinetic energy Ekh,amp (Figs 5, 6, 7, 8b).

Fig. 6.

Logarithmic plot of mean amplitude of potential energy and kinetic energy change during the oscillatory gait as a function of mass.Ep,amp (solid line and black circles) and Ekh,amp (dashed line and open circles). Errors shown are s.e.m. Filtering with the low pass 1 Hz filter in Mathematica instead of the 30 point moving average used for Fig. 6 slightly increases the coefficients and decreases the exponents (see supplementary Materials and Methods section 4 for further details).

Fig. 6.

Logarithmic plot of mean amplitude of potential energy and kinetic energy change during the oscillatory gait as a function of mass.Ep,amp (solid line and black circles) and Ekh,amp (dashed line and open circles). Errors shown are s.e.m. Filtering with the low pass 1 Hz filter in Mathematica instead of the 30 point moving average used for Fig. 6 slightly increases the coefficients and decreases the exponents (see supplementary Materials and Methods section 4 for further details).

Fig. 7.

One oscillatory step for three sea stars. (A) A 24 g sea star, data plotted indicated by grey area in Fig. 2. (B) A 41 g sea star, data plotted indicated by grey area in Fig. 3B. (C) A 34 g sea star, data plotted indicated by grey area in Fig. 5B. The steps are shown as hodographs (left): a parametric plot of vz and vxy over the time of one step, and a parametric plot of ΔEp and ΔEkh over the time of one step (right). All data shown are one second moving averages with points 1/30 s apart. Arrows indicate direction of time. The oscillations shown took 3.7 (A), 4.1 (B) and 4.4 s (C). Dotted vertical and horizontal lines indicate mean values for the corresponding variables for the corresponding cycle. Blue points (i) and (ii) in C correspond to the lowest and highest position of the sea star as indicated by the leading arm of the sea star as shown in images D and E, respectively. Yellow and orange points in A–C correspond to identical points in left and right graphs, which occur at the mean change in potential energy and kinetic energy, respectively. Colored lines in A and B plot the first 25 s of crawling data from the figures indicated above in 5 s intervals (green to purple). Contrast the hodograph pattern of the early crawling period to the clockwise ovals during the oscillatory gait. Scale bar: 3 mm.

Fig. 7.

One oscillatory step for three sea stars. (A) A 24 g sea star, data plotted indicated by grey area in Fig. 2. (B) A 41 g sea star, data plotted indicated by grey area in Fig. 3B. (C) A 34 g sea star, data plotted indicated by grey area in Fig. 5B. The steps are shown as hodographs (left): a parametric plot of vz and vxy over the time of one step, and a parametric plot of ΔEp and ΔEkh over the time of one step (right). All data shown are one second moving averages with points 1/30 s apart. Arrows indicate direction of time. The oscillations shown took 3.7 (A), 4.1 (B) and 4.4 s (C). Dotted vertical and horizontal lines indicate mean values for the corresponding variables for the corresponding cycle. Blue points (i) and (ii) in C correspond to the lowest and highest position of the sea star as indicated by the leading arm of the sea star as shown in images D and E, respectively. Yellow and orange points in A–C correspond to identical points in left and right graphs, which occur at the mean change in potential energy and kinetic energy, respectively. Colored lines in A and B plot the first 25 s of crawling data from the figures indicated above in 5 s intervals (green to purple). Contrast the hodograph pattern of the early crawling period to the clockwise ovals during the oscillatory gait. Scale bar: 3 mm.

Fig. 8.

Polar histograms for the velocity and energy phase differences of all sea stars. (A) Phase differences between vz and vxy. (B) Phase differences between Ep and Ekh. Data in each histogram are the angular means of all the videos of each individual sea star. For (A) vz preceded vxy and for (B) Ep preceded Ekh by the phase shift shown.

Fig. 8.

Polar histograms for the velocity and energy phase differences of all sea stars. (A) Phase differences between vz and vxy. (B) Phase differences between Ep and Ekh. Data in each histogram are the angular means of all the videos of each individual sea star. For (A) vz preceded vxy and for (B) Ep preceded Ekh by the phase shift shown.

During the oscillatory gait, mass of the sea stars accounted for >89% of the variation in the potential and kinetic energy amplitudes with heavier sea stars having higher energy amplitudes (log–log regressions: each F1,52>424, each P<0.0001; Fig. 6). The slope was significantly less than 1.3 for both energy amplitudes (each F1,52>15, each P<0.001). Variation in potential energy was two orders of magnitude more than variation in kinetic energy (paired t-test on log-transformed data: t53=75, P<0.0001; Figs 5, 6; mean over all sea stars of Ep,amp/Ekh,amp=185±16; mean±s.e.m.). The mean over all 128 videos of Ekh,amp/Ek,amp,total=0.97±0.002; therefore, Ekh,amp represents substantially all of the Ek,amp,total.

During the oscillatory gait, hodographs for all sea stars were clockwise, with peak vertical velocities preceding peak horizontal speeds by about a quarter of a circle (Fig. 7A). On average, vertical velocities preceded horizontal speeds by 1.58 radians (90.3 deg), which is not significantly different from 90 deg (t-test: P=0.82) (Fig. 8A).

Oscillations in potential and kinetic energy were nearly in phase, with the potential energy peak preceding the kinetic energy peak by a mean of 0.16±0.15 radians (9.6±8.7 deg; means±s.d.), which is slightly, but significantly, greater than 0 deg (sign test: P<0.001; Figs 5, 7B, 8B). This is also significantly different than the slight 1.2 deg phase shift in vx between the side and bottom cameras (Mann–Whitney U-test, P<0.001).

Gravitational versus accelerative forces

Here, we assess the magnitude of forces required to support the underwater weight of the sea stars and compare them to the magnitude of the forces accelerating the sea star in the horizontal and vertical directions. The mean of the ratios of accelerative to gravitational forces for the 54 sea stars used in the kinematic analyses are: Frz=1.8×10−3±1.1×10−4 and Frxy=3.3×10−3±1.8×10−4 (means±s.e.m.), indicating that the vertically-directed forces required to support the sea star are large compared with either the vertical or horizontal accelerative forces (see Table S1 for a comparison of results using low pass filtered data with 1,2,3 and 4 Hz cut offs).

Speed and gait

In a general invertebrate textbook, sea star locomotion is described as crawling and smooth, a result of the high number of podia each in a different phase of the power stroke (Brusca and Brusca, 2003); Pisaster giganteus in particular is described as crawling but we have seen this species using the vertically oscillating gait. Although also describing sea star locomotion as a crawl, Cole (1913) nevertheless observed that the motion was not a smooth glide but proceeded in short lunges of the body. This description concurred with Jennings (1907), who described it as a ‘crawl’ and ‘creep’ but also gave detailed observations of ‘steps’, comparing the motion to the steps of higher animals, and describing a slight elevation of the body, as the body swung over the podia. Only one paper (Montgomery and Palmer, 2012) provides direct data showing that Patiria miniata increases gradually in speed until a plateau in speed is reached – a plateau in which there is oscillating vertical motion and oscillating forward speed. Despite observing what we can recognize in hindsight as a gait transition, this transition is not specifically recognized as such in that paper. One characteristic of a gait is that it operates over a specific range of speeds. The focus of most recent literature on sea star locomotion has been on speeds in various ecological contexts. Although these contexts are sometimes classified as ‘pursuit’ or ‘escape’, these speeds are given without regard to gait. A recognition and description of gait requires a description of several elements that define a gait such as speed ranges in which a gait is used and the characteristic patterns of motion or kinematics of the gait.

Fundamental to the definition of gait, is that characteristic kinematic quantities change discontinuously as speed changes (Alexander, 1989). What we observed is a change in the pattern of motion as speed increases (Figs 2, 3B,C and 7A,B). As the speed increases, at some point vertical oscillations are established and phase relationships develop between vertical and horizontal speed oscillations. Specifically, in Fig. 7A,B the colorful squiggles represent 25 s during which the sea star is crawling and the velocity is increasing. Subsequently, during the period when vertical oscillations occur, the hodograph is a series of clockwise circles.

Gaits are typically used over a range of speeds. For instance, walk, trot and gallop are gaits that can each be used over a characteristic range of speeds in horses, although there is an energetically optimal gait within specific speed ranges (Hoyt and Taylor, 1981). Similarly, humans can walk at a variety of speeds with a variety of stride lengths and frequencies, but there is a combination of stride length and frequency that feels more natural and may be more efficient (Kuo, 2001, 2007; Kuo et al., 2005). We have no information about the energetic consequences of the crawling versus oscillatory gait for sea stars, however, the crawling gait seems to be used over a wide range of speeds; after the transition to the oscillatory gait the highest speeds are attained and variation in speed is less (Fig. 3).

What causes this change in gait? One idea is that the faster, oscillatory gait may be a necessary consequence of podial dynamics. Heydari et al. (2020) developed a model of podial actions that described an emergent patterning in the coordination of the podia that was associated with the bouncy gait. We have observed that podial action during the crawl is not obviously coordinated but that it transitions to an oscillatory gait in which there is an alternation between coordinated elongated podia at the highest point in each step followed by the much shorter or collapsed podia at the bottom of each step (see Fig. 7D,E).

How does theory inform frequency predictions?

The hallmark of the bouncy gait is the rhythmic vertical oscillation of the sea star accompanied by cyclical horizontal and vertical velocity changes. We have observed that the frequency of these oscillations is regular and specific to a sea star. That is, sequential cycles in the same sea star have similar periods, a given sea star tends to oscillate at a certain frequency, and frequency decreases with sea star mass (Figs 3, 4B, 5). In the sections below, we develop a simple abstract model and ask: what calculation predicts the observed frequencies? The goal is to develop a conceptual understanding that allows us to predict the observed frequencies and abstracts what is happening in the steps.

Model description

Consider a pendulum bob with a mass MPB at the end of a massless rod of length l with the rod rotating about a pivot point at the end of the rod opposite to the mass (Fig. 9). The rod is caused to oscillate by a moment generated by the vertical force driving the motion and the rod's position is described by the angle θ to the vertical. That vertical force Fz is the sum of forces due to podial pressure that opposes the sum of gravitational and bouyant forces (Eqn 3). For simplicity, we consider that and act at the same point. Using mass Mb for the inertial term and mass minus fluid displaced ΔMb in the Fz term and using the small angle approximation that sinθ≈θ, the moments about the pivot point are:
formula
(23)
where
formula
(24)
where P is podial pressure and A is the horizontal plane cross-sectional area of all of the actively lifting podia.
Fig. 9.

A sea urchin podium imagined as an inverted pendulum swinging a bob over the podium during a step, where the bob represents the sea star mass. Forces exerted on the bob are gravity minus buoyancy and the upward force due to podial pressure . The resultant vertical force Fz has a component, Ft in the direction tangent to the arc of the pendulum. If the forces due to pressure are larger than the forces due to gravity minus buoyancy, then a stable inverted pendulum results and a simple differential equation represents this oscillation as in Eqns 23 and 26 and can be used to predict kinematic patterns and scaling. This is our TS-IP model. The green angle represents the torque spring, which is a conceptual representation of the effect of the tangential forces exerted by the vertical forces shown. This model ignores podial length changes and podial bending, which are beyond the scope of the present manuscript.

Fig. 9.

A sea urchin podium imagined as an inverted pendulum swinging a bob over the podium during a step, where the bob represents the sea star mass. Forces exerted on the bob are gravity minus buoyancy and the upward force due to podial pressure . The resultant vertical force Fz has a component, Ft in the direction tangent to the arc of the pendulum. If the forces due to pressure are larger than the forces due to gravity minus buoyancy, then a stable inverted pendulum results and a simple differential equation represents this oscillation as in Eqns 23 and 26 and can be used to predict kinematic patterns and scaling. This is our TS-IP model. The green angle represents the torque spring, which is a conceptual representation of the effect of the tangential forces exerted by the vertical forces shown. This model ignores podial length changes and podial bending, which are beyond the scope of the present manuscript.

Eqn 23 is mathematically equivalent to placing a spring at the fulcrum that exerts a torque that counteracts the torque. Using a torque spring in a walking model is not unprecedented. Biswas et al. (2018) found that both an inverted pendulum and a SLIP model cannot model observed features of walking at low Froude numbers (<0.1) and instead incorporated an angular spring to model low Froude number walking in fruit flies. An angular spring has also been incorporated in a version of the standard SLIP model used to analyze the human walking gait (Antoniak et al., 2019). Mathematically, the angular spring has the effect of creating a stable inverted pendulum without a feedback control if the moment from the spring is larger than the moment from gravity. This condition is met if PAMbg. In that case, Eqn 24 is equivalent to a force acting on the bob of the pendulum directed upwards and opposing . The system then acts as if there was a net force field acting upwards causing the inverted pendulum to swing in a manner analogous to an ordinary pendulum in a gravitational field.

This approximation is suitable for application to sea star locomotion because the observed accelerations and accelerative forces are small compared to the gravitational force (Frxy and Frz <10−2). Since the podia are causing all forces other than , then the fact that the accelerative forces are relatively small implies that the podial forces are aligned with gravity and slightly larger than . This pattern of forces mimics the assumed forces in our model.

We can solve for angular frequency ω by rewriting Eqn 23 as:
formula
(25)
Solutions of Eqn 25 give periodic functions with frequency ω, where:
formula
(26)

There are a few unrealistic aspects to this model, which we call a torque stabilized inverted pendulum model (TS-IP). For example, real sea star podia bend and extend, whereas the podia in the model do neither. Also, real upward podial forces vary cyclically during a step, whereas upward podial forces are constant in the model, at least over one step. More involved models in the future, may incorporate those features; the simplified model nevertheless provides insights as will be shown in the following sections.

Consideration of the magnitudes of frequencies observed

In this section, we compare two hypotheses that make different predictions about the frequency of the sea star oscillatory step. The first hypothesis is that fall velocity sets the frequency and the alternative hypothesis is that frequency may be set by the podial forces, in Eqn 26, slowing the sea star's descent.

In terms of fall velocity, a person watching the oscillatory gait of a sea star might hypothesize that each underwater step consists of lifting the sea star to its highest point, followed by a free fall to the end of the step, and that falling time might set the frequency of the step. If we assume that the rising and falling halves of the step are approximately symmetrical, then the step time is twice the fall time from height l, which is:
formula
(27)

Using ΔMb/Mb=0.11 from Eqn 21 and using a maximum estimate of l as maximum m, we get τ1=0.16 s, which is an order of magnitude less than the periods we observed (Fig. 4B, range: 2.5–7.2 s).

We get a similar result if the equation is further adjusted to give the period of a pendulum underwater τu (Neill et al., 2007):
formula
(28)
where ρPB is the density of the pendulum bob, ρf is the density of the fluid, and κ is the coefficient of added mass. A further adjustment can be made for drag forces, but that adjustment is negligible for low speeds and will not be considered here. Note, however, that there may be significant damping by tissues and that such damping would have the effect of lowering and broadening the resonance peak of a damped harmonic oscillator.

Using ρPB=ρ=1150 kg m−3 (a middle value for the density of a sea star from Fig. 4A), ρf=1025 kg m−3, and κ=3 (an upper value from Daniel, 1984), then τ2=0.69 s. Although similar to our result from Eqn 27, this value is still an order of magnitude less than the observed periods. We can conclude that the periods predicted by fall time and the underwater pendulum are too short and that some other process must be driving the periodicity.

Consider the alternative hypothesis that podial forces control gait frequency. For comparison to Eqns 27 and 28, the corresponding prediction from Eqn 26 is:
formula
(29)

To the degree that PA is larger than ΔMbg, the step period will be longer than the predictions of Eqns 27 and 28, where the smaller the difference, the longer the step period. For example, if PA is 1% larger than ΔMbg, then the prediction from Eqn 29, with l=0.0035 m and ΔMb/Mb=0.11 is τ3=3.5 s, which is within the range of periods observed.

We have no direct measurement of the total podial forces, but the observed accelerations provide a parity check for the relative size of PA and ΔMbg. We can estimate the magnitude of PA using the observed accelerations of the sea stars during a step. The vertical upward force of podia on the sea star is of magnitude:
formula
(30)
Substituting Eqn 30 into Eqn 26, expressing angular frequency in terms of frequency and using the mean over all sea stars of the experimentally observed as an average vertical acceleration and as an average leg length, we get:
formula
(31)
which is a period of 7.7 s. This is of the same order of magnitude as the observed frequencies, supporting the hypothesis that podial forces are increasing the period relative to what would be expected due to free fall.

Scaling of oscillatory frequency

For sea stars, we observed that . This exponent is remarkably similar to the scaling exponent observed for terrestrial vertebrate legged locomotors (mean=−0.16; range: −0.12 to −0.22; Heglund et al., 1974; Pennycuick, 1975; Alexander et al., 1977; Alexander and Maloiy, 1984; Gatesy and Biewener, 1991; Daley and Birn-Jeffery, 2018). Our exponent is also similar to the exponent (−0.14) observed in ghost crabs moving in air (Whittemore et al., 2015). Multiple theories have been advanced to explain this scaling exponent.

Pennycuick (1992) elaborated on and compared two of these theories that predict different scaling of frequency with mass and we will consider these theories in the context of our TS-IP model and sea star data. The first theory is based on scaling arguments of Hill (1950), which focus on the role of muscle in generating force and moments tending to cause oscillations either of the legs or of the center of mass over the legs. The second theory is based on Alexander and Jayes (1983), which focuses on the role of gravitational forces in producing pendulum-like motions during legged locomotion. We will show below how hypotheses about gravitational and muscle force scalings predict frequency scaling with mass.

Consider scaling predictions using our TS-IP model for the sea star oscillatory gait (Eqn 26). In geometric similarity, Al2, Mbl3 and ΔMbMb, but pressure is independent of size. Pressure is generated by muscle stress in the ampullae, which is governed by Laplace's Law, in which the pressure across the ampullar wall ΔP:
formula
(32)
where ΔR is ampullar wall thickness, R is the radius of curvature of the ampulla, and σ is muscle stress, which does not scale with length.
Under the first theory, Hill's muscle hypothesis (Hill, 1950), as formulated by Pennycuick (1992), , so the gravitational term is negligible and angular frequency is:
formula
(33)

This predicted scaling is much steeper than we observed for the sea star oscillatory gait (Fig. 4B); it is also greater than any of the literature values for terrestrial pedestrian locomotion.

Alternatively, under the second theory, assume as we did for τ3 (Eqn 29), that PA is 1% larger than ΔMbg, i.e., that PA=1.01ΔMbg, and substituting into Eqn 26:
formula
(34)

This is the scaling predicted by Alexander and Jayes (1983). Their prediction is that this scaling is expected for geometrically similar animals locomoting at a constant Froude number, i.e., a constant ratio of inertial to gravitational forces. Our assumption that PA is 1% larger than ΔMbg is the equivalent of assuming a constant Froude number since the inertial forces (PA−ΔMbg) are also a multiple of ΔMbg under that assumption.

Our interpretation of Eqn 34 is thus that podial forces are recruited to an extent just sufficient to accelerate the sea star upwards and sideways and that this causes the podial forces to follow the size of the underwater gravitational forces as reflected by ΔMbg. This circumstance is equivalent to a constant Froude number and similar to the analysis of Alexander and Jayes (1983) for vertebrate locomotion. As also noted in their work, the observed exponent is also close to that predicted for elastic similarity by McMahon (1975, ), so we cannot rule out other possible explanations for this scaling in vertebrates or in sea stars.

Scaling of energy

In the sea star oscillatory gait, we found that the amplitude of both the change in kinetic and potential energy scaled identically with mass (Fig. 6). In the following argument, we derive the expected scaling of kinetic and potential energy for a series of geometrically similar animals using legged locomotion in a given medium.

Speed U is a function of step length λ and step time τ:
formula
(35)
with τ being 1/ω2 in Eqn 34. Further, assuming leg length l∝λ, then, Ep and Ekh are:
formula
(36)
and
formula
(37)

If we assume that the oscillations of kinetic and potential energy are proportional to energies in Eqns 36 and 37, then we expect that the slopes of the log-log plots of the rms energies as a function of mass should be straight lines both with slopes equal to . The experimentally determined slopes, both equal to 1.1, are slightly but significantly less (Fig. 6). This underscaling is probably due to the observed underscaling of podial length with mass (Eqn 22) and mean speed with mass (Fig. 4C).

Relationship between potential and kinetic energies and Froude number

Of equal interest to the scaling of energy to mass is the relationship between potential and kinetic energy. Two striking features stand out in our analysis: First, Ep,amp is two orders of magnitude greater than Ekh,amp, which precludes the possibility of efficient energy exchange between the two. This is different from the case of terrestrial vertebrate walking where kinetic and potential energy are of similar magnitude and exchange energy. Second, Ep,amp precedes Ekh,amp by 9.6 deg, and so they are substantially in phase (Figs 5,7,8b), which is a pattern more similar to terrestrial running than walking (Cavagna et al., 1964; Cavagna and Legramandi, 2020), though without an aerial phase. In the sea star bouncy gait, both due to the phase and magnitudes of the kinetic and potential energies, there is clearly no exchange of energies, although because potential energy is larger than kinetic energy, it is likely that some of the potential energy at the top of the bounce is converted to kinetic energy in the second half of each step.

In terrestrial locomotion, considerable attention is paid to the redirection of motion during the ground phase (Kuo et al., 2005; Ruina et al., 2005; Adamczyk and Kuo, 2009). This redirection is associated with energy storage in springs. In the terrestrial running gait the potential and kinetic energies exchange energy with collagen-based springs in the legs that store elastic strain energy (Alexander, 1988; Biewener, 1998; Dickinson et al., 2000). A potential candidate for such energy storage in sea stars could be the collagen fibers that wind around each podium in a cross-fiber helical array and along the long axis of each ampulla (McCurley and Kier, 1995). Our TS-IP model suggests a significant role for podia in slowing the descent of the sea star in the second half of each step, which might provide an opportunity for storage of elastic energy in the podia.

The ratio of kinetic to potential energy occurs in the Froude number. In particular, in an underwater setting, it is appropriate to write an underwater Froude number in the following way:
formula
(38)
which is approximately 10× higher in water than in air for a given pedestrian speed and leg length, and assuming animal densities similar to those measured for the sea stars in our study as well as for two species of crabs (Martinez et al., 1998; Chellapurath et al., 2020). We have observed, however, that the kinetic energy is much less than the potential energy in the sea star oscillatory gait, and therefore bouncing sea stars are operating at much lower Froude numbers, on the order of 10−2 to 10−3, than are reported for terrestrial vertebrate locomotors, which are on the order of one, or underwater crabs, which ranged from 0.4 to 3 (Martinez et al., 1998; Chellapurath et al., 2020). For terrestrial vertebrate pedestrians, across a range of species moving naturally, walking Froude numbers have been reported between 10−2 and 0.5 (Jayes and Alexander, 1978; Kramer and Sarton-Miller, 2008; Lees et al., 2016; Dececchi et al., 2020), but at those Froude numbers, one expects a walking gait, where potential and kinetic energy are out of phase. In contrast, in the bouncy sea star gait, despite the very low Froude numbers, potential and kinetic energy are in phase. Thus, the bouncy sea star gait represents a novel kinematic pattern. The only other legged locomotion for which similarly low Froude numbers have been reported is the case of slowly stepping fruit flies, where Froude number was between 0.011 and 0.0038 (Biswas et al., 2018). That study concluded that the fruit fly slow walking gait was fit poorly by a standard SLIP model and required an angular spring inverted pendulum model. Thus, studying legged locomotion at extreme Froude numbers expands the range of known legged gaits.

Hodographs and phase relationships for velocities and energies

The parametric plot of horizontal speed on the abscissa and vertical speed on the ordinal axis produces a nearly circular plot that proceeds in a clockwise fashion as in Fig. 7. This implies that the vertical speed function precedes the horizontal speed by a phase shift of about π/2 (90 deg) as was observed for all sea stars in Fig. 8 and can also be seen in the example in Fig. 2. This phase relationship between the vertical and horizontal speeds is characteristic of the running gait in terrestrial locomotors. Another feature of the sea star bouncy gait is that the fastest horizontal speeds occur when the sea star is highest as can be seen in the example in Fig. 2. This observed pattern of motion is consistent with the pattern of motion described by our proposed TS-IP model. In the following paragraphs we will describe how that pattern of motion arises.

Consider an upright pendulum. At the horizontal ends of travel of the bob, the bob is highest and the speed is zero. In contrast, at the middle of the swing, the horizontal speed is highest and the bob is at its lowest position. Furthermore, the vertical speed is zero at the ends and at the middle of the travel. Thus, for every one angular cycle of the pendulum there is one horizontal cycle of position and velocity and two vertical cycles of position and velocity. This pattern of motion causes the characteristic relationship of kinetic and potential energy in a pendulum with potential energy being highest when kinetic energy is lowest and vice versa. There is an exchange of kinetic and potential energy.

Consider as a contrast the relationships in a stably oscillating TS-IP. As in the case of the upright pendulum, there are twice as many vertical as horizontal cycles of speed and position and the number of horizontal cycles equals the number of angular cycles (Fig. 10). The horizontal speed is maximum as the bob passes the center of travel when the vertical position is highest. This causes the kinetic energy and potential energy to be in phase as is the case in running. Thus, the kinetic and potential energy pattern typical of running can be seen as a consequence of inverted pendulum motion over a single foot.

Fig. 10.

Kinematic data predicted by the TS-IP model (Eqn 25). Initial conditions rad, θ′(0)=0 rad s−1 and ω=1 rad s−1. The vertical component is cos θ(t) and the horizontal component is sin θ(t). (A) Actual pattern for vertical and horizontal speed for two steps of a sea star. The velocity data are from the same video as the energy data shown in Fig. 5A. (B–D) Predictions of patterns and phases; axes units are arbitrary. Compare the predicted pattern in B with data in A, predicted pattern in C with data in Fig. 7 and predicted pattern in D with data in Fig. 5. Note that in D, changes in kinetic energy and potential energy are exactly in phase, whereas in the sea stars changes in potential energy slightly precede changes in kinetic energy (Fig. 8B). For B and C, similar to Figs 7 and 8A, vertical speed precedes horizontal speed by π/2. Pattern of energies and speeds of the TS-IP model are similar to those observed in the sea star bouncy gait.

Fig. 10.

Kinematic data predicted by the TS-IP model (Eqn 25). Initial conditions rad, θ′(0)=0 rad s−1 and ω=1 rad s−1. The vertical component is cos θ(t) and the horizontal component is sin θ(t). (A) Actual pattern for vertical and horizontal speed for two steps of a sea star. The velocity data are from the same video as the energy data shown in Fig. 5A. (B–D) Predictions of patterns and phases; axes units are arbitrary. Compare the predicted pattern in B with data in A, predicted pattern in C with data in Fig. 7 and predicted pattern in D with data in Fig. 5. Note that in D, changes in kinetic energy and potential energy are exactly in phase, whereas in the sea stars changes in potential energy slightly precede changes in kinetic energy (Fig. 8B). For B and C, similar to Figs 7 and 8A, vertical speed precedes horizontal speed by π/2. Pattern of energies and speeds of the TS-IP model are similar to those observed in the sea star bouncy gait.

When comparing any pendulum model to a gait, there is a mismatch in that the pendulum goes back and forth, whereas the gait is a series of uni-directional swings over alternating legs. It is possible to mimic this pattern by considering the absolute value of the horizontal location and speeds. In that case the periodicity of the horizontal motion appears to double so that the vertical period is now equal to the horizontal period. In this case the phase shift between the horizontal and vertical speeds is π/2 as we observed in the sea star bouncy gait.

The patterns of motion shown by our TS-IP model mimic those of the sea star gait. Specifically, the horizontal and vertical speed patterns (Fig. 10A,B) have similar temporal patterns in each cycle, the hodograph (Fig. 10C) is similar to the hodographs in Fig. 8, and the kinetic and potential energies are in phase. The similarity between observed and predicted kinematic patterns shows that, despite its simplifying assumptions, the TS-IP model can replicate some kinematic patterns of the sea star bouncy gait.

Why does speed matter to sea stars?

It seems likely that the high-speed bouncy gait is energetically costly given the synchrony of peaks in kinetic and potential energies. Energy is the currency of evolution and the evolution of a high energy behavior may confer offsetting fitness benefits. Below we speculate on some ways the speed of the bouncy gait may do so.

In general, sea stars are not thought of as high energy creatures. They spend most of their time in sitting in a low energy state, not eating anything. However, sea stars must move to find and capture food and to escape predation. For example, we have observed that the sea star Luidia clathrata initiates the oscillatory gait when food is dropped into an aquarium with them. Perhaps these sea stars are competing to reach a prey resource first by moving the fastest. Similarly, A. forbesi sped up to 0.002±0.004 m s−1 (mean±s.e.m.) when cracked clams were placed in their aquarium (Moore and Lepper, 1997), which is similar to the speeds we observe for this species' oscillatory gait. Speed may not only help sea stars obtain food first, but also to escape becoming someone else's food. For example, faster speeds in the small, surface grazing sea star Parvulastra exigua allows it to flee from predation by the larger, slower sea star Meridiastra calcar, thereby enabling their co-existence in more resource rich areas of the intertidal (McLaren and Byrne, 2021). Similarly, the sea star Heliaster helianthus flees from predation by the sea star Meyenaster gelatinosus (Gaymer and Himmelman, 2008). Speed may generally factor into the ability of sea stars to capture not only mobile prey such as other sea stars, but also, for example, to capture other prey such as gastropod snails that escape sea star predation by fleeing (Vermeij, 2020).

We particularly thank Dr Tatsuo Motokawa, in whose lab at the Tokyo Institute of Technology, we first studied the sea star oscillatory gait in a different sea star, and Katie Guttenplan, who worked with us in Dr Motokawa's lab as a post-baccalaureate student from Bowdoin College. We also thank the undergraduate and high school students who helped with the collecting, wrangling, and filming of sea stars and tracking of videos: Rowan Etzel, Samantha Garvey, Do Yeun Kim, Madison Kuras, Nicole Morse, Salam Nassar, Kendra Novak, and Brendan Soane. In addition, we thank Robert Stevens for building the filming tanks, the staff at Bowdoin's Schiller Coastal Study Center for supporting our work at Bowdoin's marine lab, and Marko Melendy for animal care. We also thank two anonymous reviewers. This is contribution #8 from the Bowdoin Marine Laboratory at the Schiller Coastal Studies Center.

Author contributions

Conceptualization: O.E., A.S.J.; Methodology: O.E., A.S.J.; Software: O.E., M.K., A.S.J.; Validation: O.E., A.S.J.; Formal analysis: O.E., M.K., A.S.J.; Investigation: O.E., A.S.J.; Resources: O.E., A.S.J.; Data curation: O.E., M.K., A.S.J.; Writing - original draft: O.E., A.S.J.; Writing - review & editing: O.E., M.K., A.S.J.; Visualization: O.E., M.K., A.S.J.; Supervision: A.S.J.; Project administration: A.S.J.; Funding acquisition: A.S.J.

Funding

Funding for this study was provided by a Bowdoin College Andrew W. Mellon Award faculty leave supplement.

Data availability

Data are available in Dryad (Ellers et al., 2021): dryad.2547d7wrq

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Competing interests

The authors declare no competing or financial interests.

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