Intermittent locomotion composed of periods of active flapping/stroking followed by inactive gliding has been observed with species that inhabit both aerial and marine environments. However, studies on the energetic benefits of a fluke-and-glide (FG) gait during horizontal locomotion are limited for dolphins. This work presents a physics-based model of FG gait and an analysis of the associated cost of transport for bottlenose dolphins (Tursiops truncatus). New gliding drag coefficients for the model were estimated using measured data from free-swimming bottlenose dolphins. The data-driven approach used kinematic measurement from 84 h of biologging tag data collected from three animals to estimate the coefficients. A set of 532 qualifying gliding events were automatically extracted for estimation of the gliding drag coefficient. Next, data from 783 FG bouts were parameterized and used with the model-based dynamic analysis to investigate the cost benefits of FG gait. Experimental results indicate that FG gait was preferred at speeds of ∼2.2–2.7 m s−1. Observed FG bouts had an average duty factor of 0.45 and a gliding duration of 5 s. The average associated metabolic cost of transport (COT) and mechanical cost of transport (MECOT) of FG gait are 2.53 and 0.35 J m−1 kg−1, respectively, at the preferred speeds. This corresponded to a respective 18.9% and 27.1% reduction in cost when compared with model predictions of continuous fluking gait at the same average bout speed. Average thrust was positively correlated with fluking frequency and amplitude as animals accelerated during the FG bouts, whereas fluking frequency and amplitude were negatively correlated for a given thrust range. These results suggest that FG gait enhances the horizontal swimming efficiency of bottlenose dolphins and provides new insights into the gait dynamics of these animals.

Efficient locomotion is essential for animals traveling long distances, and various morphological or behavioral strategies have been observed to this end (Feldkamp, 1987; Au and Weihs, 1980; Hui, 1989; Fish and Hui, 1991; Weimerskirch et al., 2001; Williams et al., 1992; Weihs, 2004). Intermittent locomotion (Kramer and McLaughlin, 2001), where the animal employs a gait composed of periods of active flapping, stroking or bursting and then inactive gliding or coasting, is a convergent evolutionary trend (Gleiss et al., 2011) that has been observed in birds and marine animals (Tobalske, 2001; Rayner et al., 2001; Watanuki et al., 2005; Weihs, 1973; Fish et al., 1991; Ware et al., 2016; Müller et al., 2000; Gough et al., 2022; Cade et al., 2020). While the energetic benefits of intermittent locomotion have been studied for certain fishes from various perspectives (Weihs, 1973; Fish et al., 1991; Müller et al., 2000; Lighthill, 1971; Akoz and Moored, 2018; Weihs, 1974; Videler and Weihs, 1982; Ashraf et al., 2020), little work has been done with dolphins. Because dolphins are excellent swimmers and burst-and-coast swimming is not always energetically beneficial in fish (Ashraf et al., 2020), it is important to investigate the energetic trade-offs of intermittent locomotion. For cetaceans, published work has focused on how animals extend dive duration and conserve energy by taking advantage of buoyancy changes while gliding during descent or ascent in a dive (Williams, 2001; Williams et al., 2000; Skrovan et al., 1999). However, a model-based investigation of the energetic benefits of intermittent gait during horizontal locomotion, specifically a fluke-and-glide (FG) gait, is lacking for these animals. This work will investigate the following questions. (1) When is FG gait energetically beneficial? (2) Do animal speeds observed during FG gait have a reduced energetic cost when compared with continuous fluking? (3) Do the relationships between kinematic parameters (e.g. fluking frequency versus speed) observed during steady-state swimming hold for FG gait?

Here, we are interested in comparing the energetic cost between continuous and FG gait, along with the associated dynamical conditions during intermittent locomotion. Researchers have investigated the metabolic effort of dolphins during prescribed trials by estimating energetic cost using respirometry and blood lactate measurements. Previous trials have included boat following at a constant speed (Williams et al., 1992), static force generation (Williams et al., 1993), and point-to-point swimming at different speeds and levels of effort (Yazdi et al., 1999; Williams et al., 2017; van der Hoop et al., 2018, 2014). Gait during self-selected swimming spans a continuum that includes continuous and intermittent fluking. However, it is not easy to train the animals to swim with a prescribed gait pattern and experimental estimates of cost from respiratory or blood measurements can be limited because it is difficult to decouple the mixed effects of different motion modes, particularly when these modes are combined dynamically during the experimental trials. Thus, an explicit dynamic model is particularly useful for investigating a combined gait composed of multiple motion modes with varying ratios.

A model-based evaluation of energetic cost requires estimates of the hydrodynamic drag experienced by the animal during both fluking and gliding. Various methods have been used to obtain such estimations for marine animals, including theoretical computations (Lighthill, 1971; Akoz and Moored, 2018; Hertel, 1966), computational fluid dynamic simulations (Zhang et al., 2020; van der Hoop et al., 2018, 2014), towing experiments (Purves et al., 1975) and camera assisted experiments (Zhang et al., 2020; Noren et al., 2011; Feldkamp, 1987; Skrovan et al., 1999). Data from field biologging tag deployments have been used to derive speed and propulsive body acceleration during swimming and have been used to estimate the drag and energetic cost in Steller sea lions (Ware et al., 2016) and sperm whales (Miller et al., 2004). However, estimates of gliding drag over a range of speeds for bottlenose dolphins (Tursiops truncatus) are currently lacking.

In this work, we characterize the FG gait of bottlenose dolphins during self-selected swimming. We model the FG gait of bottlenose dolphins as a two-phase process and quantify the associated thrust force, the metabolic energy cost of transport (COT) and the mechanical energy cost of transport (MECOT) of the animal over a range of swimming gaits. Custom biologging tags were used to directly measure forward speed, 3-axis acceleration, orientation and depth of the animal during self-selected swimming. Gliding drag coefficients for the dolphins were experimentally estimated by fitting a drag model to speed measurements during periods of gliding (Lang, 1975; Bilo and Nachtigall, 1980; Stelle et al., 2000; McHenry and Lauder, 2005; Zhang et al., 2020). This data-driven approach leveraged information collected during self-selected swimming, resulting in estimates that were based on information from hundreds of swimming bouts. This contrasts with studies that use video data from a handful of bouts collected during structured trials to derive gliding drag coefficients (Zhang et al., 2020). Swimming gait was characterized using parameters calculated from the tag data, including bout duration, gliding duration, initial speed, maximum speed and minimum speed. These swimming characteristics informed the range of input parameters for the FG gait model used to evaluate the COT and MECOT. Finally, fluking frequency and amplitude during the accelerating fluking phase of the FG gait were extracted from tag data to investigate swimming biomechanics. New information about gliding hydrodynamics, fluking gait parameters, and the model-based estimates of COT and MECOT during horizontal intermittent locomotion provide further insight into the swimming biomechanics of bottlenose dolphins.

Data collection

Three bottlenose dolphins [Tursiops truncates (Montagu 1821)] (TT01, TT02, TT03), from the Brookfield Zoo, Chicago Zoological Society, participated in this study (age: 16, 5 and 38 years; mass: 153, 150 and 202 kg; body length: 242, 226 and 266 cm; girth: 128, 127 and 139 cm, respectively). A biologging tag was attached to each animal 20 cm behind the blowhole via four silicone suction cups (Fig. 1A) by an animal care specialist. The ESTAR Lab (University of Michigan) designed and built the biologging tag with the tag electronics (OpenTag, Loggerhead Instruments, FL, USA) encapsulated inside a 3D-printed tag housing. Sensors onboard include a 3-axis accelerometer, gyroscope, magnetometer (50 Hz), pressure sensor, temperature sensor and speed sensor (10 Hz). The speed sensor measures relative fluid speed by counting the revolving frequency of a microturbine on top of the tag, which is then mapped to the animal's forward speed in the water. A detailed presentation of the sensor calibration and verification with swimming dolphins is presented in Gabaldon et al. (2019). While fluctuations in the speed measurements are created by the pitching of the animal during fluking, Gabaldon et al. (2019) demonstrated good agreement between tag sensor measurements and speed derived from camera data during fluking. Additionally, the animal holds a relatively rigid pose during a glide, reducing pitching-related disturbance in the speed estimates during gliding events. Software from a custom gait analysis toolbox for tag data (Zhang et al., 2022) was used to calculate the orientation and important gait parameters. The animals swam freely in the main aquarium habitat [33.5 m (length)×12.2 m (width)×6.7 m (depth)] during the deployments. The tags were deployed for an average of 2 h at a time (84 h total).

Fig. 1.

Experimental environment and concepts. (A) A bottlenose dolphin with a biologging tag in situ. (B) Simplified free-body diagrams of an animal fluking (left) and then gliding (right). (C) Simplified speed model of FG gait and continuous fluking gait over two cycles. Ax, x-axis of accelerometer; Az, z-axis of accelerometer; v, speed; θpitch, pitch angle; vmax, maximum speed of the bout; vref, reference speed; vmin, minimum glide speed; vmeasure, tag measured speed of the animal; vint, initial speed; Df, duty factor; Tf, fluking duration; Tg, gliding duration.

Fig. 1.

Experimental environment and concepts. (A) A bottlenose dolphin with a biologging tag in situ. (B) Simplified free-body diagrams of an animal fluking (left) and then gliding (right). (C) Simplified speed model of FG gait and continuous fluking gait over two cycles. Ax, x-axis of accelerometer; Az, z-axis of accelerometer; v, speed; θpitch, pitch angle; vmax, maximum speed of the bout; vref, reference speed; vmin, minimum glide speed; vmeasure, tag measured speed of the animal; vint, initial speed; Df, duty factor; Tf, fluking duration; Tg, gliding duration.

Glide extraction and FG bout identification

Similarly to the flapping and gliding gaits reported for birds (Sachs, 2015), dolphins have been observed swimming with FG gait patterns. During this gait, dolphins use a series of fluke strokes to first accelerate through the water. The fluking phase (with duration Tf) is then followed by a period of gliding (Tg), where the animal assumes a hydrodynamic pose to reduce drag and maintain speed through the water (Fig. 1C). Bouts of FG gait were identified manually in the tag data using features from the calibrated speed, accelerometer, gyroscope and magnetometer data, and the estimated orientation of the animal. Orientation (pitch, roll and yaw) was calculated from sensor data using a gradient-descent-based optimization method in the quaternion space (Madgwick et al., 2011). The start of an event was identified using the fluking signature present in the pitching motion of the animal. The measured speed was identified at the start of the event (vint), and the endpoint of the glide was selected to equal the segment's starting speed (vmin= vint with a difference <0.1 m s−1). The speed measurement at the transition between fluking and gliding was identified as vmax. Fluking duration Tf, gliding duration Tg, duty factor Df = Tf /(Tf + Tg), reference speed vref = (vmin+ vmax)/2, as well as the fluking frequency and amplitude during fluking, were subsequently parameterized from the identified FG bouts. Fluking amplitude for a stroke cycle was defined as the difference between the maximum and minimum tag-measured animal pitch (i.e. peak-to-peak relative pitch difference) and the average fluking amplitude for a bout of stroke cycles was reported. The software tools used for the gait characterization (fluking frequency and amplitude) were described by Zhang et al. (2022). Speed data from the identified FG bouts is presented in Fig. 2. All bouts were aligned using the transition time between fluking and gliding (grey lines) and then averaged (black line).

Fig. 2.

Speed data from the identified FG bouts of three bottlenose dolphins. (A) Fluke-and-glide (FG) bouts (light grey) aligned at the transition time from fluking to gliding (n=783 bouts). The mean (black) and s.d. range (shaded) of the speed were computed at each time point. The initial speed (vint), maximum speed of the bout (vmax) and minimum glide speed (vmin) are identified. (B) The multi-state model (Eqn 11) was fitted to the computed mean speed.

Fig. 2.

Speed data from the identified FG bouts of three bottlenose dolphins. (A) Fluke-and-glide (FG) bouts (light grey) aligned at the transition time from fluking to gliding (n=783 bouts). The mean (black) and s.d. range (shaded) of the speed were computed at each time point. The initial speed (vint), maximum speed of the bout (vmax) and minimum glide speed (vmin) are identified. (B) The multi-state model (Eqn 11) was fitted to the computed mean speed.

A subset of speed data was selected for the experimental estimation of drag coefficients during gliding. When the animals were moving (forward speed >1.0 m s−1), three criteria were empirically used to identify gliding events for analysis. First, the glide had to occur at a depth >2.5 body diameters to avoid surface drag (Hertel, 1966). For all animals, a depth >3.5 m satisfied this condition. Second, the animal was not actively swimming or changing direction during the gliding event. The postural configuration and the use of control surfaces (fins and flukes) for maneuvering would affect drag acting on the body. The 1-norm of the jerk (i.e. the derivative of the 3-axis measurements from the accelerometer, measured in the unit of the standard gravitational acceleration for Earth: g=9.81 m s−2) ||jerk||1 was required to be less than 0.5 g s−1 to exclude active swimming. Restricted angular rates (pitch rate <20 deg s−1; yaw rate <22 deg s−1) were also used to exclude active turning. Third, to mitigate the effects of gravity and buoyancy, gliding events for analysis were restricted to constant depth (|depth rate|<0.2 m s−1 and |pitch|<17 deg). When the animal was moving at a constant depth, it was assumed that buoyancy and gravitational forces were equal and opposite and did not affect the animal's forward motion. A decision tree was employed to detect gliding points in the data that satisfied the specified conditions: (1) forward speed >1.0 m s−1; (2) depth >3.5 m; (3) |pitch rate|<20 deg s−1; (4) |yaw rate|<22 deg s−1; (5) ||jerk||1<0.5 g s−1; (6) |depth rate|<0.2 m s−1; (7) |pitch angle|<17 deg. Temporally adjacent gliding points were grouped to form a gliding event. Speed measurements from a gliding event were then used with Eqn 4, defined in the next section, to estimate the gliding drag coefficient, .

Dynamics of the FG gait

The thrust force during fluking at time instance t can be related to the hydrodynamic drag force Fdrag(t) and animal's acceleration a(t) by:
(1)
where mb=1.05m is the induced virtual mass of the animal that accounted for the animal's body mass (m) and the fluid carried along (Lang, 1975; Vogel, 1996). In this work, we also assume that the animal is neutrally buoyant and swimming straight (Fig. 1B). The drag force Fdrag(t) opposes the direction of motion and is modeled in a conventional way by:
(2)
where Cd is the dimensionless drag coefficient for the animal, Aw=0.08m0.65 is the wetted surface area of the animal (Fish, 1993), ρ is the density of seawater, and v(t) is the forward speed with respect to water at t. The drag coefficient during fluking is denoted as and during gliding as . These two parameters are critical for the estimation of drag during FG gait. Fish et al. (2014) estimated by analyzing thrust produced by swimming bottlenose dolphins using digital particle image velocimetry (DPIV) during small amplitude swimming. In this work, we use the coefficient () estimated by Fish et al. (2014) to calculate drag force when the animal is fluking.
During a glide, there is no thrust force []. Combining Eqns 1 and 2, we obtain the differential equation that describes how speed v(t) changes during a glide as:
(3)
which can be used to solve for speed v(t) as a function of time t and initial speed v0:
(4)
Eqn 4 is fitted to measured speed data during periods of gliding to identify the drag coefficient , as in Bilo and Nachtigall (1980), Lang (1975), Zhang et al. (2020). Here, we use the dual parameter sweep fitting approach, presented by Zhang et al. (2020), with the subset of the gliding speed data described in the previous section. In this approach, drag coefficients are estimated using data collected during glides that occur within a bout of FG gait and have small overall changes in speed.
Drag coefficients estimated using data at different speeds were modeled as a function of Reynolds number (Re) by Cd = bRec, where b and c are the associated coefficients (Fish et al., 2014). When taking the logarithm on both sides of the equation, the logarithmic terms of Cd and Re are related linearly: log (Cd) = c log Re + log b. The Reynolds number for each gliding event was calculated as:
(5)
where L is the body length of the animal, and ν is the kinematic viscosity of seawater. These experimentally derived estimates of were then used in our analysis when the animal was gliding.

FG gait kinematics

Measured speed for all identified periods of FG gait are presented in Fig. 2A. These data were aligned using the transition time between fluking and gliding and then averaged (black line). Measured speed during an individual bout has fluctuation created by the pitching motion of the animal, but the averaged data show a linear increase in speed during fluking that is followed by a period of deceleration during the glide. To address the fluctuation in the speed measurements, swimming kinematics were represented using a multi-state approach.

The speed profile was modeled as a period of constant acceleration during fluking, followed by deceleration during gliding (Eqn 3). Key parameters for the multi-state approach were identified from the measured speed data during bouts of FG gait: the initial speed vint, the maximum speed vmax, the minimum speed during the glide vmin, the fluking duration Tf, the gliding duration Tg, and a duty factor Df. The maximum and minimum speeds were used to define a reference speed for the overall movement bout calculated as:
(6)
Duty factor was used to relate Tg and Tf via:
(7)
The fluking duration Tf and estimated acceleration during fluking af were calculated as:
(8)
(9)
Next, the maximum and minimum speeds during the glide (i.e. vmax and vmin), along with the experimentally determined gliding drag coefficient determined in this work, were used with Eqn 4 to model the speed of the animal during the glide:
(10)
The combined speed profile of an FG gait cycle is then represented as:
(11)
with the two lines corresponding to the speed profile during fluking and gliding, respectively.

Cost of transport

Metabolic cost of transport (denoted as COT; Schmidt-Nielsen, 1972) characterizes the amount of metabolic or chemical potential energy (E, in joules) required to transport one unit of body mass (total m, in kilograms) over one unit of distance (total S, in meters):
(12)
The metabolic cost (E) can be estimated via measured oxygen consumption, heart rate, and/or plasma lactate concentration during exercise (Williams et al., 1992, 1993, 2017; Yazdi et al., 1999). In this work, we estimated E by combining estimated resting metabolic energy (Eres) with an estimate of the mechanical work (W) produced by the animal during fluking, similarly to Gabaldon et al. (2022):
(13)
where ηmp=0.85 is the estimated muscle-to-propulsion conversion efficiency for the bottlenose dolphin (Fish, 1998) and ηmm=0.25 is the estimated mammalian metabolic-to-muscle efficiency (Faraji et al., 2018). Eres was obtained by integrating an estimated resting metabolic rate (Pres) over time (van der Hoop et al., 2014):
(14)
For a bout of FG gait (Fig. 1C), we calculated the traveled distance S and estimated mechanical work W via:
(15)
(16)
(17)
where Fthrust(t) was obtained from Eqn 1 and v(t) is defined using Eqn 11. Pthrust(t) represents the estimated thrust power during an FG bout.
In addition to the metabolic cost of transport (COT), we also used mechanical energy cost of transport (MECOT) to quantify the minimum amount of mechanical energy (in joules) required to transport one unit of body weight (1 kg) over one unit of distance (1 m):
(18)

Unlike COT, MECOT only includes animal dynamics. This simplified metric was used to compare gait across individuals. This parameter could also be used for comparisons with other mechanical systems, such as a bioinspired robotic dolphin (Wu et al., 2019). The swimming kinematics, COT and MECOT were used to investigate how swimming parameters are related to cost during FG gait. Constant speed continuous fluking gait [i.e. v(t)=vref, Tg =0, Df =1] was investigated using the model for comparison.

Speed distribution and normalization

To investigate speed distribution during locomotion, we first normalize the measured speed (in m s−1) by the body length of the animal to obtain a normalized speed (in l s−1, i.e. body length per second). Now, let Pall (v) be the probability that the animal's speed is within the left-closed interval [v–0.05, v+0.05) l s−1 during all locomotion, then:
(19)
where Call (v) is the counted number of occurrences for speed that falls within the interval associated with v during all locomotion and counts the total number of occurrences for any speed during all locomotion. Similarly, we define the probability for speed during FG gait:
(20)
where Cfg (v) is the counted number of occurrences for speed to fall within the interval of v during all FG bouts and counts the total number of occurrences for any speed during all FG bouts. The normalized count for an FG gait bout is then defined as:
(21)
which represents the animal's relative preference for selecting an FG gait when at speed v. And the associated probability for the normalized FG speed is:
(22)
where ensures Pfg/all (v) sums to 1 over all speed intervals. Pfg/all (v) is a normalized ratio between speed distributions of FG bouts and all locomotion, making it an indicator of the animal's tendency to select an FG gait at speed v.

Statistical methods

A linear mixed-effects model (LMEM; Robinson, 1991; Mclean et al., 1991; Pinheiro and Bates, 1996; fitlme in MATLAB) was employed to test the relationship between drag coefficient and Reynolds number Re, under log scale, for the results from the three dolphins during gliding. Specifically, was the response variable, Re was the predictor variable, and animal identity was the grouping variable. The fixed-effects portion of the model corresponded to the slope and intercept of the global linear relationship between and Re under log scale for all animals. Random effects were associated with the (potentially correlated) slope and intercept of each animal in addition to the global trend. In total, there were two parameters for the fixed effects and six parameters for the random effects assessed in the model. The t-statistic and P-value of these parameters were evaluated against a hypothesized value of 0.

The Pearson correlation coefficients (R-value; Fisher, 1958; Kendall, 1979; corrcoef in MATLAB) together with associated P-values were computed to test for correlations between model estimated thrust and tag-measured fluking frequency and amplitude. A plane was fitted to these three variables using robust linear regression to map from measured fluking frequency and amplitude to the estimated thrust (Fisher, 1958; Huber, 1981; robustfit in MATLAB). The P-values and standard errors of the plane's coefficients were assessed simultaneously. The correlation coefficient between plane-predicted thrust and thrust from data points was evaluated to characterize the plane's effectiveness in the mapping from fluking frequency and amplitude to the thrust. Robust linear regression was also used to obtain a pair-wise relationship among thrust, fluking frequency and fluking amplitude.

FG bouts

The analysis identified 783 FG bouts in the tag data from the three dolphins. These bouts had an average fluking duration Tf of 4.3±2.1 s, gliding duration Tg of 5.0±2.0 s and duty factor, , of 0.45±0.11. Fig. 2 presents the speed data from the identified FG bouts and a comparison between the measured data and the modeled speed (Eqn 11). In the figure, the 783 individual FG bouts (light grey lines) were aligned using the transition from fluking to gliding identified using features in the data streams (dashed vertical line). Data were averaged at each time point before and after the transition event (solid black line). The standard deviation (shaded region) along with the individual bouts (dark grey lines), are presented with the average. Speed estimated using Eqn 11 compares very well to the average values presented in the figure with an overall RMS error of 0.02 m s−1.

The speed distributions of the animals while swimming (speed >0.3 m s−1) are shown in Fig. 3. Speed distributions varied between dolphins, with an overall swimming speed of ∼0.45 l s−1 being most likely during all swimming bouts for all animals. FG gait was more likely to occur at slightly higher speeds (0.6–0.9 l s−1). The normalized ratio between FG speed distribution and the general speed distribution (Eqn 22) indicates that FG bouts were more likely to be used by the animals at ∼0.9–1.1 l s−1 (2.2–2.7 m s−1 for a body length of 2.45 m). This is a value similar to the previously reported most-efficient bottlenose dolphin swimming speed of 2.1 m s−1 (Williams et al., 1992).

Fig. 3.

Normalized speed distributions of studied animals. (A–C) Normalized speed (unit in body length per second) distribution of all locomotion [Pall(v)], sampled FG bouts [Pfg(v)] and the normalized ratio between FG bouts and all locomotion [Pfg/all(v)] (see Eqn 22) from 3 individuals (TT01, TT02, TT03). (D) Normalized speed distribution of Pall(v), Pfg(v) and Pfg/all(v) from all individuals combined.

Fig. 3.

Normalized speed distributions of studied animals. (A–C) Normalized speed (unit in body length per second) distribution of all locomotion [Pall(v)], sampled FG bouts [Pfg(v)] and the normalized ratio between FG bouts and all locomotion [Pfg/all(v)] (see Eqn 22) from 3 individuals (TT01, TT02, TT03). (D) Normalized speed distribution of Pall(v), Pfg(v) and Pfg/all(v) from all individuals combined.

Gliding drag coefficient

Data from a representative dive, along with the identified gliding events, are shown in Fig. 4A–D. Eqn 10 was used with the gliding speed profiles (Fig. 4E) to calculate the gliding drag coefficient . A total of 532 gliding segments with an average duration of 1.97±1.09 s (mean±s.d.) were identified from the animals for the analysis. The 532 pairs are shown in Fig. 5. An LMEM was fitted to the values from all animals to identify the global trend as well as assess potential individual differences. The slope and intercept of the fixed effects (the global trend) were obtained as −0.6621 (t-statistic=−8.0, P<0.001) and 2.2106 (t-statistic=4.0, P<0.001), yielding the following relationship between Re and (R=0.331, P<0.001):
(23)
which can be rearranged to give:
(24)
Fig. 4.

Examples of the data and analysis used to estimate drag coefficients during gliding. Features from tag data (depth and speed; A,D,E) and signals derived from the data (pitch, yaw and jerk; B,C) used to identify periods of gliding. Jerk (C) represents the derivative of the 3-axis measurements from the accelerometer, with the axes x, y and z corresponding to the surge, sway and heave directions of the animal, respectively. Three gliding events (labeled G1, G2 and G3, respectively) were identified from the data in A,B,C. Speed data from these gliding events (D) and the proposed dynamic model were used to identify a drag coefficient for each event (E).

Fig. 4.

Examples of the data and analysis used to estimate drag coefficients during gliding. Features from tag data (depth and speed; A,D,E) and signals derived from the data (pitch, yaw and jerk; B,C) used to identify periods of gliding. Jerk (C) represents the derivative of the 3-axis measurements from the accelerometer, with the axes x, y and z corresponding to the surge, sway and heave directions of the animal, respectively. Three gliding events (labeled G1, G2 and G3, respectively) were identified from the data in A,B,C. Speed data from these gliding events (D) and the proposed dynamic model were used to identify a drag coefficient for each event (E).

Fig. 5.

A linear mixed-effects model (LMEM) fitted to the 532 pairs of gliding Reynolds number-drag coefficient {Re, Cd} values from all three dolphins. (A) LMEM fitted to 532 pairs of gliding {Re, Cd} values from all animals (TT01, TT02, TT03) under log scale. The LMEM fit estimates a lower drag coefficient during the glide than drag estimates calculated during low-amplitude dolphin fluking (Fish et al., 2014). (B) The LMEM estimate compares well to drag coefficients estimated using CFD simulations (Cd glide sim.) using dolphin models in static poses, as well as coefficients calculated from experimental data (Cd glide) (Zhang et al., 2020; Gutarra et al., 2019).

Fig. 5.

A linear mixed-effects model (LMEM) fitted to the 532 pairs of gliding Reynolds number-drag coefficient {Re, Cd} values from all three dolphins. (A) LMEM fitted to 532 pairs of gliding {Re, Cd} values from all animals (TT01, TT02, TT03) under log scale. The LMEM fit estimates a lower drag coefficient during the glide than drag estimates calculated during low-amplitude dolphin fluking (Fish et al., 2014). (B) The LMEM estimate compares well to drag coefficients estimated using CFD simulations (Cd glide sim.) using dolphin models in static poses, as well as coefficients calculated from experimental data (Cd glide) (Zhang et al., 2020; Gutarra et al., 2019).

The random effects between Re and caused by individual differences were insignificant, with parameters for the random effects that accounted for the individual differences in the LMEM all smaller than 0.001 (absolute value) and the absolute values of their 95% confidence intervals all <0.001.

This relationship was used to estimate the gliding drag coefficient for the modeling analysis. Fig. 5A presents the estimated coefficients from the 532 gliding events, the LMEM fit to those estimates, and the from Fish et al. (2014) (). The estimated drag coefficients during a glide were lower than the fluking drag coefficients across the range of observed swimming speeds. Fig. 5B presents a comparison between the gliding drag coefficients estimated here with the average experimental estimate from Zhang et al. (2020) and the estimated from computational fluid dynamic (CFD) simulations by Zhang et al. (2020) and Gutarra et al. (2019) using rigid dolphin models.

Cost of transport

The COT (Eqn 12) and MECOT (Eqn 18) values associated with the 783 FG bouts were estimated and plotted against the corresponding speed in Fig. 6 and color-coded by duty factor (Eqn 7). The estimated COT and MECOT of constant speed continuous fluking gait are also shown in the figures. The COT versus speed relationship demonstrated a decreasing-then-increasing trend, while the MECOT versus speed relationship presented a monotonically increasing trend. For a fixed speed, both COT and MECOT predicted a higher cost with increasing duty factor, where constant speed continuous fluking has the highest cost. The cost difference between continuous fluking and FG gait increased with speed. At a set of representative parameters, {vref =1 l s−1; Tg= 5 s; Df =0.45}, the COT and MECOT of an FG bout were 2.53 and 0.35 J m−1 kg−1, respectively, corresponding to 18.9% and 27.1% reductions compared with the continuous fluking gait at the same average speed.

Fig. 6.

Predicted metabolic energy cost of transport (COT) and mechanical energy cost of transport (MECOT) of FG bouts. Predicted (A) COT and (B) MECOT of the 783 identified FG bouts from 3 animals plotted over the associated speeds with color codes for their duty factors. COT and MECOT of constant speed continuous fluking gait, with an effective duty factor of 1, are compared against the FG bouts. The curves of an FG gait with a duty factor equal to 0.45 and a gliding duration equal to 5 s are also shown.

Fig. 6.

Predicted metabolic energy cost of transport (COT) and mechanical energy cost of transport (MECOT) of FG bouts. Predicted (A) COT and (B) MECOT of the 783 identified FG bouts from 3 animals plotted over the associated speeds with color codes for their duty factors. COT and MECOT of constant speed continuous fluking gait, with an effective duty factor of 1, are compared against the FG bouts. The curves of an FG gait with a duty factor equal to 0.45 and a gliding duration equal to 5 s are also shown.

Fluking dynamics

The measured relative pitch and speed, together with the estimated thrust and power of an example FG bout is presented in Fig. 7. The average model estimated thrust forces (Fthrust, Eqn 1) during the fluking phase were normalized against the animal's body weight (i.e. Fthrust/mg; where g=9.8 m s−2), color-coded by duty factor and plotted against speed in Fig. 8. In this figure, the normalized thrust is positively correlated with speed (R=0.895, P<0.001). A gait with a higher duty factor resulted in a lower average thrust during the fluking phase at a given reference speed. The normalized thrust ranged from 0.01 to 0.15 in the identified FG bouts for these animals. Additionally, the relationships between normalized thrust and the measured fluking frequency (f, in Hertz) and amplitude (α, in degree) are presented in Fig. 9. Significant positive correlations existed between thrust and fluking frequency (R=0.533, P<0.001):
(25)
as well as between thrust and fluking amplitude (R=0.445, P<0.001):
(26)
Fig. 7.

The relative pitch, speed and model estimated thrust force and power of an example FG bout. (A) Relative pitch (i.e. zero-centered pitch). (B) Speed. (C) Model estimated thrust force. (D) Model estimated power of an example FG bout. The reference speed of the bout is 0.91 l s−1, with fluking duration of 4.9 s, gliding duration of 4.5 s and duty factor 0.52. The average fluking frequency and amplitude during the fluking phase are 1.0 Hz and 27.0 deg, respectively. vmax, maximum speed of the bout; vref, reference speed.

Fig. 7.

The relative pitch, speed and model estimated thrust force and power of an example FG bout. (A) Relative pitch (i.e. zero-centered pitch). (B) Speed. (C) Model estimated thrust force. (D) Model estimated power of an example FG bout. The reference speed of the bout is 0.91 l s−1, with fluking duration of 4.9 s, gliding duration of 4.5 s and duty factor 0.52. The average fluking frequency and amplitude during the fluking phase are 1.0 Hz and 27.0 deg, respectively. vmax, maximum speed of the bout; vref, reference speed.

Fig. 8.

Predicted thrust during fluking for FG bouts. The predicted thrust is normalized to body weight and is plotted over the associated speeds with color codes for duty factor for 3 animals (TT01, TT02, TT03). The curve of an FG gait with a duty factor equal to 0.45 and a gliding duration equal to 5 s is shown, along with the constant speed continuous fluking gait.

Fig. 8.

Predicted thrust during fluking for FG bouts. The predicted thrust is normalized to body weight and is plotted over the associated speeds with color codes for duty factor for 3 animals (TT01, TT02, TT03). The curve of an FG gait with a duty factor equal to 0.45 and a gliding duration equal to 5 s is shown, along with the constant speed continuous fluking gait.

Fig. 9.

The predicted thrust during fluking for FG bouts plotted against the measured fluking frequency and peak-to-peak amplitude. (A) A plane was robustly fitted to the data points from all 3 animals (TT01, TT02, TT03) in 3D. The solid line with a gradient color is the result of the 3D plane viewed from the side. The 2D views of the data are also shown for thrust versus fluking frequency (B) and thrust versus fluking peak-to-peak amplitude (C). The purple and magenta lines represent the robust line fit to the corresponding 2D data.

Fig. 9.

The predicted thrust during fluking for FG bouts plotted against the measured fluking frequency and peak-to-peak amplitude. (A) A plane was robustly fitted to the data points from all 3 animals (TT01, TT02, TT03) in 3D. The solid line with a gradient color is the result of the 3D plane viewed from the side. The 2D views of the data are also shown for thrust versus fluking frequency (B) and thrust versus fluking peak-to-peak amplitude (C). The purple and magenta lines represent the robust line fit to the corresponding 2D data.

The robust linear regression fit between thrust, fluking frequency (f), and fluking amplitude (α) was determined to be:
(27)
with P<0.001 for all the coefficients and standard errors of 0.002, 0.0001 and 0.003, respectively, for the three coefficients. The intercept coefficient in the above relationships is dimensionless while the slope coefficient has units that are the reciprocal of the associated quantity (i.e. Hertz−1 for the slope coefficient associated with f and deg−1 for the coefficient associated with α). The root mean square error (RMSE) of the fit from the robust linear regression was 0.016 (unitless). A comparison between the estimated thrust and the thrust predicted by the regression plane in a Pearson correlation test gave an R-value of 0.682 with P<0.001, which was a tighter correlation than either the thrust-frequency relationship (R=0.533, P<0.001) or the thrust–amplitude relationship (R=0.445, P<0.001), indicating that the plane fit was better than using either frequency or amplitude alone.

For the relationship between fluking frequency and amplitude during the fluking phase of the 783 FG bouts (Fig. 10A), no significant correlation was detected between the two parameters over the entire set of data points (R=0.026, P=0.460). However, correlations emerged when a narrower set of points were analyzed. The complete set of points was split into six equal-sized bins (131 points each) based on their normalized thrust so that points in the same bin share similar thrust magnitude. Statistical results showed that five out of the six bins presented a significant negative correlation between fluking frequency and amplitude (Fig. 10B–G). The negative slope of the linear relationship became steeper as the thrust increased. And the strongest correlation occurred in the bin associated with the highest thrust range.

Fig. 10.

Relationship between fluking amplitude and frequency during FG bouts. During the fluking phase of the 783 FG bouts from the 3 animals (A, color-coded in accordance with the bottom six subplots B–G), the fluking frequency and amplitude were gathered into six equal-sized bins (131 points each, subplots B–G) based on their estimated thrust (normalized over body weight). The thrust range, together with the R-value and P-value for a Pearson correlation test, are indicated above each subplot. Five out of the six bins presented a significant negative correlation (α=0.01) between fluking frequency and amplitude. A robust linear regression line was fitted to the data from the bin that presents a significant correlation (C–G). The equation associated with the regression line is shown in each subplot.

Fig. 10.

Relationship between fluking amplitude and frequency during FG bouts. During the fluking phase of the 783 FG bouts from the 3 animals (A, color-coded in accordance with the bottom six subplots B–G), the fluking frequency and amplitude were gathered into six equal-sized bins (131 points each, subplots B–G) based on their estimated thrust (normalized over body weight). The thrust range, together with the R-value and P-value for a Pearson correlation test, are indicated above each subplot. Five out of the six bins presented a significant negative correlation (α=0.01) between fluking frequency and amplitude. A robust linear regression line was fitted to the data from the bin that presents a significant correlation (C–G). The equation associated with the regression line is shown in each subplot.

This work provides insight into the dynamics and energetics of bottlenose dolphins during FG gait, along with new estimates of gliding drag coefficients over a range of swimming speeds. As expected, drag during gliding (Fig. 5) was lower than during low amplitude fluking (Fish et al., 2014). The dynamic dorsal–ventral bending of the body used to generate propulsive force for locomotion results in body configurations that create more drag than the static pose used during a glide. The Bone–Lighthill boundary layer thinning hypothesis also suggests that undulating motion increases skin friction drag (Lighthill, 1971; Akoz and Moored, 2018). The reduction in drag is significant across the range of speeds. For example, animal TT01 experienced 104% more drag during low amplitude fluking than it did during a glide at 2.25 m s−1. This significant reduction contributes to the energetic benefits of FG gait observed in the swimming patterns of whales and dolphins (Shorter et al., 2017).

The energetic benefits of FG gait were then further investigated using the proposed model and parameters identified from observed bouts of FG gait. The parabolic trend observed in the metabolic energy cost of transport (COT; Fig. 6A) was the result of two factors. First, at lower speeds (<0.5 l s−1), the animal's resting/baseline metabolic resulted in higher COT since the animal still consumes metabolic energy (the numerator E in Eqn 12) during small displacements (the denominator S in Eqn 12). Second, the quadratic relationship between drag force and speed (Eqn 2) creates significantly larger COT at higher speed as the animal has to increase thrust production to overcome the drag. While COT is an estimate of the overall metabolic energy cost of the animal during locomotion, the mechanical energy cost of transport (MECOT) describes only the amount of mechanical energy required for locomotion. The monotonic trend in the MECOT versus speed relationship (Fig. 6B) is mainly a result of the quadratic relationship between drag force and speed. MECOT better captures the interaction between the animal, as one dynamical system, and the environment.

Despite the differences between COT and MECOT, both metrics indicate that an FG gait is more efficient than a corresponding constant speed continuous fluking gait. Results from the model at a 1.0 l s−1 swimming speed with a representative set of gait parameters showed an 18.9% reduction in the COT and a 27.1% reduction in the MECOT compared with constant speed fluking. An 18.9% saving for bottlenose dolphins is smaller than the >50% estimated savings for a fish (Weihs, 1974). Still, for dolphins that may travel more than 50 km per day (Mate et al., 1995), an 18.9% cost saving would result in a significant reduction in the foraging effort required to maintain the animal's energy budget. Furthermore, the energetic benefits of FG gait increased as swimming speed increases (Fig. 6), supporting the observation that the animals may preferentially select an FG gait at higher speeds (Fig. 3).

Additional results in Fig. 6 demonstrate that COT and MECOT of FG gait are lower when the duty factor is lower. That is, swimming efficiency improves when the animal spends more time gliding and less time fluking. While a small duty factor with as much gliding as possible leads to improved COT and MECOT, a low duty factor also requires higher acceleration, thus higher thrust force (Fig. 8), during the limited amount of fluking to build up speed for the glide. A higher thrust requirement leads to two caveats. First, the metabolic cost to support the burst could be higher than estimated. And second, the animal may have to increase its fluking amplitude to satisfy the dynamic demand of higher thrust.

To further investigate the dynamical properties during the fluking phase of the FG bouts, the relationships between average thrust force, fluking frequency and fluking amplitude were investigated (Fig. 9, Eqn 27). It was observed that thrust is positively correlated with fluking frequency and amplitude. Previous studies (Fish, 1998; Rohr and Fish, 2004) have investigated the relationship between the steady-state swimming speed of cetaceans and their fluking frequency and amplitude, and found a positive correlation between swimming speed and fluking frequency, yet an insignificant correlation between speed and amplitude. The data presented here build on these previous observations of gait kinematics during steady-state swimming (i.e. near-constant speed) by including transient state dynamics as the animals accelerate before a glide. A negative correlation was found between fluking frequency and amplitude within a given range of thrust force (Fig. 10), which expands the observation that frequency and amplitude were not correlated (Rohr and Fish, 2004) when all data were shown together.

These results indicate that the animals may modulate both frequency and amplitude to generate a given thrust force, with a higher average thrust force for acceleration requiring increased fluking frequency and amplitude. Increased force development rates would likely lead to extra metabolic costs in dolphins, as described in human biomechanics (van der Zee and Kuo, 2021), owing to the additional active calcium transport in muscle tissues. Furthermore, it has been shown that maintaining a higher force (without doing any mechanical work) also costs more metabolic energy in dolphins (Williams et al., 1993). Suggesting that using a greater muscle force induces additional metabolic costs. As such, the energy conversion ratios, ηmp and ηmm in Eqn 13, should be investigated in future work to improve the physics-based metabolic energy estimates.

Another consideration regarding the model is that the drag coefficient used during the fluking phase of gait () comes from estimates made by Fish et al. (2014) during low amplitude dolphin fluking. As such, we assume that the fluking used to accelerate the animal was low amplitude, but we did not measure fluking amplitude directly. The drag coefficient associated with larger amplitude fluking would likely be higher, making the actual mechanical work larger than estimated here if large amplitude motion was used to generate thrust for acceleration. Further investigation could be conducted into this matter too.

The benefits of the physics-based data-driven approach in this work were threefold: first, there was no need for dedicated experiments that involved divers or underwater cameras for drag estimation. Second, data were collected during the animal's daily routine, linking the theoretical model with the animals' self-selected behavior. Third, the many hours of tag data resulted in a large number of events for the analysis, helping to reduce the noise and uncertainty in the estimated parameters. While expert knowledge was used to select features from the sensor data to create a decision tree classifier and to identify the FG bouts from data, other automated machine learning algorithms could be used to identify events of interest in the data (Kabra et al., 2013; Zhang et al., 2018; Sibal et al., 2021; Zhang, 2021). These automated approaches could further reduce the expert knowledge required for signal feature selection and event detection for potentially more complicated movement patterns. However, they would still need a labeled data set to train the algorithms. As an extension of this work, the on-body tag measurements could be combined with a model of the animal's full-body kinematics to provide thrust and drag estimates under different scenarios, providing more detailed insights about the locomotion of these animals. Future work should also use tag data from dolphins in the wild to investigate the cost of transport associated with swimming gaits used in a different environmental context.

In addition, video data (e.g. Gough et al., 2022; Segre et al., 2022) could be collected to support the data-driven methods presented in this work. More specifically, the variance in the {Re, Cd} values observed in Fig. 5 could be a result of the animals using flippers or flukes to adjust trim during a glide or undetected motions of the head or tail. As shown in Fig. 5, the CFD simulations of different rigid animal models (Zhang et al., 2020; Gutarra et al., 2019) are comparable to the LMEM fit to the experimental data. As such, we believe that collecting data from hundreds or thousands of events can mitigate the impact of these motions. Furthermore, video data collected in managed settings during data collection with tags could be used to investigate the impact of body posture and control surfaces on drag.

Overall, this work presented a data-driven physics-based approach to estimate gliding drag coefficients, energetic efficiency and gait dynamics of bottlenose dolphins during horizontal intermittent locomotion. The COT and MECOT were investigated together with gait kinematics during the bursts of accelerating speed generated for the FG gait. The results indicate that horizontal FG gait, in addition to porpoising/leaping (Au and Weihs, 1980; Hui, 1989; Fish and Hui, 1991), wave riding (Williams et al., 1992), drafting (Weihs, 2004) and gliding during diving (Williams, 2001; Williams et al., 2000; Skrovan et al., 1999), is a behavioral mechanism that can enhance the locomotion efficiency of swimming dolphins.

We would like to thank the Seven Seas animal care specialists at the Brookfield Zoo who facilitated this study and the animals in their care. We would also like to thank Sarah Breen-Bartecki and Bill Zeigler for their continued support. The study protocol was approved by the Chicago Zoological Society IACUC and the University of Michigan Animal Welfare Committee (IACUC, #PRO00008825).

Author contributions

Conceptualization: D.Z., K.B., K.A.S.; Methodology: D.Z., K.A.S.; Software: D.Z., J.G.; Validation: D.Z., K.A.S.; Formal analysis: D.Z., Y.W., K.A.S.; Investigation: D.Z., Y.W., J.G., L.K.L., K.A.S.; Resources: D.Z., L.K.L., L.J.M., K.B., K.A.S.; Data curation: D.Z., Y.W., L.K.L.; Writing - original draft: D.Z., K.A.S.; Writing - review & editing: D.Z., L.K.L., L.J.M., K.B., K.A.S.; Visualization: D.Z., Y.W.; Supervision: L.K.L., L.J.M., K.B., K.A.S.; Project administration: L.K.L., L.J.M., K.B., K.A.S.; Funding acquisition: L.J.M., K.B., K.A.S.

Funding

This work is funded by the Canadian Department of Fisheries and Oceans (DFO), the US Office of Naval Research (ONR) and the US Navy's Living Marine Resources (LMR) Program.

Data availability

Data and code are available from the authors upon reasonable request.

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Competing interests

The authors declare no competing or financial interests.