Kinematic and hydrodynamic analyses of turning manoeuvres in penguins: body banking and wing upstroke generate centripetal force

Penguins (Spheniscidae) are considered to be the most specialized aquatic birds for underwater swimming. Previous studies have suggested that the basic mechanism of forward swimming in penguins includes lift-based propulsion and force generation during both the upstroke and the downstroke (Bannasch, 1994, 1995; Clark and Bemis, 1979; Harada et al., 2021; Hui, 1988). More recently, attempts have been made to estimate the wing forces of penguins using mechanical flapping-wing models (Bandyopadhyay et al., 2008; Shen et al., 2021). These hydrodynamic studies have shown that penguins control the angle of attack (AoA) of their wings and generate a forward lift force.

Although the aforementioned biomechanics studies have focused on steady forward swimming, turning manoeuvres are also important for penguins because the capacity to alter a trajectory may have an important role in survival (e.g. foraging, escaping and negotiating obstacles) in natural environments. Hui (1985) observed the turning manoeuvres of Humboldt penguins (Spheniscus humboldti) with a single video camera and showed that these penguins perform powered (with flapping) and unpowered (without flapping) turns. In addition, activation of the various body parts, such as the beak, tail, feet and wings, during the turns was visually confirmed. The mean turning radius of the five sharpest measured turns was reported to be 0.14 m [0.24 body lengths (BL)]. However, the details of the 3D motions of each body part and the resultant hydrodynamic forces and torques were not addressed. Therefore, the mechanisms of turning are largely unknown.

More studies have been conducted to understand the turning mechanisms of animal flight in air. Recently, the manoeuvring mechanisms of flying birds, bats and insects with flapping wings have been investigated using 3D motion measurements. These flying animals mainly perform turning manoeuvres using two mechanisms: (1) banking their bodies inwards (with their belly facing outwards) to use the upwards force on the wings as a centripetal force and (2) exhibiting contralateral differences in their wing motion (Hedrick and Biewener, 2007; Hedrick et al., 2007; Iriarte-Díaz and Swartz, 2008; Muijres et al., 2014, 2015; Read et al., 2016; Ros et al., 2011, 2015; Warrick and Dial, 1998). Although penguins are also birds, the underwater turning mechanism is presumably different from that in the air owing to the density difference between water and air. Water is more than 800 times as dense as air; thus, buoyancy can balance or overcome the weight of the animals, leading to different strategies for turning. Actually, in the case of forward swimming, penguins can accelerate more by upstroke than by downstroke, unlike flying birds, because a strong upstroke is beneficial for balancing the positive upwards buoyancy (Harada et al., 2021).

The aim of the present study was to reveal the turning mechanism of penguins, focusing mainly on the body posture, wing kinematics and hydrodynamic force generation on the wings. We hypothesized that penguins should alter the motion of their main propulsors, namely, the left and right flapping wings, during turning. For this purpose, we extended the kinematic and hydrodynamic analyses of forward swimming in penguins (Harada et al., 2021) to turning. Free swimming of gentoo penguins (Pygoscelis papua) was recorded by multiple underwater cameras at an aquarium. Based on the obtained turning kinematics, we estimated the wing forces using quasi-steady (QS) calculations. Our findings reveal mechanisms for controlling the turning manoeuvres of wing-propelled underwater swimming.

All experimental conditions were the same as in our previous study (Harada et al., 2021) so that data could be compared between forward and turning swimming manoeuvres. The following is a brief summary of the method.

To obtain the 3D kinematics, we recorded underwater swimming of gentoo penguins [Pygoscelis papua (Forster 1781)] at an aquarium with multiple underwater cameras and performed 3D direct linear transformation (DLT) motion analysis. The experiments were conducted twice at Nagasaki Penguin Aquarium (Nagasaki, Japan) on 7 March 2018 and 4 September 2019. The water tank had a length of 14 m, a width of 4 m and a water depth of 4 m. The morphological parameters of the penguins are summarized in Table 1. Two penguins (individuals 2 and 3) were observed in one of the two experiments, and one penguin (individual 1) was observed in both experiments. We treated individual 1 in 2018 and 2019 separately in the analysis, considering the difference in the experimental conditions. Thus, the penguin IDs were defined as 1A (2018), 2 (2018), 3 (2019) and 1B (2019). Twelve (2018) or fourteen (2019) waterproof video cameras (GoPro HERO6 Black and GoPro HERO7 Black, GoPro Inc., USA) were placed in the water tank. All videos were recorded at 60 frames s⁻¹, which enabled us to obtain approximately 30 frames per wingbeat assuming a wingbeat frequency of 2 Hz. The videos were calibrated using a cuboid-shaped calibration frame, which consisted of buoys and orange tapes attached to chains. The dimensions of the frame consisted of a length of 3.6 m, a width of 1.5 m and heights of 2.0 m (7 March 2018) or 2.5 m (4 September 2019). The target penguins swam freely in the tank with other penguins for approximately 40–60 min. All experiments were conducted in accordance with the Animal Experiment Management Regulations at the Tokyo Institute of Technology. The experimental protocol was also approved by the Nagasaki Penguin Aquarium.

After the videos were recorded, 12 tracking points on a penguin (Fig. 1A) in the videos were tracked. Four markers were on the body in the dorsal front (DF), dorsal rear (DR), ventral front (VF) and ventral rear (VR) positions. Three markers were on each wing in the wingbase (WB), leading edge (LE) and trailing edge (TE) positions. Left and right wingtips (WTs) were tracked without markers. Hereafter, the positions of the tracking points are referred to as P_DF. Each tracking point was manually tracked, and its 3D position was calculated using motion analysis software (DIPP-motion V/3D, Ditect Co., Ltd, Japan). The positional data were processed with a custom MATLAB (2020b and 2021a, MathWorks, Inc., USA) script, in which the raw time-series data were smoothed with a 17-point weighted moving average with a generalized Hamming window tuned as a low-pass filter with a cut-off frequency of 8 Hz and a gain of 1 at 0 Hz.

Assuming that a penguin is a rigid body, its motion can be described by the translation of the centre of mass (CoM) and the rotation around the CoM. In our previous work (Harada et al., 2021), we used the centre of the body (CoB; the geometric centre of the four body markers) as a representative point of the body. In the present study, we estimated the CoM from the 3D body model to conduct further analysis of the swimming dynamics.

We created a 3D body model for each penguin ID to obtain the body volume of each penguin, as described in our previous investigation (Harada et al., 2021). In the body model, the position of the CoB is calculated by averaging the four body marker positions (DF, DR, VF and VR). The volume centroid of each body model is calculated as the CoM, assuming a uniform density of the body. Then, the relative position of the CoM to the CoB can be determined. Therefore, in the motion measurement analysis, the time-varying position of the CoM can be calculated from the CoB position.

Notably, the change in the position of the CoM due to wing motion is neglected. Previously, we also created a 3D scanned wing model (Harada et al., 2021; Shen et al., 2021), and the ratio of the volume of both wings to the total volume (body+both wings) ranged from approximately 1.5% to 2.5%. Therefore, the effect of the wing mass on the position of the CoM is small.

Three coordinate systems were defined: the world coordinate system fixed to the tank, O-xyz; the body coordinate system fixed to the CoM, O_b-x_by_bz_b; and the modified world coordinate system fixed to the tank, O_m-x_my_mz_m. The xy plane is horizontal, and the z-axis is oriented vertically upwards. The x_b-, y_b- and z_b-axes of the body coordinate system represent the longitudinal (anterior–posterior), left–right and dorsoventral directions, respectively. Positive x_b, y_b and z_b indicate forward, left and dorsal directions, respectively.

Because the penguins swam freely in the water tank, the directions of the x-axis and the x_b-axis at the start of the sequence or wingbeat do not coincide. Thus, we introduced the modified world coordinate system O_m-x_my_mz_m. The modified world coordinate system was obtained by rotating the world coordinate system O-xyz around the z-axis so that the x-axis coincided with the projection of the x_b-axis at the start of the sequence or wingbeat onto the xy plane (horizontal plane). The modified world coordinate system was used to describe the trajectory and orientation of the penguins.

The definitions of the body orientation angles in this subsection and the body angles in the next subsection are based on the conventional flight dynamics theory in aircraft, which is similar to previous studies on the flight manoeuvres of insects and bats (Card and Dickinson, 2008; Iriarte-Díaz and Swartz, 2008).

Three body orientation angles, namely, bank, elevation and heading, were defined to describe the orientation of the penguins (Fig. 1B). The body orientation angles are a set of intrinsic Euler rotation angles from the modified world coordinate system to the body coordinate system. The transformation was accomplished by a ZYX rotation order: the body was first rotated by an angle Ψ about the z_m-axis, then rotated by an angle Θ about the new y_m′-axis, and finally rotated by an angle Φ about the new x_m″-axis (x_b-axis). Here, ′ denotes the axis after one rotation and ″ denotes the axis after two rotations. The first two rotation angles are called the heading angle (Ψ) and the elevation angle (Θ). The initial values of the heading angles at the start of the sequence or wingbeat are all zero because the horizontal projection of the x_b-axis at the start of the sequence or wingbeat coincides with the x_m-axis. The last rotation angle is called the bank angle (Φ).

The bearing angle is defined as the angle between the horizontal component of the CoM velocity (v_CoM,h) and the x_m axis (Fig. 1C). The bearing angle represents the direction in which the penguin was actually moving on the horizontal plane. The difference between the bearing angle and the heading angle is referred to as sideslip.

The time variation of the body angles (i.e. roll, pitch and yaw angles) from the start of the sequence or wingbeat was calculated as the cumulative sum of the angular velocities. The initial roll and pitch angles were the same as the bank and elevation angles at the start of the sequence or wingbeat, respectively. The initial yaw angle was set to zero. Note that the elevation Θ is negative when the penguin points upwards to the positive z direction (i.e. vertically upwards). The pitch angle is also negative when the penguin rotates upwards towards the positive z_b direction (i.e. dorsal side).

The body orientation angles (Euler angles: bank, elevation and heading angles) and the body angles (roll, pitch and yaw angles) are similar. However, these angles do not have a one-to-one relationship with each other. The body orientation angles represent the body direction relative to the horizontal plane and vertical axis. Note that some papers have referred to the body orientation angles in our definition as the roll, pitch and yaw (e.g. Schilstra and Van Hateren, 1999). In contrast, the body angles represent rotation around the body-centred axis. These angles are virtual displacements, which can be obtained by integrating the angular velocities.

Additionally, note that the initial values of the body orientation angles and the body angles change depending on the motion that is being considered. For example, for a swimming sequence that includes some wingbeats and glidings, the heading and yaw angles are set to zero at the start of the sequence. When we focus on a single wingbeat, the heading and yaw angles are set to zero at the start of the wingbeat.

In this subsection, the definitions of the kinematic parameters are briefly explained. For more details and schematics, please refer to our previous paper (Harada et al., 2021).

The velocity and acceleration were obtained from the time-series positional data in the world coordinate system using the second-order central difference method. We used the CoM velocity and acceleration, v_CoM and a_CoM, respectively, as the representative velocity and acceleration of the body.

To express the 3D wing kinematics with respect to the body, we defined three angles: flapping angle, β_flap; sweepback angle, β_sweep; and feathering angle, β_feather. Here, β_flap represents the power stroke in the dorsoventral direction, β_sweep indicates how the wing is swept backwards relative to the body, and β_feather represents a pronation or supination around the long axis of the wing. Positive β_flap, β_sweep and β_feather values indicate dihedral, backwards sweep and supination, respectively.

The start and end timings of the upstroke and downstroke were determined by β_flap and its derivative. The representative wingbeat period was determined by averaging the wingbeat periods for the left and right wings. In addition, the wingbeat period for each of the left and right wings is discussed in relation to the contralateral timing difference during turning.

The wing deformation angles represent out-of-plane bending and in-plane folding. The bending angle, θ_bend, is the angle between an inner wing (ΔP_WBP_LEP_TE, orange triangle in Fig. 1A) and an outer wing (ΔP_LEP_TEP_WT, green triangle in Fig. 1A). To calculate the in-plane folding, the outer wing was rotated around a P_LEP_TE line by −θ_bend so that the rotated wingtip point (P_WT′) was in the inner-wing plane. The folding angle, θ_fold, was then calculated as the angle between the line segment P_WBP_LE and the leading edge of the outer wing (line segment P_LEP_WT′). For details and schematics of the definitions of θ_bend and θ_fold, please refer to our previous paper (Harada et al., 2021).

The AoA is the angle between the wing plane and the relative flow velocity vector (opposite vector of the moving velocity vector) at a reference point. The three reference points, P_R,1, P_R,2 and P_R,3, were defined as follows: P_R,1 is the midpoint between P_WB and P_LE, and P_R,2 and P_R,3 are the 25% and 75% positions between P_LE and P_WT, respectively. In this paper, the AoA is reported at P_R,2 as a representative AoA.

For the QS calculation of the wing force, we decomposed each wing into three elements. Then, we calculated the fluid force on the wing in a QS manner using the velocity, acceleration and AoA of each wing element (for details, please see Harada et al., 2021). We measured the steady lift and drag coefficients in a water tunnel with a rigid 3D-printed wing model. Based on the steady coefficients, the lift and drag coefficients of a flapping wing were estimated by the cross-flow vortex model (Bandyopadhyay et al., 2008). The lift and drag forces acting on the wing element were calculated using the coefficients. The added-mass forces were also calculated based on the method of Walker (2002).

From the results of the force calculations, the torque of the wing force in the body coordinate system around the CoM was obtained. The wing torque was calculated by taking the cross-products of the vectors from the CoM to the point of action and the forces (lift, drag and added-mass force) and then summing them.

Each component of the inertia matrix was obtained from the 3D model of the body assuming a uniform density. Note that we ignored the inertia of the wing.

Then, the turning radius and turning rate were time-averaged for each wingbeat to provide representative values of the wingbeat (i.e. and ⁠). Finally, for turning wingbeats, the mean, standard deviation (s.d.), maximum, minimum and mean of the extreme 20% of values (i.e. the mean of the bottom 20% values of ⁠, R_bottom20, and the mean of the top 20% values of ⁠, ω_top20) are reported as metrics of the turning performance (Fish et al., 2018).

To distinguish turning from forward swimming during each wingbeat, we defined three angles, namely, the azimuth angle, γ_azm, inclination angle, γ_inc, and ascending angle, γ_asc, as in our previous paper (Harada et al., 2021). These angles are defined based on the moving direction of the CoB (not the CoM) for each wingbeat, that is, the vector connecting the position of the CoB at the start and end of the wingbeat. The azimuth angle and the inclination angle were calculated from the relationship between this vector and the coronal plane (x_by_b plane) at the start of the wingbeat. A positive azimuth angle represents a left turn. A positive inclination angle represents an upwards turn. In addition, we defined the ascending angle, which is an angle between the vector and the horizontal plane. In our previous paper (Harada et al., 2021), we analysed only forward and horizontal swimming, defined as ≤15 deg, |γ_inc| ≤20 deg and |γ_asc| ≤20 deg. In this paper, we analysed all horizontal swimming cases, defined as |γ_inc| ≤20 deg and |γ_asc| ≤20 deg. In addition, wingbeats with an absolute azimuth angle greater than 15 deg are called ‘turning wingbeats’.

In the analysis of turning wingbeats, right turns were reflected to left turns so that all turns were analysed together. For example, when calculating the average y_b velocity, the sign of the right turns was reversed, and then they were treated as left turns.

Note that each wingbeat can be classified into four cases, depending on whether the preceding and following statuses are gliding or flapping (glide–glide, glide–flap, flap–glide, flap–flap). However, we did not distinguish them in the analysis owing to sample size limitations.

The time was normalized by the wingbeat period, followed by a linear interpolation of each variable to generate 50 time-series data points per wingbeat. The normalized time is expressed as t′. The mean value and s.d. were calculated using the generated data. We calculated the mean of all penguin IDs mixed together, without distinguishing between each penguin ID. The values are presented as means±s.d.

We used linear regression analysis to examine the relationship between the mean swimming speed and each variable. A P-value of less than 0.05 was considered statistically significant.

The penguins demonstrated both powered turns with wingbeats (Figs 2 and 3; Movies 1 and 2) and unpowered turns without wingbeats (Fig. S1 and Movie 3). Most of the powered turns were observed during horizontal or descending swimming, and most of the unpowered turns were observed during ascending swimming. In an example of a turn powered by a single wingbeat shown in Fig. 2 and Movie 1, the penguin changed its bearing angle by approximately 35 deg with a single wingbeat. The mean swimming speed during the wingbeat was 1.79 m s⁻¹. The change in trajectory was larger during the upstroke than during the downstroke, and the trajectory during the downstroke was almost straight. During the turning wingbeat, the flapping angle was larger on the inside wing (i.e. the left wing, because the penguin turned to the left) than the outside wing (i.e. the right wing), and the penguin banked outwards, making its belly face inwards. Before the start of the wingbeat, the penguin had already started banking. During the downstroke of the turning wingbeat, the bank angle decreased, and the penguin returned to a neutral posture except for the sideslip (i.e. difference between the bearing and heading angles). The neck was bent in the direction of the turn, especially during the upstroke, and almost straightened at the downstroke. The tail and foot motions during this sequence were not confirmed owing to the position and angle of the cameras. Therefore, it is unclear how the tail and foot were moving during the turn.

An example of a powered turn with continuous wingbeats is shown in Fig. 3 and Movie 2. Only a single wingbeat marked by star no. 2 was classified as a turning wingbeat based on our definition. The penguin turned horizontally with a total bearing angle of approximately 160 deg. The mean swimming speed during the sequence was 0.94 m s⁻¹. Although the asymmetry of the wing motion was not as clear as in the previous example, the flapping angle on the inside wing (left wing) was also larger than that on the outside wing (right wing) in the wingbeat shown by star no. 2, demonstrating a large change in direction during the upstroke. During a whole turn, the bank angle remained positive, which indicates that the body always banked outwards, making the belly face inwards. Notably, the tail, and sometimes the neck, was observed to be flexed inwards in the middle of the turn during the sequence (Fig. 3C), indicating that the head and tail could work as rudders to generate yaw torque.

For an analysed individual, 54 sequences and 65 wingbeats were obtained. Ten wingbeats were excluded from the analysis because the absolute inclination angle or absolute ascending angle was greater than 20 deg. One horizontal forward wingbeat was excluded because we determined that the wingbeat was in the middle of descent after considering the trajectory of the body. As a result, 54 wingbeats were analysed. The trajectory of all 54 wingbeats is shown in Fig. S2. Of these, 40 were forward wingbeats, and 14 were turning wingbeats. The results of the analysis of the 40 forward wingbeats are also described in our previous paper (Harada et al., 2021). Of the 14 turning wingbeats, 10 were left turns and four were right turns. These 14 turning wingbeats were obtained from 11 sequences. The experimental results were obtained from four penguin IDs; however, the turning wingbeats were obtained from only three penguin IDs (7 from ID 1A, 3 from ID 2, 4 from ID 3).

The turning wingbeats include all four cases for the status before and after the wingbeat (i.e. glide–glide, glide–flap, flap–glide, flap–flap). There were three wingbeats with gliding before and after the wingbeat (glide–glide). There were four wingbeats with gliding before the wingbeat and flapping after the wingbeat (glide–flap) and three wingbeats with flapping before the wingbeat and gliding after the wingbeat (flap–glide). The remaining four wingbeats flapped before and after the wingbeat (flap–flap).

The wingbeat-averaged swimming speed for the turning wingbeats ranged from 0.67 to 1.79 m s⁻¹, and the pooled mean±s.d. was 1.13±0.28 m s⁻¹. The wingbeat frequency for the turning wingbeats ranged from 1.43 to 2.22 Hz, and the mean±s.d. was 1.81±0.22 Hz. The mean ratio of the downstroke duration to the wingbeat period was 0.46±0.07, which shows that the upstroke was slightly longer than the downstroke.

The timing of the turning wingbeats was slightly different between the inside and outside wings, especially at the start of the upstroke. The start of the upstroke was 4.00±5.91 frames (0.067±0.099 s) earlier on the inside wing than on the outside wing. The start of the downstroke was 1.21±1.67 frames (0.020±0.028 s) earlier, and the end of the downstroke was 0.79±2.12 frames (0.013±0.035 s) earlier.

The ensemble-averaged parameters of body translation and rotation in a normalized wingbeat cycle are shown in Fig. 4. This figure shows the average of 14 turning wingbeats, and right turns were reflected to left turns .

The ensemble-averaged travel distance, velocity and acceleration of the CoM are shown in Fig. 4A–C. Here, the distance in each body direction, p_CoM, is the virtual displacement obtained by integrating the CoM velocity in the body coordinate system (Fig. 4A). The time variation of the longitudinal (x_b) and dorsoventral (z_b) velocities during the turning wingbeats were similar to those during the forward wingbeats (Harada et al., 2021). In the turning wingbeats, a positive inward (y_b) acceleration was notable, reflecting the curvature of the body trajectory (Fig. 4C). The upstroke produced more y_b acceleration than the downstroke, implying that the upstroke contributed to the turn more than the downstroke. In contrast, x_b acceleration showed positive peaks at the mid-upstroke and mid-downstroke, and the downstroke produced a larger x_b acceleration (mean 0.38±0.66 m s⁻²) than the upstroke (mean 0.24±0.42 m s⁻²). Note that the upstroke produced a larger x_b acceleration than the downstroke in forward swimming (Harada et al., 2021).

The ensemble-averaged body orientation angles, angular velocity and angular acceleration are shown in Fig. 4D, E and F, respectively. In most turning wingbeats, the penguin banked outwards, making the belly face inwards (Fig. 4D). The mean bank angle over a wingbeat cycle was 15.3±6.9 deg. At the start of the wingbeat, the bank angle was 12.6±8.6 deg, which means that the body had already banked outwards. The bank angle was approximately constant until the mid-downstroke and then slightly decreased until the end of the downstroke. The elevation angle was positive and almost constant during a wingbeat cycle, of which the mean elevation angle was 10.8±7.6 deg [note: a positive elevation indicates that the longitudinal body direction (x_b) is downwards relative to the horizontal plane].

A large number of changes in the bearing angle occurred in the upstroke. The ensemble-averaged bearing angles at the start of the upstroke, end of the upstroke and end of the downstroke were −6.9±4.3, 28.5±5.6 and 39.4±9.8 deg, respectively. That is, the changes during the upstroke, downstroke and total wingbeat were 35.4, 11.0 and 46.4 deg, respectively. Therefore, the upstroke contributed 76% of the total changes in the bearing.

Similarly, the upstroke contributed to the changes in the heading angle more than the downstroke. The changes in the ensemble-averaged heading angles during the upstroke, downstroke and total wingbeat were 27.2, 14.5 and 41.8 deg, respectively. That is, the upstroke contributed 65% of the total changes in the heading. Sideslip (i.e. a difference between the bearing and heading angles) was observed in the first half of the upstroke: a sideslip of at most 7.2±4.2 deg outwards (i.e. rightward in the case of a left turn) just after the start of the upstroke (t′=0.06).

The ensemble-averaged body angles, angular velocity and angular acceleration are shown in Fig. 4G–I. The roll angle was almost constant at approximately 13 deg during the upstroke and decreased during the downstroke to almost zero (Fig. 4G). The pitch angle was also kept positive; it slightly increased during the upstroke and remained almost constant during the downstroke. The yaw angle increased constantly throughout the wingbeat. These angles, representing body rotation, showed similar changes to the body orientation angles. As shown in Eqn 3, the angular velocity (Fig. 4H) and angular acceleration (Fig. 4I) of the body angles are related to the body torque, and these variables were used to estimate the torques.

The inside–outside differences in wing kinematics are summarized in Fig. 5. The ensemble average is shown for the inside wing (red; left wing of the left turn and right wing of the right turn) and outside wing (blue; right wing of the left turn and left wing of the right turn). In addition, the pooled mean of the forward wingbeats for the four penguin IDs in Harada et al. (2021) (green) are shown.

The three wing angles, which explain the 3D motion of the wing relative to the body, are shown in Fig. 5A–C. The largest asymmetry was found in the flapping angle (β_flap) (Fig. 5A). From the start of the wingbeat, β_flap of the inside wing was larger than that of the outside wing during the upstroke; β_flap at the start of the upstroke was −24.2±18.8 deg (inside) and −42.3±12.6 deg (outside). At the start of the downstroke, the difference between the two decreased, with values of 48.1±12.8 deg (inside) and 33.9±19.0 deg (outside). At the end of the downstroke, the difference was small, with values of −45.1±19.3 deg (inside) and −49.7±15.5 deg (outside). The wingbeat-averaged β_flap was 5.6±11.8 deg for the inside wing and 10.0±11.7 deg for the outside wing. That is, the inside wing was elevated more than the outside wing.

A small difference was observed in the sweepback angle (β_sweep) between the inside and outside wings at the start and end of each stroke (Fig. 5B). The inside wing, however, swept back more than the outside wing in the first half of the downstroke.

Asymmetry was also observed in the feathering angle (β_feather) (Fig. 5C): β_feather of the outside wing was larger than that of the inside wing from the mid-upstroke to the mid-downstroke. At the start of the downstroke, β_feather was −23.4±15.4 deg (inside) and 6.6±11.6 deg (outside). The wingbeat-averaged β_feather was 14.8±7.1 deg for the inside wing and 5.6±4.4 deg for the outside wing, indicating that the inside wing was more pronated than the outside wing.

The wing deformation angles are shown in Fig. 5D,E. The bending angles of the inside and outside wings were sinusoidally varied, while their phases differed from each other (Fig. 5D). Owing to the phase difference during flapping (Fig. 5A), the phase difference corresponds to that of the bending angle. The folding angle of the inside wing was similar to that of forward swimming, while the folding angle of the outside wing was slightly smaller than that of the inside wing (Fig. 5E).

The representative AoA is shown in Fig. 5F. The AoA magnitude of the inside wing was larger than that of the outside wing during the upstroke: the minimum AoA was −28.1±13.0 deg for the inside wing and −18.2±7.1 deg for the outside wing. During the downstroke, the difference between the inside and outside became small: the maximum AoA was 23.4±8.1 deg for the inside wing and 19.1±6.4 deg for the outside wing. In addition, the minimum and maximum AoAs for the outside wing were similar to those for forward swimming: the minimum and maximum AoAs for forward swimming were −17.7±2.0 and 17.9±3.8 deg, respectively.

We examined the correlation between the azimuth angle (an indicator of the direction and magnitude of turning wingbeats) and the kinematic variables, as shown in Fig. 6. In this figure, the horizontal axis indicates the azimuth angle, where a positive value indicates a left turn and a negative value indicates a right turn. The vertical axis is the left–right difference in the wing angle, defined as the left-wing value–right-wing value. For reference, the regression lines for each individual penguin and for all the penguins are displayed. The correlation coefficients and the results of the linear regression analysis are summarized in Table S1.

The left–right differences in the wingbeat-averaged values are shown in Fig. 6A–C. A positive correlation was observed for the wingbeat-averaged flapping angle (β_flap) (Fig. 6A), and a negative correlation was observed for the wingbeat-averaged feathering angle (β_feather) (Fig. 6C). For β_flap (Fig. 6A), the left wing tended to be larger than the right wing during left turns. In other words, the wingbeat-averaged β_flap was larger for the inside wing of the turn. In contrast, β_feather of the right wing tended to be larger than that of the left wing during left turns (Fig. 6C). In other words, the wingbeat-averaged β_feather was larger for the outside wing of the turn. No clear trend was observed in the sweepback angle (β_sweep) (Fig. 6B). These results are consistent with the ensemble-averaged values of the wing angles shown in Fig. 5A–C. The left–right differences in the amplitude during one wingbeat cycle showed no clear trend for all three wing angles (Fig. 6D–F).

The results shown in Figs 5 and 6 can be summarized as follows. During a turning wingbeat, penguins increase the wingbeat-averaged β_flap (Figs 5A, 6A) and decrease the wingbeat-averaged β_feather (Figs 5C, 6C) of the inside wing. The inside wing is more elevated and pronated than the outside wing.

The inside–outside differences in wing force and torque are summarized in Fig. 7. In this figure, the sum of each wing (green; inside+outside) is also shown.

The wing forces in each body direction are shown in Fig. 7A–C. The force in the forward (x_b) direction was almost zero during the upstroke, while a positive x_b force was mainly generated by the inside wing during the downstroke (Fig. 7A). In contrast, the force in the left (inwards of the turn, y_b) direction was mostly produced by the outside wing during the upstroke, while the y_b forces cancelled each other out during the downstroke (Fig. 7B). A small difference was observed in the force in the dorsoventral (z_b) direction between the inside and outside wings (Fig. 7C). The z_b forces during each half-stroke almost cancelled each other out, reflecting the fact that horizontal swimming was selected in this study.

The wing torque in each body direction is shown in Fig. 7D–F. The roll torque waveform is almost symmetrical between the inside and outside wing (Fig. 7D). The sum of the inside and outside torques was positive during the wingbeat, corresponding to the fact that the roll angle was positive. For the pitch torque, the outside wing generated a positive torque during the upstroke, while the inside wing generated a positive torque during the downstroke (Fig. 7E). The sum of the inside and outside torques was mainly positive during the wingbeat. Note that a positive pitch torque indicates a head-down motion. A small positive yaw torque was observed during the upstroke, while a large negative yaw torque was generated by the inside wing during the downstroke (Fig. 7F). That is, the inside wing generates a yaw counter torque to brake yaw rotation during the downstroke.

The QS calculation and the total force and torque calculated from the translational and angular accelerations of the body are shown in Fig. S3. The waveforms of the QS force and total force are relatively similar. However, the torque waveforms of the two vary, suggesting that the torque is produced by factors other than the wings, such as the motion and flexion of the body, beak, tail and feet. In particular, the tail feathers and the webbed feet have a relatively flattened shape and perhaps generate a non-negligible hydrodynamic force during turning manoeuvres, which can produce roll, pitch and yaw torques. For pitch and yaw, the moment arm of the tail and feet from the body centre is morphologically larger than that of the wings. Instead, for roll, the wings have a larger moment arm than the tail and feet.

The wingbeat-averaged turning radius, ⁠, ranged from 0.34 m (0.50 BL) to 2.54 m (3.74 BL), and its mean±s.d. was 0.99±0.57 m (1.53±0.87 BL). The wingbeat-averaged turning rate, ⁠, ranged from 83 to 181 deg s⁻¹, and its mean±s.d. was 116±28 deg s⁻¹. The mean of the bottom 20% values of (R_bottom20) was 0.49 m (0.72 BL), and the mean of the top 20% values of (ω_top20) was 159 deg s⁻¹.

Fig. S3A–C shows the normal force, F_n, decomposed in the body coordinate system. F_n represents the total force (F_total) vector that is perpendicular to the body velocity vector. The forward component of F_n was small, and the left (inwards) and dorsoventral components of F_n were very similar to those of F_total, indicating that the penguins mainly used the left and dorsoventral components of the forces as a centripetal force.

The wingbeat-averaged centripetal acceleration ranged from 1.50 to 3.08 m s⁻², and its mean±s.d. was 2.18±0.52 m s⁻². The wingbeat-averaged centripetal force ranged from 9.48 to 20.16 N, and its mean±s.d. was 13.56±3.49 N. The centripetal acceleration and force were greater during the upstroke (mean 2.59±0.83 m s⁻², 16.09±5.47 N) than during the downstroke (mean±s.d. 1.73±0.44 m s⁻², 10.71±2.80 N).

Manoeuvrability can be evaluated as the minimum radius of the turning trajectory, R_bottom20. In contrast, agility can be measured as the maximum turning rate, ω_top20. Hui (1985) reported that Humboldt penguins (Spheniscus humboldti) can turn with a turning radius of 0.14 m (0.24 BL), whereas we obtained a value of 0.49 m (0.72 BL) in our measurements. Hui (1985) also reported that the turning rate ranged from 1.01 to 10.05 rad s⁻¹ (57.9 to 575.8 deg s⁻¹), while it ranged from 83 to 181 deg s⁻¹ in our measurements. That is, the measured performance in our experiments was not as high as that in the study by Hui (1985). Notably, Hui (1985) did not distinguish between flapping and gliding (35% of the movements in Hui's analysis were turning without flapping), whereas our analysis focused only on horizontal flapping.

Fish (2020) summarized the maximum turning rate and minimum turning radius for various autonomous underwater vehicles, biological autonomous underwater vehicles and animals. Fig. 8 shows the turning performance of various animals with our data. The black circle represents Humboldt penguins, as reported by Hui (1985), and the red star indicates the values from the present study (R_bottom20 and ω_top20). The ranges of the rate and radius for the animals are large, while the values for penguins are moderate. The maximum turning rate of our data was smaller than that of Hui's data (black circle in Fig. 8A), and the minimum turning radius relative to the body length of our data was larger than that of Hui's data (Fig. 8B). These comparisons indicate that our measurements captured gentler turns than the other studies.

In this study, we analysed swims in which the penguins were not engaging in goal-directed behaviors, such as foraging or escaping. Thus, there was no incentive for the penguins to maximize their turning performance. In the future, the video recording setup should be improved to induce tighter turns.

This subsection explains how the penguins turned by flapping their wings with outward banking (Fig. 9). The penguins prepared for the turn by changing the posture of their body and wings before the start of the wingbeat of the powered turn. The bank angle at the start of the upstroke was 12.6±8.6 deg, indicating that the penguins were already banking outwards before the turn started. The flapping and feathering angles of the inside wing were larger than those of the outside wing (Fig. 5A,C) at the start. In addition, in most turning wingbeats (10 out of 14), the start of the wingbeat of the inside wing was earlier than that of the outside wing. These facts indicate that the penguins were already changing the posture of the inside and outside wings (i.e. flapping and feathering angles) before the start.

During the upstroke, the penguins generated a force in the ventral direction with a large centripetal component (Fig. 9A,B). A centripetal force towards the inside of the turn was generated mainly on the outside wing owing to the contralateral difference in the flapping angle. The force of the outside wing also generated a positive yaw torque, which aligned the head with the direction of the turn. Throughout the upstroke, the body maintained outward banking. This banking increased the centripetal component of the wing force. The changes in the heading and bearing of the body mainly occurred during the upstroke.

During the subsequent downstroke, the penguins generated a force in the dorsal direction with a large forward component (Fig. 9C,D). The forward force was especially large in the inside wing. The inside wing moved faster than the outside wing because (i) the flapping angle at the start of the downstroke was larger for the inside wing, (ii) the flapping angle at the end of the downstroke for the inside and outside wings was similar, and (iii) the downstroke duration for the inside and outside wings was similar. Therefore, a faster inside wing generated a larger force. The difference in the forward force between the inside and outside wings generated a negative yaw torque, which aligned the head with the direction of travel.

In summary, two mechanisms were used in the powered turns of the penguins: (i) maintaining outward banking using the force in the ventral direction during the upstroke as a centripetal force and (ii) changing the motion of the inside and outside wings to generate a centripetal force in the outside wing during the upstroke and a forward force in the inside wing during the downstroke.

In controlling the direction and magnitude of the force, penguins likely adjust the AoA to an appropriate range by using the rotation of the wings around the spanwise axis (change in the feathering angle, that is, pronation or supination). For example, during the downstroke, the inside wing moves faster than the outside wing. The relative flow velocity of the wing is the vector sum of the body velocity and the downstroke (or upstroke) velocity (see fig. 8A,B in Harada et al., 2021). Therefore, if the feathering angle does not change, then the AoA must be larger when the wing moves faster. However, the contralateral difference in the AoA during the downstroke is small (Fig. 5F) owing to the pronation (i.e. negative feathering) of the inside wing (Fig. 5C). Penguins generate applicable forces by controlling the AoA through the feathering angle.

The measured penguins performed turns by banking outwards (i.e. with their belly facing inwards). This result is consistent with the results of a previous study that reported an unpowered submerged turn in Humboldt penguins (Hui, 1985).

In contrast, flying birds and bats turn by banking inwards (i.e. with their belly facing outwards) (Hedrick and Biewener, 2007; Iriarte-Díaz and Swartz, 2008; Ros et al., 2015; Usherwood et al., 2011; Warrick and Dial, 1998). Moreover, fixed-wing aircraft also turn by banking inward. The difference in the bank direction derives from the effect of buoyancy. For flying animals and aircraft in horizontal flight, the wings generate an upwards vertical force to support the weight, with negligible buoyancy in air. Thus, inward banking easily diverts the upwards vertical force to a centripetal force for turning (Fig. S4A). In contrast, the wings of penguins undergoing a horizontal dive generate a vertical downwards force to balance the buoyancy in water. Hence, penguins can divert the downwards vertical force to a centripetal force by outward banking (Fig. S4B).

In addition, penguins can generate a remarkable upwards (dorsal) force by a downstroke and a downwards (ventral) force by an upstroke (Fig. 7C); therefore, they should be able to bank inwards and take advantage of the generated upwards force to turn in the downstroke. Why did our penguins perform outward-banking turns? Our hypothesis is as follows. In our experiment, for the horizontal straight gliding at shallow depths, the wings of the penguins had a negative β_flap (anhedral angle) and generated downwards forces to balance the upwards buoyancy. In such a situation, slight outward banking can immediately generate a centripetal force to the inside of a turn. In addition, outward banking can produce a centripetal force from the downwards force by the subsequent upstroke. Therefore, penguins can easily initiate turns from straight swimming using outward banking. However, the upwards force generated by the downstroke can be used as a centripetal force by inward banking depending on the situation. Further investigation is needed.

Note that many diving animals who hold their breath frequently perform inward-banking turns, unlike our penguins. Measurements using biologging tags have demonstrated that blue whales (Balaenoptera musculus) mainly bank inwards during turning manoeuvres (Segre et al., 2019). California sea lions (Zalophus californianus) perform unpowered turns by utilizing their foreflippers and hindflippers with 90 deg inward banking (Fish, 1997, 2004; Fish et al., 2003; Leahy et al., 2021). Fish (1997) observed the turning performance of eight species of marine mammals (seven cetaceans and one pinniped) and found that inward banking was mainly observed during the unpowered turns of cetaceans. However, beluga whales (Delphinapterus leucas) performed outward banking, and Amazon River dolphins (Inia geoffrensis) showed no tendency to bank. Therefore, the bank direction may vary depending on the musculoskeletal characteristics and morphology of the species.

Although our penguins mostly turned with flapping, a single unpowered ascending turn without flapping was also measured, as shown in Fig. S1 and Movie 3. The penguin performed a 70 deg turn to the left while ascending. The mean turning radius was 2.23±3.26 m (3.23±4.73 BL), and the turning rate was 51±21 deg s⁻¹. The mean swimming speed during the sequence was 1.27±0.07 m s⁻¹. This example is a slow turn with a larger turning radius and a smaller turning rate than the powered turns.

The mean flapping angle of the inside (left) wing was larger than that of the outside (right) wing: −7.3±4.8 deg for the inside wing and −21.8±3.0 deg for the outside wing. The mean AoAs of the inside and outside wings were similar: −3.9±2.3 deg for the inside wing and −2.8±1.5 deg for the outside wing. The mean bank angle was 23.9±6.0 deg; that is, the penguin maintained outward banking as in the powered turn. The combination of outward banking and the large negative flapping angle of the outside wing caused the surface of the outside (right) wing to face in the direction of the turn (left). Thus, the lift of the outside wing with a negative AoA contributed to the centripetal force (leftward force) for the turn. The mean difference between the heading and bearing angles was 1.1±2.0 deg, so the sideslip was small throughout the turn.

In this case, the neck and tail were always flexed inwards in the middle of the turn. Note that penguins can perform tighter unpowered turns by using the motion of their body, beak and tail (see fig. 1 in Hui, 1985). Further research is needed to determine the effects of factors other than the wings.

Three-dimensional motion analysis and QS hydrodynamic force calculations of swimming penguins performing horizontal powered turns in an aquarium were conducted to reveal the mechanism of outward-banking turns. The motion analysis demonstrated that the body banked outward by approximately 15 deg during turning so that the ventral side faced the inside of the turn. The upstroke produced 76% of the total change during one wingbeat in the bearing angle of the moving velocity. The mean flapping angle of the inside wing during the upstroke was shifted to the dorsal side compared with the outside wing. The inside wing also feathered so that its angle of attack became larger than that of the outside wing during the upstroke. The QS hydrodynamic analysis revealed that the combination of outward banking and the left–right difference in the wing motion produced a centripetal force and yaw torque for the turn by the upstroke, followed by the downstroke producing forward thrust and counter yaw torque. Considering the buoyancy force relative to the weight, the outward-banking turn in penguins, in which the buoyancy force is larger than the weight, can be interpreted as the opposite of the inward-banking turn in aircraft, in which the buoyancy is negligible compared with the weight. In future studies, the force and torque generated by the body, beak, tail and feet should be considered. Moreover, the mechanisms of other various manoeuvres, such as rapid acceleration, pitch up and down, and jumping out of the water, are still unknown. The present study serves as the basis for the further understanding of more complex manoeuvres.

We sincerely thank the staff of the Nagasaki Penguin Aquarium for their warm and enthusiastic support for the video recording and 3D scanning of the penguins. We also thank Dr Takeshi Yamasaki (Yamashina Institute for Ornithology) for insightful comments on the statistical analysis, Takuma Oura for the preliminary analysis and Dr Dale M. Kikuch (Tokyo University of Agriculture) for valuable comments on the study. We also thank the anonymous reviewers for improving the paper.