Omnidirectional propulsion in a metachronal swimmer

Aquatic organisms often employ maneuverable and agile swimming behavior to escape from predators, find prey, or navigate through complex environments. Many of these organisms use metachronally coordinated appendages to execute complex maneuvers. However, though metachrony is used across body sizes ranging from microns to tens of centimeters, it is understudied compared to the swimming of fish, cetaceans, and other groups. In particular, metachronal coordination and control of multiple appendages for three-dimensional maneuvering is not fully understood. To explore the maneuvering capabilities of metachronal swimming, we combine 3D high-speed videography of freely swimming ctenophores (Bolinopsis vitrea) with reduced-order mathematical modeling. Experimental results show that ctenophores can quickly reorient, and perform tight turns while maintaining forward swimming speeds close to 70% of their observed maximum—performance comparable to or exceeding that of many vertebrates with more complex locomotor systems. We use a reduced-order model to investigate turning performance across a range of beat frequencies and appendage control strategies, and reveal that ctenophores are capable of near-omnidirectional turning. Based on both recorded and modeled swimming trajectories, we conclude that the ctenophore body plan enables a high degree of maneuverability and agility, and may be a useful starting point for future bioinspired aquatic vehicles.

The transformation between the inertial and body frames is given by  ⃗ = [] ⃗, where the transformation matrix is given by [] = cos() cos() cos() sin() − sin() sin() sin() cos() − cos() sin() sin() sin() sin() + cos() cos() sin() cos() cos() sin() cos() + sin() sin() cos() sin() sin() − sin() cos() cos() cos() (S. 1) To avoid mathematical singularities when solving 3D motion using Euler angles (ψ, θ, ϕ), we performed all calculations using Euler parameters instead: The most general rigid body rotation has only three degrees of freedom; thus, the Euler parameters are subject to the constraint  +  +  +  = 1.To calculate the Euler parameters  ⃗ , we use the Stanley method [1].The last step for a formulation based on Euler parameters is to find the relationship between the time rates of change of  ⃗ and the body angular velocities ( ⃗ ).This relationship is known as the Euler parameter kinematic differential equation: Hence, under an Euler parameter formulation, we have three (vector) governing equations: the first and second Euler's laws (equations ( 2) and (3) in the manuscript) and the Euler parameter kinematic differential equation (S.3).

Propulsion force 𝑭 ⃗ 𝒏𝒆𝒕
As described in the main manuscript, each ctene is modeled as an oscillating flat plate, whose kinematics  (),  () depend on the beating parameters and placement on the animal body where  is the cycle number ( = 0,1,2,3, … ).These equations model the oscillating flat plate kinematics with respect to its central position  .The next step is to "place" them on the body.

Force and Torque coefficients Drag coefficient for an oscillating flat plate
To model the drag of the oscillating plate while considering the correct Reynolds number range for ctene beating (1 <  < 200 ), we use the empirical expression obtained by [2], appropriate for Reynolds numbers between 1 and 1057.
where  is the period parameter and is defined as  =  ⃗ −  ⃗ ̇ /   , and the Reynolds number is defined as  =  ⃗ −  ⃗ ̇ /   .

Drag coefficient for a prolate spheroid
From [3], we obtain an expression for the drag coefficient of a prolate spheroid: where  is the spherical equivalent diameter and  depends on flow direction where  is the aspect ratio of the body,  ≡   ⁄ .

Added mass coefficients
From [4], we obtain the added mass coefficients for a spheroidal body for the axial and lateral movements ( ,  ): where  is the eccentricity  =  −   ⁄ .This approach is fully valid only for linearly superposable flows (i.e.potential flow (high ) or Stokes flow (low )) but is a good engineering approximation for intermediate Reynolds numbers [5].

Torque coefficients
To model the opposing torques, we used the numerical expressions obtained by [6], which is appropriate for rotating Reynolds numbers  = ⃗ , between 10 − 10 .
where the coefficients  depend on the rotating direction (rolling or pitch/yaw axes); see Table 1.

Solution procedure
We solved the reduced-order model using a fourth-order Runge-Kutta scheme for implicit equations, using the MATLAB function ode15i.The solution algorithm consists of the following steps: 1. Input the initial particle position, orientation, and speeds (translational and angular).

Calculate the corresponding initial values of the Euler parameters using the Stanley method
and evaluate the transformation matrix in its parametrized form using equation (S.2).

Return to step 4 and continue until the halting condition is met. Validations and Motor
Volume calculations are halted after a certain solution time; while the MAP results are halted when a steady state radius of curvature is achieved The numerical integration of the equations of motion iterated until it reached tolerances of 10 .

Swimming model verification
To confirm that our reduced-order model can estimate the forces and torques present in ctenophore swimming, we will compare the model predictions to freely swimming ctenophores.We considered two experimentally observed turning sequences: 1) mode 1, with only two active ctene rows, and 2) mode 3, with 6 rows beating at a higher frequency.According to our definition of beat frequency ( = 1/, where  is the time between two power strokes), the animals can change the beat frequency for each beat cycle; however, the reduced-order model requires a frequency input at a higher time resolution.Fig 3 shows the observed beat frequencies for the two active ctene rows for Sequence 1, measured by counting the beat period of ctenes on the three camera views (blue dots).We artificially increased the time resolution by fitting a spline to the measured beat frequencies (black line).This approach results in a continuously varying paddle speed instead of the discrete beat frequency of the animals; however, it also smooths out the rapid oscillations that appear to be present (see Fig. 3) in the recorded sequence.We note that these oscillations are largely an artefact of the way we have defined frequency, which dictate that it can only change once per beat cycle).This ensures that overall swimming trajectories remain smooth, but may miss any physical consequences of these apparent rapid changes in frequency.
The beat sequence for mode 1 has   ⁄ = 0.13 and  = 0.4 /.We run our reduced-order swimming model based on these observations and the morphometrics reported in Table 2.Both comparisons show that our highly simplified mathematical model can predict propulsion and opposing forces/torques similar to those experienced by a swimming ctenophore.Therefore, we are justified in using this model for our parametric exploration of the maneuverability and agility of the ctenophore body plan and locomotion strategy.

Fig S1 .
Fig S1.Schematic of a ctenophore's simplified geometry moving in a 3D space.The unit vectors  ,  , and

Fig
Fig S2. (A) Graphical description of the spatial asymmetry overlaid on the ctene lateral profile time series.(B) Simplified elliptical trajectory (blue line, ( ,  )) and the oscillating flat plate (green line).Dotted red lines denote stroke amplitude (Φ), and  is the ctene length.(C) Top view of a modeled ctenophore, showing the tentacular and Fig 6A show the entire solution space (0.1 <  < 0.6 and 0.1 <  < 0.6).For this case, the

Fig S3 .
Fig S3.Beat frequency measurements for the mode 1 turning trajectory.(A) Snapshot of freely swimming ctenophore and the tracked points: apical organ (red) and tentacular bulbs (blue and green).(B) and (C) show the direct frequency measurements for ctene rows 4 and 5, respectively (bottom ctene rows).Dots represent measurements, and the fitted black line is used as an input to calculate the kinematics of the oscillating plates in the mathematical model.

Fig S4 .
Fig S4.Comparison between experimental measurements (red) and mathematical predictions (blue) for the mode 1 turning trajectory.(A) shows experimental vs predicted swimming trajectories.The shaded area shows the entire spatiotemporal solution space ( − ), while the blue line is the best model prediction ( = 0.2 and  = 0.6).(B) shows experimental vs predicted swimming orientation.The red triangles show the experimental positions for the tentacular bulbs and the apical organ for different timepoints t = 0, 1.5, and 2.5s.The blue triangles are the best fit ( = 0.2 and  = 0.6) predicted positions for the same time instants.

Fig S5 .
Fig S5.Beat frequency measurements for the mode 3 turning trajectory.(A) Snapshot of freely swimming ctenophore and the tracked points: apical organ (red) and tentacular bulbs (blue and green).(B) to (I) show the direct frequency measurements for ctene rows 1 to 8. Dots represent measurements, and the fitted black line is used as an input to calculate the kinematics of the oscillating plates in the mathematical model.

Fig
Fig S6.Comparison between experimental measurements (red) and mathematical predictions (blue) for the mode 3 turning trajectory.(A) shows experimental vs predicted swimming trajectories.The shaded area shows the entire spatiotemporal solution space ( − ), while the blue line is the best model prediction ( = 0.18 and  = 0.18).(B) shows experimental vs predicted swimming orientation.The red triangles show the experimental positions for the tentacular bulbs and the apical organ for different timepointst = 0, 1.5, and 3s.The blue triangles are the best fit ( = 0.18 and  = 0.18) predicted positions for the same time instants.

Table S1 .
Values for the torque coefficient expression along the roll and pitch/yaw directions

Table S2 .
Morphometric measurements of observed animals