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First published online September 19, 2008
Journal of Experimental Biology 211, 3181-3194 (2008)
Published by The Company of Biologists 2008
doi: 10.1242/jeb.019844
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The kinematic determinants of anuran swimming performance: an inverse and forward dynamics approach

Christopher T. Richards

Concord Field Station, Department of Organismic and Evolutionary Biology, Harvard University, Bedford, MA 01730, USA


Figure 1
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  		Fig. 1. Anuran model morphology. (A) Dorsal view showing an ellipsoid body and cylindrical leg segments connected to two thin plate feet. Foot velocity components are also shown: cranio-caudal translational velocity (vt), medio-lateral translational velocity (vl) and rotational velocity (vr). The foot angle ({theta}f) is defined with respect to the body midline, and vCOM denotes the forward swimming velocity of the body (COM, center of mass). Joint angles at the hip ({theta}h), knee ({theta}k) and ankle ({theta}a) are indicated with shaded discs. (B) Posterior view showing the shape and dimensions of the feet. The coordinate system used for added mass calculations is shown by arrows for aft-directed translation (x), lateral translation (y) and a rotational axis (z) about the ankle. A separate coordinate system is used for the motion of the body. Velocity in the direction of the arrows is defined as positive in their respective coordinate systems. All dimensions shown correspond to measurements taken from frog 1. Note that all limb movements are constrained to occur in the frontal (x–y) plane. a–c, dimensions of the foot.

 

Figure 2
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  		Fig. 2. Verification of the numerical model. Observed Xenopus laevis swimming velocity (solid gray lines) and simulated velocity traces (dashed black lines) of three representative power strokes during (A) slow swimming, (B) moderate speed swimming and (C) fast swimming. Note that each stroke occurred over a different duration. (D) Relative model error [100%x(simulated velocity – observed velocity)/peak observed stroke velocity] at 10 time points of the power stroke (mean ± s.d., N=6 strokes; frog 1). Because stroke durations were variable, the 10 time points were normalized by the total duration of each power stroke. Note that all data points are, on average, slightly below zero relative error, indicating that the model generally underestimates swimming velocity at each time point during the stroke. (E) Simulated relative swimming velocity (simulated swimming velocity/observed peak stroke velocity) vs observed relative swimming velocity (observed swimming velocity/observed peak stroke velocity) for the six swimming strokes represented in D, from frog 1. Different symbols represent different strokes.

 

Figure 3
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  		Fig. 3. Representative traces of Xenopus laevis kinematics for two power strokes. Each stroke was defined as the period of positive COM acceleration (i.e. the period between the onset of swimming and peak COM velocity). Foot translational velocity (solid red line) and rotational velocity (blue line) during (A) slow swimming and (B) fast swimming. Swimming velocity traces (dashed black line) were scaled to fit the data range shown. For A and B, outlines of X. laevis are shown at various stages during the power stroke: I, onset of swimming; II, peak COM acceleration; III, peak translational foot velocity; IV, peak rotational foot velocity; V, peak COM velocity (defined as the end of the power stroke). Joint angles for the hip (solid line), knee (dashed line), ankle (dash-dot line) and foot (dotted line) are shown in the lower panels of A and B.

 

Figure 4
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  		Fig. 4. Components of thrust in Xenopus laevis swimming. Total thrust (green line), translational thrust (red line) and rotational thrust (blue line) and swimming velocity (dashed black line) during (A) slow swimming and (B) fast swimming. (C) Total thrust (green line), added mass-based thrust (red line) and drag-based thrust (blue line) during slow swimming and (D) fast swimming. (E) Total added mass-based thrust (solid red line), and translational (dashed red line) and rotational added mass-based thrust (dotted red line) during slow swimming and (F) fast swimming. (G) Total drag-based thrust (solid blue line), and translational (dashed blue line) and rotational drag-based thrust (dotted blue line) during slow swimming and (H) fast swimming. Note that the green total thrust traces are identical in A and C as well as in B and D.

 

Figure 5
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  		Fig. 5. Simulated anuran swimming. Power strokes are shown for three different conditions: (A) maximum foot rotation with no translation, (B) 50% maximum translation and rotation, (C) maximum translation with no rotation. For all conditions, top panels show swimming velocity traces (dashed black line), total thrust (green line), and translational (red line) and rotational components of thrust (blue line). Bottom panels show total thrust (green line), and added mass-based (red line) and drag-based (blue line) thrust. Translational and rotational velocities are in phase for all conditions shown. Note that total thrust traces are identical for top and bottom panels.

 

Figure 6
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  		Fig. 6. Simulated anuran swimming during maximal foot translation and rotation. The top panel shows swimming velocity traces (dashed black line), total thrust (green line), and translational (red line) and rotational components of thrust (blue line). The bottom panel shows total thrust (green line), and added mass-based (red line) and drag-based (blue line) thrust. Translational and rotational velocities are in phase for all conditions shown. Note that total thrust traces are identical for top and bottom panels.

 

Figure 7
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  		Fig. 7. Swimming performance space. Hypothetical maps of anuran swimming performance were generated from forward dynamic simulations run across a range of kinematic input conditions. Since only two input parameters were varied (amplitudes of trigonometric functions describing translational and rotational foot displacement), a 3D hypothetical space could be systematically explored by mapping a single output (such as peak simulated swimming velocity) against each of the two independently varied inputs: peak rotational (x-axis) and peak translational (y-axis) foot velocity. The initial foot angle for each simulation in the performance space was derived such that the foot is 90 deg. to flow at mid stroke (t0.5) to decouple the effects of foot translation vs rotation. The color scale (from 0 to 100%) shows two performance parameters: (A) peak stroke swimming velocity and (B) glide velocity (the velocity at the end of the power stroke). Contour lines inclined more horizontally indicate a higher dependence on translational velocity, whereas more vertical lines indicate a higher dependence on rotational velocity. Arrows in A and B show different examples in the parameter space: arrow I shows large foot translation and small rotation, whereas arrow II shows large rotation and small translation. The corresponding diagrams show the path of foot motion throughout each simulated power stroke. These examples illustrate how two contrasting stroke patterns can have identical peak velocities (50% maximum) yet different glide velocities (~20% maximum vs 40% for examples I and II, respectively).

 

Figure 8
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  		Fig. 8. Predicted mechanical work and efficiency of anuran swimming. As in Fig. 7, contour plots show peak rotational (x-axis) and peak translational velocity (y-axis) resulting from incrementally varying the input translational and rotational displacements in the numerical model. The color scale (from 0 to 100%) shows (A) net COM work (hydrodynamic and inertial forces acting on the body x total distance traveled) over each power stroke, (B) total net joint work (the sum of work produced at the hip, knee and ankle) over each power stroke, (C) net efficiency (net COM work/total net joint work) and (D) percentage deceleration [100%x(peak COM velocity – final COM velocity)/peak COM velocity].

 

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© The Company of Biologists Ltd 2008