Christopher T. Richards

Kinematics and hydrodynamics analysis of swimming anurans reveals striking inter-specific differences in the mechanism for producing thrust

SUMMARY This study aimed to compare the swimming kinematics and hydrodynamics within and among aquatic and semi-aquatic/terrestrial frogs. High-speed video was used to obtain kinematics of the leg joints and feet as animals swam freely across their natural range of speeds. Blade element analysis was then used to model the hydrodynamic thrust as a function of foot kinematics. Two purely aquatic frogs, Xenopus laevis and Hymenochirus boettgeri, were compared with two semi-aquatic/terrestrial frogs, Rana pipiens and Bufo americanus. The four species performed similarly. Among swimming strokes, peak stroke velocity ranged from 3.3±1.1 to 20.9±2.5, from 6.8±2.1 to 28.6±3.7 and from 4.9±0.5 to 20.9±4.1 body lengths per second (BL s−1) in X. laevis, H. boettgeri and R. pipiens, respectively (means ± s.d.; N=4 frogs for each). B. americanus swam much more slowly at 3.1±0.3 to 7.0±2.0 BL s−1 (N=3 frogs). Time-varying joint kinematics patterns were superficially similar among species. Because foot kinematics result from the cumulative motion of joints proximal to the feet, small differences in time-varying joint kinematics among species resulted in species-specific foot kinematics (therefore hydrodynamics) patterns. To obtain a simple measure of the hydrodynamically useful motion of the foot, this study uses ‘effective foot velocity’ (EFV): a measure of the component of foot velocity along the axis of swimming. Resolving EFV into translational and rotational components allows predictions of species-specific propulsion strategies. Additionally, a novel kinematic analysis is presented here that enables the partitioning of translational and rotational foot velocity into velocity components contributed by extension at each individual limb joint. Data from the kinematics analysis show that R. pipiens and B. americanus translated their feet faster than their body moved forward, resulting in positive net translational EFV. Conversely, translational EFV was slower than the body velocity in H. boettgeri and X. laevis, resulting in negative net translational EFV. Consequently, the translational component of thrust (caused mostly by hip, knee and ankle extension) was twofold higher than rotational thrust in Rana pipiens. Likewise, rotational components of thrust were nearly twofold higher than translational components in H. boettgeri. X. laevis, however, was the most skewed species observed, generating nearly 100% of total thrust by foot rotation generated by hip, ankle and tmt extension. Thus, this study presents a simple kinematics analysis that is predictive of hydrodynamic differences among species. Such differences in kinematics reveal a continuum of different propulsive strategies ranging from mostly rotation-powered (X. laevis) to mostly translation-powered (R. pipiens) swimming.

A bio-robotic platform for integrating internal and external mechanics during muscle-powered swimming

To explore the interplay between muscle function and propulsor shape in swimming animals, we built a robotic foot to mimic the morphology and hind limb kinematics of Xenopus laevis frogs. Four foot shapes ranging from low aspect ratio (AR = 0.74) to high (AR = 5) were compared to test whether low-AR feet produce higher propulsive drag force resulting in faster swimming. Using feedback loops, two complementary control modes were used to rotate the foot: force was transmitted to the foot either from (1) a living plantaris longus (PL) muscle stimulated in vitro or (2) an in silico mathematical model of the PL. To mimic forward swimming, foot translation was calculated in real time from fluid force measured at the foot. Therefore, bio-robot swimming emerged from muscle-fluid interactions via the feedback loop. Among in vitro-robotic trials, muscle impulse ranged from 0.12 ± 0.002 to 0.18 ± 0.007 N s and swimming velocities from 0.41 ± 0.01 to 0.43 ± 0.00 m s(-1), similar to in vivo values from prior studies. Trends in in silico-robotic data mirrored in vitro-robotic observations. Increasing AR caused a small (∼10%) increase in peak bio-robot swimming velocity. In contrast, muscle force-velocity effects were strongly dependent on foot shape. Between low- and high-AR feet, muscle impulse increased ∼50%, while peak shortening velocity decreased ∼50% resulting in a ∼20% increase in net work. However, muscle-propulsion efficiency (body center of mass work/muscle work) remained independent of AR. Thus, we demonstrate how our experimental technique is useful for quantifying the complex interplay among limb morphology, muscle mechanics and hydrodynamics.

The kinematic determinants of anuran swimming performance: an inverse and forward dynamics approach.

SUMMARY The aims of this study were to explore the hydrodynamic mechanism of Xenopus laevis swimming and to describe how hind limb kinematics shift to control swimming performance. Kinematics of the joints, feet and body were obtained from high speed video of X. laevis frogs (N=4) during swimming over a range of speeds. A blade element approach was used to estimate thrust produced by both translational and rotational components of foot velocity. Peak thrust from the feet ranged from 0.09 to 0.69 N across speeds ranging from 0.28 to 1.2 m s–1. Among 23 swimming strokes, net thrust impulse from rotational foot motion was significantly higher than net translational thrust impulse, ranging from 6.1 to 29.3 N ms, compared with a range of –7.0 to 4.1 N ms from foot translation. Additionally, X. laevis kinematics were used as a basis for a forward dynamic anuran swimming model. Input joint kinematics were modulated to independently vary the magnitudes of foot translational and rotational velocity. Simulations predicted that maximum swimming velocity (among all of the kinematics patterns tested) requires that maximal translational and maximal rotational foot velocity act in phase. However, consistent with experimental kinematics, translational and rotational motion contributed unequally to total thrust. The simulation powered purely by foot translation reached a lower peak stroke velocity than the pure rotational case (0.38 vs 0.54 m s–1). In all simulations, thrust from the foot was positive for the first half of the power stroke, but negative for the second half. Pure translational foot motion caused greater negative thrust (70% of peak positive thrust) compared with pure rotational simulation (35% peak positive thrust) suggesting that translational motion is propulsive only in the early stages of joint extension. Later in the power stroke, thrust produced by foot rotation overcomes negative thrust (due to translation). Hydrodynamic analysis from X. laevis as well as forward dynamics give insight into the differential roles of translational and rotational foot motion in the aquatic propulsion of anurans, providing a mechanistic link between joint kinematics and swimming performance.

Modulation of in vivo muscle power output during swimming in the African clawed frog (Xenopus laevis).

SUMMARY The goal of this study is to explore how swimming animals produce the wide range of performance that is seen across their natural behaviors. In vivo recordings of plantaris longus muscle length change were obtained by sonomicrometry. Simultaneous with muscle length data, force measurements were obtained using a novel tendon buckle force transducer placed on the Achilles tendon of Xenopus laevis frogs during brief accelerating bursts of swimming. In vivo work loops revealed that the plantaris generates a variable amount of positive muscle work over a range of swimming cycle durations (from 0.23 to 0.76 s), resulting in a large range of cycle power output (from 2.32 to 74.17 W kg–1 muscle). Cycle duration correlated negatively with cycle power, and cycle work correlated positively (varying as a function of peak cycle stress and, to a much lesser extent, fascicle strain amplitude). However, variation in cycle duration only contributed to 12% of variation in power, with cycle work accounting for the remaining 88%. Peak cycle stress and strain amplitude were also highly variable, yet peak stress was a much stronger predictor of cycle work than strain amplitude. Additionally, EMG intensity correlated positively with peak muscle stress (r2=0.53). Although the timing of muscle recruitment (EMG phase and EMG duty cycle) varied considerably within and among frogs, neither parameter correlated strongly with cycle power, cycle work, peak cycle stress or strain amplitude. These results suggest that relatively few parameters (cycle duration, peak cycle stress and strain amplitude) vary to permit a wide range of muscle power output, which allows anurans to swim over a large range of velocities and accelerations.

Building a robotic link between muscle dynamics and hydrodynamics.

SUMMARY This study used a novel feedback approach to control a robotic foot using force and length signals transmitted from an isolated Xenopus laevis frog muscle. The foots environment (inertial versus hydrodynamic), gearing (outlever/inlever) and size were changed to alter the muscles load. Upon nerve stimulation (250 Hz, 80 ms train duration), variation in loading generated a range of muscle stress (19.8±5.3 to 66.0±22.5 kPa), work (1.89±0.67 to 6.87±2.96 J kg–1 muscle) and power (12.4±7.5 to 64.8±28.3 W kg–1 muscle; mean ± s.d., N=6 frogs). Inertial versus hydrodynamic loading dramatically shifted contractile dynamics. With the foot in water, the muscle generated ∼30% higher force, yet shortened slower, producing lower power than inertial loading. Power increased in air from 22.6±5.8 to 63.6±27.2 W kg–1 muscle in response to doubling the gear ratio, but did not increase in water. Surprisingly, altering foot size diminished muscle performance in water, causing power to drop significantly from 41.6±11.1 to 25.1±8.0 W kg–1 muscle as foot area was doubled. Thus, morphological modifications influenced muscle dynamics independently of neural control; however, changes in loading environment and gearing affected contractile output more strongly than changes in foot size. Confirming recent theory, these findings demonstrate how muscle contractile output can be modulated solely by altering the mechanical environment.

Muscle function and hydrodynamics limit power and speed in swimming frogs

Studies of the muscle force-velocity relationship and its derived n-shaped power-velocity curve offer important insights into muscular limits of performance. Given the power is maximal at 1/3 V(max), geometric scaling of muscle force coupled with fluid drag force implies that this optimal muscle-shortening velocity for power cannot be maintained across the natural body-size range. Instead, muscle velocity may decrease with increasing body size, conferring a similar n-shaped power curve with body size. Here we examine swimming speed and muscle function in the aquatic frog Xenopus laevis. Swimming speed shows an n-shaped scaling relationship, peaking at 47.35 g. Further, in vitro muscle function of the ankle extensor plantaris longus also shows an optimal body mass for muscle power output (47.27 g), reflecting that of swimming speed. These findings suggest that in drag-based aquatic systems, muscle-environment interactions vary with body size, limiting both the muscles potential to produce power and the swimming speed.

Built for rowing: frog muscle is tuned to limb morphology to power swimming

Rowing is demanding, in part, because drag on the oars increases as the square of their speed. Hence, as muscles shorten faster, their force capacity falls, whereas drag rises. How do frogs resolve this dilemma to swim rapidly? We predicted that shortening velocity cannot exceed a terminal velocity where muscle and fluid torques balance. This terminal velocity, which is below Vmax, depends on gear ratio (GR = outlever/inlever) and webbed foot area. Perhaps such properties of swimmers are ‘tuned’, enabling shortening speeds of approximately 0.3Vmax for maximal power. Predictions were tested using a ‘musculo-robotic’ Xenopus laevis foot driven either by a living in vitro or computational in silico plantaris longus muscle. Experiments verified predictions. Our principle finding is that GR ranges from 11.5 to 20 near the predicted optimum for rowing (GR ≈ 11). However, gearing influences muscle power more strongly than foot area. No single morphology is optimal for producing muscle power. Rather, the ‘optimal’ GR decreases with foot size, implying that rowing ability need not compromise jumping (and vice versa). Thus, despite our neglect of additional forces (e.g. added mass), our model predicts pairings of physiological and morphological properties to confer effective rowing. Beyond frogs, the model may apply across a range of size and complexity from aquatic insects to human-powered rowing.

There is always a trade-off between speed and force in a lever system: comment on McHenry (2010)

In a recent Biology Letters article, McHenry [[1][1]] makes a distinction between levers that operate under ‘quasi-static’ and ‘dynamic’ conditions, concluding that ‘no trade-off between force and velocity exists in a lever with spring–mass dynamics’. As evidence, McHenry uses a

Inverse dynamic modelling of jumping in the red-legged running frog Kassina maculata

ABSTRACT Although the red-legged running frog, Kassina maculata, is secondarily a walker/runner, it retains the capacity for multiple locomotor modes, including jumping at a wide range of angles (nearly 70 deg). Using simultaneous hind limb kinematics and single-foot ground reaction forces, we performed inverse dynamics analyses to calculate moment arms and torques about the hind limb joints during jumping at different angles in K. maculata. We show that forward thrust is generated primarily at the hip and ankle, while body elevation is primarily driven by the ankle. Steeper jumps are achieved by increased thrust at the hip and ankle and greater downward rotation of the distal limb segments. Because of its proximity to the GRF vector, knee posture appears to be important in controlling torque directions about this joint and, potentially, torque magnitudes at more distal joints. Other factors correlated with higher jump angles include increased body angle in the preparatory phase, faster joint openings and increased joint excursion, higher ventrally directed force, and greater acceleration and velocity. Finally, we demonstrate that jumping performance in K. maculata does not appear to be compromised by presumed adaptation to walking/running. Our results provide new insights into how frogs engage in a wide range of locomotor behaviours and the multi-functionality of anuran limbs. Summary: Experimental data and inverse dynamic modelling demonstrate how forward thrust and elevation are produced in the frog hind limb, allowing frogs to jump at a wide range of angles.