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

Transcription

1 38 The Journal of Experimental Biology 2, Published by The Company of Biologists 28 doi:.242/jeb.9844 The kinemati determinants of anuran swimming performane: an inverse and forward dynamis approah Christopher T. Rihards Conord Field Station, Department of Organismi and Evolutionary Biology, Harvard University, Bedford, MA 73, USA Aepted 2 August 28 SUMMARY The aims of this study were to explore the hydrodynami mehanism of Xenopus laevis swimming and to desribe how hind limb kinematis shift to ontrol swimming performane. Kinematis 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 approah was used to estimate thrust produed by both translational and rotational omponents of foot veloity. Peak thrust from the feet ranged from.9 to.69 N aross speeds ranging from.28 to.2 m s. Among 23 swimming strokes, net thrust impulse from rotational foot motion was signifiantly higher than net translational thrust impulse, ranging from 6. to 29.3 N ms, ompared with a range of 7. to 4. N ms from foot translation. Additionally, X. laevis kinematis were used as a basis for a forward dynami anuran swimming model. Input joint kinematis were modulated to independently vary the magnitudes of foot translational and rotational veloity. Simulations predited that maximum swimming veloity (among all of the kinematis patterns tested) requires that maximal translational and maximal rotational foot veloity at in phase. However, onsistent with experimental kinematis, translational and rotational motion ontributed unequally to total thrust. The simulation powered purely by foot translation reahed a lower peak stroke veloity than the pure rotational ase (.38 vs.54 m s ). In all simulations, thrust from the foot was positive for the first half of the power stroke, but negative for the seond half. Pure translational foot motion aused greater negative thrust (7% of peak positive thrust) ompared 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 produed by foot rotation overomes negative thrust (due to translation). Hydrodynami analysis from X. laevis as well as forward dynamis give insight into the differential roles of translational and rotational foot motion in the aquati propulsion of anurans, providing a mehanisti link between joint kinematis and swimming performane. Key words: blade element model, forward dynami model, hydrodynamis, frog, Xenopus laevis INTRODUCTION Aquati invertebrates and vertebrates ommonly use osillating paired limbs for propulsion. For example, rowing animals suh as larval insets, fish and mammals rotate their limbs in a ranio-audal diretion to reate thrust (e.g. Blake, 985; Blake, 979; Fish et al., 997). These animals, often alled drag-based swimmers, rely on resistive hydrodynami fores to swim (Daniel, 984; Vogel, 994). Alternatively, other paired-limb swimmers generate lift-based propulsion by the use of modified morphology and kinematis allowing propulsive limbs to move dorso-ventrally (e.g. Walker and Westneat, 22; Johansson and Norberg, 23). This has stimulated studies investigating how lift- vs drag-based swimming relates to propulsive effiieny and swimming speed (Vogel, 994; Fish, 996; Walker and Westneat, 2). More reent work has proposed that frog feet generate lift; however, no evidene for lift-based propulsion has been found in swimming anurans (Johansson and Lauder, 24; Nauwelaerts et al., 25). More broadly, these studies relate the kinematis of limb motion to hydrodynamis as well as swimming performane. Using swimming frogs as a model, the urrent study expands on previous work by exploring how swimmers ontrol different omponents of motion (i.e. foot rotation and translation) in order to modulate hydrodynami fores and swimming performane. In addition to simple ranio-audal rotation in rowing, aquati tetrapod limbs use joints to further ontrol the propulsor s position with respet to the body. For example, swimming turtles use proximal joints to ontrol the medio-lateral position of the forefeet to maximize drag-based thrust during audal limb rotation, but minimize drag during the reovery stroke (Pae et al., 2). Diving grebes (Podieps ristatus) also benefit from the additional range of motion, generating lift-based thrust by using proximal joints (ausing bakward and upward foot motion) while rotating the feet at distal joints (Johansson and Lindhe Norberg, 2). Given that jointed limbs onfer diverse swimming modes among speies, is kinemati variability a means for ontrolling swimming performane within a speies? In addition, do the relative roles of limb joints shift aross different swimming behaviors to enable a broad range of performane within individuals? Studies of terrestrial loomotion have addressed how the funtions of different limb joints hange to enable inreases in speed (e.g. Dutto et al., 26), inline (e.g. Roberts and Belliveau, 25), aeleration (Roberts and Sales, 24; MGowan et al., 25) and stabilizing responses to substrate height perturbations (Daley et al., 27). Suh studies have shown that partitioning of limb funtion (e.g. mehanial work prodution, absorption, stabilization) ours aross individual limb joints. For example, in wallabies, the ankle serves to store and return elasti energy during steady speed loomotion (Biewener and Baudinette, 995). However, during aeleration the roles of hind limb joints in turkeys and wallabies hange, with the ankle providing most of the inreased mehanial

2 382 C. T. Rihards work required to inrease speed (Roberts and Sales, 24; MGowan et al., 25). Similarly, the ankle shifts from elasti energy reovery (produing little net joint work) during steady level running in guinea fowl, to energy absorption following an unexpeted drop in substrate height (Daley et al., 27). By analogy, limb joints during swimming may also have distint funtions (e.g. work prodution, energy transmission between joints, or joint stabilization). Presumably, these roles an hange aording to varying mehanial demands aross different swimming tasks (e.g. predator esape, prey apture and steady swimming). Understanding how musuloskeletal dynamis enable diverse swimming behaviors is therefore important for understanding the evolutionary and eologial diversity of aquati vertebrates. Aquati frogs are ideal models for exploring the differential use of limb joints to modulate swimming performane. For example, work by Nauwelaerts and Aerts addressed funtions of anuran hind limb joints in swimming vs jumping to explore how hind limb mehanis enable funtion aross eologial performane spae (Nauwelaerts and Aerts, 23). They used a novel and elegant approah of analyzing joint kinematis patterns as funtions of both propulsive impulse ( loomotor effort ) and loomotor mode. Their findings demonstrate that kinemati variation within a loomotor mode (explained by variation in propulsive impulse) an onfound omparisons between jumping and swimming kinematis. Consequently, their work gives ompelling evidene that anurans modulate limb kinematis to enable a range of performane within as well as between loomotor modes. However, the mehanisti link between time-varying patterns of joint motion and performane has not yet been expliitly examined in swimming frogs. Given the potential range of kinematis patterns available to frog hind limbs (Kargo and Rome, 22), resolving the funtional roles of individual joints may be a daunting task. However, frog hind limbs move mostly in the frontal plane during swimming (i.e. within the plane defined by the ranio-audal and medio-lateral axes) (Peters et al., 996). Therefore, the joint motions an be summed into three omponents: ranio-audal foot translation, medio-lateral foot translation (eah aused by hip and knee rotation) and ranio-audal foot rotation (from ankle and tarsometatarsal joint rotation). Several reent studies have speulated on the importane of translational foot motion, observing that the foot is swept through the water at nearly 9deg. to flow for most of the power stroke, with rotation delayed towards the end of limb extension (Peters et al., 996; Nauwelaerts et al., 25). This suggests that foot rotation (via ankle extension) need not diretly aid in propulsion. Instead, the role of foot rotation may be to straighten the foot parallel to flow to minimize drag just prior to the glide phase (Peters et al., 996; Johansson and Lauder, 24; Nauwelaerts et al., 25). Johansson and Lauder further suggest that foot rotation serves to shed the attahed vortex from the foot, minimizing a retarding hydrodynami fore inurred from fluid added mass as the foot deelerates late in the propulsive phase (Johansson and Lauder, 24). Building on this earlier work, my study tests the hypothesis that propulsion in Xenopus laevis is powered primarily by hip and knee extension (ausing foot translation) rather than foot rotation produed at the ankle. Inreases in speed from stroke to stroke, therefore, are expeted to be powered mainly by inreases in translational thrust from the foot. For the present study, a blade element model modified from an earlier study (Gal and Blake, 988b) was used to disset the omponents of thrust due to foot translation and rotation. Additionally, the blade element kinemati analysis was oupled with a forward dynami approah to reate a generalized anuran swimming model. This simulation allowed the modulation of swimming performane through manipulation of hind limb kinemati patterns. Along with a prior study of plantaris longus musle funtion during X. laevis swimming (Rihards and Biewener, 27), the urrent study provides a framework for interpreting the role of musle funtion in the ontext of the omplex kinematis of jointed appendages. MATERIALS AND METHODS Anuran swimming model Spatial dimensions of the anuran swimming model were based on morphologial measurements obtained from adult male Xenopus laevis (Daudin 82) frogs (25.5±3.8 g mean ± s.d. body mass; 6.±.5 m snout vent length; N=4 frogs). Frog morphology was modeled as an ellipsoid body attahed to two legs, eah with pin joints at the hip, knee and ankle onneted by two ylindrial segments (Fig. ). Foot area was alulated digitally by traing an image of individual Xenopus laevis feet (spread flat on a white surfae) using Sion Image (Sion Corporation, Frederik, MD, USA). A trapezoidal flat plate of the same foot area was used in the model to approximate the foot s shape. Model alulations onsisted of two parts: () an inverse approah, whih estimated the hydrodynami thrust fores at the feet based on presribed joint kinematis (based on X. laevis kinematis) and (2) a forward approah, whih simulated the swimming veloity profile of the frog due to the hydrodynami and inertial fores ating on the body. Inverse model: estimating propulsive fores from joint kinematis input Thrust was estimated as the sum of two independent hydrodynami fores ating at the feet: drag and added mass (Daniel, 984; Gal and Blake, 988b). In this model, the feet were the only propulsive surfaes (i.e. propulsive hydrodynami effets of the ylindrial leg segments were not onsidered). Propulsion was driven by extension of the hip and knee, ausing both lateral and aft-direted foot translation, as well as at the ankle, ausing foot rotation. All equations, therefore, ould be expressed in terms of translational veloity aft to the enter of mass (v t ), lateral translational veloity (v l ), rotational veloity about the ankle joint (v r ) and veloity of the enter of mass (v COM ) (Fig.). In the urrent study, all veloity omponents of the hind limb were defined with respet to the oordinate system illustrated in Fig.. Therefore, aft-direted foot translational veloity was positive. The drag-based thrust fore on eah foot was estimated from a blade element model modified from (Gal and Blake, 988b): T Drag =ρc D sinθ f ( b a r + a) [rv r + (v t v COM )] 2 d r, () where ρ is the water density, θ f is the foot angle measured from the body midline, r is the distane along the foot, v r and v t are the veloity omponents defined above and a, b and are dimensions of the foot (assumed to be symmetri about its mid-axis; see Fig.). Due to a lak of published literature addressing the oeffiient of drag (C D ) of a translating and rotating plate in the range of Reynolds number (Re) of Xenopus laevis feet (Re ~ to 2,), C D was set onstant at 2., the maximum value for a flat plate at 9deg. angle of attak at Re= 3 (Andersen et al., 25). A sensitivity analysis to C D was performed by running 5 model iterations while randomly varying the C D between. (Gal and Blake, 988b) and 2. at eah iteration. All other input parameters (e.g. foot kinematis, added mass oeffiients) were left unhanged. Simulated variation in C D resulted in a negligible (<2%) variation in the net drag produed during the

3 { Modeling anuran swimming performane 383 A x body B z v COM y body 3. m 6.5 m.9 m { Thigh 2.6 m θ h Hip Foot Shank a { b y foot 2.6 m θ k Knee x foot v r v l Ankle θ a θ f y 3.8 m v t x Fig.. Anuran model morphology. (A) Dorsal view showing an ellipsoid body and ylindrial leg segments onneted to two thin plate feet. Foot veloity omponents are also shown: ranio-audal translational veloity (v t ), medio-lateral translational veloity (v l ) and rotational veloity (v r ). The foot angle (θ f ) is defined with respet to the body midline, and v COM denotes the forward swimming veloity of the body (COM, enter of mass). Joint angles at the hip (θ h ), knee (θ k ) and ankle (θ a ) are indiated with shaded diss. (B) Posterior view showing the shape and dimensions of the feet. The oordinate system used for added mass alulations is shown by arrows for aft-direted translation (x), lateral translation (y) and a rotational axis (z) about the ankle. A separate oordinate system is used for the motion of the body. Veloity in the diretion of the arrows is defined as positive in their respetive oordinate systems. All dimensions shown orrespond to measurements taken from frog. Note that all limb movements are onstrained to our in the frontal (x y) plane. a, dimensions of the foot. stroke, suggesting that the findings predited by the model are insensitive to variation in C D within the range of. to 2.. From the added mass oeffiients (m), added mass thrust was alulated (see Appendix A) as: where v n is net translational veloity (v t v COM ), with total thrust produed at the feet alulated as: Forward model: simulating swimming veloity from joint kinematis input A forward dynamis approah was used to omputationally solve the time-varying aeleration and veloity of the frog body due to the time-varying thrust estimated at the foot (Eqns and 2). The following equations were used (Nauwelaerts et al., 2): given: T amass = 2( v n m + v l v r m 22 + v r m 6 ), (2) v frog = T = T Drag + T amass. (3) m frog ( + C ) amass (T + D), (4) D = 2 ρ AC D,body v frog 2, (5) where A is the area of the frog body projeted onto the animal s transverse plane, C D,body is the body oeffiient of drag, C amass is the added mass oeffiient of the body, m frog is the frog mass, T is the propulsive fore produed by both feet (Eqn 3 above), D is the drag on the frog body and v frog is the frog s simulated swimming veloity. Coeffiient values C amass =.2 and C D,frog =.4 were taken from previous studies (Nauwelaerts et al., 2; Nauwelaerts and Aerts, 23). Sine swimming aeleration and thrust are funtions of one another, the oupled ordinary differential equations (Eqns, 4 and 5) were solved simultaneously using a numerial equation solver in Mathematia 6. (Wolfram Researh, Champaign, IL, USA). Model verifiation: prediting swimming veloity from foot kinematis To verify the numerial model, Xenopus laevis swimming was reorded for four individuals aross the entire range of their performane (from slow swimming to rapid esape swimming). Joint kinematis data were measured from video sequenes filmed from a dorsal view at 25framess with a /25s shutter speed using a high speed amera (Photron, San Diego, CA, USA), as detailed in a previous study (Rihards and Biewener, 27). Small plasti markers (.5m diameter) were plaed on the snout, vent, knee, ankle and tarsometatarsal joint using a yanoarylate adhesive. Foot kinematis were digitized in Matlab (The MathWorks, Natik, MA, USA) using a ustomized routine (DLTdataviewer 2. written by Tyson Hedrik). Only strokes with a straight swimming trajetory were analyzed. Due to the large mass of the legs (~% of body mass), the position of the enter of mass (COM) was assumed to vary depending on the position of the legs behind the body. Using a dead frog, the position of the COM (relative to the snout) was measured with the ankle joint moved to various distanes audal to the vent

4 384 C. T. Rihards Swimming veloity (m s ) A Observed X. laevis swimming Simulated swimming B C Time (s) Relative error (% peak veloity) D Relative time Simulated relative swimming veloity..8.6 E.4 Stroke, r 2 =.98 Stroke 2, r 2 =.95 Stroke 3, r 2 =.92.2 Stroke 4, r 2 =.99 Stroke 5, r 2 =.99 Stroke 6, r 2 = Observed relative swimming veloity Fig. 2. Verifiation of the numerial model. Observed Xenopus laevis swimming veloity (solid gray lines) and simulated veloity traes (dashed blak lines) of three representative power strokes during (A) slow swimming, (B) moderate speed swimming and (C) fast swimming. Note that eah stroke ourred over a different duration. (D) Relative model error [% (simulated veloity observed veloity)/peak observed stroke veloity] at time points of the power stroke (mean ± s.d., N=6 strokes; frog ). Beause stroke durations were variable, the time points were normalized by the total duration of eah power stroke. Note that all data points are, on average, slightly below zero relative error, indiating that the model generally underestimates swimming veloity at eah time point during the stroke. (E) Simulated relative swimming veloity (simulated swimming veloity/observed peak stroke veloity) vs observed relative swimming veloity (observed swimming veloity/observed peak stroke veloity) for the six swimming strokes represented in D, from frog. Different symbols represent different strokes. of the body (see Walter and Carrier, 22). Using the measured relationship between the position of the ankle (relative to the vent) and the COM on the dead frog, the instantaneous COM position on swimming frogs was estimated from the known aft translational displaement of the foot. To minimize error in setting the initial onditions of the forward dynami simulation, a subset of six swimming strokes were seleted where the animal began swimming from a rest position (initial COM veloity=). Joint kinematis traes were smoothed using a seond order forward bakward 2 Hz low-pass Butterworth filter before alulating translational and rotational veloities and aelerations of the foot. All data proessing was done in Labview 7. (National Instruments, Austin, TX, USA). Hind limb kinematis (θ f, v t, v r, v l, v t, v r, v l ) for six trials were then input into the swimming model (see above) to predit the frog s swimming veloity and aeleration output (Fig. 2A C). Only the power stroke, defined as the period of positive COM aeleration (i.e. the period between the onset of swimming and peak COM veloity), was analyzed. Simulated and observed COM veloity profiles were then ompared to verify the model. Estimating net joint work and hydrodynami effiieny As an index of overall musular effort required to extend the hip, knee and ankle to power swimming, estimates of net joint work were obtained by inverse dynamis (not to be onfused with the inverse approah used to estimate hydrodynami fores from foot kinematis; see above). The net work required at a given joint is the sum of internal work (from the inertia of the segments), external work (from the thrust reation fore at the foot), and hydrodynami work (from the hydrodynami fores ating diretly on the segments; see Appendix B). The thigh and shank were modeled as ylinders (Fig. A) with uniformly distributed mass, suh that the COM lies at the enter of eah segment length. Segment masses were measured from the Xenopus laevis frog used above. Internal and external moments were alulated aording to Biewener and Full (Biewener and Full, 992). The internal moment due to inertia of the thigh and shank was alulated as follows: where I is the segment s moment of inertia, m is the segment s mass, r is the distane from the joint s enter of rotation to the segment COM and α is the segment s angular aeleration, summed over i= to n joint segments distal to the joint of interest (see Appendix B for further details). The external moment required at eah joint to resist hydrodynami fores at the foot was alulated and the total moment at eah joint was then obtained: and n M internal = I i + m i r2 i α i, (6) i= M external = F foot R (7) M total = M internal + M external + M hydrodynami, (8) where F foot is the total estimated fore on the foot from the blade element model (see Gal and Blake, 988b), R is the perpendiular distane between the F foot vetor (ating at the enter of pressure on the foot) and the joint of interest, and M hydrodynami is the moment due

5 Modeling anuran swimming performane 385 to hydrodynami fores ating diretly on the segments (see Appendix B). Instantaneous power required to overome the total joint moment was modified from Roberts and Sales (Roberts and Sales, 24): where ω is the angular veloity of the joint of interest. To follow, the net work required to move a given joint is: where t ps is the duration of the power stroke. An estimate of hydrodynami effiieny was obtained by dividing the work done to move the COM (the time-averaged fore on the body during the power stroke the total distane traveled) by the estimated net joint work summed for all joints. Hypothetial performane spae The effets of varying the relative magnitude of translational and rotational foot veloity on swimming performane (e.g. peak stroke veloity) were explored by running simulated power strokes aross the entire observed range of translational and rotational veloity in 33 inrements, generating a matrix of unique input onditions (i.e. 89 simulated swimming strokes). Partial least squares regression was used to evaluate the relative ontributions of translational or rotational veloity on swimming performane (Rihards and Biewener, 27). Simulated foot kinematis The range of input onditions in all swimming simulations was bounded by maximum translational and rotational veloities of.8ms and 6rads, respetively, obtained from Xenopus laevis foot veloity measurements of 35 swimming strokes spanning the entire range of performane of frog. Similar veloity ranges were found in the other three individuals used in this study. The power stroke of anuran swimmers is often followed by a period where the joints are held in fixed positions while the frog glides. Sine the motion of the feet is impulsive (rather than periodi), simple sine/osine funtions are inadequate to desribe the translational and rotational foot motion patterns. Hyperboli tangent funtions (as opposed to sine or osine funtions) were therefore used to approximate time-varying patterns of translational and rotational displaement of the foot: and Rotation(t) = A r P total = M total ω, (9) W total = t ps P total dt, () () Translation(t) = A t 2 2 A t tanh π (t.5t ps ) A r +θ i, (2) t ps 2 φ π (t.5t ps tanh ) 2 where A t and A r are amplitudes of translational and angular displaements and t ps is the duration of the power stroke. For eah simulation, the initial foot angle, θ i, was derived suh that the foot angle was always at 9deg. to the swimming diretion at peak rotational and translational veloity. The phase angle between rotation and translation, φ, was set to deg. for all of the simulations in this study. Translational and rotational veloities were modulated by varying the amplitudes of foot translation and angular displaement while maintaining a onstant stroke period. t ps RESULTS Verifiation of the numerial simulation The numerial model reliably predited the temporal pattern of the swimming veloity profiles of six swimming strokes (Fig. 2E); however, the magnitude of veloity was slightly underestimated in most strokes observed (Fig. 2D). At any given time point the average perentage error between the simulated profile and the observed data ranged from 2±6% at the beginning of the stroke to 6±8% at the end of the stroke (mean ± s.d., N=6 swimming strokes, frog ; Fig. 2D). Most of the simulated veloity profiles averaged within ±5% of the observed data. Xenopus laevis foot kinematis In both representative slow and fast swimming strokes, foot veloity peaked prior to COM veloity (Fig.3A,B). For the slow swimming stroke, translation and rotation were out of phase for the duration of the power stroke, with peak foot translational and rotational veloities ourring at 38% and 8% of the power stroke duration, respetively (Fig. 3A). During fast swimming, in ontrast, translational and rotational foot veloity peaked in phase (Fig.3B). For these representative strokes, peak translational and rotational foot veloities inreased.7- and 2.-fold, respetively, from the slow to fast swimming speeds. Hydrodynamis of Xenopus laevis swimming: prodution of thrust by the feet From slow to fast swimming, peak thrust inreased from.9 to.49n and net thrust impulse inreased from 9. to 2.5Nms. During both slow and fast swimming thrust developed rapidly at the onset of foot motion, aelerating the animal s COM early in the stroke (Fig. 4). Near the end of eah stroke, as the hind limb approahed peak extension, thrust dereased rapidly and beame negative as the foot deelerated in translation and rotation. In all strokes observed, thrust peaked before COM veloity. However, the timing offset of peak thrust and peak swimming veloity varied among trials, as the time delay and relative magnitudes of translational and rotational foot veloity hanged among trials (Fig. 3; Fig. 4A,B; Table ). Kinemati omponents of thrust: translational and rotational thrust In all swimming strokes observed, propulsion was predominantly powered by rotational thrust. In all strokes observed, rotational impulse aounted for 93±24% of total thrust impulse (mean ± s.d., pooled data from 23 swimming strokes, N=4 frogs; Table 2). As exemplified by the two representative strokes hosen, the timevarying patterns of translational vs rotational thrust produed by the foot differed markedly in both slow and fast swimming. This variation aounted for differenes in the relative ontributions of rotational vs translational thrust to total thrust (Fig. 4A,B). Independent of swimming speed, translational thrust developed earliest, reahing a peak prior to rotational thrust. Subsequently, translational thrust of the foot rapidly diminished, beoming negative for the remainder of the stroke, resulting in a net negative translational impulse in the representative slow stroke. In ontrast, rotational thrust of the foot was a signifiant omponent of total thrust, peaking later in the stroke and remaining positive for the duration of the power stroke. Beause of the net negative translational impulse, total thrust impulse (translational + rotational impulse) was sometimes less than rotational impulse (Table2). From slow to fast swimming, peak translational thrust inreased 2.-fold from.4 to.8 N and peak rotational thrust inreased 4.4-fold from. to.48 N. In ontrast, net translational impulse dereased

6 Joint angle (deg.) 386 C. T. Rihards A Slow swimming B Fast swimming I II III IV V I II III, IV V Translational veloity (m s ) Rotational veloity (rad s ) Translational veloity Rotational veloity Saled COM veloity Translational veloity (m s ) Rotational veloity (rad s ) Hip 8 Knee Ankle 6 Foot Time (s) Time (s). Fig. 3. Representative traes of Xenopus laevis kinematis for two power strokes. Eah stroke was defined as the period of positive COM aeleration (i.e. the period between the onset of swimming and peak COM veloity). Foot translational veloity (solid red line) and rotational veloity (blue line) during (A) slow swimming and (B) fast swimming. Swimming veloity traes (dashed blak line) were saled 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 aeleration; III, peak translational foot veloity; IV, peak rotational foot veloity; V, peak COM veloity (defined as the end of the power stroke). Joint angles for the hip (solid line), knee (dashed line), ankle (dashdot line) and foot (dotted line) are shown in the lower panels of A and B. 3.-fold from.9 to 6.3 N ms, whereas rotational impulse inreased 2.5-fold from. to 27.Nms. Hydrodynami omponents of thrust: added mass and drag Similar to the temporal pattern of translational thrust, added massbased thrust only ontributed to propulsion early in the stroke (Fig.4C,D), rising to a peak in the first half of the propulsive period then dereasing to beome negative at the end of limb extension, resulting in a net positive impulse of. vs 6.Nms in representative slow vs fast swimming strokes. Similarly, the pattern of drag-based thrust differed only slightly between slow and fast swimming strokes, peaking after mid-stroke, but beoming negative in the last % of eah stroke (Fig. 4C,D). During slow swimming, added mass-based thrust peaked 28% of the stroke earlier than drag-based thrust. However, in fast swimming added mass-based thrust shifted later and drag-based thrust shifted earlier in the stroke, being nearly in phase for the representative fast stroke. Drag-based thrust dominated for both representative swimming speeds, produing an impulse of 7.8Nms in slow and 5.Nms in fast swimming and aounting for 86% and 7% of total thrust impulse, respetively. Among all four animals observed, however, the relative ontributions of drag-based and added mass-based thrust to total thrust impulse were highly variable. For three individuals (frogs 2, 3 and 4) net added mass-based impulse was not signifiantly different from net drag-based impulse (P>.5; Table 3). Kinemati omponents of added mass-based and drag-based thrust Added mass-based thrust produed by translational motion gave a net negative impulse during the representative slow swimming stroke. Rotational added mass, however, was suffiient to overome the negative translational added-mass impulse, ausing the net added mass-based impulse to be positive. In ontrast, translational and rotational motion ontributed equally to the observed added massbased impulse during fast swimming (Fig. 4E,F). Another notable differene was the phase offset of 3% vs 5% of translational and rotational added mass-based thrust in slow vs fast swimming, respetively. Both representative slow and fast swimming strokes showed a high ontribution of rotational motion to total drag-based thrust (Fig. 4G,H). In both slow and fast strokes, rotational drag was positive for the duration of the propulsive phase. From slow to fast swimming, net drag impulse due to rotation inreased from 9.Nms to 24.2 N ms. However, translational drag was mostly negative, resulting in negative net impulses that dereased from.3 N ms to 9.8Nms in slow vs fast swimming.

7 Modeling anuran swimming performane 387 Slow swimming Fast swimming Thrust (N)..5.5 A Swimming veloity Total thrust Translational thrust Rotational thrust B Swimming veloity (m s ) Thrust (N)..5.5 C Swimming veloity Total thrust.2 Added mass-based thrust Drag-based thrust D. Added mass-based thrust (N)..5.5 E Total added mass-based thrust.2 Translational added mass Rotational added mass F. Drag-based thrust (N)..5.5 G Total drag-based thrust Translational drag Rotational drag H. Time (s) Time (s) Fig. 4. Components of thrust in Xenopus laevis swimming. Total thrust (green line), translational thrust (red line) and rotational thrust (blue line) and swimming veloity (dashed blak 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 traes are idential in A and C as well as in B and D. Simulated anuran swimming: modeling stroke-to-stroke modulation of swimming veloity Modulating the relative magnitudes of translational and rotational veloity in the numerial model, as desribed above, aused marked differenes in simulated swimming performane among power strokes (Fig. 5). The model predited a maximal swimming veloity of.2ms with maximum translational and rotational veloities (.8ms and 6rads, respetively; Fig.6). A simulation with pure rotational veloity (Fig. 5A) reahed a peak swimming veloity of.54ms (45% of maximal veloity), whereas the simulation driven by pure translation only reahed 3% of maximal veloity (Fig. 5C). In the intermediate ase, with 5% maximum translational

8 388 C. T. Rihards Table. Xenopus laevis foot kinematis data summary Number of Peak swimming Peak translational Peak rotational Rotation translation phase Phase of peak thrust Frog strokes analyzed veloity (m s ) veloity (m s ) veloity (rad s ) offset (% stroke duration)* (% stroke duration) 6.55±.26.35± ±9.28 5±2 48± ±.9.4± ±6.8 25±8 5± ±.2.44± ±7.34 3± 42± ±.34.43±.2 25.±4.8 25±2 6±2 All values are means ± s.d. *Translation rotation phase offset=% (time of peak rotational veloity time of peak translational veloity)/stroke duration. Table 2. Estimated thrust (T) and impulse (I) from Xenopus laevis feet during swimming Peak T Peak T translational Peak T rotational I total I translational I rotational I amass I drag Frog (N) (N) (N) (N s) (N s) (N s) (N s) (N s).35±.25.6±.3.34±.242.6±.7.4±.3.9±.9.4±.3.3± ±..9±.4.8±.69.37±.4.2±..±.3.4±.2.9± ±.8.33±.5.255±.5.6±.7.2±.3.4±.6.9±.4.7±.4 4.9±.3.69±.3.47±.595.±.3.5±.8.8±..4±..4±.2 Values are means ± s.d. amass is added mass. Table 3. P values from Studentʼs paired t-tests Frog Peak T translational vs peak T rotational I translational vs I rotational I drag vs I amass.278*.47*.8* 2.29*.8* *.5* *.6*.966 T is thrust, I is impulse, amass is added mass. *Signifiant (P<.5) from paired t-test. and rotational veloity, simulated swimming veloity peaked at 35% of maximal veloity (Fig. 5B). In all simulations, swimming veloity inreased to a peak (positive aeleration) then dereased (deeleration) during the power stroke. This relative slowing [% (peak COM veloity final COM veloity)/peak COM veloity] inreased from 32% to 45% to 62% as the translational veloity was inreased from to 5% to % maximum (Fig.5). Simulated variation of translational and rotational veloity also strongly affeted the thrust profile and the underlying omponents of thrust (added mass and drag). In eah power stroke model, thrust inreased in the first half of the stroke, peaked prior to peak COM veloity and fell to a negative peak near the end of the stroke (Fig.5). Comparing the pure translation to the pure rotation ase (Fig. 5A vs Fig. 5C), added mass thrust ontributed most signifiantly to overall thrust during pure translation, with a peak added mass to peak drag thrust ratio of.7, as opposed to a ratio of.5 in the pure rotational ase. Sine peak added mass thrust preeded peak dragbased thrust in all simulations, this hange in the relative ontributions of these hydrodynami omponents aused a orresponding shift in the timing of peak thrust from.33 to.36s in the pure translation vs the pure rotation model, respetively. Additionally, the ratio of peak positive thrust to peak negative thrust dereased from 2.9 to 2. to.5 as the ratio of translational to rotational veloity was inreased from : to : to :, orresponding to the inreased negative added mass thrust inurred by translational motion (Fig. 5). Hypothetial anuran swimming performane spae The dependene of two performane parameters, peak stroke veloity (the peak swimming veloity reahed in the stroke), and glide veloity (the final stroke veloity at the end of the power stroke), was tested against the relative magnitude of translational vs rotational foot veloity. As reported above, stroke veloity peaked before the end of limb extension. Consequently, in all simulations the veloity entering the glide phase was less than peak stroke veloity. Both of these parameters depended strongly on the magnitudes of translational and rotational foot veloity, with maximal performane predited at the highest translational and rotational veloity (Fig. 7). Peak stroke veloity inreased linearly (as indiated by the parallel straight diagonal ontour lines) with peak translational veloity, but more strongly with rotational veloity. Partial least squares regression indiated that 6% of the variation in peak stroke veloity (among the 89 simulated trials) was modulated by hanges in rotational veloity alone. The swimming stroke powered by maximum foot translational veloity (zero rotation) reahed 3% of the maximal veloity ahieved with full rotation and translation, whereas maximal pure rotation produed a peak stroke veloity of 45% maximum (Fig.7A). In ontrast, pure translational veloity produed a glide veloity of only 5% maximum ompared with 44% maximum with pure rotational veloity (Fig. 7B). Moreover, 76% of the variation in glide veloity (ompared with 6% of variation in peak stroke veloity) was explained by modulation of rotational veloity. Joint work, COM work and hydrodynami effiieny The simulated work done to move the animal s body was highest for the stroke powered by maximal foot translation and rotation (Fig. 8A). Similarly, the predited ombined net joint work required from musles ating at the hip, knee and ankle was also highest when the foot was driven to maximal rotation and translation (Fig.8B). For both joint and COM work, the inreasing magnitude of ontour lines was skewed toward maximal translation and rotation. Variation in COM work (among the 89 simulations) was slightly more sensitive to hanges in rotational (explaining 6% of variation in joint work) than translational foot veloity. However, translational and rotational foot veloity had nearly the same effet on simulated joint work. Simulated effiieny (COM work/net joint work) was almost entirely a funtion of rotational veloity, with maximal effiieny predited for strokes with maximum rotational veloity and ~5% maximum translational veloity (Fig. 8C). Moreover, perentage deeleration was mainly dependent on

9 Modeling anuran swimming performane 389 Thrust (N) A Maximum rotation only B 5% maximum translation and rotation C Maximum translation only Swimming veloity Total thrust Translational thrust Rotational thrust Swimming veloity (m s ) Simulation time (s) Total thrust Added mass-based thrust Drag-based thrust Fig. 5. Simulated anuran swimming. Power strokes are shown for three different onditions: (A) maximum foot rotation with no translation, (B) 5% maximum translation and rotation, (C) maximum translation with no rotation. For all onditions, top panels show swimming veloity traes (dashed blak line), total thrust (green line), and translational (red line) and rotational omponents of thrust (blue line). Bottom panels show total thrust (green line), and added massbased (red line) and drag-based (blue line) thrust. Translational and rotational veloities are in phase for all onditions shown. Note that total thrust traes are idential for top and bottom panels. rotational veloity, with the highest values predited for simulations laking rotational foot motion (Fig. 8D). DISCUSSION The hydrodynamis of Xenopus laevis swimming This study aimed to disset the relative importane of translational vs rotational foot motion in the propulsion of an obligate swimmer, Xenopus laevis. Based on observations made in ranid frogs (Peters et al., 996; Johansson and Lauder, 24; Nauwelaerts et al., 25), hydrodynami thrust was hypothesized to be produed mainly from foot translational veloity and aeleration early in the extension phase. Therefore, stroke-to-stroke inreases in translational veloity were expeted to ause inreases in peak swimming speed. Data from this study do not support either hypothesis. In all of the stroke yles observed, the thrust impulse produed by rotational motion was signifiantly higher than the translational impulse (P<.5 for eah frog; Table 3) and translational thrust impulse did not orrelate with total thrust impulse (r 2 =., P>.5, N=23 strokes pooled from four frogs). I propose that these disparities between the present findings and previous work may be attributed to differenes in joint kinematis patterns as well as leg and foot morphology among anuran speies. A generalized model for anuran swimming performane A generalized model for anuran swimming an be used to explore interations between hydrodynamis and aspets of performane (e.g. swimming effiieny and speed). Reent studies have reported low hydrodynami effiieny for rowing swimmers at high Reynolds number (Fish, 996; Walker and Westneat, 2). Consistent with these previous findings, simulations of anuran swimming from the urrent study predit that the total net work summed over all hind limb joints is high ompared with the work required to move the Thrust (N) Swimming veloity (m s ) Swimming veloity Total thrust Translational thrust Rotational thrust Fig. 6. Simulated anuran swimming during maximal foot translation and rotation. The top panel shows swimming veloity traes (dashed blak line), total thrust (green line), and translational (red line) and rotational omponents 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 veloities are in phase for all onditions shown. Note that total thrust traes are idential for top and bottom panels...5 Total thrust Added mass-based thrust Drag-based thrust Time (s).8

10 39 C. T. Rihards Peak stroke veloity (m s ) Glide veloity (m s ) Peak translational veloity (m s ) I II I II Peak rotational veloity (rad s ) % max Peak stroke veloity (m s ) Glide veloity (m s ) I t t t.5 II t t t.5 Fig. 7. Swimming performane spae. Hypothetial maps of anuran swimming performane were generated from forward dynami simulations run aross a range of kinemati input onditions. Sine only two input parameters were varied (amplitudes of trigonometri funtions desribing translational and rotational foot displaement), a 3D hypothetial spae ould be systematially explored by mapping a single output (suh as peak simulated swimming veloity) against eah of the two independently varied inputs: peak rotational (x-axis) and peak translational (y-axis) foot veloity. The initial foot angle for eah simulation in the performane spae was derived suh that the foot is 9 deg. to flow at mid stroke (t.5 ) to deouple the effets of foot translation vs rotation. The olor sale (from to %) shows two performane parameters: (A) peak stroke swimming veloity and (B) glide veloity (the veloity at the end of the power stroke). Contour lines inlined more horizontally indiate a higher dependene on translational veloity, whereas more vertial lines indiate a higher dependene on rotational veloity. Arrows in A and B show different examples in the parameter spae: arrow I shows large foot translation and small rotation, whereas arrow II shows large rotation and small translation. The orresponding diagrams show the path of foot motion throughout eah simulated power stroke. These examples illustrate how two ontrasting stroke patterns an have idential peak veloities (5% maximum) yet different glide veloities (~2% maximum vs 4% for examples I and II, respetively). COM in a single stroke (Fig.8A,B). This reflets the fat that additional work is required to overome hydrodynami resistane, as well as any work done by musles to move body segments. As a result, effiieny is predited to be low for anuran swimming over the range of kinemati onditions explored here (Fig. 8C). The model was also used to address how swimming speed may be modulated by variation of kinemati patterns of the feet. Frog hind limbs have a wide range of joint onfigurations (Kargo and Rome, 22) enabling a large repertoire of potential foot motion patterns. Using a forward dynami model, one an map the relationship between foot kinematis and swimming speed by presribing input joint kinematis and simulating the frog s swimming veloity output. This allows the examination of swimming hydrodynamis in the ontext of kinemati patterns that are anatomially possible, but not realized in atual X. laevis behavior. For example, simulations were bounded by two extreme hypothetial ases: () maximal foot translational veloity with no foot rotation (with minimal ankle ation) and (2) maximal foot rotational veloity (with no hip or knee ation). Surprisingly, swimming speed in the pure translation model was lower than in the opposite ase of pure foot rotation (.38 vs.54ms ; Fig.5A,C). There are two explanations for this result. Firstly, beause the foot rotates very rapidly in X. laevis the maximal tangential rotational veloity (foot length foot angular veloity) was muh higher than the translational foot veloity. The highest values observed in X. laevis swimming (thus the values used as maximum input values for the simulations) were 2.3 and.8ms, for tangential rotational and translational veloities, respetively (based on frog ). Aordingly, peak thrust was higher in the pure rotational vs translational simulation (.83 vs.5 N, respetively). Seondly, added mass-based and drag-based thrust are out of phase in the translational ase, whereas they are nearly oinident in the rotational ase, enhaning their umulative ontribution to total thrust (Fig. 5A,C). In addition to peak stroke veloity, predited glide distane was also onsidered an important performane parameter. Sine there is no propulsion during the glide, distane is limited by glide veloity (defined as the swimming veloity at the end of the power stroke).

11 Modeling anuran swimming performane 39 Peak translational veloity (m s ) A C Net COM work (J) Effiieny B D Total net joint work (J) % deeleration Fig. 8. Predited mehanial work and effiieny of anuran swimming. As in Fig. 7, ontour plots show peak rotational (x-axis) and peak translational veloity (y-axis) resulting from inrementally varying the input translational and rotational displaements in the numerial model. The olor sale (from to %) shows (A) net COM work (hydrodynami and inertial fores ating on the body total distane traveled) over eah power stroke, (B) total net joint work (the sum of work produed at the hip, knee and ankle) over eah power stroke, (C) net effiieny (net COM work/total net joint work) and (D) perentage deeleration [% (peak COM veloity final COM veloity)/peak COM veloity] Y Data Peak rotational veloity (rad s ) % max. COM work (J) Net joint work (J) Effiieny % deeleration e 2 2.2e 2 3.3e 2 4.4e 2 5.5e 2 6.6e 2 7.7e 2 8.8e 2 9.9e 2.e.2e In all power stroke simulations, the thrust impulse was positive for the first half of the stroke and negative for the seond half. Beause positive thrust always exeeded negative thrust, the net impulse was always positive, resulting in forward swimming veloity throughout eah stroke. In the pure foot translational simulation (Fig. 5C), peak negative thrust reahed 7% of peak positive thrust. During this stroke, aft-direted translational foot veloity exeeded forward COM veloity, so that the net translational foot veloity (foot translational veloity COM veloity) was positive throughout limb extension and no negative drag was produed. Therefore, importantly, foot orientation 9 deg. to the flow did not ause drag retarding the forward movement of the body during the power stroke. In this ase, negative thrust was produed entirely from added mass effets resulting from foot deeleration (i.e. positive, but dereasing translational veloity). In ontrast, the simulation with sub-maximal foot translational and rotational veloities (Fig. 5B) reahed a forward COM veloity that exeeded the rearward foot translational veloity, resulting in negative drag-based thrust (due to negative net translational veloity) in addition to negative added mass-based thrust (due to foot deeleration). Searhing the hypothetial performane spae between the extremes of foot motion provides additional insights into the ontrol of swimming performane. Fish inrease swimming speed by inreasing their stroke frequeny (Brill and Dizon, 979; Rome et al., 984; Altringham and Ellerby, 999; Swank and Rome, 2). Although variation in stroke frequeny also ours in frogs, the

12 392 C. T. Rihards relationship between power stroke period and performane is unlear (Nauwelaerts et al., 2). To avoid potentially onfounding effets of stroke duration, power stroke simulations were run at a onstant duration. As expeted, simulations with proportional inreases in translational and rotational amplitude (i.e. moving upwards and rightwards through the parameter spae; Fig. 7A) predit a linear inrease in peak stroke swimming veloity, as indiated by the diagonal ontour lines. However, predited glide performane did not follow the same trend (Fig. 7B). Glide veloity was disproportionately lower than peak stroke veloity in translationdominated strokes, espeially in the upper left quadrant of the performane spae. In these strokes, rotational motion is too minimal to ounterat the retarding thrust (from relatively large negative fore due to foot translational deeleration; see above). Therefore, the highest COM deeleration during the power stroke is predited to our in the absene of foot rotation (being largely independent of translational veloity; Fig. 8D). By prediting the hydrodynami roles of translational vs rotational foot motion, this forward dynami simulation provides a framework for understanding the kinemati determinants of thrust observed in frog swimming. Disseting the propulsive mehanism of a generalized Xenopus laevis swimming stroke Despite observed variation in the temporal patterns of all omponents of thrust aross 23 strokes (N=4 frogs), most propulsive strokes show two main phases. In the initial phase (Fig.3, stages I, II and III), aeleration of the COM is driven mainly by both net translational veloity and foot aeleration (both translational and rotational). Propulsion in this phase, therefore, is dominated by translational drag and total added mass-based thrust. In the final phase (Fig. 3, stages IV and V), propulsion is enhaned and sustained by rotational veloity (generating rotational drag-based thrust), whih usually peaks later than translational veloity. In all strokes observed, net translational veloity peaked in the first phase, but rapidly dereased to negative values in the seond phase of the stroke as the forward veloity of the COM exeeded the bakward translational veloity of the foot. This has two effets: () negative net translational veloity produes negative drag-based thrust and (2) translational deeleration (aused by the slowing translational foot motion toward the end of the power stroke) results in negative added massbased thrust. Therefore, the kinemati omponents of thrust have unique roles in propulsion: early translational and rotational motion aelerate the frog at the onset of swimming. As foot rotational veloity inreases later in the stroke, drag-based rotational thrust ounterats and overomes the negative omponents of thrust, ausing propulsion to ontinue until the end of the power stroke. Linking kinemati plastiity to hydrodynamis: a proposed mehanism for modulating swimming performane from stroke to stroke Xenopus laevis hind limb kinematis are highly variable, even within the behavioral subset of forward, straight and synhronous swimming. In ontrast with the reported stereotypi nature of Hymenohirus boettgeri (Gal and Blake, 988b), X. laevis modulate time-varying flexion extension patterns of the hind limb joints between sequential kiks of a single swimming burst (C.T.R., unpublished observations). Beause foot motion is the sum of motion produed at the hip, knee, ankle and tarsometatarsal joints, the relative phases and magnitudes of translational and rotational veloity vary greatly from stroke to stroke in X. laevis (Table ). Despite this variability, trends emerge. Most notably, peak stroke translational and rotational veloity are positively orrelated aross all swimming speeds and individuals (r 2 =.7, P<., pooled data from 23 swimming strokes, N=4 frogs). Further, peak COM veloity orrelates with peak translational veloity (r 2 =.8, P<.) and peak rotational veloity (r 2 =.66, P<.), as well as with net thrust impulse (r 2 =.58, P<.). Surprisingly, although peak translational and rotational veloity are orrelated, the observed inrease in thrust impulse among strokes is explained only by inreases in rotational thrust impulse (r 2 =.7, P<.). Total thrust impulse is independent of translational thrust impulse among the strokes analyzed (r 2 =., P=.63). This suggests that orrelative observations of kinematis and performane an be misleading in the absene of a more detailed hydrodynami analysis. Despite the o-variation of translational, rotational and COM veloity, the underlying drive that provides inreased net thrust from one stroke to the next is the inrease of rotational impulse alone. For any given X. laevis stroke, the time-varying pattern of thrust depends on both translational and rotational foot motion. However, the shift between slow and fast swimming appears to require only an inrease in net rotational thrust. Hydrodynami model verifiation and omments on Gal and Blakeʼs model Before applying the hydrodynami model to resolve thrust omponents in X. laevis swimming, the model was verified using a modified implementation of Gal and Blake s approah (Gal and Blake, 988b) to determine whether the model adequately estimated the temporal pattern of propulsive hydrodynami fore during Xenopus laevis swimming. Differing from that of Gal and Blake (Gal and Blake, 988b), the modified blade element model was oupled to a forward dynami model to simulate swimming veloity profiles for omparison to atual veloity profiles from analyzed video sequenes (see Materials and methods). Although this approah generally underestimated the hydrodynami fores required to propel the animal, the average error between observed and simulated swimming veloity was low (Fig. 2D,E). Although improvements to this model may be made for future studies (see below), the urrent approah is suffiient to investigate the hydrodynami mehanisms for X. laevis swimming examined here. Gal and Blake (Gal and Blake, 988a; Gal and Blake, 988b) used an elegant approah to verify their quasi-steady blade element model. From analyzed video sequenes they alulated the fore balane on the frog s COM required to generate an observed swimming aeleration pattern. They found a onsiderable disparity between thrust estimated by their blade element model and thrust predited from the fore balane on the frog body. Thrust estimated from the blade element model peaked at the beginning of limb extension and fell to a minimum (negative thrust) towards the end of the power stroke. Conversely, COM thrust alulated from the fore balane peaked at midstroke and remained positive throughout the limb extension period. Although Gal and Blake aknowledged potential error from their added mass alulations, they proposed that a jet (produed as the feet move toward the midline) may aount for thrust not predited from blade element theory (Gal and Blake, 988a; Gal and Blake, 988b). I propose two additional explanations for their findings. Firstly, Gal and Blake did not aount for the rotational omponent of added mass-based thrust, whih ontributes substantially to thrust in X. laevis (7±26% of total added mass impulse) and, therefore, may be important in Hymenohirus boettgeri. Seondly, Gal and Blake s method for measuring COM veloity may not be aurate. They used the