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

swimming

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Bioinspir. Biomim. 7 (2012) 016010 (15pp) doi:10.1088/1748-3182/7/1/016010

A bio-robotic platform for integratinginternal and external mechanics duringmuscle-powered swimming

Christopher T Richards and Christofer J Clemente

The Rowland Institute at Harvard, 100 Edwin H. Land Blvd, Cambridge, MA 02142, USA

E-mail: richards@fas.harvard.edu

Received 15 July 2011Accepted for publication 11 January 2012Published 16 February 2012Online at stacks.iop.org/BB/7/016010

AbstractTo 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 laevisfrogs. Four foot shapes ranging from low aspect ratio (AR = 0.74) to high (AR = 5) werecompared to test whether low-AR feet produce higher propulsive drag force resulting in fasterswimming. Using feedback loops, two complementary control modes were used to rotate thefoot: force was transmitted to the foot either from (1) a living plantaris longus (PL) musclestimulated in vitro or (2) an in silico mathematical model of the PL. To mimic forwardswimming, 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 sand swimming velocities from 0.41 ± 0.01 to 0.43 ± 0.00 m s−1, similar to in vivo valuesfrom 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. Incontrast, 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 demonstratehow our experimental technique is useful for quantifying the complex interplay among limbmorphology, muscle mechanics and hydrodynamics.

S Online supplementary data available from stacks.iop.org/BB/7/016010/mmedia

1. Introduction

The mechanics of aquatic locomotion is a rich field for bothengineers and biologists. For example, engineers have showngreat interest in fish fin motion (e.g. Blake 1979, Lauderet al 2006) and fin shape (e.g. Blake 1981, Combes andDaniel 2001) toward advancing the design of underwatervehicles (Kato and Liu 2003, Phelan et al 2010). Similarly,engineering approaches also inform biologists, using robotic(e.g. Tangorra et al 2007) or mathematical models (e.g. Danieland Meyhofer 1989, Tytell et al 2010), to measure parameters

(e.g. fluid dynamic or muscle force) which are difficult to

measure in vivo. Yet, despite shared motivations to understand

aquatic propulsion, neither field has fully characterized how

external properties (e.g. fin shape) couple to musculoskeletal

properties to confer effective swimming. Despite the diversity

of propulsor morphology in fishes (Thorsen and Westneat

2005) and frogs (Goldberg and Fabrezi 2008), little is known

about the influences of fin morphology on muscle mechanical

output and hydrodynamic performance. Furthermore, we lack

experimental tools for testing muscle contraction dynamics

1748-3182/12/016010+15$33.00 1 © 2012 IOP Publishing Ltd Printed in the UK & the USA

Bioinspir. Biomim. 7 (2012) 016010 C T Richards and C J Clemente

Fluidforce/MA

Muscle force

Propulsion

Muscle properties:*F-V, F-L*pennation

Fin properties: *fin size*shape*stiffness

Fluidreaction

forceSkeletal

properties:mechanical

advantage (MA)

Environmental feedback from water

Activationproperties:*duration*phase*frequency

CNS MUSCLE SKELETON LIMB (FIN) Neural activation

Muscle force * MA

Figure 1. Diagram, modified from Marsh (1999), showing the coupled relationships among internal and external forces during swimming.Bold items represent ‘tunable’ properties that may undergo selection to enhance swimming performance. Input from the central nervoussystem modulates muscle force from frequency, phase and duration of motor neuron firing. Skeletal morphology confers mechanicaladvantage (MA = 1/gear ratio) by determining the muscle force and displacement required for a given force and displacement at the outertip of the fin. For a given neural activation, muscle force develops according to activation kinetics and F–V/F–L properties. Skeletalstructures (e.g. joints, bone processes, etc) as well as muscle fiber pennation angles (Lieber 1992, Azizi et al 2008) act as a ‘transmission’ toconfer the appropriate mechanical advantage to move the limb against the load. Finally, the size, shape and material properties of finsinfluence the drag reaction force (proportional to fin velocity2).

and swimming performance in response to changes in limbmorphology.

More broadly, the limited ability to address therelationship between fin morphology and muscle dynamicsstems from a poor knowledge of how muscle developsforce against dynamic loads. Although relationships betweenelectrical stimulation and force are well established whenmuscle length is held constant in vitro (e.g. Cooper andEccles 1930), muscle mechanics during locomotion areless well understood. When a muscle shortens, its forcedepends on length (i.e. force–length ‘F–L’ effects; Gordonet al 1966) and fiber shortening velocity (force–velocity‘F–V’ effects; Hill 1938). Hence during locomotion, musclemovement depends on the force transmitted through the limbfrom the environment (Daniel 1995, Marsh 1999, Aerts andNauwelaerts 2009). A feedback loop illustrates this principle(figure 1). As muscle force develops from neural stimulation,limb motion causes reaction force resulting in ‘environmentalfeedback’ as the velocity and displacement of the loadedlimb influence muscle force via F–V and F–L effects.Because of this fluid–muscle coupling, one cannot easilypredict how changes in skeletal or limb morphology influencehydrodynamic performance without knowledge of underlyingmuscle mechanics. Likewise, without understanding externalloads, physiologists do not necessarily know, a priori, whatconditions to test with in vitro experiments (Marsh 1999).Helping to address this challenge, bio-robotic platforms arepowerful tools to isolate either morphological (e.g. Low andChong 2010), kinematic (e.g. Kato and Liu 2003, Phelanet al 2010) or neural (Ijspeert et al 2007) aspects of animalswimming. Moreover, detailed numerical simulations (e.g.Daniel and Meyhofer 1989, Tytell et al 2010) offer tremendousinsights. The current study aims to integrate aspects ofthese bio-robotic and mathematical tools to explore musclephysiology.

Adding to recent innovations in biomechatronics (Herrand Dennis 2004, Farahat and Herr 2006, Farahat andHerr 2010) and in neuromechanical manipulation (Sponberg

et al 2011a, Sponberg et al 2011b), we introduce bio-robotic approaches that help link our understanding of limbmorphology and neural control to locomotor performance.Using swimming frogs as a model, we expanded upon recentlydeveloped methods (Richards 2011) using a bio-robotic footwhich is controlled by muscle. We used two complementaryapproaches: (1) the command signal for the motors in the bio-robot was transmitted from in vitro muscle force measurementsmade from a Xenopus laevis frog muscle in real time.(2) To further isolate how intrinsic muscle properties influencecontractile output, we implemented a mathematical Hill-typemuscle model as a control algorithm for the robotic foot.

For demonstration, we explored a question that isdifficult to address with conventional methods: how does finmorphology influence the muscle mechanical requirementsfor swimming? Aspect ratio (AR = fin span2/fin area) isa key descriptor of fin shape and an important determinantof fluid-dynamic performance (Vogel 1981, Combes andDaniel 2001, Dong et al 2006). For instance, low-AR finshave been associated with drag-based ‘rowing’ locomotion(Walker and Westneat 2002, Thorsen and Westneat 2005),which produces greater hydrodynamic force at slow speeds(Walker and Westneat 2000, Kato and Liu 2003) and fasterturns as compared to lift-based ‘flapping’ with high-ARfins (Gerstner 1999). However, studies that explicitly testhow AR influences fluid-dynamic performance often focuson lift-based locomotion rather than drag-based swimmingcommon across many taxa (e.g. fish, frogs, turtles, waterfowland aquatic insects). Furthermore, comparative functionalmorphology approaches cannot discern whether performancevariation is due to AR or due to other differences in finmorphology (i.e. flexibility, surface morphology and finmusculature). Therefore, two open questions remain: whatare the hydrodynamic consequences of variation in ARamong drag-based swimmers? How does AR influence musclemechanical output?

From a simplified fluid-dynamic perspective, high-ARfins have a greater distance from the fin base to the

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Bioinspir. Biomim. 7 (2012) 016010 C T Richards and C J Clemente

Fh

Servobody

Servofoot

Servofoot controller

Servofoot Servobody

VbodyVrot

Water level

(a) (b)Servobodycontroller

Foot Aquarium

Translating sled

Hydrodynamic force sensor

Guide rails

Plastic foot (length = 2.6 cm)

Hydrodynamic sensor

Figure 2. The bio-robotic setup. (a) Photo (top view) of the robotic foot which is rotated through the water by Servofoot. Simultaneously,Servobody moves the translating sled to mimic the forward translation of a frog as it swims in response to propulsive foot rotation.(b) Schematic diagram of (a).

hydrodynamic center of pressure (R). One might thereforepredict drag-based thrust (α [rotational velocity R]2) toincrease with AR. Yet, muscle force declines sharply asshortening velocity increases (Hill 1938) indicating thatboth rotational velocity and R may change reciprocally.Furthermore, 3D aspects of fluid flow (e.g. spanwise flow;Dong et al 2006) complicate our predictions of interactionsamong muscle force, R and fin rotational velocity. Thus forsimplicity, we formulated a hypothesis consistent with bothcomparative swimming performance data and experimentalfluid dynamics. Based on inverse correlations between ARand fish turning speed (Gerstner 1999), we hypothesizedhydrodynamic force and swimming speed to be greatest atlow AR. More directly, measurements on fabricated wingsrotating at a 90◦ angle of incidence (similar to rowing)showed an inverse relationship between thrust coefficient andAR, a trend likely due to the interference of leading edgeand trailing edge vortices as fins grow thinner (Usherwoodand Ellington 2002). Using muscle-actuated fins, we testedour hypothesis with four foot shapes ranging from AR =0.74 to 4.9. We then repeated our measurements using Hill-type muscle model actuation, enabling us to dissect the relativecontributions of neural activation kinetics, F–L and F–Veffects underlying shifts in muscle dynamics. Thus, the currentstudy integrates measurements of intrinsic muscle functionand hydrodynamics to gain insight into how changing themechanical loading (via changes in fin morphology) influencesswimming performance.

2. System design and setup

2.1. Model system

Xenopus laevis, an obligatorily aquatic frog, was the modelfor the current study for three reasons: (1) the feet movein a simple rotational motion to power swimming. Unlikeother swimming frogs that use hip, knee and ankle extensionsto translate and rotate the foot backward to produce thrust,Xenopus laevis generates propulsion mainly by rotating thefoot at the ankle (Richards 2008, Richards 2010). Because

backward foot translation contributes very little to thrust inXenopus laevis, we can isolate the PL ankle extensor muscleas the main motor for propulsion, eliminating the need to modelproximal muscles and joints in our setup. (2) During swimmingXenopus laevis hind limb movements are constrained to thefrontal plane. Specifically, the limb segments only rotateanterior-posterior, rather than dorsal-ventral as in other taxa,such as ranids (Johansson and Lauder 2004, Nauwelaerts et al2005) and Bufo (Gillis and Biewener 2000). For this reason,only two degrees of freedom in our bio-robot are necessaryto model Xenopus laevis foot motion as it rotates backwardto produce thrust while advancing forward with the body.(3) The simple parallel fiber architecture of the PL in Xenopuslaevis avoids additional complexity of variable fiber pennationangles (see Azizi et al 2008) in the transmission of force to thebio-robot.

2.2. Bio-inspired robotic frog foot

Unlike previous studies that rely on dynamically scaled models(Dickson and Dickinson 2004), our goal was to developan actual scale biomechatronic instrument that simulatesthe mechanical environment of a living frog muscle. Toaccomplish this, we used a robotic foot mimicking thedimensions and range of motion of a small adult male Xenopuslaevis frog (20–50 g body mass). Importantly, the mass andmoment of inertia of the moving parts were minimized toapproximate the moment of inertia of a frog’s foot and tarsalbones (≈1 × 10−6 kg m2) such that the principle forcesacting on the foot approach physiological conditions duringswimming.

Motion of the robotic foot was driven by two motors.(1) To mimic the ankle extensor action of the PL muscle,fore-aft foot rotation was driven by a Pittman 4443S013(PittmanExpress, Harleysville, PA, USA) brushless servomotor (Servofoot) paired with an Accelus ASP-055-18 digitalservo amplifier (Copley Controls, Canton, MA, USA). (2) Anidentical motor (Servobody) drove fore-aft translation of thefoot (see below) to mimic the unsteady swimming motionof the body (figure 2). Analogue command signals for these

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Bioinspir. Biomim. 7 (2012) 016010 C T Richards and C J Clemente

Muscle force output

Muscle length input

Foot positionoutput

Torque input tofootAquarium

Servo motor controllers

Submergedrobotic foot

Servofoot

FPGA controllerX. laevisPL muscle inRinger solution

Musclepositioner

(a)

Virtual gearboxCalibration

User input:- Force and position calibration constants- Muscle moment arm (r)

degrees/voltvolts/degree

volts/NN/volt

FPGA CONTROL LOOP: Operating mode 1 for in vitro-robotic experiments

Muscle force output

Torque input tomechanical linkage

Foot position output

Muscle length input

(b)

Servomuscle

ServomuscleServofoot

Power supply

oa

1.2 k

+-

analogoutput

analoginputs

5 k

Sciaticnerve

Suctionelectrode

USBA/D

module

x r

x r

Vir

tual

leve

r

Muscle force and position signals for data recording

Figure 3. (a) Control mode 1 schematic diagram for in vitro-robotic experiments: a muscle ergometer (Servomuscle) measures muscle force inresponse to electrical stimulation from a suction electrode. The suction electrode is powered by an op–amp circuit (‘oa’; see the text). Ateach time step, muscle force measured by Servomuscle passes through an FPGA controller as a command signal for Servofoot whichinstantaneously exerts torque (recorded from the muscle) against the foot. Servofoot then measures the resulting robotic foot displacementsignal which returns through the FPGA module as a position command to Servomuscle. This forms a feedback loop enabling the muscle toshorten as if it were directly attached to the foot. (b) Detail of the servo control loop where Servomuscle and Servofoot control each other withforce and position signals, respectively. Within the software running on the FPGA controller, force and position are adjusted by theappropriate calibration constants specific to either servo motor. Additionally, increasing the software-controlled virtual muscle moment armr increases the muscle displacement–limb displacement ratio. For the current study, r was held at 5.25 mm. All operations in the control loopare executed at each 0.1 ms time step.

two motors were generated by a custom software written inLabview 10.0 (Instruments, Austin, TX, USA; see below fordetails). The current setup was capable of reproducing foot-rotational and body-translational velocities equal to (and farexceeding) in vivo values of ∼30 rad s−1 and ∼ 1 m s−1.

Depending on the nature of the experiment, the currentsetup could be used in one of two separate control modes:(1) force measured from a living frog muscle (isolated in vitro)was used as a torque command signal to the servo poweringfoot rotation (modified from Richards 2011). (2) Instead ofusing a living frog muscle, force was calculated from a Hill-type mathematical muscle model to dictate the Servofoot motortorque output. In each control mode, Servofoot mimics theaction of the PL muscle to rotate the foot through the water.

2.3. Control mode 1: in vitro-robotic feedback loop

For control mode 1, a living PL muscle was placed in aringers’ bath and mounted to a muscle ergometer 305C-

LR servo motor (Servomuscle; Aurora Scientific, Inc., Aurora,ON, Canada). The muscle was stimulated with 1 ms pulsesgenerated by an analogue output from a USB-6289 module(National Instruments) amplified by an OPA549T op-ampcircuit (Texas Instruments, Dallas, TX, USA) powered by aconventional 24 V supply (Protek 3003B; Protek Devices,Tempe, AZ, USA; see section 3 for muscle stimulationdetails). Upon electrical stimulation of the sciatic nerve,Servomuscle measures muscle force at a controlled length(figure 3). A computer-controlled feedback loop was usedto electronically link the bio-robotic foot to the ergometerenabling the living muscle to ‘control’ foot rotation. Unlikeconventional bio-robotic instruments, the position of the finwas not directly controlled. Rather, the motion of the finemerged from the force signal input to Servofoot simultaneouslymeasured from the contracting muscle. Likewise, insteadof arbitrarily controlling the muscle’s length, the muscleposition signal was given by Servofoot as it measured therobotic foot’s displacement under the applied torque. Force

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Bioinspir. Biomim. 7 (2012) 016010 C T Richards and C J Clemente

and position commands were updated every 0.1 ms on acRio9074 field programmable gate array (FPGA) ‘Real time’controller (National Instruments, Austin, TX, USA), enablingthe muscle to contract as if it were pulling directly on a skeletallever arm to mechanically rotate the foot (supplementaryfigure S1, available at stacks.iop.org/BB/7/016010). Themeasured feedback delay of <1 ms between muscle motionand robotic foot motion was dramatically improved from anearlier version of the setup (Richards 2011).

2.4. Control mode 2: in silico-robotic feedback loop

For control mode 2, a Hill-type muscle contractile elementwas integrated into an FPGA control program such that muscleforce could be calculated and transmitted to the bio-robot ateach 0.1 ms time step. Muscle model morphology was modeledbased on measurements of actual muscle used for in vitro-robotic trials. At each instant of time, muscle force depends onthe filament overlap, hence muscle length (Gordon et al 1966)as well as the instantaneous velocity of the contractile element(Hill 1938). Conventionally, these F–L and F–V properties areexpressed as ‘gain’ functions varying between 0 and 1. Basedon a prior work (Otten 1987), we used

FL gain(t) = e−∣∣[( L

Lo)b−1]/s

∣∣

a

, (1)

where L/Lo = instantaneous length/optimal length,b = −1.44, s = −0.508 and a = 1.37. Values for b, s anda were based on measurements of Xenopus laevis PL muscle(C T Richards, unpublished observations). For the currentstudy, optimal length was estimated to be 0.02 m based on invitro-robotic muscle experiments. F–V properties were definedfrom Xenopus laevis plantaris data:

FV gain(t) = 2.61 − V (t) · 2.61/vmax

V (t) + 2.61, V (t) � 0, (2)

where V(t) is the normalized shortening velocity (shorteningvelocity/optimal length) and vmax is the maximum shorteningvelocity (vmax = 15 ML s−1; C T Richards, unpublishedobservations). Finally, the activation state of the muscle neededto be modeled. Wavelet analysis of the electromyography(EMG) intensity (Wakeling et al 2002) from the Xenopus laevisPL during swimming shows a symmetric bell-shaped rise andfall of muscle fiber recruitment (C T Richards, unpublishedobservations). To fit this pattern, we used a simple cosinewaveform to mimic the rise and fall of the muscle’s activationstate (figure 4):

A(t) = S · Po

2[1 − Cos(t · 2π/dur)], (3)

where Po is the maximum isometric tension of the PLmuscle, dur is the duration of the activation and S is thestimulation constant which represents the fraction of musclefibers recruited (S = 1 is the maximal stimulation, whichwas used for all in silico-robotic experiments). Thus, A(t)represents a force envelope which dictates the force availablefor a given stimulation level, physiological cross-sectional areaand activation kinetics. Equations defining F–V and F–L (1)and (2) were assumed to be invariant across different activationlevels; thus, muscle force was expressed as

Fmuscle(t) = A(t) · FV gain(t) · FLgain(t). (4)

For simplicity, additional parameters such as in-serieselasticity and history-dependent effects were not included inthe present model, however, will be implemented in futureexperiments. The FPGA muscle model works in the followingway: in the simplest case, initially, L(t0) = 1, V(t0) = 0 andA(t0) = 0. Therefore, both F–L and F–V gains = 1, but forceoutput = 0. As activation increases to 0.01 N, for example,0.01 N is transmitted to Servofoot and exerted againstthe robotic foot causing rotation. At the following timestep, foot displacement measured by Servofoot is convertedto muscle lengths (via the muscle moment arm r)to update L(t1). Within the same time step, L(t1) isdifferentiated to update V(t1) as the virtual muscle shortens.At this point, the muscle length has decreased belowoptimal length and increased in shortening velocity; thus,F–L and F–V gains are both <1. Therefore, as activation risesfurther in subsequent time steps, force output is lower thanthe input activation waveform due to F–L and F–V effects. Atthe end of the contraction, muscle force subsides as F–L gainapproaches zero.

2.5. Musculoskeletal gearing: electronically defining themuscle moment arm

The range of muscle shortening for a given joint angularexcursion is determined by the muscle moment arm (r), whichis the perpendicular distance from the muscle’s line of action tothe center of rotation. In vertebrate musculoskeletal systems,r generally depends on the morphology of the bone, tendonand joint to which the muscle inserts. However, in our setup,there is no physical model of the skeleton (the muscle and footare directly attached to their respective actuators). To mimicthe skeletal environment of a muscle, we simply included r inour real-time control loop to calculate angular displacement ofServofoot:

�θfoot ≈ �Lmuscle · C1

r≈ d · �θmuscle, (5)

where �θ foot is the change in foot angle (= change inServofoot angle), �Lmuscle is the change in muscle length,�θmuscle is the change in Servomuscle angle and d is the leverarm length for the muscle lever connecting the motor shaft ofServomuscle to the muscle tissue. Torque was calculated by thefollowing:

Torquefoot = r · Forcemuscle · C2 = Torquemuscle, (6)

where C1 and C2 are unitless constants computedfrom experimentally determined displacement and torquecalibration coefficients for Servofoot and Servomuscle hardware.For the current study r, the muscle moment arm was set to5.25 mm in the feedback loop control software (similar tothe measurements from Xenopus laevis; C T Richards andC J Clemente, unpublished observations).

2.6. Virtual self-propulsion of the bio-robot

One challenge of modeling frog swimming is the unsteadynature of body motions. In frogs, as in many other animals, thebody does not reach a steady velocity due to the reciprocatingmovements of the limbs. Over repeated swimming strides,

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Bioinspir. Biomim. 7 (2012) 016010 C T Richards and C J Clemente

User inputs:- Force and position calibration constants- Muscle moment arm (r)- Hill coefficients (FV)- Optimal muscle length (FL)- Peak isometric tension- Muscle activation waveform Data trace outputs:- Force, FL and FV gains

FPGA CONTROL LOOP: Operating mode 2 for in silico-robotic experiments

FV gain

FLgain

Lowpass filter

Forc

e

Forc

e

Shortening velocity Muscle length

11

d dt

Shortening veloctiy

Time

Po

Muscleactivation (A)

X X = Force (N) Torque (volts)

to Servofoot

X r

Musclelength (ML)

Input from USB A/O board

Position (volts)from

Servofoot

0.05 s

Vir

tual

leve

r

volt/N

length chage/volt

I

I

O

(a)

(b)

X r

Figure 4. (a) Control mode 2 schematic diagram for in silico-robotic experiments: The force command signal for Servofoot emerges from aHill-type muscle model computed within the FPGA controller. An arbitrary muscle activation waveform (from a separate USB A/O device)is input to the FPGA in real time. At each time step, Servofoot position and velocity information are calibrated (pink boxes) to muscle lengthunits and are fed into the muscle model to output motor torque. The displacement and velocity of Servofoot against the mechanical limb arethen updated to repeat the loop for the subsequent time step. Similar to control mode 1, r controls the virtual muscle moment arm of thesetup. r = 5.25 mm for the current study. All operations in the control loop are executed at each 0.1 ms time step. (b) A typical EMG trace isshown from a Xenopus laevis PL muscle during swimming. Total myoelectric intensity (blue line) from the EMG is overlaid with theactivation profile used for the current study (black dashed line).

body velocity fluctuates sharply as animals accelerate topeak velocity then slow down to <25% of peak velocity(Nauwelaerts et al 2001, Richards and Biewener 2007).Consequently, rather than a steady velocity flume typicallyused for fish swimming, the bio-robotic platform requiredself-propulsion to accelerate and decelerate the foot throughstill water. Elegant fish-inspired bio-robotic platforms haverelied on frictionless air bearings to enable a flapping finto propel the robotic apparatus (e.g. Tangorra et al 2007,Phelan et al 2010). Regardless of friction, however, thedramatic accelerations and decelerations typical of unsteadyswimming would incur unrealistic inertial forces to acceleratethe heavy (>1 kg) robotic platform along with the servomotors and cables. To solve this problem, ‘virtual self-propulsion’ was implemented, similar to recent developmentsin robotic insect wing actuation (Dickson et al 2010). Insteadof allowing thrust from the robotic foot to overcome thefriction and inertia of the translational track, we used aservo motor (Servobody) to advance the foot forward. Toemulate self-propulsion, the normal hydrodynamic reactionforce on the foot (Fnormal) measured from a hydrodynamic forcesensor (see appendix A; supplementary figure S2, available atstacks.iop.org/BB/7/016010) was input into a mathematical

model of a swimming frog body to compute the accelerationresulting from the balance of thrust and drag on the virtualbody (see appendix B; supplementary figure S3, available atstacks.iop.org/BB/7/016010).

3. Materials and methods

3.1. In vitro-robotic experiments: animal care

Male adult Xenopus laevis (Daudin 1802) frogs (∼20 g bodymass) were purchased from Xenopus Express, Inc. (PlantCity, FL, USA) and housed in glass tanks at the RowlandInstitute at 20–22 ◦C under a 12 h:12 h light:dark cycle. Allexperimental protocols used were approved by the HarvardUniversity Institutional Animal Care and Use Committee.

3.2. In vitro-robotic experiments: tissue preparation andstimulation

Frogs were double pithed with a 21 g syringe needle andthe PL muscle was removed from the animal (Richards2011). Small sections of the femur and the proximal tarsalbones were left intact to anchor the proximal and distal

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Bioinspir. Biomim. 7 (2012) 016010 C T Richards and C J Clemente

ends of the muscle, respectively. Using 4-0 vicryl suture(Ethicon, Somerville, NJ, USA), the muscle was tied betweena stationary pin and a 0.04 m lever attached to an Aurora servomotor (Servomuscle). Muscle resting length was defined as thelongest muscle length with zero passive tension (e.g. Biewener2003). Before all trials, initial length of the muscle wasarbitrarily set by stretching the muscle slightly beyond restinglength to achieve a passive tension of 0.1 N to remove slackfrom the preparation. The muscle was continuously bathed in22 ◦C (matching the temperature of the frog housing facility)oxygenated amphibian Ringer solution (Carolina Biological,Burlington, NC, USA).

A suction electrode was used to stimulate the PLmuscle via the sciatic nerve using 1 ms width stimulationpulses generated from an A/D board (see above). Afterdetermining the maximal stimulation voltage with isometrictwitch contractions, experimental trials were stimulated atsupramaximal voltage at a spike frequency of 250 Hz(Richards, 2011). A pulse train duration of 80 ms elicitedmuscle contraction periods of ∼150–200 ms, similar tocontraction durations from in vivo plantaris measurementsfrom the PL muscle of Xenopus laevis (Richards and Biewener2007). Each trial consisted of one single muscle contraction.

3.3. In silico-robotic experiments: defining the activationwaveform

Instead of using a train of pulses to stimulate the in silicomodel (as in the in vitro muscle stimulation), a smoothcosine waveform was used for the following reason: to elicitmaximum in vitro force from a vertebrate muscle, a train ofelectrical pulses must be delivered to stimulate intracellularCa2+ release to activate the contractile filaments (Lieber 1992).However, in a mathematical muscle simulation, modelingthe cyclic release of Ca2+ due to electrical excitation isunnecessary. One may simply use a smooth rising and fallingwaveform to describe the cumulative state of activation-deactivation for all of the contractile filaments within allmuscle fibers. Therefore, if the in silico activation pattern isselected based upon real muscle data, it can simulate the time-varying activation process of the filaments which occurs (butis not measured) inside actual muscle fibers in vitro. Thus, theonly time-varying input to our muscle model was the activationwaveform (equation (3)). Activation conditions were chosenbased on in vitro muscle measurements. The cross-sectionalarea of the muscle model was set to 0.285 cm2 (based on theexample in vitro experiment of the current study) enabling apeak isometric tension of 5.7 N (assuming a peak isometrictension of 20 N cm−2; McMahon 1984) and muscle length wasset to 20 mm. These values are within the range for Xenopuslaevis PL (area = 0.33 ± 0.08 cm2; length = 22.0 ± 1.7 mm,N = 6; Richards 2011). Additionally, under the current in vitroconditions, the muscle typically produces force for ∼150–200 ms (Richards 2011). Thus, the activation waveform wasselected to be a single sinusoidal period with an amplitude of5.7 N and duration of 0.15 s. By adjusting software parametersin the control mode 2 control loop, the muscle model initiallength was set to L/Lo = 1.072 (F–L-gain = force/Po = 0.95)

to begin the simulated muscle contraction on the descendinglimb of the F–L curve (after Azizi and Roberts 2010).

3.4. Experimental design and statistics

We tested the effect of foot AR on muscular dynamicsand hydrodynamics. Digital images of Xenopus laevisfeet were traced in ImageJ (National Institutes of Health,Bethesda, MD, USA) to measure the foot area (6 cm2; AR≈ 1.4). Based on anatomical drawings of frog feet (Liu 1950,Goldberg and Fabrezi 2008), we selected two extreme footshapes to approximate the natural range of foot shape foundin aquatic frogs, represented by Xenopus sp. and Amolopsloloensis (normalized AR ≈ 1.4 and 2.9, respectively). Thesedrawings were simplified and either elongated or compressedto increase or decrease the foot AR beyond the natural rangeto give four feet (AR = 0.74, 1.4, 2.9, 4.9 for feet 1–4), wherefoot 2 (AR = 1.4) is the natural condition for Xenopus laevis.A constant total area of 6 cm2 was maintained using AdobeIllustrator CS4 (Adobe Systems Inc., San Jose, CA, UA). Feetwere cut from 1.57 mm thick PlexiglassTM using an EpilogMini 35 W laser cutter (Epilog Laser, Golden, CO, USA).

For both in vitro-robotic and in silico-robotic experiments,each robotic foot was tested using three repetitions (4 feet ×3 repetitions = 12 trials). Because of potential fatigue effectsof isolated muscle, additional caution was required for invitro-robotic experiments. To minimize the effect of trialorder, conditions (and their replicates) were performed inrandom order. Additionally, for each condition, we verified thatpeak force measurements remained within 10% of maximummeasurements indicating negligible fatigue effects.

Prior to each trial, the initial foot angle was set to 45◦

(anterior-medial). For each control mode, trace recordingsof muscle stimulation, muscle force, muscle length, normalhydrodynamic force and bio-robot swimming velocity wererecorded at a sample rate of 4 kHz by a National InstrumentsUSB-6289 A/D board. To remove electrical noise fromstrain gauge amplifiers, the hydrodynamic force signal wasfiltered with a 50 Hz second-order forward-backward low-pass Butterworth filter in Labview 2010 (National Instruments)prior to data analysis. Mechanical performance data (e.g.muscle impulse, peak bio-robot swimming velocity, etc) wereaveraged among three replicate trials. Mean values werecompared among the four foot shape types using a 1-wayANOVA with the Tukey–Kramer HSD post-hoc test performedin R (R Foundation for Statistical Computing, Vienna, AT).Simple linear correlations were used to compare results fromcontrol modes 1 and 2.

4. Results

4.1. In vitro-robotic experiments: bio-robotic swimmingperformance and muscle contractile dynamics in response tochanges in foot shape

For all trials, the muscle generated force for ∼125–150ms while shortening throughout figure 5(a). Within the first∼6–8 ms following the onset of nerve stimulation, muscleforce began to increase. Almost immediately, the muscle

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Figure 5. Muscle force, stimulation, hydrodynamic thrust, muscle shortening velocity and bio-robot swimming velocity traces for (a) invitro-robotic (control mode 1) and (b) in silico-robotic (control mode 2) experiments for feet 1 (green), 2 (gray), 3 (pink) and 4 (red). Notethat in vitro-robotic stimulation consists of a series of pulses input directly to the sciatic nerve, whereas in silico-robotic stimulation is anactivation waveform input to the mathematical model in control loop 2. All feet have the same area = 6 cm2 and all muscle stimulationconditions are identical within each control mode. Lines indicate mean values (N = 3 trials) and shading indicates mean ± s.d. Arrowsindicate where active force generation ends, marking the end of the period where all parameters (e.g. net power, impulse, etc) werecalculated.

generated enough force to accelerate the foot in rotationcausing a steep increase in hydrodynamic force. As thefoot produced hydrodynamic force, the bio-robot began toswim forward, reaching peak acceleration as the rotating footgenerated maximal hydrodynamic force at ∼40 ms. Despitethe forward motion of the bio-robot, the foot continued to rotatewith sufficient velocity to produce net thrust on the virtualbody. Consequently, the bio-robot continued to accelerate fornearly the entire period of muscle force, reaching peak velocityat ∼125 ms as hydrodynamic force fell to 0. During theremaining ∼25 ms of muscle force, the bio-robot entered a‘glide’ phase and began to decelerate due to negative net forceon the virtual body. Two sources contributed to the negativenet force on the virtual body: (1) as muscle force declined, themuscle could no longer rotate the foot faster than oncomingflow causing the foot to be ‘dragged’ along with the body.

(2) Forward swimming was slowed by virtual drag calculatedfrom the virtual body’s own forward motion.

Despite qualitative similarities among the experimentaltrials, changes in foot shape caused significant shifts in muscleF–L dynamics resulting in differences in bio-robot swimmingperformance. Most notably, muscle force increased with AR,whereas shortening velocity decreased (figures 5(a) and 6(a),table 1). From foot 1 to foot 2, muscle impulse (time integralof muscle force) increased only slightly from 0.12 ± 0.002to 0.13 ± 0.004, but significantly to 0.15 ± 0.009 (foot 2versus foot 3) and to 0.18 ± 0.007 N s (foot 3 versus foot 4;figure 6(a); table 1, P < 0.01; N = 3 trials; 1-way ANOVAwith the Tukey–Kramer HSD post-hoc test). Reciprocally,peak shortening velocity decreased significantly from6.16 ± 0.10 to 5.54 ± 0.17 to 4.4 ± 0.30, but only slightlyto 4.15 + 0.06 ML s−1 from low to high AR (P � 0.016).Perhaps due to opposing shifts in muscle force and shortening

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Figure 6. Box–whisker plots for muscle impulse, peak muscle shortening velocity, net muscle work, hydrodynamic thrust impulse, thrustimpulse: muscle impulse ratio, peak bio-robotic swimming velocity and total bio-robot distance traveled for (a) in vitro-robotic and (b) insilico-robotic experiments for feet 1 (green), 2 (gray), 3 (pink) and 4 (red). Note that all feet have the same area = 6 cm2. Horizontal bars aremedians. Boxes outline the interquartile range of the data, whereas whiskers (not discernable) represent the remaining data range. Asterisksrepresent significant differences between means (P � 0.05; 1-way ANOVA with TukeyHSD post-hoc test) of one foot compared to a lowerAR foot. Bold, black asterisks represent significant comparisons between adjacent foot types (e.g. foot 3 versus foot 2), whereas redasterisks indicate one or more significant differences between non-adjacent comparisons (e.g. foot 4 versus foot 2; see results for details).

Table 1. Summary muscle and bio-robot performance data for in vitro-robotic experiments.

Peak Net Hydrodynamic Peak TotalMuscle shortening muscle thrust swimming distanceimpulse velocity work impulse velocity traveled Muscle-propulsive

Foot (N s) (ML s−1) (J kg−1) (mN s) (m s−1) (mm) efficiency

1 0.12 ± 0.002 6.16 ± 0.10 14.2 ± 0.4 7.51 ± 0. 17 0.41 ± 0.01 21 ± 0. 4 0.15 ± 0.00422 0.13 ± 0.004 5.54 ± 0.17 14.4 ± 0.6 7.65 ± 0. 11 0.41 ± 0.00 25 ± 0. 7 0.15 ± 0.00513 0.15 ± 0.009 4.40 ± 0.30 14.2 ± 1.8 7.47 ± 0. 42 0.40 ± 0.02 27 ± 2 0.15 ± 0.0014 0.18 ± 0.007 4.15 + 0.06 17.1 ± 1.0 8.4 ± 0. 36 0.43 ± 0.00 31 ± 0. 5 0.14 ± 0.006

Values are mean ± s.d. for three replicates of data tested from one PL muscle.

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Bioinspir. Biomim. 7 (2012) 016010 C T Richards and C J Clemente

velocity, net muscle work remained at ∼14 J kg−1 increasingsignificantly to 17.14 ± 1.0 only with the highest AR foot(P = 0.048). Similarly, hydrodynamic thrust impulse onlyincreased significantly with the highest AR from ∼7.5 mNs to 8.4 ± 0.0003 (P = 0.025). Consequently, the thrustimpulse: muscle impulse ratio was significantly greater forthe low-AR feet as compared to the high-AR feet (∼0.06versus ∼0.05). Experimental variation among trial replicatesobscured differences in peak bio-robot swimming velocity(∼0.4 m s−1 for all trials). Yet at the end of trials, the finalbio-robot swimming velocity was significantly slower for thehigher AR feet (3 and 4) compared to feet 1 and 2 (P �0.01), with all other comparisons insignificant (P > 0.05).Within trials however, small differences accumulated over timecausing the total distance traveled to increase from 0.021 ±0.0004 to 0.031 ± 0.0005 m (all differences were significant,P � 0.01, except for foot 2 versus foot 3, P = 0.14).

4.2. In silico-robotic experiments: bio-robotic swimmingperformance and Hill-type muscle contractile dynamics inresponse to changes in foot shape

As can be seen qualitatively (figure 5(a) versus(figure 5(b)), in silico-robotic results were similar, but notidentical, to those obtained from in vitro-robotic experiments(see supplementary figure S4 for further comparison, availableat stacks.iop.org/BB/7/016010). However, in silico-roboticresults showed much greater differences between the footshapes when compared to in vitro-robotic experiments, dueto the low variability of the in silico-robotic method. Muscleimpulse increased significantly from 0.12 ± 0.0007 to0.17 ± 0.001 N s as AR increased from feet 1 to 4(figure 6(b), table 2; all increases were significant, P <0.05;N = 3 trials; 1-way ANOVA with the Tukey–Kramer HSDpost-hoc test). Conversely as AR increased from lowest tohighest, peak shortening velocity decreased significantly from5.42 ± 0.03 to 3.75 ± 0.02 ML s−1 (all increases weresignificant, P <0.001). Despite the opposing trends of muscleforce and velocity with increasing AR, net muscle work outputincreased from 11.53 ± 0.03 to 12.45 ± 0.04 J kg−1 (allincreases were significant, P � 0.0005, except for foot 3versus foot 4). Thrust impulse also positively correlated withAR; however, increases were only significant between thehighest and lowest AR. Similar to the in vitro-robotic results,thrust impulse: muscle impulse ratio decreased significantlyfrom 0.044 ± 0.001 to 0.034 ± 0.001 as AR increased (alldecreases were significant, P � 0.003). At the end of eachtrial, final bio-robotic swimming velocity mirrored the trendobserved for the in vitro-robotic experiments. However, unlikethe in vitro-robotic experiments, the peak bio-robot swimmingvelocity of ∼0.38 M s−1 for the high-AR feet (3 and 4) wassignificantly higher than ∼0.36 Ms−1 for the low-AR feet(1 and 2; P � 0.01). As observed or in vitro-robotic results,the total distance traveled increased significantly from 0.024± 0.0004 to 0.028 ± 0.0004 m as AR increased (all increaseswere significant except for foot 3 versus foot 4, P < 0.01).

5. Discussion

5.1. Foot AR strongly influences muscle mechanics andweakly influences bio-robot swimming performance

The current study provides the first direct measurements ofmuscle force in response to a known fluid-dynamic load.Our current setup expands upon the recent in vitro-roboticmethod (Richards 2011) with the addition of foot translation(mimicking forward animal swimming), hydrodynamic forcemeasurements and the in silico-robotic approach. Withinthe large parameter space that the current setup canexplore, we tested the influence of foot morphology onmuscle contraction dynamics, hydrodynamics and bio-robotswimming performance. Although results from in vitro-robotic experiments (control mode 1) were more variable thanfrom the in silico-robotic model (control mode 2), musclemechanical behavior was qualitatively similar between the twoapproaches. Moreover, current results were within the range ofin vivo values for hydrodynamic forces (Johansson and Lauder2004, Nauwelaerts et al 2005) and muscle stresses (Richardsand Biewener 2007) for frogs.

We changed foot AR using values both within andexceeding natural variation in frogs (Liu 1950, Goldberg andFabrezi 2008) from 0.74 to 4.9, with foot 2 (AR = 1.4)representing the natural shape for Xenopus laevis. Contraryto our expectations, hydrodynamic thrust did not diminishfrom increasing AR as observed in wing models (AR = 4.53–15.84; Usherwood and Ellington 2002). Instead, increasing ARresulted in a slight (∼10%) increase in hydrodynamic thrust,suggesting that fluid-dynamic interference between leadingand trailing edge vortices may not occur over the current rangeof AR. Thus, the current data do not support our hypothesisthat low-AR feet enhance rowing performance. In contrast,findings suggest that when all other biomechanical parametersare invariant, increasing foot AR causes a small increasein peak swimming speed and distance traveled. Moreover,changes in bio-robot swimming performance were small(∼10%) compared to the large changes in muscle mechanicsover a nearly fourfold increase in AR.

5.2. Muscle mechanics not hydrodynamics drive differencesamong fin ARs

Due to the modest differences in hydrodynamic force andbio-robotic swimming performance, current data suggest thatmuscle dynamics can dramatically shift to produce the workrequired to maintain propulsive performance across largechanges in foot morphology. Most notably, muscle forceimpulse increased by ∼50% and shortening velocity decreasedby a similar margin. Despite these reciprocal trends of forceversus velocity, net muscle work increased by ∼20% fromthe lowest to highest AR. Since muscle activation was notvaried, these shifts in muscle dynamics can only emergefrom differences in the timing of muscle shortening (F–Leffects) and velocity (F–V effects) with respect to the timingof activation. In the absence of a mathematical muscle model,temporally resolving the relative influences of activationversus F–L versus F–V effects would be difficult. During robot

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Bioinspir. Biomim. 7 (2012) 016010 C T Richards and C J Clemente

Table 2. Summary muscle and bio-robot performance data for in silico-robotic experiments.

Peak Net Hydrodynamic Peak TotalMuscle shortening muscle thrust swimming distanceimpulse velocity work impulse velocity traveled Muscle-propulsive

Foot (N s) (ML s−1) (J kg−1) (mN s) (m s−1) (mm) efficiency

1 0.12 ± 0.001 5.42 ± 0.03 11.5 ± 0.03 5.25 ± 0.11 0.36 ± 0.007 24 ± 0.4 0.11 ± 0.0032 0.13 ± 0.002 5.10 ± 0.04 11.9 ± 0.04 5.36 ± 0.16 0.37 ± 0.003 26 ± 0.6 0.10 ± 0.0063 0.15 ± 0.001 4.32 ± 0.03 12.4 ± 0.05 5.62 ± 0.08 0.38 ± 0.002 29 ± 0.4 0.11 ± 0.0014 0.17 ± 0.001 3.75 ± 0.02 12.4 ± 0.04 5.71 ± 0.11 0.37 ± 0.003 28 ± 0.4 0.11 ± 0.007

Values are mean ± s.d. for three replicates of data.

acceleration, the in silico-robotic approach demonstrates thatAR influenced muscle dynamics primarily through changesin F–V effects (figure 7). At the peak muscle activation andmuscle power (t = 75), the F–L gain only differed by ∼10%compared to the ∼40% shift in the F–V gain from low tohigh AR. As AR increased, the centroid shifted toward thetip of the foot, likely increasing the time-varying distanceto the hydrodynamic center of pressure, thus increasing theoutlever R. Consequently, the torque required to rotate thefoot (≈R Fh) increased (figures 5 and 6), requiring greaterforce at a lower shortening velocity, thus higher F–V gain.However, the F–L effects influenced trials more strongly inthe latter half of muscle contractions, where the F–L gain wasnearly threefold greater for foot 4 versus foot 1 (comparedto a ∼12% difference in the F–V gain) as activation returnedto zero (t = 150 ms). These findings indicate that for low-AR feet, the muscle ceases to generate force because themuscle has shortened far beyond its optimal length. In contrastfor high-AR feet, force reaches zero when the activationends. These findings therefore suggest that slower deactivationkinetics would lengthen the period of force generation (henceproducing greater thrust and swimming speed) for high-ARfeet, but would have a little influence for low-AR feet. Futureexperiments would be necessary to test whether the rates ofactivation and deactivation may be tuned to enhance swimmingperformance depending on the morphology of the propulsor.

5.3. Measurements of muscle–fluid interactions do not predictwhich foot shapes are optimal

An important question arises: why do frogs (and other drag-based swimmers) possess low-AR propulsors? In the currentstudy as AR increased, the muscular ‘effort’ (i.e. muscleimpulse and net work) increased more strongly than theincrease in hydrodynamic thrust. Hence, low-AR feet (feet1 and 2) generated greater thrust impulse/muscle impulse(figure 6). Despite the decreasing impulse ratio, muscle-propulsion efficiency (work done on the bio-robot’s virtualcenter of mass/net muscle work) remained nearly constant(tables 1 and 2). Such performance changes are smallcompared to the increases in speed and efficiency due todifferences in kinematics (e.g. ‘flapping’ versus ‘rowing’;Walker and Westneat 2000, Kato and Liu 2003). Consequently,within the experimental range of AR (∼0.7–5), currentfindings do not provide concrete insights to explain whyswimming frogs have low-AR feet (AR = 1–2). However,experiments performed at additional AR values (from ∼0.3

to 30) suggest that frog foot morphology falls within a broadrange of maximal thrust production (figure 8). Thus, the fluid-dynamic interference of vortices that was expected to occurwithin our initial experimental range likely only contributessignificantly at the high and low extremes (AR < 1 or AR >

5). Within this ‘optimal’ range (in which frog feet and fishpectoral fins lie), lower AR feet may be favorable for reasonsother than hydrodynamics. For example, short propulsors maybe favorable because they have a lower moment of inertia andexperience lower bending moments during swimming (Donget al 2006). Furthermore, a short–broad foot is likely moreappropriate for generating high ground reaction forces whileminimizing joint torques during jumping. Alternatively, froghind limb morphology may be developmentally constrained(Emerson 1988), possibly limiting the diversity of foot shapes.Regardless, further studies are required to determine therelevance of AR for the functional morphology of frog feet.

5.4. In silico-robotic and in vitro-robotic approaches arecomplementary not interchangeable

Given the qualitative agreement between in vitro-roboticand in silico-robotic results (figures 5 and 6), why areboth approaches necessary? In addition to the F–L andF–V effects explored above, muscle force is an intricatefunction of activation kinetics and history-dependent effects(Van Leeuwen 1992, Josephson 1999). Additionally, thesecontractile properties are coupled to the load via in-series elastic structures (both within and outside themuscle). Mathematical simulations are helpful for dissectingeither how linear versus nonlinear topologies of theF–L and F–V properties (Gerritsen et al 1998, Haeufleet al 2010) or in-series elastic properties (Lichtwark andWilson 2005) influence muscle mechanics. Yet, interactionsamong nonlinear contractile properties may become toodifficult to resolve in the case of a complex dynamicload (such as a deformable fin in fluid). Moreover, suchmodels cannot be easily verified when simulating realisticlocomotion. The current in vitro-robotic techniques enablemeasurements of muscle F–L dynamics in response toan arbitrarily complex load. Subsequently, in silico-roboticexperiments are performed to verify that the activationconditions are appropriate. Then, with knowledge of muscle’sF–L and F–V curves, one can decompose muscle forceinto length-dependent, velocity-dependent and activation-dependent components. To address in-series elastic effects (notconsidered in the current study), the time course of work done

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and muscle model power. Plot colors are identical to figure 5. As infigure 5, lines indicate mean values (N = 3 trials) and shadingindicates mean ± s.d. The gray bar indicates the period of muscleactivation with the maximum point marked by a red line. ∗If the gainratio = 0.5, the F–L and F–V gains contribute equally to muscleforce. The gain ratios above 0.5 indicate that the F–V effects aremore severe than the F–L gain and vice versa for values below 0.5.Note that for the first half of muscle activation, the F–V effectsdiminish muscle force more strongly than the F–L effects. However,the F–L effects limit muscle force more severely toward the end ofthe activation period especially for low-AR feet.

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on the load (e.g. fluid force on the foot) could be compared withthe timing of the stretch and recoil of elastic structures. Finally,comparative physiologists can ‘tune’ the intrinsic properties tobetter understand how natural variation in these propertiesleads to variation in locomotor performance. Therefore, acomplementary use of in vitro-robotic and in silico-roboticapproaches enables us both to verify a particular muscle modeland to resolve the influences of muscle properties, architectureand in-series elasticity against ‘real-world’ loads.

5.5. Summary

With the aim of linking muscle dynamics, skeletal andlimb morphology and external forces, we introduced amethod for controlling a bio-robotic limb with a muscle.Using constant muscle stimulation parameters, we comparedmuscle dynamics, hydrodynamics and bio-robot swimmingperformance during rowing propulsion with four foot shapesranging from low to high AR. We found qualitative agreementbetween data traces measured from an in vitro musclecompared to an in silico Hill-type muscle model. Furthermore,results suggest that despite large shifts in muscle F–Vdynamics, increasing foot AR only increased peak swimmingvelocity by ∼10%. Most importantly, differences in muscleloading due to fin shape were mainly caused by F–V effects asrevealed by the in silico-robotic experiments. Consequently asAR increased, muscle force and velocity changed reciprocallysuch that muscle work and power (therefore swimming speed)did not shift considerably.

Future experiments using the current setup are requiredto further investigate the effects of fin shape, such as thesteepness of fin taper or location of the centroid, on musclemechanics and swimming performance. Additionally, theinteraction between ‘optimal’ foot morphology and ‘optimal’muscle properties should be explored. For example, one mightexpect that specialized swimmers (e.g. fish) have ‘optimized’fin morphology. However, the most effective morphology mayonly be appropriate given the specific F–L, F–V and activationkinetics’ properties of a particular muscle. Likewise, certain

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muscle parameters may appear ‘optimized’ for swimming (e.g.high power, fast fiber types). Yet, these parameters may nolonger be optimal when different fin morphology is considered.Thus, the current setup is useful for exploring Marsh’sobservation (Marsh 1999) that natural selection likely operatessimultaneously on external and internal limb properties suchthat the ‘optimal design’ of one component of the locomotorsystem depends on the properties of all other components.

Acknowledgments

We are grateful to Chris Stokes for providing crucialsuggestions on the mechanical design of the translating sledas well as with the hydrodynamic force sensor. Additionally,we thank Donald Rodgers for assistance with machining. Weare thankful for assistance from two undergraduate students,Zwoisaint Mears-Clarke and Mark Moriarty, who helped inthe testing of the in silico-robotic model. We also thank JimUsherwood for insightful discussions regarding fluid dynamicsand Brooke Flammang for helpful comments during thepreparation of this paper. We thank two anonymous reviewersfor providing fair and thoughtful comments. This work wassupported by the Rowland Junior Fellows Program at HarvardUniversity.

Appendix A. Hydrodynamic force sensor

We developed a sensor to measure the normal componentof hydrodynamic force applied to the water by the foot.Unlike time-averaged force measurements from digital particleimage velocimetry (e.g. Drucker and Lauder 1999), thissensor measures instantaneous fluid reaction force normalto the foot as it moves against the water. The sensor isan aluminum framework modified from a 4-bar linkagecompound-flexure design (Moore et al 2009) which connectsthe foot to a rotating axle (supplementary figure S2 (A),available at stacks.iop.org/BB/7/016010). Within the linkage,the foot is mounted to a sliding platform which movesparallel with respect to the base. When loaded, the stiffleaf springs connecting the two bars bend to allow slightmotion of the sliding platform. Because of the bending ofthe leaf springs, medial displacement of the middle platformis exactly compensated by lateral displacement of the slidingplatform, constraining sliding to a single axis (parallel to thebase). Thus, independent of the location and orientation ofthe load, the sliding bar only displaces due to the componentof the load that is normal to the surface of the attached foot.Consequently, measuring bending displacements of the fourleaf springs enables measurements of force independent ofwhere the hydrodynamic center of pressure occurs along thelength of the foot. To measure hydrodynamic force, four straingauges in a full-bridge configuration (SR-4 120 Ohms, VishayIntertechnology, Inc., Malvern, PA, USA) were bonded usingan M-Bond adhesive resin (Type AE, Vishay Intertechnology,Inc.) onto each of the four leaf springs. The strain gaugesand contacts were then covered in three coats of air-dryingpolyurethane (M-Coat A, Vishay Intertechnology, Inc.) toprovide electrical insulation. The signal was amplified using

a Vishay amplifier (model no. 2120), and recorded using aNational Instruments A/D board (USB-6289).

To verify that the hydrodynamic sensor showed a lineardeformation in both compression and tension and thatdeformation was neither plastic nor hysteretic, the forcesensor was oscillated at 10 Hz against a calibrated Honeywellload cell (Model 31, Honeywell Automation and Controlsolutions, Columbus, OH, USA). Supplementary figure S2(B), available at stacks.iop.org/BB/7/016010, shows a linearresponse when load cell output is plotted against forcetransducer output. To calibrate the hydrodynamic sensor andconfirm its insensitivity to lever arm, we hung a series ofweights (436–5690 mg) at different lever arms (5–35 mm)along the plastic foot. Lever arm had no significant influenceon the force-calibration coefficient (supplementary figure S2(C), available at stacks.iop.org/BB/7/016010).

Appendix B. Virtual self-propulsion control loop

An additional feedback loop was added into the FPGAcontrol program to integrate the following differentialequation to obtain the body velocity Vb as a function ofhydrodynamic thrust (supplementary figure S3, available atstacks.iop.org/BB/7/016010).

dVb

dt= thrust − K · Vb(t−1)

2

mass(1 + Ca)

= 28(thrust − 0.03 · Vb(t−1)2), (B.1)

where thrust = 2 sin(foot angle) Fnormal, K = 0.5 ρ Cd Bodyfrontal area (Cd = 0.14; Nauwelaerts et al 2001; ρ = 1000kg m−3, Body frontal area = 4 cm2), mass = 30 g and Ca isthe added mass coefficient (= 0.2; Nauwelaerts et al 2001).Since the bio-robot only has a single foot, thrust was multipliedby 2 to model the typical symmetric leg motions of Xenopuslaevis. At each time step, Vb from equation (B.1) is the targetvelocity for the translational servo motor. To calculate the drag,Vb from the previous time step is fed back into the equation.As the foot rotates, it generates thrust and effectively ‘swims’forward if it were attached to a frog-sized ellipsoid body withCd = 0.14. Following the propulsive phase, the translationaltrack simulates gliding after Fnormal reaches 0 and the bodydecelerates. For either control mode 1 or control mode 2, boththe muscle feedback and the virtual self-propulsion controlloops were run in a single Labview program compiled to theFPGA controller.

Appendix C: testing the Hill-type model of thein silico-robotic method

To verify that our implementation of the Hill-type musclemodel is sufficient to mimic realistic muscle dynamics, wecompared in vitro muscle data from foot 2 (supplementaryfigure S4, available at stacks.iop.org/BB/7/016010) to datafrom the in silico-robotic method. Additionally, we performedtrials using the same foot and activation conditions afterremoving either the F–V or F–L effects using the partialHill-type models: (A) force = activation∗ F–Vgain∗ F–Lgain (the model used in the current study), (B) force =

13

Bioinspir. Biomim. 7 (2012) 016010 C T Richards and C J Clemente

activation, (C) force = activation∗ F–L gain, (D) force =activation ∗F–V gain (supplementary figure S4, availableat stacks.iop.org/BB/7/016010). These trials revealed thatmuscle models B and C cannot reproduce realistic dynamics inthe absence of the F–V function under the current conditions.However, model D produced force and velocity that nearlymatched the in vitro muscle data. Linear regression of the invitro muscle force trace versus in silico model force tracesfor model A showed a very strong correlation (in vitro muscleforce = −0.1 + 1.1 in silico model force; r2 = 0.99) comparedto models B (intercept = 0.02, slope = 0.26, r2 = 0.56), C(intercept = −0.02, slope = 0.49, r2 = 0.89) and D (intercept=−0.11, slope = 0.96, r2 = 0.92). Similarly, linear regressionsfor the shortening velocity for model A was also strong(in vitro shortening velocity = 0.05 + 1.1 in silico shorteningvelocity; r2 = 0.99) compared to models B (intercept = 0.35,slope = 0.74, r2 = 0.80), C (intercept = 0.31, slope = 0.74,r2 = 0.93) and D (intercept = 0.1, slope = 0.94, r2 = 0.95).

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15

0.25 s

5

0

10

15

1 deg

25 ms

Ang

ular

disp

lace

men

t(de

g)

Servomuscle displacement signalServofoot displacement signal

Supplementary figure S1

Supplementary figure S1. Raw traces from Servomuscle (black) and Servofoot (red) in response to arapid input force measured at Servomuscle. A small (~ 50 g) mass was dropped on the muscle lever to generate rapid motor velocity and acceleration beyond the normal conditions measured in muscle experiments. Overlapping data traces show negligible magnitude and phase error. Boxed inset shows a more detailed view during a brief period of rapid acceleration. Data were recorded at 4 kHz.

Supplementary figure S1

Supplementary figuure S1

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Hydrodynamic sensor Top View

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0.018

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nam

icse

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outp

ut(N

)

Initial foot angle

0.02 0.04 0.06 0.08

0.02

0.04

0.06

0.08

Supplementary figure S2

Supplementary figure S2

Supplementary figure S2. (A) Top view of hydrodynamic sensor, which consists of a 4 bar linkage with strain gauges [either loaded in compression (red) or tension (blue)] in full bridge configuration to transducer normal force (Fnormal) for calculating thrust (Fthrust). (B) Hydrodynamic sensor oscillated against a Honeywell force cell at 10Hz, shows linear deformation in both compression and tension over four loading-unloading cycles. (C) Force sensor was calibrated by hanging different weights (436 – 5690 mg), the force calibra on coefficient does not change at different lever arms (5 to 35 mm).

Supplementary figure S2

gaugesStrain

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platformSliding

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SSupplementary figure S2

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Hydrodynamicforce (volts)

Position (volts)from

Servofoot

N/volt

Fthrust)(-Drag+ = dt

/volt Sine( )

K x Vbody(t -1)2Vbody(t-1)

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Fnormal

2Fnormal x

Vbody(t)

FPGA CONTROL LOOP: Virtul self propulsion Supplementary figure S3

A

User inputs:- Virtual body drag and added mass coefficients- Calibration constants

Data trace outputs:-Vbody

I

I

I

O

Drag calculation

Thrust calculation

B

1/mass

Body acceleration calculation

Supplementary figure S3. (A) Virtual self propulsion control loop for both in vitro-robotic and in silico-robotic experiments. Voltage signals from the hydrodynamic sensor and Servofoot position are calibrated (pink boxes) to normal hydrodynamic force (Fnormal) and foot angle ( ) data, respectively, for calculating thrust (Fthrust =2Fnormal*Sin ). Simultaneously, body velocity (Vbody) from the previous time step is used to calculate drag (K* Vbody 2). Net force (thrust-drag) is integrated and multiplied by 1/mass to calculate Vbody as the instantaneous target velocity for the translational motor. All operations in the loop are performed in a single 0.1 ms time step. (B) Example data from the bio-robotic swimmer (top) and from a video recording of X. laevis swimming (bottom). Traces are Vbody (black, thick) and foot rotational velocity, Vrot (gray, thin).

Supplementary figure S3

50 100 150 200

0.1

0.2

0.3

0.4

50 100 150 200

0.1

0.2

0.3

0.4

5

10

15

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Foot

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tiona

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ocity

,Vro

t(ra

d/s)

Swim

min

gve

loci

ty,V

body

(m/s

)

Time (ms)

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X. laevis swimming

Bio-robot swimmingHydrodynamic

Supplementary figure SFPGA

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S3iViLOOP:CONTROLGAA irtul self propulsion B

0 4Bio-robot swimming

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footServofrom

Position (volts)

force (volts)Hydrodynamic

body(t-1)V

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I

I 2

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/volt )Sine(

body(t -1)VxK

normalFFn

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xnormal2F Sin

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50 10

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0.2

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0.4

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10

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dy

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0.3

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imeT

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0 150 200

5

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Supplementacitoboocilii

ater levelWWater level

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SSupplementary figure S3

20 40 60 80 100 120 1400

1

2

3

4

5

20 40 60 80 100 120 1400

2

4

6

In silico model B: force = activationIn silico model C: force = activation*F-L gainIn silico model D: force = activation*F-V gain

In silico model A: force = activation*F-V gain*F-L gain

In vitro muscle

Mus

cle

forc

e(N

)Sh

orte

ning

velo

city

(ML/

s)Supplementary figure S4

Supplementary figure S4

Supplementary figure S4. Testing the in silico-robotic model. Data from in vitro-robotic experiments were compared to data from in silico-robotic experiments. Using foot #2, muscle force and velocity were recorded with four partial muscle models to decomposing the effects of force-length (F-L) vs. force-velocity (F-V): Model A) force = activation*F-Vgain*F-L gain (the model used in the current study), B) force = activation, C) force = activation*F-L gain, D) force = activation *F-V gain. Note the close match between muscle model A and actual in vitro muscle data. Additionally, note how F-V effects are crucial for generating appropriate dynamics with the current model.

Supplementary figurre S4

In silico mIn silico m

model B: force = activatiomodel A: force = activatio

onon*F-V gain*F-L gain

2

3

4

5

Mus

cle

forc

e(N

)

In silico mIn silico mIn silico m

ro In vitro m

model B: force = activatiomodel C: force = activatiomodel D: force = activation*F-V

muscle

onon*F-L gain

Von*F-V gain

200

1

Ms)

40 60 80 100 120 140

2

4

6

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Supplementary figurre S4