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Determining the influence of muscle operating length on muscle performance during frog
swimming using a bio-robotic model
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2012 Bioinspir. Biomim. 7 036018
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IOP PUBLISHING
BIOINSPIRATION & BIOMIMETICS
doi:10.1088/1748-3182/7/3/036018
Bioinspir. Biomim. 7 (2012) 036018 (12pp)
Determining the influence of muscle
operating length on muscle performance
during frog swimming using a bio-robotic
model
Christofer J Clemente and Christopher Richards
Rowland Institute, Harvard University, Cambridge, MA 02142, USA
E-mail: clemente@rowland.harvard.edu, and richards@fas.harvard.edu
Received 17 January 2012
Accepted for publication 18 May 2012
Published 8 June 2012
Online at stacks.iop.org/BB/7/036018
Abstract
Frogs are capable of impressive feats of jumping and swimming. Recent work has shown that
anuran hind limb muscles can operate at lengths longer than the ‘optimal length’. To address
the implications of muscle operating length on muscle power output and swimming
mechanics, we built a robotic frog hind limb model based upon Xenopus laevis. The model
simulated the force–length and force–velocity properties of vertebrate muscle, within the
skeletal environment. We tested three muscle starting lengths, representing long, optimal and
short starting lengths. Increasing starting length increased maximum muscle power output by
27% from 98.1 W kg−1 when muscle begins shortening from the optimal length, to
125.1 W kg−1 when the muscle begins at longer initial lengths. Therefore, longer starting
lengths generated greater hydrodynamic force for extended durations, enabling faster
swimming speeds of the robotic frog. These swimming speeds increased from 0.15 m s−1 at
short initial muscle lengths, to 0.39 m s−1 for the longest initial lengths. Longer starting lengths
were able to increase power as the muscle’s force–length curve was better synchronized with
the muscle’s activation profile. We further dissected the underlying components of muscle
force, separating force–length versus force–velocity effects, showing a transition from
force–length limitations to force–velocity limitations as starting length increased.
(Some figures may appear in colour only in the online journal)
1. Introduction
have focused on muscle power (force x velocity) and work
(force x displacement) to understand how frogs jump (James
and Wilson 2008, Peplowski and Marsh 1997) and swim
(Calow and Alexander 1973, Richards and Biewener 2007).
For example, recent work has suggested that modifications in
muscle–tendon elastic properties enable muscle–tendon units
to produce enhanced force and power output for jumping
(Azizi and Roberts 2010, Roberts and Marsh 2003). Based on
these observations from prior work, the current study aims to
explore the mechanisms by which frog musculature produces
power to overcome fluid dynamic forces during swimming.
Underlying the complexity of muscle–fluid dynamic
interactions, muscle force is governed by well-established
Many anurans possess spectacular abilities to jump, swim
or both. Because either locomotor mode is crucial for
catching prey (e.g. Howard 1950, Lafferty and Page 1997)
and avoiding predators (Feder 1983, Figiel Jr and Semlitsch
1991, Jung and Jagoe 1995, Kaplan and Phillips 2006,
Raimondo et al 1998, Watkins 1996), higher speeds and
accelerations are advantageous for survival and reproduction.
Hence, the evolution and diversification of frogs were likely
influenced by mechanical specializations of the frog skeleton
(Emerson 1982, Reilly and Jorgensen 2011, Shubin and
Jenkins 1995) and musculature (Lutz and Rome 1994) to
confer locomotor performance. In particular, physiologists
1748-3182/12/036018+12$33.00
1
© 2012 IOP Publishing Ltd
Printed in the UK & the USA
Bioinspir. Biomim. 7 (2012) 036018
C J Clemente and C Richards
propulsive drag force the feet must move faster. However a
muscle’s force capacity decreases hyperbolically as muscle
velocity increases (Hill 1938). Consequently, maximum power
occurs at a shortening velocity of ∼1/3 vmax (Lutz and
Rome 1994). Thus, in addition to limits imposed by force–
length properties, force–velocity effects might represent a
unique trade-off between peak muscle force and limb velocity
in these aquatic systems. Further, since maximal power is
produced over a narrow range of muscle velocities, changes in
starting length may affect muscle velocity and therefore power
throughout the swimming cycle.
No study has explored the interacting effects of muscle
operating length and environmental mechanical feedback
while considering both force–length and force–velocity
effects. Therefore, it is unclear how strongly neural activation
versus force–length versus force–velocity effects influence
muscle dynamics in aquatic systems. Yet, determining the
limiting features of musculoskeletal systems is important
to understanding constraints imposed by intrinsic muscle
properties. However, measurement of force–velocity and
force–length effects in vivo is difficult because patterns of
neural activation and starting length cannot be controlled.
To study the interplay between hydrodynamics and muscle
dynamics we used a recently-developed robotic model of a frog
leg based upon the plantaris longus (PL) muscle in the hindlimb
of frogs (Richards and Clemente 2012). The foot was equipped
with a hydrodynamic sensor which was rotated through water
by a virtual muscle governed by activation kinetics, force–
length and force–velocity properties measured from Xenopus
laevis. Based on previous studies of muscle starting lengths in
frogs (Azizi and Roberts 2010), we predicted that increasing
both activation and starting length would increase muscle force
and power output for swimming frogs. In further analysis,
we dissected the underlying components of muscle force
to evaluate how strongly force–length versus force–velocity
effects dominate in aquatic systems.
parameters. Primarily, increased neural activation causes peak
force to rise as more muscle fibers are recruited (Adrian and
Bronk 1929). At a high level of stimulation, virtually all the
muscle fibers become active, and further increases in the
strength of the stimulation are not accompanied by further
increases in twitch force (e.g. McMahon 1984).
The force generated by active muscle fibers also depends
upon the instantaneous length of the sarcomeres, with maximal
force occurring when the thick and thin filaments (myosin
and actin) are near 100% overlap (Gordon et al 1966). For
whole muscles, the plateau of this force–length curve (i.e.
‘optimal length’) is relatively narrow, with maximal forces
only being produced within ± 5% of the optimal length (Azizi
and Roberts 2010). At lengths longer or shorter than optimal
length, the muscle is said to operate on the descending or
ascending limbs of the force–length curve respectively. The
force declines on these limbs as the number of potential crossbridge interactions declines, producing the force–length curve.
To produce sufficient work for swimming or jumping,
muscles often shorten considerably during the movement.
Measurements of muscle fascicle length support this
prediction, with some muscles shortening by 30% (Olson
and Marsh 1998, Roberts and Marsh 2003). But a muscle
shortening by 30% cannot avoid the significant effect of the
force–length relationship of muscle, especially for muscles
that operate between the plateau and the descending limb of
the force–length curve. Thus, force–length properties impose
a limit for a muscle’s capacity to perform work (Woledge et al
1985), as shortening over a broad range of the force–length
curve may produce greater work, but smaller instantaneous
forces than operating within the narrow plateau region.
Azizi and Roberts (2010) found that hind limb muscles in
bullfrogs may partially overcome the effects of this tradeoff by operating primarily on the descending limb of the
force–length curve, starting the contraction at long initial
lengths and shortening toward the muscle’s optimal length
during fixed-end contractions. This effect was attributed to
an increase, rather than a decrease, in filament overlap as
muscle force develops, because the muscle length was moving
up the descending limb. In contrast, when the muscle began
on the plateau, it shortened beyond the optimal lengths for
force production during the early period of force development.
However, these measurements on bullfrog muscle were made
at constant muscle–tendon length, thus the implications of
muscle operating length on muscle–tendon work and power
output are yet to be explored. Moreover, it is unclear the
extent to which Azizi and Roberts’ prediction can be applied in
aquatic systems where environmental influences are different.
The mechanical constraints imposed by the environment
differ drastically for jumping versus swimming. Theoretical
studies comparing the two environments have exposed a
potential trade-off within this system (Aerts and Nauwelaerts
2009). During swimming, limbs do not push off against a rigid
support (Nauwelaerts and Aerts 2003, Nauwelaerts et al 2005).
Instead during swimming, the propulsive drag force resisting
the push of the feet varies with the square of the instantaneous
velocity (Gal and Blake 1988). When velocity of the feet is low,
the force resisting them is also low. In order to produce a higher
2. Methods
We developed a robotic frog foot based on rotation at the ankle
joint of the aquatic frog Xenopus laevis. A detailed description
of the model is given elsewhere (Richards and Clemente 2012)
and only a brief description is given here (figure 1).
The model consists of a 1.57 mm thick acrylic plate
mimicking the shape of a X. laevis foot. The plate is attached
to a hydrodynamic sensor, which can be rotated and translated
through water by two identical brushless servo motors
Servofoot and Servobody (Pittman 4443S013, PittmanExpress,
Harleysville, PA, USA) each controlled by an Accelus ASP055-18 digital servo amplifier (Copley Controls, Canton, MA,
USA). The rotation of the foot and sensor through water was
powered by a computer-generated muscle model, which, using
a feedback loop at 10 kHz, simulated the force–length and
force–velocity properties of vertebrate muscle.
2
Bioinspir. Biomim. 7 (2012) 036018
C J Clemente and C Richards
(a)
(b)
(c )
(d )
Figure 1. Setup of the bio-robotic frog. (a) The robotic foot is rotated through the water by Servofoot and translated by Servobody. The
rotation of Servofoot is controlled by a muscle model FPGA controller (c) which combines the information from the activation waveform, the
force–velocity gain and the force–length gain to produce a target torque for Servofoot. The rotation of the plastic foot cause hydrodynamic
drag, which is recorded by the hydrodynamic sensor (d). The hydrodynamic force is then fed into the virtual self propulsion FPGA
controller (b), which along with the body velocity from the previous time step, calculates the target translational velocity for Servobody.
bored using a 21 gauge needle, and surgical suture (Vicryl
4–0, braided, Ethicon) was threaded though to anchor the
proximal end of the muscle. At the distal end of the PL muscle,
suture was threaded through and around the PL muscle just
proximal to the PL tendon. The muscle was then mounted
to the ergometer, with the proximal end tied to a stiff metal
pin embedded in plexiglass, and the distal suture tied to the
small hole in the lever on the servo motor (305C-LR, Aurora
Scientific Inc, Aurora, ON, Canada). This setup was then
bathed in oxygenated amphibian ringers solution (Carolina
Biological, Burlington, NC, USA) at 22 ◦ C.
The PL muscle was stimulated using plate electrodes
constructed from surgical blades. Stimulation pulses of 1 ms
width were generated by an A/D board and amplified by
an OPA549T op-amp (Texas instruments, Dallas, TX, USA)
powered by a Sorensen LS 18-5 power supply (AMETEK
Programmable Power, Inc, CA, USA). Isometric twitch
2.1. In vitro measurement of force–length and force–velocity
properties
Xenopus laevis Daudin 1802 (n = 7, body mass 23.7 ± 2.4 g),
were obtained from Xenopus Express Inc (Plant City, FL,
USA). Animals were housed in aquaria and maintained at
20–22 ◦ C under a 12:12 h light:dark cycle, and fed twice per
week.
Force–velocity properties of the PL muscle X. laevis were
investigated in three individuals (mass = 18.7 ± 2.0 g).
Frogs were double pithed with a 21 gauge syrine needle, and
the PL muscle length (Lo) was recorded by positioning the
hip, knee, ankle and metatarsal joints at 90 and measuring
the longest proportion of the muscle between the tendon
and the aponeurosis. The PL was then removed keeping the
proximal attachment at the knee intact, by cutting the femur
approximately 0.5 cm from the knee joint. This was then
3
Bioinspir. Biomim. 7 (2012) 036018
C J Clemente and C Richards
contractions were used to determine the passive force at which
twitch muscle forces peak, and this was used as an initial
starting tension for isotonic experimental contractions. For
experimental contractions the muscle was stimulated with a
150 ms pulse train at supramaximal voltage (18 V) with a
spike frequency of 250 Hz, based on in vivo EMG patterns
previously observed in the PL muscle of X. laevis (Richards
and Biewener 2007). The isotonic force was varied in 0.1 N
intervals, until the muscle was unable to move the force lever
(i.e. an isometric contraction), and both muscle force and
muscle length were recorded using a custom built Labview
script (National Instruments, Austin, TX, USA). The muscle
was allowed to rest for 10–15 min between contractions. To
calculate the vmax for each individual muscle, the velocity of
shortening (Vm) at each isotonic force (Fmuscle) was fitted using
equation (1)
bFo − aV m
Fmuscle =
(1)
Vm + b
where a and b are constants specific to the muscle, and Fo is
the maximum isometric tension (Daniel 1995). The maximum
muscle-shortening velocity (vmax) is then calculated where
the force drops to zero (vmax = bTo/a). The mean vmax for
the three individuals was used in bio-robotic simulations to
determine the force–velocity curve.
Force–velocity curve was defined from using the formula
from Richards and Clemente (2012):
2.61 − V (t ) · 2.61/vmax
FV (t ) =
, V (t ) 0,
(2)
V (t ) + 2.61
where V(t) is the normalized shortening velocity (shortening
velocity/Lo) and vmax is the maximum shortening velocity.
The coefficient 2.61 is derived from twitch properties of
Xenopus muscle fibers in Lannergren (1987).
Force–length properties were determined for four
individual X. laevis (Mass = 28.6 ± 1.3 g). The PL muscle was
prepared as above. Two small knots were stitched superficially
onto the muscle using fine suture (Vicryl 6–0, Black silk,
Ethicon) to act as landmarks for measuring muscle length.
The muscle was set to resting length as determined during
dissection, and maximum isometric force was determined via
muscle stimulation using the same stimulation pattern as for
force–velocity measurements. The muscle length was then
shortened and lengthened in 5% increments between −30%
and + 15% resting length. The force–length data was pooled
and the force–length gain function was fitted based on a prior
model (Otten 1987):
a
b
(3)
FLgain(t ) = e−|(L(t ) −1)/s|
the nervous system. The activation impulse was modeled as
a cosine waveform function (mimicking activation in vivo;
Richards and Biewener 2007):
Po
(4)
A(t ) = S · [1 − Cos(t · 2π /dur)]
2
where Po is the maximum isometric tension of the muscle
(Po = 20 N cm−2 = 10 N, McMahon 1984), dur is the duration
of the activation (0.2 s) and S is the stimulation constant which
represents the fraction of muscle fibers recruited (S = peak
activation (N)/Po). Therefore A(t) represents the active state
of the muscle which dictates the maximum force at each instant
in time; full-scale activation builds up gradually and smoothly
with Peak Activation being reached in 0.1 s. The active state
similarly declines ending at 0.2 s. The activation was then
fed into a National Instruments cRio9074 field programmable
gate array (FPGA) ‘real time’ controller (National Instruments,
Austin, TX, USA) along with the position and velocity of the
virtual muscle.
2.3. Force–length and force–velocity functions
To determine the position and velocity of the virtual muscle,
angular displacement of the rotating foot (measured directly
from Servofoot) was related to the displacement of the muscle
via the muscle moment arm (r) (equation (2)) to mimic the
skeletal environment of a muscle.
Lmuscle · C1
(5)
θfoot ≈
r
where θ foot is the change in foot angle ( = change in
Servofoot angle), Lmuscle is the change in muscle length
and C1 is a unitless calibration constant computed from
experimentally for Servofoot. The muscle moment arm (r) was
set to 0.254 mm (based on direct morphological measurements
from X. laevis; Clemente and Richards, in review). The FPGA
then used these variables to compute the resulting muscle force
based upon the virtual muscles’ position along both the force–
length and force-velocity curves, as described above.
The resulting muscle force was then multiplied by the
muscle moment arm (r) to calculate the torque on the foot
for the next time step. This foot torque was then fed back to
Servofoot, rotating the foot through the water.
2.4. Hydrodynamic force measurement
To measure hydrodynamic forces as the foot rotated through
the water, we developed the hydrodynamic force sensor
based upon a 4-bar linkage system which measures force
normal to the foot surface independent of the lever arm
(Moore et al 2009). This meant resistive hydrodynamic
forces could be measured directly without estimations of the
hydrodynamic center of pressure. To measure hydrodynamic
forces, strain gauges (SR-4 350 Ohms, Vishay Intertechnology,
Inc, Malvern, PA, USA) were placed on the 4-bar linkage in
a full-bridge configuration (figure 1(d)). The strain gauges
and contacts were then covered in three coats of air-drying
polyurethane (M-Coat A, Vishay Intertechnology, Inc) to
provide electrical insulation. The signal was amplified using
a Vishay amplifier (model no 2120), and recorded using a
where L(t) is the relative muscle length (L/Lo = instantaneous
length/optimal length). Values for b, s and a were determined
using the fitcurve function in Mathematica 8 (Wolfram
Research, Inc).
2.2. Activation of the virtual muscle
The muscle resting length (2 cm) and cross sectional area
(0.5 cm2) were based on both prior (Richards 2011, Richards
and Clemente 2012) and current in vitro studies. Each trial
began with an activation impulse simulating the impulse from
4
C J Clemente and C Richards
National Instruments A/D board (USB-6289). Further details
on the sensor are provided in prior work (Richards and
Clemente 2012).
2.5. Foot translation via ‘virtual self propulsion’
Translation of the foot through the water was determined
by the hydrodynamic thrust produced by the rotating foot
through a system termed ‘virtual self propulsion’ (see Richards
and Clemente 2012 for details). Instead of requiring thrust
from the rotating foot to overcome the friction and inertia
of the translational track, we used a servo motor (Servobody),
identical to Servofoot, to translate the foot forward. To do this
we added an additional feedback loop to the FPGA control
program which input the hydrodynamic thrust at the foot into
a mathematical model of a swimming frog to compute the
acceleration resulting from the balance of thrust and drag on
the body:
thrust − K · Vb(t−1)2
dVb
=
= 28(thrust − 0.03 · Vb(t−1)2 )
dt
mass(1+Ca )
(6)
(a)
1.0
Relative muscle
force (F/Fo)
Bioinspir. Biomim. 7 (2012) 036018
0.8
0.6
0.4
0.2
(b)
1.0
Relative muscle
force (F/Fo)
0
2
4
6
8
-1
Relative muscle velocity (ML.sec )
0.8
0.6
0.4
0.2
0.7
0.8
0.9
1.0
1.1
Relative muscle length (L/Lo)
where thrust = 2 · sin(foot angle) · Fnormal, K = 0.5 · ρ · Cd ·
body frontal area (Cd = 0.14; Nauwelaerts and Aerts 2003,
ρ = 1000 kg m−3, body frontal area = 4 cm−2), mass = 30 g and
Ca is the added mass coefficient (Ca = 0.2; Nauwelaerts et al
2001). Since the bio-robot only has a single foot, thrust was
multiplied by 2 to model the typical symmetric leg motions
of X. laevis. As for the muscle model above, the calculated
velocity of the virtual body is fed back into the model to
calculate drag for the next timestep.
Two different experimental variables were tested;
variation in muscle activation amplitude and muscle starting
length. Peak muscle activation varied from 2.5 to 7.0 N in
0.5 N intervals for each starting length of the muscle. Larger
activations were not tested due to limitations in the length
of the translational track. Three starting lengths (Lstart) were
examined relative to the resting length (Lrest) of the muscle 0.9,
1.0 and 1.1. These three starting lengths represent contractions
beginning on the ascending limb, at the plateau and on the
descending limb, respectively. Each trial consisted of a single
kick in which muscle activation caused backward foot rotation
while the body was translated forward. These movements
represent the foot and center of mass movements during a
typical X. laevis power stroke and glide phase. The recovery
stroke was not modeled in the current study.
Figure 2. In vitro muscle data for the Plantarus longus muscle from
Xenopus laevis. (a) Force–velocity data from three individual
muscles, showing the fit used in the muscle model (dashed line,
R2 = 0.97). (b) Force–length data from four individual muscles
showing the fit used in the muscle model (dashed line, R2 = 0.94).
Force–length properties of the PL muscle were also
consistent with those previously published (Lutz and Rome
1994). The coefficients best describing the data, using Otten’s
(1987) function were b = −2.529, s = −0.589 and a =
1.407. This curve closely described the raw data (R2 = 0.94,
figure 2(b)). The plateau of the force–length curve was narrow,
being less than ± 5%.
3.2. Muscular and hydrodynamic performance of robotic
swimming
Mean traces for the muscle force, hydrodynamic force, muscleshortening velocity and bio-robotic swimming velocity
throughout a trial are shown in figure 3. For all trials the
muscle force began to increase rapidly ∼20–25 ms following
the onset of activation. During this period the muscle overcame
the inertia of the foot and the added mass of the water,
and began to shorten, rotating the foot. This generates
hydrodynamic force, which rapidly increased, causing the biorobotic swimming velocity to increase. Both the muscle and
the hydrodynamic force peak at ∼60–90 ms, after activation
began but before peak activation (at 100 ms), and then
declined. Hydrodynamic force became negative toward the
end of the trial (∼200 ms). Shortening velocity peaked
much later (∼100–150 ms) causing bio-robotic swimming
velocity to increase until ∼140 ms, after which it reached
a plateau, and began to decline as the bio-robot entered the
glide phase. Notably, due to force–length and force–velocity
effects, muscle force was always lower than the maximum
force prescribed by the activation waveform.
3. Results
3.1. In vitro muscle performance
Muscle force–velocity properties of the PL agree well with
previously published results for other Xenopus muscles
(Lannergren 1987). The force decreased hyperbolically with
muscle velocity, reaching a maximum muscle velocity (vmax)
of 7.55 ± 0.71 ML s−1. The force–velocity function
(equation (2)) calculated based on Richards and Clemente
(2012), with vmax of 7.55 ML s−1 agreed closely with the
raw data (R2 = 0.97; figure 2(a)).
5
Bioinspir. Biomim. 7 (2012) 036018
C J Clemente and C Richards
High activation (7.0 N)
Muscle Force (N)
Low activation (3.5 N)
3.0
3.0
2.5
2.5
2.0
2.0
1.5
1.5
1.0
1.0
0.5
0.5
0.10
0.08
0.06
0.04
0.02
0.00
-0.02
-0.04
Muscle Shortening
-1
Velocity (ML.sec )
Hydrodynamic Force (N)
0.0
50
100
100
150
150
200
200
0.0
0.10
0.08
0.06
0.04
0.02
0.00
-0.02
-0.04
2.5
2.5
2.0
2.0
1.5
1.5
1.0
1.0
0.5
0.5
0.0
Bio-robotic swimming
velocity (m.sec-1)
50
50
100
150
200
0.0
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
50
100
150
200
0.0
50
100
150
200
50
100
150
200
50
100
150
200
50
100
150
200
Figure 3. Muscle force, hydrodynamic force, muscle-shortening velocity and bio-robotic velocity traces for the muscle model with low peak
activation of 3.5 N (left) and the muscle model with high peak activation of 7.0 N (right). Peak activation represents potential peak muscle
forces if force–length and force–velocity limitations were absent. Three muscle starting lengths are shown for each case, muscles which
began shortening at 1.1 Lstart (red), 1.0 Lstart (green) and 0.9 Lstart (gray). Solid lines indicate mean values (N = 3 trials) and shaded areas
indicates mean ± s.d.
and Lstart = 1.1 (P = 0.625). These differences were reflected
at high activation trials, with the shortest starting length
having much lower peak stress (39.66 ± 0.42 kPa) than both
Lstart = 1.0 (58.28 ± 0.31 kPa) and 1.1 (59.40 ± 0.08 kPa;
F2,8 = 3820, P < 0.001). However, post hoc tests revealed a
small but significant increase between peak muscle stress at
Lstart = 1.0 and Lstart = 1.1 (P = 0.010). The timing of peak
muscle force also differed between starting lengths. At low
activation the peak of the muscle force was 80 ms for Lstart =
0.9, decreasing to 77 ms for Lstart = 1.0 and being highest for
Lstart = 1.1 (92.6 ms). As activation increased, the time to peak
force decreased to 68.3, 70.6 and 92 ms for Lstart = 0.9, 1.0
and 1.1 respectively.
Greater differences were observed when the muscle
impulse (time integral of muscle force) was considered. At
low activations muscle impulse increased significantly (F2,8 =
871, P < 0.001) from 0.129 ± 0.003 to 0.198 ± 0.001 N s
For any given starting length, muscle forces increased with
increasing activation. Similarly, hydrodynamic forces, muscle
velocities and bio-robotic swimming velocities were higher
at higher activations. However, as discussed below, the rate of
increase for these variables with activation varied with starting
length.
3.3. Interacting effects of starting length and activation
The effects of starting length on muscle function,
hydrodynamic force and bio-robotic swimming speed for high
activation (7.0 N) and low activation (3.5 N) trials are shown
in figure 4. For low activation trials peak muscle stress at
Lstart = 0.9 was 27.33 ± 0.76 kPa (mean ± s.d.), significantly
lower than peak stress of 37.87 ± 0.31 kPa at Lstart = 1.0 or
37.01 ± 0.12 kPa at Lstart = 1.1 (F2,8 = 449, P < 0.001,
single factor ANOVA), however Tukey post hoc tests show no
significant difference between peak muscle stress at Lstart = 1.0
6
Bioinspir. Biomim. 7 (2012) 036018
C J Clemente and C Richards
0.15
0.10
0.05
0.00
0.008
0.006
0.004
0.002
0.000
-1
0.010
140
1.4
1.2
1.0
Peak muscle power (W.kg )
0.20
Av. shortening velocity (ML.sec-1)
0.25
0.012
Hydrodynamic impulse (N.sec)
Muscle impulse (N.sec)
0.30
n.s.
0.8
0.6
0.4
0.2
0.0
120
100
80
60
40
20
0
-1
Peak swimming velocity (m.sec )
Low Activation (3.5 N)
0.4
0.3
0.2
n.s.
0.1
0.0
0.15
0.10
0.05
0.00
-1
0.010
0.008
0.006
0.004
0.002
0.000
140
1.4
Peak muscle power (W.kg-1)
0.20
Av. shortening velocity (ML.sec )
0.25
0.012
Hydrodynamic impulse (N.sec)
Muscle impulse (N.sec)
0.30
1.2
1.0
0.8
0.6
0.4
0.2
0.0
Lstart = 0.9
Lstart = 1.0
120
100
80
60
40
20
0
Peak swimming velocity (m.sec-1)
High Activation (7.0 N)
0.4
0.3
0.2
0.1
0.0
Lstart = 1.1
Figure 4. Influence of activation and starting length (Lstart) on muscle impulse (time integral of muscle force), hydrodymanic impulse (time
integral of hydrodynamic force), average muscle-shortening velocity (muscle lengths per second), muscle power and peak bio-robotic
swimming velocity. Bars show means (n = 3 trials) ± s.d. All interactions significant unless shown (n.s.), based on Tukey post hoc tests.
higher for longer starting lengths, increasing from 22.91 ±
0.51 to 48.35 ± 0.31 W kg−1 for low activation trials and from
48.3 ± 0.71 to 125.1 ± 0.62 W kg−1 for high activation trials
(P < 0.001). Consequently, bio-robotic swimming velocity
was significantly higher for trials with longer initial starting
lengths, particularly at higher activations, despite a short period
at the beginning of the trial when translational velocity of
shorter initial lengths were higher (figure 3). Final swimming
velocity for Lstart = 1.1 was 0.39 m s−1, while it was lower
for starting lengths of Lstart = 1.0 being 0.27 m s−1, down to
0.15 m s−1 at Lrest = 0.9 (m s−1), for the highest activation
trials (F2,8 = 1157, P < 0.001).
for the shortest to the longest starting length respectively.
Similarly, muscle impulse increased significantly at the highest
activation (F2,8 = 4816, P < 0.001), from 0.175 ± 0.002
to 0.299 ± 0.001 N s, for the shortest to longest starting
length respectively. Hydrodynamic impulse also increased
significantly with starting length at low activation (F2,8 =
1574, P < 0.001; figure 4), with a hydrodynamic impulse
increasing from 0.0021 ± 0.0003 N s at the shortest starting
length to 0.0050 ± 0.0004 at the longest starting length. At
higher activations the trend was similar (F2,8 = 4141, P <
0.001), increasing from 0.0039 ± 0.0009 N s at Lstart = 0.9 to
0.0100 ± 0.0004 N s at Lstart = 1.1.
Following patterns for muscle impulse, average muscle
velocity increased for longer starting lengths. At low
activations average muscle velocity increased significantly
(F2,8 = 950, P < 0.001), from 0.52 ± 0.01 ML s−1 at Lstart =
0.9, to 0.84 ± 0.01 ML s−1 at Lstart = 1.1, though there was no
significant difference between average muscle velocity when
starting at optimal length (Lstart = 1.0) and longer (Lstart = 1.1)
starting lengths. At high activation, average muscle velocity
increased significantly for all comparisons (F2,8 = 5234,
P < 0.001) from 0.83 ± 0.01 ML s−1 at Lstart = 0.9, to
1.28 ± 0.01 ML s−1 at Lstart = 1.1.
As a result of both the higher net muscle force and muscleshortening velocity, peak muscle power was significantly
3.4. Muscle power production with increasing activation and
starting length
Maximal power output with increasing activation for the
three starting lengths is shown in figure 5. Although powers
increased with activation for each starting length, longer
starting lengths caused power to increase more steeply.
The contribution of both force–length gains and force–
velocity gains to power is shown in figure 6. Force–length
and force–velocity gains change as a result of the interaction
between muscle activation, and the instantaneous muscle
length and velocity, the two latter of which change due to
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Bioinspir. Biomim. 7 (2012) 036018
C J Clemente and C Richards
Figure 5. Net muscle power with increasing activation for three initial muscle length Lstart = 1.1 (red), Lstart = 1.0 (green) and Lstart =
0.9 (gray). The activation and force–length gain are given for two example points, showing better synchronization of peak activation with
optimal force–length gain at longer starting lengths.
Figure 6. Average force–length gain (left) and force–velocity gain (right) with activation for three initial muscle length Lstart = 1.1 (red),
Lstart = 1.0 (green) and Lstart = 0.9 (gray). Gains which produce higher power are shown in darker shades of orange. Maximum power is
produced at a force–length gain of 1.0 and a force–velocity gain of 0.351.
Since muscle power is the product of force and
velocity, the optimal force–velocity gain for power will
occur neither at high force–velocity gains (where velocity is
low) nor at low force–velocity gains (where force is low).
Based upon the force–velocity curve produced for the PL
muscle of X. laevis, maximum muscle power is produced when
the force–velocity gain is 0.351 at a velocity of 2.5 ML s−1
(figure 7). The average force–velocity gain for all starting
lengths decreases as activation increases, but the rate of
decrease diminishes at the highest activations (figure 6).
The longest starting length shows lower force–velocity
gains (approaching maximum power) at all but the lowest
activations, down to an average force–velocity gain of 0.60 at
the muscular–skeletal environment. Since instantaneous
muscle length and velocity are independent, muscle power
will be highest when muscle force is highest, meaning that
force–length gains closest to 1.0 will produce the most power.
For starting lengths of 1.0 Lrest and 0.9 Lrest the average
force–length gain decreases with increasing activation, as the
muscle’s operating length moves down the ascending limb
of the force–length curve. At the longest starting length, the
force–length gain initially increases as the average force–
length gain moves up the descending limb of the force–length
curve, but decreases at higher activations as the average force–
length gain moves past the optimum and down the ascending
limb.
8
0
2
4
6
Velocity (ML s-1)
8
140
120
100
80
60
40
20
0
muscle to produce peak force for an extended duration, thus
increasing impulse. Therefore, current results suggest that
when a muscle develops force on the descending limb of
the force–length curve and shortens, the muscle is capable of
generating greater hydrodynamic force for extended durations,
ultimately enabling faster swimming speeds.
Underlying the enhancement of swimming performance
due to shifts in muscle operating length, we used our muscle
model to dissect the extent to which muscle force production
is limited by force–length or force–velocity effects (figure 8).
The activation curve follows a sinusoidal pattern, reaching a
maximum at the midpoint during the trial. This means that
for starting lengths of 1.0 Lrest or 0.9 Lrest, muscle length has
moved away from the optimal region along the force–length
curve, resulting in lower force production. In contrast, at longer
starting lengths (1.1 Lrest), the force–length gain increases
initially allowing peak force–length gain to coincide with peak
activation (figure 5). This finding supports the results presented
by Azizi and Roberts (2010) for jumping in bullfrogs. Few
muscles have been shown to operate on the descending limb
of the force–length curve, since it is thought that this increases
the susceptibility of muscle fibers to damage when they are
actively stretched at longer than optimal lengths (Lieber and
Fridén 1993, Proske and Morgan 2001). However, results
from EMG studies on frog musculature show no activity in
hindlimb muscles during the landing phase of a jump or a
hop, suggesting that these muscles are rarely actively stretched
(Ahn et al 2003, Olson and Marsh 1998). Moreover, anuran
hind limb muscles are more compliant at long muscle lengths
compared to mammalian muscle, enabling frogs to stretch their
Muscle Power (W kg-1)
-1
140
120
100
80
60
40
20
0
C J Clemente and C Richards
2.5 ML.s
Muscle stress (kPa)
Bioinspir. Biomim. 7 (2012) 036018
Figure 7. The force–velocity (blue) and power–velocity curve
(orange). The force–velocity curve was measured directly from X.
laevis plantaris muscle, muscle stress is shown for the highest
activation (7 N). The power–velocity curve was calculated from the
force–velocity fit. The dashed line indicates the maximal power at a
velocity of 2.5 muscle lengths per second, and a force–velocity gain
of 0.351.
the highest activation, but never reaches the maximum power
(figure 6).
4. Discussion
4.1. Muscle operating length influences muscle mechanical
output and swimming performance
The virtual muscle-frog robot allows us to examine the
complex interaction between the hydrodynamic environment
and PL muscle mechanics while precisely controlling both
activation and starting length. Our primary finding is that,
as expected, increasing starting length (Lstart) enabled the
(a)
(b)
Figure 8. Gain function and activation over time. (a) Columns 1–3 show starting muscle lengths of 0.9 Lstart, 1.0 Lstart, and 1.1
Lstart respectively. The solid blue line represents force–length gain, the solid red line represents force–velocity gain, and the solid black line
is the multiple of these two gain functions. The broken black line represents the activation curve. Activation for all plots was 7.0 N, but for
scaling purposes the activation curve is shown to 1.0. (b) Relative gain ratio (force–length gain/(force–length gain + force–velocity gain))
for the three starting lengths.
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Bioinspir. Biomim. 7 (2012) 036018
C J Clemente and C Richards
muscles farther beyond ‘optimal length’ without incurring high
passive elastic forces (Azizi and Roberts 2010). These factors
likely allow frog muscles to operate on the descending limb
of the force–length curve, and therefore avoid the trade-off
between work and instantaneous force, since the muscle is
able to shorten considerably without reducing its potential to
develop force.
similar power at lower activations than shorter starting lengths
at higher activation, meaning longer starting lengths can
compensate for reduced activation, or produce higher power
at similar activations. For example, an activation of 3.5 N
on muscle with longer initial lengths produces more power
than 7.0 N activations at shorter start lengths (figure 5).
Perhaps, modulating muscle power by shifting Lstart may
be a mechanism for enhancing muscle efficiency. Because
X. laevis muscle fatigues rapidly over repeated work cycles
(Wilson et al 2002), increasing starting length might enable the
muscle to maintain force and power output at reduced neural
activation to avoid a fatigue-induced performance deficit.
Further investigation would be required to test how operating
length influences muscle mechanical efficiency.
4.2. Optimal starting lengths for swimming
Although the current study only tested within the range of
starting lengths observed in Rana pipiens (Lutz and Rome
1994) and Rana catesbeiana (Azizi and Roberts 2010), can
further increases in starting length lead to further increases in
power? Current findings suggest an optimal starting length
of L = 1.1 Lrest due to the synchronization of muscle
activation and force–length gain dynamics (figure 5). Hence,
we predict that further stretching of the muscle would shift
the peak force–length gain out of phase with activation,
thereby lowering muscle impulse, hydrodynamic force and
swimming velocity. Yet, our speculation of optimal starting
length assumes an activation waveform of constant duration.
In vivo, muscle activation periods are highly variable (Richards
and Biewener 2007) and may strongly influence contraction
dynamics. For example, shorter activation periods may enable
the muscle to reach peak activation while operating on
the force–length plateau, favoring shorter starting lengths.
Thus, activation dynamics and starting length likely exhibit
interesting interactions which should be explored further in
future work. However, regardless of activation, passive force
in anuran hind limb muscle rises steeply at lengths above
L = 1.1 Lrest (Azizi and Roberts 2010). Therefore, the muscle
is unlikely to operate at L > 1.1 Lrest to avoid excess work
required to relengthen the PL during cyclic contractions of
swimming.
4.4. Limits to muscle force production in aquatic systems
While effects of starting length on muscle force and power
are somewhat intuitive, the current approach revealed deeper
insights. Most notably, as Lstart was increased, the relative
importance of force–length versus force–velocity properties
shifted (figure 8). Specifically, at short starting lengths (Lstart =
0.9, figure 8(a)) the total gain follows the force–length gain,
while at the longest starting lengths (Lstart = 1.1, figure 8(a)),
the total gain appears to closely follow the force–velocity gain
function. This suggests that at shorter starting lengths, force–
length effects determine force production to a greater extent
than force–velocity effects. To further highlight the relative
importance of force–length versus force–velocity effects, we
compared the relative gain ratio (FL gain/(FL gain + FV gain))
among the three starting length conditions (figure 8(b)). During
the beginning of the trial the total gain function shows a small
force–length limitation, but during the midpoint of the trial the
total gain function begins to show a force–velocity limitation,
followed by a larger period of force–length limitation toward
the end of the trial. For longer starting lengths this curve is
shifted upward toward a force–velocity limitation, suggesting
that force–velocity effects mostly dominate the pattern of force
development at longer initial lengths (figure 8(b)). As one
might expect, these progressive shifts in muscle force primarily
emerge from shifts in the force–length gain profile, whereas
the force–velocity gain profile remained nearly invariant across
activation and Lstart conditions (figure 8(a)).
Given the influence of longer starting lengths on both
force–length and force–velocity dynamics we emphasize the
importance of the inclusion of both these parameters in future
muscle models. Previous studies on muscle shortening for
jumping frogs in vivo has suggested that limb extensors act
almost exclusively on the plateau of the force–length curve
(Lutz and Rome 1994), and this has been used as justification
for excluding force–length effects from muscle models (Aerts
and Nauwelaerts 2009). This may not be the case for models of
swimming frogs, where muscle can shorten for considerably
longer periods of time (Richards and Biewener 2007) and
therefore may not operate on the plateau of the force–length
curve. Such behavior makes the inclusion of both force–length
and force–velocity effects essential, particularly if the limits
to muscle power production are of interest.
4.3. Neural activation and muscle operating length are both
important for modulating muscle power
An important distinction from muscle force measured in
previous work in the context of jumping (Azizi and Roberts
2010) is that muscle force in an aquatic environment emerges
primarily from overcoming drag forces (α foot rotational
velocity2; Gal and Blake 1988, Richards 2008). Therefore,
a muscle’s capacity to generate high fluid dynamic forces
depends on muscle power (force x velocity) which has been
shown to correlate with swimming speed and acceleration
(Richards and Biewener 2007). We have shown that muscle
power is greatly affected by starting length, with longer
starting lengths producing higher power than shorter starting
lengths (figure 4). This increased power is likely a direct
effect of both muscle force and velocity increasing due
to hydrodynamic loading. Specifically, as Lstart increases,
the resulting increase in muscle force capacity causes the
robotic foot to accelerate more rapidly, incurring greater
hydrodynamic drag. Additionally, as Lstart shifts to longer
lengths, muscle-shortening velocity increases to approach
velocities that are favorable for producing peak power
(figure 6). Consequently, longer starting lengths can produce
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C J Clemente and C Richards
Ahn A, Monti R and Biewener A 2003 In vivo and in vitro
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Azizi E and Roberts T J 2010 Muscle performance during frog
jumping: influence of elasticity on muscle operating lengths
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4.5. Limitations and assumptions in the muscle model
The current model is by no means a complete representation of
the hydrodynamic and muscular dynamic coupling in anuran
swimming. Rather, it is meant as a simplified model to better
understand which features are essential to the observed effect
(Alexander 2003). The current representation acts as a base
model, and allows us to add levels of complexity to the
neuromuscular system in a controlled and structured fashion
to determine their influence.
Several interactions are neglected in the current model
for reasons of simplicity. One important assumption we have
made is that length effects, velocity effects and activation
are independent. However, this assumption may not hold true
for all muscle types. Brown et al (1999) have reported that
muscle activation is dependent on fascicle length in the feline
caudofemoralis muscle. Activation tends to increase at longer
muscle lengths, though this affects seems reduced at higher
activation frequencies, such as used for in vitro preparations
in the current study. Other studies have reported that the
force–velocity relationship changes as a function of length.
Peak forces occurred at increasingly shorter muscle length
as velocity increased (Baratta et al 1996, Fuglevand 1987,
Thorstensson et al 1976). However this effect seems to be
most evident at high loads, and is likely to be reduced at the
low loads used in the current study. Finally the absence of
in-series compliance is probably significant, as the potential
loading of elastic components may alter the timing of the
force gains peaks with the activation waveform, potentially
influencing the results, especially for muscles of jumping
frogs which experience greater environmental loads than
swimmers (Aerts and Nauwelaerts 2009, Nauwelaerts and
Aerts 2003). However, as our findings demonstrate, even
the current simplified model can provide valuable insights,
showing the potential for muscle models to predict the behavior
and performance of the muscular–skeletal system, which is
difficult in vivo.
Acknowledgments
We thank Chris Stokes for providing crucial suggestions on
the mechanical design of the translating sled as well as with
the hydrodynamic force sensor. Henry Astley gave useful
advice on force–velocity experiments. Additionally, we thank
Donald Rodgers for assistance with machining. We also
thank Angela Berg Robertson for useful comments during
the preparation of our manuscript. Lastly, we thank two
anonymous reviewers for providing fair and thoughtful
comments. This work was supported by the Rowland Junior
Fellows Program at Harvard University.
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