|
| |
|
|
|
|
||||
| Home Help Feedback Subscriptions Archive Search Table of Contents | ||||
First published online August 31, 2007
Journal of Experimental Biology 210, 3209-3217 (2007)
Published by The Company of Biologists 2007
doi: 10.1242/jeb.008367
An isolated insect leg's passive recovery from dorso-ventral perturbations
Department of Integrative Biology, University of California at Berkeley, Berkeley, CA 94720-3140, USA
* Author for correspondence at present address: Department of Zoology, University of British Columbia, Vancouver, BC V6T 1Z4, Canada (e-mail: dudek{at}zoology.ubc.ca)
Accepted 2 July 2007
 |
Summary |
|---|
 TOP Summary IntroductionMaterials and methodsResultsDiscussionReferences |
|---|
Key words: locomotion, biomechanics, modeling, Blaberus discoidalis
|
Introduction |
|---|
|
TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
|---|
)], and
the nervous system (Cruse,
1990; Wilson,
1966), which must accommodate, and potentially modify, the
musculo-skeletal mechanics. Since perturbations to the leg trajectories are
the norm rather than the exception, we hypothesize that passive dynamic
self-stabilization can simplify the control of rapidly traversing uneven
surfaces. By passive self-stabilization we refer to recovery back to a limit
cycle in the absence of neural feedback from reflexes. By dynamic we refer to
systems in which motion and kinetic energy must be considered.
Passive self-stabilization has been identified as an important factor in
recovering from unexpected perturbations, and can arise from both the material
properties of the joints and skeleton as well as from the dynamics of the
center of mass (COM) (Holmes et al.,
2006). The dynamics of the COM of an array of morphologically
diverse terrestrial runners can be modeled as self-stabilizing spring-mass
systems in both the sagittal and horizontal planes
(Ghigliazza et al., 2005).
Humans rapidly alter leg stiffness prior to neural reflex action to maintain
similar COM trajectories when substrate compliance changes unexpectedly
(Moritz and Farley, 2004), and
guinea fowl maintain dynamic stability when encountering an unexpected change
in substrate height due in part to changed muscle moment arms and preflexes
(Daley et al., 2007;
Daley et al., 2006). These
mechanics often act in concert with preprogrammed, feed-forward motor commands
(such as those from central pattern generators) to generate corrective forces
to external perturbations more rapidly than negative feedback alone would
allow (Raibert, 1986). Even in
the absence of neural feedback, spinalized frogs compensate for phasic
perturbations during wiping movements and reach the intended endpoint, and the
response is not degraded by deafferentation
(Richardson et al., 2005).
When reflex responses are needed to maintain stability, the passive
dynamics can simplify the type of neural feedback required. Humans recovering
from a trip-up actively control knee flexion while the hip and ankle joints
respond according to the passive mechanical interactions with the knee
(Eng et al., 1997). By
learning the optimum impedance, humans can control arm movements in an
unstable environment to achieve stable trajectories where negative feedback
alone once failed (Burdet et al.,
2001). In such movements, force output is due predominantly to the
properties of the mechanical system (Hof,
2003). To be most advantageous, therefore, neural reflexes must be
tuned to enhance the stabilizing features of the mechanical system and
preflexes, rather than work against them
(Hodgins and Raibert,
1991).
The extent to which insects rely on mechanical properties of the
musculo-skeletal system to simplify the control of locomotion is largely
unknown. Stick insects use neural feedback to stabilize limb trajectories
following perturbations in both stance
(Bartling and Schmitz, 2000;
Cruse et al., 2004) and swing
(Dean, 1984;
Dean and Wendler, 1982).
Bartling and Schmitz (Bartling and Schmitz,
2000) concluded that while walking at 34 mm s–1,
stick insects rely on negative velocity feedback to produce forces opposing
lateral or fore–aft substrate perturbations within approximately 75 ms.
Stick insects use a different type of control when responding to lateral
substrate perturbations depending on substrate stiffness, by relying on
efferent copy (Cruse et al.,
2004). They produce forces opposing horizontal perturbations
during the swing phase in as little as 20 ms, and the compensation during
active movements is larger than the resistance reflex of calmly standing
animals (Dean, 1984). All of
these neural reflexes occur during slow walking. As speed increases, feedback
control may suffer due to low reflex gains and long delays in both conduction
and force production (Klavins et al.,
2002; Rack,
1981).
Cockroaches rely on reflexes to respond to perturbations in ground position
or load when standing and walking (Quimby
et al., 2006; Ridgel et al.,
2001; Zill et al.,
2004), and are less effective at climbing inclines and crossing
obstacles when subjected to bilateral circumoesophageal connective lesions
(Ritzmann et al., 2005). As
running speed increases, however, cockroaches appear to rely less on neural
reflexes and more on mechanical feedback. For example, cockroaches and spiders
maintain high running speeds when crossing a substrate with 90% of the ground
surface removed by using distributed mechanical feedback
(Spagna et al., 2007). While
lizards decrease speed, change kinematics, and insert frequent pauses when
crossing obstacles (Kohlsdorf and
Biewener, 2006), cockroaches exhibit only minor changes in speed,
kinematics or muscle activation patterns while running over unstructured
terrain (Sponberg et al.,
2004). They rapidly recover from large lateral perturbations to
their center of mass (Jindrich and Full,
2002), and their dynamics have been modeled as passive,
dynamically self-stabilizing in the face of perturbations
(Kubow and Full, 1999). Since
both the neural control of slow cockroach locomotion
(Delcomyn, 1977;
Graham, 1985;
Wilson, 1966) and their
mechanics during high-speed running (Full
et al., 1991; Full and Tu,
1990; Ting et al.,
1994) have been well studied, cockroaches are an excellent model
for understanding how the nervous system and mechanical properties combine to
control rapid running.
The purpose of this study is to determine the role that passive mechanical
properties of the leg may play in controlling and stabilizing the rapid
running of the cockroach, Blaberus discoidalis. The passive hind leg
of Blaberus is highly damped when subjected to dynamic oscillations
(Dudek and Full, 2006). The
stiffness and damping parameters from fitting the hysteretic damping model to
dynamically oscillated legs predict a rapid recovery from an impulse
perturbation. The hysteretic damping model is a simple, linear, two-parameter
damped spring (Nashif et al.,
1985), used to describe elastomers when material properties are
largely independent of strain rate or oscillation frequency. Its two constant
parameters accurately predict the cockroach leg response to small amplitude
(0.1–1.0 mm) dynamic oscillations over a 240-fold frequency range
(0.25–60 Hz). In the present study, we subjected the hind leg to
dorso-ventral impulse perturbations to test the hypothesis that the passive
exoskeleton can aid recovery during rapid running. Trials on uncontrolled,
complex terrain have shown that perturbations from debris are common in the
dorso-ventral direction during the swing phase. We examined whether the simple
hysteretic damping model could predict the response of the leg to an impulse
perturbation. By comparing the animal's response to the model derived to
capture dynamic oscillations, we propose the next step toward modeling
non-linear, large amplitude impulse perturbations to appendages.
|
Materials and methods |
|---|
|
TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
|---|
Impulse perturbations
The response of the left meta-thoracic leg to rapid force perturbations was
analyzed to quantify its viscoelastic properties. Dorso-ventrally directed
perturbations were applied to the leg in two distinct preparations: one in
which the body–coxa joint was free to rotate (free-coxa,
Fig. 1A) and the other with the
body–coxa joint rigidly fixed (fixed-coxa). Activation of muscles that
span the body–coxa joint can stiffen the leg in the dorso-ventral
direction, so the two preparations provide bounds to the possible material
properties of the leg resulting from muscle activation, with zero activation
represented by the free-coxa preparation and maximal activation represented by
the fixed-coxa case. Since the joint axes distal to the body–coxa joint
are aligned with the perturbation direction
(Fig. 1B), muscle activation at
these joints should have minimal effect on the leg's viscoelastic
response.
|
), we removed the tarsus and fixed various distal
leg joints using cyanoacrylate. While the distal leg joints were not fixed in
the dynamic oscillation study, no rotation of these joints was observed in the
medio-lateral or antero-posterior directions. In five legs we fixed just the
femur–tibia (Fe-Ti) joint because it passively rotates and absorbs
energy if left unglued. In another five legs we fixed the
coax–trochanter (C-Tr) and trochanter–femur (Tr-Fe) joints in
addition. In this manuscript, we will refer to the C-Tr and Tr-Fe joints as a
single coxa–trochanter–femur (C-Tr-Fe) joint because the
trochanter and femur rotate about the same axis in unison relative to the coxa
(Fig. 1B). Furthermore, we
mounted the preparations with the body rolled 90° relative to its
antero-posterior axis (so that the normally horizontal dorsal plane is now
vertical) to remove the effect of gravity from the leg response
(Fig. 1A,C).
In the free-coxa preparation, the thorax was rigidly glued (5 min epoxy) to 0.95 cm thick PlexiglasTM. The abdomen was loosely pulled dorsally so that the leg's recovery from a perturbation was not affected by striking the ventral surface of the abdomen. After perturbing legs with a freely rotating body coxa joint, the legs were removed at the base of the body–coxa joint. The coxa was then rigidly glued to 0.95 cm thick PlexiglasTM (fixed-coxa preparation) in the same orientation as was done in the free-coxa preparation.
We generated perturbations using a lever arm attached to a servo-motor (300B-LR; Aurora Scientific, Aurora, ON, Canada) commanding square waves of 4 ms durations with peak forces of 25, 50 or 75 mN (approximately 1, 2 and 3 times body mass, respectively). The lever arm started in contact with the tibia, 1 mm from the distal end, where a marker (white enamel paint) was placed (Fig. 1A), so that the leg's position and recovery could be recorded at 1000 frames s–1 (MotionScopePCI; Redlake, Tucson, AZ, USA).
Controls
There was an approximately 15 min delay between euthanasia and the
beginning of tests on the free-coxa preparation. Three impulses were applied
to the leg at each of the three peak commanded forces in random order. Tests
on the fixed-coxa preparation began within 45 min post-mortem, where
three impulses each were applied again in random order at the three peak
commanded forces. The three position traces at each force level were nearly
indistinguishable from one another in both coxa preparations, indicating that
material properties were not changing over these short timescales. The traces
remained indistinguishable for at least 3 h. As a control, one leg not
included in this data set was tested in the fixed-coxa preparation without
first undergoing free-coxa perturbations. Its response was not significantly
different (P>0.05) from the responses of the other fixed-coxa
preparations.
We did not observe post-mortem reflex activity or internal
hemolymph pressure to affect hind leg properties in a previous study on
Blaberus (Dudek and Full,
2006), so we did not consider them here. Moreover, we found no
effect on leg properties when stimulating either muscle 177c, 177d, 179, 182c,
185 or 194a during a dynamic oscillation. Similarly, living cockroaches
wedging against the lever arm of the servo motor in the posterior direction
did not have significantly different leg properties compared to euthanized
animals when dorso-ventral oscillations were applied.
Data acquisition and parameter calculations
Impulse magnitude
Displacement and force signals from the servo-motor were digitized (board
AT-MIO-16E-1; National Instruments, Austin, TX, USA) at 5 kHz and stored to
the hard disk of a personal computer (Berta; Transduction Ltd, Mississauga,
ON, Canada) running analysis software (Matlab; The Mathworks, Natick, MA,
USA). The magnitude of the impulse imparted to the leg (I) was
determined by integrating the area under the force signal corresponding to
when the lever arm started moving and was in contact with the leg.
Leg position and recovery
The position of the distal end of the tibia was digitized and tracked using
commercial software (Motus, Peak Performance). To determine the rate of
recovery from the perturbation, we normalized position data in each trial by
dividing by the maximum displacement. The time and percent of perturbation
absorbed was then recorded for each peak of the damped free response, as well
as the time at which 95% and 99% of the perturbation had been damped.
Modeling leg properties
We calculated the leg's stiffness (k) and structural damping
factor (
) by fitting the recovery trace to the hysteretic damping model
(Dudek and Full, 2006;
Nashif et al., 1985). The
hysteretic damping model gets its name from the fact that it was designed to
fit materials whose hysteresis is independent of velocity, and is governed by
the equation:
 | (1) |
. The response of the model to an
idealized unit impulse function in the time domain is:
 | (2) |
is frequency.
Eqn 2 does not have a closed form
solution, but can be estimated by numerical integration of:
 | (3) |
Stiffness and damping coefficients of the leg were determined by searching
for the I, k and that minimized the sum of the squared
difference between x(t) and the position data from the
digitized video. Each leg underwent three impulse perturbations at each
commanded force, which were averaged to produce the actual position response.
The initial guess for I was therefore the mean measured impulse
imparted for each of the three perturbations. The initial guess for k
was determined by assuming the frequency of the free response was proportional
to 
, while the initial guess for
was 0.28 for the free-coxa and 0.2 for the fixed-coxa case [the average
damping coefficients determined by dynamic oscillation tests
(Dudek and Full, 2006)]. Mass
of the free-coxa legs averaged 0.097±0.020 g and the fixed-coxa legs
averaged 0.041±0.008 g. The integration range was
0–105 Hz, as recommended by Jones
(Jones, 1986).
The impulse response of the hysteretic damping model has been a source of
controversy since it is non-causal [h(t)=0 for
t<0] (Crandall,
1963; Milne,
1985). The non-causality arises due to the assumption that
k and are constants, independent of frequency. If k
and are allowed to vary with frequency, as they must for all materials
over a sufficiently large frequency range, the non-causality disappears
(Nashif et al., 1985). For
constant parameters, the error in predicted displacement due to the
non-causality is very small when <0.1. The error grows as
increases, and we predict errors of less than 1% for the fixed-coxa and less
than 7% for the free-coxa preparations [following Milne
(Milne, 1985)].
Statistics
In each preparation, only the left metathoracic limb from each animal was
tested. Three-way ANOVAs were performed to examine the relationship of the
mechanical properties to coxa preparation, status of the C-Tr-Fe joint, and
the impulse magnitude. All tests were performed using statistics software
packages (JMP; SAS Institute, Cary, NC, USA; Statistics toolbox; The
Mathworks). Unless otherwise stated, all reported values are means ±
standard deviations (s.d.).
|
Results |
|---|
|
TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
|---|
Impulse response
Video frames from a representative free-coxa trial show the typical
response to an impulse (Fig.
1C). Stiffness and damping parameters obtained by fitting the
hysteretic damping model to leg data from dynamic oscillations predicts that
legs respond to impulse perturbations with damped ringing, absorbing most of
the perturbation within four cycles (Fig.
2A, Fig. 3A). In
both preparations, the response of the distal tip of the tibia showed damped
ringing (Fig. 2B,
Fig. 3B), oscillating two to
four times before coming to rest within 0.04±0.01 mm (mean ±
s.e.m., digitizing resolution of 0.02 mm) of the original position. Since
impulse magnitude did not have a significant effect on stiffness and damping
parameters (Table 1,
Table 2), we pooled all the
data for the free- and fixed-coxa preparations to examine the time required
for the leg to recover from a perturbation
(Fig. 4). In all cases, the
fixed-coxa legs recovered faster than those with a freely rotating
body–coxa joint (t-test, P<0.0001). Legs reached
their peak displacement within 4–6 ms, and were 99% recovered within
16±3.3 ms for the fixed-coxa and within 46±6.6 ms for the
free-coxa preparations. By comparison, the average swing duration of the hind
leg while running at preferred speed is 50–60 ms.
|
|
|
|
|
The dorsally directed perturbation resulted primarily in dorsal-ventral leg motion due to rotation about the medial-laterally directed body–coxa joint axis (Fig. 1B). Peak displacement tended to increase as impulse magnitude increased (Table 3). In the free-coxa preparation, peak dorsal-ventral leg displacement was unaffected (ANOVA, F=29.73, P<0.0001) by whether or not the C-Tr-Fe joint was fixed, averaging 3.94±1.02 mm when just the femur–tibia was fixed and 3.92±1.13 mm when the C-Tr-Fe was also fixed. Peak average displacement declined in the fixed-coxa preparation, averaging 2.23±0.89 mm with the femur–tibia fixed and declining to 1.28±0.60 mm with the C-Tr-Fe joint fixed as well. There was also motion off the perturbation axis due to smaller rotations about the dorso-ventrally directed body–coxa and coxa–trochanter joint axes (Fig. 1B). While the exact amount of this displacement could not be quantified, it was clear from the videos that fixing the C-Tr-Fe joint led to an increase in off-axis motions.
|
Viscoelastic model (hysteretic damping)
Impulse response trajectories recreated using the stiffness and damping
parameters resulting from fitting a damped spring
(Eqn 3) to the data closely
matched the actual data (Fig.
2B, Fig. 3B). The
stiffness and damping parameters depended on whether the body–coxa joint
was fixed or free to rotate; they also depended on whether C-Tr-Fe joint was
fixed in addition to the femur–tibia (Tables
1,
2). Legs with a fixed coxa were
significantly stiffer and less damped
(Table 3) than those with a
freely rotating coxa (three-way ANOVA, Tukey–Kramer).
Stiffness (k)
Leg stiffness depended on whether the body–coxa joint was fixed or
free to rotate, on whether the C-Tr-Fe joint was fixed or free to rotate, and
on the interaction between these two parameters (Tables
1,
2). Stiffness was consistently
and significantly higher as more joints were fixed, increasing by a factor of
17 between the free- and fixed-coxa preparations and getting 1.5 times stiffer
when both the C-Tr-Fe and femur–tibia joints were fixed compared to just
fixing the femur–tibia joint (Table
3) alone. Stiffness was independent of the applied impulse
magnitude (Tables 1,
3).
Structural damping factor ()
The damping factor also depended on both the coxa preparation and how many
distal leg joints were fixed, with no interaction between the three grouping
variables (Table 2). Fixing the
body–coxa joint nearly halved the damping factor while fixing the
C-Tr-Fe joint in addition to the femur–tibia led to a small increase in
damping (Table 3). Damping
factor was independent of the applied impulse magnitude (Tables
2,
3).
|
Discussion |
|---|
|
TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
|---|
) accurately predict that recovery from an impulse
perturbation is greater than 90% complete within a single stance or swing
phase. This simple, linear model reveals the importance of non-linear leg
stiffness and off-axis joint motions in aiding leg recovery. Future efforts to
improve our ability to predict the effect of passive mechanical properties on
the control of locomotion should include a three-dimensional damping model
with a non-linear stiffness component.
Passive leg recovery
The extent to which an insect leg's recovery from a perturbation is
mediated by active neural feedback and passive muscle and skeletal properties
is likely to depend on speed, with neural feedback playing a larger role
during slow movements and self-stabilization arising from passive dynamics
playing a larger role as speed increases
(Full and Koditschek, 1999).
Stick insects walking at one-tenth the preferred speed of B.
discoidalis appear to rely on negative velocity feedback to control foot
position when perturbed during stance, with delays between perturbation onset
and rise in leg force of approximately 75 ms
(Bartling and Schmitz, 2000).
This is more than adequate when stance lasts 1000 ms, but would not suffice
when stance only lasts 50 ms. While the short delay of 6–15 ms between
transient substrate perturbations of 1–8 mm and tibial campaniform
sensilla firing in the cockroach, P. americana, is impressively fast
(Ridgel et al., 2001), stance
lasts only 20–25 ms at their preferred running frequency
(Full and Tu, 1991). Since
feedback has such a short time window to respond, the passive dynamics could
completely determine system behavior at such high running speeds.
During normal running, B. discoidalis lifts the distal end of the metathoracic tibia approximately 2.5 mm above the substrate (primarily due to rotation of the body–coxa joint) during each 50–60 ms swing phase. Passive recovery from large perturbations of the leg is rapid, with peak amplitudes reached in 4–6 ms and 99% recovery in less than 50 ms for the most compliant leg orientation, requiring less than 15 ms for the stiffest (z-test, z=–3.14 free body–coxa, z=–56.56 fixed body–coxa, Fig. 4). Such rapid responses may make feedback control of perturbations undesirable during rapid running.
The 6–15 ms latency observed by Ridgel et al.
(Ridgel et al., 2001) is
consistent with the 10 ms latency observed by Wilson
(Wilson, 1965). Adding to this
reflex latency is the time required for a muscle to generate force following
stimulation, which is 10 ms in Blaberus leg muscles
(Full and Meijer, 2001).
Therefore, the fastest reflex response following a leg perturbation is not
likely to occur before 16–20 ms have passed. This prediction is
supported by Schaefer et al. (Schaefer et
al., 1994), who report the latency between tactile leg stimulus
and first leg movement in P. americana at 17 ms. For the fixed-coxa
case, the leg has fully recovered by then
(Fig. 3B,
Fig. 4). With a freely rotating
body–coxa joint, more than 82% of the perturbation has been absorbed
within 20 ms (Fig. 2B). Due to
the rapid passive response and reflex delays, neural control models that fail
to account for the mechanical properties of the legs will be destabilizing,
responding to perturbations that no longer exist (or are greatly reduced).
Moreover, actively stretched muscles produce immediate resistance to
perturbations (Rack, 1981), so
the role of the nervous system in rejecting perturbations at higher speeds may
be the feedforward control of overall leg impedance. Due to these muscle
preflexes, we predict the response of the leg to a similar perturbation while
running will approach the rapid response seen in the fixed-coxa case.
Hysteretic damping model
To test the ability of the hysteretic damping model to predict the leg
response under different perturbation regimes, we predicted the leg's response
to an impulse using the stiffness and damping factors attained from dynamic
oscillations (Dudek and Full,
2006) and compared this prediction to the actual leg response. At
an amplitude of 0.5 mm and a frequency of 10 Hz, a dynamically oscillated leg
has a stiffness of 8.57±0.57 N m–1 (mean ±
s.e.m., N=13) for the free-coxa preparation and 19.72±1.24 N
m–1 (mean ± s.e.m., N=10) for the fixed-coxa.
The damping factor was 0.28±0.02 for the free-coxa and 0.20±0.02
for the fixed and the leg mass was assumed to be 0.1 g with the coxa and 0.04
g without the coxa. For the most compliant preparation, where the
body–coxa joint is free to rotate
(Fig. 2A), the damped ringing
occurs at the same frequency as the actual response and 88–92% of the
perturbation is recovered within the 50–60 ms duration of a stance or
swing phase. While the actual leg has fully recovered by this time
(Fig. 4), the model continues
ringing with very small amplitudes for another 50–60 ms. The stiffness
of the free-coxa preparation with the C-Tr-Fe joint fixed (8.03±1.93 N
m–1, Table 3)
was the only case where the model parameter from the impulse experiments
matched the prediction from dynamic oscillation experiments
(P=0.2706, z=–1.09). The predicted impulse response
for the stiffest preparation, where the body–coxa joint was rigidly
fixed (Fig. 3A), does not match
the actual response well (Fig.
3B), capturing neither the ringing frequency nor the time to
recover. The fixed-coxa preparation was more than 5 times as stiff as expected
(Table 3). Both coxa
preparations were 2–3 times as damped as expected
(P<0.0001).
Only a small portion of the discrepancy between the impulse response data and the model derived from dynamic oscillations can be accounted for by the fact that the leg is closer to a beam attached to a rotational spring than a point mass attached to a linear spring. For the linear spring, we predict the fixed-coxa leg to have a natural frequency 2.4 times greater than the free-coxa leg. Assuming the leg is a beam, we predict the fixed-coxa preparation with its increased stiffness, shorter length, and decreased mass to have a natural frequency 2.6 times greater than the free-coxa preparation. While the free-coxa leg oscillates very near to the predicted 46 Hz natural frequency, the fixed-coxa leg oscillates at 2–3 times its predicted resonant frequency of 112 Hz.
While we did not expect a precise matching of the model parameters between
the two perturbation types, we did expect to be able to predict how stiffness
and damping would be affected by the experimental manipulations. Leg stiffness
should increase as fewer joints are free to rotate because joint cuticle is
much more compliant than the cuticle of the leg segments
(Vincent, 1980). This
prediction is supported by our data, where a leg with a rigid body–coxa
joint is almost 20 times stiffer than a leg with a freely rotating
body–coxa joint and legs with a fixed C-Tr-Fe joint are 50% stiffer than
legs with a free C-Tr-Fe joint (Tables
1,
3). Conversely, the structural
damping factor should decrease as more joints are immobilized, because leg
segments are more resilient than leg joints
(Blickhan, 1986;
Dudek and Full, 2006;
Katz and Gosline, 1992). This
prediction is upheld when the body–coxa joint is immobilized and damping
is 60% greater for a leg with a freely rotating body–coxa joint (Tables
2,
3). Despite a lower damping
factor, a leg with a fixed body–coxa joint recovers faster than one with
the joint freely rotating because, while it loses less energy per oscillation,
it oscillates at a much higher frequency and losses energy at a higher rate.
Interestingly, damping increases 20% when the C-Tr-Fe joint is fixed (Tables
2,
3). This excess damping likely
arises due to greater off-axis displacements seen in legs with a rigid C-Tr-Fe
joint.
Suggested next steps in modeling
Applying the simplest, linear model to the impulse response of the
cockroach leg reveals the importance of non-linear material properties and
motions off the perturbation axis in aiding recovery. The leg undergoes
deflections following an impulse as much as four times greater than the
deflection amplitude of dynamic oscillations. Dynamic oscillations at an
amplitude of 1 mm predict a leg stiffness of approximately 20 N
m–1 (Dudek and Full,
2006), but the slope of the loading curve from 0.7–1.0 mm
indicates a stiffness of 35 N m–1 over this range. Leg
stiffness should continue to increase as displacements approach the 2–4
mm amplitudes produced by impulse perturbations. A linear model cannot take
the non-linear leg stiffness of these large deflections into account.
Additionally, a one-dimensional model cannot account for energy absorption due
to off-axis rotations. Off-axis rotations did not occur during dynamic
oscillations because the leg was attached to the lever arm. Here, the leg was
free to move in any dimension and off-axis motions could not be prevented.
This is because the leg segments are dynamically coupled, so perturbations in
one dimension create perturbations to other dimensions
(Jindrich and Full, 2002;
Kubow and Full, 1999). In this
case, the coupling benefits the passive response to a single-axis impulse
perturbation by allowing energy absorption to occur simultaneously about
multiple axes. To improve biological relevance and predictive power for a
leg's response to multiple perturbation types, a three-dimensional,
frequency-independent model with non-linear stiffness is required.
Such a model would reduce or eliminate the effects of non-causality. While
the hysteretic damping model has analytical solutions for predicting the
response to dynamic oscillations, the impulse response must be solved by
numerical integration of Eqn 3
across a broad frequency range (0–105 Hz). While many
biological materials have nearly constant properties across the range of
biologically relevant frequencies (Dudek
and Full, 2006; Fung,
1984), no material has constant parameters across five frequency
decades (Vincent, 1990). This
can be a drawback when applying the model when the material has not been
characterized across a full frequency spectrum. Using constant parameters to
predict the impulse response results in errors in predicted position,
particularly as damping factor exceeds 0.1
(Milne, 1985). The damping
factors observed in this study (Table
3) yield predicted position errors of 22% for the free-coxa
preparation and 10% for the fixed-coxa case. While off-axis motions of the leg
occur due to dynamic coupling of the segments, the single degree of freedom
model applied here lumps all energy absorption into a single damping factor. A
three-dimensional model would allow the damping to be accounted for by three
dampers, each with a lower damping factor than that of the one-dimensional
model, reducing non-causality errors to an acceptable level.
Conclusions
The hysteretic damping model can predict the timing of response to an
impulse perturbation so long as the amplitudes remain low and the leg is
compliant. To improve the biological relevance of the model and increase its
predictive ability to include larger perturbations and stiffer legs, a
three-dimensional model that accounts for off-axis energy absorption and
includes a non-linear stiffness term is needed. Due to high stiffness and
damping of energy in multiple dimensions, the passive metathoracic leg of the
cockroach recovers to within 0.04 mm of its original position following
impulsive perturbations of more than twice the leg's peak momentum during a
stride in less time than a single stance or swing phase. Such rapid energy
dissipation leads us to suggest that recovering from perturbations during
rapid running may be dominated by passive mechanical properties of the
exoskeleton, thereby simplifying the neural control required for
stability.
|
Acknowledgments |
|---|
|
References |
|---|
|
TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
|---|
Bartling, C. and Schmitz, J. (2000). Reaction to disturbances of a walking leg during stance. J. Exp. Biol. 203,1211 .[Abstract]
Blickhan, R. (1986). Stiffness of an arthropod leg joint. J. Biomech. 19,375 -384.[CrossRef][Medline]
Brown, I. E. and Loeb, G. E. (2000). A reductionist approach to creating and using neuromusculoskeletal models. In Biomechanics and Neural Control of Posture and Movement (ed. J. A. Winters and P. E. Crago), pp.148 -163. New York: Springer.
Burdet, E., Osu, R., Franklin, D. W., Milner, T. E. and Kawato, M. (2001). The central nervous system stabilizes unstable dynamics by learning optimal impedance. Nature 414, 446.[CrossRef][Medline]
Crandall, S. H. (1963). Dynamic response of systems with structural damping. In Air, Space, and Instruments, Draaper Anniversary Volume (ed. S. Lees), pp.183 -193. New York: McGraw-Hill.
Cruse, H. (1990). What mechanisms coordinate leg movement in walking arthropods? Trends Neurosci. 13, 15-21.[CrossRef][Medline]
Cruse, H., Kuhn, S., Park, S. and Schmitz, J. (2004). Adaptive control for insect leg position: controller properties depend on substrate compliance. J. Comp. Physiol. A 190,983 .[CrossRef][Medline]
Daley, M. A., Usherwood, J. R., Felix, G. and Biewener, A.
A. (2006). Running over rough terrain: guinea fowl maintain
dynamic stability despite a large unexpected change in substrate height.
J. Exp. Biol. 209,171
.
Daley, M. A., Felix, G. and Biewener, A. A.
(2007). Running stability is enhanced by a proximo-distal
gradient in joint neuromechanical control. J. Exp.
Biol. 210,383
-394.
Dean, J. (1984). Control of leg protraction in the stick insect – a targeted movement showing compensation for externally applied forces. J. Comp. Physiol. 155, 771.[CrossRef]
Dean, J. and Wendler, G. (1982). Stick insects walking on a wheel – perturbations induced by obstruction of leg protraction. J. Comp. Physiol. 148, 195.[CrossRef]
Delcomyn, F. (1977). Co-ordination of invertebrate locomotion. In Mechanics and Energetics of Animal Locomotion (ed. R. M. Alexander and G. Goldspink), pp.82 -114. London: Chapman & Hall.
Dudek, D. M. and Full, R. J. (2006). Passive
mechanical properties of legs from running insects. J. Exp.
Biol. 209,1502
-1515.
Eng, J. J., Winter, D. A. and Patla, A. E. (1997). Intralimb dynamics simplify reactive control strategies during locomotion. J. Biomech. 30, 581.[CrossRef][Medline]
Full, R. J. and Koditschek, D. E. (1999). Templates and anchors: neuromechanical hypotheses of legged locomotion on land. J. Exp. Biol. 202,3325 -3332.[Abstract]
Full, R. J. and Meijer, K. (2001). Metrics of natural muscle function. In Electro Active Polymers (EAP) as Artificial Muscles, Reality, Potential, and Challenges (ed. Y. Bar-Cohen), pp. 67-83. New York: SPIE and William Andrew/Noyes Publications.
Full, R. J. and Tu, M. S. (1990). Mechanics of
six-legged runners. J. Exp. Biol.
148,129
-146.
Full, R. J. and Tu, M. S. (1991). Mechanics of
a rapid running insect: two-,four- and six-legged locomotion. J.
Exp. Biol. 156,215
-232.
Full, R. J., Blickhan, R. and Ting, L. H.
(1991). Leg design in hexapedal runners. J. Exp.
Biol. 158,369
-390.
Fung, Y. C. (1984). Structure and stress-strain relationship of soft tissues. Am. Zool. 24, 13-22.
Ghigliazza, R. M., Altendorfer, R., Holmes, P. and Koditschek, D. (2005). A simply stabilized running model. Siam Rev. 47,519 -549.[CrossRef]
Graham, D. (1985). Pattern and control of walking in insects. Adv. Insect Physiol. 18, 31-140.
Hodgins, J. K. and Raibert, M. H. (1991). Adjusting step length for rough terrain locomotion. IEEE Trans. Rob. Autom. 7,289 .[CrossRef]
Hof, A. L. (2003). Muscle mechanics and neuromuscular control. J. Biomech. 36,1031 -1038.[CrossRef][Medline]
Holmes, P., Full, R. J., Koditschek, D. and Guckenheimer, J. (2006). The dynamics of legged locomotion: models, analyses, and challenges. Siam Rev. 48,207 -304.[CrossRef]
Jindrich, D. L. and Full, R. J. (2002). Dynamic
stabilization of rapid hexapedal locomotion. J. Exp.
Biol. 205,2803
-2823.
Jones, D. I. G. (1986). The impulse response function of a damped single degree of freedom system. J. Sound Vibration 106,353 -356.[CrossRef]
Katz, S. L. and Gosline, J. M. (1992).
Ontogenic scaling and mechanical-behavior of the tibiae of the African desert
locust (Schistocerca-Gregaria). J. Exp. Biol.
168,125
-150.
Klavins, E., Komsuoglu, H., Full, R. J. and Koditschek, D. E. (2002). The role of reflexes versus central pattern generators in dynamical legged locomotion. In Neurotechnology for Biomimetic Robots (ed. J. Ayers, J. L. Davis and A. Rudolph). Cambridge, MA: MIT Press.
Kohlsdorf, T. and Biewener, A. A. (2006). Negotiating obstacles: running kinematics of the lizard Sceloporus malachiticus. J. Zool. 270,359 -371.[CrossRef]
Kubow, T. M. and Full, R. J. (1999). The role of the mechanical system in control: a hypothesis of self-stabilization in hexapedal runners. Philos. Trans. R. Soc. Lond. B Biol. Sci. 354,849 -861.[CrossRef]
Milne, H. K. (1985). The impulse-response function of a single degree of freedom system with hysteretic damping. J. Sound Vibration 100,590 -593.[CrossRef]
Moritz, C. T. and Farley, C. T. (2004). Passive
dynamics change leg mechanics for an unexpected surface during human hopping.
J. Appl. Physiol. 97,1313
-1322.
Nashif, A., Jones, D. and Henderson, J. (1985). Vibration Damping. New York: John Wiley & Sons.
Quimby, L. A., Amer, A. S. and Zill, S. N. (2006). Common motor mechanisms support body load in serially homologous legs of cockroaches in posture and walking. J. Comp. Physiol. A 192,247 -266.[CrossRef][Medline]
Rack, P. M. H. (1981). Limitations of somatosensory feedback in control of posture and movement. In Handbook of Physiology Section 1, The Nervous System, Motor Control. Vol. II, Pts 1 and 2 (ed. V. B. Brooks), pp. 229-256. Bethesda: American Physiological Society.
Raibert, M. H. (1986). Legged Robots that Balance. Cambridge, MA: MIT Press.
Richardson, A. G., Slotine, J.-J. E., Bizzi, E. and Tresch, M.
C. (2005). Intrinsic musculoskeletal properties stabilize
wiping movements in the spinalized frog. J. Neurosci.
25,3181
-3191.
Ridgel, A. L., Frazier, S. F. and Zill, S. N. (2001). Dynamic responses of tibial campaniform sensilla studied by substrate displacement in freely moving cockroaches. J. Comp. Physiol. A 187,405 .[CrossRef][Medline]
Ritzmann, R. E., Pollack, A. J., Archinal, J., Ridgel, A. L. and Quinn, R. D. (2005). Descending control of body attitude in the cockroach Blaberus discoidalis and its role in incline climbing. J. Comp. Physiol. A 191,253 -264.[CrossRef][Medline]
Schaefer, P. L., Kondagunta, V. and Ritzmann, R. E. (1994). Motion analysis of escape movements evoked by tactile stimulation in the cockroach, Periplaneta americana. J. Exp. Biol. 190,287 -294.[Abstract]
Spagna, J. C., Goldman, D. I., Lin, P. C., Koditschek, D. E. and Full, R. J. (2007). Distributed mechanical feedback in arthropods and robots simplifies control of rapid running on challenging terrain. Bioinspiration Biomimetics 2, 9-18.[CrossRef][Medline]
Sponberg, S. N., Chang, C. and Full, R. J. (2004). Sensory independent, feedforward control of locomotion in cockroaches running over rough surfaces. Integr. Comp. Biol. 44,644 .
Ting, L. H., Blickhan, R. and Full, R. J. (1994). Dynamic and static stability in hexapedal runners. J. Exp. Biol. 197,251 -269.[Abstract]
Vincent, J. F. V. (1980). Insect cuticle: a paradigm for natural composites. Symp. Soc. Exp. Biol. 34,183 -210.[Medline]
Vincent, J. F. V. (1990). Structural Biomaterials. Princeton: Princeton University Press.
Wilson, D. M. (1965). Proprioceptive leg
reflexes in cockroaches. J. Exp. Biol.
43,397
-409.
Wilson, D. M. (1966). Insect walking. Annu. Rev. Entomol. 11,103 -123.[CrossRef][Medline]
Zill, S., Schmitz, J. and Buschges, A. (2004). Load sensing and control of posture and locomotion. Arthropod Struct. Dev. 33,273 -286.[CrossRef][Medline]
| |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
