Dynamic stabilization of rapid hexapedal locomotion
Department of Integrative Biology, University of California at Berkeley, Berkeley, CA 94720-3140, USA
* Author for correspondence at present address: Harvard School of Public Health, 665 Huntington Avenue, Boston, MA 02115, USA
Accepted 25 June 2002
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Summary |
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Key words: cockroach, Blaberus discoidalis, locomotion, mechanics, perturbation, stability, neural control
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Introduction |
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Although many types of periodic, dynamically stable motions are possible
(Guckenheimer, 1982), one
simple way of defining dynamic stability is the maintenance of an equilibrium
trajectory over time: a defined pattern of positions and velocities that
repeats with a characteristic frequency such as the stride frequency
(Full et al., 2002
). Following
perturbations, dynamically stable systems return towards an unchanged
equilibrium trajectory. Perturbations to neutrally stable systems persist in
magnitude over time, and perturbations to unstable systems grow larger over
time (Strogatz, 1994
). Even
simplified mechanical systems, such as inverted pendulum or spring-mass
systems, can act as stable systems in some directions relative to their motion
and as unstable or neutrally stable systems in others
(Bauby and Kuo, 2000
; Schmitt
and Holmes, 2000a
,
b
). A controller, such as the
nervous system in animals, is necessary to stabilize systems with unstable
components, even if the system is stable in some directions
(Bauby and Kuo, 2000
). Control
may also be required to counteract perturbations in neutrally stable
directions, such as desired movement direction or speed. Control is often
active, taking the form of negative feedback from sensors to alter the state
of a system. However, a consideration of the passive dynamic behavior of a
mechanical system is critical for interpreting the effects of a controller
(Full et al., 2002
).
Two general mechanisms are available to maintain stability during legged
locomotion. First, the initial condition of the legs at the transition from
swing to stance can stabilize locomotion. For example, foot placement can
stabilize bipedal locomotion (Townsend,
1985), and leg stiffness adjustments can compensate for substratum
changes in humans (Ferris et al.,
1999
). In insects, foot placement plays an important part in
stabilizing slow locomotion (Jander,
1985
; Zollikofer,
1994
). Control of step transitions can result in changes to leg
stance, swing or stride periods in addition to changes in phase relationships
among legs. Changes in leg placement, stepping periods and phase relationships
have been used to identify mechanisms of neural control in arthropods
(Cruse, 1990
).
Movement need not be actively controlled to exhibit dynamic stability. For
example, uncontrolled walking bipeds
(McGeer, 1990) and
sagittal-plane spring-mass systems
(Seyfarth et al., 2002
) with
discontinuous stepping events can exhibit stability. In the horizontal plane,
uncontrolled spring-mass models analogous to those of sagittal-plane running
also exhibit stability (Schmitt and Holmes,
2000a
,
b
). Parameters such as mass,
moment of inertia, segment lengths, touchdown angles and segment compliance
can determine the stability of an uncontrolled mechanical system
(Schmitt et al., 2002
;
Seyfarth et al., 2002
).
Coupled with uncontrolled, or `passive' stabilization, the action of a
controller acting at step transitions can contribute to dynamic stability.
Whereas passive mechanisms contribute to stabilizing bipedal locomotion in the
sagittal plane, humans use lateral foot placement to stabilize the unstable
lateral direction during walking (Bauby and
Kuo, 2000; Mackinnon and
Winter, 1993
). As for walking, control of leg placement and
stiffness at step transitions is an important part of one successful control
strategy used for dynamically stable three-dimensional hopping and running
robots (Raibert et al.,
1984
).
An alternative to stabilizing locomotion at step transitions is to
counteract perturbations within a step (Grillner,
1972,
1975
). Within-step changes in
joint torques could generate forces appropriate to counteract perturbations.
Humans can modulate torque production to maintain constant-speed locomotion
against an imposed force (Bonnard and
Pailhous, 1991
) and use changes in joint torques to counteract
imposed force impulses when the impulses occur early in the step cycle
(Yang et al., 1990
). These
dynamic changes in joint torques could serve to control movements about
equilibrium trajectories during locomotion.
However, as animals move faster and stride periods decrease, the time
available to recover from perturbations to movement within a step period
decreases (Alexander, 1982).
Neural delays in sensing a perturbation and in generating an appropriate motor
pattern within the nervous system to arrest the perturbation, and delays
involved in muscle activation and force generation, could limit the
effectiveness with which neural feedback systems could continuously stabilize
rapid movement (Full and Koditschek,
1999
; Hogan, 1990
;
Joyce et al., 1974
;
McIntyre and Bizzi, 1993
;
Pearson and Iles, 1973
).
Alternatively, stabilization of movement through non-neural mechanisms is
also possible. The viscoelastic properties of muscles, skeletons and
connective tissue, changing muscle moment arms and the length- and
velocity-dependence of force production in active muscle all have the
potential to contribute to the mechanical stabilization of musculoskeletal
systems (Grillner, 1975;
Seyfarth et al., 2001
;
Wagner and Blickhan, 1999
).
The potentially stabilizing properties of active muscles have been termed
`preflexes', since the stabilizing behavior of musculoskeletal systems may
appear similar to neural reflexes but has the potential to occur very quickly
before neural reflexes are able to act
(Brown and Loeb, 2000
). During
rapid locomotion, musculoskeletal `preflexes' could offer continuous
stabilization, even at very high movement frequencies, and augment reflexive
stabilization generated by the nervous system.
The goals of this study were to understand the mechanisms used by running insects to stabilize rapid locomotion. We therefore tested the following hypotheses: (i) that hexapods require step transitions to maintain stability during rapid running and are incapable of generating restoring forces to counteract perturbations within a step, and (ii) that non-neural `preflexive' mechanisms contribute to the stabilization of rapid locomotion.
To test our hypotheses, we subjected cockroaches to laterally directed perturbations using a novel apparatus, a `rapid impulsive perturbation' (RIP) device. The RIP apparatus was designed to be mounted directly above the center of mass of a freely running animal and to change the momentum of the animal's body by generating a brief force impulse. If the animal were to fail to generate an opposing force, the change in momentum caused by an impulsive perturbation would persist over time. If the animal were to generate a force impulse to oppose the perturbation and stabilize so their movements return to an equilibrium trajectory, the time necessary to stabilize to the equilibrium could be used to test whether step transitions are necessary for stability and whether musculoskeletal preflexes contribute to stabilization.
Three-dimensional kinematic measurements of body movement were recorded before, during and after perturbations as the animals ran freely on a Plexiglas track. We compared linear and rotational velocities from periods following perturbations with reference kinematics from unperturbed periods to determine the time at which recovery from perturbations occurred.
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Materials and methods |
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Rapid impulsive perturbations
Since the timing of recovery from perturbations was important for testing
our hypotheses, we designed the RIP apparatus to generate force impulses of as
short duration as possible. We constructed an apparatus which employed a
chemical propellant (black powder) to launch a small projectile, analogous to
a miniaturized cannon mounted on the running animals. The RIP apparatus
generated reaction force impulses of appropriate magnitude over a period of
less than 10 ms, or less than 20% of the stance period of a cockroach running
at its preferred speed.
To construct the RIP apparatus, we used a 2.3 cm long, 0.45 cm diameter plastic tube closed at one end. We added 6.3±1.4 mg (mean ± S.D.) of flint shavings (a low-ignition-temperature accelerant) to the tube. The flint shavings were necessary to ignite the black powder, but did not contribute substantial energy to the subsequent explosion. We then measured 3.2±0.7 mg of FFFF-grade black rifle powder (Goex, Inc.) and added it to the tube. A 0.13 g stainless-steel ball bearing was placed into the tube on top of the powder and held in place by a small piece of paper.
A spark from an ignition module (6520S0201, Harper-Wyman, Inc.) ignited the flint and black powder (Fig. 1B). Two 50 µm wires were connected to the ignition module at one end and soldered to two larger-diameter (0.6 mm) insulated wires at the other ends. The larger-diameter wires were threaded through holes in the base and side of the tube and glued to the outside of the tube with epoxy adhesive. The terminal 2 mm of the larger-diameter wires was uninsulated and served as the origin of the spark. The ignition module created sparks at approximately 3 Hz. The module and video cameras were triggered simultaneously via a relay switch.
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Calibration of the RIP apparatus
We calibrated the RIP apparatus using a miniature force platform
(Full and Tu, 1990). The RIP
apparatus was mounted to a square plastic base and attached to the surface of
the force platform using double-sided tape. We mounted the RIP apparatus
vertically on the platform so that the ball-bearing was projected upwards, and
sampled the output of the platform following an explosion at 10 kHz.
The RIP apparatus and the plastic holder weighed 14 g, and the added mass
decreased the natural frequency of the force platform from 500 Hz
(Full and Tu, 1990) to
approximately 100 Hz. The forces measured by the force platform
(Fig. 2A) are consistent with
the hypothesis that the RIP apparatus generates a force impulse of duration
less than half the period of oscillation of the platform/holder system (5 ms).
No sustained force production was evident from the force platform recordings.
Operating under this hypothesis, we considered the platform and RIP apparatus
to be an elastic system. A near-instantaneous force impulse was hypothesized
to accelerate the mass, and the platform generated a force to decelerate the
mass in a spring-like manner. In this system, the area under the force curve
between the beginning of the explosion and the time when the force begins to
decrease (the peak force) is the force impulse necessary to arrest the
momentum of the RIP apparatus. This force impulse must be equal in magnitude
and in the direction opposite to the force impulse imparted by the RIP. We
calculated the force impulse of the RIP apparatus by integrating the vertical
force from the start of the explosion until the time at which peak force was
reached (Fig. 2B). The mean
impulse from 11 calibration trials using the miniature force platform was
0.84±0.87 mN s (mean ± S.D.). Noise in data acquisition from the
force platform caused the variability in force impulse measurements. This
average impulse is approximately 85% of the linear momentum of a 2.7 g
cockroach carrying a 1.3 g RIP apparatus and running at 24 cm
s-1.
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Attachment of the RIP apparatus to animals
The center of mass of cockroaches is 46% of the distance from the head to
the tip of the abdomen, within the anterior portion of the abdomen directly
behind the thorax (Kram et al.,
1997). However, the abdomen of cockroaches is soft, and abdominal
segments can move relative to one another. We consequently chose to attach the
plastic tube to the crosspiece of a lightweight balsawood base (4.0 cm wide by
4.0 cm long; Fig. 1A) and
attach the base to the stiffer thorax (Fig.
1A,B). The crosspiece was located 1.15 cm behind the most anterior
bolt to place the centre of mass (COM) of the apparatus directly above the COM
of the animals. The balsawood base not only provided a means of attaching the
RIP apparatus to the body, but also facilitated digitization by amplifying
thoracic rotation.
We used 1.1 mm diameter, 6.15 mm long brass bolts to attach the balsawood base to the animals. Using bolts allowed the RIP apparatus to be removed from the animal to be reloaded and facilitated the collection of COM and moment of inertia (MOI) data. We glued two bolts to the mesonotum with cyanoacrylate adhesive and 60 s epoxy adhesive, and a third bolt to an abdominal tergite. The second and third bolts fitted through small slots in the balsawood base. The second bolt ensured that the base remained aligned with the body axis, and the third bolt constrained lateral movements of the abdomen while allowing vertical motion of the abdomen relative to the thorax. The balsawood base was firmly attached to the most anterior bolt on the thorax with a small hex-nut. We attached the RIP apparatus so that the ball bearing was projected laterally towards the animal's right side, causing a reaction force that accelerated the animal to its left.
The RIP apparatus, including balsawood base, powder and ball bearing, weighed 1.3 g, approximately half the body mass of the animals. Attachment of the RIP apparatus must change both the COM and MOI of the animals. To minimize any effects that changing the COM location or MOI may have had on locomotory kinematics, our analysis (see below) compared kinematics from perturbed trials with kinematics from unperturbed trials in which the RIP apparatus was also mounted onto the animals.
Running track
A Plexiglas running track 91.4 cm long x 8.25 cm wide x 10.16
cm high was constructed to allow the animals free movement within a contained
area. In its center, the track fitted over an 11 cm long x 8.25 cm wide
balsawood platform, which allowed the animals to grip the substratum with
their pretarsal claws. We did not visually observe any instances in which the
legs slipped on the platform during running during unperturbed or perturbed
trials.
Video recording
Each trial was recorded at a frame rate of 1000 Hz using a high-speed
digital video system (Motionscope, Redlake Imaging). Three synchronized
cameras focused on the space directly above the wood platform simultaneously
recorded each trial. One camera was placed directly above the platform, and
two cameras recorded from either side lateral to the average movement
direction of the animals. Video frames had a resolution of 240x210
pixels. Lateral cameras had fields of view of 11 cm, and the camera above the
platform had a field of view of 15 cm in the average movement direction.
Kinematic data analysis
During every experiment, a stationary calibration object was placed in the
field of view to allow three-dimensional calibration. The calibration object
was constructed from small plastic blocks (Lego systems, Inc.) and had
dimensions of 6.5 cmx5 cmx2.5 cm, which was large enough to fill
more than half the field of view of the lateral cameras in one dimension. The
calibration object had 33 points identifiable in all three video cameras. The
distances of 32 of the points from one point (which served as the origin) were
measured with digital calipers (Omega Scientific, Inc) to an accuracy of 0.01
mm. An image of the calibration object was recorded prior to, and following,
each experimental session. Cameras were not moved during an experimental
session. Calibration errors in position were 0.11 mm in the x
(foreaft) direction, 0.21 mm in the y (mediolateral)
direction and 0.31 mm in the z (vertical) direction.
Digital video recorded during each trial was saved to computer disk as uncompressed AVI files, which were imported into a three-dimensional video analysis system (Motus, Peak Performance Technologies, Inc.).
Trials were selected for analysis if the perturbation occurred near the
middle of the field of view of the video cameras and the animal and
perturbation apparatus did not touch the wall during the trial. Selected
trials were digitized using the video analysis system. Four points on the
balsawood base (the rear of the base, the front of the base and the two
lateral ends of the base; Fig.
1A) were digitized in two camera views (the vertical camera view
and one of the lateral camera views). Raw coordinate data were filtered using
a fourth-order zero-phase-shift Butterworth filter with a cut-off frequency of
100 Hz. Given the calibration and the filtered coordinate data, the video
analysis system was used to calculate the three-dimensional location of each
of the points relative to the origin of the calibration object using direct
linear transformation (Biewener and Full,
1992). Resulting three-dimensional position data were filtered
using a fourth-order Butterworth filter using a cut-off frequency of 50
Hz.
Experimental protocol
Prior to each experiment, we anesthetized the animals by placing them in a
refrigerated (4°C) room for 1 h. We removed the wings from the animals
(carefully cutting around the largest wing veins) using scissors and roughened
the cuticle on the mesonotum by gently rubbing it with sandpaper. We glued the
brass bolts to the mesonotum and allowed the animals to recover at room
temperature in an unsealed plastic container for at least 1 h.
For each experimental trial, we carefully bolted the RIP apparatus into place. We placed the cockroach on the running track and encouraged it to run by lightly tapping its cerci. Typically, we conducted between five and ten unsuccessful running trials before attempting to trigger the RIP. When the cockroach appeared to run at constant average speed near the center of the video field of view, we manually triggered the RIP apparatus and video collection system. Since 300 ms elapsed between triggering and when the spark occurred (causing the explosion), it was necessary to anticipate the animal's location and trigger the RIP before the animal was actually in the video field of view. After the RIP apparatus had been triggered, it was necessary to reload the apparatus with flint, black powder and the ball bearing. The RIP apparatus was carefully removed from the cockroach and reloaded between trials. The cockroach was placed in a plastic enclosure and allowed to rest for approximately 10 min.
Center of mass determination
Cockroaches were deep-frozen immediately following each experimental
session and stored in airtight plastic containers. We determined the location
of the center of mass for each animal (N=9) individually by
suspending the animal from strings attached to three different points of the
body. We filmed the animals while suspended from each location, and determined
the vertical axis in each image by digitizing small pieces of reflective tape
attached to the string. The COM lies at the intersection of the vertical axes
in the three camera views (Blickhan and
Full, 1992). We digitized the tail, head, left and right pronotum
points and the four base points. To measure changes in COM position due to the
RIP apparatus, we expressed the location of the COM in the coordinate frame
set by the tail, head and left and right pronotum points. To calculate the COM
position during running, we expressed the location of the COM in the
coordinate frame set by the four base points.
Attachment of the RIP apparatus to the cockroaches shifted the COM -1.4±1.1 mm in the foreaft direction (towards the tail), 0.6±0.6 mm in the mediolateral direction (towards the left) and 3.6±0.8 mm in the vertical direction (upwards). These shifts represent less than 5% of body length in the foreaft and mediolateral directions, but a more than 25% shift in vertical COM position.
Moment of inertia determination
Since the perturbation apparatus was mounted on the animals and could
potentially change the location of the COM and MOI of the animals, we directly
measured the COM and MOI for each animal, with and without the apparatus. We
determined the moment of inertia about the three principal axes (yaw, pitch
and roll axes) by piercing the deep-frozen animals parallel to a principal
axis with a long pin, which was balanced on razorblades and allowed to swing
freely (Kram et al., 1997). We
filmed the animals at 500 Hz, and measured the period of oscillation after
lightly tapping the animals. We digitized points on the head and the tip of
the abdomen, two points on either side of the pronotum and four points on the
base (when the base was attached to the animal). Using the center of mass
position calculated above, we determined the distance from the center of mass
to the pin (d). We calculated the moment of inertia (I)
about the given axis using the following formula derived from the parallel
axis theorem:
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Coordinate frames
Kinematic data were expressed in two inertial reference frames with the
origin at the instantaneous position of the COM, using custom-designed
programs implemented in MATLAB (The MathWorks, Inc.).
First, the position of the COM in the three-dimensional global coordinate frame was calculated from the positions of the four base points for each sampled time frame. A natural coordinate system to use to express perturbations away from the initial (assumed to be the nominal or `desired') movement direction is a rotational frame with one axis parallel to the average movement direction of the animal one stride before the perturbation and a second axis parallel to the global horizontal plane. We termed this coordinate (x,y,z) frame the `initial movement direction frame' (Fig. 3A). Body orientation was calculated using the four digitized base points expressed in the initial movement direction frame and expressed as Euler angles in the order yaw, pitch, roll (Fig. 3C-E).
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Since the animals were free to adopt a new average movement direction after being perturbed, a separate coordinate (X,Y,Z) system based on body orientation was employed to compare translational velocities and accelerations with reference data. We termed this frame a `foreaft' frame (Fig. 3B).
We differentiated the three-dimensional coordinates of the COM and the yaw,
pitch and roll Euler angles with a fourth-order difference equation
(Biewener and Full, 1992) to
yield the instantaneous translational and rotational velocities of the body
over time.
Step events
Insects typically employ an alternating tripod gait during rapid locomotion
in which the front and hind legs on one side of the body step synchronously
with the contralateral middle leg. For each stride of each trial, we recorded
the time when each of the animals' legs switched from protraction to
retraction and the time when each leg switched from retraction to protraction
by visually inspecting video recordings from the two lateral views. The tarsi
made contact with the ground (i.e. `touchdown') at approximately the same time
as the legs switched from protraction to retraction. Similarly, the tarsi left
the ground at approximately the same time as the retractionprotraction
transition (i.e. `lift-off'). Consequently, we considered the stance period to
be equal to the retraction period, and we considered the swing period to be
equal to the protraction period. At times, shadows in the video image
prevented accurate measurement of protraction, retraction, touchdown or
lift-off step events. These steps were consequently not included in the
analysis of stance and swing periods (consequently, the number of steps
reported is not the same for all legs). We measured stride periods, stance
duration, swing duration and phase relationships among legs during unperturbed
and perturbed strides. We calculated stride period as the period between
touchdown events and phase as the time of touchdown relative to the stride of
a reference leg (Jamon and Clarac,
1995). Step event data from unperturbed trials and from strides
prior to the perturbation formed an `unperturbed' data set. Step event data
from stride, stance and swing periods during which the perturbation occurred
formed a data set `during' the perturbation. Step event data from strides that
occurred after the perturbed stride formed a third data set. Unperturbed and
perturbed data were drawn from the same animals. For all data sets, we
calculated mean values for each measure (i.e. phase and stride, stance and
swing periods) for each condition and animal. Data sets from stride, stance
and swing periods during and after the perturbation were compared with those
from the unperturbed periods using an unpaired t-test implemented in
MATLAB.
Reference data sets from unperturbed strides
We collected 12 unperturbed trials from eight of the nine animals used in
the study to provide reference kinematics against which the perturbed trials
could be compared. Animals were run with the perturbation apparatus loaded and
attached to their thorax, but the RIP apparatus was not triggered during the
trial. No animal contributed more than two trials to the reference data set.
Whole-body kinematics and step event data over 1-3 strides were collected from
each of the reference trials. Animals in unperturbed reference trials ran with
an average foreaft speed of 29±9 cm s-1, within the
range 24-38 cm s-1 commonly observed in cockroaches running without
the RIP apparatus (Full et al.,
1991; Full and Tu,
1990
).
Comparison of perturbed data with reference data
Scaling of unperturbed kinematics to stride periods of perturbed
trials
To compare the kinematic data from perturbed trials with the reference
kinematics, we scaled the reference kinematics in time and then normalized for
differences in initial conditions. To scale the reference kinematics in time,
we delimited stance periods for each trial by averaging the touchdown and
lift-off times for legs of each stepping tripod. For the tripod containing the
left front (LF), right middle (RM) and left rear (LR) legs, touchdown and
lift-off times are identified as LF,RM,LR (abbreviated LF). Step events from
the opposite tripod are identified as RF,LM,RR (abbreviated RF). Since stance
and swing periods following perturbations were not significantly different
from those for unperturbed running, and phase relationships among tripods did
not differ by more than 5% from 0.5, we scaled the reference data to the
stance and swing periods of the LF tripod of the perturbed trial.
To compare kinematics from perturbed strides with unperturbed kinematics,
we first scaled the unperturbed data in time to yield the most representative
reference data set corresponding to a period equal to the period of each
perturbed step. For each stance period during or after a perturbation, we
extracted unperturbed kinematics from all stance periods corresponding to the
same tripod of legs of every reference trial. We scaled these kinematics from
each unperturbed stance period to have the same number of samples as the
selected perturbed stance period. Scaled, unperturbed (denoted by subscript U)
kinematics are referred to as U,LF and
U,RF for the LF and RF tripods,
respectively. We averaged the scaled kinematics from all reference stance
periods for each animal, resulting in eight reference stance data sets scaled
to the length of each stance period of every perturbed trial. We termed the
average of these eight unperturbed mean data sets
and
for the LF and RF tripods, respectively. Concatenating the average reference
data for alternating tripods resulted in a mean reference data set for each
perturbed trial.
After scaling in time, we normalized the reference kinematics to control for differences in position and velocity. We scaled the mean reference kinematics to have the same average position and velocity as the stride immediately before the perturbation.
Finally, we subtracted the scaled mean reference data from each normalized perturbation trial and measured the deviation in translational and rotational velocities from the reference mean. Statistical comparisons of maxima and minima of velocity deviations from the reference mean were conducted using a statistical package (JMP, the SAS Institute, Cary, NC, USA). We used a z-test to compare the measured populations of velocity deviation maxima and minima to a hypothesized mean of zero.
Criterion for recovery: deviation from the mean reference
trajectory
Locomotion can be considered to be perturbed if observed movements are
significantly outside the range observed during unperturbed locomotion. If the
`error', or the difference between a movement cycle and the mean unperturbed
movements for an equivalent cycle, lies outside the range of errors observed
during unperturbed locomotion, then the cycle can be considered to be
perturbed. We measured movement error by calculating the mean-squared
difference between movements over an entire locomotory cycle
(Schwind, 1998). For each
perturbed trial (denoted with subscript RIP), we selected one velocity
direction (such as the lateral velocity, denoted with subscript y) and formed
a vector from the velocity over one stance period. For the LF tripod, this
vector is denoted
RIP,LFy and that for
the RF tripod
RIP,RFy. The magnitude
of the error ERIP,RFy or ERIP,RFy
between this vector and an equivalent vector from the scaled mean reference
data set is:
![]() | (2) |
We compared ERIP,LFy and ERIP,RFy
with the population of errors from the unperturbed trials, which serves as an
estimate of the variability of unperturbed running. To create a population of
unperturbed errors for each scaled unperturbed trial, we calculated the error
EU,LFy or EU,RFy:
![]() | (3) |
We compared ERIP,LFy and ERIP,RFy
with the population of EU,LFy and
EU,RFy, respectively, using a z-test to determine
whether movements over the cycle of the perturbed trial were significantly
different from the mean unperturbed movements. A z-test determines
whether a value lies outside confidence limits for a population. We used a
one-tailed z-test with a significance level of 0.05 to determine
whether ERIP,LFy and ERIP,RFy fell
outside the population of EU,LFy and
EU,RFy from the unperturbed trials. If
ERIP,LFy was not significantly different from the
population of EU,LFy values,
RIP,LFy was considered to be not
significantly different from
,
with a comparable comparison for the opposite tripod.
If the perturbation resulted in a value of
RIP,LFy or
RIP,RFy that was significantly
different from the corresponding value of
or
during the stance period containing the perturbation, then the stance period
following the perturbation during which the velocity ceased to be different
from the reference mean velocity was recorded. If the velocity ceased to be
significantly different from the reference mean velocity for the stance period
immediately following the stance period containing the perturbation, then the
animal was considered to have recovered within the stance period during which
the perturbation occurred.
Time to recovery
We considered the time to recovery to be the period between a perturbation
and the time at which the velocity over an appropriate period was not
significantly different from the reference mean. We chose the mean stance
duration, a period assumed to be equal to a locomotory half-cycle, as the
appropriate period for evaluating recovery. Following each 1 ms time sample
after a perturbation, we constructed a vector for each variable (such as the
lateral velocity, y) with length equal to the mean stance period. We
compared this vector with equivalent vectors constructed from the unperturbed
trials, appropriately scaled to phase in the step cycle. We constructed error
vectors by subtracting the scaled reference mean from data from perturbed and
unperturbed trials. Errors from perturbed trials were compared with the
population of errors from unperturbed trials using a z-test with a
significance level of 0.05. We repeated this measurement for each sample
following the perturbation, sliding a window one mean stance period in length
along the data sets and testing for significant differences. The time to
recovery was considered to be the time sample after the perturbation at which
error vectors from perturbed trials first failed to be significantly different
from the reference mean. This indicates that the locomotory half-cycle
beginning at this time is not significantly different from the population of
unperturbed half-cycles of the same phase.
Values are presented as means ± S.D.
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Results |
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Effects of perturbation
Translational position and velocity
Lateral velocity increased over the reference velocity to a maximum of
21.0±6.9 cm s-1 (z-test; P<0.0001),
indicating that the perturbations imparted a force impulse that was on average
80% of the forward momentum of the animals. This impulse was not significantly
different from the mean impulse of 85% generated by the RIP (t-test;
P>0.9). Fig. 5
shows lateral velocity from a representative perturbation trial. In this
trial, the perturbation occurred during the stance phase of the LF tripod and
caused the lateral velocity to increase to a maximum value of 27 cm
s-1 in 11 ms. For this trial, lateral velocity was significantly
different from the reference velocity during the perturbed step
(z-test; P<0.05), but was not significantly different in
subsequent steps. In this trial, the lateral velocity recovered in 31 ms.
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The mean time from the onset of the perturbation until the lateral velocity error began to decrease was 13±5 ms (Table 1). A decrease in lateral velocity must be caused by a force opposing the perturbation. This indicates that cockroaches were able to begin generating forces opposing the perturbation 13 ms following the onset of the perturbation.
The perturbation to lateral velocity resulted in a mean lateral displacement of 0.46±0.2 cm relative to the initial COM position (Fig. 6B). On average, 200 ms (approximately two strides) following the perturbation, the lateral position of the COM returned to 0.17±0.66 cm from its position before the perturbation (Fig. 6B). Even though the cockroaches were not constrained to run in a particular direction, the finding that perturbations to lateral position decreased after reaching a maximum suggests that cockroaches stabilize lateral COM position in addition to velocity when confronted with perturbations.
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Lateral perturbations did not have a consistent effect on foreaft or vertical velocity (Fig. 6A,E). Clear maxima or minima in foreaft or vertical velocities were not evident. Consequently, velocity maxima are not reported in Table 1. Prior to the perturbation, animals ran at a foreaft velocity of 23.9±5.4 cm s-1. Following perturbations, the animals ran at a mean velocity of 25.5±3.6 cm s-1, an insignificant difference. Changes in vertical velocity and position were variable, and many occurred more than 100 ms following the perturbation.
Rotational position and velocity
Lateral force impulses imparted by the RIP caused changes to yaw, pitch and
roll velocity and to position (Fig.
7). Yaw velocity was affected by lateral perturbations above the
COM. On average, yaw velocity decreased relative to the reference velocity by
-451±283° s-1 (z-test; P<0.001), an
approximately threefold increase over the maximum yaw velocities during
unperturbed running (Kram et al.,
1997). Yaw velocity began to increase from a minimum 14±9
ms following the perturbation (Table
1). Following a perturbation, the animals oriented in a yaw
direction close to the orientation before the perturbation
(Fig. 7A).
|
Since the RIP apparatus was approximately 5 mm above the COM, perturbations to roll would be expected as a result of the moment arm about the roll axis, particularly considering the low moment of inertia about the roll axis (Table 2). Surprisingly, the magnitude and timing of the effects of perturbations on roll were variable (Table 1). The mean change in roll velocity following perturbations was not significantly different from zero (z-test, P=0.052). Nearly 30 ms following the perturbation, however, there was a trend for cockroaches to roll in the direction opposite to the roll expected from the perturbation itself (Fig. 7E). Cockroaches appeared to stabilize perturbations to roll, but exhibited variability in roll orientation during locomotion following perturbations.
|
Perturbations caused an immediate negative pitch velocity, causing the head
to rotate downwards. Pitch velocity decreased relative to the reference
velocity, reaching a minimum of -498±364° s-1
(z-test; P<0.001). This decrease in pitch velocity is
approximately 1-2 times the maximum pitch velocities observed during
unperturbed running (Kram et al.,
1997). Pitch velocity began to increase from this minimum
10±12 ms following the perturbation. This negative pitch velocity was
followed by a positive pitch velocity, which caused the head to rotate
upwards. Similar to changes in vertical COM position, the pitch response over
longer time scales was variable.
Effects of perturbation did not depend on tripod perturbed
The effects of perturbations on translational and rotational velocity
maxima and on the time to reach maxima did not depend on which tripod was in
stance during the perturbation (t-tests, P>0.15 for all
comparisons; Table 1). This
difference is surprising, since the lateral force generated by the side of the
body with two legs in stance is nearly twice the lateral force generated by
the side of the body with one leg in stance
(Full et al., 1991;
Full and Tu, 1990
). The LF
tripod might be expected to generate forces opposing the perturbation more
easily since the direction of the unperturbed net lateral force for this
tripod is opposite to that of the perturbation. However, no effect of tripod
was observed in our experiments.
Stride periods and leg phase relationships were not altered in
responsfe to perturbations
Changes to leg placement at step transitions could be achieved through
changes in stance or swing periods of individual legs during running. For
example, to begin a step with the legs positioned more anteriorly than normal,
the swing period could be lengthened to allow the leg to move farther forward
than during unperturbed running. This change could also alter the stride
period or the phase relationships among the legs following a perturbation.
Stride periods during and after perturbations did not differ from stride
periods during unperturbed running for any leg
(Table 3; P>0.12
for all comparisons). Stance duration during and after perturbations also did
not differ significantly from unperturbed running, although there was a trend
towards increasing stance duration in strides during and after perturbations
(Table 3; P0.05
for all comparisons). Swing periods of strides during and after perturbations
also did not differ significantly from swing periods of unperturbed strides
(Table 3; P>0.32
for all comparisons).
|
Leg phase relationships for strides during or following perturbations were not significantly different from phase relationships during unperturbed strides (Table 4; P>0.09 for all comparisons). Cockroaches maintained an alternating tripod gait during and after perturbations, resulting in phase relationships among legs that remained close to 0.5.
|
Recovery from perturbation
Translation
In all 11 trials, perturbations caused the lateral velocity to be
significantly different from the reference velocity during the perturbed step
(z-tests; P<0.05;
Table 5). One trial which did
not show recovery for any window following the perturbation, and one which
showed instantaneous recovery, were excluded from the calculation of the time
to lateral velocity recovery. For the remaining nine trials, lateral velocity
recovered in 27±12 ms. This quick recovery caused lateral velocity to
recover within the stance period during which the perturbation occurred in 45%
of the trials. In a majority (5 of 7) of the trials in which the perturbation
occurred within the first half of the stance period, the lateral velocity
recovered within the perturbed stance period. Consequently, step transitions
were not necessary to recover from lateral perturbations when the perturbation
occurred sufficiently early in the stance period. In all trials in which the
perturbation occurred in the second half of stance, the lateral velocity did
not recover within the perturbed step. The phase of the step cycle during
which the perturbation occurs appears to constrain the ability of the animals
to recover from lateral perturbations. Perturbation magnitudes were not
significantly different between trials recovering within one stance period and
trials that failed to recover within one stance period (0.84±0.30
versus 0.84±0.29 mN s; t-test,
P>0.99).
|
In six of 11 trials, perturbations caused the foreaft velocity to be significantly different from the reference velocity during the perturbed step (z-tests; P<0.05; Table 5). Foreaft velocity was significantly different from the reference velocity in the majority of steps (N=8) immediately following the perturbation and failed to recover during any window following the perturbation within the time period of the trial in four of 11 trials. In five trials, the perturbation caused a significantly different foreaft velocity during the perturbed step and the foreaft velocity recovered during the trial. In these trials, foreaft velocity recovered in a period equal to 90% of the mean stride period of 111 ms (Table 3). Cockroaches did not appear to show within-step stabilization of foreaft velocity following lateral perturbations, and we cannot reject the hypothesis that step transitions are important for maintaining foreaft velocity.
Perturbations caused vertical velocity to become different from the reference velocity in 10 of the 11 trials (z-tests; P<0.05; Table 5), of which three recovered within the perturbed step and three recovered in a subsequent step in the trial. In two trials, the animals did not recover during a subsequent step period.
Rotation
Perturbations caused yaw velocity to become different from the reference
velocity in six of 11 trials, of which one recovered within the perturbed step
and four recovered in a subsequent step in the trial (z-tests;
P<0.05; Table 5).
Three trials that were not significantly different from the reference yaw
velocity during the perturbed step became different during the step following
the perturbation. Yaw velocity recovered in approximately 40 ms.
Perturbations caused pitch velocity to become different from the reference velocity in nine of 11 trials (z-tests; P<0.05; Table 5), of which two recovered within the perturbed step and five recovered in a subsequent step in the trial. On average, pitch velocity recovered in approximately 40 ms, similar to the yaw velocity.
Perturbations caused roll velocity to become different from the reference velocity in eight of 11 trials (z-tests; P<0.05; Table 5), none of which recovered within the perturbed step and one recovered in a subsequent step in the trial. For the three trials in which roll velocity recovered during a step period window, recovery occurred in approximately 100 ms.
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Discussion |
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Mechanical coupling results in complex responses to lateral
perturbations
The complexity and non-linearity of musculoskeletal systems can cause
forces, torques and motions in different directions to be interdependent
(Zajac and Gordon, 1989). Even
in highly simplified dynamic systems, mechanical coupling can cause
perturbations to one variable to impact the dynamics of the entire system
(Kubow and Full, 1999
).
Consequently, mechanical coupling can alter the control requirements for
maintaining stability.
The laterally directed force impulses used in this study did not simply
perturb lateral velocity, but also resulted in perturbations to other movement
directions. For example, lateral perturbations caused yaw velocities to
increase (in the negative direction) substantially
(Table 1;
Fig. 7A). A potential
explanation for the coupling of yaw velocity to lateral COM velocity may be
due to the position of the COM behind the point of attachment of the legs. The
COM lies 3.5 mm behind the attachment of the hind legs to the body, 10 mm
behind the attachment of the middle legs and 16 mm behind the attachment of
the front legs (Kram et al.,
1997). The position of the COM, and consequently the RIP
apparatus, behind the legs creates a moment arm about the thorax in the
foreaft direction. This mechanical coupling may have caused the
observed perturbations to yaw.
Cockroaches were able to generate moments about the vertical axis to
recover from induced perturbations to yaw velocity. In five of six trials in
which perturbations caused the yaw velocity to be different from the reference
yaw velocity, the yaw velocity recovered within the trial
(Table 5). In contrast to their
rapid recovery in the lateral direction, however, cockroaches did not develop
yaw torques sufficient to cause recovery in the yaw direction within one
stance period in the majority of trials. This may be due to spring-like
behavior in the lateral direction (Full et
al., 2002) and to the large moment of inertia of cockroaches about
the dorsoventral axis (Table
2). The failure of yaw velocity to recover quickly from
perturbations underscores the complexity of stabilizing rapidly moving dynamic
systems. Even though the RIP apparatus directly caused perturbations to
lateral movements, yaw velocity often remained altered after the lateral
velocity had recovered.
Cockroaches did not require step transitions to recover from
perturbations
Following lateral perturbations, cockroaches were able to stabilize lateral
velocity before the transition to the next stance period occurred
(Table 5). Cockroaches did not
require step transitions to stabilize lateral velocity when subject to lateral
force impulses of a magnitude equal to 85% of their forward momentum. However,
the ability of cockroaches to recover from perturbations within a stance
period depends on the substratum on which they run. Cockroaches do not exhibit
within-step recovery on smooth surfaces (such as acetate) or on surfaces on
which they have less purchase than soft balsawood
(Jindrich, 2001). The
influence of friction and other properties of substrata on stability and
maneuverability during locomotion is an important area for research
(Alexander, 1982
).
Gait kinematics did not change in response to lateral
perturbations
Changes in stance period, swing period or phase relationships among legs,
which could indicate changes to leg kinematics at step transitions, were not
evident (Tables 3,
4). This finding supports the
hypothesis that kinematic changes at step transitions are not necessary to
maintain stability in response to lateral perturbations and underscores the
need to consider forces and inertias when studying rapid locomotion
(Jindrich and Full, 1999).
However, we did not directly measure leg or joint kinematics in response to
perturbations. Changes in foot placement, leg configuration or stiffness could
contribute to stability without requiring changes to step periods or phase
relationships. We therefore cannot rule out the possibility that kinematic
changes at step transitions augment within-step changes in force production
and may be necessary to stabilize locomotion subject to perturbations of
different magnitude or direction from those used in the present study.
Contribution of intrinsic musculoskeletal properties to stabilization
of rapid running
The velocity maxima and minima following perturbations (Figs
6,
7) must result from forces
generated by the animals. Forces opposing the perturbation are necessary to
cause decreases in velocity. The short times to maximum lateral velocity
(Table 1) indicate that the
animals were capable of generating opposing forces 10-14 ms following the
perturbation. This rapid force generation poses the question of whether neural
reflexive feedback is likely to account for the ability of cockroaches to
recover from perturbations within one step.
Neural feedback allows for precise control of leg kinematics and interleg
coordination in slowly moving insects (Cruse,
1985a,
b
,
1990
). Even during extremely
rapid movements, some insects are capable of extraordinarily rapid and precise
neural control. For example, flies are capable of cycle-by-cycle modulation of
wing kinematics when beating their wings at 140 Hz
(Tu and Dickinson, 1996
).
Reflexes involving chemical synapses can allow synaptic delays of less than 5
ms, and even faster electrical connections allow synaptic delays of less than
1 ms (Fayyazuddin and Dickinson,
1996
). The capabilities of insect nervous systems clearly allow
for the possibility that within-step stabilization is controlled by neural
feedback.
For neural feedback to cause recovery from perturbations, three sequential
events must take place. First, the perturbation must be detected by sensory
cells. Sensors on the cerci (Camhi and
Levy, 1988), the antennae
(Camhi and Johnson, 1999
), the
exoskeleton (Schaefer et al.,
1994
), within the exoskeleton
(Burrows, 1996
) or even within
the muscles (Matheson and Field,
1995
) could sense the perturbation. Antennal or cercal sensors
could detect air currents, noise or heat generated by the RIP, but these
stimuli are less likely to provide information about the specific nature of
the perturbation. Sensors on the exoskeleton such as campaniform sensilla
(Ridgel et al., 1999
) or
chordotonal organs within the exoskeleton
(Kondoh et al., 1995
) can
detect loading of the exoskeleton or joint position and velocity and are more
likely to provide the specific information about the perturbed COM velocity
necessary to generate an opposing force.
A brief force impulse does not necessarily produce an equally fast change
in exoskeletal loading, joint position or velocity. Exoskeletal strain due to
loading, which campaniform sensilla can detect, can occur when a segment is
stressed axially or when muscles generate forces on the segment
(Ridgel et al., 1999). Axial
forces along leg segments opposing a perturbation would decrease the change in
lateral momentum resulting from the perturbation. The agreement between the
average force impulses generated by the RIP (85% of forward momentum) and the
maximum lateral momentum measured following perturbations (80% of forward
momentum) suggests that little immediate force is generated by axially loading
the leg segments. Axial loading would be expected to decrease the degree to
which the force impulse generated by the RIP results in increased lateral
momentum. Substantial axial loading of the legs would generate a force impulse
counter to the impulse generated by the RIP and cause the resulting change in
lateral momentum to be smaller than the force impulse generated by the RIP.
However, without direct measurements of exoskeletal strain or sensory output,
it is not possible to exclude the possibility that campaniform sensilla can
immediately sense perturbations. In the American cockroach, Periplaneta
americana, campaniform sensilla have been shown to detect lateral
substrate displacement, with latencies of 6.1±3,5 ms (mean±S.D.,
N=61) (Ridgel et al.,
2001
).
Hair plates and chordotonal organs can sense changes in joint position or velocity. However, some time may elapse before changes in velocity exceed the threshold of the sensors. The time necessary to sense a perturbation consequently depends on the sensor threshold and the magnitude of the perturbation.
The potential for a time delay between a perturbation and when sensors could detect position changes is magnified. In the 13 ms between a perturbation and the first stabilizing acceleration, lateral velocity changes by approximately 20 cm s-1 or almost 10 times the peak lateral velocity during unperturbed running. In the same time period, lateral position changes by only 0.15 cm, or 2.5 times the peak excursions during unperturbed running (Fig. 6B). In general, large position changes will lag behind velocity changes and, if sensors show comparable relative sensitivities, the time necessary to detect changes in position may be expected to be larger than the time necessary to detect changes in velocity or force.
The second event that must occur for neural feedback to stabilize a
perturbation is that sensory information must be transmitted to the central
nervous system (CNS) and processed, and an appropriate motor output must be
sent to the muscles. Camhi and Nolen
(1981) estimated a minimum
neural latency, the time from stimulus onset to muscle stimulation, of 6.5 ms
for the stereotyped escape response of P. americana, which, with a
mass of less than 1 g, is smaller than B. discoidalis and can run at
over twice the maximum leg cycling frequency of B. discoidalis
(Full and Tu, 1991
). It is
reasonable to hypothesize that P. americana exhibits faster reflexes
than B. discoidalis. However, it is unclear whether a neural response
to a perturbation would be expected to show a shorter or longer time delay
than the escape response.
Third, once the perturbation has been sensed and the appropriate motor
output conducted to the muscles, the muscles must generate corrective forces.
Muscles take time to begin to generate force because of delays inherent in
excitationcontraction coupling. The estimate of 6.5 ms for minimum
neural latency, coupled with the observed velocity changes 13 ms following
RIPs, leaves 6.5 ms for the muscles to increase force production following
stimulation. The time to force onset for individual muscles in B.
discoidalis is almost 10 ms (Full and
Meijer, 2001). Muscle kinetics alone could account for much of the
time to recovery onset following perturbations, even if there were no neural
latencies involved with sensing, processing and generating motor output.
Moreover, the time to peak force of active, shortening muscles for B.
discoidalis is 36 ms (Ahn and Full,
2002). Similarly, K. Meijer (unpublished data) has subjected
individual legs to rapid step position changes (1 mm change in foreaft
position in less than 2 ms) and found that, in passive muscles, peak force is
developed on average 30±2 ms (mean ± S.D., N=8)
following the step length change. The lateral velocity was observed to recover
from the perturbations in slightly less than 30 ms
(Table 5). Since the kinetics
of relaxation is slower than the kinetics of force generation
(Ahn and Full, 2002
), if
muscles could be stimulated immediately following a perturbation, peak force
might be reached as much as 30 ms later. In this case, the lateral velocity
might be expected to exhibit a large decrease (i.e. `overshoot') after
neurally stimulated muscles had generated forces to arrest the velocity
imparted by the perturbation. Such an overshoot would prolong the time to
recovery. We did not observe such a decrease in the perturbation trials
(Fig. 6C).
In summary, to generate forces appropriate to counteract a perturbation, the perturbation must be sensed, sensory information must be integrated in the nervous system to generate appropriate motor output and the muscles must generate additional forces. Latencies due to sensing the perturbation, to integrating sensory information and generating motor output in the nervous system and to developing muscle forces could each separately account for a substantial proportion of the time to force onset observed following perturbations.
Several studies have directly measured the latency between the application
of a stimulus and the generation of electrical (EMG) activity at the muscles
or the onset of leg movement. This type of measurement accounts for the time
necessary for the CNS to process a stimulus and, in the case of movement
onset, some of the excitationcontraction kinetics of the muscle as
well. Studies on P. americana report a wide range of time delays.
Levi and Camhi (1996) reported
a 25-50 ms delay (depending on the muscle) between a wind stimulus and EMG
activity onset in P. americana. During walking, Camhi and Nolen
(1981
) reported a 14 ms delay
between a stimulus and movement onset for P. americana. Schaefer et
al. (1994
) reported latencies
to movement onset of 17 ms in response to a tactile stimulation, and Nye and
Ritzmann (1992
) reported a
latency of 55 ms from wind stimulation to leg movement. The fastest reflex
latencies measured for P. americana (14 ms) are longer than the mean
time to lateral velocity decrease following RIPs (13 ms;
Table 1). If B.
discoidalis could sense perturbations immediately after they occurred,
the time required to generate the observed forces would be comparable with the
very fastest reflexes measured in cockroaches.
On the whole, a reflex-based mechanism for stabilizing locomotion could account for the extremely rapid force development observed following perturbations only if the time delays introduced in sensing and the neural processing of sensory information were close to the theoretical or measured minima. The extremely quick sensing and processing required of the nervous system, however, poses the question of whether the animals could sense the results of the perturbation accurately enough immediately following the perturbation to generate an appropriate motor response. Coupled with this problem, the likely magnitudes of both neural and muscular time delays calls into question whether reflex-based mechanisms could generate appropriate responses to the brief perturbations generated by the RIP apparatus. Consequently, the extremely rapid force generation and recovery times found in cockroaches support the hypothesis that musculoskeletal `preflexes' contribute to stabilizing rapid locomotion. Measurements of EMG activity following perturbations will be an important next step to test this hypothesis.
A preflex can be considered as a `zero-delay, intrinsic response of a
neuromusculoskeletal system to a perturbation'
(Brown and Loeb, 2000).
Stabilization by musculoskeletal elements can result from the passive
properties of muscles and connective tissue contributing to joint impedance
(Brown et al., 1982
;
Esteki and Mansour, 1996
;
Hajian and Howe, 1997
) and the
length- and velocity-dependence of force production in active muscle
(Grillner, 1972
;
Rack, 1970
). For example,
increased force generation by active muscle when subjected to lengthening
could act to resist sudden length changes and counteract perturbations
(Morgan, 1990
;
Rack and Westbury, 1974
).
Passive musculoskeletal elements and dynamic muscle properties can act to
stabilize many-jointed musculoskeletal systems
(Brown and Loeb, 2000
;
Seyfarth et al., 2001
;
Wagner and Blickhan,
1999
).
The potential for musculoskeletal preflexes does not necessarily imply that
recovery from a perturbation is instantaneous. Compliance in the
musculoskeletal system can cause time delays in recovery from perturbations
similar to the time delays that would arise from neural latencies
(Campbell and Kirkpatrick,
2001). However, if the recovery from perturbations can be
explained by the viscoelastic properties of the unperturbed
neuromusculoskeletal system, it would provide additional support for the
hypothesis that preflexes contribute to stabilization.
Spring-like recovery from brief perturbations is similar to
spring-like dynamics during unperturbed running
Running animals employ a bouncing gait during unperturbed locomotion. The
COM oscillates in a spring-like manner. Consequently, running animals can be
modeled as compliant systems in the sagittal plane
(Blickhan, 1989;
Blickhan and Full, 1993
;
Farley et al., 1993
;
Full, 1989
;
McMahon and Cheng, 1990
).
Cockroaches also exhibit lateral oscillations of their COM during each stride
(Full and Tu, 1990
) and may
act like spring-mass systems in the horizontal plane. Spring-like behavior
during unperturbed locomotion may arise from the spring-like properties of
active muscle and passive skeletal compliance without requiring reactive
excitation of muscle.
Musculoskeletal properties that contribute to spring-like behavior during
unperturbed locomotion could also contribute to stability. The `lateral leg
spring' (LLS) model of legged locomotion in the horizontal plane captures many
aspects of insect locomotion without requiring a control system (Schmitt and
Holmes, 2000a,
b
). Stability in yaw velocity
and body orientation relative to the direction of COM movement emerges from
the dynamics of the LLS model without the need for explicit control
(Schmitt et al., 2002
). If
cockroaches act like spring-mass systems in the horizontal plane, then passive
dynamic behavior analogous to that observed in the LLS model may contribute to
stability. The direct lateral perturbations used in the present study provide
the opportunity to determine whether cockroaches act like horizontal
spring-mass systems.
Analysis of spring-mass systems in two dimensions is complex and non-linear
(Schmitt and Holmes, 2000a;
Schwind, 1998
). Variables such
as the length and touchdown angle of a `virtual' leg are necessary to
characterize the system, but have yet to be directly measured experimentally.
To simplify our analysis and to allow for direct comparison with the mechanics
of unperturbed locomotion (Full and Tu,
1990
), we chose to constrain our analysis to movements in the
lateral dimension. In one dimension, the frequency (
) of a spring-mass
system oscillating with a period equal to the step period (
) is:
![]() | (4) |
![]() | (5) |
![]() | (6) |
The 4 g (animals, 2.7 g; apparatus, 1.3 g) cockroaches used in this study moved using a step period of approximately 50 ms (Table 3). We would consequently predict a klat of 16 N m-1 for the weighted animals running at approximately 25 cm s-1.
To describe the mechanism used by cockroaches to recover from brief perturbations and compare it with the lateral spring-like behavior during unperturbed running, we fit a simple viscoelastic model to the kinematics of recovery from perturbations. This description can help to explain the relative contributions of position- and velocity-dependent components to the overall acceleration of the COM and to compare the recovery from perturbations with the compliance observed during unperturbed locomotion.
A `viscoelastic' kinematic model hypothesizes that recovery from
perturbations is due to position- and velocity-dependent acceleration, which
act to arrest the momentum imparted on the animal by the RIP apparatus. The
position-dependent acceleration can be considered as a `spring' and the
velocity-dependent acceleration as a `damper'. In parallel, the spring and
damper acting together is analogous to the Voigt model of viscoelasticity
(Fung, 1993;
Fig. 8A). Both the spring and
damper are here assumed to be linear functions of lateral position.
|
The equation for the Voigt model can be written as:
![]() | (7) |
For each trial, we predicted the acceleration using equation 7, the trial values of k and b and the position and velocity data for that trial. Fig. 8B compares the measured and calculated accelerations for one trial. In this trial, the Voigt model is better able to predict the measured acceleration than position-dependent (spring) or velocity-dependent (damping) components alone. By calculating the proportional root-mean-square error between the predicted and measured acceleration for each trial, we found that, over all trials, spring-dependent behavior was able to predict 44±23% of the total acceleration, damping was able to predict 30±21% of the acceleration and the full Voigt model was able to predict 74±17% of the trial accelerations following perturbations. Errors for the spring and damping models were not significantly different (t-test; P=0.12), but the Voigt model errors were significantly smaller than those for the spring and damping models alone (t-test; P<0.0001).
The mean value of k was 3800±3200 s-2, and the
mean value of b was 26±14 s-1. Taking into account
the mass of the animals, the lateral spring constant observed in response to
perturbations is 15 N m-1, in reasonable agreement with the lateral
spring constant (16 N m-1) predicted for unperturbed,
straight-ahead running calculated from the whole-body mechanics reported in
Full and Tu (1990).
Cockroaches act in a viscoelastic manner in response to brief perturbations.
This may contribute to explaining why simple, self-stabilizing
horizontal-plane spring-mass models can capture many aspects of cockroach
locomotion (Schmitt and Holmes,
2000a
,
b
). However, the ability of a
linear Voigt model to describe recovery from perturbations with errors of 74%
does not exclude the possibility that the stability characteristics of
cockroaches could be better described by more complex or non-linear
viscoelastic models.
This `spring' constant of 15 N m-1 is 5-30 times the `virtual
leg' spring constants used in the LLS model (Schmitt and Holmes,
2000a,
b
). This difference is due to
compression of the LLS `virtual leg' by movement in both the foreaft
and lateral directions. Since cockroaches move approximately 10 mm in the
foreaft direction in one step, dependence on foreaft movement
results in larger `virtual leg' spring compression than would be experienced
by a purely lateral spring, with correspondingly lower spring constants. This
difference is directly comparable with the higher vertical stiffness
(kvert) relative to leg stiffness
(kleg) observed for legged running in the sagittal plane
(Farley et al., 1993
).
Similarities between the spring-like component of the viscoelastic behavior observed during recovery from perturbations and the spring-like behavior observed during unperturbed locomotion lend additional support to the hypothesis that musculoskeletal preflexes contribute to stabilizing rapid locomotion. We hypothesize that the same spring-like properties that confer passive dynamic stability to horizontal-plane models of locomotion (J. Schmitt and P. Holmes, in preparation) contribute to the dynamic stabilization of rapid running.
Limitations
Using the RIP apparatus to perturb cockroaches during running is subject to
several limitations, which should be taken into consideration. First, although
the RIP apparatus was designed to mount to the cockroaches firmly with minimal
changes to the location of the COM, the RIP apparatus increased body weight by
approximately 50%. Increases in body weight have the potential to affect the
mechanics, energetics and control of locomotion
(Chang et al., 2000;
Farley and Taylor, 1991
;
Taylor et al., 1980
). However,
adding mass equivalent to 50% of body weight above the COM does not appear to
change locomotory kinematics substantially. The 10 Hz stride frequency
observed in the cockroaches used in the present study is comparable with the
stride frequency observed in unloaded cockroaches running at 29 cm
s-1 (Full and Tu,
1990
; Ting et al.,
1994
), and addition of load did not cause cockroaches to depart
from the alternating-tripod gait observed during unloaded locomotion. The
foreaft velocity of animals carrying the RIP apparatus fluctuated
around the average velocity by ±12%, in reasonable agreement with the
±8% fluctuations observed in unloaded cockroaches
(Full and Tu, 1990
).
Finally, in a separate study (D. L. Jindrich and R. J. Full, in
preparation), we compared detailed leg kinematics between loaded and unloaded
cockroaches. The anterior extreme positions (AEPs) and posterior extreme
positions (PEPs) of unperturbed cockroaches carrying the RIP apparatus showed
no significant differences from those employed by unloaded cockroaches. The
kinematic similarities between unloaded cockroaches and cockroaches loaded
with approximately 50% of body weight suggest that mounting the RIP apparatus
on cockroaches does not result in qualitative changes in locomotory mechanics.
Simulation studies suggest that, even if locomotory mechanics was altered by
changing body mass or moments of inertia, these changes would be likely to
decrease stability (Full et al.,
2002). We therefore consider the stability observed when using the
RIP apparatus to be a conservative estimate of the performance of these
animals.
Future directions
Recent research advances have demonstrated that the passive dynamics of
musculoskeletal systems, the properties of active muscle and neural control
must all be considered together to understand animal movement
(Dickinson et al., 2000).
Moreover, the dynamics of the entire neuromechanical system must be taken into
account when trying to understand any of its constituents. Neural motor output
must be interpreted by muscles whose response to stimulation depends on many
factors, including intrinsic muscle properties, system dynamics and previous
activity. Muscles do not act solely as power generators, but also as springs,
brakes, struts and, as our findings support, stabilizers.
We hypothesize that musculoskeletal preflexes contribute to stabilizing
rapid locomotion. However, the nervous system remains necessary for
coordinating movements and modulating the mechanical properties of the
musculoskeletal system during locomotion. When subject to large or persistent
perturbations, neurally mediated responses may be required for stability.
Continued research on the dynamic interactions between neural control and
musculoskeletal dynamics will be critical to understanding the exceptional
performance of animals in their environment. Insights gained from
neuromechanical studies of rapidly running insects have provided biological
inspiration towards the design of simple, legged robots
(Altendorfer et al., 2001).
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Acknowledgments |
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References |
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