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First published online September 5, 2008
Journal of Experimental Biology 211, 2989-3000 (2008)
Published by The Company of Biologists 2008
doi: 10.1242/jeb.014357
Running on uneven ground: leg adjustment to vertical steps and self-stability
1 Friedrich-Schiller-Universität, Institut für Sportwissenschaft,
Lehrstuhl für Bewegungswissenschaft, Seidelstraße 20, D-07749 Jena,
Germany
2 Eberhard-Karls-Universität, Institut für Sportwissenschaft,
Arbeitsbereich III, Wilhelmstraße 124, D-72074 Tübingen,
Germany
* Author for correspondence (e-mail: sten.grimmer{at}uni-jena.de)
Accepted 11 June 2008
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Summary |
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 TOP Summary INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXReferences |
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Key words: biomechanics, human locomotion, spring–mass model, leg stiffness, self-stability
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INTRODUCTION |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXReferences |
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;
Blickhan et al., 2003;
Blickhan et al., 2007).
A simple approach to describe the basic mechanics of human locomotion is
the spring–mass model (Blickhan,
1989). Since its introduction it has been used in several research
studies of hopping and running (e.g.
Farley et al., 1991;
Blickhan and Full, 1993;
Full and Koditschek, 1999;
Seyfarth et al., 2002;
Geyer et al., 2006;
Seyfarth et al., 2006) and in
recent investigations of walking (Geyer et
al., 2006). The model consists of a mass-less spring and the body
represented by a point mass and is simply described by the parameters
stiffness k, angle of attack 
0 and leg length
l0 (Blickhan,
1989; McMahon and Cheng,
1990; Geyer,
2005). Stiffness was identified as the key parameter to describe
the dynamics of running (Ferris et al.,
1998). In several investigations the model was used to describe
the dependence of stiffness on speed
(McMahon and Cheng, 1990;
Farley et al., 1993) and
hopping or stride frequency (Farley et
al., 1991; Farley and
Gonzalez, 1996). Other experiments demonstrated an adjustment of
leg stiffness to ground stiffness in hopping and running
(Ferris and Farley, 1997;
Farley et al., 1998;
Ferris et al., 1998;
Kerdok et al., 2002;
Lindstedt, 2003;
Moritz and Farley, 2004). All
these studies on the interaction of the spring–mass model with ground
compliance are based on two springs in series: one representing the leg and
the other the ground (Alexander,
1997). The main result is that total stiffness (inverted sum of
compliances) is rather constant and leg stiffness adapts to the ground
stiffness by adjusting the leg compression
(Ferris and Farley, 1997;
Ferris et al., 1998). This leg
behaviour might also be a possible strategy for varying ground levels but has
not yet been reported in humans.
In an experimental observation on birds running over a track with an
unexpected drop, it was shown that an adaptation of the angle of attack
explains most of the variation in stance-phase dynamics
(Daley and Biewener, 2006;
Daley et al., 2006). Although
stiffness varied dramatically it is not clear how this leg stiffness
adjustment contributes to match the varying surface.
Recently, simulations and analytical calculations have provided evidence
for self-stable operation of the spring–mass model for a limited range
of combinations of the angle of attack and leg stiffness
(Seyfarth et al., 2002;
Geyer et al., 2005). Under
these conditions the system is able to cope with uncertainties, such as uneven
ground, without adjusting system properties, i.e. without directly sensing the
disturbance. Self-stability was formulated as the underlying concept
(Ringrose, 1997b;
Wagner and Blickhan, 1999;
Full et al., 2002;
Blickhan et al., 2003;
Blickhan et al., 2007). Here,
neural feedback is not necessary. However, the mechanical ability depends on a
well-adjusted angle of attack and leg stiffness
(Seyfarth et al., 2002). Thus,
on uneven terrain, using self-stability might require changing leg stiffness
and angle of attack.
In part, this change can automatically be achieved by late swing leg
retraction (Seyfarth et al.,
2003). This rotational behaviour of the swing leg changes the
angle of attack depending on the flight time (the longer the flight phase, the
steeper the angle of attack). Leg retraction greatly enlarges the range of leg
stiffness and angle of attack that the model can tolerate. In simulations, by
using leg retraction, changes in ground level of 50% of the initial leg spring
length can be managed (Seyfarth et al.,
2006). Although swing-leg retraction is an important stabilizing
mechanism, it has limitations at higher speeds and cannot increase stability
by increasing the rotational velocity
(Seyfarth et al., 2003;
Seyfarth et al., 2006). Here,
alternative strategies might be crucial for stable running. However, for both
`running with fixed angle of attack' and `running with leg retraction', the
ability to cope with uneven terrain and the evidence of self-stability has
been reported.
Recently, in a commentary, Biewener and Daley suggested integrative
biomechanical approaches, i.e. a combination of modelling and experimental
techniques, to understand locomotion over a variety of conditions
(Biewener and Daley, 2007).
Such approaches need to investigate the elegant interplay of intrinsic
mechanisms (e.g. self-stability) to achieve stability and strategies of
changing parameters in an appropriate way to the environment to achieve
agility.
In this investigation, we focused on leg adjustments on uneven ground and how they are linked to the concept of self-stability. We addressed two main research questions. (1) Do humans use a control strategy mainly explained by a stiffness adjustment process? (2) Does the adjustment utilize the self-stability of the underlying mechanical system and, thus, is the adjustment of the angle of attack and a proper stiffness as predicted by the spring–mass simulations the key feature for stable running on varying ground?
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MATERIALS AND METHODS |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXReferences |
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The experiment on uneven ground
Running track and setup
A 17 m long track was instrumented with three force plates. Two smaller
ones (9282BA, Kistler, Winterthur, Switzerland) were mounted side by side and
displaced 20 cm (Fig. 1A). A
larger one (9285C, Kistler) was added at a distance of one step. Both force
plates could be hit with step lengths from 1.60 to 2.50 m. In addition, the
second plate was adjustable in height (Fig.
1B). We used the plate as an obstacle on the track.
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We used a twelve camera 3D motion capture system (240 Hz, 658 pixelsx496 pixels; Qualisys, Gothenburg, Sweden) to measure the kinematics. A rough estimate of the spatial resolution was made by determining the noise level of the coordinates of a static marker. We found a resolution of about ±0.3 mm (about factor 10 subpixel resolution). The subject's body was marked (marker size 18 mm) on the hips, knees, ankles and balls of the feet (Fig. 2) as well as on the head and vertebra T1. The head and T1 markers were only used to obtain running speed, which was provided to the subjects.
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). Subjects were informed about the risks of tripping and falling due to the experimental design. They provided written informed consent prior to their participation. The investigation was approved by the ethics review board of the University of Jena and the experiments were conducted in accordance with the Declaration of Helsinki.
Experimental protocol
After introducing the subjects to the track we started a running sequence
without any perturbation. This procedure enabled us to check all measurement
systems, gave a good warm up, and trained the subjects to maintain their
velocity. A subject began running in the pre-force plate zone, then came into
contact with the ground-level force plate (first contact). In the stride that
followed the first force plate, the subject contacted a variable height force
plate (second contact). Afterwards, the subject continued running into the
post-force plate zone, ending the trial. At the end of each trial the subject
was instructed to decelerate, turn around and slowly jog back to the beginning
of the running track. While jogging back, the investigator made a quick
decision based on the horizontal velocity of the T1 marker (available online)
to delete trials with obvious acceleration between the penultimate step before
the first force plate and the first step after the second force plate.
The experimental investigation started with subjects running on the flat track. Afterwards, we prepared the uneven track. We then began taking measurements from trials on the uneven track with the 5, 10 and 15 cm perturbation. The subjects were visually aware of the whole running track and were allowed to get familiar with each height of the perturbation through one or two initial trials. In each of the four track conditions the subject had to accomplish at least 15 trials in a row.
Data processing
From the collected data, we filtered out all those trials of each subject
that were distributed in a preferably narrow range from their preferred
running speed over all track types. The selection was realized in three
stages. First, we calculated the mean of the horizontal velocity of the T1
marker for each force plate. If these two values differed by more than 5%
within one trial then this trial was discarded. Next, we calculated a mean
overall remaining speed for both force plates and all track types for each
subject. In the last stage, we further discarded all those trials where speed
differed by more than 5% from this overall subject mean. After these selection
steps we had on average 10 trials per experimental condition and subject
(minimum three, maximum 15 trials).
The raw kinematic data were preprocessed to guarantee constant segment
lengths (Günther et al.,
2003) and filtered with a fourth-order low-pass Butterworth filter
(Winter, 2005) at 20 Hz
cut-off frequency.
The distance between the hip and ball of the foot marker is defined as leg
length lleg of the stance leg. The leg stiffness
kleg was calculated as the ratio between the peak ground
reaction force Fmax and the maximum leg compression
lleg,max=lleg,TD–min.(lleg,TD:TO)
(where TD is touch-down and TO is take-off)
(McMahon and Cheng, 1990;
Seyfarth, 2000):
 | (1) |
;
Geyer, 2005). The leg force
was normalized to subject mass m and gravity constant g:
 | (2) |
 | (3) |
 | (4) |
 | (5) |
To compare the different track conditions with the three heights of the
force plate (i=1,2,3) and the 11 subjects, we also normalized each
parameter (
,
,
)
to a subject-specific reference mean value. This mean value was extracted from
normal running without perturbations (track type i=0).
Statistics
We present the results of the investigation as descriptive statistics
(parameter values in tables with means ± s.d.) over all subjects and
parameters. An inferential statistic is only used for the mean-normalized
global parameters (stiffness, force and compression) to show the significant
change within the flight phase of the consecutive contacts. For this purpose,
we used a Friedman test for paired observations
(Friedman, 1937;
Friedman, 1939;
Siegel and Castellan, 1988).
The statistics were processed with the statistical toolbox of Matlab (release
14, Mathworks Inc., Natick, MA, USA).
Model and simulation
Model
Our simulation consisted of a conservative spring–mass model. The
elastic stance phase was modelled with a point mass attached to a mass-less
linear spring (Blickhan, 1989)
and the flight phase was a simple ballistic behaviour of the point mass with a
reset of the leg angle to the angle of attack at TD
(Seyfarth et al., 2002). We
simulated two consecutive contacts in which in the second contact we varied
the spring stiffness and angle of attack as well as the ground height.
Simulation parameter setup and analysis
The initial conditions for the first contact were about average values of
the subjects: horizontal component of the initial velocity
x,0=4.5 m s–1, initial
apex height y0=0.95 m, body mass m=80 kg, initial
leg length l0=1 m, dimensionless stiffness
=35.7 (k=28 kN
m–1), and a fixed angle of attack of TD=68
deg., which is also typical for level running
(Seyfarth et al., 2002).
In the second contact we introduced a disturbance with a ground height of
+15 cm (equal to experimental track type i=3). Here, to cover the
experimental values, we mapped the peak spring force Fmax
and maximum spring compression
for a
range of angles of attack (TD=50:75 deg.) and spring
stiffnesses (=5:50).
Simulation tools
The model was built in Simulink, Matlab R14 (Mathworks Inc.) with a
variable step-size integrator (ode45) and a relative tolerance of 1e–12.
As part of the simulation the system energy was checked and remained constant
over the simulation time (relation energy fluctuation <1e–10).
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RESULTS |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXReferences |
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Normalizing leg stiffness, leg compression and peak force to the individual
undisturbed mean values (Fig.
5, Table 3), we
identified an increase in leg stiffness of about +9% in the first contact and
a decrease of about –26% for the highest step height (track type
i=3) in the second contact. The peak GRF values varied
correspondingly. For maximum leg compression the mean values differed between
±4% in the first contact and between ±8% in the second contact
of the undisturbed mean value. The mean value of maximum leg compression for
undisturbed running in absolute units was about 9 cm
(l0=1.01±0.035 m). Our observed maximum
deviation from the reference compression was about 8%
(Table 3,
lleg,max,i=1/lleg,max,i=0=0.92).
This equals a fluctuation of leg compression of less than 1 cm during running
over a perturbation of 15 cm.
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Looking at the stride-to-stride analysis, we found that the stiffness of two subsequent contacts was equal on a track where the height of the force plate was 5 cm. Differences were found between subsequent contacts in the other track conditions with 10 and 15 cm height (i=2 and i=3, Table 3). The peak GRF values show a significant difference for all analysed track types. The maximum leg compression values remained constant except for the trials with the smallest disturbance (5 cm, track type 1).
For the angle of attack TD (i.e. leg angle at the
beginning of contact) we also found adaptations to the different track types
(Table 1). The angle of attack
varied in the first contact by about ±1 deg. and diminished in the
second from 68 to 62 deg. with the increasing vertical height of the
perturbation.
Fig. 6 presents the relationship of leg stiffness and angle of attack for a simulation based on the spring–mass model for running. The black J-shaped region represents combinations of angle of attack and leg stiffness with stable solutions of periodic movement patterns. While parameter combinations left of the black area still allow at least one more contact, not a single step is possible at the right side (simulation stops). The circles (first contact) and squares (second contact: track types 1–3) illustrate measured combinations for a typical subject with identical initial conditions used in the simulation. For both contacts the combinations of angle of attack and stiffness fall in areas where on level ground usually more than five steps are possible without further adjustment (Fig. 6). Interestingly, all experimentally measured combinations are located on the left side of the J-shaped stability boundary. However, only some combinations fall into the self-stable black area. Furthermore, a wider range of combinations of angle of attack (from 66 to 55 deg.) and leg stiffness (from 35 to 15) in the second contact is applied. Interestingly, two separate distributions of the two contacts can be identified. In the second contact flatter angles of attack as well as smaller leg stiffness values are prevalent.
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TD, ),
the peak spring force and maximum spring compression result in unique values.
In our case, the peak spring force was approximately
Fmax=3.5 bw and the maximum spring compression
=0.1. For the second contact, we first demonstrate the effect that in principle emerges from single parameter variation (see `Stiffness variation' and `Angle of attack variation' below). Here, we simply compared each of the three different values of spring stiffness and angle of attack estimated from the experimental results. Second, we systematically present the peak spring force and maximum spring compression for an expanded range of spring stiffness and angle of attack (see `Stiffness variation and angle of attack variation' below).
Stiffness variation
By using a fixed angle of attack of 61 deg. and decreasing the
dimensionless spring stiffness from 25.5 to 19.1 to 12.7, the peak spring
force decreased too (Fig. 7A).
At the same time, the maximum spring compression increased
(Fig. 7C). Furthermore, maximum
spring compression in the second contact was always larger than in the first
contact.
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Angle of attack variation
The variation in angle of attack compared with the variation in spring
stiffness showed similar behaviour in the GRFs but different behaviour in
spring compression. Here, by increasing the angle of attack from 59 to 61 to
63 deg. and using a constant dimensionless spring stiffness of 19.1, both the
GRF and the spring compression decreased
(Fig. 7B,D).
Stiffness and angle of attack variation
As shown in Fig. 8, by
varying both parameters (angle of attack and stiffness), the simulation
provided spring forces of up to 10 bw and maximum spring compressions of up to
0.5 times initial leg length. It was found that the maximum leg compression
decreased with increasing spring stiffness and a steepening angle of attack
(Fig. 8B). Likewise, the peak
spring force decreased with steepening angle of attack. In contrast to maximum
leg compression, a diminishing stiffness was needed for a decrease in peak
spring force (Fig. 8A).
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TD=63.2 deg.
Further, it is remarkable that a small area of simulated parameter
combinations of stiffness and angle of attack correspond with the
experimentally measured peak leg force of 1.5 to 3 bw and maximum leg
compression of 0.1 to 0.15 times initial leg length
(Fig. 8, arrows). |
DISCUSSION |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXReferences |
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).
Their subjects increased vertical velocity in preparation for the obstacle
height. As a result they elevated the hip position. Accordingly, we found that
the vertical TO velocity of the hip was increased (0 cm: 0.8±0.2 m
s–1; 15 cm: 1.1±0.2 m s–1) with the
subsequent height of the disturbance (Table
4, y,TO). Another possible
effect is provoked by the longer flight phase caused by the uneven ground
before hitting the first force plate. Higher landing velocities lead to higher
GRFs (Cavanagh and Lafortune,
1980; Blickhan,
1989). We found that the vertical landing velocity of the hip was
marginally increased (–0.6 vs –0.7 m
s–1) in the uneven track situations as it was on level ground
(Table 4,
y,TD). Yet, we do not know which effect
dominates and the extent to which each effect corresponds to the increase in
GRF.
In contrast to the first contact, our measured adaptations in the disturbed
second contact are more striking and systematic. With increasing vertical
height of the perturbation, leg force was decreased proportionally
(Fig. 3D). This effect can be
attributed to a lower vertical landing velocity (0 cm: –0.6±0.2 m
s–1; 15 cm: –0.1±0.1 m s–1;
Table 4,
y,TD) caused by a reduced distance between
the apex of the flight phase and the landing height at touch-down. The leg
showed a tendency to be shorter at touch-down
(Table 1,
lTD) but also to have a decreasing minimum length during
contact at higher steps (Fig.
3D). As a result, leg compression remained constant. This effect
may also rest on the initial conditions at touch-down and their effects on the
muscle (Blickhan et al., 2007).
According to the lever gear ratio of the two-segment leg
(Wagner and Blickhan, 1999),
also known as effective mechanical advantage
(Biewener, 1989;
Biewener et al., 2004), the
load lever of the knee joint at touch-down increases when the knee is more
flexed. The knee extensors are more elongated during contact and the effective
mechanical advantage is reduced. In the case of nearly constant muscle force,
the resulting leg force decreases. It is conceivable that both of these
effects (reduced leg force and vertical landing velocity) may lead to an
almost constant leg compression. Therefore, stiffness of the leg could be
altered without sensory feedback as a response to the altered loading
condition (landing velocity) and due to the changed working range of the
muscles [force–length and force–velocity curve (e.g.
Brown et al., 1996)].
Leg stiffness adjustment is well known in the case of running and hopping
on ground varying with respect to compliance
(Alexander, 1997;
Ferris and Farley, 1997;
Farley et al., 1998;
Ferris et al., 1998;
Ferris et al., 1999;
Kerdok et al., 2002;
Lindstedt, 2003). Ferris and
colleagues reported in the case of stiffened ground that the leg response is
characterized by a higher compliance, i.e. a lower stiffness (Ferris et al.,
1997; Ferris et al., 1998).
There, in contrast to our findings, the stiffer the ground the more the leg
compression was increased, whereas the peak force was found to be rather
constant. Therefore, the strategy on compliant ground is the direct opposite
to that on uneven ground with vertical steps up.
Stiffness and angle of attack adjustment are explained by a simple spring–mass simulation
In spring–mass running, the simplest strategy is running with a fixed
angle of attack and constant leg stiffness
(Seyfarth et al., 2002). Here,
with constant leg stiffness and subject-specific initial conditions,
simulations revealed periodic movement patterns as well as an ability to cope
with uncertainties such as rough terrain
(Geyer et al., 2002). As
mentioned earlier in our experiments, leg stiffness was altered. So, from a
first point of view, the strategy of maintaining stiffness and angle of attack
does not seem to apply. This is in contrast to the findings of Seyfarth and
colleagues (Seyfarth et al.,
2002). However, it has also been reported that there exists a
range of solutions of leg stiffness adjusted to the angle of attack. By
selecting one combination from the associated J-shaped area
(Fig. 6), a spring–mass
system can deliver self-stable cyclic movement
(Seyfarth et al., 2002;
Geyer et al., 2006). Yet,
switching within this area can be read as a possible control strategy
resulting in adapted leg stiffness due to an altered angle of attack. We found
that angle of attack was altered. The higher the encountered step, the flatter
the touch-down angle (Table 1).
This would be expected for the case where the runner performs flight-phase
retraction (Seyfarth et al.,
2003). Flight-phase retraction automatically shifts the leg angle
during flight, dependent on the flight-phase duration
(de Wit et al., 2000;
Günther and Blickhan,
2002; Daley and Biewener,
2006; Daley et al.,
2006) without any control. Running on uneven ground with a step up
shortens the flight phase and decreases the angle of attack. Assuming that the
change in leg stiffness is induced by the retracting leg being less extended
at touch-down (see above), retraction can be identified as a key feature for
shifting leg stiffness within the self-stable area.
As shown in Fig. 6, our
experimental results fit these assumptions. As well as the first unaltered
contact, the combinations also fit the altered disturbed second contact. Yet a
shift from combinations of first to second contact in relation to the
theoretical prediction is seen. However, not all experimental results fit to
the area of self-stability. Most combinations allow at least five subsequent
steps. This indicates that subjects chose not self-stable combinations but
rather combinations that would allow for immediate stability, knowing that
other mechanisms for stable control would engage in later stages of
locomotion. The retraction control scheme may be one such type of control
mechanism (Seyfarth et al.,
2003; Seyfarth et al.,
2006).
Force and compression can be predicted by a simple spring–mass simulation
As we found that leg stiffness and angle of attack are altered, we now
examine the possible explanations for describing the dependency of the
resulting parameters (peak force and maximum compression) within the
spring–mass model. We found that a simple spring–mass simulation
can produce the same force and compression parameters in the case of a 15 cm
perturbation (area marked with arrows in
Fig. 8) as those measured.
Furthermore, in simulations we found that for this region (arrows in Fig. 8), angles of attack between 61 and 63.2 deg. were in accordance with the experimentally observed angle of attack of 62±2.9 deg. The 63.2 deg. border in the simulation is due to the fact that the apex height is not high enough to match the landing height. By using a steeper angle of attack, the simulation stops because of stumbling. It is clear that changing the leg length at touch-down as observed in our experiment makes steeper angles of attack possible.
From the simulation results seen in Fig.
8, we derive two main outcomes of an alteration in stiffness.
First, stiffness adaptation results in a nearly constant leg force and
alteration of leg compression, which is found on compliant ground (e.g.
Ferris et al., 1998;
Ferris et al., 1999;
Kerdok et al., 2002). Second,
an altered leg force and constant leg compression can be caused by alterations
of stiffness. This is generally what we found for running on steps of
different heights. These two cases are limit cases for two different
constraints (experimental conditions) perhaps due to the same additional
movement criterion (smooth ride, see below). Deviations from these limit
cases, i.e. a mixture of altered leg compression and leg force, can occur
individually and are within the range of stable solutions predicted by the
model. These deviations are caused by respective leg stiffness (experimental
s.d.: ±6 bw l0–1) and angle of
attack (experimental s.d.: ±2.5 deg.) variations.
The conservative spring–mass model and real energy losses
In contrast to the underlying spring–mass model, the legs of human
runners show non-conservative work-loops
(Fig. 3A,B,
Table 4). In order to assess
the consequences of net work balances on the validity of the spring–mass
model, we estimated the influence of energy dissipation on the eigenfrequency
of the model.
Geyer and colleagues provided an analytical solution for the equation of
motion of the axial oscillation of a spring–mass model for running
(Geyer et al., 2005).
Therefore, two further approximations are needed: (i) leg compression is
clearly smaller than leg length
(l/l0
)
1). According to this model approach, the eigenfrequency of
the conservative spring–mass model is
where 
is the
eigenfrequency of the harmonic oscillator made of the linear leg stiffness
k (axial spring) and the body mass m, and
is the contribution of the angular momentum of the body mass pivoting on the
contact point of the spring. In humans,
=16.9
rad s–1 dominates 
SM=18.7 rad
s–1 as p=(4.5 m s–1/1
m)=4.5 rad s–1. Thus, we find that the angular momentum
biases the oscillatory behaviour of bouncing locomotor dynamics (represented
by SM) by about 10% [(18.7–16.9)/16.90.1].
The exact expression of the eigenfrequency
d(0,Eleg/Eleg)
of the damped harmonic oscillator (axial spring) in terms of its typical
energy content
and net work balance:
 | (6) |
Eleg/Eleg|=|–0.32±0.36|
(Table 4), we find that
0 itself is biased by less than 1% for
|Eleg/Eleg|=0.5
and by 3.5% for an extreme case
|Eleg/Eleg|=0.9.
Consequently, theory tells us that quasi-elastic leg operation is maintained
even in the face of the measured net work balances.
Self-stability and control
Our approach is also meant to verify whether the stiffness adaptations
found may be an indicator of self-stability in human running. To address this,
we simulated the self-stable movement of a mechanical model running across
perturbations that matched the experimental conditions. A comparison of
experimental and simulated adaptations answers the question whether running
across uneven ground might be based on self-stability, i.e. in principle
possible without any control that leads to a change in the system parameters.
However, humans have sensors (e.g. muscle spindles, visual system) that
definitely come into play. Due to the definition of `self-stability' this
seems to be an ostensible paradox. Therefore, we will first review the
emergence of the term `self-stability' in the literature. Then, we will try to
reconcile the self-stability and the control approach.
Self-stability
To our knowledge the term `self-stability' was introduced by Ringrose
(Ringrose, 1997a;
Ringrose, 1997b) in the field
of robotics: `Running motions can be self-stabilizing. That is, with proper
design the structure and motion of a robot can automatically cause it to
recover from minor disturbances even if it cannot detect them.' Recently,
Blickhan and colleagues suggested how to identify self-stability in biological
movement systems and to re-transfer these findings back to engineering
(Blickhan et al., 2007).
Before, these authors had integrated the term `self-stability' into the
framework of biomechanics (Blickhan et al.,
2003). Accordingly, a biomechanical movement is called
`self-stable' if movement stability is gained by any flow of signals
(`sensing' mechanical state variables) being exclusively coupled to forces. In
dynamics, applying forces necessarily generates a flow of mechanical energy
per time (power).
In other words, the forces acting on and within the movement system must
exclusively depend on the mechanical state variables
and the mechanical parameters
(passive mechanics and actuators) {pm} of the movement
system. Specific rheonomic constraints [`predetermined patterns'
(Ringrose, 1997b)]
(t,
{pc}), i.e. mechanical state variables, external or
internal forces, muscle activations or stimulations exclusively and explicitly
varying with time, can contribute to self-stable movement. Thus, the equations
of motion have the form:
 |
.
denotes the first time derivative of the state and the parameters
{pc} define the rheonomic constraints as an explicit
function of time. In order to decide whether a parameter is of mechanical or
non-mechanical (e.g. sensory feedback gain) character, one has to (i)
mathematically formulate a model of the movement system and (ii) suggest a
re-implementation of the model in the real world (e.g. prosthesis or robot).
According to this definition, dissipative and thermodynamically open systems
driven by internal or external energy sources may be as self-stable as closed
conservative systems.
For example, a movement generated by a `predetermined pattern' acting as
input to a musculo-skeletal system under gravity
(Ringrose, 1997b;
Aoi and Tsuchiya, 2007) might
be as self-stable as spring–mass walking
(Geyer et al., 2006) or
running (Seyfarth et al.,
2002; Ghigliazza et al.,
2005; Owaki and Ishiguro,
2007). Other examples of self-stable mechanisms are: a purely
passive robot steadily moving down a slope under gravity plus roll friction
(McGeer, 1990a;
McGeer, 1990b), insect
locomotion in the horizontal plane
(Schmitt and Holmes, 2000a;
Schmitt and Holmes, 2000b;
Schmitt and Holmes, 2001),
oscillations induced by a fixed muscle stimulation program
(Wagner and Blickhan, 1999;
Wagner and Blickhan, 2003),
robot juggling (Schaal et al.,
1996), and somersault locomotion
(Mombaur et al., 2005).
To represent a `predetermined pattern', a `central pattern generator' (CPG)
must fulfil the rheonomic requirement. This statement corresponds exactly to
Full and Koditschek (Full and Koditschek,
1999): `The Kubow and Full
(1999) dynamic, cockroach
model prescribes leg forces using a feedforward clock analogous to a central
pattern generator with no equivalent of neural feedback among any of the
components... The model self-stabilizes.' At first view, the sketched feedback
arrow that the authors present in their illustration 3 (connecting sensors to
CPG) seems to contradict their own CPG definition. Taking a closer look, the
arrow might be interpreted as the option to transmit discrete bits of
information (e.g. the event `reset the clock' or, more generally, resetting a
state variable) rather than to represent an intermittent or continuous flow of
signals being used to change system parameters dependent on an error signal
(feedback).
Note that the definition of `self-stability' does not interfere with the specific approach (e.g. using `Lyapunov-stability' or `return-maps', or finding `basins of attraction') chosen to quantify dynamic stability.
Reconciling self-stability and control
If all subsystems (e.g. muscles, neurons, joints, limbs) of a biological
movement system [an `anchor' (Full and
Koditschek, 1999)] cooperate in order to generate behaviour
represented by a model [a `template' (Full
and Koditschek, 1999)], the system benefits if self-stability is
an inherent property of the model. With self-stability, the movement system
gains energetic efficiency with increasing contributions of elastic parts. The
system also gains control efficiency (reduced signal and information flow, in
engineering the latter being equivalent to reduced `bandwidth') as, according
to the definition of self-stability, the system in principle would not need
any continuous sensor signal flow of state variables, other than from
mechanical dynamics, to generate the specific movement.
During stance phase the runner benefits from self-stability because his
take-off conditions provide excellent touch-down conditions for the following
contact phase. Neural signals may contribute to self-stability if they provide
event detection [processed signals, i.e. information about loss of ground
contact, `reset the angle' (Ghigliazza et
al., 2005)]. In contrast, continuous flow of neural signals by
definition would not contribute to self-stability. However, it contributes to
parameter tuning potentially supporting underlying self-stability
(Wagner and Giesl, 2006). For
example, non-linear characteristics on the joint level linearize the leg
force–length relationship (Seyfarth
et al., 1999; Günther and
Blickhan, 2002). These non-linearities may come from leg segment
geometry (Günther et al.,
2004), tendon properties
(Maganaris and Paul, 1999;
Ker, 2007) or
muscle–tendon complex properties
(Seyfarth et al., 1999). On
the other hand, neuro-muscular control mechanisms may linearize the muscle
force–length relationship (Hoffer
and Andreassen, 1978; Greene
and McMahon, 1979; Hoffer and
Andreassen, 1981; Maganaris,
2003).
Feedback can also be used to stabilize a movement system with an increased
number of degrees of freedom still profiting from self-stability of a lower
dimensional subsystem (Jindrich and Full,
2002; Seipel and Holmes,
2005; Seipel and Holmes,
2006). Furthermore, feedback can increase the robustness of a
mechanical system against parameter variations and/or perturbations
(Geyer et al., 2003;
Schmitt and Holmes, 2003;
Seipel and Holmes, 2007).
The simple spring–mass model is robust against small variations in
angle of attack and very robust when leg retraction is applied
(Seyfarth et al., 2003;
Seyfarth et al., 2006). As
mentioned above, the measured angle of attack accordingly decreases when step
height increases, i.e. leg retraction is unaltered. Following Seyfarth
(Seyfarth, 2002;
Seyfarth, 2003), stiffness
could be kept constant, whereas in the experiment, stiffness was reduced with
increasing step height. It seems that during flight phase anticipatory
influences become effective. The runner predicts the perturbation and changes
leg stiffness. This is feed-forward control. Feed-forward control requires
prediction based on a model, whether implicit or explicit (A. J. Ijspeert,
personal communication). For running, we suggest that the self-stabilizing
spring–mass model is suitable. Then, if leg stiffness is adjusted to the
expected angle of attack within the predicted stiffness-angle area of
stability (J-shape, Fig. 6),
the feed-forward control leads to a new self-stabilizing touch-down condition.
As even the simplest spring–mass model allows for manifold self-stable
solutions, the additional application of control (change of stiffness) can
serve multiple movement goals. Control reduces the number of self-stable
solutions. Though the cause of stiffness adaptation on uneven and compliant
ground might be different, the movement target is the same. Besides stability,
runners prefer a smooth ride of the centre of mass trajectory
(Ferris et al., 1999).
Blickhan and colleagues discussed a step-wise response to a disturbance and
suggested that strong accelerations of the head following an abrupt alteration
of the centre of mass are energetically detrimental
(Blickhan et al., 2003).
By generating biological movements in accordance with the self-stable
solutions of less complex movement systems, increasingly complex systems
inherit the underlying stability. To achieve this, control strategies and
structures have to be adjusted to the self-stable design of the less complex
system (e.g. Cham et al.,
2004).
|
APPENDIX |
|---|
|
TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXReferences |
|---|
 | (A1) |
 | (A2) |
the damping coefficient, m the
mass, k the stiffness, l the displacement, in terms
of energy loss:
 | (A3) |
/d and the undamped
eigenfrequency:
 | (A4) |
E<0 for a damped system (d>0).
The solution of the dynamic equation of motion is:
 | (A5) |
the phase. To
determine the relative energy loss E/E during one
period we assume the typical energy content of the oscillator to be estimated
by the initial kinetic energy
(in the specific case 
=0) or, equivalently, by the maximum stored
elastic energy
.
Therefore, using Eqn A5 the
left integral in Eqn A3 solves
to:
 |
 | (14) |
 | (A6) |
Taking only the first order term within the Taylor expansion of the
ln-function in Eqn A6 into
account results in:
 | (A7) |
E/E.
From Eqn A7 we find the
often-used quality factor
Q=–2(E/E)=d/2
and the linear damping coefficient
d=–(md/2)(E/E)=–(m/Td)(E/E).
LIST OF ABBREVIATIONS
Eleg
Eleg/Eleg
max
leg
leg
lleg
leg
lleg,max
leg,max
max
leg,max
x, y
x,0
|
Acknowledgments |
|---|
|
References |
|---|
|
TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXReferences |
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). Stable running requires a proper
adjustment of angle of attack to spring stiffness
(Seyfarth et al., 2002
