Swing-leg retraction: a simple control model for stable running
1 Artificial Intelligence Laboratory, Cambridge, MA 02139, USA
2 Harvard/MIT Division of Health Sciences and Technology, Cambridge, MA 02139,
USA
3 ParaCare Laboratory, Balgrist Hospital, University Zurich, CH-8008 Zurich,
Switzerland
4 Department of Physical Medicine and Rehabilitation, Harvard Medical School,
Spaulding Rehabilitation Hospital, Boston, MA 02114, USA
* Author for correspondence (e-mail: a_seyfarth{at}yahoo.com)
Accepted 22 February 2003
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Summary |
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Key words: biomechanics, legged locomotion, return map, spring-mass model, swing phase
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Introduction |
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Since its formulation the spring-mass model has served as the basis for
theoretical treatments of animal and human running, not only for the study of
running mechanics, but also stability. Kubow and Full
(1999) investigated the
stability of hexapod running in numerical simulation. At a preferred forward
velocity, a pre-defined sinusoidal pattern of each leg's ground reaction force
resulted in stable movement patterns. However, the legs could not be viewed as
entirely spring-like since their force production did not change in response
to disturbances applied to the system. Later Schmitt and Holmes
(2000
) found a lateral
spring-mass stability for hexapod running on a conservative level where total
mechanical energy is constant. However, in this study, they investigated
lateral and not sagittal plane stability in a uniform gravitational field. In
contrast, Seyfarth et al.
(2002
) investigated the
stride-to-stride sagittal plane stability of a spring-mass model. Although the
model is conservative it can distribute its energy into forward and horizontal
directions by selecting different leg angles at touch-down
(Geyer et al., 2002
).
Surprisingly, this partitioning turns out to be assymptotically stable and
predicts human data at moderate running speeds (5 m s-1). However,
model stability cannot be achieved at slow running speeds (≤3 m
s-1). Additionally, at moderate speeds (
5 m s-1), a
high accuracy of the landing angle (±1°) is required, necessitating
precise control of leg orientation.
The purpose of this study is to investigate control strategies that enhance
the stability of the spring-mass model on a conservative level. In the control
scheme of Seyfarth et al.
(2002), the angle with which
the spring-mass model strikes the ground is held constant from
stride-to-stride. In this investigation, we relax this constraint and impose a
swing-leg retraction, a behavior that has been observed in running humans and
animals (Muybridge, 1955
;
Gray, 1968
) in which the
swing-leg is moved rearward towards the ground during late swing-phase. This
controlled limb movement has been shown to reduce foot-velocity with respect
to the ground and, therefore, landing impact
(De Wit et al., 2000
).
Additionally, a biomechanical model for quadrupedal locomotion indicated that
leg retraction could improve stability in quadrupedal running
(Herr, 1998
; Herr and McMahon,
2000
,
2001
;
Herr et al., 2002
). We
hypothesize that swing-leg retraction improves the stability of the
spring-mass model by automatically adjusting the angle with which the model
strikes the ground from one stride to the next. We test this hypothesis by
imposing a constant rate of retraction throughout the second half of the swing
phase. Using a return map analysis on swing-phase apex height
(Seyfarth et al., 2002
), we
compare model stability at zero retraction velocity (constant angle of attack)
to model stability at several non-zero retraction velocities.
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Materials and methods |
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In this investigation, the effect of swing-leg retraction on the stability
of the spring-mass model is investigated. Here the orientation of the leg is
not held fixed during the swing phase, but is now considered a function of
time (t). For simplicity, we assume a linear relationship
between leg angle (measured with respect to the horizontal) and time, starting
at the apex tAPEX with an initial leg angle
R (retraction angle)
(Fig. 1):
![]() | (1a) |
![]() | (1b) |
|
Stability analysis
To evaluate the stability of potential movement trajectories, we use a
return map analysis. For legged locomotion, a return map relates the system
state at a characteristic event or moment within a gait cycle to the system
state at the same event or moment one period later. To keep the analysis as
simple as possible, we select the swing-phase apex height as the
characteristic event. At this point, the system state (x, y,
vx, vy)APEX is
uniquely identified by one variable, the apex height
yAPEX. Here, x and y are the
horizontal and vertical positions, and vx and
vy are the horizontal and vertical velocities of the
model's point mass. The system state is uniquely defined by the apex height
due to (1) the vanishing vertical velocity
vy,APEX=0 at this point, (2) the fact that
x has no influence on future periodic behavior, and (3) the
conservative nature of the spring-mass system in which total mechanical energy
is held constant.
The return map investigates how this apex height changes from step to step,
or more precisely, from one apex height (index `i') to the next one
(index `i+1') in the following flight phase (after one contact
phase). For a stable movement pattern, two conditions must be fulfilled within
this framework: (1) there must be a periodic solution (Equation 2a, called a
fixed point where is the steady
state apex height), and (2) deviations from this solution must diminish
step-by-step (Equation 2b, or an asymptotically stable fixed point).
![]() | (2a) |
![]() | (2b) |
The requirements for stable running can be checked graphically by plotting
a selected return map (e.g. for a given retraction angle R
and a given retraction velocity
R) within the
(yi, yi+1) plane and searching for
stable fixed points fulfilling both conditions defined by Equations 2a and 2b.
The first condition (Equation 2a, periodic solutions) requires that there is a
solution (i.e. a single point) of the return map
yi+1(yi) located at the diagonal
(yi+1=yi). The second condition
(Equation 2b, asymptotic stability) demands that the slope
(dyi+1/dyi) of the return map
yi+1(yi) at the periodic solution
(intersection with the diagonal) is neither steeper than 1 (higher than
45°) nor steeper than 1 (smaller than 45°).
As a consequence of the imposed leg retraction, the return map of the apex
height yi+1(yi) is determined by two
mechanisms: the control of the angle of attack
0(yi) before landing (leg retraction)
and the dynamics of the spring-mass model resulting in the next apex height
yi+1(
0, yi).
According to the definition of leg retraction (Equation 1), the analytical
relationship between the apex height yAPEX and
the landing angle of attack
0 is:
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Numerical procedure
The running model is implemented in Simulink (Mathworks) using a built-in
variable time step integrator (ode113) with a relative tolerance of
1e12. For a human-like model (point mass m=80 kg, leg length
l0=1 m) at different horizontal speeds
vx (initial conditions at apex y0,APEX
are vx,APEX=vx and
vy,APEX=0), the leg parameters
(kLEG, R,
R) for stable running are identified by scanning the
parameter space and measuring the number of successful steps. The stability of
potential solutions is evaluated using the return map
yi+1(yi) of the apex height
yAPEX of two subsequent flight phases
(i and i+1). For a given system energy E, all
possible apex heights 0≤y0,APEX ≤E/(mg) are
taken into account. For instance, for a system energy E corresponding
to an initial horizontal velocity vx=5 m s-1 at
an apex height y0,APEX=1 m, apex heights between 0 and
2.27 m are taken into account. To keep the system energy constant, the
horizontal velocity at apex v0,APEX=vx
is adjusted according to the selected apex height y0,APEX
using the equation
mgy0,APEX+m/2(v0,APEX)2=E.
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Results |
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With the swing-leg retraction control, the rotational leg velocity before
landing (retraction speed R) leads to a step-to-step
adjustment of the angle of attack
0, which gradually
converges to a final steady state angle
(dotted line in Fig. 2C). Since
the leg has a fixed angular velocity during the second half of the flight
phase, the chosen initial apex height (y0,APEX=1.25 m)
leads to a steeper landing angle compared to the steady state angle
. Consequently, the first contact phase
is asymmetric with respect to the vertical axis
(Fig. 2A,C) and therefore, the
next apex height is lower than the previous apex height. Due to the shorter
flight phase, the second angle of attack is clearly flatter (a smaller angle
of attack). Finally, the system stabilizes at the steady state angle
with a corresponding apex height
.
With leg retraction, steady-state running is achieved within approximately 2 steps, whereas the system without retraction needs approximately 8 steps (Fig. 2A). This indicates that leg retraction can improve the attraction of stable limit cycles in running.
Stability analysis for running
The influence of leg retraction on the return map of the apex height is
shown in Fig. 3. With increased
retraction speed (R=25 and 50 deg s-1) the
solutions of yi+1(yi) for different
retraction angles
R become more horizontally aligned. As a
consequence, disturbances in apex height are compensated for more rapidly
(paths indicated by the arrows in Fig.
3).Furthermore, the attraction range in
yAPEX for the stable fixed points is largely
increased (maximum increase in yAPEX:
35
cm for
R=0,
90 cm for
R=25 deg
s-1, and
120 cm for
R=50 deg s-1.
See dotted lines in Fig.
3).
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In the case of leg retraction, the control of the angle of attack
0 is shifted into a control of the retraction angle
R. For zero retraction speed (
R=0) the
retraction angle
R becomes identical to the angle of attack
0 (
R=
0,
Fig. 3A), i.e. the leg angle is
adjusted at apex height and does not change until ground contact. With
increasing retraction speed
R, the range of retraction
angles resulting in stable running is enlarged (2.6° for
R=0; 7.2° for
R=25 deg s-1;
14.6° for
R=50 deg s-1).
Running at low speeds
Spring-mass running with a fixed angle of attack is characterized by a
minimum speed required for stability (Seyfarth, 2002). In
Fig. 4, a running speed
(vX=3 m s-1) close to this minimum speed is
selected. At the given leg stiffness (kLEG=20
kN m-1) no stable fixed point exists without retraction
(Fig. 4A). Employing the leg
retraction control, stable fixed points emerge in the return map. Similar to
the finding in Fig. 3, an
increased retraction speed R leads to (1) an enlarged range
of attraction in yAPEX, (2) a faster
convergence to the stable fixed point (fewer steps), and (3) an increased
range of successful retraction angles
R for stable
running.
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Robustness with respect to leg stiffness kLEG
Spring-mass running requires a proper adjustment of leg stiffness to the
chosen angle of attack (Blickhan,
1989; McMahon and Cheng,
1990
; Herr and McMahon,
2000
,
2001
; Seyfarth, 2002).
However, even at zero retraction speed (
R=0), a range of leg
stiffness can fulfill periodic running at a given angle of attack
0 (Seyfarth, 2002). To test the robustness of spring-mass
running with respect to variations in leg stiffness, we estimate the maximum
and minimum stiffness change that could be tolerated by the system. A
stiffness change is applied during steady state running, starting from an
initial leg stiffness of 20 kN m-1
(Fig. 5A). For these numerical
experiments, the mean angles of attack (
R=67.6°,
64.4°, 60.0° in Fig.
5A,C,E) with respect to the range of all
R with
stable fixed points in Fig.
3AC are used. After the first three steps in steady state
running, leg stiffness is permanently shifted. Without retraction, variations
in leg stiffness within 18.2 and 22.4 kN m-1 are tolerated
(Fig. 5A) even without any
stride-to-stride adaptations in the angle of attack
(Fig. 5B).
|
By introducing leg retraction (Fig.
5C, R=25 deg s-1;
Fig. 5E,
R=50
deg s-1), the range of tolerated stiffness is largely increased
(16-28.8 kN m-1 for
R=25 deg s-1;
13.9-62 kN m-1 for
R=50 deg s-1).
These results show that the rotational velocity of the leg
R
inherently adapting the angle of attack
0 allows for large
variations in leg stiffness (Fig.
5D,F).
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Discussion |
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Swing-leg retraction approximates the natural angle of attack
In terms of the return map of the apex height, we can ask for an `optimal'
control strategy by imposing the constraint
yi+1(yi)=yCONTROL=constant.
Within one step this return map projects all possible initial apex heights
yi to the desired apex height
yi+1=yCONTROL.
As a consequence of the dynamics of the spring-mass system, the apex height
yi+1 is merely determined by the preceding apex height
yi and the selected angle of attack 0.
This dependency yi+1(yi,
0) can be understood as a `fingerprint of spring-like leg
operation' and is represented as a surface in
Fig. 6A. When applying any
control strategy
0(yi), this generalized
surface yi+1(yi,
0)
can be used to derive the corresponding return maps.
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For example, in the case of a `fixed angle of attack' (no retraction:
0(yi)=
R=constant) the
surface has to be scanned at lines of constant angles
0
(Fig. 6A, e.g. red line:
0=68°). These lines are projected to the left
(yi+1,yi) plane in
Fig. 6A and match the return
map in Fig. 3A.
Let us now consider the `optimal control strategy for stable running'
0(yi) fulfilling
yi+1(yi)=yCONTROL=constant.
Using the identified fingerprint, this simply requires us to search for
isolines of constant yi+1 on the generalized surface
yi+1(yi,
0), as
indicated by the green lines in Fig.
6A (yi+1=1, 1.5 and 2 m). The projection of
these isolines onto the (
0, yi) plane
represents the desired `natural' control strategy
0(yi) for spring-mass running as
depicted for yCONTROL=1, 1.5, 2 m in
Fig. 6B.
The constant-velocity leg retraction model put forward in this paper
represents a particular control strategy
0(yi) relating the angle of attack
0 to the apex height yi of the preceding
flight phase (Equation 3), as shown in Fig.
6B for different retraction speeds (
=0, 25, 50, 75 deg
s-1) and one retraction angle (
R=60°). It
turns out that this particular leg retraction model can approximate the
natural control strategy within a considerable range of apex heights if the
proper retraction parameters (
R,
R) are
selected. The value of the retraction angle
R shifts the
line of the retraction control
0(yi)
along the
0 axis, whereas the retraction speed
R determines the slope of the control line. Thus, the
retraction parameters have different qualities with respect to the control of
running; if the retraction speed
R guarantees the stability
(setting the range and the strength of attraction to a fixed point), then the
retraction angle
R selects the apex height of the
corresponding fixed point yCONTROL. Due to this
adaptability, a constant velocity leg retraction model, as evaluated in this
paper, can significantly enhance the stability of running compared to the
fixed angle control model described by Seyfarth et al.
(2002
).
Influence of speed on the stability of spring-mass running
The return maps in Figs 3
and 4 indicate that the
generalized surface yi+1(yi,
0) is a function of the forward running speed. The selected
retraction speeds in Figs 3 and
4 (
R=0, 25,
50 deg s-1) show that the slope of the return map
yi+1(yi) generally increases with (1)
decreasing running speed and (2) decreasing retraction speed
R. As a consequence, running at 3 m s-1 is not
stable using a fixed angle of attack (
R=0 in
Fig. 4A), but is stable using
non-zero retraction speeds (
R=25 and 50 deg s-1
in Fig. 4B and C,
respectively). Hence, even at slow forward running speeds (≤3 m
s-1), there exists a natural control strategy represented by the
isolines of the corresponding generalized surface with
yi+1(yi,
0)=constant. In comparison with the fixed angle of attack
control, leg retraction at constant velocity approximates this natural control
(Fig. 6B). Thus, a constant
velocity retraction is a successful strategy to stabilize running below the
critical forward running speed where stable running is not achievable using a
fixed angle control.
The fact that the spring-mass model, with retraction, is stable at slow
forward running speeds seems critical. Clearly, for a running model to be
viewed as a plausible biological representation, the model should be stable
across the full range of biological running speeds. Without swing-leg
retraction, the spring-mass model could not be stabilized at slow biological
running speeds (3 m s-1 for m=80 kg,
l0=1 m, kLEG=20 kN
m-1; Fig. 4A), but
with retraction, the spring-mass model could readily be stabilized
(Fig. 4B,C).
Swing-leg retraction in human running: preliminary experimental
results
A treadmill (Woodway, Germany) was equipped with an obstacle-machine
designed to disturb swing-phase dynamics during human running. The
obstacle-machine consisted of a cylindrical-shaped bar (2.5 cm diameter, 40 cm
length) passing from the left to the right side of the treadmill walkway (the
bar's long axis is generally perpendicular to the direction of the moving
treadmill surface). Every 9-16 s, the bar moved towards the human runner at a
speed equivalent to the treadmill surface, forcing the runner to change his
swing-phase kinematics to avoid the obstacle. The movement of the obstacle bar
was triggered by the ground reaction force F. For each experiment,
the bar was positioned 12 cm above the moving treadmill surface.
Using this apparatus, we conducted experiments on five male subjects (body
mass 79.6±5.9 kg, age 30.6±3.2 yrs)performing treadmill running
at 3 m s-1. We measured leg kinematics (leg angle, leg length)
during both the stance and swing phases. Leg angle and length
lLEG at the onset of swing-leg retraction and
at touch-down were used to characterise the kinematic leg control prior to
landing. The retraction velocity
R was estimated as the mean
angular velocity within the last 20 ms before touch-down. Furthermore, the leg
stiffness kLEG was approximated using the
maximum vertical ground reaction force FMAX and
the maximum leg compression
lMAX=max(l0l)
during stance phase with
kLEG=FMAX/
lMAX.
For undisturbed running, we found surprisingly uniform leg kinematics
during both the stance and swing phases (shown for one subject in
Fig. 7). In contrast, when
passing over the obstacle, swing-leg kinematics were altered significantly,
but stance period dynamics immediately following the obstacle were largely
unaffected. Swing-leg retraction was observed in undisturbed running with an
angular range equal to SHIFT=4.5±0.9°
(Table 1). During this period
of swing-leg retraction, only a minor change in leg length was observed
(lSHIFT=1±0.5 cm), supporting one of the
assumptions of our control model.
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We observed a significant re-adjustment of leg retraction in response to
the disturbance. Both the retraction angular range
SHIFT
(
SHIFT=4.7±3.0°,
P<0.05, paired t-test) and the retraction velocity
R (
R=22±13 deg
s-1, P<0.05) increased in response to the disturbance.
Here, the change in the angular range
SHIFT was primarily the result of a
decreased retraction angle
R
(
R=-3.0±2.5°, P=0.057) rather
than an increased angle of attack
0
(
0=1.7±2.1°, P=0.15). In
contrast, no significant change was observed in leg stiffness
(
kLEG=2.3±4.4 kN
m-1, P=0.31) or in leg length adjustment
(
lSHIFT=0±1.0 cm, P=1)
during the stance period immediately following the disturbance.
These results indicate that leg retraction is employed in human running and is even enhanced when an obstacle disturbance is applied. The data presented here support the hypothesis of the model, namely, that swing-leg retraction is a strategy used in running to select an angle of attack that sustains a desired movement pattern.
Alternative biological strategies to stabilize running
The analysis reveals that the stability of spring-mass running is highly
sensitive to the angular velocity of the leg before landing. Although
swing-leg retraction seems an important stabilizing mechanism, we cannot
ignore the importance of alternative strategies that might also be crucial for
stable running. For instance, researchers have shown that visual feedback
plays an important role in obstacle avoidance and, therefore, in stabilizing
the movement trajectory. Warren et al.
(1986) investigated regulatory
mechanisms to secure proper footing using visual perception in human running.
In their investigation, subjects ran on a treadmill across irregularly spaced
foot-targets in order to effectively modulate step length and the vertical leg
impulse during stance. Although their results suggest that vision is important
for running stability, they do not specifically address the issue of how
mechanical or neuro-muscular mechanisms may contribute when running over
ground surfaces without footing constraints.
Intrinsic or `preflex' leg stabilizing mechanisms may also be important for
running stabilization. It is well established that the intrinsic properties of
muscle leads to immediate responses to length and particularly velocity
perturbations (Humphrey and Reed,
1983; Brown et al.,
1995
). In an analytical study, Wagner and Blickhan
(1999
) showed that a
self-stabilizing oscillatory leg operation emerges if well-established muscle
properties are adopted.
Furthermore, by modeling the dynamics of the muscle-reflex system, stable,
spring-like leg operations can be achieved in numerical simulations of hopping
tasks if positive feedback of the muscle force sensory signals (simulated
Golgi organs) are employed (Geyer et al.,
in press). These results suggest that during cyclic locomotory
tasks such as walking or running, the body could counteract disturbances even
during a single stance period.
Future work
Here we argue that swing-leg retraction is one of many stabilizing
strategies used in biological running. Our research suggests that both the
control of stance leg dynamics and swing-leg movement patterns may be
critically important for overall running stability in humans and animals.
Leg retraction is a feedforward control scheme, and therefore, can neither avoid obstacles nor place the foot at desired foot-targets. Rather, the scheme provides a mechanical `background stability' that may relax the control effort for locomotory tasks. It remains for future research to understand to what extent environmental sensory information might allow for varied kinematic trajectories and an increase in the stabilizing effects of swing-leg retraction. Future investigations will also be necessary to fully understand the impact of late swing-leg retraction on running stability. To gain insight into the control scheme employed by running animals, we wish to compare the natural retraction control formulated in this paper to the actual limb movements of running animals. Furthermore, since the spring-mass model of this paper is two-dimensional, we wish to generalize retraction to three dimensions to address issues of body yaw and roll stability. And finally, we hope to test optimized retraction control schemes on legged robots to enhance their robustness to internal (leg stiffness variations) and external disturbances (ground surface irregularities).
Conclusion
In this paper we show that swing-leg retraction can improve the stability
of spring-mass running. With retraction, the spring-mass model is stable
across the full range of biological running speeds and can overcome larger
disturbances in the angle of attack and leg stiffness. In the stabilization of
running humans and animals, we believe both stance-leg dynamics and swing-leg
rotational movements are important control features.
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Acknowledgments |
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References |
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