## SUMMARY

In running, the spring-like axial behavior of stance limbs is a well-known and remarkably general feature. Here we consider how the rotational behavior of limbs affects running stability. It is commonly observed that running animals retract their limbs just prior to ground contact, moving each foot rearward towards the ground. In this study, we employ a conservative spring-mass model to test the effects of swing-leg retraction on running stability. A feed-forward control scheme is applied where the swing-leg is retracted at constant angular velocity throughout the second half of the swing phase. The control scheme allows the spring-mass system to automatically adapt the angle of attack in response to disturbances in forward speed and stance-limb stiffness. Using a return map to investigate system stability, we propose an optimal swing-leg retraction model for the stabilization of flight phase apex height. The results of this study indicate that swing-leg retraction significantly improves the stability of spring-mass running, suggesting that swing-phase limb dynamics may play an important role in the stabilization of running animals.

## Introduction

In running, kinetic and potential energy removed from the body during the first half of a running step is transiently stored as elastic strain energy and later released during the second half by elastic recoil. The mechanism of elastic recoil was first proposed in 1964, when Cavagna and collaborators noticed that the forward kinetic energy of the body's center of mass is in phase with fluctuations in gravitational potential energy (Cavagna et al., 1964). They hypothesized that humans and animals most likely store elastic strain energy in muscle, tendon, ligament and perhaps even bone to reduce fluctuations in total mechanical energy. Motivated by these energetic data, Blickhan (1989) and McMahon and Cheng (1990) proposed a simple model to describe the stance period of symmetric running gaits: a point mass attached to a massless, linear spring. Using animal data to select the initial conditions at first ground contact, they demonstrated that the spring-mass model can predict important features of stance period dynamics (Blickhan, 1989; McMahon and Cheng, 1990).

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.

## Materials and methods

### Spring-mass running with leg retraction

Running is characterized by a sequence of contact and flight phases. For the contact phase of symmetric running gaits, researchers have described the dynamics of the center of mass with a spring-mass model comprising a point mass attached to a massless, linear leg spring (Blickhan, 1989; McMahon and Cheng, 1990). To describe the dynamics of the flight phase, a ballistic representation of the body's center of mass has been used (McMahon and Cheng, 1990). In their investigation of the stability of spring-mass running, Seyfarth et al. (2002) assumed that the leg spring strikes the ground at a fixed angle with respect to the ground.

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 α(*t*). For simplicity, we assume a linear relationship
between leg angle (measured with respect to the horizontal) and time, starting
at the apex *t*_{apex} with an initial leg angleα
_{R} (retraction angle)
(Fig. 1):
1a
1b
where ω_{R} is a constant angular leg velocity (retraction
speed).

### 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,
v*_{x}, *v*_{y})_{apex} is
uniquely identified by one variable, the apex height
*y*_{apex}. Here, *x* and *y* are the
horizontal and vertical positions, and *v*_{x} and
*v*_{y} 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
*v*_{y,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
where
2b
For simplicity, the subscript apex in *y*_{i+1} and
*y*_{i} has been removed.

The requirements for stable running can be checked graphically by plotting
a selected return map (e.g. for a given retraction angle α_{R}
and a given retraction velocity ω_{R}) within the
(*y*_{i}, *y*_{i+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
*y*_{i+1}(*y*_{i}) located at the diagonal
(*y*_{i+1}=*y*_{i}). The second condition
(Equation 2b, asymptotic stability) demands that the slope
(d*y*_{i+1}/d*y*_{i}) of the return map
*y*_{i+1}(*y*_{i}) 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 *y*_{i+1}(*y*_{i}) is determined by two
mechanisms: the control of the angle of attackα
_{0}(*y*_{i}) before landing (leg retraction)
and the dynamics of the spring-mass model resulting in the next apex height
*y*_{i+1}(α_{0}, *y*_{i}).
According to the definition of leg retraction (Equation 1), the analytical
relationship between the apex height *y*_{apex} and
the landing angle of attack α_{0} is:
3
where *l*_{0} denotes the leg length at touch-down and
*g* is the vertical component of the gravitational acceleration. Merely
one branch of the quadratic function in α_{0} has to be
considered as retraction holds only for times
*t*≥*t*_{apex} according to Equation 1
(either α_{0}>α_{R} orα
_{0}<α_{R}, depending on the sign ofω
_{R}). This allows us to derive the control strategyα
_{0}(*y*_{apex}).

### Numerical procedure

The running model is implemented in Simulink (Mathworks) using a built-in
variable time step integrator (ode113) with a relative tolerance of
1e–12. For a human-like model (point mass *m*=80 kg, leg length
*l*_{0}=1 m) at different horizontal speeds
*v*_{x} (initial conditions at apex *y*0,apex
are *v*x,apex=*v*_{x} and
*v*y,apex=0), the leg parameters
(*k*_{leg}, α_{R},ω
_{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
y_{i+1}(*y*_{i}) of the apex height
*y*_{apex} of two subsequent flight phases
(*i* and *i*+1). For a given system energy *E*, all
possible apex heights 0≤*y*0,apex ≤*E*/(mg) are
taken into account. For instance, for a system energy *E* corresponding
to an initial horizontal velocity *v*_{x}=5 m s^{-1} at
an apex height *y*0,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 *v*0,apex=*v*_{x}
is adjusted according to the selected apex height *y*0,apex
using the equation
*mgy*0,apex+*m*/2(*v*0,apex)^{2}=*E*.

## Results

### Can leg retraction stabilize spring-mass running?

The kinematics of the spring-mass model are evaluated using (1) a fixed
angle of attack α_{0} and (2) the swing-leg retraction strategy
(Equation 1). The results are shown in Fig.
2. Starting at an initial apex height of 1.25 m, both control
strategies stabilize to a final limit cycle. Spring-mass running with a fixed
angle of attack α_{0} is stable if (1) the leg stiffness
*k*_{leg} and the angle of attackα
_{0} are both properly adjusted to the chosen running speed and
(2) the initial vertical position *y*0,apex is within the
range of attraction for the corresponding stable fixed point. (For more
information on spring-mass running using a fixed angle of attack, see
Seyfarth et al., 2002).

With the swing-leg retraction control, the rotational leg velocity before
landing (retraction speed ω_{R}) leads to a step-to-step
adjustment of the angle of attack α_{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 (*y*0,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 (ω_{R}=25 and 50 deg s^{-1}) the
solutions of *y*_{i+1}(*y*_{i}) for different
retraction angles α_{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
*y*_{apex} for the stable fixed points is largely
increased (maximum increase in *y*_{apex}: ∼35
cm for ω_{R}=0, ∼90 cm for ω_{R}=25 deg
s^{-1}, and ∼120 cm for ω_{R}=50 deg s^{-1}.
See dotted lines in Fig.
3).

In the case of leg retraction, the control of the angle of attackα
_{0} is shifted into a control of the retraction angleα
_{R}. For zero retraction speed (ω_{R}=0) the
retraction angle α_{R} becomes identical to the angle of attackα
_{0} (α_{R}=α_{0},
Fig. 3A), i.e. the leg angle is
adjusted at apex height and does not change until ground contact. With
increasing retraction speed ω_{R}, the range of retraction
angles resulting in stable running is enlarged (2.6° forω
_{R}=0; 7.2° for ω_{R}=25 deg s^{-1};
14.6° for ω_{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
(*v*_{X}=3 m s^{-1}) close to this minimum speed is
selected. At the given leg stiffness (*k*_{leg}=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 ω_{R} leads to (1) an enlarged range
of attraction in *y*_{apex}, (2) a faster
convergence to the stable fixed point (fewer steps), and (3) an increased
range of successful retraction angles α_{R} for stable
running.

### Robustness with respect to leg stiffness k_{LEG}

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 (ω_{R}=0), a range of leg
stiffness can fulfill periodic running at a given angle of attackα
_{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 (α_{R}=67.6°,
64.4°, 60.0° in Fig.
5A,C,E) with respect to the range of all α_{R} with
stable fixed points in Fig.
3A–C 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, ω_{R}=25 deg s^{-1};
Fig. 5E, ω_{R}=50
deg s^{-1}), the range of tolerated stiffness is largely increased
(16-28.8 kN m^{-1} for ω_{R}=25 deg s^{-1};
13.9-62 kN m^{-1} for ω_{R}=50 deg s^{-1}).
These results show that the rotational velocity of the leg ω_{R}
inherently adapting the angle of attack α_{0} allows for large
variations in leg stiffness (Fig.
5D,F).

## Discussion

Late swing-phase retraction has been observed in running animals of
different leg number and body size
(Muybridge, 1955;
Gray, 1968). Although swing-leg
retraction seems to be a general feature in biological running, few
researchers (De Wit et al.,
2000; Herr and McMahon,
2000,
2001;
Herr et al., 2002) have
studied the behavior and, consequently, its purpose is not fully understood.
In this investigation, we show that leg retraction is a simple strategy to
improve the stability of spring-mass running. By imposing a uniform retraction
velocity, we demonstrate that the stability of the spring-mass model is
increased with respect to variations in forward speed, leg angle (retraction
angle α_{R}) and leg stiffness
*k*_{leg}.

### 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
*y*_{i+1}(*y*_{i})=*y*_{control}=constant.
Within one step this return map projects all possible initial apex heights
*y*_{i} to the desired apex height
*y*_{i+1}=*y*_{control}.

As a consequence of the dynamics of the spring-mass system, the apex height
*y*_{i+1} is merely determined by the preceding apex height
*y*_{i} and the selected angle of attack α_{0}.
This dependency *y*_{i+1}(*y*_{i},α
_{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 α_{0}(*y*_{i}), this generalized
surface *y*_{i+1}(*y*_{i}, α_{0})
can be used to derive the corresponding return maps.

For example, in the case of a `fixed angle of attack' (no retraction:α
_{0}(*y*_{i})=α_{R}=constant) the
surface has to be scanned at lines of constant angles α_{0}
(Fig. 6A, e.g. red line:α
_{0}=68°). These lines are projected to the left
(y_{i+1},*y*_{i}) plane in
Fig. 6A and match the return
map in Fig. 3A.

Let us now consider the `optimal control strategy for stable running'α
_{0}(*y*_{i}) fulfilling
*y*_{i+1}(*y*_{i})=*y*_{control}=constant.
Using the identified fingerprint, this simply requires us to search for
isolines of constant *y*_{i+1} on the generalized surface
*y*_{i+1}(*y*_{i}, α_{0}), as
indicated by the green lines in Fig.
6A (*y*_{i+1}=1, 1.5 and 2 m). The projection of
these isolines onto the (α_{0}, *y*_{i}) plane
represents the desired `natural' control strategyα
_{0}(*y*_{i}) for spring-mass running as
depicted for *y*_{control}=1, 1.5, 2 m in
Fig. 6B.

The constant-velocity leg retraction model put forward in this paper
represents a particular control strategyα
_{0}(*y*_{i}) relating the angle of attackα
_{0} to the apex height *y*_{i} of the preceding
flight phase (Equation 3), as shown in Fig.
6B for different retraction speeds (ω =0, 25, 50, 75 deg
s^{-1}) and one retraction angle (α_{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 (α_{R}, ω_{R}) are
selected. The value of the retraction angle α_{R} shifts the
line of the retraction control α_{0}(*y*_{i})
along the α_{0} axis, whereas the retraction speedω
_{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 ω_{R} guarantees the stability
(setting the range and the strength of attraction to a fixed point), then the
retraction angle α_{R} selects the apex height of the
corresponding fixed point *y*_{control}. 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 *y*_{i+1}(*y*_{i},α
_{0}) is a function of the forward running speed. The selected
retraction speeds in Figs 3 and
4 (ω_{R}=0, 25,
50 deg s^{-1}) show that the slope of the return map
*y*_{i+1}(*y*_{i}) generally increases with (1)
decreasing running speed and (2) decreasing retraction speedω
_{R}. As a consequence, running at 3 m s^{-1} is not
stable using a fixed angle of attack (ω_{R}=0 in
Fig. 4A), but is stable using
non-zero retraction speeds (ω_{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
*y*_{i+1}(*y*_{i},α
_{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,
*l*_{0}=1 m, *k*_{leg}=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 α and length
*l*_{leg} 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 ω_{R} was estimated as the mean
angular velocity within the last 20 ms before touch-down. Furthermore, the leg
stiffness *k*_{leg} was approximated using the
maximum vertical ground reaction force *F*_{max} and
the maximum leg compressionΔ
*l*_{max}=max(*l*_{0}–*l*)
during stance phase with
*k*_{leg}=*F*_{max}/Δ*l*_{max}.

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 α_{shift}=4.5±0.9°
(Table 1). During this period
of swing-leg retraction, only a minor change in leg length was observed
(*l*_{shift}=1±0.5 cm), supporting one of the
assumptions of our control model.

We observed a significant re-adjustment of leg retraction in response to
the disturbance. Both the retraction angular rangeα
_{shift}
(Δα_{shift}=4.7±3.0°,
*P*<0.05, paired *t*-test) and the retraction velocityω
_{R} (Δω_{R}=22±13 deg
s^{-1}, *P*<0.05) increased in response to the disturbance.
Here, the change in the angular rangeΔα
_{shift} was primarily the result of a
decreased retraction angle α_{R}
(Δα_{R}=-3.0±2.5°, *P*=0.057) rather
than an increased angle of attack α_{0}
(Δα_{0}=1.7±2.1°, *P*=0.15). In
contrast, no significant change was observed in leg stiffness
(Δ*k*_{leg}=–2.3±4.4 kN
m^{-1}, *P*=0.31) or in leg length adjustment
(Δ*l*_{shift}=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.

- List of symbols
- E
- system energy
- F
- vertical ground reaction force
- g
- vertical component of the gravitational acceleration
- i
- index
*k*_{leg}- leg stiffness
- l
- leg length
- m
- mass
- t
- time
- v
- velocity
- x
- horizontal position
- y
- vertical position
- αR
- retraction angle
- ωR
- retraction speed

## ACKNOWLEDGEMENTS

This research was supported by an Emmy-Noether grant of the German Science Foundation (DFG) to A.S. (SE 1042/1) and a grant of the German Academic Exchange Service (DAAD) `Hochschulsonderprogramm III von Bund und Länder' to H.G. We also thank the Michael and Helen Schaffer Foundation of Boston, Massachusetts for their generous support of this research.

- © The Company of Biologists Limited 2003