Swing leg retraction helps biped walking stability
M. Wisse*, C. G. Atkeson†, D. K. Kloimwieder†
* Delft University of Technology, www.dbl.tudelft.nl, firstname.lastname@example.org
†Carnegie Mellon University, www.cs.cmu.edu/∼cga
Abstract—In human walking, the swing leg moves backward
just prior to ground contact, i.e. the relative angle between
the thighs is decreasing. We hypothesized that this swing leg
retraction may have a positive effect on gait stability, because
similar effects have been reported in passive dynamic walking
models, in running models, and in robot juggling. For this study,
we use a simple inverted pendulum model for the stance leg. The
swing leg is assumed to accurately follow a time-based trajectory.
The model walks down a shallow slope for energy input which
is balanced by the impact losses at heel strike. With this model
we show that a mild retraction speed indeed improves stability,
while gaits without a retraction phase (the swing leg keeps moving
forward) are consistently unstable. By walking with shorter steps
or on a steeper slope, the range of stable retraction speeds
increases, suggesting a better robustness. The conclusions of this
paper are therefore twofold; (1) use a mild swing leg retraction
speed for better stability, and (2) walking faster is easier.
Index Terms—Swing leg trajectory, dynamic walking, biped,
swing leg retraction
In human walking, the swing leg moves forward to maximal
extension and then it moves backward just prior to ground
contact. This backward motion is called ‘swing leg retraction’;
the swing foot stops moving forward relative to the floor
and it may even slightly move backward. In biomechanics
it is generally believed that humans apply this effect (also
called ‘ground speed matching’) in order to reduce heel strike
impacts. However, we believe that there is a different way
in which swing leg retraction can have a positive effect on
stability; a fast step (too much energy) would automatically
lead to a longer step length, resulting in a larger energy loss
at heel strike. And conversely, a slow step (too little energy)
would automatically lead to a shorter step length, resulting in
less heel strike loss. This could be a useful stabilizing effect
for walking robots.
The primary motivation to study swing leg retraction comes
from our previous work on passive dynamic walking , ,
. Passive dynamic walking  robots can demonstrate
stable walking without any actuation or control. Their energy
comes from walking downhill and their stability results from
the natural dynamic pendulum motions of the legs. Interest-
ingly, such walkers possess two equilibrium gaits, a ‘long
period gait’ and a ‘short period gait’ , . The long
period gait has a retraction phase, and this gait is the only
one that can be stable. The short period gait has no swing
leg retraction. This solution is usually dismissed, as it never
provides passively stable gaits.
More motivation stems from work on juggling and running,
two other underactuated dynamic tasks with intermittent con-
tact. The work on juggling  featured a robot that had to hit
a ball which would then ballistically follow a vertical trajectory
up and back down until it was hit again. The research showed
that stable juggling occurs if the robot hand is following a well
chosen trajectory, such that its upward motion is decelerating
when hitting the ball. The stable juggling motion required no
knowledge of the actual position of the ball. We feel that the
motion of the hand and ball is analogous to that of the swing
leg and stance leg, respectively. Also analogous is the work
on a simple point-mass running model . It was shown that
the stability of the model was significantly improved by the
implementation of a retraction phase in the swing leg motion.
It has been suggested  that this effect also appears in
In this paper we investigate the stabilizing influence of the
swing leg retraction speed just prior to heel strike impact. We
use a Poincar´ e map analysis of a simple point-mass model
(Section II). The results are shown in Section III, including a
peculiar asymmetric gait that is more stable than any of the
symmetric solutions. Section IV reports that the results are
also valid for a model with a more realistic mass distribution.
The discussion and conclusion are presented in Sections V and
II. SIMULATION MODEL AND PROCEDURE
The research in this paper is performed with an inverted
pendulum model consisting of two straight and massless legs
(no body) and a single point mass at the hip joint, see Fig. 1.
Straight legged (‘compass gait’) models are widely used as an
approximation for dynamic walking , , , , .
Fig. 1. Our inverse pendulum model, closely related to the ‘Simplest Walking
Model’ of .
A. Stance leg
The stance leg is modeled as a simple inverted pendulum
of length 1 (m) and mass 1 (kg) (Fig. 1). It undergoes
Proceedings of 2005 5th IEEE-RAS International Conference on Humanoid Robots
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Random torque level (Nm)
Retraction speed φ (rad/s)
disturbances. A negative value for˙φ indicates the presence of a retraction
phase. The results are obtained with an optimization algorithm which was
initialized with trajectories with a retraction phase for some runs and without
one for others. Irrespective of the initialization and the level of disturbances,
the optimization always settles into a trajectory with a retraction phase, i.e.˙φ
is always negative just prior to foot contact.
Retraction speed˙φ as a function of the level of random torque
a retraction phase in the swing leg motion. For illustration,
Fig 12 shows the motion of the swing foot with respect to the
floor, as measured with a motion capture system on our most
recent prototype Denise . The measurements (an average of
over 150 steps) show that there exists a clear retraction phase
just prior to heel strike.
-0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1
Marker position for motion capture
foot postion in sagittal plane (m)
in all data
existence of a retraction phase. The graphs shows a measurement of the motion
of Denise’s heel with respect to the floor. We measured over 150 steps from
several trials. The data is time-synchronized using heel strike at the end as
the reference, and the final position after heel strike is defined as 0 meters.
Then we calculated the mean and standard deviation, which are shown in the
Left: our prototype Denise. Right: the graph shows the clear
In this paper we research the effect of swing leg retraction
on gait stability. The conclusions are straightforward:
1) Walk fast; this decreases the sensitivity of the gait to
the swing leg motion just prior to heel contact.
2) Use mild swing leg retraction; by moving the swing
leg rearward just prior to heel contact, one avoids the
highly unstable effects that occur when the swing leg is
still moving forward at heel contact.
This research was supported by US National Science Foun-
dation grants ECS-0325383 and CNS-0224419 and by the
NWO, the Netherlands Organization for Scientific Research.
The authors are grateful to Seiichi Miyakoshi for discussions
about the concept of swing leg retraction. Also thanks to Brian
Moyer at Mark Redfern’s lab of University of Pittsburgh for
the motion capture measurements of robot Denise.
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