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nce on some external interactions (Xiang et al., 2010).
One of the well known models was proposed by Gar-
cia et al. (Garcia et al., 1998) based on the passive
dynamic theory of McGeer (Garcia et al., 1998): a
double articulated pendulum system for which feet
are relatively small with respect to the trunk and the
heelstrike follows a very restrictive rule. This gait de-
scription is still limited because about a 30 % of the
gait cycle that corresponds to a double stance phase,
is completely eliminated and the CoG displacement
in the lateral axis (Y ) is not considered at all. Other
approaches included the inverted pendulum represen-
tation to describe the lateral and frontal CoG motion,
but not its vertical displacement, which is assumed
constant (Komura et al., 2004).
Other recent approaches are based on optimization
techniques and control-based models. The optimiza-
tion methods include a large number of degrees-of-
freedom for producing optimal motions while joint
force profiles remain subjected to a large number of
constraints(Delp et al., 2007; Fregly, 2008; Xiang
et al., 2010). These methods require relatively few
data to simulate simple human structures and pre-
dict new motions, very useful in computer graph-
ics, robotics and animation applications.The control-
based models have been commonly used on robotics
and biomechanics for designing the real-time control
in biped walking prototipes (Trifonov and Hashimoto,
2008). The main advantage of these methods is that
they approximate the actual human control systems,
allowing to simulate both normal and pathological
gaits. The neuromotor system simulation allows the
analysis of some neurological pathologies (Komura
et al., 2004), while this is computationally more ef-
ficient than the optimization-based models. Never-
theless these methods are computationally expensive
and require specific knowlegde of the problem (Xiang
et al., 2010) whereby these strategies are highly sub-
jective(Xiang et al., 2010), and also require a large
group of experimental data to generate natural mo-
tions. This last drawback has highly limited its appli-
cation in clinical gait analysis because of the specific
requirements to obtain a stable and natural motion.
3 MATERIALS AND METHODS
The gait model herein proposed fully describes a
3D CoG trajectory of normal and pathological gaits.
The whole model is built upon simple mechanical
relationships, approximating the 3D CoG displace-
ment with a double-inverted pendulum for the sim-
ple stance phase, and a double spring-mass system
for the double stance. Using this physics-based repre-
sentation, we can animate an human leg structure and
simulate normal and pathological kinematic patterns.
The kinematic patterns for each case was calculated
with a classic method of inverse-kinematic, using the
CoG trajectory obtained and a learned heel trajectory
taken from some gait laboratory’s data as illustrated
in figure 1.
3.1 Sagital CoG Description
As a first step, the CoG sagittal trajectory was com-
puted using a physics-based gait representation. The
single stance phase (one foot supporting the body)
was represented as a double-inverted pendulum. This
representation is based on the passive dynamic (Gar-
cia et al., 1998). This model is formulated as a pair of
coupled nonlinear equations: β(1 −cos φ)(3
¨
θ −
¨
φ) −
βsinφ(
˙
φ
2
−2θ
˙
φ)+(
gsinθ
l
)(β(sin(θ −φ)−1)) = 0 and
¨
θ(β(1−cosφ))−β
¨
φ+β
˙
θ
2
sinφ +(
βg
l
)sin(θ −φ) = 0,
where β = m/M , m is the mass of each foot and
M is the body mass, θ is the angle of the support-
ing leg at a particular time t and φ is the angle be-
tween both legs. When the leg stance has been ac-
complished, the heelstrike is represented by the non
linear ruleφ(t) −2θ(t) = 0. This phase is character-
ized by the hip and knee moments generated within
this interval, important biomarkers in many abnormal
movements.
On the other hand, the double stance phase, which
starts just after the heel strike, was represented as a
double spring-mass system. This physical formula-
tion introduces attenuation as it is usually observed
in an actual CoG trajectory because of the knee rota-
tion, but it also involves an intrinsic representation of
the muscles acting during this phase. This represen-
tation is given by the following equations (Blickhan,
1989): m ¨x = Px −Q(d −x) and m ¨y = Py + Qy −mg,
where P = k(
l
0
√
x
2
+y
2
−1), Q = k(
l
0
√
(d−x)
2
+y
2
−1), m
is the body mass, x and y are the sagittal and frontal
displacement respectively, d in the step length, k is
the spring constant, l
0
is the spring lengh at rest and g
is the gravity constant (9.81). This physic-based rep-
resentation describes completely the CoG trajectory
and allows to have the flexibility needed to generate
different kind of gaits, i.e., pathological gaits.
3.2 Coronal CoG Trajectory
A main contribution of this work is a 3D gait repre-
sentation that allows to accurately mimic pathological
motions like the pelvic balancing (Garcia et al., 1998).
The CoG trajectory in the coronal view was simu-
lated using the inverted-pendulum approach. First, we
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