Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

45, 1, pp. 91-98, Warsaw 2007

ANALYSIS OF STABILITY OF THE HUMAN GAIT

Jerzy Mrozowski

Jan Awrejcewicz

Piotr Bamberski

Department of Automatics and Biomechanics, Technical University of Lodz, Poland

e-mail: j.mrozow@p.lodz.pl; awrejcew@p.lodz.pl

Analysis of the gait stability of a man moving along an even surface
with a constant velocity is presented. The stability criteria applied to
biped robots, namely: the Zero Moment Point (ZMP) and the Ground
projection of the Center Of Mass (GCOM) have been employed in the
investigations. The analysis has been carried out on the basis of me-
asurement data obtained from the human gait recorded with a digital
camera.

Key words: human gait, stability criteria

1. Introduction

The human walking is a complex activity, which consists of four main tasks:
the initiation and termination of a locomotion movement, generation of a
continuous movement, maintenance of the equilibrium, and adaptability to
meet any changes in the environment. The maintenance of the equilibrium is
one of themost important activities for twomain reasons. Firstly, the center of
mass is located in a considerable distance from the support area, and secondly,
for amajor period of thewalking cycle, the body is supported by a single limb
with the center of mass outside the base of the support. In older people, up
to 70% falls occur during locomotion. Differently from previous approaches
dealingwith the balance of the static posture only, this paper presents a study
of stability of the human gait during the walking process. The analysis makes
useof the concept of theZeroMomentPoint (ZMP)and theGroundprojection
of the Center Of Mass (GCOM) previously used to define the control and
stability of biped robots.



92 J. Mrozowski et al.

2. Stability criteria

Both biomechanics and robotics use some criteria needed to estimate the amo-
unt of stability. There are two main types of such criteria, namely:
• static stability criteria (GCOM),

• dynamic stability criteria (ZMP).

Support area

The support area in the single-support phase is approximately equal to the
area of a single foot. In the double-support phase, the area is muchwider and
becomes a convex hull of two-feet areas, as shown in Fig.1.

Fig. 1. Support polygon during single-support phase (a) and double-support
phase (b)

GCOM criterion

The biped/human gait is said to be statically stable and a human posture is
said to be balanced if the gravity line from its center of mass falls within the
convex hull of the foot support area (supportpolygon) (Goswami, 1999). If the
GCOM point is beyond the support polygon, it is equivalent to the presence
of an uncompensated moment acting on the foot, which causes it to rotate
around the point on the polygon boundary.
By the static stability margin onemeans a distance of theGCOM from the

edge of the support polygon,measured along a current vector ofmotion of the
gravity center

xGCOM =

∑
n
i=1Mxi∑
n
i=1Fxi

=

∑
n
i=1mixci∑
n
i=1mi

(2.1)

yGCOM =

∑
n
i=1Myi∑
n
i=1Fyi

=

∑
n
i=1miyci∑
n
i=1mi

where mi is themass of the ith body, xci, yci – location of the center of mass
of the ith body.



Analysis of stability of the human gait 93

ZMP criterion

The Zero Moment Point (ZMP) is a point on the ground where the total
moment generated due to gravity and inertia equals zero (Goswami, 1999;
Vukobratovic et al., 1990). The ZMP of a biped robot must be inside the
convex hull of the foot support area to ensure stability of the walk without
falling down (Goswami, 1999; Vukobratovic et al., 1990). The ZMP is also
known as the Center Of Pressure (COP).
Finally, we can obtain each component of the ZMP in the formof equation

(2.2)

Xzmp =

∑
n
i=1mi(z̈i−gz)xi−

∑
n
i=1mi(ẍi−gx)zi−

∑
n
i=1(Ṫy)i∑

n
i=1mi(z̈i+gz)

(2.2)

Yzmp =

∑
n
i=1mi(z̈i−gz)yi−

∑
n
i=1mi(ÿi−gy)zi+

∑
n
i=1(Ṫx)i∑

n
i=1mi(z̈i+gz)

where (Ṫx)i, (Ṫy)i denote x, y components of time derivatives of the ith body
angular momentum.
The dynamic stability margin is the distance between the ZMP and the

support polygon boundarymeasured along the actual ZMP velocity vector.

3. Investigations

The investigations comprised the following stages:

• assumption of a simplifiedmodel of the human body and its association
with anthropometrical data,

• recording of a trajectory of some selected points on the human body
during walking (with the help of a digital camera),

• generation of an input data file for a program for the gait stability ana-
lysis on the basis of the digital data recorded,

• writing a computer program for evaluation of the gait stability according
to the ZMP andGCOM criteria,

• calculations and visualization of the obtained results.

In Figs. 2a,b, a simplified kinematic model of the human body in the form
of a 6-element kinematical chain is presented. The element that substitutes
the pelvis ismassless. The upper part of the body is represented by element 3,
whosemass is equal to the sum of masses of the head, chest and upper limbs.
The legs are without feet, elements 1, 5 replace shanks, whereas elements 2,



94 J. Mrozowski et al.

4 replace thighs. The mass of elements 1, 5 is equal to the mass of feet and
shanks.The segments aremadeof rodsof a constant density andare connected
bymeans of spherical joints. Each segment is treated as a rodwith the center
of mass located in the geometrical center of the rod. It is assumed that the
center of mass of each element is positioned in the middle of its length.

Fig. 2. Model view in sagittal plane (a) and frontal plane (b)

4. Recording of trajectories of the gait

Two digital cameras positioned perpendicularly with respect to each other
were used for recording the gait trajectories. They traced the path of markers
that contrasted with the surrounding and were placed on the body of the
walking man. The time of shuttle opening was fixed at 1/1000s, the shots
were made at the speed of 25 exposures per second. A small part of the film
showing the central portion of the pathwas employed in the analysis. In order
to minimize the deformations, the object was filmed from a long distance by
means of telephoto lens.

Fig. 3. Arrangement of two-camera system



Analysis of stability of the human gait 95

Conversion of the recorded gait trajectories into a digital data file

Further examination was conducted with a computer program for analysis of
images that allows for spatial reconstruction of the objects recorded with two
cameras at least. The relevant markers on the human body were associated
with the markers in the program. The data concerning the position of the
markers, obtained by automatic tracing of the marked points, were recorded
in a text file.

Program for analysis of the gait stability and data filtration

The program for analysis of human gait trajectories was written by means of
theMATLABcalculation package.This package contains ahigh-level language
interpreter that allows for implementation of numerical algorithms and now
constiutes a standard in scientific and technical computations as well as in
visualization of their results.

In order to calculate accelerations in individual points of the kinematic
chain, the recorded data should be pre-filtered. It is done with low-pass filters
that cut off higher harmonics of the signal. It can also be done by means of
interpolating B-spline cubic functions.

5. Results of calculations

The stability of the human gait can be analyzed in two perpendicular planes:
the sagittal (anterior-posterior) plane and the frontal (medial-lateral) one.The
stability of the human body in one plane only is not sufficient to assume that
the human gait is stable. The recorded walking process (Fig.4) was analyzed
separately in three stages: the standing stage, transitional stage and walking
stage, and in the before mentioned planes.

Fig. 4. Recorded walking process



96 J. Mrozowski et al.

Fig. 5. Stabilogram recorded during walking stage

On the basis of the results obtained from the calculations, it is concluded
that:

• trajectories of the GCOMand ZMP are situated within the convex hull
of the foot support area,

• locations of characteristic points of the trajectories (common points of
the GCOMand ZMP, themaxima and theminima of both trajectories)
are symmetric and evenly distributed,

• the delay between the maximum of the ZMP with respect to the maxi-
mum of the GCOMhas a constant value,

• the amplitude of the body displacement in the sagittal plane is 10 times
higher than displacements recorded in the frontal plane. However, be-
cause themaximumdimension of the supportingpolygon in the anterior-
posterior direction is larger than the dimension of this polygon in the
medial-lateral direction (foot length is much larger than its width), the
stability is maintained.

6. Concluding remarks

Currently there is not any uniform standard in biomechanical laboratories
that is applied in investigations of the human gait stability. Themeasurement
procedure presented in this paper is based on methods that are employed by
designers of biped robots. It is an alternative to the platforms recording the
background reaction forces that have already been used for 50 years.

The investigations conducted and the experience gathered have allowed for
formulation of postulates concerning the modification of the method for the



Analysis of stability of the human gait 97

Fig. 6. Trajectories YGCOM , YZMP – sagittal (anterior-posterior) plane

Fig. 7. Trajectories XGCOM,XZMP – frontal (medial-lateral) plane

gait stability analysis. It seems that connection of two criteria of the stability
evaluation, namely the ZMPand theGCOM,with twoways of recording, that
is to say, bymeans of pressure sensors and an opticalmethod, can significantly
increase the accuracy of measurements, and consequently, the reliability of
estimations made.

References

1. Goswami A., 1999, Postural stability of biped robots and the foot rotation
indicator (FRI) point, International Journal of Robotics Research, 18, 6

2. Jinhyun Kim, Wan Kyun Cung, Youngil Youm, 2002, Real time ZMP
compensation method using null motion for mobile manipulators, Proc. 2002
IEEE, Washington



98 J. Mrozowski et al.

3. Kuczyński M., 2000, Regulacja pozycji pionowej człowieka: od metod oceny
do mechanizmów,Człowiek i ruch, 40, 2,Wrocław

4. Tak S., Oh-Young Song, Hyeong Seok, 2000, Motion Balance filtering,
Eurographics, 19

5. Vukobratovic, Borovac M., Surla B., 1990, Biped Locomotion: Dyna-
mics, Stability, Control and Application, Springer, Berlin

Analiza stabilności chodu człowieka

Streszczenie

Artykuł przedstawia analizę stabilności chodu człowieka poruszającego się po
równej poziomej powierzchni. W badaniach wykorzystano dwa kryteria stateczno-
ści: ZMP(Zero Moment Point) i GCOM (Ground projection of the Center Of Mass)
stosowane w odniesieniu do robotów dwunożnych. Analizę przeprowadzono na pod-
stawie zarejestrowanych za pomocą kamer cyfrowych danych dotyczących trajektorii
wybranych punktów.

Manuscript received August 3, 2006; accepted for print October 18, 2006