Engineering, 2012, 5, 20-26
doi:10.4236/eng.2012.410B006 Published Online October 2012 (http://www.SciRP.org/journal/eng)
Towards a Methodology for the Differential Analysis in
Human Locomotion: A Pilot Study on t he Participation of
Individuals with Multiple Sclerosis
Huiying Y u, Thompson Sar kodie-Gyan
Department of Electri cal Engineering, University of Texas at El Paso, Texas, USA.
Received 2012
ABSTRACT
Mult iple sclerosi s (MS) is an unpredi ctable di sease of the centr al nervous syste m that can range fro m relativel y benign to somewhat
disabling to devastating, as communication between the brain and other parts of the body is disrupted. Scientists have learned a great
deal ab out MS in recent years; yet still , its cau se remains el usive. This p aper in tend s to investigate the h ypothes is that gait dynamics
have meaning and may be useful in providing insight into the neural control of locomotion. It further seeks to explore the mutual
interactions and influences of MS functions on gait, and vice versa, in a quantitative and robust fashion. Ground reaction forces
(GRFs), muscle activities, and segmental accelerations within a gait cycle were analyzed in this study. Patterns of the signals from
six relapsing-remitting multiple sclerosis (RRMS) patients were compared with the healthy subjects. This quantitative gait analysis
aids to illuminate a better understanding of the mobility-related di sease such as RRM S characteristi cs. An outcome of thi s study is a
reproducible methodology for helping therapists make reliable and differentiable diagnosis, design a tailored therapeutic strategy, and
comfortably evaluate the follow-ups on patient’s functional recovery.
Keywords: Ground Reaction Forces; EMG Muscle Activities; Multiple Scelorsis; Segmental Accelerations; Kinetic and Kinematics
Measurement; Fuzzy Relation Matrix; Fuzzy Similarity
1. Introduction
Approximately 400,000 Americans have multiple sclerosis
(MS), an d every week abo ut 200 new cases are diagno sed [20].
Worldwide, MS affects around 2.5 million people: it is one of
the most common chronic neurological disorders and causes of
disability in young adults between 20 and 40 years of age,
especially in Europe and in N orth America [ 29].
Multiple sclerosis is a demyelinating disease of the central
nervous system (CNS) that can result in severe morbidity and
mortality. Mobility loss and walking impairment are among the
direct impact on the outcomes and the quality of life of individuals
with MS. A noticeable symptom of multiple sclerosis is tremor,
a rhythmic, involuntary oscillatory movement of an area of the
body. Tremor caused by MS can affect the head, neck, vocal
cords, trunk, and limbs, but it is most common in the arms. The
two most prevalent forms of tremor in MS are postural tremor
(present when holding a position against gravity) and intention
tremor (present during goal-directed movement, worsening as
the individual approaches a target). Tremor can be incapacitating,
seriously impairing functioning and ability to perform activities
of daily living. Other common motor symptoms of multiple
sclerosis include foot drop, muscle weakness and/or spasticity,
and altered gait rhythm. Sutliff has evaluated that maintaining
mobility was ranked as one of the highest priorities among
patients with MS, regardless of disease duration or disability
level [27]. The loss of mobility contributes to a substantial
patient burden. The statistical techniques of path analysis has
shown how difficult walking significantly affects physical
activity in patients with MS. Impaired mobility is associated
with reductions in quality of life and activities of daily living.
Intrinsic to medical practice particularly in the evaluation of
different therapeutic approaches to chronic disease, the health
of the patient in association with the individual’s quality of life
is extremely essential. In th e case of MS, a chronic debili tating
disease, the preservation of the health-related quality of life
becomes significant. Therefore, there is the need for measure-
ment tools to accurately and consistently quantify the quality of
life over the course of the disease. The reliable and efficient
measurement of the level of impairment will assist in
evaluating the variances from normal patterns through training
and other compensatory strategies. To date, the evaluation of
gait impairment in individuals with MS is typically performed
through subjective examination by the clinician. The
standardized clinical assessment tools for MS relating to gait
include MS Functional Composite (MSFC) [4,12], Expanded
Disability Status Scale (EDSS) [22], Disease Steps (DS) [9],
MS walking scale -12 (MSWS-12) [8, 19], and other tests for
balance and gait dysfunctions [1,10,15,17,18,21]. A significant
problem with the subjective clinical examination is that it may
be misleadi ng du e to an in ability to ob serve the smal l variance s
from the unquantifiable impact of the observer. Therefore,
clinical quantitative gait analysis must be used to provide
accurate scientific evaluation through measurement of kinetic,
kinematics and muscle activity.
Current clinical methods of gait analysis including the optical
motion capture system are time an d labor in tensi ve and invo lve
extensive post-hoc data analysis [2,6,11]. These limitations

H. Y. YU, T. SARKODIE-GYAN
21
reduce access to gait analysis and exclude direct application of
the patien t’s gait dat a to rehabilitati ve interven tion s in real-time.
In contrast, inertial sensor systems with its properties of small
size, light weight, lower cost, direct measurement, and their
unobtrusive effect on the human is becoming an important
methodology in biomechanics and clinical applications. Liu et
al. used the in ertial sensor system ( gyroscopes and accelerometers)
to detect the gait phases [16]. However, quantitative gait
analysis using inertial sensor system in individuals with MS has
not yet attracted any relevance in clinical application. In fact,
the most important symptoms of MS include muscle weakness
and spasticity and hence the essence of evaluating the muscle
activities in individuals with MS.
This study involved the collection of pilot data in MS pa-
tients with mild to moderate gait difficulty. The fusion of the
acquired multiple sensor data and the fuzzy inferential reason-
ing algorithm were applied to quantify and differentiate the
kinetic, kinematics and electromyographic data for correlation
with each subject’s physiological states, health conditions, and
the underlying mechanisms of disability, if any.
2. Data Acquisition and Signal Processing
2.1. Participants
Sub jects were recrui ted from th e local co mmunity. All subjects
provided informed consent prior to participation, and the Insti-
tutional Review Board at the University of Texas at El Paso
(UTEP) approved the study protocol.
Six pat ients were recrui ted, and their respective pr imary care
physicians made the diagnosis, and classified them as relaps-
ing-remitting multiple sclerosis (RRMS). Each patient had sub-
jective experience with reduced walking ability, from mild to
moderate levels; they had an expanded disability status scale
(EDSS) score of 6.0 or less. Table 1 illustrates a short descrip-
tion of each RRMS patient, and five of them had no others (non
MS-related) conditions affecting walking ability. Twenty-two
healthy subjects were recruited as control group. Twelve male
subjects with age 22.8±2.12 years old, height 1.70±7.37m,
weight 68.5±5.56 kg, BMI 23.9±2.05 kg/m2, and t readmill walk-
ing speed 0.99±0.07 m/s; ten female subjects with age 23±2.31
years old, height 1.65±4.80m, weight 58.4±6.47 kg, BMI
21.9±1.7 4 k g /m 2, and t rea dm il l walk ing speed 0. 96± 0.09 m/s .
2.2. Equipments
• Instrumented treadmill
GRFs were measured using an instrumented treadmill
(Bertec® Corporation,USA). The treadmill is a dual-belt force
sensing component, records the consecutive left and right GRFs
in three dimensions: mediolateral (Fx), anterioposterior (Fy), and
vertical (Fz). GRF is a ‘reflection of the total mass-times-
acceleration product of all body segments and therefore represents
the total of all net muscle and gravitational forceactio ns at each
instant of time over the stance phase’ [28]. Therefore, accurate
measurement on the GRF can provide significant information
for the study of normal and pathologic gait.
• Surface Electr omyography (sEMG)
Delsys surface EMG with its Myomonior® (Delsys Inc., MA,
U.S.A.) was used to measur e the muscle activities o n the lo wer
extremity. The surface electrodes were placed on the skin
overl ying the muscle, and therefo re can nonin vasively measu re
the muscle activity in dynamic motion such as walking and
running. Surface EMG (sEMG) signals are often used as a
diagnostic tool for identifying neuromuscular diseases as well
as control signals for electronic bio-devices such as prosthetic
hands, arms, and lower limbs [24].
• Accelero meters
Tri-axial ADXL330 iMEMS® accelerometers from Analog
Device Inc. were used to measure the accelerations directly at
the point of attachment. Advantages of using accelerometers
over other biomechanical devices primarily are expressed by
their miniature size, hence offer limited restrictions of their
anatomical placement and provide minimal impediment to
movement. Direct and continuing measurement of accelerations
therefore, can be used as wearable biosensors to measure gait
parameters in real world situations without the limitations
inherent in immobile laboratory approaches such as optical
motion capture systems [25].
2.3. Experimental Procedure
The complete sensor placements including force sensors, sur-
face EMG electro des an d tri-axial acceler ometers are i llust rated
in Figure 1. Dynamic left and right GRFs in three dimensions
were measured independently. The following eight muscles on
both extremities (totally 16 EMG electrode placements) were
recorded: Soleus (Sol), Tibialis Anterior (TA), Gastrocnemius
Lateralis (LG), Vastus Lateralis (VL), Rectus Femoris (RF),
Biceps Femoris (BF), Gluteus Medius (Gmed), and Erector
Spin ae (ES). Eight Tri-axial accelero meters were placed on the
lower body segments: foot (Ftx, Fty, Ftz), shank (Skx, Sky, Skz),
thigh (Thx, Thy, Thz), and hip (Hipx, Hipy, Hipz).
Tabl e 1. MS patient’s characteristics.
MS
Patient Sex M/F Age years BMI (kg/m2) Sp eed
(m/s) Observed gait impairment Medicine
1 F 55 29.4 0.65 Wobbly gait with no aids Interfer o n beta
2 F 40 22.5 0.6 5 Mild left foot drop, oc casional tremor, walking with no aids Interfero n beta
3 M 62 29.6 0.1 5 Walkin g with unilateral cane, foot drop Int erferon beta
4 M 37 39.9 0.3 5 Limp gait, walking with unilateral cane, foot drop Interfer on be ta ; Desy rel
5 M 45 39.2 0.7 5 No visible gait abn ormality, walking with no aids Copaxone
6 M 28 21.8 0.5 0 Wobbly gai t with no aids, right leg postural tremor Naltrexon
BMI – Body Mass Index (kg/m2) RRMS – Relapsing Remitting Mul tiple Sclerosis

H. Y. YU, T. SARKODIE-GYAN
22
Figure 1. Multiple Sensors Placements for Data Acquisition.
Each subject wore running shorts and comfortable T-shirt
during the experiment. Subjects were allowed to become familiar
with the walking track on the treadmill before conducting the
experiment. For patients, partial body-weight support (harness)
fixed on the instrumented treadmill was applied for safety
purp os es. The sp eed of the tr eadmil l was in cre men tall y i nc rea sed
or decr eased to determine the participant’s comfortable walking
speed. Every subject was instructed to walk continuously for
three minutes on the treadmill.
The LabVi ew software was used for th e acquisition of accel-
erations and GRFs data at sampling frequency of 100 Hz. EMG
data was acquired at sampling frequency of 1000 Hz using
Delsys EMGWorks acquisition software. The synchronization
between the acquired data was achieved by Delsys EMG trigger
module. Therefore, the GRFs, EMGs, and acceleration signals
were coll ected simultaneously.
2.4. Data Processing
White noise was observed in the force data due to the vibrations
and the motion artifact during the walking process on the
treadmill. The GRF data were smoothened using second order
Butterworth low pass filter with cut-off frequency of 20 Hz.
The second order Butterworth low pass filter with cutoff
frequency of 6 Hz was applied to reduce the noise and improve
the resolution of the acceleration data.
Since unprocessed EMG data or raw EMG data cannot be
used for quantitative comparison between subjects, the EMG
raw data were filtered by: Band Pass Filtering (second order
Butterworth band-pass filter with a frequency of 20-200 Hz),
Full Wave Rectification, and Linear Envelope (second order
Butterworth low pass digital filter with a cutoff frequency of 7
Hz).
2.5. Fuzzy Relation Matrix and Fuzzy Similarity
In this study, a cluster of data relating to the kinematics,
kinetics and muscle activities was acquired using an array of
inertial sensors (accelerometers), instrumented treadmill and
electromyographic device during normal walking tasks. These
data were treated as aggregate information granules that enabled
the eff ici e nt par tit ion of input spa ce a nd m ore rapid a na ly s is. Thi s
means that we dealt with the relationships of the kinematic,
kinetic and muscle activity functions. The relationships depict
the attributive features of the human movement and are
expressed in an implication table giving rise to a fuzzy
relational matrix, established between the dynamic activities
during the walking tasks. Therefore, we established granulated
information between 1) the acquired kinetic and kinematic data
and the gait phases; and 2) the acquired muscle activity data
and the gait phases. Since data are usually represented in a
tabular form, it is also easy to consider the table as a matrix.
Hence, a fuzzy relational matrix was used as the expression of
the strength of association or interaction amongst the elements
of the gait functio ns, and it elucidates a ru le base th at is used to
provide a model, a model of a featur e matri x.
GRF data were normalized by individual body weight, whe-
reas EMG and acceleration data were normalized by the maxi-
mum averaged values. Normalized Data were then aver- aged
from 100 strides (100 gait cycles). Averaged data within a gait
cycle with s even fun ctional gait phas es were extracted using t he
vertical GRFs [11]. The seven functional gait phases as illu-
strated in Figure 2 are: Loading Response (LR), Mid- stance
(MST), Terminal Stance (TST), P reswing (PSW), Initial Swing
(ISW), Midswing (MSW), and Terminal Swing (TSW) [28].
The mean values of the data in the seven gait phases are
represented in a matrix for m, fuzzy relational matrix.
Three types of fuzzy relational matrices were constructed
using the methods in [11]. The relatio nal matrix between GRF s
{ }
,,Fx FyFz
and the 7 gait phases
{ }
1234567
,,,,,,PPPPPPP
;
EMG
{ }
SolTALG VLRFBFGmedES
and the
seven gait p hases; the accelerati ons
{ }
,,,,,,,,, ,,
xyzxyzxyz x y z
FtFtFtSkSkSkTh ThThHipHipHip
and the seven gait phases .
Fuzzy similarity of each parameter was calculated using the
equation (3) defined in [11]. This methodology involving the
acquisition of the kinematics, kinetics and electromyographic
data of human gait enable the understanding of the neurophysi-
ology/biomechanics of gait for augmenting objective measure-
ments of mobility and functional status. This methodology may
be applied for the efficient and reliable analysis of human gait
dynamics at a level that quantifies variations in MS from mor-
phometrically adjusted normal.
3. Results
The GRFs, the EMG muscle activities, and the accelerations
within a gait cycle were analyzed in this study. Patterns of the
signals from six RRMS patients were compared with the
healthy subjects. The pathologic characteristics of multiple
sclerosis were concluded and the intelligent algorithm provided
a quantitative analysis for the clinical diagnosis for MS
patients.
In Figure 3(a), the normalizations of the 3-D GRFs (GRFs
in % BW) with respect to the gait cycle were performed for
both healthy and MS patients. Figure 3(b) illustrates the fuzzy
simi- larities between the GRFs of the MS patients and the
healthy

H. Y. YU, T. SARKODIE-GYAN
23
Fi g ure 2. Seven gait phases using % gait cycle and transitional gait
events.
subjects in three dimensions. This similarity ranges between 0
and 1, and it represents the strength of the association between
the GRFs of the healthy subjects and MS patients within the
respecti ve gait p hase.
Eight normalized EMG data within a gait cycle were
illustrated in Figure 4(a). The Figure presents the gait pattern
of the healthy subject as an asymptote to the variances in the
gait patterns of the individual patients. Figure 4(b) illustrates
the grade of similarities between the eight EMGs of the MS
patients and the healthy subjects within the seven gait phases .
Normalized 3-D acceleration data on four body segments
(foot, shank, thigh, and hip) within a gait cycle are shown in
Figure 5(a). Figure 5(b) represents the fuzzy similarities of
accelerations of the MS patients within the seven gait phases.
4. Discussion
The self-selected walking speeds of the MS patients were
generall y lower as compared with th ose of the healthy subj ects
(range 0.9~1.2 m/s). The effects can be observed from the
GRFs in all dimensions, i.e. the increased stance phase/double
stance phase, decreased swing phase. The GRF pattern has
simply shown an estimated disease-severity of the individual,
RRMS-3 subject for instance was found with more severe
symptoms along all six MS patients. This may also be due to
the age of this patient (62 years old) with other gait-related
disorders such as diabetes (information not shown in methods).
A previous research result suggested that the GRFs varied
significantly between the MS patients and the healthy subjects
in both vertical and anterior-posterior directions but showed no
differences in the mediolateral direction [23, 30]. However this
study found that the vertical GRF of the MS patients did not
depi ct the referen ce M-shaped pattern, and the lower magnitude
of the anterior-posterior forces appear in most of the MS
patients.
The result illustrated in Figure 3(a) suggests a common
feature of MS patients in th e ver tical GRF, the magn itude of the
first peak is excessi vel y high in t he earl y stance ph ase, wherea s
that of the second peak is moderate in the terminal stance. In
normal gait the magnitude of the two peaks are approximately
(a) (b)
Figure 3. Ground Reaction Force Pattern within a Gait Cycle and Fuzzy Similari tie s of the Seve n Gai t Phases.

H. Y. YU, T. SARKODIE-GYAN
24
(a) (b)
Figure 4. Ele ct romyogr ap hic a l P at tern within a G ait C y c le an d F u zzy Simi lari ties o f th e Seven Ga it Phases.
(a) (b)
Figure 5. Acceleration Pattern within a Gait Cycle and Fuzzy Similarities of the Seven Gait Phases.
equal in which the M-shape force curve illustrates the weight
transfer from the heel to the mid-foot and the mid-foot to the
ball of the foot for push-off. The fact that the vertical GRF
exceeds b ody weight in the terminal stan ce (i.e. seco nd peak of
the vertical GRF) indicates that the body weight is being
supported and some propulsion is generated successfully. On
the other hand, the low propulsive force is generated in the MS
patient, and the greater impairments (much more reduced
second peak or sometimes even less than the body weight)
indicated a more severely affected MS patient (RRMS-3). By
comparing the magnitude of the anterior- posterior force of the
healthy subjects, the MS subjects revealed much lower
magnitu de of th e force.
This study has also found different patterns in mediolateral
GRF. Apart from the RRMS-3 patient, the higher magnitudes
of the mediolateral force were found in all other five patients
during the pre- and terminal stance phase (i.e. single support
phase). The mediolateral force in normal gait is of lower
magnitude in most situations and relates to balance during
walking. The higher magnitude of the mediolateral force with
MS patients indicates that more effort was applied to keep the
balan ce, especial ly durin g t he single su pport phase.
Figure 4(a) illustrates the eight muscle activities of both the
healthy and MS subjects, respectively. The Soleus and
Gastrocnemius muscles are ankle plantarflexors, and designed
to stabilize the foot and knee during the stance phase of the gait.
However, most of the MS subjects exhibited long-latency periods
with increased activities on the Soleus and Gastrocnemius in the
stance phase.
The Tibialis Anterior (TA) is an ankle dorsiflexor and most
active at heel strike and prevents “foot-slap”. The behavior of
the TA in this study was significantly greater during the initial
contact and during the toe-off. This increased EMG activity that
was apparent in the ankle dorsiflexor muscles in the MS
subj ects are co n sidered the mechan ism for co unt eract-balancing
the deficits. The exhibited behavior may illustrate implications
relating to fatigue and spasticity [13]. In addition, the TA
activity of most of the MS patients was significantly greater
during the pre-swing phase and the early swing phase. This
activity is supposed to assist the toes in clearing the floor for
the push-off into swing.
Vastus Lateralis (VL) and Rectus Femoris (RF) muscles
depict the knee extensors/hip flexors and the muscles of
quadriceps group. Most of the MS subjects showed intense and
prolonged action of the VL and RF from the midstance through
to the initial swing. Extensor spasticity of the legs, particularly
of the quadriceps, might be considered advantageous for
standing, walking and particularly transferring, as it may
compensate for muscular weakness. Clinical research has
suggested that some structural changes of spastic muscles and
of connective tissues develop over periods of weeks to months
in MS patients, leading to changes in mechanical properties in
the leg extensors. This finding may therefore be due to
inadequate management of spasticity which leads to such
structural changes of muscles with subsequent functional
impairment [14].
Biceps Femoris (BF) muscle is the lateral hamstring, which
is activated to accelerate the knee toward flexion and the hip
toward extension during swing in normal gait. This was found
in the prolonged and increased muscle activity of BF in the
stance p hase, parti cul arly in the single-su p port phase, whereas a
delay and reduced muscle activity of BF in the midswing
through terminal swing phase. These findings suggest that the
prolonged stance activity of the upper leg muscles, parti cularly
tho se knee and hip ext ensors, could be r elated t o a more gen eral
neuromuscular strategy that serves to supply additional support.
Evidence for the use of this strategy was found in hemiparetic
stroke fro m a recent muscle actuated simulation o f hemiparetic
gait by Higginson, who showed that co-contraction of the upper
leg muscles during midstance can be used to enhance knee
stability [7]. It is suggested that this strategy is used to
compensate BF weakness in the swing phase. These spastic
upper leg extensors may in turn cause delayed Gluteus medius
(Gmed) in the loading response. Gluteus medius is the hip
abductor and pelvic stabilizer. Figure 4(a) shows the MS
subjects with greater activity of Gmed in the midstance and
midswing phases to support the body weight with single leg

H. Y. YU, T. SARKODIE-GYAN
25
over the ground.
Erector Spinae (ES) or lower back muscle monitors the
paraspinal activity of the trunk movement and stabilization. The
EMG activation level of the ES was found to be greater in the
MS than in healthy subjects at the swing-stance and
stance-swing transitions, which is likely to be a compensatory
factor in the instability during phase transition, as well as
serving to reduce the risk of falling.
The use of a ccelero mete rs is an i mportant alt ernativ e appr oach
for the acquisition of kinematic data in gait analysis techniques.
Accelerometers placed on the foot, shank, thigh, and hip reveal
both temporal aspects o f the g ait cycl e and ac celeratio n forc es at
key events in the cycle, such as heel strike, toe-off, and events
during swing phase. Study of the acceleration/deceleration
patterns in MS patients can further confirm the muscle spasticity
or muscle weakness and physiological tremor in the individual
gait. There were significant different patterns of acceleration in
MS sub jects as co mpared with the health y subjects, especi ally in
the ant e ri opost e ri or a nd mediola t eral di re c t ions .
In the medio lateral d irecti on, there was a greater a cceler ation
occurring in the initial contact during the loading response (the
fir st 10% of the gait cycle) in the MS subjects. It is thought to
be the compensatory mechanism for the stability of the ankle
and knee. The acceleration in the following gait cycle on that
direction was generally reduced in most of the MS subjects, an
expected occurrence with respect to the slow working speeds.
In the anterior-posterior d irection, a sharp accelerati on in th e
initial contact was found in most of the MS subjects, which
depicts an ind ication of faster h eel-to-toe force tran sfers at that
time, especially for patients with foot-drop. On the foot
accelerat io n patt ern in this directio n, th ere were eith er mod erate
deceleration at toe-off observed in RRMS-1 and RRMS-2, or
faster deceleration followed by decreased acceleration at the
beginning of the initial swing seen in RRMS-3 ~ RRMS-6. This
is an indication of an inadequate toe clearance in the MS
subjects, resulting in reduced forward progression of the limb
with push off into swing.
The histogram of Figure 3(b) illustrates a relational mapping
between the GRFs of the subject with respect to the seven gait
phases, for both male and female subjects. The normalized “1”
is an asymptote of the GRFs of a healthy patient and is used as
the reference p attern . The compar ative relati onsh ip between t he
healthy subject and the MS patients illustrates a functional as-
sociati on between t he elements o f the GRFs and th e gait phases.
The same illustrations shown in Figures 4(b) and 5(b) for the
muscles activities and acceleration respectively. The asymp-
totes depicted in the histogram illustrate a direct grade of ag-
gregation or association between the physiologic data of the
MS patient with respect to those of the healthy subject. This
strength of aggregation is a functional relationship that clearly
illustrates an objective and differential assessment of the effi-
cacy and outcomes of rehabilitation therapies and practices.
This will also enable the further development of precise me-
thods for measuring impairments and disabilities.
5. Conclusions
This study offers a narrative description of the dynamic phasic
patterns of GRFs, muscle activities, and accelerations in the
body segments in the MS patients. The study enables the analy-
sis of multiple components in three-dimensional plane for the
reliable differentiation of pathological and/or compensatory
functional impairments in MS and in normal gait patterns. The
fuzzy inferential reasoning algorithm provides the understand-
ing of the dynamic patterns of these components. In real-time
data acquisition and pattern analysis, this intelligent methodol-
ogy will serv e as a self-motivated tool for the individual subject
during treatment. The technology may help therapists in choos-
ing a tailored therapeutic strategy for the individual, predict
responses to treatment, or determine if maximal recovery has
occurr ed.
6. Acknowledgements
The authors wish to thank the Stern Foundation for providing
the funds for this research work.
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