Indian Institute of Information Technology - Allahabad - Disputationes

2nd Disputations on
Future Technologies for Health
Care
Allahabad (India), 18-20 September 2011
Diagnosis
Through
Gait Oscillations
Pavan Chakraborty, Soumik
Mondal, Anup Nandy, Saman
Shahid and G. C. Nandi.
Indian Institute of Information Technology, Allahabad,
Deoghat, Jhalwa, Allahabad – 211 012 India
E-mail: pavan@iiita.ac.in
Indian Institute of Information Technology - Allahabad

Diagnosis Through Gait Oscillations
Learning to walk is a daunting
task for a human baby.
It takes close to a year for a
human baby to stand on its two
legs, balance and then learn to
walk.
Indian Institute of Information Technology - Allahabad

Indian Institute of Information Technology - Allahabad

Locomotive
Balance
Static balance slower
locomotion.
Dynamic balance Faster locomotion.
Indian Institute of Information Technology - Allahabad

Our biped locomotion
was learned on Trees
Orang-utan study suggests that upright
walking may have started in the trees.
--- Branch walking easier on biped,
holding the branch above with hands.
Indian Institute of Information Technology - Allahabad

Walking: A common yet
complex process
•Human walking is
accomplished with a strategy
called the double pendulum
•During forward motion, the leg
that leaves the ground swings
forward from the hip
•Leg strikes the ground with
the heel and rolls through to
the toe in a motion
•Motion of the two legs is
coordinated so that one foot or
the other is always in contact
with the ground
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Indian Institute of Information Technology - Allahabad

Indian Institute of Information Technology - Allahabad

Foot
Pressure
Platform
Animations
Contact points of the sole of the foot during
normal weight bearing.
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Indian Institute of Information Technology - Allahabad

Indian Institute of Information Technology - Allahabad

Subject with reflective markers for motion analysis. Also
seen are the electrodes for the telemetry EMG unit
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Use of Sensor Network
To Understand
The Human Gait
Indian Institute of Information Technology - Allahabad

Objective
“The main objective of this research tends to develop a low
cost, portable, non-invasive wearable sensor based
biometric suit which could be applied to numerous different
applications.” Like,
• Human Identification
• Human Robot Interaction
• Gesture Classification
• Human Computer Interaction
• Diagnostic Gait Signature Detection
• Find the effect on gait with different age, body weight and
different diseases.
• Analysis of abnormal gait signature etc.
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The Degrees of Freedom
3
3 3
1 1
1
1 1
3 3
1
1
2
2
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θ
Potentiometer
Voltage Calibrated to Angle θ
θ
t
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θ ( t)
= −65.71
V ( t)
+ 122.9
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The actual knee
angle oscillation
of a healthy
human being
measured using
a potentiometer
circuit. The top
panel shows the
noisy data over
a multiple gait
oscillations.
Averaging multiple gaits and convolving it with a 3 value running
average in 2 iterations removes the noise (shown in the lower
panels).
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8 Analog Inputs
AVR Dev. Board
ACD, Multiplexing &
Parallel to Serial
conversion.
ATMEGA 32
RF Module Tx
RF Module Rx
Max 232
RS – 323 Port
Serial Port
Software
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Indian Institute of Information Technology - Allahabad

The HGOD Suit consisting of 8 potentiometer sensors
Indian Institute of Information Technology - Allahabad

Indian Institute of Information Technology - Allahabad

Indian Institute of Information Technology - Allahabad

Figure 6. Gait oscillations from 8 joints of the body. From the top (left and
right): Left and right Shoulders, left and right elbows, left and right hips and
left and write knees. The dotted line is a simple sine curve fit to the shoulder
elbow and hip data. The knee oscillation is complex; therefore it has not been
fitted.
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Design…
Indian Institute of Information Technology - Allahabad

Sensor and Interface
Kit Specification
• Phidget Rotation sensor (1109) which is being
integrated with Phidget interface kit (1018).
• Power supply voltage varies from 3.5VDC to 5 VDC with
10kΏ output impedance
• The rotation sensor has been opted for 0 to 300 degree
resolution
• Measurement of analog value from the rotation sensor
and produces the digital counts as output between 0 to
1000 ranges
• Analog Input update rate is 65 samples/sec
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Calibration Curve for Sensor
• Phidget interface kit has
Internal 10 bit ADC
circuit
• Equation depicting linear
relationship between the
observed data and
calibrated data with least
square fitting is : θ = 3.3335 x
count (±0.020161)
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Analysis of human gait
oscillations
• Initially, for each gait pattern of respective joint a
zero correction has already been done by collecting
α 0 represented in the form of digital counts.
• The initial digital counts are subtracted from the
current digital count which is being interpolated by
joint angle values in degree for each oscillation.
• The movement of each oscillation for a particular
joint is manipulated in terms of degree which is
being calculated by
desired angle (degree)=(current digital count-initial
count)x 300/1000.
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Gait Patterns Obtained
Gait pattern of both elbow joints
Gait pattern of both knee joints
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Gait Patterns Obtained…
Gait pattern of both hip joints
Gait pattern of both shoulder joints
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Observations
• Each gait oscillation consists of both swing and stance
phase respectively.
• The generated pattern for both left limb and right limb
implies the variation of calibrated angle values over the
period of oscillation
• The period of each oscillation for a particular pattern can
be calculated by
Total no of samples in a particular oscillation 64(approx)
T =
=
= 1sec(approx)
sampling rate
65
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Correlation and coupling of
significant joints
• Correlation and coupling of those joint
oscillations, we have compared them with an
oscillation equation of the form:
X
( t)
= asin(
ω 1t
+ ϕ)
Y
( t)
= bsin(
ω 2t)
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Coupling Observations
It has been observe from the figures that phase difference φ from the fitting
between two gait oscillation is |φ|=π and
ω1 = ω 2 = ω = 2π
/ T = 2π
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Coupling Between Elbows
The above figure shows an interesting oscillation of an envelope in the shape
of an ‘L’. This indicates when one elbow oscillates the other is practically
static. This happens because of the elbow locking in the reverse cycle.
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Coupling Between Shoulder
and Hips
It is being noted from the above figures that the coupling between both
shoulder and hip oscillations tends to an elliptical curve where phase
difference |φ|= 5π/4 .
Indian Institute of Information Technology - Allahabad

Coupling Between Shoulder
and Elbows
From the above figures it has been noticed that the movement of
shoulder oscillation arises in both directions where as the oscillation of
elbow joint belongs to in single direction
Indian Institute of Information Technology - Allahabad

Creation of a Gait Data Base
Gait Oscillations at Different Speed.
With Different Carrying Weights.
Other Biometric Information for Correlations:
Male/Female,
Weight
Height
Age
Body Shape – Front and side view photo
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Indian Institute of Information Technology - Allahabad

Indian Institute of Information Technology - Allahabad

List of Publications
• Pavan Chakraborty, Anup Nandy and G. C. Nandi, 2007.
• "Use of Sensor Network to Understand the Human Gait"
• An Invited Oral Presentation WCSN-2007 International
Conference held at IIIT-Allahabad.
• Pavan Chakraborty, G. C. Nandi and Anup Nandy, 2008.
• “Sensing the Human Locomotion”
• An Invited Oral Presentation at the Multi Discipline Expert
Panel Meeting held at ALIMCO Kanpur on January 30 and 31,
2008.
• Akshay K Singh, Ajit D Dhiwal, Pavan Chakraborty and G C
Nandi. 2008. “Designing A Full Body Human Computer
Interaction Device” Proceedings of the National Conference
CSI-RDHS 2008 Research and Development in Hardware and
Systems by Computer Society of India).
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Continued………

List of Publications
• “A framework for synthesis of human gait oscillation using intelligent gait
oscillation detector (IGOD)” In: IC3 2010. CCIS, vol. 94, pp. 340–349.
Springer, Heidelberg, 2010.
• “Gait Based Personal Recognition System Using Rotation Sensor,”
Submitted in IEEE Trans. On System, Man Cybernetics part: B (Under
Review).
• “Modeling a Central Pattern Generator to generate Biped Locomotion of a
Bipedal Robot using Rayleigh Oscillators” – In the proceedings of S. Aluru
et al. (Eds.): IC3 2011, CCIS 168, pp. 289–300, Springer-Verlag Berlin
Heidelberg, 2011. (In press)
• “A Central Pattern Generator based Nonlinear Controller to Simulate Biped
Locomotion with a Stable Human Gait Oscillation” – In International Journal
of Robotics and Automation (IJRA), vol. 2, issue 2, pp. 93-106, 2011.
Indian Institute of Information Technology - Allahabad

Indian Institute of Information Technology - Allahabad
Indian Institute of Information Technology - Allahabad

Indian Institute of Information Technology - Allahabad

Indian Institute of Information Technology - Allahabad

Indian Institute of Information Technology - Allahabad

Conclusion
• A biometric suit IGOD was build to make simultaneously measurement of the 8
major joint oscillations that plays a crucial role during human biped locomotion.
• The mechanical structure of IGOD was completed, fine-tuning and calibrations was
done.
• The concept of IGOD is extremely simple with significant biometric and robotics
application.
• Analysis of the data using correlation coupling between pair of left and right limbs
will be an important diagnostics of the bio information.
• The gait recognition study has been made using the IGOD data from various
people.
• The TS-LDA developed for the purpose provided promising results.
Indian Institute of Information Technology - Allahabad

Conclusion…
• The results were doubly checked with the well established ANN algorithm.
• Interestingly in a different work on gesture based recognition TS-LDA did not
work.
• This made us analyze the capability of TS-LDA which is well tuned to detect small
variations which is normally seen in gait recognition; it does not work well for large
variation as seen in gesture recognition.
Indian Institute of Information Technology - Allahabad

Pavan Chakraborty, Anup Nandy and G. C. Nandi, 2007.
"Use of Sensor Network to Understand the Human Gait"
An Invited Oral Presentation WCSN-2007 International Conference held
at IIIT-Allahabad.
Pavan Chakraborty, G. C. Nandi and Anup Nandy, 2008.
“Sensing the Human Locomotion”
An Invited Oral Presentation at the Multi Discipline Expert Panel Meeting
held at ALIMCO Kanpur on January 30 and 31, 2008.
Akshay K Singh, Ajit D Dhiwal, Pavan Chakraborty and G C Nandi.
2008. “Designing A Full Body Human Computer Interaction Device”
Proceedings of the National Conference CSI-RDHS 2008 Research and
Development in Hardware and Systems by Computer Society of India).
Pavan Chakraborty, Anup Nandy and G. C. Nandi,
"HGOD A Multi Sensor Network to analyze Human Gait and its stability“
Submitted to The Proceeding IEEE Conference.
Raghvendra Jain, Pavan Chakraborty, and G. C. Nandi
“Energy budgeting for a Humanoid Robot using physics properties”
Raghvendra Jain, K.V.N. Pavan, Pavan Chakraborty, and G. C. Nandi,
“Full body Human Robot real Time Gesture based interaction”
Indian Institute of Information Technology - Allahabad

Indian Institute of Information Technology - Allahabad

Indian Institute of Information Technology - Allahabad

The Degrees of Freedom
3
3 3
1 1
1
1 1
3 3
1
1
2
2
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Statically stable walking: The projection of the robot’s
center of gravity always lies within the footprint or the area
between the two footprints of the robot.
The locomotion is quite easier to perform but is very slow.
Dynamic walking: The robot’s center of gravity outside the
footprints. Corrections must be done at all times to maintain
the robot on his feet.
To achieve stability, it is mandatory to know about the robot’s
dynamics (speed and inertia of each of its parts).
This method is without a doubt the more efficient in
terms of velocity but is very tricky to implement.
Indian Institute of Information Technology - Allahabad

Passive walking: No actuators are used and the robot uses its
own weight and dynamics to walk down a slope without external
input or control.
The system has a stable limit cycle and the robots are even
able to deal with small perturbations. We can also mention the
concept of passive dynamic walkers. In this case small
power sources on their ankles and/or hips are included to
simulate gravity,
which allows the robots to walk on a level ground without
control servos.
This approach is very important since it allows to learn many
features of robot’s dynamics that are also useful for other kind
of locomotion [5, 4].
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Discussion
• According to our research works we can conclude that for distance based human
identification using gait not only depends upon the leg feature. Here hand
movement is also an important factor.
• Both the methods gives us 100% accuracy.
• This system can not be use for surveillance security purpose.
• This can be used in personal identification purpose where walker full cooperation is
needed.
• In this system mainly we are trying to find out what are the basic features to
recognize the different human walking for identification and research.
Indian Institute of Information Technology, Allahabad
Indian Institute of Information Technology - Allahabad
5

Future Work
• Interaction between humanoid robot and human being (Imitation
Learning).
• Classification of Gestures for Human Robot Interaction.
• Tuning up the mechanical design and making IGOD wires free will
improve the freedom, flexibility and movability of the subject wearing
IGOD.
• Use the suit for medical applications.
• We are also trying to train our system by the suit data and test the
system by image data for giving the freedom of walker and use this
system as a gait based human identifier.
Indian Institute of Information Technology, Allahabad
Indian Institute of Information Technology - Allahabad
5

List of My Publications…
• “Classification of Indian Sign Language In Real Time” - In the proceedings of International Journal
on Computer Engineering and Information Technology (IJCEIT), Vol. 10, No.15, pp. 52-57, January
- February 2010.
• “Recognition of Isolated Indian Sign Language gesture in Real Time” – In the proceeding of
Springer LNCS-CCIS 70, pp. 102-107, March 2010.
• “Recognizing & Interpreting Indian Sign Language Gesture for Human Robot Interaction” - In the
proceeding of ICCCT’10, IEEE Xplore Digital Library, pp. 712-717, September 2010.
• “Gesture based imitation learning for Human Robot Interaction” – In the proceeding of ICACCN’11,
pp. 287-292, 2011.
Indian Institute of Information Technology, Allahabad
Indian Institute of Information Technology - Allahabad
5

Pavan Chakraborty
Indian Institute of Information Technology, Allahabad.
A Project funded by DST,
P.I. : Prof. M.D. Tiwari, Director, IIIT-Allahabad
The Working Team: Prof G. C. Nandi (Head Robotics and AI Lab),
Mr. Anup Nandi and Advitiya Saxena (Research Scholars) & Myself.
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Indian Institute of Information Technology - Allahabad

S5
S6
S7
S8
S3
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Indian Institute of Information Technology - Allahabad

Classification of eight regions over
a gait cycle
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Gait oscillation with noise Vs gait
oscillation with filtering noise
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Indian Institute of Information Technology - Allahabad

AVR 40 Pin Development board
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Indian Institute of Information Technology - Allahabad

RF Module
• An RF modulator
(for radio
frequency
modulator) is a
device that takes a
baseband input
• signal and outputs
a radio frequencymodulated
signal.
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Hierarchical
SENSE PLAN ACT
Reactive
Hybrid deliberative / reactive
PLAN
PLAN
SENSE
ACT
SENSE
ACT
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Indian Institute of Information Technology - Allahabad

Indian Institute of Information Technology - Allahabad

2nd Disputations on
Future Technologies for Health Care
Allahabad (India), 18-20 September 2011
Indian Institute of Information Technology - Allahabad

Indian Institute of Information Technology - Allahabad

· Frames 1 - 3: The robot's momentum causes the robot to rise on its standing leg and
a motor moves the swinging leg into position
· Frame 3: The stretch sensor of the swinging leg is activated, which triggers the knee
joint to straighten
· Frames 3 - 6: The robot falls forward naturally, with no motor functions being used,
and catches itself on the next standing leg
· Frame 6: As the swinging leg touches the ground, the ground contact sensor in the
foot triggers the hip extensor and the knee joint of the standing leg and the hip and knee
joints of the swinging leg to swap roles
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Activation
of the calf
muscle
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Indian Institute of Information Technology - Allahabad

M.R. Damper
A typical Magneto-Rheological
(MR) fluid consists of 20-40% by
volume of relatively pure, 3-10
micron diameter iron particles,
suspended in a carrier liquid such
as mineral oil, synthetic oil, water
or glycol.
Iron particles in suspension align and develop a yield strength
in the presence of a magnetic field. The change from a freeflowing
liquid to a semi-solid when a magnetic field is applied is
rapid and reversible.
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Indian Institute of Information Technology - Allahabad

Indian Institute of Information Technology - Allahabad

Indian Institute of Information Technology - Allahabad

11.5mm
35mm
Bearings
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11.5mm
35mm
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Bearings

Joint surfaces move with respect to one another by
simultaneously (1) rolling, (2) gliding, and (3) spinning.
When the concave surface is
fixed and the convex surface
moves on it, the convex surface
rolls and glides in opposite
directions.
When the convex surface is fixed
and the concave surface moves
on it, the concave surface rolls
and glides in the same direction.
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Indian Institute of Information Technology - Allahabad

Hardware Circuit
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Indian Institute of Information Technology - Allahabad

T
ϕ
Assumptions:
•The period of the hip cycle is the same as that of the knee.
•The phase difference between the hip and knee oscillation
is constant.
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Measurement of Period T of
the Hip Oscillation
• The period of oscillation could be determined from the
gap between 2 maxima of consecutive oscillations
θi
V i
by the POT
V i
V 0
t
pj−1
t
pj
T
j
= t − t
pj pj
−1
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Measurement of Period T of
the Hip Oscillation
V = ∑ j
V i
, VT = ∑ j
Vi
⋅ti,
i
i
for
V i
> V 0
t
pj
=
VT
V
= ∑V
⋅t
i
i
i
∑
i
V
i
T j
T
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j
= t − t
pj pj
The period , is the time gap between consecutive
peaks j and j-1.
T j
⇒ T 0
−1

Check for Threshold
T −
j
≥ ∂
0
T t
2
Check if
threshold is crossed, then
provide new period for damping profile ( T = T j
).
0
We take
8 sections.
∂t = T 0
8
since we divide the gait cycle into
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Indian Institute of Information Technology - Allahabad
Classification of eight
regions over a gait cycle,
superposed over the actual
data of the knee angle.

RF Module
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RF Module
• An RF modulator
(for radio
frequency
modulator) is a
device that takes a
baseband input
• signal and outputs
a radio frequencymodulated
signal.
Indian Institute of Information Technology - Allahabad

Why did nature never invent
the wheel
• Well, Nature has exploited circular objects and
surfaces in many of its designs. Many
mammalian joints follow the classic ball-andsocket
design that allows circumduction (full 360
degree range of movement).
• When we try to mimic nature
(Legged motion) we get in to a lot
of problems related to stability and
energy efficiency.
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Legged locomotion
• Adaptability and maneuvrability in rough
terrain.
• Mechanical complexity.
• Less efficient in energy.
• Complexity in control.
• Nature has usually preferred legs.
• Stability is a problem.
Indian Institute of Information Technology - Allahabad

Indian Institute of Information Technology - Allahabad - Allahabad

Indian Institute of Information Technology - Allahabad - Allahabad

Indian Institute of Information Technology - Allahabad

Monoped locomotion
• Makes continuous
adjustments to leg
angle to control
the body attitude
and velocity.
• Dynamic balance
only.
(Stability)
http://www.ai.mit.edu/projects/leglab/
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Biped locomotion
• Static balance is available in a small range.
Dynamic stability required for movement.
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Outline
Objective
Contribution up to now
Design of IGOD
Analysis of human gait oscillations
Application
Conclusion and Future Work
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Indian Institute of Information Technology - Allahabad
9

The dotted line is a simple sine curve fit to the shoulder elbow
and hip oscillations.
θ
L
= θ
L
sin( ωt + δ
L
)
i 0i
i
R
R
0i
θ = sin( +
θ ωt δ
i
As expected, we notice that the periods of oscillation for the fitted joints are
the same ( T = 2 π ω =1. 57 sec). It is essential for the stability that all limbs
should oscillate at the same frequency. We also notice as expected that the
phase difference between the corresponding left and right limb is
∆δ = δ −δ
=
i
L
i
for the shoulder and hip joints. To check the correlation and coupling
between the left and right of the body during a stable gait, and the
consistency of the data, we plot the left with the right limb oscillations
L
i
π
R
i
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)

Degrees of Freedom (DOF)
The arrows represent
the unidirectional
coupling of this first
model. The values on
the arrows correspond
to the phase
differences we
imposed between the
different oscillators.
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Figure 7. Correlations and coupling between the left and right
limbs. In this figure, the fitted sine functions (in Fig. 6),
describe a Lissajous curve, shown in thick lines.
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Contribution up to now
• Rotation sensor based biometric wearable suit has been made
for capturing different human gait oscillations for analysis
and different applications.
• Introduce rotation sensor base biometric wearable suit for
human identification.
• Personal recognition system which can applicable to real
world situation.
• Better feature selection procedure which can help us to
recognize the person with giving 100% accuracy.
• Choose the real time pattern
compatible with our feature.
recognition algorithm
• Better accuracy
approaches.
has been achieved rather existing
• Introduce Tree Structured LDA (TS-LDA) classification
algorithm.
Indian Institute of Information Technology - Allahabad

Application
1. Gait Based Personal Recognition
System
2. Biological controller
development for biped robot to
generate biped locomotion
Indian Institute of Information Technology, Allahabad
Indian Institute of Information Technology - Allahabad
1

Gait Based Personal Recognition
System
To recognize different human walking in real time
for Identification and research.
Indian Institute of Information Technology, Allahabad
Indian Institute of Information Technology - Allahabad
1

Recognition Procedure
Person
Movement
Rotation sensors
based suit
( IGOD)
Capture Joint
Angle values
Create Feature
Matrix
Choose 60%
sample for
training
Recognition
using ANN/TS-
LDA
After training rest
of data taken for
testing
Result
106
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Indian Institute of Information Technology - Allahabad

Data collection and
Feature Extraction
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1

Data collection
1. We have taken 30 (25 male and 5 female) different humans 30sec
normal walking data and the data was taken in 10 different sessions.
2. The age of our volunteers is 20-30 years.
3. According to our different volunteers point of view the suit does not
affect their natural walking.
4. In each session we have taken just 30 sec of walking step in order to
offer some amount of freedom to their normal walking. It leads to an
assumption that if we put any constraint such as only 4 gait cycle
needs to be accounted while capturing walking pattern, it will affect
tremendously from his/her natural gait pattern.
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1

Phases of the gait cycle
Stance Phase
• Initial Double Support
• Single Support
• Terminal Double Support
Swing Phase
• Initial Swing
• Terminal Swing
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Indian Institute of Information Technology - Allahabad
1

Feature vector
• Data Folding is used for creating feature
vector.
• Minimum and Maximum angle value of
each joints within a particular Gait Cycle.
• Time taken by the Gait Cycle.
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Indian Institute of Information Technology - Allahabad
1

Feature vector…
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Indian Institute of Information Technology - Allahabad
1

Feature vector…
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Indian Institute of Information Technology - Allahabad
1

Gait Recognition
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Indian Institute of Information Technology - Allahabad
1

Method Used
• Artificial Neural Network (ANN)
• Tree Structured Linear Discriminant
Analysis (TS-LDA)
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Indian Institute of Information Technology - Allahabad
1

LDA
• For detecting the variables that allow the
researcher to discriminate between different
(naturally occurring) groups.
• For classifying cases into different groups with a
better accuracy.
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Indian Institute of Information Technology - Allahabad
1

LDA Numerical Example
Factory “ABC” produces very expensive and high quality chip rings that heir qualities are
measured in term of curvature and diameter. Result of quality control by experts is given in
the table below.
Curvature Diameter Quality Control Result
2.95 6.63 Passed
2.53 7.79 Passed
3.57 5.65 Passed
3.16 5.47 Passed
2.58 4.46 Not Passed
2.16 6.22 Not Passed
3.27 3.52 Not Passed
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1

Solution
X= features (or independent variables) of all data. Each row (denoted by) represents
one object; each column stands for one feature.
Y= group of the object (or dependent variable) of all data. Each row represents one
object and it has only one column.
In our example, and
2.95 6.63
2.53 7.79
X= 3.57 5.65 Y=
3.16 5.47
2.58 4.46
2.16 6.22
3.27 3.52
1
1
1
1
2
2
2
1
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Indian Institute of Information Technology - Allahabad

Solution…
X i = features data for group i. Each row represents one object; each
column stands for one feature. We separate X into several groups
based on the number of category in Y.
2.95 6.63
2.53 7.79
X 1 = X 2 =
3.57 5.65
3.16 5.47
2.58 4.46
2.16 6.22
3.27 3.52
μ i = mean of features in group i, which is average of X i
μ 1 = μ 2 =
3.05 6.38 2.67 4.73
μ = global mean vector, that is mean of the whole data set.
μ =
2.88 5.676
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1

Solution…
X’ i = mean corrected data, that is the features data for group
i => X i , minus the global mean vector μ .
0.06 0.951
- 0.357 2.109
X’ 1 = X’ 2 =
0.679 - 0.025
0.269 - 0.209
- 0.305 - 1.218
- 0.732 0.547
0.386 - 2.155
C i = [(X’ i ) T * X’ i ] / n i = covariance matrix of group i.
C 1 = C 2 =
0.166 - 0.192
- 0.192 1.349
0.259 - 0.286
-0.286 2.142
Indian Institute of Information Technology, Allahabad
Indian Institute of Information Technology - Allahabad
1

Solution…
= pooled within group
covariance matrix. It is calculated for each entry (r, s) in the matrix. In
our example,
C =
0.206 - 0.233
- 0.233 1.689
The inverse of the pooled covariance matrix is
C -1 =
5.745 0.791
0.791 0.701
Indian Institute of Information Technology, Allahabad
Indian Institute of Information Technology - Allahabad
1

Solution…
P i = prior probability vector (each row represent prior probability of
group i ). If we do not know the prior probability, we just assume it
is equal to total sample of each group divided by the total samples,
that is P i = n i / N
P =
4 / 7
3 / 7
Discriminant function
f i = μ i C -1 X k
T
– (μ i C -1 μ i
T
) / 2 +ln ( P i )
We should assign object k to group i that has maximum f i
Indian Institute of Information Technology, Allahabad
Indian Institute of Information Technology - Allahabad
1

Solution…
The discriminant function is our classification rules to assign the object into
separate group. If we input the new chip rings that have curvature 2.81 and
diameter 5.46, reveal that it does not pass the quality control.
Transforming all data into discriminant function (f 1 , f 2 ) we can draw
the training data and the prediction data into new coordinate.
Indian Institute of Information Technology, Allahabad
Indian Institute of Information Technology - Allahabad
1

Recognition Procedure with TS-
LDA
X
R 1 R 2
Max (f 1 , f 2 ) = f 1
R 3 R 4
Max (f 3 , f 4 ) = f 4
X
R 1 R 4
X
Max (f 1 , f 4 ) = f 4
X is recognized
as class 4
123
Indian Institute of Information Technology, Allahabad
Indian Institute of Information Technology - Allahabad

Result analysis
Indian Institute of Information Technology, Allahabad
Indian Institute of Information Technology - Allahabad
1

Indian Institute of Information Technology, Allahabad
Indian Institute of Information Technology - Allahabad
1

Performance Evaluation
No. of Subjects
Performance
Gafurov et al. [3][10] 21 5%-13% EER
Nakajima et al. [4] 10 85% RR
Middleton et al. [5] 15 80% RR
Derawi et al. [7], [8] 60 5.7%-20% EER
Rong et al. [9] 35 6.7% EER
Pan et al. [18] 30 96.7% RR
Moustakidis et al. [19] 40 72.89% - 98.21% RR
Our approach 30 100% RR
Indian Institute of Information Technology, Allahabad
Indian Institute of Information Technology - Allahabad
1