The document discusses Kalman filters, which are algorithms used to estimate the internal state of a system from a series of noisy measurements. Specifically: - Kalman filters were developed in the 1960s to estimate the state of dynamic systems and filter out noise from sensor measurements in a reliable way. - They work by using a system's predicted state and measurements from sensors to produce estimates of the true state that are better than either the predictions or measurements alone. - The algorithm models the dynamic behavior of the system being estimated as well as the measurement noise characteristics, and uses these to produce an optimal estimate.

1. KALMAN FILTER
E . CINTHURIYA
ME – COMMUNICATION ENGINEERING
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2. Kalman Filter
•Born 1930 in Hungary
•Studied at MIT / Columbia
•Developed filter in 1960/61
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3. Kalman Filters
 A Kalman Filter is a more
sophisticated smoothing algorithm that
will actually change in real time as the
performance of Various Sensors
Change and become more or less
reliable
 What we want to do is filter out noise
in our measurements and in our
sensors and Kalman Filter is one way
to do that reliably
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4. Noise
This is the actual path of our
UGV(Unmanned Ground Vehicle) as
it follows unwinding road.
This is the data we got
from GPS.
Notice it has the same
shape with some random
variations plus and minus.
Ok . How about if we just
average out our GPS
reading?
That’s a bit better and smoothes out some of the
bumps but it is not useful since GPS is slow already
and averaging makes it much slower
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5. Noise
 Removing Noise from Measurements by
Sensor is the purpose of Kalman Filter
 When you average the sensor signals or
readings it will slow down the system
Path of a Robot
Data We Get from SENSOR
Data after we average the Signals kalman filter 5/18

6. Kalman Filter-Basic Block
Diagram
 System state cannot be measured directly
 Need to estimate “optimally” from measurements
Measuring
Devices Estimator
Measurement
Error Sources
System State
(desired but not
known)
External
Controls
Observed
Measurements
Optimal
Estimate of
System State
System
Error Sources
System
Black Box
Sometimes the system state
and the measurement may be
two different things
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7. Error in PS(Global Positioning
System)-HISTOGRAM
 We usually get GPS from manufacturers with
specification specifying some percentage of Accuracy
with in Some meters. For example we may get 95%
accuracy with in 2 meters If we were to plot
just the error of
GPS on graph, we
would see a plot
like this
Erro
r
As a review,
Histogram gets
taller the more
measurements
are at a particular
value
Most of the time GPS is
inside this error range.
Notice that this makes a
particular distribution of
values that fall in certain
range. Let's say that as
4 meters
4m
So the Noise is
this random bits
of stuff here at
the bottom
So we could make a filter
that would get rid of this
noise.BUT we can’t
measure our own error
while we are moving
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8. Moving Robot – Estimating Error
Path followed by robot
according to law of motion
Sensor
Readings
Sensor Readings doesn’t follow
the Law of Physics or
Equilibrium of Motion which is
Y=x + vt + 1/2at2
Here’s the real path of
UGV over the ground
with boxes showing
1second interval
Now here’s the
GPS reading we
received at 1 Hz
So what if we used the
physical motion of the
vehicle to estimate where
the next measurement
should be and use that for
the error?
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9. Moving Robot – Estimating Error
Equilibrium of Motion which is
Y=x + vt + 1/2at2
X=position
V=velocity(meters per second)
a= acceleration(meters per
second squared
T=time interval
We also Know if we commanded
the vehicle to change course of
Speed- a steering input or
throttle change
With this sort of straight line motion
the state estimate can be very
accurate as the UGV is moving in a
constant direction and at constant
speed.
So we might see something like this
out of a continuously updated
estimation function
20
mph
20
mph
20
mph
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10. Kalman Filter Concept
 We can’t directly measure where the UGV(Unmanned Ground
Vehicle Robot) is- We have to use various Sensors to make
estimates.
 Each of those Sensors has a Certain amount of Accuracy and a
Certain amount of NOISE.
 We can use Equation of Motion to provide estimates on where
UGV may have moved and then see if Sensor Readings makes
sense given that estimate.
 Then we can Update our Estimates with new Sensor information
and whole cycle starts over again.
 Kalman filter is all done with matrix math
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11. The Kalman Filter
Prior Knowledge
of State
Pk-1|k-1
x̌ k-1|k-1
Prediction Step
Based on
Example Physical
Model
Pk|k-1
x̌ k|k-1
Update Step
Compare
Prediction to
Measurements
Measurement
s
yk
Pk|k
x̌ k|k
Next Time
step
k=k+1
Output Estimate
of State
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12. Set of Kalman filter Equations in
Detail
Prediction(Time Update)
1) Project the STATE
ahead
ŷk
-=Ayk-1+Buk
2) Project the Error
Covariance ahead
P-
k≡APk-1AT+Q
Correction (Measurement
Update)
1) Compute the KALMAN
GAIN
K=P-
kHT(HP-
kHT+R)-1
2) Update the estimate with
measurement zk
ŷk=ŷk
- + K (zk-H ŷk
- )
3) Update the Error Covariance
Pk=(I-KH)P-
k
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13. Kalman Filter
1. Project the STATE ahead
ŷk
-=Ayk-1+Buk
 Ŷk is the predicted state of Vehicle at time k
 A is the model(equations of motion) that predict the
new state
 yk-1 is the state of Vehicle at previous time k-1
 B is the model that predicts what changes based on
commands to the vehicle – increase in throttle or
steering
 Uk is the commanded inputs at time k
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14. Kalman Filter
2. Project the Error Covariance ahead
 We want to predict how much noise will be in our
measurements
P-
k≡APk-1AT+Q
 A, same as first equation,Model of Motion
 P,the previous error value at time k-1
 AT,the model Transposed
 Q,the covariance of Error noise – describes the
distribution of noise
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15. Kalman Filter
1. Computing the KALMAN GAIN
K=P-
kHT(HP-
kHT+R)-1
 K, the Kalman gain(How much to trust this sensor)
 P-
k, as before, the Predicted Error Covariance
 H,the model of how Sensor readings reflect the
vehicle state – a function to go from Sensor
Reading to STATE VECTOR
 R describes the noise in Sensor Measurement
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16. Kalman Filter
 Update the Error Covariance
Pk=(I-KH)P-
k
 Pk is our new Error Covariance(description of the error
noise Gaussian Curve)
 I is the Identity Matrix
 K is the Kalman Gain
 H is the measurement Model
 P-
k was the previous estimate of error noise
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17. Problems with Kalman Filters
 It is very difficult to compute the covariance matrix
of noise of various sensors and systems
 It is possible to do some trial and error fitting of
error matrix to the problem to “tune” the filter for
performance
 The filter needs to process several samples in
order to get enough iterations to produce
meaningful results – You need to discard the first
20 iterations or so when the filter first starts.
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18. What is a Kalman Filter?-
Another Interpretation
 P(H|D)=P(H) * P(D|H)
 P(H)=Probability of Hypothesis
 P(D)=Probability of Data
 Kalman Filter is based on
 P(Ht|Ht-1,Dt) => P(Ht|Ht-1,At,Dt) Where At =Action State
Trash
WALL-E
GPS
Robot
Trash
Robot Picks
up Trash
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19. What is a Kalman Filter?
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
0 0.5 1 1.5 2 2.5 3 3.5 4
Position
Time
Probability Where
Robot Is
Position Where
Robot is Sent
Where GPS
THINKS Robot is
Combination Where
Robot Actually is
Xt
µ-Motor Command
Xt-1
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20. What is a Kalman Filter?
STATE PREDICTION:
Xt=Axt-1+Bµt+€t
Where €t is Gaussian Error. It is a Linear Function
Based on Rule of Physics.
SENSOR PREDICTION:
Zt=Cxt +€t
Xest =Xt+K(Zt-Žt) WHERE
•Xt=Prediction
•K=Kalman Gain
•Zt =Actual Measurement
•Žt =Predicted Measurement
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21. What is a Kalman Filter?
STATE PREDICTION MODEL:
Xt=Axt-1+Bµt+€t
Where €t is Gaussian Error. It is a Linear Function
Based on Rule of Physics.
Pt=Pt-1+Vt-1.t+1/2 att2
Vt=Vt-1+att 
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22. What is a Kalman Filter?
STATE PREDICTION MODEL:
Pt=Pt-1+Vt-1.t+1/2 att2
Vt=Vt-1+att
Measurement Prediction:
Zt=Cxt +€t
k
tTav
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23. kalman filter
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