Kalman developed the Kalman filter in 1960-1961 to estimate the state of a dynamic system from a series of incomplete and noisy measurements. The Kalman filter uses a recursive Bayesian approach to estimate the state of a system by minimizing the mean of the squared error. It provides an efficient computational means to estimate past, present, and even future states, and can do so even when the precise nature of the modeled system is unknown.

1. Kalman Filter
•Born 1930 in Hungary
•Studied at MIT / Columbia
•Developed filter in 1960/61
•Based on Recursive Bayesian Filter

2. 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

3. 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

4. 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

5. 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

6. Error in GPS(Global Positioning
System)-HISTOGRAM
GPS ERRORS
• 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 etc…
If we were to plot
just the error of
GPS on graph, we
would see a plot
like this
Error
This is really a Good
GPS RECEIVER
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

7. 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 HzSo what if we used the
physical motion of the
vehicle to estimate where
the next measurement
should be and use that for
the error?

8. 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

9. 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

10. 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
Measurements
yk
Pk|k
x̌ k|k
Next Time step
k=k+1
Output Estimate of
State

11. 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

12. 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

13. 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

14. 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

15. Kalman Filter
• Update the estimate with measurements
from Sensor
ŷk=ŷk
- + K (zk-H ŷk
- )
 Ŷk is the state at time k and the output of the filter
 ŷk
- is the estimate of the state we did previously
 K is the Kalman Gain
 Zk is the measurements from Sensor
 H is the model of how the measurements reflect the state of
the Vehicle

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

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.

19. 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

20. What is a Kalman Filter?
-0.5
0
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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

21. 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

22. 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
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23. 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
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