The Kalman Filter (part 1) Administrative Stuff Rudolf Emil Kalman - - PowerPoint PPT Presentation

▶
the kalman filter part 1 administrative stuff rudolf emil
SMART_READER_LITE
LIVE PREVIEW

The Kalman Filter (part 1) Administrative Stuff Rudolf Emil Kalman - - PowerPoint PPT Presentation

Feb 13, 2024 768 likes •1.51k views

The Kalman Filter (part 1) Administrative Stuff Rudolf Emil Kalman [http://www.cs.unc.edu/~welch/kalman/kalmanBiblio.html] Definition A Kalman filter is simply an optimal recursive data processing algorithm Under some assumptions the

slide-1
SLIDE 1

The Kalman Filter (part 1)

slide-2
SLIDE 2

Administrative Stuff

slide-3
SLIDE 3

Rudolf Emil Kalman

[http://www.cs.unc.edu/~welch/kalman/kalmanBiblio.html]

slide-4
SLIDE 4

Definition

  • A Kalman filter is simply an optimal

recursive data processing algorithm

  • Under some assumptions the Kalman filter

is optimal with respect to virtually any criterion that makes sense.

slide-5
SLIDE 5

Definition

“The Kalman filter incorporates all information that can be provided to it. It processes all available measurements, regardless of their precision, to estimate the current value of the variables of interest.”

[Maybeck (1979)]

slide-6
SLIDE 6

Why do we need a filter?

  • No mathematical model of a real

system is perfect

  • Real world disturbances
  • Imperfect Sensors
slide-7
SLIDE 7

Application: Radar Tracking

slide-8
SLIDE 8

Application: Lunar Landing

https://github.com/chrislgarry/Apollo-11

slide-9
SLIDE 9
slide-10
SLIDE 10

Application: Missile Tracking

slide-11
SLIDE 11

Application: Sailing

slide-12
SLIDE 12

Application: Robot Navigation

slide-13
SLIDE 13

Application: Other Tracking

slide-14
SLIDE 14

Application: Head Tracking

slide-15
SLIDE 15

Face & Hand Tracking

slide-16
SLIDE 16

A Simple Recursive Example

  • Problem Statement:

Given the measurement sequence: z1, z2, …, zn find the mean

[Brown and Hwang (1992)]

slide-17
SLIDE 17

First Approach

  • 1. Make the first measurement z1

Store z1 and estimate the mean as µ1=z1

  • 2. Make the second measurement z2

Store z1 along with z2 and estimate the mean as µ2= (z1+z2)/2

[Brown and Hwang (1992)]

slide-18
SLIDE 18

First Approach (cont’d)

  • 3. Make the third measurement z3

Store z3 along with z1 and z2 and estimate the mean as µ3= (z1+z2+z3)/3

[Brown and Hwang (1992)]

slide-19
SLIDE 19

First Approach (cont’d)

  • n. Make the n-th measurement zn

Store zn along with z1 , z2 ,…, zn-1 and estimate the mean as µn= (z1 + z2 + … + zn)/n

[Brown and Hwang (1992)]

slide-20
SLIDE 20

Second Approach

  • 1. Make the first measurement z1

Compute the mean estimate as µ1=z1 Store µ1 and discard z1

[Brown and Hwang (1992)]

slide-21
SLIDE 21

Second Approach (cont’d)

  • 2. Make the second measurement z2

Compute the estimate of the mean as a weighted sum of the previous estimate µ1 and the current measurement z2:

µ2= 1/2 µ1 +1/2 z2

Store µ2 and discard z2 and µ1

[Brown and Hwang (1992)]

slide-22
SLIDE 22

Second Approach (cont’d)

  • 3. Make the third measurement z3

Compute the estimate of the mean as a weighted sum of the previous estimate µ2 and the current measurement z3: µ3= 2/3 µ2 +1/3 z3 Store µ3 and discard z3 and µ2

[Brown and Hwang (1992)]

slide-23
SLIDE 23

Second Approach (cont’d)

  • n. Make the n-th measurement zn

Compute the estimate of the mean as a weighted sum of the previous estimate µn-1 and the current measurement zn: µn= (n-1)/n µn-1 +1/n zn Store µn and discard zn and µn-1

[Brown and Hwang (1992)]

slide-24
SLIDE 24

Comparison

Batch Method Recursive Method

slide-25
SLIDE 25

Analysis

  • The second procedure gives the same

result as the first procedure.

  • It uses the result for the previous step to

help obtain an estimate at the current step.

  • The difference is that it does not need to

keep the sequence in memory.

[Brown and Hwang (1992)]

slide-26
SLIDE 26

Second Approach (rewrite the general formula)

µn= (n-1)/n µn-1 +1/n zn µn= (n/n) µn-1 - (1/n) µn-1 +1/n zn

slide-27
SLIDE 27

Second Approach (rewrite the general formula)

µn= (n-1)/n µn-1 +1/n zn µn= (n/n) µn-1 - (1/n) µn-1 +1/n zn µn= µn-1 + 1/n (zn - µn-1)

Old Estimate Difference Between New Reading and Old Estimate Gain Factor

slide-28
SLIDE 28

Second Approach (rewrite the general formula)

Old Estimate Difference Between New Reading and Old Estimate Gain Factor

slide-29
SLIDE 29

Gaussian Properties

slide-30
SLIDE 30

The Gaussian Function

slide-31
SLIDE 31

Gaussian pdf

slide-32
SLIDE 32

Properties

  • If

and

  • Then
slide-33
SLIDE 33

pdf for

slide-34
SLIDE 34

Properties

slide-35
SLIDE 35

Summation and Subtraction

slide-36
SLIDE 36

A simple example using diagrams

slide-37
SLIDE 37

Conditional density of position based on measured value of z1

[Maybeck (1979)]

slide-38
SLIDE 38

Conditional density of position based on measured value of z1

[Maybeck (1979)]

position measured position uncertainty

slide-39
SLIDE 39

Conditional density of position based on measurement of z2 alone

[Maybeck (1979)]

slide-40
SLIDE 40

Conditional density of position based on measurement of z2 alone

[Maybeck (1979)]

measured position 2 uncertainty 2

slide-41
SLIDE 41

Conditional density of position based on data z1 and z2

[Maybeck (1979)]

position estimate uncertainty estimate

slide-42
SLIDE 42

Propagation of the conditional density

[Maybeck (1979)]

slide-43
SLIDE 43

Propagation of the conditional density

[Maybeck (1979)]

movement vector expected position just prior to taking measurement 3

slide-44
SLIDE 44

Propagation of the conditional density

[Maybeck (1979)]

movement vector expected position just prior to taking measurement 3

slide-45
SLIDE 45

Propagation of the conditional density

z3 σx(t3)

measured position 3 uncertainty 3

slide-46
SLIDE 46

Updating the conditional density after the third measurement

z3 σx(t3)

position uncertainty position estimate

x(t3)

slide-47
SLIDE 47
slide-48
SLIDE 48

Questions?

slide-49
SLIDE 49

Now let’s do the same thing …but this time we’ll use math

slide-50
SLIDE 50

How should we combine the two measurements?

[Maybeck (1979)] σZ1 σZ2

slide-51
SLIDE 51

Calculating the new mean

Scaling Factor 1 Scaling Factor 2

slide-52
SLIDE 52

Calculating the new mean

Scaling Factor 1 Scaling Factor 2

slide-53
SLIDE 53

Calculating the new mean

Scaling Factor 1 Scaling Factor 2

Why is this not z1?

slide-54
SLIDE 54

Calculating the new variance

[Maybeck (1979)] σZ1 σZ2

slide-55
SLIDE 55

Calculating the new variance

Scaling Factor 1 Scaling Factor 2

slide-56
SLIDE 56

Remember the Gaussian Properties?

slide-57
SLIDE 57

Remember the Gaussian Properties?

  • If

and

  • Then

This is a2 not a

slide-58
SLIDE 58

The scaling factors must be squared!

Scaling Factor 1 Scaling Factor 2

slide-59
SLIDE 59

The scaling factors must be squared!

Scaling Factor 1 Scaling Factor 2

slide-60
SLIDE 60

Therefore the new variance is

Try to derive this on your own.

slide-61
SLIDE 61

Another Way to Express The New Position

[Maybeck (1979)]

slide-62
SLIDE 62

Another Way to Express The New Position

[Maybeck (1979)]

slide-63
SLIDE 63

Another Way to Express The New Position

[Maybeck (1979)]

slide-64
SLIDE 64

The equation for the variance can also be rewritten as

[Maybeck (1979)]

slide-65
SLIDE 65

Adding Movement

[Maybeck (1979)]

slide-66
SLIDE 66

Adding Movement

[Maybeck (1979)]

slide-67
SLIDE 67

Adding Movement

[Maybeck (1979)]

slide-68
SLIDE 68

Properties of K

  • If the measurement noise is large K is small

[Maybeck (1979)]

slide-69
SLIDE 69

The Kalman Filter (part 2)

slide-70
SLIDE 70

Example Applications

slide-71
SLIDE 71

https://www.youtube.com/watch?v=MxwVwCuBEDA https://github.com/pabsaura/Prediction-of-Trajectory-with-kalman-filter-and-open-cv

slide-72
SLIDE 72

https://www.youtube.com/watch?v=sG-h5ONsj9s https://www.myzhar.com/blog/tutorials/tutorial-opencv-ball-tracker-using-kalman-filter/

slide-73
SLIDE 73

THE END