principles of radar tracking using the kalman filter to locate targets
TRANSCRIPT
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Principles of Radar Tracking
Using the Kalman Filter to locate targets
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Abstract
Problem-Tracking moving targets, minimize radar noise
Solution-Use the Kalman Filter to largely eliminate noise when determining the velocities and distances
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Noise
• Error (noise) is described by an ellipse– Defined by variance and covariance in x
and y
• Two kinds of error– State– Measurement
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TeamsReciproverse
Brian DaiJoshua NewmanMichael Sobin
LextenStephen ChanAdam LloydJonathan MacMillanAlex Morrison
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History of the Kalman Filter
• Problem: 1960’s, Apollo command capsule
• Dr. Kalman and Dr. Bucy– Make highly adaptable iterative
algorithm– No previous data storage– Estimates non-measured quantities
(velocity)
• Later found to be useful for other applications, such as air traffic control
Dr. Kalman
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Model
kkk
kkk
rHxy
qΦxx
1
xk: position and velocity (state) of the target at time k (k+1 is next time step)Φ: state transition matrix qk: uncertainty in the state due to “noise” (e.g. wind variation and pilot error)
yk: measurement at time kH: term that gets rid of velocity in Xr: measurement noise, dictated by our devices
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Other Important Matrices
• P: error covariance matrix– Describes estimate accuracy
• K: Kalman gain matrix– Intermediate weighting factor between
measured and predicted
• I: identity matrix
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Some Matrices
y
y
x
x
x
2222
2222
2222
2222
yyyyxyx
yyyyxxy
yxyxxxx
yxxyxxx
P
m
m
y
xy
22
22
mmm
mmm
yyx
yxx
R
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Kalman Filter: Predict
kkk xΦx ˆˆ |1
QΦΦPP T
kkk |1
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Kalman Filter: Correct
1|1|| ˆˆˆ kkkkkkkk xHyKxx
1|| kkkkk PHKIP
1
1|1|
RHHPHPK Tkk
Tkkk
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Tools: Visual Basic• Matlib- an external matrix operations
library• Input format – text files, simulated
radar data• Console- data output
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Tools: Excel Track Charts
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-10
0
0 5 10 15 20 25 30 35
Truth
Our Results
Raw Data
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Tools: Excel Residual Analysis
Residual Plot
0
0.5
1
1.5
2
2.5
3
Time (hr)
Re
sid
ua
l (m
i)
Our ResidualRaw Data residual
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Filter Development: Cartesian Coordinates
• Filter Implemented• Test: Residual Analysis• Does it work?
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Cartesian Residuals
Residual, Case 2
0
0.5
1
1.5
2
2.5
3
0.02
50
0.04
17
0.05
83
0.07
50
0.09
17
0.10
83
0.12
50
0.14
17
0.15
83
0.17
50
0.19
17
0.20
83
0.22
50
0.24
17
Time (hours)
Res
idu
al (
mil
es)
Our Residual
Raw Data residual
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Filter Development: Polar Coordinates
• Prefiltering• Polar to
Cartesian conversion
• More appropriate data feed
• Error matrices– Redefine R
][2sin2
1
cossin
sincos
22222
222222
222222
mmmm
mmm
mmm
R
R
R
Ryx
Ry
Rx
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Filter Development: Multiple Radars
• Mapping coordinates to absolute coordinate plane
• Two radars means a smaller error ellipse
• Note drop in residual– Switch to second
radar
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Residual, Case 4
0
1
2
3
4
5
6
7
8
0.02
50
0.03
33
0.04
17
0.05
00
0.05
83
0.06
67
0.07
50
0.08
33
0.10
00
0.10
83
0.12
50
0.13
33
0.14
17
0.15
83
0.16
67
0.19
17
0.20
00
0.20
83
Time (hours)
Res
idu
al (
mil
es)
Our Residual
Raw Data Residual
Multiple Radar Residuals
Radar 2 starts
Radar 1
Radar 2 to end
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Maneuvering Targets
• Filter Reinitialization– 3σ error ellipse
(~98%)– If three consecutive
data points outside ellipse, reinitialize filter
– Should happen upon maneuvering
• Prevents biased prediction matrix
3σ
GOOD
Predicted point
BAD
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Maneuvering UFO
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0
10
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Truth Track
Our Data
Raw Data
Maneuvering Target Tracks
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Maneuvering Target Residuals
Residual, Case 5
0
1
2
3
4
5
6
7
8
9
0.00
00
0.03
32
0.06
64
0.09
96
0.13
28
0.16
60
0.19
92
0.23
24
0.26
56
0.29
88
0.33
20
0.36
52
0.39
84
0.43
16
0.46
48
Time (hrs)
Re
sid
ua
l (m
i)
Residual of Us
Residual of Data
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Interception
• Give interceptor path using filter– Interceptor: constant velocity– Intercept UFO
• Cross target path before designated time
• Solve using Law of Cosines
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Interception Triangles
vt (from filter)
Dist plane-UFO
630t
Intercept pt
Current plane pt
Current UFO pt
β
θ
Δy
Δx
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Interceptor Equations
vt
Dist
Current UFO pt
β
x
y
x
y
v
v
Dist
Distarctanarctan
Disty
Distx
vy
vx
Current plane pt
Intercept pt
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Interceptor Equations
vt
Dist630t
Current UFO pt
β
22
2222
22222
630
)(630cos)(cos)(
cos)(2630:Cosines of Law
v
distvdistvdistvt
distvtdisttvt
Intercept pt
Current plane pt
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Interceptor Equations
630t (course
of plane)
Intercept pt
Current plane pt
θ
Δy
Δx
x
yarctan
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Interceptor Track
Maneuvering Plane with Interceptor
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Truth
Data
Our Results
Interceptor
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Multiple Targets
• Tracking multiple targets lends itself to an object oriented approach
• Why is it useful? Collision avoidance
Target Class
Methods:
•Initialize
•Predict
•Correct
Matrices
•X
•Y
•P
•R
Target Object
Target Object
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Collision Avoidance
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Collision Avoidance Math
Express position at a future time t:
tvyy
tvxx
y
x
111
111
ˆ
ˆ
Plane 1:
tvyy
tvxx
y
x
222
222
ˆ
ˆ
Plane 2:
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Collision Avoidance Math
1ˆˆˆˆ 212
212 yyxx
12
12
12
12
yyy
xxx
vvv
vvv
yyy
xxx
Determine if planes will be within one mile at any such time:
Make some substitutions to simplify the expression:
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Collision Avoidance Math
Arrive at inequality describing dangerous time interval:
The solution to this inequality is the time intervalwhen the planes will be in danger
01 2 22222 yxtvyvxtvv yxyx
1
222
22
yxc
vyvxb
vva
yx
yx
02 cbtat
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Collision Tracks
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0
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30
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Plane 1Plane 2Estimated Collision Interval
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Conclusion
• Using the Kalman filter, we were able to minimize radar noise and analyze target tracking scenarios.
• We solved: plane collision avoidance, interception, tracking multiple aircraft
• Still relevant today: several space telescopes use the Kalman Filter as a low powered tracking device
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Acknowledgements
• Mr. Randy Heuer• Zack Vogel• Dr. Paul Quinn• Dr. Miyamoto• Ms. Myrna Papier• NJGSS ’07 Sponsors
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Works Cited
• http://www.physics.utah.edu/~detar/phycs6720/handouts/curve_fit/curve_fit/img147.gif
• http://www.afrlhorizons.com/Briefs/Mar02/OSR0106.html
• http://www.cs.unc.edu/~welch/kalman/media/images/kalman-new.jpg
• http://www.combinatorics.org/Surveys/ds5/gifs/5-VD-ellipses-labelled.gif