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0 3 2 0 1 1 NO-TRACE-STRING
0 4 3 0 1 1 NO-TRACE-STRING
0 1 4 0 1 1 NO-TRACE-STRING
0 2 5 0 1 1 NO-TRACE-STRING
0 3 6 0 1 1 NO-TRACE-STRING
0 4 0 0 1 1 NO-TRACE-STRING
0 1 1 0 1 1 NO-TRACE-STRING
0 2 2 0 1 1 NO-TRACE-STRING
0 3 3 0 1 1 NO-TRACE-STRING
0 4 4 0 1 1 NO-TRACE-STRING
0 1 5 0 1 1 NO-TRACE-STRING
0 2 6 0 1 1 NO-TRACE-STRING
0 3 0 0 1 1 NO-TRACE-STRING
0 4 1 0 1 1 NO-TRACE-STRING
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Cg a24.750000,24.750000,72
{\rtf\ansi \deff0{\colortbl;\red10\green36\blue106;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Based upon Dan Simon's 
Mathematica Kalman filtering example calculation.}
.TXT 8 -40 388 0 0
Cg a55.000000,55.000000,33
{\rtf\ansi \deff0{\colortbl;\red10\green36\blue106;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard {\fs24\b\ul Kalman 
Filter Example Calculation}}
.TXT 6 -24 43 0 0
Cg a71.000000,71.000000,8
{\rtf\ansi \deff0{\colortbl;\red10\green36\blue106;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Delta t:}
.EQN 0 8 44 0 0
{0:\Dt}NAME:0.5
.TXT 0 20 286 0 0
Cg a49.000000,49.000000,11
{\rtf\ansi \deff0{\colortbl;\red10\green36\blue106;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Iterations:}
.EQN 0 12 287 0 0
{0:n}NAME:60
.TXT 6 -40 41 0 0
Cg a71.000000,71.000000,19
{\rtf\ansi \deff0{\colortbl;\red10\green36\blue106;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Acceleration noise:}
.EQN 0 19 40 0 0
{0:a.noise}NAME:0.5
.TXT 7 -19 45 0 0
Cg a71.000000,71.000000,27
{\rtf\ansi \deff0{\colortbl;\red10\green36\blue106;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Position Measurement noise:}
.EQN 0 25 48 0 0
{0:PM.noise}NAME:10
.TXT 9 -25 28 0 0
Cg a71.000000,71.000000,18
{\rtf\ansi \deff0{\colortbl;\red10\green36\blue106;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Transition Matrix:}
.EQN 0 17 29 0 0
{0:a}NAME:({2,2}ö1ö{0:\Dt}NAMEö0ö1)
.TXT 9 -17 30 0 0
Cg a71.000000,71.000000,19
{\rtf\ansi \deff0{\colortbl;\red10\green36\blue106;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Measurement Matrix:}
.EQN 0 20 31 0 0
{0:c}NAME:({1,2}ö0ö1)
.TXT 8 -20 32 0 0
Cg a71.000000,71.000000,21
{\rtf\ansi \deff0{\colortbl;\red10\green36\blue106;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Initial State Vector:}
.EQN 0 19 33 0 0
({0:x}NAME)[(0):({2,1}ö0ö0)
.TXT 9 -19 34 0 0
Cg a71.000000,71.000000,23
{\rtf\ansi \deff0{\colortbl;\red10\green36\blue106;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Initial State Estimate:}
.EQN 0 21 35 0 0
({0:x.hat}NAME)[(0):({0:x}NAME)[(0)
.TXT 9 -21 36 0 0
Cg a71.000000,71.000000,25
{\rtf\ansi \deff0{\colortbl;\red10\green36\blue106;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Process noise covariance:}
.EQN 0 22 178 0 0
{0:Q}NAME:({0:a.noise}NAME)^(2)*({2,2}ö({0:\Dt}NAME)^(2)ö(({0:\Dt}NAME)^(3))/(2)ö(({0:\Dt}NAME)^(3))/(2)ö(({0:\Dt}NAME)^(4))/(4))
.TXT 10 -22 56 0 0
Cg a71.000000,71.000000,30
{\rtf\ansi \deff0{\colortbl;\red10\green36\blue106;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Initial estimation covariance:}
.EQN 0 24 57 0 0
({0:P}NAME)[(0):{0:Q}NAME
.TXT 6 -24 58 0 0
Cg a71.000000,71.000000,29
{\rtf\ansi \deff0{\colortbl;\red10\green36\blue106;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Measurement error covariance:}
.EQN 0 26 59 0 0
{0:R}NAME:({0:PM.noise}NAME)^(2)
.TXT 6 -26 92 0 0
Cg a71.000000,71.000000,61
{\rtf\ansi \deff0{\colortbl;\red10\green36\blue106;}{\fonttbl{\f0
\fcharset0\fnil Arial;}{\f1\fcharset2\fnil Symbol;}}\plain\cf1\fs20 
\pard Define a matrix of random numbers with mean 0 and {\f1 s} = a{
\dn noise}:}
.EQN 0 48 381 0 0
{0:randn.process}NAME:{0:rnorm}NAME({0:n}NAME+1,0,{0:a.noise}NAME)
.TXT 6 -48 94 0 0
Cg a71.000000,71.000000,62
{\rtf\ansi \deff0{\colortbl;\red10\green36\blue106;}{\fonttbl{\f0
\fcharset0\fnil Arial;}{\f1\fcharset2\fnil Symbol;}}\plain\cf1\fs20 
\pard Define a matrix of random numbers with mean 0 and {\f1 s} = PM{
\dn noise}:}
.EQN 0 49 383 0 0
{0:randn.measurement}NAME:{0:rnorm}NAME({0:n}NAME+1,0,{0:PM.noise}NAME)
.EQN 1 47 390 0 0
{0:j}NAME:0;{0:n}NAME
.TXT 2 -98 147 0 0
C x1,1,0,0
.TXT 7 2 357 0 0
Cg a71.000000,71.000000,21
{\rtf\ansi \deff0{\colortbl;\red10\green36\blue106;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Simulate the process:}
.EQN 0 21 356 0 0
({0:x}NAME)[({0:j}NAME+1):{0:a}NAME*({0:x}NAME)[({0:j}NAME)+({0:randn.process}NAME)[({0:j}NAME)*({2,1}ö{0:\Dt}NAMEö(({0:\Dt}NAME)^(2))/(2))
.TXT 7 -21 362 0 0
Cg a71.000000,71.000000,25
{\rtf\ansi \deff0{\colortbl;\red10\green36\blue106;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Simulate the measurement:}
.EQN 0 25 363 0 0
({0:z}NAME)[({0:j}NAME):{0:c}NAME*({0:x}NAME)[({0:j}NAME)+({0:randn.measurement}NAME)[({0:j}NAME)
.TXT 7 -25 364 0 0
Cg a71.000000,71.000000,33
{\rtf\ansi \deff0{\colortbl;\red10\green36\blue106;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Calculate the estimated 
position:}
.EQN 0 27 365 0 0
{0:x.hat}NAME:((((({0:x.hat}NAME)[(0)){72}(({2,1}ö0ö0))){71}((({0:P}NAME)[(0)){72}({0:Q}NAME))){71}(((((((({0:s}NAME)[({0:j}NAME)){72}({0:c}NAME*({0:P}NAME)[({0:j}NAME)*({0:c}NAME){51}+{0:R}NAME)){71}((({0:K}NAME)[({0:j}NAME)){72}(({0:a}NAME*(
{0:P}NAME)[({0:j}NAME)*({0:c}NAME){51})*((({0:s}NAME)[({0:j}NAME)))^(-1)))){71}((({0:I}NAME)[({0:j}NAME)){72}(({0:z}NAME)[({0:j}NAME)-{0:c}NAME*({0:a}NAME*({0:x.hat}NAME)[({0:j}NAME))))){71}((({0:P}NAME)[({0:j}NAME+1)){72}({0:a}NAME*({0:P}NAME)[(
{0:j}NAME)*({0:a}NAME){51}+{0:Q}NAME-({0:K}NAME)[({0:j}NAME)*{0:c}NAME*({0:P}NAME)[({0:j}NAME)*({0:a}NAME){51}))){71}((({0:x.hat}NAME)[({0:j}NAME+1)){72}({0:a}NAME*({0:x.hat}NAME)[({0:j}NAME)+({0:K}NAME)[({0:j}NAME)*({0:I}NAME)[({0:j}NAME)))){79}((
{0:j}NAME){72}(0;{0:n}NAME)))){71}({0:x.hat}NAME)
.TXT 11 -14 366 0 0
Cg a71.000000,71.000000,25
{\rtf\ansi \deff0{\colortbl;\red10\green36\blue106;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Covariance of innovation:}
.TXT 4 9 367 0 0
Cg a71.000000,71.000000,12
{\rtf\ansi \deff0{\colortbl;\red10\green36\blue106;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Gain matrix:}
.TXT 4 1 368 0 0
Cg a73.000000,73.000000,11
{\rtf\ansi \deff0{\colortbl;\red10\green36\blue106;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Innovation:}
.TXT 5 -14 369 0 0
Cg a71.000000,71.000000,31
{\rtf\ansi \deff0{\colortbl;\red10\green36\blue106;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Covariance of prediction error:}
.TXT 3 7 400 0 0
Cg a71.000000,71.000000,19
{\rtf\ansi \deff0{\colortbl;\red10\green36\blue106;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Estimated position:}
.EQN 7 80 391 0 0
({0:Pos}NAME)[({0:j}NAME):((({0:x}NAME)[({0:j}NAME)))[(0)
.EQN 0 12 392 0 0
({0:Pos.meas}NAME)[({0:j}NAME):((({0:z}NAME)[({0:j}NAME)))[(0)
.EQN 0 16 393 0 0
({0:Pos.hat}NAME)[({0:j}NAME):((({0:x.hat}NAME)[({0:j}NAME)))[(0)
.EQN 0 19 394 0 0
({0:t}NAME)[({0:j}NAME):{0:\Dt}NAME*{0:j}NAME
.TXT 1 -143 370 0 0
Cg a71.000000,71.000000,15
{\rtf\ansi \deff0{\colortbl;\red10\green36\blue106;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard State estimate:}
.EQN 1 0 376 0 0
&&(_n_u_l_l_&_n_u_l_l_)&({0:Pos}NAME)[({0:j}NAME),({0:Pos.meas}NAME)[({0:j}NAME),({0:Pos.hat}NAME)[({0:j}NAME)@&&(_n_u_l_l_&_n_u_l_l_)&({0:t}NAME)[({0:j}NAME)
0 0 1 1 1 0 0 1 1 Time (seconds)
0 0 1 1 1 0 0 1 1 
0 4 3 0 1 1 True Position
0 2 1 0 1 1 Measured Pos
0 1 0 0 1 1 Estimated Pos
0 4 3 0 1 1 NO-TRACE-STRING
0 1 4 0 1 1 NO-TRACE-STRING
0 2 5 0 1 1 NO-TRACE-STRING
0 3 6 0 1 1 NO-TRACE-STRING
0 4 0 0 1 1 NO-TRACE-STRING
0 1 1 0 1 1 NO-TRACE-STRING
0 2 2 0 1 1 NO-TRACE-STRING
0 3 3 0 1 1 NO-TRACE-STRING
0 4 4 0 1 1 NO-TRACE-STRING
0 1 5 0 1 1 NO-TRACE-STRING
0 2 6 0 1 1 NO-TRACE-STRING
0 3 0 0 1 1 NO-TRACE-STRING
0 4 1 0 1 1 NO-TRACE-STRING
1 0 2 67 38 10 0 2 Kalman Filter Performance