Local Positioning with Parallelepiped Moving Grid
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1 Local Positioning with Parallelepiped Moving Grid, WPNC , p. 1/?? TA M P E R E U N I V E R S I T Y O F T E C H N O L O G Y M a t h e m a t i c s Local Positioning with Parallelepiped Moving Grid Niilo Sirola and Simo Ali-Löytty Tampere University of Technology, Finland
2 Local Positioning with Parallelepiped Moving Grid, WPNC , p. 2/?? Personal positioning for example, a mobile handset sources of positioning information: navigation equipment: GPS, IMU cellular network, WLAN, Bluetooth,... digital compass, step counter, barometer aiding data (altitude, cell sector,... ) digital maps etc.
3 Local Positioning with Parallelepiped Moving Grid, WPNC , p. 3/?? Challenges large linearization errors scarce measurements underdetermined non-normal noise structure?
4 Local Positioning with Parallelepiped Moving Grid, WPNC , p. 3/?? Challenges large linearization errors scarce measurements underdetermined non-normal noise structure make use of all available information?
5 Local Positioning with Parallelepiped Moving Grid, WPNC , p. 3/?? Challenges large linearization errors scarce measurements underdetermined non-normal noise structure make use of all available information exploit the time dependency?
6 Local Positioning with Parallelepiped Moving Grid, WPNC , p. 3/?? Challenges large linearization errors scarce measurements underdetermined non-normal noise structure make use of all available information exploit the time dependency allow flexible measurement models?
7 Local Positioning with Parallelepiped Moving Grid, WPNC , p. 4/?? Discrete nonlinear filtering problem Distribution of initial state x 0 given. Motion model: Measurement model: x k+1 = f k (x k ) + w k y k = h k (x k ) + v k Optimal solution: recursive Bayesian filter
8 Local Positioning with Parallelepiped Moving Grid, WPNC , p. 5/?? Recursive Bayesian filter Prediction step: p(x k Y k 1 ) = p(x k x k 1 ) p(x }{{} k 1 Y k 1 ) }{{} R d motion model previous dx k 1
9 Local Positioning with Parallelepiped Moving Grid, WPNC , p. 5/?? Recursive Bayesian filter Prediction step: p(x k Y k 1 ) = p(x k x k 1 ) p(x }{{} k 1 Y k 1 ) }{{} R d Update step: motion model p(x k Y k ) p(y k x k ) p(x }{{} k Y k 1 ) measurement previous dx k 1
10 Local Positioning with Parallelepiped Moving Grid, WPNC , p. 5/?? Recursive Bayesian filter Prediction step: p(x k Y k 1 ) = p(x k x k 1 ) p(x }{{} k 1 Y k 1 ) }{{} R d Update step: motion model p(x k Y k ) p(y k x k ) p(x }{{} k Y k 1 ) measurement previous dx k 1 In general, exact solution requires infinite time and memory!
11 Numerical pdf approximations point mass grid mass true pdf Gaussian Monte Carlo Local Positioning with Parallelepiped Moving Grid, WPNC , p. 6/??
12 Numerical pdf approximations point mass grid mass true pdf Gaussian Monte Carlo Local Positioning with Parallelepiped Moving Grid, WPNC , p. 6/??
13 Local Positioning with Parallelepiped Moving Grid, WPNC , p. 7/?? Moving grid algorithm 1. prior approximation
14 Local Positioning with Parallelepiped Moving Grid, WPNC , p. 7/?? Moving grid algorithm 1. prior approximation 2. measurement likelihood
15 Local Positioning with Parallelepiped Moving Grid, WPNC , p. 7/?? Moving grid algorithm 1. prior approximation 2. measurement likelihood 3. approximate likelihood
16 Local Positioning with Parallelepiped Moving Grid, WPNC , p. 7/?? Moving grid algorithm 1. prior approximation 2. measurement likelihood 3. approximate likelihood 4. multiply to get posterior
17 Local Positioning with Parallelepiped Moving Grid, WPNC , p. 7/?? Moving grid algorithm 1. prior approximation 2. measurement likelihood 3. approximate likelihood 4. multiply to get posterior 5. propagate with motion model
18 Local Positioning with Parallelepiped Moving Grid, WPNC , p. 7/?? Moving grid algorithm 1. prior approximation 2. measurement likelihood 3. approximate likelihood 4. multiply to get posterior 5. propagate with motion model 6. repeat from step 2...
19 Local Positioning with Parallelepiped Moving Grid, WPNC , p. 8/?? Example run true track reference x 0 base station
20 Local Positioning with Parallelepiped Moving Grid, WPNC , p. 8/?? Example run x 0 true track EKF reference base station
21 Local Positioning with Parallelepiped Moving Grid, WPNC , p. 8/?? Example run moving grid x 0 true track EKF reference base station
22 Local Positioning with Parallelepiped Moving Grid, WPNC , p. 9/?? Grid accuracy error from track cell radius error / m error from reference time / s The 2D error of the moving grid filter mean estimate compared to the cell radius
23 Local Positioning with Parallelepiped Moving Grid, WPNC , p. 10/?? How to compare filters? Ideally: compare the posterior distribution to the ideal one In practise we compute the mean and covariance estimates error from true track error from reference track consistency (error vs. estimated covariance)
24 Local Positioning with Parallelepiped Moving Grid, WPNC , p. 11/?? Some numbers mean error 95 % CERP inconsistent time grid 200 m 574 m <0.1 % 100 EKF 105 m 370 m 4 % 1 SMC, 10k 75 m 260 m 1 % 30 SMC, 2M 75 m 255 m <0.1 % 10000
25 Local Positioning with Parallelepiped Moving Grid, WPNC , p. 12/?? Conclusions grid filter gives rough but reliable results fair and expressive comparison of nonlinear filters still an open problem how to weigh the overall score? compare to true track or reference track? accuracy vs. reliability accuracy vs. computation time?
26 Local Positioning with Parallelepiped Moving Grid, WPNC , p. 12/?? Conclusions grid filter gives rough but reliable results fair and expressive comparison of nonlinear filters still an open problem how to weigh the overall score? compare to true track or reference track? accuracy vs. reliability accuracy vs. computation time? Questions?
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