Local Positioning with Parallelepiped Moving Grid

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Local Positioning with Parallelepiped Moving Grid, WPNC06 16.3.2006, niilo.sirola@tut.fi 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

Local Positioning with Parallelepiped Moving Grid, WPNC06 16.3.2006, niilo.sirola@tut.fi 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.

Local Positioning with Parallelepiped Moving Grid, WPNC06 16.3.2006, niilo.sirola@tut.fi p. 3/?? Challenges large linearization errors scarce measurements underdetermined non-normal noise structure?

Local Positioning with Parallelepiped Moving Grid, WPNC06 16.3.2006, niilo.sirola@tut.fi p. 3/?? Challenges large linearization errors scarce measurements underdetermined non-normal noise structure make use of all available information?

Local Positioning with Parallelepiped Moving Grid, WPNC06 16.3.2006, niilo.sirola@tut.fi p. 3/?? Challenges large linearization errors scarce measurements underdetermined non-normal noise structure make use of all available information exploit the time dependency?

Local Positioning with Parallelepiped Moving Grid, WPNC06 16.3.2006, niilo.sirola@tut.fi 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?

Local Positioning with Parallelepiped Moving Grid, WPNC06 16.3.2006, niilo.sirola@tut.fi 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

Local Positioning with Parallelepiped Moving Grid, WPNC06 16.3.2006, niilo.sirola@tut.fi 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

Local Positioning with Parallelepiped Moving Grid, WPNC06 16.3.2006, niilo.sirola@tut.fi 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

Local Positioning with Parallelepiped Moving Grid, WPNC06 16.3.2006, niilo.sirola@tut.fi 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!

Numerical pdf approximations point mass grid mass true pdf Gaussian Monte Carlo Local Positioning with Parallelepiped Moving Grid, WPNC06 16.3.2006, niilo.sirola@tut.fi p. 6/??

Numerical pdf approximations point mass grid mass true pdf Gaussian Monte Carlo Local Positioning with Parallelepiped Moving Grid, WPNC06 16.3.2006, niilo.sirola@tut.fi p. 6/??

Local Positioning with Parallelepiped Moving Grid, WPNC06 16.3.2006, niilo.sirola@tut.fi p. 7/?? Moving grid algorithm 1. prior approximation

Local Positioning with Parallelepiped Moving Grid, WPNC06 16.3.2006, niilo.sirola@tut.fi p. 7/?? Moving grid algorithm 1. prior approximation 2. measurement likelihood

Local Positioning with Parallelepiped Moving Grid, WPNC06 16.3.2006, niilo.sirola@tut.fi p. 7/?? Moving grid algorithm 1. prior approximation 2. measurement likelihood 3. approximate likelihood

Local Positioning with Parallelepiped Moving Grid, WPNC06 16.3.2006, niilo.sirola@tut.fi p. 7/?? Moving grid algorithm 1. prior approximation 2. measurement likelihood 3. approximate likelihood 4. multiply to get posterior

Local Positioning with Parallelepiped Moving Grid, WPNC06 16.3.2006, niilo.sirola@tut.fi p. 7/?? Moving grid algorithm 1. prior approximation 2. measurement likelihood 3. approximate likelihood 4. multiply to get posterior 5. propagate with motion model

Local Positioning with Parallelepiped Moving Grid, WPNC06 16.3.2006, niilo.sirola@tut.fi 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...

Local Positioning with Parallelepiped Moving Grid, WPNC06 16.3.2006, niilo.sirola@tut.fi p. 8/?? Example run true track reference x 0 base station

Local Positioning with Parallelepiped Moving Grid, WPNC06 16.3.2006, niilo.sirola@tut.fi p. 8/?? Example run x 0 true track EKF reference base station

Local Positioning with Parallelepiped Moving Grid, WPNC06 16.3.2006, niilo.sirola@tut.fi p. 8/?? Example run moving grid x 0 true track EKF reference base station

Local Positioning with Parallelepiped Moving Grid, WPNC06 16.3.2006, niilo.sirola@tut.fi p. 9/?? Grid accuracy 2500 2000 error from track cell radius error / m 1500 1000 500 error from reference 0 0 20 40 60 80 100 120 time / s The 2D error of the moving grid filter mean estimate compared to the cell radius

Local Positioning with Parallelepiped Moving Grid, WPNC06 16.3.2006, niilo.sirola@tut.fi 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)

Local Positioning with Parallelepiped Moving Grid, WPNC06 16.3.2006, niilo.sirola@tut.fi 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

Local Positioning with Parallelepiped Moving Grid, WPNC06 16.3.2006, niilo.sirola@tut.fi 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?

Local Positioning with Parallelepiped Moving Grid, WPNC06 16.3.2006, niilo.sirola@tut.fi 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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