Fingerprint Kalman Filter in Indoor Positioning ... - Semantic Scholar
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
AP Signal strength Glass door
Mathematics
——–
Fingerprint Kalman Filter in Indoor Positioning Applications MSC 2009 Simo Ali-Löytty, Tommi Perälä, Ville Honkavirta and Robert Piché http://math.tut.fi/posgroup/ ¨ Simo Ali-Loytty – p.1/21
Outline Radio map (offline phase) State of the art methods Weighted K-Nearest Neighbor Position Kalman Filter Fingerprint Kalman Filter Test results Conclusions
¨ Simo Ali-Loytty – p.2/21
Calibration point
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
67 m
Mathematics
——–
Radio map
¨ Simo Ali-Loytty – p.3/21
The radio map is M = {Mi }, where Mi = (Bi , Ri )
Cell
Center
Bi
pi
¨ Simo Ali-Loytty – p.4/21
The radio map is M = {Mi }, where Mi = (Bi , Ri ) Ri,1. AP (WLAN access point)
0.12
Cell
Bi 0
-92
pi
-67
Ri,2. AP
0.16
Center
a ¯i,1. AP
Relative frequency
0
-91
RSSI
a ¯i,2. AP
-74
¨ Simo Ali-Loytty – p.4/21
The stem plot of the means of histograms AP Signal strength Glass door
¨ Simo Ali-Loytty – p.5/21
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 Mathematics
——–
State of the art methods
¨ Simo Ali-Loytty – p.6/21
Weighted K-Nearest Neighbor Cell
xˆWKNN =
M X i=1
wi PM
j=1
wj
Bi
pi, Center
pi
where pi is center of cell Bi and nonnegative weights wi are based on radio map.
¨ Simo Ali-Loytty – p.7/21
Measured histograms are not smooth 0.12 RSSI histogram
Probability
Problem
0 -90
RSSI
-65 ¨ Simo Ali-Loytty – p.8/21
We use histogram kernel approximation 0.12 RSSI histogram Exponential kernel, width=2
Probability
0 -90
RSSI
-65 ¨ Simo Ali-Loytty – p.9/21
Example: When y = −82.5 then wi ≈ 0.036 0.12 RSSI histogram Exponential kernel, width=2
Probability
wi ≈ 0.036
0 -90
y = −82.5
RSSI
-65 ¨ Simo Ali-Loytty – p.10/21
Special case of WKNN: Nearest Neighbor (NN)
xˆNN = pi,
where i
= argmaxi (wi ).
In our case best results produce weights [1]
1 , wi = ky − a¯i k1 where y is measurement vector and mean values of histograms are in vector a ¯i. [1] Ville Honkavirta, Tommi Perälä, Simo Ali-Löytty, and Robert Piché. A comparative survey of WLAN location fingerprinting methods. In Proceedings of the 6th Workshop on Positioning, Navigation and Communication 2009 (WPNC’09), pages 243-251, March 2009. ¨ Simo Ali-Loytty – p.11/21
Filtering approach: Kalman Filter Initial state: Motion model: Meas. model:
Prior:
− xˆk − Pk
x0 , E(x0) = xˆ0, V(x0) = P0 xk+1 = Fk xk + wk , V(wk ) = Qk yk = Hk xk + vk , V(vk ) = Rk = Fk−1xˆk−1 T = Fk−1Pk−1Fk−1 + Qk−1
Posterior: x ˆk = xˆ−k + Kk (yk − Hk xˆk )
Pk = Kk =
− (I − Kk Hk )Pk P−k HTk (Hk P−k HTk
+ Rk )−1 ¨ Simo Ali-Loytty – p.12/21
Position Kalman Filter (PKF)
PKF uses the static solutions as a measurement: PKFWKNN : y = x ˆWKNN PKFNN : y = x ˆNN We use stationary state model:
Fk = I and Hk = I
¨ Simo Ali-Loytty – p.13/21
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 Mathematics
——–
Fingerprint Kalman Filter (FKF)
¨ Simo Ali-Loytty – p.14/21
Comparison between PKF and FKF problems
PKF: Initial state: x0 , E(x0 ) = xˆ0 , V(x0 ) = P0 Motion model: xk+1 = Fk xk + wk , V(wk ) = Qk Meas. model: y k = H k xk + v k , V(vk ) = Rk
FKF: Initial state:
x0 ,
E(x0 ) = xˆ0 , V(x0 ) = P0
Motion model: xk+1 = Fk xk + wk , Meas. model:
V(wk ) = Qk
yk = hk (xk , vk ),
where hk (xk , vk ) is known only in the calibration points pi ! ¨ Simo Ali-Loytty – p.15/21
FKF is based on recursive use of BLUE
Let
E
x y
=
x˜ y˜
and V
x y
=
Pxx Pxy Pyx Pyy
BLUE (Best Linear Unbiased Estimator)
ˆ = x˜ + Pxy P−1 ˜) x yy (y − y T −1 ˆ ) (x − x ˆ ) = Pxx − Pxy Pyy Pyx E (x − x
¨ Simo Ali-Loytty – p.16/21
FKF algorithm − xˆk P−k
Prior:
= Fk−1xˆk−1 = Fk−1Pk−1FTk−1 + Qk−1
ˆk ) Posterior: x ˆk = xˆ−k + Pxyk P−1 yy k (yk − y T P Pk = Pxxk − Pxyk P−1 yy k xy k ,
where yˆk =
P
i
βi,k a¯i ,
Pxxk = Pxyk = Pyyk =
X
i X
i X i
pˆk =
P
i
βi,k pi ≈
xˆ− k,
βi,k ≈ P
βi,k Ppi + (pi − pˆk )(pi − pˆk )
T
,
T
.
x− k
∈ Ai
βi,k (pi − pˆk )(¯ ai − yˆk )T and βi,k Pai + (¯ ai − yˆk )(¯ ai − yˆk )
¨ Simo Ali-Loytty – p.17/21
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 Mathematics
——–
Test results
¨ Simo Ali-Loytty – p.18/21
One route example True FKF PKFWKNN
30 m End
Start
Cells
¨ Simo Ali-Loytty – p.19/21
Results from 4 routes
NN WKNN PKFNN PKFWKNN FKF
ME RMSE 95% Max (m) (m) (m) (m) 7.7 14.3 21.6 127.4 5.9 9.8 13.7 105.9 6.9 11.8 18.4 110.3 4.7 5.8 10.1 36.5 4.7 5.8 11.1 27.2
¨ Simo Ali-Loytty – p.20/21
Results from 4 routes
NN WKNN PKFNN PKFWKNN FKF
ME RMSE 95% Max (m) (m) (m) (m) 7.7 14.3 21.6 127.4 5.9 9.8 13.7 105.9 6.9 11.8 18.4 110.3 4.7 5.8 10.1 36.5 4.7 5.8 11.1 27.2
Note: WKNN and PKFWKNN have significantly greater computational and memory requirements than NN, PKFNN or FKF. ¨ Simo Ali-Loytty – p.20/21
Conclusions FKF outperformed PKFNN and the static estimators
¨ Simo Ali-Loytty – p.21/21
Conclusions FKF outperformed PKFNN and the static estimators FKF and PKFWKNN have similar performance
¨ Simo Ali-Loytty – p.21/21
Conclusions FKF outperformed PKFNN and the static estimators FKF and PKFWKNN have similar performance FKF has much lower computational and memory requirements than PKFWKNN
¨ Simo Ali-Loytty – p.21/21
Conclusions FKF outperformed PKFNN and the static estimators FKF and PKFWKNN have similar performance FKF has much lower computational and memory requirements than PKFWKNN True FKF PKFWKNN
http://math.tut.fi/posgroup/ 30 m End
Start
Cells
¨ Simo Ali-Loytty – p.21/21
AP Signal strength Glass door
Mathematics
——–
Fingerprint Kalman Filter in Indoor Positioning Applications MSC 2009 Simo Ali-Löytty, Tommi Perälä, Ville Honkavirta and Robert Piché http://math.tut.fi/posgroup/ ¨ Simo Ali-Loytty – p.1/21
Outline Radio map (offline phase) State of the art methods Weighted K-Nearest Neighbor Position Kalman Filter Fingerprint Kalman Filter Test results Conclusions
¨ Simo Ali-Loytty – p.2/21
Calibration point
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
67 m
Mathematics
——–
Radio map
¨ Simo Ali-Loytty – p.3/21
The radio map is M = {Mi }, where Mi = (Bi , Ri )
Cell
Center
Bi
pi
¨ Simo Ali-Loytty – p.4/21
The radio map is M = {Mi }, where Mi = (Bi , Ri ) Ri,1. AP (WLAN access point)
0.12
Cell
Bi 0
-92
pi
-67
Ri,2. AP
0.16
Center
a ¯i,1. AP
Relative frequency
0
-91
RSSI
a ¯i,2. AP
-74
¨ Simo Ali-Loytty – p.4/21
The stem plot of the means of histograms AP Signal strength Glass door
¨ Simo Ali-Loytty – p.5/21
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 Mathematics
——–
State of the art methods
¨ Simo Ali-Loytty – p.6/21
Weighted K-Nearest Neighbor Cell
xˆWKNN =
M X i=1
wi PM
j=1
wj
Bi
pi, Center
pi
where pi is center of cell Bi and nonnegative weights wi are based on radio map.
¨ Simo Ali-Loytty – p.7/21
Measured histograms are not smooth 0.12 RSSI histogram
Probability
Problem
0 -90
RSSI
-65 ¨ Simo Ali-Loytty – p.8/21
We use histogram kernel approximation 0.12 RSSI histogram Exponential kernel, width=2
Probability
0 -90
RSSI
-65 ¨ Simo Ali-Loytty – p.9/21
Example: When y = −82.5 then wi ≈ 0.036 0.12 RSSI histogram Exponential kernel, width=2
Probability
wi ≈ 0.036
0 -90
y = −82.5
RSSI
-65 ¨ Simo Ali-Loytty – p.10/21
Special case of WKNN: Nearest Neighbor (NN)
xˆNN = pi,
where i
= argmaxi (wi ).
In our case best results produce weights [1]
1 , wi = ky − a¯i k1 where y is measurement vector and mean values of histograms are in vector a ¯i. [1] Ville Honkavirta, Tommi Perälä, Simo Ali-Löytty, and Robert Piché. A comparative survey of WLAN location fingerprinting methods. In Proceedings of the 6th Workshop on Positioning, Navigation and Communication 2009 (WPNC’09), pages 243-251, March 2009. ¨ Simo Ali-Loytty – p.11/21
Filtering approach: Kalman Filter Initial state: Motion model: Meas. model:
Prior:
− xˆk − Pk
x0 , E(x0) = xˆ0, V(x0) = P0 xk+1 = Fk xk + wk , V(wk ) = Qk yk = Hk xk + vk , V(vk ) = Rk = Fk−1xˆk−1 T = Fk−1Pk−1Fk−1 + Qk−1
Posterior: x ˆk = xˆ−k + Kk (yk − Hk xˆk )
Pk = Kk =
− (I − Kk Hk )Pk P−k HTk (Hk P−k HTk
+ Rk )−1 ¨ Simo Ali-Loytty – p.12/21
Position Kalman Filter (PKF)
PKF uses the static solutions as a measurement: PKFWKNN : y = x ˆWKNN PKFNN : y = x ˆNN We use stationary state model:
Fk = I and Hk = I
¨ Simo Ali-Loytty – p.13/21
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 Mathematics
——–
Fingerprint Kalman Filter (FKF)
¨ Simo Ali-Loytty – p.14/21
Comparison between PKF and FKF problems
PKF: Initial state: x0 , E(x0 ) = xˆ0 , V(x0 ) = P0 Motion model: xk+1 = Fk xk + wk , V(wk ) = Qk Meas. model: y k = H k xk + v k , V(vk ) = Rk
FKF: Initial state:
x0 ,
E(x0 ) = xˆ0 , V(x0 ) = P0
Motion model: xk+1 = Fk xk + wk , Meas. model:
V(wk ) = Qk
yk = hk (xk , vk ),
where hk (xk , vk ) is known only in the calibration points pi ! ¨ Simo Ali-Loytty – p.15/21
FKF is based on recursive use of BLUE
Let
E
x y
=
x˜ y˜
and V
x y
=
Pxx Pxy Pyx Pyy
BLUE (Best Linear Unbiased Estimator)
ˆ = x˜ + Pxy P−1 ˜) x yy (y − y T −1 ˆ ) (x − x ˆ ) = Pxx − Pxy Pyy Pyx E (x − x
¨ Simo Ali-Loytty – p.16/21
FKF algorithm − xˆk P−k
Prior:
= Fk−1xˆk−1 = Fk−1Pk−1FTk−1 + Qk−1
ˆk ) Posterior: x ˆk = xˆ−k + Pxyk P−1 yy k (yk − y T P Pk = Pxxk − Pxyk P−1 yy k xy k ,
where yˆk =
P
i
βi,k a¯i ,
Pxxk = Pxyk = Pyyk =
X
i X
i X i
pˆk =
P
i
βi,k pi ≈
xˆ− k,
βi,k ≈ P
βi,k Ppi + (pi − pˆk )(pi − pˆk )
T
,
T
.
x− k
∈ Ai
βi,k (pi − pˆk )(¯ ai − yˆk )T and βi,k Pai + (¯ ai − yˆk )(¯ ai − yˆk )
¨ Simo Ali-Loytty – p.17/21
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 Mathematics
——–
Test results
¨ Simo Ali-Loytty – p.18/21
One route example True FKF PKFWKNN
30 m End
Start
Cells
¨ Simo Ali-Loytty – p.19/21
Results from 4 routes
NN WKNN PKFNN PKFWKNN FKF
ME RMSE 95% Max (m) (m) (m) (m) 7.7 14.3 21.6 127.4 5.9 9.8 13.7 105.9 6.9 11.8 18.4 110.3 4.7 5.8 10.1 36.5 4.7 5.8 11.1 27.2
¨ Simo Ali-Loytty – p.20/21
Results from 4 routes
NN WKNN PKFNN PKFWKNN FKF
ME RMSE 95% Max (m) (m) (m) (m) 7.7 14.3 21.6 127.4 5.9 9.8 13.7 105.9 6.9 11.8 18.4 110.3 4.7 5.8 10.1 36.5 4.7 5.8 11.1 27.2
Note: WKNN and PKFWKNN have significantly greater computational and memory requirements than NN, PKFNN or FKF. ¨ Simo Ali-Loytty – p.20/21
Conclusions FKF outperformed PKFNN and the static estimators
¨ Simo Ali-Loytty – p.21/21
Conclusions FKF outperformed PKFNN and the static estimators FKF and PKFWKNN have similar performance
¨ Simo Ali-Loytty – p.21/21
Conclusions FKF outperformed PKFNN and the static estimators FKF and PKFWKNN have similar performance FKF has much lower computational and memory requirements than PKFWKNN
¨ Simo Ali-Loytty – p.21/21
Conclusions FKF outperformed PKFNN and the static estimators FKF and PKFWKNN have similar performance FKF has much lower computational and memory requirements than PKFWKNN True FKF PKFWKNN
http://math.tut.fi/posgroup/ 30 m End
Start
Cells
¨ Simo Ali-Loytty – p.21/21
