A Modified Kalman Filter for Hybrid Positioning Tilastopivt 2007 - - PowerPoint PPT Presentation

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TAMPERE UNIVERSITY OF TECHNOLOGY Mathematics

A Modified Kalman Filter for Hybrid Positioning

Tilastopäivät 2007 Simo Ali-Löytty

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Outline

Motivation Problem Mathematical background Example Some results Conclusion

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Not enough measurements for unique solution. . .

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. . . restrictive information is necessary.

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Combine restrictive information and Kalman-type filters?

Restrictive information tells that state is inside

some area. For example sector information.

Kalman-type filters are linear filters which

approximate only first two moments of distribution.

ˆ x+ = ˆ x− + K(y − ˆ y) P+ = P− − KPyKT

For example: Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF).

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K-type filter approximates Bayes filter

Initial state: x0 Motion model: xk+1 = f(xk) + wk Measurement model: yk = h(xk) + vk

p(xk|y1:k) =

current meas.

p(yk|xk)

state model and past meas.

• p(xk|y1:k−1)
• p(yk|xk)p(xk|y1:k−1)dxk
• normalization

K-type filters xk|y1:k ≈ N(ˆ

x+

k , P+ k ).

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Posterior mean and covariance?

500 m Base station

pposterior(x) =

χA(x)pprior(x)

• χA(x)pprior(x)dx,

where xprior ∼ N(µprior, Σprior).

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Approximate restrictive area

500 m Base station

pposterior(x) ≈

χA′(x)pprior(x)

• χA′(x)pprior(x)dx,

where A ⊂ A′,

A′ = {x||Ax − Axmid| ≤ α}

and AΣpriorAT = I.

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Posterior mean and covariance

500 m Base station

µpost = µprior + ΣpriorATǫ Σpost = Σprior − ΣpriorATΛAΣprior,

where ǫ = n

i=1 ei exp „ − upi2

2

« −exp „ − lowi2

2

« √ 2π(Φ(upi)−Φ(lowi))

Λ = diag(δ) + diag(ǫ) diag(Aµpost − Axmid), δ = n

i=1 αiei exp „ − upi2

2

« +exp „ − lowi2

2

« √ 2π(Φ(upi)−Φ(lowi))

.

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Base station 500 m Start Base station 500 m Start True track EKF EKF with restrictive info

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Simulation variances:

Base station range meas.: 802 m2 Pseudorange meas.: 202 m2 Delta pseudorange meas.: 22 m2

s2

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0-2 base stations: BOX-solver is better than EKF

100 simulations: Time Err. Err. Err.

<50m

Inc. Solver

∝ µ

ref 95%

% %

EKF 1 393 358 1356 11 28.1 BOX⋆ 1 314 253 669 3 0.1 EKFBOX 2 192 126 625 19 7.7 SMC 74 145 17 436 25 0.2

⋆ uses only restrictive info.

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Suburban: EKFBOX is almost as accurate as SMC

Time Err. Err. Err.

<50m Inc.

Solver

∝ µ

ref 95%

% %

BOX 1 285 279 644 5 0.7 EKF 1 76 46 255 60 2.1 EKFBOX 2 60 23 195 63 0.5 SMC 50 53 4 166 66 0.1

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Conclusion

Base station 500 m Start Base station 500 m Start True track EKF EKF with restrictive info

The new algorithm Uses restrictive information Gives better results (accuracy, consistency) Is fast to compute

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Conclusion

Base station 500 m Start Base station 500 m Start True track EKF EKF with restrictive info

The new algorithm Uses restrictive information Gives better results (accuracy, consistency) Is fast to compute Thank you for your attention! Questions? http://math.tut.fi/posgroup/

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