Introduction to Mobile Robotics
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1 Inroducion o Mobile Roboics Bayes Filer Kalman Filer Wolfram Burgard Cyrill Sachniss Giorgio Grisei Maren Bennewiz Chrisian Plagemann
2 Bayes Filer Reminder Predicion bel p u bel d Correcion bel η p z bel
3 Gaussians : ~ π e p N p - Univariae / / : ~ e p Ν p d π Mulivariae
4 ~ ~ a b a N Y b ax Y N X Properies of Gaussians ~ ~ ~ N X p X p N X N X
5 We say in he Gaussian world as long as we sar wih Gaussians and perform only linear ransformaions. ~ ~ A A B A N Y B AX Y N X Mulivariae Gaussians ~ ~ ~ N X p X p N X N X
6 Discree Kalman Filer Esimaes he sae of a discree-ime conrolled process ha is governed by he linear sochasic difference equaion A B u ε wih a measuremen z C δ 6
7 Componens of a Kalman Filer A B C ε δ Mari nn ha describes how he sae evolves from o - wihou conrols or noise. Mari nl ha describes how he conrol u changes he sae from o -. Mari kn ha describes how o map he sae o an observaion z. Random variables represening he process and measuremen noise ha are assumed o be independen and normally disribued wih covariance R and Q respecively. 7
8 Kalman Filer Updaes in D 8
9 9 Kalman Filer Updaes in D wih Q C C C K K C I C z K bel wih obs K K z K bel
10 Kalman Filer Updaes in D R A A B u A bel ac a b u a bel
11 Kalman Filer Updaes
12 Linear Gaussian Sysems: Iniializaion Iniial belief is normally disribued: bel N ;
13 Dynamics are linear funcion of sae and conrol plus addiive noise: u B A ε Linear Gaussian Sysems: Dynamics R B u A N u p ; ; ~ ; ~ N R B u A N d bel u p bel
14 Linear Gaussian Sysems: Dynamics R A A B u A bel d B u A R B u A bel N R B u A N d bel u p bel ep ep ; ~ ; ~ η
15 Observaions are linear funcion of sae plus addiive noise: C z δ Linear Gaussian Sysems: Observaions Q C z N z p ; N Q C z N bel z p bel ; ~ ; ~ η
16 Linear Gaussian Sysems: Observaions wih ep ep ; ~ ; ~ Q C C C K K C I C z K bel C z Q C z bel N Q C z N bel z p bel η η
17 Kalman Filer Algorihm. Algorihm Kalman_filer - - u z :. Predicion: 3. A Bu 4. A A R 5. Correcion: 6. K C C C Q 7. K z C 8. I K C 9. Reurn
18 Kalman Filer Algorihm
19 Kalman Filer Algorihm Predicion Observaion Maching Correcion
20 0 he Predicion-Correcion-Cycle R A A u B A bel ac a u b a bel Predicion
21 he Predicion-Correcion-Cycle Q C C C K C K I C z K bel obs K K z K bel Correcion
22 he Predicion-Correcion-Cycle Q C C C K C K I C z K bel obs K K z K bel R A A u B A bel ac a u b a bel Correcion Predicion
23 Kalman Filer Summary Highly efficien: Polynomial in measuremen dimensionaliy k and sae dimensionaliy n: Ok.376 n Opimal for linear Gaussian sysems! Mos roboics sysems are nonlinear!
24 Nonlinear Dynamic Sysems Mos realisic roboic problems involve nonlinear funcions g u z h
25 Lineariy Assumpion Revisied
26 Non-linear Funcion
27 EKF Linearizaion
28 EKF Linearizaion
29 EKF Linearizaion 3
30 Predicion: Correcion: EKF Linearizaion: Firs Order aylor Series Epansion G u g u g u g u g u g H h h h h h
31 EKF Algorihm. Eended_Kalman_filer - - u z :. Predicion: Correcion: Reurn u g R G G Q H H H K h z K H K I u g G h H u A B R A A Q C C C K C z K C K I
32 Localizaion Using sensory informaion o locae he robo in is environmen is he mos fundamenal problem o providing a mobile robo wih auonomous capabiliies. [Co 9] Given Map of he environmen. Sequence of sensor measuremens. Waned Esimae of he robo s posiion. Problem classes Posiion racking Global localizaion Kidnapped robo problem recovery
33 Landmark-based Localizaion
34 . EKF_localizaion - - u z m: Predicion: u g V V M G G θ θ θ θ θ θ ' ' ' ' ' ' ' ' ' y y y y y y u g G v y v y v u u g V ω θ θ ω ω ' ' ' ' ' ' v v M ω α α ω α α Moion noise Jacobian of g w.r. locaion Prediced mean Prediced covariance Jacobian of g w.r. conrol
35 . EKF_localizaion - - u z m: Correcion: ˆ z z K H K I θ θ ϕ ϕ ϕ y y r r r m h H θ aan ˆ y y y y m m m m z Q H H S S H K 0 0 r r Q Prediced measuremen mean Pred. measuremen covariance Kalman gain Updaed mean Updaed covariance Jacobian of h w.r. locaion
36 EKF Predicion Sep
37 EKF Observaion Predicion Sep
38 EKF Correcion Sep
39 Esimaion Sequence
40 Esimaion Sequence
41 Comparison o Groundruh
42 EKF Summary Highly efficien: Polynomial in measuremen dimensionaliy k and sae dimensionaliy n: Ok.376 n No opimal! Can diverge if nonlineariies are large! Works surprisingly well even when all assumpions are violaed!
43 EKF Localizaion Eample Line and poin landmarks
44 EKF Localizaion Eample Line and poin landmarks
45 EKF Localizaion Eample Lines only Swiss Naional Ehibiion Epo.0 Quickime and a MPEG-4 Video decompressor are needed o see his picure.
46 Linearizaion via Unscened ransform EKF UKF
47 UKF Sigma-Poin Esimae EKF UKF
48 UKF Sigma-Poin Esimae 3 EKF UKF
49 Unscened ransform n i n w w n n w n w i c i m i i c m... for ± λ λ χ β α λ λ λ λ χ Sigma poins Weighs i i g χ ψ n i i i i c n i i i m w w 0 0 ' ' ψ ψ ψ Pass sigma poins hrough nonlinear funcion Recover mean and covariance
50 UKF_localizaion - - u z m: Predicion: v v M ω α α ω α α 0 0 r r Q a a Q M a a a a a a γ γ χ u u g χ χ χ L i i i i w c 0 χ χ L i i i w m 0 χ Moion noise Measuremen noise Augmened sae mean Augmened covariance Sigma poins Predicion of sigma poins Prediced mean Prediced covariance
51 UKF_localizaion - - u z m: Correcion: Ζ z χ h χ L i zˆ w m S i 0 L w z i 0 L i 0 i c w Ζ i Ζ z Ζ zˆ i c i ˆ i χ Ζ zˆ i i Measuremen sigma poins Prediced measuremen mean Pred. measuremen covariance Cross-covariance K S z K zˆ z Kalman gain Updaed mean K S K Updaed covariance
52 . EKF_localizaion - - u z m: Correcion: ˆ z z K H K I θ θ ϕ ϕ ϕ y y r r r m h H θ aan ˆ y y y y m m m m z Q H H S S H K 0 0 r r Q Prediced measuremen mean Pred. measuremen covariance Kalman gain Updaed mean Updaed covariance Jacobian of h w.r. locaion
53 UKF Predicion Sep
54 UKF Observaion Predicion Sep
55 UKF Correcion Sep
56 EKF Correcion Sep
57 Esimaion Sequence EKF PF UKF
58 Esimaion Sequence EKF UKF
59 Predicion Qualiy EKF UKF
60 UKF Summary Highly efficien: Same compleiy as EKF wih a consan facor slower in ypical pracical applicaions Beer linearizaion han EKF: Accurae in firs wo erms of aylor epansion EKF only firs erm Derivaive-free: No Jacobians needed Sill no opimal! 60
61 Kalman Filer-based Sysem [Arras e al. 98]: Laser range-finder and vision High precision <cm accuracy Couresy of K. Arras
62 Mulihypohesis racking
63 Localizaion Wih MH Belief is represened by muliple hypoheses Each hypohesis is racked by a Kalman filer Addiional problems: Daa associaion: Which observaion corresponds o which hypohesis? Hypohesis managemen: When o add / delee hypoheses? Huge body of lieraure on arge racking moion correspondence ec.
64 MH: Implemened Sysem Hypoheses are eraced from LRF scans Each hypohesis has probabiliy of being he correc one: H i ˆ { i i i P H} Hypohesis probabiliy is compued using Bayes rule P Hi s Hypoheses wih low probabiliy are deleed. New candidaes are eraced from LRF scans. C j P s Hi P Hi P s { zj j R} [Jensfel e al. 00]
65 MH: Implemened Sysem Couresy of P. Jensfel and S. Krisensen
66 MH: Implemened Sysem 3 Eample run # hypoheses PH bes Map and rajecory #hypoheses vs. ime Couresy of P. Jensfel and S. Krisensen
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