Probabilistic Robotics The Sparse Extended Information Filter
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1 Probabilisic Roboics The Sparse Exended Informaion Filer MSc course Arificial Inelligence 2018 hps://saff.fnwi.uva.nl/a.visser/educaion/probabilisicroboics/ Arnoud Visser Inelligen Roboics Lab Informaics Insiue Universiei van Amserdam Images couresy of Sebasian Thrun, Wolfram Burghard, Dieer Fox, Michael Monemerlo, Dick Hähnel, Pieer Abbeel and ohers.
2 Simulaneous Localizaion and Mapping A robo acquires a map while localizing iself relaive o his map. Online SLAM problem Full SLAM problem p( x, m z :, u1 : 1 p( x :, m z1:, u1 : 1 Esimae map m and curren posiion x Esimae map m and driven pah x 1: Probabilisic Roboics Course a he Universiei van Amserdam 2
3 SEIF SLAM SEIF SLAM reduces he sae vecor y again o he curren posiion x ( x m m s m m s T y = 1, x 1, y 1 N, x N, y N This is he same sae vecor y as EKF SLAM Probabilisic Roboics Course a he Universiei van Amserdam 3
4 Sae esimae SEIF SLAM requires every imesep inference o esimae he sae ~ µ = Ω ~ 1 ~ ξ The sae esimaed is also done by GraphSLAM, as a pos-processing sep. Probabilisic Roboics Course a he Universiei van Amserdam 4
5 Probabilisic Roboics Course a he Universiei van Amserdam 5 Sparseness of Informaion Marix Afer a while, all landmarks are correlaed in EKF s correlaion marix The normalized informaion marix is naurally sparse; mos elemens are close o zero (bu none is zero., ( (, ( ( 1 j i T j i m x h z Q m x h z
6 Acquisiion of he informaion marix The observaion of a landmark m 1 inroduces a consrain: The consrain is of he ype: H T Q 1 H Where h(x,m j is he measuremen model and Q he covariance of he measuremen noise. Probabilisic Roboics Course a he Universiei van Amserdam 6
7 Acquisiion of he informaion marix The observaion of a landmark m 2 inroduces anoher consrain: The informaion vecor increases wih he erm: T 1 i H Q ( z h( µ + H µ Probabilisic Roboics Course a he Universiei van Amserdam 7
8 Acquisiion of he informaion marix The movemen of he robo from x 1 o x 2 also inroduces an consrain: The consrain is now beween he landmarks m 1 and m 2 (and no beween he pah x -1 o x : Ω = [ G Ω + 1 T T 1 1 G Fx R Fx ] Which can be simplified o Ω = Φ κ Probabilisic Roboics Course a he Universiei van Amserdam 8
9 Acquisiion of he informaion marix The informaion marix can become really sparse by applying a sparsificaion sep: This is done by pariion he se of feaures ino hree disjoin subses: m = + m m m Where m - is he se of passive feaures and m + m 0 is he se of acive feaures. The number of feaures ha are allowed o remain acive (se m + is hresholded o guaranee efficiency. Probabilisic Roboics Course a he Universiei van Amserdam 9
10 Nework of feaures Approximae he sparse informaion marix wih he argumen ha no all feaures are srongly conneced: Probabilisic Roboics Course a he Universiei van Amserdam 10
11 Updaing he curren sae esimae The curren sae esimae µˆ is needed every imesep: µ ~ Ω ~ 1 = ~ ξ Ye, from he curren sae esimae only subse is needed: y ( x m m s m m s T = 2 1, x 1, y 1 2, x 2, y i.e. he robo posiion x and he locaions of he acive landmarks m +. This can be done wih an ieraive hill climbing algorihm: T 1 T ( F ΩF F [ ξ Ωµ + ΩF F µ ] µ i i i i i i Where F i is a projecion marix o exrac elemen i from marix Ω. Probabilisic Roboics Course a he Universiei van Amserdam 11
12 Full Algorihm The algorihm combines he four seps; wo updaes and wo approximaions: Algorihm SEIF_SLAM_known_correspondences( ξ,ω- 1,μ-1,u ξ, Ω, µ = SEIF_moion_updae( ξ- 1,Ω-1,μ-1,u ~ µ = SEIF_updae_sae_esimae( ξ, Ω, µ ξ SEIF_measuremen_updae( ~ Ω = ξ,ω,μ,z,c ~ ~ ξ, Ω = SEIF_sparsificaion( ξ, Ω ~ ~ reurn ξ,, ~ µ, Ω - 1,z,c Probabilisic Roboics Course a he Universiei van Amserdam 12
13 for all observed feaures SEIF_measuremen_updae endfor Calculae reurn Probabilisic Roboics Course a he Universiei van Amserdam 13
14 SEIF_sparsificaion Calculae wih reurn Probabilisic Roboics Course a he Universiei van Amserdam 14
15 SEIF_moion_updae Canonical form of reurn Probabilisic Roboics Course a he Universiei van Amserdam 15
16 SEIF_updae_sae_esimae For mos map feaures For a few feaures For he pose reurn Probabilisic Roboics Course a he Universiei van Amserdam 16
17 The effec of sparsificaion The compuaion requires consan ime: Probabilisic Roboics Course a he Universiei van Amserdam 17
18 The effec of sparsificaion The memory scales linearly: Probabilisic Roboics Course a he Universiei van Amserdam 18
19 The effec of sparsificaion The prize is less accuracy, due o he approximaion: Probabilisic Roboics Course a he Universiei van Amserdam 19
20 The degree of sparseness By choosing he number of acive feaures, accuracy can be raded agains efficiency : Probabilisic Roboics Course a he Universiei van Amserdam 20
21 Effec of approximaion The effec of sparsificaion is less links beween landmarks, more confidence, bu nearly same informaion marix: Probabilisic Roboics Course a he Universiei van Amserdam 21
22 Probabilisic Roboics Course a he Universiei van Amserdam 22 Full Algorihm To exend he algorihm for unknown correspondences, an esimae for he correspondence is needed:, ˆ,, ( argmax ˆ 1 1: 1: 1 : 1 c c c u z z p c = c dy c u z y p c y z p c = ˆ,, (, ( argmax ˆ 1 1: 1: 1 1: c c c c dy dx c u z y x p c y x z p c = ˆ,,, (,, ( argmax ˆ 1 1: 1: 1 1:
23 Esimaing he correspondence p( x, y z, u, cˆ 1 To probabiliy c can be approximaed by 1: 1 1: 1: he Markov blanke of all landmarks conneced o robo pose x and landmark y c Probabilisic Roboics Course a he Universiei van Amserdam 23
24 Correspondence es Based on he probabiliy ha m j corresponds o m k : Algorihm SEIF_correspondence_es( Ω, ξ, µ,m j,c B = B( j B( k Σ B = ( F B ΩF µ = Σ FBξ Σ B B T B = ( F ΩBF F ξ B µ = Σ T 1 1 k reurn de(2 Σ T π exp{ µ Σ µ } 2 Probabilisic Roboics Course a he Universiei van Amserdam 24
25 Resuls MIT building (muliple loops: Probabilisic Roboics Course a he Universiei van Amserdam 25
26 Resuls MIT building (muliple loops: Probabilisic Roboics Course a he Universiei van Amserdam 26
27 Resuls MIT building (muliple loops: UvA approach Q-WSM Probabilisic Roboics Course a he Universiei van Amserdam 27
28 Conclusion The Sparse Exended Informaion Filer: Solves he Online SLAM problem efficienly. Where EKF spread he informaion of each measuremen over he full map, SEIF limis he spread o acive feaures. All informaion in he sored in he canonical parameerizaion. Ye, an esimae of he mean µˆ is sill needed. This esimae is found wih a hill climbing algorihm (and no a inversion of he informaion marix. The accuracy and efficiency can be balanced by selecing an appropriae number of acive feaures. p( x, m z :, u1 : 1 Probabilisic Roboics Course a he Universiei van Amserdam 28
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