What is scan matching Andrea Censi , La Sapienza Universit y of - PowerPoint PPT Presentation

S an mat hing in a p robabilisti framew o rk Andrea Censi a master student in ontrol engineering at Universit� degli Studi di Roma La Sapienza www.dis.uniroma1.it/ ∼ acensi andrea.censi@dis.uniroma1.it Andrea Censi , “La Sapienza” Universit y of Rome S an mat hing in a p robabilisti framew o rk - p. 1/25

What is scan matching Andrea Censi , “La Sapienza” Universit y of Rome S an mat hing in a p robabilisti framew o rk - p. 2/25

What is scan matching Find a rotation ϕ and a translation t whi h maximize the overlapping of t w o sets of 2D-data. Geometric interpretation: Andrea Censi , “La Sapienza” Universit y of Rome S an mat hing in a p robabilisti framew o rk - p. 2/25

What is scan matching Find a rotation ϕ and a translation t whi h maximize the overlapping of t w o sets of 2D-data. Geometric interpretation: Find an app ro ximation to the p robabilit y distribution Probabilistic interpretation: : rob ot p ose, z : senso r reading, u : o dometry . Andrea Censi , “La Sapienza” Universit y of Rome S an mat hing in a p robabilisti framew o rk - p. 2/25 p ( t , ϕ | x t − 1 , u t , z t , z t − 1 ) x

� gpm � is a new algo rithm that � uses, soundly , an a rbitra ry evolution mo del; no random sampling Contribution of the paper required Gaussian assumption: [Minguez&al.’05]; Random sampling: MCL , [Silver&al.’04] � ha ra terizes the un ertaint y analyti ally , also in under onstrained situations Sample error function around the estimate: [Bengtsson&al.’03]; analytic, elegant but � not iterative: result do es not dep end on �rst guess W eak p oints of gpm : bounded estimate of covariance: [Pfister&al.’02] � The environment must have some regula rit y to estimate surfa es' o rientation. � It is mo re p re ise than i p , id , but not than last generation i p -lik e metho ds. [Minguez&al.’06] Andrea Censi , “La Sapienza” Universit y of Rome S an mat hing in a p robabilisti framew o rk - p. 3/25

It is a dual of Monte Ca rlo Lo alization: � In m l , pa rti les a re dra wn from the evolution mo del and w eighted GPM overview b y the observation mo del. � In gpm , pa rti les a re generated (deterministi ally) from the observation mo del and w eighted b y the evolution mo del. Summa ry of the algo rithm: 1. Extra t o rientation info rmation from the senso r data. 2. Generate a loud of pa rti les from the observations. 3. W eight ea h pa rti le a o rding to the evolution mo del. 4. T urn the pa rti les into � onstraints� to ha ra terize un ertaint y . Andrea Censi , “La Sapienza” Universit y of Rome S an mat hing in a p robabilisti framew o rk - p. 4/25

The input data a re t w o sets of �o riented� p oints { ( p i , α i , ) } , where p i is the a rtesian p oint and α i is the dire tion of the no rmal to the surfa e. Extracting the orientation Currently using linea r regression; there a re many alternatives. Andrea Censi , “La Sapienza” Universit y of Rome S an mat hing in a p robabilisti framew o rk - p. 5/25

W e reate a set of hyp otheses (pa rti les) b y onsidering all p ossible pairs of p oints ( no correspondence heuristics ). Generating the particles Invert to obtain p j = R ϕ p i + t α j = α i + ϕ Ea h hyp othesis ( ˆ is treated as a pa rti le (generated deterministically ; no ϕ ˆ = α j − α i random sampling here). Andrea Censi , “La Sapienza” Universit y of Rome S an mat hing in a p robabilisti framew o rk - p. 6/25 ˆ = p j − R ˆ ϕ p i t ϕ, ˆ t )

This is ho w the set of pa rti les lo oks lik e: Example (1) (green is one of the sensor scans; particles are red) 10 8 6 4 2 0 -2 W e onsider only the pa rti les in a �xed ball where the evolution mo del is -4 non-zero. -6 -8 -15 -10 -5 0 5 10 15 20 Andrea Censi , “La Sapienza” Universit y of Rome S an mat hing in a p robabilisti framew o rk - p. 7/25

P a rti les with | ϕ | ≤ 20 ◦ , | t | ≤ 20 cm . Example (2) 0.2 0.15 0.1 0.05 0 -0.05 -0.1 it is a pa rti le app ro ximation to p ( ϕ, t | x t − 1 , z t , z t − 1 ) -0.15 (little a rro ws rep resent ˆ ) -0.2 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 ⇒ Andrea Censi , “La Sapienza” Universit y of Rome S an mat hing in a p robabilisti framew o rk - p. 8/25 ϕ

W eight b y evolution mo del: Using the evolution model w k = p ( ϕ k , t k | x t − 1 , u t ) 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 0 0 -0.05 -0.05 b efo re w eighting after w eighting -0.1 -0.1 -0.15 -0.15 no w a pa rti le app ro ximation to p ( ϕ, t | x t − 1 , u t , z t , z t − 1 ) . -0.2 -0.2 Andrea Censi , “La Sapienza” Universit y of Rome S an mat hing in a p robabilisti framew o rk - p. 9/25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 ⇒

T o ha ra terize the un ertaint y of the pa rti les, w e onsider the info rmation useful only along the dire tion of the w all. Least squares formulation The result is a set of onstraints: a least squa res p roblem. Andrea Censi , “La Sapienza” Universit y of Rome S an mat hing in a p robabilisti framew o rk - p. 10/25

F rom pa rti les . . . Example (3) 0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 Andrea Censi , “La Sapienza” Universit y of Rome S an mat hing in a p robabilisti framew o rk - p. 11/25 -0.2 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

to onstraints . . . Example (3) . . . 0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 Andrea Censi , “La Sapienza” Universit y of Rome S an mat hing in a p robabilisti framew o rk - p. 11/25 -0.2 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

to ova rian e. Example (3) . . . 0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 Andrea Censi , “La Sapienza” Universit y of Rome S an mat hing in a p robabilisti framew o rk - p. 11/25 -0.2 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

Exp erimentally , the estimated ova rian e is signi� ant only up to a onstant (go o d �shap e�, bad �a rea�). The need for tuning Residual error on x,y 6 GPM sample Σ 4 GPM Σ 2 y (mm) 0 −2 −4 T w o reasons fo r this: � un ertaint y should b e mo deled b etter −6 GPM m=2.57 ||bias|| = 1.5mm sqrt(mse) = 4.2mm � all pa rti les onsidered indep endent (instead, the global ova rian e matrix −10 −8 −6 −4 −2 0 2 4 6 x (mm) is not diagonal) Andrea Censi , “La Sapienza” Universit y of Rome S an mat hing in a p robabilisti framew o rk - p. 12/25

Unconstrained situations Andrea Censi , “La Sapienza” Universit y of Rome S an mat hing in a p robabilisti framew o rk - p. 13/25

Unconstrained situations Andrea Censi , “La Sapienza” Universit y of Rome S an mat hing in a p robabilisti framew o rk - p. 13/25

A rob ot in a mine - thanks to Dirk Haehnel and the CMU group fo r the data �les. Mine example Senso r data gpm result Andrea Censi , “La Sapienza” Universit y of Rome S an mat hing in a p robabilisti framew o rk - p. 14/25

Eigenvalues of estimated covariance Green and red a re the eigenvalues of the estimated ova rian e matrix. � One eigenvalue is alw a ys small (left and right w alls). � The other is big at b eginning of o rrido rs (1,2,. . . ) then de reases. � O lusions a re not fatal and dete ted. Andrea Censi , “La Sapienza” Universit y of Rome S an mat hing in a p robabilisti framew o rk - p. 15/25

Eigenvalues of estimated covariance Green and red a re the eigenvalues of the estimated ova rian e matrix. � One eigenvalue is alw a ys small (left and right w alls). � The other is big at b eginning of o rrido rs (1,2,. . . ) then de reases. � O lusions a re not fatal and dete ted. Andrea Censi , “La Sapienza” Universit y of Rome S an mat hing in a p robabilisti framew o rk - p. 15/25

Eigenvalues of estimated covariance Green and red a re the eigenvalues of the estimated ova rian e matrix. � One eigenvalue is alw a ys small (left and right w alls). � The other is big at b eginning of o rrido rs (1,2,. . . ) then de reases. � O lusions a re not fatal and dete ted. Andrea Censi , “La Sapienza” Universit y of Rome S an mat hing in a p robabilisti framew o rk - p. 15/25