L07. KALMAN FILTERING FOR NON-LINEAR SYSTEMS. NA568 Mobile Robotics: Methods & Algorithms
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1 L07. KALMAN FILTERING FOR NON-LINEAR SYSTEMS NA568 Mobile Roboics: Mehods & Algorihms
2 Today s Topic Quick review on (Linear) Kalman Filer Kalman Filering for Non-Linear Sysems Exended Kalman Filer (EKF) Unscened Kalman Filer (UKF) Bes Linear Unbiased Esimaor (BLUE)
3 Minimum Mean Squared Error Esimaion Cos funcion Choose so argumen is minimised Expecaion operaor ( average ) Soln From P. Newman, Oxford
4 MMSE: Discree Kalman Filer 4 Esimaes he sae x of a discree-ime conrolled process ha is governed by he linear sochasic difference equaion wih a measuremen and saisics
5 Kalman Filer Algorihm 5 Algorihm Kalman_filer( µ -1, Σ -1, u, z ): Predicion: Σ = A Σ A + 1 T R Correcion: K Reurn µ, Σ T T = ΣC ( C ΣC + Q ) Σ = ( I K C ) Σ 1
6 The Predicion-Correcion-Cycle 6 Predicion predicion
7 The Predicion-Correcion-Cycle 7 correcion measuremen Correcion
8 The Predicion-Correcion-Cycle 8 Predicion Correcion
9 Kalman Filer Summary 9 Highly efficien: Polynomial in measuremen dimensionaliy k and sae dimensionaliy n: O(k n 2 ) MMSE opimal for linear Gaussian sysems! No oher esimaor can do beer Mos roboics sysems are nonlinear!
10 Nonlinear Dynamic Sysems 10 Mos realisic roboic problems involve nonlinear funcions
11 11 Lineariy Assumpion Revisied
12 12 Non-linear Funcion
13 Nonlinear Gaussian Filers Approach 1: Exended Kalman Filer Approximae he model! Linearize our nonlinear plan and/or observaion model(s) abou he curren mean and use he linear KF equaions.
14 14 EKF Linearizaion via Firs Order Taylor Series
15 15 EKF Linearizaion: Large Variance
16 16 EKF Linearizaion: Narrow Variance
17 17 EKF Linearizaion: Firs Order Taylor Series Expansion Predicion: Correcion:
18 18 EKF Algorihm* 1. Exended_Kalman_filer( µ -1, Σ -1, u, z ): 2. Predicion: Correcion: Reurn µ, Σ T R G G + Σ = Σ 1 1 ) ( + Σ Σ = T T Q H H H K H K I Σ = Σ ) ( T R A A + Σ = Σ 1 1 ) ( + Σ Σ = T T Q C C C K C K I Σ = Σ ) ( Linear KF * The form shown assumes addiive process and observaion model noise
19 EKF Summary 19 Highly efficien: Polynomial in measuremen dimensionaliy k and sae dimensionaliy n: O(k n 2 ) No opimal! Can diverge if nonlineariies are large! Can work surprisingly well even when all assumpions are violaed!
20 KF, EKF and UKF Kalman filer requires linear models EKF linearizes via Taylor expansion Is here a beer way o linearize? Unscened Transform Unscened Kalman Filer (UKF) Couresy: Cyrill Sachniss
21 Taylor Approximaion (EKF) Linearizaion of he non-linear funcion hrough Taylor expansion Couresy: Cyrill Sachniss
22 Unscened Transform Compue a se of (so-called) sigma poins Couresy: Cyrill Sachniss
23 Unscened Transform Transform each sigma poin hrough he non-linear funcion Couresy: Cyrill Sachniss
24 Unscened Transform Compue Gaussian from he ransformed and weighed sigma poins Couresy: Cyrill Sachniss
25 Nonlinear Gaussian Filers Approach 2: Unscened Kalman Filer Approximae he PDF! Use he full nonlinear plan and observaion models and recompue 1 s and 2 nd order saisics.
26 UKF Linearizaion via Unscened Transform EKF UKF
27 27 UKF Sigma-Poin Esimae: Large Variance EKF UKF
28 28 UKF Sigma-Poin Esimae: Narrow Variance EKF UKF
29 UKF vs. EKF Couresy: E.A. Wan and R. van der Merwe
30 Unscened Transform Overview Compue a se of sigma poins Each sigma poins has a weigh Transform he poin hrough he non-linear funcion Compue a Gaussian from weighed poins Avoids need o linearize around he mean as Taylor expansion (and EKF) does Couresy: Cyrill Sachniss
31 Sigma Poins How o choose he sigma poins? How o se he weighs? Couresy: Cyrill Sachniss
32 Sigma Poins Properies How o choose he sigma poins? How o se he weighs? Selec so ha: There is no unique soluion for Couresy: Cyrill Sachniss
33 Sigma Poins Choosing he sigma poins Firs sigma poin is he mean Couresy: Cyrill Sachniss
34 Sigma Poins Choosing he sigma poins marix square roo column vecor dimensionaliy scaling parameer Couresy: Cyrill Sachniss
35 Real Symmeric Marix Square Roo Defined as Compued via diagonalizaion Couresy: Cyrill Sachniss
36 Real Symmeric Marix Square Roo Thus, we can define so ha and have he same Eigenvecors Couresy: Cyrill Sachniss
37 Cholesky Marix Square Roo Alernaive definiion of he marix square roo Resul of he Cholesky decomposiion Numerically sable soluion Ofen used in UKF implemenaions Acually, any such square roo facorizaion is ok, e.g., could use facorizaion
38 Sigma Poins and Eigenvecors Sigma poins can bu do no have o lie on he main axes of Couresy: Cyrill Sachniss
39 Sigma Poins Example Sigma = SigmaL = SigmaS = SigmaA =
40 Sigma Poin Weighs Weigh sigma poins for compuing he mean parameers for compuing he covariance Couresy: Cyrill Sachniss
41 Recover he Gaussian Compue Gaussian from weighed and ransformed poins Couresy: Cyrill Sachniss
42 (Scaled) Unscened Transform 2n+1 +n -n Source: Van Der Merwe, Thesis
43 Example Couresy: Thrun, Burgard, Fox
44 Examples Couresy: Cyrill Sachniss
45 Unscened Transform Summary Sigma poins Weighs Couresy: Cyrill Sachniss
46 SUT Parameers Free parameers as here is no unique soluion Scaled Unscened Transform suggess Influence how far he sigma poins are away from he mean Opimal choice for Gaussians Couresy: Cyrill Sachniss
47 SUT parameers Choose 0 o guaranee posiive semi-definieness of he covariance marix. The specific value of is no criical hough, so a good defaul choice is =0. Choose 0< 1 o conrol he size of he sigma-poin disribuion and should be chosen o avoid sampling non-local effecs when he nonlineariies are srong; a defaul choice is =1. Choose 0 o incorporae knowledge of he higher-order momens of he disribuion. For example, for a Gaussian prior he opimal choice is =2. The original (un-scaled) UT ransform is equivalen o: SUT wih =1, =0, =
48 (Scaled) Unscened Transform 48 Sigma poins Weighs Pass sigma poins hrough nonlinear funcion Recover mean and covariance Sigma poin scaling Higher-order momen maching Scalar uning parameer
49 Examples Couresy: Cyrill Sachniss
50 Examples Couresy: Cyrill Sachniss
51 Malab Demo of SUT CTools/Resources/L07/su_demo.zip
52 How o apply UT o esimaion?? UKF (Unscened Kalman Filer)
53 UKF uses he Kalman Updae KF is he Bes Linear Unbiased Esimaor (BLUE) i.e., if we resric our esimaor o he class of linear esimaors, hen he KF is he bes linear MMSE esimaor* Affine funcion of z Wha should A and b be? * Noe: a nonlinear esimaor could do beer!!
54 To derive, we wan our error o be orhogonal o he measuremen space Esimaor Error Unbiased Orhogonal
55 Bes Linear Unbiased Esimaor (BLUE) Unbiased Orhogonal Esimaor Marix MSE Remarks The bes esimaor (in he MMSE sense) for Gaussian Random variables is idenical o The bes linear unbiased esimaor for arbirarily disribued random variables wih he same firs- and second-order momens.
56 EKF Algorihm* * The form shown assumes addiive process and observaion model noise Couresy: Cyrill Sachniss
57 EKF o UKF Predicion Unscened replace his by sigma poin propagaion of he moion Couresy: Cyrill Sachniss
58 UKF Algorihm Predicion* * The form shown assumes addiive process model noise Couresy: Cyrill Sachniss
59 EKF o UKF Correcion Unscened replace his by sigma poin propagaion of he moion use sigma poin propagaion for he expeced observaion and Kalman gain Couresy: Cyrill Sachniss
60 UKF Algorihm Correcion (1)* * The form shown assumes addiive observaion model noise Couresy: Cyrill Sachniss
61 UKF Algorihm Correcion (1)* (from BLUE) * The form shown assumes addiive observaion model noise
62 UKF Algorihm Correcion (2) Couresy: Cyrill Sachniss
63 UKF This version of he algorihm implicily assumes addiive zero-mean process and observaion noise Take care wih means of circular quaniies
64 Means of Circular Quaniies Trick is o map angles θ i o he uni circle Take arihmeic mean of Caresian quaniies Map back o corresponding average angle* *Noe: poor approx when θ i is widely disribued
65 Similarly Map angular differences, such as when compuing innovaion and covariance expressions, e.g.: i.e. 2π-0 = 0!!!
66 UKF vs. EKF Couresy: Thrun, Burgard, Fox
67 UKF vs. EKF (Small Covariance) Couresy: Thrun, Burgard, Fox
68 UKF vs. EKF Banana Shape EKF approximaion UKF approximaion Couresy: Cyrill Sachniss
69 UKF Summary 69 Highly efficien: Same complexiy as EKF, wih a consan facor slower in ypical pracical applicaions Beer linearizaion han EKF: Accurae in firs wo derivaives * of Taylor expansion (EKF only firs erm) Derivaive-free: No Jacobians needed Sill no opimal! * Accurae in firs hree derivaives if Gaussian prior
70 UKF vs. EKF Same resuls as EKF for linear models Beer approximaion han EKF for non-linear models Differences ofen somewha small No Jacobians needed for he UKF Same complexiy class Slighly slower han he EKF Couresy: Cyrill Sachniss
71 Lieraure Unscened Transform and UKF Thrun e al.: Probabilisic Roboics, Chaper 3.4 A New Exension of he Kalman Filer o Nonlinear Sysems by Julier and Uhlmann, 1995 Sigma-Poin Kalman Filers for Probabilisic Inference in Dynamic Sae-Space Models, PhD Thesis, Rudolph van der Merwe, 2004 Couresy: Cyrill Sachniss
72 Nex Lecure Kalman Localizaion wih Landmarks Inro o Paricle Filers
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