Probabilistic Fundamentals in Robotics. DAUIN Politecnico di Torino July 2010

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1 Probabilistic Fundamentals in Robotics Gaussian Filters Basilio Bona DAUIN Politecnico di Torino July 2010

2 Course Outline Basic mathematical framework Probabilistic models of mobile robots Mobile robot localization problem Robotic mapping Probabilistic planning and control Reference textbook Thrun, Burgard, Fox, Probabilistic Robotics, MIT Press, robotics.org/ Basilio Bona July

3 Basic mathematical framework Recursive state estimation Basic concepts in probability bbl Robot environment Bayes filters Gaussian filters (parametric filters) Kl Kalman filter Extended Kalman Filter UnscentedKalman filter Information filter Nonparametric filters Histogram filter Particle filter Basilio Bona July

4 Introduction Gaussian filters are different implementations of Bayes filters for continuous spaces, with specific assumptions on probability distributions Beliefs are represented by multi variate normal distributions Basilio Bona July

5 Multi variate Gaussian distribution Covariance matrix Mean vector Basilio Bona July

6 Examples Bi dimensional Gaussian with conditional probabilities Mixture of Gaussians Basilio Bona July

7 Covariance matrix Basilio Bona July

8 Kalman filter (1) Kalman filter (KF) [Swerling: 1958, Kalman: 1960] applies to linear Gaussian systems KF computes the belief for continuous states governed by linear dynamic state equations Beliefs are expressed by normal distributions KF is not applicable to discrete or hybrid state space systems Basilio Bona July

9 Kalman filter (2) Basilio Bona July

10 Kalman filter (3) Basilio Bona July

11 Kalman filter (4) Basilio Bona July

12 Kalman filter algorithm (1) Prediction Kalman gain Innovation (residuals) covariance Update Basilio Bona July

13 Block diagram z t u t B t μ + t + C t zˆt + At μt 1 D μt + K + t residuals Basilio Bona July

14 Kalman filter algorithm (2) visible hidden Basilio Bona July

15 Kalman filter example Initial state measurement update prediction measurement update Basilio Bona July

16 From Kalman filter to extended Kalman filter Kalman filter is based on linearity assumptions Gaussian random variables are expressed by means and covariance matrices of normal distributions Gaussian distributions are transformed into Gaussian distributions Kalman filter is optimal Kalman filter is efficient Basilio Bona July

17 Linear transformation of Gaussians Basilio Bona July

18 Extended Kalman Filter (EKF) When the linearity assumptions do not hold (as in robot motion models dlor orientation i models) dl) a closed dform solution of the predicted belief does not exists Nonlinear state & measurement equations ExtendedKalman Filter (EKF) approximates the nonlinear transformations with a linear one Linearization is performed around the most likely value: i.e., the mean value Basilio Bona July

19 EKF Example Montecarlo generated distribution Transformed mean value Approximating mean value Approximating Gaussian Approximating Gaussian uses mean and covariance of the Montecarlo generated distribution Basilio Bona July

20 EKF Example Basilio Bona July

21 EKF Example Approximating Gaussian: the normal distribution built using mean and covariance of the true nonlinear distributions Approximating Gaussian EKF Gaussian EKF Gaussian: the normal distribution built using mean and covariance of the true nonlinear distributions Basilio Bona July

22 EKF linearization Taylor expansion Depends only on the mean Basilio Bona July

23 EKF algorithm Basilio Bona July

24 KF vs EKF Basilio Bona July

25 Features EKF is a very popular tool for state estimation in robotics It has the same timecomplexity of the KF It is robust and simple Limitations: rarely state and measurement functions are linear. Goodness of linear approximation depends on Degree of uncertainty Degree of nonlinearity When using EKF the uncertainty must be kept small as much as possible Basilio Bona July

26 Uncertainty More uncertain More uncertain Less uncertain Less uncertain Basilio Bona July

27 Uncertainty More uncertain Less uncertain Basilio Bona July

28 Nonlinearity More nonlinear More linear Basilio Bona July

29 Nonlinearity More nonlinear More linear Basilio Bona July

30 Example: EKF Localization within a sensor infrastructure Fixed sensors (deployed in known positions inside the environment) Mobile bl Robot can acquire odometric measurements and distance information from sensors in known positions True position of the mobile robot t=0 i KF estimate (time zero)

31 Example: EKF Localization within a sensor infrastructure STEP 1: Acquire odometry t=0

32 Example: EKF Localization within a sensor infrastructure STEP 1: Acquire odometry Filter Prediction t=0 Luca Carlone Politecnico di Torino

33 Example: EKF Localization within a sensor infrastructure STEP 1: Acquire odometry Filter Prediction Acquire meas. t=0 Luca Carlone Politecnico di Torino

34 Example: EKF Localization within a sensor infrastructure STEP 1: Acquire odometry Filter Prediction Acquire meas. Filter Update t=0 Luca Carlone Politecnico di Torino

35 Example: EKF Localization within a sensor infrastructure STEP 1: Acquire odometry Filter Prediction Acquire meas. Filter Update STEP 2: Acquire odometry t=0 Luca Carlone Politecnico di Torino

36 Example: EKF Localization within a sensor infrastructure STEP 1: Acquire odometry Filter Prediction Acquire meas. Filter Update STEP 2: Acquire odometry Filter Prediction t=0 Luca Carlone Politecnico di Torino

37 Example: EKF Localization within a sensor infrastructure STEP 1: Acquire odometry Filter Prediction Acquire meas. Filter Update STEP 2: Acquire odometry Filter Prediction Acquire meas. t=0 Luca Carlone Politecnico di Torino

38 Example: EKF Localization within a sensor infrastructure t=0 STEP 1: Acquire odometry Filter Prediction Acquire meas. Filter Update STEP 2: Acquire odometry Filter Prediction Acquire meas. Filter Update Luca Carlone Politecnico di Torino

39 Example: EKF Localization within a sensor infrastructure STEP 1: Acquire odometry Filter Prediction Acquire meas. Filter Update STEP 2: Acquire odometry Filter Prediction Acquire meas. Filter Update t=0... Luca Carlone Politecnico di Torino

40 Unscented Kalman Filter (UKF) UKF performs a stochastic linearization based on a weighted statistical linear regression A deterministic sampling technique (the unscented transform) ) is used to pick a minimal set of sample points (sigma points) around the mean value of the normal pdf The sigma points are propagated through the nonlinear functions, and then used to compute the mean and covariance of the transformeddistribution distribution This approach removes the need to explicitly compute Jacobians, which for complex functions can be difficult to calculate produces a more accurate estimate of the posterior distribution Basilio Bona July

41 UKF Basilio Bona July

42 UKF Basilio Bona July

43 UKF Basilio Bona July

44 UKF Algorithm part a) Basilio Bona July

45 UKF Algorithm part b) Cross covariance Basilio Bona July

46 EKF vs UKF Basilio Bona July

47 EKF vs UKF Basilio Bona July

48 KF EKF UKF KF EKF UKF Basilio Bona July

49 Information filters Belief is represented by Gaussians Moments parameterization Canonical parameterization KF EKF UKF Duality IF EIF Mean Information vector Covariance Information matrix Basilio Bona July

50 Multivariate normal distribution Basilio Bona July

51 Mahalanobis distance Mahalanobis distance Same Euclidean distance Same Mahalanobis distance Basilio Bona July

52 IF algorithm Basilio Bona July

53 IF vs KF IF Prediction i step requires two matrix inversion KF Prediction i step is additive i Measurements update dt is additive Measurements update requires matrix inversion Duality Basilio Bona July

54 Extended information filter EIF It is similar to EKF and applies when state and measurement equations are nonlinear Jacobians G and H replace A, B and C matrices State estimate Basilio Bona July

55 Practical considerations IF advantages over KF: Simpler global uncertainty representation: set Ω = 0 Numerically more stable (in many but not all robotics applications) Integrates information in simpler way Is naturally fit for multi robot problems (decentralized data integration => Bayes rule => logarithmic form => addition of terms => arbitrary order) IF limitations: A state estimation is required (inversion of a matrix) Other matrix inversions are necessary (not required for EKF) Computationally inferior to EKF for high dim state spaces Basilio Bona July

56 Final comments In many problems the interaction between state variable is local l=> structure t on Ω => sparseness of Ω but not of Σ Information filters as graphs: sparse information matrix = sparse graph Such graphs are known as Gaussian Markov random fields Basilio Bona July

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75 Thank you. Any question? Basilio Bona July

76 Thrun Slides on Kalman Filters

77 PhD_course_2010 Outline.pptx Basilio Bona July

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