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Jose-Luis Blanco, Javier González, Juan-Antonio Fernández-Madrigal University of Málaga (Spain) Dpt. of System Engineering and Automation May 19-23 Pasadena, CA (USA) An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” Outline of the talk 1. Introduction 2. The proposed method 3. Experimental results 4. Conclusions

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” Outline of the talk 1. Introduction 2. The proposed method 3. Experimental results 4. Conclusions

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 1. Introduction The addressed problem: Bayesian filtering Two choices determine the tools suitable to solve this problem: The representation of the prior/posterior densities: Gaussian vs. samples. Assumptions about the form of the likelihood. p(x) : prior belief p(y|x) : observation likelihood p(x|y) : posterior belief

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 1. Introduction In this work: Representation of pdfs? Observation likelihood? Any arbitrary function (need to evaluate it pointwise) Weighted, random samples (particle filter)

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 1. Introduction The basic particle filtering algorithm: The role of the proposal distribution in particle filters p(x) : prior belief What happens to each particle?

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 1. Introduction The basic particle filtering algorithm: The role of the proposal distribution in particle filters What happens to each particle? Draw new particles from the proposal distribution:

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 1. Introduction The basic particle filtering algorithm: The role of the proposal distribution in particle filters What happens to each particle? Weights are updated, depending on: The observation likelihood, and The proposal distribution.

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 1. Introduction The basic particle filtering algorithm: The role of the proposal distribution in particle filters What happens to each particle? Weights are updated, depending on: The observation likelihood, and The proposal distribution. p(y|x) : observation likelihood

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 1. Introduction The basic particle filtering algorithm: The role of the proposal distribution in particle filters The goal To approximate as well as possible the posterior How much does the choice of the proposal distribution matter? p(x|y) : posterior belief

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 1. Introduction The basic particle filtering algorithm: The role of the proposal distribution in particle filters p(x|y) : posterior belief How much does the choice of the proposal distribution matter? q(·) : proposal distribution

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 1. Introduction The basic particle filtering algorithm: The role of the proposal distribution in particle filters p(x|y) : posterior belief How much does the choice of the proposal distribution matter? q(·) : proposal distribution For a large mismatch between proposal and posterior, the particles represent the density very poorly:

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 1. Introduction The role of the proposal distribution in particle filters The proposal distribution q(·) is the key for the efficiency of a particle filter! It is common to use the transition model as proposal: We refer to this choice as the standard proposal. It is far from optimal. [Doucet et al. 2000] introduced the optimal proposal.

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 1. Introduction Relation of our method to other Bayesian filtering approaches: Non-Linear Observation model Optimal proposal Algorithms Gaussian-Kalman Filter Gaussian-EKF, UKF Arbitrary SIR, APF, FastSLAM Gaussian FastSLAM 2.0, [Grisetti et al. 2007] Arbitrary This work

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” Outline of the talk 1. Introduction 2. The proposed method 3. Experimental results 4. Conclusions

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 2. The proposed method Our method: A particle filter based on the optimal proposal [Doucet et al. 2000]. Can deal with non-parameterized observation models, using rejection sampling to approximate the actual densities. Integrates KLD-sampling [Fox 2003] for a dynamic sample size (optional: it’s not fundamental to the approach). The weights of all the samples are always equal.

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 2. The proposed method The theoretical model for each step of our method is this sequence of operations: Duplication SIR with optimal proposal Fixed/Dyn. sample-size resampling

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 2. The proposed method The theoretical model for each step of our method is this sequence of operations: Duplication SIR with optimal proposal Fixed/Dyn. sample-size resampling

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 2. The proposed method The theoretical model for each step of our method is this sequence of operations: Duplication SIR with optimal proposal Fixed/Dyn. sample-size resampling

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 2. The proposed method The theoretical model for each step of our method is this sequence of operations: Duplication SIR with optimal proposal Fixed/Dyn. sample-size resampling

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 2. The proposed method Illustrative example of how our method works: [1][1] t–1 t [2][2] [3][3] [4][4] Particles at time t-1

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 2. The proposed method Illustrative example of how our method works: t–1 t Each particle propagates in time probabilistically: this is the reason of the duplication Group [ 1 ] [1][1] [2][2] [3][3] [4][4]

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 2. The proposed method Illustrative example of how our method works: t–1 t Each particle propagates in time probabilistically: this is the reason of the duplication Group [ 1 ] [1][1] [2][2] [3][3] [4][4] Group [ 2 ]

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 2. The proposed method Illustrative example of how our method works: t–1 t Each particle propagates in time probabilistically: this is the reason of the duplication Group [ 1 ] [1][1] [2][2] [3][3] [4][4] Group [ 2 ] Group [ 3 ]

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 2. The proposed method Illustrative example of how our method works: t–1 t Each particle propagates in time probabilistically: this is the reason of the duplication Group [ 1 ] [1][1] [2][2] [3][3] [4][4] Group [ 2 ] Group [ 3 ] Group [4]

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 2. The proposed method Illustrative example of how our method works: t–1 t The observation likelihood states what particles are really important… Group [ 1 ] [1][1] [2][2] [3][3] [4][4] Group [ 2 ] Group [ 3 ] Group [4] Too distant particles do not contribute to the posterior! Observation likelihood

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 2. The proposed method Illustrative example of how our method works: t–1 t Group [ 1 ] [1][1] [2][2] [3][3] [4][4] Group [ 2 ] Group [ 3 ] Group [4] We can predict which groups will be more important, before really generating the new samples! Observation likelihood

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 2. The proposed method The optimal proposal distribution: Importance weights update as: The weight does not depend on the actual value of the particle.

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 2. The proposed method Illustrative example of how our method works: t–1 t Group [ 1 ] [1][1] [2][2] [3][3] [4][4] Group [ 2 ] Group [ 3 ] Group [4] Group [ 1 ] 55% Group [ 2 ] 0% Group [ 3 ] 45% Group [ 4 ] 0% Observation likelihood

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 2. The proposed method Illustrative example of how our method works: t–1 t Group [ 1 ] [1][1] [2][2] [3][3] [4][4] Group [ 2 ] Group [ 3 ] Group [4] Observation likelihood Given the predictions, we draw particles according to the optimal proposal, only for those groups that really contribute to the posterior. A fixed or dynamic number of samples can be generated in this way.

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 2. The proposed method Comparison to… basic Sequential Importance Resampling (SIR)

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 2. The proposed method Comparison to… basic Sequential Importance Resampling (SIR) t–1 t [1][1] [2][2] [3][3] [4][4] Observation likelihood 1 particle 1 particle Prone to particle depletion!

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 2. The proposed method Comparison to… Auxiliary Particle Filter (APF) [Pitt & Shephard, 1999]

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 2. The proposed method t–1 t [1][1] [2][2] [3][3] [4][4] Observation likelihood 1 particle variable number of particles Propagation does not use optimal proposal! Comparison to… Auxiliary Particle Filter (APF) [Pitt & Shephard, 1999]

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” Outline of the talk 1. Introduction 2. The proposed method 3. Experimental results 4. Conclusions 3.1. Numerical simulation 3.2. Robot localization

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 3.1. Results Numerical simulations: A Gaussian model for both the filtered density and the observation model. We compare the closed form optimal solution (Kalman filter) to: PF using the “standard” proposal distribution. Auxiliary PF method [Pitt & Shephard, 1999]. This work (“optimal” PF). (Fixed sample size for these experiments)

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 3.1. Results Results from the numerical simulations, and comparison to 1D Kalman filter: x axis: particles y axis: weights Approximated pdf (histogram) from particles. Actual pdf from Kalman filter. Kullback-Leibler distance (KLD) for increasing number of samples.

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 3.1. Results Results from the numerical simulations, and comparison to 1D Kalman filter:

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” Outline of the talk 1. Introduction 2. The proposed method 3. Experimental results 4. Conclusions 3.1. Numerical simulation 3.2. Robot localization

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” Robot path during localization Start End 1 m 3.2. Results Localization with real data: Path of the robot: ground truth from a RBPF with a large number of particles.

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 3.2. Results Localization with real data: Average errors in tracking (the particles are approximately at the right position from the beginning).

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 3.2. Results Ratio of convergence from global localization: Localization with real data: Ratio of convergence success Initial sample size (particles/m 2 ) 10 1 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Our method SIR with “standard” proposal

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” 3.2. Results

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” Outline of the talk 1. Introduction 2. The proposed method 3. Experimental results 4. Conclusions

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” Conclusions A new particle filter algorithm has been introduced. It can cope with non-parameterized observation likelihoods, and a dynamic number of particles. Compared to standard SIR, it provides more robust global localization and pose tracking for similar computation times. It is a generic algorithm: can be applied to other problems in robotics, computer vision, etc.

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Jose Luis Blanco University of Málaga “An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization” Source code (MRPT C++ libs), datasets, slides and instructions to reproduce the experiments available online: http://mrpt.sourceforge.net/ papersICRA 08 Finally…

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Jose-Luis Blanco, Javier González, Juan-Antonio Fernández-Madrigal University of Málaga (Spain) Dpt. of System Engineering and Automation Thanks for your attention! An Optimal Filtering Algorithm for Non-Parametric Observation Models in Robot Localization

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