Monte Carlo Localization for Mobile Robots
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1 Monte Carlo Localization for Mobile Robots Frank Dellaert 1, Dieter Fox 2, Wolfram Burgard 3, Sebastian Thrun 4 1 Georgia Institute of Technology 2 University of Washington 3 University of Bonn 4 Carnegie Mellon University Take Home Message Representing uncertainty using samples is powerful, fast, and simple! 1
2 Outline Robot Localization Sensor? Density Representation? Monte Carlo Localization Results Outline Robot Localization Sensor? Density Representation? Monte Carlo Localization Results 2
3 Minerva Motivation Crowded public spaces Unmodified environments 3
4 Desired Location Museum Application Exhibit Global Localization Where in the world is Minerva the Robot? Vague initial estimate Noisy and ambiguous sensors 4
5 Local Tracking Sharp initial estimate Noisy and ambiguous sensors The Bayesian Paradigm Knowledge as a probability distribution 60% Rain 40% dry 5
6 Probability of Robot Location P(Robot Location) Y State space = 2D, infinite #states X Bayesian Filtering Two phases: 1. Prediction Phase 2. Measurement Phase 6
7 1. Prediction Phase u x t-1 x t P(x t ) = P(x t x t-1,u) P(x t-1 ) Motion Model 2. Measurement Phase z x t P(x t z) = k P(z x t ) P(x t ) Sensor Model 7
8 Outline Robot Localization Sensor? Density Representation? Monte Carlo Localization Results Sonar? Laser? Vision? What sensor? 8
9 Problem: Large Open Spaces Walls and obstacles out of range Sonar and laser have problems One solution: Coastal Navigation Problem: Large Crowds Horizontally mounted sensors have problems One solution: Robust filtering 9
10 Solution: Ceiling Camera Upward looking camera Model of the world = Ceiling Mosaic Global Alignment (other talk) 10
11 Ceiling Mosaic Large FOV Problems 3D ceiling -> 3D Model? Matching whole images slow 11
12 Small FOV Solution Model = orthographic mosaic No 3D Effects Very fast Vision based Sensor P(z x) z h(x) 12
13 Outline Robot Localization Sensor? Density Representation? Monte Carlo Localization Results Hidden Markov Models A B C D E A B C D E 13
14 Very powerful Gaussian, unimodal Kalman Filter sensor motion motion Under Light 14
15 Next to Light Elsewhere 15
16 Markov Localization Fine discretization over {x,y,theta} Very successful: Rhino, Minerva, Xavier Dynamic Markov Localization Burgard et al., IROS 98 Idea: use Oct-trees 16
17 Sampling as Representation P(Robot Location) Y X Samples <=> Densities Density => samples Obvious Samples => density Histogram, Kernel Density Estimation 17
18 Sampling Advantages Arbitrary densities Memory = O(#samples) Only in Typical Set Great visualization tool! minus: Approximate Outline Robot Localization Sensor? Density Representation? Monte Carlo Localization Results 18
19 Disclaimer Handschin 1970 (!) lacked computing power Bootstrap filter Monte Carlo filter Condensation 1993 Gordon et al Kitagawa 1996 Isard & Blake Added Twists Camera moves, not object Global localization 19
20 Monte Carlo Localization S k-1 S k weighted S k S k Predict Weight Resample 1. Prediction Phase u P(x t,u) Motion Model 20
21 2. Measurement Phase P(z x t ) Sensor Model 3. Resampling Step O(N) 21
22 A more in depth look Bayes Law, new look Densities: update prior p(x) to p(x z) via l(x;z) Samples update a sample from p(x) to a sample from the posterior p(x z) through l(x;z) 22
23 Bayes Law Problem We really want p(x z) samples But we only have p(x) samples! How can we upgrade p(x) to p(x z)? More General Problem We really want h(x) samples But we only have g(x) samples! How can we upgrade g(x) to h(x)? 23
24 Solution = Importance Sampling 1. generate x i from g(x) 2. calculate w i = h(x i )/g(x i ) 3. assign weight q i = w i / w i Still works if h(x) only known up to normalization factor Mean and Weighted Mean Fair sample: obtain samples x i from p(x z) E[m(x) z] ~m(x i )/N Weighted sample: obtain weighted samples (x i,q i ) from p(x z) E[m(x) z] ~q i m(x i ) 24
25 Bayes Law using Samples 1. generate x i from p(x) 2. calculate w i = l(x i ;z) 3. assign weight q i = w i / w i Indeed: w i =p(x z)/p(x) = l(x;z) p(x) /p(x) = l(x;z) 4. if you want, resample from (x i,q i ) Monte Carlo Localization S k-1 S k weighted S k S k Predict Weight Resample 25
26 Outline Robot Localization Sensor? Density Representation? Monte Carlo Localization Results Office Environment Sonar Sensors Global Localization Symmetry confusion Video A 26
27 Global Localization Global Localization (2) 27
28 Global Localization (3) Reference Path 28
29 Accuracy Smithsonian Museum of American History Ceiling Camera, Global Localization Video B 29
30 Odometry Only Using Vision 30
31 Fast Internet Morning Fast Internet Morning Odometry only Vision Laser 31
32 UW Sieg Hall Laser Video C Conclusions Monte Carlo Localization: Powerful yet efficient Significantly less memory and CPU Very simple to implement Future: discrete states, rate information, distributed 32
33 Take Home Message Representing uncertainty using samples is powerful, fast, and simple! 33
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