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1 Assignment, Project and Presentation Mobile Robot Localization by using Particle Filter by Chong Wang, Chong Fu, and Guanghui Luo. Tracking, Mapping & Localizing the iRobot: An effort using only the iRobot Creates sensors by PankajKumar Mendapara, Dibyendu Mukherjee, Ashirbani Saha Implementing the Monte Carlo Localization using Lego NXT by Yuefeng Wang Sepideh Seifzadeh Little Border Protector Guard by Mohammad Raeesi Ahmad Soleimani Autonomous Moving IRobot Based on Vision System by Thanh Nguyen, Mohammed Golam Sarwer Soccer for One by Jonathan Vermette, Jonathan Farlam, Shawn DenHartogh Slide 2 2 Assignment, Project and Presentation Thursday (Nov 20) Yuefeng Wang Sepideh Seifzadeh Tuesday (Nov 25) Mohammad Raeesi Ahmad Soleimani Chong Wang, Chong Fu, and Guanghui Luo. Thursday (Nov 27) Jonathan Vermette, Jonathan Farlam, Shawn DenHartogh PankajKumar Mendapara, Dibyendu Mukherjee, Ashirbani Saha Tuesday (Dec 2) Thanh Nguyen, Mohammed Golam Sarwer Slide 3 3 Mobile Robot Localization (ch. 7, 8) We are now back to the topic of localization after reviewing some necessary background. Mobile robot localization is the problem of determining the pose of a robot relative to a given map of the environment. Remember, in localization problem, the map is given, known, available. Is it hard? Not really, because, Slide 4 4 Mobile Robot Localization Most localization algorithms are variants of Bayes filter algorithm. However, different representation of maps, sensor models, motion model, etc lead to different variant. Slide 5 5 Mobile Robot Localization Solved already, the Bayes filter algorithm. How? The straightforward application of Bayes filters to the localization problem is called Markov localization. Here is the algorithm (abstract?) Slide 6 6 Mobile Robot Localization Algorithm Bayes_filter ( ) for all do endfor return Slide 7 7 Mobile Robot Localization Algorithm Markov Locatlization ( ) for all do endfor return The Markov Localization algorithm addresses the global localization problem, the position tracking problem, and the kidnapped robot problem in static environment. Slide 8 8 Mobile Robot Localization Revisit Figure 7.5 to see how Markov localization algorithm in working. The algorithm Markov Localization is still very abstract. To put it in work (eg. your project), we need a lot of more background knowledge to realize motion model, sensor model, etc. We studied them (motion and sensor models) in Ch 5, and 6. Put everything together in Markov Localization algorithm. Slide 9 9 Mobile Robot Localization Slide 10 10 Mobile Robot Localization Slide 11 11 Mobile Robot Localization We will discuss a few different implementations of Markov Localization algorithm based on: Kalman filter Discrete, grid representation Particle filter (MCL) Slide 12 SA-1 12 Bayes Filter Implementations (1) (Extended) Kalman Filter (Gaussian filters) (Ch.3 and 7) Page 201-220 in Ch 7 and Page 40-64 Read and Compare them Slide 13 13 Prediction Correction Bayes Filter Reminder Slide 14 14 Gaussians -- Univariate Multivariate Slide 15 15 Properties of Gaussians Slide 16 16 We stay in the Gaussian world as long as we start with Gaussians and perform only linear transformations. Multivariate Gaussians Slide 17 17 Discrete Kalman Filter Estimates the state x of a discrete-time controlled process that is governed by the linear stochastic difference equation with a measurement Slide 18 18 Components of a Kalman Filter Matrix (nxn) that describes how the state evolves from t-1 to t without controls or noise. Matrix (nxl) that describes how the control u t changes the state from t-1 to t. Matrix (kxn) that describes how to map the state x t to an observation z t. Random variables representing the process and measurement noise that are assumed to be independent and normally distributed with covariance R t and Q t respectively. Slide 19 19 Kalman Filter Updates in 1D Slide 20 20 Kalman Filter Updates in 1D Slide 21 21 Kalman Filter Updates in 1D Slide 22 22 Kalman Filter Updates Slide 23 23 Linear Gaussian Systems: Initialization Initial belief is normally distributed: Slide 24 24 Dynamics are linear function of state and control plus additive noise: Linear Gaussian Systems: Dynamics Slide 25 25 Linear Gaussian Systems: Dynamics Slide 26 26 Observations are linear function of state plus additive noise: Linear Gaussian Systems: Observations Slide 27 27 Linear Gaussian Systems: Observations Slide 28 28 Kalman Filter Algorithm 1. Algorithm Kalman_filter( t-1, t-1, u t, z t ): 2. Prediction: 3. 4. 5. Correction: 6. 7. 8. 9. Return t, t Slide 29 29 The Prediction-Correction-Cycle Prediction Slide 30 30 The Prediction-Correction-Cycle Correction Slide 31 31 The Prediction-Correction-Cycle Correction Prediction Slide 32 32 Kalman Filter Summary Highly efficient: Polynomial in measurement dimensionality k and state dimensionality n: O(k 2.376 + n 2 ) Optimal for linear Gaussian systems! Most robotics systems are nonlinear! Slide 33 33 Nonlinear Dynamic Systems Most realistic robotic problems involve nonlinear functions Slide 34 34 Linearity Assumption Revisited Slide 35 35 Non-linear Function Slide 36 36 EKF Linearization (1) Slide 37 37 EKF Linearization (2) Slide 38 38 EKF Linearization (3) Slide 39 39 Prediction: Correction: EKF Linearization: First Order Taylor Series Expansion Slide 40 40 EKF Algorithm 1.Extended_Kalman_filter ( t-1, t-1, u t, z t ): 2. Prediction: 3. 4. 5. Correction: 6. 7. 8. 9. Return t, t Slide 41 41 Landmark-based Localization Slide 42 42 Landmark-based Localization Slide 43 43 Landmark-based Localization Our EKF localization algorithm assumes that the map is represented by a collection of features. All features are uniquely identifiable. At any point in time t, the robot gets to observe a vector of ranges and bearings to nearby features: The identity of a feature is expressed by set of correspondence variables, denoted The correspondence is known. Slide 44 44 Landmark-based Localization Slide 45 45 Landmark-based Localization We have silently assumed the availability of an appropriate motion and measurement model, and have left unspecified a number of key variables in the EKF update. We will now discuss a concrete implementation of the EKF, for feature-based maps. Our feature-based maps consist of point landmarks. For such point landmarks, we will use the common measurement model discussed in Chapter 6.6 (feature based measurement model). We will also adopt the velocity motion model defined in Chapter 5.3 (velocity motion model). Slide 46 SA-1 46 a Slide 47 47 Prediction: Correction: Reminder The notations Slide 48 SA-1 48 a Slide 49 SA-1 49 a Slide 50 SA-1 50 a Slide 51 SA-1 51 a Slide 52 SA-1 52 EKF Localization with unknown Correspondence pg. 215-218 Slide 53 SA-1 53