Markov Localization & Bayes Filtering 1 with Kalman Filters Discrete Filters Particle Filters Slides adapted from Thrun et al., Probabilistic Robotics
Faculties of Engineering & Computer Science 2 Autonomous Robotics CSCI 6905 / Mech 6905 – Section 3 Dalhousie Fall 2011 / 2012 Academic Term Introduction Motion Perception Control Concluding Remarks LEGO Mindstorms Control Scheme for Autonomous Mobile Robot
Faculties of Engineering & Computer Science 3 Autonomous Robotics CSCI 6905 / Mech 6905 – Section 3 Dalhousie Fall 2011 / 2012 Academic Term Introduction Motion Perception Control Concluding Remarks LEGO Mindstorms Control Scheme for Autonomous Mobile Robot – the plan –Thomas will cover generalized Bayesian filters for localization next week –Mae sets up the background for him, today, by discussing motion and sensor models as well as robot control –Mae then follows on Bayesian filters to do a specific example, underwater SLAM
Markov Localization 4 The robot doesn’t know where it is. Thus, a reasonable initial believe of it’s position is a uniform distribution.
Markov Localization 5 A sensor reading is made (USE SENSOR MODEL) indicating a door at certain locations (USE MAP). This sensor reading should be integrated with prior believe to update our believe (USE BAYES).
Markov Localization 6 The robot is moving (USE MOTION MODEL) which adds noise.
Markov Localization 7 A new sensor reading (USE SENSOR MODEL) indicates a door at certain locations (USE MAP). This sensor reading should be integrated with prior believe to update our believe (USE BAYES).
Markov Localization 8 The robot is moving (USE MOTION MODEL) which adds noise. …
12 Recursive Bayesian Updating Markov assumption: z n is independent of z 1,...,z n-1 if we know x.
13 Putting oberservations and actions together: Bayes Filters Given: Stream of observations z and action data u: Sensor model P(z|x). Action model P(x|u,x’). Prior probability of the system state P(x). Wanted: Estimate of the state X of a dynamical system. The posterior of the state is also called Belief:
14 Graphical Representation and Markov Assumption Underlying Assumptions Static world Independent noise Perfect model, no approximation errors
15 Bayes Filters Bayes z = observation u = action x = state Markov Total prob. Markov
17 Bayes Filter Algorithm 1. Algorithm Bayes_filter( Bel(x),d ): 2. 0 3. If d is a perceptual data item z then 4. For all x do 5. 6. 7. For all x do 8. 9. Else if d is an action data item u then 10. For all x do 11. 12. Return Bel’(x)
Linear transform of Gaussians -- Univariate Gaussians
We stay in the “Gaussian world” as long as we start with Gaussians and perform only linear transformations. Multivariate Gaussians
23 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
24 Linear Gaussian Systems: Initialization Initial belief is normally distributed:
25 Dynamics are linear function of state and control plus additive noise: Linear Gaussian Systems: Dynamics
26 Observations are linear function of state plus additive noise: Linear Gaussian Systems: Observations
27 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
28 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!
29 Nonlinear Dynamic Systems Most realistic robotic problems involve nonlinear functions
35 Prediction: Correction: EKF Linearization: First Order Taylor Series Expansion
36 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
37 Localization Given Map of the environment. Sequence of sensor measurements. Wanted Estimate of the robot’s position. Problem classes Position tracking Global localization Kidnapped robot problem (recovery) “Using sensory information to locate the robot in its environment is the most fundamental problem to providing a mobile robot with autonomous capabilities.” [Cox ’91]
39 EKF Summary Highly efficient: Polynomial in measurement dimensionality k and state dimensionality n: O(k 2.376 + n 2 ) Not optimal! Can diverge if nonlinearities are large! Works surprisingly well even when all assumptions are violated!
40 [Arras et al. 98]: Laser range-finder and vision High precision (<1cm accuracy) Kalman Filter-based System [Courtesy of Kai Arras]
42 Belief is represented by multiple hypotheses Each hypothesis is tracked by a Kalman filter Additional problems: Data association: Which observation corresponds to which hypothesis? Hypothesis management: When to add / delete hypotheses? Huge body of literature on target tracking, motion correspondence etc. Localization With MHT
43 MHT: Implemented System (2) Courtesy of P. Jensfelt and S. Kristensen
46 Discrete Bayes Filter Algorithm 1. Algorithm Discrete_Bayes_filter( Bel(x),d ): 2. 0 3. If d is a perceptual data item z then 4. For all x do 5. 6. 7. For all x do 8. 9. Else if d is an action data item u then 10. For all x do 11. 12. Return Bel’(x)
1. Algorithm particle_filter( S t-1, u t-1 z t ): 2. 3. For Generate new samples 4. Sample index j(i) from the discrete distribution given by w t-1 5. Sample from using and 6. Compute importance weight 7. Update normalization factor 8. Insert 9. For 10. Normalize weights Particle Filter Algorithm
draw x i t 1 from Bel (x t 1 ) draw x i t from p ( x t | x i t 1,u t 1 ) Importance factor for x i t : Particle Filter Algorithm
68 Limitations The approach described so far is able to track the pose of a mobile robot and to globally localize the robot. How can we deal with localization errors (i.e., the kidnapped robot problem)?
69 Approaches Randomly insert samples (the robot can be teleported at any point in time). Insert random samples proportional to the average likelihood of the particles (the robot has been teleported with higher probability when the likelihood of its observations drops).
73 Summary Particle filters are an implementation of recursive Bayesian filtering They represent the posterior by a set of weighted samples. In the context of localization, the particles are propagated according to the motion model. They are then weighted according to the likelihood of the observations. In a re-sampling step, new particles are drawn with a probability proportional to the likelihood of the observation.