1 / 24

Robot Localization Using Bayesian Methods

Robot Localization Using Bayesian Methods. Stochastic Processes Mini Conference Winter 2011 EE 670 - Prof. Brian Mazzeo Amin Nazaran Stephen Quebe. Presentation Outline. Robot Localization Modeling Robot L ocalization as a Stochastic P rocess. Bayesian Estimation and Filtering.

woods
Download Presentation

Robot Localization Using Bayesian Methods

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Robot Localization Using Bayesian Methods Stochastic Processes Mini Conference Winter 2011 EE 670 - Prof. Brian Mazzeo AminNazaran Stephen Quebe

  2. Presentation Outline • Robot Localization • Modeling Robot Localization as a Stochastic Process. • Bayesian Estimation and Filtering. • The Extended Kalman Filter. • Extended Kalman Filter Simulation Results. • Conclusions.

  3. Robot Localization • In order for a mobile robot to complete many meaningful tasks, it must be able to identify and control its position in an environment. • “Using sensory information to locate the robot in its environment is the most fundamental problem in robotics [1].”

  4. The Localization Problem • Given a map of an environment and a sequence of sensor measurements and control inputs, estimate the robot’s pose.

  5. The Localization Problem Inputs Outputs • Robot initial pose. • Control inputs. • Observations. • Map feature or landmarks. • Estimated robot pose.

  6. Robot Motion and Observation Models

  7. Modeling Robot Localization as a Stochastic Process • One approach to solving this problem is by modeling the robot’s control inputs, observations using a Markov Chain.

  8. The Markov Assumption • The Markov assumption states that if we know the current state of the robot, past and future states are conditionally independent of one another. • In other words. If we know where the robot is now, then knowing where the robot was 5 minutes ago doesn’t give us any more information than we already have, regarding it’s current state. • The arrows on Dynamic Bayes Network show this conditional independence.

  9. Stochastic Motion Model • The robot motion model describes the robot’s pose as a function of it’s previous pose and control inputs. • The observation model describes the robot’s sensor measurements as a function of the robot’s position and the landmark position.

  10. Stochastic Motion Model Bayes Network

  11. Bayesian Estimation and Filtering • It is a recursive algorithm. At time t, given the belief at time t-1 belt-1(xr), the last motion control ut-1 and the last measurement zt, determine the new belief belt(xr) as follows: Motion model Measurement model

  12. Bayesian Estimation: Prediction Motion model Robot pose space Based on the total probability theorem: (discrete case) where Bi, i=1,2,... is a partition of W. In the continuous case:

  13. The Extended Kalman Filter (EKF) • The Extended Kalman Filter is one way to apply Bayesian estimation techniques to robot localization and mapping. • The Kalman filter is the optimal Least Mean Squares estimator of a linear Gaussian system. • The Extended Kalman filter is a way of using the Kalman filter with non-linear models by approximating the model.

  14. EKF Assumptions and Violations • Assumptions: • Gaussian noise and uncertainty. • Linear approximations are good. • Markov assumption or complete state assumption holds. • Violations: • Data association create Non-Gaussian uncertainties. • With large time steps or angles the linear approximation is poor. • If the estimate becomes unstable or overconfident the Markov assumption is violated by a poor estimate. • If the robot is “bumped” or moved by something not in the model, the Markov is also violated.

  15. EKF Assumptions and Violations

  16. EKF Algorithm • EKF_localization( mt-1,St-1, ut, zt,m):Prediction: Jacobian of g w.r.t location Jacobian of g w.r.t control

  17. EKF Algorithm Continued 17 Motion noise Predicted mean Predicted covariance 17

  18. EKF Measurement Update Normalizing factor Measurement model Based on the Bayes Rule: i.e. also: Taking: We have:

  19. Predicted measurement mean • EKF_localization( mt-1,St-1, ut, zt,m):Correction: Jacobian of h w.r.t location Pred. measurement covariance Kalman gain Updated mean Updated covariance

  20. EKF Simulation Results • Normal operation. • Overconfident prediction. • Overconfident measurement. • Large time steps where linearization fails. • External bump where Markov assumption fails.

  21. Simulation Results • Show simulation results in real time by opening matlab.

  22. Conclusions • The critical assumption in the stochastic model is the Markov assumption. This assumption is restrictive but probably cannot be avoided in any real world scenario. • The Extended Kalman Filter implementation is fast and remains consistent under normal conditions. • In the real world the model can be adjusted to reduce and recover from failure. • The robot must be able to recognize and recover from inevitable failure (the lost robot problem).

  23. Thank You For Your AttentionQuestions?

  24. References [1]: I.J. Cox. Blanche—an experiment in guidance and navigation of an autonomous robot vehicle. IEEE Transactions on Robotics and Automation, vol.7,NO.2 ,pp.193–204, 1991. [2] S. Thrun,W. Burgard, and D.Fox, “Probabilistic Robotics”, MIT press: Cambridge, 1967.

More Related