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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.

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Robot Localization Using Bayesian Methods

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  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.

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