Particle Filters for Mobile Robot Localization

1. 11/24/2006 Particle Filters for Mobile Robot Localization Instructor: Dr. Shiri Amirkabir University of Technology Aliakbar Gorji Roborics

2. Preface • State Space models • Bayesian Filters for State estimation • Particle Filters • Mobile Robot Localization • Particle Filters for real time localization • Conclusion

3. Nonlinear State Space Systems • A General Model: State’s Dynamic White noise with covariance Q Output Observations White noise with covariance R

4. Nonlinear State Space Systems • Ultimate Goal in modeling: • Inference (State Estimation) • Learning (Parameter Estimation) • Inference designates to the estimation of states with regard to output observations and known parameters. Parameters of f and g

5. Various inference approaches

6. Online System Identification • First Case: f and g are known. • Second Case: There is not any information about the system’s dynamic: • Proposing parametric structures for f and g. Intelligent (Neural, RBF or Fuzzy) Classic (Linear or Nonlinear)

7. What do we seek? • We consider case 1,that is f and g are known. • There is not any parametric structure , therefore, parameter estimation is eliminated. • We are seeking the estimation of states (Latent Variables) based on observations (Sensor Measurements)

8. Bayesian Filters Input and Output measurements • We want to compute: • To convert to a recursive form: • If f and g are linear, the integral is tractable and results in Kalman Filtering. Likelihood State Model

9. Bayesian Filters • If f and g are nonlinear, the density distributions are not in Gaussian form. • Extended Kalman filter: by linearization about nominal point, f and g convert to linear forms. • EKF is not applicable in many real applications such as Target Tracking. • Particle filters prove a strong tool to model the Non-Gaussian distributions.

10. What is Particle Filter? • It is the online version of Monte Carlo algorithms. • Its idea is to estimate a distribution function by sampling.

11. Particle Filter • But, sampling from posterior distribution function is intractable. • Solution: sampling from a simpler distribution function (proposal function). Proposal density function

12. What did change? • Sampling is conducted via proposal function rather than posterior density function. • Question: How can one determine proposal density function. • There are two choices. Suitable accuracy and easy to implement Good accuracy but hard to implement

13. Recursive form for weights • Usually, q is chose as: • Recursive Equation: • Now we are ready to propose Monte Carlo algorithms.

14. SIS algorithm • Draw the samples from prior density function and initialize weights. • For t=1:tmax: • For i=1:N(number of samples): • sample • Compute importance weight and normalize it: • Check the terminating condition (tmax).

15. Degeneracy Problem and SIR algorithm • During the implementation of SIS algorithm the weight of all samples approach zero and only one sample has the weight 1. • Solution: in each iteration, the weights with higher value are multiplied.

16. SIR algorithm

17. SIR algorithm

18. Some Modifications • Kernal methods: considering a Gaussian distribution for each sample.

19. KERNAL method and Hybrid SIR • To adjust the parameters of the above distribution, KALMAN Filter method is combined with SIR algorithm. • The stages of Hybrid SIR algorithm: • KALMAN Filter measurement update. • SIS algorithm to choose the new samples and computing importance weights. • Resampling stage to avoid degeneracy problem.

20. KALMAN Filter measurement update

21. SIS and Resampling stage

22. The general Particle Filter

23. The other Particle Filter Algorithms • Sequential Monte Carlo : mixing Particle Filters with common Monte Carlo methods [ De.Freits PhD thesis, University of Cambridge, 1999]. • Marginalized Particle Filters (Rao-Blackwellized Particle Filters): dividing states to linear and nonlinear ones. For linear states KALMAN Filter and for nonlinear ones Particle Filter is applied.

24. Applications • Navigation and Positioning. • Multiple Target Tracking and Data Fusion. • Financial Forecasting. • Computer Vision. • Wireless Communication and Blind Equalization problems.

25. Mobile Robots Localization • Predicting robot’s position relative to its environment map. • There are three types of positioning: • Position Tracking: the initial position of robot is known. • Global Positioning: the initial conditions are not given (initial values of states are not determined). • Multiple Robot Positioning. • Particle Filters provide satisfactory results for all of above issues.

26. Particle Filters for Mobile Robot Localization • The following points should be considered: • As the point of State Space Models, f is motion dynamic and g is Sensor characteristic and both are supposed to be known. • The following distribution are designated as: Motion Model Perceptual Likelihood

27. How can we determine each distribution? • Motion model is determined by the behaviour of values measured by odometry. • Perceptual Likelihood model is dependent to the sensor used for measurement, such as Sonar, Camera or Laser. • Usually, one sensor is used as target (the one with highest accuracy) and the others’ data are modified by the mentioned sensor. • After determining the structure of each distribution, general Particle Filter is applied for tracking.

28. Simulation • A XR400 robot is tested to be tracked in the following map.

29. Comparison With Grid-Based Markov Model Particle Filter Grid-Based

30. Comparison With Grid-Based Markov Model • A Particle Filter with 1000 to 5000 samples had a similar error compared with a Grid-Based method with resolution 4cm. • The mentioned Grid-Based is not possible to apply in real-time mode but a Particle Filter with 5000 samples is easily implemented in real-time condition.

31. Multi-Robot Particle Filters • A team of robots want to localize each other. • A difficult problem: the states of each robot are dependent to the other robots’ states. • Solution: the following dependency factor is defined:

32. Multi-Robot Particle Filters • Now, the posterior distribution function is determined as: • The recursive equation is derived as: • The above equation can be implemented by Particle Filter.

33. Conclusion • Particle Filters can estimate the wide variety of Non-Gaussian distribution functions. • In comparison with KALMAN Filters, Particle Filters have a more accurate result relative to KALMAN Filters. • Particle Filters are easily implemented and in comparison with Grid-Based methods can provide better results for mobile robot localization.

34. Some References • Dieter Fox, Particle Filters for Mobile Robot Localization. • Jo.ao F. G. de Freitas, Bayesian Methods for Neural Networks, PhD thesis, University of Cambridge. • Website of Dr. Arnaud Doucet, www.cs.ubc.ca/~arnaud/ . • Pierre Del Moral, Arnaud Doucet, ‘Sequential Monte Carlo Samplers’, J. R. Statist. Soc. B (2006). • Huosheng Hu and John Q. Gan, ‘Sensors and Data Fusion Algorithms in Mobile Robotics’, Technical Report: CSM-422, University of Essex.

35. Best Wishes The End