Report

Stochastic Processes Mini Conference Winter 2011 EE 670 - Prof. Brian Mazzeo Amin Nazaran Stephen Quebe 1 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. 2 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].” 3 The Localization Problem Given a map of an environment and a sequence of sensor measurements and control inputs, estimate the robot’s pose. 4 The Localization Problem Inputs Outputs Robot initial pose. Estimated robot pose. θ ( ,x )y Control inputs. Y Observations. O x robot's state : y θ X Map feature or landmarks. 5 Robot Motion and Observation Models θ ( ,x )y Y O x robot's state : y θ X 6 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. 7 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. 8 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. 9 Stochastic Motion Model Bayes Network 10 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 11 Bayesian Estimation: Prediction Robot pose space Motion model Based on the total probability theorem: (discrete case) where Bi, i=1,2,... is a partition of W. In the continuous case: 12 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. 13 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. 14 EKF Assumptions and Violations 15 EKF Algorithm 1. EKF_localization ( mt-1, St-1, ut, zt, m): Prediction: 3. Gt 5. Vt g (ut , mt 1 ) xt 1 x' mt 1, x y ' mt 1, x ' mt 1, x g (ut , mt 1 ) ut x' vt y ' vt ' v t x' mt 1, y y ' mt 1, y ' mt 1, y x' t y ' t ' t x' mt 1, y ' Jacobian of g w.r.t location mt 1, ' mt 1, Jacobian of g w.r.t control EKF Algorithm Continued 1 | vt | 2 | t |2 M t 0 2 3 | vt | 4 | t | mt g (ut , mt 1 ) St Gt St 1GtT Vt MtVtT 0 Motion noise 17 Predicted mean Predicted covariance 17 17 EKF Measurement Update Normalizing factor Measurement model Based on the Bayes Rule: i.e. also: Taking: We have: 18 EKF_localization ( mt-1, St-1, ut, zt, m): 1. Correction: 3. zˆ i t 2 2 mx , j mt , x my , j mt , y Predicted measurement mean atan2m m , m m m y, j t,y x, j t,x t , h( mt , m) xt rti m t ,ix t m t,x 5. H ti 6. r2 0 Qt 2 0 r 7. S ti H ti St H ti Qt T 8. K ti St H ti Sti 9. rti mt , y ti mt , y rti mt , ti mt , Jacobian of h w.r.t location T Pred. measurement covariance 1 Kalman gain mti mti Kti ( zti zˆti ) 10. St I Kti Hti St Updated mean Updated covariance 19 EKF Simulation Results Normal operation. Overconfident prediction. Overconfident measurement. Large time steps where linearization fails. External bump where Markov assumption fails. 20 Simulation Results Show simulation results in real time by opening matlab. 21 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). 22 Thank You For Your Attention Questions? 23 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. 24