1. Kalman filter and SLAM problem
Young Ki Baik
Computer Vision Lab.
Seoul National University

2. Contents References Kalman filter SLAM problem
Example (2D circular motion)
Demo
Conclusion and future work

3. References An Introduction to the Kalman Filter
G. Welch and G. Bishop (SIGGRAPH 2001)
A Solution to the Simultaneous Localization and Map Building (SLAM) problem
Gamini Dissanayake. Et. Al. (IEEE Trans. Robotics and Automation 2001)
Lessons in Estimation Theory for Signal Processing, Communications and Control
Jerry M. Mendel (1995)

4. Kalman filter What is a Kalman filter? Applications
Mathematical power tool
Optimal recursive data processing algorithm
Noise effect minimization
Applications
Tracking (head, hands etc.)
Lip motion from video sequences of speakers
Fitting spline
Navigation
Lot’s of computer vision problem

5. Kalman filter Kalman filter Example Sensor noise Measurement error
Landmark
Kalman filter
How can we obtain optimal pose of robot and landmark simultaneously?
Real location
Robot
Location with error
Movement noise
Refined location
Localizing error (Processing error)

6. Kalman filter Example (Simple Gaussian form) Assumption
All error form Gaussian noise
Estimated value
Measurement value

7. Kalman filter Example (Simple Gaussian form) Optimal variance
Optimal mean
Innovation
Kalman gain

8. Kalman filter
Example (Overall process)
Prediction
Update

9. SLAM What is SLAM problem?
Can we do localization and mapping simultaneously?
If we have the solution to the SLAM problem…
Allow robots to operate in an environment without a priori knowledge of a map
Open up a vast range of potential application for autonomous vehicles and robot
Kalman filter based approach
Research over the last decade has shown that SLAM is indeed possible

10. SLAM Kalman filter and SLAM problem
Extended Kalman filter form for SLAM
Prediction
Observation
Update
: Previous value
: Input and measure
: Function
: Computed value

11. Implementation Example (2D circular motion) x, z, L Setting of x and P
x : Position and direction of robot
and L
z : Distance and angle from robot point of view
L : Landmark position
Setting of x and P

12. Implementation Example (2D circular motion) Initial x and P
Setting of x and P with landmark
3x1
3x3
5x1
5x5

13. : Circular motion with radius 25
Implementation
Example (2D circular motion)
Control input
: 2d position and direction
: Velocity and angular velocity
: time (constant)
: Circular motion with radius 25

14. Implementation Example (2D circular motion) Real motion
: Zero mean unit variance Gaussian random value
: Control error for velocity
: Control error for angular velocity
White line : Control input motion
Pink line : Real motion
Large circle : robot

15. Implementation Example (2D circular motion)
Predicted and measured information of land mark
Prediction
Measurement
: Position of i-th landmark (x,y)
: Distance and angle (d, ) from a robot point of view r
Range r = 20.0
sensor

16. Implementation
Example (2D circular motion)
Jacobian matrix for F

17. Implementation
Example (2D circular motion)
Jacobian matrix for H

18. Implementation Example (2D circular motion) Error covariant matrix
Covariant matrix of control error
Covariant matrix of measurement error
: Control error for velocity
: Control error for angular velocity
: measurement error for distance
: Measurement error for angle

19. Implementation Kalman filter and SLAM problem
Extended Kalman filter form for SLAM
Prediction
Observation
Update
: Previous value
: Input and measure
: Function
: Computed value

20. Implementation Demo Large Circle(white, pink, yellow) : robot
White line : control input path
Pink line : real path
Small white circle : Real landmark
Yellow line : Estimated path (EKF)
Large light blue circle
: Detected (and estimated) landmark
Blue ellipse : Uncertainty boundary

21. Conclusion Conclusion Future work Simple example and demo
Possibility of solution for SLAM problem using EKF
In the limit of successive observations, the error in estimated position of landmarks become fully correlated.
Future work
Considering closing loop and kidnapping problem
Applying EKF to general structure (robot) using vision sensor