Real-Time Simultaneous Localization and Mapping with a Single Camera (Mono SLAM) 2005. 9. 26 Young Ki Baik Computer Vision Lab. Seoul National University

Contents References Kalman filter and SLAM Mono-SLAM Simulation Demo Conclusion

References Real-Time Simultaneous Localisation and Mapping with a Single Camera Andrew J. Davison (ICCV 2003) A Solution to the Simultaneous Localization and Map Building (SLAM) problem Gamini Dissanayake. Et. Al. (IEEE Trans. Robotics and Automation 2001) An Introduction to the Kalman Filter G. Welch and G. Bishop (SIGGRAPH 2001) Site for Quaternion http://www.euclideanspace.com/maths/geometry/rotations

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

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)

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

Kalman filter Example (Simple Gaussian form) Optimal variance Optimal value Innovation Kalman gain

SLAM SLAM If we have the solution to the SLAM problem… Simultaneously Localization and map building system EKF(Extended Kalman filter)-based framework 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 Research over the last decade has shown that SLAM is indeed possible

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

Mono SLAM ? What is Mono SLAM? User input EKF-SLAM framework (EKF : Extended Kalman Filter) Single camera Unknown user input User input Known control input Encoder information of robot or vehicle (odometry) ? Most case of localization system, odometry information is used as initial moving value. Mono- slam don't use odometry information and it can be new feature.

Mono SLAM World frame model Camera frame R W : World coordinate World Frame W R : Local coordinate r r : Camera position vector in W frame y : Landmark position vector in W frame y

Mono SLAM Motion model 3D position and orientation This state vector is parameters for conventional SLAM .

Mono SLAM Motion model Key difference between Mono- and conventional-SLAM In the robot case, there is in possession of the control inputs driving the motion, such as “moving forward 1m with steering angle 5 degree” In the hand helded camera case, we do not have such prior information about a person’s movement. Assumption (Mono SLAM) In the case of a camera attached to a person, it takes account of the unknown intentions of the person, but these too can be statistically modeled. Constant velocity, constant angular velocity model are chosen as initial value and added undetermined accelerations occur with a Gaussian profile.

The total dimension of state vector is 13. Mono SLAM Motion model (Mono SLAM) 3D position and orientation The total dimension of state vector is 13.

Mono SLAM Motion model (Mono SLAM) Unknown user input (or noise vector) In each time step, unknown acceleration and angular acceleration processes of zero mean and Gaussian distribution.

Mono SLAM Motion model (Mono SLAM) State update function Quaternion trivially defined by the angle-axis rotation vector : Previous value : Unknown user input

Mono SLAM Motion model (Mono SLAM) Covariance update In the EKF, the new state estimate must be accompanied by the increase in state uncertainty (process noise covariance) for the camera after this motion. Qv is found via the Jacobian calculation

Mono SLAM Motion model (Mono SLAM) Covariance of noise vector The rate of growth of uncertainty in this motion model is determined by the size of , and setting these parameters to small or large values defines the smoothness of the motion we expect. small - We expect a very smooth motion with small accelerations, and would be well placed to track motion but unable to cope with sudden rapid movements High - The uncertainty in the system increases significantly at each time step. - This can be cope with rapid accelerations.

Simulation Demo Condition 3D view 2D view Simple circular motion Five 3D landmarks Observation is 2D using projective camera model 3D view 2D view Estimated LM Real LM Estimated Pos & Covariance Projected LM

Conclusion Conclusion Simulation result Localization is possible with out control input. Simulation result 3D position can be estimated using SLAM through the projected landmark information. It needs more debuging for perfect simulation.

Mono SLAM Quaternion form for orientation (or rotation) Eular angle Arbitrary 3D rotation is equal to one rotation (by scalar angle) around an axis. The result of any sequence of rotation is equal to a single rotation around an axis. 3 degree of freedom in 3D space Gimbal lock problem

Mono SLAM Quaternion form for orientation (or rotation) Axis angle Arbitrary 3D rotation composed by 3-d unit vector and 1-d angle value 4 degree of freedom in 3D space Singularity problem

Mono SLAM Quaternion form for orientation (or rotation) Quaternion angle Arbitrary 3D rotation composed by 3-d unit vector and 1-d angle value 4 degree of freedom in 3D space Why quaternion? Simpler algebra Easy to fix numerical error No singularity and Gimbal lock problem