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Real-Time Simultaneous Localization and Mapping with a
Single Camera(Mono SLAM)
2005. 9. 26Young Ki Baik
Computer Vision Lab.
Seoul National University
Real-Time SLAM with a Single Camera
Computer Vision Lab. SNU
Contents
References
Kalman filter and SLAM
Mono-SLAM
Simulation Demo
Conclusion
Real-Time SLAM with a Single Camera
Computer Vision Lab. SNU
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
Real-Time SLAM with a Single Camera
Computer Vision Lab. SNU
Kalman filter
What is a Kalman filter?Mathematical power tool
Optimal recursive data processing algorithm
Noise effect minimization
ApplicationsTracking (head, hands etc.)
Lip motion from video sequences of speakers
Fitting spline
Navigation
Lot’s of computer vision problem
Real-Time SLAM with a Single Camera
Computer Vision Lab. SNU
Kalman filter
Example
Real location
Location with error
Measurement error
Localizing error (Processing error)
Refined location
Robot
Landmark
Movement noise
Sensor noise
How can we
obtain optimal
pose of robot and
landmark
simultaneously?
Kalman filter
Real-Time SLAM with a Single Camera
Computer Vision Lab. SNU
Kalman filter
Example (Simple Gaussian form)
Assumption
All error form Gaussian noise
Estimated value
Measurement value
2, eex 2, eexN
2, mmxN 2, mmx
Real-Time SLAM with a Single Camera
Computer Vision Lab. SNU
Example (Simple Gaussian form)
Optimal variance
Optimal value
Kalman filter
2,xN
mme
ee
me
m xxx
22
2
22
2
222
111
me
emme
ee xxxx
22
2
iKxx e Kalman gain
Innovation
Real-Time SLAM with a Single Camera
Computer Vision Lab. SNU
SLAM
SLAM
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
Real-Time SLAM with a Single Camera
Computer Vision Lab. SNU
SLAM
Kalman filter and SLAM problem
Extended Kalman filter form for SLAM
Prediction
Observation
Update
kSJkPkK THxe
1)(
xFJFx
xHJHx
kxHkz ee
kzkzki em )(
kikKkxkx e )( kKkSkKkPkP T
e )(
kQkJkPkJkP TFxFxe 1
kukxFkxe ,1
kRJkPJkS THxeHx )(
: Previous value
: Input and measure
: Function
: Computed value
iFL LFJi
Real-Time SLAM with a Single Camera
Computer Vision Lab. SNU
Mono SLAM
What is Mono SLAM?EKF-SLAM framework (EKF : Extended Kalman Filter)
Single camera
Unknown user input
User inputKnown 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.
kukxFkxe ,1 ?
Real-Time SLAM with a Single Camera
Computer Vision Lab. SNU
Mono SLAM
World frame model
WY
World Frame WWZ
WX
RY
RX
RZ
Camera
frame R W : World coordinate
R : Local coordinate
r
r : Camera position
vector in W frame
yy : Landmark position
vector in W frame
Real-Time SLAM with a Single Camera
Computer Vision Lab. SNU
Mono SLAM
Motion model
3D position and orientation
T
ZYXWR
W
p qqqqZYXq
r,,,,,, 0
x
form)n (Quaternio camera ofn orientatio:
camera ofposition -3D:
vectorstate :
W
W
p
q
r
x
This state vector is parameters for conventional SLAM .
Real-Time SLAM with a Single Camera
Computer Vision Lab. SNU
Mono SLAM
Motion modelKey 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.
Real-Time SLAM with a Single Camera
Computer Vision Lab. SNU
Mono SLAM
Motion model (Mono SLAM)
3D position and orientation
camera ofn orientatio:
camera ofposition 3D:
SLAM-mono of vector state :
W
WR
v
q
r
x
W
W
WR
W
v
w
v
q
r
x
T
ZYX qqqqZYX ,,,,,, 0
TZYXZYX wwwvvv ,,,,,
motion camera oflocity angular ve :
motion camera ofocity linear vel:W
W
w
v
The total dimension of state
vector is 13.
Real-Time SLAM with a Single Camera
Computer Vision Lab. SNU
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.
locityangular ve of difference the:
velocityof difference the:
vectornoise the:
W
W
W
V
n
t
taW
W
W
W
V
n
WaW
Real-Time SLAM with a Single Camera
Computer Vision Lab. SNU
Mono SLAM
Motion model (Mono SLAM)
State update function
function update state:
vectorstate previous:
vectorstate new :
F
x
xnew
WW
WW
WWWR
WWW
Wnew
Wnew
WRnew
Wnew
w
v
twqq
tvr
w
v
q
r
V
V
F
kkknew uxFx ,1
Quaternion trivially
defined by the
angle-axis rotation
vector tw WW
uV
n
t
taW
W
W
W
: Previous value
: Unknown user input
Real-Time SLAM with a Single Camera
Computer Vision Lab. SNU
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
n vector noise of covariance:
yuncertaint state previous:
yuncertaint state new :
n
new
P
P
P
T
nv PQ
n
F
n
F
kQkJkPkJkP vT
new FxFx 1
uxF ,
Real-Time SLAM with a Single Camera
Computer Vision Lab. SNU
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.
nP
nP
nP
Real-Time SLAM with a Single Camera
Computer Vision Lab. SNU
Simulation Demo
Condition
Simple circular motion
Five 3D landmarks
Observation is 2D using projective camera model
3D view 2D view
Estimated LM
Real LM
Estimated Pos
& CovarianceProjected LM
Real-Time SLAM with a Single Camera
Computer Vision Lab. SNU
Conclusion
Conclusion
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.
Real-Time SLAM with a Single Camera
Computer Vision Lab. SNU
Real-Time SLAM with a Single Camera
Computer Vision Lab. SNU
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
Z
Y
X
),,( Rparameter
Real-Time SLAM with a Single Camera
Computer Vision Lab. SNU
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
Z
Y
X),,( AAA ZYX
,,, AAA ZYXRparameter
Real-Time SLAM with a Single Camera
Computer Vision Lab. SNU
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
Z
Y
X),,( AAA ZYX
2sin,
2sin,
2sin,
2cos AAA ZYXR
parameter