Autonomous Navigation for Flying Robots - PowerPoint Presentation

Slide1

Autonomous Navigation for Flying RobotsLecture 3.1:3D Geometry

Jürgen

Sturm

Technische

Universität

MünchenSlide2

Points in 3D3D point

Augmented

vector

Homogeneous coordinates

Jürgen Sturm

Autonomous Navigation for Flying Robots

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Geometric Primitives in 3D3D line through pointsInfinite

line:

Line

segment joining :Jürgen Sturm

Autonomous Navigation for Flying Robots

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Geometric Primitives in 3D3D plane3D plane equationNormalized plane

with

unit normal

vector and distance Jürgen Sturm

Autonomous Navigation for Flying Robots

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3D TransformationsTranslationEuclidean transform (translation + rotation), (also called the Special Euclidean group SE(3))

Scaled rotation, affine transform, projective transform…

Jürgen Sturm

Autonomous Navigation for Flying Robots

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3D Euclidean TransformationsTranslation has 3 degrees of freedom

Rotation

has

3 degrees of freedomJürgen Sturm

Autonomous Navigation for Flying Robots

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3D RotationsA rotation matrix is a 3x3 orthogonal matrix

Also

called the special orientation group SO(3

)Column vectors correspond to coordinate axes

Jürgen Sturm

Autonomous Navigation for Flying Robots

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3D RotationsWhat operations do we typically do with rotation matrices?

Invert, concatenate

Estimate/optimize

How easy are these operations on matrices?Jürgen Sturm

Autonomous Navigation for Flying Robots

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3D RotationsAdvantage: Can be easily concatenated and inverted (how?)Disadvantage:

Over-parameterized (

9 parameters instead of 3)

Jürgen Sturm

Autonomous Navigation for Flying Robots

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Euler AnglesProduct of 3 consecutive rotations (e.g., around X-Y-Z axes)Roll-pitch-yaw convention is very common in aerial navigation (DIN 9300)

Jürgen Sturm

Autonomous Navigation for Flying Robots

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http://en.wikipedia.org/wiki/File:Rollpitchyawplain.pngSlide11

Roll-Pitch-Yaw ConventionRoll , Pitch , YawConversion

to

3x3

rotation matrix:

Jürgen Sturm

Autonomous Navigation for Flying Robots

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Roll-Pitch-Yaw ConventionRoll , Pitch , YawConversion

from

3x3

rotation matrix:

Jürgen Sturm

Autonomous Navigation for Flying Robots

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Euler AnglesAdvantage:Minimal representation (3 parameters)Easy interpretation

Disadvantages:

Many “alternative” Euler representations exist (XYZ, ZXZ, ZYX, …)

Difficult to concatenateSingularities (gimbal lock)

Jürgen Sturm

Autonomous Navigation for Flying Robots

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Gimbal LockWhen the axes align, one degree-of-freedom (DOF) is lost:

Jürgen Sturm

Autonomous Navigation for Flying Robots

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http://commons.wikimedia.org/wiki/File:Rotating_gimbal-xyz.gifSlide15

Axis/AngleRepresent rotation byrotation axis androtation angle

4 parameters

3 parameters

length is rotation anglealso called the angular velocityminimal but not unique (why?)

Jürgen Sturm

Autonomous Navigation for Flying Robots

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ConversionRodriguez’ formulaInverse

see:

An Invitation to 3D

Vision (Ma,

Soatto, Kosecka, Sastry

), Chapter 2

Jürgen Sturm

Autonomous Navigation for Flying Robots

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Axis/AngleAlso called twist coordinatesAdvantages:Minimal representation

Simple derivations

Disadvantage:

Difficult to concatenateSlow conversionJürgen Sturm

Autonomous Navigation for Flying Robots

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QuaternionsQuaternionReal and vector part

Unit

quaternions have

Opposite sign quaternions represent the same rotationOtherwise

unique

Jürgen SturmAutonomous Navigation for Flying Robots

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Richard

Szeliski

, Computer Vision: Algorithms and

Applications

http

://szeliski.org/Book/Slide19

QuaternionsAdvantage: multiplication, inversion and rotations are very efficient

Concatenation

Inverse (=flip signs of real or imaginary part)

Rotate 3D vector using a quaternion:

Jürgen Sturm

Autonomous Navigation for Flying Robots

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QuaternionsRotate 3D vector using a quaternion:Relation to

axis

/angle representation

Jürgen Sturm

Autonomous Navigation for Flying Robots

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3D OrientationsNote: In general, it is very hard to “read” 3D orientations/rotations, no matter in what representation

Observation:

They are usually easy to visualize and can then be intuitively interpreted

Advice: Use 3D visualization tools for debugging (RVIZ, libqglviewer, …)

Jürgen Sturm

Autonomous Navigation for Flying Robots

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3D to 2D Perspective ProjectionsJürgen Sturm

Autonomous Navigation for Flying Robots

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Richard

Szeliski

, Computer Vision: Algorithms and Applications

http://szeliski.org/Book/Slide23

3D to 2D Perspective ProjectionsJürgen Sturm

Autonomous Navigation for Flying Robots

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Richard

Szeliski

, Computer Vision: Algorithms and Applications

http://szeliski.org/Book/Slide24

3D to 2D Perspective ProjectionsJürgen Sturm

Autonomous Navigation for Flying Robots

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Pin-hole

camera model

Note: is

homogeneous, needs to be normalizedSlide25

Camera IntrinsicsSo far, 2D point is given in meters on image planeBut: w

e

want 2D point be measured in pixels (as the sensor does

)Jürgen Sturm

Autonomous Navigation for Flying Robots

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Richard

Szeliski

, Computer Vision: Algorithms and Applications

http://szeliski.org/Book/Slide26

Camera IntrinsicsNeed to apply some scaling/offset

Focal length

Camera center

SkewJürgen Sturm

Autonomous Navigation for Flying Robots

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Camera ExtrinsicsAssume is given in world coordinatesTransform from world to camera (also called the camera

extrinsics

)

Projection of 3D world points to 2D pixel coordinates:

Jürgen Sturm

Autonomous Navigation for Flying Robots

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Lessons Learned3D points, lines, planes

3D

transformations

Different representations for 3D orientationsChoice depends on applicationWhich representations do you remember?3D to 2D perspective projectionsYou

really have to know 2D/3D transformations by heart

(for more info, read Szeliski, Chapter 2, available online)

Jürgen Sturm

Autonomous Navigation for Flying Robots

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