Tracker Alignment - TUM · 2005. 7. 22. · Tracker Alignment using Hand-Eye Calibration Sebastian...

Tracker Alignmentusing Hand-Eye Calibration

Sebastian Grembowietz

2005-03-31 Sebastian Grembowietz [tracker alignment] 2

Outline Introduction

Tracker Alignment• Definition and Applications

Achieving Aligned Trackers Solving AX=XB

classical solution (Tsai)• Splitting and solving AX=XB

More robust solution (Daniilidis)• Complex Numbers, Dual Quaternions and Screws• Solving aq = qb

Improving accuracy

Introduction of myself Sebastian Grembowietz

23 years old studying computer science at TUM 5th semester Current focus: Augmented Reality

Tracker Alignment Hybrid trackers

Combining strengths Minimizing weaknesses

Applications More robust sensors:

ART + camera:• fewer dependence on illumination• obtaining 3d data from camera

Aurora + ART:• partly independent from line-of-view• high precision

Applications Robotics aided model generation

Robot arm and laser scanner combined

Applications Robotics aided model generation

Taking measurementsfor custom-madeclothing

Applications Automated part assembly

Automatic weldingguidedby feedbackfrom cameras

Applications Endoscopic Surgery

minimally invasive surgery

Aligning Trackers Forward Engineering

Hybrid tracker entirely designed Only one way of mounting possible Fixed alignment

Aligning Trackers Manually registering points

User points on marker Correspondences used

for tracker alignment

Aligning Trackers Automatic

alignmentfrommovements

Setup Transforms

obtained

Setup Transforms

computed

Setup Transform

wanted

Classical Solution 1989: Roger Tsai and Reimar Lenz

Splitting AX=XB AX=XB in matrix form

multiplied out

2 Phases Solving AX=XB in 2 phases

Phase 1 - computing modified Rodrigues formula

Phase 2 - computing

Phase 1 - Rotation Rodrigues formula corresponding to

Rodrigues formula computed soon

Phase 1 - Rotation Step 1 - For all movement pairs

rewritten to

and solved usinglinear least squares

Phase 1 - Computing Rotation Step 2 - obtain

Phase 2 - Translation known

Computation of from

using linear least squares

Modern Solution 1998:

KostasDaniilidis

Complex Numbers Short introduction:

Addition

Complex Numbers Multiplication

Conjugate

Complex Numbers Multiplication = Rotation in 2D

anglesadded

normsmultiplied

Quaternions Discovered by Hamilton in 1834

Quaternions extension of complex numbers to 4D

imaginaryunitsi, j, k

Quaternion Conjugate

Addition

Quaternions Multiplication

Quaternions Norm

Unit quaternions: norm of 1

Using Quaternions Unit quaternion used for rotation in 3D

Rotating x to y

Superiority of quaternions......concerning rotations: size

4 values instead of 9 interpolation

SLERP: Spherical Linear Interpolation no gimbal lock robustness thanks to easy normalization

Dual Numbers Definition

Addition

Dual Numbers Multiplication

Plücker Coordinates Dual numbers used in vectors as

representation of lines

direction u moment v

Dual Quaternions Dual Quaternion = Quaternion using dual

numbers

with q, q‘ quaternions

Line transformations Line as Plücker coordinate

with additional constraint

line transformations by

line transformationsusing dual quaternions

Line transformations Proof: From vector ...

... to quaternion ...

Line transformations

... to dual quaternions

or equivalently

Screws Rigid transformation (R, t)

equivalentto screw

rotation axis lnot throughorigin

pitch d same angle

Screw from Transform Direction l of screw axis

parallel to rotation axis of transform Pitch d

projecton of translation on rotation axis Moment m

using auxilliary point c

Screw to Dual Quaternion

Back to Tracker Alignment AX=XB with dual quaterions

Screw Congruence Theorem Pitch and angle of a screw invariant under

rigid transformations

Tracker Alignment Simplified

Note: only motion of the screw axis left

Tracker Alignment Splitting in dual and non-dual part

rearranging and using

Tracker Alignment Disregarding the scalar parts to set up

the linear system

with S being 6x8 matrix

Tracker Alignment Construct 6nx8 matrix T

for n motion pairs

Compute SVD

and obtain

Tracker Alignment Compute the lambdas using

unit quaternion criterions

Constraits result in

Tracker Alignment Assuming so that

and substituting results in

and thus two solutions for s Choose the s maximizing

and easily obtain the lambas

Tracker Alignment Convert dual quaternions to

non-dual part corresponds to rotation

translation by using

Improving accuracy Preselection by applying threshold angle

for rotations

Optimality criterion to be minimized for allmotion pairs

Using as many stations as possible RANSAC approach

Thank you...... for your attention

Tracker Alignment - TUM · 2005. 7. 22. · Tracker Alignment using Hand-Eye Calibration Sebastian...

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