Quaternion-based attitude representation by linear combination of matrix operators

8/10/2019 Quaternion-based attitude representation by linear combination of matrix operators

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Introduction

Geometrical interpretation of rotation quaternions

Conclusions

References

Quaternion-based attitude representation by linearcombination of matrix operators

Rigoberto Juarez-Salazar1

, Carlos Robledo-Sanchez2

, and W.Fermin Guerrero-Sanchez2

1Instituto Tecnologico Superior de ZacapoaxtlaDivision de Ingeniera Informatica

2Benemerita Universidad Autonoma de Puebla

Facultad de Ciencias Fsico Matematicas

XLVII Congreso Nacional de la Sociedad Matematica MexicanaDurango, Mexico. Octubre, 2014.

1 / 2 0 Rigoberto Juarez Salazar [email protected] Quaternion attitude representation by linear combination of matrix operators
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Introduction


Conclusions

References

Content

1 IntroductionOptical 3D imaging systemThe motivation problemRotation quaternions

2 Geometrical interpretation of rotation quaternionsAlternative representation of rotation quaternionsGraphic illustrationPreliminary experimental results

3 Conclusions

4 References

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Introduction


Conclusions

References

Optical 3D imaging system

The motivation problem

Rotation quaternions

Introduction Optical 3D imaging system

Figura :Optical setup of a imaging system for 3Ddataacquisition

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Introduction


Conclusions

References




Introduction Optical 3D imaging system (Cont. 1)

A 3D imaging system is very useful in many areas such as medical, industrial andengineering because of its inherent features such as noninvasive, fast, and highaccuracy.

Figura :Survey of dental apparatus.

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Introduction


Conclusions

References




Introduction Optical 3D imaging system (Cont. 4)

A 3D imaging system is very useful in many areas such as medical, industrial andengineering because of its inherent features such as noninvasive, fast, and highaccuracy.

Figura :3D imaging technology for accident scene investigation.


I t d ti
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Introduction


Conclusions

References




Introduction The motivation problem

Extrinsic camera parameters calibration

For three-dimensional surface object reconstruction, it is necessary to known theso-calledextrinsic camera parameters(spatial and angular positions).

xy

z

!

!

z y z

!

! ! !

Camera

x0

y0

z0


Introduction
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Introduction


Conclusions

References




Introduction Rotation quaternions

Unitary quaternions or rotation quaternionsare a very efficient tool to perform

three-dimensional rotations; by the simple expression

y=qxq (1)

whereyis the rotated version of the quaternion xwithqbeing the unitaryquaternion

q=cos

2+usin

2, (2)

with being the rotation angle around the rotation axis defined by the unitarythree-dimensional vectoru.

!

u

x

y

e1

e2

e3


Introduction
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Introduction


Conclusions

References




Introduction Rotation quaternions (Cont.)

Three-dimensional rotations by unitary quaternions have a simple and brief

notation. They are computationally fast and stable.

Unfortunately, for some applications the quaternion product is not intuitive and itsmathematical manipulation is not alway easy.

In this talk we will present an alternative representation of quaternion rotations by using

matrix operators.

!

u

x

y

e

1

e2

e3


Introduction
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Introduction


Conclusions

References

Alternative representation of rotation quaternions

Graphic illustration

Preliminary experimental results

Quaternion rotation

Letqbe a unitary quaternion given by

q=cos +usin , (3)

where = /2 with being the rotation angle around the unitary three-dimensionalvectoru. Alternatively,(3)can be represented as

q=

cos usin

. (4)

Then, for the three-dimensional rotation of point v,qvq, we have the first quaternionproduct

q

0v

=

u vsin

vcos +u vsin

. (5)

After that, the rotation ofvis performed by

q

0v

q =

u vsin

vcos +u vsin

cos usin

, (6)

10 / 20 Rigober to Juarez Salazar [email protected] Quaternion attitude representation by linear combination of matrix operators
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IntroductionAlt ti t ti f t ti t i


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Conclusions

References




Quaternion rotation

By using the matrix representation of both inner and cross vectorial products:

(u v)u=uuTv,

u v=uv,(11)

withu being the skew-symmetric matrix:

u =

0 u3 u2u3 0 u1u2 u1 0

, (12)

we have forw:w= Qv. (13)

Finally, we can reach

Q =2 sin2 uuT + (cos2 sin2 )I +2 sin cos u, (14)

or through a simplification procedure as:

.


IntroductionAlternative representation of rotation quaternions
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Conclusions

References




Quaternion rotation

Q =cos2 I sin2 H +2 sin cos u, (15)

where H is the Householder matrix given by

H = I 2uuT. (16)


IntroductionAlternative representation of rotation quaternions
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Conclusions

References




Geometrical interpretation of quaternion rotation

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Quaternion-based attitude representation by linear combination of matrix operators