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.
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Introduction
Geometrical interpretation of rotation quaternions
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
Geometrical interpretation of rotation quaternions
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
Geometrical interpretation of rotation quaternions
Conclusions
References
Optical 3D imaging system
The motivation problem
Rotation quaternions
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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8/10/2019 Quaternion-based attitude representation by linear combination of matrix operators
6/21
Introduction
Geometrical interpretation of rotation quaternions
Conclusions
References
Optical 3D imaging system
The motivation problem
Rotation quaternions
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.
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I t d ti
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Introduction
Geometrical interpretation of rotation quaternions
Conclusions
References
Optical 3D imaging system
The motivation problem
Rotation quaternions
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
7 / 2 0 Rigoberto Juarez Salazar [email protected] Quaternion attitude representation by linear combination of matrix operators
Introduction
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Introduction
Geometrical interpretation of rotation quaternions
Conclusions
References
Optical 3D imaging system
The motivation problem
Rotation quaternions
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
8 / 2 0 Rigoberto Juarez Salazar [email protected] Quaternion attitude representation by linear combination of matrix operators
Introduction
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Introduction
Geometrical interpretation of rotation quaternions
Conclusions
References
Optical 3D imaging system
The motivation problem
Rotation quaternions
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
9 / 2 0 Rigoberto Juarez Salazar [email protected] Quaternion attitude representation by linear combination of matrix operators
Introduction
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Introduction
Geometrical interpretation of rotation quaternions
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)
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IntroductionAlt ti t ti f t ti t i
8/10/2019 Quaternion-based attitude representation by linear combination of matrix operators
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Geometrical interpretation of rotation quaternions
Conclusions
References
Alternative representation of rotation quaternions
Graphic illustration
Preliminary experimental results
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:
.
12 / 20 Rigober to Juarez Salazar [email protected] Quaternion attitude representation by linear combination of matrix operators
IntroductionAlternative representation of rotation quaternions
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Geometrical interpretation of rotation quaternions
Conclusions
References
Alternative representation of rotation quaternions
Graphic illustration
Preliminary experimental results
Quaternion rotation
Q =cos2 I sin2 H +2 sin cos u, (15)
where H is the Householder matrix given by
H = I 2uuT. (16)
13 / 20 Rigober to Juarez Salazar [email protected] Quaternion attitude representation by linear combination of matrix operators
IntroductionAlternative representation of rotation quaternions
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Geometrical interpretation of rotation quaternions
Conclusions
References
Alternative representation of rotation quaternions
Graphic illustration
Preliminary experimental results
Geometrical interpretation of quaternion rotation
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