Coordinate transformations
Kim Mathiassen
28.09.2010
Nomenclature
Vectors
Rotation matrix
Simple rotations
Euler angles
Angle-axis
Quaternions
Euler-Rodrigues
Comparison
Angular velocity
References
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Nomenclature
~a
|~a|
a
|a|
ab
a×
A
R ab
q ab
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Geometric vector
Euclidean norm of a geometric vector
Algebraic vector
Euclidean norm of a algebraic vector
Algebraic vector represented in frame b
Skew symmetric form of a
Matrix
Rotation matrix from frame b to frame a
Quaternion representing the rotation from frame b to frame a
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Vectors
Geometric vector
~u =
n
X
ui ~ai
i=1
where
Algebraic vector


u1


u =  ... 
un
ui = ~u · ~ai
Scalar product
~u · ~v = |~u ||~v | cos θ
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~u · ~v = u T v
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Cross product
Geometric vector
~u × ~v = ~n|~u ||~v | sin θ
Algebraic vector


0
−u3 u2
0
−u1 
u × =  u3
−u2 u1
0
~a1 ~a2 ~a3 ~ = ~u × ~v = u1 u2 u3 w
v1 v2 v3 w = u×v
u × is the skew symmetric form of the vector u
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Definitions
I
Frame - Coordinate system defined with a set of basis vectors
I
Direction Cosine Matrix - Rotates between frames which the
basis vectors not necessarily are orthonormal right-handed
system
I
Rotation matrix - Rotates between frames which the basis
vectors forms a orthonormal right-handed system
Will only consider orthonormal right-handed systems
The rotation matrix is also called coordinate transformation matrix
or attitude matrix depending on the context
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Rotation matrix
To transform an algebraic vector from b to a we introduce the
rotation matrix R ab . The transformation is given by
v a = R ab v b
This matrix satisfies
det(R ab ) = 1
R ab R ba = I
R ab = (R ba )−1
R ab = (R ba )T
Combining series of rotations
R ab R bc = R ac
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Coordinate transformation
The following equation is used to transform from coordinates in
frame b to coordinates in frame a
p a = p aab + R ab p b
p a - Position coordinates in frame a
p b - Position coordinates in frame b
p aab - The position in a coordinates of the origin of frame b relative
to the origin a
R ab - Rotation matrix from frame b to frame a
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Homogeneous transformation matrices
The coordinate transformation may be expressed as a homogeneous
transformation matrices
a a R b r ab
a
Tb =
0T 1
A property of the homogeneous transformation matrix is
(T ab )−1 = T ba
To transform coordinates between multiple frames we can use
T ac = T ab T bc
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Simple rotations


1
0
0
R x (φ) =  0 cos φ − sin φ 
0 sin φ cos φ


cos θ 0 sin θ
0
1
0 
R y (θ) = 
− sin θ 0 cos θ


cos ψ − sin ψ 0
R z (ψ) =  sin ψ cos ψ 0 
0
0
1
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Euler angles
The rotation is based on the simple rotations
Roll-Pitch-Yaw
R ab = R z (ψ)R y (θ)R x (φ)
Classical Euler angles
R ab = R z (ψ)R y (θ)R z (φ)
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Angle-axis parameterization
The rotation between the frames a and b is about the vector ~k by
an angle θ. The coordinates of ~k is the same in both frames a and b
ka = kb = k
The rotation matrix as a function of k and θ is given by
R ab (k, θ) = cos θI + sin θk × + (1 − cos θ)kk T
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Euler parameters/Quaternions
Euler parameters is defined by
η = cos
θ
2
= k sin
θ
2
A quaternion is a vector defined by (defined in two different ways
depending on the author)
q = q0 + q1 i + q2 j + q3 k
= q0 i + q1 j + q2 k + q3
where qi ∈ R and i , j and k satisfy the equations
i i = j j = kk = i j k = −1
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Rotations with Euler parameters
q=
η
q=
η
where η and are Euler parameters and where q T q = 1.
The inverse unit quaternion of q is defined by
η
q̄ =
−
The rotation matrix is given as a function of Euler parameters by
R ab (η, ) = I + 2η× + 2× ×
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Quaternion product
T
The quaternion product between the quaternions q 1 = [α1 β T
1]
T T
and q 2 = [α2 β 2 ] is given by
α1 α2 − β T
β2
1
q1 ⊗ q2 =
α1 β 2 + α2 β 1 + β ×
1 β2
A rotation can be expressed as two quaternion products
0
η
0
η
=
⊗
⊗
Rv
v
−
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Rotations with quaternions
If we have three frames, a, b and c, and we know the quaternions
q ab (rotation from frame b to frame a) and q bc (rotation from frame
c to frame b). It is possible to calculate the quaternion q ac
(rotation from frame c to frame a) by using the following equation
q ac = q ab ⊗ q bc
Normally we would calculate the rotation as
v a = R ab R bc v c
By using the quaternion product this problem can also be
formulated as
0
0
a
= qc ⊗
⊗ q̄ ac
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va
vc
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Euler-Rodrigues parameters
Defined as
θ
ρ = k tan( )
2
ρ=
η
Singularities for η = 0, but no trigonometric functions. Rotation
matrix given by
R = (I + ρ× )(I − ρ× )−1
Used in numerical integrators in attitude problems
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Comparison
Param.
Matrix
Euler angles
Advantage
No singularities
no trigonometric functions
simple product rule
No redundant parameters
clear physical interpretation
Angle-axis
Clear physical interpretation
Quaternions
No singularities
no trigonometric functions
simple product rule
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Drawback
Six redundant paramters
Trigonometric functions
singularities for som angles
no easy prduct rule
One redundant parameter
undefined for sin θ = 0
trigonometric functions
One redundant parameter
no clear physical interpretation
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Angular velocity
Frames can rotate relative to each other. Let the vector ω
~ ab be the
geometric vector of ω aab . ω
~ ab is the angular velocity vector of frame
b relative to frame a. The differential equation of the rotation
matrix can be written as
R˙ab = (ω aab )× R ab = R ab (ω bab )×
For quaternions the angular velocity is given by
1 0
⊗q
q̇ =
2 ω
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References
Olav Egeland and Jan Tommy Gravdahl.
Modeling and Simulation for Automatic Control.
Tapir trykkeri, 2002.
Oddvar Hallingstad.
Matematisk modellering av dynamiske systemer.
2008.
Compendium in UNIK4540 (Norwegian).
Kim Mathiassen.
Positioning with inertial sensors and aid sensors when loosing
gps fix.
Master’s thesis, Norwegian University of Science and
Technology, 2010.
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