Axis Angle
Content
© 2006 Gerhard Besold
Axis-angle representation
of rotation matrices:
Geometric derivation
and Rodrigues’ identity
© 2006 Gerhard Besold
nxr
nxr
r||
esz
r_| θ
r
r_|’
r’
r_|’
θ
r_| cos θ
r_|
r_| sin θ
r||
r_|
r
n
n
esy
s
ex
Axis-angle
representation
of the rotation matrix:
Geometric derivation
← N ≡ skew(n) :
Nij := −εijk nk
⇒ n x r = Nr
© 2006 Gerhard Besold
Axis-angle representation of the rotation matrix:
Matrix exponential and Rodrigues’ identity
1st-order differential equation for the orientation matrix A
of a rotating rigid body:
Definition:
r(S) = A r(B)
Superscripts:
“S”: space frame,
“B”: body frame
with instantaneous angular velocity matrix Ω /
/ angular velocity vector ω (in the space frame) :
Ω = skew(ω) = skew(ω n) = ω skew(n) = ω N
(ω = |ω| and n = ω/ω is the unit vector of the rotation axis)
Consider special case: Rotation with ω = const. ⇔ Ω = const.
⇒ Solution:
with rotation matrix written as matrix exponential:
With axis-angle representation as obtained by geometric derivation:
“Rodrigues’ identity”
© 2006 Gerhard Besold
Axis-angle representation of the rotation matrix:
Matrix exponential and Rodrigues’ identity
Direct proof of Rodrigues’ identity
without the “detour” via the geometric derivation?
Yes !
• Definition of the matrix exponential:
For any real 3x3 matrix B the matrix exponential
is defined via the convergent (Taylor ) series
eB
• Sequence of powers of N is periodic :
⇔
• Analytic summation of the Taylor series of eθ N :
Infinite series in θ N reduces to 2nd-order polynomial in N with three coefficients
of which two are each a well-known ordinary Taylor series in θ :
9
© 2006 Gerhard Besold
Axis-angle representation of the rotation matrix:
Calculation of the rotation matrix when axis & angle are given
Straight-forward application of the axis-angle representation formula!
Example:
Rotation around z-axis by angle θ
• Equations:
• Calculation:
9
© 2006 Gerhard Besold
Axis-angle representation of the rotation matrix:
Determination of axis & angle from a given rotation matrix
Determination of angle θ :
© 2006 Gerhard Besold
Axis-angle representation of the rotation matrix:
Determination of axis & angle from a given rotation matrix
Determination of axis unit vector n : several cases
•
θ =0
(cosθ = 1) :
•
0<θ < π
Trivial case — no rotation at all (undefined n)
(1 > cosθ > –1) :
© 2006 Gerhard Besold
Axis-angle representation of the rotation matrix:
Determination of axis & angle from a given rotation matrix
Determination of axis unit vector n : (cont’d)
•
θ =π
(cosθ = –1) :
© 2006 Gerhard Besold
Axis-angle representation of the rotation matrix:
Determination of axis & angle from a given rotation matrix
•
θ =π
(cosθ = –1) : (cont’d)
Cases:
All three Bij > 0
Single Bij > 0
& two Bij < 0
Two Bij = 0
(↔ single Bii = 0)
All three Bij = 0
(↔ two Bii = 0)
[For θ =π : n only defined up to sign → –n is also solution]
Axis-angle representation
of rotation matrices:
Geometric derivation
and Rodrigues’ identity
© 2006 Gerhard Besold
nxr
nxr
r||
esz
r_| θ
r
r_|’
r’
r_|’
θ
r_| cos θ
r_|
r_| sin θ
r||
r_|
r
n
n
esy
s
ex
Axis-angle
representation
of the rotation matrix:
Geometric derivation
← N ≡ skew(n) :
Nij := −εijk nk
⇒ n x r = Nr
© 2006 Gerhard Besold
Axis-angle representation of the rotation matrix:
Matrix exponential and Rodrigues’ identity
1st-order differential equation for the orientation matrix A
of a rotating rigid body:
Definition:
r(S) = A r(B)
Superscripts:
“S”: space frame,
“B”: body frame
with instantaneous angular velocity matrix Ω /
/ angular velocity vector ω (in the space frame) :
Ω = skew(ω) = skew(ω n) = ω skew(n) = ω N
(ω = |ω| and n = ω/ω is the unit vector of the rotation axis)
Consider special case: Rotation with ω = const. ⇔ Ω = const.
⇒ Solution:
with rotation matrix written as matrix exponential:
With axis-angle representation as obtained by geometric derivation:
“Rodrigues’ identity”
© 2006 Gerhard Besold
Axis-angle representation of the rotation matrix:
Matrix exponential and Rodrigues’ identity
Direct proof of Rodrigues’ identity
without the “detour” via the geometric derivation?
Yes !
• Definition of the matrix exponential:
For any real 3x3 matrix B the matrix exponential
is defined via the convergent (Taylor ) series
eB
• Sequence of powers of N is periodic :
⇔
• Analytic summation of the Taylor series of eθ N :
Infinite series in θ N reduces to 2nd-order polynomial in N with three coefficients
of which two are each a well-known ordinary Taylor series in θ :
9
© 2006 Gerhard Besold
Axis-angle representation of the rotation matrix:
Calculation of the rotation matrix when axis & angle are given
Straight-forward application of the axis-angle representation formula!
Example:
Rotation around z-axis by angle θ
• Equations:
• Calculation:
9
© 2006 Gerhard Besold
Axis-angle representation of the rotation matrix:
Determination of axis & angle from a given rotation matrix
Determination of angle θ :
© 2006 Gerhard Besold
Axis-angle representation of the rotation matrix:
Determination of axis & angle from a given rotation matrix
Determination of axis unit vector n : several cases
•
θ =0
(cosθ = 1) :
•
0<θ < π
Trivial case — no rotation at all (undefined n)
(1 > cosθ > –1) :
© 2006 Gerhard Besold
Axis-angle representation of the rotation matrix:
Determination of axis & angle from a given rotation matrix
Determination of axis unit vector n : (cont’d)
•
θ =π
(cosθ = –1) :
© 2006 Gerhard Besold
Axis-angle representation of the rotation matrix:
Determination of axis & angle from a given rotation matrix
•
θ =π
(cosθ = –1) : (cont’d)
Cases:
All three Bij > 0
Single Bij > 0
& two Bij < 0
Two Bij = 0
(↔ single Bii = 0)
All three Bij = 0
(↔ two Bii = 0)
[For θ =π : n only defined up to sign → –n is also solution]
