AER 506
1)
Vector
2)
Celestial
Reference
Frames
3)
Rotation
Matrix
4)
Basic
(Principal)
Rotations
5)
Cascade of
Rotations
6)
Rotation
(Attitude)
Representation
Spacecraft Dynamics and Control
Chapter 2 - Section 1
Basics
Prof. M.R. Emami
Fall 2022
 M.R. Emami, 2022
1
Outline
1)
Vector
2)
Celestial
Reference
Frames

Vector
3)
Rotation
Matrix

Celestial Reference Frames
4)
Basic
(Principal)
Rotations

Rotation Matrix
5)
Cascade of
Rotations

Basic (Principal) Rotations
6)
Rotation
(Attitude)
Representation

Cascade of Rotations

Rotation (Attitude) Representation
 X-Y-Z Fixed (Roll-Pitch-Yaw) Angles
 Z-X-Z (Classic) Euler Angles
 Z-Y-Z Euler Angles
 Axis-Angle Representation
 M.R. Emami, 2022
2
Vector
1)
Vector
2)
Celestial
Reference
Frames
3)
Rotation
Matrix
4)
Basic
(Principal)
Rotations
5)
Cascade of
Rotations
6)
Rotation
(Attitude)
Representation
𝑧̂
 Mathematical entity with magnitude and direction, often used
for representing physical quantities that require magnitude and
direction to be fully characterized, e.g., force, velocity, flux, etc.
𝑝
𝑝̂
Reference A set of three mutually-perpendicular unit vectors (axes)
Frame
𝑥 , 𝑦 and 𝑧̂ (orthonormal), which intersect at the origin
(𝐴 , 𝐴 ) point A and follow a right-handed order (dextral).
𝑥
𝐴≡ 𝐴 ≜ 𝑦
𝑧̂
A
𝑝
𝑦
𝑝
𝑥
= 𝑥
𝑦
(vectrix)
𝑧̂
 For computational purposes, a vector 𝑝̂ can be defined (expressed) in a reference frame 𝐴
to have 3 scalers 𝑝 , 𝑝 , and 𝑝 .
𝑝
𝑥
𝑝̂ = 𝑝 𝑥 + 𝑝 𝑦 + 𝑝 𝑧̂ = 𝑝 𝑝 𝑝 𝑦 = 𝑥 𝑦 𝑧̂ 𝑝
𝑝
𝑧̂
Notation:
𝑝
𝑝≜ 𝑝
𝑝
𝑝̂ 𝑥
= 𝑝̂ 𝑦
𝑝̂ 𝑧̂
𝑥
= 𝑦
𝑧̂
𝑝̂
𝑝̂
𝑝̂
(matrix)
NOTE: Mathematically, 𝑝 is not a vector, but a column matrix (consisting of 3 scalars) that
represents the definition of vector 𝑝̂ in reference frame 𝐴 .
𝑝̂ = 𝐴
 M.R. Emami, 2022
𝑝
NOTE: Vectors by themselves can be treated (and even operated) independently of any
reference frame, but once all vectors being considered are expressed in the same
reference frame, we can then work only with their scalar components.
3
𝑧̂
Vector
Inner Product (scalar):
𝑘=𝑝
1)
Vector
2)
Celestial
Reference
Frames
3)
Rotation
Matrix
4)
Basic
(Principal)
Rotations
5)
Cascade of
Rotations
6)
Rotation
(Attitude)
Representation
𝑘=𝑝
𝑝 = 𝑝
𝑝 =
𝑝
𝑝
𝑝
cos 𝛼
𝑝
= 𝑝
𝑝
𝛼
𝑝
𝑦
𝑝
Cross Product (vector):
𝑟̂ = 𝑝 × 𝑝 = 𝑝
𝑝̂
𝑒̂
(𝛼 is the angle between 𝑝 and 𝑝 .)
𝑝
𝑝
𝑝
𝑝
=𝑝 𝑝 +𝑝 𝑝 +𝑝 𝑝
𝑝
𝑝̂
𝑥
sin 𝛼 𝑒̂
(𝑒̂ is the unit vector perpendicular to 𝑝 and 𝑝 .)
𝑝 (skew-symmetric matrix; 𝑎 = −𝑎 , 𝑎𝑏 = −𝑏𝑎)
𝑝
𝑝
0
= 𝑝
−𝑝
Example:
𝑎
𝑏 × 𝑐̂ = 𝑐̂
𝑟=
Useful
Identities
−𝑝
0
𝑝
𝑝
−𝑝
0
𝑝
𝑝
𝑝
𝑎×𝑏 =𝑏
−𝑝 𝑝 + 𝑝 𝑝
= 𝑝 𝑝 −𝑝 𝑝
−𝑝 𝑝 + 𝑝 𝑝
𝑐̂ × 𝑎
𝑎
𝑏 𝑐=
𝑐
𝑎 𝑏= 𝑏
𝑎 × 𝑏 × 𝑐̂ = 𝑎 𝑐̂ 𝑏 − 𝑎 𝑏 𝑐̂ = 𝑏 𝑎 𝑐̂ − 𝑐̂ 𝑎 𝑏
𝑎 𝑏= 𝑏 𝑎 −
𝑎
𝑏 𝐼
𝑐̃ 𝑎
(bac-cab identity)
(𝐼 is the 3 × 3 identity matrix.)
NOTE: The cross-product operation does not have the associativity property, i.e.,
𝑎 × 𝑏 × 𝑐̂ = 𝑎 × 𝑏 × 𝑐̂ + 𝑏 × 𝑎 × 𝑐̂ ≠ 𝑎 × 𝑏 × 𝑐̂
 M.R. Emami, 2022
4
Celestial Reference Frames
 Equatorial Plane: The plane normal to the Earth’s axis of rotation about itself.
 Ecliptic Plane: The orbital plane of Earth’s centre about the Sun.
1)
Vector
2)
Celestial
Reference
Frames
 Obliquity of the Ecliptic (𝜀): The angle between the Earth’s axis of rotation and normal line
to the ecliptic plane (𝜀 ≅ 23.44° ).
3)
Rotation
Matrix
4)
Basic
(Principal)
Rotations
 Vernal Equinox Line (𝛾): The line passing through the centre of Earth and the Sun on the
first day of spring on the northern hemisphere, when the noontime sun crosses the equator
from south to north. (The number of hours of daylight and darkness are equal.)
 Heliocentric-Ecliptic Reference Frame 𝐻 :
5)
Cascade of
Rotations
 Origin 𝑂 at the Sun’s centre of mass
6)
Rotation
(Attitude)
Representation
 𝑥 along vernal equinox 𝛾
(the Earth-Sun direction)
Vernal (Spring) Equinox
(first day of spring, March 21)
 𝑧̂ normal to Ecliptic plane
 𝑦 in the Ecliptic
plane, so that
𝑥 × 𝑦 = 𝑧̂
𝑧̂
Celestial
Axis
Rotation
Axis
𝑦
𝑥
Hibernal
(Winter) Solstice
(first day of winter, December 21)
Estival
(Summer) Solstice
(first day of summer, June 21)
 M.R. Emami, 2022
Autumnal (Fall) Equinox
(first day of autumn, September 23)
5
Celestial Reference Frames
 Earth-centred Inertial (Geocentric-Equatorial) Reference Frame 𝐺 :
 Origin 𝑂 at the Earth’s centre of mass
1)
Vector
2)
Celestial
Reference
Frames
3)
4)
Rotation
Matrix
Basic
(Principal)
Rotations
5)
Cascade of
Rotations
6)
Rotation
(Attitude)
Representation
 𝑥 (𝐼 ) along vernal equinox 𝛾
(the Earth-Sun direction)
 𝑧̂ (𝐾) along the Earth’s North
Pole (normal to Equatorial plane)
 𝑦 (𝐽) in the Equatorial plane, so that
𝑥 × 𝑦 = 𝑧̂
𝑧̂
OG
𝑦
Ecliptic Plane
𝑥
 Perifocal Reference Frame 𝑃 :
 Origin 𝑂 at the Earth’s centre of mass
 𝑥 (𝑝̂ ) toward the orbit perigee
𝑧̂
 𝑧̂ (𝑤 ) normal to orbital plane
 𝑦 (𝑞 ) in the orbital plane, so that
𝑥 × 𝑦 = 𝑧̂
 Orbiting Reference Frame 𝑂 :
𝑦
𝑧̂
𝑥
𝑥
OO
OP
𝑦
 Origin 𝑂 at the Earth’s centre of mass
 𝑥 toward the satellite
 𝑧̂ normal to orbit plane
 M.R. Emami, 2022
 𝑦 in the orbital plane, so that
𝑥 × 𝑦 = 𝑧̂
6
Celestial Reference Frames
 Local Vertical Local Horizontal Reference Frame 𝐿 :
 Origin 𝑂 at the satellite’s centre of mass
1)
Vector
 𝑥 along outward radial from Earth to satellite, defining direction of satellite’s local vertical
2)
Celestial
Reference
Frames
 𝑧̂ normal to the orbital plane (in the direction of satellite’s angular momentum vector)
 𝑦 in orbital plane, so that 𝑥 × 𝑦 = 𝑧̂ , defining direction of satellite’s local horizon
3)
Rotation
Matrix
4)
Basic
(Principal)
Rotations
 𝑥 pointing in the direction of (but not necessarily parallel to) satellite’s velocity vector
5)
Cascade of
Rotations
 𝑧̂ pointing toward (but not necessarily along) Earth’s centre of mass in orbital plan
and perpendicular to 𝑥
6)
Rotation
(Attitude)
Representation
 𝑦 dextral frame, so that 𝑥 × 𝑦 = 𝑧̂
 Body-fixed Reference Frame 𝐵 (attached to satellite):
 Origin 𝑂 at the satellite’s centre of mass
 Inertial Frame: Nonrotational with its
origin stationary or moving at a constant
velocity (w.r.t. an inertial frame).
NOTE:
Some of the reference frames may be
assumed as inertial if Newton’s laws hold
relative to them with a reasonable accuracy.
𝑧̂
𝑥
𝑧̂
𝑦
OL
𝑧̂
𝑧̂
𝑥
𝑥
𝑥
OO
OB
𝑦
𝑦
OP
𝑦
 Frame 𝐻 can be assumed as inertial
for orbital dynamics of Solar System.
 M.R. Emami, 2022
 Frames 𝐺 and 𝑃 can be assumed as
inertial for studying satellite orbital
dynamics.
7
Rotation Matrix
𝑝̂ = 𝐴
1)
Vector
2)
Celestial
Reference
Frames
3)
Rotation
Matrix
4)
Basic
(Principal)
Rotations
5)
Cascade of
Rotations
6)
Rotation
(Attitude)
Representation
𝐴𝐴
𝑥
𝑦
𝑧̂
𝑥
𝑥
𝑥
𝑥
𝑦
𝑧̂
𝑦
𝑦
𝑦
𝑥
𝑦
𝑧̂
𝑝=𝐵
𝑝
𝑝 = 𝐴𝐵
𝑝
𝑧̂
𝑧̂
𝑧̂
𝑥
𝑝= 𝑦
𝑧̂
𝑥
𝑦
𝑧̂
𝑦
𝑦
𝑦
𝑥
𝑦
𝑧̂
𝑧̂
𝑧̂
𝑧̂
𝑝
𝑅
1 0
0 1
0 0
0
0
1
𝑥
𝑝= 𝑦
𝑧̂
 Rotation Matrix:
𝑥
𝑥
𝑥
𝑥
𝑦
𝑧̂
𝑝= 𝑅
𝑝
𝑥
𝑅 = 𝑦
𝑧̂
=

 M.R. Emami, 2022
𝑥
𝑥
𝑥
𝑅
𝑦
𝑦
𝑦
𝑥
𝑥
𝑥
𝑥
𝑥
𝑦
𝑧̂
𝑥
𝑦
𝑧̂
𝑦
𝑧̂
𝑧̂
𝑧̂
𝑦
𝑦
𝑦
𝑝
𝑥
𝑦
𝑧̂
𝑧̂
𝑧̂
𝑧̂
𝑥
= 𝑥
𝑥
𝑧
𝑐𝑜𝑠 𝜃
𝑐𝑜𝑠 𝜃
𝑐𝑜𝑠 𝜃
= 𝑐𝑜𝑠 𝜃
𝑐𝑜𝑠 𝜃
𝑐𝑜𝑠 𝜃
𝑐𝑜𝑠 𝜃
𝑐𝑜𝑠 𝜃
𝑐𝑜𝑠 𝜃
𝑥
𝑦
𝑧̂
𝑦
𝑦
𝑦
𝑥
𝑦
𝑧̂
𝑧̂
𝑧̂
𝑧̂
𝑥
𝑦
𝑧̂
Direction Cosine
(angle between 𝑧̂ and 𝑥 )
represents the rotation of frame 𝐵 with respect to frame 𝐴 .
8
Rotation Matrix
 Some Properties:

1)
Vector
2)
Celestial
Reference
Frames
3)
4)
Rotation
Matrix
𝑅= 𝑋

Cascade of
Rotations
6)
Rotation
(Attitude)
Representation
𝑌
𝑍
Inverse
𝑅
Basic
(Principal)
Rotations
5)
Orthonormality
𝑋 𝑋=1
𝑋 𝑌=0
𝑌 𝑌=1
𝑋 𝑍=0
𝑍 𝑍=1
𝑌 𝑍=0
𝑥
𝑅 =
𝑥
𝑦
𝑦
𝑧
𝑅

=
𝑅
𝑧
1 0
= 0 1
0 0
0
0
1
= 𝑅
Determinant of a rotation matrix is +1.
Useful
Identities
𝑝= 𝑅
𝑅
𝑝 𝑅 = 𝑅
𝑝 = 𝑅
𝑝 𝑅
𝑝 𝑅
NOTE:
 A rotation matrix can also be viewed as an operator that
rotates a vector 𝑝̂ to a new vector 𝑝̂ in a given reference
frame 𝐴 .
𝑝 = 𝑅 𝑝
 M.R. Emami, 2022
 Matrix 𝑅 is the rotation matrix of a (hypothetical) frame
𝐵 with respect to 𝐴 , which can be obtained by a reverse
rotation of 𝑝̂ to 𝑝̂ .
9
Basic (Principal) Rotations
1)
Vector
2)
Celestial
Reference
Frames
3)
Rotation
Matrix
4)
Basic
(Principal)
Rotations
5)
Cascade of
Rotations
6)
Rotation
(Attitude)
Representation
 About 𝑧̂ axis:
𝑐𝛼
𝑅 = 𝑅 𝛼 = 𝑠𝛼
0
 About 𝑦 axis:
𝑐𝛽
𝑅 =𝑅 𝛽 = 0
−𝑠𝛽
 About 𝑥 axis:
 M.R. Emami, 2022
−𝑠𝛼
𝑐𝛼
0
0
0
1
0
1
0
𝑠𝛽
0
𝑐𝛽
1 0
𝑅 = 𝑅 𝛾 = 0 𝑐𝛾
0 𝑠𝛾
0
−𝑠𝛾
𝑐𝛾
10
Cascade of Rotations
 About fixed axes:
1)
Vector
2)
Celestial
Reference
Frames
3)
Rotation
Matrix
4)
Basic
(Principal)
Rotations
5)
Cascade of
Rotations
6)
Rotation
(Attitude)
Representation
order of rotations
𝑅
𝛾, 𝛽, 𝛼 = 𝑅 𝛼 ⋅ 𝑅 𝛽 ⋅ 𝑅 𝛾
 About moving axes:
order of rotations
𝑅
 M.R. Emami, 2022
𝛼, 𝛽, 𝛾 = 𝑅 𝛼 ⋅ 𝑅
NOTE:
Three rotations taken about fixed axes yield the same
final orientation as the same three rotations taken in the
opposite order about the axes of the moving frames.
𝛽 ⋅𝑅
𝑅
𝛾
𝛾, 𝛽, 𝛼 =
𝑅
𝛼, 𝛽, 𝛾
11
Rotation (Attitude) Representation
X-Y-Z Fixed (Roll-Pitch-Yaw) Angles
1)
Vector
2)
Celestial
Reference
Frames
3)
Rotation
Matrix
4)
Basic
(Principal)
Rotations
5)
Cascade of
Rotations
6)
Rotation
(Attitude)
Representation
𝑅
𝛾̅ , 𝛽̅ , 𝛼 = 𝑅 𝛼 ⋅ 𝑅 𝛽̅ ⋅ 𝑅 𝛾̅
𝑅
𝑐𝛼 𝑐𝛽̅
𝛾̅ , 𝛽̅ , 𝛼 = 𝑠𝛼 𝑐𝛽̅
−𝑠𝛽̅
𝑐𝛼
= 𝑠𝛼
0
−𝑠𝛼
𝑐𝛼
0
0 𝑐𝛽̅
0
0
1 −𝑠𝛽̅
𝑐𝛼 𝑠𝛽̅ 𝑠𝛾̅ − 𝑠𝛼 𝑐𝛾̅
𝑠𝛼 𝑠𝛽̅ 𝑠𝛾̅ + 𝑐𝛼 𝑐𝛾̅
𝑐𝛽̅𝑠𝛾̅
0
1
0
𝑠𝛽̅
0
𝑐𝛽̅
𝑅
𝑟
𝑟
𝑟
𝑟
𝑟
𝑟
𝛼 = 𝐴𝑡𝑎𝑛2
𝛾̅ = 𝐴𝑡𝑎𝑛2
𝑟
𝑟
0
𝑐𝛾̅
𝑠𝛾̅
0
−𝑠𝛾̅
𝑐𝛾̅
𝑐𝛼 𝑠𝛽̅𝑐𝛾̅ + 𝑠𝛼 𝑠𝛾̅
𝑠𝛼 𝑠𝛽̅ 𝑐𝛾̅ − 𝑐𝛼 𝑠𝛾̅
𝑐𝛽̅ 𝑐𝛾̅
𝛽̅ = 𝐴𝑡𝑎𝑛2 −𝑟 , ± 𝑟 + 𝑟
𝑟
𝛾̅ , 𝛽̅ , 𝛼 = 𝑟
𝑟
1
0
0
,
𝑐𝛽̅
,
𝑐𝛽̅
𝑟
𝑟
𝑐𝛽̅
𝑐𝛽̅
Atan2 𝑥, 𝑦 computes
tan 𝑥 ⁄𝑦 , but uses the
signs of both 𝑥 and 𝑦 to
determine the quadrant
in which the resulting
angle lies.
 A unique solution exists for −90° < 𝛽̅ < +90° .
𝛽̅ = ±90
 M.R. Emami, 2022
One degree of Convention
freedom is lost.
(degeneracy)
𝛽̅ = +90 : Set 𝛼 = 0 & Compute 𝛾̅ = Atan2 𝑟 , 𝑟
𝛽̅ = −90 : Set 𝛼 = 0 & Compute 𝛾̅ = − Atan2 𝑟 , 𝑟
.
. 12
Rotation (Attitude) Representation
Z-X-Z (Classic) Euler Angles
1)
Vector
2)
Celestial
Reference
Frames
3)
Rotation
Matrix
4)
Basic
(Principal)
Rotations
5)
Cascade of
Rotations
6)
Rotation
(Attitude)
Representation
𝑅
𝑅
𝛼, 𝛽, 𝛾 = 𝑅 𝛼 ⋅ 𝑅
𝑟
𝛼, 𝛽, 𝛾 = 𝑟
𝑟
𝑟
𝑟
𝑟
𝑟
𝑟
𝑟
𝛽 ⋅𝑅
−𝑠𝛼𝑐𝛽𝑠𝛾 + 𝑐𝛼𝑐𝛾
𝛾 = 𝑐𝛼𝑐𝛽𝑠𝛾 + 𝑠𝛼𝑐𝛾
𝑠𝛽𝑠𝛾
−𝑠𝛼𝑐𝛽𝑐𝛾 − 𝑐𝛼𝑠𝛾
𝑐𝛼𝑐𝛽𝑐𝛾 − 𝑠𝛼𝑠𝛾
𝑠𝛽𝑐𝛾
𝛽 = 𝐴𝑡𝑎𝑛2 ± 𝑟 + 𝑟 , 𝑟
𝛼 = 𝐴𝑡𝑎𝑛2 𝑟 ⁄𝑠𝛽 , − 𝑟 ⁄𝑠𝛽
𝛾 = 𝐴𝑡𝑎𝑛2 𝑟 ⁄𝑠𝛽 , 𝑟 ⁄𝑠𝛽
𝑠𝛼𝑠𝛽
−𝑐𝛼𝑠𝛽
𝑐𝛽
Atan2 𝑥, 𝑦 computes
tan 𝑥 ⁄𝑦 , but uses the
signs of both 𝑥 and 𝑦 to
determine the quadrant in
which the resulting angle
lies.
 A unique solution exists for 0 < 𝛽 < 180° .
𝛽 = 0 or 180
One degree of Convention
freedom is lost.
(degeneracy)
𝛽 = 0:
Set 𝛼 = 0 & Compute 𝛾 = Atan2 −𝑟 , 𝑟
.
𝛽 = 180 : Set 𝛼 = 0 & Compute 𝛾 = − Atan2 𝑟 , −𝑟
NOTE:
 M.R. Emami, 2022
The same final attitude can be obtained using Euler angles by a cascade of rotations
about the fixed axes (of frame 𝐴 ) if the order of rotations is reversed, i.e., first a rotation
of 𝛾 about 𝑧̂ , then a rotation of 𝛽 about 𝑥 , and then a rotation of 𝛼 about 𝑧̂ .
13
.
Rotation (Attitude) Representation
Z-Y-Z Euler Angles
1)
Vector
2)
Celestial
Reference
Frames
3)
Rotation
Matrix
4)
Basic
(Principal)
Rotations
5)
Cascade of
Rotations
6)
Rotation
(Attitude)
Representation
𝑅
𝑅
𝛼, 𝛽, 𝛾 = 𝑅 𝛼 ⋅ 𝑅
𝑟
𝛼, 𝛽, 𝛾 = 𝑟
𝑟
𝑟
𝑟
𝑟
𝑟
𝑟
𝑟
𝛽 ⋅𝑅
𝑐𝛼𝑐𝛽𝑐𝛾 − 𝑠𝛼𝑠𝛾
𝛾 = 𝑠𝛼𝑐𝛽𝑐𝛾 + 𝑐𝛼𝑠𝛾
−𝑠𝛽𝑐𝛾
−𝑐𝛼𝑐𝛽𝑠𝛾 − 𝑠𝛼𝑐𝛾
−𝑠𝛼𝑐𝛽𝑠𝛾 + 𝑐𝛼𝑐𝛾
𝑠𝛽𝑠𝛾
𝛽 = 𝐴𝑡𝑎𝑛2 ± 𝑟 + 𝑟 , 𝑟
𝛼 = 𝐴𝑡𝑎𝑛2 𝑟 ⁄𝑠𝛽 , 𝑟 ⁄𝑠𝛽
𝛾 = 𝐴𝑡𝑎𝑛2 𝑟 ⁄𝑠𝛽 , −𝑟 ⁄𝑠𝛽
𝑐𝛼𝑠𝛽
𝑠𝛼𝑠𝛽
𝑐𝛽
Atan2 𝑥, 𝑦 computes
tan 𝑥 ⁄𝑦 , but uses the
signs of both 𝑥 and 𝑦 to
determine the quadrant in
which the resulting angle
lies.
 A unique solution exists for 0 < 𝛽 < 180° .
𝛽 = 0 or 180
One degree of Convention
freedom is lost.
(degeneracy)
𝛽 = 0:
Set 𝛼 = 0 & Compute 𝛾 = Atan2 −𝑟 , 𝑟
.
𝛽 = 180 : Set 𝛼 = 0 & Compute 𝛾 = − Atan2 𝑟 , −𝑟
NOTE:
 M.R. Emami, 2022
The same final attitude can be obtained using Euler angles by a cascade of rotations
about the fixed axes (of frame 𝐴 ) if the order of rotations is reversed, i.e., first a rotation
of 𝛾 about 𝑧̂ , then a rotation of 𝛽 about 𝑦 , and then a rotation of 𝛼 about 𝑧̂ .
14
.
Rotation (Attitude) Representation
Axis-Angle Representation
1)
Vector
2)
Celestial
Reference
Frames
3)
4)
Rotation
Matrix
Basic
(Principal)
Rotations
5)
Cascade of
Rotations
6)
Rotation
(Attitude)
Representation
Euler’s (Rotation Axis) Theorem:
For any finite rotation of a rigid body (reference frame) about a
fixed point, a unique axis passing through the fixed point can be
found on which all points remain fixed during the rotation. In
other words, the rotation of the rigid body (reference frame) is
equivalent to a single rotation about such an axis. The axis is
called rotation axis 𝑒̂ , and 𝜑 is the angle (scalar) with which the
initial frame rotates to the new one.
ẑB
𝑒=
𝑅
𝑒=
Eigenvalue Problem:
𝑟 −𝜆
𝑅 −𝜆𝐼 = 𝑟
𝑟
𝜆 − 𝑡𝑟
𝑅
𝑅 −𝐼
𝑒
𝜆 + 𝑡𝑟
𝑅 − 𝜆𝐼
𝑅
𝑒=0
𝑟
=0
−𝜆
ŷ A
𝜆 −1=0
cos 
T B
B
B
B
B
B
B
 m n e  RA  m n e    sin 

 
B T
B
 0
P
P
𝑟
𝑅 = 𝑟
𝑟
𝑒̂ , 𝜑
 M.R. Emami, 2022
𝑟
𝑟
𝑟
𝑟
𝑟
𝑟
𝑒 =
𝑅 = 𝑒𝑒 + 𝐼 − 𝑒𝑒
m̂
ŷ B
𝑟
𝑟
−𝜆
𝑟
n̂
ẑ A
𝑝=0
𝑟
𝑟
ê
𝑒
𝑒 = 𝑒
𝑒
 sin  0 x̂ A
cos  0   Rz  
0
1 
𝑟 −𝑟
1
𝑟 −𝑟
=
2 𝑠𝑖𝑛 𝜑 𝑟 − 𝑟
x̂ B
;
𝜑 = 𝑐𝑜𝑠
𝑡𝑟
𝑅
𝑡𝑟
𝑐𝑜𝑠 𝜑 − 𝑒̃ 𝑠𝑖𝑛 𝜑 = 𝐼 𝑐𝑜𝑠 𝜑 + 𝑒𝑒 1 − 𝑐𝑜𝑠 𝜑 − 𝑒̃ 𝑠𝑖𝑛 𝜑
= 𝑟 +𝑟
+𝑟
𝑅 −1
2
Rodrigues
Formula
NOTE: Only three of four parameters of the axis-angle representation, 𝑒 , 𝑒 , 𝑒 , and 𝜑,
can be independent, due to the unit length relationship for the rotation axis: 𝑒 𝑒 = 1.
15