or  RRIRRRRR xxxx R  xX

Rotation

 (Euclidean) Distance-Invariant

 Finite Rotation: Matrix representation

 Orthogonality

   

x X  

 R

   

T T

T

R R

I R

R

R R

R



















1 T 2

2

or

x x

x

x

Infinitesimal Rotational Displacement

 Antisymmetric 　 Matrix

 Vector Product

x θ

x   





























0 0

0

x y

x z

y z













 







 









z y x







Finite Rotation

 Expressions: Matrix, Spinol, Quarternion

 Rotation = Matrix Operation

 Rot. Matrix = Set of Basis Vectors (= Triad)

 ^e

^e

^e



R 

Y Z

Euler’s Theorem

 Any Finite Rotation = 3 Basic Rotation

 Euler angles: 3 Angles of Basic Rotations

 ^{ ,  , } 

^{(  )}

^{(  )}

^{(  )}

ijk

R R R

R

R  

 

 ^R

^{ ,  , } 

^{ R}

^{   ,   ,   }

Basic Rotation

 Rotation around z-axis by angle 

) (

)

3 (





 _X

Y

y P x

Basic Rotation (contd.)

 Rotation around j-axis by angle 

 Inverse Rotation

) ( 

R

 ^R

  ^ 

^{ R}

^{   }

Basic Rotation Matrix

 Example: Equatorial – Ecliptic

 Obliquity of Ecliptic

 







 











1 0

0

0 cos

sin

0 sin

cos )

3

(  







R

  ^

R



Basic Rotation Matrix (contd.)

 Small Angle Approximation

 

  _ ^



 



 









 

 





 













 

j

j j j

j

j e

e













I R

I I

R_{3 3}

0 0

0

0 0

0 0

Angular Velocity

 

















 







 



 



 







 

ω e

e

j

j j

j

j j j

j j

dt d dt

d 



 R

I R

R





j

j j

dt

d e

ω 

Euler Rotation

 3x2x2 = 12 different combinations

 3-1-3 Sequence (= x-convention)

 Most popular (Euler angles)

 Used to describe rotational dynamics

 ^{  } 

      ^

^

^

313

, , R R R

R 

Euler Angles (3-1-3)

 

































































































C C

S S

S

S C C

C C S

S S

C C C

S

S S C

C S S

C S

C S C

C ,

313 , R



























































































cos cos

sin sin

sin

sin cos

cos cos

cos sin

sin sin

cos cos

cos sin

sin sin

cos cos

sin sin

cos sin

cos sin

cos cos

Euler Angles

X

Z

Y N

P







Demerit of 3-1-3 Sequence

  ^

 







 























 , , 0

313

I

R

 Degeneration in case of small angles

 Solution: 3-2-1-like Sequences

3-2-3 Sequence

 y-convention: precession

 Conic Rotation

 Rotation around

a fixed direction 























 cos

sin sin

cos sin

n

 ^{ ,  , } 

R





 R ₃₂₃





   

I+ sin 1 cos 

 n    n n 

Other Sequences

 1-3-1: Nutation

 2-1-3: Polar Motion + Sidereal Rotation

 1-2-3: Aerodynamics, Attitude Control

 Best Recommended

 

 ^{        } 

 R

, ,

N





 R ₃₁₂ , , WS

Small Angle Rotation

 





































































































































I

R R

R R

C C

S C S

S C C

S S C

C S

S S C

S

S S C

S C C

S S

S C C

C

) ( )

( )

( ,

, _{3 2 1}

123

Rotational Velocity

x v

x v

V

x X















dt R R dR

R