Euler Angles - Classical Mechanics - Lecture Slides, Slides of Classical Mechanics

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Euler

Angles

Three

Angles

•^

A^ rotation

matrix

can

be

described

with

three

free

parameters.^ –^

Select

three

separate

rotations

about

body

axes

1)^

Rotation

of

^ about

e^3

axis.

2)^

Rotation

of

^ about

e^1

axis.

3)^

Rotation

of

^ about

e^3

axis.

^

These

are

the

Euler

angles.

e^1

e^2

e^3 e^1

e^2

e^3  

^ 

Full

Rotation

•^

Any

rotation

is

then

expressed

with

the

three

angles.

lj kl ik ij^

S

S

S

S^

)( )( )(

   

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 S cos

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Matching

Terms

•^

The

terms

in^

the

^

matrix

correspond

to the

components

of

the

angular

velocity

vector.

•^

The

relationship

between

angular

velocity

and

Euler

angles

comes

from

the

S^ matrix.

1 2

1

3

2

cos

cos sin

sin

sin sin

cos 1 2 3

^ 

ml T km jkl kl jkl j^

S

S^



 ^