on the euler angles - university of rochester
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On the Euler Angles
The two tables here show the relation between the body axes and the inertial frame for each step
of the Euler angle rotation process. Table 1 gives the location of the body axes in the inertial
frame at each step, and Table 2 gives the apparent position of the inertial frame at each step. The
subscripts in each case denote the number of steps completed. The rotations and their order are
those shown in Goldstein’s Figure 4-7, which I have reproduced here.
Table 1: The Euler Rotations: Body Axes in the Inertial Frame
!
i
!
i 0
!
i
!
j
!
j 0
!
j
!
k
!
k 0
!
k
!
i
!
i 1
!
cos"i + sin"j
!
j
!
j 1
!
"sin#i + cos#j
!
k
!
k 1
!
k
!
i
!
i 2
!
cos"i + sin"j
!
j
!
j 2
!
"cos# sin$i + cos# cos$j+ sin#k
!
k
!
k 2
!
sin" sin#i $ sin" cos#j+ cos"k
!
i
!
i 3
!
cos" cos# $ cos% sin& sin"( )i + cos" sin# + cos% cos& sin"( )j+ sin% sin"k
!
j
!
j 3
!
"sin# cos$ " cos% sin& cos#( )i + "sin# sin$ + cos% cos& cos#( )j+ sin% cos#k
!
k
!
k 3
!
sin" sin#i $ sin" cos#j+ cos"k
Table 2. The Euler Rotations: Inertial Axes in the Body Frame
!
i
!
i0
!
i
!
j
!
j0
!
j
!
k
!
k0
!
k
!
i
!
i1
!
cos"i # sin"j
!
j
!
j1
!
sin"i + cos"j
!
k
!
k1
!
k
!
i
!
i2
!
cos"i # sin"j
!
j
!
j2
!
cos" sin#i + cos" cos#j $ sin"k
!
k
!
k2
!
sin" sin#i + sin" cos#j + cos"k
!
i
!
i3
!
cos" cos# $ cos% sin& sin"( )i + $cos" sin# $ cos% cos& sin"( )j + sin% sin"k
!
j
!
j3
!
sin" cos# + cos$ cos" sin#( )i + %sin# sin& + cos$ cos" cos#( )j % sin$ cos"k
!
k
!
k3
!
sin" sin#i + sin" cos#j + cos"k
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