ME 6590 Multibody Dynamics
Euler Parameters and Angular Velocity Components
Earlier we derived relationships between angular velocity components and various sets
of orientation angles and their derivatives. Here, we will derive a similar relationship
between angular velocity components and the Euler parameters and their derivatives.
Note: The presentation that follows uses a slightly different notation than that found in the text.
Angular Velocity Components in a Fixed Frame
Previously, we found that the time derivative of a transformation matrix may be written
as
[C ]  [ ][C ]
(1)
Post-multiplying the above equation by [C ]T gives
[C ][C ]T  [ ][C ][C ]T  []
[ ]  [C ][C ]T
or
(2)
Recall that the elements of [ ] are the components of RB in the fixed frame
R : ( N1, N2 , N3 ) . Using this relationship, we can derive equations relating the angular
velocity components and the Euler parameters and the Euler parameter time derivatives.
This can be done by noting that
3
1  32  
i 1
C3iCiT2
3
2  13  
i 1
C1iCiT3
3
3  21   C2iCiT1
(3)
i 1
Substituting expressions into the above equations for Cij and CijT in terms of the Euler
parameters, it can be shown that
 4
1 

 
 2
3
   2
  2
3 

 0 
 1
 3
4
1
2
 2 1   1 
1  2   2 
 
 4  3   3 

 3  4   4 
(4)
Note that the last equation is simply the derivative of the parameter constraint equation,
12   22   32   42  1.
1/3
Eq. (4) may be written in the more compact form
{}  2[ E ]{ }
(5)
where the definition of the matrix [ E ] is obvious by comparing Eqs. (4) and (5). It can be
shown that [ E ] is an orthogonal matrix. Hence, the above equation can be easily inverted.
{ }  12 [ E ]T {}
(6)
These equations are analogous to those that relate the angular velocity components to the
derivatives of a set of orientation angles. Recall, however, that in those equations,
singularities always exist. The above equation exhibits no singularities.
Angular Velocity Components in the Body-Fixed Frame
Previously, we found that the time derivative of a transformation matrix may be written
as
[C ]  [C ][]
(7)
Pre-multiplying the above equation by [C ]T gives
[C ]T [C ]  [C ]T [C][]  []
or
[]  [C ]T [C ]
(8)
Recall that the elements of [] are the components of RB in the body-fixed frame
B : (e1, e2 , e3 ) . Using this relationship, we can derive equations relating the angular velocity
components in the body-fixed reference frame and the Euler parameters and their time
derivatives. This can be done by noting that
3
1  32  
i 1
C3TiCi 2
3
2  13  
i 1
C1Ti Ci 3
3
3  21   C2TiCi1
(9)
i 1
Substituting expressions into the above equations for Cij and CijT in terms of the Euler
parameters, it can be shown that
 4
1 
 
 
 2
3
   2
 2
3 

 0 
 1
 3  2 1   1 
 4 1  2   2 
 
1  4  3   3 

 2  3  4   4 
(10)
2/3
Note that (as before) the last equation is simply the derivative of the parameter constraint
equation, 12   22   32   42  1.
Eq. (10) can be written in the more compact form
{}  2[ E]{ }
(11)
where the definition of the matrix [ E] is obvious by comparing Eqs. (10) and (11). It can
be shown that [ E] is an orthogonal matrix. Hence, Eq. (11) can be easily inverted
{ }  12 [ E]T {}
(12)
As noted above, this matrix equation exhibits no singularities.
Calculation of First Angular Velocity Component of Equation (4)
Using the first of equations (3), we can write
T
T
T
1  32  C31C12
 C32C22
 C33C32

 2(1 3   2 4 )  2(1 2   3 4 )  
d
2( 2 3  1 4 )   12   22   32   42  
dt 
d
dt
d
dt
 
2
1

  22   32   42  2( 2 3  1 4 ) 
 4(1 2   3 4 ) 1 3  1 3   2 4   2 4  


2 12   22   32   42   2 3   2 3  1 4  1 4  
2( 2 3  1 4 )  211  2 2 2  2 3 3  2 4 4 
Expanding the products in this equation and using the Euler parameter constraint
equation 12   22   32   42  1 gives the final result


1  2  41   3 2   2 3  1 4 
The development of equations like this one can be quite tedious. The development of all
of Eq. (4) using the Maple symbolic manipulation software is presented in a separate set
of notes.
3/3