Question

How would you use numerical integration to create a classical rodrigues parameter as a function of time in MATLAB?

Answer 1

CLASSICAL RODRIGUES PARAMETERS:

The Rodrigues parameters have a singularity at Φ = ±180°. This corresponds to a point on the two-dimensional unit circle. The corresponding symmetric stereographic projection has the projection point a at the origin and the mapping line at β0 = 1. It becomes evident why the classical Rodrigues parameters must go singular at Φ = ±180° when describing them as a special case of the symmetric stereographic parameters. The transformation from the Euler parameters to the Rodrigues parameters is found by setting ΦS = ±180° . The well known result is shown in below.

  i=1,2,3 (1)

The inverse transformation from the Rodrigues to the Euler parameters is found by using the same ΦS and is given as:

i=1,2,3 (2)

The differential kinematic equation in terms of the classical Rodrigues parameters is given in vector form as:

   (3)

An explicit matrix form of Eq. (3) is given below

   (4)

Using the definitions of the Euler parameters, the Rodrigues parameters can also be expressed directly in terms of the principal rotation angle Φ and the principal line of rotation .

   (5)

(5) From Eq. (5), it is obvious why the classical Rodrigues parameters go singular at ±180°. For completeness the direction cosine matrix C is given in explicit matrix form :

  (6)

and in vector form3 :

(7)

Eq. (7) and its inverse can also be written as the Cayley Transform3,5,7:

  

  

Both and the kinematic differential equation shown in Eqs. (4-5) has the “Cayley” form5 :

(8)

The tilde matrix[q] is defined by as given in Eq. (9).

   (9)

Let the vector (defined with - ) denote the shadow point of the classical Rodrigues parameters.

     i=1,2,3 (10)

shows that for the Rodrigues parameters, the shadow point vector components are identical to the original Rodrigues parameters, with identical values and properties. Therefore the shadow point concept is of no practical consequence in this case; the classical Rodrigues parameters are unique!

Having the projection point a at the origin causes this elegant, degenerate phenomenon. Illustrates why both sets of Rodrigues parameters are identical. The classical Rodrigues parameters are the only symmetric stereographic parameters which exhibit this lack of distinction between the original parameters and their shadow point counterparts. This proves simultaneously to be an advantage and a disadvantage.