In the problem yo-yo is made of two identical cylinders of radius $R$, thickness $h$ and mass $M$, and the yo-yo is let go.
In order to define the position of the yo-yo, I need as position vector and a rotation matrix.
In the example of asteroid problem in space when we find the Euler's equation of solid body motion. The $r$ is in $R^3$ and transformation $\phi$ is $\left( {\begin{array}{*{20}{c}} 0&{ - {x_3}}&{{x_2}}\\ {{x_3}}&0&{ - {x_1}}\\ { - {x_2}}&{{x_1}}&0 \end{array}} \right)$. The asteroid problem has 6 degrees of freedom.
Here comes my problem, If I assume yo-yo also rotate as the asteroid example, the problem has only two degrees of freedom, so I assign the origin of system in the center of yo-yo's first position, the position vector is just in $R^1$, which measures the length the yo-yo has gone from the original positon. The rotation of the yo-yo is just rotation around the one-point. So is it OK to just follow the asteroid problem? Is there anything I need to change?