Solving Spherical Rolling Problem - Hossein

[hmoein]

hi , every one!
I have a problem with a sphere rolling on a fixed sphere. My problem is to find relationship between coordinate of center of sphere (X,Y,Z) and orientations (alpha, beta, gamma) or Euler angles of sphere. as we know a sphere has 6 DOF in space (3 coordiantes and 3 rotation) when a sphere rolling on surface we expect that it have 3 dof beacuse of relation beween coordinate and rotation.
for example when a circle roll on a surface the x coordinate of its center is:
X=R*teta (R = radius of circle) and it has one DOF.
Like the circle rolling i want to find the relations for sphere.
thanks
hossein

 

[Ben Niehoff]

Unfortunately, the constraint for a sphere rolling on a 2-dimensional surface cannot be integrated; it is "non-holonomic". Consider, as a simple case, a sphere rolling on a flat plane without slipping.

By rolling the sphere around a closed path, back to its starting point, you can imagine that in general the sphere will not end up in exactly the same orientation as it started; it will be rotated about the normal axis. Therefore, there is not a 1-to-1 correspondence between locations on the plane and orientations of the sphere.

You can construct differential relations, though; however, they will be more difficult to use.

 

[hmoein]

thanks