I have a problem in the representation of kinematic joint with 6 DoF. The mathematical representation of single rotations and translations is no problem using homogeneous coordinates. Also the calculation of a chain of joints is no problem. However I do not know how to predict the final point, if each angle in rotation and each postponement in translation is equipped with a probability distribution. How can i predict the distribution of the final point. Does anyone know an analytical solution?
A 6 DOF joint isn't much of a joint. If, by that you mean you have a general transformation (3 rotations and 3 translations) then, noting that
the total transformation, T, could be represented by R * D, where R is the rotation homogeneous transformation, and D is the translation.
There are multiple ways to represent rotations - Euler angles, axis-angle, quaternion, etc. In 3D space, each has a set of three variables. You could pick a representation, assign random variables to each of these parameters, attach random displacements to D (without some rationale to relate the rotation model to the frame you use to define the displacements, your result might be completely ambiguous), and you can compute the resulting T.
You're on your own figuring out what that could possibly mean.
'distribution of the final point'? By the above procedure, you've calculated some kind of a random spatial transformation. If you're looking for some spatial distribution of the transformed origin, you can use Monte Carlo on the rotation and displacement parameters to generate the random transform, then pick off the random displacements from the homogeneous form.
Depending on how you represent rotations, you can get some pretty useless results from this procedure.
I think you need to work more on problem definition. I can't imagine what you're _really_ after.