[addendum, Feb-05-2002]
This equation
     dX/dt = [dA/dt Ainv] X a
also resembles a simple differential equation of the form
      dY/dt = K Y
for which the solution is known to be (Yo e^Kt). We can think of the entire X(t) curve as being the solution to this differential equation, the solution simply being e raised to the tangent operator. It's somewhat more complicated than that (first of all because A actually depends on t), but if you follow these ideas it leads to some interesting analysis. See A Mathematical Introduction to Robotic Manipulation by Richard Murray, Zexiang Li and S. Shankar Sastry (CRC Press, publisher) for more info and derivations about exponential forms for manipulator kinematics.