### Notes:

[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.