Determination of orbital elements for Trans-Neptunian Objects, how? How are orbits calculated for Trans-Neptunian Objects?  I understand there are six standard parameters for defining an orbit (link) but I am not sure how telescope observations translate into these parameters.  Telescope observations tell us where the object is on the Celestial Sphere but can't even tell us (without a good spectroscopic line to do a redshift analysis with) how fast the object is moving towards or away from us.  

Additionally, the motion of these objects on the sky is dominated by Earth's motion relative to the objects rather than the motion of the objects themselves.  This seems to me to offer additional complications.

My question is: how do 2-D observations get fit with a reasonable degree of accuracy to a 6-D parameter space with the additional complication of Earth's motion relative to the object?

Don't be afraid to be mathematical in your response.
You've asked a big question, too big perhaps for a Q&amp;A forum such as this. Your question is the sole subject of graduate level aerospace engineering classes, e.g., University of Colorado ASEN 5070, Introduction to Statistical Orbit Determination, and is the subject of multiple graduate level texts, e.g., Statistical Orbit Determination by Bob Schutz, Byron Tapley and George H. Born. To have a chance in that class, you'll need to already be well-versed in multivariate calculus, linear algebra, probability and statistics, numerical methods, and computer programming.

That said, you might want to look at Bernstein and Khushalani (2000), "Orbit fitting and uncertainties for Kuiper belt objects," The Astronomical Journal, 120.6:3323.

The preferred approach in statistical orbit determination is to collect data over multiple orbits. That's a luxury that is not possible with Trans-Neptunian Objects that have only been seen a handful of times, and only over a small arc of the multi-hundred year long orbits of those objects.

One thing Bernstein and Khushalani did to overcome this was to realize that TNOs are nearly inertial (Newton's first law) objects. Gravitation is but a small perturbation of inertial behavior at such distances. Another thing they did was to take advantage of the fact that for observations separated by a short span of time (e.g., a day or two), almost all of the apparent motion is due to the Earth rather than proper motion of the target object. This gives a good estimate of the distance to the target.

Their approach involves doing some of the regression in Cartesian space, with gravitation being a small perturbation, and then switching to orbital element space to complete the regression. Along the way, they worry about whether they have enough information to do the full orbital element space regression.