I'm working on a project where I need to model a system that is essentially comprised of a series of ball-and-socket joints attached to a base, which is attached in turn to a prismatic joint (rail).
I've read Roy Featherstone's Rigid Body Dynamics Algorithms cover-to-cover, and I've also read the Dynamics section from the Springer Handbook of Robotics (also written by Featherstone).
It took me a long time to get acclimated to using his "spatial vector" and "spatial matrix" notation, but after re-creating all of his notation by hand as an exercise it works out to just be a nice way of concatenating 3x3 and 3x1 matrices and vectors into 6x6 and 6x1 matrices and vectors. The maths he invents to perform operations can be a bit tedious to read as he hijacks some standard notation, but overall everything is very compact, very easy to implement in MATLAB.
My problem is this: How do I add actuators to the model? He walks through explicitly configuring the joint definitions, link definitions, etc., but when it comes to actuators or applied forces he says something like, "Just add a $\tau_a$ here and Bob's your uncle!" - it's not discussed at all. In the Handbook of Robotics he suggests introducing a false acceleration to the fixed base to add the gravitational force term, but doesn't show how to add it in local coordinates nor does he mention how to add the actuator input.
Any help would be greatly appreciated. I've considered starting over with a different book, but it's going to be a great expense of my time to re-acclimate myself to a different set of notation. I'd like to move forward with this, but I feel like I'm just a few inches shy of the finish line.