I made a long post, and when submitted, I wasn't logged in them after logging, the posted got cleaned up.
I will try to put my thought over the weekend, but I think there is a confusion here.
I will state the summary of my thoughts now, them go over the point I believe are kind of a condition of mixing things.
Equation
M(q)q'' + C(q,q')= F_contact + F_motor
is a reduced coordinate system model for rigid bodies. probably derived from a special case of
Lagrangian, Legendre, Hamilton or D' Lamber.
So I enter the equation in google and Got this paper.
https://scaron.info/robotics/equations-of-motion.htmlIt seems it is a D' Lamber method, although not stated clear.
I am or I was very familiar with D'Alembert methods, I used them way back in the late 80's in college for solving vibrations of static structures. steel cages, towers, that sort of things.
Going on the assumption that it is a generalized coordinates system method, the things to know is that there are some pros and const with generalized vs reduced coordinates systems.
but I will say that, the const of generalized coordinate systems far outweigh the benefits.
The reason is that the problem with generalized coordinate system are unsolvable.
the people who use them, always, end up with ad hoc iterative solutions.
one such problem is lost of Holonomicity, that is, the model assumes that the number of coordinates are independent, that is the number of equations representing a coordinate, are linearly independent, therefore the acceleration will be equal to the number of coordinates (DOF).
however, that is not the case, not even with the most trivial systems, like a ball socket.
we all know that a ball socket will has orientations where one of the one of the three dof will be lost.
so you end up with two coordinates since one will be a linear function of the other.
but, they are many more cases for example, kinematic loop.
now these problems are not unique to generalized coordinate methods, when but the solution using generalized methods does not have known direct linear system, even when you solve the Convex problem I will just be an approximation. reason been that the system is not linear.
that's not the case when using reduces methods, reduced system always decompose the problem into a set of free bodies, each with 6 dof.
them the method for constraint forces always yield a linear system that provide the unknown forces. But in this case now when using convex quadratic solver, there are linear algebra method to exclude variables that make the matrix I'll conditioned.
And since the system is linear, it does not matter what did is excluded, the other will make up.
Reduce method do have the severe problem of drifting,
however, the problem of drifting although is not solvable, can be made arbitrarily small at the expense of computer power, that is not the case with generalize methods.
there are other hard problems still, but that one is a big show stopper.
here is the Newton 4 answer to the problem.
Generalized coordinate system are attractive to students and researcher because they promise that by running the system you can always get the kinematic state of each DOF.
that is, knowing position of the state vector. you can run the system and get the velocity and acceleration.
Then you make decisions as to what forces, motors to apply to the free dofs.
In newton 4 I introduce the dModel and IkSolver objects.
this dModel is an object that let the application group a series of bodies, joints, and other things together an they get thier own simulation updates.
the order if
-collision update
-dModel update
-Constraimt solver update
-dModel post update
-Integration
with this, an application can configured an articulated robot,
and add an ikSolver.
in the update the application can Get the exact same answer that it would get if this was formulatef as a reduced system. In fact it can get far more than that, because it handle loops, and lost of DOF graciously.
but the app can also try iterative solutions for highly non linear system. and example is an articulated vehicle tire model, using a brush friction model.
anyway I will probably say more later.
but the answer is that since both generalize and reduced coordinate system are equivalent,
yes, in newton 4 we added the dModel as an answer to generalized systems.