EoM for rigid body, wrench and twist help

[Liferider]

I am currently working on a robotic manipulation problem and need to form a model for how an object responds. I start by writing up the equations of motion for the body, the motion is then constrained by an additional constraint equation. However, I am new to the notions of twists and wrenches and are afraid to make mistakes.

The position and orientation of the object are represented by
\begin{align*}
u =
\begin{bmatrix}
p \\
\epsilon
\end{bmatrix}
\end{align*}
and/or
\begin{equation*}
T^0_b =
\begin{bmatrix}
R^0_b & p \\
0 & 1
\end{bmatrix}
\end{equation*}
so 'p' is expressed in the inertial frame of reference.

The twist of the object are defined to be
\begin{equation*}
\nu =
\begin{bmatrix}
v \\
\omega
\end{bmatrix}
\end{equation*}
where the elements satisfy
\begin{align*}
v &= \dot{p} - \omega \times p \\
[\omega] &= \dot{R}^0_b (R^0_b)^T
\end{align*}
Therefore, the twist is expressed in the spatial (inertial) frame.

My problem arises when I form the equations of motion. Since I have expressed the object twist in the spatial frame, do I also have to specify the object wrench in the spatial frame?

I did that and found that due to the definition of wrenches, specifying the wrench applied at the point 'p' in the spatial frame requires a special transformation where an additional torque is applied in the spatial frame (due to moving the point of application). I find this a bit frustrating because it becomes harder to interpret the simulation data. Could I just leave the point of application as 'p' and just rotate the force and moment vectors into the inertial frame instead?

Several authors of books and articles gives the EoM in the body frame and not in the spatial frame, is there a reason for this? I mean, don't you almost always want to express the position and orientation relative to an inertial frame?

 

[Liferider]

Ok, I see now that it is a somewhat stupid problem statement, of course one has to develop the EoM for spatial wrench. One can easily be confused when one reads:

Let f ∈ R3 be the force applied to the object at the
point p and let m ∈ R3 be the applied moment. These are
combined into the object load, or wrench, vector denoted
by g = [f, m] ∈ Rnν , where f and m are expressed
in {N}. Like twists, wrenches can be referred to any
convenient frame fixed to the body. One can think of this
as translating the line of application of the force until it
contains the origin of the new frame, then adjusting the
moment component of the wrench to offset the moment
induced by moving the line of the force.

A wrench is a only a wrench if it is expressed in a frame where the point of application is the origin, it is a bit confusing when he writes "f and m are expressed in {N}", where {N} is the inertial frame.

Anyways, in the EoM I must include the applied external wrench to the object (other than contact wrenches) where I will assume that the force of gravity is the only one, what is the correct way of implementing that? I mean, the point of application is the point 'p', so if I make up a body frame that is world aligned but displaced by 'p', I can transform this gravity wrench from the world aligned body frame to the spatial frame. So,
\begin{align*}
T^0_{b'} =
\begin{bmatrix}
I & p \\
0 & 1
\end{bmatrix} \ g_{app}^{b'} &=
\begin{bmatrix}
f^{b'} \\
\tau^{b'}
\end{bmatrix} =
\begin{bmatrix}
0 \\
0 \\
-mg \\
\mathbf{0}
\end{bmatrix} \\
g_{app} &= Ad^T g_{app}^{b'} =
\begin{bmatrix}
(R^{b'}_0)^T & 0 \\
-(R^{b'}_0)^T [p] & (R^{b'}_0)^T
\end{bmatrix}g_{app}^{b'} =
\begin{bmatrix}
I & 0 \\
-[p] & I
\end{bmatrix}g_{app}^{b'} \\
&\Downarrow \\
g_{app} &=
\begin{bmatrix}
f^{b'} \\
-[p]f^{b'}
\end{bmatrix}
\end{align*}
However, this body, world aligned frame that I just created, is not the same as the general body frame that also has some orientation, so will this spatial applied wrench g_{app} be correct?