- #1
Liferider
- 43
- 0
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?
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?