relationship between geometric twist jacobian and wrench

Question

I read the robot textbook of Peter Corke and Richard M. Murray, in Peter’s, chapter 8 about jacobian, there are two jacobian, one is geometric jacobian $^0J$ from math, other is world coordinate frame jacobian $J$, I konw from $^0J$ to $J$ need transformation matrix.

My question is how to describe joint torque from the formula $\tau = J^T F$? Because there are two kind jacobian. In Peter’s book, use the geometric jacobian, as follows

Answer 1

I'd suggest thinking about the problem this way:

When you construct a Jacobian for an arm, it will typically map your joint velocities to to one of three representations of the end effector velocity:

1. The "world velocity", which is the derivative of the position coordinates of the end effector.
2. The "body velocity", which is the world velocity you would get if you put your origin at the current location of the robot (so that it measures how the end effector would perceive its own motion, like a driver thinking in terms of forward-lateral components of motion).
3. The "spatial velocity", which is the velocity of a frame rigidly attached to the end effector, but currently collocated with and aligned with the origin frame (see my answer here for a description of why the spatial velocity is useful, although unintuitive).

The relationship $\tau=J^{T}F$ works for each of these Jacobians as long as the $F$ forces are described in the same frame as the output velocity:

1. If your Jacobian outputs the world velocity, then $J^{T}$ must take in a world wrench (force components in the coordinate $x$, $y$, and $z$ directions, and moment components around the coordinate $x$, $y$, and $z$ axes).
2. If your Jacobian outputs the body velocity, then $J^{T}$ must take in a body wrench (force components in the $x$, $y$, and $z$ directions of the end-effector frame, and moment components around the $x$, $y$, and $z$ axes of the end-effector frame).
3. If your Jacobian outputs the spatial velocity, then $J^{T}$ must take in a spatial wrench. (For any wrench applied at any location in the space, you can find an equivalent wrench that could be applied at the origin, which has an extra moment component equal to the cross product between the force component and the displacement between the point where the force is applied and the origin. This equivalent wrench is the spatial wrench.)

Two notes:

1. I'm not familiar with the book you're citing, but it's not clear to me from this passage whether the use of the word "spatial" here is the true "spatial velocity" discussed in Murray's book, or whether (as is sometimes the case) it's being used to mean "through space" in what I would call the world velocity.
2. I believe that the text you highlighted "can never be singular" should be interpreted as "transposing the Jacobian is always a computable operation, even if the Jacobian is a singular matrix whose inverse cannot be computed".