Question

The transformation matrix for angular velocities from the inertial frame to the body frame

Answer 1

Answer:

There seems to be confusion between a transformation of coordinates (matrix RR) and the Jacobian (matrix JJ).

The rotation matrix transforms the components of vectors between the body frame and the inertial frame. This happens for all vectors.

v0ω0F0τ0=Rv=Rω=RF=Rτ

v0=Rvω0=RωF0=RFτ0=Rτ

The Jacobian relates the three joint speeds (ϕ˙,ψ˙,θ˙)(ϕ˙,ψ˙,θ˙) to body rotational velocity ω0ω0. For a sequence of rotations the body to inertial rotation matrix is: R=RxRyRzR=RxRyRz. Now the body rotational velocity vector is defined as follows

ω0=ı^ϕ˙+Rx(ȷ^ψ˙+Ryk^θ˙)

ω0=ı^ϕ˙+Rx(ȷ^ψ˙+Ryk^θ˙)

Do you see the pattern above? See this post as well as this post for more details.

the above is grouped together into the Jacobian as

ω0=J⎛⎝⎜⎜ϕ˙ψ˙θ˙⎞⎠⎟⎟

ω0=J(ϕ˙ψ˙θ˙)

You see, the list of joint speeds is not a vector because each joint speed is riding on a different reference frame. The columns of the Jacobian contain the orientation of each rotation axis in the inertial system

J=[ı^Rxȷ^RxRyk^]

J=[ı^Rxȷ^RxRyk^]

The matrix you describe in your post is the inverse Jacobian which relates the joint motions to the body rotational velocity

⎛⎝⎜⎜ϕ˙ψ˙θ˙⎞⎠⎟⎟=J−1ω0

(ϕ˙ψ˙θ˙)=J−1ω0

where the inverse Jacobian evaluates to

J−1=⎛⎝⎜100sinϕtanψcosϕ−sinϕ/cosψ−cosϕtanψsinϕcosϕ/cosψ⎞⎠