# How to extract the screw axis vector and the angle from the exponential coordinates?

Given the 6-dimensional vector of the exponential coordinates of the homogeneous transformation: $$S\theta$$, where $$S$$ is the screw axis consisting of the pair $$(\omega, v)$$ and both of them are $$3$$ dimensional vectors, $$\theta$$ is the angle followed by the transformation around that screw axis. Also for a screw axis, either $$v$$ or $$\omega$$ is a unit vector.

How to separate the screw axis $$S$$ and the angle of rotation $$\theta$$ from the given exponential coordinates 6-vector $$S\theta$$? Note that the unit vector is not given. it may be $$v$$ or $$\omega$$.

Is the following the correct algorithm?

\begin{align*} \theta &= \|S\theta\| &\text{#which is the magnitude of S\theta 6-vector.} \\ S &= \frac{S\theta}{\|S\theta\|} &\text{#the screw axis is the S\theta 6-vector divided by its magnitude.} \end{align*}

• Hi, welcome to MSE. I've edited your question to bring it up to the standards of the site. I would encourage you to look here for a tutorial in doing this for yourself. Plus, if you press edit, you can see how I've formatted your question. I'm not sure if you wanted $S\theta$ to look like $S_\theta$; to achieve this, write S_\theta in place of S\theta. Feb 14, 2019 at 20:59
• @TheoBendit Thank you very much for the editing. Sθ is S times θ. Feb 14, 2019 at 21:07

to answer my question, the screw axis has 2 vectors each of 3 dimensions. these vectors are ω and v. As long as we consider the screw axis and not the corresponding twist, either one of those vectors should be a unit vector.

If the given screw axis S has non-zero ω vector then the ω vector is the unit vector, if ω is a zero vector (pure translation) then the unit axis is the other vector v.

Returning to the exponential coordinates Sθ, after the unit vector of the screw axis S is determined as aforementioned above, the angle theta θ and the axis S is extracted from the 6-vector of exponential coordinates Sθ as follows,

let Sθ = (ωθ, vθ)

Hence,

θ = ∥ωθ∥ if ω is the unit vector. In other words, ωθ vector is non-zero.

OR

θ = ∥vθ∥ if v is the unit vector. In other words, ωθ vector is zero.

Consequently the screw axis S can be determined simply as,

S= Sθ / θ