# The major differences between dual-quaternions and screw theory

I am working on a project involving the motion of rigid body. From the literatures, I found two main tools, namely the dual-quaternions and screw theory. May I ask what are the major differences between them? It is appreciated that if some documents can be suggested for beginners.

## 1 Answer

From what I can tell, the analogy is roughly this:

Dual quaternions ~ screw theory : quaternions ~ module theory

So let me explain. (This relies on what I was able to teach myself in a short time from the Wiki article, which was the fastest transparent reference I could find.)

## Analogy for the basic objects

The basic objects of screw theory can be described this way. Given any ring $R$ and any $R$-module $M$, you can make a new ring $R_d$ and a new $R_d$-module $M_d$. Each element of $M_d$ is called a screw and the elements of $R_d$ are called dual scalars.

The dual scalars are merely a specially constructed ring that operates on the specially constructed module $M_d$. There's a convenient way to describe this using matrices. The dual scalars for a ring $R$ is the ring of matrices

$$R_d=\left\{\begin{bmatrix}a&0\\b&a\end{bmatrix}\mid a,b\in R\right\}$$

and if $$M_d:=\left\{\begin{bmatrix}S\\V\end{bmatrix}\mid S,V\in M\right\}$$

then $R_d$ acts on $M_d$ by matrix multiplication on the left in the way described in the wiki article:

$$\begin{bmatrix}a&0\\ b&a\end{bmatrix}\begin{bmatrix}S\\ V\end{bmatrix}=\begin{bmatrix}aS\\ bS+aV\end{bmatrix}$$

So if we think of things this way

Dual scalar rings ~ modules of screws : rings ~ modules

then one could say the dual quaternions are just a special example of a dual scalar ring. Since the quaternions are a module over the quaternions, the $R$ and $M$ above can both be taken to be $\Bbb H$, and this may be what you're thinking of when you think of "the dual quaternions": both ideas mushed together.

But it seems to me that this ring-module analogy is a clear way to think about it.

## Analogy doesn't cover ...

You could say, that screw theory really is more than just particular modules in rings. In particular, screw theory really wants to add dot and cross products akin to the ones we use in vector algebra. (And interestingly, they take the same formal shape as the matrix equations given above.)

## References

As for beginner references, the first one that looks promising that I've found is A geometrical introduction to screw theory. From the abstract:

This work introduces screw theory, a venerable but yet little known theory aimed at describing rigid body dynamics. [...] This work provides a short and rigorous introduction to screw theory intended to an undergraduate and general readership.

The exposition does not really impress me, but I guess there are better ones. If I discover any I'll try to add them here.

• Thank you. Your answer is appreciated. – Ben Jan 8 '15 at 10:42