Quaternions

• Related to Gimbal Lock, Coordinate Systems and Understanding rotational frames in 3D.
• Source: UTwente slides and Steven Gong

A quaternion is a four-part hypercomplex number used to describe 3D rotations and orientations.

Think of quaternions as 3 parameters to indicate a unit vector and 1 to indicate a rotation around it.

• In math, we have:
  • $q=q_0+q_{1}i+q_{2}j+q_{3}k$
  • $q=\left[\begin{array}{c}{q}_0\\ q_1\\ q_2\\ q_3\end{array}\right]\in R^4$ and $\mid \mid q\mid \mid =1$.

Important property: q $=$ -q

Notes from Steven

Double Cover Property (Singularity)

Quaternions have singularities in the context of representing orientation, known as the double-cover property. This means that two quaternions can represent the same orientation. Specifically, a quaternion (q) and its negation (-q) represent the same spatial orientation.

So then why are quaternions better if they also have singularities? Because

A quaternion q has a real part and three imaginary parts.

$$q=q_0+q_{1}i+q_{2}j+q_{3}k$$

The three imaginary parts of the quaternion $i,j,k$ satisfy the following relationship:

$$\{\begin{array}{l}{i}^2=j^2=k^2=-1\\ ij=k,ji=-k\\ jk=i,kj=-i\\ ki=j,ik=-j\end{array}$$

Rotation with quaternions

To rotate a point p using Rotation Matrix, we know that p’ = Rp. How do we do that with a quaternion?

We can also use a scalar and a vector to express quaternions: q $=[s,v{]}^T$ with s $=q_0\in R,v=[q_1,q_2,q_3{]}^T\in R^3$.

Let the rotation be specified by a unit quaternion q. First, we extend the 3D point to an imaginary quaternion p $=[0,x,y,z{]}^T=[0,v{]}^T$ 

We just put the three coordinates into the imaginary part and leave the real part to be zero. Then, the rotated point p′ can be expressed as such a product:

$$p^{\prime }=qp{q}^{-}1$$

The multiplication here is the quaternion multiplication, and the result is also a quaternion. Finally, we take the imaginary part of p′ and get the coordinates of the point after the rotation.