Hugo Hadfield
Cambridge University PhD student, Signal Processing and Communications Laboratory
A practical look at quaternions and dual quats
Hugo Hadfield
What is a rotation vector?
Imagine a 3D vector from the origin
Magnitude of the vector is the angle of rotation
Direction of the vector is the axis to rotate around
Minimum representation of rotation
Forms a linear (but periodic) space
What is a quaternion?
First we make some algebraic elements with specific properties:
ii = -1 \quad jj = -1 \quad kk = -1 \quad ijk = -1
\Phi = xi + yj + zk
Then write our rotation vector as:
\exp(\frac{\Phi}{2}) = \cos(\frac{\theta}{2}) + \sin(\frac{\theta}{2})\frac{\Phi}{\theta}
We pass this through the exponential function:
w coef of the quat
x,y,z coefs of the quat
How do I use a quaternion?
We apply a quaternion to a point with a double sided sandwich operation:
p = xi + yj + zk
Let's make a point:
p' = qpq^{-1}
And a quat:
q = \cos(\theta/2) + \sin(\theta/2)v
q^{-1} = \cos(\theta/2) - \sin(\theta/2)v
(In fact there is a slightly faster formula)
How to convert a quat to a rotation matrix?
Imagine some quat rotates a vector:
p' = qpq^{-1}
All we have to do to get the rotation matrix is to apply the quat to each basis vector in turn:
Q[:, 0] = qiq^{-1}\\
Q[:, 1] = qjq^{-1}\\
Q[:, 2] = qkq^{-1}
How to convert Euler angles to a quat?
To convert Euler angles to a quat we will use the fact that they compose:
q_x = \exp(ix/2)\\
q_y = \exp(jy/2)\\
q_z = \exp(kz/2)\\
q = q_zq_yq_x
Just choose your ordering to match your Euler angle convention
(note we can of course expand this formula out for efficiency)
How could we represent translation?
We know we need 6 degrees of freedom
We could stack a rotation vector with a translation vector
\begin{bmatrix}
r_i\\
r_j\\
r_k\\
t_i\\
t_j\\
t_k
\end{bmatrix}
Rotation vector
Translation vector
}
}
This is a really good way of storing a transform
It is minimal (exactly 6dof)
It is intuitive
The relationship between rotation and translation
Imagine a line in 3D space, we will call it L
This line encodes a rotation about that line with the magnitude of the line as the angle of rotation:
We are now rotating about something that does not pass through the origin
\theta = |L|
How do we write a line?
Consider a line through a point a, and with a certain direction m, we could just stack them up:
L = \begin{bmatrix}
m\\
a
\end{bmatrix}
This is not unique, any point on the line produces a different numerical L despite being the same 3D line...
L = \begin{bmatrix}
m\\
a\times m
\end{bmatrix}
Do this instead:
This representation is called Plücker coordinates!
What happens if we add two lines together?
Consider two lines and add them together:
L_1 = \begin{bmatrix}
m_1\\
a_1\times m_1
\end{bmatrix}
The pitch represents translation in the direction of the line!
L_2 = \begin{bmatrix}
m_2\\
a_2\times m_2
\end{bmatrix}
L_1 + L_2
= \begin{bmatrix}
m_1 + m_2\\
a_1\times m_1 + a_2\times m_2
\end{bmatrix}
= \begin{bmatrix}
m_3\\
a_3\times m_3
\end{bmatrix}
+
\begin{bmatrix}
0\\
hm_3
\end{bmatrix}
This thing is called a screw
Line bit (axis)
Not line bit (pitch)
We can always break up a screw into an axis and a pitch
What is a dual quaternion?
Write a screw like this:
s = ai + bj + ck + \epsilon(di + ej + fk)
ii = -1 \quad jj = -1 \quad kk = -1 \quad ijk = -1
\epsilon^2 = 0
Where:
BUT ALSO:
Again we just run it through the exponential function to get something we can apply to points:
q_\epsilon = \exp(s/2)
Why use dual quaternions?
q_{ij} = q_iq_j
They compose with a single multiplication
They are by default orthogonal
They are great for interpolation of poses
q_\alpha = \exp(s_1(1-\alpha) + s_2\alpha)
They are very easy to invert
q = \exp(s)\\
q^{-1} = \exp(-s)
They are numerically efficient
They have very neat formulations of kinematics and dynamics
Researchers love them
Dual quat to quat and translation vector
Consider a dual quat \(q_\epsilon\) and you want to break it into a quat \(q\) and a translation \(t\)
q_\epsilon =
We can read off the quat directly, q = w + xi + yj + zk
+ \epsilon(ai + bj + ck + d)
w + xi + yj + zk
q_t = q_\epsilon q^{-1} = \lambda(1 + \epsilon t/2)
To get the translation vector we then get rid of the rotation bit, grab the \(\epsilon i , \epsilon j, \epsilon k\) coefs and divide by the scalar bit
Quat and translation vector to dual quat
q_\epsilon =
q_t
q
q = w + xi + yj + zk
q_t = 1 + \epsilon(t_xi + t_yj + t_zk)/2
Make a standard quat to do the rotation
Make a dual quat that does translation only
Compose them
Summary
Rotations only:
- Rotation vectors are great
- Quats are intuitive (basically just rotation vectors)
Rotation and translation (6dof pose):
-
Rotation vector + translation is great
-
Screws are just made up of adding rotations about 3D lines
-
Dual quats are not so scary (just screws basically, and easy to convert to other formats)
Bonus: Why do the algebra rules for quats seem so arbitrary?
The quat rules are hiding the geometry of the rotation!
We can recover the quaternion multiplication rules if we compose reflections together.
Too long to go into here, but check out this video.
What is a Quaternion?
By Hugo Hadfield
What is a Quaternion?
Explore the world of quaternions and dual quats in this practical presentation. Discover how to represent rotation, translation, and more with these mathematical tools. Uncover the relationship between rotation and translation, and delve into the benefits of using dual quaternions.
