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Quaternion

Robotics / math

Quaternions are an alternate way to describe orientation or rotations in 3D space using an ordered set of four numbers. They have the ability to uniquely describe any three-dimensional rotation about an arbitrary axis and do not suffer from gimbal lock

Quaternions without math alt text

Unit quaternion

A unit quaternion like unit vector is simply a vector with length (magnitude) equal to 1, but pointing in the same direction as the original vector.

quaternion: \(\(q = w + xi + yj + zk\)\)

as a vector \(\(q = (w, x, y, z)\)\)

calc the norm \(\(\|q\| = \sqrt{w^2 + x^2 + y^2 + z^2}\)\)

calc unit quaternion \(\(q_{unit} = \frac{q}{\|q\|} = \left(\frac{w}{\|q\|}, \frac{x}{\|q\|}, \frac{y}{\|q\|}, \frac{z}{\|q\|}\right)\)\)

code example 
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Conjugate

A quaternion is: \(\(q = (x, y, z, w)\)\)

The conjugate is: \(\(q^* = (-x, -y, -z, w)\)\)

flip the signs of x, y, z, keep w the same.

Slerp

Spherical Linear intERPolation (SLERP) is a method to smoothly interpolate between two orientations (unit quaternions). - SLERP traces the shortest arc along that sphere between them

\[SLERP(q_0, q_1, t) = \frac{\sin((1-t)\theta)}{\sin(\theta)} q_0 + \frac{\sin(t\theta)}{\sin(\theta)} q_1\]
  • At t=0 → result = q0
  • At t=1 → result = q1
  • At t=0.5 → halfway rotation between them.
Demo: using scipy 
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Inverse

unit quaternion

If q is a unit quaternion (∥q∥=1), then the conjugate is also the inverse.

when not unit length

\[q^{-1} = \frac{q^*}{\|q\|^2}\]
\[q \cdot q^{-1} = 1\]

Multiple

by vector

\[\mathbf{v}' = q \, \mathbf{v} \, q^{-1}\]

we treat the vector as pure quaternion

\[\mathbf{v} = (v_x, v_y, v_z, 0)\]
Numpy Example 
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by quaternion

Quaternion multiplication is used to combine rotations in 3D space

\[q_1 = (w_1, x_1, y_1, z_1), \quad q_2 = (w_2, x_2, y_2, z_2)\]
\[q = q_2 \cdot q_1 = \Big( w_2 w_1 - x_2 x_1 - y_2 y_1 - z_2 z_1, \; w_2 x_1 + x_2 w_1 + y_2 z_1 - z_2 y_1, \; w_2 y_1 - x_2 z_1 + y_2 w_1 + z_2 x_1, \; w_2 z_1 + x_2 y_1 - y_2 x_1 + z_2 w_1 \Big)\]

Warning

\[q_2 q_1 \neq q_1 q_2\]

Reference

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