product

Quaternion.product(q: ndarray) → ndarray

Product of two quaternions.

Given two unit quaternions \(\mathbf{p}=(p_w, \mathbf{p}_v)\) and \(\mathbf{q} = (q_w, \mathbf{q}_v)\), their product is defined [Sola] [Dantam] as:

\[\begin{split}\begin{eqnarray} \mathbf{pq} & = & \big( (q_w p_w - \mathbf{q}_v \cdot \mathbf{p}_v) \; , \; \mathbf{q}_v \times \mathbf{p}_v + q_w \mathbf{p}_v + p_w \mathbf{q}_v \big) \\ & = & \begin{bmatrix}q_w p_w - \mathbf{q}_v \cdot \mathbf{p}_v \\ \mathbf{q}_v \times \mathbf{p}_v + q_w \mathbf{p}_v + p_w \mathbf{q}_v \end{bmatrix} \\ & = & \begin{bmatrix} p_w & -\mathbf{p}_v^T \\ \mathbf{p}_v & p_w \mathbf{I}_3 + \lfloor \mathbf{p}_v \rfloor_\times \end{bmatrix} \begin{bmatrix} q_w \\ \mathbf{q}_v \end{bmatrix} \\ & = & \begin{bmatrix} p_w & -p_x & -p_y & -p_z \\ p_x & p_w & -p_z & p_y \\ p_y & p_z & p_w & -p_x \\ p_z & -p_y & p_x & p_w \end{bmatrix} \begin{bmatrix} q_w \\ q_x \\ q_y \\ q_z \end{bmatrix} \\ & = & \begin{bmatrix} p_w q_w - p_x q_x - p_y q_y - p_z q_z \\ p_w q_x + p_x q_w + p_y q_z - p_z q_y \\ p_w q_y - p_x q_z + p_y q_w + p_z q_x \\ p_w q_z + p_x q_y - p_y q_x + p_z q_w \end{bmatrix} \end{eqnarray}\end{split}\]

where \(\lfloor \mathbf{a} \rfloor_\times\) represents the skew-symmetric matrix of \(\mathbf{a}\).

Parameters:

q (numpy.ndarray, Quaternion) – Quaternion to multiply with.

Returns:

pq – Product of quaternions.

Return type:

numpy.ndarray

Examples

>>> p = Quaternion(random=True)
>>> p.view()
Quaternion([ 0.22091606,  0.94554179, -0.23723731,  0.02941561])
>>> q = Quaternion(random=True)
Quaternion([-0.12430979,  0.83988925, -0.39229689,  0.35388736])
>>> p.product(q)
array([-0.92508969, -0.0044107 , -0.3670832 , -0.09715728])

It holds with the result after the cross and dot product definition

>>> rw = p.w*q.w - np.dot(p.v, q.v)
>>> rv = p.w*q.v + q.w*p.v + np.cross(p.v, q.v)
>>> rw, rv
(-0.9250896906548645, array([-0.0044107 , -0.3670832 , -0.09715728]))

Can multiply with a normalized quaternion in an array …

>>> p.product([-0.12430979, 0.83988925, -0.39229689, 0.35388736])
array([-0.92508969, -0.0044107 , -0.3670832 , -0.09715728])

but it ALSO MULTIPLIES with non-unitary quaternions (assumed to be simple vectors with 4 elements in them.)

>>> r = np.random.random(4) * 10
>>> r.view()
array([8.48031045, 2.49690044, 5.78466679, 6.30034199])
>>> p.product(r)
array([ 0.69952346,  6.90525759, -6.61770901,  7.70330242])

The result of this latter operation does NOT yield a unitary quaternion. Thus, the user must normalize the array, if a unitary quaternion is desired.