dair_pll.quaternion
Quaternion-based \(SO(3)\) operations.
- class dair_pll.quaternion.DataType
Static type for both supported types of quaternion/vector representations:
Tensorandndarray.alias of TypeVar(‘DataType’, ~torch.Tensor, ~numpy.ndarray)
- class dair_pll.quaternion.TensorCallable(*args, **kwargs)[source]
Bases:
ProtocolStatic type for callable mapping from list of
Tensors toTensor.
- class dair_pll.quaternion.NdarrayCallable(*args, **kwargs)[source]
Bases:
ProtocolStatic type for callable mapping from list of
ndarrays tondarray.
- dair_pll.quaternion.operation_selector(tensor_operation, ndarray_operation, *args)[source]
Helper function which selects between Pytorch and Numpy implementations of a quaternion operation.
- Parameters:
tensor_operation (
TensorCallable) –Tensor-backed implementation.ndarray_operation (
NdarrayCallable) –ndarray-backed implementation.*args (
~DataType) – Arguments to pass to implementation.
- Return type:
- Returns:
Operation’s return value, same type as arguments.
- dair_pll.quaternion.inverse(q)[source]
Quaternion inverse function.
For input quaternion \(q = [q_w, q_{xyz}]\), returns the inverse quaternion \(q^{-1} = [-q_w, q_{xyz}]\).
- dair_pll.quaternion.multiply_torch(q, r)[source]
Tensorimplementation ofmultiply().- Return type:
- dair_pll.quaternion.multiply_np(q, r)[source]
ndarrayimplementation ofmultiply().
- dair_pll.quaternion.multiply(q, r)[source]
Quaternion multiplication.
Given 2 quaternions \(q = [q_w, q_{xyz}]\) and \(r = [r_w, r_{xyz}]\), performs the quaternion multiplication via the formula
\[\begin{split}q \times r = \begin{bmatrix} q_w r_w - q_{xyz} \cdot r_{xyz} \\ q_w r_{xyz} + r_w q_{xyz} + q_{xyz} \times r_{xyz} \end{bmatrix}\end{split}\]This formula was taken from the following address: https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation#Quaternion-derived_rotation_matrix
- dair_pll.quaternion.rotate(q, p)[source]
Quaternion rotation.
Given a quaternion \(q = [q_w, q_{xyz}]\) and vector \(p\), produces the \(q\)-rotated vector \(p'\) via the formula
\[p' = (q_{xyz} \cdot p) q_{xyz} + 2 q_w (q_{xyz} \times p) + q_w^2 p + q_{xyz} \times (q_{xyz} \times p)\]This formula was taken from the following address: https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation#Quaternion-derived_rotation_matrix
- dair_pll.quaternion.sinc(x)[source]
Elementwise \(\mathrm{sinc}\) function.
Given a tensor \(x\), applies the elementwise-mapping
\[\begin{split}x \to \begin{cases}1 & x =0 \\ \frac{\sin(x)}{x} & x \neq 0\end{cases}.\end{split}\]
- dair_pll.quaternion.log(q)[source]
Transforms quaternion into logarithmic coordinates.
Given a quaternion
\[q = [\cos(\theta/2), \hat u \sin(\theta/2)] = [q_w, q_{xyz}],\]returns the corresponding logarithmic coordinates (rotation vector) \(r = \theta\hat u\).
This computation is evaluated via the pertations
\[\begin{split}\begin{align} \theta(q) &= 2\mathrm{atan2}(||q_{xyz}||_2, q_w), \\ r &= \begin{cases} \frac{\theta(q)}{\sin(\theta(q)/2)} q_{xyz} & \sin( \theta(q)/2) \neq 0,\\ 0 & \sin(\theta(q)/2) = 0.\end{cases} \end{align}\end{split}\]This function inverts
exp().
- dair_pll.quaternion.exp(r)[source]
Transforms logarithmic coordinates into quaternion.
Given logarithmic coordinates representation (rotation vector) \(r = \theta\hat u\), returns the corresponding quaternion
\[q = [\cos(\theta/2), \hat u \sin(\theta/2)] = [q_w, q_{xyz}].\]This computation is evaluated via the operations
\[\begin{split}\begin{align} \theta(r) &= ||r||_2, \\ q &= \begin{bmatrix}\cos(\theta(r)/2) \\ \frac{1}{2} r \mathrm{sinc}(\theta(r)/2) \end{bmatrix}. \end{align}\end{split}\]This function inverts
log().