quaternion.quaternion_time_series
Class appending_array
appending_array(shape, dtype=<class 'numpy.float64'>, initial_array=None)
Source: quaternion/quaternion_time_series.py
Methods
append
append(self, row)
Source: quaternion/quaternion_time_series.py
a
a
Source: quaternion/quaternion_time_series.py
angular_velocity
angular_velocity(R, t)
Source: quaternion/quaternion_time_series.py
Approximate angular velocity of a rotating frame
Parameters

R
:array_like
Quaternionvalued function of time evaluated at a set of times. This represents the quaternion that rotates the standard (x,y,z) frame into the moving frame at each instant.

t
:array_like
Times at which
R
is evaluated.
Returns

Omega
:array_like
The angular velocity (threevector) as a function of time
t
. A vector that is fixed in the moving frame rotates with this angular velocity with respect to the inertial frame.
!!! note
The angular velocity at each instant is given by 2 * (dR/dt) / R. This function approximates the input R
using a cubic spline, and differentiates it as such.
integrate_angular_velocity
integrate_angular_velocity(Omega, t0, t1, R0=None, tolerance=1e12)
Source: quaternion/quaternion_time_series.py
Compute frame with given angular velocity
Parameters

Omega
:callable
,tuple
Angular velocity from which to compute frame. Can be 1) a 2tuple of float arrays (t, v) giving the angular velocity vector at a series of times, 2) a function of time that returns the 3vector angular velocity, or 3) a function of time and orientation (t, R) that returns the 3vector angular velocity In case 1, the angular velocity will be interpolated to the required times. Note that accuracy is poor in case 1.

t0
:float
Initial time

t1
:float
Final time

R0
:optional
,quaternion
Initial frame orientation. Defaults to 1 (the identity orientation).

tolerance
:optional
,float
Absolute tolerance used in integration. Defaults to 1e12.
Returns

t
:float array

R
:quaternion array
minimal_rotation
minimal_rotation(R, t, iterations=2)
Source: quaternion/quaternion_time_series.py
Adjust frame so that there is no rotation about z' axis The output of this function is a frame that rotates the z axis onto the same z' axis as the input frame, but with minimal rotation about that axis. This is done by precomposing the input rotation with a rotation about the z axis through an angle gamma, where dgamma/dt = 2*(dR/dt * z * R.conjugate()).w This ensures that the angular velocity has no component along the z' axis. Note that this condition becomes easier to impose the closer the input rotation is to a minimally rotating frame, which means that repeated application of this function improves its accuracy. By default, this function is iterated twice, though a few more iterations may be called for.
Parameters

R
:quaternion array
Time series describing rotation

t
:float array
Corresponding times at which R is measured

iterations
:int [defaults to 2]
Repeat the minimization to refine the result
slerp
slerp(R1, R2, t1, t2, t_out)
Source: quaternion/quaternion_time_series.py
Spherical linear interpolation of rotors
This function uses a simpler interface than the more fundamental slerp_evaluate
and slerp_vectorized
functions. The latter are fast, being implemented at the C level, but take input tau
instead of time. This function adjusts the time accordingly.
Parameters

R1
:quaternion
Quaternion at beginning of interpolation

R2
:quaternion
Quaternion at end of interpolation

t1
:float
Time corresponding to R1

t2
:float
Time corresponding to R2

t_out
:array
offloats
,float
Times to which the rotors should be interpolated
squad
squad(R_in, t_in, t_out, unflip_input_rotors=False)
Source: quaternion/quaternion_time_series.py
Spherical "quadrangular" interpolation of rotors with a cubic spline
This is typically the best way to interpolate rotation timeseries. It uses the analog of a cubic spline, except that the interpolant is confined to the rotor manifold in a natural way. Alternative methods involving interpolation of other coordinates on the rotation group or normalization of interpolated values give bad results. The results from this method are continuous in value and first derivative everywhere, including around the sampling locations. The input R_in
rotors are assumed to be reasonably continuous (no sign flips), and the input t
arrays are assumed to be sorted. No checking is done for either case, and you may get silently bad results if these conditions are violated. The first dimension of R_in
must have the same size as t_in
, but may have additional axes following. This function simplifies the calling, compared to squad_evaluate
(which takes a set of four quaternions forming the edges of the "quadrangle", and the normalized time tau
) and squad_vectorized
(which takes the same arguments, but in array form, and efficiently loops over them).
Parameters

R_in
:array
ofquaternions
A timeseries of rotors (unit quaternions) to be interpolated

t_in
:array
offloat
The times corresponding to R_in

t_out
:array
offloat
The times to which R_in should be interpolated

unflip_input_rotors
:optional
,bool
If True, this function calls
unflip_rotors
on the input, to ensure that the rotors are more continuous than not. Defaults to False.
unflip_rotors
unflip_rotors(q, axis=1, inplace=False)
Source: quaternion/quaternion_time_series.py
Flip signs of quaternions along axis to ensure continuity
Quaternions form a "double cover" of the rotation group, meaning that if q
represents a rotation, then q
represents the same rotation. This is clear from the way a quaternion is used to rotate a vector v
: the rotated vector is q * v * q.conjugate()
, which is precisely the same as the vector resulting from (q) * v * (q).conjugate()
. Some ways of constructing quaternions (such as converting from rotation matrices or other representations) can result in unexpected sign choices. For many applications, this will not be a problem. But if, for example, the quaternions need to be interpolated or differentiated, the results may be surprising. This function flips the signs of successive quaternions (along some chosen axis, if relevant), so that successive quaternions are as close as possible while still representing the same rotations.
Parameters

q
:array_like
Quaternion array to modify

axis
:optional
,int
Axis along which successive quaternions will be compared. Default value is the last axis of the quaternion array.

inplace
:optional
,bool
If True, modify the data in place without creating a copy; if False (the default), a new array is created and returned.
Returns

q_out
:array_like
An array of precisely the same shape as the input array, differing only by factors of precisely 1 in some elements.