def test_q_inv(self):
result = quat.q_mult(self.qz, quat.q_inv(self.qz))
correct = array([1., 0., 0., 0.])
error = norm(result - correct)
self.assertTrue(error < self.delta)
result = quat.q_mult(self.quatMat, quat.q_inv(self.quatMat))
correct = array([[1., 0., 0., 0.], [1., 0., 0., 0.]])
error = norm(result - correct)
self.assertTrue(error < self.delta)
result = quat.q_mult(self.q3x, quat.q_inv(self.q3x))
correct = array([0., 0., 0.])
error = norm(result - correct)
self.assertTrue(error < self.delta)
def test_q_inv(self):
result = quat.q_mult(self.qz, quat.q_inv(self.qz))
correct = array([ 1., 0., 0., 0.])
error = norm(result - correct)
self.assertTrue(error < self.delta)
result = quat.q_mult(self.quatMat, quat.q_inv(self.quatMat))
correct = array([[ 1., 0., 0., 0.],
[ 1., 0., 0., 0.]])
error = norm(result - correct)
self.assertTrue(error < self.delta)
result = quat.q_mult(self.q3x, quat.q_inv(self.q3x))
correct = array([ 0., 0., 0.])
error = norm(result - correct)
self.assertTrue(error < self.delta)
def calc_position(q, accel, initialVelocity, initialPosition, timeVector):
g_v = np.r_[0, 0, constants.g]
accReSensor = accel - vector.rotate_vector(g_v, quat.q_inv(q))
accReSpace = vector.rotate_vector(accReSensor, q)
# Position and Velocity through integration, assuming 0-velocity at t=0
vel = np.nan * np.ones_like(accReSpace)
pos = np.nan * np.ones_like(accReSpace)
for ii in range(accReSpace.shape[1]):
vel[:, ii] = cumtrapz(accReSpace[:, ii],
x=timeVector,
initial=initialVelocity[ii])
pos[:, ii] = cumtrapz(vel[:, ii],
x=timeVector,
initial=initialPosition[ii])
avevel = np.mean(vel, axis=0)
aveaccel = np.mean(accel, axis=0)
print("Average Accel: {}, Average Velocity: {}, Time taken: {}".format(
np.sqrt(aveaccel[0]**2 + aveaccel[1]**2 + aveaccel[2]**2),
np.sqrt(avevel[0]**2 + avevel[1]**2 + avevel[2]**2),
timeVector[-1] * 1000))
return pos, vel
def calc_position(self):
'''Calculate the position, assuming that the orientation is already known.'''
initialPosition = self.pos_init
# Acceleration, velocity, and position ----------------------------
# From q and the measured acceleration, get the \frac{d^2x}{dt^2}
g = constants.g
g_v = np.r_[0, 0, g]
accReSensor = self.acc - vector.rotate_vector(g_v, quat.q_inv(
self.quat))
accReSpace = vector.rotate_vector(accReSensor, self.quat)
# Position and Velocity through integration, assuming 0-velocity at t=0
vel = np.nan * np.ones_like(accReSpace)
pos = np.nan * np.ones_like(accReSpace)
for ii in range(accReSpace.shape[1]):
vel[:, ii] = cumtrapz(accReSpace[:, ii],
dx=1. / np.float(self.rate),
initial=0)
pos[:, ii] = cumtrapz(vel[:, ii],
dx=1. / np.float(self.rate),
initial=initialPosition[ii])
self.pos = pos
def rotate_vector(vector, q):
'''
Rotates a vector, according to the given quaternions.
Note that a single vector can be rotated into many orientations;
or a row of vectors can all be rotated by a single quaternion.
Parameters
----------
vector : array, shape (3,) or (N,3)
vector(s) to be rotated.
q : array_like, shape ([3,4],) or (N,[3,4])
quaternions or quaternion vectors.
Returns
-------
rotated : array, shape (3,) or (N,3)
rotated vector(s)
.. image:: ../docs/Images/vector_rotate_vector.png
:scale: 33%
Notes
-----
.. math::
q \\circ \\left( {\\vec x \\cdot \\vec I} \\right) \\circ {q^{ - 1}} = \\left( {{\\bf{R}} \\cdot \\vec x} \\right) \\cdot \\vec I
More info under
http://en.wikipedia.org/wiki/Quaternion
Examples
--------
>>> mymat = eye(3)
>>> myVector = r_[1,0,0]
>>> quats = array([[0,0, sin(0.1)],[0, sin(0.2), 0]])
>>> quat.rotate_vector(myVector, quats)
array([[ 0.98006658, 0.19866933, 0. ],
[ 0.92106099, 0. , -0.38941834]])
>>> quat.rotate_vector(mymat, [0, 0, sin(0.1)])
array([[ 0.98006658, 0.19866933, 0. ],
[-0.19866933, 0.98006658, 0. ],
[ 0. , 0. , 1. ]])
'''
vector = np.atleast_2d(vector)
qvector = np.hstack((np.zeros((vector.shape[0], 1)), vector))
vRotated = quat.q_mult(q, quat.q_mult(qvector, quat.q_inv(q)))
vRotated = vRotated[:, 1:]
if min(vRotated.shape) == 1:
vRotated = vRotated.ravel()
return vRotated
def kalman(rate,
acc,
omega,
mag,
D=[0.4, 0.4, 0.4],
tau=[0.5, 0.5, 0.5],
Q_k=None,
R_k=None,
initialPosition=np.zeros(3),
initialVelocity=np.zeros(3),
initialOrientation=np.zeros(3),
timeVector=np.zeros((5, 1)),
accMeasured=np.column_stack((np.zeros((5, 2)), 9.81 * np.ones(5))),
referenceOrientation=np.array([1., 0., 0., 0.])):
'''
Calclulate the orientation from IMU magnetometer data.
Parameters
----------
rate : float
sample rate [Hz]
acc : (N,3) ndarray
linear acceleration [m/sec^2]
omega : (N,3) ndarray
angular velocity [rad/sec]
mag : (N,3) ndarray
magnetic field orientation
D : (,3) ndarray
noise variance, for x/y/z [rad^2/sec^2]
parameter for tuning the filter; defaults from Yun et al.
can also be entered as list
tau : (,3) ndarray
time constant for the process model, for x/y/z [sec]
parameter for tuning the filter; defaults from Yun et al.
can also be entered as list
Q_k : None, or (7,7) ndarray
covariance matrix of process noises
parameter for tuning the filter
If set to "None", the defaults from Yun et al. are taken!
R_k : None, or (7,7) ndarray
covariance matrix of measurement noises
parameter for tuning the filter; defaults from Yun et al.
If set to "None", the defaults from Yun et al. are taken!
Returns
-------
qOut : (N,4) ndarray
unit quaternion, describing the orientation relativ to the coordinate
system spanned by the local magnetic field, and gravity
pos : (N,3) ndarray
Position array
vel : (N,3) ndarray
Velocity
Notes
-----
Based on "Design, Implementation, and Experimental Results of a Quaternion-
Based Kalman Filter for Human Body Motion Tracking" Yun, X. and Bachman,
E.R., IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, 1216-1227 (2006)
'''
numData = len(acc)
# Set parameters for Kalman Filter
tstep = 1. / rate
# check input
assert len(tau) == 3
tau = np.array(tau)
# Initializations
x_k = np.zeros(7) # state vector
z_k = np.zeros(7) # measurement vector
z_k_pre = np.zeros(7)
P_k = np.eye(7) # error covariance matrix P_k
Phi_k = np.eye(7) # discrete state transition matrix Phi_k
for ii in range(3):
Phi_k[ii, ii] = np.exp(-tstep / tau[ii])
H_k = np.eye(7) # Identity matrix
D = np.r_[0.4, 0.4, 0.4] # [rad^2/sec^2]; from Yun, 2006
if Q_k is None:
# Set the default input, from Yun et al.
Q_k = np.zeros((7, 7)) # process noise matrix Q_k
for ii in range(3):
Q_k[ii,
ii] = D[ii] / (2 * tau[ii]) * (1 -
np.exp(-2 * tstep / tau[ii]))
else:
# Check the shape of the input
assert Q_k.shape == (7, 7)
# Evaluate measurement noise covariance matrix R_k
if R_k is None:
# Set the default input, from Yun et al.
r_angvel = 0.01
# [rad**2/sec**2]; from Yun, 2006
r_quats = 0.0001
# from Yun, 2006
r_ii = np.zeros(7)
for ii in range(3):
r_ii[ii] = r_angvel
for ii in range(4):
r_ii[ii + 3] = r_quats
R_k = np.diag(r_ii)
else:
# Check the shape of the input
assert R_k.shape == (7, 7)
# Calculation of orientation for every time step
qOut = np.zeros((numData, 4))
for ii in range(numData):
accelVec = acc[ii, :]
magVec = mag[ii, :]
angvelVec = omega[ii, :]
z_k_pre = z_k.copy(
) # watch out: by default, Python passes the reference!!
# Evaluate quaternion based on acceleration and magnetic field data
accelVec_n = vector.normalize(accelVec)
magVec_hor = magVec - accelVec_n * (accelVec_n @ magVec)
magVec_n = vector.normalize(magVec_hor)
basisVectors = np.column_stack(
[magVec_n, np.cross(accelVec_n, magVec_n), accelVec_n])
quatRef = quat.q_inv(rotmat.convert(basisVectors, to='quat')).ravel()
# Calculate Kalman Gain
# K_k = P_k * H_k.T * inv(H_k*P_k*H_k.T + R_k)
K_k = P_k @ np.linalg.inv(P_k + R_k)
# Update measurement vector z_k
z_k[:3] = angvelVec
z_k[3:] = quatRef
# Update state vector x_k
x_k += np.array(K_k @ (z_k - z_k_pre)).ravel()
# Evaluate discrete state transition matrix Phi_k
Delta = np.zeros((7, 7))
Delta[3, :] = np.r_[-x_k[4], -x_k[5], -x_k[6], 0, -x_k[0], -x_k[1],
-x_k[2]]
Delta[4, :] = np.r_[x_k[3], -x_k[6], x_k[5], x_k[0], 0, x_k[2],
-x_k[1]]
Delta[5, :] = np.r_[x_k[6], x_k[3], -x_k[4], x_k[1], -x_k[2], 0,
x_k[0]]
Delta[6, :] = np.r_[-x_k[5], x_k[4], x_k[3], x_k[2], x_k[1], -x_k[0],
0]
Delta *= tstep / 2
Phi_k += Delta
# Update error covariance matrix
P_k = (np.eye(7) - K_k) @ P_k
# Projection of state
# 1) quaternions
x_k[3:] += tstep * 0.5 * quat.q_mult(x_k[3:], np.r_[0,
x_k[:3]]).ravel()
x_k[3:] = vector.normalize(x_k[3:])
# 2) angular velocities
x_k[:3] -= tstep * tau * x_k[:3]
qOut[ii, :] = x_k[3:]
# Projection of error covariance matrix
P_k = Phi_k @ P_k @ Phi_k.T + Q_k
# Calculate Position from Orientation
# pos, vel = calc_position(qOut, acc, initialVelocity, initialPosition, timeVector)
# Make the first position the reference position
qOut = quat.q_mult(qOut, quat.q_inv(referenceOrientation))
return qOut
def analytical(
R_initialOrientation=np.eye(3),
omega=np.zeros((5, 3)),
initialPosition=np.zeros(3),
initialVelocity=np.zeros(3),
accMeasured=np.column_stack((np.zeros((5, 2)), 9.81 * np.ones(5))),
rate=100):
''' Reconstruct position and orientation with an analytical solution,
from angular velocity and linear acceleration.
Assumes a start in a stationary position. No compensation for drift.
Parameters
----------
R_initialOrientation: ndarray(3,3)
Rotation matrix describing the initial orientation of the sensor,
except a mis-orienation with respect to gravity
omega : ndarray(N,3)
Angular velocity, in [rad/s]
initialPosition : ndarray(3,)
initial Position, in [m]
accMeasured : ndarray(N,3)
Linear acceleration, in [m/s^2]
rate : float
sampling rate, in [Hz]
Returns
-------
q : ndarray(N,3)
Orientation, expressed as a quaternion vector
pos : ndarray(N,3)
Position in space [m]
vel : ndarray(N,3)
Velocity in space [m/s]
Example
-------
>>> q1, pos1 = analytical(R_initialOrientation, omega, initialPosition, acc, rate)
'''
if omega.ndim == 1:
raise ValueError('The input to "analytical" requires matrix inputs.')
# Transform recordings to angVel/acceleration in space --------------
# Orientation of \vec{g} with the sensor in the "R_initialOrientation"
g = constants.g
g0 = np.linalg.inv(R_initialOrientation).dot(np.r_[0, 0, g])
# for the remaining deviation, assume the shortest rotation to there
q0 = vector.q_shortest_rotation(accMeasured[0], g0)
q_initial = rotmat.convert(R_initialOrientation, to='quat')
# combine the two, to form a reference orientation. Note that the sequence
# is very important!
q_ref = quat.q_mult(q_initial, q0)
# Calculate orientation q by "integrating" omega -----------------
q = quat.calc_quat(omega, q_ref, rate, 'bf')
# Acceleration, velocity, and position ----------------------------
# From q and the measured acceleration, get the \frac{d^2x}{dt^2}
g_v = np.r_[0, 0, g]
accReSensor = accMeasured - vector.rotate_vector(g_v, quat.q_inv(q))
accReSpace = vector.rotate_vector(accReSensor, q)
# Make the first position the reference position
q = quat.q_mult(q, quat.q_inv(q[0]))
# compensate for drift
#drift = np.mean(accReSpace, 0)
#accReSpace -= drift*0.7
# Position and Velocity through integration, assuming 0-velocity at t=0
vel = np.nan * np.ones_like(accReSpace)
pos = np.nan * np.ones_like(accReSpace)
for ii in range(accReSpace.shape[1]):
vel[:, ii] = cumtrapz(accReSpace[:, ii],
dx=1. / rate,
initial=initialVelocity[ii])
pos[:, ii] = cumtrapz(vel[:, ii],
dx=1. / rate,
initial=initialPosition[ii])
return (q, pos, vel)
def kalman(rate, acc, omega, mag):
'''
Calclulate the orientation from IMU magnetometer data.
Parameters
----------
rate : float
sample rate [Hz]
acc : (N,3) ndarray
linear acceleration [m/sec^2]
omega : (N,3) ndarray
angular velocity [rad/sec]
mag : (N,3) ndarray
magnetic field orientation
Returns
-------
qOut : (N,4) ndarray
unit quaternion, describing the orientation relativ to the coordinate system spanned by the local magnetic field, and gravity
Notes
-----
Based on "Design, Implementation, and Experimental Results of a Quaternion-Based Kalman Filter for Human Body Motion Tracking" Yun, X. and Bachman, E.R., IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, 1216-1227 (2006)
'''
numData = len(acc)
# Set parameters for Kalman Filter
tstep = 1. / rate
tau = [0.5, 0.5, 0.5] # from Yun, 2006
# Initializations
x_k = np.zeros(7) # state vector
z_k = np.zeros(7) # measurement vector
z_k_pre = np.zeros(7)
P_k = np.matrix(np.eye(7)) # error covariance matrix P_k
Phi_k = np.matrix(np.zeros(
(7, 7))) # discrete state transition matrix Phi_k
for ii in range(3):
Phi_k[ii, ii] = np.exp(-tstep / tau[ii])
H_k = np.matrix(np.eye(7)) # Identity matrix
Q_k = np.matrix(np.zeros((7, 7))) # process noise matrix Q_k
D = 0.0001 * np.r_[0.4, 0.4, 0.4] # [rad^2/sec^2]; from Yun, 2006
# check 0.0001 in Yun
for ii in range(3):
Q_k[ii,
ii] = D[ii] / (2 * tau[ii]) * (1 - np.exp(-2 * tstep / tau[ii]))
# Evaluate measurement noise covariance matrix R_k
R_k = np.matrix(np.zeros((7, 7)))
r_angvel = 0.01
# [rad**2/sec**2]; from Yun, 2006
r_quats = 0.0001
# from Yun, 2006
for ii in range(7):
if ii < 3:
R_k[ii, ii] = r_angvel
else:
R_k[ii, ii] = r_quats
# Calculation of orientation for every time step
qOut = np.zeros((numData, 4))
for ii in range(numData):
accelVec = acc[ii, :]
magVec = mag[ii, :]
angvelVec = omega[ii, :]
z_k_pre = z_k.copy(
) # watch out: by default, Python passes the reference!!
# Evaluate quaternion based on acceleration and magnetic field data
accelVec_n = vector.normalize(accelVec)
magVec_hor = magVec - accelVec_n * accelVec_n.dot(magVec)
magVec_n = vector.normalize(magVec_hor)
basisVectors = np.vstack((magVec_n, np.cross(accelVec_n,
magVec_n), accelVec_n)).T
quatRef = quat.q_inv(quat.rotmat2quat(basisVectors)).flatten()
# Update measurement vector z_k
z_k[:3] = angvelVec
z_k[3:] = quatRef
# Calculate Kalman Gain
# K_k = P_k * H_k.T * inv(H_k*P_k*H_k.T + R_k)
# Check: why is H_k used in the original formulas?
K_k = P_k * np.linalg.inv(P_k + R_k)
# Update state vector x_k
x_k += np.array(K_k.dot(z_k - z_k_pre)).ravel()
# Evaluate discrete state transition matrix Phi_k
Phi_k[3, :] = np.r_[-x_k[4] * tstep / 2, -x_k[5] * tstep / 2,
-x_k[6] * tstep / 2, 1, -x_k[0] * tstep / 2,
-x_k[1] * tstep / 2, -x_k[2] * tstep / 2]
Phi_k[4, :] = np.r_[x_k[3] * tstep / 2, -x_k[6] * tstep / 2,
x_k[5] * tstep / 2, x_k[0] * tstep / 2, 1,
x_k[2] * tstep / 2, -x_k[1] * tstep / 2]
Phi_k[5, :] = np.r_[x_k[6] * tstep / 2, x_k[3] * tstep / 2,
-x_k[4] * tstep / 2, x_k[1] * tstep / 2,
-x_k[2] * tstep / 2, 1, x_k[0] * tstep / 2]
Phi_k[6, :] = np.r_[-x_k[5] * tstep / 2, x_k[4] * tstep / 2,
x_k[3] * tstep / 2, x_k[2] * tstep / 2,
x_k[1] * tstep / 2, -x_k[0] * tstep / 2, 1]
# Update error covariance matrix
#P_k = (eye(7)-K_k*H_k)*P_k
# Check: why is H_k used in the original formulas?
P_k = (H_k - K_k) * P_k
# Projection of state quaternions
x_k[3:] += quat.q_mult(0.5 * x_k[3:], np.r_[0, x_k[:3]]).flatten()
x_k[3:] = vector.normalize(x_k[3:])
x_k[:3] = np.zeros(3)
x_k[:3] = tstep * (-x_k[:3] + z_k[:3])
qOut[ii, :] = x_k[3:]
# Projection of error covariance matrix
P_k = Phi_k * P_k * Phi_k.T + Q_k
# Make the first position the reference position
qOut = quat.q_mult(qOut, quat.q_inv(qOut[0]))
return qOut
print('Input:')
pprint(q_vec)
# Unit quaternion
q_unit = quat.unit_q(q_vec)
print('\nUnit quaternions:')
pprint(q_unit)
# Also add a non-unit quaternion
q_non_unit = np.r_[1, 0, np.sin(alpha_rad[0] / 2), 0]
q_data = np.vstack((q_unit, q_non_unit))
print('\nGeneral quaternions:')
pprint(q_data)
# Inversion
q_inverted = quat.q_inv(q_data)
print('\nInverted:')
pprint(q_inverted)
# Conjugation
q_conj = quat.q_conj(q_data)
print('\nConjugated:')
pprint(q_conj)
# Multiplication
q_multiplied = quat.q_mult(q_data, q_data)
print('\nMultiplied:')
pprint(q_multiplied)
# Scalar and vector part
q_scalar = quat.q_scalar(q_data)