def test_qinverse():
# Takes sequence
iq = tq.qinverse((1, 0, 0, 0))
# Returns float type
assert iq.dtype.kind == 'f'
for M, q in eg_pairs:
iq = tq.qinverse(q)
iqM = tq.quat2mat(iq)
iM = np.linalg.inv(M)
assert np.allclose(iM, iqM)
def _distance_from_fixed_angle(angle, fixed_angle):
"""Designed to be mapped, this function finds the smallest rotation between
two rotations. It assumes a no-symmettry system.
The algorithm involves converting angles to quarternions, then finding the
appropriate misorientation. It is tested against the slower more complete
version finding the joining rotation.
Parameters
----------
angle : np.array()
Euler angles representing rotation of interest.
fixed_angle : np.array()
Euler angles representing fixed reference rotation.
Returns
-------
theta : np.array()
Rotation angle between angle and fixed_angle.
"""
angle = angle[0]
q_data = euler2quat(*np.deg2rad(angle), axes="rzxz")
q_fixed = euler2quat(*np.deg2rad(fixed_angle), axes="rzxz")
if np.abs(2 * (np.dot(q_data, q_fixed))**2 - 1) < 1: # arcos will work
# https://math.stackexchange.com/questions/90081/quaternion-distance
theta = np.arccos(2 * (np.dot(q_data, q_fixed))**2 - 1)
else: # slower, but also good
q_from_mode = qmult(qinverse(q_fixed), q_data)
axis, theta = quat2axangle(q_from_mode)
theta = np.abs(theta)
return np.rad2deg(theta)
def diff_drive2(self, robot, index, target_pose, js1, joint_target, js2):
"""
This is a hackier version of the diff_drive
It uses specified joints to achieve the target pose of the end effector
while asking some other specified joints to match a global pose
"""
pf = robot.get_compute_functions()['passive_force'](True, True, False)
max_v = 0.1
max_w = np.pi
qpos, qvel, poses = robot.get_observation()
current_pose: Pose = poses[index]
delta_p = target_pose.p - current_pose.p
delta_q = qmult(target_pose.q, qinverse(current_pose.q))
axis, theta = quat2axangle(delta_q)
if (theta > np.pi):
theta -= np.pi * 2
t1 = np.linalg.norm(delta_p) / max_v
t2 = theta / max_w
t = max(np.abs(t1), np.abs(t2), 0.001)
thres = 0.1
if t < thres:
k = (np.exp(thres) - 1) / thres
t = np.log(k * t + 1)
v = delta_p / t
w = theta / t * axis
target_qvel = robot.get_compute_functions()['cartesian_diff_ik'](
np.concatenate((v, w)), 9)
for j, target in zip(js2, joint_target):
qpos[j] = target
robot.set_action(qpos, target_qvel, pf)
def diff_drive(self, robot, index, target_pose):
"""
this diff drive is very hacky
it tries to transport the target pose to match an end pose
by computing the pose difference between current pose and target pose
then it estimates a cartesian velocity for the end effector to follow.
It uses differential IK to compute the required joint velocity, and set
the joint velocity as current step target velocity.
This technique makes the trajectory very unstable but it still works some times.
"""
pf = robot.get_compute_functions()['passive_force'](True, True, False)
max_v = 0.1
max_w = np.pi
qpos, qvel, poses = robot.get_observation()
current_pose: Pose = poses[index]
delta_p = target_pose.p - current_pose.p
delta_q = qmult(target_pose.q, qinverse(current_pose.q))
axis, theta = quat2axangle(delta_q)
if (theta > np.pi):
theta -= np.pi * 2
t1 = np.linalg.norm(delta_p) / max_v
t2 = theta / max_w
t = max(np.abs(t1), np.abs(t2), 0.001)
thres = 0.1
if t < thres:
k = (np.exp(thres) - 1) / thres
t = np.log(k * t + 1)
v = delta_p / t
w = theta / t * axis
target_qvel = robot.get_compute_functions()['cartesian_diff_ik'](
np.concatenate((v, w)), 9)
robot.set_action(qpos, target_qvel, pf)
def getRelativeQuaternionAtoB(quat_a, quat_b):
"""
Given two quaternions A and B, determines the quaternion that rotates A to B
(Multiplying A by q_rel will give B)
"""
q_rel = qmult(qinverse(quat_a), quat_b)
return q_rel
def obtain_v_dqs(ndat, delta, q):
out=np.zeros((ndat,3))
for i in range(ndat):
j=i+delta
#Since everything is squared does quat_reduce matter? ...no, but do so anyway.
dq=qs.quat_reduce(qops.qmult( qops.qinverse(q[i]), q[j]))
out[i]=dq[1:4]
return out
def compute_imu_orientation_from_world(robot_quat_in_world):
# imu orientation has roll and pitch relative to gravity vector. yaw in world frame
# get global yaw
yrp_world_frame = quat2euler(robot_quat_in_world, axes='szxy')
# remove global yaw rotation from roll and pitch
yaw_quat = euler2quat(yrp_world_frame[0], 0, 0, axes='szxy')
rp = rotate_vector((yrp_world_frame[1], yrp_world_frame[2], 0), qinverse(yaw_quat))
# save in correct order
return [rp[0], rp[1], 0], yaw_quat
def residuals(self, poses, vos, L_ax, L_aq, L_rx, L_rq):
"""
computes the residuals
:param poses: N x 7
:param vos: (N-1) x 7
:param L_ax: 3 x 3
:param L_aq: 4 x 4
:param L_rx: 3 x 3
:param L_rq: 4 x 4
:return:
"""
r = np.zeros((0, 1))
# unary residuals
L = np.zeros((7, 7))
L[:3, :3] = L_ax
L[3:, 3:] = L_aq
for i in range(self.N):
rr = self.z[7 * i:7 * (i + 1)] - np.reshape(poses[i], (-1, 1))
r = np.vstack((r, np.dot(L, rr)))
# pairwise residuals
k = 0
for i in range(self.N):
for j in range(i + 1, self.N):
# translation residual
v = self.z[7 * j:7 * j + 3, 0] - self.z[7 * i:7 * i + 3, 0]
q = txq.qinverse(self.z[7 * i + 3:7 * i + 7, 0])
rt = txq.rotate_vector(v, q)
rt = rt[:, np.newaxis] - vos[k, :3].reshape((-1, 1))
# rt = self.z[7*(i+1) : 7*(i+1)+3] - self.z[7*i : 7*i+3] - \
# vos[i, :3].reshape((-1, 1))
r = np.vstack((r, np.dot(L_rx, rt)))
# rotation residual
q0 = self.z[7 * i + 3:7 * i + 7].squeeze()
q1 = self.z[7 * j + 3:7 * j + 7].squeeze()
qvo = txq.qmult(txq.qinverse(q0), q1).reshape((-1, 1))
rq = qvo - vos[k, 3:].reshape((-1, 1))
r = np.vstack((r, np.dot(L_rq, rq)))
k += 1
return r
def get_allocentric_poses(self):
poses = self.get_poses()
poses_allocentric = []
for pose in poses:
dx = np.arctan2(pose[0], -pose[2])
dy = np.arctan2(pose[1], -pose[2])
quat = euler2quat(-dy, -dx, 0, axes='sxyz')
quat = qmult(qinverse(quat), pose[3:])
poses_allocentric.append(np.concatenate([pose[:3], quat]))
#print(quat, pose[3:], pose[:3])
return poses_allocentric
def get_distance_between_two_angles_longform(angle_1, angle_2):
"""
Using the long form to find the distance between two angles in euler form
"""
q1 = euler2quat(*np.deg2rad(angle_1), axes='rzxz')
q2 = euler2quat(*np.deg2rad(angle_2), axes='rzxz')
# now assume transform of the form MODAL then Something = TOTAL
# so we want to calculate MODAL^{-1} TOTAL
q_from_mode = qmult(qinverse(q2), q1)
axis, angle = quat2axangle(q_from_mode)
return np.rad2deg(angle)
def rot_diff(self, other):
'''Compute rotation btw frames
Arguments:
other {Frame} -- target frame
Returns:
np.array, float -- axis and angle
'''
dq = quaternions.qmult(other.q, quaternions.qinverse(self.q))
axis, angle = quaternions.quat2axangle(dq)
return axis, angle
def get_imu_quaternion(self):
# imu orientation has roll and pitch relative to gravity vector. yaw in world frame
_, robot_quat_in_world = self.get_robot_pose()
# change order to transform3d standard
robot_quat_in_world = (robot_quat_in_world[3], robot_quat_in_world[0],
robot_quat_in_world[1], robot_quat_in_world[2])
# get global yaw
yrp_world_frame = quat2euler(robot_quat_in_world, axes='szxy')
# remove global yaw rotation from roll and pitch
yaw_quat = euler2quat(yrp_world_frame[0], 0, 0, axes='szxy')
rp = rotate_vector((yrp_world_frame[1], yrp_world_frame[2], 0),
qinverse(yaw_quat))
# save in correct order
rpy = [rp[0], rp[1], 0]
# convert to quaternion
quat_wxyz = euler2quat(*rpy)
# change order to ros standard
return quat_wxyz[1], quat_wxyz[2], quat_wxyz[3], quat_wxyz[0]
def rotmatrix_to_quaternion(time, matrix, bInvert=False):
"""
Converts a matrix to quaternion, with a reverse if necessary
"""
nPts = len(time)
if nPts != len(matrix):
print >> sys.stderr, "= = = ERROR in rotmatrix_to_quaternion: lengths are not the same!"
return
out = np.zeros((5, nPts))
for i in range(nPts):
out[0, i] = time[i]
if bInvert:
out[1:5, i] = qops.qinverse(qops.mat2quat(matrix[i]))
else:
out[1:5, i] = qops.mat2quat(matrix[i])
return out
def diff_drive(self,
robot: PandaArm,
index,
target_pose,
max_v=0.1,
max_w=np.pi,
active_joints_ids=[],
override=None):
obs = robot.observation
qpos, qvel, poses = obs['qpos'], obs['qvel'], obs['poses']
current_pose: Pose = poses[index]
delta_p = target_pose.p - current_pose.p
delta_q = quat.qmult(target_pose.q, quat.qinverse(current_pose.q))
axis, theta = quat.quat2axangle(delta_q)
if theta > np.pi:
theta -= np.pi * 2
t1 = np.linalg.norm(delta_p) / max_v
t2 = theta / max_w
t = max(np.abs(t1), np.abs(t2), 0.01)
thres = 0.1
if t < thres:
k = (np.exp(thres) - 1) / thres
t = np.log(k * t + 1)
v = delta_p / t
w = theta / t * axis
index = index if index >= 0 else len(robot._robot.get_joints()) + index
cal_qvel = robot.get_compute_functions()['cartesian_diff_ik'](
np.concatenate((v, w)), index, active_joints_ids)
target_qvel = np.zeros_like(qvel)
indx = np.arange(
len(qvel)) if active_joints_ids == [] else active_joints_ids
target_qvel[indx] = cal_qvel
pf = robot.get_compute_functions()['passive_force']()
if override is not None:
override_indx, override_qpos, override_qvel = override
qpos[override_indx] = override_qpos
target_qvel[override_indx] = override_qvel
robot.set_action(qpos, target_qvel, pf)
def optimize_poses(pred_poses,
vos=None,
fc_vos=False,
target_poses=None,
sax=1,
saq=1,
srx=1,
srq=1):
"""
optimizes poses using either the VOs or the target poses (calculates VOs
from them)
:param pred_poses: N x 7
:param vos: (N-1) x 7
:param fc_vos: whether to use relative transforms between all frames in a fully
connected manner, not just consecutive frames
:param target_poses: N x 7
:param: sax: covariance of pose translation (1 number)
:param: saq: covariance of pose rotation (1 number)
:param: srx: covariance of VO translation (1 number)
:param: srq: covariance of VO rotation (1 number)
:return:
"""
pgo = PoseGraphFC() if fc_vos else PoseGraph()
if vos is None:
if target_poses is not None:
# calculate the VOs (in the pred_poses frame)
vos = np.zeros((len(target_poses) - 1, 7))
for i in range(len(vos)):
vos[i, :3] = target_poses[i + 1, :3] - target_poses[i, :3]
q0 = target_poses[i, 3:]
q1 = target_poses[i + 1, 3:]
vos[i, 3:] = txq.qmult(txq.qinverse(q0), q1)
else:
print
'Specify either VO or target poses'
return None
optim_poses = pgo.optimize(poses=pred_poses,
vos=vos,
sax=sax,
saq=saq,
srx=srx,
srq=srq)
return optim_poses
def orientationEstimaton():
#import the data
filename = "imu/imuRaw1.mat"
viconFileName = "vicon/viconRot1.mat"
Ax, Ay, Az, Wx, Wy, Wz, ts = importData(filename)
#intialize the P estimate covariance matrix (3x3)
Pk_minus = np.zeros((3, 3))
Pk_minus[0, 0] = 0.277
Pk_minus[1, 1] = 0.277
Pk_minus[2, 2] = 0.4
#initialize a Q matrix (3x3)
Q = np.diag(np.ones(3) * .1)
#intialize state
xk_minus = np.array([1, 0, 0, 0])
#matrix for final output
allEstimates = np.empty((len(Ax), 3))
#run the measurement model for each of the pieces of data we have
for i in range(0, len(Ax) - 1):
epsilon = 5
c = epsilon * np.eye(3)
#find the S matrix (3x3)
S = np.linalg.cholesky((2 * len(Q)) * (Pk_minus + Q + c))
#create the Wi matrix
Wi = np.hstack((S, -S)) #3x6
#convert W to quaternion representation
Xi = np.empty((6, 4))
for j in range(0, len(Wi)):
newQuat = axisAngleToQuat(Wi[j, :])
Xi[j, :] = quat.qnorm(newQuat)
#add qk-1 to the Xi
for j in range(0, len(Xi)):
if i == 0:
xk_minusAsQuat = np.array([1, 0, 0, 0])
else:
xk_minusAsQuat = axisAngleToQuat(xk_minus)
newQuat = quat.qmult(xk_minusAsQuat, Xi[j, :])
Xi[j, :] = quat.qnorm(newQuat)
#get the process model and convert Xi to Yi
qDelta = processModel(Wx[i], Wy[i], Wz[i], ts[i + 1] - ts[i])
Yi = np.empty((6, 4))
for j in range(0, len(Xi)):
newQuat = quat.qmult(Xi[j, :], qDelta)
Yi[j, :] = quat.qnorm(newQuat)
Yi = np.nan_to_num(Yi)
#get the priori mean and covariance
[xk_minus_quat, allErrorsX, averageErrorX] = findQuaternionMean(Xi)
[n, _] = Xi.shape
xk_minus = quatToAxisAngle(xk_minus_quat)
#3x3 matrix in terms of axis angle r vectors
Pk_minus = 1.0 / (2 * n) * np.dot(
np.array(allErrorsX - averageErrorX).T,
np.array(allErrorsX - averageErrorX))
Pk_minus = np.nan_to_num(Pk_minus)
#this is where the update model starts
g = measurementModel(Ax[i], Ay[i], Az[i])
Zi = np.empty((6, 4))
for j in range(0, len(Yi)):
vector = np.nan_to_num(Yi[j, :])
if (np.array_equal(vector, np.array([0, 0, 0, 0]))):
inverse = vector
else:
inverse = np.nan_to_num(quat.qinverse(vector))
lastTwo = np.nan_to_num(quat.qmult(g, inverse))
newQuat = np.nan_to_num(quat.qmult(vector, lastTwo))
Zi[j, :] = np.nan_to_num(quat.qnorm(newQuat))
#remove all the nans
Zi = np.nan_to_num(Zi)
#find the mean and covariance of Zi
[zk_minus, allErrorsZ, averageErrorZ] = findQuaternionMean(Zi)
# 6x6 matrix in terms of axis angle r vectors
Pzz = 1.0 / (2 * n) * np.dot(
np.array(allErrorsZ - averageErrorZ).T,
np.array(allErrorsZ - averageErrorZ))
#find the innovation vk
zk_plus = np.array([Ax[i], Ay[i], Az[i]])
zk_minusVector = quatToAxisAngle(zk_minus)
vk = zk_plus - zk_minusVector
#find the expected covariance
R = np.diag(np.ones(3) * .40)
Pvv = Pzz + R
Pvv = np.nan_to_num(Pvv)
#find Pxz, the cross-correlation matrix
Pxz = 1.0 / (2 * n) * np.dot(
np.array(allErrorsX - averageErrorX).T,
np.array(allErrorsZ - averageErrorZ))
Pxz = np.nan_to_num(Pxz)
#find the Kalman gain matix
Kk = Pxz * np.linalg.inv(Pvv)
Kk = np.nan_to_num(Kk)
#find the posteriori mean, which is the updated estimate of the state
xk = xk_minus + np.dot(Kk, vk)
xk = np.nan_to_num(xk)
#find the posteriori variation, which is the updated variation
Pk = Pk_minus - Kk * Pvv * Kk.T
Pk = np.nan_to_num(Pk)
#set the new xk to xk_minus and the new Pk to the Pk_minus
xk_minus = xk
Pk_minus = Pk
#add to a matrix and return
allEstimates[i] = xk_minus
#visualize the results
visualizeResults(allEstimates, viconFileName)
#create the panorama using my own data
rotsInMatrix = np.empty((3, 3, len(allEstimates)))
for est in range(0, len(allEstimates)):
rot = axisAngletoMat(allEstimates[est])
rotsInMatrix[:, :, est] = rot
createPanorama(rotsInMatrix, ts)
return allEstimates
Copyright (C) 2018 NVIDIA Corporation. All rights reserved. Licensed under the CC BY-NC-SA 4.0 license (https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode). """ import numpy as np import transforms3d.quaternions as txq import transforms3d.euler as txe # predicted poses tp1 = np.random.rand(3) tp2 = np.random.rand(3) qp1 = txe.euler2quat(*(2 * np.pi * np.random.rand(3))) qp2 = txe.euler2quat(*(2 * np.pi * np.random.rand(3))) # relatives t_rel = txq.rotate_vector(v=tp2 - tp1, q=txq.qinverse(qp1)) q_rel = txq.qmult(txq.qinverse(qp1), qp2) # vo poses trand = np.random.rand(3) qrand = txe.euler2quat(*(2 * np.pi * np.random.rand(3))) tv1 = txq.rotate_vector(v=tp1, q=qrand) qv1 = txq.qmult(qrand, qp1) tv2 = txq.rotate_vector(v=t_rel, q=qv1) + tv1 qv2 = txq.qmult(qv1, q_rel) # aligned vo voq = txq.qmult(txq.qinverse(qv1), qv2) vot = txq.rotate_vector(v=tv2 - tv1, q=txq.qinverse(qv1)) vot = txq.rotate_vector(v=vot, q=qp1)
q_A_C = quat.axangle2quat(r2_A, -theta2)
print("q_A_B: ")
print(q_A_B)
print("q_A_C: ")
print(q_A_C)
# 2. Vector coordinate transformation
xA_A = np.array([1.0, 0.0, 0.0]) # A's x axis in A
xA_B = quat.rotate_vector(xA_A, q_A_B) # A's x axis in B
xA_C = quat.rotate_vector(xA_A, q_A_C) # A's x axis in C
print("xA_B: ")
print(xA_B)
print("xA_C: ")
print(xA_C)
q_B_C = quat.qmult(q_A_C, quat.qinverse(q_A_B))
xB_B = np.array([1.0, 0.0, 0.0]) # B's x axis in B
xB_C = quat.rotate_vector(xB_B, q_B_C) # B's x axis in C
print("q_B_C: ")
print(q_B_C)
print("xB_C: ")
print(xB_C)
# 3. Quatrenion and rotational matrix conversion
M_A_B_1 = quat.quat2mat(q_A_B)
M_A_B_2 = np.eye(3)
yA_B = np.array([0.0, -1.0, 0.0])
zA_B = np.array([0.0, 0.0, 1.0])
M_A_B_2[:, 0] = xA_B
M_A_B_2[:, 1] = yA_B
M_A_B_2[:, 2] = zA_B
def egocentric2allocentric(qt, T):
dx = np.arctan2(T[0], -T[2])
dy = np.arctan2(T[1], -T[2])
quat = euler2quat(-dy, -dx, 0, axes='sxyz')
quat = qmult(qinverse(quat), qt)
return quat
def q2q_to_rot(q1, q2):
q1inverted = qinverse(q1)
##hopefully this next line is equivalent to q2*q1' = r where r is the rotation quaternion`
rotation_quaternion = qmult(q2, q1inverted)
rotation_matrix = quat2mat(rotation_quaternion)
return rotation_matrix
q_A_C = quat.axangle2quat(r2_A, -theta2)
print("q_A_B: ")
print(q_A_B)
print("q_A_C: ")
print(q_A_C)
# 2. Vector coordinate transformation
xA_A = np.array([1.0, 0.0, 0.0]) # A's x axis in A
xA_B = quat.rotate_vector(xA_A, q_A_B) # A's x axis in B
xA_C = quat.rotate_vector(xA_A, q_A_C) # A's x axis in C
print("xA_B: ")
print(xA_B)
print("xA_C: ")
print(xA_C)
q_B_C = quat.qmult(q_A_C, quat.qinverse(q_A_B))
xB_B = np.array([1.0, 0.0, 0.0]) # B's x axis in B
xB_C = quat.rotate_vector(xB_B, q_B_C) # B's x axis in C
print("q_B_C: ")
print(q_B_C)
print("xB_C: ")
print(xB_C)
# 3. Quatrenion and rotational matrix conversion
M_A_B_1 = quat.quat2mat(q_A_B)
M_A_B_2 = np.eye(3)
yA_B = np.array([1.0, 0.0, 0.0])
zA_B = np.array([0.0, 0.0, 1.0])
M_A_B_2[:, 0] = xA_B
M_A_B_2[:, 1] = yA_B
M_A_B_2[:, 2] = zA_B
