File size: 23,752 Bytes
823807d |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 |
import numpy as np
import scipy.linalg as linalg
from visualization import Animation
from visualization import AnimationStructure
from visualization.Quaternions import Quaternions
class BasicInverseKinematics:
"""
Basic Inverse Kinematics Solver
This is an extremely simple full body IK
solver.
It works given the following conditions:
* All joint targets must be specified
* All joint targets must be in reach
* All joint targets must not differ
extremely from the starting pose
* No bone length constraints can be violated
* The root translation and rotation are
set to good initial values
It works under the observation that if the
_directions_ the joints are pointing toward
match the _directions_ of the vectors between
the target joints then the pose should match
that of the target pose.
Therefore it iterates over joints rotating
each joint such that the vectors between it
and it's children match that of the target
positions.
Parameters
----------
animation : Animation
animation input
positions : (F, J, 3) ndarray
target positions for each frame F
and each joint J
iterations : int
Optional number of iterations.
If the above conditions are met
1 iteration should be enough,
therefore the default is 1
silent : bool
Optional if to suppress output
defaults to False
"""
def __init__(self, animation, positions, iterations=1, silent=True):
self.animation = animation
self.positions = positions
self.iterations = iterations
self.silent = silent
def __call__(self):
children = AnimationStructure.children_list(self.animation.parents)
for i in range(self.iterations):
for j in AnimationStructure.joints(self.animation.parents):
c = np.array(children[j])
if len(c) == 0: continue
anim_transforms = Animation.transforms_global(self.animation)
anim_positions = anim_transforms[:, :, :3, 3]
anim_rotations = Quaternions.from_transforms(anim_transforms)
jdirs = anim_positions[:, c] - anim_positions[:, np.newaxis, j]
ddirs = self.positions[:, c] - anim_positions[:, np.newaxis, j]
jsums = np.sqrt(np.sum(jdirs ** 2.0, axis=-1)) + 1e-10
dsums = np.sqrt(np.sum(ddirs ** 2.0, axis=-1)) + 1e-10
jdirs = jdirs / jsums[:, :, np.newaxis]
ddirs = ddirs / dsums[:, :, np.newaxis]
angles = np.arccos(np.sum(jdirs * ddirs, axis=2).clip(-1, 1))
axises = np.cross(jdirs, ddirs)
axises = -anim_rotations[:, j, np.newaxis] * axises
rotations = Quaternions.from_angle_axis(angles, axises)
if rotations.shape[1] == 1:
averages = rotations[:, 0]
else:
averages = Quaternions.exp(rotations.log().mean(axis=-2))
self.animation.rotations[:, j] = self.animation.rotations[:, j] * averages
if not self.silent:
anim_positions = Animation.positions_global(self.animation)
error = np.mean(np.sum((anim_positions - self.positions) ** 2.0, axis=-1) ** 0.5)
print('[BasicInverseKinematics] Iteration %i Error: %f' % (i + 1, error))
return self.animation
class JacobianInverseKinematics:
"""
Jacobian Based Full Body IK Solver
This is a full body IK solver which
uses the dampened least squares inverse
jacobian method.
It should remain fairly stable and effective
even for joint positions which are out of
reach and it can also take any number of targets
to treat as end effectors.
Parameters
----------
animation : Animation
animation to solve inverse problem on
targets : {int : (F, 3) ndarray}
Dictionary of target positions for each
frame F, mapping joint index to
a target position
references : (F, 3)
Optional list of J joint position
references for which the result
should bias toward
iterations : int
Optional number of iterations to
compute. More iterations results in
better accuracy but takes longer to
compute. Default is 10.
recalculate : bool
Optional if to recalcuate jacobian
each iteration. Gives better accuracy
but slower to compute. Defaults to True
damping : float
Optional damping constant. Higher
damping increases stability but
requires more iterations to converge.
Defaults to 5.0
secondary : float
Force, or bias toward secondary target.
Defaults to 0.25
silent : bool
Optional if to suppress output
defaults to False
"""
def __init__(self, animation, targets,
references=None, iterations=10,
recalculate=True, damping=2.0,
secondary=0.25, translate=False,
silent=False, weights=None,
weights_translate=None):
self.animation = animation
self.targets = targets
self.references = references
self.iterations = iterations
self.recalculate = recalculate
self.damping = damping
self.secondary = secondary
self.translate = translate
self.silent = silent
self.weights = weights
self.weights_translate = weights_translate
def cross(self, a, b):
o = np.empty(b.shape)
o[..., 0] = a[..., 1] * b[..., 2] - a[..., 2] * b[..., 1]
o[..., 1] = a[..., 2] * b[..., 0] - a[..., 0] * b[..., 2]
o[..., 2] = a[..., 0] * b[..., 1] - a[..., 1] * b[..., 0]
return o
def jacobian(self, x, fp, fr, ts, dsc, tdsc):
""" Find parent rotations """
prs = fr[:, self.animation.parents]
prs[:, 0] = Quaternions.id((1))
""" Find global positions of target joints """
tps = fp[:, np.array(list(ts.keys()))]
""" Get partial rotations """
qys = Quaternions.from_angle_axis(x[:, 1:prs.shape[1] * 3:3], np.array([[[0, 1, 0]]]))
qzs = Quaternions.from_angle_axis(x[:, 2:prs.shape[1] * 3:3], np.array([[[0, 0, 1]]]))
""" Find axis of rotations """
es = np.empty((len(x), fr.shape[1] * 3, 3))
es[:, 0::3] = ((prs * qzs) * qys) * np.array([[[1, 0, 0]]])
es[:, 1::3] = ((prs * qzs) * np.array([[[0, 1, 0]]]))
es[:, 2::3] = ((prs * np.array([[[0, 0, 1]]])))
""" Construct Jacobian """
j = fp.repeat(3, axis=1)
j = dsc[np.newaxis, :, :, np.newaxis] * (tps[:, np.newaxis, :] - j[:, :, np.newaxis])
j = self.cross(es[:, :, np.newaxis, :], j)
j = np.swapaxes(j.reshape((len(x), fr.shape[1] * 3, len(ts) * 3)), 1, 2)
if self.translate:
es = np.empty((len(x), fr.shape[1] * 3, 3))
es[:, 0::3] = prs * np.array([[[1, 0, 0]]])
es[:, 1::3] = prs * np.array([[[0, 1, 0]]])
es[:, 2::3] = prs * np.array([[[0, 0, 1]]])
jt = tdsc[np.newaxis, :, :, np.newaxis] * es[:, :, np.newaxis, :].repeat(tps.shape[1], axis=2)
jt = np.swapaxes(jt.reshape((len(x), fr.shape[1] * 3, len(ts) * 3)), 1, 2)
j = np.concatenate([j, jt], axis=-1)
return j
# @profile(immediate=True)
def __call__(self, descendants=None, gamma=1.0):
self.descendants = descendants
""" Calculate Masses """
if self.weights is None:
self.weights = np.ones(self.animation.shape[1])
if self.weights_translate is None:
self.weights_translate = np.ones(self.animation.shape[1])
""" Calculate Descendants """
if self.descendants is None:
self.descendants = AnimationStructure.descendants_mask(self.animation.parents)
self.tdescendants = np.eye(self.animation.shape[1]) + self.descendants
self.first_descendants = self.descendants[:, np.array(list(self.targets.keys()))].repeat(3, axis=0).astype(int)
self.first_tdescendants = self.tdescendants[:, np.array(list(self.targets.keys()))].repeat(3, axis=0).astype(
int)
""" Calculate End Effectors """
self.endeff = np.array(list(self.targets.values()))
self.endeff = np.swapaxes(self.endeff, 0, 1)
if not self.references is None:
self.second_descendants = self.descendants.repeat(3, axis=0).astype(int)
self.second_tdescendants = self.tdescendants.repeat(3, axis=0).astype(int)
self.second_targets = dict([(i, self.references[:, i]) for i in range(self.references.shape[1])])
nf = len(self.animation)
nj = self.animation.shape[1]
if not self.silent:
gp = Animation.positions_global(self.animation)
gp = gp[:, np.array(list(self.targets.keys()))]
error = np.mean(np.sqrt(np.sum((self.endeff - gp) ** 2.0, axis=2)))
print('[JacobianInverseKinematics] Start | Error: %f' % error)
for i in range(self.iterations):
""" Get Global Rotations & Positions """
gt = Animation.transforms_global(self.animation)
gp = gt[:, :, :, 3]
gp = gp[:, :, :3] / gp[:, :, 3, np.newaxis]
gr = Quaternions.from_transforms(gt)
x = self.animation.rotations.euler().reshape(nf, -1)
w = self.weights.repeat(3)
if self.translate:
x = np.hstack([x, self.animation.positions.reshape(nf, -1)])
w = np.hstack([w, self.weights_translate.repeat(3)])
""" Generate Jacobian """
if self.recalculate or i == 0:
j = self.jacobian(x, gp, gr, self.targets, self.first_descendants, self.first_tdescendants)
""" Update Variables """
l = self.damping * (1.0 / (w + 0.001))
d = (l * l) * np.eye(x.shape[1])
e = gamma * (self.endeff.reshape(nf, -1) - gp[:, np.array(list(self.targets.keys()))].reshape(nf, -1))
x += np.array(list(map(lambda jf, ef:
linalg.lu_solve(linalg.lu_factor(jf.T.dot(jf) + d), jf.T.dot(ef)), j, e)))
""" Generate Secondary Jacobian """
if self.references is not None:
ns = np.array(list(map(lambda jf:
np.eye(x.shape[1]) - linalg.solve(jf.T.dot(jf) + d, jf.T.dot(jf)), j)))
if self.recalculate or i == 0:
j2 = self.jacobian(x, gp, gr, self.second_targets, self.second_descendants,
self.second_tdescendants)
e2 = self.secondary * (self.references.reshape(nf, -1) - gp.reshape(nf, -1))
x += np.array(list(map(lambda nsf, j2f, e2f:
nsf.dot(linalg.lu_solve(linalg.lu_factor(j2f.T.dot(j2f) + d), j2f.T.dot(e2f))),
ns, j2, e2)))
""" Set Back Rotations / Translations """
self.animation.rotations = Quaternions.from_euler(
x[:, :nj * 3].reshape((nf, nj, 3)), order='xyz', world=True)
if self.translate:
self.animation.positions = x[:, nj * 3:].reshape((nf, nj, 3))
""" Generate Error """
if not self.silent:
gp = Animation.positions_global(self.animation)
gp = gp[:, np.array(list(self.targets.keys()))]
error = np.mean(np.sum((self.endeff - gp) ** 2.0, axis=2) ** 0.5)
print('[JacobianInverseKinematics] Iteration %i | Error: %f' % (i + 1, error))
return self.animation
class BasicJacobianIK:
"""
Same interface as BasicInverseKinematics
but uses the Jacobian IK Solver Instead
"""
def __init__(self, animation, positions, iterations=10, silent=True, **kw):
targets = dict([(i, positions[:, i]) for i in range(positions.shape[1])])
self.ik = JacobianInverseKinematics(animation, targets, iterations=iterations, silent=silent, **kw)
def __call__(self, **kw):
return self.ik(**kw)
class ICP:
def __init__(self,
anim, rest, weights, mesh, goal,
find_closest=True, damping=10,
iterations=10, silent=True,
translate=True, recalculate=True,
weights_translate=None):
self.animation = anim
self.rest = rest
self.vweights = weights
self.mesh = mesh
self.goal = goal
self.find_closest = find_closest
self.iterations = iterations
self.silent = silent
self.translate = translate
self.damping = damping
self.weights = None
self.weights_translate = weights_translate
self.recalculate = recalculate
def cross(self, a, b):
o = np.empty(b.shape)
o[..., 0] = a[..., 1] * b[..., 2] - a[..., 2] * b[..., 1]
o[..., 1] = a[..., 2] * b[..., 0] - a[..., 0] * b[..., 2]
o[..., 2] = a[..., 0] * b[..., 1] - a[..., 1] * b[..., 0]
return o
def jacobian(self, x, fp, fr, goal, weights, des_r, des_t):
""" Find parent rotations """
prs = fr[:, self.animation.parents]
prs[:, 0] = Quaternions.id((1))
""" Get partial rotations """
qys = Quaternions.from_angle_axis(x[:, 1:prs.shape[1] * 3:3], np.array([[[0, 1, 0]]]))
qzs = Quaternions.from_angle_axis(x[:, 2:prs.shape[1] * 3:3], np.array([[[0, 0, 1]]]))
""" Find axis of rotations """
es = np.empty((len(x), fr.shape[1] * 3, 3))
es[:, 0::3] = ((prs * qzs) * qys) * np.array([[[1, 0, 0]]])
es[:, 1::3] = ((prs * qzs) * np.array([[[0, 1, 0]]]))
es[:, 2::3] = ((prs * np.array([[[0, 0, 1]]])))
""" Construct Jacobian """
j = fp.repeat(3, axis=1)
j = des_r[np.newaxis, :, :, :, np.newaxis] * (
goal[:, np.newaxis, :, np.newaxis] - j[:, :, np.newaxis, np.newaxis])
j = np.sum(j * weights[np.newaxis, np.newaxis, :, :, np.newaxis], 3)
j = self.cross(es[:, :, np.newaxis, :], j)
j = np.swapaxes(j.reshape((len(x), fr.shape[1] * 3, goal.shape[1] * 3)), 1, 2)
if self.translate:
es = np.empty((len(x), fr.shape[1] * 3, 3))
es[:, 0::3] = prs * np.array([[[1, 0, 0]]])
es[:, 1::3] = prs * np.array([[[0, 1, 0]]])
es[:, 2::3] = prs * np.array([[[0, 0, 1]]])
jt = des_t[np.newaxis, :, :, :, np.newaxis] * es[:, :, np.newaxis, np.newaxis, :].repeat(goal.shape[1],
axis=2)
jt = np.sum(jt * weights[np.newaxis, np.newaxis, :, :, np.newaxis], 3)
jt = np.swapaxes(jt.reshape((len(x), fr.shape[1] * 3, goal.shape[1] * 3)), 1, 2)
j = np.concatenate([j, jt], axis=-1)
return j
# @profile(immediate=True)
def __call__(self, descendants=None, maxjoints=4, gamma=1.0, transpose=False):
""" Calculate Masses """
if self.weights is None:
self.weights = np.ones(self.animation.shape[1])
if self.weights_translate is None:
self.weights_translate = np.ones(self.animation.shape[1])
nf = len(self.animation)
nj = self.animation.shape[1]
nv = self.goal.shape[1]
weightids = np.argsort(-self.vweights, axis=1)[:, :maxjoints]
weightvls = np.array(list(map(lambda w, i: w[i], self.vweights, weightids)))
weightvls = weightvls / weightvls.sum(axis=1)[..., np.newaxis]
if descendants is None:
self.descendants = AnimationStructure.descendants_mask(self.animation.parents)
else:
self.descendants = descendants
des_r = np.eye(nj) + self.descendants
des_r = des_r[:, weightids].repeat(3, axis=0)
des_t = np.eye(nj) + self.descendants
des_t = des_t[:, weightids].repeat(3, axis=0)
if not self.silent:
curr = Animation.skin(self.animation, self.rest, self.vweights, self.mesh, maxjoints=maxjoints)
error = np.mean(np.sqrt(np.sum((curr - self.goal) ** 2.0, axis=-1)))
print('[ICP] Start | Error: %f' % error)
for i in range(self.iterations):
""" Get Global Rotations & Positions """
gt = Animation.transforms_global(self.animation)
gp = gt[:, :, :, 3]
gp = gp[:, :, :3] / gp[:, :, 3, np.newaxis]
gr = Quaternions.from_transforms(gt)
x = self.animation.rotations.euler().reshape(nf, -1)
w = self.weights.repeat(3)
if self.translate:
x = np.hstack([x, self.animation.positions.reshape(nf, -1)])
w = np.hstack([w, self.weights_translate.repeat(3)])
""" Get Current State """
curr = Animation.skin(self.animation, self.rest, self.vweights, self.mesh, maxjoints=maxjoints)
""" Find Cloest Points """
if self.find_closest:
mapping = np.argmin(
(curr[:, :, np.newaxis] -
self.goal[:, np.newaxis, :]) ** 2.0, axis=2)
e = gamma * (np.array(list(map(lambda g, m: g[m], self.goal, mapping))) - curr).reshape(nf, -1)
else:
e = gamma * (self.goal - curr).reshape(nf, -1)
""" Generate Jacobian """
if self.recalculate or i == 0:
j = self.jacobian(x, gp, gr, self.goal, weightvls, des_r, des_t)
""" Update Variables """
l = self.damping * (1.0 / (w + 1e-10))
d = (l * l) * np.eye(x.shape[1])
if transpose:
x += np.array(list(map(lambda jf, ef: jf.T.dot(ef), j, e)))
else:
x += np.array(list(map(lambda jf, ef:
linalg.lu_solve(linalg.lu_factor(jf.T.dot(jf) + d), jf.T.dot(ef)), j, e)))
""" Set Back Rotations / Translations """
self.animation.rotations = Quaternions.from_euler(
x[:, :nj * 3].reshape((nf, nj, 3)), order='xyz', world=True)
if self.translate:
self.animation.positions = x[:, nj * 3:].reshape((nf, nj, 3))
if not self.silent:
curr = Animation.skin(self.animation, self.rest, self.vweights, self.mesh)
error = np.mean(np.sqrt(np.sum((curr - self.goal) ** 2.0, axis=-1)))
print('[ICP] Iteration %i | Error: %f' % (i + 1, error))
import torch
from torch import nn
class InverseKinematics:
def __init__(self, rotations: torch.Tensor, positions: torch.Tensor, offset, parents, constrains):
self.rotations = rotations.cuda()
self.rotations.requires_grad_(True)
self.position = positions.cuda()
self.position.requires_grad_(True)
self.parents = parents
self.offset = offset.cuda()
self.constrains = constrains.cuda()
# hyper-param to tune
self.optimizer = torch.optim.AdamW([self.position, self.rotations], lr=5e-2, betas=(0.9, 0.999))
self.crit = nn.MSELoss()
self.weights = torch.ones([1,22,1]).cuda()
self.weights[:, [4, 8]] = 0.8
self.weights[:, [1, 5]] = 2.
def step(self):
self.optimizer.zero_grad()
glb = self.forward(self.rotations, self.position, self.offset, order='', quater=True, world=True)
# weighted joint position mse
loss = self.crit(glb*self.weights, self.constrains*self.weights)
# regularization term
loss += 0.5 * self.crit(self.rotations[1:, [3, 7, 12, 16, 20]], self.rotations[:-1, [3, 7, 12, 16, 20]]) + 0.1 * self.crit(self.rotations[1:], self.rotations[:-1])
loss.backward()
self.optimizer.step()
self.glb = glb
return loss.item()
def tloss(self, time):
return self.crit(self.glb[time, :], self.constrains[time, :])
def all_loss(self):
res = [self.tloss(t).detach().numpy() for t in range(self.constrains.shape[0])]
return np.array(res)
'''
rotation should have shape batch_size * Joint_num * (3/4) * Time
position should have shape batch_size * 3 * Time
offset should have shape batch_size * Joint_num * 3
output have shape batch_size * Time * Joint_num * 3
'''
def forward(self, rotation: torch.Tensor, position: torch.Tensor, offset: torch.Tensor, order='xyz', quater=False,
world=True):
'''
if not quater and rotation.shape[-2] != 3: raise Exception('Unexpected shape of rotation')
if quater and rotation.shape[-2] != 4: raise Exception('Unexpected shape of rotation')
rotation = rotation.permute(0, 3, 1, 2)
position = position.permute(0, 2, 1)
'''
result = torch.empty(rotation.shape[:-1] + (3,), device=position.device)
norm = torch.norm(rotation, dim=-1, keepdim=True)
rotation = rotation / norm
# if quater:
transform = self.transform_from_quaternion(rotation)
# else:
# transform = self.transform_from_euler(rotation, order)
offset = offset.reshape((-1, 1, offset.shape[-2], offset.shape[-1], 1))
result[..., 0, :] = position
for i, pi in enumerate(self.parents):
if pi == -1:
assert i == 0
continue
result[..., i, :] = torch.matmul(transform[..., pi, :, :], offset[..., i, :, :]).squeeze()
transform[..., i, :, :] = torch.matmul(transform[..., pi, :, :].clone(), transform[..., i, :, :].clone())
if world: result[..., i, :] += result[..., pi, :]
return result
@staticmethod
def transform_from_axis(euler, axis):
transform = torch.empty(euler.shape[0:3] + (3, 3), device=euler.device)
cos = torch.cos(euler)
sin = torch.sin(euler)
cord = ord(axis) - ord('x')
transform[..., cord, :] = transform[..., :, cord] = 0
transform[..., cord, cord] = 1
if axis == 'x':
transform[..., 1, 1] = transform[..., 2, 2] = cos
transform[..., 1, 2] = -sin
transform[..., 2, 1] = sin
if axis == 'y':
transform[..., 0, 0] = transform[..., 2, 2] = cos
transform[..., 0, 2] = sin
transform[..., 2, 0] = -sin
if axis == 'z':
transform[..., 0, 0] = transform[..., 1, 1] = cos
transform[..., 0, 1] = -sin
transform[..., 1, 0] = sin
return transform
@staticmethod
def transform_from_quaternion(quater: torch.Tensor):
qw = quater[..., 0]
qx = quater[..., 1]
qy = quater[..., 2]
qz = quater[..., 3]
x2 = qx + qx
y2 = qy + qy
z2 = qz + qz
xx = qx * x2
yy = qy * y2
wx = qw * x2
xy = qx * y2
yz = qy * z2
wy = qw * y2
xz = qx * z2
zz = qz * z2
wz = qw * z2
m = torch.empty(quater.shape[:-1] + (3, 3), device=quater.device)
m[..., 0, 0] = 1.0 - (yy + zz)
m[..., 0, 1] = xy - wz
m[..., 0, 2] = xz + wy
m[..., 1, 0] = xy + wz
m[..., 1, 1] = 1.0 - (xx + zz)
m[..., 1, 2] = yz - wx
m[..., 2, 0] = xz - wy
m[..., 2, 1] = yz + wx
m[..., 2, 2] = 1.0 - (xx + yy)
return m |