Repository navigation
Expand file tree
/
Copy pathtwist.py
More file actions
1734 lines (1288 loc) · 50.5 KB
/
Copy pathtwist.py
File metadata and controls
1734 lines (1288 loc) · 50.5 KB
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
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
# Part of Spatial Math Toolbox for Python
# Copyright (c) 2000 Peter Corke
# MIT Licence, see details in top-level file: LICENCE
import numpy as np
from spatialmath.pose3d import SO3, SE3
from spatialmath.pose2d import SE2
from spatialmath.geom3d import Line3
import spatialmath.base as base
from spatialmath.baseposelist import BasePoseList
class BaseTwist(BasePoseList):
"""
Superclass for 3D and 2D twist objects
Subclasses are:
- ``Twist3`` representing rigid-body motion in 3D as a 6-vector
- ``Twist2`` representing rigid-body motion in 2D as a 3-vector
A twist is the unique elements of the logarithm of the corresponding SE(N)
matrix.
Arithmetic operators are overloaded but the operation they perform depend
on the types of the operands. For example:
- ``*`` will compose two instances of the same subclass, and the result will be
an instance of the same subclass, since this is a group operator.
These classes all inherit from ``UserList`` which enables them to
represent a sequence of values, ie. a ``Twist3`` instance can contain
a sequence of twists. Most of the Python ``list`` operators
are applicable:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> x = Twist3() # new instance with zero value
>>> len(x) # it is a sequence of one value
>>> x.append(x) # append to itself
>>> len(x) # it is a sequence of two values
>>> x[1] # the element has a 4x4 matrix value
>>> x[1] = SE3.Rx(0.3).Twist3() # set an elements of the sequence
>>> x.reverse() # reverse the elements in the sequence
>>> del x[1] # delete an element
:References:
- "Mechanics, planning and control"
Park & Lynch, Cambridge, 2016.
This class is subclassed for the 3D and 2D cases
.. inheritance-diagram:: spatialmath.twist.Twist3 spatialmath.twist.Twist2
:top-classes: collections.UserList
:parts: 2
"""
def __init__(self):
super().__init__() # enable UserList superpowers
@property
def S(self):
"""
Twist as a vector (superclass property)
:return: Twist vector
:rtype: ndarray(N)
- ``X.S`` is a 3-vector if X is a ``Twist2`` instance, and a 6-vector if
X is a ``Twist3`` instance.
.. note::
- the vector is the unique elements of the se(N) representation.
- the vector is sometimes referred to as the twist coordinate vector.
- if ``len(X)`` > 1 then return a list of vectors.
"""
# get the underlying numpy array
if len(self.data) == 1:
return self.data[0]
else:
return self.data
@property
def isprismatic(self):
r"""
Test for prismatic twist (superclass property)
:return: Whether twist is purely prismatic
:rtype: bool
A prismatic twist has :math:`\vec{\omega} = 0`.
Example:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> x = Twist3.Prismatic([1,2,3])
>>> x.isprismatic
>>> x = Twist3.Revolute([1,2,3], [4,5,6])
>>> x.isprismatic
"""
if len(self) == 1:
return base.iszerovec(self.w)
else:
return [base.iszerovec(x.w) for x in self.data]
@property
def isrevolute(self):
r"""
Test for revolute twist (superclass property)
:return: Whether twist is purely revolute
:rtype: bool
A revolute twist has :math:`\vec{v} = 0`.
Example:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> x = Twist3.Prismatic([1,2,3])
>>> x.isrevolute
>>> x = Twist3.Revolute([1,2,3], [0,0,0])
>>> x.isrevolute
"""
if len(self) == 1:
return base.iszerovec(self.v)
else:
return [base.iszerovec(x.v) for x in self.data]
@property
def isunit(self):
r"""
Test for unit twist (superclass property)
:return: Whether twist is a unit-twist
:rtype: bool
A unit twist is one with a norm of 1, ie. :math:`\| S \| = 1`.
Example:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> S = Twist3([1,2,3,4,5,6])
>>> S.isunit()
>>> S = Twist3.Revolute([1,2,3], [4,5,6])
>>> S.isunit()
"""
if len(self) == 1:
return base.isunitvec(self.S)
else:
return [base.isunitvec(x) for x in self.data]
@property
def theta(self):
"""
Twist angle (superclass method)
:return: magnitude of rotation (1x1) about the twist axis in radians
:rtype: float
"""
if self.N == 2:
return abs(self.w)
else:
return base.norm(np.array(self.w))
def inv(self):
"""
Inverse of Twist (superclass method)
:return: inverse
:rtype: Twist instance
Compute the inverse of each of the values within the twist instance.
The inverse is the negative of the twist vector.
Example:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> S = Twist3(SE3.Rand())
>>> S
>>> S.inv()
>>> S * S.inv()
"""
return self.__class__([-t for t in self.data])
def prod(self):
r"""
Product of twists (superclass method)
:return: Product of elements
:rtype: Twist2 or Twist3
For a twist instance with N values return the matrix product of those
elements :math:`\prod_i=0^{N-1} S_i`.
Example:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> S = Twist3.Rx([0.2, 0.3, 0.4])
>>> len(S)
>>> S.prod()
>>> Twist3.Rx(0.9)
"""
if self.N == 2:
log = base.trlog2
exp = base.trexp2
else:
log = base.trlog
exp = base.trexp
twprod = exp(self.data[0])
for tw in self.data[1:]:
twprod = twprod @ exp(tw)
return self.__class__(log(twprod))
def __eq__(left, right): # lgtm[py/not-named-self] pylint: disable=no-self-argument
"""
Overloaded ``==`` operator (superclass method)
:return: Equality of two operands
:rtype: bool or list of bool
``S1 == S2`` is True if ``S1` is elementwise equal to ``S2``.
Example:
.. runblock:: pycon
>>> from spatialmath import Twist2
>>> S1 = Twist3([1,2,3,4,5,6])
>>> S2 = Twist3([1,2,3,4,5,6])
>>> S1 == S2
>>> S2 = Twist3([1,2,3,4,5,7])
>>> S1 == S2
:seealso: :func:`__ne__`
"""
if type(left) != type(right):
raise TypeError('operands to == are of different types')
return left.binop(right, lambda x, y: all(x == y), list1=False)
def __ne__(left, right): # lgtm[py/not-named-self] pylint: disable=no-self-argument
"""
Overloaded ``!=`` operator (superclass method)
:rtype: bool
``S1 == S2`` is True if ``S1` is not elementwise equal to ``S2``.
Example:
.. runblock:: pycon
>>> from spatialmath import Twist2
>>> S1 = Twist([1,2,3,4,5,6])
>>> S2 = Twist([1,2,3,4,5,6])
>>> S1 != S2
>>> S2 = Twist([1,2,3,4,5,7])
>>> S1 != S2
:seealso: :func:`__ne__`
"""
if type(left) != type(right):
raise TypeError('operands to != are of different types')
return left.binop(right, lambda x, y: not all(x == y), list1=False)
def __truediv__(left, right): # lgtm[py/not-named-self] pylint: disable=no-self-argument
if base.isscalar(right):
return type(left)(left.S / right)
else:
raise ValueError('Twist /, incorrect right operand')
# ======================================================================== #
class Twist3(BaseTwist):
r"""
3D twist class
A Twist class holds the parameters of a twist, a representation of a
3D rigid body transformation which is the unique elements of the Lie
algebra se(3) of the corresponding SE(3) matrix.
:References:
- **Robotics, Vision & Control**, Corke, Springer 2017.
- **Modern Robotics, Lynch & Park**, Cambridge 2017
.. note:: Compared to Lynch & Park this module implements twist vectors
with the translational components first, followed by rotational
components, ie. :math:`[\omega, \vec{v}]`.
"""
def __init__(self, arg=None, w=None, check=True):
"""
Construct a new 3D twist object
- ``Twist3()`` is a Twist3 instance representing null motion -- the
identity twist
- ``Twist3(S)`` is a Twist3 instance from an array-like (6,)
- ``Twist3(v, w)`` is a Twist3 instance from a moment ``v`` (3,) and
direction ``w`` (3,)
- ``Twist3([S1, S2, ... SN])`` where each ``Si`` is a numpy array (6,)
- ``Twist3(X)`` is a Twist3 instance with the same value as ``X``, ie.
a copy
- ``Twist3([X1, X2, ... XN])`` where each Xi is a Twist3 instance, is a
Twist3 instance containing N motions
"""
super().__init__()
if w is None:
# zero or one arguments passed
if super().arghandler(arg, check=check):
return
elif isinstance(arg, SE3):
self.data = [arg.twist().A]
elif w is not None and base.isvector(w, 3) and base.isvector(arg,3):
# Twist(v, w)
self.data = [np.r_[arg, w]]
return
else:
raise ValueError('bad value to Twist constructor')
# ------------------------ SMUserList required ---------------------------#
@staticmethod
def _identity():
return np.zeros((6,))
def _import(self, value, check=True):
if isinstance(value, np.ndarray) and self.isvalid(value, check=check):
if value.shape == (4,4):
# it's an se(3)
return base.vexa(value)
elif value.shape == (6,):
# it's a twist vector
return value
elif base.ishom(value, check=check):
return base.trlog(value, twist=True, check=False)
raise TypeError('bad type passed')
@staticmethod
def isvalid(v, check=True):
"""
Test if matrix is valid twist
:param x: array to test
:type x: ndarray
:return: Whether the value is a 6-vector or a valid 4x4 se(3) element
:rtype: bool
A twist can be represented by a 6-vector or a 4x4 skew symmetric matrix,
for example:
.. runblock:: pycon
>>> from spatialmath import Twist3, base
>>> import numpy as np
>>> Twist3.isvalid([1, 2, 3, 4, 5, 6])
>>> a = base.skewa([1, 2, 3, 4, 5, 6])
>>> a
>>> Twist3.isvalid(a)
>>> Twist3.isvalid(np.random.rand(4,4))
"""
if base.isvector(v, 6):
return True
elif base.ismatrix(v, (4, 4)):
# maybe be an se(3)
if not base.iszerovec(v.diagonal()): # check diagonal is zero
return False
if not base.iszerovec(v[3, :]): # check bottom row is zero
return False
if check and not base.isskew(v[:3, :3]):
# top left 3x3 is skew symmetric
return False
return True
return False
# ------------------------ properties ---------------------------#
@property
def shape(self):
"""
Shape of the object's internal array representation
:return: (6,)
:rtype: tuple
"""
return (6,)
@property
def N(self):
"""
Dimension of the object's group
:return: dimension
:rtype: int
Dimension of the group is 3 for ``Twist3`` and corresponds to the
dimension of the space (3D in this case) to which these
rigid-body motions apply.
Example:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> x = Twist3()
>>> x.N
"""
return 3
@property
def v(self):
"""
Moment vector of twist
:return: Moment vector
:rtype: ndarray(3)
``X.v`` is a 3-vector representing the moment vector of the twist.
Example:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> t = Twist3([1, 2, 3, 4, 5, 6])
>>> t.v
"""
return self.data[0][:3]
@property
def w(self):
"""
Direction vector of twist
:return: Direction vector
:rtype: ndarray(3)
``X.w`` is a 3-vector representing the direction vector of the twist.
Example:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> t = Twist3([1, 2, 3, 4, 5, 6])
>>> t.w
"""
return self.data[0][3:6]
# -------------------- variant constructors ----------------------------#
@classmethod
def UnitRevolute(cls, a, q, pitch=None):
"""
Construct a new 3D rotational unit twist
:param a: Twist axis or line of action
:type a: array_like(3)
:param q: Point on the line of action
:type q: array_like(3)
:param p: pitch, defaults to None
:type p: float, optional
:return: a rotational or helical twist
:rtype: Twist instance
A revolute twist with a line of action in the z-direction and passing
through (1, 2, 0) would be:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> Twist3.Revolute([0, 0, 1], [1, 2, 0])
"""
w = base.unitvec(base.getvector(a, 3))
v = -np.cross(w, base.getvector(q, 3))
if pitch is not None:
v = v + pitch * w
return cls(v, w)
@classmethod
def UnitPrismatic(cls, a):
"""
Construct a new 3D unit prismatic twist
:param a: Twist axis or line of action
:type a: array_like(3)
:return: a prismatic twist
:rtype: Twist instance
A prismatic twist with a line of action in the z-direction would be:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> Twist3.Prismatic([0, 0, 1])
"""
w = np.r_[0, 0, 0]
v = base.unitvec(base.getvector(a, 3))
return cls(v, w)
@classmethod
def Rx(cls, theta, unit='rad'):
"""
Create a new 3D twist for pure rotation about the X-axis
:param θ: rotation angle about X-axis
:type θ: float
:param unit: angular units: 'rad' [default], or 'deg'
:type unit: str
:return: 3D twist vector
:rtype: Twist3 instance
- ``Twist3.Rx(θ)`` is an SE(3) rotation of θ radians about the x-axis
- ``Twist3.Rx(θ, "deg")`` as above but θ is in degrees
If ``θ`` is an array then the result is a sequence of rotations defined
by consecutive elements.
Example:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> Twist3.Rx(0.3)
>>> Twist3.Rx([0.3, 0.4])
:seealso: :func:`~spatialmath.base.transforms3d.trotx`
:SymPy: supported
"""
return cls([np.r_[0,0,0,x,0,0] for x in base.getunit(theta, unit=unit)])
@classmethod
def Ry(cls, theta, unit='rad', t=None):
"""
Create a new 3D twist for pure rotation about the Y-axis
:param θ: rotation angle about X-axis
:type θ: float
:param unit: angular units: 'rad' [default], or 'deg'
:type unit: str
:return: 3D twist vector
:rtype: Twist3 instance
- ``Twist3.Ry(θ)`` is an SO(3) rotation of θ radians about the y-axis
- ``Twist3.Ry(θ, "deg")`` as above but θ is in degrees
If ``θ`` is an array then the result is a sequence of rotations defined
by consecutive elements.
Example:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> Twist3.Ry(0.3)
>>> Twist3.Ry([0.3, 0.4])
:seealso: :func:`~spatialmath.base.transforms3d.troty`
:SymPy: supported
"""
return cls([np.r_[0,0,0,0,x,0] for x in base.getunit(theta, unit=unit)])
@classmethod
def Rz(cls, theta, unit='rad', t=None):
"""
Create a new 3D twist for pure rotation about the Z-axis
:param θ: rotation angle about Z-axis
:type θ: float
:param unit: angular units: 'rad' [default], or 'deg'
:type unit: str
:return: 3D twist vector
:rtype: Twist3 instance
- ``Twist3.Rz(θ)`` is an SO(3) rotation of θ radians about the z-axis
- ``Twist3.Rz(θ, "deg")`` as above but θ is in degrees
If ``θ`` is an array then the result is a sequence of rotations defined
by consecutive elements.
Example:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> Twist3.Rz(0.3)
>>> Twist3.Rz([0.3, 0.4])
:seealso: :func:`~spatialmath.base.transforms3d.trotz`
:SymPy: supported
"""
return cls([np.r_[0,0,0,0,0,x] for x in base.getunit(theta, unit=unit)])
@classmethod
def Tx(cls, x):
"""
Create a new 3D twist for pure translation along the X-axis
:param x: translation distance along the X-axis
:type x: float
:return: 3D twist vector
:rtype: Twist3 instance
`Twist3.Tx(x)` is an se(3) translation of ``x`` along the x-axis
Example:
.. runblock:: pycon
>>> Twist3.Tx(2)
>>> Twist3.Tx([2,3])
:seealso: :func:`~spatialmath.base.transforms3d.transl`
:SymPy: supported
"""
return cls([np.r_[_x,0,0,0,0,0] for _x in base.getvector(x)], check=False)
@classmethod
def Ty(cls, y):
"""
Create a new 3D twist for pure translation along the Y-axis
:param y: translation distance along the Y-axis
:type y: float
:return: 3D twist vector
:rtype: Twist3 instance
`Twist3.Ty(y) is an se(3) translation of ``y`` along the y-axis
Example:
.. runblock:: pycon
>>> Twist3.Ty(2)
>>> Twist3.Ty([2, 3])
:seealso: :func:`~spatialmath.base.transforms3d.transl`
:SymPy: supported
"""
return cls([np.r_[0,_y,0,0,0,0] for _y in base.getvector(y)], check=False)
@classmethod
def Tz(cls, z):
"""
Create a new 3D twist for pure translation along the Z-axis
:param z: translation distance along the Z-axis
:type z: float
:return: 3D twist vector
:rtype: Twist3 instance
`Twist3.Tz(z)` is an se(3) translation of ``z`` along the z-axis
Example:
.. runblock:: pycon
>>> Twist3.Tz(2)
>>> Twist3.Tz([2, 3])
:seealso: :func:`~spatialmath.base.transforms3d.transl`
:SymPy: supported
"""
return cls([np.r_[0,0,_z,0,0,0] for _z in base.getvector(z)], check=False)
@classmethod
def Rand(cls, *, xrange=(-1, 1), yrange=(-1, 1), zrange=(-1, 1), N=1): # pylint: disable=arguments-differ
"""
Create a new random 3D twist
:param xrange: x-axis range [min,max], defaults to [-1, 1]
:type xrange: 2-element sequence, optional
:param yrange: y-axis range [min,max], defaults to [-1, 1]
:type yrange: 2-element sequence, optional
:param zrange: z-axis range [min,max], defaults to [-1, 1]
:type zrange: 2-element sequence, optional
:param N: number of random transforms
:type N: int
:return: SE(3) matrix
:rtype: SE3 instance
Return an SE3 instance with random rotation and translation.
- ``SE3.Rand()`` is a random SE(3) translation.
- ``SE3.Rand(N=N)`` is an SE3 object containing a sequence of N random
poses.
Example:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> Twist3.Rand(N=2)
:seealso: :func:`~spatialmath.quaternions.UnitQuaternion.Rand`
"""
X = np.random.uniform(low=xrange[0], high=xrange[1], size=N) # random values in the range
Y = np.random.uniform(low=yrange[0], high=yrange[1], size=N) # random values in the range
Z = np.random.uniform(low=yrange[0], high=zrange[1], size=N) # random values in the range
R = SO3.Rand(N=N)
def _twist(x, y, z, r):
T = base.transl(x, y, z) @ base.r2t(r.A)
return base.trlog(T, twist=True)
return cls([_twist(x, y, z, r) for (x, y, z, r) in zip(X, Y, Z, R)], check=False)
# ------------------------- methods -------------------------------#
def printline(self, **kwargs):
return self.SE3().printline(**kwargs)
def unit(self):
"""
Unit twist
- ``S.unit()`` is a Twist2 objec3 representing a unit twist aligned with the
Twist ``S``.
Example:
.. runblock:: pycon
>>> from spatialmath import SE3, Twist3
>>> T = SE3(1, 2, 0.3)
>>> S = Twist3(T)
>>> S.unit()
"""
if base.iszerovec(self.w):
# rotational twist
return Twist3(self.S / base.norm(S.w))
else:
# prismatic twist
return Twist3(base.unitvec(self.v), [0, 0, 0])
def ad(self):
"""
Logarithm of adjoint of 3D twist
:return: logarithm of adjoint matrix
:rtype: ndarray(6,6)
``S.ad()`` is the 6x6 logarithm of the adjoint matrix of the
corresponding homogeneous transformation.
For a twist representing motion from frame {B} to {A}, the adjoint will
transform a twist relative to frame {A} to one relative to frame {B}.
Example:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> S = Twist3.Rx(0.3)
>>> S.ad()
.. note:: An alternative approach to computing the adjoint is to exponentiate this 6x6
matrix.
:seealso: :func:`Twist3.Ad`
"""
return np.block([
[base.skew(self.w), base.skew(self.v)],
[np.zeros((3, 3)), base.skew(self.w)]
])
def Ad(self):
"""
Adjoint of 3D twist
:return: adjoint matrix
:rtype: ndarray(6,6)
``S.Ad()`` is the 6x6 adjoint matrix of the corresponding
homogeneous transformation.
For a twist representing motion from frame {B} to {A}, the adjoint will
transform a twist relative to frame {A} to one relative to frame {B}.
Example:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> S = Twist3.Rx(0.3)
>>> S.Ad()
.. note:: This method computes the equivalent SE(3) matrix, then the adjoint
of that.
:seealso: :func:`Twist3.ad`, :func:`Twist3.SE3`, :func:`Twist3.exp`
"""
return self.SE3().Ad()
def skewa(self):
"""
Convert 3D twist to se(3)
:return: An se(3) matrix
:rtype: ndarray(4,4)
``X.skewa()`` is the twist as a 4x4 augmented skew-symmetric matrix
belonging to the group se(3). This is the Lie algebra of the
corresponding SE(3) element.
Example:
.. runblock:: pycon
>>> from spatialmath import Twist3, base
>>> S = Twist3.Rx(0.3)
>>> se = S.skewa()
>>> se
>>> base.trexp(se)
"""
if len(self) == 1:
return base.skewa(self.S)
else:
return [base.skewa(x.S) for x in self]
@property
def pitch(self):
"""
Pitch of a 3D twist
:return: the pitch of the twist
:rtype: float
``X.pitch()`` is the pitch of the twist as a scalar in units of distance
per radian.
If we consider the twist as a screw, this is the distance of
translation along the screw axis for a one radian rotation about the
screw axis.
Example:
.. runblock:: pycon
>>> from spatialmath import SE3, Twist3
>>> T = SE3(1, 2, 3) * SE3.Rx(0.3)
>>> S = Twist3(T)
>>> S.pitch
"""
return np.dot(self.w, self.v)
def line(self):
"""
Line of action of 3D twist as a Plucker line
:return: the 3D line of action
:rtype: Line instance
``X.line()`` is a Plucker object representing the line of the twist axis.
Example:
.. runblock:: pycon
>>> from spatialmath import SE3, Twist3
>>> T = SE3(1, 2, 3) * SE3.Rx(0.3)
>>> S = Twist3(T)
>>> S.line()
"""
return Line3([Line3(-tw.v - tw.pitch * tw.w, tw.w) for tw in self])
@property
def pole(self):
"""
Pole of a 3D twist
:return: the pole of the twist
:rtype: ndarray(3)
``X.pole()`` is a point on the twist axis. For a pure translation
this point is at infinity.
Example:
.. runblock:: pycon
>>> from spatialmath import SE3, Twist3
>>> T = SE3(1, 2, 3) * SE3.Rx(0.3)
>>> S = Twist3(T)
>>> S.pole
"""
return np.cross(self.w, self.v) / self.theta
def SE3(self, theta=1, unit='rad'):
"""
Convert 3D twist to SE(3) matrix
:return: an SE(3) representation
:rtype: SE3 instance
``S.SE3()`` is an SE3 object representing the homogeneous transformation
equivalent to the Twist3. This is the exponentiation of the twist vector.
Example:
.. runblock:: pycon
>>> from spatialmath import Twist3
>>> S = Twist3.Rx(0.3)
>>> S.SE3()
:seealso: :func:`Twist3.exp`
"""
theta = base.getunit(theta, unit)
if base.isscalar(theta):
# theta is a scalar
return SE3(base.trexp(self.S * theta))
else:
# theta is a vector
if len(self) == 1:
return SE3([base.trexp(self.S * t) for t in theta])
elif len(self) == len(theta):
return SE3([base.trexp(S * t) for S, t in zip(self.data, theta)])
else:
raise ValueError('length of twist and theta not consistent')
return SE3(self.exp(theta))
def exp(self, theta=1, unit='rad'):
"""
Exponentiate a 3D twist
:param theta: rotation magnitude, defaults to None
:type theta: float, optional
:param units: rotational units, defaults to 'rad'
:type units: str, optional
:return: SE(3) matrix
:rtype: SE3 instance
- ``X.exp()`` is the homogeneous transformation equivalent to the twist,
:math:`e^{[S]}`
- ``X.exp(θ) as above but with a rotation of ``θ`` about the twist axis,
:math:`e^{\theta[S]}`
If ``len(X)==1`` and ``len(θ)==N`` then the resulting SE3 object has
``N`` values equivalent to the twist :math:`e^{\theta_i[S]}`.
If ``len(X)==N`` and ``len(θ)==1`` then the resulting SE3 object has
``N`` values equivalent to the twist :math:`e^{\theta[S_i]}`.
If ``len(X)==N`` and ``len(θ)==N`` then the resulting SE3 object has
``N`` values equivalent to the twist :math:`e^{\theta_i[S_i]}`.
Example:
.. runblock:: pycon
>>> from spatialmath import SE3, Twist3
>>> T = SE3(1, 2, 3) * SE3.Rx(0.3)
>>> S = Twist3(T)
>>> S.exp(0)
>>> S.exp(1)
.. notes::
- For the second form, the twist must, if rotational, have a unit
rotational component.
:seealso: :func:`spatialmath.base.trexp`
"""
theta = np.r_[base.getunit(theta, unit)]