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Check the section C-API incompatibility at the Troubleshooting ImportError section at https://numpy.org/devdocs/user/troubleshooting-importerror.html#c-api-incompatibility for indications on how to solve this problem.FATAL: module compiled as unknown endianFATAL: module compiled as little endian, but detected different endianness at runtime'%.200s' object is unsliceablescipy.spatial.transform._rotation._empty1scipy.spatial.transform._rotation._cross3Acquisition count is %d (line %d)scipy.spatial.transform._rotation._compose_quat_singlescipy.spatial.transform._rotation._empty2scipy.spatial.transform._rotation._get_anglesBuffer acquisition failed on assignment; and then reacquiring the old buffer failed too!scipy.spatial.transform._rotation._compute_euler_from_matrixscipy.spatial.transform._rotation._compute_davenport_from_quatscipy.spatial.transform._rotation._compose_quatscipy.spatial.transform._rotation.compose_quat%.200s() takes %.8s %zd positional argument%.1s (%zd given)scipy.spatial.transform._rotation.Rotation.from_quatscipy.spatial.transform._rotation.Rotation.__len__scipy.spatial.transform._rotation._format_anglesscipy.spatial.transform._rotation.Rotation.__init__scipy.spatial.transform._rotation.Rotation.__getstate__too many values to unpack (expected %zd)scipy.spatial.transform._rotation.Rotation.__setstate__need more than %zd value%.1s to unpackscipy.spatial.transform._rotation.Rotation.from_matrixscipy.spatial.transform._rotation._zeros2scipy.spatial.transform._rotation._make_elementary_quatscipy.spatial.transform._rotation.Rotation.from_rotveclocal variable '%s' referenced before assignmentfree variable '%s' referenced before assignment in enclosing scopescipy.spatial.transform._rotation.Rotation.from_mrpscipy.spatial.transform._rotation.Rotation.as_quatscipy.spatial.transform._rotation.Rotation.from_euler.genexprscipy.spatial.transform._rotation._elementary_quat_composescipy.spatial.transform._rotation.Rotation.from_eulerscipy.spatial.transform._rotation.Rotation.from_davenportscipy.spatial.transform._rotation._empty3scipy.spatial.transform._rotation.Rotation.as_matrixOut of bounds on buffer access (axis %d)scipy.spatial.transform._rotation.Rotation.as_mrpscipy.spatial.transform._rotation.Rotation.as_rotvecscipy.spatial.transform._rotation.Rotation._as_euler_from_matrixscipy.spatial.transform._rotation.Rotation.as_eulerscipy.spatial.transform._rotation.Rotation._compute_euler.genexprscipy.spatial.transform._rotation._compute_euler_from_quatscipy.spatial.transform._rotation.Rotation._compute_eulerscipy.spatial.transform._rotation.Rotation.as_davenportscipy.spatial.transform._rotation.Rotation.concatenate.genexprscipy.spatial.transform._rotation.Rotation.concatenatescipy.spatial.transform._rotation.Rotation.applyscipy.spatial.transform._rotation.Rotation.approx_equalscipy.spatial.transform._rotation.Rotation.__mul__scipy.spatial.transform._rotation.Rotation.invCannot convert %.200s to %.200sscipy.spatial.transform._rotation.Rotation.magnitudescipy.spatial.transform._rotation.Rotation.meanscipy.spatial.transform._rotation.Rotation.__pow__scipy.spatial.transform._rotation.Rotation.create_groupscipy.spatial.transform._rotation.Rotation.randomscipy.spatial.transform._rotation.Rotation.reduce.split_rotationscipy.spatial.transform._rotation.Rotation.reducescipy.spatial.transform._rotation.Rotation.__getitem__scipy.spatial.transform._rotation.Rotation.identityscipy.spatial.transform._rotation.Rotation.align_vectorsscipy.spatial.transform._rotation.Rotation.__setitem__scipy.spatial.transform._rotation.Rotation.__repr__'%.200s' object does not support slice %.10sstrings are too large to concatscipy.spatial.transform._rotation.Rotation.__reduce_cython__scipy.spatial.transform._rotation.Rotation.__setstate_cython__scipy.spatial.transform._rotation.Slerp.__init__scipy.spatial.transform._rotation.Slerp.__call__Subscript deletion not supported by %.200sInterpreter change detected - this module can only be loaded into one interpreter per process.'NoneType' object is not subscriptablescipy.spatial.transform._rotation.__pyx_unpickle_Rotation__set_statescipy.spatial.transform._rotation.__pyx_unpickle_Rotationint (struct __pyx_array_obj *)struct __pyx_array_obj *(PyObject *, Py_ssize_t, char *, char const *, char *)PyObject *(PyObject *, int, int, __Pyx_TypeInfo const *)struct __pyx_memoryview_obj *(struct __pyx_memoryview_obj *, PyObject *)int (__Pyx_memviewslice *, Py_ssize_t, Py_ssize_t, Py_ssize_t, int, int, int *, Py_ssize_t, Py_ssize_t, Py_ssize_t, int, int, int, int)char *(Py_buffer *, char *, Py_ssize_t, Py_ssize_t)PyObject *(__Pyx_memviewslice, int, PyObject *(*)(char *), int (*)(char *, PyObject *), int)__Pyx_memviewslice *(struct __pyx_memoryview_obj *, __Pyx_memviewslice *)void (struct __pyx_memoryview_obj *, __Pyx_memviewslice *)PyObject *(struct __pyx_memoryview_obj *)PyObject *(struct __pyx_memoryview_obj *, __Pyx_memviewslice *)char (__Pyx_memviewslice *, int)Py_ssize_t (__Pyx_memviewslice *, int)Py_ssize_t (Py_ssize_t *, Py_ssize_t *, Py_ssize_t, int, char)void *(__Pyx_memviewslice *, __Pyx_memviewslice *, char, int)int (int, Py_ssize_t, Py_ssize_t)int (PyObject *, PyObject *, int)int (__Pyx_memviewslice, __Pyx_memviewslice, int, int, int)void (__Pyx_memviewslice *, int, int)void (__Pyx_memviewslice *, int, int, int)void (char *, Py_ssize_t *, Py_ssize_t *, int, int)void (__Pyx_memviewslice *, int, size_t, void *, int)void (char *, Py_ssize_t *, Py_ssize_t *, int, size_t, void *)PyObject *(__Pyx_TypeInfo const *)__name__ must be set to a string object__qualname__ must be set to a string objectfunction's dictionary may not be deletedsetting function's dictionary to a non-dict__annotations__ must be set to a dict object__defaults__ must be set to a tuple objectchanges to cyfunction.__defaults__ will not currently affect the values used in function calls__kwdefaults__ must be set to a dict objectchanges to cyfunction.__kwdefaults__ will not currently affect the values used in function callsUnexpected format string character: '%c'Buffer dtype mismatch, expected %s%s%s but got %sBuffer dtype mismatch, expected '%s' but got %s in '%s.%s'Expected a dimension of size %zu, got %zuExpected %d dimensions, got %dPython does not define a standard format string size for long double ('g')..Buffer dtype mismatch; next field is at offset %zd but %zd expectedShared Cython type %.200s is not a type objectShared Cython type %.200s has the wrong size, try recompiling%s() got multiple values for keyword argument '%U'%.200s() keywords must be stringsunbound method %.200S() needs an argument while calling a Python objectNULL result without error in PyObject_Callraise: arg 3 must be a traceback or Noneinstance exception may not have a separate valuecalling %R should have returned an instance of BaseException, not %Rraise: exception class must be a subclass of BaseExceptionBig-endian buffer not supported on little-endian compilerBuffer acquisition: Expected '{' after 'T'Cannot handle repeated arrays in format stringDoes not understand character buffer dtype format string ('%c')Expected a dimension of size %zu, got %dExpected a comma in format string, got '%c'Expected %d dimension(s), got %dUnexpected end of format string, expected ')'Buffer has wrong number of dimensions (expected %d, got %d)Item size of buffer (%zd byte%s) does not match size of '%s' (%zd byte%s)cannot fit '%.200s' into an index-sized integer'%.200s' object is not subscriptable%s() got an unexpected keyword argument '%U'join() result is too long for a Python stringgenerator raised StopIterationArgument '%.200s' has incorrect type (expected %.200s, got %.200s)Module '_rotation' has already been imported. Re-initialisation is not supported.scipy.spatial.transform._rotationcompile time Python version %d.%d of module '%.100s' %s runtime version %d.%d../../../../../../usr/lib/python3/dist-packages/numpy/__init__.cython-30.pxdinit scipy.spatial.transform._rotationqualified name of the generatorobject being iterated by 'yield from', or Nonesend(arg) -> send 'arg' into generator, return next yielded value or raise StopIteration.throw(typ[,val[,tb]]) -> raise exception in generator, return next yielded value or raise StopIteration.close() -> raise GeneratorExit inside generator.Whether this instance represents a single rotation._cython_3_1_6._common_types_metatype_cython_3_1_6.cython_function_or_methodscipy.spatial.transform._rotation.__pyx_scope_struct_4_genexprscipy.spatial.transform._rotation.__pyx_scope_struct_3_genexprscipy.spatial.transform._rotation.__pyx_scope_struct_2__compute_eulerscipy.spatial.transform._rotation.__pyx_scope_struct_1_genexprscipy.spatial.transform._rotation.__pyx_scope_struct__from_eulerscipy.spatial.transform._rotation.RotationRotation in 3 dimensions. This class provides an interface to initialize from and represent rotations with: - Quaternions - Rotation Matrices - Rotation Vectors - Modified Rodrigues Parameters - Euler Angles - Davenport Angles (Generalized Euler Angles) The following operations on rotations are supported: - Application on vectors - Rotation Composition - Rotation Inversion - Rotation Indexing Indexing within a rotation is supported since multiple rotation transforms can be stored within a single `Rotation` instance. To create `Rotation` objects use ``from_...`` methods (see examples below). ``Rotation(...)`` is not supposed to be instantiated directly. Attributes ---------- single Methods ------- __len__ from_quat from_matrix from_rotvec from_mrp from_euler from_davenport as_quat as_matrix as_rotvec as_mrp as_euler as_davenport concatenate apply __mul__ __pow__ inv magnitude approx_equal mean reduce create_group __getitem__ identity random align_vectors See Also -------- Slerp Notes ----- .. versionadded:: 1.2.0 Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np A `Rotation` instance can be initialized in any of the above formats and converted to any of the others. The underlying object is independent of the representation used for initialization. Consider a counter-clockwise rotation of 90 degrees about the z-axis. This corresponds to the following quaternion (in scalar-last format): >>> r = R.from_quat([0, 0, np.sin(np.pi/4), np.cos(np.pi/4)]) The rotation can be expressed in any of the other formats: >>> r.as_matrix() array([[ 2.22044605e-16, -1.00000000e+00, 0.00000000e+00], [ 1.00000000e+00, 2.22044605e-16, 0.00000000e+00], [ 0.00000000e+00, 0.00000000e+00, 1.00000000e+00]]) >>> r.as_rotvec() array([0. , 0. , 1.57079633]) >>> r.as_euler('zyx', degrees=True) array([90., 0., 0.]) The same rotation can be initialized using a rotation matrix: >>> r = R.from_matrix([[0, -1, 0], ... [1, 0, 0], ... [0, 0, 1]]) Representation in other formats: >>> r.as_quat() array([0. , 0. , 0.70710678, 0.70710678]) >>> r.as_rotvec() array([0. , 0. , 1.57079633]) >>> r.as_euler('zyx', degrees=True) array([90., 0., 0.]) The rotation vector corresponding to this rotation is given by: >>> r = R.from_rotvec(np.pi/2 * np.array([0, 0, 1])) Representation in other formats: >>> r.as_quat() array([0. , 0. , 0.70710678, 0.70710678]) >>> r.as_matrix() array([[ 2.22044605e-16, -1.00000000e+00, 0.00000000e+00], [ 1.00000000e+00, 2.22044605e-16, 0.00000000e+00], [ 0.00000000e+00, 0.00000000e+00, 1.00000000e+00]]) >>> r.as_euler('zyx', degrees=True) array([90., 0., 0.]) The ``from_euler`` method is quite flexible in the range of input formats it supports. Here we initialize a single rotation about a single axis: >>> r = R.from_euler('z', 90, degrees=True) Again, the object is representation independent and can be converted to any other format: >>> r.as_quat() array([0. , 0. , 0.70710678, 0.70710678]) >>> r.as_matrix() array([[ 2.22044605e-16, -1.00000000e+00, 0.00000000e+00], [ 1.00000000e+00, 2.22044605e-16, 0.00000000e+00], [ 0.00000000e+00, 0.00000000e+00, 1.00000000e+00]]) >>> r.as_rotvec() array([0. , 0. , 1.57079633]) It is also possible to initialize multiple rotations in a single instance using any of the ``from_...`` functions. Here we initialize a stack of 3 rotations using the ``from_euler`` method: >>> r = R.from_euler('zyx', [ ... [90, 0, 0], ... [0, 45, 0], ... [45, 60, 30]], degrees=True) The other representations also now return a stack of 3 rotations. For example: >>> r.as_quat() array([[0. , 0. , 0.70710678, 0.70710678], [0. , 0.38268343, 0. , 0.92387953], [0.39190384, 0.36042341, 0.43967974, 0.72331741]]) Applying the above rotations onto a vector: >>> v = [1, 2, 3] >>> r.apply(v) array([[-2. , 1. , 3. ], [ 2.82842712, 2. , 1.41421356], [ 2.24452282, 0.78093109, 2.89002836]]) A `Rotation` instance can be indexed and sliced as if it were a single 1D array or list: >>> r.as_quat() array([[0. , 0. , 0.70710678, 0.70710678], [0. , 0.38268343, 0. , 0.92387953], [0.39190384, 0.36042341, 0.43967974, 0.72331741]]) >>> p = r[0] >>> p.as_matrix() array([[ 2.22044605e-16, -1.00000000e+00, 0.00000000e+00], [ 1.00000000e+00, 2.22044605e-16, 0.00000000e+00], [ 0.00000000e+00, 0.00000000e+00, 1.00000000e+00]]) >>> q = r[1:3] >>> q.as_quat() array([[0. , 0.38268343, 0. , 0.92387953], [0.39190384, 0.36042341, 0.43967974, 0.72331741]]) In fact it can be converted to numpy.array: >>> r_array = np.asarray(r) >>> r_array.shape (3,) >>> r_array[0].as_matrix() array([[ 2.22044605e-16, -1.00000000e+00, 0.00000000e+00], [ 1.00000000e+00, 2.22044605e-16, 0.00000000e+00], [ 0.00000000e+00, 0.00000000e+00, 1.00000000e+00]]) Multiple rotations can be composed using the ``*`` operator: >>> r1 = R.from_euler('z', 90, degrees=True) >>> r2 = R.from_rotvec([np.pi/4, 0, 0]) >>> v = [1, 2, 3] >>> r2.apply(r1.apply(v)) array([-2. , -1.41421356, 2.82842712]) >>> r3 = r2 * r1 # Note the order >>> r3.apply(v) array([-2. , -1.41421356, 2.82842712]) A rotation can be composed with itself using the ``**`` operator: >>> p = R.from_rotvec([1, 0, 0]) >>> q = p ** 2 >>> q.as_rotvec() array([2., 0., 0.]) Finally, it is also possible to invert rotations: >>> r1 = R.from_euler('z', [90, 45], degrees=True) >>> r2 = r1.inv() >>> r2.as_euler('zyx', degrees=True) array([[-90., 0., 0.], [-45., 0., 0.]]) The following function can be used to plot rotations with Matplotlib by showing how they transform the standard x, y, z coordinate axes: >>> import matplotlib.pyplot as plt >>> def plot_rotated_axes(ax, r, name=None, offset=(0, 0, 0), scale=1): ... colors = ("#FF6666", "#005533", "#1199EE") # Colorblind-safe RGB ... loc = np.array([offset, offset]) ... for i, (axis, c) in enumerate(zip((ax.xaxis, ax.yaxis, ax.zaxis), ... colors)): ... axlabel = axis.axis_name ... axis.set_label_text(axlabel) ... axis.label.set_color(c) ... axis.line.set_color(c) ... axis.set_tick_params(colors=c) ... line = np.zeros((2, 3)) ... line[1, i] = scale ... line_rot = r.apply(line) ... line_plot = line_rot + loc ... ax.plot(line_plot[:, 0], line_plot[:, 1], line_plot[:, 2], c) ... text_loc = line[1]*1.2 ... text_loc_rot = r.apply(text_loc) ... text_plot = text_loc_rot + loc[0] ... ax.text(*text_plot, axlabel.upper(), color=c, ... va="center", ha="center") ... ax.text(*offset, name, color="k", va="center", ha="center", ... bbox={"fc": "w", "alpha": 0.8, "boxstyle": "circle"}) Create three rotations - the identity and two Euler rotations using intrinsic and extrinsic conventions: >>> r0 = R.identity() >>> r1 = R.from_euler("ZYX", [90, -30, 0], degrees=True) # intrinsic >>> r2 = R.from_euler("zyx", [90, -30, 0], degrees=True) # extrinsic Add all three rotations to a single plot: >>> ax = plt.figure().add_subplot(projection="3d", proj_type="ortho") >>> plot_rotated_axes(ax, r0, name="r0", offset=(0, 0, 0)) >>> plot_rotated_axes(ax, r1, name="r1", offset=(3, 0, 0)) >>> plot_rotated_axes(ax, r2, name="r2", offset=(6, 0, 0)) >>> _ = ax.annotate( ... "r0: Identity Rotation\n" ... "r1: Intrinsic Euler Rotation (ZYX)\n" ... "r2: Extrinsic Euler Rotation (zyx)", ... xy=(0.6, 0.7), xycoords="axes fraction", ha="left" ... ) >>> ax.set(xlim=(-1.25, 7.25), ylim=(-1.25, 1.25), zlim=(-1.25, 1.25)) >>> ax.set(xticks=range(-1, 8), yticks=[-1, 0, 1], zticks=[-1, 0, 1]) >>> ax.set_aspect("equal", adjustable="box") >>> ax.figure.set_size_inches(6, 5) >>> plt.tight_layout() Show the plot: >>> plt.show() These examples serve as an overview into the `Rotation` class and highlight major functionalities. For more thorough examples of the range of input and output formats supported, consult the individual method's examples. __int__ returned non-int (type %.200s). 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Expected %zd from C header, got %zd from PyObjectinvalid vtable found for imported typememviewslice is already initialized!Item size of buffer (%zu byte%s) does not match size of '%s' (%zu byte%s)Buffer is not indirectly contiguous in dimension %d.Buffer and memoryview are not contiguous in the same dimension.C-contiguous buffer is not contiguous in dimension %dC-contiguous buffer is not indirect in dimension %dBuffer exposes suboffsets but no stridesBuffer not compatible with direct access in dimension %d.Buffer is not indirectly accessible in dimension %d.value too large to convert to intgenerator ignored GeneratorExit%.200s does not export expected C function %.200sC function %.200s.%.200s has wrong signature (expected %.500s, got %.500s)?Hz>-DT! @-DT! -DT!@-DT!-DT!??@@@MbP?H@@@@(@:0yE>-q=ؗҜ<@??BB~BgBPB00010203040506070809101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899zzzztz?~h``````````````````````````````````````````````````````````````/ ``````` ```````````````````' ` `````px`p@@-@+3555555555055055555555555555555503555555355555555555555555|350n33zwzetazerosz2zyzywy2yxzxyz,ix,jy,kz^[xyz]{1,3}$xyxwx2xwhereweights_sec`weights` must be non-negative.`weights` may not contain negative valuesweights_inf_zeroweightsweight_is_infwarningswarnw2wvhvectorsvalue must be a Rotation objectvaluevuse_setstateupdateutranspose_transition_to_rngtolerance`times` must be at most 1-dimensional.timestimedeltathrowtheta2theta__test__swapaxessvdsumstatestacklevelsqrtsplit_rotation__spec__sizesingle_vectorsingle_timesinglesin_termsinsignshape__setstate_cython____setstate____set_name__seqsensitivitysendselfsearchsortedscipy/spatial/transform/_rotation.pyxscipy.spatial.transform._rotationscipy._lib._utilscalescalar_firstsamplesrvrssdrsrotvecsrotvec`rotations` must be a sequence of at least 2 rotations.`rotations` must be a `Rotation` instance.rotations_rotation_groupsrollrngright_bestrightreturn_sensitivityreturn_indicesretresultreshaperepeatreducedreduce..split_rotation__reduce_ex____reduce_cython__reduce__reduce__rerangerandom_staterandomrad2degrquatsquat__qualname__qsqiq__pyx_vtable____pyx_unpickle_Rotation__pyx_type__pyx_state__pyx_result__pyx_checksum__pyx_PickleErrorpvps__prepare__position_numpoppicklepiphipotherorder should be 'e'/'extrinsic' for extrinsic sequences or 'i'/'intrinsic' for intrinsic sequences, got {}orderonesnumpy._core.umath failed to importnumpy._core.multiarray failed to importnumpynum_rotationsnum_axesnumnpnormalizenormalnormnextnewaxis__new__ndim__name__n_vectorsn_rotationsn_infn3n2n1mrpsmrp_squared_plus_1mrpmoveaxismodulus not supported__module____metaclass__meanmax_indmatrixmatmulmatch__main__magnitudelvlslowerlogical_orlinalgleft_bestleftkappakji,jk->ikjix,jx,kix,j,kxitemsizeisposinfisenablediscloseis_singleis_orthogonal_is_coroutineinverseinvalidinvintrinsicinput must contain Rotation objects only_initializing__init__indices_to_orthogonalizeindijk,ik->ijij,ij->iignoreidentityi,jx,kxi,j,kigroupgramians__getstate__genexprgc__func__from_rotvecfrom_quatfrom_mrpfrom_matrixfrom_euler..genexprfrom_eulerfrom_davenport_format_anglesformateyeextrinsic__exit__errstate__enter__encodeenableeinsumeighedtypedot__doc__dividedisablediff_dict__dict__detsdetdenominatordegreesdeg2raddecisiondcrotveccross_normcrosscreate_groupcos_termcopyconcatenate..genexprconcatenatecompute_times_compute_euler..genexpr_compute_eulercompose_quatcmrpcmatrixclsclosecline_in_traceback__class_getitem____class__choicecheck_random_statecanonical__call__c_secb_secb_pri_normb_prib_originalbaxisaxesatol must be set to use the degrees flag, defaulting to 1e-8 radians.atolatleast_2datleast_1dasyncio.coroutinesasarrayas_rotvecas_quatas_mrpas_matrix_as_euler_from_matrixas_euleras_davenportarrayargminargmaxapprox_equalapplyanyanglesangle2anglealphaallalign_vectorsalgorithmadd_noteabsa_seca_pri_norma_pria_originala_esta_?) : Z^[XYZ]{1,3}$VtValueErrorUTypeErrorTimes must be in strictly increasing order.TSpherical Linear Interpolation of Rotations. The interpolation between consecutive rotations is performed as a rotation around a fixed axis with a constant angular velocity [1]_. This ensures that the interpolated rotations follow the shortest path between initial and final orientations. Parameters ---------- times : array_like, shape (N,) Times of the known rotations. At least 2 times must be specified. rotations : `Rotation` instance Rotations to perform the interpolation between. Must contain N rotations. Methods ------- __call__ See Also -------- Rotation Notes ----- .. versionadded:: 1.2.0 References ---------- .. [1] https://en.wikipedia.org/wiki/Slerp#Quaternion_Slerp Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> from scipy.spatial.transform import Slerp Setup the fixed keyframe rotations and times: >>> key_rots = R.random(5, random_state=2342345) >>> key_times = [0, 1, 2, 3, 4] Create the interpolator object: >>> slerp = Slerp(key_times, key_rots) Interpolate the rotations at the given times: >>> times = [0, 0.5, 0.25, 1, 1.5, 2, 2.75, 3, 3.25, 3.60, 4] >>> interp_rots = slerp(times) The keyframe rotations expressed as Euler angles: >>> key_rots.as_euler('xyz', degrees=True) array([[ 14.31443779, -27.50095894, -3.7275787 ], [ -1.79924227, -24.69421529, 164.57701743], [146.15020772, 43.22849451, -31.34891088], [ 46.39959442, 11.62126073, -45.99719267], [-88.94647804, -49.64400082, -65.80546984]]) The interpolated rotations expressed as Euler angles. These agree with the keyframe rotations at both endpoints of the range of keyframe times. >>> interp_rots.as_euler('xyz', degrees=True) array([[ 14.31443779, -27.50095894, -3.7275787 ], [ 4.74588574, -32.44683966, 81.25139984], [ 10.71094749, -31.56690154, 38.06896408], [ -1.79924227, -24.69421529, 164.57701743], [ 11.72796022, 51.64207311, -171.7374683 ], [ 146.15020772, 43.22849451, -31.34891088], [ 68.10921869, 20.67625074, -48.74886034], [ 46.39959442, 11.62126073, -45.99719267], [ 12.35552615, 4.21525086, -64.89288124], [ -30.08117143, -19.90769513, -78.98121326], [ -88.94647804, -49.64400082, -65.80546984]]) Slerp.__init__Slerp.__call__SlerpSingle rotation is not subscriptable.Single rotation has no len().Rotation.reduceRotation.random (line 3243)Rotation.randomRotation.mean (line 2913)Rotation.meanRotation.magnitude (line 2816)Rotation.magnitudeRotation.inv (line 2775)Rotation.invRotation.identityRotation.from_rotvec (line 1206)Rotation.from_rotvecRotation.from_quat (line 914)Rotation.from_quatRotation.from_mrp (line 1593)Rotation.from_mrpRotation.from_matrix (line 1014)Rotation.from_matrixRotation.from_matrix(Rotation.from_euler (line 1316)Rotation.from_eulerRotation.from_davenport (line 1437)Rotation.from_davenportRotation.create_groupRotation.concatenate (line 2400)Rotation.concatenateRotation.as_rotvec (line 1895)Rotation.as_rotvecRotation.as_quat (line 1695)Rotation.as_quatRotation.as_mrp (line 2321)Rotation.as_mrpRotation.as_matrix (line 1790)Rotation.as_matrixRotation.as_euler (line 2086)Rotation.as_eulerRotation.as_davenport (line 2175)Rotation.as_davenportRotation.approx_equal (line 2860)Rotation.approx_equalRotation.apply (line 2459)Rotation.applyRotation.align_vectors (line 3304)Rotation.align_vectorsRotation._compute_eulerRotation._as_euler_from_matrixRotation.__setstate_cython__Rotation.__setstate__Rotation.__reduce_cython__Rotation.__pow__ (line 2694)Rotation.__mul__ (line 2613)Rotation.__getstate__Rotation.__getitem__ (line 3117)RotationRepresent as rotation vectors. A rotation vector is a 3 dimensional vector which is co-directional to the axis of rotation and whose norm gives the angle of rotation [1]_. Parameters ---------- degrees : boolean, optional Returned magnitudes are in degrees if this flag is True, else they are in radians. Default is False. .. versionadded:: 1.7.0 Returns ------- rotvec : ndarray, shape (3,) or (N, 3) Shape depends on shape of inputs used for initialization. References ---------- .. [1] https://en.wikipedia.org/wiki/Axis%E2%80%93angle_representation#Rotation_vector Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np Represent a single rotation: >>> r = R.from_euler('z', 90, degrees=True) >>> r.as_rotvec() array([0. , 0. , 1.57079633]) >>> r.as_rotvec().shape (3,) Represent a rotation in degrees: >>> r = R.from_euler('YX', (-90, -90), degrees=True) >>> s = r.as_rotvec(degrees=True) >>> s array([-69.2820323, -69.2820323, -69.2820323]) >>> np.linalg.norm(s) 120.00000000000001 Represent a stack with a single rotation: >>> r = R.from_quat([[0, 0, 1, 1]]) >>> r.as_rotvec() array([[0. , 0. , 1.57079633]]) >>> r.as_rotvec().shape (1, 3) Represent multiple rotations in a single object: >>> r = R.from_quat([[0, 0, 1, 1], [1, 1, 0, 1]]) >>> r.as_rotvec() array([[0. , 0. , 1.57079633], [1.35102172, 1.35102172, 0. ]]) >>> r.as_rotvec().shape (2, 3) Represent as rotation matrix. 3D rotations can be represented using rotation matrices, which are 3 x 3 real orthogonal matrices with determinant equal to +1 [1]_. Returns ------- matrix : ndarray, shape (3, 3) or (N, 3, 3) Shape depends on shape of inputs used for initialization. References ---------- .. [1] https://en.wikipedia.org/wiki/Rotation_matrix#In_three_dimensions Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np Represent a single rotation: >>> r = R.from_rotvec([0, 0, np.pi/2]) >>> r.as_matrix() array([[ 2.22044605e-16, -1.00000000e+00, 0.00000000e+00], [ 1.00000000e+00, 2.22044605e-16, 0.00000000e+00], [ 0.00000000e+00, 0.00000000e+00, 1.00000000e+00]]) >>> r.as_matrix().shape (3, 3) Represent a stack with a single rotation: >>> r = R.from_quat([[1, 1, 0, 0]]) >>> r.as_matrix() array([[[ 0., 1., 0.], [ 1., 0., 0.], [ 0., 0., -1.]]]) >>> r.as_matrix().shape (1, 3, 3) Represent multiple rotations: >>> r = R.from_rotvec([[np.pi/2, 0, 0], [0, 0, np.pi/2]]) >>> r.as_matrix() array([[[ 1.00000000e+00, 0.00000000e+00, 0.00000000e+00], [ 0.00000000e+00, 2.22044605e-16, -1.00000000e+00], [ 0.00000000e+00, 1.00000000e+00, 2.22044605e-16]], [[ 2.22044605e-16, -1.00000000e+00, 0.00000000e+00], [ 1.00000000e+00, 2.22044605e-16, 0.00000000e+00], [ 0.00000000e+00, 0.00000000e+00, 1.00000000e+00]]]) >>> r.as_matrix().shape (2, 3, 3) Notes ----- This function was called as_dcm before. .. versionadded:: 1.4.0 Represent as quaternions. Rotations in 3 dimensions can be represented using unit norm quaternions [1]_. The 4 components of a quaternion are divided into a scalar part ``w`` and a vector part ``(x, y, z)`` and can be expressed from the angle ``theta`` and the axis ``n`` of a rotation as follows:: w = cos(theta / 2) x = sin(theta / 2) * n_x y = sin(theta / 2) * n_y z = sin(theta / 2) * n_z There are 2 conventions to order the components in a quaternion: - scalar-first order -- ``(w, x, y, z)`` - scalar-last order -- ``(x, y, z, w)`` The choice is controlled by `scalar_first` argument. By default, it is False and the scalar-last order is used. The mapping from quaternions to rotations is two-to-one, i.e. quaternions ``q`` and ``-q``, where ``-q`` simply reverses the sign of each component, represent the same spatial rotation. Parameters ---------- canonical : `bool`, default False Whether to map the redundant double cover of rotation space to a unique "canonical" single cover. If True, then the quaternion is chosen from {q, -q} such that the w term is positive. If the w term is 0, then the quaternion is chosen such that the first nonzero term of the x, y, and z terms is positive. scalar_first : bool, optional Whether the scalar component goes first or last. Default is False, i.e. the scalar-last order is used. Returns ------- quat : `numpy.ndarray`, shape (4,) or (N, 4) Shape depends on shape of inputs used for initialization. References ---------- .. [1] https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np A rotation can be represented as a quaternion with either scalar-last (default) or scalar-first component order. This is shown for a single rotation: >>> r = R.from_matrix(np.eye(3)) >>> r.as_quat() array([0., 0., 0., 1.]) >>> r.as_quat(scalar_first=True) array([1., 0., 0., 0.]) When multiple rotations are stored in a single Rotation object, the result will be a 2-dimensional array: >>> r = R.from_rotvec([[np.pi, 0, 0], [0, 0, np.pi/2]]) >>> r.as_quat().shape (2, 4) Quaternions can be mapped from a redundant double cover of the rotation space to a canonical representation with a positive w term. >>> r = R.from_quat([0, 0, 0, -1]) >>> r.as_quat() array([0. , 0. , 0. , -1.]) >>> r.as_quat(canonical=True) array([0. , 0. , 0. , 1.]) Represent as Modified Rodrigues Parameters (MRPs). MRPs are a 3 dimensional vector co-directional to the axis of rotation and whose magnitude is equal to ``tan(theta / 4)``, where ``theta`` is the angle of rotation (in radians) [1]_. MRPs have a singularity at 360 degrees which can be avoided by ensuring the angle of rotation does not exceed 180 degrees, i.e. switching the direction of the rotation when it is past 180 degrees. This function will always return MRPs corresponding to a rotation of less than or equal to 180 degrees. Returns ------- mrps : ndarray, shape (3,) or (N, 3) Shape depends on shape of inputs used for initialization. References ---------- .. [1] Shuster, M. D. "A Survey of Attitude Representations", The Journal of Astronautical Sciences, Vol. 41, No.4, 1993, pp. 475-476 Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np Represent a single rotation: >>> r = R.from_rotvec([0, 0, np.pi]) >>> r.as_mrp() array([0. , 0. , 1. ]) >>> r.as_mrp().shape (3,) Represent a stack with a single rotation: >>> r = R.from_euler('xyz', [[180, 0, 0]], degrees=True) >>> r.as_mrp() array([[1. , 0. , 0. ]]) >>> r.as_mrp().shape (1, 3) Represent multiple rotations: >>> r = R.from_rotvec([[np.pi/2, 0, 0], [0, 0, np.pi/2]]) >>> r.as_mrp() array([[0.41421356, 0. , 0. ], [0. , 0. , 0.41421356]]) >>> r.as_mrp().shape (2, 3) Notes ----- .. versionadded:: 1.6.0 Represent as Euler angles. Any orientation can be expressed as a composition of 3 elementary rotations. Once the axis sequence has been chosen, Euler angles define the angle of rotation around each respective axis [1]_. The algorithm from [2]_ has been used to calculate Euler angles for the rotation about a given sequence of axes. Euler angles suffer from the problem of gimbal lock [3]_, where the representation loses a degree of freedom and it is not possible to determine the first and third angles uniquely. In this case, a warning is raised, and the third angle is set to zero. Note however that the returned angles still represent the correct rotation. Parameters ---------- seq : string, length 3 3 characters belonging to the set {'X', 'Y', 'Z'} for intrinsic rotations, or {'x', 'y', 'z'} for extrinsic rotations [1]_. Adjacent axes cannot be the same. Extrinsic and intrinsic rotations cannot be mixed in one function call. degrees : boolean, optional Returned angles are in degrees if this flag is True, else they are in radians. Default is False. Returns ------- angles : ndarray, shape (3,) or (N, 3) Shape depends on shape of inputs used to initialize object. The returned angles are in the range: - First angle belongs to [-180, 180] degrees (both inclusive) - Third angle belongs to [-180, 180] degrees (both inclusive) - Second angle belongs to: - [-90, 90] degrees if all axes are different (like xyz) - [0, 180] degrees if first and third axes are the same (like zxz) References ---------- .. [1] https://en.wikipedia.org/wiki/Euler_angles#Definition_by_intrinsic_rotations .. [2] Bernardes E, Viollet S (2022) Quaternion to Euler angles conversion: A direct, general and computationally efficient method. PLoS ONE 17(11): e0276302. https://doi.org/10.1371/journal.pone.0276302 .. [3] https://en.wikipedia.org/wiki/Gimbal_lock#In_applied_mathematics Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np Represent a single rotation: >>> r = R.from_rotvec([0, 0, np.pi/2]) >>> r.as_euler('zxy', degrees=True) array([90., 0., 0.]) >>> r.as_euler('zxy', degrees=True).shape (3,) Represent a stack of single rotation: >>> r = R.from_rotvec([[0, 0, np.pi/2]]) >>> r.as_euler('zxy', degrees=True) array([[90., 0., 0.]]) >>> r.as_euler('zxy', degrees=True).shape (1, 3) Represent multiple rotations in a single object: >>> r = R.from_rotvec([ ... [0, 0, np.pi/2], ... [0, -np.pi/3, 0], ... [np.pi/4, 0, 0]]) >>> r.as_euler('zxy', degrees=True) array([[ 90., 0., 0.], [ 0., 0., -60.], [ 0., 45., 0.]]) >>> r.as_euler('zxy', degrees=True).shape (3, 3) Represent as Davenport angles. Any orientation can be expressed as a composition of 3 elementary rotations. For both Euler angles and Davenport angles, consecutive axes must be are orthogonal (``axis2`` is orthogonal to both ``axis1`` and ``axis3``). For Euler angles, there is an additional relationship between ``axis1`` or ``axis3``, with two possibilities: - ``axis1`` and ``axis3`` are also orthogonal (asymmetric sequence) - ``axis1 == axis3`` (symmetric sequence) For Davenport angles, this last relationship is relaxed [1]_, and only the consecutive orthogonal axes requirement is maintained. A slightly modified version of the algorithm from [2]_ has been used to calculate Davenport angles for the rotation about a given sequence of axes. Davenport angles, just like Euler angles, suffer from the problem of gimbal lock [3]_, where the representation loses a degree of freedom and it is not possible to determine the first and third angles uniquely. In this case, a warning is raised, and the third angle is set to zero. Note however that the returned angles still represent the correct rotation. Parameters ---------- axes : array_like, shape (3,) or ([1 or 2 or 3], 3) Axis of rotation, if one dimensional. If two dimensional, describes the sequence of axes for rotations, where each axes[i, :] is the ith axis. If more than one axis is given, then the second axis must be orthogonal to both the first and third axes. order : string If it belongs to the set {'e', 'extrinsic'}, the sequence will be extrinsic. If if belongs to the set {'i', 'intrinsic'}, sequence will be treated as intrinsic. degrees : boolean, optional Returned angles are in degrees if this flag is True, else they are in radians. Default is False. Returns ------- angles : ndarray, shape (3,) or (N, 3) Shape depends on shape of inputs used to initialize object. The returned angles are in the range: - First angle belongs to [-180, 180] degrees (both inclusive) - Third angle belongs to [-180, 180] degrees (both inclusive) - Second angle belongs to a set of size 180 degrees, given by: ``[-abs(lambda), 180 - abs(lambda)]``, where ``lambda`` is the angle between the first and third axes. References ---------- .. [1] Shuster, Malcolm & Markley, Landis. (2003). Generalization of the Euler Angles. Journal of the Astronautical Sciences. 51. 123-132. 10.1007/BF03546304. .. [2] Bernardes E, Viollet S (2022) Quaternion to Euler angles conversion: A direct, general and computationally efficient method. PLoS ONE 17(11): e0276302. 10.1371/journal.pone.0276302 .. [3] https://en.wikipedia.org/wiki/Gimbal_lock#In_applied_mathematics Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np Davenport angles are a generalization of Euler angles, when we use the canonical basis axes: >>> ex = [1, 0, 0] >>> ey = [0, 1, 0] >>> ez = [0, 0, 1] Represent a single rotation: >>> r = R.from_rotvec([0, 0, np.pi/2]) >>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True) array([90., 0., 0.]) >>> r.as_euler('zxy', degrees=True) array([90., 0., 0.]) >>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True).shape (3,) Represent a stack of single rotation: >>> r = R.from_rotvec([[0, 0, np.pi/2]]) >>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True) array([[90., 0., 0.]]) >>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True).shape (1, 3) Represent multiple rotations in a single object: >>> r = R.from_rotvec([ ... [0, 0, 90], ... [45, 0, 0]], degrees=True) >>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True) array([[90., 0., 0.], [ 0., 45., 0.]]) >>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True).shape (2, 3) R_secR_priR_optR_flipRPickleErrorOptimal rotation is not uniquely or poorly defined for the given sets of vectors.Only one infinite weight is allowedNote that Cython is deliberately stricter than PEP-484 and rejects subclasses of builtin types. If you need to pass subclasses then set the 'annotation_typing' directive to False.NotImplementedErrorNon-positive determinant (left-handed or null coordinate frame) in rotation matrix NMean of an empty rotation set is undefined.KInvert this rotation. Composition of a rotation with its inverse results in an identity transformation. Returns ------- inverse : `Rotation` instance Object containing inverse of the rotations in the current instance. Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np Inverting a single rotation: >>> p = R.from_euler('z', 45, degrees=True) >>> q = p.inv() >>> q.as_euler('zyx', degrees=True) array([-45., 0., 0.]) Inverting multiple rotations: >>> p = R.from_rotvec([[0, 0, np.pi/3], [-np.pi/4, 0, 0]]) >>> q = p.inv() >>> q.as_rotvec() array([[-0. , -0. , -1.04719755], [ 0.78539816, -0. , -0. ]]) Interpolation times must be within the range [{}, {}], both inclusive.Initialize from rotation vectors. A rotation vector is a 3 dimensional vector which is co-directional to the axis of rotation and whose norm gives the angle of rotation [1]_. Parameters ---------- rotvec : array_like, shape (N, 3) or (3,) A single vector or a stack of vectors, where `rot_vec[i]` gives the ith rotation vector. degrees : bool, optional If True, then the given magnitudes are assumed to be in degrees. Default is False. .. versionadded:: 1.7.0 Returns ------- rotation : `Rotation` instance Object containing the rotations represented by input rotation vectors. References ---------- .. [1] https://en.wikipedia.org/wiki/Axis%E2%80%93angle_representation#Rotation_vector Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np Initialize a single rotation: >>> r = R.from_rotvec(np.pi/2 * np.array([0, 0, 1])) >>> r.as_rotvec() array([0. , 0. , 1.57079633]) >>> r.as_rotvec().shape (3,) Initialize a rotation in degrees, and view it in degrees: >>> r = R.from_rotvec(45 * np.array([0, 1, 0]), degrees=True) >>> r.as_rotvec(degrees=True) array([ 0., 45., 0.]) Initialize multiple rotations in one object: >>> r = R.from_rotvec([ ... [0, 0, np.pi/2], ... [np.pi/2, 0, 0]]) >>> r.as_rotvec() array([[0. , 0. , 1.57079633], [1.57079633, 0. , 0. ]]) >>> r.as_rotvec().shape (2, 3) It is also possible to have a stack of a single rotation: >>> r = R.from_rotvec([[0, 0, np.pi/2]]) >>> r.as_rotvec().shape (1, 3) Initialize from rotation matrix. Rotations in 3 dimensions can be represented with 3 x 3 orthogonal matrices [1]_. If the input is not orthogonal, an approximation is created by orthogonalizing the input matrix using the method described in [2]_, and then converting the orthogonal rotation matrices to quaternions using the algorithm described in [3]_. Matrices must be right-handed. Parameters ---------- matrix : array_like, shape (N, 3, 3) or (3, 3) A single matrix or a stack of matrices, where ``matrix[i]`` is the i-th matrix. Returns ------- rotation : `Rotation` instance Object containing the rotations represented by the rotation matrices. References ---------- .. [1] https://en.wikipedia.org/wiki/Rotation_matrix#In_three_dimensions .. [2] https://en.wikipedia.org/wiki/Orthogonal_Procrustes_problem .. [3] F. Landis Markley, "Unit Quaternion from Rotation Matrix", Journal of guidance, control, and dynamics vol. 31.2, pp. 440-442, 2008. Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np Initialize a single rotation: >>> r = R.from_matrix([ ... [0, -1, 0], ... [1, 0, 0], ... [0, 0, 1]]) >>> r.single True >>> r.as_matrix().shape (3, 3) Initialize multiple rotations in a single object: >>> r = R.from_matrix([ ... [ ... [0, -1, 0], ... [1, 0, 0], ... [0, 0, 1], ... ], ... [ ... [1, 0, 0], ... [0, 0, -1], ... [0, 1, 0], ... ]]) >>> r.as_matrix().shape (2, 3, 3) >>> r.single False >>> len(r) 2 If input matrices are not special orthogonal (orthogonal with determinant equal to +1), then a special orthogonal estimate is stored: >>> a = np.array([ ... [0, -0.5, 0], ... [0.5, 0, 0], ... [0, 0, 0.5]]) >>> np.linalg.det(a) 0.125 >>> r = R.from_matrix(a) >>> matrix = r.as_matrix() >>> matrix array([[ 0., -1., 0.], [ 1., 0., 0.], [ 0., 0., 1.]]) >>> np.linalg.det(matrix) 1.0 It is also possible to have a stack containing a single rotation: >>> r = R.from_matrix([[ ... [0, -1, 0], ... [1, 0, 0], ... [0, 0, 1]]]) >>> r.as_matrix() array([[[ 0., -1., 0.], [ 1., 0., 0.], [ 0., 0., 1.]]]) >>> r.as_matrix().shape (1, 3, 3) Notes ----- This function was called from_dcm before. .. versionadded:: 1.4.0 Initialize from quaternions. Rotations in 3 dimensions can be represented using unit norm quaternions [1]_. The 4 components of a quaternion are divided into a scalar part ``w`` and a vector part ``(x, y, z)`` and can be expressed from the angle ``theta`` and the axis ``n`` of a rotation as follows:: w = cos(theta / 2) x = sin(theta / 2) * n_x y = sin(theta / 2) * n_y z = sin(theta / 2) * n_z There are 2 conventions to order the components in a quaternion: - scalar-first order -- ``(w, x, y, z)`` - scalar-last order -- ``(x, y, z, w)`` The choice is controlled by `scalar_first` argument. By default, it is False and the scalar-last order is assumed. Advanced users may be interested in the "double cover" of 3D space by the quaternion representation [2]_. As of version 1.11.0, the following subset (and only this subset) of operations on a `Rotation` ``r`` corresponding to a quaternion ``q`` are guaranteed to preserve the double cover property: ``r = Rotation.from_quat(q)``, ``r.as_quat(canonical=False)``, ``r.inv()``, and composition using the ``*`` operator such as ``r*r``. Parameters ---------- quat : array_like, shape (N, 4) or (4,) Each row is a (possibly non-unit norm) quaternion representing an active rotation. Each quaternion will be normalized to unit norm. scalar_first : bool, optional Whether the scalar component goes first or last. Default is False, i.e. the scalar-last order is assumed. Returns ------- rotation : `Rotation` instance Object containing the rotations represented by input quaternions. References ---------- .. [1] https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation .. [2] Hanson, Andrew J. "Visualizing quaternions." Morgan Kaufmann Publishers Inc., San Francisco, CA. 2006. Examples -------- >>> from scipy.spatial.transform import Rotation as R A rotation can be initialzied from a quaternion with the scalar-last (default) or scalar-first component order as shown below: >>> r = R.from_quat([0, 0, 0, 1]) >>> r.as_matrix() array([[1., 0., 0.], [0., 1., 0.], [0., 0., 1.]]) >>> r = R.from_quat([1, 0, 0, 0], scalar_first=True) >>> r.as_matrix() array([[1., 0., 0.], [0., 1., 0.], [0., 0., 1.]]) It is possible to initialize multiple rotations in a single object by passing a 2-dimensional array: >>> r = R.from_quat([ ... [1, 0, 0, 0], ... [0, 0, 0, 1] ... ]) >>> r.as_quat() array([[1., 0., 0., 0.], [0., 0., 0., 1.]]) >>> r.as_quat().shape (2, 4) It is also possible to have a stack of a single rotation: >>> r = R.from_quat([[0, 0, 0, 1]]) >>> r.as_quat() array([[0., 0., 0., 1.]]) >>> r.as_quat().shape (1, 4) Quaternions are normalized before initialization. >>> r = R.from_quat([0, 0, 1, 1]) >>> r.as_quat() array([0. , 0. , 0.70710678, 0.70710678]) Initialize from Modified Rodrigues Parameters (MRPs). MRPs are a 3 dimensional vector co-directional to the axis of rotation and whose magnitude is equal to ``tan(theta / 4)``, where ``theta`` is the angle of rotation (in radians) [1]_. MRPs have a singularity at 360 degrees which can be avoided by ensuring the angle of rotation does not exceed 180 degrees, i.e. switching the direction of the rotation when it is past 180 degrees. Parameters ---------- mrp : array_like, shape (N, 3) or (3,) A single vector or a stack of vectors, where `mrp[i]` gives the ith set of MRPs. Returns ------- rotation : `Rotation` instance Object containing the rotations represented by input MRPs. References ---------- .. [1] Shuster, M. D. "A Survey of Attitude Representations", The Journal of Astronautical Sciences, Vol. 41, No.4, 1993, pp. 475-476 Notes ----- .. versionadded:: 1.6.0 Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np Initialize a single rotation: >>> r = R.from_mrp([0, 0, 1]) >>> r.as_euler('xyz', degrees=True) array([0. , 0. , 180. ]) >>> r.as_euler('xyz').shape (3,) Initialize multiple rotations in one object: >>> r = R.from_mrp([ ... [0, 0, 1], ... [1, 0, 0]]) >>> r.as_euler('xyz', degrees=True) array([[0. , 0. , 180. ], [180.0 , 0. , 0. ]]) >>> r.as_euler('xyz').shape (2, 3) It is also possible to have a stack of a single rotation: >>> r = R.from_mrp([[0, 0, np.pi/2]]) >>> r.as_euler('xyz').shape (1, 3) Initialize from Euler angles. Rotations in 3-D can be represented by a sequence of 3 rotations around a sequence of axes. In theory, any three axes spanning the 3-D Euclidean space are enough. In practice, the axes of rotation are chosen to be the basis vectors. The three rotations can either be in a global frame of reference (extrinsic) or in a body centred frame of reference (intrinsic), which is attached to, and moves with, the object under rotation [1]_. Parameters ---------- seq : string Specifies sequence of axes for rotations. Up to 3 characters belonging to the set {'X', 'Y', 'Z'} for intrinsic rotations, or {'x', 'y', 'z'} for extrinsic rotations. Extrinsic and intrinsic rotations cannot be mixed in one function call. angles : float or array_like, shape (N,) or (N, [1 or 2 or 3]) Euler angles specified in radians (`degrees` is False) or degrees (`degrees` is True). For a single character `seq`, `angles` can be: - a single value - array_like with shape (N,), where each `angle[i]` corresponds to a single rotation - array_like with shape (N, 1), where each `angle[i, 0]` corresponds to a single rotation For 2- and 3-character wide `seq`, `angles` can be: - array_like with shape (W,) where `W` is the width of `seq`, which corresponds to a single rotation with `W` axes - array_like with shape (N, W) where each `angle[i]` corresponds to a sequence of Euler angles describing a single rotation degrees : bool, optional If True, then the given angles are assumed to be in degrees. Default is False. Returns ------- rotation : `Rotation` instance Object containing the rotation represented by the sequence of rotations around given axes with given angles. References ---------- .. [1] https://en.wikipedia.org/wiki/Euler_angles#Definition_by_intrinsic_rotations Examples -------- >>> from scipy.spatial.transform import Rotation as R Initialize a single rotation along a single axis: >>> r = R.from_euler('x', 90, degrees=True) >>> r.as_quat().shape (4,) Initialize a single rotation with a given axis sequence: >>> r = R.from_euler('zyx', [90, 45, 30], degrees=True) >>> r.as_quat().shape (4,) Initialize a stack with a single rotation around a single axis: >>> r = R.from_euler('x', [90], degrees=True) >>> r.as_quat().shape (1, 4) Initialize a stack with a single rotation with an axis sequence: >>> r = R.from_euler('zyx', [[90, 45, 30]], degrees=True) >>> r.as_quat().shape (1, 4) Initialize multiple elementary rotations in one object: >>> r = R.from_euler('x', [90, 45, 30], degrees=True) >>> r.as_quat().shape (3, 4) Initialize multiple rotations in one object: >>> r = R.from_euler('zyx', [[90, 45, 30], [35, 45, 90]], degrees=True) >>> r.as_quat().shape (2, 4) Initialize from Davenport angles. Rotations in 3-D can be represented by a sequence of 3 rotations around a sequence of axes. The three rotations can either be in a global frame of reference (extrinsic) or in a body centred frame of reference (intrinsic), which is attached to, and moves with, the object under rotation [1]_. For both Euler angles and Davenport angles, consecutive axes must be are orthogonal (``axis2`` is orthogonal to both ``axis1`` and ``axis3``). For Euler angles, there is an additional relationship between ``axis1`` or ``axis3``, with two possibilities: - ``axis1`` and ``axis3`` are also orthogonal (asymmetric sequence) - ``axis1 == axis3`` (symmetric sequence) For Davenport angles, this last relationship is relaxed [2]_, and only the consecutive orthogonal axes requirement is maintained. Parameters ---------- axes : array_like, shape (3,) or ([1 or 2 or 3], 3) Axis of rotation, if one dimensional. If two dimensional, describes the sequence of axes for rotations, where each axes[i, :] is the ith axis. If more than one axis is given, then the second axis must be orthogonal to both the first and third axes. order : string If it is equal to 'e' or 'extrinsic', the sequence will be extrinsic. If it is equal to 'i' or 'intrinsic', sequence will be treated as intrinsic. angles : float or array_like, shape (N,) or (N, [1 or 2 or 3]) Euler angles specified in radians (`degrees` is False) or degrees (`degrees` is True). For a single axis, `angles` can be: - a single value - array_like with shape (N,), where each `angle[i]` corresponds to a single rotation - array_like with shape (N, 1), where each `angle[i, 0]` corresponds to a single rotation For 2 and 3 axes, `angles` can be: - array_like with shape (W,) where `W` is the number of rows of `axes`, which corresponds to a single rotation with `W` axes - array_like with shape (N, W) where each `angle[i]` corresponds to a sequence of Davenport angles describing a single rotation degrees : bool, optional If True, then the given angles are assumed to be in degrees. Default is False. Returns ------- rotation : `Rotation` instance Object containing the rotation represented by the sequence of rotations around given axes with given angles. References ---------- .. [1] https://en.wikipedia.org/wiki/Euler_angles#Definition_by_intrinsic_rotations .. [2] Shuster, Malcolm & Markley, Landis. (2003). Generalization of the Euler Angles. Journal of the Astronautical Sciences. 51. 123-132. 10.1007/BF03546304. Examples -------- >>> from scipy.spatial.transform import Rotation as R Davenport angles are a generalization of Euler angles, when we use the canonical basis axes: >>> ex = [1, 0, 0] >>> ey = [0, 1, 0] >>> ez = [0, 0, 1] Initialize a single rotation with a given axis sequence: >>> axes = [ez, ey, ex] >>> r = R.from_davenport(axes, 'extrinsic', [90, 0, 0], degrees=True) >>> r.as_quat().shape (4,) It is equivalent to Euler angles in this case: >>> r.as_euler('zyx', degrees=True) array([90., 0., -0.]) Initialize multiple rotations in one object: >>> r = R.from_davenport(axes, 'extrinsic', [[90, 45, 30], [35, 45, 90]], degrees=True) >>> r.as_quat().shape (2, 4) Using only one or two axes is also possible: >>> r = R.from_davenport([ez, ex], 'extrinsic', [[90, 45], [35, 45]], degrees=True) >>> r.as_quat().shape (2, 4) Non-canonical axes are possible, and they do not need to be normalized, as long as consecutive axes are orthogonal: >>> e1 = [2, 0, 0] >>> e2 = [0, 1, 0] >>> e3 = [1, 0, 1] >>> axes = [e1, e2, e3] >>> r = R.from_davenport(axes, 'extrinsic', [90, 45, 30], degrees=True) >>> r.as_quat() [ 0.701057, 0.430459, -0.092296, 0.560986] Incompatible checksums (0x%x vs (0x0fbb6f7, 0x22c69a8, 0x14ed78b) = (_quat, _single))ImportErrorGimbal lock detected. Setting third angle to zero since it is not possible to uniquely determine all angles.Get the mean of the rotations. The mean used is the chordal L2 mean (also called the projected or induced arithmetic mean) [1]_. If ``A`` is a set of rotation matrices, then the mean ``M`` is the rotation matrix that minimizes the following loss function: .. math:: L(M) = \sum_{i = 1}^{n} w_i \lVert \mathbf{A}_i - \mathbf{M} \rVert^2 , where :math:`w_i`'s are the `weights` corresponding to each matrix. Parameters ---------- weights : array_like shape (N,), optional Weights describing the relative importance of the rotations. If None (default), then all values in `weights` are assumed to be equal. Returns ------- mean : `Rotation` instance Object containing the mean of the rotations in the current instance. References ---------- .. [1] Hartley, Richard, et al., "Rotation Averaging", International Journal of Computer Vision 103, 2013, pp. 267-305. Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> r = R.from_euler('zyx', [[0, 0, 0], ... [1, 0, 0], ... [0, 1, 0], ... [0, 0, 1]], degrees=True) >>> r.mean().as_euler('zyx', degrees=True) array([0.24945696, 0.25054542, 0.24945696]) Get the magnitude(s) of the rotation(s). Returns ------- magnitude : ndarray or float Angle(s) in radians, float if object contains a single rotation and ndarray if object contains multiple rotations. The magnitude will always be in the range [0, pi]. Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np >>> r = R.from_quat(np.eye(4)) >>> r.as_quat() array([[ 1., 0., 0., 0.], [ 0., 1., 0., 0.], [ 0., 0., 1., 0.], [ 0., 0., 0., 1.]]) >>> r.magnitude() array([3.14159265, 3.14159265, 3.14159265, 0. ]) Magnitude of a single rotation: >>> r[0].magnitude() 3.141592653589793 Generate uniformly distributed rotations. Parameters ---------- num : int or None, optional Number of random rotations to generate. If None (default), then a single rotation is generated. rng : `numpy.random.Generator`, optional Pseudorandom number generator state. When `rng` is None, a new `numpy.random.Generator` is created using entropy from the operating system. Types other than `numpy.random.Generator` are passed to `numpy.random.default_rng` to instantiate a `Generator`. Returns ------- random_rotation : `Rotation` instance Contains a single rotation if `num` is None. Otherwise contains a stack of `num` rotations. Notes ----- This function is optimized for efficiently sampling random rotation matrices in three dimensions. For generating random rotation matrices in higher dimensions, see `scipy.stats.special_ortho_group`. Examples -------- >>> from scipy.spatial.transform import Rotation as R Sample a single rotation: >>> R.random().as_euler('zxy', degrees=True) array([-110.5976185 , 55.32758512, 76.3289269 ]) # random Sample a stack of rotations: >>> R.random(5).as_euler('zxy', degrees=True) array([[-110.5976185 , 55.32758512, 76.3289269 ], # random [ -91.59132005, -14.3629884 , -93.91933182], [ 25.23835501, 45.02035145, -121.67867086], [ -51.51414184, -15.29022692, -172.46870023], [ -81.63376847, -27.39521579, 2.60408416]]) See Also -------- scipy.stats.special_ortho_group Found zero norm quaternions in `quat`.Extract rotation(s) at given index(es) from object. Create a new `Rotation` instance containing a subset of rotations stored in this object. Parameters ---------- indexer : index, slice, or index array Specifies which rotation(s) to extract. A single indexer must be specified, i.e. as if indexing a 1 dimensional array or list. Returns ------- rotation : `Rotation` instance Contains - a single rotation, if `indexer` is a single index - a stack of rotation(s), if `indexer` is a slice, or and index array. Raises ------ TypeError if the instance was created as a single rotation. Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> rs = R.from_quat([ ... [1, 1, 0, 0], ... [0, 1, 0, 1], ... [1, 1, -1, 0]]) # These quats are normalized >>> rs.as_quat() array([[ 0.70710678, 0.70710678, 0. , 0. ], [ 0. , 0.70710678, 0. , 0.70710678], [ 0.57735027, 0.57735027, -0.57735027, 0. ]]) Indexing using a single index: >>> a = rs[0] >>> a.as_quat() array([0.70710678, 0.70710678, 0. , 0. ]) Array slicing: >>> b = rs[1:3] >>> b.as_quat() array([[ 0. , 0.70710678, 0. , 0.70710678], [ 0.57735027, 0.57735027, -0.57735027, 0. ]]) List comprehension to split each rotation into its own object: >>> c = [r for r in rs] >>> print([r.as_quat() for r in c]) [array([ 0.70710678, 0.70710678, 0. , 0. ]), array([ 0. , 0.70710678, 0. , 0.70710678]), array([ 0.57735027, 0.57735027, -0.57735027, 0. ])] Concatenation of split rotations will recover the original object: >>> R.concatenate([a, b]).as_quat() array([[ 0.70710678, 0.70710678, 0. , 0. ], [ 0. , 0.70710678, 0. , 0.70710678], [ 0.57735027, 0.57735027, -0.57735027, 0. ]]) Expected `weights` to have number of values equal to number of input vectors, got {} values and {} vectors.Expected `weights` to have number of values equal to number of rotations, got {} values and {} rotations.Expected `weights` to be 1 dimensional, got shape {}.Expected up to 3 axes, got {}Expected times to be specified in a 1 dimensional array, got {} dimensions.Expected `rot_vec` to have shape (3,) or (N, 3), got {}Expected `quat` to have shape (4,) or (N, 4), got Expected number of rotations to be equal to number of timestamps given, got {} rotations and {} timestamps.Expected `mrp` to have shape (3,) or (N, 3), got {}Expected `matrix` to have shape (3, 3) or (N, 3, 3), got {}Expected inputs `a` and `b` to have same shapes, got {} and {} respectively.Expected input of shape (3,) or (P, 3), got {}.Expected input `b` to have shape (3,) or (N, 3), got {}Expected input `a` to have shape (3,) or (N, 3), got {}Expected float, 1D array, or 2D array for parameter `angles` corresponding to `seq`, got shape {}.Expected equal numbers of rotations and vectors , or a single rotation, or a single vector, got {} rotations and {} vectors.Expected equal number of rotations in both or a single rotation in either object, got {} rotations in first and {} rotations in second object.Expected consecutive axes to be different, got {}Expected axis specification to be a non-empty string of upto 3 characters, got {}Expected axes from `seq` to be from ['x', 'y', 'z'] or ['X', 'Y', 'Z'], got {}Expected angles to have shape (num_rotations, num_axes), got {}.Expected `angles` to be at most 2-dimensional with width equal to number of axes specified, got {} for shapeExpected `angles` parameter to have shape (N, 1), got {}.Expected 3 axes, got {}.Estimate a rotation to optimally align two sets of vectors. Find a rotation between frames A and B which best aligns a set of vectors `a` and `b` observed in these frames. The following loss function is minimized to solve for the rotation matrix :math:`C`: .. math:: L(C) = \frac{1}{2} \sum_{i = 1}^{n} w_i \lVert \mathbf{a}_i - C \mathbf{b}_i \rVert^2 , where :math:`w_i`'s are the `weights` corresponding to each vector. The rotation is estimated with Kabsch algorithm [1]_, and solves what is known as the "pointing problem", or "Wahba's problem" [2]_. There are two special cases. The first is if a single vector is given for `a` and `b`, in which the shortest distance rotation that aligns `b` to `a` is returned. The second is when one of the weights is infinity. In this case, the shortest distance rotation between the primary infinite weight vectors is calculated as above. Then, the rotation about the aligned primary vectors is calculated such that the secondary vectors are optimally aligned per the above loss function. The result is the composition of these two rotations. The result via this process is the same as the Kabsch algorithm as the corresponding weight approaches infinity in the limit. For a single secondary vector this is known as the "align-constrain" algorithm [3]_. For both special cases (single vectors or an infinite weight), the sensitivity matrix does not have physical meaning and an error will be raised if it is requested. For an infinite weight, the primary vectors act as a constraint with perfect alignment, so their contribution to `rssd` will be forced to 0 even if they are of different lengths. Parameters ---------- a : array_like, shape (3,) or (N, 3) Vector components observed in initial frame A. Each row of `a` denotes a vector. b : array_like, shape (3,) or (N, 3) Vector components observed in another frame B. Each row of `b` denotes a vector. weights : array_like shape (N,), optional Weights describing the relative importance of the vector observations. If None (default), then all values in `weights` are assumed to be 1. One and only one weight may be infinity, and weights must be positive. return_sensitivity : bool, optional Whether to return the sensitivity matrix. See Notes for details. Default is False. Returns ------- rotation : `Rotation` instance Best estimate of the rotation that transforms `b` to `a`. rssd : float Stands for "root sum squared distance". Square root of the weighted sum of the squared distances between the given sets of vectors after alignment. It is equal to ``sqrt(2 * minimum_loss)``, where ``minimum_loss`` is the loss function evaluated for the found optimal rotation. Note that the result will also be weighted by the vectors' magnitudes, so perfectly aligned vector pairs will have nonzero `rssd` if they are not of the same length. So, depending on the use case it may be desirable to normalize the input vectors to unit length before calling this method. sensitivity_matrix : ndarray, shape (3, 3) Sensitivity matrix of the estimated rotation estimate as explained in Notes. Returned only when `return_sensitivity` is True. Not valid if aligning a single pair of vectors or if there is an infinite weight, in which cases an error will be raised. Notes ----- The sensitivity matrix gives the sensitivity of the estimated rotation to small perturbations of the vector measurements. Specifically we consider the rotation estimate error as a small rotation vector of frame A. The sensitivity matrix is proportional to the covariance of this rotation vector assuming that the vectors in `a` was measured with errors significantly less than their lengths. To get the true covariance matrix, the returned sensitivity matrix must be multiplied by harmonic mean [4]_ of variance in each observation. Note that `weights` are supposed to be inversely proportional to the observation variances to get consistent results. For example, if all vectors are measured with the same accuracy of 0.01 (`weights` must be all equal), then you should multiple the sensitivity matrix by 0.01**2 to get the covariance. Refer to [5]_ for more rigorous discussion of the covariance estimation. See [6]_ for more discussion of the pointing problem and minimal proper pointing. References ---------- .. [1] https://en.wikipedia.org/wiki/Kabsch_algorithm .. [2] https://en.wikipedia.org/wiki/Wahba%27s_problem .. [3] Magner, Robert, "Extending target tracking capabilities through trajectory and momentum setpoint optimization." Small Satellite Conference, 2018. .. [4] https://en.wikipedia.org/wiki/Harmonic_mean .. [5] F. Landis Markley, "Attitude determination using vector observations: a fast optimal matrix algorithm", Journal of Astronautical Sciences, Vol. 41, No.2, 1993, pp. 261-280. .. [6] Bar-Itzhack, Itzhack Y., Daniel Hershkowitz, and Leiba Rodman, "Pointing in Real Euclidean Space", Journal of Guidance, Control, and Dynamics, Vol. 20, No. 5, 1997, pp. 916-922. Examples -------- >>> import numpy as np >>> from scipy.spatial.transform import Rotation as R Here we run the baseline Kabsch algorithm to best align two sets of vectors, where there is noise on the last two vector measurements of the ``b`` set: >>> a = [[0, 1, 0], [0, 1, 1], [0, 1, 1]] >>> b = [[1, 0, 0], [1, 1.1, 0], [1, 0.9, 0]] >>> rot, rssd, sens = R.align_vectors(a, b, return_sensitivity=True) >>> rot.as_matrix() array([[0., 0., 1.], [1., 0., 0.], [0., 1., 0.]]) When we apply the rotation to ``b``, we get vectors close to ``a``: >>> rot.apply(b) array([[0. , 1. , 0. ], [0. , 1. , 1.1], [0. , 1. , 0.9]]) The error for the first vector is 0, and for the last two the error is magnitude 0.1. The `rssd` is the square root of the sum of the weighted squared errors, and the default weights are all 1, so in this case the `rssd` is calculated as ``sqrt(1 * 0**2 + 1 * 0.1**2 + 1 * (-0.1)**2) = 0.141421356237308`` >>> a - rot.apply(b) array([[ 0., 0., 0. ], [ 0., 0., -0.1], [ 0., 0., 0.1]]) >>> np.sqrt(np.sum(np.ones(3) @ (a - rot.apply(b))**2)) 0.141421356237308 >>> rssd 0.141421356237308 The sensitivity matrix for this example is as follows: >>> sens array([[0.2, 0. , 0.], [0. , 1.5, 1.], [0. , 1. , 1.]]) Special case 1: Find a minimum rotation between single vectors: >>> a = [1, 0, 0] >>> b = [0, 1, 0] >>> rot, _ = R.align_vectors(a, b) >>> rot.as_matrix() array([[0., 1., 0.], [-1., 0., 0.], [0., 0., 1.]]) >>> rot.apply(b) array([1., 0., 0.]) Special case 2: One infinite weight. Here we find a rotation between primary and secondary vectors that can align exactly: >>> a = [[0, 1, 0], [0, 1, 1]] >>> b = [[1, 0, 0], [1, 1, 0]] >>> rot, _ = R.align_vectors(a, b, weights=[np.inf, 1]) >>> rot.as_matrix() array([[0., 0., 1.], [1., 0., 0.], [0., 1., 0.]]) >>> rot.apply(b) array([[0., 1., 0.], [0., 1., 1.]]) Here the secondary vectors must be best-fit: >>> a = [[0, 1, 0], [0, 1, 1]] >>> b = [[1, 0, 0], [1, 2, 0]] >>> rot, _ = R.align_vectors(a, b, weights=[np.inf, 1]) >>> rot.as_matrix() array([[0., 0., 1.], [1., 0., 0.], [0., 1., 0.]]) >>> rot.apply(b) array([[0., 1., 0.], [0., 1., 2.]]) Determine if another rotation is approximately equal to this one. Equality is measured by calculating the smallest angle between the rotations, and checking to see if it is smaller than `atol`. Parameters ---------- other : `Rotation` instance Object containing the rotations to measure against this one. atol : float, optional The absolute angular tolerance, below which the rotations are considered equal. If not given, then set to 1e-8 radians by default. degrees : bool, optional If True and `atol` is given, then `atol` is measured in degrees. If False (default), then atol is measured in radians. Returns ------- approx_equal : ndarray or bool Whether the rotations are approximately equal, bool if object contains a single rotation and ndarray if object contains multiple rotations. Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np >>> p = R.from_quat([0, 0, 0, 1]) >>> q = R.from_quat(np.eye(4)) >>> p.approx_equal(q) array([False, False, False, True]) Approximate equality for a single rotation: >>> p.approx_equal(q[0]) False Consecutive axes must be orthogonal.Concatenate a sequence of `Rotation` objects into a single object. This is useful if you want to, for example, take the mean of a set of rotations and need to pack them into a single object to do so. Parameters ---------- rotations : sequence of `Rotation` objects The rotations to concatenate. If a single `Rotation` object is passed in, a copy is returned. Returns ------- concatenated : `Rotation` instance The concatenated rotations. Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> r1 = R.from_rotvec([0, 0, 1]) >>> r2 = R.from_rotvec([0, 0, 2]) >>> rc = R.concatenate([r1, r2]) >>> rc.as_rotvec() array([[0., 0., 1.], [0., 0., 2.]]) >>> rc.mean().as_rotvec() array([0., 0., 1.5]) Concatenation of a split rotation recovers the original object. >>> rs = [r for r in rc] >>> R.concatenate(rs).as_rotvec() array([[0., 0., 1.], [0., 0., 2.]]) Note that it may be simpler to create the desired rotations by passing in a single list of the data during initialization, rather then by concatenating: >>> R.from_rotvec([[0, 0, 1], [0, 0, 2]]).as_rotvec() array([[0., 0., 1.], [0., 0., 2.]]) Notes ----- .. versionadded:: 1.8.0 Compose this rotation with the other. If `p` and `q` are two rotations, then the composition of 'q followed by p' is equivalent to `p * q`. In terms of rotation matrices, the composition can be expressed as ``p.as_matrix() @ q.as_matrix()``. Parameters ---------- other : `Rotation` instance Object containing the rotations to be composed with this one. Note that rotation compositions are not commutative, so ``p * q`` is generally different from ``q * p``. Returns ------- composition : `Rotation` instance This function supports composition of multiple rotations at a time. The following cases are possible: - Either ``p`` or ``q`` contains a single rotation. In this case `composition` contains the result of composing each rotation in the other object with the single rotation. - Both ``p`` and ``q`` contain ``N`` rotations. In this case each rotation ``p[i]`` is composed with the corresponding rotation ``q[i]`` and `output` contains ``N`` rotations. Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np Composition of two single rotations: >>> p = R.from_quat([0, 0, 1, 1]) >>> q = R.from_quat([1, 0, 0, 1]) >>> p.as_matrix() array([[ 0., -1., 0.], [ 1., 0., 0.], [ 0., 0., 1.]]) >>> q.as_matrix() array([[ 1., 0., 0.], [ 0., 0., -1.], [ 0., 1., 0.]]) >>> r = p * q >>> r.as_matrix() array([[0., 0., 1.], [1., 0., 0.], [0., 1., 0.]]) Composition of two objects containing equal number of rotations: >>> p = R.from_quat([[0, 0, 1, 1], [1, 0, 0, 1]]) >>> q = R.from_rotvec([[np.pi/4, 0, 0], [-np.pi/4, 0, np.pi/4]]) >>> p.as_quat() array([[0. , 0. , 0.70710678, 0.70710678], [0.70710678, 0. , 0. , 0.70710678]]) >>> q.as_quat() array([[ 0.38268343, 0. , 0. , 0.92387953], [-0.37282173, 0. , 0.37282173, 0.84971049]]) >>> r = p * q >>> r.as_quat() array([[ 0.27059805, 0.27059805, 0.65328148, 0.65328148], [ 0.33721128, -0.26362477, 0.26362477, 0.86446082]]) Compose this rotation with itself `n` times. Composition of a rotation ``p`` with itself can be extended to non-integer ``n`` by considering the power ``n`` to be a scale factor applied to the angle of rotation about the rotation's fixed axis. The expression ``q = p ** n`` can also be expressed as ``q = Rotation.from_rotvec(n * p.as_rotvec())``. If ``n`` is negative, then the rotation is inverted before the power is applied. In other words, ``p ** -abs(n) == p.inv() ** abs(n)``. Parameters ---------- n : float The number of times to compose the rotation with itself. modulus : None This overridden argument is not applicable to Rotations and must be ``None``. Returns ------- power : `Rotation` instance If the input Rotation ``p`` contains ``N`` multiple rotations, then the output will contain ``N`` rotations where the ``i`` th rotation is equal to ``p[i] ** n`` Notes ----- For example, a power of 2 will double the angle of rotation, and a power of 0.5 will halve the angle. There are three notable cases: if ``n == 1`` then the original rotation is returned, if ``n == 0`` then the identity rotation is returned, and if ``n == -1`` then ``p.inv()`` is returned. Note that fractional powers ``n`` which effectively take a root of rotation, do so using the shortest path smallest representation of that angle (the principal root). This means that powers of ``n`` and ``1/n`` are not necessarily inverses of each other. For example, a 0.5 power of a +240 degree rotation will be calculated as the 0.5 power of a -120 degree rotation, with the result being a rotation of -60 rather than +120 degrees. Examples -------- >>> from scipy.spatial.transform import Rotation as R Raising a rotation to a power: >>> p = R.from_rotvec([1, 0, 0]) >>> q = p ** 2 >>> q.as_rotvec() array([2., 0., 0.]) >>> r = p ** 0.5 >>> r.as_rotvec() array([0.5, 0., 0.]) Inverse powers do not necessarily cancel out: >>> p = R.from_rotvec([0, 0, 120], degrees=True) >>> ((p ** 2) ** 0.5).as_rotvec(degrees=True) array([ -0., -0., -60.]) Cannot return sensitivity matrix with an infinite weight or one vector pairCannot align zero length primary vectorsCBAxes must be vectors of length 3.AssertionErrorApply this rotation to a set of vectors. If the original frame rotates to the final frame by this rotation, then its application to a vector can be seen in two ways: - As a projection of vector components expressed in the final frame to the original frame. - As the physical rotation of a vector being glued to the original frame as it rotates. In this case the vector components are expressed in the original frame before and after the rotation. In terms of rotation matrices, this application is the same as ``self.as_matrix() @ vectors``. Parameters ---------- vectors : array_like, shape (3,) or (N, 3) Each `vectors[i]` represents a vector in 3D space. A single vector can either be specified with shape `(3, )` or `(1, 3)`. The number of rotations and number of vectors given must follow standard numpy broadcasting rules: either one of them equals unity or they both equal each other. inverse : boolean, optional If True then the inverse of the rotation(s) is applied to the input vectors. Default is False. Returns ------- rotated_vectors : ndarray, shape (3,) or (N, 3) Result of applying rotation on input vectors. Shape depends on the following cases: - If object contains a single rotation (as opposed to a stack with a single rotation) and a single vector is specified with shape ``(3,)``, then `rotated_vectors` has shape ``(3,)``. - In all other cases, `rotated_vectors` has shape ``(N, 3)``, where ``N`` is either the number of rotations or vectors. Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np Single rotation applied on a single vector: >>> vector = np.array([1, 0, 0]) >>> r = R.from_rotvec([0, 0, np.pi/2]) >>> r.as_matrix() array([[ 2.22044605e-16, -1.00000000e+00, 0.00000000e+00], [ 1.00000000e+00, 2.22044605e-16, 0.00000000e+00], [ 0.00000000e+00, 0.00000000e+00, 1.00000000e+00]]) >>> r.apply(vector) array([2.22044605e-16, 1.00000000e+00, 0.00000000e+00]) >>> r.apply(vector).shape (3,) Single rotation applied on multiple vectors: >>> vectors = np.array([ ... [1, 0, 0], ... [1, 2, 3]]) >>> r = R.from_rotvec([0, 0, np.pi/4]) >>> r.as_matrix() array([[ 0.70710678, -0.70710678, 0. ], [ 0.70710678, 0.70710678, 0. ], [ 0. , 0. , 1. ]]) >>> r.apply(vectors) array([[ 0.70710678, 0.70710678, 0. ], [-0.70710678, 2.12132034, 3. ]]) >>> r.apply(vectors).shape (2, 3) Multiple rotations on a single vector: >>> r = R.from_rotvec([[0, 0, np.pi/4], [np.pi/2, 0, 0]]) >>> vector = np.array([1,2,3]) >>> r.as_matrix() array([[[ 7.07106781e-01, -7.07106781e-01, 0.00000000e+00], [ 7.07106781e-01, 7.07106781e-01, 0.00000000e+00], [ 0.00000000e+00, 0.00000000e+00, 1.00000000e+00]], [[ 1.00000000e+00, 0.00000000e+00, 0.00000000e+00], [ 0.00000000e+00, 2.22044605e-16, -1.00000000e+00], [ 0.00000000e+00, 1.00000000e+00, 2.22044605e-16]]]) >>> r.apply(vector) array([[-0.70710678, 2.12132034, 3. ], [ 1. , -3. , 2. ]]) >>> r.apply(vector).shape (2, 3) Multiple rotations on multiple vectors. Each rotation is applied on the corresponding vector: >>> r = R.from_euler('zxy', [ ... [0, 0, 90], ... [45, 30, 60]], degrees=True) >>> vectors = [ ... [1, 2, 3], ... [1, 0, -1]] >>> r.apply(vectors) array([[ 3. , 2. , -1. ], [-0.09026039, 1.11237244, -0.86860844]]) >>> r.apply(vectors).shape (2, 3) It is also possible to apply the inverse rotation: >>> r = R.from_euler('zxy', [ ... [0, 0, 90], ... [45, 30, 60]], degrees=True) >>> vectors = [ ... [1, 2, 3], ... [1, 0, -1]] >>> r.apply(vectors, inverse=True) array([[-3. , 2. , 1. ], [ 1.09533535, -0.8365163 , 0.3169873 ]]) .Interpolate rotations. Compute the interpolated rotations at the given `times`. Parameters ---------- times : array_like Times to compute the interpolations at. Can be a scalar or 1-dimensional. Returns ------- interpolated_rotation : `Rotation` instance Object containing the rotations computed at given `times`. Rotation.align_vectors(cls, a, b, weights=None, return_sensitivity=False) Estimate a rotation to optimally align two sets of vectors. Find a rotation between frames A and B which best aligns a set of vectors `a` and `b` observed in these frames. The following loss function is minimized to solve for the rotation matrix :math:`C`: .. math:: L(C) = \frac{1}{2} \sum_{i = 1}^{n} w_i \lVert \mathbf{a}_i - C \mathbf{b}_i \rVert^2 , where :math:`w_i`'s are the `weights` corresponding to each vector. The rotation is estimated with Kabsch algorithm [1]_, and solves what is known as the "pointing problem", or "Wahba's problem" [2]_. There are two special cases. The first is if a single vector is given for `a` and `b`, in which the shortest distance rotation that aligns `b` to `a` is returned. The second is when one of the weights is infinity. In this case, the shortest distance rotation between the primary infinite weight vectors is calculated as above. Then, the rotation about the aligned primary vectors is calculated such that the secondary vectors are optimally aligned per the above loss function. The result is the composition of these two rotations. The result via this process is the same as the Kabsch algorithm as the corresponding weight approaches infinity in the limit. For a single secondary vector this is known as the "align-constrain" algorithm [3]_. For both special cases (single vectors or an infinite weight), the sensitivity matrix does not have physical meaning and an error will be raised if it is requested. For an infinite weight, the primary vectors act as a constraint with perfect alignment, so their contribution to `rssd` will be forced to 0 even if they are of different lengths. Parameters ---------- a : array_like, shape (3,) or (N, 3) Vector components observed in initial frame A. Each row of `a` denotes a vector. b : array_like, shape (3,) or (N, 3) Vector components observed in another frame B. Each row of `b` denotes a vector. weights : array_like shape (N,), optional Weights describing the relative importance of the vector observations. If None (default), then all values in `weights` are assumed to be 1. One and only one weight may be infinity, and weights must be positive. return_sensitivity : bool, optional Whether to return the sensitivity matrix. See Notes for details. Default is False. Returns ------- rotation : `Rotation` instance Best estimate of the rotation that transforms `b` to `a`. rssd : float Stands for "root sum squared distance". Square root of the weighted sum of the squared distances between the given sets of vectors after alignment. It is equal to ``sqrt(2 * minimum_loss)``, where ``minimum_loss`` is the loss function evaluated for the found optimal rotation. Note that the result will also be weighted by the vectors' magnitudes, so perfectly aligned vector pairs will have nonzero `rssd` if they are not of the same length. So, depending on the use case it may be desirable to normalize the input vectors to unit length before calling this method. sensitivity_matrix : ndarray, shape (3, 3) Sensitivity matrix of the estimated rotation estimate as explained in Notes. Returned only when `return_sensitivity` is True. Not valid if aligning a single pair of vectors or if there is an infinite weight, in which cases an error will be raised. Notes ----- The sensitivity matrix gives the sensitivity of the estimated rotation to small perturbations of the vector measurements. Specifically we consider the rotation estimate error as a small rotation vector of frame A. The sensitivity matrix is proportional to the covariance of this rotation vector assuming that the vectors in `a` was measured with errors significantly less than their lengths. To get the true covariance matrix, the returned sensitivity matrix must be multiplied by harmonic mean [4]_ of variance in each observation. Note that `weights` are supposed to be inversely proportional to the observation variances to get consistent results. For example, if all vectors are measured with the same accuracy of 0.01 (`weights` must be all equal), then you should multiple the sensitivity matrix by 0.01**2 to get the covariance. Refer to [5]_ for more rigorous discussion of the covariance estimation. See [6]_ for more discussion of the pointing problem and minimal proper pointing. References ---------- .. [1] https://en.wikipedia.org/wiki/Kabsch_algorithm .. [2] https://en.wikipedia.org/wiki/Wahba%27s_problem .. [3] Magner, Robert, "Extending target tracking capabilities through trajectory and momentum setpoint optimization." Small Satellite Conference, 2018. .. [4] https://en.wikipedia.org/wiki/Harmonic_mean .. [5] F. Landis Markley, "Attitude determination using vector observations: a fast optimal matrix algorithm", Journal of Astronautical Sciences, Vol. 41, No.2, 1993, pp. 261-280. .. [6] Bar-Itzhack, Itzhack Y., Daniel Hershkowitz, and Leiba Rodman, "Pointing in Real Euclidean Space", Journal of Guidance, Control, and Dynamics, Vol. 20, No. 5, 1997, pp. 916-922. Examples -------- >>> import numpy as np >>> from scipy.spatial.transform import Rotation as R Here we run the baseline Kabsch algorithm to best align two sets of vectors, where there is noise on the last two vector measurements of the ``b`` set: >>> a = [[0, 1, 0], [0, 1, 1], [0, 1, 1]] >>> b = [[1, 0, 0], [1, 1.1, 0], [1, 0.9, 0]] >>> rot, rssd, sens = R.align_vectors(a, b, return_sensitivity=True) >>> rot.as_matrix() array([[0., 0., 1.], [1., 0., 0.], [0., 1., 0.]]) When we apply the rotation to ``b``, we get vectors close to ``a``: >>> rot.apply(b) array([[0. , 1. , 0. ], [0. , 1. , 1.1], [0. , 1. , 0.9]]) The error for the first vector is 0, and for the last two the error is magnitude 0.1. The `rssd` is the square root of the sum of the weighted squared errors, and the default weights are all 1, so in this case the `rssd` is calculated as ``sqrt(1 * 0**2 + 1 * 0.1**2 + 1 * (-0.1)**2) = 0.141421356237308`` >>> a - rot.apply(b) array([[ 0., 0., 0. ], [ 0., 0., -0.1], [ 0., 0., 0.1]]) >>> np.sqrt(np.sum(np.ones(3) @ (a - rot.apply(b))**2)) 0.141421356237308 >>> rssd 0.141421356237308 The sensitivity matrix for this example is as follows: >>> sens array([[0.2, 0. , 0.], [0. , 1.5, 1.], [0. , 1. , 1.]]) Special case 1: Find a minimum rotation between single vectors: >>> a = [1, 0, 0] >>> b = [0, 1, 0] >>> rot, _ = R.align_vectors(a, b) >>> rot.as_matrix() array([[0., 1., 0.], [-1., 0., 0.], [0., 0., 1.]]) >>> rot.apply(b) array([1., 0., 0.]) Special case 2: One infinite weight. Here we find a rotation between primary and secondary vectors that can align exactly: >>> a = [[0, 1, 0], [0, 1, 1]] >>> b = [[1, 0, 0], [1, 1, 0]] >>> rot, _ = R.align_vectors(a, b, weights=[np.inf, 1]) >>> rot.as_matrix() array([[0., 0., 1.], [1., 0., 0.], [0., 1., 0.]]) >>> rot.apply(b) array([[0., 1., 0.], [0., 1., 1.]]) Here the secondary vectors must be best-fit: >>> a = [[0, 1, 0], [0, 1, 1]] >>> b = [[1, 0, 0], [1, 2, 0]] >>> rot, _ = R.align_vectors(a, b, weights=[np.inf, 1]) >>> rot.as_matrix() array([[0., 0., 1.], [1., 0., 0.], [0., 1., 0.]]) >>> rot.apply(b) array([[0., 1., 0.], [0., 1., 2.]])Rotation.random(cls, num=None, rng=None) Generate uniformly distributed rotations. Parameters ---------- num : int or None, optional Number of random rotations to generate. If None (default), then a single rotation is generated. rng : `numpy.random.Generator`, optional Pseudorandom number generator state. When `rng` is None, a new `numpy.random.Generator` is created using entropy from the operating system. Types other than `numpy.random.Generator` are passed to `numpy.random.default_rng` to instantiate a `Generator`. Returns ------- random_rotation : `Rotation` instance Contains a single rotation if `num` is None. Otherwise contains a stack of `num` rotations. Notes ----- This function is optimized for efficiently sampling random rotation matrices in three dimensions. For generating random rotation matrices in higher dimensions, see `scipy.stats.special_ortho_group`. Examples -------- >>> from scipy.spatial.transform import Rotation as R Sample a single rotation: >>> R.random().as_euler('zxy', degrees=True) array([-110.5976185 , 55.32758512, 76.3289269 ]) # random Sample a stack of rotations: >>> R.random(5).as_euler('zxy', degrees=True) array([[-110.5976185 , 55.32758512, 76.3289269 ], # random [ -91.59132005, -14.3629884 , -93.91933182], [ 25.23835501, 45.02035145, -121.67867086], [ -51.51414184, -15.29022692, -172.46870023], [ -81.63376847, -27.39521579, 2.60408416]]) See Also -------- scipy.stats.special_ortho_groupRotation.identity(cls, num=None) Get identity rotation(s). Composition with the identity rotation has no effect. Parameters ---------- num : int or None, optional Number of identity rotations to generate. If None (default), then a single rotation is generated. Returns ------- identity : Rotation object The identity rotation.Rotation.create_group(cls, group, axis='Z') Create a 3D rotation group. Parameters ---------- group : string The name of the group. Must be one of 'I', 'O', 'T', 'Dn', 'Cn', where `n` is a positive integer. The groups are: * I: Icosahedral group * O: Octahedral group * T: Tetrahedral group * D: Dicyclic group * C: Cyclic group axis : integer The cyclic rotation axis. Must be one of ['X', 'Y', 'Z'] (or lowercase). Default is 'Z'. Ignored for groups 'I', 'O', and 'T'. Returns ------- rotation : `Rotation` instance Object containing the elements of the rotation group. Notes ----- This method generates rotation groups only. The full 3-dimensional point groups [PointGroups]_ also contain reflections. References ---------- .. [PointGroups] `Point groups `_ on Wikipedia.Rotation.reduce(self, left=None, right=None, return_indices=False) Reduce this rotation with the provided rotation groups. Reduction of a rotation ``p`` is a transformation of the form ``q = l * p * r``, where ``l`` and ``r`` are chosen from `left` and `right` respectively, such that rotation ``q`` has the smallest magnitude. If `left` and `right` are rotation groups representing symmetries of two objects rotated by ``p``, then ``q`` is the rotation of the smallest magnitude to align these objects considering their symmetries. Parameters ---------- left : `Rotation` instance, optional Object containing the left rotation(s). Default value (None) corresponds to the identity rotation. right : `Rotation` instance, optional Object containing the right rotation(s). Default value (None) corresponds to the identity rotation. return_indices : bool, optional Whether to return the indices of the rotations from `left` and `right` used for reduction. Returns ------- reduced : `Rotation` instance Object containing reduced rotations. left_best, right_best: integer ndarray Indices of elements from `left` and `right` used for reduction.Rotation.mean(self, weights=None) Get the mean of the rotations. The mean used is the chordal L2 mean (also called the projected or induced arithmetic mean) [1]_. If ``A`` is a set of rotation matrices, then the mean ``M`` is the rotation matrix that minimizes the following loss function: .. math:: L(M) = \sum_{i = 1}^{n} w_i \lVert \mathbf{A}_i - \mathbf{M} \rVert^2 , where :math:`w_i`'s are the `weights` corresponding to each matrix. Parameters ---------- weights : array_like shape (N,), optional Weights describing the relative importance of the rotations. If None (default), then all values in `weights` are assumed to be equal. Returns ------- mean : `Rotation` instance Object containing the mean of the rotations in the current instance. References ---------- .. [1] Hartley, Richard, et al., "Rotation Averaging", International Journal of Computer Vision 103, 2013, pp. 267-305. Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> r = R.from_euler('zyx', [[0, 0, 0], ... [1, 0, 0], ... [0, 1, 0], ... [0, 0, 1]], degrees=True) >>> r.mean().as_euler('zyx', degrees=True) array([0.24945696, 0.25054542, 0.24945696])Rotation.approx_equal(self, Rotation other, atol=None, degrees=False) Determine if another rotation is approximately equal to this one. Equality is measured by calculating the smallest angle between the rotations, and checking to see if it is smaller than `atol`. Parameters ---------- other : `Rotation` instance Object containing the rotations to measure against this one. atol : float, optional The absolute angular tolerance, below which the rotations are considered equal. If not given, then set to 1e-8 radians by default. degrees : bool, optional If True and `atol` is given, then `atol` is measured in degrees. If False (default), then atol is measured in radians. Returns ------- approx_equal : ndarray or bool Whether the rotations are approximately equal, bool if object contains a single rotation and ndarray if object contains multiple rotations. Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np >>> p = R.from_quat([0, 0, 0, 1]) >>> q = R.from_quat(np.eye(4)) >>> p.approx_equal(q) array([False, False, False, True]) Approximate equality for a single rotation: >>> p.approx_equal(q[0]) FalseRotation.magnitude(self) Get the magnitude(s) of the rotation(s). Returns ------- magnitude : ndarray or float Angle(s) in radians, float if object contains a single rotation and ndarray if object contains multiple rotations. The magnitude will always be in the range [0, pi]. Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np >>> r = R.from_quat(np.eye(4)) >>> r.as_quat() array([[ 1., 0., 0., 0.], [ 0., 1., 0., 0.], [ 0., 0., 1., 0.], [ 0., 0., 0., 1.]]) >>> r.magnitude() array([3.14159265, 3.14159265, 3.14159265, 0. ]) Magnitude of a single rotation: >>> r[0].magnitude() 3.141592653589793Rotation.inv(self) Invert this rotation. Composition of a rotation with its inverse results in an identity transformation. Returns ------- inverse : `Rotation` instance Object containing inverse of the rotations in the current instance. Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np Inverting a single rotation: >>> p = R.from_euler('z', 45, degrees=True) >>> q = p.inv() >>> q.as_euler('zyx', degrees=True) array([-45., 0., 0.]) Inverting multiple rotations: >>> p = R.from_rotvec([[0, 0, np.pi/3], [-np.pi/4, 0, 0]]) >>> q = p.inv() >>> q.as_rotvec() array([[-0. , -0. , -1.04719755], [ 0.78539816, -0. , -0. ]])Rotation.apply(self, vectors, inverse=False) Apply this rotation to a set of vectors. If the original frame rotates to the final frame by this rotation, then its application to a vector can be seen in two ways: - As a projection of vector components expressed in the final frame to the original frame. - As the physical rotation of a vector being glued to the original frame as it rotates. In this case the vector components are expressed in the original frame before and after the rotation. In terms of rotation matrices, this application is the same as ``self.as_matrix() @ vectors``. Parameters ---------- vectors : array_like, shape (3,) or (N, 3) Each `vectors[i]` represents a vector in 3D space. A single vector can either be specified with shape `(3, )` or `(1, 3)`. The number of rotations and number of vectors given must follow standard numpy broadcasting rules: either one of them equals unity or they both equal each other. inverse : boolean, optional If True then the inverse of the rotation(s) is applied to the input vectors. Default is False. Returns ------- rotated_vectors : ndarray, shape (3,) or (N, 3) Result of applying rotation on input vectors. Shape depends on the following cases: - If object contains a single rotation (as opposed to a stack with a single rotation) and a single vector is specified with shape ``(3,)``, then `rotated_vectors` has shape ``(3,)``. - In all other cases, `rotated_vectors` has shape ``(N, 3)``, where ``N`` is either the number of rotations or vectors. Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np Single rotation applied on a single vector: >>> vector = np.array([1, 0, 0]) >>> r = R.from_rotvec([0, 0, np.pi/2]) >>> r.as_matrix() array([[ 2.22044605e-16, -1.00000000e+00, 0.00000000e+00], [ 1.00000000e+00, 2.22044605e-16, 0.00000000e+00], [ 0.00000000e+00, 0.00000000e+00, 1.00000000e+00]]) >>> r.apply(vector) array([2.22044605e-16, 1.00000000e+00, 0.00000000e+00]) >>> r.apply(vector).shape (3,) Single rotation applied on multiple vectors: >>> vectors = np.array([ ... [1, 0, 0], ... [1, 2, 3]]) >>> r = R.from_rotvec([0, 0, np.pi/4]) >>> r.as_matrix() array([[ 0.70710678, -0.70710678, 0. ], [ 0.70710678, 0.70710678, 0. ], [ 0. , 0. , 1. ]]) >>> r.apply(vectors) array([[ 0.70710678, 0.70710678, 0. ], [-0.70710678, 2.12132034, 3. ]]) >>> r.apply(vectors).shape (2, 3) Multiple rotations on a single vector: >>> r = R.from_rotvec([[0, 0, np.pi/4], [np.pi/2, 0, 0]]) >>> vector = np.array([1,2,3]) >>> r.as_matrix() array([[[ 7.07106781e-01, -7.07106781e-01, 0.00000000e+00], [ 7.07106781e-01, 7.07106781e-01, 0.00000000e+00], [ 0.00000000e+00, 0.00000000e+00, 1.00000000e+00]], [[ 1.00000000e+00, 0.00000000e+00, 0.00000000e+00], [ 0.00000000e+00, 2.22044605e-16, -1.00000000e+00], [ 0.00000000e+00, 1.00000000e+00, 2.22044605e-16]]]) >>> r.apply(vector) array([[-0.70710678, 2.12132034, 3. ], [ 1. , -3. , 2. ]]) >>> r.apply(vector).shape (2, 3) Multiple rotations on multiple vectors. Each rotation is applied on the corresponding vector: >>> r = R.from_euler('zxy', [ ... [0, 0, 90], ... [45, 30, 60]], degrees=True) >>> vectors = [ ... [1, 2, 3], ... [1, 0, -1]] >>> r.apply(vectors) array([[ 3. , 2. , -1. ], [-0.09026039, 1.11237244, -0.86860844]]) >>> r.apply(vectors).shape (2, 3) It is also possible to apply the inverse rotation: >>> r = R.from_euler('zxy', [ ... [0, 0, 90], ... [45, 30, 60]], degrees=True) >>> vectors = [ ... [1, 2, 3], ... [1, 0, -1]] >>> r.apply(vectors, inverse=True) array([[-3. , 2. , 1. ], [ 1.09533535, -0.8365163 , 0.3169873 ]])Rotation.concatenate(cls, rotations) Concatenate a sequence of `Rotation` objects into a single object. This is useful if you want to, for example, take the mean of a set of rotations and need to pack them into a single object to do so. Parameters ---------- rotations : sequence of `Rotation` objects The rotations to concatenate. If a single `Rotation` object is passed in, a copy is returned. Returns ------- concatenated : `Rotation` instance The concatenated rotations. Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> r1 = R.from_rotvec([0, 0, 1]) >>> r2 = R.from_rotvec([0, 0, 2]) >>> rc = R.concatenate([r1, r2]) >>> rc.as_rotvec() array([[0., 0., 1.], [0., 0., 2.]]) >>> rc.mean().as_rotvec() array([0., 0., 1.5]) Concatenation of a split rotation recovers the original object. >>> rs = [r for r in rc] >>> R.concatenate(rs).as_rotvec() array([[0., 0., 1.], [0., 0., 2.]]) Note that it may be simpler to create the desired rotations by passing in a single list of the data during initialization, rather then by concatenating: >>> R.from_rotvec([[0, 0, 1], [0, 0, 2]]).as_rotvec() array([[0., 0., 1.], [0., 0., 2.]]) Notes ----- .. versionadded:: 1.8.0Rotation.as_mrp(self) Represent as Modified Rodrigues Parameters (MRPs). MRPs are a 3 dimensional vector co-directional to the axis of rotation and whose magnitude is equal to ``tan(theta / 4)``, where ``theta`` is the angle of rotation (in radians) [1]_. MRPs have a singularity at 360 degrees which can be avoided by ensuring the angle of rotation does not exceed 180 degrees, i.e. switching the direction of the rotation when it is past 180 degrees. This function will always return MRPs corresponding to a rotation of less than or equal to 180 degrees. Returns ------- mrps : ndarray, shape (3,) or (N, 3) Shape depends on shape of inputs used for initialization. References ---------- .. [1] Shuster, M. D. "A Survey of Attitude Representations", The Journal of Astronautical Sciences, Vol. 41, No.4, 1993, pp. 475-476 Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np Represent a single rotation: >>> r = R.from_rotvec([0, 0, np.pi]) >>> r.as_mrp() array([0. , 0. , 1. ]) >>> r.as_mrp().shape (3,) Represent a stack with a single rotation: >>> r = R.from_euler('xyz', [[180, 0, 0]], degrees=True) >>> r.as_mrp() array([[1. , 0. , 0. ]]) >>> r.as_mrp().shape (1, 3) Represent multiple rotations: >>> r = R.from_rotvec([[np.pi/2, 0, 0], [0, 0, np.pi/2]]) >>> r.as_mrp() array([[0.41421356, 0. , 0. ], [0. , 0. , 0.41421356]]) >>> r.as_mrp().shape (2, 3) Notes ----- .. versionadded:: 1.6.0Rotation.as_davenport(self, axes, order, degrees=False) Represent as Davenport angles. Any orientation can be expressed as a composition of 3 elementary rotations. For both Euler angles and Davenport angles, consecutive axes must be are orthogonal (``axis2`` is orthogonal to both ``axis1`` and ``axis3``). For Euler angles, there is an additional relationship between ``axis1`` or ``axis3``, with two possibilities: - ``axis1`` and ``axis3`` are also orthogonal (asymmetric sequence) - ``axis1 == axis3`` (symmetric sequence) For Davenport angles, this last relationship is relaxed [1]_, and only the consecutive orthogonal axes requirement is maintained. A slightly modified version of the algorithm from [2]_ has been used to calculate Davenport angles for the rotation about a given sequence of axes. Davenport angles, just like Euler angles, suffer from the problem of gimbal lock [3]_, where the representation loses a degree of freedom and it is not possible to determine the first and third angles uniquely. In this case, a warning is raised, and the third angle is set to zero. Note however that the returned angles still represent the correct rotation. Parameters ---------- axes : array_like, shape (3,) or ([1 or 2 or 3], 3) Axis of rotation, if one dimensional. If two dimensional, describes the sequence of axes for rotations, where each axes[i, :] is the ith axis. If more than one axis is given, then the second axis must be orthogonal to both the first and third axes. order : string If it belongs to the set {'e', 'extrinsic'}, the sequence will be extrinsic. If if belongs to the set {'i', 'intrinsic'}, sequence will be treated as intrinsic. degrees : boolean, optional Returned angles are in degrees if this flag is True, else they are in radians. Default is False. Returns ------- angles : ndarray, shape (3,) or (N, 3) Shape depends on shape of inputs used to initialize object. The returned angles are in the range: - First angle belongs to [-180, 180] degrees (both inclusive) - Third angle belongs to [-180, 180] degrees (both inclusive) - Second angle belongs to a set of size 180 degrees, given by: ``[-abs(lambda), 180 - abs(lambda)]``, where ``lambda`` is the angle between the first and third axes. References ---------- .. [1] Shuster, Malcolm & Markley, Landis. (2003). Generalization of the Euler Angles. Journal of the Astronautical Sciences. 51. 123-132. 10.1007/BF03546304. .. [2] Bernardes E, Viollet S (2022) Quaternion to Euler angles conversion: A direct, general and computationally efficient method. PLoS ONE 17(11): e0276302. 10.1371/journal.pone.0276302 .. [3] https://en.wikipedia.org/wiki/Gimbal_lock#In_applied_mathematics Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np Davenport angles are a generalization of Euler angles, when we use the canonical basis axes: >>> ex = [1, 0, 0] >>> ey = [0, 1, 0] >>> ez = [0, 0, 1] Represent a single rotation: >>> r = R.from_rotvec([0, 0, np.pi/2]) >>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True) array([90., 0., 0.]) >>> r.as_euler('zxy', degrees=True) array([90., 0., 0.]) >>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True).shape (3,) Represent a stack of single rotation: >>> r = R.from_rotvec([[0, 0, np.pi/2]]) >>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True) array([[90., 0., 0.]]) >>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True).shape (1, 3) Represent multiple rotations in a single object: >>> r = R.from_rotvec([ ... [0, 0, 90], ... [45, 0, 0]], degrees=True) >>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True) array([[90., 0., 0.], [ 0., 45., 0.]]) >>> r.as_davenport([ez, ex, ey], 'extrinsic', degrees=True).shape (2, 3)Rotation.as_euler(self, seq, degrees=False) Represent as Euler angles. Any orientation can be expressed as a composition of 3 elementary rotations. Once the axis sequence has been chosen, Euler angles define the angle of rotation around each respective axis [1]_. The algorithm from [2]_ has been used to calculate Euler angles for the rotation about a given sequence of axes. Euler angles suffer from the problem of gimbal lock [3]_, where the representation loses a degree of freedom and it is not possible to determine the first and third angles uniquely. In this case, a warning is raised, and the third angle is set to zero. Note however that the returned angles still represent the correct rotation. Parameters ---------- seq : string, length 3 3 characters belonging to the set {'X', 'Y', 'Z'} for intrinsic rotations, or {'x', 'y', 'z'} for extrinsic rotations [1]_. Adjacent axes cannot be the same. Extrinsic and intrinsic rotations cannot be mixed in one function call. degrees : boolean, optional Returned angles are in degrees if this flag is True, else they are in radians. Default is False. Returns ------- angles : ndarray, shape (3,) or (N, 3) Shape depends on shape of inputs used to initialize object. The returned angles are in the range: - First angle belongs to [-180, 180] degrees (both inclusive) - Third angle belongs to [-180, 180] degrees (both inclusive) - Second angle belongs to: - [-90, 90] degrees if all axes are different (like xyz) - [0, 180] degrees if first and third axes are the same (like zxz) References ---------- .. [1] https://en.wikipedia.org/wiki/Euler_angles#Definition_by_intrinsic_rotations .. [2] Bernardes E, Viollet S (2022) Quaternion to Euler angles conversion: A direct, general and computationally efficient method. PLoS ONE 17(11): e0276302. https://doi.org/10.1371/journal.pone.0276302 .. [3] https://en.wikipedia.org/wiki/Gimbal_lock#In_applied_mathematics Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np Represent a single rotation: >>> r = R.from_rotvec([0, 0, np.pi/2]) >>> r.as_euler('zxy', degrees=True) array([90., 0., 0.]) >>> r.as_euler('zxy', degrees=True).shape (3,) Represent a stack of single rotation: >>> r = R.from_rotvec([[0, 0, np.pi/2]]) >>> r.as_euler('zxy', degrees=True) array([[90., 0., 0.]]) >>> r.as_euler('zxy', degrees=True).shape (1, 3) Represent multiple rotations in a single object: >>> r = R.from_rotvec([ ... [0, 0, np.pi/2], ... [0, -np.pi/3, 0], ... [np.pi/4, 0, 0]]) >>> r.as_euler('zxy', degrees=True) array([[ 90., 0., 0.], [ 0., 0., -60.], [ 0., 45., 0.]]) >>> r.as_euler('zxy', degrees=True).shape (3, 3)Rotation._as_euler_from_matrix(self, seq, degrees=False) Represent as Euler angles. Any orientation can be expressed as a composition of 3 elementary rotations. Once the axis sequence has been chosen, Euler angles define the angle of rotation around each respective axis [1]_. The algorithm from [2]_ has been used to calculate Euler angles for the rotation about a given sequence of axes. Euler angles suffer from the problem of gimbal lock [3]_, where the representation loses a degree of freedom and it is not possible to determine the first and third angles uniquely. In this case, a warning is raised, and the third angle is set to zero. Note however that the returned angles still represent the correct rotation. Parameters ---------- seq : string, length 3 3 characters belonging to the set {'X', 'Y', 'Z'} for intrinsic rotations, or {'x', 'y', 'z'} for extrinsic rotations [1]_. Adjacent axes cannot be the same. Extrinsic and intrinsic rotations cannot be mixed in one function call. degrees : boolean, optional Returned angles are in degrees if this flag is True, else they are in radians. Default is False. Returns ------- angles : ndarray, shape (3,) or (N, 3) Shape depends on shape of inputs used to initialize object. The returned angles are in the range: - First angle belongs to [-180, 180] degrees (both inclusive) - Third angle belongs to [-180, 180] degrees (both inclusive) - Second angle belongs to: - [-90, 90] degrees if all axes are different (like xyz) - [0, 180] degrees if first and third axes are the same (like zxz) References ---------- .. [1] https://en.wikipedia.org/wiki/Euler_angles#Definition_by_intrinsic_rotations .. [2] Malcolm D. Shuster, F. Landis Markley, "General formula for extraction the Euler angles", Journal of guidance, control, and dynamics, vol. 29.1, pp. 215-221. 2006 .. [3] https://en.wikipedia.org/wiki/Gimbal_lock#In_applied_mathematicsRotation._compute_euler(self, seq, degrees, algorithm)Rotation.as_rotvec(self, degrees=False) Represent as rotation vectors. A rotation vector is a 3 dimensional vector which is co-directional to the axis of rotation and whose norm gives the angle of rotation [1]_. Parameters ---------- degrees : boolean, optional Returned magnitudes are in degrees if this flag is True, else they are in radians. Default is False. .. versionadded:: 1.7.0 Returns ------- rotvec : ndarray, shape (3,) or (N, 3) Shape depends on shape of inputs used for initialization. References ---------- .. [1] https://en.wikipedia.org/wiki/Axis%E2%80%93angle_representation#Rotation_vector Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np Represent a single rotation: >>> r = R.from_euler('z', 90, degrees=True) >>> r.as_rotvec() array([0. , 0. , 1.57079633]) >>> r.as_rotvec().shape (3,) Represent a rotation in degrees: >>> r = R.from_euler('YX', (-90, -90), degrees=True) >>> s = r.as_rotvec(degrees=True) >>> s array([-69.2820323, -69.2820323, -69.2820323]) >>> np.linalg.norm(s) 120.00000000000001 Represent a stack with a single rotation: >>> r = R.from_quat([[0, 0, 1, 1]]) >>> r.as_rotvec() array([[0. , 0. , 1.57079633]]) >>> r.as_rotvec().shape (1, 3) Represent multiple rotations in a single object: >>> r = R.from_quat([[0, 0, 1, 1], [1, 1, 0, 1]]) >>> r.as_rotvec() array([[0. , 0. , 1.57079633], [1.35102172, 1.35102172, 0. ]]) >>> r.as_rotvec().shape (2, 3)Rotation.as_matrix(self) Represent as rotation matrix. 3D rotations can be represented using rotation matrices, which are 3 x 3 real orthogonal matrices with determinant equal to +1 [1]_. Returns ------- matrix : ndarray, shape (3, 3) or (N, 3, 3) Shape depends on shape of inputs used for initialization. References ---------- .. [1] https://en.wikipedia.org/wiki/Rotation_matrix#In_three_dimensions Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np Represent a single rotation: >>> r = R.from_rotvec([0, 0, np.pi/2]) >>> r.as_matrix() array([[ 2.22044605e-16, -1.00000000e+00, 0.00000000e+00], [ 1.00000000e+00, 2.22044605e-16, 0.00000000e+00], [ 0.00000000e+00, 0.00000000e+00, 1.00000000e+00]]) >>> r.as_matrix().shape (3, 3) Represent a stack with a single rotation: >>> r = R.from_quat([[1, 1, 0, 0]]) >>> r.as_matrix() array([[[ 0., 1., 0.], [ 1., 0., 0.], [ 0., 0., -1.]]]) >>> r.as_matrix().shape (1, 3, 3) Represent multiple rotations: >>> r = R.from_rotvec([[np.pi/2, 0, 0], [0, 0, np.pi/2]]) >>> r.as_matrix() array([[[ 1.00000000e+00, 0.00000000e+00, 0.00000000e+00], [ 0.00000000e+00, 2.22044605e-16, -1.00000000e+00], [ 0.00000000e+00, 1.00000000e+00, 2.22044605e-16]], [[ 2.22044605e-16, -1.00000000e+00, 0.00000000e+00], [ 1.00000000e+00, 2.22044605e-16, 0.00000000e+00], [ 0.00000000e+00, 0.00000000e+00, 1.00000000e+00]]]) >>> r.as_matrix().shape (2, 3, 3) Notes ----- This function was called as_dcm before. .. versionadded:: 1.4.0Rotation.as_quat(self, canonical=False, *, scalar_first=False) Represent as quaternions. Rotations in 3 dimensions can be represented using unit norm quaternions [1]_. The 4 components of a quaternion are divided into a scalar part ``w`` and a vector part ``(x, y, z)`` and can be expressed from the angle ``theta`` and the axis ``n`` of a rotation as follows:: w = cos(theta / 2) x = sin(theta / 2) * n_x y = sin(theta / 2) * n_y z = sin(theta / 2) * n_z There are 2 conventions to order the components in a quaternion: - scalar-first order -- ``(w, x, y, z)`` - scalar-last order -- ``(x, y, z, w)`` The choice is controlled by `scalar_first` argument. By default, it is False and the scalar-last order is used. The mapping from quaternions to rotations is two-to-one, i.e. quaternions ``q`` and ``-q``, where ``-q`` simply reverses the sign of each component, represent the same spatial rotation. Parameters ---------- canonical : `bool`, default False Whether to map the redundant double cover of rotation space to a unique "canonical" single cover. If True, then the quaternion is chosen from {q, -q} such that the w term is positive. If the w term is 0, then the quaternion is chosen such that the first nonzero term of the x, y, and z terms is positive. scalar_first : bool, optional Whether the scalar component goes first or last. Default is False, i.e. the scalar-last order is used. Returns ------- quat : `numpy.ndarray`, shape (4,) or (N, 4) Shape depends on shape of inputs used for initialization. References ---------- .. [1] https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np A rotation can be represented as a quaternion with either scalar-last (default) or scalar-first component order. This is shown for a single rotation: >>> r = R.from_matrix(np.eye(3)) >>> r.as_quat() array([0., 0., 0., 1.]) >>> r.as_quat(scalar_first=True) array([1., 0., 0., 0.]) When multiple rotations are stored in a single Rotation object, the result will be a 2-dimensional array: >>> r = R.from_rotvec([[np.pi, 0, 0], [0, 0, np.pi/2]]) >>> r.as_quat().shape (2, 4) Quaternions can be mapped from a redundant double cover of the rotation space to a canonical representation with a positive w term. >>> r = R.from_quat([0, 0, 0, -1]) >>> r.as_quat() array([0. , 0. , 0. , -1.]) >>> r.as_quat(canonical=True) array([0. , 0. , 0. , 1.])Rotation.from_mrp(cls, mrp) Initialize from Modified Rodrigues Parameters (MRPs). MRPs are a 3 dimensional vector co-directional to the axis of rotation and whose magnitude is equal to ``tan(theta / 4)``, where ``theta`` is the angle of rotation (in radians) [1]_. MRPs have a singularity at 360 degrees which can be avoided by ensuring the angle of rotation does not exceed 180 degrees, i.e. switching the direction of the rotation when it is past 180 degrees. Parameters ---------- mrp : array_like, shape (N, 3) or (3,) A single vector or a stack of vectors, where `mrp[i]` gives the ith set of MRPs. Returns ------- rotation : `Rotation` instance Object containing the rotations represented by input MRPs. References ---------- .. [1] Shuster, M. D. "A Survey of Attitude Representations", The Journal of Astronautical Sciences, Vol. 41, No.4, 1993, pp. 475-476 Notes ----- .. versionadded:: 1.6.0 Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np Initialize a single rotation: >>> r = R.from_mrp([0, 0, 1]) >>> r.as_euler('xyz', degrees=True) array([0. , 0. , 180. ]) >>> r.as_euler('xyz').shape (3,) Initialize multiple rotations in one object: >>> r = R.from_mrp([ ... [0, 0, 1], ... [1, 0, 0]]) >>> r.as_euler('xyz', degrees=True) array([[0. , 0. , 180. ], [180.0 , 0. , 0. ]]) >>> r.as_euler('xyz').shape (2, 3) It is also possible to have a stack of a single rotation: >>> r = R.from_mrp([[0, 0, np.pi/2]]) >>> r.as_euler('xyz').shape (1, 3)Rotation.from_davenport(cls, axes, order, angles, degrees=False) Initialize from Davenport angles. Rotations in 3-D can be represented by a sequence of 3 rotations around a sequence of axes. The three rotations can either be in a global frame of reference (extrinsic) or in a body centred frame of reference (intrinsic), which is attached to, and moves with, the object under rotation [1]_. For both Euler angles and Davenport angles, consecutive axes must be are orthogonal (``axis2`` is orthogonal to both ``axis1`` and ``axis3``). For Euler angles, there is an additional relationship between ``axis1`` or ``axis3``, with two possibilities: - ``axis1`` and ``axis3`` are also orthogonal (asymmetric sequence) - ``axis1 == axis3`` (symmetric sequence) For Davenport angles, this last relationship is relaxed [2]_, and only the consecutive orthogonal axes requirement is maintained. Parameters ---------- axes : array_like, shape (3,) or ([1 or 2 or 3], 3) Axis of rotation, if one dimensional. If two dimensional, describes the sequence of axes for rotations, where each axes[i, :] is the ith axis. If more than one axis is given, then the second axis must be orthogonal to both the first and third axes. order : string If it is equal to 'e' or 'extrinsic', the sequence will be extrinsic. If it is equal to 'i' or 'intrinsic', sequence will be treated as intrinsic. angles : float or array_like, shape (N,) or (N, [1 or 2 or 3]) Euler angles specified in radians (`degrees` is False) or degrees (`degrees` is True). For a single axis, `angles` can be: - a single value - array_like with shape (N,), where each `angle[i]` corresponds to a single rotation - array_like with shape (N, 1), where each `angle[i, 0]` corresponds to a single rotation For 2 and 3 axes, `angles` can be: - array_like with shape (W,) where `W` is the number of rows of `axes`, which corresponds to a single rotation with `W` axes - array_like with shape (N, W) where each `angle[i]` corresponds to a sequence of Davenport angles describing a single rotation degrees : bool, optional If True, then the given angles are assumed to be in degrees. Default is False. Returns ------- rotation : `Rotation` instance Object containing the rotation represented by the sequence of rotations around given axes with given angles. References ---------- .. [1] https://en.wikipedia.org/wiki/Euler_angles#Definition_by_intrinsic_rotations .. [2] Shuster, Malcolm & Markley, Landis. (2003). Generalization of the Euler Angles. Journal of the Astronautical Sciences. 51. 123-132. 10.1007/BF03546304. Examples -------- >>> from scipy.spatial.transform import Rotation as R Davenport angles are a generalization of Euler angles, when we use the canonical basis axes: >>> ex = [1, 0, 0] >>> ey = [0, 1, 0] >>> ez = [0, 0, 1] Initialize a single rotation with a given axis sequence: >>> axes = [ez, ey, ex] >>> r = R.from_davenport(axes, 'extrinsic', [90, 0, 0], degrees=True) >>> r.as_quat().shape (4,) It is equivalent to Euler angles in this case: >>> r.as_euler('zyx', degrees=True) array([90., 0., -0.]) Initialize multiple rotations in one object: >>> r = R.from_davenport(axes, 'extrinsic', [[90, 45, 30], [35, 45, 90]], degrees=True) >>> r.as_quat().shape (2, 4) Using only one or two axes is also possible: >>> r = R.from_davenport([ez, ex], 'extrinsic', [[90, 45], [35, 45]], degrees=True) >>> r.as_quat().shape (2, 4) Non-canonical axes are possible, and they do not need to be normalized, as long as consecutive axes are orthogonal: >>> e1 = [2, 0, 0] >>> e2 = [0, 1, 0] >>> e3 = [1, 0, 1] >>> axes = [e1, e2, e3] >>> r = R.from_davenport(axes, 'extrinsic', [90, 45, 30], degrees=True) >>> r.as_quat() [ 0.701057, 0.430459, -0.092296, 0.560986]Rotation.from_euler(cls, seq, angles, degrees=False) Initialize from Euler angles. Rotations in 3-D can be represented by a sequence of 3 rotations around a sequence of axes. In theory, any three axes spanning the 3-D Euclidean space are enough. In practice, the axes of rotation are chosen to be the basis vectors. The three rotations can either be in a global frame of reference (extrinsic) or in a body centred frame of reference (intrinsic), which is attached to, and moves with, the object under rotation [1]_. Parameters ---------- seq : string Specifies sequence of axes for rotations. Up to 3 characters belonging to the set {'X', 'Y', 'Z'} for intrinsic rotations, or {'x', 'y', 'z'} for extrinsic rotations. Extrinsic and intrinsic rotations cannot be mixed in one function call. angles : float or array_like, shape (N,) or (N, [1 or 2 or 3]) Euler angles specified in radians (`degrees` is False) or degrees (`degrees` is True). For a single character `seq`, `angles` can be: - a single value - array_like with shape (N,), where each `angle[i]` corresponds to a single rotation - array_like with shape (N, 1), where each `angle[i, 0]` corresponds to a single rotation For 2- and 3-character wide `seq`, `angles` can be: - array_like with shape (W,) where `W` is the width of `seq`, which corresponds to a single rotation with `W` axes - array_like with shape (N, W) where each `angle[i]` corresponds to a sequence of Euler angles describing a single rotation degrees : bool, optional If True, then the given angles are assumed to be in degrees. Default is False. Returns ------- rotation : `Rotation` instance Object containing the rotation represented by the sequence of rotations around given axes with given angles. References ---------- .. [1] https://en.wikipedia.org/wiki/Euler_angles#Definition_by_intrinsic_rotations Examples -------- >>> from scipy.spatial.transform import Rotation as R Initialize a single rotation along a single axis: >>> r = R.from_euler('x', 90, degrees=True) >>> r.as_quat().shape (4,) Initialize a single rotation with a given axis sequence: >>> r = R.from_euler('zyx', [90, 45, 30], degrees=True) >>> r.as_quat().shape (4,) Initialize a stack with a single rotation around a single axis: >>> r = R.from_euler('x', [90], degrees=True) >>> r.as_quat().shape (1, 4) Initialize a stack with a single rotation with an axis sequence: >>> r = R.from_euler('zyx', [[90, 45, 30]], degrees=True) >>> r.as_quat().shape (1, 4) Initialize multiple elementary rotations in one object: >>> r = R.from_euler('x', [90, 45, 30], degrees=True) >>> r.as_quat().shape (3, 4) Initialize multiple rotations in one object: >>> r = R.from_euler('zyx', [[90, 45, 30], [35, 45, 90]], degrees=True) >>> r.as_quat().shape (2, 4)Rotation.from_rotvec(cls, rotvec, degrees=False) Initialize from rotation vectors. A rotation vector is a 3 dimensional vector which is co-directional to the axis of rotation and whose norm gives the angle of rotation [1]_. Parameters ---------- rotvec : array_like, shape (N, 3) or (3,) A single vector or a stack of vectors, where `rot_vec[i]` gives the ith rotation vector. degrees : bool, optional If True, then the given magnitudes are assumed to be in degrees. Default is False. .. versionadded:: 1.7.0 Returns ------- rotation : `Rotation` instance Object containing the rotations represented by input rotation vectors. References ---------- .. [1] https://en.wikipedia.org/wiki/Axis%E2%80%93angle_representation#Rotation_vector Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np Initialize a single rotation: >>> r = R.from_rotvec(np.pi/2 * np.array([0, 0, 1])) >>> r.as_rotvec() array([0. , 0. , 1.57079633]) >>> r.as_rotvec().shape (3,) Initialize a rotation in degrees, and view it in degrees: >>> r = R.from_rotvec(45 * np.array([0, 1, 0]), degrees=True) >>> r.as_rotvec(degrees=True) array([ 0., 45., 0.]) Initialize multiple rotations in one object: >>> r = R.from_rotvec([ ... [0, 0, np.pi/2], ... [np.pi/2, 0, 0]]) >>> r.as_rotvec() array([[0. , 0. , 1.57079633], [1.57079633, 0. , 0. ]]) >>> r.as_rotvec().shape (2, 3) It is also possible to have a stack of a single rotation: >>> r = R.from_rotvec([[0, 0, np.pi/2]]) >>> r.as_rotvec().shape (1, 3)Rotation.from_matrix(cls, matrix) Initialize from rotation matrix. Rotations in 3 dimensions can be represented with 3 x 3 orthogonal matrices [1]_. If the input is not orthogonal, an approximation is created by orthogonalizing the input matrix using the method described in [2]_, and then converting the orthogonal rotation matrices to quaternions using the algorithm described in [3]_. Matrices must be right-handed. Parameters ---------- matrix : array_like, shape (N, 3, 3) or (3, 3) A single matrix or a stack of matrices, where ``matrix[i]`` is the i-th matrix. Returns ------- rotation : `Rotation` instance Object containing the rotations represented by the rotation matrices. References ---------- .. [1] https://en.wikipedia.org/wiki/Rotation_matrix#In_three_dimensions .. [2] https://en.wikipedia.org/wiki/Orthogonal_Procrustes_problem .. [3] F. Landis Markley, "Unit Quaternion from Rotation Matrix", Journal of guidance, control, and dynamics vol. 31.2, pp. 440-442, 2008. Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np Initialize a single rotation: >>> r = R.from_matrix([ ... [0, -1, 0], ... [1, 0, 0], ... [0, 0, 1]]) >>> r.single True >>> r.as_matrix().shape (3, 3) Initialize multiple rotations in a single object: >>> r = R.from_matrix([ ... [ ... [0, -1, 0], ... [1, 0, 0], ... [0, 0, 1], ... ], ... [ ... [1, 0, 0], ... [0, 0, -1], ... [0, 1, 0], ... ]]) >>> r.as_matrix().shape (2, 3, 3) >>> r.single False >>> len(r) 2 If input matrices are not special orthogonal (orthogonal with determinant equal to +1), then a special orthogonal estimate is stored: >>> a = np.array([ ... [0, -0.5, 0], ... [0.5, 0, 0], ... [0, 0, 0.5]]) >>> np.linalg.det(a) 0.125 >>> r = R.from_matrix(a) >>> matrix = r.as_matrix() >>> matrix array([[ 0., -1., 0.], [ 1., 0., 0.], [ 0., 0., 1.]]) >>> np.linalg.det(matrix) 1.0 It is also possible to have a stack containing a single rotation: >>> r = R.from_matrix([[ ... [0, -1, 0], ... [1, 0, 0], ... [0, 0, 1]]]) >>> r.as_matrix() array([[[ 0., -1., 0.], [ 1., 0., 0.], [ 0., 0., 1.]]]) >>> r.as_matrix().shape (1, 3, 3) Notes ----- This function was called from_dcm before. .. versionadded:: 1.4.0Rotation.from_quat(cls, quat, *, scalar_first=False) Initialize from quaternions. Rotations in 3 dimensions can be represented using unit norm quaternions [1]_. The 4 components of a quaternion are divided into a scalar part ``w`` and a vector part ``(x, y, z)`` and can be expressed from the angle ``theta`` and the axis ``n`` of a rotation as follows:: w = cos(theta / 2) x = sin(theta / 2) * n_x y = sin(theta / 2) * n_y z = sin(theta / 2) * n_z There are 2 conventions to order the components in a quaternion: - scalar-first order -- ``(w, x, y, z)`` - scalar-last order -- ``(x, y, z, w)`` The choice is controlled by `scalar_first` argument. By default, it is False and the scalar-last order is assumed. Advanced users may be interested in the "double cover" of 3D space by the quaternion representation [2]_. As of version 1.11.0, the following subset (and only this subset) of operations on a `Rotation` ``r`` corresponding to a quaternion ``q`` are guaranteed to preserve the double cover property: ``r = Rotation.from_quat(q)``, ``r.as_quat(canonical=False)``, ``r.inv()``, and composition using the ``*`` operator such as ``r*r``. Parameters ---------- quat : array_like, shape (N, 4) or (4,) Each row is a (possibly non-unit norm) quaternion representing an active rotation. Each quaternion will be normalized to unit norm. scalar_first : bool, optional Whether the scalar component goes first or last. Default is False, i.e. the scalar-last order is assumed. Returns ------- rotation : `Rotation` instance Object containing the rotations represented by input quaternions. References ---------- .. [1] https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation .. [2] Hanson, Andrew J. "Visualizing quaternions." Morgan Kaufmann Publishers Inc., San Francisco, CA. 2006. Examples -------- >>> from scipy.spatial.transform import Rotation as R A rotation can be initialzied from a quaternion with the scalar-last (default) or scalar-first component order as shown below: >>> r = R.from_quat([0, 0, 0, 1]) >>> r.as_matrix() array([[1., 0., 0.], [0., 1., 0.], [0., 0., 1.]]) >>> r = R.from_quat([1, 0, 0, 0], scalar_first=True) >>> r.as_matrix() array([[1., 0., 0.], [0., 1., 0.], [0., 0., 1.]]) It is possible to initialize multiple rotations in a single object by passing a 2-dimensional array: >>> r = R.from_quat([ ... [1, 0, 0, 0], ... [0, 0, 0, 1] ... ]) >>> r.as_quat() array([[1., 0., 0., 0.], [0., 0., 0., 1.]]) >>> r.as_quat().shape (2, 4) It is also possible to have a stack of a single rotation: >>> r = R.from_quat([[0, 0, 0, 1]]) >>> r.as_quat() array([[0., 0., 0., 1.]]) >>> r.as_quat().shape (1, 4) Quaternions are normalized before initialization. >>> r = R.from_quat([0, 0, 1, 1]) >>> r.as_quat() array([0. , 0. , 0.70710678, 0.70710678])Composition of quaternions.hkA ^1!!yy{{|881A|7!*!;nA 1A$ (!1 =b *AQm6A+Qa b Qd(/1 1N#Tqa 2T"KqBcRs!4|2Q *AQ8?q $F!4t61rV1F"D !1$ja,at81E5A 1 V1Aq^A 4z+Q )1A 9HCs!;c *AQ(!1 5c *AQ*1%q 5as#S *AQ18 #1LV1A IQ M5 2T$kA *AQ M#Q LJd#RyZqc &avQBTQ G1F,avWA!qqq)Qg[q)Qg[A)B!N RvQcqRvQcq Bk! Bk! 1F"Cs! *AQ-4AZq 1F"Cs! *AQ-4AZq 1G3aq *AQ 55F#Q d*AT7%q qyaq 3a HIQ #Q 89A BfBcA #SQas#U!1Cs%q #SQas#U!1Cs&  A Eq Eq Eq Rt1BgQit4t1gQkTQgQkTQgQkTQgQ.cT Rxq)1D5&a "G1BhauCqe5HCs!1XRs!1 4t1 4q 4uA E! %r2Q 4q d*AWHBat:Q  ! 1A 1 uCqA vSQ 9Kq 1d"AX 4vV1Cs! *AQ 83a bQc! b wfCqj,31G1 wfAS3aqj>EQ$+6$c! rQhbjr$a Bd!82TQ 4rQat:QauD !aA:HqN 51 qQa 1  2XQa&$d#Zqwb,A> "Q(F!1 q G5 !7"Be1F!4qe3at1E 4q 6! 2XQaAB rqHE Dq Dq Dq 4q 4qt:QfJgU!AJA%&l "HAQ 7&"CwfBcA *AQ&-QgQ"'s!$E)EQaa *AQ%,AQcq N!89A'q7$ha 1 3at1D 'a 3avZwe1d$ A*+@ A81HF! 1 Rxq 6xs#SfASwas#Q *AQ07qa 6A fAQ  a(vQa! G5 F!7!1 vSrBgRs"G2WBa1F"Cr e67!5 e67!5 e67!5 e3avRq 1 3at1D 'a 3avZwe1cAZ A6(& F&3a &AV7!4t3a *AQ07qa 6Cq V1Ba  rAQ 2T%s! "F!52Qb *AQH#6! 7"Bj#S$ar*Bd!4uA &c#2V1A^1A 3a)1 s%rAV1A !5Rr !(vQa"'! " G5 AU'%s! AU'%s! AU'%s! AU'%s#RwauCq'%s! XQa wcARrBaRrBaAU%r81Cr2WAU#QAU%wauCs"G1EAAU%wauCs"G1EAAU%wauCs"G1EAAU%wauCs"G1EAAU%wauCs"G1EAAU%wauCs"G1EAAU%r81A qAQ 1 3at1D 'a 3avZwe1b2A-.@ s!66arAiq ITa KqmAr$hfHD!js Rxqaq81Ay1 6s! V82S  6A V4q 6Bd&b3a *AQ.5QfA 61 *AQ,31F! 6xs#SfBcA *AQ,3!6 6s! V1A vV3b6rCqj-4AV1 81 =AB +Qaxq 1EQfAA Q ) 'a %Set rotation(s) at given index(es) from object. Parameters ---------- indexer : index, slice, or index array Specifies which rotation(s) to replace. A single indexer must be specified, i.e. as if indexing a 1 dimensional array or list. value : `Rotation` instance The rotations to set. Raises ------ TypeError if the instance was created as a single rotation. Notes ----- .. versionadded:: 1.8.0 Rotation.__getitem__(self, indexer) Extract rotation(s) at given index(es) from object. Create a new `Rotation` instance containing a subset of rotations stored in this object. Parameters ---------- indexer : index, slice, or index array Specifies which rotation(s) to extract. A single indexer must be specified, i.e. as if indexing a 1 dimensional array or list. Returns ------- rotation : `Rotation` instance Contains - a single rotation, if `indexer` is a single index - a stack of rotation(s), if `indexer` is a slice, or and index array. Raises ------ TypeError if the instance was created as a single rotation. Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> rs = R.from_quat([ ... [1, 1, 0, 0], ... [0, 1, 0, 1], ... [1, 1, -1, 0]]) # These quats are normalized >>> rs.as_quat() array([[ 0.70710678, 0.70710678, 0. , 0. ], [ 0. , 0.70710678, 0. , 0.70710678], [ 0.57735027, 0.57735027, -0.57735027, 0. ]]) Indexing using a single index: >>> a = rs[0] >>> a.as_quat() array([0.70710678, 0.70710678, 0. , 0. ]) Array slicing: >>> b = rs[1:3] >>> b.as_quat() array([[ 0. , 0.70710678, 0. , 0.70710678], [ 0.57735027, 0.57735027, -0.57735027, 0. ]]) List comprehension to split each rotation into its own object: >>> c = [r for r in rs] >>> print([r.as_quat() for r in c]) [array([ 0.70710678, 0.70710678, 0. , 0. ]), array([ 0. , 0.70710678, 0. , 0.70710678]), array([ 0.57735027, 0.57735027, -0.57735027, 0. ])] Concatenation of split rotations will recover the original object: >>> R.concatenate([a, b]).as_quat() array([[ 0.70710678, 0.70710678, 0. , 0. ], [ 0. , 0.70710678, 0. , 0.70710678], [ 0.57735027, 0.57735027, -0.57735027, 0. ]])Rotation.__pow__(self, float n, modulus) Compose this rotation with itself `n` times. Composition of a rotation ``p`` with itself can be extended to non-integer ``n`` by considering the power ``n`` to be a scale factor applied to the angle of rotation about the rotation's fixed axis. The expression ``q = p ** n`` can also be expressed as ``q = Rotation.from_rotvec(n * p.as_rotvec())``. If ``n`` is negative, then the rotation is inverted before the power is applied. In other words, ``p ** -abs(n) == p.inv() ** abs(n)``. Parameters ---------- n : float The number of times to compose the rotation with itself. modulus : None This overridden argument is not applicable to Rotations and must be ``None``. Returns ------- power : `Rotation` instance If the input Rotation ``p`` contains ``N`` multiple rotations, then the output will contain ``N`` rotations where the ``i`` th rotation is equal to ``p[i] ** n`` Notes ----- For example, a power of 2 will double the angle of rotation, and a power of 0.5 will halve the angle. There are three notable cases: if ``n == 1`` then the original rotation is returned, if ``n == 0`` then the identity rotation is returned, and if ``n == -1`` then ``p.inv()`` is returned. Note that fractional powers ``n`` which effectively take a root of rotation, do so using the shortest path smallest representation of that angle (the principal root). This means that powers of ``n`` and ``1/n`` are not necessarily inverses of each other. For example, a 0.5 power of a +240 degree rotation will be calculated as the 0.5 power of a -120 degree rotation, with the result being a rotation of -60 rather than +120 degrees. Examples -------- >>> from scipy.spatial.transform import Rotation as R Raising a rotation to a power: >>> p = R.from_rotvec([1, 0, 0]) >>> q = p ** 2 >>> q.as_rotvec() array([2., 0., 0.]) >>> r = p ** 0.5 >>> r.as_rotvec() array([0.5, 0., 0.]) Inverse powers do not necessarily cancel out: >>> p = R.from_rotvec([0, 0, 120], degrees=True) >>> ((p ** 2) ** 0.5).as_rotvec(degrees=True) array([ -0., -0., -60.])Rotation.__mul__(self, Rotation other) Compose this rotation with the other. If `p` and `q` are two rotations, then the composition of 'q followed by p' is equivalent to `p * q`. In terms of rotation matrices, the composition can be expressed as ``p.as_matrix() @ q.as_matrix()``. Parameters ---------- other : `Rotation` instance Object containing the rotations to be composed with this one. Note that rotation compositions are not commutative, so ``p * q`` is generally different from ``q * p``. Returns ------- composition : `Rotation` instance This function supports composition of multiple rotations at a time. The following cases are possible: - Either ``p`` or ``q`` contains a single rotation. In this case `composition` contains the result of composing each rotation in the other object with the single rotation. - Both ``p`` and ``q`` contain ``N`` rotations. In this case each rotation ``p[i]`` is composed with the corresponding rotation ``q[i]`` and `output` contains ``N`` rotations. Examples -------- >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np Composition of two single rotations: >>> p = R.from_quat([0, 0, 1, 1]) >>> q = R.from_quat([1, 0, 0, 1]) >>> p.as_matrix() array([[ 0., -1., 0.], [ 1., 0., 0.], [ 0., 0., 1.]]) >>> q.as_matrix() array([[ 1., 0., 0.], [ 0., 0., -1.], [ 0., 1., 0.]]) >>> r = p * q >>> r.as_matrix() array([[0., 0., 1.], [1., 0., 0.], [0., 1., 0.]]) Composition of two objects containing equal number of rotations: >>> p = R.from_quat([[0, 0, 1, 1], [1, 0, 0, 1]]) >>> q = R.from_rotvec([[np.pi/4, 0, 0], [-np.pi/4, 0, np.pi/4]]) >>> p.as_quat() array([[0. , 0. , 0.70710678, 0.70710678], [0.70710678, 0. , 0. , 0.70710678]]) >>> q.as_quat() array([[ 0.38268343, 0. , 0. , 0.92387953], [-0.37282173, 0. , 0.37282173, 0.84971049]]) >>> r = p * q >>> r.as_quat() array([[ 0.27059805, 0.27059805, 0.65328148, 0.65328148], [ 0.33721128, -0.26362477, 0.26362477, 0.86446082]])Rotation.__len__(self) Number of rotations contained in this object. Multiple rotations can be stored in a single instance. Returns ------- length : int Number of rotations stored in object. 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