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H@dL$%(LeIHEЃteH{@HCHHtHC@xtOHuHL0H}Ht xt:HUdH+%(u8H [A\]fHUI$fDfEEff.UHAUIATSHG|G|Lg@HMA$=wA$LH{@HCHHtHC@x A$x A$tatE1LH(tsI}H;=C|H1[A\A]]HH8LUU܅t@UUfkC|HxpIHtHtH_L(H~H0H9t I9ID$p]RbHDxt[HdIEH5%LH8-C|H[A\A]]fH9H5H8IEfHC@HXHtKHLA1Muff.HT H9I9HI9uBHt/HQH01DHH9L;l uHI9HuL;-HtI|$pID$pHlLHHHH9;HuH\H9&HI9HuI9dUHSHHdH%(HE1HGH9HEH5xHUH}H]HtbfHuHH)Ex H1ۅx0t]HEdH+%(H]{H}Hud1fDHuHEHtxuH@HHEtHEhs)1g-fff.UHHdH%(HE1HuHEDt?H}Ht xtHrHUdH+%(uÐf1UHAWAVAUATSHWxdH%(HE1I9HXpIH@pHL{A=wALc(MtA$=wA$HuLHE^H}Htxu f.Ht L9c(I~pI^pHtx MtAx AMtA$x A$t=HEdH+%(H[A\A]A^A_]AExtE1E1+DHEdH+%(uYHL[A\A]A^A_]CL3L(hCfDLHff.UHSHHHHtHCx .H{8HtHC8x H{@HCHHtHC@x H{ HtHC x H{hHtHChx H{pHtHCpx H{PHtHCPx H{XHtHCXx H{`HtHC`xt H]@H]fDfD{fDkfD[fDKfD;*fD+__setstate_cython__tupleExpected %s, got %.200s__pyx_unpickle_RigidTransform'bool''char''signed char''unsigned char''short''unsigned short''int''unsigned int''long''unsigned long''long long''unsigned long long''complex float''float''complex double''double''complex long double''long double'a structPython objecta pointera stringunparsable format string__loader__loader__file__origin__package__parent__path__submodule_search_locationsneeds an argument%.200s() %stakes no keyword argumentstakes no arguments%.200s() %s (%zd given)takes exactly one argumentcannot pickle '%.200s' object_cython_3_1_6Bad call flags for CyFunctionkeywords must be strings__pyx_capi__cannot import name %Sbuiltinscython_runtime__builtins__does not match__len____getitem____setitem____mul____pow__numpyflatiterbroadcastndarraygenericnumberunsignedintegerinexactcomplexfloatingflexiblecharacterufuncscipy._cyutilitymemoryview_allocate_bufferarray_cwrappermemoryview_cwrappermemview_sliceslice_memviewslicepybuffer_indexint (__Pyx_memviewslice *)transpose_memslicememoryview_fromsliceget_slice_from_memviewslice_copymemoryview_copymemoryview_copy_from_sliceget_best_orderslice_get_sizefill_contig_strides_arraycopy_data_to_temp_err_extents_err_dimint (PyObject *, PyObject *)_errint (void)_err_no_memorymemoryview_copy_contentsbroadcast_leadingrefcount_copyingrefcount_objects_in_slice_slice_assign_scalarbuffer dtypean integer is requiredthrow__module__func_doc__doc__func_name__name____qualname__func_dict__dict__func_globals__globals__func_closure__closure__func_code__code__func_defaults__defaults____kwdefaults____annotations___is_coroutine__dictoffset____vectorcalloffset____weaklistoffset____reduce__name of the generatorgi_frameFrame of the generatorgi_runninggi_yieldfromgi_codesendclose__reduce_ex__single_cython_3_1_6.generator_rigid_transform%.200s() takes %.8s %zd positional argument%.1s (%zd given)scipy/spatial/transform/_rigid_transform.pyxscipy.spatial.transform._rigid_transform.RigidTransform.from_matrixscipy.spatial.transform._rigid_transform._create_transformation_matrixscipy.spatial.transform._rigid_transform.RigidTransform.__repr__'%.200s' object does not support slice %.10sstrings are too large to concatscipy.spatial.transform._rigid_transform.RigidTransform.__init__Acquisition count is %d (line %d)_ARRAY_API is not PyCapsule objectmodule compiled against ABI version 0x%x but this version of numpy is 0x%xmodule was compiled against NumPy C-API version 0x%x (NumPy 1.23) but the running NumPy has C-API version 0x%x. 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../../../../../../usr/lib/python3/dist-packages/numpy/__init__.cython-30.pxdscipy.spatial.transform._rigid_transform._create_skew_matrixscipy.spatial.transform._rigid_transform._compute_se3_exp_translation_transformscipy.spatial.transform._rigid_transform._normalize_dual_quaternionscipy.spatial.transform._rigid_transform.normalize_dual_quaterniontoo many values to unpack (expected %zd)need more than %zd value%.1s to unpackscipy.spatial.transform._rigid_transform.RigidTransform.from_rotationscipy.spatial.transform._rigid_transform.RigidTransform.identityscipy.spatial.transform._rigid_transform.RigidTransform.from_translationscipy.spatial.transform._rigid_transform.RigidTransform.from_componentsscipy.spatial.transform._rigid_transform.RigidTransform.from_exp_coordsscipy.spatial.transform._rigid_transform.RigidTransform.from_dual_quatscipy.spatial.transform._rigid_transform.RigidTransform.concatenate.genexprscipy.spatial.transform._rigid_transform.RigidTransform.concatenatelocal variable '%s' referenced before assignmentgenerator raised StopIterationscipy.spatial.transform._rigid_transform.RigidTransform.as_componentsscipy.spatial.transform._rigid_transform.RigidTransform.as_matrixscipy.spatial.transform._rigid_transform._compute_se3_log_translation_transformscipy.spatial.transform._rigid_transform.RigidTransform.as_exp_coordsscipy.spatial.transform._rigid_transform.RigidTransform.as_dual_quatscipy.spatial.transform._rigid_transform.RigidTransform.rotation.__get__scipy.spatial.transform._rigid_transform.RigidTransform.translation.__get__scipy.spatial.transform._rigid_transform.RigidTransform.__len__scipy.spatial.transform._rigid_transform.RigidTransform.__getitem__scipy.spatial.transform._rigid_transform.RigidTransform.__setitem__Subscript deletion not supported by %.200sArgument '%.200s' has incorrect type (expected %.200s, got %.200s)scipy.spatial.transform._rigid_transform.RigidTransform.__mul__scipy.spatial.transform._rigid_transform.RigidTransform.__pow__scipy.spatial.transform._rigid_transform.RigidTransform.invscipy.spatial.transform._rigid_transform.RigidTransform.applyscipy.spatial.transform._rigid_transform.RigidTransform.__reduce_cython__'NoneType' object is not subscriptablescipy.spatial.transform._rigid_transform.__pyx_unpickle_RigidTransform__set_statescipy.spatial.transform._rigid_transform.RigidTransform.__setstate_cython__scipy.spatial.transform._rigid_transform.__pyx_unpickle_RigidTransformscipy.spatial.transform._rigid_transform.RigidTransform.__pow__() takes 3 arguments but 2 were given__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 objectInterpreter change detected - this module can only be loaded into one interpreter per process.__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 callsShared Cython type %.200s is not a type objectShared Cython type %.200s has the wrong size, try recompilingUnexpected 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 expectedBig-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 ')'%s() got multiple values for keyword argument '%U'%.200s() keywords must be stringsbase class '%.200s' is not a heap typeextension type '%.200s' has no __dict__ slot, but base type '%.200s' has: either add 'cdef dict __dict__' to the extension type or add '__slots__ = [...]' to the base type%.200s.%.200s is not a type object%.200s.%.200s size changed, may indicate binary incompatibility. Expected %zd from C header, got %zd from PyObjectinvalid vtable found for imported typeunbound method %.200S() needs an argument%.200s does not export expected C function %.200sC function %.200s.%.200s has wrong signature (expected %.500s, got %.500s)Unable to initialize pickling for %.200s 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 BaseExceptioncannot 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 stringModule '_rigid_transform' has already been imported. Re-initialisation is not supported.scipy.spatial.transform._rigid_transformcompile time Python version %d.%d of module '%.100s' %s runtime version %d.%dint (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 *)init scipy.spatial.transform._rigid_transformBuffer has wrong number of dimensions (expected %d, got %d)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.memviewslice is already initialized!__int__ returned non-int (type %.200s). The ability to return an instance of a strict subclass of int is deprecated, and may be removed in a future version of Python.__int__ returned non-int (type %.200s)generator ignored GeneratorExitqualified 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.Return the rotation component of the transform. A transform is a composition of a rotation and a translation, such that when applied to a vector, the vector is first rotated and then translated. This property returns the rotation part of the transform. Returns ------- rotation : `Rotation` instance A single rotation or a stack of rotations. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np The rotation component is extracted from the transform: >>> t = np.array([1, 0, 0]) >>> r = R.random(3) >>> tf = Tf.from_components(t, r) >>> np.allclose(tf.rotation.as_matrix(), r.as_matrix()) True Return the translation component of the transform. A transform is a composition of a rotation and a translation, such that when applied to a vector, the vector is first rotated and then translated. This property returns the translation part of the transform. Returns ------- translation : numpy.ndarray, shape (N, 3) or (3,) A single translation vector or a stack of translation vectors. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np The translation component is extracted from the transform: >>> t = np.array([[1, 0, 0], [2, 0, 0], [3, 0, 0]]) >>> r = R.random() >>> tf = Tf.from_components(t, r) >>> np.allclose(tf.translation, t) True Whether this instance represents a single transform. Single transforms are not subscriptable, and do not have a length. Returns ------- single : bool True if this instance represents a single transform, False otherwise. _cython_3_1_6._common_types_metatype_cython_3_1_6.cython_function_or_methodscipy.spatial.transform._rigid_transform.__pyx_scope_struct__genexprscipy.spatial.transform._rigid_transform.RigidTransformRigid transform in 3 dimensions. This class provides an interface to initialize from and represent rigid transforms (rotation and translation) in 3D space. In different fields, this type of transform may be referred to as "*pose*" (especially in robotics), "*extrinsic parameters*", or the "*model matrix*" (especially in computer graphics), but the core concept is the same: a rotation and translation describing the orientation of one 3D coordinate frame relative to another. Mathematically, these transforms belong to the Special Euclidean group SE(3), which encodes rotation (SO(3)) plus translation. The following operations on rigid transforms are supported: - Application on vectors - Transformation composition - Transformation inversion - Transformation indexing Note that coordinate systems must be right-handed. Because of this, this class more precisely represents *proper* rigid transformations in SE(3) rather than rigid transforms in E(3) more generally [1]_. Indexing within a transform is supported since multiple transforms can be stored within a single `RigidTransform` instance. To create `RigidTransform` objects use ``from_...`` methods (see examples below). ``RigidTransform(...)`` is not supposed to be instantiated directly. For rigorous introductions to rigid transforms, see [2]_, [3]_, and [4]_. Attributes ---------- single rotation translation Methods ------- __len__ __getitem__ __mul__ __pow__ from_matrix from_rotation from_translation from_components from_exp_coords from_dual_quat as_matrix as_components as_exp_coords as_dual_quat concatenate apply inv identity References ---------- .. [1] https://en.wikipedia.org/wiki/Rigid_transformation .. [2] https://motion.cs.illinois.edu/RoboticSystems/CoordinateTransformations.html .. [3] https://www.brainvoyager.com/bv/doc/UsersGuide/CoordsAndTransforms/SpatialTransformationMatrices.html .. [4] Kevin M. Lynch and Frank C. Park, "Modern Robotics: Mechanics, Planning, and Control" Chapter 3.3, 2017, Cambridge University Press. https://hades.mech.northwestern.edu/images/2/25/MR-v2.pdf#page=107.31 .. [5] Paul Furgale, "Representing Robot Pose: The good, the bad, and the ugly", June 9, 2014. https://rpg.ifi.uzh.ch/docs/teaching/2024/FurgaleTutorial.pdf Notes ----- .. versionadded:: 1.16.0 Examples -------- A `RigidTransform` 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. **Notation Conventions and Composition** The notation here largely follows the convention defined in [5]_. When we name transforms, we read the subscripts from right to left. So ``tf_A_B`` represents a transform A <- B and can be interpreted as: - the coordinates and orientation of B relative to A - the transformation of points from B to A - the pose of B described in A's coordinate system .. parsed-literal:: :class: highlight-none tf_A_B ^ ^ | | | --- from B | ----- to A When composing transforms, the order is important. Transforms are not commutative, so in general ``tf_A_B * tf_B_C`` is not the same as ``tf_B_C * tf_A_B``. Transforms are composed and applied to vectors right-to-left. So ``(tf_A_B * tf_B_C).apply(p_C)`` is the same as ``tf_A_B.apply(tf_B_C.apply(p_C))``. When composed, transforms should be ordered such that the multiplication operator is surrounded by a single frame, so the frame "cancels out" and the outside frames are left. In the example below, B cancels out and the outside frames A and C are left. Or to put it another way, A <- C is the same as A <- B <- C. .. parsed-literal:: :class: highlight-none ----------- B cancels out | | v v tf_A_C = tf_A_B * tf_B_C ^ ^ | | ------------ to A, from C are left When we notate vectors, we write the subscript of the frame that the vector is defined in. So ``p_B`` means the point ``p`` defined in frame B. To transform this point from frame B to coordinates in frame A, we apply the transform ``tf_A_B`` to the vector, lining things up such that the notated B frames are next to each other and "cancel out". .. parsed-literal:: :class: highlight-none ------------ B cancels out | | v v p_A = tf_A_B.apply(p_B) ^ | -------------- A is left **Visualization** >>> from scipy.spatial.transform import RigidTransform as Tf >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np The following function can be used to plot transforms with Matplotlib by showing how they transform the standard x, y, z coordinate axes: >>> import matplotlib.pyplot as plt >>> colors = ("#FF6666", "#005533", "#1199EE") # Colorblind-safe RGB >>> def plot_transformed_axes(ax, tf, name=None, scale=1): ... r = tf.rotation ... t = tf.translation ... loc = np.array([t, t]) ... 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 + t ... ax.text(*text_plot, axlabel.upper(), color=c, ... va="center", ha="center") ... ax.text(*tf.translation, name, color="k", va="center", ha="center", ... bbox={"fc": "w", "alpha": 0.8, "boxstyle": "circle"}) **Defining Frames** Let's work through an example. First, define the "world frame" A, also called the "base frame". All frames are the identity transform from their own perspective. >>> tf_A = Tf.identity() We will visualize a new frame B in A's coordinate system. So we need to define the transform that converts coordinates from frame B to frame A (A <- B). Physically, let's imagine constructing B from A by: 1) Rotating A by +90 degrees around its x-axis. 2) Translating the rotated frame 2 units in A's -x direction. From A's perspective, B is at [-2, 0, 0] and rotated +90 degrees about the x-axis, which is exactly the transform A <- B. >>> t_A_B = np.array([-2, 0, 0]) >>> r_A_B = R.from_euler('xyz', [90, 0, 0], degrees=True) >>> tf_A_B = Tf.from_components(t_A_B, r_A_B) Let's plot these frames. >>> fig, ax = plt.subplots(subplot_kw={"projection": "3d"}) >>> plot_transformed_axes(ax, tf_A, name="tfA") # A plotted in A >>> plot_transformed_axes(ax, tf_A_B, name="tfAB") # B plotted in A >>> ax.set_title("A, B frames with respect to A") >>> ax.set_aspect("equal") >>> ax.figure.set_size_inches(6, 5) >>> plt.show() Now let's visualize a new frame C in B's coordinate system. Let's imagine constructing C from B by: 1) Translating B by 2 units in its +z direction. 2) Rotating B by +30 degrees around its z-axis. >>> t_B_C = np.array([0, 0, 2]) >>> r_B_C = R.from_euler('xyz', [0, 0, 30], degrees=True) >>> tf_B_C = Tf.from_components(t_B_C, r_B_C) In order to plot these frames from a consistent perspective, we need to calculate the transform between A and C. Note that we do not make this transform directly, but instead compose intermediate transforms that let us get from C to A: >>> tf_A_C = tf_A_B * tf_B_C # A <- B <- C Now we can plot these three frames from A's perspective. >>> fig, ax = plt.subplots(subplot_kw={"projection": "3d"}) >>> plot_transformed_axes(ax, tf_A, name="tfA") # A plotted in A >>> plot_transformed_axes(ax, tf_A_B, name="tfAB") # B plotted in A >>> plot_transformed_axes(ax, tf_A_C, name="tfAC") # C plotted in A >>> ax.set_title("A, B, C frames with respect to A") >>> ax.set_aspect("equal") >>> ax.figure.set_size_inches(6, 5) >>> plt.show() **Transforming Vectors** Let's transform a vector from A, to B and C. To do this, we will first invert the transforms we already have from B and C, to A. >>> tf_B_A = tf_A_B.inv() # B <- A >>> tf_C_A = tf_A_C.inv() # C <- A Now we can define a point in A and use the above transforms to get its coordinates in B and C: >>> p1_A = np.array([1, 0, 0]) # +1 in x_A direction >>> p1_B = tf_B_A.apply(p1_A) >>> p1_C = tf_C_A.apply(p1_A) >>> print(p1_A) # Original point 1 in A [1 0 0] >>> print(p1_B) # Point 1 in B [3. 0. 0.] >>> print(p1_C) # Point 1 in C [ 2.59807621 -1.5 -2. ] We can also do the reverse. We define a point in C and transform it to A: >>> p2_C = np.array([0, 1, 0]) # +1 in y_C direction >>> p2_A = tf_A_C.apply(p2_C) >>> print(p2_C) # Original point 2 in C [0 1 0] >>> print(p2_A) # Point 2 in A [-2.5 -2. 0.8660254] Plot the frames with respect to A again, but also plot these two points: >>> fig, ax = plt.subplots(subplot_kw={"projection": "3d"}) >>> plot_transformed_axes(ax, tf_A, name="tfA") # A plotted in A >>> plot_transformed_axes(ax, tf_A_B, name="tfAB") # B plotted in A >>> plot_transformed_axes(ax, tf_A_C, name="tfAC") # C plotted in A >>> ax.scatter(p1_A[0], p1_A[1], p1_A[2], color=colors[0]) # +1 x_A >>> ax.scatter(p2_A[0], p2_A[1], p2_A[2], color=colors[1]) # +1 y_C >>> ax.set_title("A, B, C frames and points with respect to A") >>> ax.set_aspect("equal") >>> ax.figure.set_size_inches(6, 5) >>> plt.show() **Switching Base Frames** Up to this point, we have been visualizing frames from A's perspective. Let's use the transforms we defined to visualize the frames from C's perspective. Now C is the "base frame" or "world frame". All frames are the identity transform from their own perspective. >>> tf_C = Tf.identity() We've already defined the transform C <- A, and can obtain C <- B by inverting the existing transform B <- C. >>> tf_C_B = tf_B_C.inv() # C <- B This lets us plot everything from C's perspective: >>> fig, ax = plt.subplots(subplot_kw={"projection": "3d"}) >>> plot_transformed_axes(ax, tf_C, name="tfC") # C plotted in C >>> plot_transformed_axes(ax, tf_C_B, name="tfCB") # B plotted in C >>> plot_transformed_axes(ax, tf_C_A, name="tfCA") # A plotted in C >>> ax.scatter(p1_C[0], p1_C[1], p1_C[2], color=colors[0]) >>> ax.scatter(p2_C[0], p2_C[1], p2_C[2], color=colors[1]) >>> ax.set_title("A, B, C frames and points with respect to C") >>> ax.set_aspect("equal") >>> ax.figure.set_size_inches(6, 5) >>> plt.show() ?@@@8@?@UUUUUU?^@@@C@ @UUUUUU?@@MbP??O "     G 2 * 00000000010000000000000000000000000000001100000000000 322 322222$44224224442222222222222222 3 3424.4$44224222 342 3 411 41111104`5114114441111111111111111 4 4`41M4:404`5114111 441 47999999999K599K5999999999999999999K579999996899999999999999999 89E577zerosxwherevector_singlevectorvalue must be a RigidTransform objectvalueuse_setstateupdatetranslationstranslationtransforms in second object. transforms in first and transforms to be exactly [0, 0, 0, 1], got throw__test__tant_invswapaxessumstate__spec__singlesinshape__setstate_cython____setstate____set_name__sendselfscipy/spatial/transform/_rigid_transform.pyxscipy.spatial.transform._rigid_transformscalar_firstrotmatrotationsrotation_matrices_rotationrotationrot_vecrollresrepeat__reduce_ex____reduce_cython____reduce__real_partsreal_partrealranger_inv__qualname____pyx_vtable____pyx_unpickle_RigidTransform__pyx_type__pyx_state__pyx_result__pyx_checksum__pyx_PickleErrorpure_translation_quatspoppickleonesnumpy._core.umath failed to importnumpy._core.multiarray failed to importnumpynum_translationsnum_transformsnumnpnormalize_dual_quaternionnormalizenormnextnewaxis__new__ndim__name__ms__module___matrixmatrixmatmul__main__mlinalgisenabled_is_coroutineinverseinvinput must contain RigidTransform objects only_initializingijk,ik->ijidentityhstack__getstate__genexprgc__func__from_translationfrom_rotvecfrom_rotationfrom_quatfrom_matrixfrom_exp_coordsfrom_dual_quatfrom_componentseyeexp_coordsenableemptyeinsumdual_quatsdual_quatdual_partsdual_partdualdtypedisable_dict__dict___create_transformation_matrixcoscopyconcatenate..genexprconcatenatecompose_quatclsclosecline_in_traceback__class_getitem____class__axisatleast_2dasyncio.coroutinesasarrayas_rotvecas_quatas_matrixas_exp_coordsas_dual_quatas_componentsarrayapplyanyalladd_note?) ValueErrorTypeErrorThe number of rotation matrices and translations must be the same.Single transform is not subscriptable.Single transform has no len().Set transform(s) at given index(es) in this object. Parameters ---------- indexer : int or slice or array_like Specifies which transform(s) to replace. A single indexer must be specified, i.e. as if indexing a 1 dimensional array or list. value : `RigidTransform` instance The transform(s) to set. Raises ------ TypeError If the transform is a single transform. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> t = [[0, 0, 0], [1, 0, 0], [2, 0, 0]] # 3 translations >>> tf = Tf.from_translation(t) Set a single transform: >>> tf[0] = Tf.from_translation([9, 9, 9]) >>> tf.translation array([[9., 9., 9.], [1., 0., 0.], [2., 0., 0.]]) RotationRigidTransform.translation.__get__ (line 1883)RigidTransform.rotation.__get__ (line 1850)RigidTransform.inv (line 1667)RigidTransform.invRigidTransform.identity (line 1001)RigidTransform.identityRigidTransform.from_translation (line 671)RigidTransform.from_translationRigidTransform.from_rotation (line 601)RigidTransform.from_rotationRigidTransform.from_matrix (line 522)RigidTransform.from_matrixRigidTransform.from_matrix(RigidTransform.from_exp_coords (line 823)RigidTransform.from_exp_coordsRigidTransform.from_dual_quat (line 927)RigidTransform.from_dual_quatRigidTransform.from_components (line 756)RigidTransform.from_componentsRigidTransform.concatenate (line 1077)RigidTransform.concatenateRigidTransform.as_matrix (line 1115)RigidTransform.as_matrixRigidTransform.as_exp_coords (line 1232)RigidTransform.as_exp_coordsRigidTransform.as_dual_quat (line 1272)RigidTransform.as_dual_quatRigidTransform.as_components (line 1170)RigidTransform.as_componentsRigidTransform.apply (line 1733)RigidTransform.applyRigidTransform.__setstate_cython__RigidTransform.__setitem__ (line 1423)RigidTransform.__reduce_cython__RigidTransform.__pow__ (line 1555)RigidTransform.__mul__ (line 1465)RigidTransform.__getitem__ (line 1364)RigidTransformReturn the translation component of the transform. A transform is a composition of a rotation and a translation, such that when applied to a vector, the vector is first rotated and then translated. This property returns the translation part of the transform. Returns ------- translation : numpy.ndarray, shape (N, 3) or (3,) A single translation vector or a stack of translation vectors. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np The translation component is extracted from the transform: >>> t = np.array([[1, 0, 0], [2, 0, 0], [3, 0, 0]]) >>> r = R.random() >>> tf = Tf.from_components(t, r) >>> np.allclose(tf.translation, t) True Return the translation and rotation components of the transform, where the rotation is applied first, followed by the translation. 4x4 rigid transformation matrices are of the form: .. [R | t] [0 | 1] Where ``R`` is a 3x3 orthonormal rotation matrix and ``t`` is a 3x1 translation vector ``[tx, ty, tz]``. This function returns the rotation corresponding to this rotation matrix ``r = Rotation.from_matrix(R)`` and the translation vector ``t``. Take a transform ``tf`` and a vector ``v``. When applying the transform to the vector, the result is the same as if the transform was applied to the vector in the following way: ``tf.apply(v) == translation + rotation.apply(v)`` Returns ------- translation : numpy.ndarray, shape (N, 3) or (3,) The translation of the transform. rotation : `Rotation` instance The rotation of the transform. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np Recover the rotation and translation from a transform: >>> t = np.array([2, 3, 4]) >>> r = R.from_matrix([[0, 0, 1], ... [1, 0, 0], ... [0, 1, 0]]) >>> tf = Tf.from_components(t, r) >>> tf_t, tf_r = tf.as_components() >>> tf_t array([2., 3., 4.]) >>> tf_r.as_matrix() array([[0., 0., 1.], [1., 0., 0.], [0., 1., 0.]]) The transform applied to a vector is equivalent to the rotation applied to the vector followed by the translation: >>> r.apply([1, 0, 0]) array([0., 1., 0.]) >>> t + r.apply([1, 0, 0]) array([2., 4., 4.]) >>> tf.apply([1, 0, 0]) array([2., 4., 4.]) Return the rotation component of the transform. A transform is a composition of a rotation and a translation, such that when applied to a vector, the vector is first rotated and then translated. This property returns the rotation part of the transform. Returns ------- rotation : `Rotation` instance A single rotation or a stack of rotations. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np The rotation component is extracted from the transform: >>> t = np.array([1, 0, 0]) >>> r = R.random(3) >>> tf = Tf.from_components(t, r) >>> np.allclose(tf.rotation.as_matrix(), r.as_matrix()) True Return the exponential coordinates of the transform. This implements the logarithmic map that converts SE(3) to 6-dimensional real vectors. This is an inverse of `from_exp_coords` where details on the mapping can be found. Returns ------- exp_coords : numpy.ndarray, shape (N, 6) or (6,) A single exponential coordinate vector or a stack of exponential coordinate vectors. The first three components define the rotation and the last three components define the translation. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> import numpy as np Get exponential coordinates of the identity matrix: >>> Tf.identity().as_exp_coords() array([0., 0., 0., 0., 0., 0.]) Return the dual quaternion representation of the transform. Unit dual quaternions encode orientation in a real unit quaternion and translation in a dual quaternion. There is a double cover, i.e., the unit dual quaternions q and -q represent the same transform. Parameters ---------- scalar_first : bool, optional Whether the scalar component goes first or last in the two individual quaternions that represent the real and the dual part. Default is False, i.e. the scalar-last order is used. Returns ------- dual_quat : numpy.ndarray, shape (N, 8) or (8,) A single unit dual quaternion vector or a stack of unit dual quaternion vectors. The real part is stored in the first four components and the dual part in the last four components. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> import numpy as np Get identity dual quaternion (we use scalar-last by default): >>> Tf.identity().as_dual_quat() array([0., 0., 0., 1., 0., 0., 0., 0.]) When we want to use the scalar-first convention, we use the argument: >>> Tf.identity().as_dual_quat(scalar_first=True) array([1., 0., 0., 0., 0., 0., 0., 0.]) Return a copy of the matrix representation of the transform. 4x4 rigid transformation matrices are of the form: .. [R | t] [0 | 1] where ``R`` is a 3x3 orthonormal rotation matrix and ``t`` is a 3x1 translation vector ``[tx, ty, tz]``. Returns ------- matrix : numpy.ndarray, shape (4, 4) or (N, 4, 4) A single transformation matrix or a stack of transformation matrices. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np A transformation matrix is a 4x4 matrix formed from a 3x3 rotation matrix and a 3x1 translation vector: >>> t = np.array([2, 3, 4]) >>> r = R.from_matrix([[0, 0, 1], ... [1, 0, 0], ... [0, 1, 0]]) >>> tf = Tf.from_components(t, r) >>> tf.as_matrix() array([[ 0., 0., 1., 2.], [ 1., 0., 0., 3.], [ 0., 1., 0., 4.], [ 0., 0., 0., 1.]]) >>> Tf.identity(2).as_matrix() array([[[1., 0., 0., 0.], [0., 1., 0., 0.], [0., 0., 1., 0.], [0., 0., 0., 1.]], [[1., 0., 0., 0.], [0., 1., 0., 0.], [0., 0., 1., 0.], [0., 0., 0., 1.]]]) PickleErrorNote 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.Invert this transform. Composition of a transform with its inverse results in an identity transform. A rigid transform is a composition of a rotation and a translation, where the rotation is applied first, followed by the translation. So the inverse transform is equivalent to the inverse translation followed by the inverse rotation. Returns ------- `RigidTransform` instance The inverse of this transform. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np A transform composed with its inverse results in an identity transform: >>> rng = np.random.default_rng(seed=123) >>> t = rng.random(3) >>> r = R.random(rng=rng) >>> tf = Tf.from_components(t, r) >>> tf.as_matrix() array([[-0.45431291, 0.67276178, -0.58394466, 0.68235186], [-0.23272031, 0.54310598, 0.80676958, 0.05382102], [ 0.85990758, 0.50242162, -0.09017473, 0.22035987], [ 0. , 0. , 0. , 1. ]]) >>> (tf.inv() * tf).as_matrix() array([[[1., 0., 0., 0.], [0., 1., 0., 0.], [0., 0., 1., 0.], [0., 0., 0., 1.]]]) The inverse rigid transform is the same as the inverse translation followed by the inverse rotation: >>> t, r = tf.as_components() >>> r_inv = r.inv() # inverse rotation >>> t_inv = -t # inverse translation >>> tf_r_inv = Tf.from_rotation(r_inv) >>> tf_t_inv = Tf.from_translation(t_inv) >>> np.allclose((tf_r_inv * tf_t_inv).as_matrix(), ... tf.inv().as_matrix(), ... atol=1e-12) True >>> (tf_r_inv * tf_t_inv * tf).as_matrix() array([[[1., 0., 0., 0.], [0., 1., 0., 0.], [0., 0., 1., 0.], [0., 0., 0., 1.]]]) Initialize from exponential coordinates of transform. This implements the exponential map that converts 6-dimensional real vectors to SE(3). An exponential coordinate vector consists of 6 elements ``[rx, ry, rz, vx, vy, vz]``. The first 3 encode rotation (and form a rotation vector used in `Rotation.from_rotvec`) and the last 3 encode translation (and form a translation vector for pure translations). The exponential mapping can be expressed as matrix exponential ``T = exp(tau)``, where ``T`` is a 4x4 matrix representing a rigid transform and ``tau`` is a 4x4 matrix formed from the elements of an exponential coordinate vector:: tau = [ 0 -rz ry vx] [ rz 0 -rx vy] [-ry rx 0 vz] [ 0 0 0 1] Parameters ---------- exp_coords : array_like, shape (N, 6) or (6,) A single exponential coordinate vector or a stack of exponential coordinate vectors. The expected order of components is ``[rx, ry, rz, vx, vy, vz]``. The first 3 components encode rotation and the last 3 encode translation. Returns ------- transform : `RigidTransform` instance A single transform or a stack of transforms. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> import numpy as np Creating from a single 6d vector of exponential coordinates: >>> tf = Tf.from_exp_coords([ ... -2.01041204, -0.52983629, 0.65773501, ... 0.10386614, 0.05855009, 0.54959179]) >>> tf.as_matrix() array([[0.76406621, 0.10504613, -0.63652819, -0.10209961], [0.59956454, -0.47987325, 0.64050295, 0.40158789], [-0.2381705, -0.87102639, -0.42963687, 0.19637636], [0., 0., 0., 1.]]) >>> tf.single True A vector of zeros represents the identity transform: >>> tf = Tf.from_exp_coords(np.zeros(6)) >>> tf.as_matrix() array([[1., 0., 0., 0.], [0., 1., 0., 0.], [0., 0., 1., 0.], [0., 0., 0., 1.]]) The last three numbers encode translation. If the first three numbers are zero, the last three components can be interpreted as the translation: >>> tf_trans = Tf.from_exp_coords([0, 0, 0, 4.3, -2, 3.4]) >>> tf_trans.translation array([4.3, -2., 3.4]) The first three numbers encode rotation as a rotation vector: >>> tf_rot = Tf.from_exp_coords([0.5, 0.3, 0.1, 0, 0, 0]) >>> tf_rot.rotation.as_rotvec() array([0.5, 0.3, 0.1]) Combining translation and rotation preserves the rotation vector, but changes the last three components as they encode translation and rotation: >>> (tf_trans * tf_rot).as_exp_coords() array([0.5, 0.3, 0.1, 3.64305882, -1.25879559, 4.46109265]) Initialize from a unit dual quaternion. Unit dual quaternions encode orientation in a real unit quaternion and translation in a dual quaternion. There is a double cover, i.e., the unit dual quaternions q and -q represent the same transform. Unit dual quaternions must have a real quaternion with unit norm and a dual quaternion that is orthogonal to the real quaternion to satisfy the unit norm constraint. This function will enforce both properties through normalization. Parameters ---------- dual_quat : array_like, shape (N, 8) or (8,) A single unit dual quaternion or a stack of unit dual quaternions. The real part is stored in the first four components and the dual part in the last four components. scalar_first : bool, optional Whether the scalar component goes first or last in the two individual quaternions that represent the real and the dual part. Default is False, i.e. the scalar-last order is used. Returns ------- transform : `RigidTransform` instance A single transform or a stack of transforms. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> import numpy as np Creating from a single unit dual quaternion: >>> tf = Tf.from_dual_quat([ ... 0.0617101, -0.06483886, 0.31432811, 0.94508498, ... 0.04985168, -0.26119618, 0.1691491, -0.07743254]) >>> tf.as_matrix() array([[0.79398752, -0.60213598, -0.08376202, 0.24605262], [0.58613113, 0.79477941, -0.15740392, -0.4932833], [0.16135089, 0.07588122, 0.98397557, 0.34262676], [0., 0., 0., 1.]]) >>> tf.single True Initialize from a translation numpy array, without a rotation. When applying this transform to a vector ``v``, the result is the same as if the translation and vector were added together. If ``t`` is the displacement vector of the translation, then: ``Tf.from_translation(t).apply(v) == t + v`` Parameters ---------- translation : array_like, shape (N, 3) or (3,) A single translation vector or a stack of translation vectors. Returns ------- transform : `RigidTransform` instance Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> import numpy as np Creating a transform from a single translation vector: >>> t = np.array([2, 3, 4]) >>> t + np.array([1, 0, 0]) array([3, 3, 4]) >>> tf = Tf.from_translation(t) >>> tf.apply([1, 0, 0]) array([3., 3., 4.]) >>> tf.single True The top 3x1 points in the rightmost column of the transformation matrix is the translation vector: >>> tf.as_matrix() array([[1., 0., 0., 2.], [0., 1., 0., 3.], [0., 0., 1., 4.], [0., 0., 0., 1.]]) >>> np.allclose(tf.as_matrix()[:3, 3], t) True Creating multiple transforms from a stack of translation vectors: >>> t = np.array([[2, 3, 4], [1, 0, 0]]) >>> t + np.array([1, 0, 0]) array([[3, 3, 4], [2, 0, 0]]) >>> tf = Tf.from_translation(t) >>> tf.apply([1, 0, 0]) array([[3., 3., 4.], [2., 0., 0.]]) >>> np.allclose(tf.as_matrix()[:, :3, 3], t) True >>> tf.single False >>> len(tf) 2 Initialize from a rotation, without a translation. When applying this transform to a vector ``v``, the result is the same as if the rotation was applied to the vector. ``Tf.from_rotation(r).apply(v) == r.apply(v)`` Parameters ---------- rotation : `Rotation` instance A single rotation or a stack of rotations. Returns ------- transform : `RigidTransform` instance Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np Creating a transform from a single rotation: >>> r = R.from_euler("ZYX", [90, 30, 0], degrees=True) >>> r.apply([1, 0, 0]) array([0. , 0.8660254, -0.5 ]) >>> tf = Tf.from_rotation(r) >>> tf.apply([1, 0, 0]) array([0. , 0.8660254, -0.5 ]) >>> tf.single True The upper 3x3 submatrix of the transformation matrix is the rotation matrix: >>> np.allclose(tf.as_matrix()[:3, :3], r.as_matrix(), atol=1e-12) True Creating multiple transforms from a stack of rotations: >>> r = R.from_euler("ZYX", [[90, 30, 0], [45, 30, 60]], degrees=True) >>> r.apply([1, 0, 0]) array([[0. , 0.8660254 , -0.5 ], [0.61237244, 0.61237244, -0.5 ]]) >>> tf = Tf.from_rotation(r) >>> tf.apply([1, 0, 0]) array([[0. , 0.8660254 , -0.5 ], [0.61237244, 0.61237244, -0.5 ]]) >>> tf.single False >>> len(tf) 2 Initialize from a 4x4 transformation matrix. Parameters ---------- matrix : array_like, shape (4, 4) or (N, 4, 4) A single transformation matrix or a stack of transformation matrices. Returns ------- transform : `RigidTransform` instance Notes ----- 4x4 rigid transformation matrices are of the form: .. [R | t] [0 | 1] where ``R`` is a 3x3 rotation matrix and ``t`` is a 3x1 translation vector ``[tx, ty, tz]``. As rotation matrices must be proper orthogonal, the rotation component is orthonormalized using singular value decomposition before initialization. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> import numpy as np Creating a transform from a single matrix: >>> m = np.array([[0, 1, 0, 2], ... [0, 0, 1, 3], ... [1, 0, 0, 4], ... [0, 0, 0, 1]]) >>> tf = Tf.from_matrix(m) >>> tf.as_matrix() array([[0., 1., 0., 2.], [0., 0., 1., 3.], [1., 0., 0., 4.], [0., 0., 0., 1.]]) >>> tf.single True Creating a transform from a stack of matrices: >>> m = np.array([np.eye(4), np.eye(4)]) >>> tf = Tf.from_matrix(m) >>> tf.as_matrix() array([[[1., 0., 0., 0.], [0., 1., 0., 0.], [0., 0., 1., 0.], [0., 0., 0., 1.]], [[1., 0., 0., 0.], [0., 1., 0., 0.], [0., 0., 1., 0.], [0., 0., 0., 1.]]]) >>> tf.single False >>> len(tf) 2 Matrices with a rotation component that is not proper orthogonal are orthogonalized using singular value decomposition before initialization: >>> tf = Tf.from_matrix(np.diag([2, 2, 2, 1])) >>> tf.as_matrix() array([[1., 0., 0., 0.], [0., 1., 0., 0.], [0., 0., 1., 0.], [0., 0., 0., 1.]]) Initialize an identity transform. Composition with the identity transform has no effect, and applying the identity transform to a vector has no effect. Parameters ---------- num : int, optional Number of identity transforms to generate. If None (default), then a single transform is generated. Returns ------- transform : `RigidTransform` instance The identity transform. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np Creating a single identity transform: >>> tf = Tf.identity() >>> tf.as_matrix() array([[1., 0., 0., 0.], [0., 1., 0., 0.], [0., 0., 1., 0.], [0., 0., 0., 1.]]) >>> tf.single True The identity transform can be applied to a vector without effect: >>> tf.apply([1, 2, 3]) array([1., 2., 3.]) The identity transform when composed with another transform has no effect: >>> rng = np.random.default_rng(123) >>> t = rng.random(3) >>> r = R.random(rng=rng) >>> tf = Tf.from_components(t, r) >>> np.allclose((Tf.identity() * tf).as_matrix(), ... tf.as_matrix(), atol=1e-12) True Multiple identity transforms can be generated at once: >>> tf = Tf.identity(2) >>> tf.as_matrix() array([[[1., 0., 0., 0.], [0., 1., 0., 0.], [0., 0., 1., 0.], [0., 0., 0., 1.]], [[1., 0., 0., 0.], [0., 1., 0., 0.], [0., 0., 1., 0.], [0., 0., 0., 1.]]]) >>> tf.single False >>> len(tf) 2 Initialize a rigid transform from translation and rotation components. When creating a rigid transform from a translation and rotation, the translation is applied after the rotation, such that ``tf = Tf.from_components(translation, rotation)`` is equivalent to ``tf = Tf.from_translation(translation) * Tf.from_rotation(rotation)``. When applying a transform to a vector ``v``, the result is the same as if the transform was applied to the vector in the following way: ``tf.apply(v) == translation + rotation.apply(v)`` Parameters ---------- translation : array_like, shape (N, 3) or (3,) A single translation vector or a stack of translation vectors. rotation : `Rotation` instance A single rotation or a stack of rotations. Returns ------- `RigidTransform` If rotation is single and translation is shape (3,), then a single transform is returned. Otherwise, a stack of transforms is returned. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np Creating from a single rotation and translation: >>> t = np.array([2, 3, 4]) >>> r = R.from_euler("ZYX", [90, 30, 0], degrees=True) >>> r.as_matrix() array([[ 0. , -1., 0. ], [ 0.8660254, 0., 0.5 ], [-0.5 , 0., 0.8660254 ]]) >>> tf = Tf.from_components(t, r) >>> tf.rotation.as_matrix() array([[ 0. , -1., 0. ], [ 0.8660254, 0., 0.5 ], [-0.5 , 0., 0.8660254 ]]) >>> tf.translation array([2., 3., 4.]) >>> tf.single True When applying a transform to a vector ``v``, the result is the same as if the transform was applied to the vector in the following way: ``tf.apply(v) == translation + rotation.apply(v)`` >>> r.apply([1, 0, 0]) array([0. , 0.8660254, -0.5 ]) >>> t + r.apply([1, 0, 0]) array([2. , 3.8660254, 3.5 ]) >>> tf.apply([1, 0, 0]) array([2. , 3.8660254, 3.5 ]) Incompatible checksums (0x%x vs (0x20fa94b, 0x51a5e1e, 0xe043981) = (_matrix, _single))ImportErrorExtract transform(s) at given index(es) from this object. Creates a new `RigidTransform` instance containing a subset of transforms stored in this object. Parameters ---------- indexer : int or slice or array_like Specifies which transform(s) to extract. A single indexer must be specified, i.e. as if indexing a 1 dimensional array or list. Returns ------- transform : `RigidTransform` instance Contains - a single transform, if `indexer` is a single index - a stack of transform(s), if `indexer` is a slice, or an index array. Raises ------ TypeError If the transform is a single transform. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> t = [[0, 0, 0], [1, 0, 0], [2, 0, 0]] # 3 translations >>> tf = Tf.from_translation(t) A single index returns a single transform: >>> tf[0].as_matrix() array([[1., 0., 0., 0.], [0., 1., 0., 0.], [0., 0., 1., 0.], [0., 0., 0., 1.]]) A slice returns a stack of transforms: >>> tf[1:3].translation array([[1., 0., 0.], [2., 0., 0.]]) An index array returns a stack of transforms: >>> tf[[0, 2]].translation array([[0., 0., 0.], [2., 0., 0.]]) Expected vector to have shape (N, 3), or (3,), got Expected `translation` to have shape (3,), or (N, 3), got Expected `rotation` to be a `Rotation` instance, got Expected `matrix` to have shape (4, 4), or (N, 4, 4), got Expected last row of transformation matrix Expected `exp_coords` to have shape (6,), or (N, 6), got Expected equal number of transforms in both or a single transform in either object, got Expected `dual_quat` to have shape (8,), or (N, 8), got Concatenate a sequence of `RigidTransform` objects into a single object. Parameters ---------- transforms : sequence of `RigidTransform` If a single `RigidTransform` instance is passed in, a copy of it is returned. Returns ------- transform : `RigidTransform` instance The concatenated transform. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> tf1 = Tf.from_translation([1, 0, 0]) >>> tf2 = Tf.from_translation([[2, 0, 0], [3, 0, 0]]) >>> Tf.concatenate([tf1, tf2]).translation array([[1., 0., 0.], [2., 0., 0.], [3., 0., 0.]]) Compose this transform with the other. If `p` and `q` are two transforms, then the composition of 'q followed by p' is equivalent to `p * q`. In terms of transformation matrices, the composition can be expressed as ``p.as_matrix() @ q.as_matrix()``. In terms of translations and rotations, the composition when applied to a vector ``v`` is equivalent to ``p.translation + p.rotation.apply(q.translation) + (p.rotation * q.rotation).apply(v)``. This function supports composition of multiple transforms at a time. The following cases are possible: - Either ``p`` or ``q`` contains a single or length 1 transform. In this case the result contains the result of composing each transform in the other object with the one transform. If both are single transforms, the result is a single transform. - Both ``p`` and ``q`` contain ``N`` transforms. In this case each transform ``p[i]`` is composed with the corresponding transform ``q[i]`` and the result contains ``N`` transforms. Parameters ---------- other : `RigidTransform` instance Object containing the transforms to be composed with this one. Returns ------- `RigidTransform` instance The composed transform. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np Compose two transforms: >>> tf1 = Tf.from_translation([1, 0, 0]) >>> tf2 = Tf.from_translation([0, 1, 0]) >>> tf = tf1 * tf2 >>> tf.translation array([1., 1., 0.]) >>> tf.single True When applied to a vector, the composition of two transforms is applied in right-to-left order. >>> t1, r1 = [1, 2, 3], R.from_euler('z', 60, degrees=True) >>> t2, r2 = [0, 1, 0], R.from_euler('x', 30, degrees=True) >>> tf1 = Tf.from_components(t1, r1) >>> tf2 = Tf.from_components(t2, r2) >>> tf = tf1 * tf2 >>> tf.apply([1, 0, 0]) array([0.6339746, 3.3660254, 3. ]) >>> tf1.apply(tf2.apply([1, 0, 0])) array([0.6339746, 3.3660254, 3. ]) When at least one of the transforms is not single, the result is a stack of transforms. >>> tf1 = Tf.from_translation([1, 0, 0]) >>> tf2 = Tf.from_translation([[0, 2, 0], [0, 0, 3]]) >>> tf = tf1 * tf2 >>> tf.translation array([[1., 2., 0.], [1., 0., 3.]]) >>> tf.single False >>> len(tf) 2 Compose this transform with itself `n` times. A rigid transform `p` when raised to non-integer powers can be thought of as finding a fraction of the transformation. For example, a power of 0.5 finds a "halfway" transform from the identity to `p`. This is implemented by applying screw linear interpolation (ScLERP) between `p` and the identity transform, where the angle of the rotation component is scaled by `n`, and the translation is proportionally adjusted along the screw axis. ``q = p ** n`` can also be expressed as ``q = RigidTransform.from_exp_coords(p.as_exp_coords() * n)``. If `n` is negative, then the transform 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 transform with itself. Returns ------- `RigidTransform` 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 ----- There are three notable cases: if ``n == 1`` then a copy of the original transform is returned, if ``n == 0`` then the identity transform is returned, and if ``n == -1`` then the inverse transform 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 RigidTransform as Tf >>> import numpy as np A power of 2 returns the transform composed with itself: >>> tf = Tf.from_translation([1, 2, 3]) >>> (tf ** 2).translation array([2., 4., 6.]) >>> (tf ** 2).as_matrix() array([[1., 0., 0., 2.], [0., 1., 0., 4.], [0., 0., 1., 6.], [0., 0., 0., 1.]]) A negative power returns the inverse of the transform raised to the absolute value of `n`: >>> (tf ** -2).translation array([-2., -4., -6.]) >>> np.allclose((tf ** -2).as_matrix(), (tf.inv() ** 2).as_matrix(), ... atol=1e-12) True A power of 0 returns the identity transform: >>> (tf ** 0).as_matrix() array([[1., 0., 0., 0.], [0., 1., 0., 0.], [0., 0., 1., 0.], [0., 0., 0., 1.]]) A power of 1 returns a copy of the original transform: >>> (tf ** 1).as_matrix() array([[1., 0., 0., 1.], [0., 1., 0., 2.], [0., 0., 1., 3.], [0., 0., 0., 1.]]) A fractional power returns a transform with a scaled rotation and translated along the screw axis. Here we take the square root of the transform, which when squared recovers the original transform: >>> tf_half = (tf ** 0.5) >>> tf_half.translation array([0.5, 1., 1.5]) >>> (tf_half ** 2).as_matrix() array([[1., 0., 0., 1.], [0., 1., 0., 2.], [0., 0., 1., 3.], [0., 0., 0., 1.]]) Apply the transform to a vector. If the original frame transforms to the final frame by this transform, 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 transformation of a vector being glued to the original frame as it transforms. In this case the vector components are expressed in the original frame before and after the transformation. In terms of rotation matrices and translation vectors, this application is the same as ``self.translation + self.rotation.as_matrix() @ vector``. Parameters ---------- vector : array_like, shape (N, 3) or (3,) A single vector or a stack of vectors. inverse : bool, optional If True, the inverse of the transform is applied to the vector. Returns ------- transformed_vector : numpy.ndarray, shape (N, 3) or (3,) The transformed vector(s). Shape depends on the following cases: - If object contains a single transform (as opposed to a stack with a single transform) and a single vector is specified with shape ``(3,)``, then `transformed_vector` has shape ``(3,)``. - In all other cases, `transformed_vector` has shape ``(N, 3)``, where ``N`` is either the number of transforms or vectors. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np Apply a single transform to a vector. Here the transform is just a translation, so the result is the vector added to the translation vector. >>> t = np.array([1, 2, 3]) >>> tf = Tf.from_translation(t) >>> t + np.array([1, 0, 0]) array([2, 2, 3]) >>> tf.apply([1, 0, 0]) array([2., 2., 3.]) Apply a single transform to a stack of vectors: >>> tf.apply([[1, 0, 0], [0, 1, 0]]) array([[2., 2., 3.], [1., 3., 3.]]) Apply the inverse of a transform to a vector, so the result is the negative of the translation vector added to the vector. >>> -t + np.array([1, 0, 0]) array([0, -2, -3]) >>> tf.apply([1, 0, 0], inverse=True) array([0., -2., -3.]) For transforms which are not just pure translations, applying it to a vector is the same as applying the rotation component to the vector and then adding the translation component. >>> r = R.from_euler('z', 60, degrees=True) >>> tf = Tf.from_components(t, r) >>> t + r.apply([1, 0, 0]) array([1.5, 2.8660254, 3. ]) >>> tf.apply([1, 0, 0]) array([1.5, 2.8660254, 3. ]) When applying the inverse of a transform, the result is the negative of the translation vector added to the vector, and then rotated by the inverse rotation. >>> r.inv().apply(-t + np.array([1, 0, 0])) array([-1.73205081, -1. , -3. ]) >>> tf.apply([1, 0, 0], inverse=True) array([-1.73205081, -1. , -3. ]) .RigidTransform.apply(self, vector, inverse=False) Apply the transform to a vector. If the original frame transforms to the final frame by this transform, 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 transformation of a vector being glued to the original frame as it transforms. In this case the vector components are expressed in the original frame before and after the transformation. In terms of rotation matrices and translation vectors, this application is the same as ``self.translation + self.rotation.as_matrix() @ vector``. Parameters ---------- vector : array_like, shape (N, 3) or (3,) A single vector or a stack of vectors. inverse : bool, optional If True, the inverse of the transform is applied to the vector. Returns ------- transformed_vector : numpy.ndarray, shape (N, 3) or (3,) The transformed vector(s). Shape depends on the following cases: - If object contains a single transform (as opposed to a stack with a single transform) and a single vector is specified with shape ``(3,)``, then `transformed_vector` has shape ``(3,)``. - In all other cases, `transformed_vector` has shape ``(N, 3)``, where ``N`` is either the number of transforms or vectors. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np Apply a single transform to a vector. Here the transform is just a translation, so the result is the vector added to the translation vector. >>> t = np.array([1, 2, 3]) >>> tf = Tf.from_translation(t) >>> t + np.array([1, 0, 0]) array([2, 2, 3]) >>> tf.apply([1, 0, 0]) array([2., 2., 3.]) Apply a single transform to a stack of vectors: >>> tf.apply([[1, 0, 0], [0, 1, 0]]) array([[2., 2., 3.], [1., 3., 3.]]) Apply the inverse of a transform to a vector, so the result is the negative of the translation vector added to the vector. >>> -t + np.array([1, 0, 0]) array([0, -2, -3]) >>> tf.apply([1, 0, 0], inverse=True) array([0., -2., -3.]) For transforms which are not just pure translations, applying it to a vector is the same as applying the rotation component to the vector and then adding the translation component. >>> r = R.from_euler('z', 60, degrees=True) >>> tf = Tf.from_components(t, r) >>> t + r.apply([1, 0, 0]) array([1.5, 2.8660254, 3. ]) >>> tf.apply([1, 0, 0]) array([1.5, 2.8660254, 3. ]) When applying the inverse of a transform, the result is the negative of the translation vector added to the vector, and then rotated by the inverse rotation. >>> r.inv().apply(-t + np.array([1, 0, 0])) array([-1.73205081, -1. , -3. ]) >>> tf.apply([1, 0, 0], inverse=True) array([-1.73205081, -1. , -3. ])RigidTransform.inv(self) Invert this transform. Composition of a transform with its inverse results in an identity transform. A rigid transform is a composition of a rotation and a translation, where the rotation is applied first, followed by the translation. So the inverse transform is equivalent to the inverse translation followed by the inverse rotation. Returns ------- `RigidTransform` instance The inverse of this transform. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np A transform composed with its inverse results in an identity transform: >>> rng = np.random.default_rng(seed=123) >>> t = rng.random(3) >>> r = R.random(rng=rng) >>> tf = Tf.from_components(t, r) >>> tf.as_matrix() array([[-0.45431291, 0.67276178, -0.58394466, 0.68235186], [-0.23272031, 0.54310598, 0.80676958, 0.05382102], [ 0.85990758, 0.50242162, -0.09017473, 0.22035987], [ 0. , 0. , 0. , 1. ]]) >>> (tf.inv() * tf).as_matrix() array([[[1., 0., 0., 0.], [0., 1., 0., 0.], [0., 0., 1., 0.], [0., 0., 0., 1.]]]) The inverse rigid transform is the same as the inverse translation followed by the inverse rotation: >>> t, r = tf.as_components() >>> r_inv = r.inv() # inverse rotation >>> t_inv = -t # inverse translation >>> tf_r_inv = Tf.from_rotation(r_inv) >>> tf_t_inv = Tf.from_translation(t_inv) >>> np.allclose((tf_r_inv * tf_t_inv).as_matrix(), ... tf.inv().as_matrix(), ... atol=1e-12) True >>> (tf_r_inv * tf_t_inv * tf).as_matrix() array([[[1., 0., 0., 0.], [0., 1., 0., 0.], [0., 0., 1., 0.], [0., 0., 0., 1.]]])RigidTransform.as_dual_quat(self, *, scalar_first=False) Return the dual quaternion representation of the transform. Unit dual quaternions encode orientation in a real unit quaternion and translation in a dual quaternion. There is a double cover, i.e., the unit dual quaternions q and -q represent the same transform. Parameters ---------- scalar_first : bool, optional Whether the scalar component goes first or last in the two individual quaternions that represent the real and the dual part. Default is False, i.e. the scalar-last order is used. Returns ------- dual_quat : numpy.ndarray, shape (N, 8) or (8,) A single unit dual quaternion vector or a stack of unit dual quaternion vectors. The real part is stored in the first four components and the dual part in the last four components. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> import numpy as np Get identity dual quaternion (we use scalar-last by default): >>> Tf.identity().as_dual_quat() array([0., 0., 0., 1., 0., 0., 0., 0.]) When we want to use the scalar-first convention, we use the argument: >>> Tf.identity().as_dual_quat(scalar_first=True) array([1., 0., 0., 0., 0., 0., 0., 0.])RigidTransform.as_exp_coords(self) Return the exponential coordinates of the transform. This implements the logarithmic map that converts SE(3) to 6-dimensional real vectors. This is an inverse of `from_exp_coords` where details on the mapping can be found. Returns ------- exp_coords : numpy.ndarray, shape (N, 6) or (6,) A single exponential coordinate vector or a stack of exponential coordinate vectors. The first three components define the rotation and the last three components define the translation. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> import numpy as np Get exponential coordinates of the identity matrix: >>> Tf.identity().as_exp_coords() array([0., 0., 0., 0., 0., 0.])RigidTransform.as_components(self) Return the translation and rotation components of the transform, where the rotation is applied first, followed by the translation. 4x4 rigid transformation matrices are of the form: .. [R | t] [0 | 1] Where ``R`` is a 3x3 orthonormal rotation matrix and ``t`` is a 3x1 translation vector ``[tx, ty, tz]``. This function returns the rotation corresponding to this rotation matrix ``r = Rotation.from_matrix(R)`` and the translation vector ``t``. Take a transform ``tf`` and a vector ``v``. When applying the transform to the vector, the result is the same as if the transform was applied to the vector in the following way: ``tf.apply(v) == translation + rotation.apply(v)`` Returns ------- translation : numpy.ndarray, shape (N, 3) or (3,) The translation of the transform. rotation : `Rotation` instance The rotation of the transform. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np Recover the rotation and translation from a transform: >>> t = np.array([2, 3, 4]) >>> r = R.from_matrix([[0, 0, 1], ... [1, 0, 0], ... [0, 1, 0]]) >>> tf = Tf.from_components(t, r) >>> tf_t, tf_r = tf.as_components() >>> tf_t array([2., 3., 4.]) >>> tf_r.as_matrix() array([[0., 0., 1.], [1., 0., 0.], [0., 1., 0.]]) The transform applied to a vector is equivalent to the rotation applied to the vector followed by the translation: >>> r.apply([1, 0, 0]) array([0., 1., 0.]) >>> t + r.apply([1, 0, 0]) array([2., 4., 4.]) >>> tf.apply([1, 0, 0]) array([2., 4., 4.])RigidTransform.as_matrix(self) Return a copy of the matrix representation of the transform. 4x4 rigid transformation matrices are of the form: .. [R | t] [0 | 1] where ``R`` is a 3x3 orthonormal rotation matrix and ``t`` is a 3x1 translation vector ``[tx, ty, tz]``. Returns ------- matrix : numpy.ndarray, shape (4, 4) or (N, 4, 4) A single transformation matrix or a stack of transformation matrices. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np A transformation matrix is a 4x4 matrix formed from a 3x3 rotation matrix and a 3x1 translation vector: >>> t = np.array([2, 3, 4]) >>> r = R.from_matrix([[0, 0, 1], ... [1, 0, 0], ... [0, 1, 0]]) >>> tf = Tf.from_components(t, r) >>> tf.as_matrix() array([[ 0., 0., 1., 2.], [ 1., 0., 0., 3.], [ 0., 1., 0., 4.], [ 0., 0., 0., 1.]]) >>> Tf.identity(2).as_matrix() array([[[1., 0., 0., 0.], [0., 1., 0., 0.], [0., 0., 1., 0.], [0., 0., 0., 1.]], [[1., 0., 0., 0.], [0., 1., 0., 0.], [0., 0., 1., 0.], [0., 0., 0., 1.]]])RigidTransform.concatenate(cls, transforms) Concatenate a sequence of `RigidTransform` objects into a single object. Parameters ---------- transforms : sequence of `RigidTransform` If a single `RigidTransform` instance is passed in, a copy of it is returned. Returns ------- transform : `RigidTransform` instance The concatenated transform. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> tf1 = Tf.from_translation([1, 0, 0]) >>> tf2 = Tf.from_translation([[2, 0, 0], [3, 0, 0]]) >>> Tf.concatenate([tf1, tf2]).translation array([[1., 0., 0.], [2., 0., 0.], [3., 0., 0.]])RigidTransform.identity(cls, num=None) Initialize an identity transform. Composition with the identity transform has no effect, and applying the identity transform to a vector has no effect. Parameters ---------- num : int, optional Number of identity transforms to generate. If None (default), then a single transform is generated. Returns ------- transform : `RigidTransform` instance The identity transform. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np Creating a single identity transform: >>> tf = Tf.identity() >>> tf.as_matrix() array([[1., 0., 0., 0.], [0., 1., 0., 0.], [0., 0., 1., 0.], [0., 0., 0., 1.]]) >>> tf.single True The identity transform can be applied to a vector without effect: >>> tf.apply([1, 2, 3]) array([1., 2., 3.]) The identity transform when composed with another transform has no effect: >>> rng = np.random.default_rng(123) >>> t = rng.random(3) >>> r = R.random(rng=rng) >>> tf = Tf.from_components(t, r) >>> np.allclose((Tf.identity() * tf).as_matrix(), ... tf.as_matrix(), atol=1e-12) True Multiple identity transforms can be generated at once: >>> tf = Tf.identity(2) >>> tf.as_matrix() array([[[1., 0., 0., 0.], [0., 1., 0., 0.], [0., 0., 1., 0.], [0., 0., 0., 1.]], [[1., 0., 0., 0.], [0., 1., 0., 0.], [0., 0., 1., 0.], [0., 0., 0., 1.]]]) >>> tf.single False >>> len(tf) 2RigidTransform.from_dual_quat(cls, dual_quat, *, scalar_first=False) Initialize from a unit dual quaternion. Unit dual quaternions encode orientation in a real unit quaternion and translation in a dual quaternion. There is a double cover, i.e., the unit dual quaternions q and -q represent the same transform. Unit dual quaternions must have a real quaternion with unit norm and a dual quaternion that is orthogonal to the real quaternion to satisfy the unit norm constraint. This function will enforce both properties through normalization. Parameters ---------- dual_quat : array_like, shape (N, 8) or (8,) A single unit dual quaternion or a stack of unit dual quaternions. The real part is stored in the first four components and the dual part in the last four components. scalar_first : bool, optional Whether the scalar component goes first or last in the two individual quaternions that represent the real and the dual part. Default is False, i.e. the scalar-last order is used. Returns ------- transform : `RigidTransform` instance A single transform or a stack of transforms. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> import numpy as np Creating from a single unit dual quaternion: >>> tf = Tf.from_dual_quat([ ... 0.0617101, -0.06483886, 0.31432811, 0.94508498, ... 0.04985168, -0.26119618, 0.1691491, -0.07743254]) >>> tf.as_matrix() array([[0.79398752, -0.60213598, -0.08376202, 0.24605262], [0.58613113, 0.79477941, -0.15740392, -0.4932833], [0.16135089, 0.07588122, 0.98397557, 0.34262676], [0., 0., 0., 1.]]) >>> tf.single TrueRigidTransform.from_exp_coords(cls, exp_coords) Initialize from exponential coordinates of transform. This implements the exponential map that converts 6-dimensional real vectors to SE(3). An exponential coordinate vector consists of 6 elements ``[rx, ry, rz, vx, vy, vz]``. The first 3 encode rotation (and form a rotation vector used in `Rotation.from_rotvec`) and the last 3 encode translation (and form a translation vector for pure translations). The exponential mapping can be expressed as matrix exponential ``T = exp(tau)``, where ``T`` is a 4x4 matrix representing a rigid transform and ``tau`` is a 4x4 matrix formed from the elements of an exponential coordinate vector:: tau = [ 0 -rz ry vx] [ rz 0 -rx vy] [-ry rx 0 vz] [ 0 0 0 1] Parameters ---------- exp_coords : array_like, shape (N, 6) or (6,) A single exponential coordinate vector or a stack of exponential coordinate vectors. The expected order of components is ``[rx, ry, rz, vx, vy, vz]``. The first 3 components encode rotation and the last 3 encode translation. Returns ------- transform : `RigidTransform` instance A single transform or a stack of transforms. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> import numpy as np Creating from a single 6d vector of exponential coordinates: >>> tf = Tf.from_exp_coords([ ... -2.01041204, -0.52983629, 0.65773501, ... 0.10386614, 0.05855009, 0.54959179]) >>> tf.as_matrix() array([[0.76406621, 0.10504613, -0.63652819, -0.10209961], [0.59956454, -0.47987325, 0.64050295, 0.40158789], [-0.2381705, -0.87102639, -0.42963687, 0.19637636], [0., 0., 0., 1.]]) >>> tf.single True A vector of zeros represents the identity transform: >>> tf = Tf.from_exp_coords(np.zeros(6)) >>> tf.as_matrix() array([[1., 0., 0., 0.], [0., 1., 0., 0.], [0., 0., 1., 0.], [0., 0., 0., 1.]]) The last three numbers encode translation. If the first three numbers are zero, the last three components can be interpreted as the translation: >>> tf_trans = Tf.from_exp_coords([0, 0, 0, 4.3, -2, 3.4]) >>> tf_trans.translation array([4.3, -2., 3.4]) The first three numbers encode rotation as a rotation vector: >>> tf_rot = Tf.from_exp_coords([0.5, 0.3, 0.1, 0, 0, 0]) >>> tf_rot.rotation.as_rotvec() array([0.5, 0.3, 0.1]) Combining translation and rotation preserves the rotation vector, but changes the last three components as they encode translation and rotation: >>> (tf_trans * tf_rot).as_exp_coords() array([0.5, 0.3, 0.1, 3.64305882, -1.25879559, 4.46109265])RigidTransform.from_components(cls, translation, rotation) Initialize a rigid transform from translation and rotation components. When creating a rigid transform from a translation and rotation, the translation is applied after the rotation, such that ``tf = Tf.from_components(translation, rotation)`` is equivalent to ``tf = Tf.from_translation(translation) * Tf.from_rotation(rotation)``. When applying a transform to a vector ``v``, the result is the same as if the transform was applied to the vector in the following way: ``tf.apply(v) == translation + rotation.apply(v)`` Parameters ---------- translation : array_like, shape (N, 3) or (3,) A single translation vector or a stack of translation vectors. rotation : `Rotation` instance A single rotation or a stack of rotations. Returns ------- `RigidTransform` If rotation is single and translation is shape (3,), then a single transform is returned. Otherwise, a stack of transforms is returned. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np Creating from a single rotation and translation: >>> t = np.array([2, 3, 4]) >>> r = R.from_euler("ZYX", [90, 30, 0], degrees=True) >>> r.as_matrix() array([[ 0. , -1., 0. ], [ 0.8660254, 0., 0.5 ], [-0.5 , 0., 0.8660254 ]]) >>> tf = Tf.from_components(t, r) >>> tf.rotation.as_matrix() array([[ 0. , -1., 0. ], [ 0.8660254, 0., 0.5 ], [-0.5 , 0., 0.8660254 ]]) >>> tf.translation array([2., 3., 4.]) >>> tf.single True When applying a transform to a vector ``v``, the result is the same as if the transform was applied to the vector in the following way: ``tf.apply(v) == translation + rotation.apply(v)`` >>> r.apply([1, 0, 0]) array([0. , 0.8660254, -0.5 ]) >>> t + r.apply([1, 0, 0]) array([2. , 3.8660254, 3.5 ]) >>> tf.apply([1, 0, 0]) array([2. , 3.8660254, 3.5 ])RigidTransform.from_translation(cls, translation) Initialize from a translation numpy array, without a rotation. When applying this transform to a vector ``v``, the result is the same as if the translation and vector were added together. If ``t`` is the displacement vector of the translation, then: ``Tf.from_translation(t).apply(v) == t + v`` Parameters ---------- translation : array_like, shape (N, 3) or (3,) A single translation vector or a stack of translation vectors. Returns ------- transform : `RigidTransform` instance Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> import numpy as np Creating a transform from a single translation vector: >>> t = np.array([2, 3, 4]) >>> t + np.array([1, 0, 0]) array([3, 3, 4]) >>> tf = Tf.from_translation(t) >>> tf.apply([1, 0, 0]) array([3., 3., 4.]) >>> tf.single True The top 3x1 points in the rightmost column of the transformation matrix is the translation vector: >>> tf.as_matrix() array([[1., 0., 0., 2.], [0., 1., 0., 3.], [0., 0., 1., 4.], [0., 0., 0., 1.]]) >>> np.allclose(tf.as_matrix()[:3, 3], t) True Creating multiple transforms from a stack of translation vectors: >>> t = np.array([[2, 3, 4], [1, 0, 0]]) >>> t + np.array([1, 0, 0]) array([[3, 3, 4], [2, 0, 0]]) >>> tf = Tf.from_translation(t) >>> tf.apply([1, 0, 0]) array([[3., 3., 4.], [2., 0., 0.]]) >>> np.allclose(tf.as_matrix()[:, :3, 3], t) True >>> tf.single False >>> len(tf) 2RigidTransform.from_rotation(cls, rotation) Initialize from a rotation, without a translation. When applying this transform to a vector ``v``, the result is the same as if the rotation was applied to the vector. ``Tf.from_rotation(r).apply(v) == r.apply(v)`` Parameters ---------- rotation : `Rotation` instance A single rotation or a stack of rotations. Returns ------- transform : `RigidTransform` instance Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np Creating a transform from a single rotation: >>> r = R.from_euler("ZYX", [90, 30, 0], degrees=True) >>> r.apply([1, 0, 0]) array([0. , 0.8660254, -0.5 ]) >>> tf = Tf.from_rotation(r) >>> tf.apply([1, 0, 0]) array([0. , 0.8660254, -0.5 ]) >>> tf.single True The upper 3x3 submatrix of the transformation matrix is the rotation matrix: >>> np.allclose(tf.as_matrix()[:3, :3], r.as_matrix(), atol=1e-12) True Creating multiple transforms from a stack of rotations: >>> r = R.from_euler("ZYX", [[90, 30, 0], [45, 30, 60]], degrees=True) >>> r.apply([1, 0, 0]) array([[0. , 0.8660254 , -0.5 ], [0.61237244, 0.61237244, -0.5 ]]) >>> tf = Tf.from_rotation(r) >>> tf.apply([1, 0, 0]) array([[0. , 0.8660254 , -0.5 ], [0.61237244, 0.61237244, -0.5 ]]) >>> tf.single False >>> len(tf) 2RigidTransform.from_matrix(cls, matrix) Initialize from a 4x4 transformation matrix. Parameters ---------- matrix : array_like, shape (4, 4) or (N, 4, 4) A single transformation matrix or a stack of transformation matrices. Returns ------- transform : `RigidTransform` instance Notes ----- 4x4 rigid transformation matrices are of the form: .. [R | t] [0 | 1] where ``R`` is a 3x3 rotation matrix and ``t`` is a 3x1 translation vector ``[tx, ty, tz]``. As rotation matrices must be proper orthogonal, the rotation component is orthonormalized using singular value decomposition before initialization. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> import numpy as np Creating a transform from a single matrix: >>> m = np.array([[0, 1, 0, 2], ... [0, 0, 1, 3], ... [1, 0, 0, 4], ... [0, 0, 0, 1]]) >>> tf = Tf.from_matrix(m) >>> tf.as_matrix() array([[0., 1., 0., 2.], [0., 0., 1., 3.], [1., 0., 0., 4.], [0., 0., 0., 1.]]) >>> tf.single True Creating a transform from a stack of matrices: >>> m = np.array([np.eye(4), np.eye(4)]) >>> tf = Tf.from_matrix(m) >>> tf.as_matrix() array([[[1., 0., 0., 0.], [0., 1., 0., 0.], [0., 0., 1., 0.], [0., 0., 0., 1.]], [[1., 0., 0., 0.], [0., 1., 0., 0.], [0., 0., 1., 0.], [0., 0., 0., 1.]]]) >>> tf.single False >>> len(tf) 2 Matrices with a rotation component that is not proper orthogonal are orthogonalized using singular value decomposition before initialization: >>> tf = Tf.from_matrix(np.diag([2, 2, 2, 1])) >>> tf.as_matrix() array([[1., 0., 0., 0.], [0., 1., 0., 0.], [0., 0., 1., 0.], [0., 0., 0., 1.]])Create a matrix from translations and rotation matrices. Parameters ---------- translations : array_like, shape (N, 3) or (3,) A stack of translation vectors. rotation_matrices : array_like, shape (N, 3, 3) or (3, 3) A stack of rotation matrices. single : bool Whether the output should be a single matrix or a stack of matrices. Returns ------- matrix : numpy.ndarray, shape (N, 4, 4) A stack of transformation matrices. Normalize dual quaternion.00010203040506070809101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899__pyx_fatalerrorhkA ^1!!{{}}~>!|7!01B.PQ 1SM,AV1"@ T4q G1F,avWA!qqq/t1G;gQ/t1G;aA[A$%p 81HF! 6xs#SfBcA *AQ'vQ 6s! 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This is implemented by applying screw linear interpolation (ScLERP) between `p` and the identity transform, where the angle of the rotation component is scaled by `n`, and the translation is proportionally adjusted along the screw axis. ``q = p ** n`` can also be expressed as ``q = RigidTransform.from_exp_coords(p.as_exp_coords() * n)``. If `n` is negative, then the transform 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 transform with itself. Returns ------- `RigidTransform` 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 ----- There are three notable cases: if ``n == 1`` then a copy of the original transform is returned, if ``n == 0`` then the identity transform is returned, and if ``n == -1`` then the inverse transform 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 RigidTransform as Tf >>> import numpy as np A power of 2 returns the transform composed with itself: >>> tf = Tf.from_translation([1, 2, 3]) >>> (tf ** 2).translation array([2., 4., 6.]) >>> (tf ** 2).as_matrix() array([[1., 0., 0., 2.], [0., 1., 0., 4.], [0., 0., 1., 6.], [0., 0., 0., 1.]]) A negative power returns the inverse of the transform raised to the absolute value of `n`: >>> (tf ** -2).translation array([-2., -4., -6.]) >>> np.allclose((tf ** -2).as_matrix(), (tf.inv() ** 2).as_matrix(), ... atol=1e-12) True A power of 0 returns the identity transform: >>> (tf ** 0).as_matrix() array([[1., 0., 0., 0.], [0., 1., 0., 0.], [0., 0., 1., 0.], [0., 0., 0., 1.]]) A power of 1 returns a copy of the original transform: >>> (tf ** 1).as_matrix() array([[1., 0., 0., 1.], [0., 1., 0., 2.], [0., 0., 1., 3.], [0., 0., 0., 1.]]) A fractional power returns a transform with a scaled rotation and translated along the screw axis. Here we take the square root of the transform, which when squared recovers the original transform: >>> tf_half = (tf ** 0.5) >>> tf_half.translation array([0.5, 1., 1.5]) >>> (tf_half ** 2).as_matrix() array([[1., 0., 0., 1.], [0., 1., 0., 2.], [0., 0., 1., 3.], [0., 0., 0., 1.]])RigidTransform.__mul__(self, RigidTransform other) Compose this transform with the other. If `p` and `q` are two transforms, then the composition of 'q followed by p' is equivalent to `p * q`. In terms of transformation matrices, the composition can be expressed as ``p.as_matrix() @ q.as_matrix()``. In terms of translations and rotations, the composition when applied to a vector ``v`` is equivalent to ``p.translation + p.rotation.apply(q.translation) + (p.rotation * q.rotation).apply(v)``. This function supports composition of multiple transforms at a time. The following cases are possible: - Either ``p`` or ``q`` contains a single or length 1 transform. In this case the result contains the result of composing each transform in the other object with the one transform. If both are single transforms, the result is a single transform. - Both ``p`` and ``q`` contain ``N`` transforms. In this case each transform ``p[i]`` is composed with the corresponding transform ``q[i]`` and the result contains ``N`` transforms. Parameters ---------- other : `RigidTransform` instance Object containing the transforms to be composed with this one. Returns ------- `RigidTransform` instance The composed transform. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> from scipy.spatial.transform import Rotation as R >>> import numpy as np Compose two transforms: >>> tf1 = Tf.from_translation([1, 0, 0]) >>> tf2 = Tf.from_translation([0, 1, 0]) >>> tf = tf1 * tf2 >>> tf.translation array([1., 1., 0.]) >>> tf.single True When applied to a vector, the composition of two transforms is applied in right-to-left order. >>> t1, r1 = [1, 2, 3], R.from_euler('z', 60, degrees=True) >>> t2, r2 = [0, 1, 0], R.from_euler('x', 30, degrees=True) >>> tf1 = Tf.from_components(t1, r1) >>> tf2 = Tf.from_components(t2, r2) >>> tf = tf1 * tf2 >>> tf.apply([1, 0, 0]) array([0.6339746, 3.3660254, 3. ]) >>> tf1.apply(tf2.apply([1, 0, 0])) array([0.6339746, 3.3660254, 3. ]) When at least one of the transforms is not single, the result is a stack of transforms. >>> tf1 = Tf.from_translation([1, 0, 0]) >>> tf2 = Tf.from_translation([[0, 2, 0], [0, 0, 3]]) >>> tf = tf1 * tf2 >>> tf.translation array([[1., 2., 0.], [1., 0., 3.]]) >>> tf.single False >>> len(tf) 2RigidTransform.__setitem__(self, indexer, value) Set transform(s) at given index(es) in this object. Parameters ---------- indexer : int or slice or array_like Specifies which transform(s) to replace. A single indexer must be specified, i.e. as if indexing a 1 dimensional array or list. value : `RigidTransform` instance The transform(s) to set. Raises ------ TypeError If the transform is a single transform. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> t = [[0, 0, 0], [1, 0, 0], [2, 0, 0]] # 3 translations >>> tf = Tf.from_translation(t) Set a single transform: >>> tf[0] = Tf.from_translation([9, 9, 9]) >>> tf.translation array([[9., 9., 9.], [1., 0., 0.], [2., 0., 0.]])RigidTransform.__getitem__(self, indexer) Extract transform(s) at given index(es) from this object. Creates a new `RigidTransform` instance containing a subset of transforms stored in this object. Parameters ---------- indexer : int or slice or array_like Specifies which transform(s) to extract. A single indexer must be specified, i.e. as if indexing a 1 dimensional array or list. Returns ------- transform : `RigidTransform` instance Contains - a single transform, if `indexer` is a single index - a stack of transform(s), if `indexer` is a slice, or an index array. Raises ------ TypeError If the transform is a single transform. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> t = [[0, 0, 0], [1, 0, 0], [2, 0, 0]] # 3 translations >>> tf = Tf.from_translation(t) A single index returns a single transform: >>> tf[0].as_matrix() array([[1., 0., 0., 0.], [0., 1., 0., 0.], [0., 0., 1., 0.], [0., 0., 0., 1.]]) A slice returns a stack of transforms: >>> tf[1:3].translation array([[1., 0., 0.], [2., 0., 0.]]) An index array returns a stack of transforms: >>> tf[[0, 2]].translation array([[0., 0., 0.], [2., 0., 0.]])RigidTransform.__len__(self) Return the number of transforms in this object. Multiple transforms can be stored in a single instance. Returns ------- length : int The number of transforms in this object. Raises ------ TypeError If the transform is a single transform. Examples -------- >>> from scipy.spatial.transform import RigidTransform as Tf >>> tf = Tf.identity(3) >>> len(tf) 3 >>> tf = Tf.from_translation([1, 0, 0]) >>> len(tf) # doctest: +IGNORE_EXCEPTION_DETAIL Traceback (most recent call last): ... TypeError: Single transform has no len().Initialize from a 4x4 transformation matrix. Rotations are not meant to be initialized directly. Please use one of the `from_...` methods instead. Parameters ---------- matrix : array_like, shape (4, 4) or (N, 4, 4) A single transformation matrix or a stack of transformation matrices. normalize : bool, optional If True, orthonormalize the rotation matrix using singular value decomposition. If False, the rotation matrix is not checked for orthogonality or right-handedness. copy : bool, optional If True, copy the input matrix. If False, a reference to the input matrix is used. If normalize is True, the input matrix is always copied regardless of the value of copy. 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