2.3.3.1.53. NXtransformations

Status:

base class, extends NXobject

Description:

Collection of axis-based translations and rotations to describe a geometry. ...

Collection of axis-based translations and rotations to describe a geometry. May also contain axes that do not move and therefore do not have a transformation type specified, but are useful in understanding coordinate frames within which transformations are done, or in documenting important directions, such as the direction of gravity.

A nested sequence of transformations lists the translation and rotation steps needed to describe the position and orientation of any movable or fixed device.

There will be one or more transformations (axes) defined by one or more fields for each transformation. Transformations can also be described by NXlog groups when the values change with time. The all-caps name AXISNAME designates the particular axis generating a transformation (e.g. a rotation axis or a translation axis or a general axis). The attribute units="NX_TRANSFORMATION" designates the units will be appropriate to the transformation_type attribute:

  • NX_LENGTH for translation

  • NX_ANGLE for rotation

  • NX_UNITLESS for axes for which no transformation type is specified

This class will usually contain all axes of a sample stage or goniometer or a detector. The NeXus default McSTAS coordinate frame is assumed, but additional useful coordinate axes may be defined by using axes for which no transformation type has been specified.

Transformation chain

The entry point of a chain of transformations is a field called depends_on will be outside of this class and points to a field in here. Following the chain may also require following depends_on links to transformations outside, for example to a common base table. If a relative path is given, it is relative to the group enclosing the depends_on specification.

For a chain of three transformations, where \(T_1\) depends on \(T_2\) which in turn depends on \(T_3\), the final active transformation matrix \(T_f\) is

\[T_f = T_3 . T_2 . T_1\]

For example when positioning a flat detector, its pixel positions in the laboratory reference frame (McSTAS coordinate frame by default) can be calculated by

\[X_\text{lab} = T_f . X_\text{pixel} = T_3 . T_2 . T_1 . X_\text{pixel}\]

Note that \(T_1\) comes first in the depends-on chain and is also applied first to the pixel coordinates.

When we say transformation \(A\) depends on transformation \(B\) we mean that the physical motor that realizes \(A\) is stacked on top of the motor that realizes \(B\). So the activate coordinate transformation \(A\) needs to be applied to a coordinate before applying \(B\). In other words \(X' = B . A . X\).

Transformation matrix

In explicit terms, the transformations are a subset of affine transformations expressed as 4x4 active transformation matrices that act on homogeneous coordinates, \(X=[x,y,z,1]^T\).

For rotation and translation,

\[\begin{split}T_r &= \begin{pmatrix} R & o \\ 0_3 & 1 \end{pmatrix} \\ T_t &= \begin{pmatrix} I_3 & t + o \\ 0_3 & 1 \end{pmatrix}\end{split}\]

where \(R\) is the usual 3x3 rotation matrix, \(o\) is an offset vector, \(0_3\) is a row of 3 zeros, \(I_3\) is the 3x3 identity matrix and \(t\) is the translation vector.

\(o\) is given by the offset attribute, \(t\) is given by the vector attribute multiplied by the field value, and \(R\) is defined as a rotation about an axis in the direction of vector, of angle of the field value.

Usage

One possible use of NXtransformations is to define the motors and transformations for a diffractometer (goniometer). Such use is mentioned in the NXinstrument base class. Use one NXtransformations group for each diffractometer and name the group appropriate to the device. Collecting the motors of a sample table or xyz-stage in an NXtransformations group is equally possible.

Following the section on the general description of axis in NXtransformations is a section which documents the fields commonly used within NeXus for positioning purposes and their meaning. Whenever there is a need for positioning a beam line component please use the existing names. Use as many fields as needed in order to position the component. Feel free to add more axis if required. In the description given below, only those atttributes which are defined through the name are spcified. Add the other attributes of the full set:

  • vector

  • offset

  • transformation_type

  • depends_on

as needed.

Example 1: goniometer

Position a sample mounted on a goniometer in the McSTAS coordinate frame.

The sample is oriented as follows

\[X_\text{lab} = R(\vec{v}_\omega, \omega) . T(\vec{v}_z, \text{sam}_z) . T(\vec{v}_y, \text{sam}_y) . T(\vec{v}_x, \text{sam}_x) . R(\vec{v}_\chi, \chi) . R(\vec{v}_\varphi, \varphi) . X_s\]

where

  • \(R(\vec{v},a)\) is a rotation around vector \(\vec{v}\) with angle \(a\)

  • \(T(\vec{u},t)\) is a translation along vector \(\vec{u}\) over a distance \(t\)

  • \(X_s\) a coordinate in the sample reference frame.

entry:NXentry
  sample:NXsample
    depends_on=transformations/phi
    transformations:NXtransformations
      phi=0
        @depends_on=chi
        @transformation_type=rotation
        @vector=[-1 -0.0037 -0.002]
        @units=degrees
      chi=0
        @depends_on=sam_x
        @transformation_type=rotation
        @vector=[0.0046 0.0372 0.9993]
        @units=degrees
      sam_x=0
        @depends_on=sam_y
        @transformation_type=translation
        @vector=[1 0 0]
        @units=mm
      sam_y=0
        @depends_on=sam_z
        @transformation_type=translation
        @vector=[0 1 0]
        @units=mm
      sam_z=0
        @depends_on=omega
        @transformation_type=translation
        @vector=[0 0 1]
        @units=mm
      omega=174
        @depends_on=.
        @transformation_type=rotation
        @vector=[-1 0 0]
        @units=degrees

Example 2: different coordinate system

Define a laboratory reference frame with the X-axis along the beam and the Z-axis opposite to the direction of gravity. Three point detectors are positioned in this reference:

  • transmission:

    • point detector in the beam

    • 20 cm downstream from the sample (the origin of the reference frame)

  • vertical:

    • point detector 10 cm downstream from the sample

    • making an angle of 5 degrees with the beam w.r.t. the sample

    • positioned in the XZ-plane above the XY-plane

  • horizontal:

    • point detector 11 cm downstream from the sample

    • making an angle of 6 degrees with the beam w.r.t. the sample

    • positioned in the XY-plane above the XZ-plane

The coordinates of the point detectors in the laboratory reference frame are

  • transmission: \(X_\text{lab} = T_x(20) . X_d\)

  • vertical: \(X_\text{lab} = R_y(-5) . T_x(10) . X_d\)

  • horizontal: \(X_\text{lab} = R_x(-90) . R_y(-6) . T_x(11) . X_d\)

where

  • \(T_x\), \(T_y\), \(T_z\): active transformation matrices for translation along the X, Y and Z axes

  • \(R_x\), \(R_y\), \(R_z\): active transformation matrices for rotation around the X, Y and Z axes

  • \(X_d\) is a coordinate in the detector reference frame.

Note that as these are point detectors, we only have one coordinate \(X_d=[0,0,0,1]^T\).

entry:NXentry
  instrument:NXinstrument
    vertical:NXdetector
      depends_on=position/distance
      position:NXtransformations
        distance=10         # move downstream from the sample
          @depends_on=polar
          @units=cm
          @vector=[1 0 0]
        polar=5             # title above the horizontal plane
          @depends_on=azimuth
          @units=degrees
          @vector=[0 -1 0]
        azimuth=0         # stay in the vertical plane
          @depends_on=/entry/coordinate_system/beam
          @units=degrees
          @vector=[-1 0 0]
    horizontal:NXdetector
      depends_on=position/distance
      position:NXtransformations
        distance=11         # move downstream from the sample
          @depends_on=polar
          @units=cm
          @vector=[1 0 0]
        polar=6             # title above the horizontal plane
          @depends_on=azimuth
          @units=degrees
          @vector=[0 -1 0]
        azimuth=90           # rotate to the horizontal plane
          @depends_on=/entry/coordinate_system/beam
          @units=degrees
          @vector=[-1 0 0]
    transmission:NXdetector
      depends_on=position/distance
      position:NXtransformations
        distance=20         # move downstream from the sample
          @depends_on=/entry/coordinate_system/beam
          @units=cm
          @vector=[1 0 0]
  coordinate_system:NXtransformations
    beam=NaN               # value is never used
      @depends_on=gravity
      @vector=[1 0 0]     # X-axis points in the beam direction
    gravity=NaN             # value is never used
      @depends_on=.         # end of the chain
      @vector=[0 0 -1]       # Z-axis points up

Symbols:

No symbol table

Groups cited:

none

Structure:

@default: (optional) NX_CHAR

Declares which child group contains a path leading ...

Declares which child group contains a path leading to a NXdata group.

It is recommended (as of NIAC2014) to use this attribute to help define the path to the default dataset to be plotted. See https://www.nexusformat.org/2014_How_to_find_default_data.html for a summary of the discussion.

AXISNAME: (optional) NX_NUMBER {units=NX_TRANSFORMATION}

Units need to be appropriate for translation or rotation ...

Units need to be appropriate for translation or rotation

The name of this field is not forced. The user is free to use any name that does not cause confusion. When using more than one AXISNAME field, make sure that each field name is unique in the same group, as required by HDF5.

The values given should be the start points of exposures for the corresponding frames. The end points should be given in AXISNAME_end.

@transformation_type: (optional) NX_CHAR

The transformation_type may be ``translation``, in which case the ...

The transformation_type may be translation, in which case the values are linear displacements along the axis, rotation, in which case the values are angular rotations around the axis.

If this attribute is omitted, this is an axis for which there is no motion to be specifies, such as the direction of gravity, or the direction to the source, or a basis vector of a coordinate frame.

Any of these values: translation | rotation

@vector: (optional) NX_NUMBER

Three values that define the axis for this transformation. ...

Three values that define the axis for this transformation. The axis should be normalized to unit length, making it dimensionless. For rotation axes, the direction should be chosen for a right-handed rotation with increasing angle. For translation axes the direction should be chosen for increasing displacement. For general axes, an appropriate direction should be chosen.

@offset: (optional) NX_NUMBER

A fixed offset applied before the transformation (three vector components). ...

A fixed offset applied before the transformation (three vector components). This is not intended to be a substitute for a fixed translation axis but, for example, as the mechanical offset from mounting the axis to its dependency.

@offset_units: (optional) NX_CHAR

Units of the offset. Values should be consistent with NX_LENGTH.

@depends_on: (optional) NX_CHAR

Points to the path of a field defining the axis on which this instance of ...

Points to the path of a field defining the axis on which this instance of NXtransformations depends or the string “.”. It can also point to an instance of NX_coordinate_system if this transformation depends on it.

@equipment_component: (optional) NX_CHAR

An arbitrary identifier of a component of the equipment to which ...

An arbitrary identifier of a component of the equipment to which the transformation belongs, such as ‘detector_arm’ or ‘detector_module’. NXtransformations with the same equipment_component label form a logical grouping which can be combined together into a single change-of-basis operation.

AXISNAME_end: (optional) NX_NUMBER {units=NX_TRANSFORMATION}

``AXISNAME_end`` is a placeholder for a name constructed from the actual ...

AXISNAME_end is a placeholder for a name constructed from the actual name of an axis to which _end has been appended.

The values in this field are the end points of the motions that start at the corresponding positions given in the AXISNAME field.

AXISNAME_increment_set: (optional) NX_NUMBER {units=NX_TRANSFORMATION}

``AXISNAME_increment_set`` is a placeholder for a name constructed from the ac ...

AXISNAME_increment_set is a placeholder for a name constructed from the actual name of an axis to which _increment_set has been appended.

The value of this optional field is the intended average range through which the corresponding axis moves during the exposure of a frame. Ideally, the value of this field added to each value of AXISNAME would agree with the corresponding values of AXISNAME_end, but there is a possibility of significant differences. Use of AXISNAME_end is recommended.

Hypertext Anchors

List of hypertext anchors for all groups, fields, attributes, and links defined in this class.

NXDL Source:

https://github.com/FAIRmat-NFDI/nexus_definitions/tree/fairmat/base_classes/NXtransformations.nxdl.xml