Geometries with orientation support
126 Rockwell Automation Publication MOTION-UM002F-EN-P - February 2018
As a result, it is necessary to convert target point specified in XYZRxRyRz user
format to its equivalent transform point represented by the 4 x 4 transform matrix.
The transform point along with other transforms that map for instance tool tip
with respect to the end of arm is used to set up motion of Robot manipulator
through its work envelope in Cartesian or joint space to achieve the specified
motion.
Conversion from Transform Point to XYZRxRyRz
It is also then necessary to transform the points in the 4 x 4 transform matrix
format to the user XYXRxRyRz format for user reference, teaching and display
purpose.
Transforming between the frames is complex and sometimes has limitations on
computational solutions available. For the XYZ fixed format that get used by the
Logix firmware, there are points with Ry rotation of 90
that has multiple
solutions. This condition is described as gimbal lock condition which occurs at Ry
equal to +/- 90 . The system has to handle this condition by picking a solution
out of the multiple possible solutions.
Also, solutions are not available when Ry rotates beyond 90
.
A rotation matrix can be used to rotate Rx, Ry or Rz to any value in the range of
+/-180
and obtain the rotation matrix around the base axis. Trigonometric
equations can rotate beyond 180
in either direction. They flip to the positive or
negative side at the boundary condition of 180
. This behavior is followed in the
Logix firmware for Rx and Rz rotations. The Ry rotation needs to follow a
different behavior.
Transforming between the frames sometimes has limitations on computational
solutions available. For the XYZ fixed format used by the Logix firmware, certain
orientations, such as Ry rotation of 90
or -90 , can result in multiple solutions
known as singularity. Also, solutions are not available when Ry rotates beyond
90 . As a result, Ry is restricted to +/-90 and has four regions as shown in the
RxRyRz, flip, mirror flip
condition
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