Transform Tutorial

Background

The specification of the location of an object in three dimensional space implies specifying its position and its orientation (also called attitude) with respect to a reference position and orientation.Three-dimensional coordinate frame transformations are widely used in robotics to specify locations of objects. Detailed descriptions of coordinate frame transformations are found in a number of introductory robotics textbooks. CLARAty has a number of C++ software classes for representing and operating on coordinate transforms. We provide an overview of the use of the CLARAty transform module here.

The axes of a coordinate frame are typically labeled X, Y and Z and are arranged as shown on the figure below. To specify location of an object, it is convenient to also specify a local coordinate frame attached to a fixed location on the object. The specification of the object's location can then be precisely stated as the location of its local coordinate frame with respect to the reference coordinate frame. To facilitate the discussion, we will give labels to the different coordinate frames. So, for example, we call the local coordinate frame C₂ and the reference coordinate frame C₀. A convenient approach to represent the general transformation of a coordinateframe from C₀ to C₂ is to partition the transformation into its pure translation and pure rotation components. The location of the object is obtained by a pure translation of the coordinate frame C₀to C₁ followed by a pure rotation of C₁ to C₂.

Pure translations in three dimensional space are represented by three dimensional vectors. The three elements of the vector are used to represent the components of the translation along the X, Y and Z axes of the coordinate frame respectively. In the figure above, the green vector drawn from the origin of coordinate frame C₀ to the origin of coordinate frameC₁ represents the translation.

The specification of orientation is more complex. The figure above illustrates the problem. The origins of the two coordinate frames C₁and C₂ are co-incident but the frames are in different orientations. The problem is to specify the orientation of C₂ with respect to C₁.The blue curved arrows show how the X, Y and Z axes must simultaneously rotate (while satisfying the constraint that they are attached to a rigid body) to transform the orientation of the coordinate frame from C₁ to C₂. A number of conventions and numerical constructs have been developed to represent this change in orientation. Most introductory robotics textbooks describe in detail the approaches used to represent rotation transformations. CLARAty provides two alternatives for specifying orientations: three-by-three orthonormal rotation matrices and quaternions.

The combination of a translation vector and a rotation matrix or a quaternion gives us a quantity that can represent general three dimensional transformations. CLARAty provides a number of options to represent general transformations. The CLARAty classes used for representing threedimensional transformations are listed below.

CLARAty Transform Classes

Using CLARAty Transform Classes

A demonstration, contained in a file demo_transform.cc, is included with the transform module to illustrate usage of CLARAty's transform classes. The following are code snippets taken from demo_transform.cc.

Create a Point and compute its distance from another Point

Create an RMatrix using roll, pitch yaw parameters then extract the roll, pitch, yaw parameters from the RMatrix

Create a Quaternion and convert it into an RMatrix.

Create a Trans with RMatrix then apply a rotate transform then a translate transform to it.

Create a Trans with Quaternion then apply a rotate transform then a translate transform to it.

We can perform similar operations with HTrans

And QTrans

Many more functions are available in the CLARAty transform classes. The user is encouraged to review the header files and the demo and utest files within the transform module to see how these classes can be used.