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Page 8 -- 4x4 Transformation Matrix

A 4x4 matrix is a tool used by kinematicians for simplification of coordinate transformations. It can be formatted to include information about translation and rotation than can be discerned from a quick glance as described in the following...

STRUCTURE OF A 4x4 TRANSFORMATION MATRIX

The first row of a 4x4 transformation matrix is a placemarker vector which simplifies matrix manipulations, by ensuring that the matrix is square and invertible.

A typical 4x4 transformation matrix would fit the following form:

Equation

where the position vector P represents the translation from the global to the local coordinate system, and the rotation submatrix R denotes the rotations of each axis in body 1 about body 2. Note that the rotation submatrix is a multiplication matrix of the dot products of the unit vectors of the two body coordinate systems, therefore each column of R indicates the orientation of the local axis with respect to the corresponding global axis.

MANIPULATION OF TRANSFORMATION MATRICES

The general goal of transformation algorithms is to convert the motion of global 3D coordinates to meaningful relative rotations of two bodies. Some tools are needed to ease the manipulation of the transformation matrices:

  • Position Matrix: a transformation from local (body 1 or 2) to global coordinates

    Equation
  • Local Transformation Matrix: a transformation in local coordinates from one body to another

    Equation
  • Displacement Matrix: a transformation in global coordinates from one body to another

    Equation

Three traditional problems are encountered which require matrix manipulation using the tools defined above:

  1. Given global coordinates of two bodies, find relative position in local reference frame
  2. Given global coordinates of one body and its relative position to another body, find global coordinates of second body
  3. Given global coordinates of two bodies, find displacement matrix between bodies


Solution 1:
By definition,

Equation

Inverting,
Equation


Solution 2:
Simply,

Equation


Solution 3:
First, let's introduce position vectors, and define a single point at two moments in time:

Equation

Since these equations describe the local position of a single point, which does not change with time,
Equation

Therefore, inverting and substituting,
Equation

and the displacement matrix is defined,
Equation

Note the difference between the displacement matrix and the local transformation matrix is merely a change of sequence of rotations. This order of matrix multiplication corresponds to the switch from global to local coordinates or vice-versa.