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Page 9 -- Applications of Coordinate Transformations

This website has been designed to provide a progressive understanding of three dimensional rigid body kinematics. Coverage of this material is thought to be important to students of biomechanics because the calculation of rigid body kinematics is rapidly becoming transparent to the user in many of the automated motion analysis systems.

Transformations can be used to describe the position of a body in global and local coordinates or the relative position of any rigid body from one instant to the next. These 4x4 matrices have special applications in biomechanics.

Photogrammetry techniques are often used to describe the complex motion of a human subject. Using a motion analysis system, we can track the positions of carefully placed markers attached to the subject during locomotion or other activity. Once the markers are placed at points of interest (e.g. joints), the global locations can be recorded during the experimental data capture session.

Traditionally, two or more synchronized cameras capture the movement of a subject wearing reflective markers at the body landmarks of interest. These videos are manually digitized and a complex set of transformations are performed to configure the data into three dimensional global coordinates, as described in this tutorial.

Other types of photogrammetry techniques involve the use of light emitting diodes which transmit a signal from the point of interest on the body to an IR-signal sensing camera which is able to reconstruct the signal origin location in space. State-of-the-art motion analysis systems require minimal user intervention for determination of 3D global coordinates.

Coordinate transformations are an integral part of data capture for motion analysis. The global positions of reflective markers are irrelevant for kinematic studies unless they can be described relative to anatomical landmarks. Possible applications of 4x4 matrices include:

  • Anatomical Calibration: Location of anatomical axes of rotation relative to global marker coordinates can be the result of careful calibration.
  • Joint Rotation: Rotation of the knee, for example, can be described by the knee joint center plus the motion of the shank relative to motion of the thigh.
  • Virtual Points: It may be impossible to place markers at all the key locations (e.g. the hip joint center), therefore a calibration procedure facilitates hidden landmark identification.

Demonstrations of coordinate transformations can be found for...

  • Relative Coordinate Systems
  • Anatomical Calibration
  • Relative Motion of Two Rigid Bodies
  • 3D Global Positions as a Function of Time