HHD Site



Health and Human Development Home
  Dept of Kinesiology

Abstract presented at the American Society of Biomechanics Annual Meeting
Clemson, SC.
September 25, 1997


J. K. Startzell, H. J. Sommer1, D. R. Lemmon, and P. R. Cavanagh

Center for Locomotion Studies, Penn State University, University Park, PA 16802
1Department of Mechanical Engineering, Penn State University, University Park, PA 16802


A website is described which provides the student of biomechanics with a progressive understanding of three dimensional rigid body kinematics.  Starting with global positions of targets on two rigid bodies, the user is shown the stages necessary for anatomical calibration and calculation of the angles of a joint coordinate system (JCS).  Following the initial presentation, data files from a variety of movements are available to the user who can then progress from an understanding of the calculations to an interpretation of the movement being studied.


Basic linear algebra can be used to define relative translation and rotation between two rigid bodies in 2D and 3D space.  A 4x4 transformation matrix contains a location vector, which gives the location of a local body relative to a global coordinate frame, and a rotation submatrix, which gives the attitude of the local axes relative to the global axes.  This matrix can be decomposed to determine the orientation of the local axes with respect to the global axes and the angles of rotation about each axis.

The joint  coordinate system (JCS) proposed by Grood and Suntay (1983) defines relative rotation of two bodies about two segment-fixed axes and a floating axis.  The temporal sequence dependency of Euler angle techniques is eliminated provided that the body-fixed axes are selected wisely.

The terminology and nomenclature recommended by the International Society of Biomechanics are used throughout the website (//www.kin.ucalgary.ca/isb/standards/).  The recommendations of Cole et al. (1993) regarding identification of the three JCS axes have also been followed.

MATLAB is a user-friendly software package designed for matrix manipulation.  It is an ideal environment in which to perform geometric transformations and angle decomposition for configuration of specified input data.


Basic Linear Algebra
A review of vector algebra and matrix algebra is first presented with simple examples where a physical process (translation or rotation) is related to the mathematics.  A transformation matrix is constructed which represents the relative position and orientation of two bodies.

Anatomical Calibration
A calibration procedure is described which facilitates anatomical landmark identification.  Through the use of a calibrated wand containing markers at known distances from the tip placed at an anatomical landmark, the global coordinates of this landmark can immediately be identified.  This method eliminates the need for additional anatomical markers to be affixed to the skin.

Motion Analysis and Joint Coordinate Systems
The global positions of targets on a single rigid segment are used in conjunction with anatomical calibrations to determine the body-fixed axes about which segment rotation is described.  The relative position of two bodies in motion can be defined by a joint coordinate system including two segment-fixed axes and a mutually orthogonal floating axis.

Integration and Application
Both graphical and analytical presentations are accessible on the website.  Simple MATLAB routines for such vector operations as unit vector calculation, dot and cross products, and matrix transformations are provided.  In addition, data files from a variety of movements provide the user with a transition from theory to application of kinematic analysis.

Interactive Aspects
A JAVA applet has been implemented to provide a user interface for demonstration purposes.  The student can identify angles of rotation and view the resulting rigid body movement and the corresponding transformation matrix.

The following pages are available on the site:

• Basic vector algebra - with examples
• Basic matrix algebra - with examples
• 2D transformations
• 3D transformations
• 4x4 matrices
• 3D global positions of markers from a motion analysis system
• Anatomical Calibration using a wand
• Local and Anatomical systems
• Relative position of two anatomical coordinate systems
• Joint coordinate systems and their decomposition matrices
• Example data configurations
• Examples of JCS angles for entire movements

The site can be accessed at: http://www.biomechanics.psu.edu/publications/kinematics.html
(relocated 7/29/04 NG)


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.  The site is not intended to provide software for routine analysis of data but rather to offer a user-friendly environment in which the basic concepts of 3D kinematic analysis can be mastered.  The advantage of using the Internet rather than a textbook for such education is that interactive simulations can be performed by the student who can then see both graphical and numerical feedback of specified input.


Buczek F. Three-Dimensional Kinematics and Kinetics of the Ankle and Knee Joints During Uphill, Level, and Downhill Walking, PhD Dissertation, Penn State University 1990.

Cole G., et al. Trans. ASME , 115: 344-349, 1993.

Craig. Introduction to Robotics, Addison Wesley, 1986.

Grood E., Suntay W. J Biomech Eng.  105:136-144, 1983.

Nigg B., Herzog W.  Biomechanics of the Musculo-skeletal System, 264-277. John Wiley & Sons, 1994.

Sommer HJ. ASB Tutorial: Primer on 3-D Kinematics, 1991.