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Calculating Euler angles in 3D lower extremity kinematics and kinetics


Jan C. Br�nd and Gideon Ariel, Ph.D.



Evaluation of human lower extremity movement using 3D kinetics and kinematics is becoming more and more widespread. In the literature several methods for solving the mathematics have been proposed and a commonly used method is covered in the book Dynamics of Human Gait by Kit Vaughan and co. workers. The book can be purchased with a DOS program.

Implementing the method in MATLAB, a discontinuity at pi/2 was found calculating the euler angles. The method in the book suggests a filtering of the euler angles, removing most of the huge spikes appearing in the second derivatives used in the calculation of the angular momentum.

The convention used for the definition of the Euler angles is based on old classical mechanics textbooks (Synge ad Griffith 1959, Goldstein 1965).

The following mathematics are from the book Dynamics of human gait by Kit Vaughan.

To solve the problem with the discontinuity at pi/2 a method for calculating euler angles proposed in the chapter Kinetics theory by G. Wu in the book Gait Analysis was modified and used.


firsteulerp.gif (9387 bytes)

Fig 1. This figure shows a comparison of the actual euler angle and the first and second derivatives. Left is the method used by Kit Vaughan in the book Dynamics of human gait and right the new proposed method.

secondeulerp.gif (10419 bytes)

Fig 2. This figure shows a comparison of the angular velocity, acceleration and momentum. Left is the method used by Kit Vaughan in the book Dynamics of human gait and right the new proposed method.

Figure 1 and 2 clearly shows that using the modified method for calculating the euler angles result in a complete different shape and a reduction in the peak to peak values of all the angular parameters for the thigh and calf.

Analysis of the foot segment shows that the angular parameters is not affected. This is due to a foot segment attitude rotation about the k axis in the local reference system compared to the thigh and calf.
The impact of the new euler angles on resultant joint moment output at the knee and hip is more or less insignificant but suggests that a standalone evaluation of human lower extremity movement using angular parameters using the method proposed by Kit Vaughan and co. workers should be done with caution.


Vaughan, C.L, Davis, B.L., O'Conners, J.C. (1992) Dynamics of human gait. Human Kinetics publishers, Champaign.

Wu, G. (199X) Kinetics theory in Gait Analysis.

Goldstein, H. (1965) Classical mechanics. Reading, MA: Addison-Wesley.

Synge, J.L, Griffith, B.A. (1959) Principles of mechanics. New York: McGraw Hill.


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