Biomechanics - Introduction to Biomedical Engineering - page 150

% rotation of a system of coordinates by Euler's
% angles thetax, thetay and thetaz, measured in degrees.

% Now the first rotation is about the x-axis, so we use eqs. 4.24-26

¼

A [1 0 0

0 cosd(thetax) sind(thetax)

0 -sind(thetax) cosd(thetax) ];

% Next is the y-axis rotation (eqs. 4.21-23)

B ¼ [ cosd(thetay) 0 -sind(thetay)

0 1 0

sind(thetay) 0 cosd(thetay) ];

% Finally, the z-axis rotation (eqs. 4.27-29)

¼

C [ cosd(thetaz) sind(thetaz) 0

-sind(thetaz) cosd(thetaz) 0
0 0 1 ];
% Multiplying rotation matrices C, B and A as in Eq. 4.30 gives the
solution:

¼

D C*B*A;

Now use this function to compute the numerical transformation matrix:

>>

eulangle(30,20,10)
ans

¼
0.9254

0.3188 -0.2049

-0.1632 0.8232 0.5438

0.3420 -0.4698 0.8138

This matrix can be used to convert any point in the initial coordinate system (premaneuver) to
its position after the roll, pitch, and yaw maneuvers have been executed.

4.2.3 Static Equilibrium
Newton's equations of motion applied to a structure in static equilibrium reduce to the
following vector equations:

X F

¼ 0 ð 4 : 35 Þ

X M

¼ 0 ð 4 : 36 Þ

These equations are applied to biological systems in the same manner as standard
mechanical structures. Analysis begins with a drawing of the free-body diagram of the body
segments of interest with all externally applied loads and reaction forces at the supports.
Orthopedic joints can be modeled with appropriate ideal joints, such as hinge, ball-and-
socket, and so forth, as discussed in Chapter 3 (see Figure 3.33).