Biomedical Engineering Reference
In-Depth Information
k
000
i
¼
cos y
x
sin y
y
2
4
3
5
k
000
i
cos y
x
ð
4
:
33
Þ
y
y
¼
arcsin
i
000
j
¼
sin y
z
cos y
x
2
4
3
5
i
000
j
cos y
x
ð
4
:
34
Þ
y
z
¼
arcsin
Walking produces tri-planar hip, knee, and ankle motion: flexion/extension (FE), abduction/
adduction (AA), and internal/external transverse rotation (TR). Euler angles offer an oppor-
tunity to quantify these coordinatedmotions. The order of the Euler angle rotation sequence
FE-AA-TR, corresponding to a
axis rotation sequence, or tilt-obliquity-rotation, was
chosen to correlate to the largest to smallest joint excursions during walking [6]. More
recently, research suggests that for pelvic motion, an Euler angle
y-x-z
rotation sequence
corresponding to rotation-obliquity-tilt is more consistent with clinical observations [1].
z-x-y
EXAMPLE PROBLEM 4.3
Write the Euler angle transformation matrices for the
-
-
rotation sequence using the MATLAB
y
x
z
symbolic math toolbox.
Solution
The following MATLAB script, or m-file, is a collection of MATLAB commands that can be run
by invoking the m-file name “eulerangles” in the command line.
% eulerangles.m
%
% Euler angles for y-x-z rotation sequence
% using MATLAB symbolic math toolbox
%
% x, y and z are thetax, thetay and thetaz, respectively
% First define them as symbolic variables
syms x y z
% Writing equations 4.21-23 as a matrix A
¼
A
[ cos(y), 0, -sin(y);
0,
1,
0;
sin(y), 0,
cos(y)]
% equations 4.24-26 as matrix B
¼
B
[ 1 0, 0;
0, cos(x), sin(x);
0, -sin(x), cos(x)]
