Simulation and estimation - Biomedical Engineering Desk Reference - page 49

R leg ¼ R thigh $ ð X leg ; motion A Þ $ ð X leg ; initial Þ 1

26.8353
15.3501

(2.2.8)

Rotation matrices should be orthogonal, since the trans-
formation does not change the magnitude or length of the
thigh or lower leg. A matrix R is orthogonal if R T R is the
identity matrix I

For which the MATLAB implementation is

% Find final position of knee
KneePos ¼ ThighLen;
FinalKneePos ¼ (RotThigh*KneePos) þ HipPos;
% Find rotation of leg about knee
% First determine local coordinates of leg
markers
% with respect to the knee
% The rotation is in the local coordinate
system of the Knee
LegFinalLocal ¼ LegFinalGlobal-
[FinalKneePos FinalKneePos
FinalKneePos];
CombinedRotLeg ¼ LegFinalLocal*
inv(LegRestLocal);
RotLeg ¼ inv(RotThigh)*CombinedRotLeg;

>> RotThigh'*RotThigh
ans ¼
0.9999 -0.0011 -0.0017

-0.0011 0.9990 0.0013

-0.0017 0.0013 1.0008

>> RotLeg '* RotLeg
ans ¼
1.0007 0.0007 -0.0020
0.0007 1.0035 -0.0003
-0.0020 -0.0003 1.0036
which are both the identity matrix I to within numerical
error. Check the condition number of each matrix, and
check how error is propagated when computing an
inverse.

Lastly, the position of the ankle has to be computed,
since there is no marker at the ankle. The global position
after motion is given by rotating the end of the lower leg
(position B in Fig. 2.2-3 ) and translating it with respect to
the position of the knee:

2.2.7 PHYSBE Simulations

PHYSBE is a model of the circulatory system for simu-
lating the flow of oxygen, nutrients, heat, or chemical
tracers within the bloodstream (McCleod, 1966, 1968).
Although the fundamental work on PHYSBE dates from
the 1960s, PHYSBE has since been implemented in
Simulink,making it accessible for educational and research
uses. The Simulink implementation of PHYSBE can be
downloaded from the MathWorks MATLAB Central Web
site. This section includes several examples of how to use
PHYSBE to model cardiovascular system pathologies and
predict the result on blood flow.

B[R thigh $R leg $ ½ 18 : 13 0 0 T þ A (2.2.9)

which is, in MATLAB,

% Coordinates of the ankle are
% Ankle ¼ Knee þ Rot*Ankle at Rest
% Ankle is in the local coordinates of the
% Knee, which is the center of rotation
AnklePos ¼ LegLen;
FinalAnklePos ¼ FinalKneePos þ
CombinedRotLeg*AnklePos;

Example 2.2.7 Normal PHYSBE operation.

The rotation matrices are

Use a procedure to install and run one simulation of
PHYSBE.
Start the PHYSBE simulation by running the
MATLAB script pctrl2 to open the PHYSBE control
panel ( Fig. 2.2-19 ). Do not change any of the default
values, but be sure to save the parameter file.
Next, start the PHYSBE Simulink model ( Fig. 2.2-20 )
and start the simulation.
The simulation results which are displayed on
a Simulink scope are the heart pressure ( Fig. 2.2-21 ) and
heart volume ( Fig. 2.2-22 ).
Both are easiest to differentiate by color. There is also
a floating scope, which can be tied to other system pa-
rameters of interest.
These normal results will be compared to the pressure
and volume relationships in each of the examples below.

>> RotThigh
RotThigh ¼
0.6633 -0.7000 -0.2675

0.5567 0.6975 -0.4500

0.5000 0.1500 0.8525

>> RotLeg
RotLeg ¼
0.4233 -0.8933 0.1581

0.9063 0.4183 -0.0749

-0.0011 0.1747 0.9864

and the position of the ankle is

>> FinalAnklePos
FinalAnklePos ¼ 7.6203