S. M. MEGAHED, A. A. BALBOLA

Copyright © 2013 SciRes. ENG

( )

( )

13

.

01

tt t

bacdbabccdba ba

AF

RRrR PPRP

T

×

+−

=

(41)

( )

(1 3)

.

01

tt

cdbabccdba ba

EF

RrR PPRP

T

×

+−

=

(42)

where TBA is the HTM of the right laparoscopic tool, TAB

is the inverse of the HTM of the right laparoscopic tool,

TCD is the HTM of the left laparoscopic tool, TBC is HTM

between the right laparoscopic tool fixed coordinates at

point B and the left laparoscopic tool f ixed coordinates at

C. TAD is the homogenous transformation matrix between

the movable coordinates of the right laparoscopic tool at

point A and the movable coordinates the left laparoscopic

tool at D.

TEF is the HTM of the laparoscope, TEB is the HTM

between the laparoscope fixed coordinates at E and the

right laparoscopic tool fixed coordinates at B. It’s as-

sumed that the coordinates in the same direction so the

rotation matrix is unity matrix and there are only transla-

tion in x, y, z directions. TAF is the HTM between the

movable right laparoscopic tool coordinates at A and the

coordinates at the target point (F) but the translation

vector is a ratio (r) of the translation vector of TAD. The

orientation and position (DKPM) of the laparoscope

Camera are calculated directly from TEF. The laparoscope

parameters are determined in terms of those of the lapa-

roscopi c tools [6].

4. Simulation

During surgery operation, the laparoscopic tool is moved

from one point to another on a certain trajectory. Free

and guided trajectories may be selected to avoid ob-

stacles inside the human body. The laparoscopic tools’

models are executed using any desired time function for

the path of its TCP to perform a certain task. The ob-

tained mathematical models are simulated to show the

best positioning of the laparoscope in a surgery environ-

ment. The spline trajectory is used to simulate these

models [6]. The simulation results show that the lapa-

roscope movements are smooth so the monitor’s vision

will be stable as shown in Figure 4.

5. Conclusion

In this paper, D-H parameters and HTM technique are

used to obtain the expressions of the DKPM and IKPM

models of two laparoscopic tools and one laparoscope

used for its autonomous positioning in laparoscopic sur-

gery. The laparoscopic tools and laparoscope models are

used to simulate and align the laparoscope camera with

the surgeon’s laparoscopic tools’ TCPs. The obtained

results show a smooth positioning of the laparoscope

Figure 4. (a) Relation between laparoscope parameters and

time ratio; (b) Relation between its derivative and time

ratio; (c) Relation between its second derivative and time

ratio.

camera for better visualization in laparoscopic surgery

environments.

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