Modeling and Simulation of Laparoscopic Tools for Autonomously Positioning Laparoscope in Laparoscopic Surgery

doi:10.4236/eng.2013.510B017

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Engineering, 2013, 5, 85-89

http://dx.doi.org/10.4236/eng.2013.510B017 Published Online October 2013 (http://www.scirp.org/journal/eng)

Modeling and Simulation of Lapar o scopic Tools for

Autonomously Positioning Lapar oscope in

Lapar oscopic Surgery

S. M. Megahed, A. A. Balbola

Mechanical Design & Production Engineering Department, Faculty of Engineering, Giza, Egypt

Email: [email protected], [email protected]. eg

Received December 2012

ABSTRACT

In laparoscopic surgery, the surgeons are equipped with the suitable tools for the surgery, while the laparoscope is used

to capture the operation environment and displays it on a monitor. This paper presents the mathematical kinematic posi-

tion modeling of the laparoscopic tools used for autonomous positioning of a laparoscope in such operations. These

models are obtained using Denavit-Hartenberg (D-H) Notations and Homogenous Transformation Matrix (HTM). The

laparoscopic tools are considered as six degrees of freedom (DOF) mechanisms while the laparoscope has four DOF.

The 3D loop closure equation is used to obtain the laparoscope kinematic position models in terms of those of the lapa-

roscopic tools. These models are used to simulate and align the laparoscope camera with the surgeon’s laparoscopic

Tools Center Points (TCP). The obtained results show the smooth positioning of the laparoscope camera for better visu-

alization of laparoscopic surgery environments.

Keywords: Laparosc opi c Surgery; Homogeno us T ransformation Matrix; Denavit-Hartenberg Notations

1. Introduction

In laparoscopic surgery, two main problems challenge

the surgeons. The first is increasing field of vision capac-

ity and controlling system to carry out complex opera-

tional gestures without using high body mobility. The

second is the tools flexibilities inside the human body to

do more complex in vivo healthcare operations. Now

most Laparoscopic Surgeries are done using 6 DOF la-

paroscopic tools to increase their flexibility inside the

human body. Surgeons visualize the surgery environment

using a laparoscope. Any wrong movement of the lapa-

roscope interrupts the operation and the surgeon stops

until controlling the laparoscope focus again on the oper-

ation area. Many researches are done for autonomously

positioning the laparoscope instead of the laparoscope

manual positioning by an operator. Halin et al. [1] de-

veloped a novel a laparoscope which detect and to follow

the movements of the surgeon’s head. Terry et al. [2]

built a port camera integrated with the laparoscopic tool.

Weede et al. [3] designed an endoscopic guidance system

to autonomously align the laparoscope with the TCP of

the surgeon’s instruments which uses the information on

the movements of the instruments from former interven-

tions to predict the laparoscope motion. Terry et al. also

[4] designed, built, and tested a novel single-port access

laparoscopic surgery specific camera system (magnet

camera). Xu et al. [5] derives the analytical solution for a

5-DOF manipulator to follow a given trajectory.

In this paper, the direct and inverse kinematic position

models of two laparoscopic tools and one laparoscope

are derived for autonomous positioning of the laparos-

cope camera. After an introduction, the paper is orga-

nized in three main sections. In Section 1, the kinematic

position models of the two 6 DOF laparoscopic tools and

one laparoscope are presented. In Section 2, the laparos-

cope models are developed as function of the parameters

of the two laparoscopic tools. The obtained simulation

results using the developed models are presented in Sec-

tion 3.

2. Kinematic Modeling of 6 DOF

Laparoscopic Tools

In laparoscopic surgery, Surgeons use two 6 DOF lapa-

roscopic tools (right & left) to carry out complex opera-

tions. The four Denavit-Hartenberg (D-H) parameters (θi,

ri, ai, and αi for i = 1, 2



6) are used for generating the

elementary HTM’s (Ti,i+1) of the laparoscopic tools (see

Figure 1, and Table 1) [6].

By substituting the D-H par ameters in Ta ble 1 into (1)

the 6 elementary HTM’s (Ti,i+1) for i = 1, 2



6 are de-

termined which are used in (2) to obtain the generalized

HTM (T0,7).

S. M. MEGAHED, A. A. BALBOLA

Figure 1. Laparoscopic tool kinematic diagram.

Table 1. D-H parameters.

Joint i 0 1 2 3 4 5 6

Parameter s

- 0 0 0 1 0 0

0 0 0 0

0 0

0 0 0 0 0 a5 a6

−

This generalized HTM is used with (3) representing

the desired laparoscopic tool generalized HTM (T0,7) to

obtain the expressions of the laparoscopic tool direct

kinematic position model (DKPM) and the position and

orientation of its TCP are obtained. The inverse kine-

matic position model (IKPM) is also derived using the

same generalized HTM. In the next equations S = sin and

C = cos.

Elementary HTM:

0001

iiiiii i

iiii iii

ii i ii

CCSSCCSSSSa CC

SSCCCCCCSSa SS

TSSCC r

θθ αθαθ

θθαθαθ

αα

−





−−



=





(1)

Generalized HTM:

0,7 1,

−

=∏

(2)

11 1213

21 2223

0,7 31 3233

0001

rrrn

rr rm

Trrr l





=





(3)

DKPM & IKPM: An algebraic approach is used to

calculate the laparoscopic tool DKPM and IKPM as fol-

lows:

First (2) is pre-multiplied by the inverse of the HTM

(T0,3) as presented in (4) resulting 6 Equations (5, 6, 7, 8,

9, and 10). Second (2) is pre-multiplied by the inverse of

the HTM (T0,2) and post-multiplied by the inverse of the

HTM (T6,7) as presented in (11) resulting 2 Equations (12,

13). The obtained eight equations are used to calculate

the expressions of the laparoscopic tool DKPM which

are solved to calculate its IKPM.

( )

0,11,2 2,30,73,4 4,55,6 6,7

TTTT TTTT

−

⋅⋅ ⋅⋅⋅⋅=

(4)

21131135636

rSqrCqSqCq CqCqSq−= −⋅ ⋅⋅⋅⋅

(5)

221 32135

rSqrCqSq Cq⋅⋅ ⋅−=

(6)

( )

63 5 63653 5

mSqn Cq

aSq CqCqCqSqaSq Cq⋅ ⋅⋅

⋅ −⋅

=+ ⋅− ⋅⋅

(7)

( )

1122 21131156

rCqSqrCqrSqSq Cq− +=⋅⋅⋅ ⋅⋅

(8)

( )

1322231 33156

rCqSqrCqrSqSq Sq−++ =⋅⋅⋅ ⋅⋅

(9)

( )

221 1

6 565 54

lCqSqm CqnSq

aSq CqaSqq

−+

⋅⋅⋅ ⋅

+⋅ −⋅⋅ (10)

( )

0,11,20,7 6,72,33,4 4,55,6

TTTT TTTT

−−

⋅⋅⋅=⋅ ⋅⋅

(11)

( )

621 131 1

6231 33135

CqrSqr Sq

SqrSqrCqSq Cq

⋅⋅ ⋅

⋅ =⋅−⋅⋅

−

(12)

( )

621 131 1

6231 33135

SqrSqrSq

CqrSqrCqSq Sq

−

+ −=−

⋅⋅ ⋅

⋅⋅ ⋅⋅

(13)

The following is the IK PM of the 6 DOF laparoscopic

tools

22 22

12121 2223

22 2

55 5

a am

ar rrr

aa a



=+− −+



⋅ −⋅





⋅

(14)

( )

121 313121

22 2

55 5

22 3223 33

221. ...

2. 2.

br rmrnr

aa a

rr rr



=−− −++





−+

(15)

S. M. MEGAHED, A. A. BALBOLA

22 22

13131 32 33

22 2

55 5

a an

cr rrr

aa a



= +−−+−





⋅



(16)

1 111

b bac

q atanatana



−± −



=



(17)

61231321

drSqr Cq−⋅⋅=

(18)

62211 311

drSqrCq−⋅⋅=

(19)

61321 221

cr CqrSq−⋅⋅=

(20)

116 12

62 5

mSqnCqad

⋅⋅ ⋅

−−

= (21 )

661 6261

662 6162

Cqd dc

Sqd dc

−

 

 

−

 

(22)

( )

6 66

2,qatanSq Cq=

(23)

562

qatan d



−





(24)

3 562

35 6662

Sq Cqc

CqCq CqSqd

−





−



(25)

( )

3 33

tan2 ,qaSq Cq=

(26)

4111

dm CqnSq⋅+⋅=

(27)

422 4141

ql CqSqdc⋅++⋅= −

(28)

4165655

caSq CqaSq⋅⋅ ⋅= +

(29)

( )

4263563 6

535

caCq Cq CqSqSq

a Cq Cq

= +

⋅+

⋅ ⋅⋅⋅

⋅

(30)

2414 41

241 42

Sqdlq c

Cqldc

−

−−

 

 

 

(3 1)

( )

3 33

tan2 ,qaSq Cq=

(32)

3. Positioning of a Robot Assisted Surgery

Laparoscope

It’s supposed that the laparoscope is always tracking the

midpoint between the TCPs of the right and left lapa-

roscopic tools. The 3D loop closure theory is used to

get the HTM of the laparoscope in terms of the left and

righ t laparoscopic tools joint angles [7]. Figures 2 and

3 present the closed loops formed by the HTM of the

right, left laparoscopic tool, and laparoscope. The prod-

uct of these HTMs is equal to a unity matrix as in (33) and

(39).

ADAB BC CD

T TTT=

(33)

Figure 2. Laparoscopic surgery environment.

Figure 3. Surgery area.

(33)(3 1)

(1 3)

ba ba

××



=



(34)

(33)(3,3)(3 1)

(1 3)

baba ba

R RP

××



−

=





(35)

()()

33 31

(1 3)

cd cd

××



=





(36)

(1 3)

ΙP







(37)

( )

tt t

bacdbabccdbaba

RRR PPRP



+−

=





(38)

EFEB BA AF

T TTT= ( 39)

(3 1)

(1 3)

T×







(40)

S. M. MEGAHED, A. A. BALBOLA

( )

tt t

bacdbabccdba ba

RRrR PPRP



+−





=



(41)

( )

(1 3)

cdbabccdba ba

RrR PPRP



+−





=



(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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