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Needle Steering Force Model and Trajectory Planning
Reporter : Pan Li
Supervisors:
Prof. Zhiyong Yang, A.P. Shan Jiang
Content
1 Needle Insertion Force Model
2 Trajectory Planning for insertion needle
Future Works3
Experimental designPVA Phantom Ultrasonic
motorSurgical needle
6-axis capacity ATI
force/torque sensor
Fig.1 Visual image acquisition hardware and software systems
Fig.2 The devices for force measuring experiments
High speed camera
Force sensor
Deformation
Insertion
Extraction
A
B
C
D
Experimental procedure
1) Deformation phase (from A to B): Tissue deformation occurs.2) Insertion phase (from B to C): The needle penetrates into the soft tissue. 3) Extraction phase (from C to D): The needle is exacted from the soft tissue.
The force acting on a needle is written as: ( ) ( ) ( ) ( )s f cF x f x f x f x F x
)()( xfxF s)()()( xfxfxF cf
)()( xfxF f
Stiffness Force
1
Friction and Cutting Force
2Friction Force
3
Material Relaxation and Friction
Fig.3 Insertion force-time profile Fig.4 Insertion force-deep profile
Stiffness ForceThe stiffness force corresponds to viscoelastic interaction and can be described as prepuncture force.
Needle Insertion Force Model
Surface of soft tissue
Soft tissue
Needle base
Needle shaft
Needle tip
Deformation h
r
y
o
a
)(rfy
2)tan(2
hrEf stiffness
bebFzf
axaxazf
dxastiffness
stiffness
)(
012
2
0)()(
)(
指数函数模型:
型:模式项多polynomial model:
exponential function model:
Fig.5 Five experiments of force curve profile Fig.7 Comparison between stiffness model and experimental dataFig.6 Error comparison between polynomial and exponential model
1
0)()( dttaEzf rstiffness dx
x
xfh
1
0 21
)( dxx
xfxaEf rstiffness
1
0 2
2
1
)(2
Fig.8 Stiffness model schematic diagram
cot)( axxf
Stiffness Force2)tan(
2hrEf stiffness
Stiffness force:
Equivalent modulus:2
22
1
21 111
EEEr
Parameter acquisition:
Fig.9 Parameter acquisition experiment diagram of insertion needle
7.0od mm 4.0id mm
4244 10052.1)(64
mmddI io
Moment of inertia:
Deflection equation: )3(6 1
2
max xlIE
Fxw
Strain energy density expression:
Five experiment result: 1E =14.1±1.6GPa
)3()3( 2211 ICICW
Deformation tenser invariant: 23
22
211 I
213
232
2212 )()()( I
Mooney-Rivlin model experiment:
21
1
21
1
1 1
)1
(2CC
t
Optimization toolbox by ANSYS:
2E = 116KPa
Geometry of needle :
(a) Impale the soft tissue
Friction ForceThe friction force occurs along the length of the needle inside the tissue and is due to Coulomb friction, tissue adhesion and damping.
(b) Withdrawing of the needle (c) Two consecutive insertions
Measurement of friction force:
Fig.10 Measurement of friction force
Friction Force
Soft tissue
Foundation
Needle base
Needle shaft
z
k
h
Surface of soft tissue
nF
frictionf
Foundation modulus: 12
1
42
22
2
1
65.0
IE
bEEk
Friction force model: h
IE
DEEDf friction 12
1
42
22
22 )(
1
65.0
2
Normal force: kAFn
Contact area: DhA nfriction Ff Coulomb friction
Fig.11 Modified Winker foundation model sketch diagram
Fig.12 Comparison between force model and experimental data
Cutting Force
Cfcutting
The cutting force ranges from 0.04N to 0.1N periodically, and the fluctuation can disappear when the insertion speed is equal to 3mm/s and the insertion forces of two consecutive insertions are parallel to each other.
Constant
(1) :
Complete force model
DCfriction
CBcuttingfriction
BArstiffness
needle
zzzzIE
DEEDf
zzzCzIE
DEEDff
zzzzEf
zf
12
1
42
22
22
12
1
42
22
22
2
)(
1
65.0
2
)(
1
65.0
2
)tan(2
)(
The overall shape of the model is coherent to the experimental results, although the fluctuations in the insertion phase make a perfect match impossible.
Comparison: )()( )(mod)( xfxf elsmeasures
?(2) :
?(3) :
)(and)( )(mod)( xfxf elfmeasuref
)(and)( )(mod)( xfxf elcmeasurec
Trajectory Planning for insertion needle
Key technology :Avoiding vital tissues;Implanting into tumor accurately;Shorten trajectory path;Reducing implant errors.
Anterior
Posterior
LeftRight
Bladder
Rectum UrethraTumor
Pelvis
Prostate
Superior
Inferior
Posterior Anterior
Artificial potential field?Local minimum in concave curved surface model
G
O
P
3D trajectory planning result of the spiral pipe model
3D trajectory planning result of the concave curved surface model
3D trajectory planning result of local minimum
minp
)()( attatt pp UF
T
attattattatt ,,)(
zyxp
UUUU
)(2
2
1)(
goalattatt pkp U
goalattatt )( ppkp
F
0)(
0)(0)(
rep
r
0
11
2
1
)(
p
ppep
if
ifkpU
0)(
0)()(
obstacle
)(2
0)(rep
rep
0
111
)(
p
pppp
if
ifpp
kp
F
Relationship of force and potential value: Attractive potential value and force: Repulsive potential value and force:
Improved conjugate gradient method
Escaping from Local minimum
)1( kS
kk gS )(
)2( kS
2 kg
)()( kk S1 kg
)1( kp)(kp
)2( kp
e 3l
2l
1l
s
ig
minp
Optimization of trajectory planning
Select as the initial point
k=0
Calculate the new search point
No
Yes
Calculate the gradient of the new
search point
Select the first search point and preset the deflection angle of needle tip
Start
171 ~ gg
minp
Calculate the gradient of the first point kg
Select the negative direction of the gradient )(ks
cos2
))1(( kpU
YesCompute the optimal
step size )( k
)1( kp
Escape from the local minimum?
1kg
Compute the value of parameter
Confirm the search direction of next
step )1( ks
k=k+1 Select the deflecting
direction Angle of needle tip
As next search Direction angle
k=k+1
Input the first insertion point s
Define four cubic elements around s and mark 17 nodes with letter and subscript number 171 ~gg
Calculate composition gradient field of each node
Search the node that gradient decent most quickly
ig
Connect initial point and
Reach local minimum?
Yes
No
Select as the initial point
s
No
Reach the target?
YesEnd
No
ig
s ig
APF
ICGA
(a) , is selected as search direction angle
(b) , is selected as search direction angle
kkk gpUS )( )()(
)(1
)()1()1( )( kk
kkk SgSpUS
2
2
111
][
][
k
k
kT
k
kT
k
g
g
gg
gg
?Judgment cos)(
2)1( kpU
)()()( )(min kkk SpU
)()()()1( kkkk Spp
Modeling interactive forces during needle insertion and experiment validation.
Modeling and simulation of tissue deformation.
Modeling needle deflection during insertion.
Trajectory planning considering tissue deformation and needle deflection.
Needle guidance and steering with experiment validation.
Future works
Fig.1 Accurate robotic needle steering requires a model that predicts needle motion within the tissue. Both a stochastic motion planner (used pre- and/or intra-operatively) and an image-guided model-based feedback controller can use the mechanics-based model
Insertion model
Mechanics model and insertion
model
Modeling interactive forces
Trajectory planning
Needle guidance and steering
Tissue deformation and needle deflection
Stiffness force:
Modeling interactive forces during needle insertion
(a) Symmetric needle tip
(b) Asymmetric (bevel) needle tip
Fig.2 Asymmetry of the bevel tip produces a resultant transverse load which causes a flexible needle to naturally bend
Surface shape function :
xxf )( cota
cot)( axxf
Stiffness force:
1
0 2
2
1
)(2 dx
x
xfxaEf rstiffness Simplified into:
2)(tan2
hEf rstiffness
Whereas:
cot)( axxf
To asymmetric needle tip:
Surface shape function:
To symmetric needle tip:
1
0 2
2
1
)(2 dx
x
xfxaEf rstiffness Deduction: ?
Friction force: Coulomb friction
Fig.3 Overall slip model
Friction force:
None
Macroslip
nK
Nf {
Fig.4 Partial slip model
Elastic deformation phase
Partial slip phase
Entire slip phase
s
l
s s
lll
s
ll
dsr
dsrdsrrK
dsrrK
f
slip Entire)(
slip Partial)()()(
n deformatio Elastic)()(
1 2
Friction force:
Evolutionary:
Friction force:
others)(sgn
&0
0)(
es
seef
FF
FFvF
vvF
F
veNvvF
i
sv
v
ccf
)()(sgn)( 0
Stribeck friction model:
: the damping coefficient associated with
LuGre friction model:
0
( )
0
( , )( )
( ) ( , )L t
f
vdzt v z
dt g v
F t dF t
( ) ( ) vc p cg v e
0 1 2( , ) ( , ) ( , ) ( , )n
zdF t z t t v dF t
t
Where v is the contact velocity of each differential element and L is the incision length; and
( , ) ( ) ( )n ndF t dF f d
0 :stiffness coefficient of the microscopic deformations
1 z2 : the viscous relative dampingc and are the normalized coulomb and stictions
( )
0 1 20( ) ( , ) ( , ) ( )
L t
f n
zF t z t t v f d
t
Karnopp friction model:
2/)(
2/0),min(
02/),max(
2/)(
),(
vvvbvgc
vvfd
vvfd
vvvbvgc
fvf
rrprp
rsp
rsn
rrnrn
srfriction
frictioninertiameasured fff
))()(( nnnnppppmeasured vbvgcvbvgclmaf
))()(( nnnnppppfriction vbvgcvbvgclf
Fig.5 Stribeck model Fig.6 LuGre friction model
Fig.7 Karnopp model
Modeling and simulation of needle insertion into soft tissue
Modeling needle deflection during insertion
Fig.8 Distributed compressive load acting on a needle shaft as it interacts with an elastic medium.
Rayleigh-Ritz: energy
workinput
input
energy
1 )(11
WSN EE
work
Q
energy
2 )()(22 RPEE WWWSN
load axialbending needle pure
ABE UUN Sum of energy:
Needle-tissue interaction energy:
ninteractioncompressio
TCE UUS
ilPW inputinput Insertion force work:
Transverse tip load: ))()(( 1Q iiiii lvlvQW
Axial tip load: ))0()((P iiii luPW Rupture the elastic medium:
iclaGW R
Fig.9 Free-body diagram of the forces acting on the needle tip as it interacts with the elastic medium.
Trajectory planning considering tissue deformation and needle deflection
Trajectory planning considering tissue deformation and needle deflection
Fig.10 Artificial potential field
Fig.11 Numerical analysis method Fig.12 Inverse Kinematics solution
Considering needle deflection?Neglecting tissue deformation
(c) needle deflection and local changes in needle orientation.
(a)Tissue deformation (b) trajectory changes
mathematic(al) model?physiological anatomical model
(a) Basic concept
(b)Subdivided sampling time for higher update rate of the MPC-based feedback controller.
Fig.13. Concept diagram of a model predictive controller.
Needle guidance and steering with experiment validation
Needle guidance and steering with experiment validation
Input constraints:
6.1
2.1
6.1
8.0maxmin
State constraints:
4040- maxmin
Model predictive control:
Discrete control inputs
TTp
TT NkukukukU )1(...)1()()(
Fig. 14. The reference flexible probe, the measured real flexible probe, and their local coordinates.
Trrrrr yxq ][
Tyxq ][
Linearized tracking error model
THANK YOU