Viscoelastic Model Based Force Tracking Control for

Robotic-Assisted Surgery

Chao Liu, Pedro Moreira and Philippe Poignet

Abstract— Most traditional force control methods for roboticsystem are based on the assumption that the interaction modelis purely elastic. However, in the scenario of robotic-assistedsurgery, it has been shown that the interaction between roboticinstrument and the soft human tissue exhibits much morecomplex behavior involving change rate of contact position andforce etc [1]. In this work, an adaptive force tracking controlalgorithm has been developed based on a viscoelastic model(Kelvin-Boltzmann) and it can online adapt to interaction modelparameter estimation errors. Physiological motion of tissue isalso considered in the control design. The force tracking erroris guaranteed to converge asymptotically even with parametermismatches in the interaction model. Simulation studies werecarried out to show performance improvement of the developedadaptive force tracking control algorithm over control methodwithout adaptation to model parameter uncertainties.

I. INTRODUCTION

Past few decades witness tremendous innovations in med-

ical interventions. Laparoscopic surgery, also called Mini-

mally Invasive Surgery (MIS), has been widely adopted in

medical intervention around the world due to its advantages

over traditional open surgery. Recent progresses also see

successful applications of Single Port Laparoscopy (SPL) in

which the surgeon operates almost exclusively through a sin-

gle entry point and Natural Orifice Translumenal Endoscopic

Surgery (NOTES) which performs scarless abdominal opera-

tions with an endoscope passed through a natural orifice. The

new surgical techniques provide great benefits to the patients.

On the other hand, with the new intervention techniques the

surgeons face new challenges like limited motion space and

view scope, indirect feeling of contact force, and moreover

they have to handle dedicated medical instruments whose

mechanical design is sometimes non-intuitive. To help the

surgeons better fit this situation and hence augment the new

intervention techniques, robotic-assisted systems have been

employed with the most famous and commercially successful

representative of da Vinci system by Intuitive Surgical Inc.

Robotic-assisted surgical systems help surgeons tackle

problems like reduced manipulation dexterity and lost of

depth perception, in addition they provide surgeons func-

tions that can overcome human physical constrains during

operation like tremor, accuracy, motion bandwidth etc. Nev-

ertheless, due to absence of direct physical link between

surgeon and the tissue under operation, haptic or force

feedback as an very important supplementary information

to the surgeon in tradition surgery is unfortunately missing

Chao Liu*, Pedro Moreira and Philippe Poignet are with the Depart-ment of Robotics, LIRMM (CNRS-UM2), UMR5506, Montpellier, France.liu, moreira, poignet@lirmm.fr

in most robotic-assisted surgical systems including the da

Vinci system. This results in an inability to accurately judge

how much force is being applied to tissue as well as a risk

of damaging tissue by applying more force than necessary

[2]. To address this problem, considerable research efforts

have been devoted to developing techniques which render

surgeon haptic feeling of presence or exert desired force on

tissue [3]–[5] etc. These works present detailed study of the

force interaction model with soft organ tissue, and different

modeling methods have been proposed. In [6], a force control

method for beating heart surgery was proposed based on

a simple elastic model which is adopted in most industrial

and other traditional robot force control applications in the

literature. But as shown through experimental study in [1]

biological tissues are not elastic and the history of strain

affects the stress. In [1], a quasi-linear viscoelastic function is

proposed to represent the stress-strain relationship. Accurate

models for soft tissue simulations could be obtained using

Finite Element Method (FEM) [7] but it is computationally

time-consuming, especially for dynamic simulation and real-

time applications. In [4], a polynomial function of second

order model is used to describe pre-puncture phase of needle

insertion, and for the same application purpose a model

based on nonlinear Kelvin model is developed in [8]. A

compact dynamic force model is presented in [9] where

force is modeled using a nonlinear dynamic model. In

[10], a viscoelastic model based on fractional derivative is

presented which is quite accurate especially for relaxation

phenomenon. The fractional derivative model is also used

in [11] to improve haptic feedback control for tele-operated

surgical robotic systems.

However, it is noticed that the models as proposed in

the aforementioned works [1], [3]–[5], [7]–[10] are mainly

focused on off-line analysis or interaction force control with

static or slow motion tissue. Hence it is not sure that they

will work for real-time force control in surgical operations

involving fast dynamics such as beating heart surgery due

to their heavy computation time and control design com-

plexity. In [12], the first viscoelastic interaction model based

force regulation method for robotic-assisted beating heart

surgery is proposed in literature using Kelvin-Boltzmann

model whose effectiveness has been identified through in-

vitro experiments among several other viscoelastic models. In

[15], it is confirmed through experimental comparisons that

control method using Kelvin-Boltzmann model outperforms

control method assuming traditional pure elastic interaction

model for force regulation on soft tissue involving fast

dynamics.

The Fourth IEEE RAS/EMBS International Conferenceon Biomedical Robotics and BiomechatronicsRoma, Italy. June 24-27, 2012

978-1-4577-1198-5/12/$26.00 ©2012 IEEE 1199

The work in [12] has the limitations that first of all

it’s only for constant force control and thus leaving more

challenging time-variant force tracking an open problem

which is important for robot-surgeon co-manipulation and

automatic surgical operation, secondly fixed set of estimated

model parameters, whose values in general vary from patient

to patient, are used in the control design which may not

present a big issue for constant force regulation but could

be very disturbing or even cause instability in force tracking

tasks. In this work, we develop a force control method to

exert desired time-variant force on soft tissue for robotic-

assisted surgery based on Kelvin-Boltzmann model whose

estimated initial model parameters are updated online to

adapt to change of tissue-instrument interaction properties.

Simulation studies were carried out based on a real robot

platform. Force tracking performance and comparison with

non-adaptive force control method are provided to show

its efficiency and capability to deal with interaction model

parameter uncertainty.

II. INTERACTION MODEL FOR SOFT TISSUE FORCE

CONTROL

To choose the proper interaction model for force control

design, three main criteria are to be met considering robotic-

assisted surgery scenario: accuracy, complexity (computation

and design), transient performance.

In vitro experimental studies as reported in recent work

[12], [15] show the efficiency of using Kelvin-Boltzmann

model to describe the interaction relationship between robot

and soft tissue.

Fig. 1. Kelvin-Boltzmann Model

The Kelvin-Boltzmann model consists a Newtonian

damper and Hookean elastic spring connected in parallel

with another spring in series as illustrated in Fig. 1. The

interaction force F(t) is described as:

F(t) = γx(t)+ β x(t)−αF(t) (1)

where k1, k2 represent the spring stiffness constants and b

the damping factor, α = bk1+k2

, β = bk2

k1+k2, γ = k1k2

k1+k2.

Fig. 2 shows an in vitro experimental result reported in

[12] which illustrates the superior performance of Kelvin-

Boltzmann model in describing the interaction with soft

tissue compared with several other viscoelastic models

(Maxwell, Kelvin-Volgt, Fractional derivative).

From the experimental results, it is seen that Kelvin-

Boltzmann model outperforms other viscoelastic models in

terms of both transient performance and accuracy. Moreover,

it should be noted that, compared with other potentially

more accurate but more complex models (FEM, nonlinear

viscoelastic models), the Kelvin-Boltzmann model is struc-

turally simple and hence facilitates the control design and

causes lighter computation burden which is essential for real

time implementation.

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

For

ce [N

]

Time [s]

0 0.5 1 1.5 2 2.5 3−0.5

0

0.5

1

1.5

2

2.5

|For

ce e

rror

| [N

]

Time [s]

Maxwell

Kelvin−Boltzmann

Kelvin−Voigt

Real Force

Fractional

Maxwell

Kelvin−Boltzmann

Kelvin−Voigt

Fractional

Fig. 2. In vitro relaxation test [12]

III. VISCOELASTIC MODEL BASED FORCE TRACKING

CONTROL

A. System Dynamics

The joint space dynamic model of robot in contact with

the environment is written as [13]:

M(q)q +C(q, q)q+ g(q) = τ − JT (q)Fe (2)

where q represents the generalized joint variable, M(q) is the

inertia matrix, C(q, q)q is Coriolis and centripetal force, g(q)stands for the gravity force. τ is the joint torque generated by

the joint actuator, Fe is the interaction force due to contact

with the environment. J(q) is the Jacobian matrix mapping

joint space velocity q to Cartesian space velocity x, with x

being the Cartesian coordinate of the robot end-effector.

In most applications of surgical robot, it is more conve-

nient to express the dynamic model directly in operational

(Cartesian) space for purpose of control and also for fa-

cilitating incorporation of other supportive functions (e.g.

teleoperation, virtual reality, etc). From (2) the Cartesian

space dynamic equation of the robot can be written as [13]:

Mx(x)x +Cx(x, x)x + gx(x) = Fa −Fe (3)

where

Mx(x) = [J(q)M−1(q)JT (q)]−1,

Cx(x, x)x = Mx(x)J(q)M−1(q)C(q, q)q−Mx(x)J(q)q,

gx(x) = Mx(x)J(q)M−1(q)g(q),

and Fa = J−T (q)τ denotes the actuated robot end-effector

force.

1200

B. Force Control Design and Stability Analysis

With the force feedback signal Fe obtained through the

force sensor, the nonlinear dynamic system (3) could be

linearized by designing the robot force Fa as

Fa = Fe + Mx(x)u + Cx(x, x)x+ gx(x) (4)

where u represents the auxiliary control signal.

Here estimations of dynamic matrices Mx(x), Cx(x, x) and

gx(x) are used which could be achieved through offline

pre-calibration or online adaptive identification techniques.

It should be noted that in the specific scenario of robotic

surgery, both the operating room (OR) environment and the

robot setup are well organized, thus the dynamic parameters

could be obtained with high accuracy through fine calibra-

tion.

Hence the dynamic system after linearization is expressed

as [14]:

x = u (5)

which represents a unity mass system decoupled along each

Cartesian dimension.

Considering both the motion of the surgical robot x and

the physiological motion of the tissue xt due to respiration or

heart beating, the interaction model (1) along one Cartesian

axis could be rewritten as

Fe = γxc + β xc−αFe (6)

where xc = x + xt . For simplicity of technical development

and without loss of generality, x(0) and xt(0) are defined as

0 at the initial contact state with Fe = 0.

Differentiating both sides of above equation, we have

αFe + Fe = γ xc + β xc (7)

and using the linearized system (5), it has

αFe + Fe = γ xc + β (u + xt) (8)

which could be written as:

θ1Fe + θ2Fe = θ3xc + u + xt (9)

where θ1 = αβ

, θ2 = 1β

, θ3 = γβ

.

Since the model parameters (α , β , γ) vary from patient to

patient and are difficult to get precisely pre-operation, only

best-guesses θi (i=1,2,3) are available as initial values for

control design with estimation errors.

Assume that the desired time-variant force reference Fd(t)is differentiable up to second order, then we can define the

following auxiliary variable:

Fs = Fd(t)−a(Fe−Fd(t)) = Fd(t)−a∆F (10)

where ∆F = Fe −Fd(t), a is a positive control gain. From

(10), it has

Fs = Fd(t)−a∆F. (11)

Define a sliding variable s as:

s = Fe − Fs = ∆F + a∆F (12)

such that

s = Fe − Fs = ∆F + a∆F. (13)

With the estimated uncertain model parameters θ1, θ2 and

θ3, the auxiliary control input u is designed as

u = −xt − θ3(x + xt)+ θ1Fs + θ2Fs − ks (14)

with the model parameters updated by

˙θ1 = −L1sFs, (15)˙θ2 = −L2sFs, (16)˙θ3 = L3sxc, (17)

where k, L1, L2, L3 are positive constant gains.

Remark 1: Tissue motion information xt could be obtained

resorting to artificial markers [16], exploring natural surface

textures [17] or utilizing other kinds of sensors. A latest real

time 3D tissue motion tracking technique using stereo camera

could be found in [18], [19]. △

The following conclusion could be drawn:

Theorem: The contact force Fe converges to desired force

Fd(t) asymptotically with the proposed control Force Fa in

(4), auxiliary control input u in (14) and parameter updating

laws (15) - (17).

Proof:

Substituting the auxiliary control input u into interaction

dynamics equation (9), the closed loop dynamics equation

is obtained as:

θ1Fe − θ1Fs + θ2Fe − θ2Fs −θ3xc + θ3xc + ks = 0 (18)

which can be written as

θ1s+(θ2 + k)s+(θ1 − θ1)Fs +(θ2 − θ2)Fs − (θ3 − θ3)xc = 0

(19)

and

θ1s+(θ2 + k)s+ ∆θ1Fs + ∆θ2Fs −∆θ3xc = 0 (20)

where ∆θi = θi − θi (i=1,2,3).

A Lyapunov function candidate V is proposed as:

V =θ1

2s2 +

1

2L1

∆θ 21 +

1

2L2

∆θ 22 +

1

2L3

∆θ 23 (21)

which is positive definite in s and ∆θi.

The time derivative of V can be obtained as

V = sθ1s−1

L1

∆θ1˙θ1 −

1

L1

∆θ2˙θ2 −

1

L1

∆θ3˙θ3. (22)

Substituting (20) and updating laws (15) - (17) into (22),

it has

V = −(θ2 + k)s2−∆θ1sFs −∆θ2sFs + ∆θ3sxc

−1

L1

∆θ1˙θ1 −

1

L1

∆θ2˙θ2 −

1

L1

∆θ3˙θ3

= −(θ2 + k)s2. (23)

It’s clear that V is negative definite in s and hence V

is bounded as well as s and ∆θi. From (15)-(20), it can

be seen that s is bounded, thus V is bounded and V is

1201

uniformly continuous. According to Barbalat’s Lemma, s =∆F + a∆F converges to 0 asymptotically which means that

∆F = Fe − Fd(t) converges to 0 asymptotically too. This

completes the proof. �

Remark 2: If accurate interaction model parameters are

available, i.e. no parameter estimation errors, and physio-

logical tissue motion is not considered, the control input u

can be written as

u = θ1Fs + θ2Fs − ks (24)

which reduces to the classical force tracking controller as

found in many traditional robot applications. And the closed

loop dynamics is written instead of (20) as:

θ1s+(θ2 + k)s = 0 (25)

which is a first order linear system with s converges to 0

exponentially. Considering s = ∆F +a∆F, ∆F also converges

exponentially at a rate adjustable by changing control pa-

rameters a and k. Therefore the proposed controller is con-

ceptually coherent with classical force tracking algorithms

and does not increase much complexity of overall control

structure. △

IV. PERFORMANCE EVALUATION AND COMPARISON

THROUGH SIMULATION STUDIES

To evaluate the control performance of the proposed

force tracking control method and the systems capability to

handle model parameter mismatches, simulation studies have

been carried out based on the D2M2 robot model with all

dynamic and kinematic parameters calibrated from the real

robot which has five degrees of freedom with direct drive

technology providing fast dynamics and low friction (shown

in Fig. 3).

Fig. 3. D2M2 robot

0 5 10 15−5

0

5x 10

−3

X a

xis

0 5 10 15−1

0

1

2x 10

−3

Y a

xis

0 5 10 15

−5

0

5

x 10−3

Time (s)

Z a

xis

Fig. 4. 3D tissue physiological motion (xt )

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 104

1

2

3

4

5

Time (ms)

Des

ired

For

ce (

N)

Fig. 5. Desired Force (Fd)

A. Simulation Setup

It is simulated that the tissue in contact undergoes a three

dimensional motion xt due to respiration and heat beating. In

this simulation xt follows a beating heart motion as recorded

in vivo through da Vinci system (Fig. 4) and it’s assumed to

be available by real time tracking techniques.

Since the robot end-effector just touches a specified point

on the beating heart, no torsion is involved in the force

measurement hence it is reasonable to assume that the

force measurements for three axes are uncoupled. Also,

the linearized robotic system is decoupled along different

Cartesian axis, thus in this simulation only a desired force

trajectory along Z axis is defined for tracking control. The

desired force is defined as:

Fd(t) = 3 + 1.2sin(1.5t)+ 0.5sin(t)+0.8sin(2t) (26)

whose plot is as in Fig. 5.

According to the experimental identification reported in

[12], the true Kelvin-Boltzmann model used in the simulation

study is set as

Fe = 100.65xc + 30.705xc−0.0567Fe (27)

hence θ1 = 0.0567/30.705 = 0.00185, θ2 = 1/30.705 =0.0326 and θ3 = 100.65/30.705 = 3.28. In order to evaluate

the performance of proposed control method against model

parameter mismatches (which could be quite large depending

1202

on different subjects), the model parameters in the controller

are intentionally set with large errors as θ1 = 0.04, θ2 =0.5 and θ3 = 50. In addition, white noise of ±0.01 N is

introduced to the measured force feedback and a standard

Kalman filter is used in simulation to process the noisy

measurement.

B. Performance Evaluation and Comparison

To evaluate the capability of developed method in dealing

with model parameter uncertainty, force tracking perfor-

mances are compared between tracking controller with fixed

mismatched model parameters and tracking controller with

parameter adaptation. In both cases, the control parameters

k, a are set to k = 3 and a = 0.5.

1) Force Tracking without Adaptation to Parameter Mis-

matches: First of all, force control with parameter adaptation

switched off (L1 = L2 = L3 = 0) is tested. The tracking

performance is shown in Fig 6.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 104

0

1

2

3

4

5

6

Time (ms)

Desir

ed v

.s. A

ctua

l For

ce (N

)

Fd

Fe

(a) Force Tracking Performance

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 104

−0.5

0

0.5

1

1.5

2

2.5

Time (ms)

Forc

e Tr

ackin

g Er

ror (

N)

(b) Force Tracking Error

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 104

−0.4

−0.2

0

0.2

0.4

0.6

Time (ms)

Forc

e Tr

ackin

g Er

ror (

N)

(c) Force Tracking Error (zoom in)

Fig. 6. Force Tracking Performance without Adaptation

It is observed that, without adaptation to the uncertain

interaction model parameters, although the system is still

stable there remains a periodic tracking error of around

±0.5 N which is quite annoying in the specific scenario

of medical surgery operations and may even cause safety

problem.

2) Force Tracking with Adaptation to Parameter Mis-

matches: In the second simulation study, the interaction

model parameters are online updated by setting L1 =0.2, L2 = 0.2, L3 = 1.5.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 104

0

1

2

3

4

5

6

Time (ms)

Desir

ed v

.s. A

ctua

l For

ce (N

)

Fd

Fe

(a) Force Tracking Performance

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 104

−0.5

0

0.5

1

1.5

2

2.5

Time (ms)

Forc

e Tr

ackin

g Er

ror (

N)

(b) Force Tracking Error

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 104

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Time (ms)

Forc

e Tr

ackin

g Er

ror (

N)

(c) Force Tracking Error (zoom in)

Fig. 7. Force Tracking Performance with Adaptation

It can be observed that with online adaptation to the

parameter mismatches the force tracking error converges to

0 as time goes on, which can be seen more clearly in Fig.

7(c) compared with Fig. 6(c).

The simulation studies confirm that the proposed tracking

control method handles well interaction model parameter

mismatches and guarantees the convergence of force tracking

error.

C. Discussions

It should be noted that although the desired force contains

3 different frequencies which could excite up to 6 param-

eters to converge for linear system, in this simulation the

convergence of the uncertain model parameters to their true

values are not guaranteed due to the nonlinearity nature of

the overall system. The evolution of the updated parameters

θi (i=1,2,3) is shown in Fig. 8. It is seen that the updated

parameters did not converge to their true values but settle

down to certain values as the force tracking errors converge

to 0.

History of auxiliary control input u for the adaptive force

controller is illustrated in Fig. 9. Due to bounded model

parameter adaptation, it is seen that control input remains

bounded during whole control process which is crucial for

safe surgical operation.

1203

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 104

−0.2

−0.1

0

0.1

0.2

0.3

0.4

(a) θ1(t) v.s. θ1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 104

−0.2

0

0.2

0.4

0.6

0.8

1

(b) θ2(t) v.s. θ2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 104

−10

0

10

20

30

40

50

60

(c) θ3(t) v.s. θ3

Fig. 8. Evolution of Updated Model Parameters

0 1 2 3 4

x 104

−10

−5

0

5

10

Time (ms)

Con

trol

Inpu

t u

Fig. 9. Control input u

V. CONCLUSIONS AND FUTURE WORKS

This study addresses the problem of force tracking control

on soft tissue in robotic-assisted surgery which remained

as an open problem. The force control is based on Kelvin-

Boltzmann model which represents a good balance between

modeling accuracy and computational efficiency. The pro-

posed control method deals with parameter estimation errors

of the interaction model and also considers physiologi-

cal motion of the tissue under contact. Simulation studies

were carried out to justify the effectiveness of proposed

method in handling interaction model uncertainty and it is

shown that the force tracking error converges asymptotically

compared to the residual periodic error of control method

without parameter adaptation. Next step of work will be

carrying out in vitro and/or in vivo experimental tests of

the developed force control method. Considering that this

work addresses the problem of parameter mismatch of the

viscoelastic interaction model, research work will be done to

develop an adaptive robust control method that can also deal

with model structure mismatch between currently employed

Kelvin-Boltzmann interaction model and the more complex

real one. Uncertainty in measured tissue motion is to be taken

into account as well.

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