adaptive observer-based controller design for a class of nonlinear
TRANSCRIPT
-
8/12/2019 Adaptive Observer-based Controller Design for a Class of Nonlinear
1/6
Adaptive Observer-based Controller Design for a Class of Nonlinear
Systems with Application to Image Guided Control of Steerable Needles
M. Motaharifar, H.A. Talebi, A. Afshar, and F. Abdollahi
Abstract Flexible needles with a bevel tip (steerable needles)promise to enhance targeting accuracy and maneuver inside thehuman body in order to avoid collision with delicate organs.Contributing image feedback to needle insertion tasks greatlyimproves such objectives. An important issue in 2D motionplanning tasks is stabilizing the needle in a desired plane. Anydivergence from the plane leads to the inefficiency of the motionplanning scheme. Hence, a control scheme is proposed in thispaper which guides the needle to a desired plane. The systemof such task is subject to parametric uncertainty. Although theoriginal system is linearly parametrized, the feedback linearizedform is not, which prevents the application of conventionaladaptive control schemes. Moreover, all state variables of thesystem could not be measured and a nonlinear observer isnecessary to observe the system states. In this paper, thepreviously proposed adaptive state feedback controller for suchsystems is modified to an adaptive output feedback controllerand the proposed scheme is applied to the problem of needleguidance. Simulation results are presented to illustrate theenhanced performance of the proposed controller methodologyas compared to previously proposed feedback linearizationscheme.
I. INTRODUCTION
Needle insertion into soft tissue is a widespread surgical
technique that has numerous applications in medical tasks
such as brachytherapy, anesthesia, biopsy, etc. Accurately
placing the needle is a key factor that determines the effec-
tiveness of the treatment. Moreover, the needle should avoidcollision with some sensitive organs such as nerves, bones
or vessels to prevent subsequent complications.
The traditional rigid needles have a little manoeuvrability
in the tissue. Motion planning for such needles just involve
optimization of initial parameters such as needle starting
insertion point, heading angle, etc [1],[2]. On the other hand,
flexible needles have more manoeuvrability than the rigid
ones and can be manipulated during the insertion to guide
through tissue. A class of flexible needles with a bevel-tip
have attracted great attention during the past several years.
Indeed, the asymmetry of the tip causes the needle to bend
and follow a circular arc with constant curvature[3]. There-
fore, with appropriate rotation of the needle from the base,the desired trajectory could be achieved. A mathematical
nonholonomic needle steering model has been developed
and validated in [4] for such needles. The proposed model
was a generalization of the standard 3 DOF nonholonomic
unicycle and bicycle models to 6 DOF using Lie group
theory. Based on this model, some motion planning algo-
rithms have been proposed for this class of needles using
The authors are with Electrical Engineering Department, Amirkabiruniversity of Technology , Tehran, Iran, email: {md.motaharifar,alit, aafshar, f abdollahi}@aut.ac.ir
Stochastic Motion Roadmap [5], inverse kinematics [6],
screw-based motion planning [7], and Rapidely-exploring
Random Trees(RRT) [8]. In [9], a motion planning approach
was proposed based on fast duty cycle spinning of the needle
in order to remove the limitation of a fixed curvature path.
Since a continuous spinning increases patient trauma, this
method is not appropriate for clinical diagnosis. On the other
hand, manual needle insertion using master-slave devices
is another area of research. In [10], the performance of
manual teleoperation scheme was compared to that of an
automatic needle insertion technique and it was shown that
the hybrid control provided improved accuracy. A review onsome research of needle insertion strategies was given in
[11].
Guiding and stabilizing a flexible needle to a desired plane
is a first step in 2D motion planning. In [12], a feedback
linearization controller was employed to do such task. The
proposed controller was based on the reduced order system of
the nonholonomic needle steering model introduced in [4].
Such model, however, is subject to parametric uncertainty
and the feedback linearization controller cannot tolerate
uncertainties. In [13], a way of combining such controllers
with a previously proposed motion planning algorithm has
been presented. Moreover, this class of controllers have the
ability to combine with manual teleoperated insertions.In this paper, an adaptive output feedback controller is
proposed for guiding flexible needles to a desired plane.
The needle steering model considered here belongs to a
special class of nonlinear systems which the original system
is linearly parametrized but the canonical form is not. The
traditional approaches for adaptive output feedback controller
design (e. g. [14], [15]) cannot be applied to a system
unless its canonical form model is linearly parametrized.
Moreover, the unmeasured states of the system have to be
estimated. Some adaptive state feedback controllers were
proposed for this class of nonlinear systems (e. g. [16],[17]).
However, those adaptive control schemes rely on full state
measurement and to the best of our knowledge no work withadaptive output feedback controller has been proposed for
this class of nonlinear systems. The proposed controller is
a combination of the state feedback controller proposed in
[16] with high gain observers after some modifications. As
will be explained in Section III, some complexities emerge in
case of designing an adaptive output feedback version based
on the approach stated in [16]. First, this method is based on
an auxiliary dynamic equation with some properties which
simplify the design. In case of existing unmeasured state
variables, the auxiliary dynamic does not have the simplifier
2012 American Control Conference
Fairmont Queen Elizabeth, Montral, Canada
June 27-June 29, 2012
978-1-4577-1096-4/12/$26.00 2012 AACC 4849
-
8/12/2019 Adaptive Observer-based Controller Design for a Class of Nonlinear
2/6
properties which complicates the design. Moreover, some
complexities occur in dealing with the lyapunov function.
The applied controller has the ability in online estimation of
needle curvature. This parameter depends on the needle and
the tissue. The previous work [4] estimates this parameter
off-line using trial and error strategy with some experiments.
Since inserting the needle into the patients body for so many
times is dangerous for the patients health, this method is not
applicable for clinical activities. In brief, the contribution
of this paper is twofold. First designing an adaptive output
feedback for a class of nonlinear systems which the original
(not canonical form) model is linearly parametrized and
output feedback linearizable. Second, estimation of needle
curvature in medical tasks online.
The rest of this paper is organized as follows. In Section
II, the System dynamics is described. Our proposed adaptive
observer-based controller is presented in Section III. Section
IV depicts simulation results of the proposed methodology.
Finally, the conclusions are stated in Section V.
I I . SYSTEM D ESCRIPTION ANDM ODEL
A flexible bevel-tip needle can be steered by rotation
and insertion at the base outside the body of the patient.
Such a needle bends as it is inserted into tissue at a
constant curvature, which is a property of the needle and
tissue. Hence, by rotating the needle from the base, different
trajectories could be achieved.
A kinematic bicycle model is developed for such a needle
in [4]. Based on this model, a reduced order model is
extracted for stabilizing the needle to a desired plane in [12].
This model is reproduced here for reader convenience.
Fig. 1 shows the kinematic bicycle model. In this model,
frame A is the reference frame and frames B and C areattached to the two wheels of the bicycle. Utilizing Lie-
group theory, a coordinate-free differential kinematic model
is found [4].
v
= u1V1+ u2V2 (1)
where v, R3 denote the linear and angular velocities ofthe needle tip, respectively, written relative to frame A. u1and u2 are the insertion and rotation speed of the needle,
and
V1 = e3
e1 and V2 = 033
e3 (2)
The unit vectors ei, i = 1, 2, 3 are the standard basis. Letq= [x,y,z,,, ]be the position and orientation vector ofthe needle tip where x,y,and z are position relative to the
reference frame, is the yaw of the needle in the plane, is
the pitch of the needle out of the plane, and is the roll of
the needle. Moreover, denotes the curvature which needle
follows. Body frame velocity may be expressed as
v
= Jq (3)
Fig. 1. Kinematic bicycle model [4] used with permission from the authors
where
J=
RTAB 033033 S
S=
coscos sin 0 cossin cos 0
sin 0 1
(4)
where RAB is the rotation matrix between frames A and
B. Now, using (1) and (3) the flexible bevel-tip needle model
is
q= J1V1u1+ J1V2u2 =
sin 0 cossin 0
cos cos 0 cossec 0 sin 0
cos tan 1
u1u2
(5)In order to stabilize the needle to the yz plane, the statesy,z, and need not be controlled. Moreover, these states do
not affect the dynamics of the remaining states. Hence, we
can define pT = [p1, p2, p3] = [x,,] as the state vectorof the reduced order system, which can be represented as
follows.
p= f(p)u1+ g(p)u2
=
sin(p2) sin(p3) cos(p3) tan(p2)
u1+
00
1
u2 (6)
r= h(p) = p1 (7)
Note that pT
=
0 0 0
is the desired equilibrium pointof the system which corresponds with placing the needle in
y z plane. we divide both sides of (8) by u1. Hence, thesystem is reparametrized in terms of insertion distance, l.
Note that wherever we write p, we mean dpdl
. In other words,
the insertion distance is substituted for t as the independent
variable and its derivative u1 is no longer an input signal.
Indeed, The resulted system is [12]
p=
sin(p2) sin(p3) cos(p3) tan(p2)
+
00
1
u (8)
4850
-
8/12/2019 Adaptive Observer-based Controller Design for a Class of Nonlinear
3/6
where u = u2u1
r= h(p) = p1 (9)
In essence, we can only measure p1 = x by imageprocessing and the other state variables should be estimated.
Furthermore, the parameter is the unknown curvature of
the needle which should be estimated.
The system (8) and (9) can be transformed into output
feedback linearized form using the following transformations[19]
w= [h(p), Lfh(p), L2fh(p)]
= [p1, sinp2, cosp2sinp3] (10)
v= L3fh(p) + LgL2fh(p)u
= 2 sinp2+ cosp2cosp3u (11)
where Lfh(p) is the Lie derivative ofh with respect to f,defined by [19]
Lfh(p) = h(p)
p f(p) (12)
Indeed, this is the familiar notation of the derivative of h
along the trajectories of the system p= f(p). Moreover, ina similar manner we have
LgLfh(p) =Lfh(p)
p g(p) (13)
Lkfh(p) =Lk1f h(p)
p f(p) (14)
Now, the transformed system is
w = Aw + Bv =
0 1 00 0 1
0 0 0
w+
00
1
v (15)
r= Cw =
1 0 0w (16)
Obviously, the system (15) and (16) is not linearly
parametrized. Therefore, the traditional Model Reference
Adaptive Control (MRAC) [20] cannot be applied to it.
III . THE P ROPOSEDS CHEME
Consider a nonlinear system, subjected to parametric un-
certainty, described as
x= f(x,u,) = f0(x, u) + fT1 (x, u) (17)
y= h(x, u) (18)
where x Rn is the state vector, u Rm is the control
input vector, and Rp is the parameter vector. Moreover,f0 and f1 are known nonlinear functions. Obviously, the
system (8) and (9) can be stated as (17) and (18). Our
objective is to design an adaptive controller-observer pair
for the above system such that the stability is preserved
and tracking a reference signal xd(t) is achieved in thepresence of unknown parameter vector . In order to do so,
the adaptive controller given in [16] is combined with a
nonlinear high gain observer with some modifications.
Assumption:There exist
(a)a Hurwitz matrix A
(b)an open set Dx Rn containingxd(t) for all t
(c)an open set D Rp containing
(d)a family of parametrized diffeomorphisms
W :Dx Rn :z = W(x,) (19)
exists where is an estimation of and such that thefollowing implicit equation in the unknown u
W
x(x,)[f0(x, u) + f
T1 (x, u)]
= W
x(xd,)xd A[W(xd,) W(x,)] (20)
has a unique bounded solution u= ua(x, xd,) for all x Dx. Now, taking the derivative ofz we have
z= W
x(x,)[f0(x, u) + f
T1 (x, u)] +
W
(x,) (21)
The previous equation is equal to
z= g0(x,u,) + gT1(x,u,
)+ g2(x,) (22)
where
g0(x,u,) = W
x(x,)f0 (23)
gT1(x,u,) = W
x(x,)fT1 (24)
gT2(x,) = W
(x,) (25)
Transformation (19) should convert the system (17) and
(18) to normal form. Indeed,
z = z+ B(x, u)y = Cz
(26)
where
=
0 1 0...
.... . .
...
0 0 10 0 0
BT =
0 0 1
C=
1 0 0
(27)
A high gain observer can be designed for the above system
as [19]z= z+ B0(x, u) +H(y y)y= Cz
(28)
wherezis the estimation ofz. The observer gainHis chosen
as
HT =
1
22
nn
(29)
where is a positive constant to be specified and the positive
constantsi are chosen such that the roots of
sn + 1sn1 + + n1s+ n = 0 (30)
are in the left-half plane. The function 0(x, u) is a nominalmodel of(x, u). The observer error is in the following form
z= z+ B(x, u) HCz (31)
4851
-
8/12/2019 Adaptive Observer-based Controller Design for a Class of Nonlinear
4/6
wherez = zzand (x, u) = (x, u)0(x, u). The aboveequation is equal to
z= Ao,z+ B(x, u) (32)
where
Ao,=
1
1 0
......
. . ....
n1n1 0 1n
n 0 0
(33)
Lemma 1 [18]: For the matrix E= diag(1,,...,n1), wehave the following facts:
(a)
Ao, = 1
E1AoE, B=
1E1B (34)
where
Ao =
1 1 0
......
. . ....
n1 0 1n 0 0
BT =
0 0 n
(35)
(b) Let the positive definite matrix S be the solution of
AToS+ SAo = Q, then
ATo,S+ SAo, = 1ETQE (36)
where S= ETSE
In order to find an adaptation law, the dynamic equation
(22) should be stated in terms of observed states.
z = g0(x,u,)+gT1(x,u,
)+g2(x,)+1(x,x, u) (37)
where the definition ofgi for i = 0, 1, 2 is similar to thosestated in (23) , (24) and (25) and 1(x,x, u) is a bounded
uncertainty. Adaptation law can be stated as follows
= g0(x,u,) + gT1(x,u,
)
+[gT2(x,)g1(x,u,)P ][z ] (38)
= g1(x,u,)P[W(x, ) ] (39)
whereis the auxiliary variable. Moreover, is an arbitraryHurwitz matrix and the positive definite symmetric gain
matrix Pis the solution of the following Lyapunov equation.
TP+ P = Q (40)
where Q is an arbitrary positive definite matrix.
The special statement of (37) and definition (38) will make
the proof of the proposed approach possible. Now, the error
system can be written as
z
=
g
T1(x ,u,) 0
g1(x,u,)P 0 00 0 A0
z
+
1(x,x, u)0
2(x,x, u)
(41)
where =z , =
Assumption: It is assumed that
1 = sup[1(x,x, u)] (42)
2 = sup[2(x,x, u)] (43)
Theorem 1: For the error system (41), , and z arebounded. Moreover, the tracking error is bounded.
Proof: The following Lyapunov function is considered for
the system
V( ,,z) = TP+T+ zTSz (44)
By computing the derivative of (44) and using lemma 1 we
have
V = TQ+ 2TP1(x,x, u) 1zTETQEz
+2
(x,x, u)BT Sz (45)
then we can state
V ||||2(min(Q) 2||P||1||||
)
||z||2(min(Q)
2n2||S||||z|| )
(46)
wherez = Ez. For any positive that ||z|| there existsa positive in the following range
n < min(Q)22||S||
(47)
such that the derivative of the lyapunov function is negative,
provided that
|||| 2 ||P||1
min(Q)
(48)
Since the derivative of the Lyapunov function is negative
outside a region, the system response cannot go outside of
this region. Indeed, the error is ultimately bounded.
Now, let e = zdz be the tracking error. It can be simplyfounded that e is the output of the following filter
= A + ( A+ 3(x,x ,u,)) (49)
e= (50)
The uncertain term 3(x,x,u,)) come from the obser-
vation error and the other terms that do not cancel by thecontrol signal. It is assumed that this term is bounded. Since
the filter (49) and (50) is stable and its input is bounded, the
tracking error, which is the output of the filter, is bounded.
In brief, in order to utilize the proposed control method-
ology, the following steps are required. First, The system
should be stated as (17) and (18). Then, an appropriate
diffeomorphism should be found using nonlinear control
theories to transform the system in the form of (22). Finally,
the observer and adaptation law are designed using (28), and
(38) and (39), respectively.
4852
-
8/12/2019 Adaptive Observer-based Controller Design for a Class of Nonlinear
5/6
(a) First state variable (p1 = x).
(b) Second state variable (p2 = ).
(c) Third state variable (p3 = ).
Fig. 2. Simulation results of the needle guidance problem with previouslyproposed output feedback linearization method of [12].
(a) First state variable (p1 = x).
(b) Second state variable (p2 = ).
(c) Third state variable (p3 = ).
Fig. 3. Simulation results of the needle guidance problem with the outputfeedback adaptive control method proposed in this paper.
4853
-
8/12/2019 Adaptive Observer-based Controller Design for a Class of Nonlinear
6/6
Fig. 4. The estimation of needle curvature ()
IV. SIMULATION R ESULTS
The proposed adaptive output feedback controller is em-
ployed to solve the problem of guiding a flexible bevel-tipneedle into a desired plane. The response of the system
is studied for two cases, first by applying the observer-
based controller proposed in [12] and second by applying the
proposed controller. In both cases the real needle curvature
is = 0.06 but our knowledge about this parameter is notexact. The known needle curvature is = 0.04. The initialconditions for the both cases are X0 = [0.1, 0.2, 0.8]
T.
In Fig. 2, the responses of the system with observer-
based controller [12] are shown. Since, with this controller
no parameter uncertainty is tolerated, the system response
should not be acceptable. Simulation results prove this
fact. In Fig. 3, the responses of the system with proposed
controller are plotted. This figure shows the good stabilityand convergence of the adaptive output feedback controller.
All state variables go near zero after a transient state. The
estimated needle curvature is shown in Fig. 4. Although this
parameter converges to a value, the converged value is not
the real one. This problem is not surprising, in that it can
be predicted theoretically from the given facts of Section
III. Since the error is ultimately bounded, we know that the
parameters just converge to a value. It is not any necessity
that the converged value be the real parameter.
V. CONCLUSIONS ANDF UTURE W ORKS
In this paper a novel observer-based controller is pro-posed for a class of nonlinear systems which are linearly
parametrized and feedback linearizable. The proposed strat-
egy is a modified version of a previously proposed adaptive
control scheme using high gain observer. The proposed
methodology is employed to guide a flexible bevel-tip needle
into a desired plane. A nonholonomic reduced order model
is considered for the needle which the needle curvature is
its only parameter. By utilizing the adaptive control, it is
possible to estimate the needle curvature in medical tasks
online. Through simulation results, it was demonstrated that
the proposed approach is quite effective for steering medical
needles into a desired plane.
Our next step is to evaluate the proposed methodology
with real data. One formidable barrier to achieve a good
performance in image guided tasks is measurement noise.
A more improved methodology can be proposed which
tolerates this barrier. The presented approach can also be
used with the automatic or manual path planning schemes.
REFERENCES
[1] R. Alterovitz, K. Goldberg ,J. Pouliot, R. Taschereau, and I. C. Hsu,Sensorless planning for medical needle insertion procedures, in Proc.
IEEE/RSJ Int. Conf. Intell. Robot. Syst. (IROS), 2003, vol. 3, pp. 3337-3343.
[2] E. Dehghan, and S. E. Salcudean, Needle Insertion Parameter Opti-mization for Brachytherapy, IEEE Trans. Robot. ,vol. 20, no. 2, pp.303-315, 2009.
[3] R. J. Webster III, J. Memisevic, and A. M. Okamura, Designconsideration for robotic needle steering, in Proc. IEEE Int. Conf.
Robot. Autom. (ICRA), Barcelona, Spain, 2005, pp 3588-3594.[4] R. J. Webster III, J. S. Kim, N.J. Cowan, G. S. Chirikjian and A.
M. Okamura Nonholonomic Modelling of Needle Steering, Int. J.Robot. Res., vol. 25, no. 5-6, pp. 509-525, 2006.
[5] R. Alterovitz, T. Simon, and K. Goldberg, The Stochastic Motion
Roadmap: A Sampling Framework for Planning with Markov MotionUncertainty, in Proc. Robotics: Science and Systems, 2007.
[6] V. Duindam, J. Xu, R. Alterovitz, S. Sastry, K. Goldberg, 3D MotionPlanning Algorithms for Steerable Needles Using Inverse Kinematics, in Workshop on the Algorithmic Foundations of Robotics, 2008.
[7] V. Duindam, R. Alterovitz, S. Sastry, K. Goldberg, Screw-BasedMotion Planning for Bevel-Tip Flexible Needles in 3D Environmentswith Obstacles, in Proc. IEEE Int. Conf. Robot. Autom. (ICRA),Pasadena, CA, USA, 2008, pp. 2483-2488.
[8] J. Xu, V. Duindam, R. Alterovitz, and K. Goldberg, Motion planningfor steerable needles in 3D environments with obstacles using rapidly-exploring random trees and backchaining, in Proc. IEEE Conf. Autom.Science and Engineering (CASE), 2008, pp. 41-46.
[9] D. S. Minhas,J. A. Engh, M. M. Fenske, and C. N. Riviere, Modellingof needle steering via duty-cycled spinning, in Proc. Int. Conf. IEEE
EMBS Cite Internationale, pp. 27562759, 2007.[10] J. M. Romano, R. J. Webster III, and A. M. Okamura, Teleoperation
of steerable needles, in Proc IEEE Int. Conf. Robot. Autom. (ICRA),2007, pp 934-939.
[11] N. Abolhassani, R. Patel and M. Moallem, Needle Insertion into SoftTissue, A Survey, Med. Eng. Phys., vol. 29, pp. 413-431, 2007.
[12] V. Kallem, and N. J. Cowan, Image Guidance of Flexible Tip-Steerable Needles, IEEE Trans. Robot., vol. 25, no. 1, pp. 67-78,2009.
[13] K. B. Reed, V. Kallem, R. Alterovitz, K. Goldberg, A.M. Okamura,and N.J. Cowan. Integrated Planning and Image-Guided Control forPlanar Needle Steering, in Proc. 2nd Biennial IEEE/RAS-EMBS Intl.Conf. Biomedical Robot. and Biomechateronics, Scottsdale, Arizona,2008.
[14] H. K. Khalil, Adaptive output feedback control of nonlinear systemsrepresented by input-output models, IEEE Trans. Automat. Contr.,vol. 41, no. 2, pp. 177-188, 1996.
[15] K. W. Lee, and H. K. Khalil, Adaptive output feedback control ofrobot manipulators using high-gain observer, Int. J. Contr, vol. 67,
no. 6, pp. 869-886, 1997.[16] G. Campion and G. Bastin, Indirect adaptive state feedback control
of linearly parametrized nonlinear systems, Int. J. Adaptive Controland Signal Processing, vol. 4, 345-358, 1990.
[17] M. R. Rokui, and K. Khorasani, An indirect adaptive control forfully feedback lunearizable discrete-time nonlinear systems, Int. J.
Adaptive Control and Signal Processing, vol. 11, 665-680 (1997)[18] A. Tornambe, High-gain observers for non-linear systems, Int. J.
Syst. Sci., vol. 23,no. 9, 1475-1489, 1992.[19] H.K. Khalil, Nonlinear Systems. , Prentic Hall, Third Edition, 2002.[20] J. J. E. Slotine, W. Li,Applied Nonlinear Control.Prentice Hall, 1991.
4854