Globally Linearizing Control for a Magnetic Microrobot Navigating

Within a Blood Vessel

NACERA ICHEDDADENE1, MEZIANE LARBI2, AHMED MAIDI1, KARIM BELHARET2

1L2CSP Laboratory, Mouloud MAMMERI University, 15 000 Tizi-Ouzou, ALGERIA

2HEI Campus Centre, PRISME EA 4229, Chateauroux, FRANCE.

Abstract: - In this paper, the globally linearizing control scheme is employed to guide an endovascular magnetic

microrobot navigating within a blood vessel with the objective of reaching a desired target following a trajectory

generated via a joystick device. First, we derive the 1D nonlinear dynamical model for the magnetic microrobot.

Subsequently, a stabilizing state feedback is designed based on the relative degree from geometric control, result-

ing in a closed-loop linear system. To ensure the tracking of a time-varying trajectory and reject disturbances, an

external proportional-integral controller with a bias is used to define the external variable of the resulting linear

system. The performance of the GLC is evaluated via numerical simulations. The obtained results demonstrate

the output tracking and disturbance rejection capabilities of the GLC scheme.

Key-Words: - Magnetic microrobot, geometric control, relative order, globally linearizing control, output

tracking.

Received: July 15, 2023. Revised: July 13, 2024. Accepted: August 12, 2024. Published: September 27, 2024.

1 Introduction

Due to their small size, wireless control, and power

capabilities, microrobots have successfully been ap-

plied in biomedical settings to perform tasks with re-

duced invasiveness, [1]. Among these applications,

one can cite targeted drug delivery, brachytherapy,

tissue reconstruction, diagnosis and hyperthermia, to

name a few, [2], [3], [4].

To achieve these tasks with high accuracy, partic-

ularly in complex environments, control techniques

play a key role. Indeed, to successfully reach the tar-

get in environments with bifurcations and to access

the most critical confined areas requiring treatment,

precise trajectory tracking is crucial. Thus, numer-

ous open-loop and closed-loop control strategies have

been developed in the literature; a comprehensive re-

view can be found in [2], [5], [6]. Nevertheless, due

to nonlinearities, internal and external disturbances,

uncertainties, and noise, the performance of open-

loop control is limited and may become unstable, [2].

Therefore, to achieve precise motion despite distur-

bances and uncertainties, the use of closed-loop con-

trol techniques is required.

Model-based techniques have been extensively

studied in the literature, using either a linear or a non-

linear model of the microrobot. When blood velocity

is assumed to be zero, a linear model can be used. In

this case, several well-established control strategies

have been employed. These include the use of a PID

controller, [7], a model predictive controller (MPC),

[8], a tolerant ISS-based control approach, [9], robust

control, [10], observer-based control, [11], and slid-

ing mode control, [12].

Actually, since blood velocity is variable, a non-

linear model is required to capture the dynamic be-

havior of the microrobot. However, designing easy-

to-implement controls using a nonlinear model that

achieve precise tracking trajectories despite distur-

bances and uncertainties is a complex task. Few ap-

proaches have been explored in the literature, includ-

ing backstepping control, [13], [14], and observer-

based control, [15], [16]. This observation motivates

the application of alternative techniques to improve

microrobot control performance in dynamic environ-

ments.

Globally linearizing control (GLC), [17], [18],

[19], is an interesting alternative control approach for

nonlinear systems with a relative degree equal to its

order, which applies to the microrobot. In this case, a

stabilizing state feedback can be designed within the

framework of geometric control, resulting in a closed-

loop linear system. Then, to address disturbances and

uncertainties, an external controller is used to define

the external input of the resulting linear system, [17],

[19]. Many successful applications of GLC for highly

nonlinear systems have been reported in the litera-

ture, [18], [19], [20], [21], [22].

In this work, the GLC is applied to solve the out-

put tracking of a microrobot navigating within a blood

vessel. The control objective is to steer the micro-

robot from its initial position to the desired target po-

sition following a time-varying trajectory generated

by an operator via a joystick, despite environmental

disturbances. The control design is achieved using

a nonlinear model. A modified GLC that combines

an external controller and a reference differentiator is

adopted to achieve precise trajectory tracking. This

study introduces the application of a modified GLC,

marking a first in the field. By utilizing GLC with

a nonlinear model, our work advances nonlinear con-

trol strategies for microrobots, demonstrating not only

the efficacy but also the significant potential of GLC

in enhancing microrobot navigation.

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The manuscript is outlined as follows: Section 2

focuses on the modeling of the microrobot, presenting

a comprehensive 1D nonlinear dynamic model that

captures its behavior within a blood vessel environ-

ment. In Section 3, the paper delves into the design of

the GLC for the microrobot, aiming to achieve precise

and robust trajectory tracking despite environmental

disturbances. Section 4 presents the simulation re-

sults, showcasing the effectiveness and performance

of the GLC approach in terms of trajectory track-

ing accuracy and robustness against disturbances. Fi-

nally, Section 5 concludes the paper.

2 Microrobot modeling

Consider a spherical magnetic microrobot of mass

mand a radius Rthat navigates within a cylindrical

blood vessel along the i-axis (Figure 1). In this sec-

tion, a one-dimensional (1D) state-space model de-

scribing the behavior of the microrobot is presented.

Thus, the position and velocity along the i-axis are

denoted by prand vr, respectively. It is assumed that

the microrobot is affected by the drag force Fdand the

magnetic force Fm. The magnetic force Fmis used to

move the microrobot by manipulating the magnetic

field gradient B, [23].

2.1 Magnetic and drag forces

2.1.1 Magnetic force Fm

The microrobot is propelled through the blood vessels

by the magnetic force Fm, induced by the magnetic

coils of the electromagnetic actuation system (EMA)

by manipulating the magnetic field gradient. The Fm

force is defined by [11]:

Fm=m

ρr

M∇B(1)

where Mand ρrrepresent the magnetization and the

density of the microrobot, respectively. The ∇de-

notes the gradient operator, and Bstands for the mag-

netic field; hence, ∇Bis the magnetic field gradient,

which is used as the control variable.

Figure 1: Forces acting on a microrobot navigating

through a blood vessel.

2.1.2 Drag force Fd

The microrobot moving in a static or fluid environ-

ment experiences a hydrodynamic drag force Fdthat

opposes its displacement. The drag force Fdacting on

a spherical microrobot with a frontal area Afis given

by [13]:

Fd=−1

2ρf(vr−vf)2AfDc

vr−vf

∥vr−vf∥(2)

where ρfand vfare the density and velocity of the

fluid, respectively, and Dcrepresents the drag coeffi-

cient.

Assuming that the microrobot navigates in a dy-

namic fluid, in this case the drag coefficient Dcis de-

fined as [24]:

Dc=24

1 + √Re

+ 0.4(3)

where the Reynolds number Rethat determines the

flow regime of the fluid is given by [13]:

Re=2ρf(vr−vf)R

η(4)

where ηis the fluid viscosity.

Combining Equations (2), (3) and (4), it follows

that

Fd=−6π R η (vr−vf) + ρfπ R2(vr−vf)2×







0.2 + 3

1 + 2ρf(vr−vf)R







(5)

2.2 Microrobot model

Using Newton’s fundamental law of dynamics leads

to the ordinary differential equation, which describes

the dynamic behavior of the microrobot and is given

by:

m˙vr=Fm+Fd(6)

Substituting Fmand Fdwith their expressions

from Equations (1) and (5), Equation (5) reduces to

˙vr=α1(vr−vf) + α2(vr−vf)2

+α3

(vr−vf)2

1 + α4(vr−vf)+α5∇B(7)

with

α1=−9η

2R2ρr

, α2=−0.15 ρf

R ρr

, α3=−2.25 ρf

R ρr

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α4=2R ρf

η, α5=M

ρr

Defining the control variable u=∇Band the

state variables as the position and the velocity of the

microrobot, i.e., x1=prand x2=vr(vr= ˙pr),

the state-space model of the microrobot obtained from

Equation (7) is

˙x(t) = f(x(t)) + g(x(t)) u(t)(8)

y(t) = h(x(t)) (9)

where tis the time variable, x= [x1x2]Tis the state

vector, and the vector functions f,g, and hare defined

as follows:

f(x(t)) = 



x2(t)

α1w(t) + w2(t)(α2+α3

1 + α4√w(t))



(10)

g(x(t)) = 0

α5(11)

h(x(t)) = x1(t)(12)

with w(t) = x2(t)−vf.

3 Globally linearizing control design

The objective consists in designing a control law u

(magnetic field gradient) that moves the microrobot

from a known initial position yito a desired target yf

in the blood vessel following a time-variable desired

trajectory yd. For this purpose, it is proposed to use

the GLC strategy, [17], [18], [19], to solve this output

tracking problem.

The GLC scheme consists of two control loops,

[18]. The inner loop uses state feedback that yields in

the closed loop a linear system v−y. Then, for ro-

bustness and disturbance rejection purposes, an outer

loop is used to define the external variable vby means

of the external controller.

The GLC design involves the following two steps,

[17], [18], [19]:

1. Design of the linearizing state feedback in the

framework of geometric control,

2. Design of the external control using the resulting

linear system v−y.

3.1 Design of the state feedback

The design of the state feedback is carried out in the

frame work of geometric control based on the con-

cept of the relative degree, [25], [26]. The relative

order σrefers to the minimum number of times that

the output yneeds to be differentiated to directly re-

late it to the input u. Thus, using the Lie deriva-

tive, [19], [25], [26], the time derivatives of the output

ycan be expressed as follows:

˙y(t) = Lfh(x)

=x2(t)(13)

¨y(t) = L2

fh(x) + LgLfh(x)u(t)

=α1w(t) + w2(t)φ(w(t)) + α5u(t)(14)

with

φ(w(t)) = α2+α3

1 + α4w(t)(15)

From (14), it can be seen that the control uappears

linearly in the second time derivative of the output

y, that is, LgLfh(x)= 0, hence the relative degree

σ= 2. As σis the same as the order n= 2 of the mi-

crorobot, the microrobot can be fully linearized, and

the GLC can be successfully applied. Consequently,

an output stabilizing state feedback can be designed

that achieves in closed loop the stable linear system

v−ygiven by

τ2¨y(t) + τ1˙y(t) + y(t) = v(t)(16)

where vis an external input, and τ1and τ2are tuning

parameters. Therefore, using Equations (13) and (14),

the linear system (16) reduces to

τ2α1w+w2φ(w)+τ2α5u+τ1x2+y=v

(17)

Then, solving Equation (17) with respect to the

control uyields the following output stabilizing state

feedback.

u(t) = 1

τ2α5v(t)−y(t)−τ1x2(t)−τ2(α1w(t)

+w2(t)φ(w(t))(18)

3.2 Design of the external controller

The state feedback (18) is designed by assuming that

there are no disturbances. Consequently, this control

law will be incapable of rejecting the disturbance. To

overcome this problem, the external input v(t)must

be defined by an external controller.

Assumption 1 The desired trajectory yd(t)is a

known twice differentiable function, i.e., yd(t)∈ C2

(Cbeing the space of twice differentiable functions).

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To achieve disturbance rejection and robustness

against modeling errors, the solution consists of defin-

ing the external variable vby means of an external

controller of the general form [18]

v(t) = b(t) + t

c(t−τ)yd(τ)−y(τ)dτ (19)

where cis the inverse of a given transfer function, and

b(t)is the external controller bias given by

b(t) = Dyd(t)(20)

where Dis the differential operator defined as fol-

lows:

D(.) = τ2

d2(.)

dt2+τ1

d(.)

dt + (.)(21)

In this work, a PI controller is used to define the

external variable, i.e.,

v(t) = Dyd(t) + Kce(t) + 1

Tit

e(ξ)dξ (22)

where Kcand Tiare the tuning parameters of the PI

controller, and e(t) = yd(t)−y(t)is the tracking

error. The GLC scheme is depicted in Figure 2.

Remark 1 For a constant set-point yd(t), the exter-

nal controller bias b(t) = yd(t)since d2yd(t)/dt2=

dyd(t)/dt = 0, i.e., D= 1.

For the tuning of the PI controller, the internal

model control-based tuning method is used, [27]. The

transfer function of the linear closed loop system (16)

is given by

G(s) = Y(s)

V(s)=1

τ2s2+τ1s+ 1 (23)

where Yand Vare the Laplace transforms of yand v,

respectively, and sis the Laplace variable. Thus, by

choosing the tuning parameters τ1and τ2as follows:

τ1=γ1+γ2(24)

τ2=γ1γ2(25)

e(t)

yd(t)

v(t)u(t)

y(t)

x(t)

b(t)Linear system v−y

D(.)

−

Figure 2: GLC scheme for the microrobot.

The transfer function (23) takes the following form:

Y(s)

V(s)=1

(γ1s+ 1) (γ2s+ 1) (26)

and using the IMC-based tuning method yields the

following PI controller tuning parameters, [27]

Kc=γ1+γ2

τ(27)

Ti=γ1γ2(28)

where τis the desired closed loop time constant.

4 Numerical simulation

In this section, the output tracking performance of the

GLC is assessed through numerical simulation. Ta-

ble 1 and Table 2 provide the microrobot parameters

and the tuning parameter of GLC, respectively. It

is assumed that the velocity of the blood vfis time-

varying, indicating an internal disturbance. The ex-

pression of vfis [13]:

vf(t) = 0.035 (1 + 1.15 sin(2 π t)) (29)

Parameter Value

Microrobot radius R= 250 ×10−6[m]

Fluid density ρf= 1060 [Kg/m3]

Fluid viscosity η= 16 ×10−3[Pa.s]

Microrobot density ρr= 7500 [Kg/m3]

Microrobot Magnetization M= 1.23 ×106[A/m]

Table 1: Microrobot parameters, [15].

State feedback PI controller

γ1= 0.7692 sKc= 3.01

γ2= 5.2631 sTi= 6.03 s

Table 2: GLC tuning parameters.

Evaluations of the performance of the GLC are

conducted both with and without disturbance.

The desired reference ydis generated by an oper-

ator using a joystick. The whole control scheme is

summarized by Figure 3.

4.1 Results

For the first simulation run, the control objective con-

sists of moving the microrobot from its initial posi-

tion to a desired target following the generated trajec-

tory ydwhile the disturbance vfis maintained con-

stant. The obtained results are given by Figure 4. In

the second simulation run, the microrobot is tasked

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Joys�ck

Device

posi�on

PI Controller

NOVINT falcon

Dhd.lib

Posi�on data

Input/Output

Lineariza�on

state feedback

𝒚𝒔𝒑

Reference trajectory

Desktop PC +

MATLAB + LabVIEW

𝐉𝐨𝐲𝐬𝐭𝐢𝐜𝐤+ Operator

𝑷𝒙

𝒆(𝒕)𝒚(𝒕)

𝒖(𝒕)

𝑨(𝒕)

Microrobot

Figure 3: GLC scheme based on a joystick for a microrobot.

with reaching the desired target following the trajec-

tory provided by the joystick, even with the sudden

variation of the fluid velocity. To evaluate the per-

formance in this situation, the following variation is

assumed for the blood velocity:

vf(t) = 





0.035 (1 + 1.15 sin(2 π t)) t < 12 s

vf(t)

2t≥12 s

(30)

Figure 5 gives the obtained results.

4.2 Discussion

From Figure 4, it can be observed that despite

the time-varying internal disturbance vf, the GLC

scheme allows the microrobot to reach the desired tar-

get following the trajectory generated by the joystick.

This good tracking is supported by the position track-

ing error given in Figure 4-c. Additionally, Figure 4-d

clearly shows reasonable moves of the magnetic field

gradient, i.e., the control variable does not exceed the

authorized value of 10−2T/m.

From Figure 5, it can be seen that the GLC suc-

cessfully rejects the disturbance and forces the micro-

robot to regain its imposed trajectory. This result is

corroborated by the evolution of the position track-

ing error (Fig. 5-c). Additionally, the magnetic field

gradient remains within physically acceptable limits

(Fig. 5-d).

The simulation results clearly demonstrate the ef-

fectiveness of the GLC in achieving precise output

tracking for the microrobot, both in the absence and

presence of disturbances.

5 Conclusion

In the present work, the GLC scheme is adopted to

guide a microrobot navigating in a blood vessel to

reach a desired target following a trajectory generated

by an operator via a joystick. First, the 1D nonlin-

ear dynamical model of the magnetic microrobot is

derived using Newton’s fundamental law of dynam-

ics. Then, as the relative degree is equal to the or-

der of the microrobot, a stabilizing state feedback is

designed within the framework of geometric control.

This state feedback yields a linear system in a closed

loop. Hence, for robustness and disturbance rejection,

the external variable involved in the state feedback is

defined by an external PI controller. To address the

tracking problem even in the case of a time-varying

desired trajectory, the external variable is adjusted by

adding a bias that represents the differentiation of the

desired trajectory.

The output tracking and disturbance rejection per-

formance of the GLC are evaluated through numerical

simulation. The obtained results demonstrate the ef-

fectiveness of the GLC in achieving precise trajectory

tracking for the microrobot.

Based on the findings of this study, we recommend

extending the GLC method to more complex nonlin-

ear models in 2D and 3D environments, developing

an observer to estimate the microrobot velocity for

practical implementation, and conducting real-world

testing to validate the simulation results.

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-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05

-0.05

0.05

0 2 4 6 8 10 12 14 16 18

Time [s]

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Position [m]

0 5 10 15 20

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-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05

-0.05

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Time [s]

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Blood vessel

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(a) (c)

(b) (d)

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(b) Time evolutions of the actual and desired trajectories. (c) Position tracking error. (d) Magnetic field gradients.

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Ahmed Maidi, Karim Belharet

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DOI: 10.37394/23203.2024.19.27

Nacera Icheddadene, Meziane Larbi,

Ahmed Maidi, Karim Belharet

E-ISSN: 2224-2856

254

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