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2D Robust Magnetic Resonance Navigation of aFerromagnetic Microrobot using Pareto Optimality

David Folio, Antoine Ferreira

To cite this version:David Folio, Antoine Ferreira. 2D Robust Magnetic Resonance Navigation of a Ferromagnetic Micro-robot using Pareto Optimality. IEEE Transactions on Robotics, Institute of Electrical and ElectronicsEngineers (IEEE), 2017, pp.1-11. �10.1109/TRO.2016.2638446�. �hal-01446482�

2D Robust Magnetic Resonance Navigation of aFerromagnetic Microrobot using Pareto Optimality

David Folio and Antoine Ferreira

Abstract—This paper introduces a two-dimensional au-tonomous navigation strategy of a 750 µm steel microrobot alonga complex fluidic vascular network inside the bore of a clinical3.0T magnetic resonance imaging (MRI) scanner. To ensuresuccessful magnetic resonance navigation (MRN) of a micro-robot along consecutive channels, the design of autonoumousnavigation strategy is needed taking into account the majorMRI technological constraints and physiological perturbations,e.g. non-negligible pulsatile flow, limitations on the magneticgradient amplitude, MRI overheating, susceptibility artifacts un-certainties, and so on. An optimal navigation planning frameworkbased on Pareto optimality is proposed in order to deal with thismultiple-objective problem. Based on these optimal conditions,a dedicated control architecture has been implemented in aninterventional medical platform for real-time propulsion, controland imaging experiments. The reported experiments suggest thatthe likelihood of controlling autonomously untethered 750 µmmagnetic microrobots is rendered possible in a complex two-dimensional centimeter-sized vascular phantom. The magneticmicrorobot traveled intricate paths at a mean velocity of about4mms−1 with average tracking errors below 800 µm with limitedmagnetic gradients ±15mTm−1 compatible with clinical MRIscanners. The experiments demonstrate that it is effectivelypossible to autonomously guide a magnetic microrobot using aconventional MRI scanner with only a software upgrade.

Index Terms—Microrobotics, Magnetic Resonance Imaging,Magnetic Resonance Navigation, Multi-Ojective Planning.

I. INTRODUCTION

M ICROROBOTICS is a promising solution for biomedi-cal application innovation, including targeted therapies,

diagnosis, drug delivery, minimally invasive surgery, or cellstransportation [1], [2]. To control microrobotic devices withinthe human body, magnetic actuation is till now the mostadvanced solution [2]. Currently, different concepts have beenproposed using specific magnetic manipulation systems [3],[4] or upgraded clinical Magnetic Resonance Imaging (MRI)scanner [5]–[11]. The former usually provides up to 5 degreesof freedom with great steering accuracy. However, they havelimited medical application as they provide low magneticfield and require an external vision system. In contrast, thesecond approach requires only an upgrade of current softwareand/or hardware capabilities of clinical MRI systems. Initiallydevoted to medical imaging, MRI scanners are nowadaysinvestigated to become an assisting tool for navigation controlin vascular networks. In cancer therapy, therapeutic agentsare injected using a magnetic catheter to impose a startingpoint for the navigable path of the agents to be located withinan artery. As shown in Fig. 1, to reach a tumor, therapeutic

D. Folio and A. Ferreira are with the INSA Centre Val de Loire,Universite d’Orleans, PRISME EA 4229, Bourges, France. (e-mail:david.folio@insa-cvl.fr; antoine.ferreira@insa-cvl.fr)

broken up

∇b

Catheter Magnetic microrobots

Targeted deep location(tumor, stenosis...)

BalloonClinical MRI

Magnetic gradient coils

1mm

10µm

Fig. 1. Representation of the Magnetic Resonance Navigation (MRN):magnetic microrobots are released in artery from a catheter. By correctlyapplying magnetic gradients ∇b from the MRI scanner, microrobots followan optimal planned path. In tiny vessels, microrobots are broken up and drugsare released to a targeted area.

microagents are transiting through arteries (cm-wide), nar-rower arterioles before reaching the tumoral regions accessiblethrough the capillaries (µm-wide) forming the angiogenesisnetwork. To achieve such a goal, robust and safe endovascularnavigation presents challenging issues in terms of modelingand control [8].

Actually, the main research efforts are carried out in the de-sign of robust and autonomous magnetic resonance navigation(MRN) strategies easily implementable in clinical MRI scan-ners. Basically, MRN can be performed using two fundamentalapproaches namely, Imaging Gradient Coil (IGC) or SteeringGradient Coil (SGC). The former consists of upgrading aclinical MRI scanner with the software only. A breakthroughin interventional MRI procedures demonstrated that real-timeIGC systems can offer a well-suited environment for theimaging, tracking, and control of a magnetic microrobot,which was done with a pig [5] and a rabbit [9]. The sole MRIsoftware upgrade does not imply any major modifications, andallows keeping it for conventional operations. In this context,the MRN relies on the use of gradient coils of the MRI forimaging and propulsion. By alternating these two sequencesit has been demonstrated that it is possible to guide magneticmicrorobots along a simple path using a clinical MRI scanner[6], [11]. These successful in vivo experiments pointed outcritical challenges in terms of real-time imaging and tracking,together with optimal navigation strategy for future therapeuticapplications [10]. Furthermore, the gradients used for thewhole body imaging are relatively low (e.g. < 80mT m−1) forconventional scanners. In contrast, the SGC implies integratingadditional coils capable to generate higher gradients, typicallymore than 300mT m−1 while improving the magnetic steeringand imaging capacities. The key to this innovative system isthe capability to concurrently use powerful insert coils withthe body magnet’s gradient coils. As illustration, the authorsin [12] developed the highest strength insert imaging gradient

coil that allows peak strength values of 600mT m−1 with fastrise time of 5 ms. Similarly, Martel et al. proposed to improvethe MRI hardware by adding additional electromagnetic coils[9] delivering gradients of 300mT m−1 allowing MR-imagingor MR-tracking in a time-multiplexed way. However, suchsupplementary gradient inserts lead to a reduction of theMRI bore, and usually limit the MRN application to humanmembers or small animals. New versions of ultra-high gradi-ents scanners for whole body MRN are actually developed.The possibility to perform MRN within a whole-body MRIhas been recently demonstrated by modifying a MangetomSkyra 3T magnet [13]. For these upgraded steering gradientcoil setups, technological constraints related to cooling timeof the steering gradient coils lead to limited duty cyclesof the imaging-propulsion sequences that greatly affects therobustness and stability of the real-time control loop.

Our approach in this work is to improve the standard IGC-based MRN strategy in complex phantom vascular modelsusing a standard clinical MRI scanner without hardwaremodifications. Although weaker driving gradient forces aregenerated, the propulsion of magnetic microrobot to counteractstrong pulsatile flows or to navigate in small vessels, can beovercomed by using intra-arterial balloon or upsized magneticmicrocarriers composed of aggregated nanoparticles, as illus-trated in Fig. 1. Until now, the proposed closed-loop MRNstrategies consider only simple Y-bifurcations [6] or whenmore complex phantoms are used, real-time imaging is basedon MR-compatible vision cameras inserted in the MRI bore[10]. The design of a robust navigation strategy in complexphantoms should take into account the MRI technologicalconstraints and physiological perturbations, e.g. non-negligiblepulsatile flow, saturation of magnetic gradient amplitudes,limited duty cycle of imaging-propulsion sequences and MRIoverheating [7]. To deal with these MRN constraints differentnavigation planning approaches have been proposed, such asusing potential fields and the breadth-first search algorithm[6], or fast marching method (FMM) [14] considering thecenterline [7], the uncertainty [11] or the power efficiency[15] objectives. However, each proposed navigation strategydoes not solve all above constraints together. That is why,these aspects should be anticipated at the early navigationplanning stage [15]. In this paper, we address these conflictingmulti-objective issues by proposing a robust and autonomousnavigation design based on a Pareto optimality framework.

The paper is organized as follows.The MRN system with itsbasic components (including sensing and actuation modules)are first introduced in Section II. Then, Section III presents theoptimal navigation planning frameworks that allows dealingwith multiple-objective problems. In Section IV some pre-liminary experiments are conducted to evaluate the MRNfunctionalities capabilities. Based on this preoperative results,an autonomous 2D MRN on an in vitro vascular phantomresult is presented in Section V, validating the proposed MRN.

II. THEORETICAL BACKGROUND

A. MRI PropulsionThe overall concept of the magnetic resonance navigation

(MRN) relies on the use of a clinical magnetic resonance

z-axiscoils

y-axis coils

x-axiscoils

y

x

b0

z

Helmoltz coils for the uniform magnetic field b0

Fig. 2. Representation of the MRI magnetic gradient coils set configuration:x and y axes are pairs of saddle coils and z-axis are Maxwell coils pair.

imaging (MRI) scanner to both track and propel a therapeuticagent (termed microrobot). Basically, as depicted in Fig. 2, aMRI scanner comprises one pair of Helmholtz coils generatinga strong static uniform magnetic field b0, and is aligned tothe symmetry z-axis of the MRI bore. In order to generatethe propulsion force of the microrobot, a magnetic gradientfield in the 3D-space of the MRI bore is generated by twopairs of Golay coils and one pair of Maxwell coils in anorthogonal arrangement. The magnetic field b is governed bythe following Maxwell equations:

∇× bg(r) = 0;∇.bg(r) = 0 (1)

This coil configuration is axis-symmetric and generates (at thecenter of the MRI bore) a magnetic field given by :

bg =

−0.5gzx+ gxz

−0.5gzy + gyz

gxx+ gyy + gyy

(2)

where

∇bg =(gx gy gz

)T=(∂bx∂z

∂by∂z

∂bz∂z

)T(3)

are the linearly independent magnetic gradients. The x, y, zcoordinates originate at the center of the MRI bore where z isthe direction of the bore axis of symmetry. Besides the highmagnetic field, uniform and linear magnetic gradients can becreated within a 40-to-60-cm diameter spherical volume. Forinstance, the Magnetom c© Verio 3T is capable to generate anuniform magnetic field of b0 = 3 T in the z-axis direction,and a magnetic gradient magnitude up to 45mT m−1 in thex-y-z-axes directions. Placed in the MRI bore, the magneticmaterial volume Vm of a microrobot is subject to a totalmagnetic field b = b0 + bg . In such strong magnetic field,the magnetic material reaches the saturation magnetizationmsat = (0, 0,msz)

T , and is mainly aligned with b0, asbg � b0. In this study, only spherical beads are considered,and thus, only a magnetic force fm is induced, and is expressedhere as:

fm = Vmmsz∇bg, (4)

Only bg contributes to the force generation since b0 is homo-geneous and its spatial variation is zero. The equation states

that three orthogonal magnetic forces act on the magneticmicrorobot. This means that magnetic gradients of the MRIcould induce 3-DOF propulsion only in the center of the bore,within a workspace of approximately 40×40×40cm3.

B. Endovascular Modeling

The modeling of the microrobot immersed in a microfluidicenvironment is necessary for the both navigation planning andcontrol strategies. This modeling is illustrated in Fig. 3, and thecorresponding dynamic model is briefly presented hereafter.

2R2r

v

fd

fg

fm

fe+fv∇bx

∇bz

fm(∇b)

g

Fig. 3. Representation of the magnetic microrobot (left) in microfluidicenvironment (right) with the applied forces.

Actuated by magnetic gradients ∇bg in a microfluidicenvironment, the microrobot dynamics could be expressed as:

(ρV ) v =∑

f = fm(∇bg) + fExt(v,x) (5)

with ρ, V and v the microrobot density, volume, and ve-locity respectively. fExt are all external forces such as theapparent weight (fg), electrostatic (fe), van der Waals (fv)and hydrodynamic drag (fd) microforces (see Fig. 3). At theconsidered scale, and when the microrobot navigates closeto the centerline, it has been shown that the most dominantexternal force is the hydrodynamic drag force fd [15], [16].In such case, a spherical microrobot of radius r will thenexperience mainly the magnetic pulling (4), and the dragforces, that are commonly defined as:

fd = −6πηr · v (6)

where η is the flow viscosity.Let (v,x)

T denotes the system state vector, defined bythe microrobot velocity v and location x. This position x ismeasured thanks to the developed real-time artifact trackingprocedure [11]. The input vector is defined by the magneticgradient, that is u = ∇bg . From the above dynamic modelwe get the following state-space representation (S∇):

(S∇) =

(

x

v

)= A

(x

v

)+ Bu +

(∆γc∆γp

)y = C x + ∆γb

(7)

with

A =

(0 I20 −4.5 η

ρr2 I2

); B =

(0

mszτmρ I2

); C = 1 (8)

and I2 the identity matrix, and τm = Vm/V the volumetricmagnetization rate.

The system (S∇) models the robot dynamics, and has to becontrolled along a reference trajectory subjected to uncertaintyerrors. The terms ∆γc, ∆γp and ∆γb represent the uncertainparameters related to the influence of operating conditions,

i.e. the external forces fExt affecting the microrobot’s dy-namics [16], the saturation of the MRI actuator coils [10]and the 2-D position measurement errors related to MRImagnetic signature sensing [11], respectively. Because safetyand accuracy are of critical importance, these uncertaintieswill have significant influence on the choice of the best pathfor MRN procedure. Different works show how to incorporateuncertainty and state uncertainty into the control framework.As example, predictive controller [17], adaptive controller [18]or H∞ controller [19] proved to be efficient as proof ofconcept but fail when considering multiple uncertainties inrealistic and complex vascular navigation configurations underpulsatile flow.

The challenge we discuss in the following section is toprecisely quantify these uncertainties in advance, such as thebest path can be selected in realistic MRN conditions.

III. OPTIMAL MRI-BASED NAVIGATION FRAMEWORK

A. Navigation Planning

1) Problem Statement: A navigation path between a start-ing point x0 ∈ C to a targeted point x ∈ C, could be definedas a curve p on the domain C ⊂ Rd as follows:

p : l ∈ [0, 1] 7→ p(l) ∈ C, with x0 = p(0),x = p(1) (9)

where d = 2, 3, 4 . . . is the path state-space dimension, andl the arc-length parameter. In a Riemann manifold the pathlength could be defined from an energy functional weightedby a metric w : x ∈ C 7→ w(x) ∈ R+ as follows:

Lw(p) =

∫ 1

0

w (p(l)) · ‖p(l)‖ dl (10)

with ‖·‖ the Euclidean norm. Classically, the navigation plan-ning goal is to find a minimal path (also referred as a geodesic)that globally minimizes the path length Lw(p), that is:

γ = arg minp

Lw(p) (11)

The geodesic distance between two points (x0,x) is the lengthLw(γ) wrt. the metric w. In this context, previous studies [11],[15], [17] have shown that the fast marching method (FMM),introduced by Sethian [14], offers an interesting framework.

2) Fast Marching Path Planning: In the FMM framework,the minimum path finding problem (11) is solved through thecomputation of the minimal distance map U : C 7→ R+ usingthe Eikonal equation:{

||∇U(x)|| = w(x), ∀x ∈ CU(x0) = 0

(12)

The key issue of the FMM is to define an appropriate metricw(x) for retrieving a suitable navigation path based on thedesired criteria.

3) Minimal Centerline: A first objective is to design themetric on spatial consideration. A basic idea is to use a unitmetric, where w1(x) = 1 for all free location in the domainC, and 0 in forbidden location. Such unit metric computesfrom Eq. (10) the Euclidean path arc-length L1 = Lw1

(γ1),and thus leads to the shortest path. However, this solution

commonly implies contouring the walls. In previous work [17]the following spatial cost function has been proposed:

wc : x ∈ C 7→ Enhance(x) ∈ R+ (13)

where Enhance(x) is a vesselness enhancement functiondesigned to give their maximal response in the center of thethoroughfare. Such filtering technique leads to find the center-line geodesic γc [17]. In particular, it has been shown that themicrorobot dynamics is less affected by the surface forces (thecontact, electrostatic and van der Waals microforces) when itnavigates close to the centerline [15], [16]. For such objective,γc is then an optimal solution. However, designed solely onspatial consideration, the centerline geodesic does not takeinto account neither the physiological perturbations, the MRItechnological constraints nor the sensing uncertainties.

4) Minimum-Power Navigation Path: To ensure that ageodesic is really achievable by the MRN system the navi-gation path has to reduce the maximum pulling gradients (e.g.up to 80 mT m−1), the operating time (e.g. ≤ 30 min), andalso maintain the rising temperature of the electromagneticcoils within operating limits (e.g. ≤ 50 ◦C) [10]. To overcomesuch issues, in [15] an anisotropic cost function is proposedto minimize the energy expenditure, and is expressed as:

wp : (x, f) ∈ C × F 7→ wp(x, f) ∈ R+ (14)

where the anisotropy cost is extended with the vector f of afield of force F . The proposed idea is to use a classical powerfunctional expression [15]:

wp(x, f) = arg maxx∈C,∇b∈B

〈f(x,∇b) · v(x)〉 (15)

where 〈·〉 is the inner product of Rd, and f(x,∇b) is theapplied resultant force that embeds the controlled magneticgradient ∇b ∈ B. Using model of the system dynamics (7),the anisotropic metric wp allows considering the major physio-logical perturbations (e.g. flow viscosity, time-varying pulsatileflow). Furthermore, the magnetic action set B is designed wrt.the MRI available magnetic actuation capabilities. The metricwp allows obtaining a power efficient geodesic γp effectivelyfeasible by the MRN platform.

5) Planning with MRN Uncertainties: Numerous sources ofuncertainties, such as in MRI data acquisition and processing,carrier model and position tracking, must be considered inMRN planning. Otherwise, the microrobot may be trapped inthe wrong bifurcation or cause some unwanted embolies. Ex-plicitly modeling sensing uncertainties in the planning strategyis a first main step to robustness [11]. It has been recognizedthat it could be defined some maps which describe the qualityof the sensed information obtained at each configuration usedby the system [20], [21]. In previous study [11], the expectedinformation gain I(x;y) is proposed to design this mapping,and the corresponding metric is defined by

wb : x ∈ C 7→ I−1(x;y) ∈ R+ (16)

The expected information gain quantifies the microdevice’sability to characterize its localization at different positionsx. I(x;y) is computed, given an observed data y, fromthe difference in Shannon entropy of the prior and posterior

distributions: I(x;y) = H(x) − H(x|y). The prior entropyH(x) measures the carrier’s belief of its position x, beforethe sensory input y is received. The conditional entropyH(x|y) denotes the expected entropy change after measure-ment data y. The computed geodesic γb from the metricwb (16) maximizes the expected information gain I(x;y), andthus minimizes the uncertainties along the navigation path.

B. Navigation Planning with Multiple ObjectivesPrevious proposed solutions provide an optimal naviga-

tion pathway wrt. their corresponding metric. The considerednavigation planning strategy is then a basic single-objectiveoptimization problem (11). When more than one goal has tobe addressed, the design problem becomes multi-objective. Inthis context, a geodesic is commonly obtained from:

γ = arg minp

{Lw1(p), . . . Lwk(p), . . . Lwn(p)} (17)

As the different metrics wk are usually conflicting, the scalarconcept of ”optimality” does not apply directly in the multi-objective design. This situation gives rise to the concept of aPareto solution, and the use of Pareto optimality. Especially,a feasible path p? is said to be Pareto optimal for a multi-objective problem if there are no other feasible path p suchthat: Lwk(p?) ≤ Lwk(p), ∀k = 1..n, and Lwk(p?) < Lwk(p)for at least one k = 1..n. The family of Pareto solutions formsthe so-called Pareto front.

The most widely-used approach for multi-objective opti-mization is the weighted sum technique (also referred asscalarization method), where multiple metrics are aggregatedin a single cost function as follows [22]:

wλ =

n∑k=1

λkwk (18)

If∑k λk = 1 and ∀λk ∈ [0, 1] the weighted sum is said

to be a convex combination of cost functions. Any weightset {λk} generates a single cost function wλ that determinesdifferent Pareto optimal geodesics. In particular, minimizing(18) is sufficient for Pareto optimality [22]. Testing all possiblevalues of the weight set would construct the convex hull of thePareto front. It can be shown that the navigation path computedfrom the single-cost function wλ is a Pareto optimum geodesicfor the original multi-objective problem (17).

However, usually there is no correspondence between a set{λk} and a geodesic: it is up to the end-user to choose theappropriate weights. Expert knowledge is then needed to selectthe weights while the conditions of the system changed. Topropose a weight set, an idea is to use the utopian objective(also referred as ideal point) defined by:

L? = infpLwk(p),∀k = 1..n (19)

Nevertheless, this utopian objective L? is often unattainableas the costs are conflicting. Next idea is to consider the set{λk} that generates the closest solution to utopian L?. Suchapproach leads to seek for the Pareto optimal compromisegeodesic:

γ = arg minp,{λk}

‖Lwλ(p)− L?‖ (20)

IV. PREOPERATIVE EXPERIMENTAL RESULTS

A. Experimental Conditions

The experiments are conducted using a clinical Magnetom c©

Verio 3T MRI scanner (Siemens, Germany) providing real-time capabilities. Magnetic microrobots are made of steel beadof different sizes. They are placed either in an acrylic boxfor sensing characterization, or in a vascular phantom forMRN evaluation. The phantom is made of two PMMA platesthermally bonded together, and has been divided into differentchannels with multiple bifurcations. The general characteris-tics are given in table I. The vascular phantom was filled withdeionized water, and placed inside the linear gradient volumeof the MRI bore on the x-z (coronal) plane, as shown in Fig. 4.The Table II summarizes the relevant parameters considered inthe experiments.

w

l

x

z300mm

(a) (b)

(c)

x

z

y

y

Fig. 4. Experimental setup that comprises: (a) Magnetom Verio 3T MRIscanner (Siemens, Germany) with the acrylic box which contains the vascularphantom (b). The inset (c) illustrates a typical MR-imaging of the phantom.

TABLE IGEOMETRIC CHARACTERISTICS OF THE VASCULAR PHANTOM.

Size (mm) w (mm) l (mm) Depth (mm)300× 300 8.1–12.3 30–300 6.0

TABLE IIEXPERIMENTAL CONDITIONS.

Parameter ValuesMicrorobot radii r 750 µm(steel ball) density ρ 7850 kgm−3

magnetization msz 1.6× 106 Am−1

Water viscosity η 1.005× 10−3 Pa s

Fig. 5 illustrates the gradient pulse generated during apropulsion step in the x-direction followed by a slice acqui-sition using a fast low angle shot (FLASH) sequence. Thepropulsion gradient is set to ∇bg = (20, 0, 0)T mT m−1 forTProp = 640 ms. The acquisition parameters are a repetitiontime of Tr = 7.5 ms and echo time of Te = 4.8 ms, with animage resolution of 256 × 256 and a halfed k-space, leadingto an acquisition time of TAcq = 960 ms.

To magnetically propel the microrobot to its destination,different control strategies could be considered such as simple

-20

-10

0

10

20

-20

-10

0

10

20

0 0.5 1 1.5-20

-10

0

10

20

(mT/m

)(m

T/m

)(m

T/m

)

Tprop Tsync Tacq

Gz

Gy

Gx

Gslice

Gphase

Greadout

Time (sec)

Fig. 5. Magnetic gradient sequences for propulsion and acquisition duringone time cycle TCycle. The x-y-z magnetic gradient components duringthe propulsion (TProp), synchronization (TSync) and acquisition (TAcq)sequences are reported.

proportional-integral-derivative (PID) [6], adaptive backstep-ping control schemes [18] or predictive [17]. The capabilitiesof the MRI scanner to actuate properly a given magneticmicrorobot are related to different factors. First, the dutycycle τ =

TProp

TCycleof the time multiplexing scheme directly

influences the microrobot motion and then the controllerperformance. In a previous study [17], we have shown thatan optimum duty cycle is obtained for τ = 40%, whichis similar to the one reported in [6]. The microrobot mag-netic properties also influence the MRI propulsion capabilitiesthrough the magnetic force given by Eq. (4). Moreover, theimaging part of the MRI sequence also produces magneticgradient pulses which do not necessarily sum up to zero. Forthe previous example depicted in Fig. 5, the mean magneticgradient values are (31.20, 0, 0)TmT m−1 for the propulsionphase TProp, whereas during the imaging phase TAcq it was(−2.49, 0.21, 1.65)TmT m−1. The gradient fields generatedduring the sensing phase does not appear negligible. However,as the imaging sequence does not change significantly duringthe overall navigation process, the induced force can be treatedas a constant external perturbation in the control scheme.

B. MRI Sensing Capability

Here, to compute the Shannon entropy H(x|y), we assumethat the sensing process could be modeled by a white bivariate-Gaussian distribution: N (0,Σy), with zero mean and Σy

the covariance matrix (see [11]). This covariance matrix iscomputed from the standard deviation considering:

Σy = diag(σ2x, σ

2z) (21)

where σx and σz are the standard deviations along the x andz-axis respectively. To consider microrobots of any size, it isnecessary to develop an analytical model of the covariancematrix that fits the measured sensing capability (see [11]).To do so, Fig. 6 shows a third order polynomial interpolationfitting suitably the measured standard deviations (circle and

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

Microrobot radius (mm)

STD

(m

m)

Meas. σx

Meas. σz

Interp σx

Interp σz

Fig. 6. Polynomial interpolation of the standard deviations (std) of the MR-tracking of steel ball microrobots using a FLASH sequence.

square markers) wrt. the microrobot size. It should be noticedthat the standard deviations scale favorably until to reach alow plateau value when the microrobot size is below 1 mm.

C. MRI Steering Capability Evaluation

To evaluate the MRN propulsion capability, an imaging andpropulsion sole sequence is applied on a spherical micro-robot with a radius of r = 750 µm. As in Section II-A, thesteering gradient has been set to ∇b = (20, 0, 0)T mT m−1

for TProp = 640 ms. Similarly, the sensing process usesthe same optimized FLASH sequence as it provides a goodcompromise in terms of accuracy and acquisition time. Fig. 5depicts the applied magnetic gradient sequence for propulsionand acquisition for one cycle time TCycle. Fig. 7 shows thecorresponding velocities of the magnetic microrobot that hastraveled a distance of 243.4 mm during TCycle. Moreover,during the acquisition step, the velocity of the microrobotdoes not appear to be affected by the MR-imaging sequence.This result demonstrates the effectiveness of the trackingand propulsion sequence of a magnetic microrobot througha clinical MRI scanner.

D. Preoperative MRN Planning

1) Single Objective Navigation Planning: The proposedmagnetic resonance navigation (MRN) planning frameworkhas been first evaluated considering each presented singleobjective. More precisely, the goal here is to retrieve thecenterline, γc, power efficient, γp, and the belief, γb, geodesics.Furthermore, the sensing uncertainty model is designed uponFLASH sequences as it provides the best sensing capabilities.

Firstly, a highly magnetic stain-less steel bead microrobotwith a radius of r = 750 µm is assumed to navigate inthe vascular phantom filled with water (ie., a static flow),with the experimental condition parameters summarized inTable II. The corresponding metric maps (wc , wp and wb)are calculated and are illustrated in Fig.8. Fig.9a illustrates anavigation case starting from x03 and ending to x3 locations.Table III presents the geodesic lengths Lw(γ), computed fromEq (10), for each navigation path. considering a unit (i.e., thatallows retrieving the Euclidean path arc-length L1), centerline

Tprop Tsync Tacq Tsync

v (

mm

/s)

Time (sec)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0

20

40

60

80

100

120

140

1600

(a)

(b) (c)

Fig. 7. Time-multiplexing and MRI-based actuation: (a) velocities of aspherical magnetic microrobot with a radius r = 750 µm; (b) initial and(c) final position of the magnetic microrobot.

Low High

(a) (b) (c)

Fig. 8. The calculated metrics: (a) the centerline (wc); (b) the power (wp);and (c) the sensing uncertainties (wb).

x3 γcγpγb

x03

(a)

γcγpγbstreamline

x03

x3

(b)

Fig. 9. Optimal navigation paths with the centerline (γc), the power efficient(γp), and the belief (γb) metrics. (a) A steel ball with a radius of r = 750 µmnavigating in a static flow; and (b) a steel ball with a radius of r = 1500 µmin a continuous flow with a velocity ‖v‖ = 20mms−1.

wc, power wp, and belief wb metrics. As expected, each

0.25 0.5 0.75 1 1.25 1.5 1.75 20

101520

30

40

50

60

70

80

90

Ball radius (mm)

L ωb

Failure

(a)

10 15 20 25 30 35 40 45 5040

50

60

70

80

90

100r=0.50 mm

r=0.75 mm

r=1.00 mm

r=1.25 mm

r=1.50 mm

r=1.75 mm

r=2.00 mm

Flow velocity (mm/s)

L ωp

(b)

0.4 0.5 0.6 0.7 0.8 0.9 120

22

24

26

28

30

32

34

36

Uncluttered

L ωc

(c)

Fig. 10. Evolution of the metrics: (a) belief energy functional Lwb as function of the steel ball radius. (b) Power efficiency energy functional Lwp as functionof the flow magnitude for different steel ball radii. (c) Centerline efficiency energy functional Lwp as function of the uncluttered environment.

geodesic provides the minimal length with its own metricagainst the others objective functions. In the case illustratedin Fig.9b, the geodesics are calculated when a r = 1500 µmsteel bead microrobot is navigating in presence of a steadyflow (‖v‖ = 20 mm s−1) simulating the arterial blood velocity.Fig.9b depicts the relevant streamline of this virtual flow. Asexpected, solely based on spatial information, the centerline,wc, metric gives exactly the same solution as the previouscase. In contrast, the belief, γb, and the power efficient, γp,geodesics provide new solutions. It should be noticed that thechange of the size of the microrobot leads to a change of thecomputed information gain which is related to the standarddeviation of the MRI sensing process (see also Fig. 6), to-gether with a change in its location belief. Actually, when alarger microrobot is considered the environment uncertainty isincreasing, especially in smaller channels. Considering this,we investigated experimentally the evolution of the energyfunctional Lwb

as function of the microrobot’s size (Fig.10a).The belief geodesic γb solution scales down favorably forrobot sizes less than 1mm but is strongly limited for robotsizes bigger than 2mm due to the strong artifact signature.The circulating fluid flow plays an important role on the powergeodesic, γp. The navigation pathway tends to follow thecirculating flow in order to minimize the energy expenditure.As illustration, Fig. 10b shows the power efficiency energyfunctional, Lwp

with respect to the flow magnitude for differentmicrorobot sizes. As the robot size scales down, the requiredpower increases drastically in presence of fluid flow failing tofind a geodesic, γp, solution for flow velocities values greaterthan 50 mm s−1. The power efficiency energy functional γpvalues are found nearly constant for velocity values rangingfrom 10 mm s−1-to-50 mm s−1 for robot sizes greater than1mm. Finally, the euclidean shortest path shown in Fig.10cis generally obtained using the centerline efficiency energyfunctional, γc, as the corresponding geodesic is less sensitiveto wall effects [15]. In conclusion, optimal navigation path-ways can be predicted when considering a tradeoff betweenthe efficiencies of different cost functions according to thecorresponding MRN experimental conditions.

2) Multi-Objective Planning:

TABLE IIITHE LENGTHS OF EACH GEODESICS CONSIDERING THE UNIT (w1),

CENTERLINE (wc), POWER (wp) AND BELIEF (wb) METRICS, FOR THEDIFFERENT CASES SHOWN IN FIG.9.

Cases L1 (mm) Lwc Lwp Lwb

Fig.9a γc 383.621 29.560 21.081 18.152

x03 → x3γp 472.331 35.567 19.930 21.759

γb 319.869 33.936 20.488 15.425

Fig.9b γc 383.621 29.560 126.164 17.143

x03 → x3γp 725.456 95.623 45.506 38.384

γb 357.458 43.534 103.567 10.238

a) Bi-objective Optimization: The simplest multi-objective case is to consider two objectives that have to beoptimized. Using the weighted sum method, the cost functionsare simply aggregated as follows:

wλ = (1− λ)wx + λwy, ∀λ ∈ [0; 1] (22)

with wx and wy one of proposed metric, that is wc, wp or wb

(see also Table IV).Fig. 12 shows the Pareto front between the considered cost

functions. For the sake of consistent comparison betweenobjectives, each geodesic length Lw are here normalized. Inparticular, a cost of 1 is the worst solution whereas 0 leadsto the best one. Fig. 12a presents the Pareto front related tothe normalized length of geodesics starting from point x01 totargeted location x1 corresponding to the case (1) in Fig.11.Each bi-objective problem seems to provide a convex Paretofront, but with different curvatures. Recalling that all solutionson the Pareto frontier (the circles in Fig. 12) are optimalaccording to the objective wλ (22), one can choose any ofthem. These results show that objectives of centerline wc andbelief wb have the most influence against power metric.

Moreover, in the case (2), x02 → x2, illustrated in Fig.11objective wc provides a different solution than other costfunctions. The distinctive solutions appear clearly in Fig. 12bfor centerline wc vs. belief wb with a jump in the Paretofront. This gap is due to the fact that the geodesic couldbe below (the group {1} in Fig. 12b) or above (the group{2} in Fig. 12b) the obstacle. For centerline wc vs. power

x1γcγpγb

(1)

γcγpγb

(2) x01

x02

x2

Fig. 11. Optimal navigation path with centerline (γc), the power efficient(γp), and the belief (γb) metrics considering different starting (x01, x02) andending (x1, x2) locations

0 0.5 10

0.5

1

0 0.5 10

0.5

1

0 0.5 10

0.5

1

Belief

Belief

Power

Compromise

utopia Centerline Power Centerline(a)

utopia Centerline Power Centerline

Belief

Belief

Power

0 0.5 10

0.5

1

0 0.5 10

0.5

1

0 0.5 10

0.5

1

gap

0 0.05

0

0.05

0.1{2}

{1}

gap

{2}{1}

(b)

Fig. 12. Bi-objective Pareto Front: (left) centerline wc vs. belief wb; (center)power wp vs. belief wb; and (right) centerline wc vs. power wp objectives,from (a) x01 to x1 and (b) x02 to x2. Circles depict the Pareto non-dominated solutions, and the square box the compromise solution wrt. theutopian solution (diamond).

wp objectives, it is more difficult to characterize the switchbetween the solutions. In-depth analysis brings up a gap froma weight of λ > 0.7, and is illustrated in the zooming inset inFig. 12b. This kind of switch is also related to a non-convexityin the Pareto front. Moreover, as mentioned, among the Paretonon-dominated solutions, the compromise weight, λ, is theclosest to the utopia point (0, 0). Table IV summarizes thenormalized length of the compromise navigation path. Thesesolutions could offer a suitable choice as it entails minimizingthe difference between potential optimal points.

TABLE IVNORMALIZED LENGTH OF THE COMPROMISE GEODESIC FOR

BI-OBJECTIVE OPTIMIZATION.

wλ λ Lwc Lwp Lwb

(a) (1− λ)wc + λwb 0.825 0.385 0.859 0.278

x01 → x1 (1− λ)wp + λwb 0.95 0.741 0.088 0.060

(1− λ)wc + λwp 0.855 0.216 0.181 0.495

(b) (1− λ)wc + λwb 0.85 0.010 0.828 0.132

x02 → x2 (1− λ)wp + λwb 0.75 0.220 0.098 0.106

(1− λ)wc + λwp 0.617 0.001 0.002 0.261

b) Three-objective Problems: The three proposed metricscan be aggregated together as follows:

wλ = (1− λ2)(1− λ1)wc + λ2(1− λ1)wp + λ1wb (23)

with λ1 and λ2 ∈ [0; 1]. When considering more than twoobjectives, the non-dominated solutions is the convex hull ofthe Pareto optimal surface that is explored as the weight set{λ1, λ2} is varied. Actually, with more than two objectivesthe Pareto frontier is more difficult to handle and to illustrateeasily. The compromise solution is located at λ1 = 0.52 andλ2 = 0.95, and provides the following normalized geodesiclength: Lwc

= 0.362, Lwp= 0.106 and Lwb

= 0.288. Thecorresponding navigation path is then the one that minimizesthe geodesic distance to the utopian point, and thus allows tobest fit the three objectives.

3) Discussions: From the multi-objective optimizationproblems considering the case Fig.9(a), x01 → x1, it appearstwo distinct interpretations. First, the compromise weight λ2is close to 1. It supposes that the centerline metric wc does nothave a great influence in the three-objective problem given in(23). This interpretation appears to be consistent wrt. resultsfrom bi-objective problems, where the compromise weights λare also close to 1. From the compromise weights analysisit appears that the centerline objective has less significancewrt. the other metrics. Actually, results from single objectiveshow that globally each geodesic remains not far from thecenterline pathway. Furthermore, this interpretation shows thatthe belief objective, wb, is the fully satisfied. Secondly, fromthe investigation of the normalized geodesic length at thecompromise weight, λ, location, it appears that the powercost function wp seems satisfied, as it has a low geodesiclength, Lwp

= 0.106. In contrast, centerline objective has thelargest length, Lwc

= 0.362, which could be interpreted by alower satisfaction than the other objectives at the compromisesolution. In addition, objectives of power and centerline appearclose to each other, since their compromise solution is closeto the utopia. A similar analysis between objectives of powerand belief allows us to consider that power objective is theless relevant for the case (1), x01 → x1, shown in Fig.11.Actually, in the considered multi-objective cases, the magneticmicrorobot navigates in a steady flow, and thereby the dragforce does not induce a significant constraint, particularlyagainst the uncertainty constraint. In such context, the mainobjectives are to satisfy first the metrics of belief, and thenthe centerline goals where the fExt can be neglected [15]. Thepower metric has a more important impact when the fluid

flow implies significant forces on the microrobot dynamics(see Fig.9b).

The presence of distinctive solutions while the weight isvarying (such as in the case (2) x02 → x2 depicted in Fig.11)is related to a non-convexity in the Pareto frontier. It is thenmore difficult to fully handle an ideal solution. Especially, thewell-known drawbacks of the weighted sum method (18), arei) that the optimal solution distribution is not uniform; and ii)that optimal solutions in non-convex regions are not detected.More advanced multi-objective optimization techniques haveto be used [22].

V. MAGNETIC RESONANCE NAVIGATION EXPERIMENTS

The proposed MRN framework has been experimentallyvalidated using the Magnetom c© Verio 3T clinical MRI plat-form (see Fig. 4). The considered MRN is defined as a motionaround the vascular phantom. A spherical microrobot of radiusr = 750 µm has been introduced into the phantom setupand placed in the focus of the MRI bore. Propulsion andsensing MRI sequences are similar to the paragraph IV-C.As mentioned, uncertainties are here the most significantconstraints. Therefore, the MRN planning is reduced to thebi-objective problem (22) considering the centerline wx = wc

and belief wy = wb metrics. Moreover, a compromise weightof λ = 0.825 is used to compute automatically the navigationpath, as it is the one that best meets the two objectives. Togenerate a complex pathway, several waypoints are defined inthe vascular-like environment. Then, the magnetic microrobothas to follow the predefined reference path using a simpleproportional-integral (PI) controller. To guide the microrobotto follow the reference path the magnetic gradient depicted inFig. 13 is generated with the coils of the MRI scanner. Forthe sake of clarity only the propulsion gradient sequences arereported here. As shown in Fig. 14 and in the supplementaryvideo, the MRN system is able to steer efficiently the micro-robot along the reference path. First, from its initial position(xinit) the device has to converge to the reference path. Oncethe geodesic is reached, the microrobot remains globally closeto it. These results, with very few user interaction, validate theproposed framework as a step towards autonomous MRN.

Moreover, let us recall that the beliefs metric wb is ameasurement of the uncertainties on the microrobot tracking.Hence, wb helps to predict the covariance matrix Σy in thesensing process. The ellipses in Fig. 14 describe the predictionof the Σy along the measured position from the sensingmodules. Red ellipse is related to high uncertainty, while blueellipse to low uncertainty. Based on the dynamic model (7) (cf.Section II-B), a Kalman filter (KF) can be used to estimatethe microrobot position. Particularly, in robotics, the KF isthe most suited for tracking, localization, and navigationproblems. Indeed, the KF works best i) with well-defined statedescriptions, where observation and time-propagation modelsare well understood; and ii) where all errors are Gaussiandistributions. Fig. 14 also illustrates the measured position andthe KF corrected localization. Errors between the referencepath and measurements, as well as wrt. the KF-tracked positionare presented in Fig. 15. With the MR-tracked measured

∇bx

∇bz

∇b

(m

T/m

)

Time (sec)0 5 10 15 20 25 30

-15

-10

-5

0

5

10

15

Fig. 13. Generated magnetic gradient to steer the magnetic microrobot alongthe reference path.

Ref

Waypoints

xinit

Measured

σx

σz

Σy

2

2

KF Tracked

High

Low

x0

End

(Ref. start)

Com

pro

mis

e m

etr

ic ω

λ

Fig. 14. 2D Magnetic Resonance Navigation along a reference path of aspherical magnetic microrobot of radius r = 750 µm. The image backgroundshows the compromise metric: (blue) low metric value and (red) high value.

Err

or

(m

m)

Time (sec)0 5 10 15 20 25 30

0

0.5

1

1.5

2

2.5

||Ref-Meas.||

||Ref - KF||

||Meas.-KF||

|Σ|

Conve

rgence

Fig. 15. Errors between the reference path and the measurements, and againstthe Kalman filtered localization.

value the average error is 1.249 mm, whereas consideringKF-correction the mean error is 0.851 mm. These resultssuggest that the KF correction provides a good estimate ofthe microrobot localization thanks to a suitable predictionof the microrobot’s belief in its location. The microrobottracking error appears greater than those considered in thesensing capability evaluation (cf. paragraph IV-B). Actually,

0

2

4

6

8

10

0 5 10 15 20 25 30

KF observ.

KF tracking

Meas.

Time (sec)

||v||

(m

m/s

)

Fig. 16. The magnetic microrobot velocities.

the tracking algorithm is here designed upon a template stackof static artifacts. However, when the magnetic microrobot ismoving, the artifact signature leaves a magnetic remnant oftrail, as shown in Fig. 7c.

In addition, according to the system dynamic model (7) theKF allows the observation of the entire systems state (v,x)

T

which includes the microrobot velocity v. Fig. 16 shows themotion of the microrobot computed from the measurements,the KF-corrected localization and the KF predicted velocitystate. Once again, the use of the system’s model togetherwith an a priori knowledge of the uncertainty map improvessubstantively the MRN efficiency. Therefore, this could be afirst step towards the design of an MRN-based medical SLAM(simultaneous localization and mapping) approach.

VI. CONCLUSIONS

In this study, an optimal magnetic resonance navigationframeworks integrated within a clinical MRI scanner arepresented. The basic sensing and actuation modules are firstevaluated to exhibit the MRN system capabilities to trackand magnetically steer through the magnetic field gradientgenerated from the coils of a conventional MRI. From preop-erative MRI data and system models, comprising the sensinguncertainties and the system dynamics, a robust and optimalnavigation path is planned by addressing multi-objective prob-lem. A deep analysis of the different objective functions and ofthe multi-objective navigation planning problem is presented.Especially, the weight set could be automatically selected byconsidering the compromise solution. The reliable pathwayis then used to efficiently guide a magnetic microrobot to atargeted location. The overall MRN architecture functionalitieshas been experimentally verified using an integrated clinicalMRI system. These experimental realization demonstrate thatit is effectively possible to autonomously guide a magneticmicrorobot using a conventional MRI scanner with only asoftware upgrade. Further investigation will address 3D andin vivo experimental validation.

VII. ACKNOWLEDGMENTS

The authors would like to acknowledge the Dr A. Kluge(Pius Hospital, Oldunburg, Germany) and Prof. S. Fatikov(Division Microrobotics and Control Engineering, Universityof Oldenburg, Germany) for their supports on the use of the

clinical MRI data. This work was supported by EuropeanUnions 7th Framework Program and its research area ICT-2007.3.6 Micro/nanosystems under the project NANOMA(Nano-Actuactors and Nano-Sensors for Medical Applica-tions).

REFERENCES

[1] P. Dario, M. C. Carrozza, A. Benvenuto, and A. Menciassi, “Micro-systems in biomedical applications,” J. of Micromechanics and Micro-engineering, vol. 10, no. 2, p. 235, 2000.

[2] B. J. Nelson, I. K. Kaliakatsos, and J. J. Abbott, “Microrobots forminimally invasive medicine,” Annual rev. of biomed. eng., vol. 12, pp.55–85, 2010.

[3] M. P. Kummer, J. J. Abbott, B. E. Kratochvil, R. Borer, A. Sengul, andB. J. Nelson, “Octomag: An electromagnetic system for 5-dof wirelessmicromanipulation,” IEEE Trans. Robot., vol. 26, no. 6, pp. 1006–1017,2010.

[4] Z. Zhang, F. Long, and C.-H. Menq, “Three-dimensional visual servocontrol of a magnetically propelled microscopic bead,” IEEE Trans.Robot., vol. 29, no. 2, pp. 373–382, 2013.

[5] S. Martel, J.-B. Mathieu, O. Felfoul, A. Chanu, E. Aboussouan,S. Tamaz, P. Pouponneau, L. Yahia, G. Beaudoin, G. Soulez et al.,“Automatic navigation of an untethered device in the artery of aliving animal using a conventional clinical magnetic resonance imagingsystem,” Appl. Phy. Lett., vol. 90, no. 11, p. 114105, 2007.

[6] S. Tamaz, A. Chanu, J.-B. Mathieu, R. Gourdeau, and S. Martel,“Real-time MRI-based control of a ferromagnetic core for endovascularnavigation,” IEEE Trans. Biomed. Eng., vol. 55, no. 7, pp. 1854–1863,Jul. 2008.

[7] K. Belharet, D. Folio, and A. Ferreira, Real-time software platform forin vivo navigation of magnetic micro-carriers using MRI system, ser.Biomaterials. Cambridge: Woodhead Publishing, Oct. 2012, no. 51,ch. 11.

[8] S. Martel, “Magnetic navigation control of microagents in the vascularnetwork: Challenges and strategies for endovascular magnetic navigationcontrol of microscale drug delivery carriers,” IEEE Contr. Syst., vol. 33,no. 6, pp. 119–134, 2013.

[9] P. Pouponneau, G. Bringout, and S. Martel, “Therapeutic magnetic mi-crocarriers guided by magnetic resonance navigation for enhanced liverchemoembilization: A design review,” Annals of biomedical engineering,vol. 42, no. 5, pp. 929–939, 2014.

[10] A. Bigot, C. Tremblay, G. Soulez, and S. Martel, “Magnetic resonancenavigation of a bead inside a three-bifurcation pmma phantom using animaging gradient coil insert,” IEEE Trans. Robot., vol. 30, no. 3, pp.719–727, Jun. 2014.

[11] C. Dahmen, K. Belharet, D. Folio, A. Ferreira, and S. Fatikow, “MRI-based dynamic tracking of an untethered ferromagnetic microcapsulenavigating in liquid,” Int. J. of Optomechatronics, vol. 10, no. 2, pp.73–96, 2016.

[12] B. A. Chronik, A. Alejski, and B. K. Rutt, “Design and fabrication of athree-axis edge ROU head and neck gradient coil,” Magnetic resonancein medicine, vol. 44, no. 6, pp. 955–963, 2000.

[13] J. A. McNab, B. L. Edlow, T. Witzel, S. Y. Huang, H. Bhat, K. Heberlein,T. Feiweier, K. Liu, B. Keil, J. Cohen-Adad et al., “The human con-nectome project and beyond: Initial applications of 300mt/m gradients,”NeuroImage, vol. 80, pp. 234–245, 2013.

[14] J. A. Sethian, Level set methods and fast marching methods: evolvinginterfaces in computational geometry, fluid mechanics, computer vision,and materials science. Cambridge university press, 1999, vol. 3.

[15] K. Belharet, D. Folio, and A. Ferreira, “Simulation and planning of amagnetically actuated microrobot navigating in arteries,” IEEE Trans.Biomed. Eng., vol. 60, no. 4, pp. 994–1001, Apr. 2013.

[16] L. Arcese, M. Fruchard, and A. Ferreira, “Endovascular Magnetically-Guided Robots: Navigation Modeling and Optimization,” IEEE Trans.Biomed. Eng., vol. 59, no. 4, pp. 977–987, 2012.

[17] K. Belharet, D. Folio, and A. Ferreira, “Three-dimensional controlledmotion of a microrobot using magnetic gradients,” Adv. Robotics,vol. 25, no. 8, pp. 1069–1083, 2011.

[18] L. Arcese, M. Fruchard, and A. Ferreira, “Adaptive controller andobserver for a magnetic microrobot,” IEEE Trans. Robot., vol. 29, no. 4,pp. 1060–1067, Aug. 2013.

[19] H. Marino, C. Bergeles, and B. J. Nelson, “Robust electromagneticcontrol of microrobots under force and localization uncertainties,” IEEETrans. Autom. Sci. Eng., vol. 11, no. 1, pp. 310–316, 2014.

[20] H. Takeda, C. Facchinetti, and J.-C. Latombe, “Planning the motionsof a mobile robot in a sensory uncertainty field,” IEEE Trans. PatternAnal. Mach. Intell., vol. 16, no. 10, pp. 1002–1017, 1994.

[21] N. Roy and S. Thrun, “Coastal navigation with mobile robots,” Adv. inNeural Processing Syst., vol. 12, no. 12, pp. 1043–1049, 1999.

[22] R. T. Marler and J. S. Arora, “Survey of multi-objective optimizationmethods for engineering,” Structural and multidisciplinary optimization,vol. 26, no. 6, pp. 369–395, 2004.