non-ideal behaviors of magnetically driven screws in soft ... · non-ideal behaviors of...

Non-ideal Behaviors of Magnetically Driven Screws in Soft Tissue

Arthur W. Mahoney, Nathan D. Nelson, Erin M. Parsons, and Jake J. Abbott

Abstract— This paper considers the requirements for stablecontrol of simple devices that consist of a permanent magnetrigidly attached to a screw operating in soft tissue. Placingthese devices in a rotating magnetic field causes the device’spermanent magnet to rotate synchronously with the appliedfield, making the attached screw generate forward propulsion.Although there has been significant research in the area ofmagnetic screws, this paper describes magnetic phenomonenathat have not been addressed in prior work, yet will be criticalin the design of future steering and control strategies formedical applications. Using a time-averaged model of magnetictorque, we analyze the magnetic torque present when inten-tionally steering the microrobot or due to naturally occurringdisturbances in the microrobot’s heading. We predict andexperimentally demonstrate the existence of magnetic torqueswhich simultaneously act to stabilize and destabilize screw-likemicrorobots operating in tissue. We find that there exists anapplied field rotation speed, above which screw-like microrobotswill become unstable, and we find that the interaction betweenthe stabilizing and destabilizing magnetic torques may causepotentially undesired but predictable behavior while steering.

I. INTRODUCTION

Microrobots have become an active area of research in thefield of minimally invasive medicine due to their potentialfor locomotion and manipulation in hard-to-reach areas ofthe human body [1]. Many applications of interest willrequire the microrobot to locomote through soft tissue.These may include radioactive seed positioning for prostatebrachytherapy, positioning a microrobot within the brain forhyperthermia treatments, and navigating through the vitreoushumor of the eye to perform therapy of the retina such astargeted drug delivery [2]. These devices may range in sizedepending on the needs of the application, however we usethe term “microrobot” herein.

One form of microrobot that is showing promise for med-ical applications are those actuated using forces and torquesproduced by magnetic fields. The helical microswimmer,designed to mimic the motion of bacterial flagella, whichconsists of a rigid helix attached to a magnetic element,is one such device [3], [4]. When placed in a rotatingmagnetic field, a torque is applied to the device through themagnetic head, which causes the attached helix to rotate andproduce fluidic propulsion. Helical microswimmers operatewell in the low-Reynolds-number regime where viscous dragdominates inertia [5]. Because the helix of the microswim-mer does not displace the medium around the magnetic

This material is based upon work supported by the National ScienceFoundation under Grant Nos. 0952718 and 0654414.

A. W. Mahoney is with the School of Computing, and N. D. Nelson,E. M. Parsons, and J. J. Abbott are with the Department of MechanicalEngineering, University of Utah, UT 84112, USA.

email: [email protected]

head, helical microswimmers are not well suited for movingthrough soft tissue. Instead, similar devices that consist ofa magnetic element attached to a solid screw, rather than ahelix, have been proposed for use in tissue [6]–[10].

In [6], the authors introduce a device of 11.5 mm in lengthconsisting of a 7.5 mm-long diametrically magnetized perma-nent magnet attached to a 4.0 mm-long brass wood screw.Operating in agar gel to simulate tissue, the authors showthat the device’s forward velocity is linearly related to therotation frequency of the applied field, and they demonstratethat their device can be used in real tissue by driving itthrough a 25 mm-thick sample of bovine tissue. In [7], theauthors demonstrate that by pitching the rotation axis of therotating applied field out of alignment with the principal axisof the microrobot, the devices can be made to steer toward adesired direction in agar gel. In their experiments, they findthat the smallest turning radius can be obtained with theirshortest device when the applied field rotates at 0.1 Hz andthe rotation axis of the applied field is kept pitched awayfrom the principal axis of their screw microrobot by 60

toward the direction of the desired turn. The authors showthat the average magnetic torque causing their device to turnis maximized in this configuration, and that applying torquein this manner generates turns of highest curvature. Otherresearch on screw-like microrobots has explored variousscrew-tip cutting mechanisms [8], trailing a wire [9], localhyperthermia treatment using inductive heating [10], andusing screw devices as pH sensors [11].

This paper continues on the work of [6] and [7] towardstable control of screw-like microrobots in soft tissue. In thisstudy, we numerically analyze the time averaged magnetictorque that causes the microrobot to pitch in the direction ofthe applied field’s axis of rotation, present when intentionallysteering the microrobot or due to naturally occurring distur-bances in the microrobot’s heading. We show that not only isthere a stabilizing magnetic torque that causes the microrobotto realign with the rotation axis of the applied field asdesired, but there also always exists a competing undesireddestabilizing magnetic torque that simultaneously causes themicrorobot to turn perpendicular to the desired direction ofalignment. We demonstrate that there exists a rotation speedof the applied field above which the destabilizing torque willalways be greater in magnitude than the stabilizing torque.This can cause the microrobot to become unstable anddiverge away from the desired trajectory. This phenomenonis not documented in prior literature and must be taken intoaccount when designed future steering and control strategiesfor medical applications.

CONFIDENTIAL. Limited circulation. For review only.

Preprint submitted to 2012 IEEE/RSJ International Conferenceon Intelligent Robots and Systems. Received March 9, 2012.

B

m Ω

Ψ

x

y

z

φ

θ

pitc

h

yaw

Fig. 1. The principal axis of the microrobot is constrained to lie parallel tothe z robot axis making the dipole moment of the microrobot m lie in thex-y microrobot plane. The applied magnetic field (magnetic flux density) Brotates around and is perpendicular to the rotation axis Ω. The microrobotis steered in space by pitching Ω away from the microrobot’s principal axisby the angle ψ toward the desired direction of steering.

II. MODELING MAGNETIC CONTROL

Let the applied magnetic field B (magnetic flux density)rotate with angular velocity Ω, whose direction and mag-nitude define the applied field’s axis of rotation and speed,and where B is perpendicular to Ω. Define a robot coordinateframe with axes x,y,z as shown in Fig. 1, where the z axisis parallel to the microrobot’s principal axis, the x axis liesin the plane spanned by the z axis and Ω, and the y axis liesin the direction of z× x. This definition breaks down whenz is parallel to Ω, however, this paper is concerned with thecase when z is not parallel to Ω.

The instantaneous magnetic torque τ produced on a dipolemoment m (i.e., the small permanent magnet attached to themicrorobot) by B at the location of m causes m to align withB and is expressed by τ = m×B. For simplicity, we assumethat the principal axis of the microrobot lies parallel to thez robot axis (Fig. 1). We denote the components of τ thatlie parallel to the y and x robot axis to be the stabilizing τ yand destabilizing τx torques, respectively. The componentin the robot z direction, τ z , causes the microrobot to rolland is referred to as the propulsion torque. When Ω isaligned with the principal axis of the microrobot the magnetictorque is purely composed of the propulsion torque andthe components representing the stabilizing and destabilizingmagnetic torques have zero magnitude. The result is a lineartrajectory in the direction of the microrobot’s principal axis.

The microrobot is steered through space by pitching therotation axis Ω of the applied field in the direction of thedesired turn. By the definition of the robot coordinate frame,Ω is rotated about the y robot axis by angle ψ, which werefer to as the “pitch angle.” When Ω is not aligned withthe principal axis of the microrobot, the stabilizing τ y anddestabilizing τx magnetic torques cause the microrobot toalign with and deviate from Ω, respectively. In Newtonian

fluids, the microrobot responds by aligning its principal axiswith Ω over time [3]. In viscoelastic environments, however,the simultaneous presence of stabilizing and destabilizingmagnetic torques introduces complex behavior.

We simplify analysis of the microrobot’s turning behavior,by examining the magnitude of the numerically averagedmagnetic torque in the stabilizing and destabilizing direc-tions over one revolution of the applied field. Averagingis performed using the model discussed herein, which as-sumes the net torque acting upon the microrobot consistsof the magnetic torque generated by the applied field anda linear damping torque. This assumption is often madewhen analyzing microrobots in low-Reynolds-regime fluidswhere fluidic drag dominates inertia [5]. Here, we apply thesame assumption under the belief that any inertial effectsare dominated by the drag caused by friction between themicrorobot and the tissue. There may, in fact, be Coulomb-like friction as well [7], which is unmodeled in this paper.Aside from torque, forces that act on the microrobot in tissueinclude the force due to gravity and any forces acting on themicrorobot due to the mechanics of the tissue itself. Dueto the force acting on the microrobot from the tissue, themicrorobot typically does not sink under the force of gravityas it would in Newtonian fluids [4] and we assume its effectis negligible. The full effect of the forces caused by tissuemechanics remains to be studied.

The model is derived by constraining the microrobot asif it were placed within a lumen such that the stabilizingand destabilizing torques have no effect on the microrobot’sheading and the microrobot only rotates about its principalaxis. The model is simplified in this way because in practice,the microrobot rotates much faster about its principal axisthan it changes its heading when operating in tissue. Whenaveraging over one revolution of the applied field, even withthe presence of the stabilizing and destabilizing torques,the actual change in heading due to these torques is smalland will not significantly alter the analysis of the averagedtorques themselves. If θ and φ are the rotation angles ofthe applied field and microrobot dipole moment in theirrespective planes measured as shown in Fig. 1, then therepresentations of the applied field and microrobot dipolemoment in the robot frame are given by

B = |B|

cosψ cos θsin θ

sinψ cos θ

(1)

m = |m|

cosφsinφ

0

(2)

With any given pitch angle, orientation of the applied field,and orientation of the microrobot, the magnetic torque actingon the microrobot can by found with τ = m × B, whichproduces

τ = |m||B|

sinψ cos θ sinφ− sinψ cos θ cosφ

sin θ cosφ− cosψ cos θ sinφ

(3)



Under the assumption that torque due to inertia is neg-ligible, the net torque acting upon the microrobot along itsprincipal axis consists of a linear damping torque causedby friction with coefficient b that has magnitude |τ f | = bφ,where φ is the angular velocity of the microrobot, along withpropulsive torque τ z , whose magnitude is found in (3). Sincethe net torque is zero, the differential equation governingthe rotation of the microrobot about its principal axis can befound:

φ =|m||B|b

(sin θ cosφ− cosψ cos θ sinφ) (4)

If we assume that Ω remains constant over time, then (4) canbe easily solved numerically given initial conditions φ(0) andθ(0), since the constant ψ and the variable θ(t) = |Ω|t areknown for all t.

After solving (4) numerically to produce φ(t), the magni-tude of the torque about the y axis and the x axis averagedover one turn of the applied field is produced by integratingthe x and y components of (3) using the numerical solutionsfor φ(t) and θ(t) to obtain φ(θ), which is used in theintegrations

τx =|m||B|

2πsinψ

∫ 2π(k+1)

2πk

cos θ sin (φ(θ)) dθ (5)

τ y = −|m||B|2π

sinψ

∫ 2π(k+1)

2πk

cos θ cos (φ(θ)) dθ (6)

respectively, where k ∈ Z+ is selected to reject any transientresponse due to the initial conditions when solving (4).Although we have observed the initial transient response tobe insignificant after two rotations of the applied field, forthis study we have chosen k = 9 as a conservative lowerlimit of integration with initial conditions φ(0) = 0 andθ(0) = 90.

Fig. 2(a) and Fig. 2(b) show contour plots of the averagestabilizing and destabilizing torques scaled by the maximumpossible net torque |m||B|, with ψ ∈ [0, 90) and the nor-malized rotation speed (defined as |Ω| divided by the step-outspeed when ψ = 0) of the applied field ranging from 1% to100% of the microrobot’s step-out speed when ψ = 0. Thestep-out speed is the maximum speed the applied field canrotate for which the applied field and microrobot will alwaysremain synchronized. This speed changes with the pitchangle of the applied field and is given by |m||B| cos(ψ)/b,with the fastest step-out speed of |m||B|/b occurring whenψ = 0. When the applied magnetic field is rotated at speedsabove step-out, the microrobot tends to exhibit chaotic andunpredictable behavior. Fig. 2(a) and Fig. 2(b) were producedby pitching Ω upward (i.e., ψ > 0). When Ω is pitched down(i.e., ψ < 0), the results of Fig. 2(a) and Fig. 2(b) changesign but Fig. 2(c) is unchanged.

Fig. 2(a) shows the average stabilizing torque that causesthe microrobot to turn toward Ω increases with ψ up tothe curve labeled “Max Torque Ridge” for any normalizedapplied field rotation speed with the global scaled maximumof 0.35 occurring at normalized rotation speed of 0.13 andpitch angle of 60.0 (which corresponds to the same result

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

0.34

0.29

0.24

0.19

0.140.090.04

Step-out Speed

Max Stab. TorqueMax Torque Ridge

Normalized Applied Field Speed,|m||B||Ω|b

Pitc

h A

ngle

, Ψ

(de

g)

(a) Average scaled stabilizing torque, τy/|m||B|. The maximumstabilizing torque occurs at a normalized rotation speed of 0.13 andpitch angle 60.0. The curve “Max Torque Ridge” describes the pitchangle necessary to maximize stabilizing torque for each normalizedapplied field rotation speed.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

Normalized Applied Field Speed,

0.05

0.10

0.15

0.20

0.25

0.30

|m||B||Ω|b

Step-out SpeedMax Stab. Torque

Pitc

h A

ngle

, Ψ

(de

g)

(b) Average scaled destabilizing torque, τx/|m||B|.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

0.33 0.66 1.00 1.33 1.66 2.00

Step-out SpeedMax Stab. Torque

Normalized Applied Field Speed,|m||B||Ω|b

Unstable

Pitc

h A

ngle

, Ψ

(de

g)

(c) Ratio of the average destabilizing torque to average stabilizingtorque, τx/τy . The region where the microrobot becomes unstable islabeled and shaded.

Fig. 2. The average stabilizing torque (a), average destabilizing torque(b), and the ratio between them (c), as a function of the pitch angle andnormalized applied field rotation speed. The point where the maximumstabilizing torque occurs is shown in all three plots for reference.



5.6mm

1.4mm

(a) (b) (c)

x z

y

Fig. 3. The Helmholtz coils (a), the screw microrobot (b), and the container(c) used for experiments presented in this paper. The microrobot is 5.6 mmlong and 1.4 mm in diameter and consists of a 1 mm cube permanentmagnet rigidly attached to brass screw with 0.87 mm pitch. The static worldcoordinate frame used in experiments is shown in (c).

found by [7]). As the average stabilizing torque increases,Fig. 2(b) shows that the average destabilizing torque tendsto increase as well, and can become larger than the stabilizingtorque. The region of Fig. 2(c) where the ratio betweenthe average destabilizing and stabilizing torques is largerthan one corresponds to configurations where the averagedestabilizing torque is larger than the average stabilizingtorque. When operating in this region, the magnetic torquewill produce more rotation of the microrobot away from thantoward Ω. If the microrobot’s heading is deviated from Ω,due to naturally occurring disturbances, then the microrobotmay become unstable when the average destabilizing torqueis large. Operating the microrobot so that the average stabi-lizing torque is larger than the average destabilizing torquewill enable the microrobot to reject disturbances, realigningfaster when the stabilizing torque dominates the destabilizingtorque.

From Fig. 2(b), the averaged destabilizing torque is onlyzero when ψ = 0 or |Ω| = 0. This means that while steeringthe microrobot by increasing or decreasing ψ, there willalways be a destabilizing torque present that causes the mi-crorobot to yaw in the direction perpendicular to the desiredturn. When the desired turn direction is up (i.e., ψ > 0), thedestabilizing torque will cause the microrobot to yaw aroundthe positive x robot axis according to the right-hand rule.Without loss of generality, when the desired turn directionis down in Fig. 1 (i.e., ψ < 0), the destabilizing torquecreates yaw around the negative x robot axis. Although themicrorobot will always have the tendency to yaw, its affectcan be reduced by operating in regions of Fig. 2(b) wherethe average destabilizing torque is small (i.e. at small anglesof ψ and/or at slow speeds). When operated in this manner,the resulting trajectories tend to have have uniform curvature[7].

III. EXPERIMENTAL RESULTS

The experimental setup used to generate controlled mag-netic fields consists of three nested sets of Helmholtz coils(described in detail in [4] and shown in Fig. 3(a)) arrangedorthogonally such that the magnetic field in the center of thecoils can be assigned arbitrarily, with each Helmholtz paircorresponding to one basis direction of the magnetic field.Each set of Helmholtz coils generates a magnetic field that isapproximately uniform in the center of the workspace, with

5 10 15 20 25 30 35 40 455

10

15

20

25

0 4 8

x, m

m

z, mm y, mm

Ω

12 16

(a) Normalized applied field speed 0.20

5 10 15 20 25 30 35 40 455

10

15

20

25

0 4 8z, mm

x, m

m

y, mm

Ω

12 16

(b) Normalized applied field speed 0.85

Fig. 4. Trajectories of the microrobot at two applied field rotation speedsin agar gel (0.5 wt. %) with Ω parallel to the z world axis. The exit pointsof each trajectory are shown on the right side of the figure. Each trajectoryis obtained using the computer vision system. Each exit point is measuredby hand.

a maximum field strength of 10 mT. Computer vision formicrorobot tracking was performed using a Basler A602FCcamera fitted with a Computar MLH-10X macro zoom lenswith the image plane parallel to the x-z plane of a staticworld coordinate frame, defined in Fig. 3(c).

Similarly to [6] and [7], agar gel was used to simulatesoft tissue for our experiments. Agar gel is commonly usedin research as an inexpensive and semi-transparent tissuephantom. For example, agar gel has been used to simulate thevitreous humor of the eye [12], brain tissue [13], and humanbreast tissue [14]. The agar gel used herein consisted of0.5 wt. % culinary agar to unfiltered water, which we found tohandle well for our initial experiments. For a comparison, ithas been found that 0.6 wt. % agar closely models propertiesof human brain tissue [13]. Our agar gel was manually cutinto blocks fitting a 45 mm×16 mm×38 mm acrylic con-tainer, which was positioned in the common center of theHelmholtz coils. The microrobot was placed into the con-tainer through a small section of soft plastic tubing epoxiedwithin a hole drilled into the container’s side to ensure thatthe microrobot was oriented close to horizontal prior to thestart of each experiment. The microrobot used in this study is5.6 mm long and 1.4 mm in diameter and consists of a 1 mmcube Grade-N50 permanent magnet rigidly attached to brasswood screw with 0.87 mm pitch. During experimentation, thestep-out frequency of the microrobot at ψ = 0 ranged from7.3 Hz to 8.0 Hz.

The ability of the microrobot to reject disturbances inorientation is demonstrated in the results presented in Fig.4, which shows the trajectory data of several runs of themicrorobot as viewed from the imaging system with Ωaligned with the world’s z axis throughout the durationof each run. Four typical trajectories when operating at



Top Top

Front Front

0s

0s 0s10s 20s 30s

10s 20s30s

0s

9s 18s27s

9s 18s27s

(a) ψ = 20° (b) ψ = 40°

(c) ψ = -20° (d) ψ = -40°Top Top

Front Front

0s

0s

9s 18s27s

9s18s

27s0s 9s 18s

27s

9s0s 18s 27s

Fig. 5. Front and top views of microrobot trajectories demonstratingthe microrobot’s tendency to yaw while turning. The angle between themicrorobot’s principal axis and Ω (as viewed from the x-z world pane) ismaintained at ψ = ±20 and ±40. The field is rotated at a normalizedapplied field rotation speed of 0.20 in all four cases.

normalized applied field rotation speeds of 0.20 and 0.85are shown on the left side of Fig. 4(a) and Fig. 4(b),respectively, along with the points where each trajectory exitsthe agar sample plotted on the right. At normalized speed0.20, the average stabilizing torque is sufficient to overcomethe average destabilizing torque and return the orientation ofthe microrobot to Ω resulting in a tight clustering of exitpoints near the opposite position of the entry point. Withnormalized speed of 0.85, the average destabilizing torqueovercomes the stabilizing torque and perpetually increasesthe magnitude of naturally occurring disturbances resultingin unstable behavior and large variation in the position of theexit points as shown. It is important to note, however, thatthe microrobot only naturally rejects disturbances in heading,not position. The position of the microrobot may drift off thedesired trajectory without some form of feedback control.

We demonstrate the microrobot’s tendency to yaw whilesteering by pitching Ω an angle of ψ measured from themicrorobot’s principal axis as seen from the vision system.As the microrobot turns, Ω was kept in the x-z world plane,but manually adjusted to maintain a constant ψ using visualfeedback with the assistance of guidelines drawn on realtimevideo of the microrobot presented to the user. Fig. 5 showsa front view (in the x-z world plane) and a top view (inthe y-z world plane taken with a Canon PowerShot G10) ofthe resulting trajectories with ψ = 20 and 40 in Figs. 5(a)and 5(b), and ψ = −20 and −40 in Figs. 5(c) and 5(d),

0 100 200 300 400 500 600t = 0 s

t = 600 s5

15

25

35

Time, s

Ang

le, d

eg

(a) Normalized applied field speed 0.001 and ψ = 10.0

t = 15.5 s

t = 0 s

Applied Field Revolutions (#)

Hor

izon

tal A

ngle

(de

g)

5

15

25

35

10 20 300

(b) Normalized applied field speed 0.3 and ψ = 10.0

Fig. 6. Angle of the microrobot’s principal axis measured from horizontalas a function of the number of applied field revolutions while rotating ata normalized rotation speed of 0.001 (a) and 0.3 (b). The right-hand sideshows images of the microrobot in the agar at the beginning and end of theexperiment.

respectively. In all four cases, the applied field is rotated at anormalized rotation speed of 0.20 and each trajectory stopswhen the microrobot either collides with the container orexits the top of the agar sample. When ψ > 0, Fig. 2 predictsthat the stabilizing torque will cause the microrobot to turn inthe direction of Ω but the destabilizing torque will cause themicrorobot to rotate about the x axis of the robot frame. Inthe world frame, the microrobot will tend to simultaneouslyturn about the y world axis and the z world axis as shownin Figs. 5(a) and 5(b). When ψ < 0, the microrobot will tendto simultaneously turn about the −y world axis and the −zworld axis as shown in Figs. 5(c) and 5(d). This tendency toyaw while steering will need to be taken into account whilenavigating screw-like microrobots.

In Sec. II, we applied the assumption that the microrobot’sprincipal axis is constrained during one complete rotation ofthe applied field. This tends to be a valid assumption whendriving the microrobot through tissue because the headingof the principal axis typically does not change significantlyover one revolution of the applied field. This assumptionbreaks down when rotating the applied field very slowly.At slow rotation speeds, the instantaneous magnetic torque(3) causes the microrobot’s heading to vary significantlythroughout one cycle. This is demonstrated in Fig. 6, whichshows the measured heading of the microrobot (obtainedusing the computer vision system) as a function of timewhen driving the microrobot with a normalized applied fieldrotation speed of 0.001 and 0.3, with the pitch angle ψ thesame in both cases. When rotating the applied field slowly,the microrobot’s heading tends to oscillate at a magnitudeof 7 throughout one applied field revolution, whereas themicrorobot’s heading tends to oscillate at a magnitude of1 when rotating the applied field rapidly. Without anyexplicit understanding of the microrobot-tissue interaction,



the drop in magnitude may indicate that the oscillations ofthe microrobot exceed the inherent bandwidth of the tissue.If the oscillation of the microrobot is undesired (e.g., dueto potential tissue damage), then the microrobot should bedriven at faster rotation speeds, which may come at the costof increasing the undesired yaw while steering.

IV. DISCUSSION

Various microrobot geometries have been proposed fordriving through soft tissue. In [6], the authors propose asimilar device to that used in our experiments, where apermanent magnet is rigidly attached to a single screw; in[7] and [8], the authors control a screw-like device consistingof a helical thread wrapped around a cylindrical permanentmagnet (diametrically magnetized); and in [11], the authorspropose a device consisting of screws rigidly attached toboth ends of a cylindrical permanent magnet. The resultspresented in Sec. II are purely an analysis of the interactionbetween the microrobot’s dipole moment and the appliedmagnetic field, therefore, it applies to screw-like microrobotsof any geometry. Because the magnetic torque applied to themicrorobot is a pure torque (i.e., not an immediate resultof an applied force), its affect on the microrobot (as a freebody) is invariant to where it is applied on the device itself.As a result, the microrobot used herein would exhibit thesame behavior if the cube magnet were positioned in themiddle of the screw as it does with the magnet positioned atthe rear. We do expect, however, that the microrobot’s lengthand the density of the tissue will influence the magnitude ofthe behavior demonstrated herein.

The microrobot used herein is 5.6 mm long. It is likely thatthe microrobot will need to be scaled down for many medicalapplications. We do not yet have a good understanding ofmicrorobot-tissue interaction. However, we can approximatethe role of friction as the microrobot is scaled if we model themicrorobot as if it were a cylinder and if we assume that thefriction between the tissue and the microrobot is proportionalto the surface area. Applying these assumptions, the surfacearea of the microrobot is A = 2πdl where d and l are themicrorobot’s diameter and length. If d and l are scaled by afactor s, then the surface area becomes A = πdls2 and thusscales with s2. If we scale the dimensions of the permanentmagnet rigidly attached to the screw by the same factor s,then the magnet’s volume scales with s3. The magnitude ofthe microrobot’s dipole moment m is proportional to themagnet’s volume causing m, and subsequently the appliedmagnetic torque τ (given by (3)), to also scale with s3. Thisimplies that as the microrobot is scaled down, the surfacefriction decreases at a slower rate than the magnetic torqueand the applied magnetic field magnitude |B| may need toincrease to generate effectively the same behavior presentedherein. Fully understanding how the microrobot will scale tosmaller sizes will require the microrobot-tissue interaction tobe better understood. This will be the subject of future work.

V. CONCLUSION

In this study, we have analyzed the stabilizing and destabi-lizing magnetic torques present when intentionally steeringthe microrobot or due to naturally occurring disturbancesin the microrobot’s heading. We have shown theoreticallyand experimentally that there exists an applied field rotationspeed, above which screw-like microrobots will becomeunstable. We have also demonstrated that complex interactionbetween the stabilizing and destabilizing magnetic torquescan cause a screw-like microrobot to turn away from desireddirections of steering in a predictable way. These phenomenaare not reported in prior literature and they must be taken intoaccount when designing future steering and control strategiesfor medical applications.

ACKNOWLEDGEMENT

The authors wish to thank Jeremy Greer and Dr. EberhardBamberg for fabricating the microrobot pictured in Fig. 3.

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