Purcell magneto-elastic swimmer controlled by an external magnetic field

Proceedings of the 20th World Congress The International Federation of Congress Automatic Control Proceedings of the 20th World The International Federation of Congress Automatic Control Proceedings of the 20th9-14, World Toulouse, France, July 2017 The International Federation of Automatic Control Available online at www.sciencedirect.com Toulouse, France, July 9-14, 2017 The International Federation of Automatic Control Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017

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IFAC PapersOnLine 50-1 (2017) 4120–4125 Purcell magneto-elastic swimmer controlled Purcell magneto-elastic swimmer controlled Purcell magneto-elastic swimmer controlled Purcellby magneto-elastic swimmer controlled an external magnetic field by an external magnetic field by an external magnetic field by∗ an external magnetic field ∗∗ ∗∗∗ ∗∗∗∗

F. L. Giraldi M. Zoppello F. Alouges Alouges ∗ A. A. DeSimone DeSimone ∗∗ Giraldi ∗∗∗ Zoppello ∗∗∗∗ ∗∗∗ ∗∗ L. ∗∗∗ M. ∗∗∗∗ F. Alouges ∗∗∗ A. DeSimone ∗∗ L. Giraldi M. Zoppello ∗∗∗∗ ∗∗ ∗∗∗ ∗∗∗∗ F. Alouges A. DeSimone L. Giraldi M. Zoppello ∗ ´ Polytechnique CNRS, Route de Saclay, ∗ CMAP UMR 7641 Ecole ´ CMAP UMR 7641 Ecole Polytechnique CNRS, Route de Saclay, ∗ ´ ∗ 7641 Palaiseau Ecole Polytechnique CNRS, Route de Saclay, 91128 Cedex (e-mail: ∗ CMAP UMR ´ Palaiseau Cedex France France (e-mail: CMAP UMR91128 7641 Ecole Polytechnique CNRS, Route de Saclay, 91128 Palaiseau Cedex France (e-mail: [email protected]) [email protected]) 91128 Palaiseau Cedex France (e-mail: ∗∗ [email protected]) ∗∗ Scuola Internazionale Superiore di Studi Avanzati via Bonomea 265 Internazionale Superiore di Studi Avanzati via Bonomea 265 [email protected]) ∗∗ ∗∗ Scuola Superiore di Studi Avanzati via Bonomea 265 I-34136 Italy [email protected]) ∗∗ Scuola Internazionale I-34136 Trieste TriesteSuperiore Italy (e-mail: (e-mail: [email protected]) Scuola Internazionale di LJAD, Studi Avanzati via Bonomea 265 ∗∗∗ e Cˆ o te d’Azur, CNRS, INRIA I-34136 Trieste Italy (e-mail: [email protected]) ∗∗∗ Universit´ Universit´ e Cˆ ote d’Azur, CNRS, LJAD, INRIA Sophia Sophia Antipolis Antipolis I-34136 Trieste Italy (e-mail: [email protected]) ∗∗∗ ∗∗∗ Universit´ e Cˆ o te d’Azur, CNRS, LJAD, INRIA Sophia Antipolis Mditerrane Team/´ eequipe McTAO B.P. 93 06902 ∗∗∗ Mditerrane Team/´ quipe McTAO B.P. 93 – –INRIA 06902 Sophia Antipolis Universit´ e Cˆ o te d’Azur, CNRS, LJAD, Mditerrane Team/´ equipe (e-mail: [email protected]) B.P. 93 – 06902 Sophia Antipolis cedex – France cedex – France (e-mail: [email protected]) Mditerrane Team/´ equipe McTAO B.P.via 93 – 0690263, Sophia Antipolis ∗∗∗∗ cedex – France (e-mail: [email protected]) a studi di ∗∗∗∗ Universit´ a –degli degli studi(e-mail: di Padova, Padova, via Trieste Trieste 63, 35121 35121 Padova Padova cedex France [email protected]) ∗∗∗∗ ∗∗∗∗ Universit´ Universit´ a degli studi di Padova, via Trieste 63, 35121 Padova Italy (e-mail: [email protected]) ∗∗∗∗ (e-mail: [email protected]) Universit´ aItaly degli studi di Padova, via Trieste 63, 35121 Padova Italy (e-mail: [email protected]) Italy (e-mail: [email protected]) Abstract: Abstract: This This paper paper focuses focuses on on the the mechanism mechanism of of propulsion propulsion of of aa Purcell Purcell swimmer swimmer whose whose Abstract:areThis paper focuses on the mechanism of propulsion of a Purcell swimmer whose segments magnetized and react to an external magnetic field applied into the fluid. an segments areThis magnetized and react to an external magnetic field of applied into the fluid. By By an Abstract: paper focuses on the mechanism of propulsion a Purcell swimmer whose segments areanalysis, magnetized and react toisan external magnetic field applied into the fluid. By an asymptotic we prove that it possible to steer the swimmer along a chosen direction asymptotic analysis, we prove that it is possible to steer the swimmer along a chosen direction segments are magnetized and react to an external magnetic field applied into the fluid. By an asymptotic analysis, we prove that it is possible to steer the swimmer alongwea discuss chosen direction when the control functions are prescribed as an oscillating field. Moreover, what are when the control functions are that prescribed as an oscillating field. Moreover, wea discuss what are asymptotic analysis, we prove it is possible to steer the swimmer along chosen direction when the obstructions control functions are prescribed as anget oscillating field. Moreover,result we discuss what are the main to in classical for the main to overcome overcome in order order to to get classical controllability controllability result for this this system. system. when the obstructions control functions are prescribed oscillating field. Moreover,result we discuss are the main obstructions to overcome in orderastoanget classical controllability for thiswhat system. the main obstructions to overcome to get classical controllability thisreserved. system. © 2017, IFAC (International Federationinof order Automatic Control) Hosting by Elsevier result Ltd. Allfor rights Keywords: Keywords: Application Application of of nonlinear nonlinear analysis analysis and and design, design, Tracking, Tracking, Control Control in in system system biology biology Keywords: Application of nonlinear analysis and design, Tracking, Control in system biology Keywords: Application of nonlinear analysis and design, Tracking, Control in system biology 1. as 1. INTRODUCTION INTRODUCTION as the the one one proposed proposed in in Alouges Alouges et et al. al. (2015). (2015). In In what what 1. INTRODUCTION as the one proposedisincomposed Alouges by et only al. (2015). In what follows, the swimmer 33 links which are follows, the swimmer is composed by only links which are 1. INTRODUCTION as the one proposed in Alouges et al. (2015). In what follows, the swimmer is composed by only 3 linked links which are supposed to be uniformly magnetized and together supposed to be uniformly magnetized and linked together follows, the swimmer is composed by only 3 links which are In to be uniformly magnetized and linked together with rotational springs. an magnetic field In the the last last decade, decade, micro-motility micro-motility has has become become a a subject subject supposed with rotational springs. When When an external external magnetic field In the last interest, decade, both micro-motility has become a subject with supposed to be uniformly magnetized and linked together of growing for the biological understanding rotational springs. When an external magneticafield is applied the is to deof growing interest, both for the biological understanding In growing the last interest, decade, and micro-motility hasapplications. become a subject is applied the swimmer swimmer is expected expected to experience experience afield deof both for the biological understanding with rotational springs. When antoexternal magnetic micro-organisms technological In the is applied the swimmer is expected to experience a deformation which hopefully leads a global displacement. of growing micro-organisms and technological applications. In the interest, both for addresses the biological understanding formation which hopefullyis leads to a global displacement. of micro-organisms and technological applications. In the formation is applied the swimmer expected to experience a delatter direction, the topic several challenges which hopefullymagnetic leads to field a global displacement. the the is to latter direction, theand topic addressesapplications. several challenges of micro-organisms technological In the In In the rest, rest,which the external external magnetic field is assumed assumed to be be latter direction, theconception topic addresses several challenges formation hopefully leads to a global displacement. as for instance the of artificial self-propelled In the rest, the external magnetic field is assumed to be the control function. In model, the behavas for instance the conception of artificial self-propelled latter theconception topicmicroscopic addresses several challenges the control function. In our our model,field the ismagnetic magnetic behavas for direction, instance the of artificial self-propelled In the rest, the external magnetic assumed to be and/or easily controllable robots. Such kind control function. isIn modeled our model, the magnetic behavior of the by assuming that their and/or easily controllable microscopic robots. Such kind the as devices for instance the conception of biomedical artificial self-propelled ior of the segments segments isIn modeled by the assuming thatbehavtheir and/or easily controllable microscopic robots. Such kind ior the of control function. our model, magnetic of could revolutionize the applications the segments is modeled by assuming that their magnetization is always parallel to the segment with fixed of devices could revolutionize the biomedical applications and/or easily controllable microscopic robots. Such kind magnetization is always parallel to the segment with fixed of devices could revolutionize ior of the segments isfields, modeled that their Peyer et (2013) as it useful the biomedical applications is always parallel tobytheassuming segment with fixed magnitude and especially magnetic interacPeyer et al. al. (2013) as for for instance instance it could could be be useful to to magnetization of devices could revolutionize theoperations biomedical applications magnitude andis stray stray fields, especially magnetic interacPeyer et al. (2013) as for instance it could Nelson be useful to magnitude magnetization always parallel to the segment with fixed minimize invasive microsurgical et al. and stray fields, especially magnetic interacbetween different segments are Only the minimize invasive microsurgical operations Nelson et al. Peyer et al. (2013)microsurgical as for instance it could Nelson be useful to tions tions between different segments are neglected. neglected. Only the minimize invasive operations et al. magnitude anddifferent stray fields, especially magnetic interac(2010). tions between segments are neglected. Only the magnetic torque induced by the external magnetic field on (2010). minimize invasive microsurgical operations Nelson et al. magnetic torque induced by the external magnetic field on (2010). tions between different segments are neglected. Only the magnetic torque induced by the external magnetic field on each segment is considered. We adopt the same approach One of the few possibilities recently studied in the litera(2010). each segment is considered. We adopt the same approach magnetic torque induced by the external magnetic field on One of the few possibilities recently studied in the litera- each segment is considered. Wethe adopt the same approach Alouges et (2015) equations of of One of the few possibilities recently studied in the literature, is to an swimmer that aa of of Alouges et al. al. (2015) to to get get the equations of motion motion of each segment is considered. We adopt the same approach ture, is the to consider consider an artificial artificial swimmer thatinpossesses possesses One of few possibilities recently studied the literaof Alouges et al. (2015) to get the equations of motion of magneto-elastic swimmer, that turn out to be governed ture, is toflexible considertail, an and artificial swimmer that possesses a the magnetic use an external magnetic field the magneto-elastic swimmer, that turn out to be governed of Alouges et al. (2015) to get the equations of motion of magnetic flexible tail, and use an external magnetic field ture, is toflexible considertail, an and artificial swimmer that possesses a the swimmer, that turncontrol out to system be governed by aamagneto-elastic system which affine with magnetic use an external magnetic field to act this Dreyfus et al. Gao al. by system of of ODEs ODEsswimmer, which is is an an affine control system with the magneto-elastic that turn out to be governed to act on on flexible this flagellum flagellum Dreyfus etexternal al. (2005); (2005); Gao et etfield al. by magnetic tail, and use an magnetic a system ofmagnetic ODEs which is anthe affine control system with to field. micro-scale, the is to act on this Pak flagellum Dreyfus etThis al. (2005); Gaodesign et al. respect (2010, 2012); et particular respect to the the magnetic field. At At the micro-scale, the flow flow is by a system ofby ODEs which anthe affine control system with (2010, 2012); Pak et al. al. (2011). (2011). This particular design to influenced act on this by flagellum al.spermatozoa (2005); Gaodesign et al. respect to the magnetic field.isAt micro-scale, theassume flow is characterized small Reynolds number. Thus, we (2010, 2012); Pak et locomotion al.Dreyfus (2011).etThis particular is the of which characterized by small Reynolds number. Thus, we assume respect to the magnetic field. At the micro-scale, the flow is is influenced by the locomotion of spermatozoa which (2010, 2012); Pak et locomotion al. by (2011). This particular design characterized by smallfluid Reynolds number. Thus, we assume that the surrounding is governed by Stokes equations is influenced by the of spermatozoa which achieves their propulsion propagating travelling wave that the surrounding fluid is governed by Stokes equations characterized by small Reynolds number. Thus, we assume achieves their propulsion by propagating travelling wave is influenced by the locomotion of spermatozoa the surrounding fluid is governed by Stokes equations which are linear respect to distribution achieves their propulsion by propagating travellingwhich wave that along their flagellum. which aresurrounding linear with withfluid respect to the the velocity velocity distribution thatthe the is governed by Stokes equations along their flagellum. achieves their propulsion by propagating travelling wave which are linear with respect to the velocity distribution on boundary (rates of deformation and displacement). along their flagellum. on the are boundary (ratesrespect of deformation and displacement). which linear with to the velocity distribution On the other hand, there exists now a quite wide literature along their flagellum. the boundary (rates of deformation and displacement). In that case, Force (RFT) (see On the other hand, there exists now a quite wide literature on In that case, the the Resistive Resistive Force Theory Theory (RFT) (see Gray Gray on that the boundary (ratesprovides of deformation and displacement). On the otherahand, there exists now problem a quite wide literature In that makes connection between of case, the Resistive ForceaTheory (RFT) (see Gray and Hancock (1955)) simple and concise way that makes ahand, connection between problem of swimming swimming On the other there exists now a quite wide literature and Hancock (1955)) provides aTheory simple (RFT) and concise way In that case, the Resistive Force (see Gray that makes a connection between problem ofcontrol swimming at the micro-scale, and the mathematical theHancock (1955)) provides a simple and conciseforces way to compute aa local approximation of hydrodynamic at the micro-scale, and the mathematical control the- and that makes a connection between problem of swimming to compute local approximation of hydrodynamic forces and Hancock (1955)) provides a simple and concise way at the micro-scale, andpioneering the mathematical control theory. Starting from the work of Shapere and compute a local approximation of hydrodynamic forces involved in system. ory. Starting from the pioneering work of Shapere and to at the micro-scale, and the mathematical control theinvolved in our our system. to compute a local approximation of hydrodynamic forces ory. Starting Wilczek Shapere and Wilczek (1989), and Montgomery from the pioneering work of Shapere and involved in our system. Wilczek Shapere and Wilczek (1989), and Montgomery ory. Starting fromand the pioneering workand of Shapere and Section is devoted to involved22in system. Wilczek Shapere Wilczek (1989), Montgomery Montgomery (2002), the dynamics of microSection is our devoted to recall recall the the equation equation of of motion. motion. As As in in Montgomery (2002), the dynamics of self-propelled self-propelled micro- Section Wilczek Shapere and Wilczek (1989), and Montgomery 2and is devoted to recall the equation ofan motion. As in Or (2014), Section 3 provides estimate of Montgomery (2002), the dynamics of self-propelled micro- Gutman scopic artificial swimmers has been considered for instance Gutman and Or (2014), Section 3 provides an estimate of Section 2 is devoted to recall the equation of motion. As in scopic artificial swimmers has been of considered for instance Montgomery (2002), the2008); dynamics self-propelled micro- Gutman and Or (2014), Section 3 provides an estimate of the displacement of the swimmer with respect to small scopic artificial swimmers has been considered for instance in Alouges et al. (2013, Alouges and Giraldi (2013); the displacement of the swimmer with respect to small Gutman and Or (2014), Section 3 provides an estimate of in Alouges et al. (2013, 2008); Alouges and Giraldi (2013); scopic artificial swimmers has been considered for instance the displacement of the generalizing swimmer with respect to small amplitude of field, the approach of in et al. (2013, 2008); Alouges and Giraldi (2013); G´ eeAlouges rard-Varet and Giraldi (2015)) where the rate shape amplitude of the the of field, generalizing the respect approach of E. E. the displacement the swimmer with to small G´ rard-Varet and(2013, Giraldi (2015)) where theGiraldi rate of of(2013); shape amplitude in Alouges et al. 2008); Alouges and of the field, generalizing the approach of E. G´ erard-Varet and Giraldi (2015)) where the rate of shape changes of is as control. changes of the the swimmer swimmer is considered considered as a a natural natural control. G´erard-Varet Giraldi where rate of shape amplitude of the field, generalizing the approach of E. changes of theand swimmer is (2015)) considered as athe natural control. The aim of the present is mechanism of changes is considered as athe natural control. The aim of of the the swimmer present paper paper is to to study study the mechanism of The aim of of theapresent paper is toartificial study the mechanism of propulsion magneto-elastic micro-swimmer, propulsion of apresent magneto-elastic artificial micro-swimmer, The aim of the paper is to study the mechanism of propulsion of a magneto-elastic artificial micro-swimmer, propulsion of a magneto-elastic artificial micro-swimmer,

Copyright © 2017, 2017 IFAC 4193Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 4193 Copyright 2017 responsibility IFAC 4193Control. Peer review©under of International Federation of Automatic Copyright © 2017 IFAC 4193 10.1016/j.ifacol.2017.08.798

Proceedings of the 20th IFAC World Congress F. Alouges et al. / IFAC PapersOnLine 50-1 (2017) 4120–4125 Toulouse, France, July 9-14, 2017

Gutman and Y. Or, to the case of a 3-link swimmer. 1 This result implies that small prescribed oscillating magnetic field produces a net motion along a chosen direction. The latter Section does not focus on the controllability of the system as it was done in Giraldi and Pomet (2016) for the magneto-elastic 2-link swimmer. Indeed, in the last section 4 of the paper, we discuss the main obstructions to generalize the 2-link controllability result, presented inGiraldi and Pomet (2016), to the magneto-elastic Purcell swimmer. Thus, the full controllability problem for the 3link swimmer remains open. 2. MODELING The model of the magneto-elastic N -link swimmer was already introduced in Alouges et al. (2015). The two link one was studied in Gutman and Or (2014); Giraldi and Pomet (2016).Here, we recall briefly the equation of motion focusing on the case N = 3. We consider a magneto-elastic 3-link Purcell swimmer moving in a plane. This two-dimensional setting is suitable for the study of slender, essentially one-dimensional swimmers exploring planar trajectories as explained in Alouges et al. (2013, 2015). The swimmer consists of 3 rigid segments, each of length L with articulated joints at their ends (see Fig. 1), moving in the horizontal 2d-plane of the lab-frame. Because of the symmetric geometry of the swimmer, we use slightly different notation and variables than in Alouges et al. (2015). Indeed we call x = (x, y) the coordinates of the central point of the second segment, θ the angle that it forms with the x-axis, α2 the relative angle between the first and second segments and finally α3 the relative between the third and the second segments (see Fig 1). Therefore the position and the orientation of the swimmer are characterized by the triplet (x, y, θ), while its shape is given by (α2 , α3 ). As in Alouges et al. (2015) the three segments are uniformly magnetized and linked together with torsional springs, with elastic constant K, that tend to align the segments one with another. Those produce torques when the segments are not fully  aligned. In what follows we Hx (t) assume that H(t) := Hy (t) is horizontal in such a way 0 that the motion holds in the horizontal plane. A4 • α3 • A1 α 2 ey

• • A3 x • A2

H(t)

θ ex

Fig. 1. The magneto-elastic Purcell swimmer of shape (α2 , α3 ) at the position x in the plane subject to an external magnetic fields H. 1

The 3-link swimmer was already proposed in Purcell (1977) and further considered in Becker et al. (2003); Giraldi et al. (2013); Passov and Or (2012); Tam and Hosoi (2007) for instance.

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2.1 Equations of motion As it was noticed in Alouges et al. (2015) the equations which govern the dynamics of the swimmer form a system of ODEs, which is affine with respect to the magnetic field H(t). The hydrodynamic forces acting on the i-th link, are approximated using RTF, with parallel (resp. perpendicular) drag coefficients ξi (resp. ηi ). In our particular case this system describes the evolution of the position and the shape variables and thus consists of five equations. Those are obtained by writing the balance of forces for the whole swimmer and the balance of torques for the subsystems consisting of the three, two and one rightmost segments. We call those subsystems S1 , S2 , S3 respectively (S1 is therefore the whole system). The motion of the swimmer holds in the horizontal plane since only horizontal forces and vertical torques apply. The final system reads:  Fh =  0,     A A A   ez · Th 1 + Te 1 + Tm1 = 0 ,   A2 A2 A2 = 0, · T + T + T e  z e m h       A3 A3 A3 e z · T h + Te + Tm = 0 .

(1)

Here, Fh denotes the total hydrodynamic force acting on Ai Ai i the swimmer, TA h (resp. Te and Tm ) is the hydrodynamic (resp. elastic and magnetic) torque with respect to Ai (see Fig. 1), acting on the subsystem Si . Following the construction made in Alouges et al. (2015) system (1) becomes     0 x˙  0    ˙ Mh (θ, α2 , α3 )  θ  = −K  0  α˙ 2 α  2 (2) α˙ 3 α 3

− Mxm (θ, α2 , α3 ) Hx (t) − Mym (θ, α2 , α3 ) Hy (t) , with Mh 5 × 5 matrix, Mxm and Mym vectors in R5 all depending on (θ, α2 , α3 ). All these matrices can be computed explicitly following the approach given in Alouges et al. (2015). Finally system (2) can be rewritten as   x˙  θ˙    =f0 (θ, α2 , α3 ) + fx (θ, α2 , α3 )Hx (t) α˙ 2 (3) α˙ 3 + fy (θ, α2 , α3 )Hy (t) where,   0 0   f0 = −M−1 h K 0  (4) α  2 α3 j M j = x, y. fj = −M−1 m, h Notice however that the dynamics of (θ, α2 , α3 ) is independent of x and can be decoupled. Indeed by block decomposing the matrix Mh as   A h Bh Mh = (5) BTh Ch

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(Ah , Bh and Ch being respectively 2 × 2, 2 × 3 and 3 × 3 matrices), and considering the first two rows of the system (2), we can solve for x˙ as  ˙ θ α˙ 2 α˙ 3 = G1 (θ, α2 , α3 ) θ˙ + G2 (θ, α2 , α3 ) α˙2 + G3 (θ, α2 , α3 ) α˙3 .

(θ, α2 , α3 ) Bh (θ, α2 , α3 ) x˙ = −A−1 h

(6)

Moreover, the subsystem associated with the shape and the orientation of the swimmer   becomes θ˙  ˙ 2 = (−BTh A−1 h Bh + C h ) α α˙ 3 (7)   0 ˜ y Hy (t) , ˜ x Hx (t) − M − K α2 − M m m α3 that we rewrite inverting the left hand side matrix as   θ˙ α˙ 2  = g0 (θ, α2 , α3 )+ (8) α˙ 3 + gx (θ, α2 , α3 )Hx (t) + gy (θ, α2 , αN )Hy (t) . The whole dynamical system (3) is an affine control system with drift where the controls are the two components of the magnetic field. The explicit expression of the dynamics are formally computed by using a symbolic computation software as Mathematica. 3. STEERING ALONG ONE DIRECTION WITH SMALL SINUSOIDAL MAGNETIC FIELDS As it was noticed by experiments in Poper et al. (2006), in the rest we show, that a swimmer with an initial symmetric shape, remains symmetric. Moreover by using an asymptotic expansion we provide an estimate of the swimmer displacement for a prescribed small sinusoidal magnetic field.

about the center O of the lab-frame of the whole system (swimmer and magnetic field). Notice that this rotation changes x to −x, leaves θ invariant, interchanges α2 and α3 , and reverses the magnetic field H and the magnetization along the swimmer. Therefore, if the function t → (x(t), θ(t), α2 (t), α3 (t)) is solution of the system (3) magnetized in one direction (say +), for an external magnetic field H(t), then the trajectory (−x(t), θ(t), α3 (t), α2 (t)) is the solution corresponding to the magnetic field −H(t) of the system magnetized in the opposite direction (say −). We summarize this by saying that R

(x(t), θ(t), α2 (t), α3 (t), H(t), +) −→ (−x(t), θ(t), α3 (t), α2 (t), −H(t), −) (9)

where the last component corresponds to the direction of the magnetization along the swimmer. Similarly, we consider a second transformation T which reverses only the magnetization and the external magnetic field. We remark that the equations of motion (3) depend only on the product M ek ∧ H(t), if the function t → (x(t), θ(t), α2 (t), α3 (t)) is solution of (3) with a prescribed magnetic field and magnetization, then it remains a solution with opposite magnetic field and magnetization, so that T

(x(t), θ(t), α2 (t), α3 (t), H(t), +) −→ (x(t), θ(t), α2 (t), α3 (t), −H(t), −) (10)

The geometric transformations R and T are sketched in Fig. 2. Checking more formally those symmetry properties of the systems can of course be done on (2) but is left to the reader. Now, composing R and T we have, using the notation above T ◦R

(x(t), θ(t), α2 (t), α3 (t), H(t), +) −−−→ (−x(t), θ(t), α2 (t), α3 (t), H(t), +) (11)

which means that by uniqueness of the solution of (3), a swimmer starting at position x(0) = 0 with a symmetric shape (α2 (0) = α3 (0)), under whatever driving magnetic field H(t) verifies x(t) = −x(t) , and α2 (t) = α3 (t) .

3.1 A symmetry obstruction In this part, we assume that the drag coefficients of the 3 links are identical and that they have all a uniform magnetization, ξi = ξ and ηi = η , Mi = M ∀i ∈ {1, 2, 3} . These assumptions are suitable to describe the behavior of a magnetic filament. We consider the set of symmetric shapes   S := (α2 , α3 ) ∈ [0, 2π]2 , α2 = α3 , (see Fig. 2). As it was experimentally observed in Poper et al. (2006), the following Proposition 1 proves that for a swimmer which possesses an initial shape belonging to S, its shape remains symmetric, regardless of the applied magnetic field. Proposition 1. Let T > 0, if at the initial time α2 (0) = α3 (0) then for any magnetic field t → (Hx (t), Hy (t)), applied to the system, the shape of the swimmer remains symmetric i.e., α2 (t) = α3 (t) , ∀t ∈ [0, T ] . Proof. This result is based on a symmetry argument. The dynamics must be invariant by the rotation R of angle π

It hence experiences no displacement and stays symmetric.

Fig. 2. The two transformations R and T of the system. Remark 3.1. Similar argument holds for a swimmer composed by an odd number of links.

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˜  θ(t)

3.2 Small oscillating magnetic field In this subsection we focus on a swimmer satisfying η2 = η3 = η ξ2 = ξ3 = ξ (12) with η1 = η ξ1 = ξ These latter assumptions allow to overcome the previous symmetry obstruction. They are suitable in the case of a swimmer with an head. Starting from a swimmer with a horizontal shape (θ = α2 = α3 = 0), we can use the horizontal component of the external magnetic field as a “stabilizer” whereas the oscillating vertical component produces the shape deformation and the motion. In order to understand further what happens when such a field is applied we make the following perturbation analysis. We assume that (13) (Hx (t), Hy (t)) = (1,  sin(ωt)) , and compute the asymptotic expansion of the swimmer displacement with respect to small  after a period 2π w . Linearizing the system of equations (8) for small angles (θ, α2 , α3 ), to first order in , i.e.,     θ θ˜ α 2 =  α ˜ 2  + o() α3 α ˜3   θ˜ we get that the triplet α ˜ 2  satisfies the equation, α ˜3

with

   ˙ θ˜ θ˜ α ˜ 2  + b sin(ωt) ˜˙ 2  = A α α ˜3 α ˜˙ 3

(14)

(15) A = ∇ (g0 + gx ) (0, 0, 0) , b = gy (0, 0, 0) . Here, A is the 3 × 3 matrix which depends on the drag coefficients, (η1 , η) and (ξ1 , ξ), on the magnetization M and on the elastic constant K. We find that its explicit expression is given by

A=δ where a1 1 a1 2 a1 3 a2 1 a2 2 a2 3 a3 1 a3 2 a3 3 and δ =



a1 1 a1 2 a1 3 a2 1 a2 2 a2 3 a3 1 a3 2 a3 3



α ˜ 2 (t) α ˜ 3 (t)

=

 1  + A (ω) exp (iωt)b − A− (ω) exp (−iωt)b 2i

2

= −4(K + M )η − (42K + 23M )η1 η − = −(28K + 9M )ηη1 − (5K + 3M ) − η12 = −6M (2ηη1 + η12 ) = −4(7K + 3M )ηη1 − 5Kη12 = −16(2K + M )ηη1 − (16K + 11M )η12

(16)

+ c(ω) exp (At)

where A± (ω) := (−A ± iωI)−1 and c(ω) = A− (ω)b − A+ (ω)b. The first part of the solution corresponds to a periodic solution, while the last is an exponentially decaying perturbation, as we shall see now. Indeed, by applying RouthHurwitz criterion on the characteristic polynomial of A, we prove that the real part of its eigenvalues are all negative. This provides the stability of the asymptotic periodic solution Proposition 2. The steady-state solution, called S ∞ , of (14)  1  + ∞ − S

(t) =

2i

A (ω) exp (iωt)b − A (ω) exp (−iωt)b ,

(17)

is stable.

Proof. The characteristic polynomial of A reads pA (λ) = det(A − λId) = a3 λ3 + a2 λ2 + a1 λ + a0 , where   432M 3K 2 + 4KM + M 2 (2η + η1 ) a0 = detA = − < 0, d    36 a1 = − (M 2 10η 2 + 28ηη1 + η12 d   + K 2 16η 2 + 64ηη1 + η12    2 2 + KM 31η + 98ηη1 + 3η1 ) < 0 ,    −12 a2 = trA = M 2η 2 + 17ηη1 + 5η12 d    < 0, + K 2η 2 + 37ηη1 + 9η12 a3 = −1 < 0   d = L9 η 2 8ηη1 + 7η12 and a3 a0 −a2 a1 =

−1 L9 η 3 η12 (8η + 7η1 )2

+ 31KM + 10M 2 )η 4 +

= M η1 (5η + η1 ) = (19K + 9M )ηη1 + 2Kη12 = 2(8K + 3M )ηη1 + (5K + 3M )η12   = −M 4η 2 + 13η1 η + η12

4123







432 2(K + M )(16K 3



+ 5 144K 3 + 339K 2 M + 217KM 2 + 42M 3 η 3 η1





+ 2514K 3 + 5015K 2 M + 2867KM 2 + 506M 3 η 2 η12 +





+ 613K 3 + 1309K 2 M + 802KM 2 + 150M 3 ηη13



+ (9K + 5M ) K 2 + 3KM + M 2

2Kη12

6 L3 ηη1 (8η+7η1 ) .

The solution of the system (14) is thus given by



η14



< 0.

Using the Routh-Hurwitz criterion, Gantmacher (1959), we have that the real part of the eigenvalues of A is negative and therefore he steady state of the equation (14) is stable. A a consequence, the solution of (16) exponentially converges to the periodic solution (17) and in particular θ ∼ θ˜∞ oscillates around 0 indicating that the swimmer stays nearly horizontal, stabilized by the horizontal component of the magnetic field. Similarly, the fact that the shape 4196

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variables (α2 , α3 ) are periodic (and small) indicates that the swimmer stays nearly straight. In order to go further, and compute the (asymptotic) net displacement of the swimmer after one period of the oscillating external field, we linearize as well the equation (6) to first order in (θ, α2 , α3 ) near (0, 0, 0). Noting,

 θ˙ x˙ = Gx (θ, α2 , α3 )·α˙ 2  α˙ 3 

and

Thus, to ensure that (19) is not null, it is sufficient to prove that the three of vectors {u, A− (ω)b, A+ (ω)b} are independent. But, for large frequencies ω, we can expand the matrix A± (ω) as A I 1 A± (ω) = ± − 2 + o( 2 ) . (20) iω ω ω and

 θ˙ y˙ = Gy (θ, α2 , α3 )·α˙ 2  α˙ 3 

det(u, A− (ω)b, A+ (ω)b) = det(u,



b Ab 1 ) + o( 4 ) , ω ω2 w

= − L9 ω 3 η 3 η12 (8η + 7η1 )2 (3ηη1 − 2ηξ1 − η1 ξ)

−1

× where G (resp. G ) is the 1 × 3 matrix composed of  (Gi · ex )i=1,···3 (resp. (Gi · ey )i=1,···3 ), we obtain ∆x as    2π ηη1 (113K + 38M ) + 216M 2 (2η + η1 ) η 2 ξ1 4η 2 (5K + 2M ) ω  x   G (0, 0, 0) +  t S ∞ (t )∇Gx (0, 0, 0) S ∞ (t )dt +o(2 ) .    2  2 2 x

y

+ η1 (29K + 8M ) + ηη1 (η1 − η) K 4η + 37ηη1 + 13η1

0

∞

Since, t → S (t) is periodic, the latter equality reads  2π ω t ∞  S (t )∇Gx (0, 0, 0)S ∞ (t )dt +o(2 ) , (18) ∆x = 2





+ 6η1 M (2η + η1 ) − η1 ξ 8η 3 (2K + M )



+ 2η 2 η1 (40K + 13M ) + ηη12 (53K + 14M ) + η13 (13K + 6M )

0

and a straight forward computation leads to express



.

This determinant does not vanish identically and we then obtain that by prescribing an oscillating magnetic field 2(η − η1 ) −η1 η as (13), the magneto-elastic Purcell swimmer moves along − (6ηη1 − 4ηξ + η1 ξ1 − η1 (2η + ξ1 ) − η(η1 − ξ1 )  the x-axis. Notice that here the assumption on the drag L   (2ξ + ξ1 ) (2ξ + ξ1 ) (2ξ + ξ1 )  . 2 (2η + η1 )  (2η 2 + 4ηη − 3η ξ) coefficients (12) is crucial. η1 (η − ξ) η(η + η1 + ξ) 1 1

∇Gx (0, 0, 0) =





(2ξ + ξ1 )

(2ξ + ξ1 )

(2ξ + ξ1 )

Similarly, the same formula holds for ∆y by substituting Gx for Gy and in this case,   000 y ∇G (0, 0, 0) = 0 0 0 , 000

thus, ∆y = o(2 ). It follows that the leading term, with respect to small angles, of the trajectory of the swimmer along the y-axis is negligible after one period of the oscillating fields compare to the one along the x-axis. From now on, we focus on the x-displacement of the swimmer, ∆x and we prove that the leading term of order 2 does not vanish after one period of the oscillating fields. Plugging (17) into (18) and noting that the two terms vanish because of periodicity,  2π ω  +   +  t x 

A (ω) exp (iωt)b ∇G (0, 0, 0) A (ω) exp (iωt)b = 0 ,

0

2π ω

0

t









A− (ω) exp (−iωt)b ∇Gx (0, 0, 0) A− (ω) exp (−iωt)b = 0

we obtain ∆x as



2π ω

 ω t t + b A (ω)NA− (ω)b , (19) 4 0 where N is the 3×3 matrix (∇Gx (0, 0, 0) − t ∇Gx (0, 0, 0)). 2

∆x = 

Moreover, the 3 × 3 matrix N is skew-symmetric and not null. Therefore, 0 is an eigenvalue of multiplicity 1. Let us denote by u its associated eigenvector. A direct computation, still using Mathematica, leads to   η1 ξ + ηξ1 − 2ηη1 u =  2η 2 + 4η1 η − 2ξη − ξ1 η − 3η1 ξ  . 6ηη1 − 2ξη1 − 4ηξ1

Notice also that since limω→0 A+ (ω) = limω→0 A− (ω) = −A−1 , ∆x tends to 0 when ω tends to 0. Moreover, ∆x tends to 0 as ω → ∞, because by using (20), limω→∞ ω4 (t bt A+ (ω)NA− (ω)b) = 0. Thus, a very low and hight frequencies produce no net motion (at order 2 ), even after one period. This suggests the existence of an optimal frequency to drive the swimmer as was already observed in Alouges et al. (2015) (see Fig. 5 and 7). 4. DISCUSSION

This section underlines the challenge that we have to face in order to control this magnetic micro-swimmer. The previous result indicates that with a small sinusoidal magnetic field we are able to control the direction of the swimmer’s displacement, but it does not imply neither global or local controllability properties. The latter local property is classically obtained by verifying the Kalman condition at an equilibrium point. Thus let us consider the system (3), around (Xe , ue ) := ((x, 0, 0, 0), (0, 0)), which is an equilibrium point. At such a point, the swimmer is aligned with the horizontal field and thus, the torque due to the horizontal field vanishes leading to fx (Xe ) = 0. Moreover, the system being invariant under translations, f0 does not depend on (x, y), from which we deduce that the matrix range R of the matrix ∇f0 (Xe ) is at most of dimension 3. The Lie bracket adf0 (fx )(Xe ) = [f0 , fx ](Xe ) vanishes since f0 (Xe ) = 0 and fx (Xe ) = 0. By induction, for k > 1 adkf0 (fx )(Xe ) = [f0 , adk−1 f0 (fx )](Xe ) = 0 . As far as adkf0 (fy )(Xe ) is concerned, we have, still by induction, for all k > 1 adkf0 (fy )(Xe ) = [f0 , adk−1 f0 (fy )](Xe ) ∈ R .

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Therefore   Span adkf0 (fx )(Xe ), adkf0 (fy )(Xe ), k ≥ 0 =   Span fy (Xe ), adkf0 (fy )(Xe ), k ≥ 0

⊂ Span (fy (Xe )) + R is at most of dimension 4. Thus the Kalman condition is not satisfied. It turns out that for non horizontal straight swimmers, i.e. Xe = (x, θ, 0, 0) with θ = 0, the same situation occurs due to the fact that fx (Xe ) = − tan(θ)fy (Xe ). Moreover let us notice that 3 f (Xe ) [fx , fy ](Xe ) = fy,X e y

3 is the third component of the vector fy (Xe ). where fy,X e Therefore the Lie algebra Lie{f0 , fx , fy }|Xe span a vector space of dimension 4. By changing the reference frame, a similar argument holds for all equilibrium points such as {(x, θ, 0, 0), θ ∈ [0, 2π]}. This means that also the classical LARC condition is not satisfied and then the Sussmann condition does not hold (see Coron (2007)). Here we have underlined the fact that the the magnetoelastic Purcell swimmer model is singular at the straight position which makes hard to get a local controllability result. Of course if instead the swimmer starts at a non straight position, thanks to the boundedness of the drift, the magnetic field can compensate it and drive the swimmer.

5. CONCLUSION In this paper, by prescribing a particular oscillating field and using an asymptotic expansion of the displacement of the swimmer, we prove that this particular field allows to steer the swimmer along one direction. Moreover we highlight the difficulties to get controllability result by showing that the classical conditions are not satisfied. It indicates that sophisticated techniques are required to obtain such controllability result. REFERENCES Alouges, F., DeSimone, A., Giraldi, L., and Zoppello, M. (2013). Self-propulsion of slender micro-swimmers by curvature control: N-link swimmers. Journal of NonLinear Mechanics. Alouges, F., DeSimone, A., Giraldi, L., and Zoppello, M. (2015). Can magnetic multilayers propel microswimmers mimicking sperm cells? Soft Robotics, 2(3), 117–128. Alouges, F., DeSimone, A., and Lefebvre, A. (2008). Optimal strokes for low Reynolds number swimmers : an example. Journal of Nonlinear Science, 18, 277–302. Alouges, F. and Giraldi, L. (2013). Enhanced controllability of low Reynolds number swimmers in the presence of a wall. Acta Applicandae Mathematicae. Becker, L.E., Koehler, S.A., and Stone, H.A. (2003). On self-propulsion of micro-machines at low Reynolds number: Purcell’s three-link swimmer. J. Fluid Mech. Coron, J.M. (2007). Control and Nonlinearity. American Mathematical Society.

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