Parking 3-sphere swimmer. I. Energy minimizing strokes

The paper is about the parking 3-sphere swimmer ($\text{sPr}_3$). This is a low-Reynolds number model swimmer composed of three balls of equal radii. The three balls can move along three horizontal axes (supported in the same plane) that mutually meet at the center of $\text{sPr}_3$ with angles of $120^{\circ}$ . The governing dynamical system is introduced and the implications of its geometric symmetries revealed. It is then shown that, in the first order range of small strokes, optimal periodic strokes are ellipses embedded in 3d space, i.e. closed curves of the form $t\in [0,2\pi] \mapsto (\cos t)u + (\sin t)v$ for suitable orthogonal vectors $u$ and $v$ of $\mathbb{R}^3$. A simple analytic expression for the vectors $u$ and $v$ is derived. The results of the paper are used in a second article where the real physical dynamics of $\text{sPr}_3$ is analyzed in the asymptotic range of very long arms.


Introduction
The problem of swimming at low Reynolds number is a question of considerable biological and biomedical relevance which deserves also great interest from the point of view of fundamental science. Starting from the pioneering works of Taylor [Tay51], Lighthill [Lig52] and Purcell [Pur77], the problem has received a lot of attention in recent years 1 . This problem is both surprising and attractive due to the fact that one focuses on swimmers of very small size: Indeed as pointed out by Taylor [Tay51] and Purcell [Pur77] the physics of swimming at length scales of a few micrometers is very dierent from our common macro-scale experience. The most striking dierence is reected by the low value of the Reynolds number. The Reynolds number Re = uL/ gives an estimate for the relative importance of inertial to viscous forces for an object of typical length scale L moving at speed u through a Newtonian uid of density and dynamic viscosity . In applications the velocity u can rarely exceed a few body lengths per second and therefore, if one focuses on the swimming processes taking place in a given uid, the Reynolds number is entirely controlled by L. At small L, inertial forces are negligible and, in order to move, micro-swimmers can only exploit the viscous resistance of the surrounding uid.
This implies that microorganisms, such as bacteria, are required to take swimming strategies completely dierent from those employed by larger organisms, such as sh. In particular, the observation that, in a ow regime obeying Stokes equations, a scallop cannot advance through the reciprocal motion of its valves is encapsulated by Purcell's scallop theorem [Pur77]. The mathematical explanation for this is in the symmetry of the Stokes equations under time reversal: whatever forward motion will be produced by closing the valves, it will be exactly cancelled by a backward motion upon reopening them. This leads to the investigation of the simplest mechanisms capable of self-propulsion at small spatial scales. By this, we mean the ability to advance by performing a cyclic shape change a stroke in the absence of external forces. Several proposals have been put forward and analysed (see, e.g. [Tay51], [Pur77], [NG04], [AGK04], [BKS03] and [LM09]).
The basic problem of swimming can be stated as follows: given a periodic history log of shape changes of a swimmer (a stroke), predict the corresponding history of positions and orientations in space. A related question concerns its controllability i.e., given an arbitrary initial position and orientation, its possibility to achieve any prescribed position and orientation in space by the means of a suitable sequence of strokes. Indeed, the unusualness of low Re swimming is in that, since inertia forces are negligible, reciprocal shape changes do not contribute to a net motion, and the question of controllability is far from being trivial when the swimmer is constrained, like in the case of a scallop, to few degrees of freedom to modify its shape. In that respect, the scallop theorem [Pur77] is precisely a result of non-controllability for swimmers having only one degree of freedom at their disposal.
Once controllability is known, it is then natural to investigate how to reach this conguration change at minimal energetic cost. This is a question of optimal control from the mathematical point of view, and a question of natural selection from the point of view of fundamental science, which has a fundamental role in the engineering design of micro-swimmers. Indeed, as pointed out in [AGK04]: Microbots must swim much faster than bacteria if they are to interface with the macroscopic world. A micron-size robot swimming 100 times as fast as a bacterium, at the modest speed of 1mm per second, has Reynolds number Re = O(10 ¡3 ) and, since power scales like u 2 , consumes 10 4 more power than a bacterium. Micro-swimmers must therefore attempt to swim as eectively as possible, and the problem we address is to look for optimal swimming styles.
In this paper, we focus on a very interesting micro-swimmer introduced in [LM09], hereinafter referred to as the parking 3-sphere swimmer (sPr 3 ), which can be thought as the complementary version of Purcell's three sphere rotator introduced in [DBS05]. Full controllability of sPr 3 , as well as of a wide class of other model swimmers, has been rigorously proved in [ADH+13], where also a method to numerically address the optimal control problem has been presented. Analytical investigations of the optimal control problem have only been faced for micro-swimmers which, although resembling a 3d object, are constrained to live in a 1d-like world because capable to perform a net displacement of their center of mass only along a given (1d) direction [ADL08, ADL09]. On the other hand, as in the case of sPr 3 , when the net displacement of the swimmer can take place in a plane, a greater number of control variables is present and the analysis is more involved.
The aim of this paper is to analytically address the optimal control problem for sPr 3 in the range of small strokes. In that respect the main result of the paper is Theorem 12 which reveals the complete structure of the optimal stroke in terms of mechanical energy which produces a given displacement, both in translation and rotation. They turn out to be planar ellipses.
The rest of the paper is organized as follows: in section 2 we give both a geometric and a kinematic description of parking 3-sphere swimmer (sPr 3 ); we then introduce the control system object of this paper. In section 3 we investigate the geometric structure of the control system by exploiting the symmetries it has to satisfy due to the governing Stokes equations. In section 4 we reveal the structure of control system in the range of small strokes. Finally, section 5 is devoted to the characterization of the energy minimizing strokes.
The balls do not rotate around their axes so that the shape of the swimmer is characterized by the three lengths 1 ; 2 ; 3 of its arms, measured from c to the center b i of each ball. However, the swimmer can freely rotate around c in the horizontal plane containing the jacks. Eventually, due to the symmetries of the system, the swimmer stays in the horizontal plane.
Thus, the geometrical congurations assumed by the swimmer can be described by two set of variables: The vector of shape variables := ( 1 ; 2 ; 3 ) 2 M := ¡ 2a / 3 p ; 1 3 R + 3 from which relative distances (b ij ) i;j 2N3 between the balls are obtained, the lower bound in M being chosen in order to avoid overlaps of the balls.
The vector of positions variables, denoted by p = (c; ) 2 R 2 R which describe the global position and orientation in space of the swimmer.
More precisely, we consider the reference equilateral triangle (convexly) spanned by the unit vectors z 1 ; z 2 ; z 3 2 R 2 , with z 1 := (1; 0), z 2 := R T (2 / 3)z 1 , z 3 := R(2 / 3)z 1 where R() stands for the planar rotation of angle given by the matrix: cos ¡sin sin cos : (1) Position and orientation in the plane are described by the coordinates of the center c 2 R 2 and the angle that one arm, say arm number 1 (here and hence after denoted by k 1 ), makes with the xed direction z 1 (cf. Figure 1). Therefore we place the center of the ball Figure 1. The swimmer sPr 3 is composed of three spheres of equal radii. The three spheres can move along three horizontal axes that mutually meet at c with angle 2 /3. The spheres do not rotate around their axes so that the shape of the swimmer is characterized by the three lengths 1 ; 2 ; 3 of its arms, measured from the origin to the center of each ball. However, the swimmer may freely rotate around c in the horizontal plane.
The swimmer is fully described by the parameters (; p) 2 M R 3 . Indeed, once denoted by B a the ball of R 3 centered at the origin and of radius a, for every r 2 @B a , the position of the current point on the i-th sphere of the swimmer in the state ( ; p) is given, for every (; p; r) 2 M R 3 @B a , by the function (2) Note that the functions (r i ) i2N3 are analytic in M R 3 , and we use them to compute the instantaneous velocity on the i-th sphere B i , which for every ( ; p; r) 2 M R 3 @B a and every i 2 N 3 reads as with z i ? := R( /2)z i and r ? := R( /2)r.
It has been proved in [ADH+13] that it is possible to control the state of the system sPr 3 (i.e. both the shape and the position p) using as controls only the rate of shape changes _ . To achieve this, one has to understand the way p varies when one changes _ . This is done by assuming selfpropulsion and that swimmer's inertia is negligible, which imply that the total viscous force and torque exerted by the surrounding uid on the swimmer must vanish. More precisely, and we refer to [ADH+13] for the details, the control system can be written under the control form where we have used the standard notation e 3 := (0; 0; 1) 2 R 3 and denoted by := pe 3 , the angle that the arm k 1 make with the xed direction z 1 := (0; 1). Let us note that the control system F does not depend on c because of translational invariance of Stokes problem. On the other hand, translational invariance is just one of the symmetries which sPr 3 is subject to. Aim of the next section is to reveal the structure of the control system F as a consequence of the symmetries it must satisfy being governed by Stokes equations.

Symmetries
For any initial condition p 0 := (x 0 ; y 0 ; 0 ) and for any control curve : I R ! M , with I neighbourhood of zero, we denote by (p 0 ; ): I ! R 3 the solution associated to the dynamical system

Rotational invariance
Let us denote by R() the rotation matrix that rotates by an angle about the e 3 := (0; 0; 1) axis: Rotational invariance of Stokes equations entails that the solution of the dynamical system (5) must be invariant with respect to planar rotations, i.e. that for any 2 R where we have denoted by P : (x; y; ) 7 ! (x; y; 0) the projection of the generic point p of the state space, onto the plane generated by the vectors e 1 and e 2 . Once introduced the operator A() := [Id ¡ R()]P , the condition of rotational invariance can be stated in the form: ) is a solution of the control system (5) then so is (p 0 + e 3 ; ) and we can write (9) in the following form which will be useful later Remark 2. Here and in the sequel, we have preferred to state the symmetry relations satised by sPr 3 as hypotheses on the solution . The rationale behind this is in that the results work for any control system possessing the same symmetries of sPr 3 , i.e. regardless of whether these hypotheses are guaranteed, as in the case of sPr 3 , from the invariance of Stokes equations under a certain group of transformations.
Proof. Let (p 0 + e 3 ; ) be a solution of the control system (5). For every 2 R we have On the other hand, by rst using (9) and then (5), we get be arbitrarily chosen and therefore F (p 0 e 3 + ; 0 ) = R()F (p 0 e 3 ; 0 ) for every 2 R. Eventually, referring to the initial (angular) position p 0 e 3 = 0 we get (11).

Interchanging two arms
Let us consider the control system (5) which, due to the rotational invariance, can be written in the form (cf. (11)): We want to investigate the eect that a swap of the arms has on the generic solution (p 0 ; ). This amounts to understand how the solution behaves with respect to the action of the symmetric group S 3 , as far as we identify its elements with the arms of sPr 3 . Since the two transpositions (k 1 !k 2 ) and (k 2 !k 3 ) generate S 3 , it is sucient to consider them. In what follows we focus on the transposition (k 1 !k 2 ). For the transposition (k 2 !k 3 ) we just state the result, the proof being identical. Let us denote by S( ) the (orientation reversing) symmetric matrix associated to a reection of a vector v 2 R 2 f0g with respect to a horizontal line of R 2 f0g which passes through the origin and make an angle with the e 1 axis. Let us also denote by S ? ( ) the orientation preserving counterpart. In coordinates Next, let us set L := [e 2 |e 1 |e 3 ]. Note that when the reection L is applied to the geometric domain described by a swimmer in the initial angular state 0 = ¡ /6, the result (up to a translation) is just a swap of the arms k 1 and k 2 of sPr 3 , the arm k 3 remaining (still up to a translation) the same (cf. Figure 2). Stokes equations then justify the following Condition 4. (Swap (k 1 !k 2 )) Let the initial position be p 0 := (x 0 ; y 0 ; ¡ /6). If (p 0 ; L) is a solution of the control system (5), then so is (S( /2)p 0 ; ) and the following relation holds Remark 5. Physically speaking, the previous hypothesis is a consequence of the invariance of Stokes equations with respect to the observation point (cf. Figure 2). Indeed, an observer watching the dynamics (p 0 ; ) of sPr 3 projected on a glass, imposes to another observer, lying on the other side of the glass, to watch the dynamics (p 0 ; L) of a micro-swimmer obtained from sPr 3 by inverting arms k 1 and k 2 .

The Control System in the range of small strokes
Let us restart from (14). The response of the control system is governed by the matrix valued function F : R M ! R 33 which, due to Proposition 3, can be factorised as: In what follows we suppose that := 0 + with 0 2 M having all its components equal, and we set F 0 () := F ( 0 + ). Since F is an analytic function (cf. [ADH+13]) we can write the rst order expansion 2 with F 0 := F ( 0 ) 2 R 33 and H 0 2 L(R 3 R 3 ; R 3 ) representing the rst order derivative of F 0 at = 0. The aim of this section is to reveal the structure of the zeroth and rst order terms of the expansion (26), in view of the symmetry conditions that F 0 must satisfy due to Propositions 6 and 8, i.e: Let us prove the following Lemma 9. Let us suppose that for some matrices A and S A we have F 0 (A) = S A F 0 ()A for every 2 M. Then necessarily and H 0 ((A) ) = S A H 0 ( (A)) 8 ; 2 R 3 : Proof. Evaluating the condition F 0 (A) = S A F 0 ()A at = 0, we get (28). Next, By setting := A into the expansion (26) we get Therefore from (26) and (30) we get 2. Here, for notational convenience, we make use of the universal factorization property of tensor spaces in terms of multilinear maps (cf. [Sch75]). and hence (29).

The zeroth order term F 0
Applying Lemma 9 to the two matrix actions A := L and A := M (cf. (27)) we respectively get the two conditions F 0 = S L F 0 L and F 0 = S M F 0 M , which constitute a linear system of matrix equations in the unknown F 0 . A direct computation shows that the space of solutions is one-dimensional, and has the following structure: In connection with optimality questions, it is convenient to introduce the following orthogonal basis of R 3 : 1 := (0; ¡1; 1) ; 2 := 1 3 p (¡2; 1; 1) ; 3 := (1; 1; 1): The matrix F 0 then reads as F 0 = a 3 p

The rst order term H H H H H H H H H 0
Evaluating (29) on the basis (e i e j ) i; j 2N3 we get H 0 ((Ae i ) e j ) = S A H 0 (e i (Ae j )) for every i; j 2 N 3 . In particular, if the action of A on the ordered basis (e 1 ; e 2 ; e 3 ) consists in a permutation of the ordered basis, i.e. if Ae i = e (i) for every i 2 N 3 and a suitable permutations 2 N 3 ! N 3 , then the structural condition (29) in Lemma 9 reads as In particular we have: Proof. It is sucient to recall that F 0 must satisfy relations (27) and that Le i = e L (i) , Me i = e M (i) for every i 2 N 3 .
To reveal the structure of H 0 it is necessary to compute the solution space of the system of 18 vector equations (i.e. of 54 scalar equations) given by (35)-(36). This laborious task can be addressed via a symbolic mathematical computation program or, for the brave, with the aid of some clever observation. The rst thing to observe is that what we are really interested in, is the structure of the matrices A k dened for any k 2 N 3 by the position A k := (H 0 (e i e j ) e k ) i;j 2N 3 . Indeed, for any ; 2 R 3 the vector H 0 ( ) 2 R 3 is given by Next we observe that since S L and S M are idempotent matrices multiplying both members of (35) and (36) by S L and S M we respectively get the relations H 0 (e i e L (j) ) = S L H 0 (e L (i) e j ) and as well as := 3 2 H 0 (e 3 e 2 ) e 1 and := H 0 (e 1 e 1 ) e 1 , one gets and In particular, the skew-symmetric parts (denoted by M 1 ; M 2 ; M 3 ) of the matrices A 1 ; A 2 ; A 3 which, as we shall see, are the only ones to contribute to the net displacement of the micro-swimmer, are given by Let us note that the orthogonal basis (33) is orientation-reversing and that the matrices in (41) can be characterized by the actions

The linearized control equations
In what follows we denote by I the closed interval  (37), if we set = 0 + , the behaviour of the system around = 0, up to higher order terms, is given by In particular, denoting by (c; ) 2 R 2 R the components of p and taking into account that F 0 T e 3 = 0 because of (32), we get which can be easily integrated: Hence the net angular displacement corresponding to a unit stroke is given by Since the dynamics of does not depend on the one of c, it is convenient to consider the 2d projections of the matrices R() and F 0

T
. In what follows we denote by ê 1 ; ê 2 the standard basis of R 2 and to lighten notation, since no confusion may arise, we still denote by R() and F 0 the 2d projections so that the dynamics of c is described by the system Next, we observe that the rst order expansion of the 2d rotation matrix R() around = 0 gives and therefore, up to higher order terms in , we get In physical terms: in the limit of small strokes around a constant reference shape 0 the net displacement is given by (50).
Proof. We rst note that the term hF 0 _ i is zero by periodicity. Next we observe that is sucient to prove the estimate for the scalar terms of the form (; ) _ i and h(; )A j _ i with i 2 N 3 , j 2 N 2 and Let us focus on the terms of the form (; ) _ i , the estimate concerning the other ones can be treated in exactly the same way. We have Z The Sobolev-Morrey embedding H ] gives the existence of a c S > 0 such that k k 1 6 c S k k H ] 1 for every 2 H ] 1 (I ; R 3 ). Therefore for some c > 0 depending on M 3 only, we get Z The proof is complete.
Collecting (44) and (50) we may therefore assume the net displacement p := p(2) ¡ p(0), undergone by the position p of sPr 3 in correspondence to a small stroke , to be given by (cf. (42)) where in writing the previous expression we have taken into account that only the skew-symmetric part of the matrices A k contribute to the displacement. Indeed, if we denote by A k sym the symmetric part of the matrix A k , integrating by parts we get hA k sym _ i = hA k sym _ i = ¡ hA k sym _ i and therefore hA k sym _ i = 0 for any 2 H ] 1 (I ; R 3 ).

Energy minimizing strokes
Following the swimming eciency suggested by Lighthill [Lig52], we adopt the following notion of optimality: energy minimizing strokes are the ones that minimize the kinematic energy dissipated while trying to reach a given net displacement p 2 R 3 in one stroke. In mathematical terms, the total energy dissipation due to a stroke 2 H ] 1 (I ; R 3 ), : I ! M , can be evaluated by the means of a suitable quadratic energy functional in which the energy density g 2 C 1 (R 3 ) is a function taking values in the space of symmetric and denite positive matrices of R 33 (cfr. [DAL12]). In the limit of small strokes one considers the energy G as arising from the approximation g() = g(0) + O(1), in which g(0) 2 R 33 is a symmetric and denite positive matrix. Namely where we have denoted by Q g () the denite positive quadratic form given by Q g () := g(0) . For the same symmetry reasons discussed in section 3, for every 2 R 3 the function Q g must satisfy the relations Q g (L) = Q g (M) = Q g (); in which L = [e 2 |e 1 |e 3 ] and M = [e 1 |e 3 |e 2 ]. A straightforward computation shows that the previous conditions imply the existence of two parameters h and > max (h; ¡2h), such that the symmetric and denite positive matrix G which represents Q g is given by Let us note that one has G 1 = ( ¡ h) 1 , G 2 = ( ¡ h) 2 and G 3 = ( + 2h) 3 , so that, up to a rescaling, the orthogonal basis ( 1 ; 2 ; 3 ) is invariant and orthogonal with respect to G. It is convenient to denote by g 1 := g 2 := ( ¡ h) and g 3 := ( + 2h) the eigenvalues of G. Since G is symmetric and denite positive, the coecients (g 1 ; g 2 ; g 3 ) can be interpreted as the metric coecients of the inner product dened for any a; b 2 R 3 by (a; b) g := 2 g a b, with g := diag(g i ).
Let us observe that the metric coecients (g i ) i2N3 permit to diagonalize G in the form with ( 1 ; 2 ; 3 ) being the normalization of ( 1 ; 2 ; 3 ) with respect to the usual Euclidean metric. It is worth noting that the basis ( 1 ; 2 ; 3 ) is both orthogonal and g-orthogonal.
After that we can write the following equivalent expression of the functional G: We aim at minimizing G in H ] 1 (I ; R 3 ) subject to a prescribed net displacement p 2 R 3 , i.e. (cf. (53)) subject to the constraint X with h 1 /j 1 j = h 2 /j 2 j = and h 3 /j 3 j = . More precisely, we are going to prove the following As an immediate consequence of Proposition 15 we get Corollary 16. The minimization problem for F in`2(R 3 ) `2(R 3 ), subject to the constraint (79), is equivalent to the minimization in R 3 R 3 of the function f (u; v) := 1 2 juj 2 + 1 2 jv j 2 (85) subject to the constraint v u = !: Proof. It is sucient to observe that if we denote by V ! the subset of (u; v) 2`2(R 3 ) `2(R 3 ) satisfying the constraint relation (79), and by V ! the subset of (u; v) 2 R 3 R 3 such that v u = !, then from Proposition 15 we have with e 1 = ( n 1 ) n2N and hence e 1 u = ( n 1 u) n2N .

The nite dimensional minimization: minimization of f
Proposition 17. Any couple of vectors (u ? ; v ? ) 2 R 3 R 3 minimizing the function f given by (85) and subject to the constraint (86), is characterized by the following conditions ju ? j 2 = jv ? j 2 = j!j and u ? v ? = 0: Therefore, for any 2 R 3 such that ! = 0 and j j 2 = j!j, the couple is a (global) constrained minimizer for f. Here, as in the previous section, we have denoted by ! the unit vector associated to !.

Proof.
To identify the minimizers of the problem (85)-(86), we note that for any ! = / 0, the constraint relation v u = ! implies the existence of a 2 I := (0; ) such that juj jv j sin = j!j. Therefore the constrained minimization for f is equivalent to the unconstrained minimization of the function f: (u; ) 7 ! 1 2 juj 2 + 1 2 j!j 2 juj 2 (sin ) 2 ; (90) whose stationary points (u ? ; ? ) 2 R 3 I are characterized by the conditions ? = 2 and ju ? j 2 = j! j. Therefore if (u ? ; v ? ) 2 R 3 is a minimizer for the function f expressed by (85), then necessarily ju ? j 2 = jv ? j 2 = j!j, u ? v ? = 0. That this condition is also sucient is an immediate consequence of the fact that for any such points one has f(u ? ; ? ) = j!j. Indeed, for any couple (u; ) 2 R 3 I one has f(u; ) > 1 2 juj 4 + j!j 2 juj 2 = j!j + 1 2 (j!j ¡ juj 2 ) 2 juj 2 > j!j = f(u ? ; ? ): This concludes the proof of the statement. For the second part we simply note that any (u ? ; v ? ) 2 R 3 can be characterized in terms of a vector orthogonal to !. Precisely, we set ! := ! /j!j and consider a vector 2 R 3 such that ! = 0 and j j 2 = j!j: