 Survey
 Open Access
SubRiemannian geometry, Hamiltonian dynamics, microswimmers, copepod nauplii and copepod robot
 Bernard Bonnard^{1, 2},
 Monique Chyba^{4}Email author,
 Jéremy Rouot^{3} and
 Daisuke Takagi^{4}
https://doi.org/10.1186/s4073601800369
© The Author(s) 2018
Received: 7 February 2018
Accepted: 2 May 2018
Published: 22 June 2018
Abstract
The objective of this article is to present the seminal concepts and techniques of SubRiemannian geometry and Hamiltonian dynamics, complemented by adapted software to analyze the dynamics of the copepod microswimmer, where the model of swimming is the slender body approximation for Stokes flows in fluid dynamics. In this context, the copepod model is a simplification of the 3link Purcell swimmer and is relevant to analyze more complex microswimmers. The mathematical model is validated by observations performed by Takagi’s team of Hawaii laboratory, showing the agreement between the predicted and observed motions. SubRiemannian geometry is introduced, assuming that displacements are minimizing the expanded mechanical energy of the microswimmer. This allows to compare different strokes and different microswimmers and minimizing the expanded mechanical energy of the microswimmer. The objective is to maximize the efficiency of a stroke (the ratio between the displacement produced by a stroke and its length). Using the Maximum Principle in the framework of SubRiemannian geometry, this leads to analyze family of periodic controls producing strokes to determine the most efficient one. Graded normal forms introduced in SubRiemannian geometry to evaluate spheres with small radius is the technique used to evaluate the efficiency of different strokes with small amplitudes, and to determine the most efficient stroke using a numeric homotopy method versus standard direct computations based on Fourier analysis. Finally a copepod robot is presented whose aim is to validate the computations and very preliminary results are given.
Keywords
Introduction
SubRiemannian (SR) geometry in the framework of geometric optimal control was first explored in the seminal article [17]. This article also contains the main geometric ingredients developed in our article in relation with microswimmers : the HeisenbergBrockettDido model and the evaluation of conjugatecut loci and small SRspheres using normal coordinates. These techniques were developed later in the context of singularity theory to obtain more precise computations of asymptotics of the conjugate and cut loci, in a series of articles dealing with the socalled contact Darboux case [19] and Martinet case [1]. A consequence of SRgeometry is that the singularities of the exponential mapping accumulate for small lengths and can be estimated. Clearly, this is the starting point to produce numerical computations for larger lengths by using continuation and numerical methods. Also, early on in the analysis of SRgeometries, complicated singularities (not in the “analytic” category) were recognized due to the existence of the socalled abnormal geodesics. This was a major technical problem to further developments for the computational techniques. Note that the similar problem clearly stopped the activity in the fifties of the standard calculus of variations [10].
An application of SRgeometry was identified in [33] which describes the geodesics motions of a charged particle in a 2DRiemannian manifold under the influence of a magnetic field. In this framework, closed geodesics calculations correspond precisely to stroke’s computations for microswimmers. This is a generalization of the Dido problem in calculus of variations and a very technical study [2] using tools developed in [19], has provided the asymptotics of the conjugate and cut loci. A pause for those computations based on symbolic software was observed but a revival is now motivated with the development of a specific software (see [22]) based on numerical continuation methods in optimal control [3]. This has already many applications in the areas of aerospace or quantum control problems [15, 47]. Our study is a further step in this direction.
Microswimmers were popularized by the seminal presentation [38] and were analyzed very recently using optimization techniques in a series of articles [4, 5], motivated in particular by industrial applications in robotics to design microrobots whose control is based on the swimming mechanisms of biological microswimmers. In particular a recent model was proposed in [41] to analyze the observed motions of an abundant variety of zooplanktons called copepods. Assuming the motion is performed to minimize the mechanical energy dissipated by the swimmer, the problem of determining the most efficient stroke can be analyzed in the framework of SRgeometry.
It can be compared with the standard methods in fluid dynamic, using direct optimization methods, and based on Fourier expansions to represent strokes [42]. Our method of analysis developed for the copepod model is entirely different and uses an indirect optimization method. The candidates as minimizers are parameterized as geodesics of the associated SRproblem using the Maximum Principle, and already a discrimination is obtained between the normal geodesics associated to smooth periodic controls and a unique abnormal geodesic whose shape corresponds to a triangle and observed in [41] as a pattern of the copepod strokes. The concept of efficiency can be reformulated in SRgeometry as the ratio between the displacement produced by a stroke and its length. A consequence of the Maximum Principle and the socalled transversality condition is that the most efficient stroke can be computed using a numerical shooting. It can be also determined using numerical continuation techniques starting from a center of swimming (a center of swimming being a point from which are emanating strokes with small amplitudes). The latter is computed as an invariant of SRgeometry. This study is related to the conjugate and cut loci computation [2] which are also important to analyze optimality and convergence of the numerical optimization methods. Direct optimization methods implemented in the Bocop software [11] were applied in [8] to initialize the continuation starting near the abnormal stroke with maximum amplitude. Note that our approach is closely related to the seminal work in celestial mechanics by Poincaré [36] combining already direct and indirect approaches in variational analysis. In particular our result is reminiscent of the PoincaréLyapunov theorem where periodic trajectories are obtained by continuation as a one parameter family of periodic trajectories with small amplitudes emanating from a center [32].
Structure of the paper. In section 2, the various swimmer’s models are presented using Resistive Force Theory approximation [24], ([21], Chapter 5) governing the swimmer mechanism at low Reynolds number where the interaction with the fluid is reduced to a drag force [6] and the swimmer is represented using a slender body approximation. The remaining of the section is to describe the geometric framework of the problem and the associated optimal control problem. The section 3 is a selfcontained presentation of SRgeometry to describe the concepts and technical tools necessary to our study. The following concepts are borrowed from geometric optimal control: Maximum Principle, Second order necessary optimality conditions (Jacobi equation, notion of conjugate point). The remaining ones are specific to SRgeometry: nilpotent approximation, generic graded normal forms and applications to estimate the SRspheres with small radius. Section 4 is devoted to the complete analysis of the copepod swimmer, applying the mathematical tools mentioned above supplemented with numerical simulations to compute the center of swimming for strokes with small amplitudes and to eventually determine the most efficient stroke. The final section is devoted to present the agreement between control practitioners and experiments. It is realized in two steps. First of all, to validate the model of swimming at low Reynolds number (using slender body theory for stokes flow), we present the agreement between predicted and observed displacements of the copepod nauplii (a three pair of symmetric links swimmers). Secondly to compare different stroke and different swimmers a macroscopic copepod robot is finally described to validate in a next future the agreement between control computation, simulations and experiments. Preliminary results are presented.
Microswimmers and the geometric framework
This section is a selfcontained presentation of microswimmers related to our study and the standard geometric properties contained in the literature which allows for a neat analysis in the framework of direct optimization methods.
2.1 nlinks swimmers
2.1.1 Stokes model
Definition 1
A (smooth) stroke is a periodic motion t↦θ(t) produced by a (smooth) periodic control t↦u(t), with period T>0.
For n=1, the model depicts a scallop where each stroke produces a zero displacement. This is known in the literature as the famous scallop theorem.
Hence at least two legs, n≥2, are required to produce a positive displacement and the copepod observed in Fig. 1 corresponds to n=3 for which the swimming mechanism is described and analyzed in [29, 41].
Our objective is to make a complete analysis of the case n=2, in the framework of SRgeometry. Note that this case was also obtained in [5] as a limit case of the symmetric Purcell swimmer, with equal armslegs lengths and zero central link length. It was analyzed by direct optimization methods.
Models of symmetric swimmer encountered in the literature
∙ Scallop  \(\dot x_{0} = \frac {\dot \theta \,\sin \theta }{2\cos ^{2}\theta }\)  
∙ Symmetric 2links Copepod  \(\dot x_{0} = \frac {\sum _{i=1}^{2} \dot \theta _{i} \sin \theta _{i}}{\sum _{i=1}^{2} \left (1+\sin ^{2}\theta _{i}\right)}\)  
∙ Symmetric Takagi nlinks equal lengths L=1  \(\dot x_{0} = \frac {(1+2r)\,\sum _{i=1}^{n} \dot \theta _{i} \sin \theta _{i}}{3/2\,r\log (2/r)\sum _{i=1}^{n} \left (1+\sin ^{2}\theta _{i}\right)}\)  
∙ Symmetric Purcell (AvronRaz),  \(\mathrm {d} x_{0} = \frac {\mathrm {d} \xi _{1}+\mathrm {d} \xi _{2}}{4+c\xi _{1}^{2}\xi _{2}^{2}}\)  
2links equal lengths L=1  \(\xi _{i} = \cos \theta _{i},\; c=l_{0}/2\)  
Limit case: c=r=0⇒ Takagi = Purcell. 
2.1.2 Physical limitations and state constraints
A singularity of the models occurs when two of the links are colliding. To avoid this issue, we impose the following state constraints defined by the triangle of constraints: \({\mathcal {T}} = \{\theta,\; 0\le \theta _{1}\le \theta _{2} \le \pi \}\). The physical interpretation is well described in the literature. It is the socalled “Coxtheory”: to neglect the physical interaction between the links they must be at a minimal distance from each other [23]. From the mathematical point of view, (1) can be analytically extended to the 2dimensional plane where \(\mathbb {R}^{2}\) is taken as the covering space of the θspace: the torus \(\mathbb {T}^{2}\). Hence \({\mathcal {T}}\) is considered as a state constraints and it will be shown that this constraint is not active in our study, i.e. we will demonstrate that an optimized stroke satisfies the Cox conditions by estimating the distance to the triangle.
Note also that the extension of the dynamics to \(\mathbb {R}^{2}\) leads to preserve symmetries with respect to the sides of the triangles generated by σ_{1}:(θ_{1},θ_{2})↦(θ_{2},θ_{1}), σ_{2}:(θ_{1},θ_{2})↦(−θ_{1},θ_{2}). Another observed symmetry preserving the triangle constraint is given by: σ_{3}:(θ_{1},θ_{2})↦(π−θ_{2},π−θ_{1}). The group generated by such symmetries is denoted Σ.
2.1.3 Riemannian metrics in the shape variables
2.2 Geometric framework
Having introduced the model, some insights about the problem can be derived from the geometric framework that we present next. This is crucial to understand the problem in relation with standard similar studies, see for instance [5, 33], for more details.
2.2.1 The swimming curvature
Lemma 1
 1
dω=−f(θ)dθ_{1}∧dθ_{2}
 2
dω<0 in the interior of the triangle \(\mathcal {T}\), and dω vanishes on the boundary of \(\mathcal {T}\).
This leads to introduce the following concepts.
Definition 2
 1
A geometric 2link microswimmer is defined by (dω,g).
 2The swimming curvature is defined as the ratio:$$SK = \frac{\mathrm{d} \omega}{\omega_{g}} = \frac{f(\theta)}{\sqrt{E(\theta)F(\theta)G(\theta)^{2}}}. $$
 3The geometric efficiency of a stroke γ is the ratio between the displacement and the length l:$${\mathcal{E}}(\gamma) =(x_{0}(T)x_{0}(0))/l(\gamma). $$
2.2.2 Geometric optimal control problems
Problem 1 The first problem which which can be phrased in the framework of SRgeometry is to fix the initial condition q(0)=(x_{0}(0)=0,θ(0)) and compute the SRspheres of radii r to identify the closed curves corresponding to smooth periodic controls producing a desired displacement x_{0}(T). This formulation leads to the AmbroseSinger theorem in relation with the Chow theorem in control theory, see [33].
Problem 2 The second problem is to compute the most efficient stroke.
Both problems lead to Mayer problems in optimal control theory.
Problem 1 The cost is taken as C=q_{0}(T) and we have x_{0}(T)=x_{ T } where x_{ T } is given.
Problem 2 The cost is taken as C=−x_{0}(T)/q_{0}(T).
Remark 1
In the standard literature [30,42] the efficiency is the ratio between the square of the displacement and the energy (vs length). Parameterizing by arclength leads to proportional quantities and similar minimizers. It allows for different kind of strokes for one species (e.g. copepod) or different species to determine the time minimizer (winning the competition).
Program The work is clear, we must conduct a careful analysis in the framework of SRgeometry based on the mathematical analysis of the geodesics equation. It needs to be supplemented by (simple) numerical simulations for solving problems 1 and 2.
A review of SRgeometry in relation with microswimmers
SRgeometry is a very active area of research and we refer the reader to [25] for a recent and more complete presentation. Our task is limited to a specific problem and we shall restrict our presentation to the useful concepts and results of this large area. The main concepts and seminal results were already available at the end of the nineties and general and useful references are [7,27]. Other tools are borrowed from singularity theory, we refer the reader to [31] for a general reference and to [45] for the application to Legendrian and Lagrangian singularities.
3.1 General results and concepts in SRgeometry
We can choose (locally) an orthonormal frame {F_{1},…,F_{ m }} for the distribution D so that S=Id, which from the control point of view means to apply a feedback u=β(q)v where β is a (smooth) invertible matrix.
Candidates as minimizers can be selected using the Maximum principle [37] which we recall in the next section and which we apply to our minimization problems.
3.2 Maximum principle
We refer the reader to [43] for a complete presentation. For our purpose, we need the following framework.
3.2.1 Optimal control formulation and geometric concepts
We denote by \(t\mapsto \tilde q(t,\tilde q(0),u)\) the solution associated to the control u(·) and initiated from (q_{0},0), and we assume it is defined on [0,T]. The extremity mapping is defined as the map: \(E:u(\cdot) \in {\mathcal {U}} \mapsto \tilde q(T,\tilde q(0),u)\) where \(T,\tilde q(0)\) are fixed. The image of E is the accessibility set: \(A(\tilde q(0),T) = \underset {u\in {\mathcal {U}}}{\cup } \tilde q(T,\tilde q(0),u)\).
Next we recall the necessary optimality conditions associated to the Mayer problem.
3.2.2 Weak Maximum principle and transversality conditions
Extremality conditions. The first conditions express the fact that the solution \(\tilde q(T)\) associated to a minimizing control u(·) belongs to the boundary of the accessibility set and corresponds to a singularity of the extremity mapping. The result in the optimal control theory literature is known as the weak Maximum Principle [14] and is an Hamiltonian formulation of the Lagrange multiplier rule in the classical calculus of variations [10].
Proposition 1
If (u(·),q(·)) is a controltrajectory minimizer on [0,T], then there exist \(\tilde p=(p,{\lambda _{0}}) \in \mathbb {R}^{n}\times \mathbb {R} \setminus 0\) such that the (absolutely continuous) curve t↦z(·)=(q(·),p(·)) satisfies a.e. the equations:
Definition 3
H(z,u) is called the pseudoHamiltonian and \(\tilde p = (p,\lambda _{0})\neq {(0,0)}\) is the (cost extended) adjoint vector. A trajectorycontrol pair (z,u) is called an extremal and its projection q on the state space is called a geodesic.
where N_{ K } is the (limiting) normal cone to K and λ≥0.
Definition 4
Condition (7) is called the transversality condition. An extremal satisfying the boundary conditions and the transversality condition is called a BCextremal.
3.2.3 Application to the microswimmers.
Notice first that according to the weak maximum principle, we have two types of distinct extremals in SRgeometry.
Normal case. If λ_{0}≠0, it can be normalized to −1/2 (corresponding to minimizing the energy). Introducing H_{ i }(q,p)=〈p,F_{ i }(q)〉 and using \(\frac {\partial H}{\partial u}=0\), we obtain u_{ i }=H_{ i }(q,p). Substituting back this expression for these extremal controls into the pseudoHamiltonian H gives the (true) normal Hamiltonian \(H_{n}{(q,p)} = 1/2 \sum _{i=1}^{m} H_{i}^{2}{(q,p)}\). The corresponding extremals solution of \(\overrightarrow {H_{n}} = \left (\frac {\partial H}{\partial p},\frac {\partial H}{\partial q}\right)\) are called normal and their qprojections are called normal geodesics.
Abnormal case. If λ_{0}=0, the associated extremal control is defined by the (implicit) relations H_{ i }(q,p)=0,i=1,…,m. The corresponding extremals are called abnormal and their qprojections are called abnormal geodesics. A normal geodesic is called strict if it is not the projection of an abnormal geodesic.
Geometric remark. The abnormal extremals correspond to singularities of the extremity mapping associated to the control system \(\dot q = F(q,u)\) and do not depend on the cost.

Periodicity. This is expressed as θ(0)=θ(T) and it leads to the condition:$$ p_{\theta}(0) = p_{\theta}(T) $$(8)
to produce a smooth stroke.

Efficiency maximization. As a consequence of the Maupertius principle, and assuming that x_{0}(0)=0, we can suppose that the efficiency is expressed as \({\mathcal {E}}' = x_{0}(T)^{2}/E(\gamma)\) for a given stroke γ. If h=−x_{0}(T)/q^{0}(T), then the transversality condition (7) becomes:$$ (p_{0},\lambda_{0}) = \lambda \frac{\partial h}{\partial (x_{0},q^{0})} $$(9)
at the final point (x_{0}(T),q^{0}(T)).
This has the following interpretation: at the final point, the adjoint vector is normal to the level set h=c, where c is the maximal efficiency.
3.2.4 Chow and HopfRinow theorems
Proposition 2
 1
For each q_{0},q_{1}∈U there exists a piecewise smooth horizontal curve joining q_{0} to q_{1} corresponding to a piecewise constant control.
 2
Sufficiently near points q_{0},q_{1}∈U can be joined by a minimizing geodesic.
Application The first assertion is known as Chow’s theorem and can be found in [33]. Assuming the microswimmer starts at q(0)=(x_{0}(0),θ(0)) and that we fix the desired displacement to x_{0}(T)=x_{ d } in U. Then, there exists a piecewise constant control such that the microswimmer can reach the configuration (x_{ d },θ(0)). By construction this produces a closed curve in the θspace where T is a period (not necessary minimal). The second assertion is a local version of the standard HopfRinow existence theorem. It can be easily globalized under standard (completness) assumptions. Hence in our study we can restrict our analysis to (normal and abnormal) geodesic curves. The existence theorem is easily deduced when dealing with the maximal efficiency. Indeed, our state domain is bounded by the triangle \({\mathcal {T}}\) and a direct computation shows that for strokes with “small amplitudes”, the efficiency goes to zero with the amplitude A. A straightforward computation demonstrates that the triangle stroke corresponds to a low efficiency. Therefore, there exists a solution of the problem of maximizing efficiency.
3.2.5 Spheres with small radii and nilpotent approximations.
Definition 5
The SRsphere of center q_{0} with radius r is denoted by S(q_{0},r)={q; d_{ SR }(q_{0},q)=r}.
An important result in SRgeometry is the construction of the socalled privileged coordinates to estimate the size of the sphere with small radius [7,27].
Definition 6
Let D_{1}= span{F_{1},…,F_{ m }}, we define recursively D_{ k }=D_{k−1}+ span{[D_{1},D_{k−1}]} with n_{ k }(q_{0}) be the rank of D_{ k } at q_{0}. Assume that the rank condition holds: \(\dim \, D_{L.A.}(q_{0}) = n (=\dim \, T_{q_{0}}U)\) for each q_{0}. Consider the flag \(D_{1}(q_{0})\subset D_{2}(q_{0})\subset \ldots \subset D_{n_{r}}(q_{0})=D_{L.A.}(q_{0})\). Then n_{ r }(q_{0}) is called the degree of nonholonomy and the sequence (n_{1}(q_{0}),…,n_{ r }(q_{0})) is called the growth vector of the distribution D at q_{0}.
Using [27], we introduce the following notions.
Definition 7

\(\mu (f) = \min \, \{k,\, \mid \, \exists X_{1},\ldots, X_{k} \in D\; \text { such\; that } L_{X_{1}},\ldots, L_{X_{k}} f(q_{0})\neq 0\}\) where L_{ X }f denotes the Lie derivative of f w.r.t. X: \(L_{X}f = \frac {\partial f}{\partial q}\cdot X(q)\).

if f(q_{0})≠0, μ(f)=0 and μ(0)=+∞.
Definition 8
Let f be a germ of smooth function at q_{0}, f is called privileged at q_{0} if \(\mu _{f} = \min \, \{k;\,\mathrm {d} f_{q_{0}}(D^{k}(q_{0}))\neq 0\}\).
A coordinate system (q_{1},…,q_{ n }) defined on an open subset of U at q_{0}, identified to 0, is called privileged if the coordinates functions q_{ i }, i=1≤i≤n are privileged at x_{0}. If w_{ i } is the weight of q_{ i } at q_{0}=0, the induced weight of \(\frac {\partial }{\partial q_{i}}\) is −w_{ i }, and the weight of the dual variable p_{ i } in T^{⋆}U is −w_{ i }.
The following theorem can be found in [7].
Theorem 1
Let {F_{1},…,F_{ m }} be an orthonormal frame for the pair (D,g). Fix q_{0}∈U and let (q_{1},…,q_{ n }) be a privileged coordinates system at q_{0}=0, with weights w_{1},…,w_{ n }. Then, one can expand F_{ i } as \(\sum _{j\ge 1} F_{i}^{j}\), where \(F_{i}^{j}\) are homogeneous vector fields (for the weight systems) with degree ≥−1. Denoting \(\hat F_{i} = F_{i}^{1}\), the family \(\hat F_{i}\) generates a nilpotent Lie algebra with similar growth vector (at q_{0}=0). Moreover, for small q it gives the following estimate of the SRdistance: \(\phantom {\dot {i}\!}B(q_{1}^{1/w_{1}} + \ldots +q_{n}^{1/w_{n}}) \le d_{SR}(0,q) \le A(q_{1}^{1/w_{1}} + \ldots +q_{n}^{1/w_{n}})\), where A,B are constants.
3.2.6 Singularities of SRspheres with small radius
Definition 9
Let \(H_{n} {(q,p)}= 1/2 \sum _{i=1}^{m} H_{i}^{2}{(q,p)}\) the normal Hamiltonian and let \(\exp t\overrightarrow {H_{n}}\) denote the localone parameter group associated to \(\overrightarrow {H_{n}}\) with Π:(q,p)↦q be the standard projection. Assume q_{0} is fixed, the exponential mapping is given by the map: \(\exp _{q_{0}} : (t,p)\mapsto \Pi \left (\exp t\overrightarrow {H_{n}}(q_{0},p)\right)\).
Definition 10
Let γ(·) be a reference (normal or abnormal) geodesic defined on [0,T]. The time t_{ c } is called the cut time if γ is optimal up to t_{ c } but no longer optimal for t>t_{ c }, and q(t_{ c }) is called the cut point. Considering all geodesics starting from q_{0}, the set of cut points forms the cut locus denoted by C_{ cut }(q_{0}). The time t_{1c} is called the first conjugate time if the reference geodesic γ is optimal up to t_{1c} and no longer optimal for t>t_{1c} for the C^{1}topology on the set of horizontal curves, and the point γ(t_{1c}) is called the first conjugate point. The set of first conjugate points calculated over all geodesics forms the (first) conjugate locus and is denoted by C(q_{0}).
Conjugate points can be computed (under suitable assumptions) in the normal and abnormal case. In our study, we can restrict the analysis to the normal case and we have.
Proposition 3
A property of SRgeometry is the following.
Proposition 4
There exist conjugate points arbitrarily closed to q_{0}, and a consequence is that SRspheres with arbitrary small radius have singularities.
3.3 SRgeometry in dimension 3
Motivated by our microswimmer study, in this section we recall refined results related to singularities of threedimensional SRspheres with small radius, that is explicit conjugate and cut loci computation. Those results are the consequence of intense research activities in SRgeometry at the end of the nineties, see [19] for the contact case and [1] for the Martinet case. Here U is assume to be a neighbourhood of q_{0} identified to 0, and (D,g) is defined by the choice of an orthonormal frame {F_{1},F_{2}}. The distribution can be represented as D= kerω, where ω is a welldefined (up to a factor) oneform.
A first geometric result comes from [46].
3.3.1 Local oneform models

ContactDarboux case (Dido). In this case, the normal form is expressed as:$$\omega = \mathrm{d} z + (x\mathrm{d} y  y \mathrm{d} x). $$

Martinet case. The normal form is:$$\omega = \mathrm{d} z  \frac{y^{2}}{2}\, \mathrm{d} x. $$
3.3.2 Associated (graded) local model of SRmetric

local diffeomorphisms Q=φ(q) preserving zero,

feedback u=β(q)v where β(q) is restricted to the orthogonal group O(2) (so that \(u_{1}^{2}+u_{2}^{2} \mapsto v_{1}^{2}+v_{2}^{2}\)).
A normal form is a section on the set of orbits for the \({\mathcal {G}}\)actions, and Theorem 2 states that it can be performed either on the set of SRmetrics or on the set of (normal) Hamiltonians.
A standard method in singularity theory [31] is to linearize the calculations by working on the jet spaces and restricting to homogeneous transformations. This can be also be performed using a graded system of coordinates as the privileged coordinates in SRgeometry to obtain graded normal forms. Different algorithms exist in the literature, see [19] for the contact case, and [1] for the Martinet case.
We recall the results in the contact and Martinet case.

⋆ Nilpotent model. (HeisenbergBrockettDido). This is a model of order −1 (Dido form) and it is given by the orthonormal frame:$$\hat F_{1} = \frac{\partial}{\partial x} + y\, \frac{\partial}{\partial z}, \quad \hat F_{2} = \frac{\partial}{\partial y}  x\, \frac{\partial}{\partial z}. $$

⋆ Model of order zero. Keeping all the terms of order ≤0, we have Theorem 3.
Theorem 3
([17]) In the contact case, the model of order 0 is similar to the model of order −1.

⋆ Model of order 1. Keeping the terms of order ≤1, the model in [19] is given by:with w=(x,y) and Q is a quadratic form: Q=α x^{2}+2β xy+γ y^{2} where α,β,γ are parameters.$$F_{1} = \hat F_{1} + yQ(w)\frac{\partial}{\partial z},\quad F_{2} = \hat F_{2} xQ(w)\frac{\partial}{\partial z} $$
If H_{ F }(q,p)=〈p,F(q)〉, H_{ G }(q,p)=〈p,G(q)〉 one has {H_{ F },H_{ G }}(q,p)=〈p,[F,G](q)〉.
To compute the geodesics in the HeisenbergBrockettDido case we complete \(F_{1}=\hat F_{1},\, F_{2} = \hat F_{2}\) by \(F_{3} = \frac {\partial }{\partial z}\) to form a frame. We denote H_{ i }(q,p)=〈p,F_{ i }(q)〉, i=1,2,2 and instead of the symplectic coordinates (x,y,z,p_{ x },p_{ y },p_{ z }) we use (x,y,z,H_{1},H_{2},H_{3}).
since the Lie brackets of length ≥3 are zero.

λ=0. In this case z=0 and the geodesics contained in the plane (x,y) are lines.

λ≠0. An easy integration shows that in that case the geodesics are given by:$$ \begin{aligned} x(t) &=\frac{A}{\lambda} \left(\sin(\lambda t+\phi)  \sin \phi \right) \\ y(t) &= \frac{A}{\lambda} \left(\cos(\lambda t + \phi)  \cos\phi \right) \\ z(t) &= \frac{A^{2}}{\lambda} t  \frac{A^{2}}{\lambda^{2}} \sin(\lambda t) \end{aligned} $$(10)
with \(A = \sqrt {H_{1}^{2}+H_{2}^{2}}\) and ϕ is the angle of the vector \((\dot x,\dot y)\) at the origin.
In particular we can deduce the following geometric properties.
Proposition 5
 (1)
All the controls for λ≠0 are periodic with period 2π/λ.
 (2)
The corresponding (x,y) projections will form families of circles that are invariant by any rotation along the zaxis.
Interpreting these geodesics as strokes for the microswimmer (and z is taken the displacement variable). The displacement associated to a stroke being given by dz=−2dx∧dy and is proportional to the standard volume form in \(\mathbb {R}^{2}\).
Conjugate and cut loci. They can be easily computed from (10) and according to [19] they can be calculated restricting the exponential mapping to the (x,y)projection. We can prove the following proposition.
Proposition 6
If λ≠0, the first conjugate time occurs at 2π/λ and corresponds to the cut point. Hence, it occurs exactly at the period and the projection of the cut locus in the (x,y)plane degenerates into the origin.
Generalized Dido case. Conjugate and cut loci computations in the small radius case where generalized in [2] and this study is relevant in our problem. The main features are the following. In the Dido problem, due to the zsymmetry of revolution the projection of the conjugate and cut loci in the (x,y)plane is reduced to a single point. In the generalized Dido case, the SRproblem leads to compute conjugate and cut loci for Riemannian metrics on the sphere. This is related to the seminal result from [35].
Theorem 4
Let g be an analytic Riemannian metric on the 2sphere S^{2}. Then the cut locus of a point is a finite tree, whose branches extremities are cusp points of the conjugate locus. Each ramification counts the number of intersecting minimizing geodesics.
∙ Martinet case. We use the classification from [1,14]. We denote by q=(x,y,z) the privileged coordinates, and we have that x,y are of weight 1 and z is of weight 3. The distribution D is normalized to the Martinet form: kerω, ω=dz−y^{2}/2 dx.
Hence we obtain the following result.
Proposition 7
In the normal form, the abnormal curves are contained in the Martinet surface and are lines parallel to the xaxis. Starting from the origin (0,0,0), it is given by the abnormal curve γ_{ a }:t↦(t,0,0).

Model of order −1 (Martinet flat case). It corresponds to g=dx^{2}+dy^{2}.

Model of order 0. The metric is of the form g=(1+dy)^{2}dx^{2}+(1+βx+γy)^{2}dy^{2}.
In particular, we have H_{3}=p_{ z } is constant (isoperimetric situation). Using the normal form of order 0, we can deduce the following proposition.
Proposition 8
 (1)
If β=0, x is an additional cyclic coordinate and the geodesic flow is Liouville integrable.
 (2)
The abnormal geodesic γ_{ a }:t↦(t,0,0) is strict if and only if α≠0. If α=0, it is solution of (12) for each choice of p_{ z }=H_{3}=λ.
where E is a constant. If α=0, (14) reduces to the standard pendulum equation. We refer the reader to [1,14], for a detailed analysis but we can deduce already some geometric fact about the Martinet case versus the contact case of order zero.
In the Martinet case, they are many microlocal different geodesics, in particular if β=0 we can have the oscillating or rotating cases in the pendulum equations. To parameterize the geodesics in this case, we need at least the complexity of elliptic functions [28] and only a small number of specific geodesics can be parametrized by periodic controls. In particular, it is related to the Euler elastica [26] to parameterize strokes for the microswimmers. Indeed, besides the simple strokes related to the linear pendulum, we can construct eight shapes strokes corresponding to Bernoulli lemniscates.
Application: geometric and numerical study of the copepod swimmer
The aim of this section is to provide a complete analysis of the copepod swimmer. We will start this section by introducing the numerical tools.
4.1 Numerical methods

Bocop. The socalled direct approach transforms the infinite dimensional control problem into a finite dimensional problem. This is done by a discretization in time, applied to the state and control variables. Direct methods are usually less precise than indirect methods which are based on the Maximum Principle, but more robust with respect to the initialization. It can be used to initialize an indirect method. In the swimmer problem the Bocop’s software allows us to account for the triangle state constraints and to generate a stroke with large amplitude in the triangle interior.

HamPath. This software is based upon indirect methods: in a nutshell, the Maximum Principle leads to a shooting equation which is implemented using either simple or multiple shootings. It is complemented by discrete or differential continuation (homotopy) methods to evaluate the solution, when starting initially from a known solution. In our case, it can be done with the Bocop software starting from strokes with large amplitude or by the mathematical evaluations of stroke of small amplitudes using nilpotent SRmodels. This software uses the Jacobi fields to compute the differential of the shooting equation and is suitable to check second order necessary optimality conditions corresponding to conjugate points computation.
4.2 Lie brackets and geodesics computation
We observe in particular that f vanishes on the edges of the triangle \({\mathcal {T}}: 0\le \theta _{1}\le \theta _{2} \le \pi \).
According to our terminology previously introduced, we have the following result.
Proposition 9
 1
All interior points of the triangle \(\mathcal {T}\) are contact points.
 2
The triangle \({\mathcal {T}}\) represents the only (piecewise smooth) abnormal stroke, and each point – vertices excluded – is a Martinet point. It is a geodesic triangle in the Euclidean case.
The geodesics equations can be written in the coordinates (q,H), H=(H_{1},H_{2},H_{3}) and we complete the vector fields F_{1},F_{2} with \(F_{3} = \frac {\partial }{\partial x_{0}}\) to form a frame.
4.3 Numerical simulations and geometric comments in the copepod case
4.3.1 Complexity of normal strokes
4.3.2 Complexity of normal strokes constrained to the triangle \({\mathcal {T}}\)
Geometric explanation. At an interior point of the triangle, simple strokes are predicted by the nilpotent model. A limaçon can occur also by perturbing a simple stroke followed twice, which is clear from the numerical simulation. Eight shaped curves can appear only on the sides of the triangle as predicted by the Bernoulli lemniscate associated to a periodic inflexional Euler elastica. Note also the role of the symmetry group Σ in the construction.
Conjugate points computation. The HamPath software allows to compute easily conjugate points. They appear for the limaçon and the eight shaped curves. Hence, simple closed curve are the only candidates as minimizers. This result is obtained as a numerical evidence of our approach versus using calculations based on Green’s theorem. Note that for limaçons with small amplitude they are produced by perturbing a simple closed curve of the Dido model followed twice, and this gives a rigorous proof of the existence of a conjugate point since for the nilpotent model they appear exactly at the period.
4.4 Numerical computation of the center of swimming strokes and SRinvariant computation in the copepod case
First, we need the following concept observed in numerical simulations and reminiscent of the socalled LyapunovPoincaré theorem in celestial mechanics [32].
Definition 11
A center of swimming, denoted by C, is a point in the θshape space from which we can observe a one parameter family {γ_{ λ }; λ≥0} of simple strokes emanating from C which degenerates into C when λ→0. Moreover, we impose that for λ small enough each of stroke in the one parameter family is length minimizing (for fixed displacement).
4.4.1 Numerical simulations
In both cases, the centers of swimming are on the line Σ:θ_{2}=π−θ_{1}, thanks to the symmetry of the geodesic flow with respect to the symmetry σ_{3}:(θ_{1},θ_{2})↦(π−θ_{2},π−θ_{1}).
Geometric efficiency for the abnormal stroke and different normal strokes with the Euclidean cost
Types of stroke  x_{0}(T)  l(γ)  x_{0}(T)/l(γ) 

Simple loops  5.50×10^{−2}  1.98  2.52×10^{−2} 
1.40×10^{−1}  3.79  3.70×10^{−2}  
1.70×10^{−1}  4.34  3.92×10^{−2}  
2.00×10^{−1}  4.95  4.04×10^{−2}  
2.10×10^{−1}  5.11  4.11×10^{−2}  
Optimal stroke  2.17×10 ^{ −1 }  5.18  4.19×10 ^{ −2 } 
Fig. 14 (top)  
2.20×10^{−1}  5.35  4.11×10^{−2}  
2.30×10^{−1}  5.62  4.09×10^{−2}  
2.50×10^{−1}  6.31  3.97×10^{−2}  
2.74×10^{−1}  9.05  3.03×10^{−2}  
Abnormal  2.74×10^{−1}  10.7  2.56×10^{−2} 
Limaçon  2.00×10^{−1}  6.15  3.25×10^{−2} 
Geometric efficiency for the anormal stroke and different normal strokes with the mechanical cost
Types of strokes  x_{0}(T)  l(γ)  x_{0}(T)/l(γ) 
Simple loops  0.50.10^{−1}  0.994  5.03×10^{−2} 
1.50.10^{−1}  1.86  8.06×10^{−2}  
1.70×10^{−1}  2.02  8.41×10^{−2}  
2.00×10^{−1}  2.28  8.77×10^{−2}  
2.10×10^{−1}  2.50  8.84×10^{−2}  
2.20×10^{−1}  2.47  8.89×10^{−2}  
Optimal stroke  2.23×10 ^{ −1 }  2.56  8.90×10 ^{ −2 } 
Fig. 14 (bottom)  
2.30×10^{−1}  2.59  8.90×10^{−2}  
2.50×10^{−1}  2.85  8.76×10^{−2}  
2.60×10^{−1}  3.04  8.54×10^{−2}  
Abnormal  2.742×10^{−1}  4.93  5.56×10^{−2} 
Limaçon  2.500×10^{−1}  3.35  7.46×10^{−2} 
4.5 Algorithm to compute the centers of swimming
Next, we present as an application of the previously developed normal form the construction of the center of swimming. To simplify the computations, we shall restrict to the Euclidean case.
Lemma 2
The calculation of the privileged coordinates (x,y,z) near \((\theta _{1}(0),\theta _{2}(0),0)\in \text { Interior }({\mathcal {T}} \times \mathbb {R})\) with respective weight (1,1,2) provides the link between the physical coordinates and the coordinates of the normal form. In particular, the displacement variable x_{0} cannot be identified to the zvariable since for the HeisenbergBrockettDido model we have that \(\dot z>0\) and hence z is always increasing, contrary to the copepod swimmer where one stroke produces always forward and backward displacement.
Proof
To guarantee that the geometric analysis preserves the distinction between shape and displacement variables, we must restrict the calculations to the subgroup \({\mathcal {G}}'\) where local diffeomorphisms φ are preserving the θspace.
A tedious but straightforward computation leads to the following result.
Proposition 10
Moreover (θ_{1}(0),π−θ_{1}(0)) corresponds to the center of swimming of Fig. 13, (left, Euclidean case).
Remark 2
Proposition 11
5 3links copepod, theory and experimental observations
5.1 Physical model
In Section 2 we introduced several models of microswimmers. In [20] the model was generalized to allow asymmetry, leading to a wider class of swimmers that can translate and rotate freely and corresponding to generalization of the original Purcell swimmer. However, in these earlier models the governing equations can change when adjacent legs come together and form a bundle of legs. For mathematical convenience we avoid any possibility of bundling by considering the pairs of legs to be sufficiently far apart as formulated below.
5.2 3Links
where Δ(θ)=l+3+ sin2θ_{1}+ sin2θ_{2}+ sin2θ_{3}.
Below we analyze the abnormal geodesics and correlate our results with observations on the locomotion of the nauplius made in a laboratory setting.
5.2.1 Abnormal geodesics
Proposition 12

Two legs are fixed and one is moving$$t\rightarrow \left\{\begin{array}{ccc} (0,0,t)\\ (0,t,\pi)\\ (t\pi,\pi) \end{array}\right.$$

One leg is fixed and two are moving simultaneously \(t\rightarrow \left \{\begin {array}{cc} (0,t,t)\\ (t,t,\pi) \end {array}\right.\)

Three legs are moving simultaneously t→(t,t,t)
5.2.2 Experimental observations
Figure 22 displays over a 1.5 cycles of swim sequence the appendage angles of what is refereed to in [29] the power (from 15 to about 21 seconds) and return strokes (from 21 to about 24 seconds). As noted before, the appendages on the back (Md) display a physical constraint restricting their amplitude to [110°,160°], respectively θ_{2}∈[40°,135°] for (A2) and ∈[5°,130°] for (A1). However, observations on predator escape show the ability for the nauplius to extremely rapidly change its orientation and overcome the limitations on the angular variables stated here. It can be observed that the back appendages starts the power stroke to move toward 180° at first while the other two pairs of legs position themselves to maximize the amplitude they will use. Once they reach their constraint (first for the second pair of legs) they start moving toward the back of the nauplius. The phase shift created during this power stroke between the three pairs of appendages maximizes displacement forward. The return stroke objective is to minimize backward displacement to obtain the best net displacement, this is done by coordinating the three pair of legs together.
5.3 Robotics copepod
In this section we present some preliminary results on a robotics copepod. The main challenge is to mimic the low Reynolds number conditions, and therefore the characteristics of the nauplius environment, while rescaling it to a macroscopic scale. Toward this goal, the experiments presented here are conducted in silicone oil, which is a liquid polymerized siloxane with organic side chains. The robotics copepod is designed for onedimensional displacement only and displays two pairs of legs. The main objective is to build a mechanical device and setup that demonstrates the need for decoupled swimming strategies to produce horizontal displacement.
The main features that we tried to keep with the robotic device is the low Reynolds number assumption (met by using a special oil), as well as the one regarding the slim legs to minimize fluid interaction between them. The primary difficulty is to prevent the electronic to get in contact with the oil as it would get damaged permanently. For this reason, and after several trials and iterations, the model has been designed to accommodate a horizontal rail crossing through the body to guarantee its stability on the water.
5.3.1 Experiments
Conclusion
The aim of this short survey article is to present the combination of mathematical and numeric tools recently introduced in (geometric) optimal control and applicable to analyze the problem of swimming at low Reynolds number, using the slender body theory for Stokes flow. From this point of view, the simplest model of microswimmers is the socalled “copepod model” which can be observed as the copepod nauplii, an abundant variety of zooplankton and realized as a twolink swimmerrobot. The observation of the biological species allows to validate the adequation between the displacement predicted by the model with the measured displacement. SubRiemannian geometry is introduced in the analysis by assuming that microswimmers motion are performed minimizing the expanded mechanical energy. This allows for one species to compare the efficiency of different strokes or to compare the efficiency for different species, using the developments of computational methods of SRgeometry. In particular we use estimates of geodesics based on graded normal form, to show the existence of a one parameter family of simple strokes and in this family, only one stroke with a given amplitude is shown to be most efficient. This can be compared to similar result in the literature using a direct approach based on curvature control analysis and Fourier expansion to compute strokes, both approaches are shown to be complementary. The mathematical analysis is neat, abnormal and normal geodesics strokes being related to observed strokes corresponding to “sinusoidal” and sequential paddlings. A further step is to validate the mathematical results using a copepod robot built at the macroscopic scale, to validate the model and the robot observations. Preliminary experimented results are presented based on the abnormal (triangle) stroke and paved the road to further experiments dealing with the most efficient stroke.
The SRgeometry associated to the copepod is related to the well studied 3Dmodels (Contact, Martinet). Further studies are necessary to analyze the behaviours near the triangle vertices. This is the basis to understand more complicated models, e.g. the Cartan case related to the Purcell swimmer. In this framework this leads to embed the ContactMartinet model in the Cartan case, that is to a more intricate microlocal analysis. Also in the frame of optimal control, models taking in account inertia or experimental dissipation can be investigated.
Declarations
Acknowledgements
We would like to thank Rintaro Hayashi and Amandin Chyba Rabeendran for the pictures and data regarding the robotics copepod.
Funding
DT is partially supported by NSF grant: CBET1603929. MC is partially supported by the Simons Foundation, award # 359510. BB is partially supported by the ANR Project  DFG Explosys (Grant No. ANR14CE35001301;GL203/91). JR is supported by the European Research Council (ERC) through an ERCAdvanced Grant for the TAMING project.
Authors’ contributions
All authors contributed to the writing of the paper. BB and JR major contribution is the theoretical and numerical analysis of the results. MC and DD major contribution is the 3links analysis, experimental observations and robotics device. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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