How should a snake turn on ie: A ase study of the asymptoti isoholonomi problem Jianghai Hu, Slobodan N. Simić, and Shankar Sastry Department of Eletrial Engineering and Computer Sienes University of California Berkeley, CA 947 {jianghai,simi,sastry}@ees.berkeley.edu Abstrat It is a lassial result that solutions to the isoperimetri problem, i.e., finding the planar urves with a fixed length that enlose the largest area, are irles. As a generalization, we study an asymptoti version of the dual isoholonomi problem in a Eulidean spae with a o-dimension one distribution. We propose the onepts of asymptoti rank and effiieny, and ompute these quantities as well as the effiieny-ahieving urves in several speial ases. In partiular, an example of a snake moving on ie is worked out in detail to illustrate the results. I. INTRODUCTION As the dual to the isoperimetri problem, the isoholonomi problem has appliation in a variety of fields, for example, ontrol theory [3], the falling at problem [6], the swimming miroorganism at low Reynolds number [9], and the Berry phase in quantum mehanis [7], et. Another referene an be found in []. In this paper, we will formulate and study an asymptoti version of the isoholonomi problem for whih analyti solutions are available in ertain ases. To define the problem, we shall start from a motivating example, whih an be thought of as the planar version of the falling at problem. q θ q θ Fig.. q 3 O θ 3 A snake. A. Motivating Example: Snake on Ie Consider the following model of a snake moving on a horizontal plane. The snake onsists of n + unit point masses (nodes) whose positions are denoted by q,..., q R, respetively. These nodes are then onneted subsequently by n + rigid bars of unit length and zero weight, forming a kinemati hain. Figure shows an example when n = 3. This researh was funded by the National Siene Foundation under Grant No. EIA-599. q 4 θ 4 q 5 Suppose that the plane is ideal ie (i.e., fritionless) so that the snake annot push off the ground to gain loomotion. Then sine the snake is a losed mehanial system, its total linear momentum and total angular momentum are both onserved. So for an initially stationary snake, we must have q i, () q i q i. () Without loss of generality, we assume that the snake is initially entered at the origin, i.e., q i() =. Condition () then implies that q i. The onfiguration of the snake is uniquely determined by the angles θ,..., θ, where θ i is the angle q i+ q i makes with the positive x-axis for i =,..., n +. Eah θ i takes values in R modulo π, namely, the -torus T = R/πZ, so (θ,..., θ ) takes values in the (n + )-torus T, whih is the onfiguration spae of the snake. For given θ,..., θ, q,..., q an be reovered by q = (n + j)(os θ j, sin θ j ) t, (3) n + j= i q i = q + (os θ j, sin θ j ) t, i =,..., n +, (4) j= Equation (3) and (4) together define an embedding of the onfiguration spae T into R n+4. Thus T inherits isometrially via this embedding a riemannian metri from the standard metri on R n+4. After some alulation, this metri, an be determined as g ij, = ij os(θ i θ j ), i, j n +, (5) θ i θ j where ij are onstants defined by { i( j) ij =, if i < j, ( i)j, if i j. Suppose that the trajetory of the snake over an interval I = [, ] is given by the urve γ in T. Unless otherwise
stated, we assume that all the urves in this paper are defined on I. Define L(γ) = γ dt and E(γ) = γ dt as the length and the energy of γ, respetively, where is the norm orresponding to,. From the definition of,, if the positions of the nodes of the snake orresponding to γ are given by q,..., q, respetively, then L(γ) = ( ) / q i dt, E(γ) = q i dt. The problem we study in this paper an be roughly stated as: how an the snake turn most effiiently? One possible formulation is desribed in the following. Suppose that the snake starts from onfiguration (θ,..., θ ) at time, and wishes to retain the shape but with different orientation, for example, it tries to reah onfiguration (θ +θ,..., θ +θ) at time. Among all suh trajetories γ, we want to find the one with minimal length L(γ) (or energy E(γ), whih are equivalent), subjet to the onstraint () that the total angular momentum are zero at all time. B. Solutions as Sub-Riemannian Geodesis Without the onstraint (), the above problem beomes finding geodesis in T with the riemannian metri,, whih is studied in [5]. However, with the addition of the onstraint (), the problem beomes one of finding subriemannian geodesis in a ertain sub-riemannian geometry. In fat, let γ = (θ,..., θ ) be a urve in T, and let q,..., q be the orresponding positions of its nodes. Then areful alulations show that () is equivalent to i,j= ij os(θ i θ j ) θ j =. (6) In other words, if we define a one-form on T by α = i,j= ij os(θ i θ j )dθ j, (7) then ondition () is equivalent to α( γ) =, i.e., γ ker α. Note that H = ker α is a o-dimension one distribution on T. Hene γ must be a horizontal urve for this distribution. Moreover, the restrition of, to H defines a subriemannian metri, H. In this sub-riemannian geometry, the sub-riemannian length of the horizontal urve γ oinides with its riemannian length L(γ). Therefore, the problem stated above is to find the shortest horizontal urves in T onneting (θ,..., θ ) to (θ + θ,..., θ + θ), whih is a distane-minimizing subriemannian geodesi. Compared with general sub-riemannian geometries, however, the one defined above belongs to a very speial ategory. In fat, T is a prinipal T-bundle over T n with distribution H and sub-riemannian metri, H that are ompatible with the ation of the struture group T. To see this, onsider the following ation R of T on T : for eah θ T, R θ : T T is defined by R θ (θ,..., θ ) = (θ + θ,..., θ + θ). The onfiguration of the snake orresponding to R θ (θ,..., θ ) is obtained from that orresponding to (θ,..., θ ) by a rotation of θ ounterlokwise. In fat, the set of all shapes of the snake orresponds in a one-to-one way to the set of T-orbits of T, whih an be identified as T n {(θ,..., θ ) T : θ +... + θ = mod π}. We shall all T n the shape spae. As the notation suggests, T n is topologially an n-torus. There is a natural projetion π : T T n defined P θi P θi by π(θ,..., θ ) = (θ,..., θ ), suh that for eah (θ,..., θ ) T n its inverse image under π is exatly the T-orbit in T passing through it. In the terminology of prinipal bundles, π makes T into a prinipal bundle with base spae T n and struture group T. Eah fiber of this bundle onsists of all onfigurations of the snake with a fixed shape but different orientations. Note that in the definitions (5) and (7), the terms involving θ i s are of the form θ i θ j, whih remain unhanged under the ation R. Hene the horizontal distribution H and the sub-riemannian metri, H are both invariant under the ation R. Suh distributions and sub-riemannian metris are alled ompatible (with the ation of the struture group T). In this perspetive, the problem is to determine the shortest horizontal urve from a onfiguration (θ,..., θ ) to a new onfiguration R θ (θ,..., θ ) in the same fiber. C. Asymptoti Problem Due to the global nature of the problem proposed above, its solutions are usually impossible to obtain analytially. In this paper, we shall study an asymptoti version of the problem: what is the most effiient way of turning if the snake an only exert an inreasingly smaller amount of energy? Besides giving approximate solutions to the global problem when the snake is onfined to a neighborhood of the urrent onfiguration, solutions to the asymptoti problem an be employed repetitively for the snake to turn a signifiant angle, whih makes sense if the snake has to take a breath at its original shape from time to time. The paper is organized as follows. First, some notions in sub-riemannian geometry are reviewed in Setion II. In Setion III, we formulate the problem of asymptoti isoholonomy by proposing the onepts of rank and effiieny. The rank two ase is solved in Setion IV, and the rank three ase with n = in Setion V. The results are then illustrated in Setion VI for the snake example. II. BASIC SETUP Sine we are onerned with loal solutions, we onsider R instead of T. The projetion π : (x,..., x ) R (x,..., x n ) R n defines R as a bundle
over R n whose fiber over eah point m R n is given by π (m) R, making π : R R n a prinipal R-bundle. A. Co-Dimension One Distribution on R A o-dimension one distribution H on R is defined by the vanishing of a one-form n α = α i dx i dx, (8) where α,..., α n are C funtions on R. At eah point q R, the horizontal spae H q is the kernel of α q in T q R R, thought of as an n-dimensional subspae of R, namely, H q = {(v,..., v ) R : n α i(q)v i v = }. A horizontal urve γ in R is an absolute ontinuous urve in R whose tangent vetor γ(t) wherever it exists belongs to H γ(t). Write γ = (γ,..., γ ) in oordinates and note that H = ker α, we have that γ is horizontal if and only if γ = n α i γ i, a.e. For a urve in R n starting from m, let q = (m, h) π (m) be arbitrary. The horizontal lift of based at q is the unique horizontal urve γ in R starting from q and satisfying π(γ) = at all time. If in partiular is a loop in R n based at m, then its horizontal lift γ must start and end in the same fiber π (m), i.e., the end point of γ has the same x,..., x n oordinates as m. The differene in their x oordinates is alled the holonomy of the loop (based at q), whih in general depends on the hoie of q π (m). B. Compatible Distribution The distribution H is alled ompatible (with the bundle struture π : R R n ) if it is invariant under the ation of the struture group R, namely, translations along the x - axis. In other words, H is ompatible if and only if the horizontal spaes H q, thought of as n-dimensional subspaes in R, are the same for q in the same fiber π (m). In terms of equation (8), this is equivalent to the funtions α,..., α are independent on x. (9) Beause of (9), we an defined a one-form on R n as n Θ = α i dx i, () whih is alled the onnetion form of H. The urvature form of H is the two-form on R n defined as Ω = dθ. () An important impliation of ompatible distributions is that the holonomy of a loop in R n based at m is independent of the starting point q π (m) of its horizontal lift, thus an be simply denoted by h(). An alternative interpretation of h() is the following. Let : I R n be a loop based at m, and γ be its horizontal lift based at an arbitrary point in π (m). Then h() = γ () γ () = γ dt = n α i γ i dt = Θ. Now we find a two-dimensional submanifold S immersed in R n whose boundary S is exatly under the anonial orientation of R n. Then by the Stokes equations, h() = Θ = dθ = Ω. () S Equation () expresses the holonomy h() as an integral of the urvature form Ω over an arbitrary surfae enirled by. This relation will be useful later. C. Sub-Riemannian Metri A sub-riemannian metri on H is an assignment of inner produts to horizontal spaes H q that varies smoothly with q R. One often denotes these inner produts by, H and their orresponding norms by H. The existene of a sub-riemannian metri enables one to measure the length of horizontal urves: for a horizontal urve γ in R, its length is given by L(γ) = γ dt. It should be emphasized here that the length of a non-horizontal urve is in general not defined. The sub-riemannian distane between two arbitrary points q and q in R is the infimum of the length of all horizontal urves onneting them. With this distane, the distribution H and the sub-riemannian metri, H together speify a sub-riemannian geometry on R. D. Compatible Metri For a ompatible distribution H, a sub-riemannian metri is alled ompatible (with the bundle struture π : R R n ) if it is invariant under the ation of the struture group R. Hene translations along the x -axis are isometries for the sub-riemannian geometry. Compatible sub-riemannian metris on H orresponds in a one-to-one way to riemannian metris on the base spae R n. To see this, we first define the horizontal lift operator. For eah pair of m R n and q π (m), the horizontal lift h q : T m R n H q is a linear map that maps v T m R n to the unique u H q satisfying dπ q (u) = v. Thus h q is the inverse of the map dπ q restrited on H q, whih is an isomorphism by our hoie of α in (8). Now starting from a ompatible sub-riemannian metri, H, there is a unique riemannian metri, R n on R n that makes all horizontal lifts isometries. In fat,, R n is defined by S u, v R n = h q (u), h q (v) H, u, v T m R n, (3) whih is independent of the hoie of q π (m) sine, H is ompatible. Conversely, a riemannian metri, R n on R n indues a ompatible sub-riemannian metri on H as u, v H = dπ q (u), dπ q (v) R n, u, v H q. (4) A. Asymptoti Holonomy III. PROBLEM FORMULATION Suppose that H = ker α is a o-dimension one distribution on R, where α is given in (8), and, H is a subriemannian metri on H. Fix a pair (m, q), where m R n, S
q R, and q π (m). Let be a non-trivial loop in R n based at m, and let γ be its horizontal lift in R based at q. Denote by h() the holonomy of based at q and, with some abuse of notation, by L() the length of γ (the length of is in general not defined). Note that L() > sine is non-trivial. For eah ɛ >, denote by ɛ the loop in R n obtained by saling by a fator of ɛ towards m (a more aurate notation should be m + ɛ( m)). Thus h(ɛ) and L(ɛ) are similarly defined. It is easy to show that as ɛ, L(ɛ) is of the order of ɛ, while h(ɛ) is of the order of ɛ r for some integer r. We all this integer the rank of the loop, and denote it by r q (). The (asymptoti) effiieny of is defined as h(ɛ) η q () lim ɛ L r (ɛ). (5) The rank and the (asymptoti) effiieny at q are defined as r(q) min r q (), η(q) sup{η q () : suh that r q () = r(q)}. (6) Remark Both r q () and η q () depend on q π (m) sine h() and L() in general vary with q. However, if the distribution H and the sub-riemannian metri, H are both ompatible, then h() and L() are the same as q varies in the fiber π (m). Therefore, in this ase one an write r m (), η m (), r(m), and η(m) instead. We shall simply all r(m) and η(m) the rank and the effiieny at m, respetively. Example (Heisenberg Geometry) Consider the following ompatible distribution H and sub-riemannian metri, H on R 3. Suppose that H is given by the vanishing of α = (x dx x dx ) dx 3, and that, H is indued by the anonial riemannian metri on R. Then the holonomy h() of a loop in R is the signed area it enloses, and L() is simply its length in R. Thus by the well-known isoperimetri theorem, the rank at any m R is two, and the effiieny η(m) = /4π, both realized when is a irle of arbitrary radius passing through m. The following lemmas are diret onsequene of the above definitions. We omit their proofs here. Lemma For a loop in R n based at m, and q π (m), (invariane to saling) r q () = r q (λ) and η q () = η q (λ) for any λ > ; (invariane to reparameterization) r q ( ρ) = r q () and η q ( ρ) = η q () for any diffeomorphism ρ : I I. Obviously, the rank at q R depends only on the horizontal distribution H near q, and is independent of the sub-riemannian metri, H. For ompatible H, we have Lemma Suppose H is ompatible with onnetion form Θ and urvature form Ω, respetively. Write Ω in oordinates as Ω = i,j n Ω ij dx i dx j, where Ω ij = Ω ji. Then at any m R n, r(m) is equal to the smallest integer k for whih there exist some i, j suh that at least one k-th order partial derivative of Ω ij at m is nonzero. On the other hand, although η(q) does depend on, H, due to the smoothly varying nature of, H near q, we have Lemma 3 η(q) depends on the sub-riemannian metri, H only through its restrition on H q. By Lemma 3, to the effet of studying η(q), we an modify, H on other horizontal spaes so that for any q R, dπ q : H q T π(q )R n is an isometry, (7) where the metris on all T π(q )R n are given by the same positive definite n-by-n matrix A. For ompatible H,, H thus hosen is also ompatible. By a transformation of oordinates within R n, in the rest of the paper we shall assume that (7) holds with A = I. Hene L() is simply the ar length of as a urve in R n with the anonial metri. B. Equations of Effiieny-Ahieving Loops To find the effiieny η(q) at q R and the (family of) loops based at m = π(q) for whih η q () = η(q), one needs to solve the following variational problem. Problem Let be an arbitrary loop in R n based at m, and let h() be its holonomy. Then solving for η(q) is equivalent to one of the following problems: ) minimize L() = ċ dt, subjet to h() = ; ) maximize h(), subjet to L() = ; 3) minimize E() = ċ dt, subjet to h() = ; 4) maximize h() subjet to E() =. Formulation 3 is adopted here sine it avoids the ambiguity of reparameterizations (solutions in formulations 3 are solutions in formulation parameterized with unit speed). Using the Lagrangian multiplier approah [8], solutions are given by: = λ iċω. (8) Here, iċω = Ω(ċ, ) is a one-form on R n that we identify as a vetor via the anonial metri on R n. We are interested only in those solutions to (8) that start and end in the origin. The onstraint that h() = an be ignored if one is only interested in the shape of suh solutions. Equation (8) desribes the motions of a partile of unit mass and unit harge moving in a magneti field on R n given by Ω in the ase of n =, 3, and in general admits no analyti solution. IV. RANK TWO CASE Let H be a ompatible o-dimension one distribution on R n with onnetion form Θ and urvature form Ω, and let, H be a riemannian metri on H obtained by lifting the anonial metri on R n. In this setion, we try to determine η(m) at a point m R n with rank r(m) =. Without loss we assume that m = R n and q = R. By Lemma, if Ω = i,j n Ω ij dx i dx j in oordinates, then Ω ij () for at least some i, j. Define Ω
i,j n Ω ij() dx i dx j. The for any two dimensional submanifold S in R n enirled by a loop based at, we have h(ɛ) = ɛs Ω ɛs Ω = ɛ S Ω, as ɛ. Hene in determining the effiieny at, we may as well assume that H has the urvature form Ω. Under this assumption, η () = lim ɛ h(ɛ) L (ɛ) = lim ɛ η() = sup η () = sup S Ω L () ɛ RS Ω ɛ L () = = sup Define an n-by-n skew-symmetri matrix R S Ω, so L () S Ω E(). (9) Z = [Ω ij ()] i,j n. () Z an be transformed into the following standard form: ([ ] [ ] ) σ σl Z = Q diag,...,,,..., Q t, σ σ l where Q R n n is orthonormal, l > is an integer with l n, and σ σ l >. In fat, σ,..., σ l eah repeated twie are the nonzero singular values of Z. Now perform the following oordinate transformation: (y,..., y n ) = (x,..., x n )Q. () Then Ω = (σ dy dy + + σ l dy l dy l ). So for S in R n enirled by a loop based at with oordinates y,..., y n, we have S Ω = l σ i S dy i dy i. Note that S dy i dy i is the area of the projetion of S onto the plane Π i spanned by the y i and y i axes. By the lassial isoperimetri theorem [4], S dy i dy i 4π (ẏ i + ẏ i ) dt, with equality if and only if (y i, y i ) for t I draws a irle of arbitrary radius through the origin in Π i. Summing up, we have S Ω l σ i S dy l i dy i σ 4π (ẏ i + ẏi σ ) dt π E(), where the equality holds if and only if for all i =,..., l suh that σ i = max i l σ i, the projetion of onto Π i is a irle of arbitrary radius through, and the projetion of to all other axes are all zero. Therefore, Theorem Suppose that H is a ompatible distribution on R n with urvature form Ω, and, H is a sub-riemannian metri on H suh that dπ is an isometry from H to T R n with the anonial metri. Let σ σ l eah repeated twie be the nonzero singular values of the matrix Z defined in (). Then the effiieny at is η() = σ π. () Moreover, let Π i be the subspae spanned by the y i and y i axes under the oordinate transformation (). Then for a loop in R n based at to have effiieny η(), its projetion onto eah plane Π i with σ i = σ must be a irle, and its projetions on all other axes are zero. y.6.4...4.6.5 y Fig.. A solution urve to problem (3). V. RANK THREE CASE In this setion, we study the effiieny at point m = R n for a ompatible horizontal distribution H on R with rank r() = 3. So the urvature form Ω satisfies Ω = and Ω Ω ij () i,j,k n x k x k dx i dx j. By our previous disussions, we an replae Ω by Ω. We study the simplest ase n = only. So Ω = ( Ω() x x + Ω() x x )dx dx. Define κ = [ Ω() x ] + [ Ω() x ], whih is nonzero by assumption. Perform the oordinate transformation (y, y ) = (x, x )Q, where Q is the orthonormal matrix defined by Q = κ [ Ω() Ω () x x Ω() Ω () x x Then Ω is now Ω = κy dy dy. For a nontrivial loop in R based at enlosing the (oriented) area S, we have h(ɛ) Ω = ɛ 3 Ω = ɛ 3 κy dy. ɛs Hene η () = lim ɛ S h(ɛ) L 3 (ɛ) = κ R y dy L 3 () ]., and κ η() = sup η () = sup y dy L 3 = κ sup y dy () E 3/ (). To determine the for whih η() is ahieved, one needs to solve the following variational problem: find that minimizes E() suh that y dy =. (3) Problem (3) an be solved using the Maximal Priniple. The details are omitted here, and an be found in an upoming paper. Figure plots one solution urve (up to a saling) to problem (3), whih should be parameterized so that it proeeds ounterlokwise with onstant speed. Remark The effiieny-ahieving loop alulated in this setion is part of the trajetory of a harged partile moving in a magneti field B with linear omponents on R, referred to as a grad B drift sine it exhibits an overall drift orthogonal to the diretion grad B ([]).
.5.5.5.5 t= t=/.5.5.5.5.5.5 t=/6 t=/3 t=.5.5.5.5 t=/3 t=5/6 VI. BACK TO THE SNAKE Suppose that in the snake example we have n =. So the snake onsists of three rigid segments, whose orientations are given by the angles θ i, i =,, 3. The onfiguration spae is T 3, with a Riemannian metri given by (5), where = ( ij ) i,j 3 = 3 4 4. By (7), the o-dimension one 3 distribution H is given by the vanishing of the one-form α = 3 i,j= ij os(θ i θ j )dθ j. Suppose now that the snake is at the initial onfiguration q orresponding to θ = θ = θ 3 =, i.e. the three segments of the snake are all aligned in the positive horizontal diretion. To ompute the rank at q, we perform the following oordinate transformation in a neighborhood of q: φ = 5 3 φ φ 3 5 3 5 3 θ, (4) The hoie of suh a transformation serves several purposes. First, φ 3 = θ + θ + θ 3 is the diretion along the fibers of T 3 under the ation of T as desribed in Setion I. Seond, the plane Π spanned by the φ and φ axes is transversal to the φ 3 axis, hene an be regarded as the base spae for the prinipal bundle T 3, at least loally around the origin. Third, the projetion dπ : H q T (,) Π is an isometry if Π is equipped with the anonial Eulidean metri. Thus ondition (7) is satisfied with A = I. In the new oordinates, q orresponds to the origin φ = φ = φ 3 =, and the ondition that α = is equivalent to the vanishing of a suitable onnetion form Θ defined on Π R. After a areful alulation, the urvature form Ω is Ω = dθ = f(φ, φ )dφ dφ, where f(φ, φ ) is given by 5 f(φ, φ ) = 3 sin( 3 φ ) 5f (φ, φ ) θ θ 3 { 4 os( φ ) + f (φ, φ ) f (φ, φ ) f (φ, φ ) = 5 + 4 os( 3 5 φ ) os( φ ) + os φ, f (φ, φ ) = os( 3 5 φ )[ sin ( φ ) 6 os ( φ )] + 4 sin ( φ ) os( φ ). f f One an verify that f(, ) =, φ (, ), φ (, ) =. As a result of Proposition, the rank at q is three. To ompute the effiieny at q, we an replae Ω by its first order approximate Ω = 5 5 dφ dφ. In partiular, maximum effiieny at q is ahieved by the horizontal lifting of a urve in the (φ, φ ) plane plotted in Figure. Horizontally lifting to a urve γ in (φ, φ, φ 3 ) oordinates, transforming γ bak to the (θ, θ, θ 3 ) oordinates using (4), and finally using equation (3), we obtain the motions for the snake to turn with the least energy asymptotially starting from the aligned initial position. Figure 3 shows the snapshots of the motions at different time instanes obtained numerially. },.5.5.5.5 Fig. 3..5.5.5.5.5.5 Snapshots of snake turning..5.5.5.5 Remark 3 In a neighborhood of the origin in the (φ, φ ) oordinates, the zero-th order approximate of the urvature form Ω vanishes if and only if φ =, i.e., if and only if θ +θ 3 θ =. So the rank is three at points satisfying this ondition and two otherwise. As a result, for snake starting from a shape lose to the aligned one, it is more diffiult for the snake to turn if its initial shape satisfies the ondition θ + θ 3 θ =. VII. REFERENCES [] A. A. Agrahev and J.-P. Gauthier. On the dido problem and plane isoperimetri problems. Ata Appl. Math., 57:87 338, 999. [] H. Alfven. Cosmial Eletrodynamis. Oxford University Press, 95. [3] R. W. Brokett. Control theory and singular riemannian geometry. In P. J. Hilton and G. S Young, editors, New Diretions in Applied Mathematis, pages 7. Springer-Verlag, 98. [4] H. Howards, M. Huthings, and F. Morgan. The isoperimetri problem on surfaes. Amer. Math. Monthly, 6(5):43 439, 999. [5] J. Hu, M. Prandini, and S. Sastry. Optimal ollision avoidane and formation swithing on Riemannian manifolds. submitted to SIAM J. on Control and Optimization, 3. [6] T. R. Kane and M. P. Sher. A dynamial explanation of the falling at phenomenon. International J. Solids and Strutures, (5):663 67, 969. [7] R. Montgomery. The isoholonomi problem and some appliations. Comm. Math. Phys., (8):565 59, 99. [8] R. Montgomery. A Tour of Subriemannian Geometries, Their Geodesis and Appliations. Amer. Math. Soiety, Providene, RI,. [9] A. Shapere and F. Wilzek. Self-propulsion at low reynolds number. Phys. Rev. Lett., (58):5 54, 987.