On symmetries of a control problem with growth vector (4,7)
OON SYMMETRIES OF A CONTROL PROBLEM WITHGROWTH VECTOR (4 , Jaroslav Hrdina a , Aleˇs N´avrat a , Lenka Zalabov´a b a Institute of Mathematics,Faculty of Mechanical Engineering, Brno University of Technology,Technick´a 2896/2, 616 69 Brno, Czech Republic,[email protected], [email protected] b Institute of Mathematics, Faculty of Science, University of South Bohemia,Braniˇsovsk´a 1760, ˇCesk´e Budˇejovice, 370 05, Czech Republic, andDepartment of Mathematics and Statistics, Faculty of Science, Masaryk University,Kotl´aˇrsk´a 2, Brno, 611 37, Czech Republic,[email protected]
Abstract.
We study the action of symmetries on geodesics of a control prob-lem on a Carnot group with growth vector (4 , SO (3) and a set of points in the Carnotgroup with a nontrivial stabilizer of this action. We prove that each geodesiceither lies in this set or do not intersect this set. In the former case the optimal-ity of geodesics is solved completely through the identification of the quotientwith Heisenberg group. Introduction
This paper is motivated by the paper [8] where the first and third authors studylocal control of a planar mechanism with seven–dimensional configuration spacewhose movement is restricted by three non–holonomic conditions. The mechanismthat the authors deal with is a modification of a planar mechanism generally knownas trident snake robot. Trident snake robot consists of a root block in the shapeof an equilateral triangle together with three one–link branches each of which isconnected to one vertex of the root block via revolute joint and has a passive wheelat its very end, [9, 6]. The generalized trident snake robot introduced in [8] is aplanar mechanism consisting of a root block in the shape of an equilateral triangletogether with three one–link branches that have passive wheels at their ends, too.However, each of the branches is connected to one vertex of the root block viaprismatic joint and one of the joints is simultaneously revolute joint, see Figure 1.It turns out that under the assumption that the movement of the mechanism iswith no slipping, its local control is described by a control problem with growthvector (4 , Mathematics Subject Classification.
Key words and phrases. local control and optimality, Carnot groups, symmetries, sub–Riemannian geodesics.The first and second authors was supported by the grant no. FSI-S-20-6187. Third author issupported by grant no. 20-11473S Symmetry and invariance in analysis, geometric modeling andcontrol theory from the Czech Science Foundation. We thank to Luca Rizzi for useful discussionsduring Winter School Geometry and Physics, Srn´ı, 2020. a r X i v : . [ m a t h . DG ] F e b ON SYMMETRIES OF (4 , Figure 1.
Generalized trident snake robot motion in seven–dimensional configuration space in coordinates ( x, (cid:96) , (cid:96) , (cid:96) , y, θ, ϕ ).solution space of the corresponding system forms a four–dimensional distributionin the seven–dimensional configuration space. However, this control problem ishighly non–linear. Thus it is natural to approximate the problem and swap to itsnilpotent approximation, [2].One can see from Figure 1 that there are three canonical generators of thedistribution that correspond to prismatic joints and reflect prolongations of thebranches. In particular, corresponding movements commute. The most impor-tant role then plays the additional field that describes the action of revolute andprismatic joints together. On the level of nilpotent approximation, we find fourvector fields N , N , N , N generating the distribution N such that the only non–trivial Lie brackets are the brackets [ N , N ], [ N , N ] and [ N , N ]. In particular,these brackets are not contained in the distribution and the system is controllableaccording to the Chow–Rashevskii theorem, [10].The aim of this paper is to study the corresponding nilpotent control problem indetail. Let us note that the nilpotent approximation constructed in [8, Section 3]leads to slightly different vector fields than the ones we use in Section 2. However,both define equivalent left–invariant control problems on isomorphic Carnot groups.We give explicit form of the vector fields N , N i , N i , i = 1 , , N in Section 2.1. Then we write associated control problemas ˙ q = u N + u N + u N + u N for t > q ∈ N and controls u = ( u , u , u , u ) with boundary condition q (0) = q , q ( T ) = q for fixed q , q ∈ N , where we minimize the cost functional J = 12 (cid:90) T ( u + u + u + u ) dt. In terms of the sub–Riemannian geometry, the solution of this problem defines asub–Riemannian geodesics, i.e. a local minimizers of the sub–Riemannian distance.We use the Hamiltonian concept to approach this problem, [1]. We describe theadmissible controls in Proposition 1 and the geodesics in Proposition 2.Sub–Riemannian geodesics are not optimal for all times in general and eachgeodesic carries a point where it stops to be optimal. One of natural ways to findsuch points is to use symmetries of the system. Indeed, if there is a symmetrypreserving starting point and some other point of the geodesic, then the action of
N SYMMETRIES OF (4 , suitable symmetry can generate one–parametric family of geodesics of the samelength, so all the geodesics meet in the cut point. This happens e.g. in the case of(3 ,
6) in [12, 11] and is generalized for general free distributions in [14].Let us emphasize that there is a natural decomposition of the control distribu-tion N into one–dimensional distributions generated by N and three–dimensionalinvolutive distribution generated by N , N , N . Let us note that such geomet-ric structure is called generalized path geometry in dimension 7, [4]. We use thisdecomposition to find symmetries of the control system in Section 3. We show inProposition 3 that the symmetry algebra is a semidirect sum of a seven–dimensionalalgebra of transvections that correspond to right–invariant vector fields on N andthree–dimensional stabilizer of the origin isomorphic to so (3). Then fixed points ofsymmetries of so (3) form the set C n of potential cut–points. However, it turns outthat our (non–free) (4 , − − case behaves very differently.We study in Section 4 the action of symmetries of so (3) on geodesics and theoptimality of geodesics. In particular, we find suitable representatives of orbitsof the action on geodesics in Proposition 4 and we use them to prove our mainresults. We show in Theorem 1 that each geodesic starting at the origin either donot intersect the set C n or is contained in this set for all times. Thus geodesics areof two very different types. We relate the geodesics contained in C n to geodesicsin the Heisenberg group to find their cut–time in Theorem 2. Finally, we commenton sub–Riemannian sphere and the optimality of geodesics outside of the set C n .2. Nilpotent control problem
By nilpotent control problems we mean invariant control problems on Carnotgroups. There are several types of non–free (4 , Left–invariant control problem.
Let us consider four vector fields givenon R with local coordinates ( x, (cid:96) , (cid:96) , (cid:96) , y , y , y ) as N = ∂ x − (cid:96) ∂ y − (cid:96) ∂ y − (cid:96) ∂ y ,N = ∂ (cid:96) + x ∂ y , N = ∂ (cid:96) + x ∂ y , N = ∂ (cid:96) + x ∂ y , (1)where the symbol ∂ stands for partial derivative. The only nontrivial brackets are N = [ N , N ] = ∂ y , N = [ N , N ] = ∂ y , N = [ N , N ] = ∂ y . (2)These fields then determine a 2–step nilpotent Lie algebra n with the multiplicativetable given in Table 1.There is a Carnot group N such that the fields N , N , N , N , N , N , N are left–invariant for the corresponding group structure. The group structure on N , when identified with R = R ⊕ R , reads as follows x(cid:96) (cid:96) (cid:96) y y y · ˜ x ˜ (cid:96) ˜ (cid:96) ˜ (cid:96) ˜ y ˜ y ˜ y = x + ˜ x(cid:96) + ˜ (cid:96) (cid:96) + ˜ (cid:96) (cid:96) + ˜ (cid:96) y + ˜ y + ( x ˜ (cid:96) − ˜ x(cid:96) ) y + ˜ y + ( x ˜ (cid:96) − ˜ x(cid:96) ) y + ˜ y + ( x ˜ (cid:96) − ˜ x(cid:96) ) . (3) ON SYMMETRIES OF (4 , N N N N N N N N N N N N − N N − N N − N N N N Table 1.
Lie algebra n In particular, N = (cid:104) N , N , N , N (cid:105) forms a 4–dimensional left–invariant distribu-tion on N . Moreover, there is a natural decomposition N = (cid:104) N (cid:105) ⊕ (cid:104) N , N , N (cid:105) (4)of N into 1–dimensional distribution and 3–dimensional involutive distribution,both left–invariant. We define the left–invariant sub–Riemannian metric r on N by declaring N , N , N , N orthonormal.The left–invariant sub–Riemannian structure ( N, N , r ) is related to left–invariantoptimal control problem written in coordinates ( x, (cid:96) , (cid:96) , (cid:96) , y , y , y ) as˙ q ( t ) = u − (cid:96) − (cid:96) − (cid:96) + u x + u x + u x (5)for t > q in N and the control u = ( u , u , u , u ) ∈ R with the boundarycondition q (0) = q , q ( T ) = q for fixed points q , q ∈ N , where we minimize (cid:82) T ( u + u + u + u ) dt .2.2. Local control and Pontryagin’s maximum principle.
We follow here [1,Sections 7 and 13]. Left–invariant vector fields N , N , N , N , N , N , N form a basis of T N and determine left–invariant coordinates on N . Then we definecorresponding left–invariant coordinates h , h i and w i , i = 1 , , T ∗ N by h ( λ ) = λ ( N ) , h i ( λ ) = λ ( N i ), w i ( λ ) = λ ( N i ) for arbitrary 1–form λ on N .Thus we can use ( x, (cid:96) i , y i , h , h i , w i ) as global coordinates on T ∗ N .A geodesic is an admissible curve parametrized by constant speed whose suffi-ciently small arcs are length minimizers. It turns out that the geodesics are ex-actly projections on N of normal Pontryagin extremals, i.e. integral curves ofleft–invariant normal Hamiltonian H = 12 ( h + h + h + h ) , (6)since there are no strict abnormal extremals for 2–step Carnot groups. Assume λ ( t ) = ( x ( t ) , (cid:96) i ( t ) , y i ( t ) , h ( t ) , h i ( t ) , w i ( t )) in T ∗ N is a normal extremal. Thencontrols u j , j = 0 , , , u j ( t ) = h j ( λ ( t )) and the basesystem takes form ˙ q = h N ( q ) + h N ( q ) + h N ( q ) + h N ( q )(7) N SYMMETRIES OF (4 , where q = ( x, (cid:96) i , y i ). Using u j ( t ) = h j ( λ ( t )) and the equation ˙ λ ( t ) = (cid:126)H ( λ ( t )) fornormal extremals, we can write the fiber system as˙ h j = − (cid:88) l =1 3 (cid:88) k =1 c kjl h l w k , j = 0 , , , , ˙ w i = 0 , i = 1 , , , (8)where c kjl are structure constants of the Lie algebra n for the basis N , N , N , N , N , N , N , see Table 1.2.3. Solutions of fiber system.
We see immediately from (8) that solutions w i , i = 1 , , w = K , w = K , w = K (9)for suitable constants K i . Hence the second part of the fiber system (8) takes theexplicit matrix form ˙ h = − Ω w h, where h = ( h , h , h , h ) t andΩ w = (cid:32) K K K − K − K − K (cid:33) . (10)Its solution is given by h ( t ) = e − t Ω w h (0), where h (0) is the initial value of vector h in the origin. If K = K = K = 0 then h ( t ) = h (0) is constant and the geodesic( x ( t ) , (cid:96) i ( t ) , y i ( t )) is a line in N such that y i ( t ) = 0. In next we assume that thevector ( K , K , K ) is non–zero and we denote by K = (cid:112) K + K + K its length. Proposition 1.
The general solutions of (8) satisfying (9) for non–zero K takeform h = K ( − C sin( Kt ) + C cos( Kt )) , (cid:18) h h h (cid:19) = ( C cos( Kt ) + C sin( Kt )) (cid:18) K K K (cid:19) + (cid:18) − C K − C K C K C K (cid:19) , (11) where C , C , C , C are real constants.Proof. The solution of the system (8) is given by exponential of the matrix Ω w from (10). We need to analyze its eigenvalues and eigenvectors. It follows thatthere are (complex conjugated) imaginary eigenvalues ± iK both of multiplicityone and the eigenvalue 0 of multiplicity two. The corresponding eigenspace of iK is generated by complex eigenvector v that decomposes into real and complexcomponent as (cid:60) ( v ) = (0 , K , K , K ) t and (cid:61) ( v ) = ( K, , , t . In the basis formedby these two vectors together with any basis of the two–dimensional eigenspacecorresponding to the eigenvalue 0, the matrix Ω w has zeros at all positions exceptpositions (Ω w ) = − (Ω w ) = K . Then we get in this eigenvector basis e − t Ω w = cos( Kt ) sin( Kt ) 0 0 − sin( Kt ) cos( Kt ) 0 00 0 1 00 0 0 1 . (12)For the choice of the basis of the eigenspace corresponding to the eigenvalue 0given as (0 , − K , , K ) t and (0 , − K , K , t , the solution can be written as the ON SYMMETRIES OF (4 , combination (cid:32) h h h h (cid:33) = ( C cos( Kt ) + C sin( Kt )) (cid:18) K K K (cid:19) + ( − C sin( Kt ) + C cos( Kt )) (cid:18) K (cid:19) + C (cid:18) − K K (cid:19) + C (cid:18) − K K (cid:19) with coefficients C , C , C , C . Then the formula (11) follows. (cid:3) Let us emphasize that the choice C = C = 0 gives constant solutions that arenot relevant as control functions. Thus we assume that at least one of the constants C , C is non–zero.2.4. Solutions of base system.
The base system (7) takes the explicit form˙ x = h , ˙ (cid:96) = h , ˙ (cid:96) = h , ˙ (cid:96) = h , ˙ y = 12 ( xh − h (cid:96) ) , ˙ y = 12 ( xh − h (cid:96) ) , ˙ y = 12 ( xh − h (cid:96) ) . (13)As discussed above, we are interested in solutions going through the origin, i.e. weimpose the initial condition x (0) = 0 , (cid:96) i (0) = 0 , y i (0) = 0, i = 1 , , Proposition 2.
The sub–Riemannian geodesics on Carnot group N satisfying theinitial condition x (0) = 0 , (cid:96) i (0) = 0 , y i (0) = 0 , i = 1 , , are either lines of theform ( x, (cid:96) , (cid:96) , (cid:96) , y , y , y ) t = ( C t, C t, C t, C t, , , t (14) parameterized by constants C , C , C , C satisfying C + C + C + C = 1 , or theyare curves given by equations x = C cos( Kt ) + C sin( Kt ) − C , (15) (cid:18) (cid:96) (cid:96) (cid:96) (cid:19) = 1 K ( C sin( Kt ) − C cos( Kt ) + C ) (cid:18) K K K (cid:19) + t (cid:18) − C K − C K C K C K (cid:19) , (16) (cid:16) y y y (cid:17) = 12 K ( C + C )( tK − sin( Kt )) (cid:18) K K K (cid:19) + 12 K ((2 C − C Kt ) sin( Kt ) − ( C Kt + 2 C ) cos( Kt ) + 2 C − tC K ) (cid:18) − C K − C K C K C K (cid:19) , (17) parameterized by constants C , C , C , C and K , K , K satisfying K ( C + C ) + ( C K + C K ) + C K + C K = 1 . (18) Proof.
The line (14) corresponds to K = 0 and thus h ( t ) = h (0) is constant anddefines the vector of constants ( C , C , C , C ). The length of this vector is equalto one on the level set . If K (cid:54) = 0 we obtain x, (cid:96) , (cid:96) , (cid:96) by direct integrationof the first part of (13) and involving the initial condition, where h ( t ) is givenby (11). Substituting the results into the second part of (13) we get y , y , y byintegration. The solutions define family of curves starting at the origin such that N SYMMETRIES OF (4 , the Hamiltonian H is constant along them. Geodesics correspond to the level set H = , i.e. h + h + h + h = 1. According to Proposition 1 this restriction readsas (18). (cid:3) Symmetries of the control system
Symmetries of the control system (5) coincide with the symmetries of the cor-responding left–invariant sub–Riemannian structure ( N, N , r ). These are preciselyautomorphisms of N preserving the distribution N together with its decomposition(4) and sub–Riemannian metric r . By infinitesimal symmetry we mean a vectorfield such that its flow is a symmetry at each time, [4].3.1. Infinitesimal symmetries.
Infinitesimal symmetries of the sub–Riemannianstructure ( N, N , r ) are vector fields v such that L v r = 0 and L v ( (cid:104) N (cid:105) ) ⊂ (cid:104) N (cid:105) and L v ( (cid:104) N , N , N (cid:105) ) ⊂ (cid:104) N , N , N (cid:105) . Proposition 3.
Infinitesimal symmetries of the control system (5) form a Liealgebra generated by ∂ x + (cid:96) ∂ y + (cid:96) ∂ y + (cid:96) ∂ y ,∂ (cid:96) − x ∂ y , ∂ (cid:96) − x ∂ y , ∂ (cid:96) − x ∂ y ,∂ y , ∂ y , ∂ y , (19) together with v = (cid:96) ∂ (cid:96) − (cid:96) ∂ (cid:96) + y ∂ y − y ∂ y ,v = (cid:96) ∂ (cid:96) − (cid:96) ∂ (cid:96) + y ∂ y − y ∂ y ,v = (cid:96) ∂ (cid:96) − (cid:96) ∂ (cid:96) + y ∂ y − y ∂ y . (20) Vector fields (19) are right–invariant and generate all tranvections that form anilpotent algebra with growth (4 , . Vector fields (20) generate so (3) that forms thestabilizer algebra at the origin.Proof. The statement follows by direct computation. We do not give here alldetails but rather explain main ideas. We start with general vector field v = (cid:80) j =0 f j N j + (cid:80) i =1 g i N i for functions f j , g i on N . The condition L v ( r ) = 0 impliesthat functions f j do not depend on y i and are linear in x and (cid:96) i . The correspond-ing system has solution f = c (cid:96) + c (cid:96) + c (cid:96) + c , f = − c x + c (cid:96) + c (cid:96) + c , f = − c x − c (cid:96) + c (cid:96) + c and f = − c x − c (cid:96) − c (cid:96) + c for constants c i .Annihilator of (cid:104) N , N , N (cid:105) is generated by forms − x d(cid:96) i + dy i for i = 1 , , dx . Annihilator of (cid:104) N (cid:105) is generated by forms (cid:96) i dx + dy i and d(cid:96) i for i = 1 , ,
3. Substituting the above functions f j into v , evaluating the aboveforms on it and putting them equal to zero gives a system whose solution space isgenerated by (19 , ∂ x + (cid:80) i (cid:96) i ∂ y i , ∂ (cid:96) i − x ∂y i ] = − ∂y i for i = 1 , , v i , i = 1 , , v , v ] = − v , [ v , v ] = v , [ v , v ] = − v and thus form an algebra isomorphic to so (3). It clearly preserves origin and formsthe stabilizer subalgebra at the origin. (cid:3) ON SYMMETRIES OF (4 , The result is not surprising. The Lie algebra contains transvections given byright–invariant vector fields due to the fact that we deal with a Lie group N .Moreover, there can be stabilizer at the origin which generally is a subalgebra of so (4), because it particularly preserves the metric r and distribution N . Since ourtransformations generically preserve the decomposition (4), they preserve N andthus we find so (3) generated by v , v , v acting on (cid:104) N , N , N (cid:105) .3.2. Fixed–point sets.
One can see from (20) that the action of the Lie group SO (3) on N generated by infinitesimal symmetry a v + a v + a v ∈ so (3) isgiven by simultaneous rotations on (cid:96) i and y i , i = 1 , , a =( a , a , a ), while the coordinate x is invariant. Since all invariants of a rotation aremultiples of its axis, the fixed points of the symmetry generated by a v + a v + a v form the set { ( x, ka , ka , ka , la , la , la ) : x, k, l ∈ R } . (21)These observations following from Proposition 3 are summarized in the followingtwo consequences, where we denote (cid:96) = ( (cid:96) , (cid:96) , (cid:96) ) t and y = ( y , y , y ) t . Corollary 1.
For each R ∈ SO (3) the map ( x, (cid:96), y ) (cid:55)→ ( x, R(cid:96), Ry )(22) maps geodesics starting at the origin to geodesics starting at the origin. Corollary 2.
Set of points that are fixed by some isotropy symmetry is the unionof sets (21) over all axes ( a , a , a ) C n = { ( x, (cid:96), y ) ∈ R × R × R : ∀ (cid:96) (cid:54) = 0 , ∃ λ ∈ R such that y = λ(cid:96) } , (23) i.e. the points such that the vectors (cid:96) and y are collinear. Remark 1.
One can see from the group structure (3) that the multiplication re-stricts correctly to C n , so this set in fact forms a subgroup of N . Moreover, C n isclearly invariant with respect to the action (22) . Optimality of geodesics
The set of points where geodesics intersect each other and the correspondinggeodesic segments have equal length is called the Maxwell set. Conjungate pointsare defined as critical points of the exponential map. It is proved that the normalextremal trajectory that does not contain pieces of abnormal geodesics loses itsoptimality in the conjungate point or in the Maxwel point, [1]. The points wheregeodesics lose their optimality are called cut points, where cut locus is the set ofcut points. In many cases, it contains sets of fixed points of symmetries. Indeed, ifa geodesic meets a fixed point of a symmetry, then the action of the symmetry cangive such set of geodesics, [14]. We discuss here geodesics starting at the origin,however, the results hold for geodesics starting at arbitrary point thanks to theleft–invariancy and action of vector fields (19).4.1.
Factor space.
Each choice of coefficients C , C , C , C ∈ R and K , K , K ∈ R that satisfy (18) gives geodesics ( x ( t ) , (cid:96) ( t ) , y ( t )) as described in the Proposition2. According to (13) and (17), (cid:96) ( t ) and y ( t ) are linear combinations of the vectors z = (cid:18) K K K (cid:19) , z = (cid:18) − C K − C K C K C K (cid:19) N SYMMETRIES OF (4 , for any t >
0. The vectors z and z are orthogonal with respect to the Euclideanmetric on R by definition. So, there is an orthogonal matrix R ∈ SO (3) that alignsvectors z and z with the suitable multiples of the first two vectors of the standardbasis of R . Thus we get z = R (cid:16) K (cid:17) , z = R (cid:16) C (cid:17) , where K = | z | is the length of z and we denote C = | z | the length of z . Thismatrix defines a representative of the geodesic class( x ( t ) , ¯ (cid:96) ( t ) , ¯ y ( t )) = ( x ( t ) , R t (cid:96) ( t ) , R t y ( t )) . The equations for this representative geodesics simplify remarkably. Namely x ( τ ) = C (cos τ −
1) + C sin τ, ¯ (cid:96) ( τ ) = C sin τ + C (1 − cos τ ) , ¯ (cid:96) ( τ ) = ¯ C τ, ¯ (cid:96) ( τ ) = 0 , ¯ y ( τ ) = ( C + C )( τ − sin τ ) , ¯ y ( τ ) = ¯ C [ C (2 sin τ − τ cos τ − τ ) + C (2 − τ − τ sin τ )] , ¯ y ( τ ) = 0 , (24)where τ = Kt and ¯ C = C/K . The level set equation (18) reads as K ( C + C + ¯ C ) = 1(25)and determines K >
N/SO (3) defined by the action (22) of SO (3) on N ∼ = R isdetermined by natural invariants x, ( (cid:96), (cid:96) ) , ( (cid:96), y ) , ( y, y ) , where ( , ) stands for theEuclidean scalar product on R . Proposition 4.
Each geodesic starting at the origin defines a curve in the factorspace
N/SO (3) given by a curve in invariants x = C (cos τ −
1) + C sin τ, ( (cid:96), (cid:96) ) = ( C sin τ + C (1 − cos τ )) + ( ¯ C τ ) , ( (cid:96), y ) = ( C + C )( C sin τ + C (1 − cos τ ))( τ − cos τ )+ ¯ C τ [ C (2 sin τ − τ cos τ − τ ) + C (2 − τ − τ sin τ )] , ( y, y ) = ( C + C ) ( τ − cos τ ) + ¯ C [ C (2 sin τ − τ cos τ − τ )+ C (2 − τ − τ sin τ )] . (26) Proof.
Follows directly from (24). (cid:3)
Subgroup C n . Let us recall that the subgroup C n , defined by (23), consistsof points in N that are stabilized by some non–trivial R ∈ SO (3) for the actionfrom Corollary 1. Note the similarity of C n to the set P from nilpotent (3 , P there exists a one–parameterfamily of geodesics of equal length intersecting at this point. However, the situationin our nilpotent (4 ,
7) problem is very different. , Theorem 1.
Sub–Riemannian geodesics starting at the origin either do not inter-sect C n or they lie in C n for all times.Proof. Suppose there is an intersection of the set C n with a sub–Riemannian geo-desic ( x ( t ) , (cid:96) ( t ) , y ( t )) emanating from the origin. So there is an point of intersection( x, ¯ (cid:96), ¯ y ) of the set C n with a sub–Riemannian geodesic ( x ( t ) , ¯ (cid:96) ( t ) , ¯ y ( t )) since C n isinvariant with respect to the action (22) of SO (3). At this intersection ( x, ¯ (cid:96), ¯ y ), thecollinearity of ¯ (cid:96) and ¯ y is described by vanishing of the determinant¯ (cid:96) ¯ y − ¯ (cid:96) ¯ y = 0 . The geodesics are given by equations (24) and the determinant can be writtenexplicitly as ¯ C ( d C + 2 d C C + d C ) , (27)where d = − τ − τ sin τ cos τ − τ + 2 ,d = − τ (2 cos τ − τ sin τ ) ,d = 2 cos τ + τ cos τ sin τ − τ − τ + 2 . We show that the function in the bracket of (27) is never zero (unless C = C = 0,which is irrelevant) by showing that its discriminant d of this quadratic equationis negative for all positive times. This implies that the colinearity condition (27)is equivalent to ¯ C = 0. Then ¯ (cid:96) ( τ ) = ¯ y ( τ ) = 0 by (24) and thus geodesic( x ( τ ) , ¯ (cid:96) ( τ ) , ¯ y ( τ )) belongs to C n for all τ > d is negative for all positive times we compute d = − τ ( τ − sin τ )( τ + τ sin τ + 4 cos τ − , hence it is sufficient to prove f ( τ ) = τ + τ sin τ + 4 cos τ − > τ . This can be done by combining ”local” and ”global”estimations of this function. The local estimation is obtained by the estimation ofgoniometric functions sin τ, cos τ by Taylor series. By evaluating the Taylor seriesof f in zero, we see we need to use the Taylor polynomial of degree seven and six,respectively. Then we get the estimation f ( τ ) > − τ ( τ − f on the interval (0 , √ τ, cos τ ≥ − f ( τ ) > τ − τ − f on the interval ( √ , ∞ ). The twointervals overlap and thus f is positive for all τ > (cid:3) In the Figure (2) we present both local and global estimation of f ( τ ). Remark 2.
The positivity of function f ( τ ) from the proof above can be shownalternatively by proving the positivity of its derivative. Indeed, the derivative isgiven by f (cid:48) ( τ ) = (2 + cos τ ) (cid:16) τ − τ τ (cid:17) and τ τ is concave. N SYMMETRIES OF (4 , Figure 2.
Estimation of f ( τ )4.3. Geodesics in C n . We study properties of geodesics contained in C n . Accord-ing to (27), this happens if and only if ¯ C = 0. Then the non–zero parts of geodesics(24) are x ( t ) = C (cos( Kt ) −
1) + C sin( Kt ) , ¯ (cid:96) ( t ) = C sin( Kt ) + C (1 − cos( Kt )) , ¯ y ( t ) = ( C + C )(( Kt ) − sin( Kt )) , (28)and level set condition (18) reads as K ( C + C ) = 1 . We show that these geodesics are preimages of geodesics in Heisenberg geometryand their optimality is well known, [1, 7, 13]. Thus, we get the following statement.
Theorem 2.
The vertical set { (0 , , y ) ∈ C n : y ∈ R } is the set where the geodesicsin C n starting at the origin lose their optimality. These points are Maxwell pointsand for geodesics defined by parameters C and C the cut time is t cut = 2 π (cid:113) C + C . Proof.
Since ¯ (cid:96) = (cid:112) ( (cid:96), (cid:96) ) and ¯ y = (cid:112) ( y, y ) are invariants, see (26), the expres-sion (28) defines a curve in the factor space C n /SO (3). For the choice of polarcoordinates in the plane (cid:104) C , C (cid:105) we get the standard description of geodesics onthree–dimensional Heisenberg group H . Indeed, the tangent space to the subgroup C n is generated by pushout vectors¯ N = ∂ x − ¯ (cid:96) ∂ ¯ y , ¯ N = ∂ ¯ (cid:96) + x ∂ ¯ y that are standard generators of Heisenberg Lie algebra. The group law (3) onthe subgroup C n defines an isomorphism C n /SO (3) ∼ = H . The cut locus of theHeisenberg group consists of the set of points { (0 , y ) ∈ R ⊕ ∧ R : y (cid:54) = 0 } . Namely, any geodesic from the origin loses its optimality at the point where it meetsthe vertical axis (0 , , ¯ y ) for the first time. These points are Maxwell points andthe corresponding time equals to t cut = πK . Sub–Riemannian geodesics in C n goingfrom the origin to the point ( x, (cid:96), λ(cid:96) ) form a preimage of the Heisenberg geodesic in , C n /SO (3) going from the origin to the point ( x, | (cid:96) | , λ | (cid:96) | ), where | (cid:96) | = ( (cid:96), (cid:96) ). Thesegeodesics have the same length and they lose their optimality at the same time. (cid:3) Since the geodesics in C n are preimages of Heisenberg geodesics under the SO (3)–action, we can visualize them in the same way. On the left hand side of Figure 3,there is so–called Heisenberg sub–Riemannian sphere. On the right side of the samefigure, there is a half–sphere with a family of geodesics from origin to the sphere. Figure 3.
Heisenberg sub–Riemannian sphere and geodesics fromorigin to the points of the sphere4.4.
Sub-Riemannian geodesics out of C n . Any geodesic from the origin to thepoint q = ( x, (cid:96), y ) defines a curve in N/SO (3), see Proposition 4. This curve can beseen as factorized exponential mapping exp : R → R which is given by (26). Theoptimality of the geodesic is guaranteed up to the first critical point of this map,i.e. the point with zero determinant of Jacobi matrix. If the determinant is non–zero, we find the parameters C , C , ¯ C and τ in terms of invariants of endpoint q .Substituting C , C , ¯ C and τ = tK into (24), where K is given by (25), we get ageodesic ( x ( t ) , ¯ (cid:96) ( t ) , ¯ y ( t )) from the origin to the endpoint ¯ q = ( x, ¯ (cid:96), ¯ y ) with the sameinvariants (26). So endpoints q and ¯ q lie in the same class of N/SO (3) and there isa unique transformation R ∈ SO (3) such that (cid:96) = R ¯ (cid:96) and y = R ¯ y , see Corollary 1.Then the geodesic from the origin to point q is given by t (cid:55)→ ( x ( t ) , R ¯ (cid:96) ( t ) , R ¯ y ( t )) , where t ∈ (cid:104) , T (cid:105) and T = τ (cid:112) C + C + ¯ C is the length of this sub–Riemaniangeodesic.The formulae for geodesics in Proposition 2 in the time t = 1 define the unit six–dimensional sub–Riemannian sphere parameterized by C , C , C , C and K , K , K .It is a disjoint union of the part in C n and outside of C n . In the Figure 4 one cansee two explicit cross sections of projections of the sub–Riemannian sphere withrespect to parameters ( x, (cid:96) , y ) and a family of geodesics going from the origin tothe sphere. N SYMMETRIES OF (4 , Figure 4.
Two different cross sections of a projection of the sub–Riemannian sphere with a family of geodesics.
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