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NEW HIGH ORDER SUFFICIENT CONDITIONS FOR CONFIGURATION TRACKING 5 1 M.BARBERO-LINÃŃ, M.SIGALOTTI 0 2 n a J Abstract. Inthispaper,weproposenewconditionsguaranteeingthatthetrajectoriesof 6 amechanical controlsystemcantrackanycurveontheconfiguration manifold. Wefocus 1 on systems that can be represented as forced affine connection control systems and we generalize the sufficient conditions for tracking known in the literature. The new results ] are proved by a combination of averaging procedures by highly oscillating controls with C thenotionofkinematicreduction. O . h t a m 1. Introduction [ New geometric techniques are used to generalize tracking conditions known in the litera- 1 ture [1, 2, 3]. The tracking problem plays a key role in the performance of robots and me- v chanical systems such as submarines and hovercrafts in order to avoid obstacles, stay nearby 6 a preplanned trajectory, etc. 2 Mechanicalcontrolsystemsarecontrol-affinesystemsonthetangentbundleoftheconfigu- 0 ration manifold Q. In order to simplify the motion planning tasks for these control systems, 4 0 a useful tool has been introduced in the geometric control literature, namely, the notion . of kinematic reduction. Such a procedure consists in identifying a control-linear system on 1 Q whose trajectory mimic those of the mechanical system. This approach has been useful 0 5 to describe controllability, planning properties [1] and optimality [4] of mechanical systems. 1 However, as described in [1], kinematic reduction is not always possible, some conditions v: relatedtothe symmetricclosureofthe controlvectorfieldsofbothsystemsunderstudymust i be satisfied. X In our previous work [2] we extended the first-order sufficient conditions for tracking pro- r posedin[1]by usingdifferentfamilies ofvectorfields, possiblyofinfinite cardinality. Related a constructions to generate admissible directions for tracking have been proposed in [5, 6] (see also [7, 8]). Our first goalin this currentpaper is to establish a relationshipbetween families of vector fields defined pointwise and sets of sections of the tangent bundle defined in a recurrent way, similarly to the classical Malgrange theorem [9]. This new pointwise characterizationof families of vector fields used in [2] allows to use kinematic reduction in order to obtain more generalsufficienttrackingconditions. Asaresult,itcanbeprovedthatanunderwatervehicle with a natural choice of control vector fields is always trackable, even in the most symmetric case (see the example in Section 4.4 for more details). Thepaperisorganizedasfollows. Section2containsallthenecessarybackgroundinforced affine connectioncontrolsystems [1]. Section 3 defines the notionoftrackability under study. After recalling in Section 4.1 the high-order tracking conditions known in the literature and obtainedby averagingtheory[2],kinematic reductionis usedto obtainmore generaltracking conditionsinSection4.3. Thefullcharacterizationofthetrackabilityofthesystemdescribing the motion of an underwater vehicle is achieved in Section 4.4, concluding the study started in [3] and continued in [2]. 1 2 M.BARBERO-LINÃŃ, M.SIGALOTTI 2. Notation and preliminaries Denote by N the set of positive natural numbers and write N for N∪{0}. Fix n ∈ N. 0 From now on, Q is a n–dimensional smooth manifold and X(Q) denotes the set of smooth vectorfieldsonQ. AllvectorfieldsareconsideredsmoothasfunctionsonQ,unlessotherwise stated. Let τ : TQ → Q be the canonical tangent projection. A vector field along τ is a Q Q mapping X: TQ → TQ such that τ ◦X = τ . We denote by I a compact interval of the Q Q type [0,τ], τ >0. 2.1. Affine connection control systems. The trajectories γ : I → Q of a Lagrangian mechanical systems on a manifold Q are minimizers of the action functional A (γ)= L(t,γ˙(t))dt L Z I associated with a Lagrangianfunction L: R×TQ→R. Thesolutionstothisvariationalproblemmustsatisfythewell-knownEuler–Lagrangeequa- tions, d ∂L ∂L − =0, i=1,...,n, (1) dt(cid:18)∂vi(cid:19) ∂qi where (qi,vi) are local coordinates for TQ. Here we consider controlled Euler–Lagrange equations obtained by modifying the right-hand side on the above equation, as follows: k d ∂L ∂L − = u Yi, i=1,...,n, dt(cid:18)∂vi(cid:19) ∂qi a a Xa=1 with u : I →R, Yi: Q→R. a a When the manifold Q is endowed with the Riemannian structure given by a Riemannian metric g and the Lagrangian function L (v ) = 1g(v ,v ) is considered, the solutions to (1) g q 2 q q turn out to be the geodesics of the Levi–Civita affine connection ∇g associated with the Riemannian metric. (See [1] for more details and for many examples of mechanical control systems that fit in this description.) When control forces are added to the geodesic equations we obtain an affine connection control system k ∇g γ˙(t)= u (t)Y (γ(t)), γ˙(t) a a Xa=1 where Y ,...,Y are vector fields on Q. 1 k The notion of affine connection control system can be extended without the need of the Levi–Civita connection. Definition 2.1. An affine connection is a mapping ∇: X(Q)×X(Q) −→ X(Q) (X,Y) 7−→ ∇(X,Y)=∇ Y, X satisfying the following properties: (1) ∇ is R–linear on X and on Y; (2) ∇ Y =f∇ Y for every f ∈C∞(Q); fX X (3) ∇ fY = f∇ Y +(Xf)Y, for every f ∈ C∞(Q). (Here Xf denotes the derivative X X of f in the direction X.) The mapping ∇ Y is called the covariant derivative of Y with respect to X. Given local X coordinates (qi) on Q, the Christoffel symbols for the affine connection in these coordinates are given by n ∂ ∂ ∇ = Γi , j,r =1,...,n. ∂∂qj ∂qr Xi=1 jr∂qi NEW HIGH ORDER SUFFICIENT CONDITIONS FOR CONFIGURATION TRACKING 3 From the properties of the affine connection, we have n ∂Yi ∂ ∇ Y = Xj +Γi XjYr , X (cid:18) ∂qj jr (cid:19)∂qi i,jX,r=1 where X = n Xi∂/∂qi and Y = n Yi∂/∂qi. i=1 i=1 P P Definition 2.2. A forced affine connection control system (FACCS) is a control mechanical system given by Σ=(Q,∇,Y,Y,U) where • Q is a smooth n–dimensional manifold called the configuration manifold, • Y is a smooth time-dependent vector field along the projection τ : TQ → Q, affine Q with respect to the velocities, • Y is a set of k control vector fields on Q, and • U is a measurable subset of Rk. A trajectory γ: I ⊂R→Q is admissible for Σ if γ˙: I →TQ is absolutely continuous and there exists a measurable and bounded control u: I →U such that the dynamical equations of the control system Σ k ∇ γ˙(t)=Y(t,γ˙(t))+ u (t)Y (γ(t)), (2) γ˙(t) a a Xa=1 are fulfilled (for almost every t∈I). The vector field Y includes all the non-controlled external forces; e.g., the potential and the non-potentialforces. TheassumptionthatY is affinewithrespecttothe velocitiesmeans that, in every local system of coordinates (qi,vi) on TQ, Y can be written as n Y(t,v )=Y (t,q)+ viYi(t,q). q 0 Xi=1 Equation (2) can be rewritten as a first-order control-affine system on TQ, k Υ˙(t)=Z(Υ(t))+YV(t,Υ(t))+ u (t)YV(Υ(t)), (3) a a Xa=1 where Υ: I →TQ is such that τ ◦Υ=γ, Z is the geodesic spray associated with the affine Q connectiononQand,foreveryX ∈X(Q), XV denotestheverticalliftofX (see[10]formore details). Apartfromthe usualLie bracketthat providesX(Q)with aLie algebrastructure,one can associate with ∇ the following product in X(Q). Definition 2.3. The symmetric product is the map h·: ·i: X(Q)×X(Q) −→ X(Q) (X,Y) 7−→ ∇ Y +∇ X. X Y It can be proved that [YV,[Z,YV]]=hY : Y iV (4) a b a b (see [1]). 3. Tracking problem Weconsiderheretheproblemarisingwhenonetriestofollowaparticulartrajectoryonthe configurationmanifold,calledreference ortarget trajectory,whichisingeneralnotasolution of the FACCS considered. A trajectory is successfully tracked if there exist solutions to the FACCS that approximate it arbitrarily well. Consider any distance d: Q×Q→R on Q whose correspondingmetric topologycoincides with the topology on Q. 4 M.BARBERO-LINÃŃ, M.SIGALOTTI Definition 3.1. A curve γ: I →Q of class C1 is trackable for the FACCS Σ if, for every strictly positive tolerance ǫ, there exist a control uǫ ∈ L∞(I,U) and a solution ξǫ: I → Q to Σ corresponding to uǫ such that ξǫ(0)=γ(0) and d(γ(t),ξǫ(t))<ǫ for every t ∈ I. The trajectory is said to be strongly trackable for Σ if, in addition to the above requirements, for every ǫ > 0 the approximating trajectory ξǫ may be found also satisfying ξ˙ǫ(0)=γ˙(0). A control system Σ satisfies the configuration tracking property (CTP) (respectively, the strong configuration tracking property (SCTP)) if every curve on Q of class C1 is trackable (respectively, strongly trackable) for Σ. Remark 3.2. Since any C1 curve can be uniformly approximated, with arbitrary precision, byasmoothcurvehavingthesametangentvectoratitsinitialpoint,thenΣsatisfiestheCTP (respectively,theSCTP)ifandonlyifeverycurveonQofclassC∞ istrackable(respectively, strongly trackable) for Σ. ⋄ 3.1. Tracking results for control-linear systems. A control-linear system (also called driftless kinematicsystem)onQisatriple(Q,X,U)whereX isafinitesubset{X ,...,X } 1 m of X(Q) and U is a measurable subset of Rm, identified with the control system m γ˙(t)= u (t)X (γ(t)), γ(t)∈Q, a a Xa=1 where u (·),...,u (·) are measurable and bounded functions with (u (t),...,u (t))∈U for 1 m 1 m every t. Proposition 3.3 (See[11,12]). Let X ,...,X besmooth vector fields on Qand take κ∈N. 1 m Let {X ,...,X } be the set of all Lie brackets of the vector fields X ,...,X of length less 1 mˆ 1 m than or equal to κ. Assume that γ :I →Q is a C∞ curve such that mˆ γ˙(t)= w (t)X (γ(t)), a a Xa=1 with w : I → Rmˆ smooth. Then, for every ǫ > 0 there exists a solution γ of the control- ǫ linear system (Q,{X ,...,X },Rm) with smooth control u : I → Rm and initial condition 1 m ǫ γ (0)=γ(0) such that d(γ(t),γ (t))<ǫ for every t∈I. ǫ ǫ From the above proposition we deduce the following result. (Similar arguments can be found in [13].) Corollary 3.4. If the Lie algebra Lie(X ,...,X ) generated by X ,...,X has constant 1 m 1 m rank on Q, then for every smooth curve γ :I →Q and for every ǫ>0 there exists a solution γ of the control-linear system (Q,{X ,...,X },Rm) with smooth control u : I → Rm and ǫ 1 m ǫ initial condition γ (0)=γ(0) such that d(γ(t),γ (t))<ǫ for every t∈I. ǫ ǫ Proof. Theproofworksbycoveringthecompactsetγ(I)byfinitelymanyopensetsΩ ,...,Ω 1 K ofQsuchthatforeveryj =1,...,KthereexistsonΩ abasisofthedistributionLie(X ,...,X ) j 1 m made of Lie brackets of X ,...,X . Let κ be the maximum of the length of the brackets 1 m used to constructsuch bases and let {X ,...,X } be the set ofall Lie bracketsof the vector 1 mˆ fields X ,...,X of length less than or equal to κ. Then 1 m mˆ γ˙(t)= w (t)X (γ(t)), a a Xa=1 with w:I →Rmˆ smooth and we conclude by Proposition 3.3. NEW HIGH ORDER SUFFICIENT CONDITIONS FOR CONFIGURATION TRACKING 5 3.2. Previousstrongconfigurationtrackingresults. ConditionsguaranteeingtheSCTP havebeenobtainedin[2],generalizingpreviousresultspresentedin[1](inparticularTheorem 12.26) and in [3]. We recall them here below in a versionadapted to what follows. The main difference of these statements from the ones of Theorem 4.4 and Corollary 4.7 in [2] is that here we focus onthe strongconfigurationtrackabilityof agiventrajectory,insteadof looking at the SCTP. The proof is however exactly the same, since the proof proposed in [2] is based on an argument where the target trajectory is fixed. Proposition3.5. LetΣ=(Q,∇,Y,Y,Rk)beaFACCS.Constructthefollowingsetofvector fields on Q: K = span Y, 0 C∞(Q) (5) Kl = Kl−1−co{hZ: Zi|Z ∈L(Kl−1)}, for l ∈N, where, for A⊂X(Q), L(A)=A∩(−A), co(A) denotes the convex hull of A, and A is the closureof A in X(Q) with respect tothetopology of theuniform convergence on compact sets. Fix a smooth reference trajectory γ : I → Q of class C∞. Assume that there exist ref l,N ∈N, λ ,...,λ ∈C∞(I,[0,+∞)), and Z ,...,Z ∈K such that 1 N 1 N l N ∇ γ˙ (t)−Y(t,γ˙ (t))= λ (t)Z (γ (t)), ∀t∈I. γ˙ref(t) ref ref a a ref Xa=1 Then γ is strongly trackable. ref Proposition 3.6. Let Σ=(Q,∇,Y,Y,Rk) be a FACCS. Define the following sets of vector fields for l ∈N, Z =Y , 0 Zl =Zl−1∪{hZa: Zbi|Za,Zb ∈Zl−1}. (6) Assume that there exists l ∈ N such that for each i ∈ {0,...,l − 1}, for each Z ∈ Z , i hZ: Zi∈spanC∞(Q)Zi. Let Z = {Z ,...,Z }. Fix a smooth reference trajectory γ : I → Q of class C∞. If l 1 N ref there exist λ ,...,λ ∈C∞(I,R) such that 1 N N ∇ γ˙ (t)−Y(t,γ˙ (t))= λ (t)Z (γ (t)), ∀t∈I, γ˙ref(t) ref ref a a ref Xa=1 then γ is strongly trackable. ref 4. A generalization of tracking conditions Before introducing the new results about tracking, we need some technical lemmas described in Section 4.1 and to define kinematic reduction in Section 4.2. All that is necessary to prove the new theorem about trackability in Section 4.3. 4.1. Pointwise and sectionwise characterisation of K . The results in this section, in l the spirit of the classical Malgrange theorem (see [9]), aim at characterizing the sets K of l sections of TQ, introduced above, in terms of iterated computations of subsets of TQ. Let us then associate with a family Y = {Y ,...,Y } ⊂ X(Q), in addition to the family K , the 1 k l family of subsets of TQ defined pointwise, for every q ∈Q, as K = span Y(q), (7) 0,q R c Kl,q = Kl−1,q−co hZ: Zi(q)|Z ∈X(Q), Z(q′)∈L(Kl−1,q′)∀q′ ∈Q , (8) n o where for acny A⊂cX(Q) we write A(q)={Y(q)|Y ∈A}⊆T Q.c q 6 M.BARBERO-LINÃŃ, M.SIGALOTTI Recall that each K is a convex cone in X(Q) for the C0 topology (see [2, Proposition l loc 4.1]). It is also clear that the recursive definition of K describes a closed convex cone of l,q T Q. q c We need a preliminary result to establish the equivalence between the two definitions. Lemma 4.1. Let H be a C0 -closed set of X(Q) and assume that H is closed with respect loc to finite linear combinations with coefficients in C∞(Q,[0,∞)). Then H ={V ∈X(Q)|V(q)∈H(q) ∀q ∈Q}. (9) Proof. The inclusion H ⊆{V ∈X(Q)|V(q)∈H(q) ∀q ∈Q} is trivial and we are left to prove the opposite one. If V ∈ X(Q) and V(q) ∈ H (q) for all q ∈ Q, then for every q ∈ Q there exists Wq ∈ H such that Wq(q)=V(q). For all ǫ>0 there exists a closed neighbourhood Ωq,ǫ such that kWq−Vk∞,Ωq,ǫ ≤ǫ, wherek·k∞,Ωq,ǫ isthesupremumnormrestrictedtoΩq,ǫ,withrespecttoanyfixedRiemannian structure on Q. For every ǫ > 0, {Ωq,ǫ}q∈Q is an open covering of Q. Let (Qn)n∈N be an increasing sequence of compact subsets whose interiors coverQ. For every n∈N there exists a finite covering Ωq1,1/n,...,Ωqrn,1/n of Qn and a partition of unity a1,...,arn subordinated to {Ωqi,1/n}rn such that i=1 rn 1 a Wqi −V ≤ . (cid:13) i (cid:13) (cid:13)(cid:13)Xi=1 (cid:13)(cid:13)∞,Qn n In particular, the sequence ( ri=n(cid:13)(cid:13)1aiWqi)n∈N is(cid:13)(cid:13)contained in H and converges uniformly to V on compact sets. As H isPclosed, it follows that V ∈H. Proposition 4.2. For every integer l ≥0, let K and K be the convex cones defined in (5) l l,q and (8) respectively. Then c K ={V ∈X(Q)|V(q)∈K (q)} (10) l l and K (q)=K , for every q ∈Q. (11) l l,q c Proof. We first prove (10) by induction on l. According to Lemma 4.1, it is enough to prove that if V ∈ K and a ∈ C∞(Q,[0,∞)), then aV ∈ K . The step l = 0 is trivial. Let l ≥ 1 l l and assume that the property is true for l−1. Since Kl−1 −co{hZ: Zi|Z ∈L(Kl−1)} is C0 -dense in K and {a2 | a ∈ C∞(Q,R)} is C0 -dense in C∞(Q,[0,∞)), it is enough to loc l loc prove that a2V ∈ Kl for every V ∈ Kl−1 −co{hZ: Zi|Z ∈L(Kl−1)} and a ∈ C∞(Q,R). Write V = W − Jj=1λjhZj: Zji with W ∈ Kl−1, λ1,...,λJ > 0 with Jj=1λj = 1 and Z1,...,ZJ ∈ L(KPl−1). By induction hypothesis a2W ∈ Kl−1. Moreover,PL(Kl−1) is also a C0 -closed set of X(Q), closed with respect to finite linear combinations with coefficients in loc C∞(Q,[0,∞)). Hence, applying Lemma 4.1 to H = L(Kl−1) we deduce that aZ1,...,aZJ belong to L(Kl−1). It can be easily proved using (4) that haZ : aZ i=a2hZ : Z i+b Z j j j j j j for some smooth function bj. By induction hypothesis, bjZj is in Kl−1. Hence, −a2hZj: Zji lies in Kl−1 −co{hZ: Zi|Z ∈L(Kl−1)}, concluding the proof of the identity Kl = {V ∈ X(Q)|V(q)∈K (q)}. l NEW HIGH ORDER SUFFICIENT CONDITIONS FOR CONFIGURATION TRACKING 7 As a consequence, if Z(q)∈L(K (q)) for all q ∈Q, then Z ∈L(K ), which implies that l l L(K )={Z ∈X(Q)|Z(q)∈L(K (q)) ∀q ∈Q}. (12) l l Let us now prove, again by induction on l, that (11) is true. The case l = 0 is trivial. Let us assume that (11) holds for l−1, and let us prove it for l. According to (12) and the induction hypothesis, L(Kl−1)={Z ∈X(Q)|Z(q)∈L(Kl−1,q) ∀q ∈Q}. The definition of K then gives c l,q c Kl,q =Kl−1(q)−co{hZ: Zi(q)|Z ∈L(Kl−1)} which gives the result wchen compared with (5). 4.2. Kinematic reduction. It is already known in the literature that to perform certain motion planning tasks it is useful to reduce a mechanical control system to a control-linear system in such a way that there exist relationships between the trajectories of both control systems. Before proceeding, we introduce some necessary definitions. Let X = {X ,...,X } ⊂ X(Q) and consider the control-linear system (Q,X,Rm) (de- 1 m fined in Section 3.1). Let us introduce the notations Sym(0)(Y ) = span Y(q), q R Sym(1)(Y ) = Sym(0)(Y ) +span {hW: Zi(q)|W,Z ∈Y }. q q R Definition 4.3. Let Σ = (Q,∇,0,Y,Rk) be a FACCS. A driftless kinematic system Σ = kin (Q,X,Rm) is a kinematic reduction of Σ if for every controlled trajectory (γ,u ) of kin Σ with u smooth there exists u smooth such that (γ,u) is a controlled trajectory for Σ. kin kin Let us recall the following result from [1]. Theorem 4.4 ([1, Theorem 8.18]). Let Σ and Σ be as in Definition 4.3. Assume that X kin and Y generate constant-rank distributions. Then Σ is a kinematic reduction of Σ if and kin only if Sym(1)X ⊂span Y(q) for every q ∈Q. q R 4.3. A new criterion for trackability. Let us now generalize the sufficient conditions for tracking given in Proposition 3.6. Theorem 4.5. Let Σ = (Q,∇,Y,Y,Rk) be a FACCS. Define the families Z , i ∈ N, of i vector fields on Q as in (6). Assume that there exists l∈N such that (1) Y(t,p)∈span Z (τ (p)) for every p∈TQ; R l Q (2) the distributions spanRZl−1, spanRZl, and Lie(Zl−1) have constant rank; (3) for all i∈{0,...,l−1} and Z ∈Zi, hZ: Zi∈spanC∞(Q)Zi. Fix a smooth reference trajectory γref :I →Q of class C∞. If γ˙ref(t)∈Lieγref(t)(Zl−1) for every t ∈ I, then γref is trackable. In particular, if Lieq(Zl−1) = TqQ for every q ∈ Q then the CTP holds. Proof. Let l be as in the statement of the theorem and consider the FACCS Σ =(Q,∇,0,Z,Rml), l l where m is the cardinality of Z . l l As recalled in Theorem 4.4, Σl−1,kin =(Q,Zl−1,Rml−1) is a kinematic reduction of Σl, where ml−1 is the cardinality of Zl−1, since Sym(1)(Zl−1)q = spanRZl(q) forevery q ∈Q. Hence, everycontrolledtrajectoryofΣl−1,kin is alsoa controlled trajectory of Σ . l 8 M.BARBERO-LINÃŃ, M.SIGALOTTI Since γref is tangent to the distribution Lie(Zl−1), we deduce from Corollary 3.4 that γref can be tracked with arbitrary precision by trajectories of Σl−1,kin. Hence, given a positive tolerance ǫ, there exists a controlled trajectory (γ1,u1) of Σl−1,kin (still defined on the time-interval I) with γ1(0) = γref(0), u1 : I → Rml−1 smooth, and such that d(γ (t),γ (t))<ǫ/2 ref 1 for every time t∈I. Now, by kinematic reduction, there exists u2 : I → Rml smooth such that (γ1,u2) is a controlled trajectory of Σ . Since the distribution generated by Z has constant rank, we l l canrepresentY(t,γ (t)) as alinear combinationofZ (γ (t)),...,Z (γ (t)) with coefficients 1 1 1 ml 1 depending smoothly on the time. We recover that ml ∇ γ˙ (t)−Y(t,γ (t))= λ (t)Z (γ (t)), ∀t∈I, γ˙1(t) 1 1 a a 1 Xa=1 with λ ,...,λ ∈C∞(I,R). 1 ml ApplyingnowProposition3.6,wehavethatγ isstronglytrackableforΣ,andinparticular 1 there exists u :I →Rk suchthat the trajectoryγ of Σ correspondingto u andwith initial 3 3 3 condition γ˙ (0)=γ˙ (0) satisfies 3 1 d(γ (t),γ (t))<ǫ/2 1 3 for every t∈I. We then conclude that γ is trackable for Σ. ref In order to generalize the argument to situations in which the hypothesis that hZ: Zi ∈ span Z for every Z ∈Z cannot be assumed, we introduce in the theorem below a new C∞(Q) i i requirement on the linearity of the cones K . i Theorem 4.6. Let Σ = (Q,∇,Y,Y) be a FACCS. Define the families K , i ∈ N, of vector i fields on Q as in (5). Assume that there exists l∈N such that • Y(t,p)∈K (τ (p)) for every p∈TQ; l Q • for all q ∈Q, L(Kl−1(q))=Kl−1(q) and L(Kl(q))=Kl(q); • the distributions Kl−1 and Kl and Lie(Kl−1) have constant rank. Fix a smooth reference trajectory γref :I →Q. If γ˙ref(t)∈Lieγref(t)(Kl−1) for every t∈I, then γ is trackable. ref In particular, if Lieq(Kl−1)=TqQ for every q ∈Q then the CTP holds. Proof. The reasoning works similarly to the one used in the proof of Theorem 4.5. The first step is then to check that Sym(1)(Kl−1) ⊆ Kl, allowing kinematic reduction arguments. By definition of Kl (see (5)), we know that −hZ: Zi lies in Kl for each Z ∈L(Kl−1). Moreover, we deduce from (10) in Proposition 4.2 and the hypothesis L(K (q))=K (q) for q in Q and j j j = l −1,l, that L(K ) = K for j = l −1,l. Hence, ±hZ: Zi lies in K for every Z in j j l Kl−1. As the symmetric product of any vector field can be written as a linear combination of symmetric products of vector fields with themselves, 1 hZ: Wi= (hZ+W: Z+Wi−hZ: Zi−hW: Wi), 2 we conclude that Sym(1)(Kl−1)⊆Kl. LetV ={V1,...,Vm}beasetofgeneratorsofthedistributionq 7→Kl−1(q)alongγref,i.e., Kl−1(γref(t))=span{V1(γref(t)),...,Vm(γref(t))}foreveryt∈I. ItfollowsfromCorollary3.4 that the trajectories of Σl−1,kin =(Q,V,Rm) can track γref with arbitrary precision. Hence, givenapositivetoleranceǫ,thereexistsacontrolledtrajectory(γ1,u1)ofΣl−1,kin(stilldefined on the time-interval I) with γ (0)=γ (0) such that u is smooth and 1 ref 1 d(γ (t),γ (t))<ǫ/2 ref 1 for every time t∈I. NEW HIGH ORDER SUFFICIENT CONDITIONS FOR CONFIGURATION TRACKING 9 Let U ={U ,...,U } be a set of generators of K along γ . Since 1 r l 1 Sym(1)(V)q ⊆Sym(1)(Kl−1)q ⊆Kl(q)=spanR{U1(q),...,Ur(q)} inaneighbourhoodofthecurveγ ,wededucefromTheorem4.4thatthereexistsu :I →Rr ref 2 such that (γ ,u ) is a controlled trajectory of Σ =(Q,∇,0,U,Rr). 1 2 l Hence r r ∇ γ˙ (t)= η+(t)U (γ (t))+ η−(t)(−U (γ (t))), ∀t∈I, γ˙1(t) 1 a a 1 a a 1 aX=1 Xa=1 with η+,...,η+,η−,...,η− ∈C∞(I,[0,+∞)). 1 r 1 r SinceK hasconstantrank,wecanrepresentY(t,γ˙ (t))asalinearcombinationofU (γ (t)), l 1 1 1 ...,U (γ (t)) with coefficients depending smoothly on the time. We recover that r 1 r r ∇ γ˙ (t)−Y(t,γ˙ (t))= λ+(t)U (γ (t))+ λ−(t)(−U (γ (t))), ∀t∈I, γ˙1(t) 1 1 a a 1 a a 1 Xa=1 Xa=1 with λ+,...,λ+,λ−,...,λ− ∈C∞(I,[0,+∞)). 1 r 1 r ApplyingnowProposition3.5,wehavethatγ isstronglytrackableforΣ,andinparticular 1 there exists u :I →Rk suchthat the trajectoryγ of Σ correspondingto u andwith initial 3 3 3 condition γ˙ (0)=γ˙ (0) satisfies 3 1 d(γ (t),γ (t))<ǫ/2 1 3 for every t∈I, and we conclude that γ is trackable for Σ. ref 4.4. Example. In this section we apply Theorem 4.5 to a control system studied in [2] and [3], completing the discussion on its trackability by tackling a case which was not covered by previously known criteria. The system models a neutrally buoyant ellipsoidal vehicle immersed in a infinite volume fluid that is inviscid, incompressible and whose motion is irrotational. The dynamics are obtained through Kirchhoff equations [14] and have a particularly simple form due to some symmetry assumption on the distribution of mass (see [3] for details and also [15] for general overview of control motion in a potential fluid). Consider the coordinates (ω,v) for the angular and linear velocity of the ellipsoid with respect to a body-fixed coordinate frame. Then the impulse (Π,P) of the system is given by Π ω =M (cid:18)P(cid:19) (cid:18)v(cid:19) where, under the symmetry assumptions mentioned above, M=diag(J ,J ,J ,M ,M ,M ), 1 2 3 1 2 3 diag(J ,J ,J )is the usualinertia matrix, andM ,M ,M take into accountthe mass ofthe 1 2 3 1 2 3 submarine and the added masses due to the action of the fluid. The configurationmanifoldQ forthis problemisthe SpecialEuclideangrouporthe group of rigid motions SE(3), which is homeomorphic to SO(3)×R3. Let (A,r) ∈ SE(3) be the attitude and the position of the ellipsoid. Denote by S: R3 → so(3) the linear bijection between R3 and the linear algebra so(3) of SO(3) such that 0 −x x 3 2 S(x1,x2,x3)= x3 0 −x1. −x x 0 2 1   The dynamics of the controlled system are given by dA dr =AS(ω), =Av, (13) dt dt 10 M.BARBERO-LINÃŃ, M.SIGALOTTI and u 0 dΠ 1 dP =Π×ω+P ×v+u2, =P ×ω+0. (14) dt dt 0 u 3     The controls correspondto a linear accelerationalong one of the three axes of the submarine and to two angular accelerations around the other two axes. It was proven in [3] that if M 6=M then system (13)–(14) satisfies the SCTP. In [2], moreover,based on generalquan- 1 2 titative estimates of the convergence yielding the sufficient conditions for tracking recalledin Proposition 3.6, an explicit tracking algorithm was proposed. The Lie group structure of the configuration manifold can be exploited to compute Lie bracketsandsymmetricproductsofleft-invariantvectorfields,andinparticularofthecontrol vector fields. (Otherwise, one can directly apply (4).) It turns out (see [1, 16] for details) S(w ) v S(w ) v that for η = 1 1 , η = 2 2 ∈ se(3), identified with the corresponding 1 (cid:18) 0 0(cid:19) 2 (cid:18) 0 0(cid:19) left-invariant vector fields, [S(w ),S(w )] S(w )v −S(w )v [η ,η ]= 1 2 1 2 2 1 . (15) 1 2 (cid:18) 0 0 (cid:19) Let {e ,...,e } be a basis adapted to the coordinates(A,r) so that η =ηie for a=1,2. It 1 6 a a i is then possible to compute the symmetric product as follows: hη : η i=−M−1(ad∗ Mη +ad∗ Mη ), (16) 1 2 η1 2 η2 1 where (ad∗α)ξ =α(ad ξ) for η,ξ ∈se(3), α∈se∗(3). η η The structural constants with respect to the basis {e ,...,e } are defined as 1 6 ck =[e ,e ]k, γk =he : e ik. ij i j ij i j By the expressions of the Lie bracket and the symmetric product given in (15) and (16), respectively, it follows that γk =−Mhk(M cl +M cl ), ij il jh jl ih being Mhk the entries of the inverse matrix of M. One can easily compute that c1 =c2 =c3 =c4 =c4 =c5 =c5 =c6 =c6 =1, 23 31 12 26 53 34 61 15 42 c1 =c2 =c3 =c4 =c4 =c5 =c5 =c6 =c6 =−1, 32 13 21 62 35 43 16 51 24 and J −J M −M γ1 =γ1 = 3 2, γ1 =γ1 = 3 2, 32 23 J 56 65 J 1 1 J −J M −M γ2 =γ2 = 1 3, γ2 =γ2 = 1 3, 31 13 J 46 64 J 2 2 J −J M −M γ3 =γ3 = 2 1, γ3 =γ3 = 2 1, 21 12 J 45 54 J 3 3 M M γ4 =γ4 = 3, γ4 =γ4 =− 2, 26 62 M 35 53 M 1 1 M M γ5 =γ5 =− 3, γ5 =γ5 = 1, 16 61 M 34 43 M 2 2 M M γ6 =γ6 = 2, γ6 =γ6 =− 1, 15 51 M 24 42 M 3 3 while all other structural constants are equal to zero. Notice that the control vector fields are the vertical lift to TSE(3) of 1 1 1 Y = e , Y = e , Y = e , 1 1 2 2 3 6 J J M 1 2 3