Explicit solutions of the kinetic and potential matching conditions of the energy shaping method
EExplicit solutions of the kinetic and potentialmatching conditions of the energy shapingmethod
Sergio D. GrilloInstituto Balseiro, UNCuyo-CNEASan Carlos de Bariloche, R´ıo Negro, Rep´ublica ArgentinaLeandro M. SalomoneCMaLP, Facultad de Ciencias Exactas, UNLPLa Plata, Buenos Aires, Rep´ublica ArgentinaMarcela ZuccalliCMaLP, Facultad de Ciencias Exactas, UNLPLa Plata, Buenos Aires, Rep´ublica ArgentinaOctober 2, 2018
Abstract
In this paper we present a procedure to integrate, up to quadratures,the matching conditions of the energy shaping method. We do that in thecontext of underactuated Hamiltonian systems defined by simple Hamil-tonian functions. For such systems, the matching conditions split intotwo decoupled subsets of equations: the kinetic and potential equations.First, assuming that a solution of the kinetic equation is given, we findintegrability and positivity conditions for the potential equation (becausepositive-definite solutions are the interesting ones), and we find an ex-plicit solution of the latter. Then, in the case of systems with one degreeof underactuation, we find in addition a concrete formula for the generalsolution of the kinetic equation. An example is included to illustrate ourresults.
The energy shaping method is a technique for achieving (asymptotic) stabiliza-tion of underactuated Lagrangian and Hamiltonian systems. See Ref. [10] for areview on the subject and [9, 19] for more recent works. In this paper, we shallconcentrate on Hamiltonian systems only. We shall represent underactuatedHamiltonian systems by pairs ( H, W ), where H : T ∗ Q → R is a Hamiltonian1 a r X i v : . [ m a t h - ph ] O c t unction on a finite-dimensional smooth manifold Q , and W is a (proper) sub-bundle of the vertical bundle of T ∗ Q , containing the actuation directions.The energy shaping method is based on the idea of feedback equivalence [10], and its purpose is to construct, for a given pair ( H, W ), a state feedbackcontroller and a Lyapunov function ˆ H : T ∗ Q → R for the resulting closed-loop system. To do that, a set of partial differential equations (PDEs), knownas matching conditions , must be solved. Such PDEs have the pair ( H, W ) asdatum and the aforementioned Lyapunov function ˆ H as their unknown.In the Chang version of the method [6, 7, 8, 9], on which we shall focus, onlysimple functions H and ˆ H are considered (i.e. functions with the kinetic pluspotential energy form), and the subbundle W is assumed to be the vertical liftof a subbundle W ⊆ T ∗ Q . In such a case, the matching conditions decomposeinto two subsets: the kinetic and potential matching conditions , or simply, the kinetic and potential equations. In canonical coordinates ( q , p ), if we write(sum over repeated indices is assumed) H ( q , p ) = 12 p i H ij ( q ) p j + h ( q ) (1)and ˆ H ( q , p ) = 12 p i ˆ H ij ( q ) p j + ˆ h ( q ) , (2)such equations read [see Ref. [13], Eqs. (56) and (57)] (cid:32) ∂ ˆ H ij ( q ) ∂q k H kl ( q ) − ∂ H ij ( q ) ∂q k ˆ H kl ( q ) (cid:33) p i p j p l = 0 (3)(the kinetic equation) and (cid:32) ∂ ˆ h ( q ) ∂q k H kl ( q ) − ∂h ( q ) ∂q k ˆ H kl ( q ) (cid:33) p l = 0 (4)(the potential equation), and they must hold for all p ∈ W ⊥ q (the orthogonalmust be calculated with respect to the metric defined by ˆ H ). Once a solution( ˆ H , ˆ h ) of these equations is found, the method gives a prescription to constructa state feedback controller. So, we can say that the core of the method consistsof solving above PDEs.Let us mention that one looks for a solution ˆ h of the potential equationwhich is positive-definite around some critical point of h . This insures that thefunction ˆ H is a Lyapunov function for the resulting closed-loop system and thegiven critical point.In Reference [16], necessary and sufficient conditions were given for the exis-tence of a solution ˆ h of the potential equation, once a solution ˆ H of the kineticequation is given (note that ˆ H can be seen as a datum for the potential equation).This was done within the framework of the Goldschmidt’s integrability theoryfor linear partial differential equations [12], which only works in the analytic2ategory. Nevertheless, no conditions have been presented in order to ensurethe existence of positive-definite solutions. Also, no general recipe to constructan explicit solution ˆ h has been developed. The first goal of the present paper isthree-fold: • to extend the results of [16] to the C ∞ category, • to include positivity conditions, • and to present a systematic procedure to integrate the potential equationup to quadratures.Regarding the kinetic equation, few general results are known about the ex-istence of solutions. In the case of underactuated systems with one degree ofunderactuation, the problem was completely solved in References [6, 7, 14, 11].However, a general prescription for finding explicit solutions is still lacking. Thesecond goal of this paper is, for one degree of underactuaction, to give such aprescription.The paper is organized as follows. In Section § H , ˆ h ) of the matching conditions. This gives rise to a new set of equations,in terms of which we shall study a particular subclass of solutions of the ki-netic equation. Given ˆ H inside the mentioned subclass, in Section § h of the potential equation.Also, sufficient conditions to ensure positive-definiteness of ˆ h are given. Allof these conditions together give rise to a procedure that enable us to find anexplicit expression for local solutions of the potential equation (up to quadra-tures). Finally, we devote Section § Basic notation and definitions.
Along all of the paper, Q will denote asmooth connected manifold of dimension n and ( T Q, τ, Q ) and ( T ∗ Q, π, Q ) thetangent bundle and its dual bundle, respectively. As it is customary, we denoteby (cid:104)· , ·(cid:105) the natural pairing between T q Q and T ∗ q Q at every q ∈ Q and by X ( Q )and Ω ( Q ) the sheaves of sections of τ and π , respectively. If F : Q → P is asmooth function between differentiable manifolds, we denote by F ∗ and F ∗ thepush-forward map and its transpose, respectively.Consider a local chart ( U, ϕ ) of Q , with ϕ : U → R n . Given q ∈ U , wewrite ϕ ( q ) = (cid:0) q , . . . , q n (cid:1) = q . For the induced local chart (cid:16) T ∗ U, ( ϕ ∗ ) − (cid:17) on T ∗ Q , i.e. the canonical coordinates of the cotangent bundle, we write, for all α ∈ T ∗ U , ( ϕ ∗ ) − ( α ) = (cid:0) q , . . . , q n , p , . . . , p n (cid:1) = ( q , p ) ,
3r simply (cid:0) ϕ ∗ q (cid:1) − ( α ) = p . By a quadratic form on a subbundle V ⊆ T ∗ Q we shall understand a function H : V → R given by the formula H ( α ) = 12 (cid:10) α, ρ (cid:93) ( α ) (cid:11) , ∀ α ∈ V, (5)where ρ is a fibered inner product on V ∗ , ρ (cid:91) : V ∗ → V is given by (cid:68) ρ (cid:91) ( u ) , v (cid:69) = ρ ( u, v ) , and ρ (cid:93) : V → V ∗ is the inverse of ρ (cid:91) . Given a subbundle W ⊆ V , by W ⊥ we shall denote the orthogonal complement w.r.t. ρ and by W (cid:93) the subbundle ρ (cid:93) ( W ) ⊆ V ∗ . It is easy to see that W ⊥ = ρ (cid:93) ( W ◦ ) = ( W ◦ ) (cid:93) . (6)Here, by W ◦ ⊆ V ∗ we are denoting the annihilator of W . We shall also say that W ⊥ is the orthogonal complement of W with respect to H .The vertical subbundle associated to any vector bundle is given by the kernelof the push-forward of the bundle projection, and every element (resp. smoothsection) of this subbundle is called vertical (resp. vertical vector field). Inparticular, for the cotangent bundle, it is given by ker π ∗ ⊂ T T ∗ Q . If α ∈ T ∗ q Q ,there is a canonical way to identify ker π ∗ ,α and T ∗ q Q . This can be done throughthe vertical lift map vlift πα : T ∗ q Q → ker π ∗ ,α , defined by vlift πα ( β ) = dd t ( α + tβ ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 . (7)This map is in fact an isomorphism of vector spaces for every α ∈ T ∗ Q .Given a smooth function f : T ∗ Q → R , we define the fiber derivative of f asthe vector bundle morphism F f : T ∗ Q → T Q such that, for every α, β ∈ T ∗ q Q , (cid:104) β, F f ( α ) (cid:105) = dd t f ( α + tβ ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = (cid:104) d f ( α ) , vlift πα ( β ) (cid:105) . (8)In canonical coordinates, (cid:10) d q i , F f (cid:0) p k d q k (cid:1)(cid:11) = ∂f ◦ ( ϕ ∗ ) − ∂p i ( q , p ) . (9)For a quadratic form H : T ∗ Q → R [see Eq. (5)], it can be shown that F H = ρ (cid:93) , (10)and consequently H ( α ) = 12 (cid:104) α, F H ( α ) (cid:105) , ∀ α ∈ T ∗ Q. (11)4enote by ω the canonical symplectic 2-form on T ∗ Q . Given two functions f, g : T ∗ Q → R , its canonical Poisson bracket is the function { f, g } := (cid:10) d f, ω (cid:93) (d g ) (cid:11) , (12)where ω (cid:93) is the inverse of ω (cid:91) : T Q → T ∗ Q and the latter is given by the equation (cid:68) ω (cid:91) ( u ) , v (cid:69) = ω ( u, v ) . In canonical coordinates, omitting ( ϕ ∗ ) − for simplicity, { f, g } ( q , p ) = (cid:18) ∂f∂q i ∂g∂p i − ∂g∂q i ∂f∂p i (cid:19) ( q , p ) . (13)Throughout all of the paper, we will use the following convention for indices latin indices i, j, k, l = 1 , . . . , n ;latin indices a, b, c, d = 1 , . . . , m ;greek indices µ, ν, σ, ρ = 1 , . . . , n − m. Suppose that we have an underactuated Hamiltonian system ( H, W ) with H = H + h ◦ π, (14)where H : T ∗ Q → R is a quadratic form [see Eq. (5)] and h : Q → R isan arbitrary smooth function. [In canonical coordinates, this means that H isgiven by Eq. (1)]. In other words, we are assuming that H is simple . Notethat F H = F H = ρ (cid:93) [see Eqs. (8) and (10)]. Suppose also that there exists asubbundle W of T ∗ Q of rank m such that [see (7)] W α = vlift πα ( W π ( α ) ) , ∀ α ∈ T ∗ Q. (15) Remark . Note that ( H, W ) can be described by the triple ( H , h, W ).In such a case, according to Ref. [13], the matching conditions of the Chang’sversion of the energy shaping method, for a simple unknown ˆ H = ˆ H + ˆ h ◦ π [seeEq. (2) for a local expression], are given by (cid:110) ˆ H , H (cid:111) ◦ F ˆ H − ( v ) = 0 , ∀ v ∈ W ◦ , (16)the kinetic equation , and (cid:16) dˆ h ◦ F H − d h ◦ F ˆ H (cid:17) ◦ F ˆ H − ( v ) = 0 , ∀ v ∈ W ◦ , (17)the potential equation [see Eqs. (73) and (74) and Remark 18 of Ref. [13]]. This will be the general setting and notation from now on. emark . In order to compare (16) and (17) with Eqs. (73) and (74) of Ref.[13], we must use that F ˆ H ( W ◦ ) = W ⊥ [see Eqs. (6) and (10)].Here {· , ·} denotes the canonical Poisson bracket on T ∗ Q [see Eq. (12)].Equations above are the intrinsic counterpart of Eqs. (3) and (4). Their dataare given by the triple ( H , h, W ), and their unknown by the pair ( ˆ H , ˆ h ). In thefollowing, we are going to re-write (16) and (17) by redefining the unknown. Lemma 1.
Given a subbundle W ⊆ T ∗ Q , the set of quadratic forms on T ∗ Q are in bijection with the triples ( ˆ W , K , L ) , where ˆ W is a complement of W , K is a quadratic form on ˆ W , and L is a quadratic form on W .Proof. To any quadratic form ˆ H : T ∗ Q → R , we can assign a triple ( ˆ W , K , L )with ˆ W := W ⊥ , K := ˆ H (cid:12)(cid:12)(cid:12) ˆ W and L := ˆ H (cid:12)(cid:12)(cid:12) W . (18)Here, W ⊥ means the orthogonal of W w.r.t. ˆ H . Reciprocally, to any triple( ˆ W , K , L ) as described in the lemma, we can assign the quadratic formˆ H := K ◦ ˆ p + L ◦ p , (19)where ˆ p : T ∗ Q → ˆ W and p : T ∗ Q → W (20)are the projections related to the decomposition T ∗ Q = ˆ W ⊕ W . It is clear thatthe map defined by Eq. (18) is the inverse of the map given by (19). Remark . If ˆ H and ( ˆ W , K , L ) are related as in the previous proof, then ˆ W isalways the orthogonal complement of W with respect to ˆ H . Proposition 1.
Fix a subbundle W ⊆ T ∗ Q and a quadratic form H : T ∗ Q → R .If a second quadratic form ˆ H : T ∗ Q → R is a solution of (16) , then K := ˆ H (cid:12)(cid:12)(cid:12) ˆ W is a solution of [see Eq. (20) ] { K ◦ ˆ p , H } ( σ ) = 0 , ∀ σ ∈ ˆ W , (21) where ˆ W := W ⊥ (the orthogonal complement of W w.r.t. ˆ H ). On the otherhand, given a complement ˆ W of W and its related projections ˆ p and p [seeEq. (20) again], if a quadratic form K : ˆ W → R satisfies (21) then, for everyquadratic form L : W → R , the function ˆ H given by (19) satisfies (16) .Proof. Given both a quadratic form ˆ H and a triple ( ˆ W , K , L ) related as in theprevious lemma, let us show that, for all σ ∈ ˆ W , { L ◦ p , H } ( σ ) = 0 . (22)6o do that, fix a coordinate chart (cid:0) U, (cid:0) q , . . . , q n (cid:1)(cid:1) of Q and a local basis { ξ , . . . , ξ m } ⊆ Ω ( U ) of the subbundle W . Define H ij ( q ) := (cid:68) d q i (cid:12)(cid:12) q , F H (cid:16) d q j (cid:12)(cid:12) q (cid:17)(cid:69) , q ∈ U, (23)and write p (cid:16) d q k (cid:12)(cid:12) q (cid:17) = P ka ( q ) ξ a ( q ) . Note that, using (11), H (cid:16) p k d q k (cid:12)(cid:12) q (cid:17) = 12 (cid:68) p k d q k (cid:12)(cid:12) q , F H (cid:16) p l d q l (cid:12)(cid:12) q (cid:17)(cid:69) = 12 p k p l H kl ( q ) . (24)On the other hand, if λ is the fibered inner product defining L , consider thematrix L ab ( q ) := (cid:10) ξ a ( q ) , λ (cid:93) ( ξ b ( q )) (cid:11) , q ∈ U. Then, omitting the dependence on q , just for simplicity, we have that L ◦ p (cid:0) p k d q k (cid:1) = 12 (cid:10) p (cid:0) p k d q k (cid:1) , λ (cid:93) (cid:0) p (cid:0) p l d q l (cid:1)(cid:1)(cid:11) = 12 p k p l P ka P lb L ab , and consequently [using (13)] { L ◦ p , H } (cid:0) p k d q k (cid:1) = (cid:18) ∂∂q k (cid:0) P ia P jb L ab (cid:1) H kl − ∂ H ij ∂q k P ka P lb L ab (cid:19) p i p j p l . In addition, since p k d q k ∈ ˆ W if and only if0 = p (cid:0) p k d q k (cid:1) = p k P ka ξ a , which in turn is equivalent to p k P ka = 0 for all a , the Eq. (22) is immediate.To end the proof, it is enough to note that (19) and (22) imply the equality (cid:110) ˆ H , H (cid:111) ( σ ) = { K ◦ ˆ p , H } ( σ ) , ∀ σ ∈ ˆ W , from which the proposition easily follows.So, we can replace the kinetic equation by Eq. (21), whose unknown is apair ( K , ˆ W ): ˆ W is a complement of W and K : ˆ W → R is a quadratic form.Now, let us study the potential equation. Given β ∈ T ∗ Q and σ ∈ ˆ W , onthe same fiber, it follows that (cid:104) β, F ( L ◦ p ) ( σ ) (cid:105) = dd t L ◦ p ( σ + tβ ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = dd t L ◦ p ( tβ ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = L ◦ p ( β ) dd t t (cid:12)(cid:12)(cid:12)(cid:12) t =0 = 0 , p ( σ ) = 0 and L is a quadratic form. So, F ( L ◦ p ) ( σ ) = 0 for all σ ∈ ˆ W ,and the potential equation (17) can be written (cid:16) dˆ h ◦ F H − d h ◦ F ( K ◦ ˆ p ) (cid:17) ( σ ) = 0 , ∀ σ ∈ ˆ W , (25)where only ˆ W and K are involved (the quadratic form L plays no role in eitherof the two matching conditions). As a consequence, instead of (16) and (17), wecan think of the matching conditions as the Eqs. (21) and (25) for the unknown( K , ˆ h, ˆ W ), and we shall do it from now on. For reasons that will be clear later, we shall concentrate on those solutions( K , ˆ W ) of the kinetic equations for which ˆ W (cid:93) = F H ( ˆ W ) is integrable. The factthat this is alway possible, unless locally, is proved in the next lemma. Lemma 2.
Given a subbundle W ⊆ T ∗ Q of rank m and a quadratic form H : T ∗ Q → R , around every point q ∈ Q we can construct, by using onlyalgebraic manipulations, a local coordinate chart (cid:0) U, (cid:0) q , . . . , q n (cid:1)(cid:1) such that ˆ W := F H − (cid:0) span (cid:8) ∂ / ∂q , . . . , ∂ / ∂q n − m (cid:9)(cid:1) (26) is a complement of W along U . In particular, ˆ W (cid:93) is an integrable subbundle.Proof. Given q ∈ Q , consider a coordinate neighborhood (cid:0) V, (cid:0) q , . . . , q n (cid:1)(cid:1) anda local basis { X , . . . , X m } ⊆ X ( V ) of W (cid:93) around q . It is clear that X i ( q ) = n (cid:88) j =1 C ij ( q ) ∂∂q j (cid:12)(cid:12)(cid:12)(cid:12) q , q ∈ V, being C ij ( q ) the coefficients of an m × n matrix C ( q ) of maximal rank. Reorder-ing the coordinate functions q i ’s, if necessary, we can ensure that the m × m sub-matrix S ( q ), given by the last m columns of C ( q ), is non-singular. Then,the vectors (cid:40) ∂∂q (cid:12)(cid:12)(cid:12)(cid:12) q , . . . , ∂∂q n − m (cid:12)(cid:12)(cid:12)(cid:12) q (cid:41) define a complement of W (cid:93)q and, by continuity, the first n − m coordinatevector fields (cid:8) ∂ / ∂q , . . . , ∂ / ∂q n − m (cid:9) span a complement of W (cid:93) along the openneighborhood U ⊆ V of q where S ( q ) is non-singular. As a consequence, thesubbundle ˆ W given by (26) is a complement of W along U . Remark . By Frobenius theorem, if ˆ W (cid:93) is an integrable subbundle, then ˆ W islocally given by (26) for some coordinate chart.8ow, let us fix a complement ˆ W of W such that ˆ W (cid:93) = F H ( ˆ W ) is integrableand, given q ∈ Q , consider a coordinate chart (cid:0) U, ϕ = (cid:0) q , . . . , q n (cid:1)(cid:1) around q where ˆ W is locally given by (26) (as in the last lemma). We want to findthe form that the matching conditions adopt in such coordinates. Consider thematrix with entries H ij given by (23) and define H ij := (cid:28) F H − (cid:18) ∂∂q i (cid:19) , ∂∂q j (cid:29) . (27)We are omitting the dependence on q , just for simplicity. Clearly, H ij H jk = δ ik . (28)Also, the co-vectors σ µ := F H − (cid:18) ∂∂q µ (cid:19) = H µk d q k ∈ ˆ W , µ = 1 , . . . , n − m, give a basis for ˆ W and we can writeˆ p (cid:0) d q k (cid:1) = ˆ P kµ σ µ . (29)Note that, since ˆ p ( σ µ ) = σ µ , H τk ˆ P kµ = δ µτ . (30) Remark . If the (local) forms ξ a := ϑ ai d q i , for a = 1 , . . . , m , give a (local)basis for W , since ker ˆ p = W , we also have the identity ϑ ai ˆ P iµ = 0 . (31)As a consequence, the matrix ˆ P is univocally determined by the (30) and (31).On the other hand, if κ denotes the fibered inner product on ˆ W defining K ,consider the matrix K µν := (cid:10) σ µ , κ (cid:93) ( σ ν ) (cid:11) . With this notation, we have that K ◦ ˆ p (cid:0) p k d q k (cid:1) = 12 (cid:10) ˆ p (cid:0) p k d q k (cid:1) , κ (cid:93) (cid:0) ˆ p (cid:0) p l d q l (cid:1)(cid:1)(cid:11) = 12 p k p l ˆ P kµ ˆ P lν K µν . (32)Thus, using (13), (24) and (32), the Eq. (21) translates to (cid:18) ∂∂q k (cid:16) ˆ P iµ ˆ P jν K µν (cid:17) H kl − ∂ H ij ∂q k ˆ P kµ ˆ P lν K µν (cid:19) p i p j p l = 0 . In addition, since a generic element of ˆ W has the form a τ σ τ = a τ H τk d q k , i.e. p k = a τ H τk , using the identities (28) and (30) we have that the kinetic equationreads (cid:18) ∂ K τ τ ∂q τ + G µντ τ τ K µν (cid:19) a τ a τ a τ = 0 , (33)9ith G µντ τ τ := ˆ P kµ δ ντ ∂ H τ τ ∂q k + ∂ (cid:16) ˆ P iµ ˆ P jν (cid:17) ∂q τ H τ i H τ j . (34)Now, let us study the potential equation in the above coordinates. From(23) we know that F H (cid:0) p k d q k (cid:1) = p k H kl ∂∂q l and, if p k = a τ H τk [see (28)], F H (cid:0) a τ H τk d q k (cid:1) = a τ ∂∂q τ . (35)On the other hand, using Eqs. (9) and (32) we have that F ( K ◦ ˆ p ) (cid:0) p k d q k (cid:1) = p k ˆ P kµ ˆ P lν K µν ∂∂q l and, again, if p k = a τ H τk , F ( K ◦ ˆ p ) (cid:0) a τ H τk d q k (cid:1) = a µ ˆ P lτ K τµ ∂∂q l . (36)Thus, from (35) and (36), the potential equation (25) reads (cid:32) ∂ ˆ h∂q µ − ∂h∂q k ˆ P kτ K τµ (cid:33) a µ = 0 , which is equivalent to ∂ ˆ h∂q µ − ∂h∂q k ˆ P kτ K τµ = 0 , µ = 1 , . . . , n − m. (37)[For simplicity, we are identifying h (resp. ˆ h ) with its local representative h ◦ ϕ − (resp. ˆ h ◦ ϕ − )].Summarizing the results of the entire section, we have the next two theorems. Theorem 1.
Consider an underactuated Hamiltonian system satisfying (14) and (15) , i.e. one defined by a triple ( H , h, W ) (see Remark 1). Then, every(simple) solution ˆ H = ˆ H + ˆ h ◦ π of the matching conditions (16) and (17) isunivocally described by: ( i ) a subbundle ˆ W complementary to W , ( ii ) a quadraticform K : ˆ W → R solving (21) , ( iii ) a solution ˆ h of (25) , and ( vi ) a quadraticform L : W → R . Theorem 2.
Let ˆ W be a complement of W such that ˆ W (cid:93) = F H ( ˆ W ) is integrable.Then, in a coordinate chart (cid:0) U, (cid:0) q , . . . , q n (cid:1)(cid:1) of Q satisfying (26) , the Equations (21) and (25) translate to (33) and (37) , respectively. In this section, given a solution of the kinetic equation (21), we shall study underwhich conditions a positive solution ˆ h of the potential equation (25) does exist.10n fact, we will develop a sistematic procedure to find (unless locally) an explicitsolution of this equation (up to quadratures). In addition, we will providenecessary and sufficient conditions to ensure the positivity of the solution. Suppose that the conditions in Theorems 1 and 2 hold, and that a solution K of (21) is given. We want to find a (local) solution ˆ h of the potential equa-tion (25) where ˆ p and K are considered as datum. Given q ∈ Q , denote by (cid:0) U, ϕ = (cid:0) q , . . . , q n (cid:1)(cid:1) the coordinate chart in which ˆ W is given by (26). In suchcoordinates, according to Theorem 2, Eq. (25) translates to Eq. (37). As it iswell-known, the necessary and sufficient conditions to integrate these equationsare ∂∂q µ (cid:18) ∂h∂q k ˆ P kτ K τν (cid:19) = ∂∂q ν (cid:18) ∂h∂q k ˆ P kτ K τµ (cid:19) , (38)for all µ, ν ≤ n − m . In global terms they say thatd (cid:0) d h ◦ F ( K ◦ ˆ p ) ◦ F H − (cid:1) ( u , v ) = 0 , ∀ u , v ∈ ˆ W (cid:93) . (39)In such a case, the solution not only exists but, furthermore, it can be computedup to quadratures. We shall give a formula for ˆ h at the end of this section [see(48)]. Remark . If n − m = 1, i.e. if the degree of underactuation is one, the sub-bundle ˆ W (cid:93) is always integrable because of dimensional reasons. In addition, thecondition (38) reduces to a single equation for µ = ν = 1, that immediatelyholds. Therefore, if we find a solution K of the kinetic equation for a systemwith one degree of underactuation, not only there exists a solution of the singlepotential equation (as it is already known in the literature), but even more,we can construct such a solution (in an appropriate coordinate chart) up toquadratures. Suppose that q is a critical point of h and, for simplicity, assume that theabove given coordinate chart ( U, ϕ ) is centered at q , i.e. ϕ ( q ) = (0 , . . . ,
0) =: . As we have mentioned in the Introduction, one is actually interested in asolution ˆ h of (37) which is positive-definite around q . Identifying ˆ h with itslocal representative ˆ h ◦ ϕ − , this is the same as saying that:1. is a critical point of ˆ h and2. the Hessian matrix of ˆ h at is positive-definite.If (38) holds, then a solution ˆ h of (37) exists and we can use the Methodof Characteristics to construct it. To do that, we must impose a boundarycondition, for instance, along the subset S := ϕ ( U ) ∩ ( { (0 , . . . , } × R m ) ⊆ R n . S the conditionˆ h (0 , . . . , , s , . . . , s m ) = (cid:36) m (cid:88) a =1 ( s a ) , (40)for some constant (cid:36) >
0. It follows from (37), the boundary condition aboveand the fact that is a critical point of h that ∂ ˆ h/∂q i ( ) = 0 for all i = 1 , . . . , n .Then, is a critical point of ˆ h . On the other hand, the Hessian of ˆ h at ,Hess(ˆ h ) ij ( ) = ∂ ˆ h∂q i ∂q j ( ) , (41)can be written Hess(ˆ h )( ) = (cid:20) M AA t (cid:36) I m (cid:21) , where: • I m is the m × m identity matrix; • A is the (( n − m ) × m )-matrix with entries A µa := (cid:16) Hess( h ) n − m + a,k ˆ P kτ K τµ (cid:17) ( )= ∂∂q n − m + a (cid:18) ∂h∂q k ˆ P kτ K τµ (cid:19) ( ) , µ ≤ n − m and a ≤ m (42)and • M is the square matrix of dimension n − m with M µν := (cid:16) Hess( h ) µk ˆ P kτ K τν (cid:17) ( ) = ∂∂q µ (cid:18) ∂h∂q k ˆ P kτ K τν (cid:19) ( ) . (43) Remark . The entries of the matrices A and M are obtained just by differ-entiating the potential equation (37) and using Eqs. (40) and (41) and thecriticallity of for h . Proposition 2.
Consider a local solution ˆ h of (37) . If Hess (ˆ h )( ) is positive-definite, then the matrix M , given by (43) , is positive-definite. Now, supposethat ˆ h satisfies (40) . If M is positive-definite, then there exists a constant (cid:36) such that Hess (ˆ h )( ) is positive-definite too.Proof. The first part of the proposition easily follows from the fact that M is anupper-left corner square sub-matrix of Hess(ˆ h )( ). So, let us prove the secondpart. Assume that M is positive-definite and take u ∈ R n − m and w ∈ R m .Then ( u , w ) (cid:20) M AA t (cid:36) I (cid:21) (cid:20) u t w t (cid:21) = u M u t + 2 u A w t + (cid:36) (cid:107) w (cid:107) = (cid:107) u (cid:107) M + 2 u A w t + (cid:36) (cid:107) w (cid:107) ≥ (cid:107) u (cid:107) M − (cid:107) u (cid:107) (cid:107) A (cid:107) (cid:107) w (cid:107) + (cid:36) (cid:107) w (cid:107) , (cid:107) · (cid:107) denotes the Euclidean norm in the corresponding vector space, and (cid:107) · (cid:107) M is the norm associated with M . In particular [recall Eq. (42)], (cid:107) A (cid:107) = n − m (cid:88) µ =1 m (cid:88) a =1 | A µa | = n − m (cid:88) µ =1 m (cid:88) a =1 (cid:12)(cid:12)(cid:12)(cid:12) ∂∂q n − m + a (cid:18) ∂h∂q k ˆ P kτ K τµ (cid:19) ( ) (cid:12)(cid:12)(cid:12)(cid:12) . (44)Since every norm on a finite-dimensional vector space is equivalent to the Eu-clidean norm, there exists a positive constant α such that (cid:107) u (cid:107) M ≥ α (cid:107) u (cid:107) , ∀ u ∈ R n − m . The constant α may be computed as α = min (cid:107) u (cid:107) =1 √ u M u t = (cid:113) λ M min , (45)where λ M min is the least eigenvalue of M . Then,( u , w ) (cid:20) M AA t (cid:36) I (cid:21) (cid:20) u t w t (cid:21) ≥ α (cid:107) u (cid:107) − (cid:107) u (cid:107) (cid:107) A (cid:107) (cid:107) w (cid:107) + (cid:36) (cid:107) w (cid:107) = α (cid:18) (cid:107) u (cid:107) − (cid:107) u (cid:107) (cid:107) A (cid:107) (cid:107) w (cid:107) α + (cid:36) (cid:107) w (cid:107) α (cid:19) . If A = , this expression is clearly nonnegative and vanishes only when u and w vanish. Suppose now that A (cid:54) = . Defining β = (cid:107) A (cid:107) α , we have( u , w ) (cid:20) M AA t (cid:36) I (cid:21) (cid:20) u t w t (cid:21) ≥ α β (cid:18) (cid:107) u (cid:107) β − (cid:107) w (cid:107) α (cid:107) u (cid:107) β + (cid:36)β (cid:107) w (cid:107) α (cid:19) = α β (cid:34)(cid:18) (cid:36)β − (cid:19) (cid:18) (cid:107) w (cid:107) α (cid:19) + (cid:18) (cid:107) w (cid:107) α − (cid:107) u (cid:107) β (cid:19) (cid:35) . Hence, if we choose (cid:36) > β , i.e. (cid:36) > (cid:18) (cid:107) A (cid:107) α (cid:19) , (46)it follows that Hess(ˆ h )( ) is positive-definite.In other words, the previous proposition says that we can find a solution ˆ h positive-definite around if and only if the matrix M is also positive-definite. Remark . Since q is a critical point of the potential term h , we can find acoordinate-free expression of the matrix M using the covariant Hessian tensor ∇∇ h , given by ∇∇ h ( X, Y ) = X ( Y h ) − (cid:104) d h, ( ∇ X Y ) (cid:105) , We can also use the operator norm for the matrix A . q is exactly Hessian matrix of h . Indeed, it iseasy to see that M µν = M (cid:32) ∂∂q µ (cid:12)(cid:12)(cid:12)(cid:12) q , ∂∂q ν (cid:12)(cid:12)(cid:12)(cid:12) q (cid:33) , where M ( u , v ) = ∇∇ h (cid:0) F ( K ◦ ˆ p ) ◦ F H − ( u ) , v (cid:1) . (47) We shall now condense the results of the previous subsections in the followingtheorem. Consider again an underactuated system defined by a triple ( H , h, W ). Theorem 3.
Let q be a critical point of h and ( U, ϕ ) a coordinate neighborhoodof q such that the subbundle ˆ W given by (26) is a complement of W . Let K be asolution of the kinetic equation (21) for ˆ W . Then, a solution ˆ h of the potentialequation (25) exists around q if and only if the condition d (cid:0) d h ◦ F ( K ◦ ˆ p ) ◦ F H − (cid:1)(cid:12)(cid:12) ˆ W (cid:93) × ˆ W (cid:93) = 0 , holds [see Eq. (39) ]. Moreover, in such a case, ˆ h can be found up to quadratures.On the other hand, ˆ h can be chosen positive-definite around q if and only if thebilinear form [see Eq. (47) ] M = ∇∇ h ◦ (cid:0) F ( K ◦ ˆ p ) ◦ F H − × id T Q (cid:1)(cid:12)(cid:12) ˆ W (cid:93) × ˆ W (cid:93) . is positive-definite at that point. Gathering all the results we have presented so far, we can state a procedureto explicitly construct local solutions of the potential equation that are positive-definite around q , provided a solution of the kinetic equation is given. We mustproceed as follows:1. find coordinates ( q , . . . , q n ) centered at q such thatˆ W := F H − (cid:0) span (cid:8) ∂ / ∂q , . . . , ∂ / ∂q n − m (cid:9)(cid:1) is a complement of W (which can be done just reordering an arbitrarycoordinate system centered at q , as mentioned in the proof of Lemma 2);2. consider a (local) solution K of the kinetic equation (21) for ˆ W ;3. in the coordinates of the step 1, define the functions u µ := ∂h∂q k ˆ P kτ K τµ ,for µ = 1 , . . . , n − m ;4. verify that ∂u ν ∂q µ = ∂u µ ∂q ν for all µ, ν ≤ n − m [see Eq. (38)];14. define ˆ h asˆ h ( q , . . . , q n ) := (cid:80) n − mµ =1 (cid:82) q µ u µ (0 , . . . , , t, q µ +1 , . . . , q n ) d t + (cid:36) (cid:80) ma =1 ( q n − m + a ) (48)for some constant (cid:36) ;6. check that the matrix M [recall Eq. (43)] is positive-definite, i.e. checkthat λ M min > (cid:36) such that [recall Eqs. (44), (45) and (46)] (cid:36) > (cid:80) n − mµ =1 (cid:80) ma =1 ( ∂u µ /∂q n − m + a ( )) λ M min . The idea of studying the potential equation assuming we have a solution ofthe kinetic equation appeared also in Reference [16]. In such work, the authorfinds integrability conditions for the potential matching conditions using Gold-schmidt’s integrability theory (see [12]). Then, assuming such conditions holdand supposing all the objects involved belong to the category C ω , it is provedthat there exists indeed a solution of these equations. However, the positivityof such solutions is not analyzed. Here, although our integrability conditionsare similar to those found in [16], our approach is valid in the category C ∞ .Moreover, we give necessary and sufficient conditions in order to ensure thatthe solution is positive-definite and, in addition, we show how to build suchsolution by computing ordinary integrals in an appropriate coordinate chart. In this section we are going to apply the steps above to a particular subclass ofunderactuated systems.
Assume that ( H, W ) has one degree of underactuation, i.e. W is given by avector subbundle W ⊂ T ∗ Q of rank m = n −
1. Suppose that the step 1 wasalready performed. In such a case, we can explicitly find a solution K of thelocal kinetic equation (33) (corresponding to the step 2). Let us see that. Tosimplify the calculations, we will write q = x, q a = y a , a = 1 , . . . , n − , and K = K, G = G, and ˆ P k = ˆ P k , k = 1 , . . . , n. G = ∂ H ∂x ˆ P + ∂ H ∂y a ˆ P a + ∂ (cid:16) ˆ P i ˆ P j (cid:17) ∂x H i H j . (49)Under this notation, Eq. (33) reads ∂K∂x + G K = 0 , whose general solution is K ( x, y ) = ξ ( y ) e − (cid:82) x G ( t, y ) d t . (50)In order for K to define a quadratic form, we must ask ξ to be a positive function.Following with the step 3, define u := u = (cid:18) ∂h∂x ˆ P + ∂h∂y a ˆ P a (cid:19) K. (51)It is clear that the step 4 is trivial in this case. According to step 5, we haveˆ h ( x, y ) = (cid:90) x u ( t, y ) d t + (cid:36) n − (cid:88) a =1 ( y a ) . (52)The step 6 reduces to check that the number M = λ M min = ∂u∂x ( )is positive. Finally, step 7 says that we must take (cid:36) > (cid:80) n − a =1 ( ∂u/∂y a ( )) ∂u/∂x ( ) . (53) Let us end our work with a concrete example. Consider the manifold Q = S × S and let ( U, ( ψ, ϕ )) be a system of angular coordinates. Consider on Q the simpleHamiltonian function H with H ( ψ, ϕ, p ψ , p ϕ ) = 12 m ( p ψ , p ϕ ) (cid:18) C − B cos( ψ − ϕ ) − B cos( ψ − ϕ ) A (cid:19) (cid:18) p ψ p ϕ (cid:19) and h ( ψ, ϕ ) = D cos ( ψ ) + D cos ( ϕ ) , where m := AC − B cos ( ψ − ϕ ) and A, B, C, D and D are positive constants.This Hamiltonian corresponds to the system depicted in Figure 1, the (planar) inverted double pendulum , for appropriate values of the constants A, B, C, D D . Consider in addition the subbundle W ⊆ T ∗ Q generated by the 1-form d ϕ . The latter, together with H , define an underactuated system with oneactuator, which produces a torque around the coordinate ϕ . To find a solutionof the matching conditions for ( H , h, W ), let us follow the steps above.1. Since W = (cid:104) d ϕ (cid:105) along U , then W (cid:93) = F H ( (cid:104) d ϕ (cid:105) ) = (cid:28) − B cos( ψ − ϕ ) ∂∂ψ + A ∂∂ϕ (cid:29) there. Accordingly, the subbundle (cid:68) ∂∂ψ − γ ∂∂ϕ (cid:69) is complementary to W (cid:93) (shrinking U around ψ = ϕ = 0, if needed) if we choose the constant γ (cid:54) = AB . In what follows we will assume that γ is a generic constant which,in the end, we will choose to fulfill our requirements.Define the coordinates x := ψ and y := ϕ + γ ψ, we have that ∂∂x = ∂∂ψ − γ ∂∂ϕ , ∂∂y = ∂∂ϕ , and consequently ˆ W := F H − (cid:18)(cid:28) ∂∂x (cid:29)(cid:19) In fact, if we consider massless bars of lengths L and L with particles of masses m and m attached to the ends, the values of these constants are A = m L + m L , B = m L L , C = m L , D = m gL and D = ( m + m ) gL , where g is the acceleration ofgravity.
17s complementary to W (along U ). In this way, the first step is done.Let us mention that the matrix H − representing F H − [see Eq. (27)] inthe new coordinates ( x, y ) is H − = (cid:20) A − bγ + Cγ b − γCb − γC C (cid:21) , (54)where b := B cos((1 + γ ) x − y ); (55)while the matrix ˆ P of the projection ˆ p , using σ := F H − (cid:0) ∂∂x (cid:1) as a basisfor ˆ W [see Eq. (29)], is given by the column vectorˆ P = 1 A − γb (cid:18) γ (cid:19) . (56)Also, the potential energy h in these coordinates reads h ( x, y ) = D cos ( x ) + D cos ( γx − y ) . (57)2. The general solution of the kinetic equation is given by (50), where, ac-cording to (49), (54), (55) and (56), G ( x, y ) = − γ b x (1 + γ )( A − bγ ) , where we have considered γ (cid:54) = −
1. Concretely, K ( x, y ) = ξ ( y ) ( A − bγ ) − γ γ , being ξ a positive function.3. In this case, the function u defined by (51) is given by [see (56) and (57)] u ( x, y ) = − D sin ( x ) A − bγ K ( x, y ) .
4. Nothing to do.5. According to (52),ˆ h ( x, y ) = − (cid:90) x D sin ( t ) A − γB cos((1 + γ ) t − y ) K ( t, y ) d t + (cid:36) y .
6. Since M = ∂u∂x (0 ,
0) = − D A − γB K (0 , K (0 ,
0) and D are positive, in order for M to be positive, we mustchoose γ such that A − γB < . γ > AB > . In particular, observe that the value γ (cid:54) = − ∂u∂y (0 ,
0) = 0 , we must take [see Eq. (53)] (cid:36) > . S. Grillo and L. Salomone thank CONICET for its financial support.
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