MANEUVER REGULATION, TRANSVERSE FEEDBACK LINEARIZATION, AND ZERO DYNAMICS

MANEUVER REGULATION, TRANSVERSE FEEDBACK LINEARIZATION, AND ZERO DYNAMICS Chris Nielsen,1 Manredi Maggiore,1 Department o Electrical and Computer Engineering, University o Toronto, Toronto, ON, Canada {nielsen,maggiore}@controlutorontoca Abstract: This paper presents necessary and suicient conditions or the local linearization o dynamics transverse to an embedded submaniold o the state space The main ocus is on output maneuver regulation where stabilizing transverse dynamics is a key requirement Keywords: Maneuver regulation, path ollowing, eedback linearization, zero dynamics, non-square systems, output stabilization 1 INTRODUCTION The maneuver regulation (or path ollowing) problem entails designing a smooth eedback making the trajectories o a system ollow a prespeciied path, or maneuver Unlike a tracking controller, a maneuver regulation controller drives the trajectories o a system to a maneuver up to time-reparameterization This dierence is crucial in robotics and aereospace applications where the system dynamics impose constraints on the time parameterization o easible maneuvers This paper continues a research program initiated in Nielsen and Maggiore (2004a,b) and presents an approach to solving maneuver regulation problems inspired by the work o Banaszuk and Hauser (1995) There, the authors consider periodic maneuvers in the state space and present necessary and suicient conditions or eedback linearization o the associated transverse dynamics Feedback linearization is a natural ramework or maneuver 1 This work was supported by the Natural Sciences and Engineering Research Council o Canada (NSERC) Chris Nielsen was partially supported by the Ontario Graduate Scholarship und (OGS) regulation, as evidenced by the body o work on path ollowing which employs this approach (see or example Altaini (2002), Altaini (2003), Gillespie et al (2001), Bolzern et al (2001), Hauser and Hindman (1997), Coelho and Nunes (2003)) In all these papers, the maneuver regulation problem is converted to an input output eedback linearization problem with respect to a suitable output This motivates our interest in establishing a general ramework or doing this More speciically, the work presented here and in Nielsen and Maggiore (2004a,b) investigates systems with outputs and extends the results o Banaszuk and Hauser (1995) to the case o non-periodic maneuvers deined in the output space (rather than periodic maneuvers in the state space) This work treats the maneuver regulation problem as an output stabilization problem We solve an output stabilization problem which, under appropriate conditions, also solves a maneuver regulation problem The main challenge here lies in inding conditions or eedback linearization o dynamics transverse to an embedded submaniold o the state space whose dimension is not

restricted to be one A natural way to study this problem is to use the notion o zero dynamics In Nielsen and Maggiore (2004a,b), we presented necessary and suicient conditions or global transverse eedback linearization (TFL) Here, we ocus on deriving necessary and suicient conditions or local TFL (Theorems 3-4), as well as a suicient condition or a system not to be transversely eedback linearizable (Corollary 3) We also show (Lemmas 4 and 5) that, in the case o state maneuvers, our results recover the results o Banaszuk and Hauser (1995) The ollowing notation is used throughout the paper We denote by Φ v t (x) the low o a smooth vector ield v We let col(x 1,, x k ) := [x 1 x n ] and, given two column vectors a and b, we let col(a, b) := [a b ] Given a smooth distribution D, we let inv (D) be its involutive closure (the smallest involutive distribution containing D) and D be its annihilator For brevity, the term submaniold is used in place o embedded submaniold o R n throughout 2 PROBLEM FORMULATION Consider the smooth dynamical system ẋ = (x) + g(x)u y = h(x) (1) deined on R n, with h : R n R p (p 2) o class C r (r 1), and u R Given a smooth parameterized curve σ : D R p, where D is either R or S 1, the maneuver regulation problem entails inding a smooth control u(x) driving the output o the system to the set σ(d) and making sure that the curve is traversed in one direction When D = S 1, σ(d) is a periodic curve Banaszuk and Hauser (1995) provide a solution to this problem in the special case when D = S 1 and h(x) = x Notice that one particular instance o maneuver regulation is the case when a controller is designed to make y(t) asymptotically track a speciic time parameterization o the curve σ(t) (Hauser and Hindman, 1995) Thus asymptotic tracking and maneuver regulation are closely related problems In some cases, however, it may be undesirable or even impossible to pose a maneuver regulation problem as one o tracking (consider, or instance, the problem o maneuvering a wheeled vehicle with bounded translational speed by means o steering) We impose geometric restrictions on the class o curves σ( ) Assumption 1 The curve σ : D R p enjoys the ollowing properties (i) σ is C r, (r 1) (ii) σ is regular, ie, σ 0 (iii) σ : D σ(d) is injective (when D = S 1 we instead require σ to be a Jordan curve) (iv) σ is proper, ie or any compact K R p, σ 1 (K) is compact (automatically satisied when D = S 1 ) Assumption 1 guarantees that σ(d) is a submaniold o R p with dimension 1 Assumption 2 There exists a C 1 map γ : R p R p 1 such that 0 is a regular value o γ and σ(d) = γ 1 (0) Moreover, the lit o γ 1 (0) to R n, Γ := (γ h) 1 (0), is a submaniold o R n A suicient condition or Γ = {x : γ 1 (h(x)) = = γ p 1 (h(x)) = 0} (2) to be a submaniold o R n is that h be transversal to γ 1 (0), ie, (Abraham and Robbin, 1967; Guillemin and Pollack, 1974) ( x Γ) Im(dh) x + T h(x) γ 1 (0) = R p The codimension o Γ is equal to the codimension o γ 1 (0) Thereore dim Γ = n p + 1 (Consolini and Tosques, 2003) The problem o maneuvering y to γ 1 (0) is thus equivalent to maneuvering x to Γ and can be cast as an output stabilization problem or the system ẋ = (x) + g(x)u y = (γ h)(x) (3) In general one may be able to maneuver x to the subset o Γ which can be made invariant by a suitable choice o the control input Accordingly, let Γ be the largest controlled invariant submaniold o Γ under (1) and let n = dimγ (n dimγ = n p + 1) Further, let u be a smooth eedback rendering Γ invariant and deine := ( + gu ) Γ Assumption 3 Γ is a closed connected submaniold (with n 1) and the ollowing conditions hold (i) ( ǫ > 0)( x Γ ) L h(x) > ǫ (ii) : Γ TΓ is complete In Banaszuk and Hauser (1995), Γ = Γ = σ(s 1 ), and it is assumed that (x) 0 on Γ Thus in that work Assumption 3 is automatically satisied (the completeness o ollows rom the periodicity o σ(s 1 )) We irst ocus on the well-deiniteness part o the assumption In order to derive conditions guaranteeing that Γ is well deined, associate with each constraint in (2) the single input, single output system {, g, γ i h} where i {1,,p 1} and a corresponding zero dynamics maniold Γ i

Lemma 1 I k Γ k is a closed, controlled invariant submaniold, then Γ exists and it is given by Γ = Γ k k Proo : ( ) Choose any point x Γ Since Γ Γ, ( k {1,,p 1}) γ k (h(x)) = 0 This, together with the act that, by deinition, Γ is locally invariant around x, implies that ( k {1,,p 1}) x Γ k or x k Γ k ( ) Since k Γ k is controlled invariant and output zeroing, and since Γ k Γ k, one has that, by the maximality o Γ, Γ = k Γ k Let r i be the relative degree o system {, g, γ i h} and deine H i : x col(γ i h(x), L (γ i h(x)),, L ri 1 (γ i h)(x)) I {r 1,,r p 1 } is well-deined (uniorm), one has that each Γ i is globally deined and given by Γ i = H 1 i (0) Even i k Γ k is nonempty, it may not be a submaniold A suicient condition or the intersection Γ i Γ j, i j, to be a submaniold is that (Guillemin and Pollack, 1974) ( x Γ i Γ j) T x Γ i + T x Γ j = R n or, equivalently, ker(dh i ) x + ker(dh j ) x = R n Using the act that T x (Γ i Γ j ) = T xγ i T xγ j one easily arrives at the ollowing result Corollary 1 Γ is a globally well deined closed submaniold i there exists a point x 0 Γ around which each system {, g, γ i h}, i {1 p 1} has a uniorm relative degree r i and, i p > 2, the ollowing conditions are satisied or k = 1,,p 2 (i) For k = 1,,p 2, ( k+1 x where H k x j=1 Γ j ) H k x +ker(dh k+1 ) x = R n, is deined recursively as Hx 1 := ker(dh 1 ) x, k = 1 := Hk 1 x ker(dh k ) x, k > 1 H k x (ii) Letting u k := (u 1 ) i Γ i Lr k (γ k h) L gl r k 1 (γ k h), 1 k p 1, = = (u p 1 ) In this case, n = n p 1 i=1 r i i Γ i Remark 1 Rather than using transversality to derive the suicient conditions o Corollary 1, one can employ a slight modiication o the zero dynamics algorithm o Isidori and Moog (1988) (see also Isidori (1995)) or the constrained dynamics algorithm presented in Nijmeijer and van der Schat (1990) In both cases a easible initial condition or the algorithm should be deined to be any point x 0 Γ such that (x 0 ) + g(x 0 )u 0 T x0 Γ or some real number u 0 I the suicient conditions o Corollary 1 are not satisied, the zero dynamics algorithm may still ind a locally maximal controlled invariant submaniold o Γ The condition, in Assumption 3, that L h(x) > ǫ on Γ implies that there are no equilibria on Γ and that, whenever x Γ, ẏ = L h(x) > ǫ This condition ensures that the output o (1) traverses the curve σ(d) We are now ready to ormulate the main problems investigated in this paper The ollowing are a direct generalization o analogous statements ound in Banaszuk and Hauser (1995) Problem 1: Find, i possible, a single coordinate transormation T : x (z, ξ) Γ R n n valid in a neighborhood N o Γ such that in (z, ξ) coordinates (i) Γ = {(z, ξ) Γ R n n : ξ = 0} (ii) The dynamics o system (1) take the orm ż = 0 (z, ξ) ξ 1 = ξ 2 ξ n n 1 = ξ n n ξ n n = b(z, ξ) + a(z, ξ)u where a(z, ξ) 0 in N The ollowing is the local version o Problem 1 (4) Problem 2: For some x 0 Γ, ind, i possible, a transormation T 0 : x ( z 0, ξ 0) Γ R n n valid in a neighborhood U 0 o x 0 Γ such that in ( z 0, ξ 0) coordinates properties (i) and (ii) o Problem 1 are satisied in U 0 Remark 2 It is clear that i one can solve Problem 1 or 2, then the smooth eedback u = 1 (b(z, ξ) + Kξ) (5) a(z, ξ) achieves local stabilization to Γ and traversal o σ(d) in output coordinates However, (5) does not prevent the closed-loop system rom exhibiting inite escape time (ie, the entire σ(d) is traversed in inite time), even though the vector ield o the closed-loop system is complete on Γ A similar problem is encountered in eedback linearization when stabilizing a minimum phase system in normal orm There are various ways to modiy (5) to avoid inite escape time Discussing them is beyond the scope o this paper

Remark 3 I Assumtion 3(i) does not hold, then we have that the path is not traversed and the path ollowing problem cannot be solved in this manner In these cases, solving Problems 1 and 2 results in the solution to an output stabilization problem or system (3) In Nielsen and Maggiore (2004a,b) we provided necessary and suicient conditions to solve Problem 1 In this paper we present necessary and suicient conditions to solve Problem 2 (Theorem 3), as well as a suicient condition or Problems 1 and 2 to be unsolvable We also show that in the special case when D = S 1 and y = x in (1) our conditions are equivalent to those presented in Banaszuk and Hauser (1995) 3 SOLUTION TO PROBLEM 1 For the sake o illustration, we begin our discussion by summarizing, without proo, the main results in Nielsen and Maggiore (2004a,b) Theorem 1 Problem 1 is solvable i and only i there exists a unction : R n R such that (1) Γ {x R n : (x) = 0} (2) yields a uniorm relative degree n n over Γ The conditions in Theorem 1, although rather intuitive, are diicult to check in practice In what ollows we present suicient conditions or the existence o a solution to Problem 1 which are easier to check Corollary 2 I one o the constraints in (2), γ k h, yields a relative degree n n then Problem 1 is solved by setting = γ k h Remark 4 The smooth eedback u := Ln n L g L n n 1 makes Γ an invariant submaniold o (1) Lemma 2 I there exists a unction : R n R which satisies the conditions o Theorem 1, then or each x Γ T x Γ + span{g,,ad n n 1 g}(x) = R n (6) Remark 5 Condition (6) is a generalization o the notion o transverse linear controllability to the case o controlled invariant submaniolds o any dimension It is useul in deriving checkable suicient conditions or the existence o a solution to Problem 1 The notion o transverse linear controllability was originally introduced in Nam and Arapostathis (1992) and later used in Banaszuk and Hauser (1995) or transverse eedback linearization In both papers, n = 1, D = S 1, and T x Γ = span { (x)} Theorem 2 Problem 1 is solvable i (1) Γ is parallelizable (TΓ = Γ R n ) (2) T x Γ +span {g ad n n 1 g}(x) = R n on Γ (3) The distribution span {g ad n n 2 g} is involutive 4 SOLUTION TO PROBLEM 2 The ollowing is an obvious result in the light o Theorem 1 Theorem 3 Problem 2 is solvable i and only i there exists a unction : R n R deined in a neighborhood U 0 o some x 0 Γ such that (1) Γ U 0 {x U 0 : (x) = 0} (2) yields a relative degree n n at x 0 Proo : ( ) Let = ξ1 0, conditions (1) and (2) ollow ( ) Let ξ1 0 = (x) A partial coordinate transormation on U 0 is given by ξ 0 k = Lk 1, k {1 n n } We seek n more independent unctions to complete the transormation and yield the correct orm This can always be done (Isidori, 1995, Propostion 413) From the proo o Theorem 1 we have that the zero dynamics o the resulting normal orm coincide, on U 0, with Γ Lemma 3 I there exists a unction : R n R which satisies the conditions o Theorem 3, then T x 0Γ + span{g,,ad n n 1 g}(x 0 ) = R n Proo : The proo is almost identical to the proo o Lemma 2, which is available in Nielsen and Maggiore (2004b) We report it here or the sake o illustration The existence o implies that one can locally transorm the system dynamics into the orm (4) and speciically that Γ U 0 is locally controlled invariant Let := ( + gu ) Γ, with u deined as in Remark 4 Then, or all x Γ U 0, (x) T x Γ Also, span{ } is a one dimensional, hence involutive, distribution These acts imply that, on Γ U 0, span{ } = (T x Γ ) + span {dφ 2 dφ n }

By Assumption 3, 0 Deine the map t Φ t (x 0 ) and its inverse φ 1 : Γ U 0 φ 1 (Γ ) The map Φ t (x 0 ) is a dieomorphism o φ 1 (Γ U 0 ) onto Γ U 0 By construction L φ 1 = 1 on Γ U 0, implying that dφ 1 (x 0 ) / span{(x 0 )} and thus that (R n ) = span{ (x 0 )} span {dφ 1 (x 0 )} or, equivalently, R n = T x 0Γ ( span{dφ 1 dφ 2 dφ n }(x 0 ) ) (7) Consider a set o linearly independent vectors {v 1,, v n } spanning T x 0Γ, let V = [ v 1 v n ], and deine a matrix S as ollows (Banaszuk and Hauser, 1995) dφ 1 dφ 2 [ ] dφ n V g ad n n 1 g S(x 0 ) = = dl n n 1 [ LV φ(x 0 ) 0 (x 0 ) (all vector ields and 1-orms are evaluated at x 0 ) where {L V φ} ij = L vj φ i, i, j = 1,,n, and R n n n n is upper triangular with non-zero diagonal (this ollows rom condition (2) in Theorem 1) It is clear that i the matrix L V φ(x 0 ) is nonsingular then S is nonsingular as well, implying that Im([V g(x) ad n n 1 g(x)]) = T x Γ + span{g,, ad n n 1 g}(x) = R n and the proo is complete To prove that L V φ(x 0 ) is nonsingular, we use the act that the product o two matrices AB is ull rank i and only i ImB kera = 0 In this case we must show that Im V ker ( col(dφ 1 (x 0 ),, dφ n (x 0 )) ) = 0 or, equivalently, T x 0Γ ker ( col(dφ 1 (x 0 ),, dφ n (x 0 )) ) = 0, and this ollows directly rom (7) Let D = span {g ad n n 2 g} (8) Theorem 2 proves that involutivity o D, together with transverse linear controllability, are suicient conditions or the existence o a unction satisying conditions (1) and (2) in Theorem 1 and hence solving Problem 1 When the involutive closure o D, inv(d), is regular at x 0 Γ, the next result provides necessary and suicient conditions to solve Problem 2 These conditions are easier to check than those in Theorem 3 ] Theorem 4 Assume that inv(d) is regular at x 0 Γ Then Problem 2 is solvable i and only i (1) T x 0Γ + span{g,,ad n n 1 g}(x 0 ) = R n (2) ad n n 1 g(x 0 ) / inv(d)(x 0 ) Proo : ( ) Assume that the conditions o Theorem 3 hold By Lemma 3, (1) holds By deinition o relative degree, in a neighborhood Γ U 0, d D and L ad n n 1 g 0 Recall that d D implies d (inv D) Since L ad n n 1 g 0, one has that ad n n 1 g / span{d} and thus also ad n n 1 g / inv(d), showing that (2) holds ( ) Assume inv(d) is regular at x 0 and conditions (1) and (2) hold This part o the proo closely ollows the idea o the proo o Theorem 23 in Banaszuk and Hauser (1995) Notice that n n 2 dim(inv D) n 1 I dim(inv D) = n n 2 then essentially the same proo o Theorem 2 applies and we are done Hence, we ocus on the case n n 1 dim(inv D) n 1 As in the proo o Theorem 2, let {v 1,, v n } be a set o vector ields deined on Γ such that T x Γ = span{v 1,,v n }(x), and generate s-coordinates by lowing along the vector ields v 1,, v n, ad n n 1 g,,g with times s 1,, s n, respectively By condition (1), there exists a neighborhood U o x 0 such that the map F deined as s Φ g s n Φ adn n s n +1 1 g Φ v n s n Φ v1 s 1 (x 0 ), is a dieomorphism o F 1 (U) onto U Deine the set M := {x U : s n +2(x) = = s n (x) = 0} which is a submaniold o U containing Γ U o dimension n + 1 The submaniold M is the set o points reachable rom Γ by lowing along g Since, by assumption, inv (D) is regular at x 0, it ollows that inv (D) generates a oliation by integral submaniolds, S, in a neighborhood o x 0 which, without loss o generality, we can take to be U Let S x denote a lea o the oliation passing through x Γ U On ad n n 1 Γ U, T x M = T x Γ +span{ad n n 1 g} By condition (1) T x M + D = R n, implying that T x M + inv(d) = R n or, equivalently, T x M + T x S x = R n This shows that, on Γ U, M is transversal to S and T x M T x S x = T x Γ inv(d)(x) is a regular distribution Let ˆn be its dimension Since we are considering the case n n 1 dim(inv D) n 1, we have that 1 ˆn n Making, i needed, M U smaller, let {ˆv 1,, ˆvˆn } be a set o vector ields deined on M U spanning T x M inv(d) on M U Choose additional n ˆn vector ields {ˆvˆn+1,, ˆv n } deined on Γ U such that T x Γ = span{ˆv 1,, ˆv n }(x) x Γ U Then, by condition (1),

{ˆv 1,, ˆvˆn, ˆvˆn+1,, ˆv n, g,,ad n n 1 g} is a set o independent vector ields on Γ U Moreover, inv(d) = span{ˆv 1,, ˆvˆn } + D By making, i necessary, M U smaller we can assume that the vector ields {ˆv 1,, ˆvˆn, g,,ad n n 1 g} are independent on M U The domain o deinition o the vector ields involved in our construction is summarized as: {ˆvˆn+1,, ˆv n } on Γ U {ˆv 1,, ˆvˆn } on M U {g,,ad n n 1 g} on U We use these vector ields to deine the map G : G 1 (U 0 ) U 0 (U 0 U is a neighborhood o x 0 ), p Φ g p n Φ adn n 2 Φ adn n p n ˆn+1 1 g p n +2 Φˆvˆn pn +1 Φˆv1 Φˆv n p n ˆn Φˆvˆn+1 p 1 (x 0 ) p n ˆn+2 Let P 1 = (p 1,, p n ˆn), P 2 = (p n ˆn+1,, p n +1), P 3 = (p n +2,, p n ), and deine G P 1 1 (x0 ) := Φˆv n p n ˆn Φˆvˆn+1 p 1 (x 0 ) 1 G P 2 2 (x1 ) := Φˆvˆn pn +1 Φˆv 1 p n ˆn+2 Φ adn n g p n ˆn+1 (x 1 ) 3 (x2 ) := Φ g p n Φ ad n n 2 p n +2 (x 2 ), G P 3 so that G(p) = G P3 3 G P2 2 G P1 1 (x0 ) For a ixed x 0, x 1 Γ U 0, and x 2 M U 0, each G Pi i is a dieomorphism onto its image, thus G is a dieomorphism onto U 0 This can be most easily seen by examining the order in which the various lows are composed In particular, since ˆvˆn+1,, ˆv n are independent on Γ, the set o points reached by lowing along these vector ields is a submaniold, S, o dimension n ˆn, contained in Γ Next, since ad n n 1 g, ˆv 1,, ˆvˆn are independent on M U 0, the set o points reachable rom S by lowing along these vector ields is precisely M U 0 Thus M U 0 = {x U 0 : P 3 (x) = 0} and Γ U 0 = {x U 0 : p n ˆn+1(x) = 0, P 3 (x) = 0} Finally, the set o points reachable rom M U 0 by lowing along g,,ad n n 2 g is the entire U 0 Choose (x) = p n ˆn+1(x) Then Γ U 0 {x U 0 : (x) = 0} and thus condition (1) in Theorem 3 is satisied By the involutivity o inv(d) = span{ˆv 1,, ˆvˆn } + D, the vector ields ad i g, i = 0,, n n 2, in s-coordinates have the orm (Su and Hunt, 1986, Lemma 4) 0 ad i g = 0 row n ˆn + 2, and thus, on U 0, L ad i g = 0, i = 0,,n n 2 It is also clear that L ad n n 1 g 0 on U0 Thus the assumptions o Theorem 3 are satisied Corollary 3 I dim(inv D) = n, then Problems 1 and 2 are unsolvable Corollary 4 Assume that inv D is regular on Γ and that (1) T x Γ + span{g,,ad n n 1 g}(x) = R n on Γ (2) ad n n 1 g(x) / inv(d)(x) on Γ Then there exists an open covering {U (i) } o Γ and a collection o transormations {T (i) }, with T (i) : x (z (i), ξ (i) ) Γ U (i) R n n such that Γ U (i) = {ξ (i) = 0} and in (z (i), ξ (i) ) coordinates the systems has the orm (4) 5 STATE MANEUVERS In this section, we show that when y = x in (1) and D = S 1 the results obtained thus ar are equivalent to the results presented in Banaszuk and Hauser (1995) See also Nijmeijer and Campion (1993) The conditions presented in (Banaszuk and Hauser, 1995, Theorem 21) or Problem 1 to be solvable are ( ) (a) dim span {, g,,ad n 2 g} = n on Γ (b) There exists a unction : R n R such that (i) d 0 on Γ (ii) = 0 on Γ (iii) L ad i g = 0 near Γ or i = 0 n 3 Lemma 4 Conditions (a) and (b) above hold i and only i the conditions o Theorem 1 hold Proo : ( ) Assume conditions (a) and (b) hold Condition (bii) is the same as condition (1) in Theorem 1 Next, since is by deinition tangent to Γ, condition (bii) implies that L = 0 on Γ This, together with condition (biii), implies that, on Γ, span{d} = span{, g,,ad n 3 g} By condition (a), necessarily L ad n 3 g 0 on Γ This, together with condition (biii) shows that yields a relative degree n 1, which is precisely condition (2) in Theorem 1 ( ) Assume the conditions o Theorem 1 hold Condition (a) holds by Lemma 2 Condition (bii) is identical to condition (1) in Theorem 1 Finally, since yields a relative degree n 1 (recall that here n = 1), conditions (bi) and (biii) are satisied As or Problem 2, consider the distribution D, in (8), with n = 1 The conditions presented

in (Banaszuk and Hauser, 1995, Theorem 24) or Problem 2 to be solvable are ( ) (a) dim span {, g,,ad n 2 g} = n on Γ (b) The distribution D is either (i) involutive or (ii) dim(inv D) = n 1 in a neighborhood o Γ and inv D on Γ Lemma 5 Conditions (a) and (b) above hold i and only i the conditions o Corollary 4 hold Proo : ( ) Assume (a) and (b) above hold Then we just have to show that condition (2) o Corollary 4 holds I D is involutive then (a) immediately gives (2) Otherwise, since inv D on Γ, condition (a) implies condition (2) ( ) Obvious A key dierence between the normal orm presented in this paper (4) and the one presented in Banaszuk and Hauser (1995) lies in the structure given to the vector ield 0 in (4) In the case n = 1 the ollowing procedure illustrates how to obtain the normal orm presented in Banaszuk and Hauser (1995) Fix a point x 0 Γ and deine the map t Φ t (x 0 ) and its inverse ϕ : Γ ϕ (Γ ) Note that, by Assumption 3(ii), ϕ is globally deined and that, when D = S 1, ϕ (Γ ) = S 1 By construction L ϕ = 1 on Γ Let z = ϕ (x) and let ξ i = L i 1, i = 1 n 1 With this transormation, together with the eedback u = Ln 1 + v L g L n 2 one obtains ż = 1 + 1 (z, ξ) + g 0 (z, ξ)v ξ 1 = ξ 2 ξ n 2 = ξ n 1 ξ n 1 = v, (9) ( 1 (z, 0) = 0) which is the normal orm as presented in Banaszuk and Hauser (1995) It is interesting to note that the normal orm o Banaszuk and Hauser (1995) is also valid when D = R (in such a case, the domain o z is ϕ (Γ ) = R rather than S 1 ) When n > 1, the normal orm (9), could perhaps be generalized by inding a partial coordinate transormation z = ϕ(x) yielding ż = col(1, 0,,0) on Γ This is always possible locally Doing so globally amounts to inding a global rectiication or a vector ield on a maniold REFERENCES R Abraham and J Robbin Transversal Mappings and Flows WA Benjamin Inc, New York, 1967 C Altaini Following a path o varying curvature as an output regulation problem IEEE Trans on Automatic Control, 47(9):1551 1556, 2002 C Altaini Path ollowing with reduced otracking or multibody wheeled vehicles IEEE Trans on Control Systems Technology, 11(4): 598 605, 2003 A Banaszuk and J Hauser Feedback linearization o transverse dynamics or periodic orbits Systems and Control Letters, 26:95 105, 1995 P Bolzern, RM DeSantis, and A Locatelli An input - output linearization approach to the control o an n-body articulated vehicle Journal o Dynamic Systems, Measurement, and Control, 123(3):309 316, 2001 P Coelho and U Nunes Lie algebra application to mobile robot control: a tutorial Robotica, 21 (5):483 493, 2003 L Consolini and M Tosques Local path ollowing or time-varying nonlinear control aine system In Proc: American Control Conerence, 2003 RB Gillespie, JE Colgate, and A Peshkin A general ramework or cobot control IEEE Trans on Robotics and Automation, 17(4):391 401, 2001 V Guillemin and A Pollack Dierential Topology Prentice Hall, New Jersey, 1974 J Hauser and R Hindman Maneuver regulation rom trajectory tracking: Feedback linearizable systems In 3rd IFAC Symposium on Nonlinear Control Systems Design, Tahoe City, 1995 J Hauser and R Hindman Agressive light maneuvers In Proc: 36th CDC, San Diego, 1997 A Isidori Nonlinear Control Systems 3 rd ed Springer, London, 1995 A Isidori and C H Moog On the nonlinear equivalent o the notion o transmission zeros In CI Byrnes and A Kurzhanski, editors, Modelling and Adaptive Control, volume 105 o Lect Notes Contr In Sci, pages 146 158 Springer Verlag, 1988 K Nam and A Arapostathis A suicient condition or local controllability o nonlinear systems along closed orbits IEEE Trans on Automatic Control, 37:378 380, 1992 C Nielsen and M Maggiore Maneuver regulation via transverse eedback linearization: Theory and examples In Proceedings o the IFAC Symposium on Nonlinear Control Systems (NOL- COS), Stuttgart, Germany, September 2004a C Nielsen and M Maggiore On global transverse eedback linearization Technical Report 0401, Dept o Electrical Engineering, University o Toronto, 2004b URL

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