MANEUVER REGULATION VIA TRANSVERSE FEEDBACK LINEARIZATION: THEORY AND EXAMPLES

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1 MANEUVER REGULATION VIA TRANSVERSE FEEDBACK LINEARIZATION: THEORY AND EXAMPLES Chris Nielsen,1 Manredi Maggiore,1 Department o Electrical and Computer Engineering, University o Toronto, Toronto, ON, Canada. nielsen,maggiore@control.utoronto.ca Abstract: This paper presents a methodology to solve maneuver regulation problems which is based on transverse eedback linearization. Necessary and suicient conditions or the linearization o dynamics transverse to an embedded submaniold o the state space are presented. Various examples illustrate the main eatures o this ramework. Keywords: Maneuver regulation, path ollowing, eedback linearization, zero dynamics, output stabilization, approximate eedback linearization. 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 aerospace applications where the system dynamics impose constraints on the time parameterization o easible maneuvers. This paper, together with the work in Nielsen and Maggiore (2004a,b,c), initiates a line o research 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 maneu- 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). ver regulation, as evidenced by the body o work on path ollowing which employs this approach (see or example Altaini (2002), Gillespie 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. Ater reviewing necessary and suicient conditions or global and local transverse eedback linearization (TFL) (see also Nielsen and Maggiore (2004a,b,c)), we apply our results to a kinematic unicycle (Section 5.1), a rear wheel drive car (Section 5.2) and a trailer system (Section 5.3). We show that the latter system is not transversely eedback linearizable but its transverse dynamics possess a robust relative degree and hence approximate linearization is possible. 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

2 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, i.e., 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, i.e. 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), i.e., (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) which implies dimγ = 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 )). The requirement, in Assumption 3, that Γ be a closed connected submaniold can be checked using conditions presented in Nielsen and Maggiore (2004a,c). 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 (4)

3 where a(z, ξ) 0 in N. The ollowing is the local version o Problem 1. 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. 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 output stabilization o (3) and hence local stabilization o (1) to Γ (resp., Γ U 0 ). In turn, stabilization to Γ implies, by Assumption 3(i), traveral o σ(d) in output coordinates. In light o this, the main ocus in Problems 1 and 2 is the output stabilization o (3). We begin by reviewing necessary and suicient conditions to solve Problem 1 (Theorem 1) and Problem 2 (Theorem 3). For the sake o brevity we do not include the proos which can be ound in Nielsen and Maggiore (2004c) or Nielsen and Maggiore (2004b) (Problem 1) and Nielsen and Maggiore (2004a) (Problem 2). 3. SOLUTION TO PROBLEM 1 Theorem 1. Problem 1 is solvable i and only i there exists a unction α : R n R such that (1) Γ x R n : α(x) = 0 (2) α yields a uniorm relative degree n n on Γ. The unction α is used to generate the eedback (5) by setting a(t(x)) = L g L n n 1 α(x) b(t(x)) = L n n α(x). 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 1. I one o the path constraints in (2), γ k h, yields a relative degree n n then Problem 1 is solved by setting α = γ k h. Thus, it may be possible to solve Problem 1 by perorming input-output linearization choosing as output one o the path constraints. However (Nielsen and Maggiore (2004c)), Problem 1 may be solvable even when none o the path constraints yields a well-deined relative degree. Lemma 1. I there exists a unction α : R n R which satisies the conditions o Theorem 1, then or all x Γ T x Γ + spang,..., ad n n 1 g(x) = R n. (6) 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 α : R n R deined in a neighborhood U 0 o some x 0 Γ such that (1) Γ U 0 x U 0 : α(x) = 0 (2) α yields a relative degree n n at x 0. Lemma 2. I there exists a unction α : R n R which satisies the conditions o Theorem 3, then T x 0Γ + spang,...,ad n n 1 g(x 0 ) = R n. Let D = span g... ad n n 2 g. Theorem 2 proves that involutivity o D, together with transverse linear controllability, are suicient conditions or the existence o a unction α 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 + spang,...,ad n n 1 g(x 0 ) = R n (2) ad n n 1 g(x 0 ) / inv(d)(x 0 ). Corollary 2. I dim(inv D) = n, then Problems 1 and 2 are unsolvable. Corollary 3. Assume that inv D is regular on Γ and that

4 (1) T x Γ + spang,...,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). 5.1 Kinematic Unicycle 5. EXAMPLES Consider the kinematic model o a unicycle with ixed translational speed v 0 ẋ = v cosx 3 v sin x u 0 1 y = col (x 1, x 2 ). We now solve the problem o maneuvering the output o the unicycle to any curve σ(d) satisying Assumption 1 with r 2. We irst show, by means o Theorem 2, that the problem is always solvable. Later, we show that the unction α yielding the solution can always be chosen as the path constraint. Note that Assumption 2 is not needed. Speciically, we do not need to assume that there exists a submersion γ such that σ(d) = γ 1 (0) because the lit o σ(d), Γ, is always an embedded submaniold o dimension 2 (a generalized cylinder Γ = σ(d) R) and Assumption 3 is always satisied. To see why the latter is true, reer to Figure 1 and observe that the nonholonomic constraint o the unicycle yields Γ = x R 3 : x = (σ(t), arctan2( σ(t))), t D, where arctan2 : R R R denotes the smooth arctangent unction. Hence, Γ is a well deined closed submaniold o dimension 1. To show that is complete, assume without loss o generality that σ is unit speed, i.e., σ = 1. Then, the low o, t Φ t = (σ(vt), arctan2(v σ(vt))) is well deined or all t D. Next, the conditions o Theorem 2 reduce to T x Γ + spang, ad g = R 3 on Γ, where ad g = v sin x 3 x 1 v cosx 3 x 2. Simple geometric considerations (see Figure 1) show that, or all x Γ, and T x Γ = span, g(x) = T x Γ + spang(x) spanad g(x) = (T x Γ). Theorem 2 can thus be applied to conclude that Problem 1 has a solution. Here Theorem 2 allows us to recover the well known act that unicycles with constant orward velocity can ollow any smooth regular curve on the plane. 2 Now we show that the path constraint deining Γ can always be used to deine the unction α in Theorem 1. Lemma 3. I the curve σ satisies Assumption 1 with r 2 then Corollary 1 applies to the unicycle system. Proo : We have already seen that Assumptions 2 and 3 are automatically satisied. This implies that there exists a map γ : R 2 R such that σ(d) = (y 1, y 2 ) : γ(y 1, y 2 ) = 0. The lit o this path is a generalized cylinder given by Γ = x : γ(x 1, x 2 ) = 0. By checking the conditions o Corollary 1 using the standard deinition o relative degree we get L g γ = 0 γ γ L g L γ = v cosx 3 v sinx 3. x 2 x 1 We thus ail to achieve the desired relative degree o 2 at any x R 3 such that ( γ x 3 = arctan / γ ), (7) x 2 x 1 which cannot occur x Γ (see Figure 2). γ = 0 y 2 = x 2 x 3 ( ) γ x 1, γ x 2 y 1 = x 1 Fig. 2. Geometric interpretation o condition (7). Figure 3 depicts a ew phase curves o the unicycle approaching and ollowing a unit circle. The controller is designed using α = x x Rear-wheel driving car-like robot Consider the kinematic model o a rear-wheel drive car-like robot with ixed translational speed v 0 and steering angle in ( π 2, π 2 ). v cosx 3 0 v sin x 3 ẋ = v l tan x u (8) 0 1 y = col (x 1, x 2 ). 2 Actually, Theorem 2 partially recovers this well-known property o unicycles in that it requires that the curve σ has no sel-intersections (see Assumption 1(iii)).

5 y 2 x 3 Γ lit g01 Γ σ(t) σ(t) x 3 (t) (sin x 3, cos x 3 ) T x Γ ad g x 2 y 1 Fig. 1. Maneuver regulation or the unicycle with orward velocity v = σ(t) x 1 σ(t) (sin x 3, cos x 3 ) since [g, ad g] span g, ad g, the conditions o Theorem 2 are satisied and thereore we conclude that Problem 1 has a solution. As in the unicycle example, we now show that the path constraint deining Γ can always be used to deine the unction α in Theorem 1. Lemma 4. I the curve σ in (8) satisies Assumption 1 with r 3 then Corollary 1 applies to the car system (8). Fig. 3. Unicycle approaching and ollowing the unit circle. Consider again the problem o maneuvering the output o the car to an arbitrary curve σ(d) satisying Assumption 1 with r 3. Following the same reasoning used in Section 5.1, it is easy to veriy that Assumptions 2 and 3 are satisied and Γ = x R 4 : x = (σ(s), ϕ(s), ( ) l arctan v ϕ(s), s D, where ϕ(s) = arctan2( σ(s)) is the angle o σ(s) with respect to the positive y 1 axis. In checking the conditions o Theorem 2, we require that T x Γ + span g, ad g, ad 2 g = R4, where ad g = v ( sec 2 ) x 4 l x 3 ad 2 g = v2 l sin x ( 3 sec 2 ) x 4 x 1 + v2 l cosx ( 3 sec 2 ) x 4. x 2 The condition above is satisied provided x 4 ± π 2 + 2kπ which does not occur on Γ. Finally, Proo : Similar to the proo o Lemma 3. The points at which we ail to achieve the required relative degree o n n = 3 have the same geometric interpretation as in the unicycle example (i.e., a singularity occurs when the car is perpendicular to the curve). Let us now apply this result to a more speciic and useul application example. We attempt to make the car ollow an approximation o the Toronto Indy race track, generated by a 4 th order spline. Splines are useul in path generation since they can be utilized to model arbitrary paths to any degree o accuracy. An n th order spline is class C n 1. Here, we use 4 th order splines to obtain C 3 curves. Let K represent an ordered collection o knots and let I represent an index set o all piecewise polynomials. Introduce the ollowing notation to represent the spline we wish to ollow, which may have an arbitrary number o knots ( σ : λ R λ, 4 (a i j(λ K i ) j ) i I j=0 ) (9) (1(λ K i ) 1(λ K i+1 )) where 1(t) is the unit step unction. We enorce that the spline be the graph o a unction in R 2. This guarantees injectivity and properness and thereore this class o curves satisy Assumption 1.

6 Let σ i represent a single segment o the spline. Then, Assumption 2 is also satisied since σ(d) = y R 2 : y 2 i I 4 (a i j (y 1 K i ) j ) j=0 (1(y 1 K i ) 1(y 1 K i+1 )) = 0. By Lemma 4, in each knot interval we can use the sole constraint deining Γ, α i (x) = x 2 4 j=1 ai j (x 1 K i ) j, and apply Corollary 1 Deine a coordinate transormation in each knot interval (K i, K i+1 ), T i : x col(ϕ i (x), α i (x), L α i (x), L 2 α i(x)) yielding the normal orm (4). The smooth linearizing eedback solving the maneuver regulation problem is u i = L3 α i k 1 α i k 2 L α i k 3 L 2 α i L g L α i with k 1, k 2, k 3 > 0. In order to traverse the entire path, the controller must switch rom u i to u i+1 when x 1 switches rom knot interval (K i, K i+1 ) to (K i+1, K i+2 ). The controller u i is a unction o the path constraint and its irst 3 derivatives. Thus, in order or the switching to be bumpless, we require a 4 th order or higher spline. Figure 4 shows the car ollowing a 4 th order spline consisting o 38 knots or various initial conditions. y 2 = x Desired Spline Output Trajectories y = x 1 1 Fig. 4. The car ollowing the Toronto Indy race track trailer Systems Consider the kinematic model o a 1-trailer system with a unicycle as the tractor or lead vehicle, see Figure 5. v cosx 3 0 v sin x 3 ẋ = 0 v u (10) L sin (x 3 x 4 ) 0 y = col (x 1 L cosx 4, x 2 L sinx 4 ). This model is a slight variation o the standard trailer model (see Fliess et al. (1995); Samson x 3 x 2 * y 2 * y 1 Fig. 5. Kinematic model o the Standard 1-trailer. (1995)). The subsequent discussion applies to that model as well. Our goal is to orce the output o system (10) (i.e. the position o the last trailer) to ollow a sinusoid σ : λ col ( λ, 100 cos ( λ 100 cos ( y L x 4 x 1 100)). The curve σ satisies Assumption 1 and the irst part o Assumption 2 with σ(d) = y R 2 : y 2 ) = 0. The lit Γ = (γ h) 1 (0) = x R 4 : x 2 L sinx 4 ( ) cos 100 (x 1 L cosx 4 ) = 0 (11) is an embedded submaniold and thus Assumption 2 is satisied. We require various Lie brackets: ad g =(v sin x 3 ) + ( v cosx 3 ) + x 1 x 2 ( v ) (12) L cos(x 3 x 4 ) x 4 ad 2 g = v2 L 2 x 4. (13) To check i Assumption 3 is satisied, we must irst characterize Γ. It is well-known that the standard n-trailer system is lat Fliess et al. (1995): speciying a path or the inal vehicle completely determines the state o the system and the control input. This leads to the strong suspicion that n = 1 and thus we expect to be able to characterize Γ as the zero level set o n n = 3 unctions. We characterize Γ using the zero dynamics algorithm o Isidori (1995); Isidori and Moog (1988). We initialize the algorithm at any x Γ. Let u 0 R be such that (x )+g(x )u 0 T x Γ. At the irst step o the algorithm we have M 0 = Γ = (γ h) 1 (0) = M c 0, H 0(x) := γ h(x), where M0 c denotes the connected component o M 0 containing x. The next step o the algorithm yields M 1 = x M0 c : (x) span g(x)+t x M0 c n where T x M c 0 = ker(dh 0 ). Equivalently,

7 M 1 = x M0 c : dh 0, (x) + dh 0, g(x) u = 0 is solvable or u. At this step, L g H 0 0 and L H 0 0, so letting H 1 = col(h 0, L H 0 ) we have M 1 = M1 x c = R 4 : H 1 (x) = 0. The next iteration yields M 2 = x M1 c : dh 1, (x) + dh 1, g(x) u = 0 is solvable or u, (14) where L g H 1 = col(0, L g L H 0 ). Due to the complexity o the expression, it is not immediately clear whether L g H 1 = 0 on M1 c. I L gl H 0 is nonzero on M1 c then it would ollow that (14) can be solved or u leading to the conclusion that M 2 = M1 c and the algorithm terminates yielding n = 2. Otherwise, the algorithm continues. Suspecting that n = 1, we continue the algorithm under the assumption that L g L H 0 (x) = 0 on M1 c, later, we veriy that this assumption is valid. Let H 2 = col(h 1, L H 1 ) = col(h 1, L 2 H 0). The next iteration in the algorithm yields M 3 = x M2 c : dh 2, (x) + dh 2, g(x) u = 0. We are now set to show, numerically, the ollowing two acts (1) ( x M c 1 U) L g L H 0 (x) = 0 (2) ( x M c 2 U) L gl 2 H 0 0 (where U is some open set containing x ) which allow us to conclude that the assumption above is in act correct and that the algorithm terminates with n = 1 and Γ U = H2 1 (0) U. We do that by numerically generating a uniorm orthogonal grid o M1 c (a two dimensional maniold) and M2 c (a one dimensional maniold) and checking whether properties (1) and (2) hold at the resulting grid points. To this end, given an n- dimensional submaniold M in R m, expressed as M = H 1 (0), where H = col(h 1,..., H m n ) : R m R m n, we introduce the ollowing Procedure uniorm grid 1. Find a basis v 1,..., v n o T x M = ker(dh x ) 2. Apply Gram-Schmidt orthonormalization to get ṽ 1,...,ṽ n 3. Let G = m n i=1 H2 i and let ṽ i µ G i G ǫ ˆv i = G ṽ i µ G i G < ǫ. ǫ where µ, ǫ > Choose x 0 near M and numerically integrate the ˆv i to generate a grid. The procedure above uses a continuous approximation to the gradient vector ield G to make M attractive. The parameter µ controls the speed o convergence, here we set µ = 1. The value o ǫ should be signiicantly larger than the integration tolerance used in step 4. We use a tolerance o and ǫ = Figure 6 illustrates the concept o gridding or the case dimm = 2. The t 2 φˆv 2 t 2 φˆv 1 t 1 (x 0 ) t 1 Fig. 6. Gridding a two dimensional maniold. M x 0 M U map generated by the gridding algorithm is isometric, i.e., unit time intervals are mapped to unit length intervals in the grid. The vector ields ˆv i are continuous and the maniold M is attractive or each o them. This is useul or numerical stability o the procedure: i the algorithm is initialized at a point x 0 which is not exactly on M, the procedure generates grid points that get closer and closer to M. Finally, the lows associated to the vector ields ˆv i commute, i.e., starting rom a point x M, φˆvj t j φˆvi t i (x) = φˆvi t i φˆvj t j (x). We apply the procedure above to M c 1 (choosing x 0 = x ) to determine i L g L H 0 = 0 at each point o the grid. The result, shown in Figure 7, is that L g L H 0 < and validates our conjecture (1). Next, we apply the gridding algorithm x t 1 (seconds) t 2 (seconds) Fig. 7. The value o L g L H 0 over a uniorm 2- dimensional grid o M c 1. to M2 c (choosing x0 = x ). Figure 8 shows that L g L 2 H 0 0 on M2 c thus validating conjecture (2). We have thus numerically shown that n = 1, as expected, and Γ = M2 c near x. In particular, we have seen that L g H 0 0, L g L H 0 = 0 on Γ, and L g L 2 H 0 0 on Γ. It can be shown that L g L H 0 changes sign in any neighborhood o Γ, implying that H 0 (x) = γ h(x) does not yield a well-deined relative degree and thus Corollary 1 does not apply. Actually, using symbolic mathematics sotware by Kugi et al. (2003)), one inds that inv(d) = inv(span g, ad g, ad 2 g) = R4, 3 4 5

8 replacemen Time (seconds) Fig. 8. The value o L g L 2 h on Mc 2. and thus, by Corollary 2, the 1-trailer system is not transversely eedback linearizable. However, the discussion above immediately gives that γ h yields a robust relative degree 3 on Γ, and thus one can perorm approximate input-output linearization on the transverse dynamics o system (10) (Hauser et al. (1992)). We say that the system is approximately transversely eedback linearizable. Figure 9 shows various simulation results or initial conditions on and o the desired sinusoid. The trailer system ollows the path or initial conditions suiciently close to Γ. y 2 = x Tractor Trailer Desired Path y 1 = x 1 Fig. 9. Approximate transverse eedback linearization o the trailer system REFERENCES C. Altaini. Following a path o varying curvature as an output regulation problem. IEEE Trans. on Automatic Control, 47(9): , A. Banaszuk and J. Hauser. Feedback linearization o transverse dynamics or periodic orbits. Systems and Control Letters, 26:95 105, P. Coelho and U. Nunes. Lie algebra application to mobile robot control: a tutorial. Robotica, 21 (5): , L. Consolini and M. Tosques. Local path ollowing or time-varying nonlinear control aine system. In Proc: American Control Conerence, M. Fliess, J. Lévine, P. Martin, and P. Rouchon. Flatness and deect o nonlinear systems: introductory theory and examples. International. Journal o Control, 61(6): , R.B. Gillespie, J.E. Colgate, and A. Peshkin. A general ramework or cobot control. IEEE Trans. on Robotics and Automation, 17(4): , V. Guillemin and A. Pollack. Dierential Topology. Prentice Hall, New Jersey, J. Hauser and R. Hindman. Maneuver regulation rom trajectory tracking: Feedback linearizable systems. In 3rd IFAC Symposium on Nonlinear Control Systems Design, Tahoe City., J. Hauser and R. Hindman. Agressive light maneuvers. In Proc: 36th CDC, San Diego., J. Hauser, S. Sastry, and P. Kokotović. Nonlinear control via approximate input-output linearization: The ball and beam example. IEEE Transactions on Automatic Control, 37(3): , A. Isidori. Nonlinear Control Systems 3 rd ed. Springer, London, A. Isidori and C. H. Moog. On the nonlinear equivalent o the notion o transmission zeros. In C.I. Byrnes and A. Kurzhanski, editors, Modelling and Adaptive Control, volume 105 o Lect. Notes Contr. In. Sci., pages Springer Verlag, A. Kugi, K. Schlacher, and R. Novak. Aine Input Systems Package or Maple. Department o Automatic Control and Control Systems Technology at the Johannes Kepler University, Linz, Austria, 2 nd edition, K. Nam and A. Arapostathis. A suicient condition or local controllability o nonlinear systems along closed orbits. IEEE Trans. on Automatic Control, 37: , C. Nielsen and M. Maggiore. Maneuver regulation, transverse eedback linearization, and zero dynamics. In Proceedings o the 16 th International Symposium on Mathematical Theory o Networks and Systems (MTNS 2004), Leuven, Belgium, July 2004a. C. Nielsen and M. Maggiore. On global transverse eedback linearization. Technical Report 0401, Dept. o Electrical Engineering, University o Toronto, 2004b. URL rep orts/rept0401.pd. C. Nielsen and M. Maggiore. Output stabilization and maneuver regulation: A geometric approach. Systems Control Letters, submitted, 2004c. C. Samson. Control o chained systems application to path ollowing and time-varying pointstabilization o mobile robots. IEEE Transactions on Automatic Control, 40(1):64 77, Jan