Constructive Feedback Linearization of Mechanical Systems with ...

Constructive Feedback Linearization Of Mechanical Systems With Friction and Underactuation Degree One ´ Acosta* and M. L´opez-Mart´ınez* J. A. Abstract— This paper presents a constructive methodology to control a class of unstable underactuated mechanical systems with underactuation degree one, by means of classical feedback linearization and Lyapunov design. The design is presented proposing a dummy output that allows to redesign it in a constructive way to solve the problem. We prove that the proposed output solves the problem yielding an explicit and compact control law. The stability of the obtained solution does not change taking into account the friction effects even in the unactuated coordinate. The new result is applied to obtain an (almost) globally stabilizing scheme for the vertical takeoff and landing aircraft with strong input coupling and a controller for the Inertia Wheel pendulum with friction.

I. I NTRODUCTION This paper presents a methodology for underactuated mechanical systems with underactuation degree one, to construct a dummy output through which it is possible to achieve the asymptotic and locally exponentially stabilization at the origin of the system. We prove that the proposed output solves the problem yielding an explicit and compact control law, which is based on a classical partial state feedback linearization with an external controller designed by Lyapunov and yields excellent results in the theoretical and experimental issues. From the theoretical point of view and in spite of that the feedback linearization is a classical approach, for the applications proposed the methodology yields relations with other control techniques. We start, motivated by a cyclic coordinate, choosing as the fictitious output a function quite similar to the conjugate momentum. Then, this could have a relation between the chosen output and the phase shift [13] and, therefore, with the Controlled Lagrangians method [5]. Since with this output, we cannot fulfil the objective of stabilizing asymptotically the zero dynamics, we redesign this output, using the constructive approach proposed, adding a new control term that makes the zero dynamics asymptotically stable and faster. From the applicability issues the proposed technique was motivated by [1], and is applied to two typical examples: the control of a VTOL aircraft where the roll angle and rate and the horizontal and vertical velocities are the output feedback variables [4], [9], [14]; and the Inertial Wheel Pendulum, where the friction in the unactuated coordinate has been taken into account. The solution given in [17] did *The authors are with Dept. de Ingenier´ıa de Sistemas y Autom´atica, Escuela Superior de Ingenieros, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092, Sevilla, Spain. {jaar,mlm}@esi.us.es This work was supported by the Spanish Ministry of Science and Technology under grants DPI2006-07338 and DPI2004-06419.

not take into account the friction and the implementation of the controller did not succeed as expected. On the other hand, in [7] the importance of the friction to stabilize this system was shown. Now, we would like to clarify some aspects referring to previous works. On the one hand, the work given in [2] and where we solved some examples in an intuitive way, gave us some insight for completing this work. In this way, here we generalize and formalize the results for a class of underactuation degree one, but taking into account the friction forces even in the unactuated coordinates. The methodology gives rise to an explicit controller without solving a set of partial differential equations, which with this approach it is not necessary at all. On the other hand, the problem addressed in [3] was not the same addressed here. Firstly, it can be easily checked that the solution given in [3] is not the one given in this work. Secondly, in [3] the friction/drag forces were not taken into account, and further, if the friction (physical damping) had been considered in [3] the additional passivation condition given in [7] would need to be satisfied. The outline of the paper is: Section II describes the class of underactuated mechanical systems, in Section III we propose the constructive dummy output for the class. In section IV, we state the main stabilization results. In Section V the well– known applications are solved and also the simulations are given at the end of this section. Finally, a conclusion section. II. T HE CLASS OF UNDERACTUATED MECHANICAL SYSTEMS

In order to present clearly the class of systems we address in this paper, we will start from a classical Lagrange’s formulation for a general underactuated mechanical system, and after, some assumptions related to the class will be imposed. Thus, let (q, q) ˙ ∈ Rn × Rn be the generalized coordinates and velocities, respectively. We address the control problem of underactuated mechanical systems, i.e. there are fewer control inputs than degrees of freedom. The Lagrange’s equations read M (q)¨ q + C(q, q) ˙ q˙ + ∇U (q) + D(q) ˙ = [O Im ]> τ n×n

(1)

where M ∈ R is the symmetric and positive definite inertia matrix, U ∈ R is the potential function, the matrix C ∈ Rn×n contains the Coriolis and centrifugal forces, and D ∈ Rn×n represents the viscous friction forces, and τ ∈ Rm the number of independent control inputs. We now proceed to define the class of mechanical systems for which we can explicitly solve the control problem. In fact, the class

considered refers to systems with underactuation degree one, i.e. m = n − 1, m ≥ 1. Then, we partition intuitively the set of generalized coordinates q = (z, X) ∈ R × Rn−1 , where the unactuated degree of freedom is represented by the z– coordinate and the actuated ones by the set of the generalized X–coordinates. After this partition the Lagrange’s equations of motion can be written as 

m11 m12

     F (q, q) ˙ 0 m> 12 q¨ + 1 = τ, F2 (q, q) ˙ In−1 M22

(2)

where now m11 ∈ R, m12 ∈ Rn−1 , M22 ∈ R(n−1)×(n−1) and we have introduced the scalar and vector functions F1 (q, q) ˙ ∈ R and F2 (q, q) ˙ ∈ Rn−1 , respectively. The first step of the approach presented here is to linearize partially the equations of motion (1), as done in [18] where it was called collocated partial feedback (see also [8]). Indeed, after some simple calculations mimicking the ones given in [18] it is easy to see that the partially feedback–linearized system takes the affine in control form:  q¨ =

 " m> (q) # 12 − m111 (q) F1 (q, q) ˙ + − m11 (q) u, O In−1

(3)

where the scalar function F1 (q, q) ˙ was defined as F1 (q, q) ˙ , e> ˙ q˙ + ∇U (q) + D(q)], ˙ 1 [C(q, q)

(4)

where, as usual, ei denotes the i–th vector of the n– dimensional Euclidean basis. The inner loop reads " τ

=

m12 .. − . Im m11

#

   m12 m> F1 (q, q) ˙ 12 + M22 − u. F2 (q, q) ˙ m11 (5)

Remark 1: On the one hand, the matrix m11 need to be uniformly positive definite, and then invertible. On the other hand, the elements of the vector m12 are different from zero at least in a neighborhood at the origin. In this way, if it is not difficult to see that if rank(m12 ) = 0 then the system is not controllable, i.e. the Brockett’s necessary condition would not be satisfied. This fact was defined in [18] as Strongly Inertially Coupled rank condition and so we assume rank(m12 ) 6= 0. Now, we are in position to write down the general assumption for the definition of the class of underactuated mechanical system considered in the present paper: Assumption A.1 The elements of the inertia matrix m11 and m12 do not depend on the actuated coordinates, the matrix M22 satisfies that either it is a constant matrix or a function of the actuated coordinates, the potential function is of the form U (q) , V (z) + Φ(X),

and the viscous friction forces applied on the system can be decomposed as D(q) ˙ , [D1 D2 ]> where the viscous friction force applied on the unactuated coordinate, D1 , satisfies one of these two conditions a) D1 = 0, there is no friction force applied on the unactuated coordinate. b) There exists a positive constant d1 such that kD1 k ≤ d1 kzk, ˙ ∀ z˙ ∈ Ω ⊆ R, i.e. the friction force applied on the unactuated coordinate depends only on z˙ and it is bounded by a linear growth condition.

Notice that the Assumption A.1 splits the class of systems depending on the case to study. Throughout the paper we will refer to this Assumption as A.1a or A.1b. It is worth mentioning that the class considered by Assumption A.1, contains several practically relevant examples, with four of them given in last Section V. Assumption A.1 essentially imposes that the unactuated elements of M depend on the unactuated coordinate. It is satisfied by many well–known physical examples, for instance, the VTOL Aircraft and the Inertia wheel pendulum. III. C ONTROL DESIGN In this section, we show the main stability result for the class of underactuated mechanical systems, in the intuitive way presented in [2]. This result is the natural generalization of the work presented in [2] with the stability result for the whole class added. To do so, we divide the solution in several steps: firstly we propose the constructive output and we develop a controller from this output; secondly, since the controller does not fulfill our expectations we redesign the output proposed. This redesign yields a nonlinear controller for the zero dynamics which we only need to extend with an external controller. The main contribution is that the output redesigned yields controllers derived from intuitive energetic Lyapunov functions. A. The proposed constructive output Let us define the free smooth vector function ξ(z) ∈ Rn−1 , which is at least C 2 . Thus, given the equations of motion for an underactuated mechanical system (3) that fulfill the Assumption A.1, then the “constructive” output is defined as η0 (z, q) ˙ , [ξ(z) In−1 ] q, ˙

η0 (·) ∈ Rn−1 .

(6)

Notice that the output is a nonlinear composition of the generalized velocities. To get a full–rank vector ξ, at least in a neighborhood at z = 0, we will take into account this fact later. Now we derive the zero dynamics with respect to this output, following the usual procedure of calculating the consecutive derivatives yielding η˙ 0

= =

˙ z˙ + [ξ(z) In−1 ] q¨ ξ(z)   dξ 2 F1 ξ m> z˙ − − ξ 12 − In−1 u = ν dz m11 m11 | {z }

(7)

∆(z)

where we have used the equations of motion (3) in the last identity and defined the small–rank adjustment matrix ∆(z) ∈ R(n−1)×(n−1) , and ν stands for the external controller to be defined later which will stabilize the output dynamics. Notice that, the relative degree of the output is one and then, we get a second order zero dynamics. Thus, throughout the paper, let denote the state space of these zero dynamics as Z = (z, z) ˙ ∈ R2 . The linearizing controller is obtained isolating the input control u and zeroing the output η and its derivatives. Thus, the linearizing controller u = u0 becomes u0 = ∆−1



 ˙ dξ 2 F1 (q, q) z˙ − ξ−ν , dz m11

(8)

where the matrix ∆(z) is nonsingular in the set F ≡ {z ∈ R : det(∆) 6= 0}, yielding det(∆) = m> 12 /m11 ξ − 1. Note that ξ is our free vector function to tune the set F and, by construction, the det(∆) 6= 0 at least in a neighborhood of z = 0, which can be seen from the series expansion. The set F is usually called the feasibility region [12]. The region of attraction will be composed by the intersection of the feasibility region and the largest compact subset of a Lyapunov function including the desired equilibrium. In the following sub-section we will choose the free vector function ξ(z) in order to stabilize the equilibrium of the zero dynamics associated to the proposed output (6). B. Defining the free smooth vector ξ(z)

1ˆ ∆(z)z˙ 2 + V0 − V (z), 2

(9)

where V0 is the constant reference for the potencial function and we define, for compactness, the scalar function ˆ ˆ , m11 det(∆). This energy function will ∆(z) ∈ R, as ∆ be used later as a Lyapunov function candidate, for the zero dynamics, and then, we will define the vector function ξ(z) in a right way. To this end, we calculate the derivative of this energy function (9) along the system trajectories and then this derivative will give us the key to choose ξ(z). Now we can state our first general result in the following lemma: Lemma 1 (Geometric condition): Consider the dynamical system given by equations (3) under Assumption A.1a, and the static feedback controller given by the equation (8). Then, the function (9) is a constant of motion along the system trajectories, i.e. E˙ 0 (t) = 0, ∀t ≥ 0, if the free vector function, ξ(z), is defined as ξ(z) , K1 m12 (z),

(10)

with K1 = K1> , K1 ∈ R(n−1)×(n−1) , a constant full–rank ˆ matrix. Moreover, let us define the set Ω , {z ∈ R| ∆(z) > 0}. Now, choose K1 > 0 and some V0 > 0, then the function (9) is a positive definite function in Ω, and it can be seen as an energy–like function for the zero dynamics in that set. Proof: We only need to calculate the derivative with respect to the time of the energy function (9), but first we calculate the equations for the zero dynamics which will be necessary. Thus, the zero dynamics are defined, as usual, for η0 = η˙ 0 = 0, which imply ν = 0. Then substituting the control function (8) in the first equation of (3), some length but straightforward calculations show that z¨

= = = =

1 (F1 (q, q) ˙ + m> 12 u0 ) m11   > F1 (q, q) ˙ ˙ m dξ 2 F1 (q, q) − − 12 ∆−1 z˙ − ξ m11 m11 dz m11   > > F1 (q, q) ˙ m m dξ − 1 − 12 ∆−1 ξ − 12 ∆−1 z˙ 2 m11 m11 m11 dz   dξ 2 1 F1 (q, q) ˙ − m> z ˙ , (11) 12 ˆ dz ∆

−

E˙ 0

= = = = =

In this section, we completely define the free vector function ξ(z) of the output. For, we propose a scalar function, namely E0 (Z) ∈ R, as an energy function for the zero dynamics of the output (6). Thus, let us define this function as E0 (Z) =

ˆ > = m11 m> ∆, in where we have used the identity ∆m 12 12 the last line. Then using (11), the derivative of the energy function reads ˆ 2 1 d∆ ˆ z − dV z˙ + ∆¨ 2 dz dz

! z˙

! ˆ 2 1 d∆ dV > dξ 2 z˙ + F1 (q, q) ˙ − m12 z˙ − z˙ 2 dz dz dz ! ˆ 2 1 d∆ > dξ 2 z˙ + e> z˙ z˙ 1 C q˙ − m12 2 dz dz     1 dm> dξ dm11 12 ξ − m> − z˙ 2 + e> 12 1 C q˙ z˙ 2 dz dz dz    1 dm> 1 > dξ 2 12 ξ − m12 z˙ z˙ = − [m12 , ξ] z˙ 3 2 dz dz 2

where we: used (11) in the second line; expanded the ˆ ∆ function F1 (q, q) ˙ using (4) in the third line; calculated ddz in the fourth line; expanded the C q–term ˙ in the fifth line and compacted the expression using the definition of the Lie bracket in the last one. Now, using (10) the Lie–bracket term in the last identity is zero, because the Lie–bracket of two parallel vectors is zero and then E0 is a constant of motion for the zero dynamics. To prove the last claim, recall the remark 1 together with (10) and notice that the matrix K1 is full–rank. Then, by construction for any K1 > 0 there exists, at least a neighborhood N of z = 0 such that ˆ ∆(z) > 0, ∀z ∈ N , by continuity. Thus, we can choose the constant V0 positive such that E0 > 0, ∀z ∈ N . To extend the argument to the whole set Ω, simply notice that, since (10) makes to the set Ω depend on a quadratic form associated to the positive definite matrix K1 , we can increase the value of this form by increasing K1 and so the proof is completed. C. Output redesign Once we have defined the free vector function ξ(z), we address the control problem to do the origin of the zero dynamic asymptotically stable. From the previous design step, it easy to see that using the function (9) as a Lyapunov function candidate we only could prove the (local) stability, in the sense of Lyapunov, at the origin Z = {O} of the zero dynamics. To achieve the (local) asymptotical stability we need to modify the output proposed (6). Since after proposition 3, we see the function (9) as an energy function for the zero dynamics, these zero dynamics could be the dynamical equation of a simple mechanical system then we only need to feedback the corresponding passive output [16], in the way to do semi–negative definite the derivative of this energy function along the trajectories. Thus, we split the control action into two terms, i.e. u0 + up , being up the corresponding term added to the controller (8). Otherwise, we could get this new control term, redesigning the proposed output (6). We call to the redesigned output η. Then, once the vector function ξ(z) was defined, the corresponding output to the passive output for the energy function (9), is z

Z η(z, q) ˙ , η0 + K 2

m12 (µ)dµ,

η(·) ∈ Rn−1 ,

(12)

0

where K2 ∈ R(n−1)×(n−1) , is a diagonal and positive definite constant matrix. Now, it is easy to derive the “new”

controller mimicking the before calculations, and then we get η(z, ˙ q) ˙ = η˙ 0 + K2 m12 (z)z, ˙

(13)

and therefore the redesigned controller from (8) reads u =

= (r.h.s. of (8)) + ∆−1 K2 m12 z˙     dm12 2 m12 ∆−1 K1 z˙ − F1 + K2 m12 z˙ − ν dz m11 (14)

IV. M AIN STABILIZATION RESULT Before stating the main results, regarding to the stability conditions, we will clarify two points regarding to the class of systems. The first one deals with the Assumption A.1a. This condition plays a critical role in the stabilization problem and we propose to take a brief pause to analyze it. After lemma 1, it looks like the class of systems such that the Assumption A.1a should be satisfied. In fact, we could call to the class of systems which satisfies this condition the “constructive class”, because of the lemma 1 result. The lemma 1 together with the redesigned output given by equation (12) will yield the two following results: the energy function (9) is an Lyapunov function candidate for the zero dynamics corresponding to the output (12); and, the origin of the zero dynamics, will be asymptotically stable. In spite of these promising results regarding to the called constructive class, the requirement imposed by the Assumption A.1a is too much restrictive. In fact, it is well–known that once the origin of the zero dynamics, corresponding to a defined output, is (locally) asymptotically stable, then a linear external controller will stabilize, at least locally, asymptotically the whole system [10]. On the other hand, as we will show later the proposed controller makes the origin of the zero dynamics not only asymptotically stable but also exponentially stable. In this way we split the main stabilization result in two sections: the first one regarding to this constructive class which address the (almost) global stabilization of the zero dynamics; and the second one, we extend the solution to another class, supporting in a semi– global stabilization of the zero dynamics. In the last section devoted to applications we give examples regarding to the first case, the VTOL aircraft and the Inertia Wheel Pendulum with friction. On the other hand, the potential energy, uniquely determined by the open–loop system, plays also a critical role in the stabilization problem. We consider a class where the point Z = {O} corresponds to an equilibrium that will be open–loop unstable therefore the open–loop potential energy function V (z) will have typically a maximum. Thus, we need to modify the forces induced by the unactuated coordinates of the potential energy through some “coupling term”, which ˆ Since our interest in this paper is to give a constructive is ∆. solution to the stabilization problem for the class of unstable underactuated mechanical systems, we make the additional to be able to explicitly solve the problem: 2 Assumption A.2 ddzV2 (0) < 0 and dV dz 6= 0, ∀z ∈ Ω \ {0}, which means that the undesired equilibrium is isolated in Ω.

Otherwise, we know the well–known Isidori’s approach (see [10]) which state that given an output whose associated zero dynamics have an asymptotically stable equilibrium, in the first approximation, the whole nonlinear system can be locally stabilized by means of output feedback. That means that a linear, i.e. proportional to the output, external controller will stabilize locally asymptotically the origin of the whole system [10]. This approach only was stated for local validity of the differential geometry approach. We design here a non– linear controller which yields larger regions of attraction than the linear one. Furthermore, a very simple explicit expression for the control law is given. Finally, recall that the final objective of the controller proposed is a to maneuver manually the set point in aircraft flying, i.e. a pilot manual operation. Then our desired equilibrium is (Z, η˜) = (O, O) where η˜ = η − ηref and ηref is the set point desired. A. The case with D1 = 0. In the following proposition we state the main stability conditions to stabilize the origin of desired equilibrium using the output (12). Furthermore, a very simple explicit expression for the control law is given for the class. Proposition 1: The underactuated mechanical system given by equations (3), under Assumptions A.1a–A.2 and with the lemma 1 taking effect. Then, the smooth static feedback controller given by (14) together with the external controller given by ν = −K3 (m12 z˙ + P η˜) ,

(15)

for any diagonal full-rank constant matrix K2 > 0, as defined in (12), and the diagonal matrix K3 > 0, K3 ∈ R(n−1)×(n−1) , ensures that (i) the closed–loop system is partially state feedback input–output linearizable in the set Ω through the output (12); (ii) the (locally) positive definite and bounded from below function W (Z, η) ∈ R given by W (Z, η˜) = E0 (Z) +

1 > η˜ P η˜, 2

(16)

with E0 defined in (1) and a constant matrix P = P > > 0, P ∈ R(n−1)×(n−1) , is a Lyapunov function in the set Ω. (iii) the equilibrium (Z, η˜) = (O, O) of the closed– loop dynamics is asymptotically (and locally exponential) stable, and so it is a minimum phase system; Let Ωc be the compact sub–level set defined as Ωc = {(Z, η˜) ∈ Ω × Rn | W (·) ≤ c},

(17)

with c a positive constant, then an estimate of the domain of attraction is given by the largest Ωc . Proof: The first claim follows directly from the previous derivations from (7), (13) and (14). Let us prove now the asymptotical stability of the origin, and so the second and third claim. Recall from lemma 1 the function (9) is positive definite in Ω and so (16) is also positive definite. Thus, from the derivations of the lemma 1 the derivative with respect to the time of (16) along the system trajectories becomes 2 ˙ = −m> W ˜)> ν, 12 K2 m12 z˙ + (m12 z˙ + P η

(18)

˙ ≤ 0. with the external controller ν given by (15) we get W Since the derivative is only semi–definite then we prove, ˙ = 0} from La Salle that the largest limit set in the set {W is the desired equilibrium. Then, remembering the remark 1 the case to study is the residual dynamics with z˙ = 0 and η˜ = O, and so ν = 0. Thus, by the definition of η from (12), together with z˙ = 0 we get the n − 1 constraints given by X˙ = −K2

Z

z

m12 (µ)dµ.

(19)

0

If we prove now that the unactuated z–coordinate goes to zero, i.e. the origin of the zero dynamics is asymptotically stable, then from the last identity the vector X˙ will goes to zero, which is equivalent to η˜ = O. To this end, some tedious but straigthforward calculations show that the internal dynamics are governed by the equation (11) with ξ given in the lemma 1 by (10) and F1 by (4). Now setting z˙ = 0 the 1 dV (11) becomes 0 = ∆ ˆ dz . Recalling from the lemma 1 that, ˆ by construction, ∆(z) > 0, ∀z ∈ Ω and by Assumption A.2 dV dz = 0 only at the equilibrium z = 0 and different from zero elsewhere in Ω \ {0}, then the largest invariant set in ˙ = 0} is the desired equilibrium (Z, η˜) = (O, O) the set {W and so it is asymptotically stable. The fact that the equilibrium is also locally exponentially stable is straightforward from the linearization. Thus the linearization of the whole closed–loop dynamics reads    0 Z˙  ˙η˜ = −c1 O

1 −c2 −c4

   O Z  , −c> 3 η˜ −C5 z=0

ˆ −1 and c2 = m> (K2 + where c1 = −d2 V /dz 2 ∆ 12 −1 ˆ ˆ −1 and c4 = K3 K2 )m12 ∆ are scalars, c3 = K3 P m12 ∆ K3 K2 m12 vectors and C5 = K3 P a matrix. Recalling ˆ K1 , and P are positives and K2 , K3 diagonal and that ∆, positives, it is straightforward for the exponential claim to check that all the eigenvalues have negative real part. For the sake of simplicity and without loss of generality we also fix the free matrix P as diagonal and so C5 = c5 In−1 where c5 > 0 is a constant. In this case the characteristic polynomial is given by

B. The case with D1 6= 0. We have proven in the previous subsection that the nonlinear linearizing controller given by (14) stabilizes the zero dynamics for a class of systems where the Assumption A.1a hold. Now, we prove in this subsection that the same controller (14)–(15) can also stabilize the zero dynamics for another different sub–class. In fact we will use now the Assumption A.1b. We will show that the controller proposed could achieve asymptotical stability for the desired equilibrium for systems satisfying the Assumption A.1b. Proposition 2: Consider the underactuated mechanical system given by equations (3), under Assumptions A.1b– A.2 and with the lemma 1 taking effect. Then, the smooth static feedback controller given by (14) together with the external controller given by (15). Thus, for any diagonal full–rank constant matrix K2 > 0, as defined in (12), and such that satisfies that m> 12 K2 m12 > d1 , ∀z ∈ Ω with d1 the positive constant defined in Assumption A.1b, and the diagonal matrix K3 > 0, K3 ∈ R(n−1)×(n−1) , ensures the same three claims stated at Proposition 1. Proof: The first claim follows directly from the previous derivations from (7), (13) and (14). Let us prove now the asymptotical stability of the origin, and so the second and third claim. Recall from lemma 1 the function (9) is positive definite in Ω and so (16) is also positive definite. Thus, from the derivations of the lemma 1 the derivative with respect to the time of (16) along the system trajectories becomes from (18) 2 ˙ = −m> W ˜)> ν, 12 K2 m12 z˙ + D1 z˙ + (m12 z˙ + P η and from Assumption A.1b, kD1 k ≤ d1 kzk, ˙ yields 2 ˙ ≤ −(m> W K m − d ) z ˙ + (m12 z˙ + P η˜)> ν, 2 12 1 12

(λ + c5 )n−2 (λ3 + (c2 + c5 )λ2 + (c1 + c2 c5 − c> 3 c4 )λ + c1 c5 ) = 0.

˙ and taking into account that m> 12 K2 m12 > d1 then W ≤ 0. Following the same steps that in the proof of proposition 1, the equilibrium (Z, η˜) = (O, O) is asymptotically stable. The third claim, related to local exponential stability of the equilibrium, is straightforward from the linearization, as in the proof of proposition 1, just imposing m> 12 K2 m12 > d1 .

By applying the Routh-Hurwitz criterion, the following conditions have to be satisfied

V. A PPLICATIONS

c1 + c2 c5 − c> 3 c4 > 0, c2 (c1 + c2 c5 + c25 ) − (c5

+ c2 )(c> 3 c4 ) > 0.

(20) (21)

On one hand, substituting the inequality (20) in (21) yields c2 c5 − c> 3 c4 =



K3 P m> 12 K2 m12 ˆ ∆

 > 0, z=0

which is always satisfied by definition. On the other hand, since c1 > 0 the inequality (20) is automatically satisfied and, then the matrix is Hurwitz and so the origin is locally exponentially stable. The last claim follows from the fact that the origin of the closed–loop system is locally exponentially stable, and a necessary condition for this fact it is that the origin of the zero dynamics was locally exponentially stable and so it is a minimum phase system. Finally, the estimate of the domain of attraction follows from the fact that Ωc is the largest bounded sub–level set of W and LaSalle’s Principle.

In the following section two applications are presented: the Vertical Take-Off Landing (VTOL) aircraft and the Inertia Wheel pendulum with friction. Table I presents a summary of the relevant physical parameters of the systems related to the Potencial function and the Inertia matrix. In the notation m represents a mass, l a length, g the gravitational constant, J an inertia moment, α an angle and I the identity matrix. In the following a table is presented for each example describing the dummy output η, the energy function E0 , the external controller ν and the linearizing control law u. Since the change of coordinates from (Z, η˜) to the original physical coordinates is a (global) diffeomorphism we present the examples in the more intuitive physical coordinates. A. Vertical Take-Off Landing Aircraft (VTOL) A detailed description of the VTOL model can be found in [15], [6]. In table I the parameters of the system with

Dynamical System

DOF n

VTOL Aircraft Inertia Wheel Pend.

3 2

Potential function U (q) V (z) Φ(X) mgl tan(α)

cos(z) mgl cos(z)

mgx2 -

m11 (z)

Inertia Matrix M (q) m> 12 (z)

M22 (q)

J J1 + J2

ml − tan(α) [cos(z) sin(z)] J2

mI J2

TABLE I P HYSICAL PARAMETERS OF THE S YSTEM

respect to potential and Inertial terms. With this information and applying equations (12), (9), (15) and (14), the output η, the Energy function E0 , the external controller ν and the linearizing control u have been attained respectively, and have been summarized in the following table, where for the η E0 ν

ˆ1 [cos z, sin z]> z˙ − k ˆ2 [sin z, 1 − cos z]> X˙ −k  1 2 ˆ z˙ k1 − ε + g (1 − cos z) 2   ˆ3 Pˆ η − z˙ [cos z, sin z]> −k  1ˆ ˆ1 sin2 z ε−k 2 k1 sin 2z 2 1ˆ ˆ k sin 2z ε − k cos z 1  2 1    − sin z cos z ˆ1 ˆ2 ν+k z˙ 2 + k z˙ + cos z sin z 1 ˆ ε−k 1

u



ˆ k 1 ε



1 2

sin 2z sin2 z



TABLE II VTOL S OLUTION

sake of compactness, the following constants have been α ml i = 1, 2, 3, Pˆ = P tan and defined as kˆi = ki tan α ml tan α ε = J ml . It can be noticed that the law is well-defined for all ˙ and stabilizes almost globally and asymptotically (z, z, ˙ X) the system at the origin. With this law the origin is also locally exponentially stable. B. Inertia Wheel Pendulum (IWP) This system is thoroughly described in [17], where the friction is not taken into account. In this section the model friction is included. Following the same procedure as in the preceding example, and substituting for the physical parameters in equations (12), (9), (15) and (14), the following table is presented, η E0 ν u

x˙ + k1 J2 z˙ + k2 J2 z 1 z˙ 2 2

 2 − (J + J ) + mgl(1 − cos z) k1 J2 1 2

−k3 (P η + J2 z) ˙  ν + k1

 2   J2 J2 −mgl sin z + kf z˙ − k2 J2 z˙ 1 − k1 J1 +J2 J1 +J2

!−1

TABLE III IWP S OLUTION

In this example the solution of the control law is also welldefined for all values of (z, z, ˙ x). ˙ However, the existence of several equilibria which are not physically equivalent makes the origin only locally asymptotically stable in the set {(z, z, ˙ |x|) ˙ ∈ (−π, π) × R × (|x| ˙ max )} for a value of k2 > k2∗ . The constant k2∗ will be designed such that k2∗ = |x| ˙ max < ∞. The Lyapunov function guarantees that z˙ and η 2π converge asymptotically to the origin. So a maximum value |x| ˙ max exists and therefore k2∗ . The origin is also locally exponentially stable.

VI. C ONCLUSIONS This paper presents a constructive methodology to control a class of unstable underactuated mechanical systems with underactuation degree one taking into account the friction effects. The approach is based on classical feedback linearization and Lyapunov design. The control design is based on a proposed fictitious output that can be redesigned in a constructive way. The designed controller is successfully applied to control the VTOL aircraft and to the Inertia Wheel pendulum with friction. R EFERENCES ˚ ˚ om, “Safe manual control on the Furuta pen[1] J. Akesson and K.J. Astr¨ dulum”, in the Proc. of the IEEE Conference on Control Applications, pags. 890–895, 2001. ´ Acosta and M. L´opez-Martinez, “Constructive feedback lineariza[2] J. A. tion of underactuated mechanical systems with 2 DOF”, CDC&ECC, 2005, Seville, Spain. ´ Acosta and R. Ortega and A. Astolfi, “Interconnection and [3] J. A. damping assignment passivity–based control of mechanical systems with underactuation degree one”, IEEE Transactions on Automatic Control, Dec. 2005. [4] D. Bates and I. Postlethwaite, “Robust Multivariable Control of Aerospace Systems”. Delft University Press. 2002. [5] A.M. Bloch, N.E. Leonard and J.E. Marsden, “Controlled Lagrangians and the stabilization of mechanical systems I: The first matching theorem”. IEEE Transactions on automatic control, vol 45, no. 12, 2000. [6] I. Fantoni and R. Lozano, “Non-linear Control for Underactuated Mechanical Systems”. London. Springer, cop. 2002 ´ Acosta, “Passivation [7] F. Gomez-Estern and A. V. der Shaft and J. A. of underactuated systems with physical damping”. Proceedings of the Nonlinear Control Systems, 2004. [8] J. W. Grizzle, C. H. Moog, and C. Chevallereau, Nonlinear control of mechanical systems with an unactuated cyclic variable, Transactions on Automatic Control, Vol. 50, Issue 5, pp. 559- 576, May 2005. [9] R.A. Hyde, “H∞ aerospace control design”. Advances in industrial control series. Springer-Verlag. 1995. [10] A. Isidori, “Nonlinear Control Systems”, Springer-Verlag, London, 1996 [11] H. Khalil, “Nonlinear control systems”, Third Edition, 2001. London, UK.. [12] Zhong-Hua Li and Miroslav Krsti´c, Maximizing regions of attraction via backstepping and CLFs with singularities, System & Control Letters, Vol. 30, No. 1, 1997, pp. 195–207. [13] J. E. Marsden and T. S. Ratiu, “Introduction to mechanics and symmetry”. Ed. Springer, 1999. [14] R.W. Pratt, “Flight Control Systems: Practical issues in design and implementation”. The institution of Electrical Engineers. 2000. [15] S. Sastry, “Nonlinear Systems: Analysis, Stability and Control”, Springer-Verlag, New York, 1999 [16] A. J. van der Schaft, “L2 –Gain and Passivity Techniques in Nonlinear Control”. Ed. Springer-Verlag. Berlin (1999). [17] M.W. Spong, P.Corke y R.Lozano, “Nonlinear control of the Reaction Wheel Pendulum”. Automatica 37 (2001) 1845-1851. [18] M. W. Spong,“Energy based control of a class of underactuated mechanical systems”, in Proc. IFAC World Congress, San Francisco, CA,1996, pp. 431435.