Ihab Haidar
email: ihab.haidar@ensea.fr

Florentina Nicolau
email: florentina.nicolau@ensea.fr

Jean-Pierre Barbot
email: barbot@ensea.fr

Woihida Aggoune
email: woihida.aggoune@ensea.fr

Input-output linearization of nonlinear time-varying delay systems: the single-input single-output case

Keywords: Nonlinear control systems, delay systems, input-output linearization, Lie derivative

This paper deals with the input-output linearization of nonlinear timevarying delay systems. We introduce an extension of the Lie derivative for time-varying delay systems and derive sufficient conditions for the existence of a causal and bounded nonlinear feedback linearizing the inputoutput behavior of the system. Sufficient conditions ensuring the internal stability after output stabilization are also presented. Finally, several examples illustrating our main results are discussed.

Introduction

The problem of static linearization is an important tool in nonlinear control theory and consists in transforming the nonlinear system dynamics into a linear one such that linear control laws can be used in order to achieve the desired control properties. Two types of feedback linearization can be distinguished: input-state linearization and input-output linearization. In the first case, we need a state transformation (playing the role of a change of coordinates) and a static feedback transformation (that changes coordinates in the control space in a way which is state dependent) that bring the original system into an equivalent linear one. In the case of input-output linearization, the output is connected to the control only indirectly through the state and to achieve input-output linearization, a direct relation between the input and the output of the system must be found and, after the application of a suitable feedback transformation, the input-output map of the feedback modified systems has to be linear.

The problems of input-state and input-output linearization are well know for nonlinear control systems without delays (see, e.g., [START_REF] Hunt | Linear equivalents of nonlinear time varying systems[END_REF][START_REF] Jakubczyk | On linearization of control systems[END_REF], for input-state linearization, and [START_REF] Ha | A complete characterization of decoupling control laws for a general class of nonlinear systems[END_REF][START_REF] Descusse | Decoupling with dynamic compensation for strong invertible affine non-linear systems[END_REF][START_REF] Isidori | Nonlinear decoupling via feedback: a differential geometric approach[END_REF], for input-output decoupling and linearization). Various aspects of those problems have been studied in the literature using different approaches and some of those approaches have been extended to encompass nonlinear control systems with constant delays (see, e.g., [START_REF] Baibeche | Input-state feedback linearization for a class of single-input nonlinear time-delay systems[END_REF][START_REF] Califano | Extended Lie brackets for nonlinear time-delay systems[END_REF], for the algebraic approach, and [START_REF] Germani | Input-output linearization with delay cancellation for nonlinear delay systems: the problem of the internal stability[END_REF][START_REF] Oguchi | Input-output linearization of non-linear systems with time-delays in state variables[END_REF][START_REF] Oguchi | Input-output linearization of retarded non-linear systems by using an extension of Lie derivative[END_REF], for the geometric one).

However in the particular case of nonlinear time-varying delay control systems, which is the subject of this paper, the problem is still largely open. Our goal is thus to propose a solution for the problem of input-output linearization for single-input single-output nonlinear time-varying delay systems, to discuss how our solution differs and how it reminds that for the constant-delay case and finally, to understand the problems that may arise when constructing a feedback transformation for delay systems. The multi-input multi-output case is more involved and will be considered in a future paper (see [START_REF] Nicolau | Input-output decoupling and linearization of nonlinear two-input two-output time-varying delay systems[END_REF] for the case of two-input two-output time-varying delay systems). Preliminary results leading to this paper appeared in [START_REF] Haidar | Further remarks on input-output linearization of siso time-varying delay systems[END_REF][START_REF] Haidar | Input-output linearization of siso nonlinear time-varying delay systems[END_REF].

Like for systems without delays, the input-output linearization of time-delay systems can be accomplished by successive differentiation of the output until the input appears in the resulting derivative equation (the number of times that we need to differentiate will be called relative degree) and, then, by applying a feedback transformation for which the input-output map of the feedback modified systems is linear. The particularity of time-delay systems resides in the resulting input-output relation which is a recursive equation involving (in addition to the current and past values of the state) not only the input at the current time, but also its past values. Thus, the main difficulty is the construction of a causal and bounded linearizing feedback. We give sufficient conditions ensuring those properties, explain how to verify them, and compare them with those presented in the literature for the constant-delay case [START_REF] Germani | Input-output linearization with delay cancellation for nonlinear delay systems: the problem of the internal stability[END_REF][START_REF] Germani | Local asymptotic stability for nonlinear state feedback delay systems[END_REF][START_REF] Oguchi | Input-output linearization of non-linear systems with time-delays in state variables[END_REF][START_REF] Oguchi | Input-output linearization of retarded non-linear systems by using an extension of Lie derivative[END_REF]. The interest of the paper is also to understand the role played by the time-varying delay. An extension of the Lie derivative to the time-varying case has to be defined. The particularity is that the consecutive derivatives of the considered time-varying delay appear explicitly in the Lie derivatives as well as in the expression of the linearizing feedback.

Input-output linearization is useful to study the asymptotic stability of timevarying delay systems. But, unlike for systems without delay, even if a full input-output linearization has been achieved (i.e., the relative degree equals the state dimension), the internal stability is not guaranteed after output stabilization. This problem is deeply studied in [START_REF] Germani | Local asymptotic stability for nonlinear state feedback delay systems[END_REF] in the case of constant-delay systems. In that paper, the authors prove that, in the case of fully linearization and under some assumptions on the internal state dynamics, when the output is asymptotically driven to zero, both the state and the control tend to zero asymptotically. Here, we develop sufficient conditions ensuring the internal stability in both cases (of a complete, respectively, of a partial input-output linearization).

The paper is organized as follows. In Section 2, we present some notations and recall the definition of the Lie derivative for time-varying delay systems (introduced by the authors in [START_REF] Haidar | Input-output linearization of siso nonlinear time-varying delay systems[END_REF]). In Section 3, we present our main results and compare them to the existing ones for the constant-delay case. Several examples are discussed in Section 4. The proofs are given in Section 5.

Notations, definitions and problem statement

Throughout, R n denotes the n-dimensional Euclidean space with norm • and R + the set of non-negative real numbers. Given θ > 0 and q ∈ N, we denote by

C = (C([-q θ, 0], R n ), • ∞ ) the Banach space of continuous functions from [-q θ, 0] into R n , with the norm • ∞ defined by ϕ ∞ = sup θ∈[-q θ,0] ϕ(θ) , ϕ ∈ C.
We denote by U = U([-θ, +∞), R) the set of measurable inputs from [-θ, +∞) into R.

Definition 1 (δ-operators and their properties). Let θ > 0 and θ : R → (0, θ] be a sufficiently smooth function which is supposed to be known1 and satisfying dθ dt ≤ 1. over R + and satisfies dθ dt < 1. Consider the recursive relation

τ i+1 (t) = τ i (t) -θ • τ i (t), for i ≥ 0,
where τ 0 (t) = t. We denote by δ i the time delay operator that shifts the time from t to τ i (t) and which is defined as

δ 0 σ(t) = σ(t) and δ i σ(t) = σ(τ i (t)), for i ≥ 0,
where σ is a function defined on an interval containing [t-i θ, t]. The application of δ i on a composed function is given by

δ i ϕ(t, σ(t)) = ϕ(τ i , δ i σ(t)) = ϕ(τ i , σ(τ i )), for i ≥ 0. (1)
Applied on the product of two functions, this delay operator acts as the following

δ i ϕ(t) • σ(t) = (δ i ϕ(t)) • (δ i σ(t)), for i ≥ 0, (2)
i.e., the delay spreads to the right. If brackets are present, i.e., we have (δ i ϕ(t))σ(t), then the delay affects only the first function (here ϕ).

We introduce the δ and δ ≥i operators which are defined, respectively, by

δσ(t) = (σ(t), δ 1 σ(t), • • • , δ q σ(t)), δ ≥i σ(t) = (δ i σ(t), • • • , δ q σ(t)), (3)
where 0 ≤ i < q and q is the maximal order of the delay operator acting on σ.

Finally, we introduce the advance operator denoted by δ -i that shifts the time from t to τ -1 i (t) and which is defined by

δ -i σ(t) = σ(τ -1 i (t)), for i ≥ 0.
Similarly to the δ-operator, we define

δ -σ(t) = (δ -1 σ(t), • • • , δ -q σ(t)). (4)
Remark 1. If the delay operator δ i is applied on the composition of two functions of one variable, relation (1) writes

δ i ϕ • σ(t) = δ i ϕ(σ(t)) = ϕ(δ i σ(t)), for i ≥ 0.
Remark 2. The condition dθ dt < 1 on the derivative of the delay function is important for causality reasons. Without that assumption, when dealing with time-varying delay systems, we would obtain a contradiction with the fact that the information available at t is not available at t + . Usually, in the literature, this condition is stated as dθ dt ≤ 1. In this paper, the case dθ dt = 1 is also excluded. Condition dθ dt < 1, together with the fact that the delay function is completely known, allows to define the inverse function τ -1 i of τ i over R + , for i ≥ 0. This is a "future function" which can be interpreted as follows: τ -1 i (t 1 ), for some fixed t 1 ≥ 0, is the future instant t 2 verifying τ i (t 2 ) = t 1 (and we clearly have t 2 > t 1 ).

In this paper, we study input-output linearization of single-input singleoutput nonlinear time-varying delay systems of the form

ẋ(t) = f (δx(t), t) + g 0 (δx(t), t)δ 0 u(t) + g 1 (δx(t), t)δ 1 u(t), ∀t ≥ 0, y(t) = h(δx(t), t), x(s) = x 0 (s), ∀s ∈ [-q θ, 0], u(s) = u 0 (s), ∀s ∈ [-θ, 0], (5)
where x(t) ∈ R n , the vector fields f, g 0 , g 1 : R n(q+1) × R + → R n and the function h : R n(q+1) × R + → R are sufficiently smooth. The integer q corresponds to the maximal delay order explicitly involved in f , g 0 , g 1 and h (but it does not mean that f , g 0 , g 1 and h have necessarily the same delay orders). According to (3), δx(t) = (δ 0 x(t), • • • , δ q x(t)) denotes the δ-operator associated to a sufficiently smooth time-varying delay function θ : R → (0, θ] satisfying d dt θ(t) < 1, for all t ∈ R, and where θ is a positive real number. The initial condition x 0 belongs to C([-q θ, 0], R n ) and the input u : [-θ, +∞) → R is a Lebesgue measurable function. We also assume that system (5) is forward complete; this assumption guarantees the existence of solutions on [0, +∞) for each fixed determined u.

To achieve input-output linearization (the problem that we are studying here), we must find a direct relation between the input and the output of the system by successive differentiation of the output until the input appears in the resulting derivative equations. An important tool when differentiating is the Lie derivative. Let us next define the Lie derivative for time-varying delay systems which is a generalization of that presented in [START_REF] Fliess | Controllability and observability of linear delay systems : an algebraic approach[END_REF][START_REF] Oguchi | Input-output linearization of non-linear systems with time-delays in state variables[END_REF] for constant-delay systems.

Definition 2 (Lie derivative). Let f : R n(q+1) × R + → R n be a smooth vector field whose components are functions of (δx(t), t), and h : R n(q+1) × R + → R a real valued function of (δx(t), t). The Lie derivative of h along f at (δx(t), t) is defined as

L f h(δx(t), t) = q i=0 ∂h ∂δ i x τi δ i f (δx(t), t) + ∂h ∂t (δx(t), t). (6)
Notice that ( 6) is also an extension of the Lie-Backlünd derivative introduced in [START_REF] Fliess | A Lie-Backlünd approach to equivalence and flatness of nonlinear systems[END_REF] in a control theory context. The difference resides in our consideration of time-varying delays which leads to multiplicative coefficients (the time-derivative of the recursive delay function τ i ) affecting the operator δ i .

Observe that, by taking the Lie derivative of h along f , we introduce new delays (via the term δ i f ). In the above definition q denotes the maximal delay order explicitly involved in f or h, but the number of new delays introduced in L f h is related to the maximal delay order appearing in h only. Since L f h is a real-valued function with delays, the above operation can be recursively repeated for higher order as

L k f h(δx(t), t) = kq i=0 ∂L k-1 f h ∂δ i x τi δ i f (δx(t), t) + ∂L k-1 f h ∂t (δx(t), t), for k ≥ 2.
We also need to define a reduced Lie derivative.

Definition 3 (Reduced Lie derivative). Let g : R n(q+1) × R + → R n be a smooth vector field and h : R n(q+1) × R + → R a real valued function. The reduced Lie derivative of h along g is defined as

L g h(δx(t), t) = q i=0 ∂h ∂δ i x τi δ i g(δx(t), t). (7)
Contrary to [START_REF] Fridman | On input-to-state stability of systems with time-delay: A matrix inequalities approach[END_REF], the partial derivative of h with respect to t is not present in [START_REF] Germani | Local asymptotic stability for nonlinear state feedback delay systems[END_REF]. If h does not depend explicitly on t, then L g h coincide with L g h. Notice also that, in the case of time-varying delay systems, the Lie derivatives L f h and L g h will always depend explicitly on t through the terms τi .

As indicated by the notations, in our study the reduced Lie derivative (7) will always be computed for the control vector fields g 0 and g 1 , while the Lie derivative [START_REF] Fridman | On input-to-state stability of systems with time-delay: A matrix inequalities approach[END_REF] will always be associated to the drift f .

Similarly to control systems without delays, the time-derivative of L k f h, for k ≥ 0, can be written as follows:

dL k f h dt = m i=0 ∂L k f h ∂δ i x dδ i x dt + ∂L k f h ∂t = L k+1 f h + L g0 L k f hu + L g1 L k f hδ 1 u, ( 8
)
where m is the highest delay order of L k f h (in this case m = kq). Note that, in order to be able to compute the time-derivatives of δ i x, for 0 ≤ i ≤ m, equation (8) should be defined only for t ≥ τ -1 m (0). Recall that the operator δ spreads to the right (see equation ( 2)), thus L g0 L k f hu involves as many delayed controls as the delays present in the function L k f h. Indeed, according to the definition of the reduced Lie derivative, we have:

L g0 L k f hu = m i=0 ∂L k f h ∂δ i x τi (δ i g 0 )) • (δ i u).
A similar remark can be made for L g1 L k f hδ 1 u, but now the number of delayed controls is no longer m, but m + 1 (because L g1 L k f h acts on a control that is already delayed):

L g1 L k f hδ 1 u = m i=0 ∂L k f h ∂δ i x τi (δ i g 1 ) • (δ i+1 u).
Therefore, the reduced Lie derivatives L g0 L k f h and L g1 L k f h associated to the control vector fields g 0 and g 1 , respectively, can be seen as δ-polynomials and can be developed with respect to the operator δ i as follows:

L g0 L k f h(δx(t), t) = m i=0 a i 0 (δx(t), t)δ i , L g1 L k f h = m+1 i=1 a i 1 (δx(t), t)δ i ,
where

a i 0 (δx(t), t) = ∂L k f h ∂δ i x τi (δ i g 0 (δx(t), t)), 0 ≤ i ≤ m, and
a i 1 (δx(t), t) = ∂L k f h ∂δ i-1 x τi-1 (δ i-1 g 1 (δx(t), t)), 1 ≤ i ≤ m + 1.
Combining these two δ-polynomials, we define

a(δ] = a 0 (δx(t), t)δ 0 + • • • + a m+1 (δx(t), t)δ m+1 , (9)
where a 0 = a 0 0 , a i = a i 0 + a i 1 , for 1 ≤ i ≤ m, and a m+1 = a m+1

1

. The δpolynomial a(δ] is an element of the ring of differential time-varying delay operators (see [START_REF] Quadrat | Algebraic analysis for the Ore extension ring of differential time-varying delay operators[END_REF] for more details about the algebraic properties of that ring). Equation ( 8) can then be rewritten as follows:

dL k f h dt = L k+1 f (δx(t), t) + a(δ]u(t), t ≥ τ -1 m (0). (10)
Before formalizing the problem of input-output linearization, a definition of the relative degree in the case of time-varying delay systems is needed. This can be done in the same way as for the finite dimensional case (see, e.g., [START_REF] Isidori | Nonlinear control systems[END_REF]): the relative degree of an output h corresponds to the number of times that we have to differentiate h with respect to time t before the input or the delayed input appears.

In this paper, we do not deal with singularities and we make the following assumption:

Remark 3. When we say that a function does not vanish, we mean that it is nonzero for any t ≥ 0. Definition 4 (Relative degree). The relative degree of the output h of system (5) is the positive integer ρ defined as follows:

L g0 L k f h(δx(t), t) = L g1 L k f h(δx(t), t) = 0, 0 ≤ k < ρ -1, (L g0 L ρ-1 f h, L g1 L ρ-1 f h)(δx(t), t) = (0, 0), for any (δx(t), t) ∈ R n × R nq(ρ+1) × R + .
Since, in the time-varying delay case, the Lie derivatives depend explicitly on time, the relative degree may also be time-dependent. Here, due to Remark 3, this cannot occur (i.e., the relative degree is uniform with respect to t).

From now on, we denote by a(δ] the δ-polynomial associated to the ρ-th time-derivative of h, that is

h (ρ) = L ρ f h + a(δ]u, where a(δ] = L g0 L ρ-1 f h + L g1 L ρ-1 f h.
Definition 5 (Minimal and maximal degree). We call minimal degree of a(δ] the order of its first coefficient non identically zero, i.e., the integer j such that

a j (δx(t), t) = 0, (11)
a i (δx(t), t) ≡ 0, ∀ i < j.
The highest delay order of a(δ] is called the maximal degree and is denoted by m + 1 (where the integer m can be expressed in function of ρ and the highest delay order q involved in the system: m = qρ).

Observe that h (ρ) is actually defined for t ≥ τ -1 m (0). Throughout, we will use the notation t 0 = τ -1 m (0). Definition 6 (Input-output linearization). We say that the problem of inputoutput linearization is solvable for system (5) if the output h admits a finite relative degree ρ and if there exists a causal and bounded feedback u verifying

a(δ]u(t) = -L ρ f h(δx(t), t) + δ j v(t), ∀ t ≥ τ -1 j (t 0 ), ( 12
)
where j is the minimal degree of a(δ], and v is the new control (assigned with respect to the properties that we want to achieve). If such u exists, then the feedback modified system satisfies:

y (ρ) (t) = δ j v(t), ∀ t ≥ t 0 .
Moreover, the system is said input-output linearizable with delay if j > 0 (respectively, without delay if j = 0).

The input-output linearization usually leads to the existence of an unobservable part (called zero-dynamics) of system [START_REF] Fliess | Controllability and observability of linear delay systems : an algebraic approach[END_REF]. Thus, a condition on the zero-dynamics is needed in order to guarantee the internal stability of the system. We stress that even in the case of a complete linearization (i.e., ρ = n), we may also have an implicit relation between u and δ i u, for j ≤ i ≤ m, (relation which is called internal input dynamics in the literature, see [START_REF] Germani | Input-output linearization with delay cancellation for nonlinear delay systems: the problem of the internal stability[END_REF]). By consequence, two main problems may arise when constructing a feedback u from an equation of form [START_REF] Hunt | Linear equivalents of nonlinear time varying systems[END_REF]. The first problem is the boundedness of u which comes from the zero-dynamics or the internal input dynamics or both at once. The second one is the causality of u that stems from the fact that the drift may involve more delays than the control vector fields or vice versa. In the next section we give sufficient conditions allowing to solve these problems. Under those conditions, we design new coordinates allowing the transformation of (a part of) system ( 5) into the ρ-th order linear input-output system

żi (t) = z i+1 (t), 1 ≤ i ≤ ρ -1, żρ (t) = δ j v(t), (13)
where

z i = L i-1 f h, for 1 ≤ i ≤ ρ.
The fact that in [START_REF] Hunt | Linear equivalents of nonlinear time varying systems[END_REF] we consider δ j v (instead of v, which would also lead to a linear input-output map, but without delay) will be justified in the sequel. If we replace δ j v by v in [START_REF] Hunt | Linear equivalents of nonlinear time varying systems[END_REF], i.e., u should be computed from

a(δ]u(t) = -L ρ f h(δx(t), t) + v(t), (14)
then the control would act instantaneously on the system and no predictor would be needed for the construction of a feedback v for the z-system (for instance, if the aim is to design a stabilizing feedback for ( 13)), but when designing a feedback u from equation ( 14), a j-order predictor (i.e., a predictor over [t 0 , τ -1 j (t 0 )]) would have to be implemented for the x-system. Here, we simply chose to make the predictor over the linear z-system (rather than on the nonlinear x-system).

In that case, we suppose that the system is in free evolution over [t 0 , τ -1 j (t 0 )] and that the feedback v (which has to be computed for t ≥ t 0 ) affects the system starting from the instant τ -1 j (t 0 ) only. This remark explains two points: first, the reason for which [START_REF] Hunt | Linear equivalents of nonlinear time varying systems[END_REF] is defined for t ≥ τ -1 j (t 0 ), and second, why condition ( 11) is needed starting from τ -1 j (t 0 ) only. Remark that, even if we place the predictor over the z-system, the computation of a feedback u may request a 1-order predictor (over [t 0 , τ -1 1 (t 0 )]) on the x-system (this may happen, for instance, when g 1 = 0). Finally, notice that if trajectory planning is intended, the artificial delay in the control v does not generate any technical problem because, in that case, we dispose entirely of the desired trajectory.

3 Main results

Input-output linearization

In this section, we give the main result of the paper: we design new coordinates and a causal and bounded feedback that linearizes the input-output behavior of the original system. To that end, we introduce the following conditions, that will be discussed in details after stating our main theorem: (C1) -Relative degree: The output h has a finite relative degree ρ ≤ n.

(C2) -New coordinates: There exists a smallest integer 0 ≤ ≤ m such that the matrix

∂L k f h ∂δ xi , for 0 ≤ k ≤ ρ -1 and 1 ≤ i ≤ n, is of full rank (equal to ρ) at any (δx(t), t) ∈ R n × R nq(ρ+1) × R + .
Introduce the z-variables z i (t) = L i-1 f h(δx(t), t), for t ≥ t 0 and 1 ≤ i ≤ ρ, and complete them (which is always possible) by n -ρ functions ξ i (t) = φ i (δx(t), t), for ρ + 1 ≤ i ≤ n, such that the Jacobian matrix ∂Φ ∂δ x , where

Φ = (h, . . . , L ρ-1 f h, φ ρ+1 , . . . , φ n ), is of full rank (equal to n) at any (δx(t), t) ∈ R n × R nq(ρ+1) × R + .

(C3) -Causal and bounded inversion:

There exist ξ-coordinates, a continuous function Ψ such that δ x(t) = Ψ(δz(t), δ -z(t), δξ(t), δ -ξ(t), δ ≥ +1 x(t), t), ∀t ≥ t 0 , (15)
and a constant µ > 0 such that if δ ≥ +1 x(t) < µ, then

δ x(t) ≤ α 1 (δz(t), δ -z(t)) + α 2 (δξ(t), δ -ξ(t)) + α 3 δ ≥ +1 x(t) , ∀t ≥ t 0 , (16)
where α 1 > 0, α 2 > 0 and 0 < α 3 < 1/(q(ρ + 1) -). Recall that δ -and δ ≥ +1 were defined in Section 2.

(C4) -Causal feedback: The Lie derivative L ρ f h and the coefficients of the δ-polynomial a(δ] verify

∂L ρ f h ∂δ i x ≡ 0 and ∂a k ∂δ i x ≡ 0, (17)
for j ≤ k ≤ m+1 and 0

≤ i ≤ j-1 if g 0 = 0 (respectively, 0 ≤ i ≤ j-2 if g 0 ≡ 0). (C5) -Bounded feedback: Let v : [τ j (t 0 ), +∞) → R be such that -L ρ f h(δx(t), t) + δ j v(t) a j (δx(t), t) is bounded over [τ -1 j (t 0 ), +∞) (18)
and u be a solution of [START_REF] Hunt | Linear equivalents of nonlinear time varying systems[END_REF] for that v. Suppose that u is bounded over [τ m+1-j (t 0 ), t 0 ] and that there exists a constant c > 1 such that sup

t≥τ -1 j (t0) a i (δx(t), t) a j (δx(t), t) ≤ 1 c(m + 1 -j) , ∀i > j. ( 19
)
Theorem 1. Consider system (5) and suppose that conditions (C1)-(C5) are satisfied. Then, system (5) is input-output linearizable via the causal and bounded feedback transformation [START_REF] Hunt | Linear equivalents of nonlinear time varying systems[END_REF] and (a part of ) it can be transformed into the ρ-th order linear input-output system [START_REF] Isidori | Nonlinear control systems[END_REF]. Moreover, if ρ = n, then the system is fully input-output linearizable with delay and if, in addition, j = 0, then the system is fully input-output linearizable without delay.

Theorem 1 gives only sufficient conditions. We do not claim that our conditions are also necessary, see Section 4, where we give several examples that do not satisfy the conditions of Theorem 1.

Conditions discussion

We next discuss the role and the importance of each condition. Condition (C1) requires that the relative degree of h is smaller or equal to n. It is clear that if this is not the case, system (5) is not input-output linearizable (in fact, a part of the system is not controllable).

Contrary to the cases of control systems without delays and constant-delay control systems (see Section 3.3, where we give a comparison of our results with the constant-delay case), condition (C1) may not be sufficient to define a local change of coordinates and additional conditions (given by (C2) and (C3)) may be needed.

It is important to note that by a change of coordinates (or a coordinate transformation), we do not mean (like for systems without delays) a diffeomorphism between x and the new state (z, ξ), see condition (C2). In fact the coordinate transformation map Φ : R n × R nq(ρ+1) × R + → R n of (C2), that allows us to construct the new variables z transforming a part of system (5) into the ρ-th order linear system [START_REF] Isidori | Nonlinear control systems[END_REF], is never a diffeomorphism. However, condition (C2), together with (C3), enables us to come back from the (z, ξ)-state space into the x-state space by expressing the delayed states δ x i , for a certain delay order 0 ≤ ≤ m, as functions of the new state, their delays and advances, and possible delayed state δ p x, where p ≥ + 1 (hence the name "causal inversion") and to preserve (in the x-state space) properties (like boundedness) that hold in the (z, ξ)-state space. As we will see below, conditions (C2) and (C3) play a crucial role in the verification of (C5) ensuring the boundedness of the input-output linearizing feedback.

A less restrictive condition than (C2) would be to assume the existence of delay orders 1 , . . . , n such that 0 ≤ i ≤ m and the rank of the matrix ∂φ k ∂δ i xi , for 1 ≤ k ≤ ρ, 1 ≤ i ≤ n, equals ρ, and for each δ i x i we have a causal inversion (i.e., in ( 15)-( 16), we should take δ ≥L+1 x, where L = max 1≤i≤n i instead of δ x). This would considerably complicate the notations and in order to simplify the understanding of the paper, we assume that the more restrictive condition (C2) holds.

We stress that condition (C3) (in particular relation [START_REF] Jakubczyk | On linearization of control systems[END_REF] which should hold for all t ≥ t 0 ) supposes indirectly that z and ξ are completely known over [τ q(ρ+1) (t 0 ), t 0 ] and that their maximum delay order cannot exceed q(ρ + 1). This does not necessarily mean that all their components should be known over [τ q(ρ+1) (t 0 ), t 0 ]. Throughout this paper, we suppose that the maximum delayadvance order in (z, ξ) cannot exceed q(ρ + 1).

Conditions (C4) and (C5) deal with the construction of a causal and bounded feedback. (C4) should be understood as follows: at an instant t ≥ t 0 , we have to assign to system (5) either u(t), if g 0 = 0, or δ 1 u(t), if g 0 ≡ 0, and, in both cases, the construction of u is obtained from ( 12) by applying the advance operator δ -j and thus shifting the coefficients of the δ-polynomial as well as L ρ f h of order j in the future. In the case when u(t) acts on the system (i.e., g 0 = 0), the causality condition [START_REF] Khalil | Nonlinear systems[END_REF] guarantees that the value of u(t) depends only on the current and past values of the state; thus no causality problem can be produced when introducing in (5) the control u so obtained. In the case when only the delayed input appears (i.e., g 0 ≡ 0 and, of course, g 1 = 0), the causality condition provides that the value of u(t), computed from the feedback equation, depends on the past, current and 1-advance values of the state. Again, when introducing in (5) the associated δ 1 u(t), no causality problem appears. Nevertheless, in this later case, a predictor is needed in order to compute the control as a function of the future values of the state. Finally, observe that when j = 0, there is no causality problem (and thus there is no condition to be checked).

We will see, when proving Theorem 1, that condition (C5) guarantees that the feedback u stays bounded when property [START_REF] Nicolau | Input-output decoupling and linearization of nonlinear two-input two-output time-varying delay systems[END_REF] holds. Thus an important issue is how to verify this property and next we explain how to do it. In fact, [START_REF] Nicolau | Input-output decoupling and linearization of nonlinear two-input two-output time-varying delay systems[END_REF] can be verified in the (z, ξ)-coordinates. This does not necessarily implies its validity in the x-coordinates (recall that, in the time-varying delay context, the change of coordinates map does not define a diffeomorphism). Nevertheless, thanks to property [START_REF] Karafyllis | Input-to-output stability for systems described by retarded functional differential equations[END_REF] given by (C3), under some additionnal assumptions (see Proposition 1 below), the boundedness in (z, ξ)-coordinates implies the boundedness in x-coordinates. In the following proposition, we use the notation

ε = 1 -α 3 (q(ρ + 1) -) (2q(ρ + 1) + 1) max{α 1 , α 2 } µ, (20)
where the parameters , µ, α 1 , α 2 , and α 3 are given by (C2)-(C3).

Proposition 1. Suppose that conditions (C1)-(C3) hold. If

sup t≥τ q(ρ+1) (t0)
{ z(t) , ξ(t) } < ε and sup τ q(ρ+1) (t0)≤t≤t0

x(t) < µ, then x(•) stays bounded by µ over [t 0 , +∞).

Stability of the closed loop system

In this section, we study the stability of the closed loop system. In addition to (C1)-(C5), we suppose that the following conditions are satisfied:

(C6) -Equilibrium: System (5) satisfies f (0, t) = 0, and h(0, t) = 0, ∀t ≥ 0.

where the 0-argument of f and h corresponds to the (q + 1)-tuple (0, • • • , 0).

(C7) -Total causal inversion: We suppose that (C3) holds, with the ξ-coordinates and the function Ψ such that

δ x(t) = Ψ(δz(t), δ -z(t), δξ(t), δ -ξ(t), t), ∀t ≥ t 0 , (21)
and that the dynamics of ξ do not contain advances in the ξ-states, i.e., ξ is of the form ξ(t) = G(δz(t), δ -z(t), δξ(t), δv(t), t).

Condition (C7) is quite restrictive: it supposes that all components of the state δ x(t) are explicit functions of δz(t), δ -z(t), δξ(t), δ -ξ(t), t), for all t ≥ t 0 . This condition is important when dealing with the stability of the zero dynamics, but it is not necessary in the sens that only the x-components present in the functions φ i (δx(t), t), for ρ + 1 ≤ i ≤ n, defining the ξ-variables (see condition (C2)) have to be explicit functions of δz(t), δ -z(t), δξ(t), δ -ξ(t), t), for all t ≥ t 0 (see Section 4, for an example illustrating the nonnecessity of (C7)).

In the case when ρ < n, by applying feedback [START_REF] Hunt | Linear equivalents of nonlinear time varying systems[END_REF] and introducing the new (z, ξ)-coordinates, system (5) is decomposed into two parts: a ρ-th order linear z-subsystem and a nonlinear ξ-subsystem (zero-dynamics) which may contain delayed and advanced states. Under assumptions (C1)-(C7), system (5) can be transformed, for t ≥ t 0 , as follows:

       ż(t) = Az(t) + Bδ j v(t), ∀t ≥ t 0 ξ(t) = G(δz(t), δ -z(t), δξ(t), δv(t), t), ∀t ≥ t 0 y(t) = Cz(t), (22)
where the matrices A, B and C are associated to form [START_REF] Isidori | Nonlinear control systems[END_REF]. Note that the initial condition of ( 22) is defined through that of the initial problem (5) (see the remark explaining condition (C3) after the statement of Theorem 1). Let

δ j v(t) = k 1 z 1 (t) + • • • + k ρ z ρ (t), t ≥ τ -1 j (t 0 ), 0 t 0 ≤ t ≤ τ -1 j (t 0 ), (23)
where k i , for i = 1, . . . , ρ, are chosen in such a way that A + BK has ρ different eigenvalues with strictly negative real parts, where K = (k 1 , . . . , k ρ ). In this case, the resulting closed loop subsystem

ż(t) = (A + BK)z(t), (24)
is asymptotically stable. Therefore, under this choice of v, the state z(t) tends asymptotically to zero when t tends to +∞. Even if the input-output behavior can be stabilized by a feedback, the internal dynamics may be unstable and the global system cannot be stabilized.

In order to characterize the internal stability of the closed loop system, we have to recall the definition of input-to-state stability for time-delay systems (see, e.g., [START_REF] Fridman | On input-to-state stability of systems with time-delay: A matrix inequalities approach[END_REF][START_REF] Karafyllis | Input-to-output stability for systems described by retarded functional differential equations[END_REF][START_REF] Sontag | Input to state stability: Basic concepts and results[END_REF]). We will also need the following function classes: a function γ : R + → R + is said to be of class K ∞ if it is continuous, strictly increasing and γ(0) = 0. A continuous function β :

R + × R + → R + is said to be of class KL if β(•, t) is of class K ∞ for each t ≥ 0, β(s,
•) is decreasing to zero for each s > 0 and β(s, t) → 0 as t → +∞. Definition 7. Consider the time-delay system

ξ(t) = G(δξ(t), w(t)), ∀t ≥ 0, ( 25
)
where ξ(t) ∈ R n is the state of the system, w(t) ∈ R m is an exogenous input, the dynamics G is continuously differentiable and such that G(0, 0) = 0. System (25) is said to be uniformly input-to-state stable if there exist a KL function β and a K ∞ function γ such that, for any initial time t 0 ≥ 0, any initial state ξ t0 = ϕ ∈ C([-q θ, 0], R n ) and any measurable, locally essentially bounded input w, the solution ξ(t) exists for all t ≥ t 0 and furthermore, it satisfies

ξ(t) ≤ β( ξ t0 ∞ , t -t 0 ) + γ( w [t0,t) ∞ ), ∀t ≥ t 0 .
In our case, we apply the above definition to the zero-dynamics ξ(t) = G(δz(t), δ -z(t), δξ(t), δv(t)) of ( 22), for which the exogenous input is w(t) = (δz(t), δ -z(t), δv(t)). Notice, however, that w(t) is not exogenous for the global system.

Proposition 2. Consider system [START_REF] Sontag | Input to state stability: Basic concepts and results[END_REF]. Let v, given by equation (23), be such that the matrix A + BK in (24) is Hurwitz. If the zero-dynamics is uniformly input-to-state stable then system [START_REF] Sontag | Input to state stability: Basic concepts and results[END_REF] is locally asymptotically stable around the origin.

Proof. The proof derives straightforwardly from Definition 7.

Remark 4. The uniform input-to-state stability of the zero dynamics is not necessary if only the boundedness of the input is requested. For example, if the unstable part of the zero dynamics does not affect (or at least not significantly) the feedback control, which linearizes the system, then the input-output behavior is stable. Such a remark relaxes the conditions of Proposition 2, but this is far from the scope of the paper.

The following proposition gives sufficient conditions guaranteeing the local asymptotic stability of the equilibrium point in x-coordinates. Proposition 3. Consider system (5) and suppose that conditions (C1)-(C7) are satisfied. Let u verifying equation [START_REF] Hunt | Linear equivalents of nonlinear time varying systems[END_REF] with v, given by (23), and such that the matrix A + BK in (24) is Hurwitz. Suppose that the zero-dynamics is uniformly input-to-state stable. If z(t) < ε 1 over [τ q(ρ+1) (t 0 ), τ -1 j (t 0 )], and ξ(t) < ε 1 and x(t) < µ over [τ q(ρ+1) (t 0 ), t 0 ], for a suitable constant ε 1 , then lim t→+∞ x(t) = 0.

The constant ε 1 of the above proposition depends on the parameters defined by conditions (C2) and (C3), see the proof of Proposition 3.

Constant-delay case: comparison with the existent literature

A first attempt to extend the geometric tools of Lie derivative and Lie bracket to constant-delay systems has been given in [START_REF] Oguchi | Input-output linearization of non-linear systems with time-delays in state variables[END_REF][START_REF] Oguchi | Input-output linearization of retarded non-linear systems by using an extension of Lie derivative[END_REF], where sufficient inputoutput linearizing conditions have been developed for single-input single-output constant-delay systems. Some of those conditions have been relaxed in [START_REF] Germani | Local asymptotic stability for nonlinear state feedback delay systems[END_REF][START_REF] Germani | Input-output linearization with delay cancellation for nonlinear delay systems: the problem of the internal stability[END_REF], in the case of fully linearizable time-delay systems.

It may be interesting to give a comparison of our conditions (simplified for the constant-delay case and for autonomous vector fields f, g 0 , g 1 and output h) with the results of the above cited papers. We will next briefly recall their results, but let us first underline that in [START_REF] Germani | Local asymptotic stability for nonlinear state feedback delay systems[END_REF][START_REF] Germani | Input-output linearization with delay cancellation for nonlinear delay systems: the problem of the internal stability[END_REF][START_REF] Oguchi | Input-output linearization of non-linear systems with time-delays in state variables[END_REF][START_REF] Oguchi | Input-output linearization of retarded non-linear systems by using an extension of Lie derivative[END_REF], only the case of nondelayed input (i.e., g 1 ≡ 0) has been considered.

In addition to the existence of a relative degree ρ ≤ n, conditions like

∂L ρ-1 f h ∂x g 0 (δx(t)) = 0, and ∂L ρ-1 f h ∂δ i x δ i g 0 (δx(t)) = 0, ∀i ≥ 1, ( 26
)
for any δx(t) ∈ R nq are required in [START_REF] Oguchi | Input-output linearization of non-linear systems with time-delays in state variables[END_REF][START_REF] Oguchi | Input-output linearization of retarded non-linear systems by using an extension of Lie derivative[END_REF]. It is evident that under (26), the boundedness and causality properties of the control u (discussed after Definition 6) are no longer problematic. Indeed, if (26) holds, then [START_REF] Hunt | Linear equivalents of nonlinear time varying systems[END_REF] gives

u(t) = -L ρ f h(δx(t), t) + v(t) a 0 (δx(t)) , t ≥ t 0 , (27)
which is clearly causal, and bounded when the right hand side of ( 27) is bounded. Condition (26) has been relaxed in [START_REF] Germani | Input-output linearization with delay cancellation for nonlinear delay systems: the problem of the internal stability[END_REF], where only the less-restrictive relation

∂L ρ-1 f h ∂x g 0 (δx(t)) = 0, ( 28
)
for any δx(t) ∈ R nq , has been required. In this case, the control u is given by a recursive equation like [START_REF] Hunt | Linear equivalents of nonlinear time varying systems[END_REF] with j = 0. Thus, the boundedness (but not the causality) of the control u becomes questionable. Under condition (28) and in the case of fully linearization (i.e., ρ = n), the internal stability (i.e., the state and input stability), is then deeply studied and solved in [START_REF] Germani | Input-output linearization with delay cancellation for nonlinear delay systems: the problem of the internal stability[END_REF]. Unlike for delay-free systems, even if a fully input-output linearization has been achived, the internal stability is not guaranteed after output stabilization. Sufficient conditions ensuring that property (after output stabilization) are given in [START_REF] Germani | Local asymptotic stability for nonlinear state feedback delay systems[END_REF][START_REF] Germani | Input-output linearization with delay cancellation for nonlinear delay systems: the problem of the internal stability[END_REF]. These conditions are similar to what is requested by the statement of Proposition 3, but without any additional condition on the boundedness of the input. In fact, in [START_REF] Germani | Input-output linearization with delay cancellation for nonlinear delay systems: the problem of the internal stability[END_REF], the authors show that, in the case of fully linearization and under some assumptions on the internal state dynamics, when the output is asymptotically driven to zero, both the state and the control tend to zero asymptotically. The proof of this result lies particularly in the fact that the delay is constant. The possibility of extension to the time-varying delay case is not so evident and will be treated elsewhere.

We will next explain why our conditions are less-restrictive comparing to what is done in [START_REF] Germani | Input-output linearization with delay cancellation for nonlinear delay systems: the problem of the internal stability[END_REF][START_REF] Germani | Local asymptotic stability for nonlinear state feedback delay systems[END_REF][START_REF] Oguchi | Input-output linearization of non-linear systems with time-delays in state variables[END_REF][START_REF] Oguchi | Input-output linearization of retarded non-linear systems by using an extension of Lie derivative[END_REF]. Condition (C1) demands that either L g0 L ρ-1 f h = 0 or L g1 L ρ-1 f h = 0, that is to only have (compare with (26)-( 28)):

∂L ρ-1 f h ∂δ i x δ i g j (δx(t)) = 0, for some i ≥ 0 and 0 ≤ j ≤ 1.
Of course, in addition to (C1), further conditions like (C3)-(C5) are needed. Indeed, as explained in Section 2, in general, condition (C1) alone does not allow the construction of a bounded and causal feedback u from ( 12) that input-output linearizes the system. Conditions (C4) and (C5) guaranty the boundedness (additional condition on the zero-dynamics is needed in the case of partial linearization) and causality properties of u, while (C2) and (C3) are used in order to verify (C5). Another important point is that, in the case of constant delays, conditions like (C2) are not needed since, in that case, (C1) is sufficient for the existence of a local coordinate transformation. In fact, it is easy to prove that in that case and for autonomous vector fields f, g 0 , g 1 and output h, condition (C2) is a straightforward consequence of (C1), see [START_REF] Oguchi | Input-output linearization of non-linear systems with time-delays in state variables[END_REF][START_REF] Oguchi | Input-output linearization of retarded non-linear systems by using an extension of Lie derivative[END_REF] for a related result. For time-varying delays, the fact that the Lie derivatives are not autonomous (due to the presence of the terms τi ) complicates the proof.

Examples

In this section, we discuss several examples illustrating Theorem 1. The presented examples are separated into two categories: the first one showing that our conditions are sufficient (Examples 1, 2 and 3) and the second one proving that they are not necessary (Example 4) .

Example 1 (satisfying Theorem 1). Consider the nonlinear time-varying delay system

ẋ1 (t) = x 2 (t) + 1 3 δ 1 x 2 (t) ẋ2 (t) = δ 1 x 2 3 (t) + δ 1 u(t) ẋ3 (t) = -x 3 (t) -x 1 (t) h(t) = δ 1 x 1 (t), ∀t ≥ 0, (29)
with initial condition x 0 ∈ C([-θ, 0], R 3 ) and u ∈ U. The delay function θ(•) is chosen in such a way that | θ| < 1 4 . Here, we have g 0 = (0, 0, 0) T , g 1 = (0, 1, 0) T , and

f = (x 2 + 1 3 δ 1 x 2 , δ 1 x 2 3 , -x 3 -x 1 ) T ,
and the maximum delay order is q = 1. We can easily verify that all assumptions of Theorem 1 are all fulfilled. Indeed: Condition (C1): A straightforward computation gives

L g1 h = 0, L f h = τ1 δ 1 x 2 + 1 3 τ1 δ 2 x 2 , L g1 L f h = τ 2 1 δ 2 + 1 3 τ1 τ2 δ 3 and L 2 f h = τ1 δ 1 x 2 + 1 3 τ1 δ 2 x 2 + τ 2 1 δ 2 x 2 3 + 1 3 τ1 τ2 δ 3 x 2 3 .
Thus, condition (C1) is satisfied and the relative degree is ρ = 2. Condition (C2): By introducing

z 1 = δ 1 x 1 , z 2 = τ1 δ 1 x 2 + 1 3 τ1 δ 2 x 2 , (30)
and choosing

ξ = δ 1 x 3 , (31)
the rank of ∂Φ ∂δ 1 x is equal to 3 at any (δx(t), t) ∈ R 9 × R + , thus (C2) holds with = 1.

Condition (C3): From (30)-(31), we obtain

δ 1 x 1 = z 1 , δ 1 x 2 = 1 τ1 z 2 - 1 3 τ1 δ 2 x 2 , δ 1 x 3 = ξ, ∀ t ≥ t 0 , (32)
with t 0 = τ -1 2 (0). In addition, for any µ > 0, we have

δ 1 x(t) ≤ α 1 z(t) + α 2 ξ(t) + α 3 δ 2 x(t) , ∀t ≥ t 0 ,
with α 1 = 3, α 2 = 1 and α 3 = 4 9 , and (C3) is valid. Condition (C4): The feedback equation is given by

a(δ]u = τ 2 1 δ 2 u + k τ1 τ2 δ 3 u = -L 2 f h + δ 2 v(t), t ≥ τ -1 2 (t 0 ).
We are in the case where j = 2 and g 0 ≡ 0. Since the coefficients of the δ-polynomial do not depend on the state and that

∂L 2 f h ∂x ≡ 0, it follows that condition (C4) is valid.
Condition (C5): One can easily verify that the a(δ]-polynomial satisfies condition [START_REF] Oguchi | Input-output linearization of non-linear systems with time-delays in state variables[END_REF]. Thus, it remains to check the boundedness of

-L 2 f h + δ 2 v(t) τ 2 1 ∀ t ≥ τ -1 2 (t 0 ). (33)
As mentioned in Section 3, the boundedness will be verified in the (z, ξ)-coordinates.

By consequence, we have to check if

-τ1 τ1 z 2 -τ 2 1 δ 1 ξ 2 -k τ1 τ2 δ 2 ξ 2 + δ 2 v(t) τ 2 1
, stays bounded for every t ≥ τ -1 2 (t 0 ). System (29) can be written in the (z, ξ)coordinates as follows

           ż1 (t) = z 2 (t) ż2 (t) = δ 2 v(t) ξ(t) = -τ1 (t)ξ(t) -τ1 (t)z 1 (t) y(t) = z 1 (t), ∀t ≥ t 0 .
Let v be as in equation (23), thus guaranteeing that the state (z 1 (t), z 2 (t)) converges asymptotically to zero when t tends to +∞. Since τ1 is bounded over [0, +∞), it follows that w(t) = -τ1 (t)z 1 (t) (which is seen as an external exogenous input of the zero-dynamics ξ(t) = -τ1 (t)δξ(t) -w(t), for t ≥ t 0 ) tends to 0 when t tends to +∞. Knowing that the zero-dynamics is uniformly input-tostate stable (because the unforced zero-dynamics ξ(t) = -τ1 (t)ξ(t) has a globally exponentially stable equilibrium point at ξ = 0 and G(ξ, z 1 , t) = -τ1 (t)ξ -τ1 (t)z 1 is continuously differentiable and globally Lipschitz, uniformly in t, see [START_REF] Khalil | Nonlinear systems[END_REF]), then ξ(t) tends to 0 when t tends to +∞. Since (C3) is valid for any arbitrary value of µ, by Proposition 1, we have that x stays bounded for t ≥ t 0 . It follows that (33) is bounded in the x-coordinates as well and, finally, that (C5) holds.

Example 2 (not satisfying the causality condition (C4)). Consider the following time-varying delay system

ẋ1 (t) = x 2 2 (t)u(t) ẋ2 (t) = u(t) ẋ3 (t) = x 1 (t) h(t) = x 3 (t) + δ 4 x 2 (t), (34)
for which x 0 ∈ C([-4 θ, 0], R 3 ) and u 0 ∈ U are given. We will show that (34) is not input-output linearizable via a causal and bounded feedback and that it is not in conformity with Theorem 1. We have g 0 = (x 2 2 , 1, 0) T , g 1 ≡ 0, f = (0, 0, x 1 ) T and the δ-polynomial is given by

a(δ] = τ4 δ 4 . ( 35
)
Thus the feedback u has to verify the following relation

x 1 (t) + τ4 (t)δ 4 u(t) = δ 4 v(t), t ≥ τ -1 4 (0). ( 36
)
It is clear that condition (C4) is violated since ∂L f h/∂x 1 = 1 = 0 and the causality problem becomes now clear. Indeed, the computation of u(t) from (36) depends on the future of x 1 . However, from the first equation of (34), we see that u acts instantaneously on x 1 which prevents us to assign u(t) computed by (36) to (34).

Example 3 (not satisfying the bounded feedback condition (C5)). This example shows that if inequality (19) of condition (C5) is not satisfied, then even for a very simple choice of a bounded v, there is no bounded feedback u satisfying [START_REF] Hunt | Linear equivalents of nonlinear time varying systems[END_REF]. For simplicity, we suppose that the delay is constant and equal to 1.

Consider ẋ1 (t) = x 3 (t) -2x 2 (t) ẋ2 (t) = (1 + x 2 1 (t))δ 1 u(t) ẋ3 (t) = (1 + x 2 1 (t))u(t) h(t) = x 1 (t), (37)
with x(0) = 0 and u(t) = 1, for t ∈ [-θ, 0]. Notice that in this example only the control is delayed. We have

a(δ]u(t) = (1 + x 2 1 (t))u(t) -2(1 + x 2 1 (t))δ 1 u(t). (38)
Remark from the latter equation, that [START_REF] Oguchi | Input-output linearization of non-linear systems with time-delays in state variables[END_REF] is not satisfied. Now, suppose that we want to stay at x 1 (t) = 0 for all t ≥ 0. Hence u has to verify

u(t) -2δ 1 u(t) = 0, t ≥ 0, (39)
i.e., in this case v = 0. It follows from (39) that, over [0, θ], the control u should be equal to 2. Similarly, we should have u(t) = 4 over [ θ, 2 θ]. By repeating this reasoning, we find recursively that over the interval [n θ, (n + 1) θ], the control u should be equal to 2 n+1 . Then, there is no bounded u satisfying (39).

Example 4 (Non-necessity of conditions (C3) and (C7)). The non-necessity of condition (C7) can be easily seen through Example 1. Indeed, even if we cannot express δ 1 x as a pure function of (z, ξ), like in expression [START_REF] Quadrat | Algebraic analysis for the Ore extension ring of differential time-varying delay operators[END_REF] required by (C7), we can always write the (z, ξ)-dynamics. In fact, condition (C7) can be relaxed by supposing that only the x-states which appear in the dynamics of ξ should respect [START_REF] Quadrat | Algebraic analysis for the Ore extension ring of differential time-varying delay operators[END_REF]. Concerning condition (C3) we give the following example.

For simplicity, we consider a constant-delay nonlinear system:

ẋ1 (t) = 2δ 1 x 2 (t) + δ 1 u(t) ẋ2 (t) = arctan(x 1 (t)) -u(t) y(t) = x 1 (t) + δ 1 x 2 (t), ∀t ≥ 0,
with initial conditions x 0 ∈ C([-θ, 0], R 2 ) and u 0 ∈ U. We have g 0 = (0, -1) T ,

g 1 = (1, 0) T , f = (2δ 1 x 2 , arctan(x 1 )) T . A straightforward computation shows that L g0 h = L g1 h = 0, L f h = arctan(δ 1 x 1 ) + 2δ 1 x 2 , a(δ] = L g0 L f h + L g1 L f h = -2δ 1 + 1 1+(δ 1 x1) 2 δ 2 and L 2 f h = 2 arctan(δ 1 x 1 ) + 2 δ 2 x2 1+(δ 1 x1) 2 .
Condition (C1) is satisfied, with ρ = 2, the feedback u has to verify the following relation

-2δ 1 u(t)+ 1 1 + (δ 1 x 1 ) 2 δ 2 u(t) = -2 arctan(δ 1 x 1 )-2 δ 2 x 2 1 + (δ 1 x 1 ) 2 +δ 1 v(t), t ≥ 3 θ, ( 40
) and it is immediate that the causality condition (C4) is valid. The new coordinates

z 1 = x 1 + δ 1 x 2 , z 2 = arctan(δ 1 x 1 ) + 2δ 1 x 2 , ∀t ≥ t 0 ,
satisfies condition (C2) with = 1. Remark that from the above two equations, we can never express δ 1 x in function of δz(t), δ -z(t) and δ ≥2 x(t), thus condition (C3) is not satisfied. Finally, notice that even if (C3) does not hold, the boundedness condition (C5) is satisfied for this example and its verification is obvious.

Proofs

Proof of Theorem 1

We assume that system (5) satisfies conditions (C1)-(C5). Introduce, as explained when stated (C2), the new variables z i = L i-1 f h, for 1 ≤ i ≤ ρ. For t ≥ t 0 , we obtain:

żi = z i+1 , 1 ≤ i ≤ ρ -1, żρ = L ρ f h + L g0 L ρ-1 f hu + L g1 L ρ-1 f hδ 1 u = L ρ f h + a(δ]u(t), System (5) is input-output linearizable if the feedback u verifying a(δ]u(t) = -L ρ f h(δx(t), t) + δ j v(t), t ≥ τ -1 j (t 0 ),
is bounded and causal. Those properties are guaranteed by the two lemmas stated and proven below. According to Lemma 1, if condition (C5) is verified, then the feedback u stays bounded when property (18) is satisfied. Lemma 2 states that (C4) is actually a sufficient condition for the causality of u. Therefore, by applying [START_REF] Hunt | Linear equivalents of nonlinear time varying systems[END_REF], the input-output map becomes z (ρ) = h (ρ) = δ j v(t) and systems ( 5) is thus input-output linearizable by a causal and bounded feedback. If ρ = n, then the system is fully input-output linearizable. If, moreover, j = 0, it is fully input-output linearizable without delay (with delay, otherwise).

Lemma 1. Suppose that condition (C5) holds. Then, for every t ≥ t 0 , we have

u(t) ≤ 1 c -1 sup τm+1-j (t0)≤s≤t0 u(s) + c c -1 sup s≥τ -1 j (t0)
-L ρ f h(δx(s), s) + δ j v(s) a j (δx(s), s) .

Proof. Throughout the proof, by abuse of notation, we write a i (t) (respectively L ρ f (h)(t)) instead of a i (δx(t), t), for 0 ≤ i ≤ m + 1, (respectively L ρ f h(δx(t), t)). If the minimal degree j of the δ-polynomial a(δ] is such that j ≥ 1, we start by introducing a new time-scale t that allows us to shift the time in such a way that δ j u(t) becomes u( t). In order to do that, remark that τ i (•) is a strictly increasing function of t, for 1 ≤ i ≤ m + 1. Indeed, it is easy to verify that τi+1 (t) = τi (t)(1 -θ(τ i (t))), for 1 ≤ i ≤ m + 1, from which we can deduce, by induction, that τi+1 (t) > 0, for every t ≥ 0. This allows us to define a new time-scale t = τ j (t), for t ≥ 0.

By introducing t = τ j (t), equation [START_REF] Hunt | Linear equivalents of nonlinear time varying systems[END_REF] reads

ãj ( t)u( t) + • • • + ãm ( t)δ m+1-j u( t) = -Lρ f h( t) + v( t), ∀ t ≥ t 0 , (41)
where Lρ

f h( t) = L ρ f h(τ -1 j ( t)) and ãi ( t) = a i (τ -1 j ( t)), for j ≤ i ≤ m + 1, and condition (19) becomes sup t≥t0 ãi ( t) ãj ( t) ≤ 1 c(m + 1 -j) , j < i ≤ m + 1. (42)
From the assumptions made on the delay function θ(•) (in particular, θ(t) > 0 and θ(t) < 1, for every t ≥ 0), one can easily prove (the proof is left to the reader) the existence of a strictly increasing sequence (T k ) k≥0 such that

T 0 = t 0 , T k = T k+1 -θ(T k+1 ), ∀k ≥ 0,
and T k → +∞ with k.

The idea of the proof of Lemma 1 is to give a bound for u( t), for t belonging to the sequence of intervals [T k , T k+1 ], and, show that, when k goes to infinity, this bound is finite. Denote

M = sup t≥t0 -Lρ f h( t) + v( t) ãj ( t) . Since τ i , 1 ≤ i ≤ m + 1
, is an increasing function (as we have just proven), over [T 0 , T 1 ], we have (recall that

T 0 = T 1 -θ(T 1 ) = τ 1 (T 1 )): τ 1 ( t) ∈ [τ 1 (T 0 ), τ 1 (T 1 )] = [τ 1 (T 0 ), T 0 ],
and we can easily verify that

τ i ( t) ∈ [τ i (T 0 ), τ i (T 1 )] ⊂ [τ i (T 0 ), T 0 ], ∀1 ≤ i ≤ m + 1 -j.
Then, from relation (41), it follows that, for every t ∈ [T 0 , T 1 ], we have

u( t) ≤ ãj+1 ãj u(τ 1 ( t)) + • • • + ãm+1 ãj u(τ m+1-j ( t)) + M ≤ ãj+1 ãj + • • • + ãm+1 ãj sup τm+1-j (T0)≤s≤T0 u(s) + M.
Now, recall that according to our assumptions, there exists a constant c > 1 such that relation (42) holds for all t ≥ t 0 , thus for t ∈ [T 0 , T 1 ] also. Hence,

u( t) ≤ 1 c sup τm+1-j (T0)≤s≤T0 u(s) + M.
The latter inequality allows to estimate u( t) over [T 0 , T 1 ]. As above, over [T 1 , T 2 ], we have τ

1 ( t) ∈ [τ 1 (T 1 ), τ 1 (T 2 )] = [τ 1 (T 1 ), T 1 ],
and, in the general case,

τ i ( t) ∈ [τ i (T 1 ), τ i (T 2 )] ⊂ [τ i (T 1 ), T 1 ], for all 1 ≤ i ≤ m + 1 -j. Then, for every t ∈ [T 1 , T 2 ], we obtain u( t) ≤ ãj+1 ãj + • • • + ãm+1 ãj sup τm+1-j (T1)≤s≤T1 u(s) + M ≤ 1 c sup τm+1-j (T0)≤s≤T1 u(s) + M ≤ 1 c sup τm+1-j (T0)≤s≤T0 u(s) + sup T0≤s≤T1 u(s) + M ≤ 1 c 1 + 1 c sup τm+1-j (T0)≤s≤T0 u(s) + 1 + 1 c M.
By an induction argument, for k ≥ 2, we deduce that over [T k , T k+1 ], we have

u( t) ≤ 1 c i=k i=0 1 c i sup τm+1-j (T0)≤s≤T0 u(s) + M i=k i=0 1 c i .
Recall that the constant c is such that c > 1. Thus, for all t ≥ t 0 we have

u( t) ≤ 1 c -1 sup τm+1-j (T0)≤s≤T0 u(s) + c c -1 M.
Lemma 2. If condition (C4) holds then the feedback u, solution of (12), is causal.

Proof. Let us first suppose that g 0 = 0. Applying the operator δ -j on both sides of (12), gives

(δ -j a j (t))u(t)+• • •+(δ -j a m (t))δ m+1-j u(t) = -δ -j L ρ f h(t)+v(t), ∀t ≥ t 0 . ( 43
)
As for the proof of Lemma 1, for simplicity of notation, we omit the state dependency of the coefficients a i as well as of L ρ f h. From equation (43), the input that we have to assign to system (5) is given by

u(t) = -δ -j L ρ f h(t) + v(t) δ -j a j (t) - m+1 k=j+1 δ -j a k (t) δ -j a j (t) δ k-j u(t), ∀t ≥ t 0 .
We see clearly, that condition (17) avoids the existence of advanced states in the later equation, and then no causality problem can appear in this case.

Let us now assume that g 0 ≡ 0 (and g 1 = 0). In this case, we have to assign δ 1 u(t) to system (5), for t ≥ t 0 . Thus, applying the operator δ -j+1 on both sides of (12), we obtain that

δ 1 u(t) = -δ -j+1 L ρ f h(t) + δ 1 v(t) δ -j+1 a j (t) - m+1 k=j+1 δ -j+1 a k (t) δ -j+1 a j (t) δ k-j+1 u(t), ∀t ≥ τ -1 1 (t 0 ).
Condition [START_REF] Khalil | Nonlinear systems[END_REF] avoids the existence of advanced states in the later equation.

Proof of Proposition 1

Let (T k ) k≥0 be the time sequence introduced in the proof of Lemma 1. Since τ i is an increasing function, for i ≥ 0, then over [T 0 , T 1 ] we have

τ i (t) ∈ [τ i (T 0 ), τ i (T 1 )] ⊂ [τ q(ρ+1) (T 0 ), T 0 ], ∀ + 1 ≤ i ≤ q(ρ + 1),
where the integer is defined by (C2). Knowing that over [τ q(ρ+1) (T 0 ), T 0 ] we have x(t) < µ, then, thanks to inequality ( 16) of (C3), for every t ∈ [T 0 , T 1 ], we have:

δ x(t) ≤ α 1 (δz(t), δ -z(t) + α 2 δξ(t), δ -ξ(t) + α 3 δ ≥ +1 x(t) .
From which we obtain, knowing that sup

t≥τ q(ρ+1) (T0) { z(t) , ξ(t) } < ε,
where ε is given by ( 20), together with the fact that the maximum delay-advance order in (z, ξ) cannot exceed q(ρ + 1) (see the discussion of condition (C3) after the statement of Theorem 1), that δ x(t) < (2q(ρ + 1) + 1) max{α 1 , α 2 }ε + α 3 (q(ρ + 1) -)µ.

Hence, δ x(t) < µ, ∀t ∈ [T 0 , T 1 ].

Using the same reasoning, we can straightforwardly prove that δ x(t) < µ over [T k , T k+1 ], for every k ≥ 1. Therefore, δ x(t) = x(τ (t)) < µ, for t ≥ t 0 , and since τ (t) < t, it follows immediately that the trajectories of system (5) lie inside the ball B(0, µ) for every t ≥ t 0 .

Proof of Proposition 3

The proof is inspired from [START_REF] Germani | Local asymptotic stability for nonlinear state feedback delay systems[END_REF]. Let λ 1 , • • • , λ ρ be the eigenvalues of the matrix A + BK associated to (24). Recall that we have supposed that all of them are different. In addition, assume, without any loss of generality, that λ 1 is the largest eigenvalue. Let P be a change of basis matrix diagonalizing A + BK and denote M = P P -1 + 1. Let ε 1 be any positive constant such that max{M ε 1 , β(ε 1 , 0) + γ(M ε 1 )} ≤ 1 -α 3 (q(ρ + 1) -) (2q(ρ + 1) + 1) max{α 1 , α 2 } µ, where l, µ, α 1 , α 2 and α 3 are given by conditions (C2)-(C3) and β(•) and γ(•) are given by Definition 7 applied to the zero-dynamics (which is assumed to be uniformly input-to-state stable). For t ≥ τ -1 j (t 0 ), the dynamics of the z-subsystem is given by ż = (A + BK)z and is governed by a Hurwitz matrix. Thus, for t ≥ τ -1 j (t 0 ), we have z(t) = e (A+BK)(t-τ -1 j (t0)) z(τ -1 j (t 0 )) = P e D(t-τ -1 j (t0)) P -1 z(τ -1 j (t 0 )),

where D is the diagonal matrix associated to A + BK, and we deduce z(t) ≤ (M -1)e λ1(t-τ -1 j (t0)) z(τ -1 j (t 0 )) ≤ (M -1) z(τ -1 j (t 0 )) , ∀t ≥ τ -1 j (t 0 ). Now recall (as explained at the end of Section 2) that the z-subsystem is in free evolution over [t 0 , τ -1 j (t 0 )] and the feedback v(t) = ρ i=1 k i z i (t) starts acting on the system from the instant τ -1 j (t 0 ). Thus, for every 1 ≤ i ≤ q(ρ + 1), the delayed z-states verify the following inequality

δ i z(t) ≤ max    sup [τ q(ρ+1) (t0),τ -1 j (t0)]
z(t) , (M -1) z(τ -1 j (t 0 ))

   , ∀t ≥ t 0 .
Since, by hypothesis, z(t) < ε 1 over [τ q(ρ+1) (t 0 ), τ -1 j (t 0 )], then

δ i z(t) < M ε 1 , ∀t ≥ t 0 ,
for all -q(ρ + 1) ≤ i ≤ q(ρ + 1). By consequence, for every t ≥ t 0 , we have (δz(t), δ -z(t)) = z(t), δ 1 z(t), • • • , δ q(ρ+1) z(t), δ -1 z(t), • • • , δ -q(ρ+1) z(t)

≤ (2q(ρ + 1) + 1) max -q(ρ+1)≤i≤q(ρ+1) δ i z(t)

≤ (2q(ρ + 1) + 1)M ε 1 .

(44) On other hand, since the zero-dynamics is input-to-state stable for the (exogenous) input w(t) = (δz, δ -z), according to Definition 7, for every t ≥ t 0 , we have ξ(t) ≤ β( ξ t0 ∞ , t -t 0 ) + γ( z [τ q(ρ+1) (t0),t) ∞ )

≤ β(ε 1 , 0) + γ(M ε 1 ). (45)

Thus, for every 1 ≤ i ≤ q(ρ + 1), the delayed ξ-states verify the following inequality for all t ≥ t 0

δ i ξ(t) ≤ max{ ξ t0 ∞ , β( ξ t0 ∞ , 0) + γ( z [τ q(ρ+1) (t0),t) ∞ )} ≤ max{ε 1 , β(ε 1 , 0) + γ(M ε 1 )}. ( 46
)
By consequence, for every t ≥ t 0 , we have (δξ(t), δ -ξ(t)) = ξ(t), δ 1 ξ(t), • • • , δ q(ρ+1) ξ(t), δ -1 ξ(t), • • • , δ -q(ρ+1) ξ(t)

≤ (2q(ρ + 1) + 1) max{ε 1 , β(ε 1 , 0) + γ(M ε 1 )}.

(47) Consider the same time intervals (T k ) k≥0 as in Lemma 1 and using the same reasoning as in the proof of Proposition 1, we have (thanks to condition (C3)) that for every t ∈ [T 0 , T 1 ] δ x(t) ≤ α 1 (δz(t), δ -z(t) + α 2 δξ(t), δ -ξ(t) + α 3 δ ≥ +1 x(t) ≤ (2q(ρ + 1) + 1) max{α 1 , α 2 } max{M ε 1 , β(ε 1 , 0) + γ(M ε 1 )} +α 3 (q(ρ + 1) -)µ < µ.

Using the same reasoning, we can straightforwardly prove that δ x(t) < µ over [T k , T k+1 ], for every k ≥ 1. Thus, under the assumption given by ( 16), the trajectories of system (5) lie inside the ball B(0, µ) for every t ≥ τ (t 0 ). Thus, lim Using the fact that α 3 < 1 (recall that α 3 is such that 0 < α 3 < 1/(q(ρ + 1) -), see condition (C3)), we have lim t→+∞ x(t) = 0.

Conclusion

In this paper, we have considered the problem of input-output linearization of single-input single-output nonlinear systems with time-varying delays appearing in the state, the input and the output. Sufficient conditions have been developed in order to guarantee the existence of a causal and bounded feedback linearizing the input-output map of the feedback modified system. We have discussed and presented sufficient conditions ensuring the stability of the feedback modified system. In a future work, the input-output linearization and decoupling of multi-input multi-output systems will be studied.

  t→+∞ x(t) ≤ µ. Knowing the exponential stability of the z-subsystem, we have lim t→+∞ z(t) = 0. On the other hand, from the input-to-state stability of the zero dynamics, lim t→+∞ ξ(t) = 0. Thus, lim t→+∞ x(t) ≤ α 3 lim t→+∞ x(t) .

This assumption could be weaken by supposing that the delay function has an observable dynamics.


Jonatan Alavrez Muños

Nicolas Marchand

J F Guerrero-Castellanos

Jose Juan Téllez-Guzmán

Juan Antonio Escareno

Micky Rakotondrabe

J U Alvarez-Muñoz

J J Téllez-Guzmán

Rotorcraft with a 3DOF Rigid Manipulator: Quaternion-Based Modeling and Real-time Control Tolerant to Multi-Body Couplings

Keywords: Observer-based control, Quaternion and Newton-Euler Modeling, Bounded-input control, Aerial Manipulator

come

Introduction

Aerial manipulation has been an active area of research in recent times for Unmanned Aerial Vehicles (UAVs) since it increases the application in both the military and civilian sectors. Unlike fixed-wing UAV configurations, whose flight profile lacks hovering flight, VTOL (Vertical Take-Off and Landing) rotorcrafts with three, four, six, or eight rotary propellers are well-suited for aerial manipulation operations. However, numerous scientific challenges, technological and theoretical, remain open. The main issue arises from their limited payload capacity. Thus, alternatively, multiple robots can carry heavier payloads using cables or grippers [START_REF] Mellinger | Cooperative Grasping and Transport using Multiple Quadrotors[END_REF] which feature light and dexterous end-effectors. Furthermore, the dynamics of the robot is significantly altered while shifting and/or carrying payloads, due to the center of gravity shifting and external disturbances. Indeed, this is also an attraction in assembly because aerial robots can use this to sense disturbance forces and moments, as in [START_REF] Mohammadi | Cooperative aerial tele-manipulation with haptic feedback[END_REF]. Moreover, the performance of the aerial manipulation task relies on effective estimation of dynamic couplings for compensation purposes. Numerous approaches have been proposed to deal with such problems. In [START_REF] Orsag | Lyapunov based model reference adaptive control for aerial manipulation[END_REF], a Lyapunov based model Reference Adaptive Control is used to stabilize a quadrotor with a multi degree of freedom (DoF) manipulator. However, the stability analysis is carried out with a linear approach and only the dynamics of the quadrotor was concerned due to the Regular paper This work was supported in part by CONACYT-Mexico, LabEx PERSYVAL-Lab (ANR-11-LABX-0025) and Equipex ROBOTEX (ANR-10-EQPX-44-01). complexity of the system. [START_REF] Jimenez-Cano | Control of an aerial robot with multi-link arm for assembly tasks[END_REF] presents a Newton-Euler approach to model and control a quadrotor through a Variable Parameter Integral Backstepping (VPIB) approach. However, the parametrization of the system is made through Euler angles, which present attitude estimation singularities. [START_REF] Kim | Aerial manipulation using a quadrotor with a two DOF robotic arm[END_REF] presents aerial manipulation using a quadrotor and a 2-DoF robot arm. The dynamic model of the system is obtained by the Euler-Lagrange formulation. Then, an adaptive sliding mode controller is designed. The effectiveness of the proposed method is showed experimentally by picking up and delivering an object. In [START_REF] Thomas | Toward autonomous avian-inspired grasping for micro aerial vehicles[END_REF] the problem is solved through an autonomous avianinspired grasping method, however the design of the controller is formulated in the vertical plane, which supposes a limitation to the full 3d space. [START_REF] Sun | Amplitude-Saturated Nonlinear Output Feedback Antiswing Control for Underactuated Cranes With Double-Pendulum Cargo Dynamics[END_REF] presents an amplitudesaturated nonlinear strategy for underactuated cranes with double-pendulum dynamics. Such a strategy is twice validated: theoretically and practically. The experimental platform consists of a car and a double-pendulum crane, where the coupling between the two systems (car and pendulum) is taken into account and a dynamic model is obtained. However, compared to an aerial vehicle, the attitude and position dynamics must be taken into account. Finally, in [START_REF] Yuksel | A Differential flatness and control of protocentric aerial manipulators with any number of arms and mixed rigid-/elastic-joints[END_REF] a new class of aerial manipulator is presented. It consists in a PVTOL (Planar Vertical Take-Off and Landing) equipped with parallel manipulator arms attached to the Center of Mass (CoM) of the aerial vehicle, which is called protocentric. A control law has been proposed for the case of rigid joints and validated through simulations.

The contribution of the present paper is centered on a strategy that combines an alternative torque compensation approach with a nonlinear control design. Specifically, a mathematical model is presented in detail using quaternions and it takes into account the coupling between the two systems, quadcopter and manipulator. Unlike the research previously cited, the design of the attitude control law uses the quaternion parametrization, which avoids the presence of singularities.

With quaternion parametrization one proposes a constructive control law for the attitude and position stabilization. First, the design of a smooth, almost globally asymptotical control law for attitude stabilization which takes into account the dynamics of the robotic arm, is carried out. After that, a globally asymptotical nonlinear controller for the translational dynamics is proposed. In general the control law is based in the usage of a sum of nested saturation functions in order to take into account the actuator's limitation. Real-time experimental results and a statistical study of the obtained results validate the proposed strategy.

The paper is structured as follows. In section 2, the attitude model of the quadcopter with the manipulator arm is presented. Then, the attitude control design is formulated in section 3. Section 4 gives a strategy to estimate the angular position of each link in the robot manipulator through a Luenberger observer. Section 5 is devoted to the design of the position control law. Section 6 presents the hardware setup, experimental results, as well as a statistical study. Finally, some conclusions are presented in section 7.

System Modeling

Unit quaternion and attitude kinematics

Consider two orthogonal right-handed coordinate frames: the body coordinate frame, B(x b , y b , z b ), located at the center of mass of the rigid body and the inertial coordinate frame, N (xn, yn, zn), located at some point in the space (for instance, the earth NED frame). The rotation of the body frame B with respect to the fixed frame N is represented by the attitude matrix

R ∈ SO(3) = {R ∈ R 3×3 : R T R = I, det R = 1}.
The cross product between two vectors ξ, ∈ R 3 is represented by a matrix multiplication [ξ × ] = ξ × , where [ξ × ] is the well known skew-symmetric matrix.

The n-dimensional unit sphere embedded in R n+1 is denoted as S n = {x ∈ R n+1 : x T x = 1}. Members of SO(3) are often parameterized in terms of a rotation β ∈ R about a fixed axis ev ∈ S 2 by the map U : R × S 2 → SO(3) defined as

U(β, ev) := I3 + sin(β)[e × v ] + (1 -cos(β))[e × v ] 2 (1)
Hence, a unit quaternion, q ∈ S 3 , is defined as

q := cos β 2 ev sin β 2 = q0 qv ∈ S 3 (2)
where qv = (q1 q2 q3) T ∈ R 3 and q0 ∈ R are known as the vector and scalar parts of the quaternion respectively. The quaternion q represents an element of SO(3) through the map R : S 3 → SO(3) defined as

R := I3 + 2q0[q × v ] + 2[q × v ] 2 (3)
Remark 2.1 R = R(q) = R(-q) for each q ∈ S 3 , i.e. even quaternions q and -q represent the same physical attitude.

Denoting by ω = (ω1 ω2 ω3) T the angular velocity vector of the body coordinate frame, B relative to the inertial coordinate frame N expressed in B, the kinematics equation is given by

q0 qv = 1 2 -q T v I3q0 + [q × v ] ω = 1 2 Ξ(q) ω (4)
The attitude error is used to quantify mismatch between two attitudes. If q defines the current attitude quaternion and q d the desired quaternion, i.e. the desired orientation, then the error quaternion that represents the attitude error between the current orientation and the desired one is given by qe := q -1 d ⊗ q = (qe 0 q T ev ) T

where q -1 is the complementary rotation of the quaternion q which is given by q -1 := (q0 -q T v ) T and ⊗ denotes the quaternion multiplication [START_REF] Shuster | A survey of attitude representations[END_REF].

Model of a quadcopter carrying a manipulator arm

The attitude dynamics and kinematics for the quadcopter have been reported in many works e.g. [START_REF] Castillo | Real-time stabilization and tracking of a four-rotor mini rotorcraft[END_REF][START_REF] Guerrero-Castellanos | Bounded attitude control of rigid bodies: Real time experimentation to a quadrotor mini-helicopter[END_REF][START_REF] Alaimo | Mathematical modeling and control of a hexacopter[END_REF]. In these works the quadcopter mass distribution is considered to be symmetric. However, the mass distribution of a quadcopter with a manipulator is no longer symmetrical and varies with the movement of the arm. Consider a quadcopter with a manipulator arm with n links attached to its lower part. If the dynamics of the arm is neglected, the attitude kinematics and dynamics is given by

q0 qv = 1 2 Ξ(q) ω (6)
J ˙ ω = -ω × J ω + ΓT [START_REF] Sun | Amplitude-Saturated Nonlinear Output Feedback Antiswing Control for Underactuated Cranes With Double-Pendulum Cargo Dynamics[END_REF] where J ∈ R 3×3 is the symmetric positive definite constant inertial matrix of the rigid body expressed in the body frame B and ΓT ∈ R 3 is the vector of applied torques. ΓT depends on the (control) couples generated by the actuators, the aerodynamic couples such as gyroscopic couples, the gravity gradient or, as in the case of the present work, the couple generated by the movement of a robot manipulator placed under the body. Here, only the control couples, gyroscopic couples and the couple generated by the manipulator is considered in the control design. Consequently,

ΓT = Γ + Γarm + ΓG (8)
where Γ and ΓG will be described in section 2.3. On the other hand, the vector Γarm is the torque generated by the total propulsive force being applied at the quadcopter geometric center which is displaced from the center of mass.

Taking the robot manipulator as a physical pendulum attached to the fuselage of the aerial vehicle and following a similar process to the one in [START_REF] Sangbum | Dynamics and control of underactuated multibody spacecraft[END_REF], the manipulator torque can be computed by:

Γarm dyn = magζ cd × R(q)e3 (9)
where Γarm dyn is the manipulator torque taking into account the dynamics of each servomotor in the robot manipulator; ma = Σ n i=1 mmi + m l is the total mass of the manipulator plus the load and

ζ cd = (ζ cxd ζ cy d ζ cz d ) T ∈ R 3
is the position of center of mass of the quadrotor with respect to the pivot point. Then, the center of mass can be computed by

ζ cd = 1 ma n i=1 mmi id + m l ld ( 10
)
where id and ld are the position vectors of each link of the manipulator and the load, respectively, both with respect to the reference body frame given by the quadrotor. In this case, let us consider the scheme in Fig. 1, which shows an anthropomorphic arm manipulator. This system has three degrees of freedom and then, the corresponding id , where i = {1, 2, 3}, is given by

1d = [ 0 0 -lc1 ] T 2d = [lc2 sin θa2 m cos θa1 m lc2 sin θa2 m sin θa1 m -(l1 + lc2) cos θa2 m ] T 3d = [(l2 sin θa2 m + lc3 sin(θa2 m + θa3 m )) cos θa1 m (l2 sin θa2 m + lc3 sin(θa2 m + θa3 m )) sin θa1 m -(l1 + l2) cos θa2 m -lc3 cos(θa2 m + θa3 m )] T (11)
where lc1, lc2 and lc3 are the distances from the respective joint axes to the center of mass of each link, l1, l2 and l3 are the total length of the links, and θai m measures the angular displacement from z and x axes. Then, since servomotors are used as actuators for the manipulator arm, these can be easily considered as first order systems. For this, a parameter identification is performed in order to know the different constant values of the motors. In general, the found system has the form:

θai m = Ku(t) -θaim a ( 12
)
where θai m is the angular position of the servo output shaft, θaim is the angular velocity of the servo, a is a time constant linked to the time response of the servo, K is the gain of the system and u(t) is the input.

Actuator model

The collective input (or throttle input) is the sum of the thrusts of each rotor f1, f2, f3, f4. Therefore, the reactive couple Qj generated in the free air by rotor j due to the motor drag and the total thrust T produced by the four rotors can be, respectively, approximated by

Qj = ks 2 j (13) T = 4 j=1 fj = b 4 j=1 s 2 j ( 14
)
where sj represents the rotational speed of rotor j. k > 0 and b > 0 are two parameters which depend on the density of air, the radius, the shape, the pitch angle of the blade and other factors [START_REF] Alaimo | Mathematical modeling and control of a hexacopter[END_REF]. The vector of gyroscopic couples ΓG is a consequence of the simultaneous rotation of the structure of the quadrotor and the high-speed rotation of the actuators, and it is given by

ΓG = 4 j=1 Jr( ω × z b )(-1) j+1 sj ( 15
)
where Jr is the inertia of the so-called rotor (composed of the motor rotor itself with the gears). The components of the control torque Γ ∈ R 3 generated by the rotors are given by Γ = [Γ1 Γ2 Γ3] T , with

Γ1 = d(f3 -f4) = db(s 2 3 -s 2 4 ) (16) Γ2 = d(f1 -f2) = db(s 2 1 -s 2 2 ) (17) Γ3 = -Q1 -Q2 + Q3 + Q4 = k(-s 2 1 -s 2 2 + s 2 3 + s 2 4 ) ( 18
)
where d is the distance between the rotor and the center of gravity of the quadrotor.

3 Attitude Control Design

Problem statement

The objective is to design a control law which drives the quadcopter to attitude stabilization under the torques and moments exerted to this from the movement of a manipulator arm attached to its lower part. In other words, let q d denote the constant quadcopter stabilization orientation, then the control objective is described by the following asymptotic conditions

q → [±1 0 0 0] T , ω → 0 as t → ∞ (19)
Furthermore, it is known that actuator saturation reduces the benefits of the feedback. When the controller continuously outputs infeasible control signals that saturate the actuators, system instability may follow. Then, besides the asymptotic stability, the control law also takes into account the physical constraints of the control system, in order to apply only feasible control signals to the actuators.

Attitude control with manipulator arm

In this subsection, a control law that stabilizes the system described by ( 6) and ( 7) is proposed. The goal is to design a control torque that is bounded. Definition 3.1 Given a positive constant M , a continuous, nondecreasing function σM : R → R is defined by

(1)σM = s if |s| < M ; (2)σM = sign(s)M elsewhere. ( 20
)
Note that the components of Γarm i are always bounded, i.e. | Γarm i |< δi. Then, one has the following result.

Theorem 3.2 Consider a rigid body rotational dynamics described by ( 6) and ( 7) with the following bounded control inputs Γ = (Γ1 Γ2 Γ3) T such that

Γi = -σM i2 (Γarm i + σM i1 (λi[ωi + ρiqi])) (21)
with i ∈ {1, 2, 3} and where σM i1 and σM i2 are saturation functions. Assuming δi < Mi2 -Mi1 and Mi1 ≥ 3λiρi. λi and ρi are positive parameters. Then the inputs (21) asymptotically stabilize the rigid body to the origin (1 0 T 0 T ) T (i.e. q0 = 1, qv = 0 and ω = 0) with a domain of attraction equal to

S 3 × R 3 \ (-1 0 T 0 T ) T .
The proof of this Theorem is given in Appendix A.

Remark 3.3 Note that the stability analysis has been carried out considering the asymptotic condition q → q d = [±1 0 0 0] T . In the case where the asymptotic condition q → q d with q d = [±1 0 0 0] T is considered, the control law applied will be

Γi = -σM i2 (Γarm i + σM i1 (λi[ωi + ρiqe i ])) ( 22
)
where qe represents the attitude error between the current orientation and the desired one.

4 Manipulator links angular position estimation

Problem statement

The objective is to design a strategy for the estimation of the angles on each link in the manipulator arm, combining the data coming from the first order model of the manipulator actuators and the data coming from the end effector position tracked by the VICON system (motion capture system). Since the first order model does not fully describe the behavior of the arm manipulator (non-modeled dynamics, actuators malfunction, etc.). In order to know the angular positions of the manipulator links with respect to the base body (quadcopter), the inverse kinematics of the manipulator arm is used.

Manipulator links observer design

In this subsection a Luenberger observer is designed to estimate the angles on each link in the manipulator. For this, the end-effector position is computed through the arm inverse kinematics to know the angle on each link. The expression that describes a link angle is given by:

θaV = θa + µV ( 23
)
where θaV is the estimated angle computed with the inverse kinematics, θa is the real angle and µV is a noise of minimal value. In addition, the observer allows the computation of the angular velocity. The expressions that represent the observer are given by:

θδ = a θδ + Ku δ + L(θaV -θ) (24) θ = θV ( 25
)
where θδ is the estimated angle on a link in the manipulator, a and K are parameters of the first order system previously presented and L is a positive tuning parameter. Now, given the expression (24), where θδ and θδ were estimated, it is possible to compute the manipulator torque from [START_REF] Alaimo | Mathematical modeling and control of a hexacopter[END_REF] and use this new term as Γarm into the attitude control law (21).

Position Control Design

Problem statement

The objective is to design a control law which stabilizes the quadcopter to a desired position, thereby solving the attitude stabilization problem. In other words, once the control law has stabilized the attitude of the system, limt→∞(q, ω) = (q d , 0), the position control law should stabilize the quadcopter in a desired position, limt→∞( p, v) = ( p d , 0). This stabilization must be ensured even under the disturbances from the manipulator arm. The schematic representation of a quadcopter carrying a manipulator arm can be seen in Fig. 2, where the inertial reference frame N (xn, yn, zn), the body reference frame B(x b , y b , z b ), the force u (thrust) and the weight vector m g are depicted. The dynamics of the whole system is obtained with the Newton-Euler formalism and the kinematics is represented using the quaternions formalism, given by ΣT :

Position stabilization strategy

         ˙ p = v mT ˙ v = -mT g + R    0 0 u    (26) ΣO : q = 1 2 Ξ(q) ω J ˙ ω = -ω × J ω + ΓT ( 27
)
where p and v are linear position and velocity vectors, mT is the total mass of the system (the quadcopter and the manipulator), g is the acceleration due to gravity, and R is the rotation matrix, given in (3). Note that the rotation matrix R can also be given as a function of the Euler angles, that is

R(φ, θ, ψ) =    Cψ Cθ Sψ Cθ -Sθ Cψ Sθ Sφ -Sψ Cθ Sφ Sθ Sψ + Cψ Cφ Cθ Sφ Cψ Cφ Sθ + Sψ Sφ Sθ Sψ Cφ -Cψ Sφ Cθ Cφ    , (28)
Taking into account the equations ( 26) and (27), this system can be seen as a cascade system, where the translational dynamics (26) depend on the attitude ( 27), but the attitude dynamics do not depend on the translational one. This property will be used to design the control law. Now, assume that using the control law (21) one can stabilize the yaw dynamics, that is ψ = 0, then after a sufficiently long time, system (26) becomes:

   ṗx ṗy ṗz    =    vx vy vz    , (29)
   vx vy vz    =    -u m T sin θ u m T sin φ cos θ u m T cos φ cos θ -g    (30)
With an appropriate choice of target configuration, it will be possible to transform (29)-(30) into three independent linear triple integrators. For this, take

φ d := arctan r2 r3 + g , θ d := arcsin -r1 r 2 1 + r 2 2 + (r3 + g) 2 (31)
where r1, r2 and r3 will be defined after. Then, choose as positive thrust the input control

u = mT r 2 1 + r 2 2 + (r3 + g) 2 (32)
Let the state be p = (p1, p2, p3, p4, p5, p6, p7, p8, p9) = ( px, px, vx, py, py, vy, pz, pz, vz), then (29)-(30) becomes:

Σx :      ṗ1 = p2 ṗ2 = p3 ṗ3 = r1 Σy :      ṗ4 = p5 ṗ5 = p6 ṗ6 = r2 Σz :      ṗ7 = p8 ṗ8 = p9 ṗ9 = r3 (33)
Note that u will be always positive, and u ≥ mg, in order to compensate the system's weight. Since the chains of integrators given in (33) have the same form, a control law can be proposed as in [START_REF] Cruz-José | Estabilización global de mini naves aéreas tipo VTOL[END_REF], and can be established by the next theorem: Theorem 5.1 Consider the quadcopter translational dynamics expressed in (29-30). Then, the thrust input u given by (32) with r1, r2, r3 as in (34), where σM 1 (•) is defined in [START_REF] Alvarez-Munoz Received The | Electronics, from the Autonomous University of Puebla, Mexico and the Ph.D. degree in 2017 in Automatic Control from GIPSA-Lab from the University of[END_REF] with M1 = 1 and ςi are given by ( 35 Furthermore, if none of the σM 1 are saturated, the poles of the linearized closed-loop for the subsystems (33) reside at b,c)1,b,c)2,b,c)3, respectively.

r1 := -ς1{a3σM1[ 1 ς1 (a2p1 + p2 + p3)] + a2σM1[ 1 ς1 (a1p2 + p3)] + a1σM1[ 1 ς1 (p3)]}, r2 := -ς2{b3σM1[ 1 ς1 (b2p4 + p5 + p6)] + b2σM1[ 1 ς2 (b1p5 + p6)] + b1σM1[ 1 ς2 (p6)]}, r3 := -ς3{c3σM1[ 1 ς1 (c2p7 + p8 + p9)] + c2σM1[ 1 ς3 (c1p8 + p9)] + c1σM1[ 1 ς3 (p9)]} (34) ς1 = r1/(a1 + a2 + a3), ς2 = r2/(b1 + b2 + b3), ς3 = r3/(c1 + c2 + c3) (35)
Due to space constraints, the proof of this Theorem is not presented here, but it can be easily derived from the seminal work of [START_REF] Marchand | Global stabilization of multiple integrators with bounded controls[END_REF], [START_REF] Teel | Global stabilization and restricted tracking for multiple integrators with bounded controls[END_REF] and [START_REF] Johnson | Nested saturation with guaranteed real poles[END_REF].

Remark 5.2 In the above Theorem, the stabilization goal is the origin. In the case where the asymptotic condition is different from the origin, the variables p2, p5, p8 should be replaced in the control law (34) by e1 = p2-p d

x , e2 = p5-p d y , e3 = p8 -p d z , respectively. In this case p d x , p d y , p d z represent the desired position in the space.

Experimental Validation

Hardware setup

In order to test the effectiveness of the proposed control law, a set of experiments were performed. The aerial system Fig. 3: Mini-quadcopter with its manipulator arm in flight. consists of a home-made quadcopter and arm manipulator, see Fig. 3. Both structures were specially designed and built for this project. The characteristics and parameters of each system are described in the TABLE 1. The total weight of the quadcopter with its arm is 315g and its carrying capacity is about 50g.

The attitude control law (21) for the quadcopter was programmed in a Microwii Copter board, which has gyros, accelerometers and the ATMega32u4 as processor. Then, a ground station estimates the position and attitude of the hexacopter using the Vicon Tracker system and T40s cameras [18]. With this system it is possible to compute the position and attitude up to 100Hz. The estimated states are sent to MATLAB/Simulink through a UDP frame every 2ms. The position control algorithm is implemented in real-time at 200Hz on a computer using xPC target toolbox [19]. The control variables are finally sent back to the quadcopter on the Microwii, through a GIPSA-lab's built-in bridge to DSM2 protocol. For this, a radio-frequency emitter is used. Furthermore, the manipulator arm is controlled by the DS35 Digital Super Sub-Micro Servo [START_REF] Alvarez-Munoz Received The | Electronics, from the Autonomous University of Puebla, Mexico and the Ph.D. degree in 2017 in Automatic Control from GIPSA-Lab from the University of[END_REF]. To transmit the control signals to the manipulator, Spektrum DSM2 transmitters are connected to the ground station through another built-in bridge. An overview of the whole hardware architecture is presented in Fig. 4.

Experimental scenario

A set of experiments is carried out in order to compare the performance of the proposed control with and without accounting for the torque generated by the manipulator, scenario 1 and scenario 2, respectively. The parameters of

Experimental results in scenario 1

Fig. 5 shows the general performance of the aerial system under the disturbances coming from the manipulator arm when the arm torque estimation is not taken into account.

The figure shows the angular and linear positions of the quadcopter during the experiment. Note that even when the quaternion parametrization is considered, Euler angles given in (28) are used in order to have a better perspective of the behavior of the system. In this case, attitude stabilization is guaranteed, but the movement of the manipulator causes stability issues for both attitude and linear position.

Experimental results in scenario 2

Fig. 6 shows the angular and linear positions of the quadrotor as well as the computed manipulator torques. The arm perturbations are compensated through the dynamic model and it results in a general stabilization improvement of the quadrotor compared to the precedent approach. In addition, the importance of this approach is that the precision of the angular position knowledge is enhanced, which guarantees a better torque estimation.

Results analysis and statistical study

In order to compare the obtained results of the proposed methods and the flight performance of the aerial vehicle under the disturbances coming from the movement of the manipulator arm, a statistical error study is carried out. For this, the experiments described before were performed 8 times.

Fig. 8, Fig. 9 and Fig. 7 show the linear position errors, the Normal Gaussian distribution errors and the attitude errors for scenarios 1 and 2, respectively.

In order to calculate the attitude error, 2 arccos q0 was used, where • represents the norm and q0 was defined as before. The obtained results show that the average attitude error when the dynamic method estimation is applied is reduced, compared with that obtained when the classical approach is implemented.

Since the attitude stabilization of the aerial robot is enhanced with the usage of the proposed method, then the linear position stabilization is equally improved, as it is shown in the linear position errors computation.

To have another perspective of the obtained results, the Normal Gaussian distribution errors are computed. For this, the linear position errors data were used. Then, from the obtained distributions it is clear that when the dynamic method estimation for the arm manipulator torque is used, the average error value is closer to 0. Furthermore, the area covered by the distributions when the classical approach is used is bigger, consequently, the probability of error increases.

In general, the stabilization of the system is improved and guaranteed with the use of the dynamic model and the nonlinear observer. Table 2 shows the different average error values for the set of experiments. The first column shows the attitude average value error, the second, third and fourth columns show the average error value for the x, y and z -axis. Since the experiment was repeated 8 times using the different approaches, this Table gives a better perspective of the stabilization improvement for both quadrotor attitude and linear positions, showing the effectiviness of the proposed approach.

Conclusions

In this paper, a novel model for a quadcopter carrying a manipulator arm was proposed. In addition, a control law Fig. 6: General behaviour of the system during the experiment using dynamic method estimation torque compensation and the nonlinear observer.

was designed to asymptotically stabilize the attitude and position of the system. Moreover, this work has presented a method for aiding the solution through the design of a feed-forward term which allows the estimation of the moments and torques exerted by the manipulator. Since input constraints exist in the actuators, the control law takes into account the actuator's saturations. Experimental results and a statistical study show the effectiveness of the proposed control law in facing the disturbances coming from the manipulator. In a future work, experimental mass estimation and outdoor picking up and delivery of an object will be pursued.

APPENDIX 8 Proof of the Theorem

Consider the candidate Lyapunov function V , which is positive definite. where J is defined as before, and κ > 0 must be determined. The derivative of (36) after using ( 6) and ( 7) is given by V = ω T J ˙ ω -2κ q0

V = 1 2 ω T J w + κ((1 -q0) 2 + q T v qv) = 1 2 ω T J w + 2κ(1 -q0) (36)
= ω T (-ω × J ω + Γ + Γarm + ΓG) + κq T v ω = w1(Γ1 + Γarm 1 ) + κq1ω1 The same argument is applied to V2 and V3, (37) becomes

V = V1 + V2 + V3 = -(λ1ω 2 1 + λ2ω 2 2 + λ3ω 2 3 ) ≤ 0 (47)
In order to complete the proof, the LaSalle Invariance principle is invoked. All the trajectories converge to the largest invariant set Ω in Ω = {(qv, ω) : V = 0} = {(qv, ω) : ω = 0}. In the invariant set, J ω = -[λ1ρ1q1 λ2ρ2q2 λ3ρ3q3] T = 0 that is, Ω is reduced to the origin. This ends the proof of the asymptotic stability of the closed loop system.

Fig. 1 :

 1 Fig. 1: Manipulator arm with three degrees of freedom.

Fig. 2 :

 2 Fig. 2: Schematic configuration of a quadrotor carrying a manipulator arm.

  ), a (1,2,3) , b (1,2,3) , c (1,2,3) > 0 tuning parameters such that (a, b, c)1 > (a, b, c)2 + (a, b, c)3, (a, b, c)2 >(a, b, c)3, stabilizes globally and asymptotically the quadcopter translational dynamics at the origin.

  the control laws are selected according to the characteristics of the quadrotor actuators and the condition for the manipulator arm. For the attitude control law (21) and the position control law given in (34), where max|Γarmi| = 0.0858N m and δi = 0.1, the next parameters values are proposed: M11,21,31 = 0.1, M12,22,32 = 0.5, λ1,2 = 0.015, λ3 = 0.013, ρ1,2 = 10.5 and ρ3 = 11. For the control (34), a1 = b1 = 2.3, c1 = 1.65, a2 = b2 = 1.2, c2 = 0.55, a3 = b3 = 0.1, c3 = 0.015 and r1,2,3 = 5. The experiments consist in two parts. First, the links of the manipulator arm are initialized to θai = (0 • 90 • 0 • ) T and the quadrotor is driven to the position p d = (0 0 1) T . At time 20s, θai = (40 • 30 • 0 • ) T . At time 25s, θai = (-30 • 70 • 25 • ) T . At time 30s, θai = (10 • 20 • 35 • ) T . Finally, at time 35s, θai = (0 • 90 • 60 • ) T and the quadrotor lands.

Fig. 4 :Fig. 5 :

 45 Fig. 4: Block diagram of the system.

Fig. 7 :

 7 Fig. 7: Attitude error and attitude average error value during the different experiments.

Fig. 8 :

 8 Fig. 8: Position errors during the different experiments.

Fig. 9 :

 9 Fig. 9: Error Gaussian Normal distributions.

V1+

  w2(Γ2 + Γarm 2 ) + κq2ω2 V2 + w3(Γ3 + Γarm 3 ) + κq3ω3 V3 (37) V is the sum of the three terms ( V1, V2, V3). First V1 is analyzed. From Γ1 in (21) and equation (37), one gets V1 = ω1(-σM 12 (Γarm 1 + σM 11 (λ1[ω1 + ρ1q1])) + Γarm 1 ) + κq1ω1 (38) if we choose δ1 < M12 -M11, σM 12 is always operating in its linear region so V1 becomes V1 = -ω1σM 11 (λ1[ω1 + ρ1q1]) + κq1ω1 (39) Assume that |ω1| > 2ρ1, that is ω1 ∈]2ρ1, +∞[. Since |q1| ≤ 1, it follows that |ω1 + ρ1q1| ≥ ρ1 + for any > 0 sufficiently small. Therefore, ω1 + ρ1q1 has the same sign as ω1. From equation (39) and the norm condition on the quaternion, V1 takes the following form V1 = -ω1σM 11 (λ1[ω1 + ρ1q1]) + κω1q1 ≤ -|ω1|σM 11 (λ1(ρ1 + )) + κ|ω1| (40) Taking κ < min(M11, λ1ρ1 + ) (41) one can assure the decrease of V1, i.e. V1 < 0. Consequently, ω1 enters Φ1 = {ω1 : |ω1| ≤ 2ρ1} in finite time t1 and remains in it thereafter. In this case, (ω1 + ρ1q1) ∈ [-3ρ1, 3ρ1]. Let M11 verify the next inequality M11 ≥ 3λ1ρ1, equation (41) then becomes: κ < λ1ρ1 + (42) For t2 > t1, the argument of σM 11 will be bounded as follows |λ1(ω1 + ρ1q1)| ≤ 3λ1ρ1 ≤ M11 (43) Consequently, σM1 operates in a linear region Γ1 = -λ1[ω1 + ρ1q1]

Table 1 :

 1 Characteristics and parameters of the nanohexacopter and the manipulator.

	System	Description	Value	Units
		Mass (m)	280	g
		Distance (d)	10.7	cm
		Battery	7.4	V
		Carrying capacity	80	g
	Quadcopter	Inertial moment x (J φ )	0.0056	Kg • m 2
		Inertial moment y (J θ )	0.0056	Kg • m 2
		Inertial moment z (J ψ )	0.0087	Kg • m 2
		Constant (b)	2615.23	N/s
		Constant (k)	257.80	N/s
		Mass manipulator ma	35	g
	Manipulator	Length 1st link l1	5	cm
		Length 2nd link l2	5	cm
		Length 3rd link l3	8.4	cm

Table 2 :

 2 Average error values for the experiments.

	Average error value	Orientation	Pos x	Pos y	Pos z
		1.5312	0.0375	0.0777	0.0468
		1.5411	0.0361	0.0762	0.0690
	Scenario 1	1.4901	0.0341	0.0681	0.0335
		1.4320	0.0683	0.0728	0.0552
		1.3941	0.0525	0.0679	0.0592
		1.4480	0.0507	0.0538	0.0347
		1.4175	0.0489	0.0674	0.0404
		1.4520	0.0542	0.0702	0.0532
	Total average error	1.4632	0.0477	0.0692	0.049
		0.6223	0.0294	0.0283	0.0230
		0.6034	0.0231	0.0344	0.0239
	Scenario 2	0.6197	0.0251	0.0323	0.0278
		0.5714	0.0214	0.0234	0.0229
		0.6524	0.0269	0.0289	0.0189
		0.6277	0.0247	0.0277	0.0245
		0.5998	0.0225	0.0239	0.0239
		0.6308	0.0279	0.0302	0.0246
	Total average error	0.6159	0.0251	0.0286	0.0236

International Journal of Automation and Computing X(X), X X