This paper addresses adaptive control of specific Euler-Lagrange systems: rigidbody attitude control, and the n-link robot manipulator. For each problem, the model parameters are unknown but the lower bound of the smallest eigenvalue of the inertia matrix is assumed to be known. The dynamic scaling Immersion and Invariance (I&I) adaptive controller is proposed to stabilize the system without employing a filter for the regressor matrix. A scalar scaling factor is instead implemented to overcome the integrability obstacle that arises in I&I adaptive control design. First, a filter-free controller is proposed for the attitude problem such that the rate feedback gain is proportional to the square of the scaling factor in the tracking error dynamics. The gain is then shown to be bounded through state feedback while achieving stabilization of the tracking error. The dynamic scaling factor increases monotonically by design and may end up at a finite but arbitrarily large value. However, by introducing three more dynamic equations, the non-decreasing scaling factor can be removed from the closed-loop system. Moreover, the behavior of dynamic gain is dictated by design parameters so that its upper bound is limited by a known quantity and its final value approaches the initial value. A similar approach for the dynamic gain design is also applied to a filter-dependent controller where a filter for the angular rate is utilized to build a parameter estimator. Unlike the filter-free design, the filter-dependent controller admits a constant gain for the rate feedback while the dynamic scaling factor rather appears in the filter. Finally, the proposed design is applied to robot manipulator systems. Spacecraft attitude and 2-link planar robot tracking problems are considered to demonstrate the performance of the controllers through simulations.

I. Introduction

In most nonlinear control problems, designing a satisfactory controller to force system states to track prescribed reference trajectories usually requires model parameters. Closed-loop performance depends largely on how well the mathematical model and the physical plant match. To improve the quality of the controllers, system identification can be performed to improve plant model before controller design, but it is not always perfect. Adaptive control has been extensively studied as a way of dealing with parameter uncertainties [START_REF] Åström | Theory and applications of adaptive control -a survey[END_REF][START_REF] Narendra | Stable adaptive systems[END_REF]. The philosophy of adaptive control is to control dynamic systems using system parameters estimated online so that robustness to parameter uncertainties automatically follows.

Classical adaptive control design obeys the so-called Certainty Equivalence (CE) principle.

First, a controller is designed with system parameters assumed to be known. Then the parameters are replaced by estimated values. A parameter update law is determined through the judicious use of a Lyapunov(-like) function such that a tracking objective is fulfilled. The estimate errors remain bounded as the error terms in the time derivative of the Lyapunov(-like) function are canceled by the update law. Departing from the CE philosophy, a new methodology called Immersion and Invariance (I&I) adaptive control was recently proposed promising better performance and flexibility of parameterization structure [START_REF] Astolfi | Immersion and invariance: a new tool for stabilization and adaptive control of nonlinear systems[END_REF][START_REF] Astolfi | Nonlinear and adaptive control with applications[END_REF]. Unlike CE, the estimates do not come directly from the update law. Instead, a function which satisfies a certain partial differential equation (PDE) is combined with the states of the update law to form the estimates. The norm of the estimation error monotonically decreases, and thus, the estimated parameters approach the true values as time evolves-a feature not observed in CE-based controllers. Also, the resulting estimate error dynamics allow us to dominate the perturbation term in the Lyapunov(-like) function derivative without the fragile cancellation operation. This new methodology allows non-affine parameterization for some applications and has been shown to outperform CE-based controllers in transient performance [START_REF] Ortega | Immersion and invariance adaptive control of linear multivariable systems[END_REF][START_REF] Astolfi | Two solutions to the adaptive visual servoing problem[END_REF][START_REF] Karagiannis | Two results for adaptive output feedback stabilization of nonlinear systems[END_REF][START_REF] Zachi | Dynamic control of uncertain manipulators through immersion and invariance adaptive visual servoing[END_REF][START_REF] Seo | High-performance spacecraft adaptive attitude-tracking control through attracting-manifold design[END_REF][START_REF] Seo | Non-certainty equivalent adaptive control for robot manipulator systems[END_REF][START_REF] Lee | Multi-input noncertainty-equivalent adaptive control of an aeroelastic system[END_REF].

When the I&I method was first proposed, applications were limited because designing an I&I adaptive controller typically leads to solving a partial differential equation (PDE) [START_REF] Astolfi | Immersion and invariance: a new tool for stabilization and adaptive control of nonlinear systems[END_REF][START_REF] Astolfi | Nonlinear and adaptive control with applications[END_REF]. The solution to the PDE always exists for autonomous single-input systems. Existence of solutions to such PDEs is not guaranteed for general multi-input systems-a condition commonly known as the "integrability obstacle". However, by introducing low-pass filters for the state and regressor matrix, this obstacle is shown to be circumvented. Various I&I adaptive controllers have been proposed for linear and nonlinear systems as a result [START_REF] Ortega | Immersion and invariance adaptive control of linear multivariable systems[END_REF][START_REF] Zachi | Dynamic control of uncertain manipulators through immersion and invariance adaptive visual servoing[END_REF][START_REF] Lee | Multi-input noncertainty-equivalent adaptive control of an aeroelastic system[END_REF].

Recently, an alternate method to avoid the integrability obstacle was proposed in [START_REF] Karagiannis | Dynamic scaling and observer design with application to adaptive control[END_REF]. Aided by a state filter with a dynamic scaling factor appearing in the gain, the PDE becomes solvable. A scaled manifold is proven to be globally stable via a certain choice of the scaling factor dynamics.

In [START_REF] Sonneveldt | Immersion and invariance based nonlinear adaptive flight control[END_REF] and [START_REF] Ji | Immersion and invariance based nonlinear adaptive control of hypersonic vehicles[END_REF], the dynamic scaling method is applied to aircraft flight control systems. In the special case where the system is monotonizable and parameters have known bounds, it is shown that the parameter estimates converge to the true values exponentially fast and the scaling factor returns to unity as the tracking error goes to zero [START_REF] Liu | On adaptive control of nonlinearly parameterized nonlinear systems: Towards a constructive procedure[END_REF]. Compared to the existing method that requires both state and regressor filters, this approach achieves similar performance as the lowerdimensional dynamic extensions.

It is possible to bring the dynamic gain down to the initial value when the dynamic scaling approach is used taking advantage of the structural properties like observability [START_REF] Karagiannis | Dynamic scaling and observer design with application to adaptive control[END_REF]. However, in most adaptive control design cases the scaling factor does not have a convergent term in its dynamics and grows monotonically to some unknown constant. This feature can be a drawback because we do not know how large the scaling factor becomes, although it is finite, or whether it can be reset. This could result in "high-gain" control action and cause undesirable transient behavior of the closed-loop system.

In the literature, many model reference adaptive controllers have been developed for the spacecraft attitude control and robot manipulator systems as their dynamic structures admit affineparameterization for the inertia parameters [START_REF] Junkins | Nonlinear adaptive control of spacecraft maneuvers[END_REF][START_REF] Schaub | Adaptive control of nonlinear attitude motions realizing linear closed loop dynamics[END_REF][START_REF] Costic | Quaternion-based adaptive attitude tracking controller without velocity measurements[END_REF][START_REF] Slotine | On the adaptive control of robot manipulators[END_REF][START_REF] Ortega | Adaptive motion control of rigid robots: A tutorial[END_REF][START_REF] Sadegh | Stability and robustness analysis of a class of adaptive controllers for robotic manipulators[END_REF]. Unfortunately, ideal I&I adaptive controllers do not exist as no solutions to the PDE of these systems exist. This is due to the nonlinear gyroscopic term in the momentum-level equations of motion. Only the filter approach was applied to the attitude control and robot manipulator systems respectively in [START_REF] Seo | High-performance spacecraft adaptive attitude-tracking control through attracting-manifold design[END_REF] and [START_REF] Seo | Non-certainty equivalent adaptive control for robot manipulator systems[END_REF] to overcome the integrability obstacle. In this work, we apply the dynamic scaling method to the quaternion-based attitude tracking control problem. Also, we modify the dynamic scaling factor so that it does not increase monotonically but instead converges to the vicinity of the initial value by introducing three scalar dynamics. Moreover, based on the local Lipschitz condition of the regressor matrix, a filter-free controller is proposed that achieves even lower dimensional dynamic extensions. Due to the structural analogy between the attitude and Euler-Lagrange dynamics, the proposed design scheme is also applied to n-link robot manipulator systems. The preliminary works are reported in [START_REF] Yang | Dynamic Scaling Based Adaptive Controllers for Aeroelastic Systems[END_REF][START_REF] Yang | Adaptive Control Boosted through Dynamically Adjusted Feedback Gains: Applications to Controlling Euler-Lagrange Systems[END_REF][START_REF] Yang | Dynamically Scaled Immersion and Invariance Approach for Spacecraft Attitude Tracking Control[END_REF]. The performances of the tracking controllers for both attitude and 2-link planar robot arm dynamics are demonstrated and compared through numerical simulations.

This paper is organized as follows: The governing equations of a rigid-body rotational motion are described in Section II. Both filter-free and filter-dependent controllers are proposed with stability proofs in Section III. In Section IV, the proposed design methods are shown to be applicable to a subset of Euler-Lagrange mechanical systems covering n-link robot manipulators. Numerical simulations of spacecraft attitude and 2-link planar robot controllers are carried out in Section V.

Finally, concluding remarks are presented in Section VI.

II. Model Description

A. Dynamic Equations

Rigid-spacecraft attitude motion is governed by Euler's rotational dynamics. Let J 2 R 3⇥3 be a positive definite matrix that represents the inertia of a spacecraft. In the body-fixed frame of the spacecraft, let !(t) 2 R 3 and u 2 R 3 be the angular rate and the external control torque respectively.

Then the governing equation is described as

J !(t) = S (!(t))J!(t) + u(t) , (1)
where S : R 3 ! R 3⇥3 is the skew-symmetric matrix operator such that S (v)w = v ⇥ w for any v, w 2 R 3 . We adopt the quaternion vector

q(t) 2 Q = {q = (q 0 , q v ) 2 R ⇥ R 3 | q T q = 1} (2)
to represent the body attitude, the orientation of the body-fix frame F B with respect to the inertial frame F I . The corresponding kinematics are then given by

q(t) = 1 2 E(q(t))!(t) (3)
with the function E : Q ! R 4⇥3 defined as E(q) = 2 6 6 6 6 6 6 6 6 6 4 q T v S (q v ) + q 0 I 3 3 7 7 7 7 7 7 7 7 7 5 ,

where I 3 2 R 3⇥3 is the identity matrix. Note that the corresponding direction cosine matrix is obtained from the identity [START_REF] Schaub | Analytical mechanics of space systems[END_REF] C(q) = I 3 2q 0 S (q v ) + 2S 2 (q v ) .

Let the reference quaternion q r (t) represent the rotation of the reference frame F R from the inertial frame F I and the reference rate ! r (t) be described in the frame F R . Then the tracking errors q(t) 2 Q and !(t) 2 R 3 are defined as [START_REF] Seo | High-performance spacecraft adaptive attitude-tracking control through attracting-manifold design[END_REF] C( q(t)) = C(q(t))C T (q r (t)) ,

!(t) = !(t) ⌦(t) , (6)
where ⌦(t) = C( q(t))! r (t) is the reference rate in the body-fixed frame F B . Using the kinematic equation for the rotation matrix

d dt C(q(t)) = S (!(t))C(q(t)) (8)
and the matching conditions

qr (t) = 1 2 E(q r (t))! r (t) !r (t) = f(t) , (9)
where the forcing term f(t) 2 R 3 is assumed to be a predefined function such that ! r , f, ḟ 2 L 1 , the error dynamics are obtained as [START_REF] Seo | High-performance spacecraft adaptive attitude-tracking control through attracting-manifold design[END_REF]

q(t) = 1 2 E( q(t)) !(t) J !(t) = S (!(t))J!(t) + u(t) + J S (!(t))⌦(t) C( q(t))f(t) . (10)

B. Parameterization

Let us define the model parameter vector

✓ ⇤ = (✓ 1 , ✓ 2 , ✓ 3 , ✓ 4 , ✓ 5 , ✓ 6 ) 2 R
The parameters are assumed to be unknown but the lower bound of the smallest eigenvalue of J denoted as J m is to be known. It is known for any x, !, ⌦ 2 R 3 that the regressor matrix (x, !, ⌦)

is defined such that Jx S (!)J! + JS (!)⌦ = (x, !, ⌦)✓ ⇤ . (12)
Also, (x, !, ⌦) is decomposed as

(x, !, ⌦) = 1 (x) + 2 (!, ⌦) (13)
such that 1 (x)✓ ⇤ = Jx and 2 (!, ⌦)✓ ⇤ = S (!)J! + JS (!)⌦. For x = (x 1 , x , x 3 ), ! = (! 1 , ! 2 , ! 3 ) and ⌦ = (⌦ 1 , ⌦ 2 , ⌦ 3 ), the explicit forms of 1 (x) and 2 (!, ⌦) are given by

1 (x) = 2
! 2 ⌦ 3 ! 3 ⌦ 2 ! 3 ! 1 ! 1 ! 2 ! 2 ! 3 ! 3 ⌦ 1 ! 1 ⌦ 3 ! 1 ! 2 ! 2 ! 3 ! 3 ! 1 ! 1 ⌦ 2 ! 2 ⌦ 1 ! 2 3 ! 2 2 ! 1 ⌦ 2 ! 2 ⌦ 1 + ! 1 ! 2 ! 3 ⌦ 1 ! 1 ⌦ 3 ! 3 ! 1 ! 1 ⌦ 2 ! 2 ⌦ 1 ! 1 ! 2 ! 2 1 ! 2 3 ! 2 ⌦ 3 ! 3 ⌦ 2 + ! 2 ! 3 ! 3 ⌦ 1 ! 1 ⌦ 3 + ! 3 ! 1 ! 2 ⌦ 3 ! 3 ⌦ 2 ! 2 ! 3 ! 2 2 ! 2 1 T . (15)
Once the target dynamics are set to

!(t) = k p q v (t) k v (t) !(t) , (16)
where k p > 0 is the constant scalar gain for the vector part of the error quaternion and k v (t) > 0 is the scalar gain for the rate error, which can be static or dynamic depending on parameter estimator design, Eq. ( 10) is rewritten as

! = k p q v k v ! + J 1 h u + J k p q v + k v ! C( q)f S (!)J! + JS (!)⌦ i = k p q v k v ! + J 1 u + (x, !, ⌦)✓ ⇤ , (17)
where the vector x is specifically defined here as

x = k p q v + k v ! C( q)f . (18)
In the above equations, the time argument for the states and control is omitted for notational simplicity. From here on, function arguments are often omitted whenever no confusion can arise in the context.

The objective is to design I&I adaptive controllers which achieve ( q v , !) ! (0, 0) as t ! 1

while estimating the parameter ✓ ⇤ .

III. I&I Adaptive Controller Design

In this section, two types of I&I adaptive controllers are proposed based on the dynamic scaling method. We first consider the filter-free controller where the scaling factor appears in the feedback gain k v (t) so the control effort is directly affected by the dynamic gain. When the gain k v (t) is desired to be static, we can introduce a filter for the state !(t) so that the dynamic scaling factor is implemented only in the filter. In the second case, we fix the rate feedback gain k v (t) = k v and apply the design procedure proposed in [START_REF] Karagiannis | Dynamic scaling and observer design with application to adaptive control[END_REF] in a slightly different way.

The control law for the both cases is proposed as

u(t) = (x(t), !(t), ⌦(t)) ✓(t) + (t) , (19)
where ✓ + 2 R 6 is the estimate for ✓ ⇤ to be determined. So the closed-loop rate dynamics become

!(t) = k p q v (t) k v (t) !(t) J 1 (x(t), !(t), ⌦(t)) ✓(t) , (20)
where

✓(t) = ✓(t) + (t) ✓ ⇤ .
Before estimator design, the common blocks for both methods are built in the following. Let

y = k p q v k v ⌦ C( q)f . (21)
Then by linearity of 1 , we have

1 (x) = 1 (y + k v !) = 1 (y) + k v 1 (!) . (22)
Since, for all i, j 2 {1, 2, 3},

@ 1i @! j = @ 1 j @! i , (23)
where 1i and 1 j are the ith and jth columns of T 1 (x) respectively, there is µ 1 such that

@µ 1 @! = T 1 (x) . (24)
Let

µ ⇤ 1 (!) =  1 2 ! 1 1 2 ! 2 1 2 ! 3 ! 2 ! 3 ! 3 ! 1 ! 1 ! 2 T . (25)
Then a solution to Eq. ( 24) is given by

µ 1 = T 1 (y)! + k v µ ⇤ 1 (!) . (26)
However, for 2 (!, ⌦), there is no µ 2 satisfying

@µ 2 @! = T 2 (!, ⌦) (27)
because

@ 2i @! j 6 = @ 2 j @! i (28)
for all i, j 2 {1, 2, 3}, where 2i and 2 j are the ith and jth columns of T 2 (!, ⌦). This is known as the integrability obstacle and is the reason why the I&I adaptive control design method cannot be directly applied to the rigid-body attitude control problem [START_REF] Astolfi | Immersion and invariance: a new tool for stabilization and adaptive control of nonlinear systems[END_REF]. So we consider a solvable PDE by introducing a function (!, ⌦) satisfying

@ 2 i @! j + @ i @! j = @ 2 j @! i + @ j @! i (29)
for all i, j 2 {1, 2, 3}, where the subscripts i and j for indicate the ith and jth columns of the matrix T , i.e., there is a solution to the PDE

@µ 2 @! = T 2 (!, ⌦) + T (!, ⌦) . ( 30
)
The particular choice

µ 2 (!, ⌦) = 2
! 1 (! 2 ⌦ 3 ! 3 ⌦ 2 ) ! 2 (! 3 ⌦ 1 ! 1 ⌦ 3 ) ! 3 (! 1 ⌦ 2 ! 2 ⌦ 1 ) (! 2 3 ! 2 2 )(! 1 + 1 2 ⌦ 1 ) (! 2 1 ! 2 3 )(! 2 + 1 2 ⌦ 2 ) (! 2 2 ! 2 1 )(! 3 + 1 2 ⌦ 3 ) 3
(! 3 + ⌦ 3 ) ! 1 (! 2 + ⌦ 2 ) ! 2 (! 3 + ⌦ 3 ) 0 ! 2 (! 1 + ⌦ 1 ) ! 3 (! 2 + ⌦ 2 ) ! 3 (! 1 + ⌦ 1 ) 0 0 ! 1 (3! 2 + ⌦ 2 ) ! 1 (3! 3 + ⌦ 3 ) ! 2 (3! 1 + ⌦ 1 ) 0 ! 2 (3! 3 + ⌦ 3 ) ! 3 (3! 1 + ⌦ 1 ) ! 3 (3! 2 + ⌦ 2 ) 0 3
For any

⇠ = (⇠ 1 , ⇠ 2 , ⇠ 3 ) 2 R 3 , we define ˜ (!, ⌦, ⇠) = (!, ⌦) (⇠, ⌦) . ( 33
)
After some tedious yet straightforward algebra, we can find another convenient expression for ˜

given by

˜ (!, ⌦, ⇠) = H(!, ⌦, ⇠) I 3 ⌦ (! ⇠) , (34)
where the notation ⌦ represents the Kronecker product and H(!, ⌦, ⇠) is defined as .

H(!, ⌦, ⇠) = 2
(! 3 + ⌦ 3 ) 0 ⇠ 1 0 0 0 ⇠ 3 0 (! 1 + ⌦ 1 ) 3
(3! 2 + ⌦ 2 ) 3⇠ 1 0 0 0 0 0 3⇠ 3 (3! 2 + ⌦ 2 )
For notational simplicity, we additionally define

µ(k v , y, !, ⌦) = µ 1 (k v , y, !) + µ 2 (!, ⌦) (36)
and

μ(k v , y, !, ⌦, ⇠) = µ(k v , y, !, ⌦) µ(k v , y, ⇠, ⌦) . ( 37
)
Note that μ becomes zero when ⇠ converges to ! by definition.

A. Filter-Free Method

Let us propose

= ⇣ μ(k v , y, !, ⌦, ⌦) T (⌦, ⌦) ! ⌘ , (38)
where 2 R 6⇥6 is a positive definite matrix. The corresponding parameter update law is then given by

✓ = h µ 0 + T (⌦, ⌦) ⌦ ˙ T (⌦, ⌦) ! + ⇣ T (x, !, ⌦) + ˜ T (!, ⌦, ⌦) ⌘ ⇣ k p q v k v ! + ⌦⌘i , (39)
where

µ 0 = μ @µ(k v , y, !, ⌦) @! ! (40)
so that the time derivative of the parameter estimate error becomes

✓ = ✓ + ˙ = ( + ˜ ) T J 1 ✓ . ( 41
)
Remark 1. A systematic way of constructing is suggested in [START_REF] Karagiannis | Dynamic scaling and observer design with application to adaptive control[END_REF]. This work, however, suggests another way by defining the regressor matrix as a sum of solvable and non-solvable parts such that only the non-solvable part creates the error ˜ . It is beneficial because the structure of is simplified so that it does not depend on other variables like k v and q for this particular application.

To deal with the perturbation ˜ , we introduce the scaled estimate error defined as

z(t) = 1 r(t) ✓(t) . ( 42
)
The scaling factor r 2 R is the state of the differential equation

ṙ(t) = k r M k ˜ (!(t), ⌦(t), ⌦(t))k 2 2 r(t), r(0) = 1 , (43)
where k r > 1/(4J m ) is the gain and M is the largest eigenvalue of . Since r(t) is monotonically increasing and strictly positive for all t 0, Eq. ( 42) is a valid definition. Note that Eq. ( 43) is implementable as J m is assumed to be known. By differentiating Eq. ( 42) with respect to time along the trajectories of Eqs. ( 41) and (43), we obtain

ż = ⇣ T + ˜ T ⌘ J 1 z k r M k ˜ k 2 2 z . (44)
Now, consider a Lyapunov-like function

V 1 (t) = 1 2 z T (t) 1 z(t) . ( 45
)
Utilizing the inequalities

k ˜ k 2 2 J m kJ 1/2 ˜ k 2 2 (46)
and

z T 1 z 1 M kzk 2 2 , (47)
the upper bound for the time derivative of V 1 is obtained as is a positive definite matrix. Observe that the equilibrium z = 0 of Eq. ( 44) is uniformly globally stable and accordingly z(t) is uniformly bounded for all t 0.

V1  2
Next, consider the Lyapunov-like function

V 2 (t) = k p h q T v (t) q v (t) + 1 q 0 (t) 2 i + 1 2 ! T (t) !(t) + ⇣V 1 (t) , ( 50
)
where ⇣ is a positive constant to be determined. Taking the time derivative with respect to time yields

V2 (t) = k v (t) ! T (t) !(t) r(t) ! T (t)J 1 (t)z(t) + ⇣ V1 (t) . (51)
For the second cross term in the right-hand side of V2 to be dominated by the first term, we propose the dynamic gain of the form

k v (t) = k⇢(t)r 2 (t) , ( 52
)
where k is a positive scalar and ⇢(t) > 0 is a monotonically decreasing function that is lower bounded by the positive constant ⇢ ⇤ for all t 0 such that lim t!1 ⇢(t) = ⇢ ⇤ . Once the proposed dynamic gain is plugged in, the upper bound for V2 is obtained as

V2  k⇢ ⇤ r 2 2 k !k
If we choose ⇣ > 2k r J m /[k⇢ ⇤ (4k r J m 1)], Q becomes positive definite, and thus, V 2 is said to be non-increasing. At this point, however, we cannot conclude anything meaningful because the boundedness of r(t) is not established yet. In order to establish r 2 L 1 , let us recall Eq. ( 43) and consider the solution r(t) given by

r(t) = exp k r M Z t 0 k (!( ), ⌦( )) (⌦( ), ⌦( ))k 2 2 d ! . (55)
************* Suppose r(t) has a finite escape time at t = t ⇤ > 0. Then

sup t2[0,t ⇤ ) Z t 0 || ˜ ( )|| 2 2 d = 1 . (56)
Since r(t) r(0) = 1, V2 is upper bounded by

V2 (t)  k⇢ ⇤ 2 k !(t)k 2 2 ( 57
)
for t 2 [0, t ⇤ ). Integrating both sides yields

Z t 0 k !( )k 2 2 d  2 k⇢ ⇤ V 2 (0) V 2 (t) . ( 58
) Since V 2 is nonnegative, Z t 0 k !( )k 2 2 d  2 k⇢ ⇤ V 2 (0) . (59)
Also, since !(t) is concluded to be bounded for t 2 [0, t ⇤ ] and ⌦(t) is bounded by the assumption on the reference signals, there is a positive number L such that

kH(!, ⌦, ⌦)k 2  L ( 60
)
for t 2 [0, t ⇤ ]. However, the inequalities

Z t 0 k ˜ ( )k 2 2 d  Z t 0 kH( )k 2 2 k !( )k 2 2 d  2L 2 V 2 (0) k⇢ ⇤ (61)
hold for all t 2 [0, t ⇤ ). This contradict the hypothesis (56).

*********************** Thus, r(t) has neither a finite escape time nor diverges to infinity in the limit. Now Eq. ( 53) is valid for all t 0, and there exist positive constants c 1 and c 2 such that

V2  c 1 krJ 1/2 !k 2 2 c 2 kJ 1/2 zk 2 2 . ( 62
) Since r 2 L 1 , it is concluded that ✓ 2 L 1 , ! 2 L 2 \ L 1 and ! 2 L 1 .
Then invoking Barbalat's lemma allows us to conclude

lim t!1 !(t) = 0 . (63)
Lastly, let us show the convergence of q v to zero by partially stritifying the function V 2 (t). [START_REF] Mazenc | Quaternion-based stabilization of attitude dynamics subject to pointwise delay in the input[END_REF] Consider the strictified function

V 3 (t) = cV 2 (t) + q T v (t) !(t) , (64)
where

c > max 8 > > < > > : s 1 2k p , sup t 0 (k p + k 2 v )kJk 2 2k p r 2 c 1 , sup t 0 r 2 2k p J m c 2 9 > > = > > ;
.

Since c > q 1 2k p , V 3 is lower bonded by the following non-negative function:

V 3 2
Using the inequalities

k v q T v !  k p 2 k q v k 2 2 + k 2 v 2k p k !k 2 2 (67)
and

kS (q v ) + q 0 I 3 k 2  kE(q)k 2 , (68)
the upper bound of V3 is obtained as

V3  c V2 k p 2 k q v k 2 2 + k p + k 2 v 2k p ! k !k 2 2 + krJ 1/2 q v k 2 kJ 1/2 zk 2 . ( 69
)
Furthermore, aided by the inequalities

1 kJk 2 kvk 2 2  v T J 1 v  1 J m kvk 2 2
for any v 2 R 3 , we finally have

V3  c 1 c (k p + k 2 v )kJk 2 2k p r 2 c 1 ! krJ 1/2 !
q v 2 L 2 \ L 1 and qv 2 L 1 .
Therefore, by Barbalat's lemma,

lim t!1 q v (t) = 0 . ( 72
)
To complete our design, let us propose the following scalar differential equations for ⇢(t):

⇢1 (t) = 2k r M ✓ k 1 + (⌫ + 1)k ˜ (t)k 2 2 ◆ ⇢ 1 (t) 1 r 2 (t) ✏⇢ 2 (t) ! , (73)
⇢2 (t) = k 2 ⇢ 2 3 (t)⇢ 2 (t) , (74)
⇢3 (t) = k 3 ⇢ 2 3 (t) (75)
with the initial conditions ⇢ 1 (0 the choice ⇢(t) = ⇢ 1 (t) will result in lim t!1 ( q v (t), !(t)) = (0, 0). Alternatively, one can propose ⇢(t) = 1 for all t 0 since a constant function satisfies the condition on ⇢(t) so that only one dimensional dynamic extension is required, but the gain k v (t) increases monotonically.

) = 1 + ✏ 2 , ⇢ 2 (0) = ✏ and ⇢ 3 (0) > 0,
We can also remove the monotonically increasing scaling factor r(t) from the closed-loop dy-namics by introducing the following transformations:

⌘ 1 (t) = ⇢ 1 (t)r 2 (t), (76)
⌘ 2 (t) = ⇢ 2 (t)r 2 (t), (77)
⌘ 3 (t) = ⇢ 3 (t) (78)
with the initial conditions ⌘ 1 (0

) = 1 + ✏ 2 , ⌘ 2 (0) = ✏ and ⌘ 3 (0) > 0.
Then the extended dynamics become

⌘1 (t) = 2k r M h⇣ k 1 + ⌫k ˜ (t)k 2 2 ⌘ ⌘ 1 (t) ⇣ k 1 + (⌫ + 1)k ˜ (t)k 2 2 ⌘ ⇣ 1 + ✏⌘ 2 (t) ⌘i , (79)
⌘2 (t) = 2k r M ⇣ k 2 ⌘ 2 3 (t) k ˜ (t)k 2 2 ⌘ ⌘ 2 (t) , (80)
⌘3 (t) = k 3 ⌘ 2 3 (t) , (81)
and consequently, the dynamic gain is expressed as

k v (t) = k⌘ 1 (t) . (82)
Choosing proper values for k 2 and k 3 such that the first term in the right hand side of Eq. (80)

dominates the other term enables ⌘ 2 (t) to converge to a sufficiently small positive number. Then ⌘ 1 (t) will converge close to the unity. In other words, we can select k 2 and 1/k 3 to be arbitrarily large to render ⌘ 2 (t) arbitrarily small and still ensure the boundedness of r(t). Now k 2 and k 3 are another parameters we can tune to control the convergence of k v (t). Observe that since

1  ⌘ 1 (t)  ⇣ k 1 + (⌫ + 1)k ˜ (t)k 2 2 ⌘ ⇣ 1 + ✏⌘ 2 (t) ⌘ k 1 + ⌫k ˜ (t)k 2 2  ⌫ + 1 ⌫ ⇣ 1 + ✏⌘ 2 (t) ⌘ ( 83
)
for all t 0, the upper bound of the dynamic gain k v (t) can be adjusted by the parameter ⌫ as long as ⌘ 2 converges to a small number.

Remark 2. Practically, the initial condition for ⌘ 3 (t) can be reset such that k 2 ⌘ 2 3 (t) > k ˜ (t)k 2 2 at t = t ⇤ for any t ⇤ 0 whenever needed so that ⌘ 2 (t) decreases. Since the infinite tail of the signal ⌘ 3 (t) still satisfies the L 2 condition, we can reset the dynamics finitely many times. Remark 3. To ensure the boundedness of r(t), k v (t) must be proportional to r 2 (t). Ideally, only one dimensional dynamic extension for r(t) achieves the I&I adaptive tracking control. However, the dynamic gain includes r 2 (t), and thus, it is possible for the gain to grow to an arbitrarily large number with a constant ⇢. This is undesirable for both the implementation of the scaling factor dynamics and the robustness properties of the closed-loop system. To prevent the high gain feedback control, three more dynamics (73), ( 74) and ( 75) are introduced so that monotonically increasing r(t) is removed from the closed-loop states giving us more flexibility to tune the range of the dynamic gain. Overall, only 3 more dynamics are required to overcome the integrability obstacle, which is a significant order-reduction compared to the existing I&I controller with an 18-dimensional regressor filter and a 3-dimensional state filter. [START_REF] Seo | High-performance spacecraft adaptive attitude-tracking control through attracting-manifold design[END_REF]

B. Filter Method

The gain k v (t) described in the filter-free method can be set to a constant if a filter for !(t) is introduced. Let !(t) be the filter state whose dynamics are to be determined such that lim t!1 !(t) = 0 and ✓ 2 L 1 , where !(t) = !(t) !(t). Previously, the perturbation term ˜ (t) was created using the reference rate ⌦(t). Once a filter is considered, !(t) can replace ⌦(t) so that we have

˜ (t) = ˜ (!(t), ⌦(t), !(t)) = (!(t), ⌦(t)) ( !(t), ⌦(t)) . (84)
The estimator design process is similar to the filter-free method. We first propose

= ⇣ μ(k v , y, !, ⌦, !) T ( !, ⌦) !⌘ (85)
and

✓ = h µ 0 + T ( !, ⌦) ! ˙ T ( !, ⌦) ! + ⇣ T (x, !, ⌦) + ˜ T (!, ⌦, !) ⌘ ⇣ k p q v k v ! + ⌦⌘i , (86)
where

µ 0 = μ @µ(k v , y, !, ⌦) @! ! (87)
so that the parameter error dynamics become

✓ = ( + ˜ ) T J 1 ✓ . ( 88
)
The dynamic equation for the scaling factor r(t) is then changed to

ṙ(t) = k r M k ˜ (!(t), ⌦(t), !(t))k 2 2 r(t), r(0) = 1 . (89)
to obtain

ż = ⇣ T + ˜ T ⌘ J 1 z k r M k ˜ k 2 2 z . (90)
Next, propose the filter dynamics

!(t) = k f (t) !(t) k p q v (t) k v !(t) + ⌦(t) , (91)
where the filter gain k f (t) > 0 contains the scaling factor. Since the dynamic gain does not appear in the control law, the local Lipschitz condition for cannot be used to show the boundedness of r(t). So we choose

k f (t) = ⇣ k + k H kH(!, ⌦, !)k 2 2 ⌘ ⇢(t)r 2 (t) (92)
and consider the Lyapunov-like function

W 1 (t) = 1 2 !T (t) !(t) + ⇣ 2 z T (t) 1 z(t) + k H ⇢ ⇤ k r M r 2 (t) , (93)
where the gain k and k H are positive constants [START_REF] Karagiannis | Dynamic scaling and observer design with application to adaptive control[END_REF]. When we differentiate W 1 (t) with respect to time, the upper bound for Ẇ1 (t) is obtained as

Ẇ1  k⇢ ⇤ r 2 2 k !
kJ 1/2 zk 2 kJ 1/2 ˜ k 2 kzk 2 . ( 94
)
The dynamic gain with an extra term directly ensures r 2 L 1 in this case. Consequently, by Barbalat's lemma, the convergence of the filter error to zero is guaranteed, i.e.,

lim t!1 !(t) = 0 . ( 95
)
The convergence of the tracking errors to zero is proved by considering the function

W 2 (t) = k p a 1 h q T v (t) q v (t) + 1 q 0 (t) 2 i + a 1 2 ! T (t) !(t) + a 2 W 1 (t) + q T v (t) !(t) (96)
for sufficiently large a 1 , a 2 > 0. Here we omit the proof since the procedure is almost the same as that in the filter-free case. Also, r-dynamics can be removed through the dynamic extensions (79), ( 80) and (81). Then the gain is given by

k f (t) = ⇣ k + k H kH(!, ⌦, !)k 2 2 ⌘ ⌘ 1 (t) . ( 97
)
Remark 4. The boundedness of r(t) is established by adding an extra term in the dynamic gain. If we apply this approach to the filter-free case, the dynamic gain would be of the form

k v (t) = ⇣ k + k H kH(!, ⌦, ⌦)k 2 2 ⌘ ⌘ 1 (t) . ( 98
)
However, this is not applicable since µ 0 (t) defined in Eq. (40) includes kv (t), and therefore, requires the unavailable state !(t) for the estimator implementation.

Remark 5. The advantage of the filter method is that we can always choose !(0) = !(0). So the transient response of the closed-loop system would be better than filter-free method when the initial tracking error is large. Moreover, since the high gain term only appears in the filter dynamics, the required control effort can be less than that of the filter-free controller.

IV. Application to Robot Manipulator Control

The same design process can be applied to Euler-Lagrange equations, which describe the evolution of a mechanical system subject to holonomic constraints [START_REF] Spong | Robot modeling and control[END_REF]. In fact, the dynamic model for the rigid-body rotational motion with the quaternion kinematics can be converted to a similar form of the Euler-Lagrange equations by differentiating the following equation with respect to time [START_REF] Costic | Quaternion-based adaptive attitude tracking controller without velocity measurements[END_REF]:

qv (t) = 1 2 F( q(t)) !(t) . ( 99
)
In this section, motivated by this analogy, we further develop and generalize the theory for nlink robot manipulator systems. Instead of following the entire design procedure suggested in the previous section, we only introduce necessary conditions to apply the dynamic scaling based I&I methodology.

A. Dynamic Equations

Let us consider the equations of motion for an n-link robot given by

ẋ1 (t) = x 2 (t) M(x 1 (t))ẋ 2 (t) = u(t) C(x 1 (t), x 2 (t))x 2 (t) g(x 1 (t)) , (100)
where x 1 (t), x 2 (t) 2 R n are the joint angle and rate, and u(t) 2 R n is the control torque at each joint.

M(x 1 (t)) is the inertia matrix, C(x 1 (t), x 2 (t)) is the centripetal-Coriolis matrix and g(x 1 (t)) is the gravity vector satisfying the following three important structural properties [START_REF] Spong | Robot modeling and control[END_REF]:

Property 1 (Skew-symmetry property). The matrix Ṁ(x 1 ) 2C(x 1 , x 2 ) is skew-symmetric, that is, the equality

⇠ T ⇣ Ṁ(x 1 ) 2C(x 1 , x 2 ) ⌘ ⇠ = 0 (101)
holds for all ⇠ 2 R n .

Property 2 (Bounds on the inertia matrix). The inertia matrix M(x 1 ) is positive definite and its eigenvalues are bounded for any fixed value of x 1 . Let 0 < 1 (x 1 )  ...  n (x 1 ) denote the n eigenvalues of M(x 1 ). Then the inertia matrix satisfies

1 (x 1 )I n  M(x 1 )  n (x 1 )I n , ( 102
)
where I n is the n-dimensional identity matrix and the matrix inequality B  A means A B is positive semidefinite. If x 1 2 L 1 or all joints are revolute, M(x 1 ) is uniformly bounded by positive constants M m and M M such that

M m I n  M(x 1 )  M M I n . ( 103
)
Property 3 (Linearity in the parameters). The dynamic equations are linear in the inertial parameters such that there exists an n ⇥ p regressor function Y(x 1 , x 2 , ẋ2 ), which is completely known and satisfies

M(x 1 )ẋ 2 + C(x 1 , x 2 )x 2 + g(x 1 ) = Y(x 1 , x 2 , ẋ2 )✓ ⇤ , (104)
where ✓ ⇤ 2 R p is the parameter vector.

The objective is to design a tracking adaptive controller when the system parameters ✓ ⇤ are uncertain. Let x r (t) = (x r1 (t), x r2 (t)) 2 R 2n be the reference state to be tracked with the matching condition

ẋr1 (t) = x r2 (t) ẋr2 (t) = f(t, x r (t)) , (105)
where f : [0, 1) ⇥ R 2n ! R n is a predefined smooth function that produces bounded values for bounded arguments. It is also assumed that the time derivative of the forcing function f(t, x r (t)) is bounded. If error states are defined as

e 1 (t) = x 1 (t) x r1 (t) e 2 (t) = x 2 (t) x r2 (t) , (106)
then the error dynamics are obtained as

ė1 = e 2 ė2 = M 1 (x 1 ) ⇥ u C(x 1 , x 2 )x 2 g(x 1 ) M(x 1 )f ⇤ . (107)
Let the target dynamics for e 2 (t) be given by

ė2 (t) = k p e 1 (t) k v (t)e 2 (t) , (108)
where k p > 0 is the static feedback gain for the position error and k v (t) > 0 is the static or dynamic feedback gain for the velocity error. Then the error dynamics for the velocity error in Eq. ( 107) can be rewritten as

ė2 = k p e 1 k v e 2 + M 1 (x 1 ) h u + M(x 1 ) k p e 1 + k v e 2 f C(x 1 , x 2 )x 2 g(x 1 ) i . (109)

B. Parameterization

By Property 3, there exist functions 1 (y, x 1 ), 2 (x 1 , x 2 ), 3 (x 1 ) 2 R n⇥p such that

1 (y, x 1 )✓ ⇤ = M(x 1 )y (110) 2 (x 1 , x 2 )✓ ⇤ = C(x 1 , x 2 )x 2 (111) 3 (x 1 )✓ ⇤ = g(x 1 ) , (112)
where y = k p e 1 + k v e 2 f. Once the regressor matrix (y, x 1 , x 2 ) is defined as

(y, x 1 , x 2 ) = 1 (y, x 1 ) + 2 (x 1 , x 2 ) + 3 (x 1 ) , (113)
the form of the dynamic equations we are interested in is given by

ė1 = e 2 ė2 = k p e 1 k v e 2 + M 1 (x 1 ) ⇥ u + (y, x 1 , x 2 )✓ ⇤ ⇤ . (114)

C. Partial Differential Equations for I&I Design

Suppose there exists µ(s, x 1 , x 2 ) 2 R p for some variable s independent of x 2 , which satisfies the

PDE @µ @x 2 = T (y, x 1 , x 2 ) . (115)
Then the integrability condition is said to be satisfied, and thus, there is an I&I adaptive controller [START_REF] Astolfi | Immersion and invariance: a new tool for stabilization and adaptive control of nonlinear systems[END_REF]. However, the existence of µ is not always guaranteed unless n = 1. Let i be the ith column of the matrix T (y, x 1 , x 2 ) and x j be the jth component of the vector x 2 2 R n . In order for µ(s,

x 1 , x 2 )
to exist satisfying Eq. (115), the condition

@ i @x j @ j @x i = 0 , (116)
where i, j 2 {1, 2, ..., n} must hold for each i and j.

In our formulation, we show that there exists

µ 1 (k v , ȳ, x 1 , x 2 ) 2 R p such that @µ 1 @x 2 = T 1 (y, x 1 ) , (117)
where ȳ = k p e 1 k v x r2 f. To find a solution to Eq. ( 117), let us establish the following lemma.

Lemma 1. Consider the parametrization defined in Eq. (110). Then, there exists µ ⇤ (x 1 , x 2 ) 2 R p such that @µ ⇤ /@x 2 = T 1 (x 2 , x 1 ).

Proof. By Property 3, the element of the ith row and jth column of M(x 1 ) is a inner product of the parameter vector ✓ ⇤ and some vector function f i j (x 1 ) 2 R p , i.e.,

M i j = p X k=1 ✓ k f i jk (x 1 ) (118)
with f i jk (x 1 ) = f jik (x 1 ), where ✓ k and f i jk (x 1 ) are the kth elements of ✓ ⇤ and f i j . Define ⇠ = M(x 1 )x 2 .

Let x 2i and ⇠ i be the ith elements of the vector x 2 and ⇠. Then, since

⇠ i = n X j=1 p X k=1 ✓ k f i jk (x 1 )x 2 j = p X k=1 ✓ k n X j=1 f i jk (x 1 )x 2 j , (119)
the entry in the kth row and ith column of T 1 (x 2 , x 1 ) denoted as [ T 1 ] ki is obtained as

[ T 1 ] ki = n X j=1 f i jk (x 1 )x 2 j . (120)
Applying the integrability condition (116) to the ith and lth columns of T 1 (x 2 , x 1 ), 1i and 1 j respectively, yields

@ 1i @x l @ 1l @x i = 0 , (121)
which proves the existence of µ ⇤ .

Since 1 (y, x 1 ) is linear in the second argument, it is decomposed as

1 (y, x 1 ) = 1 (ȳ, x 1 ) + k v 1 (x 2 , x 1 ) . (122)
Then, by Lemma 1, there is µ ⇤ (x 1 , x 2 ) such that @µ ⇤ /@x 2 = T 1 (x 2 , x 1 ), and thus,

µ 1 (k v , ȳ, x 1 , x 2 ) = T 1 (ȳ, x 1 )x 2 + k v µ ⇤ (x 1 , x 2 ) , (123)
is a solution to Eq. ( 117). Also, we can choose

µ 3 (x 1 , x 2 ) = T 3 (x 1 )x 2 (124) to have @µ 3 @x 2 = T 3 (x 1 ) . (125)
However, the existence of µ 2 2 R p such that @µ 2 /@x 2 = T 2 (x 1 , x 2 ) is not guaranteed. So we consider a solvable PDE by introducing the function (x 1 , x 2 ) satisfying

@ j @x i @ i @x j = @ 2 i @x j @ 2 j @x i , (126)
where the subscripts i and j for and 2 indicate the ith and jth columns of the matrices T and T 2 respectively. Then the PDE is modified as follows:

@µ 2 (x 1 , x 2 ) @x 2 = T 2 (x 1 , x 2 ) + T (x 1 , x 2 ) (127)
with a solution µ 2 . Such (x 1 , x 2 ) satisfying Eq. (127) always exists because, for example, we can always select

(x 1 , x 2 ) = 2 (x 1 , x 2 ) (128)
or alternately as

(x 1 , x 2 ) = " @ @x 2 ⇣ T 2 (x 1 , x 2 )x 2 ⌘ # T 2 (x 1 , x 2 ) (129)
so that Eq. ( 126) is satisfied. We can now apply the dynamic scaling I&I design method so long as the following assumptions are satisfied.

Assumption 1 (A1

). There exists a function µ 2 : D ⇢ R 2n ! R p such that Eq. (127) holds, where D is a compact set, and (x 1 , x 2 ) 2 R n⇥p is smooth and locally Lipschitz for any (x 1 , x 2 ) in D.

Thus, if $ 1 2 R 2n and $ 2 2 R 2n are bounded then there is a positive constant L such that The goal is to remove the effect of (x 1 , x 2 ) in parameter estimation by adopting a dynamic scaling factor. By the CE principle, we first propose our control law as

k ($ 1 ) ($ 2 )k 2  L k$ 1 $ 2 k 2 . ( 130
)
u = (y, x 1 , x 2 )( ✓ + ) (131)
and then design an update law in a similar way suggested in the previous section. The details of the final expression for the dynamic scaled adaptive controller are presented in appendix, which are omitted here in the interest of brevity.

V. Numerical Simulations

A. Attitude Tracking Control

Consider the rigid-body attitude dynamics [START_REF] Åström | Theory and applications of adaptive control -a survey[END_REF], where the inertia matrix is given by [9] with its smallest eigenvalue bound J m = 14.2672. The body attitude is initially given by q v (0) = (0.1826, 0.1826, 0.1826) and q 0 (0) = p 1 q T v (0)q v (0) with zero rate. The initial reference quaternion is set to q r (0) = (1, 0, 0, 0) and the corresponding reference rate profile is given by ! r (t) = (! r (t), ! r (t), ! r (t)), where ! r (t) = 0.3(1 e 0.01t 2 ) cos t + te 0.01t 2 (0.08⇡ + 0.006 sin t) .

J = 2
The initial parameter estimates are chosen such that ✓(0) = (1, 1, 1, 1, 1, 1). We simulate the closedloop system using ⌘ 3 (0) = 1 and the other design parameters:

k p = k = 0.7, k r = 0.2, k 1 = k 2 = k 3 = 1, = 50I 3 , ⌫ = 0.5, ✏ = 0.00001
for the filter-free case and

k p = k v = 0.7, k r = 0.2, k = k H = 0.01, k 1 = k 2 = k 3 = 1, = 50I 3 , ⌫ = 0.5, ✏ = 0.00001
for the filter case. The tracking errors for both cases converge to zero with good transient performances shown in Fig. 1. Even though there are some differences in ⌘ 1 (t) and ⌘ 2 (t) as seen in Fig. 2, the control effort and the convergence properties of the parameter estimate errors are almost identical as illustrated in Fig. 3. Also, the time histories of the error signals and control effort are very similar to those in [START_REF] Seo | High-performance spacecraft adaptive attitude-tracking control through attracting-manifold design[END_REF], which shows outperformance of non-CE controllers. To see the effect

v (t)k 2 , k !(t)k 2 and k !(t)k 2 .
of the initial rate tracking error !(0), we set the initial rate as !(t) = (1, 1, 1) and simulate the closed-loop system using the same reference trajectories. To ensure ⌘ 2 (t) < 1 for all t 0, we only change the value of k 2 to 70 for the filter-free controller. In Fig. 4, the filter-free controller shows a faster convergence in the parameter estimation, but relatively large control effort is applied in the transient region. Thus, when the control power is limited or initial tracking errors are not small enough, the filter method would be desirable.

B. 2-Link Planar Robot Tracking Control

We consider the same dynamic model appearing in the recent filter-based I&I results [START_REF] Seo | Non-certainty equivalent adaptive control for robot manipulator systems[END_REF]. The parameters for the proposed controller are selected such that tracking performance is similar to that from the reference.

We consider the 2-link planar robot arm as depicted in Fig. 5. The inertia and centripetal-

When a filter for x 2 is adopted for the filter method, we use

kH(x 1 , x 2 , x2 )k 2 2 = 4 sin 2 x 12 h (x 21 + x 22 + x21 ) 2 + x2 21 i , (138)
where x = ( x21 , x22 ) 2 R 2 is the filter state. The initial conditions for the states x 1 (0) and x 2 (0) are set to zero, and the reference position is given by x r1 (t) = (cos t + 2, sin t + 2). The parameter estimate vector is initially set to ✓(0) = (2.6, 0, 0) and the controller parameters are selected as

k p = k = 1, k r = 2, k 1 = 0.01, k 2 = 4, k 3 = 0.2, = 7.5I 3 , ⌫ = 0.5, ✏ = 0.00001
for the filter-free case and

k p = kv = 1, k = 1, k H = 0.1, k r = 2, k 1 = 0.01, k 2 = 4, k 3 = 0.2, = 7

VI. Conclusion

Novel dynamically scaled Immersion and Invariance (I&I) adaptive controllers are proposed for the rigid-body attitude and the n-link robot manipulator control problems. For each problem, we assume that the inertia parameters are unknown but a lower bound of the smallest eigenvalue of the inertia matrix is available. A dynamic scaling factor is employed to circumvent the integrability obstacle typically arising in I&I adaptive control design. Based on where the dynamic scaling factor appears, both filter-free and filter-dependent controllers are designed. Notably, by introducing three scalar dynamics, the monotonically increasing scaling factor is eliminated from the closedloop states. In addition, we have more control knobs to adjust and tune the range of the dynamic gain that contains the scaling factor. Depending on the choices of design parameters, the dynamic gain is shown to be pulled down to the vicinity of the initial value as the tracking errors converge to zero. The performances of the controllers are also analyzed through numerical simulations. The spacecraft and 2-link planar manipulator are considered. Particularly, for the attitude control case, the simulation results of the filter-free and filter-dependent controllers are compared when the initial tracking error of the rate state is intentionally chosen to be a large value. Further research will focus on removing the need of the inertia matrix bound or applying a projection mechanism to the partially known inertia parameters. To find better ways to restrict behavior of the dynamic scaling factor would be a challenging problem.

Appendix: Summary of I&I Estimators for the Robot Manipulator Problem

Suppose that A1 and A2 are satisfied, and that appropriate µ 1 , µ 2 and µ 3 are obtained so that we can define µ(k v , ȳ, x 1 , x 2 ) = µ 1 (k v , ȳ, x 1 , x 2 ) + µ 2 (x 1 , x 2 ) + µ 3 (x 1 , x 2 ) .

(139)

For any ⇠ 2 R n , we also define

μ(k v , ȳ, x 1 , x 2 , ⇠) = µ(k v , ȳ, x 1 , x 2 ) µ(k v , ȳ, x 1 , ⇠) , (140)
˜ (x 1 , x 2 , ⇠) = (x 1 , x 2 )

(x 1 , ⇠) (141)
and

µ 0 = μ @µ(k v , ȳ, x 1 , x 2 ) @x 2 ẋ2 . (142)
Since (x 1 , x 2 ) is assumed to be smooth, there exists a function H(x 1 , x 2 , ⇠) such that

k ˜ (x 1 , x 2 , ⇠)k 2  kH(x 1 , x 2 , ⇠)k 2 kx 2 ⇠k 2 . ( 143
)
In the I&I estimator design, ⇠ = x r2 for the filter-free estimator and ⇠ = x2 for the filter-dependent estimator. For the dynamic gains, we use the same dynamic extension defined in Eqs. ( 79), ( 80) and (81) with k r > 1/(4M m ).

A. Filter-Free Estimator

The update law is proposed as

✓ = h µ 0 + T (x 1 , x r2
where k f is a dynamic gain defined as

k f = ⇣ k + k H kH(x 1 , x 2 , x2 )k 2 2 ⌘ ⌘ 1 (149)
with k, k H > 0.

Figure 1 .

 1 Figure 1. Time histories of k q v (t)k 2 , k !(t)k 2 and k !(t)k 2 .

Figure 2 .Figure 3 .

 23 Figure 2. Time histories of ⌘ 1 (t) and ⌘ 2 (t)

Figure 4 .Figure 5 .

 45 Figure 4. Time histories of ku(t)k 2 and k ✓(t)k 2 with large !(0)

.5I 3 ,

 3 ⌫ = 0.5, ✏ = 0.00001 for the filter case. The results are shown in Figs. 6, 7 and 8. The tracking errors converge to zero with an acceptable transient. Also the parameter estimates converge to the true values due to the persistently exciting reference signals. Since k 2 ⌘ 2 3 (t) dominates over k ˜ (t)k 2 2 , ⌘ 2 (t) approaches close to zero as the tracking errors converge to zero as illustrated in Fig. 7b. As a result, the dynamic gain returns

Figure 6 .

 6 Figure 6. Time histories of k q v (t)k 2 , k !(t)k 2 and k !(t)k 2 .

Figure 7 .Figure 8 .

 78 Figure 7. Time histories of ⌘ 1 (t) and ⌘ 2 (t)

⇣ f k p e 1 k v e 2 ⌘i⇣ f k p e 1 k v e 2 ⌘i

 22 )f ˙ T (x 1 , x r2 )e 2 + ⇣ T (y, x 1 , x 2 ) + ˜ T (x 1 , x 2 , x r2 ) ⌘ v , ȳ, x 1 , x 2 , x r2 ) T (x 1 , x r2 )e 2 ⌘ ,(145)where = T > 0 and k v = k⌘ 1 with k > 0.B. Filter-Dependent ControllerLet x2 = x 2 x2. Then the update law is proposed as✓ = h µ 0 + T (x 1 , x2 ) ẋ2 ˙ T (x 1 , x2 )x 2 + ⇣ T (y, x 1 , x 2 ) + ˜ T (x 1 , x 2 , x2 ) ⌘ v , ȳ, x 1 , x 2 , x2 ) T (x 1 , x2 )x 2 ⌘ ,(147)where = T > 0 and k v > 0. The corresponding filter is given by ẋ2 = k f x2 k p e 1 k v e 2 + f ,

  definite matrix. Again, since r(t) is bounded and J is positive definite, we can conclude

			k 2 2	2 6 6 6 6 6 6 6 6 6 4	krJ 1/2 q v k 2 kJ 1/2 zk 2	3 7 7 7 7 7 7 7 7 7 5 T	R	2 6 6 6 6 6 6 6 6 6 4	krJ 1/2 q v k 2 kJ 1/2 zk 2	3 7 7 7 7 7 7 7 7 7 5	,	(70)
	where	R =	2 6 6 6 6 6 6 6 6 6 4	k p J m 2r 2 1 2	1 2 cc 2	3 7 7 7 7 7 7 7 7 7 5				(71)
	is a positive

  Assumption 2 (A2). The smallest eigenvalue of M(x 1 ) is uniformly lower bounded and the bound denoted as M m is known. Remark 6. Since 2 (x 1 , x 2 ) is, in general, smooth and locally Lipschitz for all(x 1 , x 2 ) 2 R 2n , the choice (x 1 , x 2 ) = 2 (x 1 , x 2 )is always feasible. Note that only filter-free controllers require this local Lipschitz condition for implementation. Moreover, since mass matrices are uniformly bounded for any robot systems whose joints are all revolute as mentioned in Property 2, our assumptions are not very restrictive. It is also important to point out that introducing the functions µ 1 , µ 3 and µ 2 that satisfy Eqs. (117), (125) and (127) gives a general way to construct as mentioned in Remark 1.

  2✓ 3 cos x 12 ✓ 2 + ✓ 3 cos x 12 = (x 11 , x 12 ) 2 R 2 and x 2 = (x 21 , x 22 ) 2 R 2 are the states and ✓ ⇤ = (✓ 1 , ✓ 2 , ✓ 3 ) 2 R 3 is the system parameter vector. The effect of gravity and friction is neglected and the true values for the parameters are set to ✓ ⇤ = (3.6, 0.2, 0.15). The lower bound of the smallest eigenvalue of M(x 1 ) is known to be M m = 0.1672 and the following µ 2 (x 1 , x 2 ) is chosen such that A1 is satisfied:

	Coriolis matrices are given by M(x 1 ) = and C(x 1 , x 2 ) = 2 6 6 6 6 6 6 6 6 6 4 The corresponding (x 1 , x 2 ) is then given by 2 6 6 6 6 6 6 6 6 6 4 ✓ 1 + ✓ 2 + ✓ 3 cos x 12 ✓ 3 x 22 sin x 12 ✓ 3 x 21 sin x 12 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 x 21 x 22 (x 21 + x 22 ) sin x 12 ✓ 2 ✓ 3 (x 21 + x 22 ) sin x 12 3 7 7 7 7 7 7 7 7 7 5 0 0 0 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 . where x 1 µ 2 (x 1 , x 2 ) = (x 1 , x 2 ) = 2 6 6 6 6 6 6 6 6 6 4 0 0 0 0 0 2x 21 (x 21 + x 22 ) sin x 12 3 7 7 7 7 7 7 7 7 7 5 .	3 7 7 7 7 7 7 7 7 7 5	,	(134) (135) (136)

Acknowledgment

The results reported in this work were supported in part by National Aeronautics and Space Administrations Johnson Space Center through Grant Number NNX14AK46A (Technical Manager: Dr. Christopher N. D'Souza).


Keywords: Simulation, génération de plates-formes, Grid'5000

Extraction à partir de Grid'5000 de plates-formes multi-grappes synthétiques pour la conduite d'expériences de simulation

Résumé : Ce rapport présente un catalogue de plates-formes de calcul distribué synthétiques mais néanmoins réalistes. Ces configurations sont destinées à des expériences de simulations dans le cadre de l'étude d'applications parallèles sur des plates-formes multi-grappes.

Introduction

The advent of parallel scientific computing of multi-cluster distributed platforms that span multiple organizations, often termed "grid computing", over the last decade has generated tremendous research activity as several applications can take advantage of such platforms. One key challenge is the development of techniques for maximizing application performance of these multi-cluster platforms, by redesigning the applications and/or by using novel scheduling algorithms that can account for the composite and heterogeneous nature of the platform. Most research works in this area are based on simulation, as it allows repeatable results, makes it possible to explore various platform scenarios at will, is not as labor-intensive or as costly as running experiments on a real platform, and often makes it possible to run enormous numbers of experiments quickly. Several toolkits exist for running these simulations, with two recent and popular ones being SimGrid [START_REF] Legrand | Scheduling Distributed Applications: The SimGrid Simulation Framework[END_REF] and GridSim [START_REF] Buyya | GridSim: A Toolkit for the Modeling and Simulation of Distributed Resource Management and Scheduling for Grid Computing[END_REF]. Every researcher is then faced with the question: "which platform configurations should I simulate?" One approach is to generate random platform configurations using simple uniform probability distributions (e.g., for the number of clusters, for the number of nodes per cluster, for the nodes' compute speeds). While this approach is simple and makes it possible to generate large numbers of platform configurations it is not clear that many of the generated platforms are representative of the real world. A better approach consists in building statistical models of real-world clusters and to use these models to generate cluster configurations based one a large set of real-world cluster configurations [START_REF] Kee | Realistic Modeling and Synthesis of Resources for Computational Grids[END_REF]. Although these models should lead to more representative platform configurations, many researchers opt for using real-world platform configurations directly. The drawback is that typically only a few such configurations are constructed. Instead, we propose the use of several configurations that correspond to subsets of one of the most prominent large-scale multi-cluster platform deployed today, Grid'5000 [START_REF]RT n 0341 Unité de recherche INRIA Lorraine LORIA, Technopôle de Nancy-Brabois -Campus scientifique 615, rue du Jardin Botanique -BP 101 -54602 Villers-lès-Nancy Cedex[END_REF]. This report presents an overview of the performance characteristics of Grid'5000 and of its clusters, and then present a compendium of platform configurations with specific characteristics, to be used for simulation experiments.

The Grid'5000 Clusters

The Grid'5000 platform consists of 20 clusters available in 9 sites in France: Bordeaux, Grenoble, Lille, Lyon, Nancy, Orsay, Rennes, Sophia, and Toulouse. Table 1 shows the number of cluster per site. Each cluster is identified by a name, and the table shows the architecture of it processors, the number of such processors, and the processor compute speed in GFlop/s as measured with the HPLinpack benchmark.

RT n 0341

Note that the table omits speed measurements for the two clusters available at the Grenoble site as we were unable to launch the HPLinpack benchmark on these clusters. To help appreciate the range of processor speeds in the Grid'5000 platform Figure 1 shows processor speed (on the x-axis) for the processors/cores of the 18 clusters on which we were able to run the benchmark. We see that the platform is fairly heterogeneous, with the fastest processors computing about 45% faster than the slowest processors. HPLinpack performance per proc/core (in Gflop/s) In the next two sections we define collections of subsets of the Grid'5000 platform, in a view to spanning a sound spectrum of platform charateristics and thus to being useful for driving sound simulation experiments. One important characteristics is the degree of processor heterogeneity in the platorm. We define the degree of heterogeneity, h, as 100 × (s max /s min -1), where s max (resp. s min ) is the maximum (resp. minimum) processor speed (in GFlop/s) in the platform. A value of 0 indicates a perfectly homogeneous system. The h value denotes the maximum relative difference in compute speed in the platform, in percentage. Given our 18 clusters in Grid'5000, no multi-cluster platform is homogeneous. We thus consider what we term "almost homogeneous" platforms, that is platforms with a low h value (h < 10).

Grelon

Deriving Almost Homogeneous Platforms

Out of our 18-cluster Grid'5000 platform we can extract multi-cluster configurations that are almost homogeneous (in terms of compute speed) with various numbers of clusters. These configurations, which are described below, for 2, 4, and 8 clusters.

With 2 Clusters -Table 2 shows 10 almost homogeneous 2-cluster configurations, named "HOM 2.x", for x = 1, . . . , 10. For each configuration the table shows which two clusters are used in the configuration, showing the number of nodes in these clusters. The last two columns of the table show the total number of processors in the configuration, as well as the configuration's degree of heterogeneity, h. Recall that for all platforms in the section h ≤ 10%.

With 4 Clusters -Table 2 is similar to Table 2, but shows 4-cluster configurations. There are 10 such configurations as well, names "HOM 4.x", for x = 1, . . . , 10.

With 8 Clusters -Table 3 is similar to the previous two tables, but shows 8-cluster configurations. There are 10 such configurations as well, names "HOM 8.x", for x = 1, . . . , 10.

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Bordeaux

Deriving Heterogeneous platforms

With 2 Clusters-When deriving heterogeneous 2-cluster platforms, we define the heterogeneity factor as h = ((f astest/slowest) -1) × 100, where f astest is the compute speed of the fastest of the two clusters, and slowest is the compute speed of the slowest of the two clusters.

Table 5 shows 3 such configurations, named "HET 2 40.x", for x = 1, . . . , 3, for which the heterogeneity factor is h ≥ 40. Table 6 shows 15 configurations, named "HET 2 30.x", for x = 1, . . . , 15, for which 40 > h ≥ 30. Table 7 shows 31 configurations, named "HET 2 20.x", for x = 1, . . . , 31, for which the heterogeneity factor is 30 > h ≥ 20. Table 8 shows 33 configurations, named "HET 2 10.x", for x = 1, . . . , 33, for which the heterogeneity factor is 20 > h ≥ 10.

For platforms that contain more than 2 clusters, or in other words platforms that consist of some "fast" clusters and of some "slow" clusters, we define h min and h max as follows. h min is the heterogeneity factor, as defined above for 2-cluster platforms, between the fastest of the slow clusters and the slowest of the fast clusters. h max is simply defined as the heterogeneity factor between the slowest cluster and the fastest cluster. Also, we limit the number of platorm configurations whenever applicable. For instance, given the clusters in Grid'5000, when trying to generate configurations with 1 fast cluster and 3 slow clusters, we could end up choosing the 3 slow clusters among 7 possibilities, for a total of 210 platform configurations, with many of these configurations virtually identical. Therefore, we chose to ignore many of these possibilities in order to keep the number of platform configurations reasonably low.

With 1 Fast and 3 Slow Clusters-Table 9 shows 14 platform configurations, named "HET 4 1f 3s 30.x", for x = 1, . . . , 14, for which the heterogeneity factor is h ≥ 30. Table 10 shows 22 configurations, named "HET 4 1f 3s 20.x", for x = 1, . . . , 22, for which 30 > h ≥ 10. Table 11 shows 13 configurations, named "HET 4 1f 3s 10.x", for x = 1, . . . , 13, for which 20 > h ≥ 10.

With 2 Fast and 2 Slow Clusters-Table 12 shows 14 platform configurations, named "HET 4 2f 2s 30.x", for x = 1, . . . , 14, for which the heterogeneity factor is 40 > h ≥ 30. Table 13 shows 24 configurations, named "HET 4 2f 2s 20.x", for x = 1, . . . , 24, for which 30 > h ≥ 10. Table 14 shows 16 configurations, named "HET 4 2f 2s 10.x", for x = 1, . . . , 16, for which 20 > h ≥ 10. 15 shows all the platform configurations comprising 3 fast and 1 slow clusters. The first 4 configurations, named "HET 4 3f 1s 30.x", for x = 1, . . . , 4, have an heterogeneity factor h ≥ 30, the next 13 configurations, named "HET 4 3f 1s 20.x", for x = 1, . . . , 13, for which 30 > h ≥ 10, and the last 13 configurations, named "HET 4 3f 1s 10.x", for x = 1, . . . , 13, for which 20 > h ≥ 10.

With 3 Fast and 1 Slow Clusters-Table

For the 8-cluster configurations hereafter we do not separate them in different tables depending on the value of h, but simply enforce that h > 20 and refer the reader to the referenced tables for specific h values for each configuration.

With 1 Fast and 7 Slow Clusters-Table 16 shows 20 platform configurations, named "HET 8 1f 7s.x", for x = 1, . . . , 20.

With 2 Fast and 6 Slow Clusters-Table 17 shows 33 platform configurations, named "HET 8 2f 6s.x", for x = 1, . . . , 33.

With 3 Fast and 5 Slow Clusters-Table 18 shows 25 platforms, named "HET 8 3f 5s.x", for x = 1, . . . , 25.

With 4 Fast and 4 Slow Clusters-Table 19 shows 11 platforms, named "HET 8 4f 4s.x", for x = 1, . . . , 11. RT n 0341

With 5 Fast and 3 Slow Clusters-

Conclusion

In this report we have listed a number of multi-cluster platform configurations extracted from a real-world grid deployment, Grid'5000, for a total of 356 individual configurations. These configurations contain 1, 2, 4, or 8 clusters, and can be categorized as homogeneous or heterogeneous, as summarized in Table 21. We argue that these configurations provide a sound basis for driving simulation experiments that explore the execution of parallel applications on multi-cluster platforms.

Figure 1 :

 1 Figure 1: Grid'5000 Processor Speeds (GFlop/s).

Table 1 :

 1 The Grid'5000 clusters.

	site	cluster	Processors	#proc Gflop/s
	Bordeaux IBM	Opteron 248	48	3.542
		Dell	Xeon 3GHz	51	3.464
	Grenoble IDPOT	Xeon 2.4GHz	32	N/A
		Icluster2	Itanium 2	103	N/A
		e326	Opteron 248	53	3.647
	Lille	e326mMC	Opteron 252	20	4.311
		e326mDC	Opteron 285	26	4.384
	Lyon	Capricorne Opteron 246	56	3.254
		Sagittaire	Opteron 250	70	3.865
	Nancy	Grillon	Opteron 246	47	3.379
		Grelon	Xeon 5110	120	3.185
	Orsay	GDX	Opteron 246	216	3.388
		GDX2	Opteron 250	126	4.040
		Paravent	Opteron 246	99	3.364
	Rennes	Parasol	Opteron 248	64	3.573
		Paraquad	Xeon 5148LV	66	4.603
		Azur	Opteron 246	74	3.258
	Sophia	Helios	Opteron 275	56	3.675
		Sol	Opteron 2218	50	4.389
	Toulouse	Sun	Opteron 248	58	3.586
				1435

Table 2 :

 2 Almost homogeneous platforms with 2 clusters.

				Lille	Lyon	Nancy	Orsay	Rennes	Sophia	Toulouse
		C1	C2	C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17	C18	Total	h
	HOM 2.1				56				74	0.12
	HOM 2.2			26					50	76	0.11
	HOM 2.3					47	216		0.27
	HOM 2.4							64	58	0.36
	HOM 2.5					47		99	0.45
	HOM 2.6						216	99	0.71
	HOM 2.7			53					56	0.77
	HOM 2.8	48						64	0.88
	HOM 2.9			53					58	1.7
	HOM 2.10 48	51						99	2.25
		Bordeaux	Lille	Lyon	Nancy	Orsay	Rennes	Sophia	Toulouse
		C1	C2	C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17	C18	Total	h
	HOM 4.1		51			47	216	99	2.97
	HOM 4.2			53				64	56	58	2.85
	HOM 4.3	48		53				64	58	2.96
	HOM 4.4				56	47		99	74	3.84
	HOM 4.5					47	216	99	74	3.99
	HOM 4.6				56		216	99	74	4.12
	HOM 4.7				56	47	216		74	4.12
	HOM 4.8				56	47	216	99	4.12
	HOM 4.9	48	51					64	58	3.52
	HOM 4.10 48	51			47	216		4.82

Table 3 :

 3 Almost homogeneous platforms with 4 clusters.

		Bordeaux	Lille	Lyon	Nancy	Orsay	Rennes	Sophia	Toulouse
		C1	C2	C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17	C18	Total	h
	HOM 8.1	48

Table 5 :

 5 Table 20 shows 4 platforms, named "HET 8 5f 3s.x", for x = 1, . . . , 4. Heterogeneous platforms with 2 clusters (h ≥ 40).

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Table 7 :

 7 Heterogeneous platforms with 2 clusters (30 > h ≥ 20).

	Bordeaux		Lille	Lyon	Nancy	Orsay	Rennes	Sophia	Toulouse
	C1	C2	C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17	C18	Total	h
	HET 10.1					70			66	136	19.09
	HET 10.2								126	66	192	13.94
	HET 10.3								56	50	106	19.43
	HET 10.4					70			50	120	13.56
	HET 10.5			26					56	82	19.29
	HET 10.6			26		70			96	13.43
	HET 10.7		53	20					73	18.21
	HET 10.8			20					56	76	17.31
	HET 10.9			20		70			90	11.54
	HET 10.10						47		126	173	19.56
	HET 10.11							216	126	342	19.24
	HET 10.12	51							126	177	16.63
	HET 10.13 48								126	174	14.06
	HET 10.14								126	64	190	13.07
	HET 10.15								126	58	184	12.66
	HET 10.16		53						126	179	10.78
	HET 10.17				56	70			126	18.78
	HET 10.18					70			74	144	18.63
	HET 10.19					70			99	169	14.89
	HET 10.20					70	47		117	14.38
	HET 10.21					70		216	286	14.08
	HET 10.22	51				70			121	11.58
	HET 10.23						120		56	176	15.38
	HET 10.24				56				56	112	12.94
	HET 10.25								74	56	130	27.58
	HET 10.26		53				120		173	27.24
	HET 10.27		53		56				109	24.45
	HET 10.28		53						74	127	21.71
	HET 10.29						120		58	178	20.65
	HET 10.30				56				58	114	20.22
	HET 10.31								74	58	132	10.07
	HET 10.32						120		64	184	13.43
	HET 10.33 48						120		168	18.21

Table 8 :

 8 Heterogeneous platforms with 2 clusters (20 > h ≥ 10).

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Table 9 :

 9 Heterogeneous platforms with 4 clusters (1 fast and 3 slow -h ≥ 30).

		Bordeaux	Lille	Lyon	Nancy	Orsay	Rennes	Sophia	Toulouse
		C1	C2	C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17	C18	Total hmax hmin
	HET 4 1f 3s 20.1	48						64	66	58	29.95 28.36
	HET 4 1f 3s 20.2			53				64	66	58	28.83 26.21
	HET 4 1f 3s 20.3			53					66	56	58	28.36 26.21
	HET 4 1f 3s 20.4		51			47	216		50	29.89	26.7
	HET 4 1f 3s 20.5	48	51				216		50	29.55	26.7
	HET 4 1f 3s 20.6	48	51					64	50	26.7	23.91
	HET 4 1f 3s 20.7	48						64	50	58	23.91 22.84
	HET 4 1f 3s 20.8			53				64	50	58	22.84 20.35
	HET 4 1f 3s 20.9		51		26	47	216		29.74 26.56
	HET 4 1f 3s 20.10 48	51		26		216		29.4	23.77
	HET 4 1f 3s 20.11 48	51		26			64	26.56	22.7
	HET 4 1f 3s 20.12 48			26			64	58	23.77 22.25
	HET 4 1f 3s 20.13			53	26			64	58	22.7	20.21
	HET 4 1f 3s 20.14			20		47	216	99	27.58 27.24
	HET 4 1f 3s 20.15		51	20		47	216		27.24 24.45
	HET 4 1f 3s 20.16 48	51	20			216		24.45 21.71
	HET 4 1f 3s 20.17 48	51	20				64	21.71 20.65
	HET 4 1f 3s 20.18 48		20				64	58	20.65 20.22
	HET 4 1f 3s 20.19				56	120	126		74	26.84	24
	HET 4 1f 3s 20.20				56	120	126	99	26.84	20.1
	HET 4 1f 3s 20.21					120	126	99	74	26.84	20.1
	HET 4 1f 3s 20.22				56		126	99	74	24.15	20.1

Table 10 :

 10

		Bordeaux		Lille	Lyon	Nancy	Orsay	Rennes	Sophia	Toulouse
		C1	C2	C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17	C18	Total hmax hmin
	HET 4 1f 3s 10.1			53	20	70			56	18.21 11.54
	HET 4 1f 3s 10.2		51				47	216	126	19.56 16.63
	HET 4 1f 3s 10.3	48	51					216	126	19.24 14.06
	HET 4 1f 3s 10.4	48	51						126	64	16.63 13.07

Heterogeneous platforms with 4 clusters (1 fast and 3 slow -30 > h ≥ 20).

Table 18 :

 18 Heterogeneous platforms with 8 clusters (3 fast and 5 slow -40 > h ≥ 20).

Table 21 :

 21 Summary of platform configurations extracted from Grid'5000.

	type	1 cluster 2 clusters 4 clusters 8 clusters total
	homogeneous	18	10	10	10	48
	heterogeneous	-	82	133	93	308
	Total	18	92	143	103	356

 HET 4