Martin Brossard

Silvère Bonnabel

Jean-Philippe Condomines

Unscented Kalman Filtering on Lie Groups

Keywords: Lie groups, unscented Kalman filter, robot localization

In this paper, we first consider a simple Bayesian fusion problem in a matrix Lie group, and propose to tackle it using the unscented transform. The method is then leveraged to derive two simple alternative unscented Kalman filters on Lie groups, for both cases of noisy partial measurements of the state, and full state noisy measurements of the state on the group. The general method is applied to a robot localization problem, and results based on experimental data combined with extensive Monte-Carlo simulations at various noise levels illustrate the superiority of the approach over the standard UKF.

I. INTRODUCTION

The problem of robot localization (and mapping) based on the fusion of various sensors has long been an important field. With the advent of probabilistic robotics methods within the field of mobile robotics, the role of accurate statistics for localization (and mapping) has been increasingly recognized. The Unscented Kalman Filter (UKF) introduced by roboticists [START_REF] Julier | A New Extension of the Kalman Filter to Nonlinear Systems[END_REF][START_REF] Julier | Unscented Filtering and Nonlinear Estimation[END_REF] has become prevalent as an alternative to the Extended Kalman Filter (EKF) that may improve estimation in various cases and spares the practitioner the computation of Jacobians. Besides, there has been various recent works that have evidenced the fact that the Lie group structure of the configuration space SE(3) plays a prominent role in probabilistic robotics, see [START_REF] Chirikjian | Stochastic Models, Information Theory, and Lie Groups[END_REF]- [START_REF] Hertzberg | Integrating Generic Sensor Fusion Algorithms with Sound State Representations Through Rncapsulation of Manifolds[END_REF]. It has also been shown recently that a Lie group underlies the state space in SLAM, and that using a (Lie group) Invariant EKF [START_REF] Barrau | Intrinsic Filtering on Lie Groups with Applications to Attitude Estimation[END_REF], the consistency issues of the EKF-SLAM can be fixed [START_REF] Barrau | An EKF-SLAM Algorithm with Consistency Properties[END_REF].

In this paper we consider systems whose state is an unimodular matrix Lie group. Using exponential coordinates to define Gaussians on the group, we derive a Lie group version of the unscented Kalman filter. We consider two distinct problems: the problem of noisy partial measurement of the state that live in a vector space, and the problem of full state noisy measurement which is to be treated slightly differently, as the measurement then lives in the Lie group. For each type of measurement, the proposed UKF on Lie Groups consists of two different variants: one based on left multiplications, and the other based on right multiplications. It turns out the variant that works best depends on the problem at hand, and both should be tested when one has no clear insight on which should be the best.

A. Links and Differences with Previous Literature

The original UKF and its square-root implementation [START_REF] Der Merwe | The square-root unscented kalman filter for state and parameter-estimation[END_REF] have been extensively adapted and used for attitude estimation, that is, for filtering on the Lie group SO [START_REF] Chirikjian | Stochastic Models, Information Theory, and Lie Groups[END_REF]. An UKF called USQUE [START_REF] Crassidis | Unscented Filtering for Spacecraft Attitude Estimation[END_REF] has become a standard filter for attitude estimation. The filter does not fully use the Lie group structure of the state space, notably the Lie exponential map, but rather takes advantage of the quaternion representation of SO(3), which does not carry over to general Lie groups. Recent works have also advocated the use of particular probability densities on SO(3), the so-called Fisher distributions, as an interesting alternative to the Lie exponential coordinates of [START_REF] Chirikjian | Stochastic Models, Information Theory, and Lie Groups[END_REF], and have brought them to bear for unscented attitude estimation [START_REF] Lee | Global Unscented Attitude Estimation via the Matrix Fisher Distributions on SO(3)[END_REF].

Another line of research uses the unscented transform on Lie groups and Lie exponential coordinates to derive uncertainty ellipsoids that are proved to contain with certainty the state, when faced with bounded sensor errors, see [START_REF] Bohn | Unscented State Estimation for Rigid Body Motion on SE(3)[END_REF][START_REF] Bohn | Unscented State Estimation for Rigid Body Attitude Motion with a Finite-time Stable Observer[END_REF]. [START_REF] Condomines | Nonlinear State Estimation Using an Invariant Unscented Kalman Filter[END_REF][START_REF] Condomines | Pi-Invariant Unscented Kalman Filter for Sensor Fusion[END_REF] also introduced an Invariant UKF, as an UKF capable of taking into account the symmetries of the system's equations, for state spaces that are generally not Lie groups.

Regarding SE(3) and more general Lie groups, one can essentially follow two slightly different routes to describe Gaussians in exponential coordinates. The first consists in directly defining a density on the group, that can be referred to as concentrated Gaussian, as in [START_REF] Wang | Error Propagation on the Euclidean Group with Applications to Manipulator Kinematics[END_REF][START_REF] Long | The Banana Distribution is Gaussian: A Localization Study with Exponential Coordinates[END_REF]. Those distributions were recently used for extended Kalman filtering on Lie groups by [START_REF] Bourmaud | Continuous-Discrete Extended Kalman Filter on Matrix Lie Groups Using Concentrated Gaussian Distributions[END_REF]. The second one consists in assuming the distribution to be normal in the Lie algebra, and then to map it to the group through the exponential map, as advocated by [START_REF] Barfoot | Associating Uncertainty with Three-Dimensional Poses for Use in Estimation Problems[END_REF] for robotics applications, and at the heart of the Invariant EKF of [START_REF] Barrau | Intrinsic Filtering on Lie Groups with Applications to Attitude Estimation[END_REF][START_REF] Barrau | Intrinsic Filtering on SO(3) with Discretetime Observations[END_REF]. In the present paper, we follow this second method. This yields a propagation step of the UKF that is identical to the method of [START_REF] Barfoot | Associating Uncertainty with Three-Dimensional Poses for Use in Estimation Problems[END_REF] to compound poses. However, [START_REF] Barfoot | Associating Uncertainty with Three-Dimensional Poses for Use in Estimation Problems[END_REF] does not treat Bayesian estimation on SE(3), which is necessary to update the state of the UKF when given a novel measurement.

The very recent paper [START_REF] Loianno | Visual Inertial Odometry for Quadrotors on SE(3)[END_REF] also proposes an UKF on the Lie group SE(3) for a navigation application. Beyond the fact it is only concerned with SE(3) whereas the present paper deals with general Lie groups, the method is different since it uses concentrated Gaussians on Lie groups of [START_REF] Wang | Error Propagation on the Euclidean Group with Applications to Manipulator Kinematics[END_REF]. As a result, the method can be related to the general unscented Kalman filtering on manifolds of [START_REF] Hauberg | Unscented Kalman Filtering on Riemannian Manifolds[END_REF], where the sigma points are mapped onto the manifold, and then the logarithm map is used to map them back in some tangent space where averages and covariances can be computed, whereas the sigma points in our method below live in a vector space, and are never mapped onto the group itself (regarding partial measurements). Last but not least, the method of [START_REF] Loianno | Visual Inertial Odometry for Quadrotors on SE(3)[END_REF] is based on left multiplications, notably on the left-invariant connection of SE(3), whereas, as emphasized in the present paper, right multiplications allow defining an alternative UKF. This is important, as simulations of Section V-D indicate the right multiplications based UKF precisely suits best measurements in the body frame, such as those considered in [START_REF] Loianno | Visual Inertial Odometry for Quadrotors on SE(3)[END_REF].

B. Paper's Organization and Contributions

The paper is divided into six sections as follows. Section II contains mathematical preliminaries. Each of the following three sections constitutes a contribution of the present paper. In Section III, we design different Bayesian estimators on Lie groups based on the unscented transform for various kinds of measurement. The method is applied in Section IV to derive two UKF on Lie groups. Section V considers 2D robot localization problem that uses experimental data, and illustrates the superiority of the proposed UKF over the standard UKF and the Invariant EKF.

II. MATHEMATICAL PRELIMINARIES

A. Lie Groups

In this section we recall the definitions and basic properties of unimodular matrix Lie groups, Lie algebra and random variables on Lie groups. A matrix Lie group G ⊂ R N ×N is a set of square invertible matrices that is a group, i.e., the following properties

I ∈ G; ∀ χ ∈ G, χ -1 ∈ G; ∀ χ 1 , χ 2 ∈ G, χ 1 χ 2 ∈ G (1)
hold. Locally about the identity matrix I, the group G can be identified with an Euclidean space R q using the matrix exponential map exp m (.), where q = dim G. Indeed, to any ξ ∈ R q one can associate a matrix ξ ∧ of the tangent space of G at I, called the Lie algebra g. We then define the exponential map exp : R q → G for Lie groups as

exp (ξ) = exp m (ξ ∧ ) , (2)
Locally, it is a bijection, and one can define the Lie logarithm map log : G → R q as the exponential inverse, leading to

log (exp (ξ)) = ξ. (3)

B. Uncertainties on Lie Groups

To define random variables on Lie groups, we cannot apply the usual approach of additive noise for χ 1 , χ 2 ∈ G as G is not a vector space, i.e., generally χ 1 + χ 2 / ∈ G does not hold. In contrast, we adopt the framework of [START_REF] Barfoot | Associating Uncertainty with Three-Dimensional Poses for Use in Estimation Problems[END_REF], see also [START_REF] Barrau | Intrinsic Filtering on Lie Groups with Applications to Attitude Estimation[END_REF][START_REF] Barrau | Intrinsic Filtering on SO(3) with Discretetime Observations[END_REF], which is slightly different from the pioneering approach of [START_REF] Chirikjian | Stochastic Models, Information Theory, and Lie Groups[END_REF][START_REF] Chirikjian | Stochastic Models, Information Theory, and Lie Groups[END_REF]. Indeed, we define the probability distribution χ ∼ N L ( χ, P) for the random variable χ ∈ G as

χ = χ exp (ξ) , ξ ∼ N (0, P) , (4)
where N (., .) is the classical Gaussian distribution in Euclidean space and P ∈ R q×q is a covariance matrix. In the sequel, we will refer to (4) as the left-equivariant Gaussian distribution on G, due to its compatibility with left multiplications. In (4), the noise-free quantity χ is viewed as the mean, and the dispersion arises through left multiplication with the exponential of a Gaussian random variable. Similarly, the distribution χ ∼ N R ( χ, P) can be defined through right multiplication as

χ = exp (ξ) χ, ξ ∼ N (0, P) . (5)
We stress that we have defined these probability density functions directly in the vector space R q such that both N L (., .) and N R (., .) are not Gaussian distributions.

III. BAYESIAN ESTIMATION ON LIE GROUPS USING THE UNSCENTED TRANSFORM

Consider a random variable χ ∈ G with prior probability distribution p ( χ ). Suppose we obtain some additional information about χ through a measurement y. The goal of Bayesian estimation is to compute an estimate of the posterior distribution p( χ |y). In this section, we derive various algorithms for various types of measurements.

A. Partial Measurement with Left-equivariant Prior Uncertainty (4) Consider a generic vector measurement of the form

y = h ( χ ) + v, (6)
where h(.) : G → R k represents the observation function and v ∼ N (v, R) is a Gaussian random noise in R k with known characteristics. The problem of Bayesian estimation we consider is as follows:

1) consider the prior distribution follows (4), i.e.,

p( χ ) ∼ N L ( χ, P), (7)
with known parameters χ and P; 2) assume one measurement y given by ( 6) is available; 3) approximate the posterior distribution as

p( χ |y) ≈ N L ( χ+ , P + ), (8)
and compute the estimates of both posterior parameters χ+ and P + .

To attack this problem, one can resort the unscented transform of [START_REF] Julier | A New Extension of the Kalman Filter to Nonlinear Systems[END_REF]. First, and contrarily to [START_REF] Loianno | Visual Inertial Odometry for Quadrotors on SE(3)[END_REF], we use the fact that the problem is amenable to nonlinear filtering in a vector space, as we have ξ ∼ N (0, P) with nonlinear measurement [START_REF] Barrau | Intrinsic Filtering on Lie Groups with Applications to Attitude Estimation[END_REF]. The UKF thus allows us to approximate the posterior p(ξ|y) for ξ as follows: we compute a finite number of samples α T j = ξ T j v T j , j = 0, . . . , 2l, with l = q + k, and then pass each of these so-called sigma points through the measurement function

y j = h ( χ exp (ξ j )) + v j , j = 0, . . . , 2l, (9)
and we then compute successively the measurement mean ȳ, the measurement covariance P yy and the cross-covariance P αy . We thus approximate the posterior for ξ in R q as p(ξ|y) ∼ N ξ, P + , where

ξ * = P αy P -1 yy (y -ȳ) and ( 11)

P + = P -P αy P αy P -1 yy T . ( 12
)
The unscented approximation to the posterior p(ξ|y) is thus the distribution of a Gaussian ξ + ξ + with ξ + ∼ N (0, P + ). Back to the Lie group, this means we approximate the posterior distribution through the variable χ exp ξ + ξ + . The Baker-Campbell-Hausdorff (BCH) formula provides a simple (firstorder) approximation as exp ξ + ξ + ≈ exp ξ exp (ξ + ). This readily yields an approximate χ+ exp(ξ + ) to the posterior of the form (4), with

χ+ = χ exp ξ , ξ + ∼ N 0, P + , ( 13
)
that concludes the Bayesian estimation, which is summarized in Algorithm 1 1 .

B. Partial Measurement with Right-equivariant Prior Uncertainty (5)

In the latter subsection, we considered that both the prior and posterior follow the distribution of the form (4). If we assume the prior has the form (5), then it is natural to assume that the posterior is also of the form [START_REF] Barfoot | Associating Uncertainty with Three-Dimensional Poses for Use in Estimation Problems[END_REF]. In this case, we substitute the computed measurements y j , j = 0, . . . , 2l in [START_REF] Hertzberg | Integrating Generic Sensor Fusion Algorithms with Sound State Representations Through Rncapsulation of Manifolds[END_REF] as

y j = h (exp (ξ j ) χ) + v j ( 14
)
and the posterior mean [START_REF] Lee | Global Unscented Attitude Estimation via the Matrix Fisher Distributions on SO(3)[END_REF] as

χ+ = exp ξ χ. ( 15
)
The difference between both methods may prove nonnegligible as will be illustrated in Section V-D.

C. Full State Measurement: Bayesian Pose Fusion

Assume we have a prior of the form p( χ ) ∼ N L ( χ, P) and a measurement of the type

Y = χ V, ( 16
)
where χ is the true state and V a noise of the form

V = exp (v) , v ∼ N (0, R) . (17)
This kind of measurement happens for instance in SE(3), when an algorithm is used to compute a pose directly from perception sensors such as, stereo images, or scan matching algorithms with point clouds, and when a covariance can be associated to the computed pose, see e.g., [START_REF] Hervier | Accurate 3D maps from depth images and motion sensors via nonlinear Kalman filtering[END_REF].

1) Proposed Method: in this case, we need to approximate the posterior p( χ |Y) ∼ N L ( χ+ , P + ), which appears to be nontrivial. Indeed, in [START_REF] Condomines | Nonlinear State Estimation Using an Invariant Unscented Kalman Filter[END_REF], Y is a matrix measurement, and this matrix belongs to G which is not a vector space, so both the definition and the computation of the covariance of Y is not straightforward. For such measurement, we propose using the logarithm map to obtain

y = log χ-1 Y = log χ-1 χ V , (18)
= log (exp (ξ) V) .

Thanks to this transformation, the computed transformed output y belongs to vector space R q , and we can then apply the proposed Bayesian estimation of Section III-A, with

y j = log (exp (ξ j ) exp (v j )) , j = 0, . . . , 2l, (20)
to obtain the posterior p( χ |Y) ∼ N L ( χ+ , P + ).

2) Fusion with Right-equivariant Prior Uncertainty (5): assume that we have a prior of the form (5), or by symmetry (inverting all quantities), that we have a left prior of the form (4) but with an output of the form Y = V χ . We then 1 According to [START_REF] Julier | A New Extension of the Kalman Filter to Nonlinear Systems[END_REF][START_REF] Dunik | Unscented Kalman Filter: Aspects and Adaptive Setting of Scaling Parameter[END_REF], we set unscented transform parameters to κ = 0 and β = 2. α keeps a free-parameter chosen by the practitioner, which must be small (α = 10 -3 in our applications). The operator diag(.) put in a block diagonal matrix its arguments in Step 3, the square-root matrix in step 4 is obtained from Cholewski decomposition and col(P) j represents the j-th column of P.

Algorithm 1: Bayesian estimation on Lie groups Input: χ, P, y, v, R, α;

1 λ = α 2 -1 l ; // scale parameter 2 W 0 s = λ λ+l , W 0 c = λ λ+l + 3 -α 2 , W j s = W j c = 1/2
λ+l , j = 1, . . . , 2l ; // weights 3 P aug = diag (P, R); // augmented covariance matrix 4 ᾱ = 0 T vT T , α j = ᾱ + col (l + λ) P aug j , j = 1, . . . , l,

α j = ᾱ -col (l + λ) P aug j-l
, j = l + 1, . . . , 2l;

5 ξ T j v T j = α T j , j = 0, . . . , 2l; 6 y j = h ( χ exp (ξ j )) + v j , j = 0, . . . , 2l; 7 ȳ = 2l j=0 W j s y j ; 8 P yy = 2l j=0 W j c (y j -ȳ) (y j -ȳ) T ; 9 P αy = 2l j=0 W j c (α j -ᾱ) (y j -ȳ) T ;
10 Extract ξ from (11);

11 χ+ = χ exp ξ ;

12 P + = P -P αy P αy P -1 yy T ;

Output:

χ+ , P + ; propose to write Y = V χ exp(ξ) = χ χ-1 V χ exp(ξ) = χ χ-1 exp(v) χ exp(ξ) = χ exp(Ad χ-1 v) exp(ξ)
where Ad is the adjoint operator of the group. We can thus compute the modified output

y = log χ-1 Y = log exp Ad χ-1 v exp (ξ) . ( 21
)
Thanks to this transformation, we recover a measurement similar to [START_REF] Long | The Banana Distribution is Gaussian: A Localization Study with Exponential Coordinates[END_REF], but with noise having covariance matrix

Ad χ-1 R Ad χ-1
T . The method above can then be readily applied.

3) Rationale for the Transformation of the Output: consider output [START_REF] Condomines | Nonlinear State Estimation Using an Invariant Unscented Kalman Filter[END_REF]. One could argue that y = log (Y) is already a vector, and a function of ξ so that an UKF estimate can readily be built along the lines of Section III-A. However, when χ is far from the identity, a large distortion is induced by the logarithm map. But ξ and v being moderate zerocentered noises in applications, we have using BCH formula log (exp (ξ) V) = ξ + v + o ( ξ , v ). Thus up to first order terms the transformed output y of ( 19) is a true Gaussian, and the posterior output by the UKF will be very close to the true one, no matter χ, whereas log (Y) can be dramatically non-Gaussian, preventing the UKF estimate from being accurate. 4) Links with the Pose Fusion Method of [START_REF] Barfoot | Associating Uncertainty with Three-Dimensional Poses for Use in Estimation Problems[END_REF]: [START_REF] Barfoot | Associating Uncertainty with Three-Dimensional Poses for Use in Estimation Problems[END_REF] proposes an algorithm to fuse various measurements of the form [START_REF] Condomines | Nonlinear State Estimation Using an Invariant Unscented Kalman Filter[END_REF]. With two measurements only, it boils down to estimating an unknown group element χ from measurements Y 1 = χ exp (v 1 ) and Y 2 = χ exp (v 2 ), where v 1 and v 2 are centered Gaussians in R q with known covariance matrices. In the Euclidean case, the corresponding problem is to estimate some quantity µ from two noisy measurements y 1 = µ + n 1 and y 2 = µ + n 2 where n 1 and n 2 are centered Gaussian noises with known covariance matrices. This is a classical estimation problem, that is optimally handled through least squares, and does not imply the Kalman update equation (Gaussian conditioning). The problem considered in the present section is quite different, since it deals with Bayesian fusion of poses, which implies conditional probabilities. In our case χ is known, and we want to refine our statistics on the random quantity χ exp (ξ), given that we have measured, e.g., Y = χ exp (ξ) exp (v).

IV. APPLICATION TO UNSCENTED KALMAN FILTERING ON LIE GROUPS

A. System Model

In the rest of this paper we consider a discrete dynamic system of the form

χ n = χ n-1 exp (ω n + w n ) , (22)
where the state χ n lives in G, ω n is a known input variable and w n ∼ N (0, Q n ) is a white Gaussian noise. We design two different filters for generic discrete measurements of the form

y n = h ( χ n ) + v n , (23)
where v n ∼ N (v n , R n ) is a white Gaussian noise with known characteristics, and also for measurements of the form

Y n = χ n exp (v n ) , (24)
where

v n ∼ N (v n , R n ).

B. Unscented Kalman Filter on Lie Groups with Leftequivariant Uncertainties (4)

For the model [START_REF] Loianno | Visual Inertial Odometry for Quadrotors on SE(3)[END_REF] with measurements given by ( 23), we can model the state as χ n ∼ N L ( χn , P n ), that is, using the left-equivariant formulation (4) of the uncertainties. The goal is to compute both estimates of χn and P n at each time. The proposed Left-Unscented Kalman Filter on Lie Groups (Left-UKF-LG) acts in two steps: propagation and update. We now develop these two steps.

1) Propagation: starting from the prior distribution

p ( χ n-1 ) ∼ N L ( χn-1 , P n-1 ) , (25)
with χn-1 and P n-1 known, we seek to approximate the propagated state distribution

p ( χ n | χ n-1 ) ∼ N L ( χn , P n ) , (26)
i.e., we search to compute both χn and P n .

This step can be viewed as a straightforward application of the method of [START_REF] Barfoot | Associating Uncertainty with Three-Dimensional Poses for Use in Estimation Problems[END_REF] to compound poses using the unscented transform. The state mean estimate is propagated using the unnoisy state model, leading to

χn = χn-1 Ω n , (27)
with Ω n = exp (ω n ), According to [START_REF] Barfoot | Associating Uncertainty with Three-Dimensional Poses for Use in Estimation Problems[END_REF] this approximation of the mean is valid up to the second order, so there is no need to compute it through the unscented transform (the calculations of [START_REF] Barfoot | Associating Uncertainty with Three-Dimensional Poses for Use in Estimation Problems[END_REF] deal with SE( 3), but we have checked they carry over to the general Lie group case). We then resort to the unscented transform to determine the covariance P n by first approximating the joint Gaussian variable using a finite number of samples ξ j , w j , j = 1, . . . , 4q. To propagate theses samples, we inject ( 4) in [START_REF] Loianno | Visual Inertial Odometry for Quadrotors on SE(3)[END_REF] as

χn exp (ξ n ) = χn-1 exp (ξ n-1 ) exp (ω n + w n ) , (28)
which simplifies as follows

exp (ξ n ) = χ-1 n χn-1 exp (ξ n-1 ) exp (ω n + w n ) , (29) exp
(ξ n ) = Ω -1 n exp (ξ n-1 ) exp (ω n + w n ) , (30)
by use of [START_REF] De Luca | Feedback Control of a Nonholonomic Car-like Robot[END_REF]. Algorithm 2 summarizes this propagation step.

2) Update: this step consists in incorporating the information coming from the measurement. As concerns measurements y n of the form [START_REF] Hauberg | Unscented Kalman Filtering on Riemannian Manifolds[END_REF], it boils down to a Bayesian estimation problem, where we search to approximate the posterior probability distribution as

p ( χ n |y n , y n-1 , . . . , y 1 ) ∼ N L χ+ n , P + n , (31)
with available prior

p ( χ n |y n-1 , . . . , y 1 ) ∼ N L ( χn , P n ) . ( 32
)
We apply for the update estimates directly Bayesian estimation as developed in Section III-A, yielding both χ+ n and P + n . To update the filter with a measurement of the type [START_REF] Chirikjian | Stochastic Models, Information Theory, and Lie Groups[END_REF], it suffices to apply in a similar manner the method of Section III-C.

C. Unscented Kalman Filter on Lie Groups with Rightequivariant Uncertainties (5)

For the model [START_REF] Loianno | Visual Inertial Odometry for Quadrotors on SE(3)[END_REF] with measurements given by ( 23), we can model the state as χ n ∼ N R ( χn , P n ), that is, using the right-equivariant formulation (5) of the uncertainties. The filter is defined in an analogous way to the previous Section IV-B. The resulting filter is yet different, and we call it Right-Unscented Kalman Filter on Lie Groups (Right-UKF-LG). Section V-D evidences that large discrepancies in performance between the Left and Right UKF-LGs can occur, so that both variants should generally be tested when facing a novel estimation problem on Lie groups.

V. RESULTS FOR EXPERIMENTAL DATA

To demonstrate the performances of the proposed method, we first apply the proposed Left-UKF-LG to the following nonlinear standard model of the two-dimensional differential drive car [START_REF] De Luca | Feedback Control of a Nonholonomic Car-like Robot[END_REF], and compare it to various algorithms of the literature. The velocity u n ∈ R is given by an odometer and we measure the angular velocity ω n ∈ R, e.g., by a differential odometer or a gyrometer. Taking into account the possible noise in the measurements, we get the discrete model

θ n = θ n-1 + ω n + w θ n , x 1 n = x 1 n-1 + cos (θ n ) u n + w l n -sin (θ n ) w tr n , x 2 n = x 2 n-1 + sin (θ n ) u n + w l n + cos (θ n ) w tr n , (33)
where w θ n , w l n and w tr t represent, respectively, the differential odometry error, the longitudinal odometry error and the transversal shift, whose introduction is thus physically motivated. We assume the vehicle obtains some noisy measurements of its position of the form

ỹn = x n + ṽn , ṽn ∼ N 0, σ 2 0 0 σ 2 , (34)
which typically models a GPS position in an outdoor application. The system can be embedded in the matrix Lie group = SE(2) as explained in Appendix A, using χ n as the state, and

ω n = ω n u n 0 and w n =   w θ n w l n w tr n   . ( 35
)
The equation governing the noisy system evolution then becomes and the observations have the equivalent form

χ n = χ n-1 exp (ω n + w n ) (36)
y n = x n + ṽn 1 = χ n 0 1 h( χ n ) + ṽn 0 vn . (37)

A. Experimental Setting

We then compare the various filters for the model described in Section V on real data obtained in an experiment conducted at the Centre for Robotics, MINES ParisTech. We used a socalled Wifibot, which is a small wheeled robot equipped with independent odometers on the left and right wheels, see Fig. 1. A set of seven highly precise cameras, the OptiTrack motion capture system, provide the reference trajectory (ground truth) θ n and x n with sub-millimeter precision at a rate of 120 Hz. We display the experimental trajectory in Fig. 2, corresponding to a random motion of 45 seconds. From this trajectory, we can determine the odometer noise characteristics to parametrize correctly the covariance matrix Q n .

The four filters to be compared are:

1) the standard EKF using the model original equations (33) with state variables x n and θ n ; 2) the standard UKF using equations (33) with state variables x n and θ n ; 3) the recent Invariant-EKF (IEKF), an EKF variant on Lie groups that comes with guaranteed convergence properties, see [START_REF] Barrau | The Invariant Extended Kalman Filter as a Stable Observer[END_REF] for its application to system (33)-(34), and with consistency properties [START_REF] Barrau | An EKF-SLAM Algorithm with Consistency Properties[END_REF]; 4) the proposed Left-UKF-LG of Section IV using the Lie group embedding (36) and state variable χ n .

B. Results

To compare the proposed Left-UKF-LG 2 with other methods for different levels of noise, we provide each filter with the raw odometer inputs ω n , and we artificially add Gaussian noise to the ground truth position measurements to simulate noisy position measurements, delivered to the filter at a rate of 1 Hz. Initially, we set for the filters both incorrect heading θ0 ∼ N 0, (π/2) 2 and position x1 0 ∼ N (0, 1/8), x2 0 ∼ N (0, 1/8). We then run 500 Monte-Carlo simulations for different levels of measurement noise σ 2 = 10 -5 , 10 -1 m 2 and compare the (average) Root Mean Square Errors w.r.t. the ground truth over the whole trajectory. Results are displayed in figures Fig. 3 and Fig. 4.

C. Discussion

For this trajectory, Left-UKF-LG and IEKF provide best position estimates at low noise. Regarding the heading, IEKF is slightly better than Left-UKF-LG at low noise but Left-UKF-LG becomes the most efficient filter when noise is moderate. We thus see that the Left-UKF-LG takes advantage of both the geometry-based structure of the IEKF, and the robustness of the unscented method. The following points seem important to us:

• The problem at hand (33) can be treated without the Lie group machinery, using x n and θ n as vector state 2 We similarly implemented the Right-UKF-LG but we show only Left-UKF-LG which is the most efficient for the measurement (34). variables. However, we see that embedding it into a Lie group framework yields more powerful variants of the Kalman filter (the IEKF and the Left-UKF-LG). • One could think the standard UKF systematically performs better than the standard EKF, especially when the measurement noise is high, due to its second order properties. This is generally true, but not always, as evidenced by the considered experiment. • On the other hand, it seems the Lie group counterpart of the UKF improves the Lie group counterpart of the EKF, that is, the IEKF.

D. Differences Between Left-UKF-LG and Right-UKF-LF

In the latter section, we focused on the comparison between Left-UKF-LG and some existing filters. We illustrate in this section that there may be discrepancies between the performances of Left-UKF-LG and Right-UKF-LF. To do so we still consider the system model (36) but with alternative measurements Y j n = R (θ n ) x n -p j + v j n , j = 1, . . . , J, where (38)

R (θ n ) = cos (θ n ) -sin (θ n ) sin (θ n ) cos (θ n ) , (39)
which represent a range-and-bearing observation of a sequence of known features located at p j ∈ R 2 for j = 1, . . . , J, and where we let R (θ) be the planar rotation of angle θ.

In our setting, we define J = 3 features p 1 = [1 2] T , p 2 = [-0.5 0] T and p 3 = [0 1] T . We then run 200 Monte-Carlo simulations for different levels of measurement noise σ 2 = 10 -5 , 10 -1 m 2 and compared the Root Mean Square Error w.r.t. the ground truth. Results are displayed in figures Fig. 5 and Fig. 6. They clearly reveal that the Right-UKF-LG outperforms both Left-UKF-LG and UKF in this particular case. In the absence of further insight, we thus recommend to the practitioner to design, implement and compare both UKF-LG variants, and then select the most efficient of them.

VI. CONCLUSION

We presented in this paper different solutions to Bayesian estimation problems on matrix Lie groups based on the UKF methodology, when the measurements consist of noisy partial measurements of the state, or full state noisy measurements of the state. We then applied these estimators to derive an UKF on Lie groups that comes in two variants. Experimental data combined with extensive Monte-Carlo simulations at various measurement noise levels illustrated the systematic superiority of the approach over the standard UKF on a robot localization example. Future work involves applications to drone navigation and to SLAM.

ACKNOWLEDGMENT

The authors would like to thank Martin BARCZYK, Tony NO ËL for their precious help with the experiments.

APPENDIX A SPECIAL EUCLIDEAN 2D GROUP

The Special Euclidean 2D SE(2) group represents rigid transformations in 2D space and has q = 3 parameters: heading θ and position x = x 1 x 2 T . The elements χ ∈ SE(2), ξ ∧ ∈ g, are given, respectively, as The exponential and logarithm maps are obtained as

exp (ξ) =   R ξ 1 V ξ ξ 2 ξ 3 0 1   , log ( χ ) = V -1 χ x θ , (41)
where R ξ 1 is defined in (39),

V ξ = 1 ξ 1 sin ξ 1 -1 + cos ξ 1 1 -cos ξ 1
sin ξ 1 and (42)

Vχ = 1 θ sin (θ) -1 + cos (θ) 1 -cos (θ) sin (θ) , (43)
and see [START_REF] Eade | Lie Groups for 2D and 3D Transformations[END_REF] for adjoint.

Fig. 1 :

 1 Fig. 1: Testing arena with Wifibot robot in the foreground of the picture. We can also see two of the seven Optitrack cameras in the background.

Fig. 2 :

 2 Fig. 2: Ground truth trajectory starting from the position (0, 0) and estimated trajectories with wrong initial positions. The noise is moderate to high (σ 2 = 10 -2 m 2 ). The standard UKF and EKF encounter serious difficulties at the beginning due to initial dispersion encoded in P0.

Fig. 3 :Fig. 4 :

 34 Fig. 3: Monte-Carlo average of the Root Mean Square Error on (θn) 1≤n≤N over the whole trajectory, as a function of the noise measurement variance σ 2 . Standard UKF and EKF achieve identical RMSE.

Fig. 5 :Fig. 6 :

 56 Fig.5: Root Mean Square Error on (θn) 1≤n≤N as a function of the noise measurement variance σ 2 . We clearly remark the superiority of the Right-UFK-LG compared to both Left-UFK-LG and UKF.

  ) -sin (θ) x 1 sin (θ) cos (θ)

  Algorithm 2: Unscented Kalman Filter on Lie Groups Input: χn-1 , P n-1 , Ω n , Q n , y n , vn , R n , α;

		Propagation
	1	λ = α 2 -1 2q ;	// scale parameter
	2 3	W j c = 1/2 λ+2q , j = 1, . . . , 4q ; P aug = diag (P n-1 , Q n );	// weights // augmented
		covariance matrix
		α j = col	(2q + λ) P aug	j	, j = 1, . . . , 2q,
		α j = -col	(2q + λ) P aug	j-2q	,
		j = 2q + 1, . . . , 4q;
	4 5	ξ T j χn = χn-1 Ω n ; w T j = α T j , j = 1, . . . , 4q; // mean update
	6 7	Ξ j = Ω -1 n exp (ξ j ) exp (ω n + w j ) , j = 1, . . . , 4q; P n = 4q j=1 W j c log (Ξ j ) log (Ξ j ) T ;
		Update
		Compute	χ+ n , P + n from Algorithm 1 with χn , P n ;
		Output:	χ+ n , P + n ;


Xavier Nicolas
email: xavier.nicolas@u-pem.fr

Eric Chénier

Chahinez Tchekiken

Guy Lauriat

GAS FLOWS WITH HEAT TRANSFER IN MICRO CHANNELS: CLARIFICATIONS ABOUT THE NUSSELT NUMBER

 are explained for the first time.

INTRODUCTION

Due to the increasing development of MEMS (Micro Electro Mechanical Systems), the study of liquid or gas flows and heat transfer in channels or ducts whose hydraulic diameter, D h , is of the order of a few microns (say 1 to 100 µm), has given rise to a considerable amount of works over the past twenty years. For gas flows at the microscopic scale, some observed phenomena require specific models [Gadel Hak 1999[START_REF] Karniadakis | Microflows and Nanoflows -Fundamentals and Simulation[END_REF]. In a layer called the Knudsen layer, whose thickness is of the order of O(λ) from the wall, where λ is the mean free path of the gas molecules, a state of local thermodynamic non-equilibrium exists. It results in non-linear mean velocity profiles and relations between stress and strain rates. From the continuum mechanics point of view, at the micro channel scale, when the Knudsen number is such that 0.001 < Kn = λ/D h < 0.1, these phenomena translate into a non-vanishing slip velocity, a temperature jump at the wall and a phenomenon called "thermal creep" [START_REF] Sharipov | Data on the velocity slip and temperature jump on a gas-solid interface[END_REF]]. The consequences of these phenomena on the macroscopic quantities such as the mass flow rate, the friction factor, the bulk temperature and the wall heat flux can be significant [START_REF] Lockerby | The usefulness of higher-order constitutive relations for describing the Knudsen layer[END_REF] and must be taken into account in the modelling of the convective heat transfer in MEMS with gas flows because they may have antagonistic effects on heat transfer.

For gas flows in micro channels of large aspect ratio, L/D h , and typical hydraulic diameter 1 µm ≤ D h ≤ 10 µm and length 100 µm ≤ L ≤ 1 mm, submitted to a moderate heating of the walls and to pressure variations between the inlet and the outlet of the channel of the order of one bar to a few bars, the conversion of the mechanical work of the viscous forces into internal energy and heat is very important. It can also be shown that the Mach number, Ma = u/(γrT) 1/2 , and the Brinkman number, Br = µu 2 /k∆T, can reach or exceed 1 and the Reynolds number, Re = uD h /ν, and Péclet number, Pe = uD h /α, can vary between 10 -2 and 10 2 . Accordingly, modelling this type of flow requires to take into account a wide range of physical phenomena: dynamic slip, temperature jump, thermal creep, gas compressibility, pressure work, viscous dissipation, work of the viscous forces at the wall, variation of the physical properties with temperature, conjugate heat transfer, relative roughness of the walls … A bibliographic review of the numerous studies on this subject has shown that all these effects are never taken into account simultaneously in the numerical models used.

One objective of this paper is thus to study the influence of different phenomena described above in the case of gas flows in 2D micro channels, heated or not. The analysis will particularly focus on the thermal aspects: we will analyse the pressure work (PW) and viscous dissipation (VD) in the energy equation and the power of the viscous stress (VSP) in the wall thermal boundary conditions on the evolution of the temperature field and the Nusselt number. A dimensional and asymptotic analysis will be performed to highlight the influence of each of these terms and potentially to simplify the mathematical model. Then the analytical solution of the simplified model established by Caï et al. [2007] for compressible gas convection in isothermal wall micro-channels is reminded. It is used to explain the vanishing values of the Nusselt number measured in the experiments by [START_REF] Demsis | Experimental determination of heat transfer coefficient in the slip regime and its anomalously low value[END_REF]Demsis et al. ( , 2010) [8,9] ) [8,9] and not explained till now. The analytical solutions of this simplified formulations is then compared with the numerical solutions of the full model.

GEOMETRY AND MATHEMATICAL MODEL

Geometry We consider gas flows between two parallel smooth walls of length L in the x-direction, separated by a distance H in the transverse y-direction. The hydraulic diameter is D h =2H. The flow is generated by a pressure difference, p in -p out > 0, between the inlet and outlet sections. The fluid is considered as an ideal gas with variable physical properties. The flow is assumed to be twodimensional, steady, compressible, laminar and symmetrical through the horizontal mid-plane (see Figure 1).

Governing equations

The governing equations are the mass, momentum and energy conservation equations in which the volume forces are neglected with respect to the viscous forces because the surface effects prevail in microfluidic. By denoting (u, v) the velocity components in directions (x, y) and p and T the local the pressure and temperature, the equations in Cartesian coordinates write:

(1)

(2) (3) (4)
where ρ is the density, µ the dynamic viscosity, C p the specific heat, k the thermal conductivity and the viscous dissipation is equal to:

(5)

The ideal gas law closes this system, with r the specific ideal gas constant: On the gas side of the impermeable wall at y=H/2 (denoted by the subscript g below), zero normal velocity is applied and, for the tangential velocity, the rarefaction effects are modelled using the firstorder Maxwell slip velocity and thermal creep boundary conditions (see [START_REF] Zhang | A review on slip models for gas microflows[END_REF] and [START_REF] Kandlikar | Heat transfer in microchannels -2012 status and research needs[END_REF]): (10) (11) where λ g is the mean free path at the wall defined as: (12) and σ µ and σ θ are the viscous and thermal slip coefficients [START_REF] Sharipov | Data on the velocity slip and temperature jump on a gas-solid interface[END_REF]]. In the present numerical applications, σ µ = σ θ = 1 which corresponds to fully accommodating walls (diffuse scattering walls).

For a channel with a uniform wall temperature T w , the thermal boundary condition writes:

(13) where ξ T is the temperature jump coefficient [START_REF] Sharipov | Data on the velocity slip and temperature jump on a gas-solid interface[END_REF]]. In the present applications, ξ T = 2.

When the Knudsen number and the wall slip are not zero, the total heat flux q w at the walls must be calculated as: (14) where k g (x) = k(T g (x)) and µ g (x) = µ(T g (x)) are the gas conductivity and viscosity at T g (x). The expression ( 14) of the wall heat flux was presumably introduced by [START_REF] Maslen | On heat transfer in slip flow[END_REF] then used by [START_REF] Sparrow | Laminar heat transfer in tubes under slip-flow conditions[END_REF]. Since then, only a few authors have taken into account the power of the viscous stress at the wall (second R.H.S. term) in their analysis of heat transfer. So, as most of the authors neglected this contribution, it was eventually forgotten. [START_REF] Hong | Some considerations on thermal boundary condition of slip flow[END_REF] reiterated its importance in a recent paper. We are going to analyse this contribution and show that it allows to recover the experimental results on very low values of the Nusselt numbers by [START_REF] Demsis | Experimental determination of heat transfer coefficient in the slip regime and its anomalously low value[END_REF][START_REF] Demsis | Heat transfer coefficient of gas flowing in a circular tube under rarefied condition[END_REF].

DIMENSIONAL ANALYSIS

Reference quantities The equations are made dimensionless using the following variables:

(15) (16)
where , ρ in , p in and T in are the mean velocity, density, pressure and temperature in the inlet section. µ in , C p,in and k in are the inlet viscosity, specific heat and thermal conductivity evaluated at T in . In micro channels, the streamwise pressure gradient balances the viscous term, so the equality of these two terms in the Navier-Stokes equation allows defining the reference pressure difference as: (17) where ε=D h /L is a small parameter for most micro channels. For a uniform wall temperature, T w , the reference temperature difference is chosen as:

∆T ref = T w -T in ( 18
)
Dimensionless equations and boundary conditions After introducing the dimensionless variables, the governing equations ( 1)-( 6) take the following form:

(21) (22) (23) (24)
where Φ v is the dimensionless form of the viscous dissipation term (Eq. ( 5)):

(25)

The equation of state (6) becomes: Dimensional analysis of PW, VD, VSP and TC The order of magnitude of the pressure work (PW), viscous dissipation (VD), viscous stress power (VSP) and thermal creep (TC) terms in the mathematical model can be evaluated as a function of the rarefaction effect (represented by Kn in ) and the compressibility effect (represented by Ma in ), for different heating rates (represented by ∆T ref /T in ). This analysis is limited to subsonic low rarefied gas flows: Pr in ∼ 1, Ma in < 1 and 10 -3 ≤ Kn in ≤ 0.1, for relative pressure and temperature differences such that ∆P ref /p in ≤ 1 and ∆T ref /T in ≤ 1, where ∆P ref and ∆T ref are defined by Eqs. ( 17) and ( 18). The typical micro-flows considered here are gas flows through long micro channels of hydraulic diameter D h ∼ 1 to 10 µm and length L ∼ 0.1 to 10 mm. The inverse of the longitudinal aspect ratio is therefore a small parameter: 10 -4 ≤ ε = D h /L ≤ 0.1.

As already discussed in Eqs. ( 24)-( 25), PW and VD are both of the order of Ec in /Re in . In Eq. ( 30), VSP is of the order of Ec in /Re in and, in Eq. ( 28), TC is of the order of εRe in Kn in 2 /Ec in but with a quite small constant in factor of this term since 2σ θ (γ-1)/πγ ∼ 0.1 to 0.2. Thus, by comparing the order of magnitude of these different terms (in terms of power of the small parameterε) with the other terms in the equation in which they appear, it can be shown that, whatever

Kn in for O(ε 2 ) ≤ Kn in ≤ O(ε 1/2 ),

ANALYTICAL SOLUTION OF THE ASYMPTOTIC MODEL

Solutions for the velocity and pressure fields In this section, following the asymptotic analyses by [START_REF] Arkilic | Gaseous slip flow in long microchannels[END_REF] and [START_REF] Cai | Gas flows in microchannels and microtubes[END_REF], analytical solutions of simplified asymptotic models for long micro channels are calculated in micro channels in the case of isothermal walls. The asymptotic analysis is carried out for Ma in ∼ O(ε) and ∆T ref /T in ≤ O(1), for Kn in ≤ O(ε 1/2 ). It can be shown from Eqs. ( 31) and ( 33) that this corresponds to O(ε 1/2 ) ≤ Re in ≤ O(ε -1 ) and ∆P ref /p in ≤ O(ε 1/2 ). This parameter range is chosen because, from the previous section, the TC, PW, VD and VSP terms could potentially be negligible and simplify the solution.

Considering weakly rarefied flows with small temperature variations, the dimensionless quantities can be written as asymptotic expansion of ε as follows:

(35) For isothermal walls, the dimensionless temperature is supposed constant, equal to θ = 1, at the zero order of ε. The temperature expansions are thus compatible with flows at constant or slowly varying temperature. As a consequence and to simplify the analytical solution of the asymptotic equations, the physical properties are considered as constant:

(36) Next, the asymptotic expansions (35) are injected into the dimensionless conservation equations and boundary conditions ( 21)-(30). Then they can be written at the zero or first order in ε, and simplified by accounting for the ranges of variations of the dimensionless parameters expressed as a function of ε. It can be shown that the asymptotic model to compute the velocity and pressure fields of the present problem is nearly identical to the asymptotic model in the case of a fully isothermal flow solved by [START_REF] Arkilic | Gaseous slip flow in long microchannels[END_REF]. Thus by noting (37) the state equation for the ideal gas becomes:

That is, for rarefied forced convection in micro-channels at constant wall temperature (for Kn in > 0), even if the conduction heat flux and the power of the viscous stress at the wall do not vanish, the total heat flux and the Nusselt number always vanish. This is valid far from the thermal entrance effects that can exist on a very short length. This asymptotic analysis could then explain the very small values of the Nusselt numbers measured by [START_REF] Demsis | Experimental determination of heat transfer coefficient in the slip regime and its anomalously low value[END_REF][START_REF] Demsis | Heat transfer coefficient of gas flowing in a circular tube under rarefied condition[END_REF] in their experiments. For four rarefied gas flows (nitrogen, oxygen, argon and helium) in the ranges of Knudsen number 0.0022-0.032 and Reynolds numbers 0.13-14.7, the measured Nusselt numbers vary between 6.2×10 -5 and 2.8×10 -2 which is four to five orders of magnitude smaller than the corresponding values in the continuum regime (when convection prevails). Till now, these very small values of the Nusselt number had never been explained theoretically because only the contribution of the diffusion heat flux at the wall was considered in the Nusselt number definition.

COMPARISON OF THE ANALYTICAL AND NUMERICAL SOLUTIONS

Numerical method. An in-house finite volume code is used to solve the full model (Eqs. (1-14)) discretized on a Cartesian grid. A second order centered scheme is used for the diffusive and convective terms because the maximum Reynolds number is Re max <15 and the cell Reynolds number is Re ∆x <1 (no numerical oscillation is observed in the discrete solutions). The discrete nonlinear steady equations are solved in a coupled way by Newton's algorithm. The mesh size on each space direction is N x ×N y = 6000×60 for a 100-aspect-ratio computational domain. The mesh is uniform on x-direction and refined near the wall on y-direction, with a size ratio of two successive cells equal to 0.97. A sensitivity analysis to mesh refinement has shown that the solutions are well converged in all simulated cases. The code has been validated through comparisons with experimental and numerical results [START_REF] Tchekiken | Modélisation et simulations numériques d'écoulements compressibles dans des micro-conduites planes[END_REF][START_REF] Tchekiken | Simulation numérique d'écoulements anisothermes de gaz en micro canal : effets de compressibilité et de glissement, 21 ème Congrès Français de Mécanique[END_REF][START_REF] Tchekiken | Modélisation et simulations numériques d'écoulements compressibles dans des micro-conduites planes[END_REF].

Studied cases

To compare the analytical and numerical solutions, the numerical simulations are performed for a nitrogen flow at T in =270 K in a micro-channel of length L=100 µm and height H=3 µm, with walls at T w = 300 K. The physical properties of the gas are considered as constant. Their values considered at T w are gathered in Table 1. Three cases, presented in Table 2, are simulated:

-case (a) is at low pressure and corresponds to a quasi incompressible flow but with a strong slip velocity at the walls all along the channel (relatively large Kn values at inlet and outlet); -case (b) is at a large pressure difference between inlet and outlet and corresponds to a strongly compressible flow (Ma out =0.6554) and a strong slip at channel outlet (Kn out =0.1135); -case (c) is at high pressure and corresponds to an incompressible flow with a very low slip along the channel (small Kn values at inlet and outlet).

Table 1 Nitrogen physical properties at

T w =300 K k [W/m.K] C p [J/kg.K] µ.10 5 [Pa.s] Pr r [J/kg.K] γ=C p /C v σ µ σ θ ξ T 0.

Comparisons of the velocity and temperature fields

The comparison of the analytical and numerical solutions of the horizontal, u, and transversal, v, velocity components and the temperature, T, are presented in Figures 234for the cases (a)-(c) respectively. Table 3 presents the L 2 -norm of the relative error on u, p, and T between the analytical and numerical solutions. Globally, the analytical and numerical solutions for u, p and T are in a good accordance since the maximum relative error is 6.3% on u in case (b). The absolute values and errors on v are very small in the present problem. The errors are globally located at the channel entrance and exit regions: in the former case it is due to the analytical solution that does not take into account the entrance effects and, in the latter case, it is due to the zero gradient outflow boundary condition applied in the numerical simulation which is not adapted for small Reynolds numbers. is not satisfied and the analytical solution is no longer accurate. In the former case, there is a thermal entrance length in the numerical solution since the inlet temperature is T in =270 K, while this thermal entrance is not taken into account by the analytical solution. Using Figures 234and the Nusselt number profiles, it is observed that the length of the thermal entrance is very short, equal to 3H at minimum in the low Reynolds number case (a) (Figure 5) and to 5H at maximum in the higher Reynolds number case (c) (Figure 7).

Analysis of the Nusselt number

The local wall Nusselt number, Nu = Nu diff + Nu VSP , is defined by Eq. ( 55). It is the sum of two contributions: the first R.H.S term of Eq. ( 55), Nu diff , is due to the diffusive flux in the wall boundary layer and the second R.H.S term, Nu VSP , is due to the power of the viscous stress at the walls. The average of Nu along the wall is denoted by Nu av and its values are given in Table 2. They are very low (Nu av ~ 0.1) compared with the well-known value of Nu=7.541 calculated for a fully-established incompressible flow at large Péclet numbers between two isothermal parallel plates.

Figures 567show that the absolute values of the magnitudes of both Nu diff and Nu VSP increase by increasing the Knudsen number (see the downstream part of Figure 5 where Kn increases and compare Figures 5 and7) and increase with increasing Mach numbers (see the downstream part of Figure 6 where Ma increases and compare Figures 6 and7). On the other hand, the total Nusselt number, Nu = Nu diff + Nu VSP , is very small in the whole channel except in the entrance where the flow is not locally thermally-established and close to the outlet section due to the outlet B.C. perturbations. In the channel core, Nu nearly vanishes because Nu VSP is nearly the opposite of Nu diff (note that it is "-Nu diff " that is plotted on Figures 567). These graphs could then explain why a vanishing Nusselt number is measured in the experiments by [START_REF] Demsis | Experimental determination of heat transfer coefficient in the slip regime and its anomalously low value[END_REF][START_REF] Demsis | Heat transfer coefficient of gas flowing in a circular tube under rarefied condition[END_REF] in case of rarefied flows at small Reynolds numbers.

CONCLUSION

In this communication, the mathematical model usually used for weakly rarefied flows with first order slip and thermal jump at the walls has been analysed in the case of the forced convection of gas in long micro channels between two isothermal infinite plates. The order of magnitude of the pressure work (PW) and viscous dissipation (VD) in the bulk flow and the order of magnitude of the power of the viscous stress and thermal creep at the walls have been determined with respect to the flow parameters (Re, Ma, Kn) and the small parameter ε=D h /L. It has been shown that PW and VD have the same order of magnitude and cannot be neglected when the Mach number is high and the wall slip is important.

The analytical asymptotic solutions of the flow and thermal fields established by [START_REF] Cai | Gas flows in microchannels and microtubes[END_REF] for isothermal wall micro-channels is compared with the solution of the full model by a finite volume method. This numerical solution takes into account the first order slip and thermal jump conditions at the walls, the power of the viscous forces, the thermal creep, the pressure work and the viscous dissipation. Three different flow types have been simulated: the first one is very slipping, the second highly compressible and the last one incompressible and non-slipping. In the three cases the analytical and simulated solutions agree well except the spanwise velocity due to entrance and outflow effects.

The analytical and numerical solutions are used to analyse the Nusselt number values and profiles. A thorough analysis of the total heat flux at the walls shows that the contribution of the power of the viscous stress at the walls must not be neglected because it is opposite to and of the order of magnitude of the diffusive flux at the walls. As a consequence, the total wall heat flux and Nusselt number tend to zero all the more so as rarefaction effects are important and the Reynolds number is small. This could then explain the very small values of the Nusselt number measured in the experiments by [START_REF] Demsis | Experimental determination of heat transfer coefficient in the slip regime and its anomalously low value[END_REF][START_REF] Demsis | Heat transfer coefficient of gas flowing in a circular tube under rarefied condition[END_REF].

Figure 1 .

 1 Figure 1. Geometry and boundary conditions for gas flows in 2D heated micro channels

  Boundary conditionsThe associated boundary conditions are the following. At the inlet section x=0, uniform pressure and temperature are applied: (7) At the outlet section x=L, the pressure is fixed and fully developed flow conditions are applied: (8) Symmetry boundary conditions are applied on the channel axis at y=0: (9)

  these equations, Re in = ρ in D h /µ in , Pr in = µ in C p,in /k in and Ec in = /C p,in ∆T ref are the Reynolds, Prandtl and Eckert numbers at the inlet section. The energy equation (24) shows that the pressure work (PW) and viscous dissipation terms (VD) (the last two terms) are both in the order of Ec in /Re in , considering that, from the continuity equation (21), V∼ε in the PW term. The ratio PW/VD is therefore of the order of -1 since PW is negative and VD is positive. Furthermore, by introducing the Mach number Ma in 2 = ρ in /γp in = /γrT in and the Knudsen number Kn in = λ in /D h at inlet, the sum of PW and VD is proportional to the product Ma in Kn in since it can be shown that: (27) Thus PW and VD can be neglected at the same time only when the gas flow is incompressible or weakly compressible (Ma in <<1) and for no slip gas flows (Kn in <<1). The dimensionless form of the boundary conditions is: number is a function of the Reynolds and Knudsen numbers : (33) then the number of independent parameters in the above dimensionless conservation equations and boundary conditions is 10: (34) Note that due to the conservation of the mass flow rate, the Reynolds number is constant in the whole channel: Re=Re in . On the other hand, Ma and Kn vary along the flow such as, from Eq. (33): Ma/Ma in = Kn/Kn in .

Figs 2 -

 2  show a good agreement between the solutions except at the inlet for the temperature field and at the outlet in the compressible case (b). In the latter case, as the Reynolds and Mach number are large (Re in =7.8 and Ma out = 0.66), the validity domain of the asymptotic analysis (Re in ~ O(1), Ma in ~ O(ε)) is not satisfied and the analytical solution is no longer accurate. In the former case, there is a thermal entrance length in the numerical solution since the inlet temperature is T in =270 K, while this thermal entrance is not taken into account by the analytical solution. Using Figures2-4and the Nusselt number profiles, it is observed that the length of the thermal entrance is very short, equal to 3H at minimum in the low Reynolds number case (a) (Figure5) and to 5H at maximum in the higher Reynolds number case (c) (Figure7).

Figure 2 .

 2 Figure 2. Comparison of the analytical solution (black isolines) and numerical solution (color fields) for u, v and T, in case (a).

Figure 3 .

 3 Figure 3. Same as Figure 2 but for case (b) and with a zoom on the outlet zone for each field (see coordinates on the abscissa axes).

Figure 4 .

 4 Figure 4. Same as Figure 2 but for case (c).

Figure 5 .

 5 Figure 5. Profiles of the total (Nu), diffusive (Nu diff ) and VSP (Nu VSP ) contributions of the Nusselt number with respect to X=x/L, simulated in case (a) and zoom on the Nu profile (on the right).

Figure 6 .

 6 Figure 6. Profiles of the total (Nu), diffusive (Nu diff ) and VSP (Nu VSP ) contributions of the Nusselt number with respect to X=x/L, simulated in case (b) and zoom on the Nu profile (on the right).

Figure 7 .

 7 Figure 7. Profiles of the total (Nu), diffusive (Nu diff ) and VSP (Nu VSP ) contributions of the Nusselt number with respect to X=x/L, simulated in case (c) and zoom on the Nu profile (on the right).

  PW, VD and VSP are negligible for Ma in ≤ O(ε); this approximately corresponds to Re in ≤ O(1). On the other hand, TC is negligible for Ma in ≥ O(ε), that is for approximately Re in > O(1).

Table 3 L

 3 2 -norm of the relative errors on u, p, and T between the analytical and numerical solutions

	024712 1032.48	1.6588	0.693	296.8	1.4	1	1	2
					Table 2
	Imposed inlet conditions and computed outlet conditions of each simulated case
	Case	p in	p out	∆p io	Kn in	Re in =	Ma out	Kn out	Nu av
		[bar]	[bar]	[bar]		Re out
	(a)	0.5	0.1	0.4 0.0196 1.038 0.0886 0.1031 0.108
	(b)	1.5	0.1	1.4 0.00654 7.833 0.6554 0.1135 0.147
	(c)	5	4.6	0.4 0.00196 12.05 0.0272 0.00225 0.221

 (38)The analytical solutions for the velocity and pressure fields are V 0 =0 and:

(39) (40) (41) with :

(42) (43) To allow the integration of the x-momentum equation to get the solution (39-43), the convective term and the thermal creep term were neglected. The asymptotic analysis indicates that the first approximation is a priori valid for Re in ≤ O(1).

Solution for the temperature field

The asymptotic solution of the temperature field for an isothermal wall micro channel was computed by [START_REF] Cai | Gas flows in microchannels and microtubes[END_REF]. This solution is adapted here to the present nondimensional formalism. For Re in ≤ O(1), it can be shown that the convective term and the PW, VD and VSP terms are all at least one order smaller than the diffusive term of the energy equation. However, the simplified problem, reduced to a pure diffusion problem is not compatible with the boundary conditions and cannot be solved. So the asymptotic thermal problem is relaxed by taking into account the PW, VD and VSP terms. Thus the energy equation for θ 1 (X, Y) is taken as follows:

(44)

The symmetry boundary condition is considered on channel axis (at Y=0) and the temperature jump condition at the wall (at Y=1/4):

Then the temperature solution is given by: where, from Eq. ( 30), the heat flux at the walls is given by: ( 53) and ( 54) Therefore (55) From Eq. ( 46), it can be shown that the conductive part of Nu(X) is equal to:

(56) and, from Eq. ( 39), the VSP part of Nu(X) is equal to:

(57) It is remarkable that the viscous stress power at the wall is the opposite of the conduction heat flux (compare Eqs. ( 56) and ( 57)): both terms have opposite behaviours or vanish at the same time. This can be explained by the simplification of the energy equation ( 44) resulting from the asymptotic analysis. The convective term is not considered in (44) and, as a consequence, the thermal entrance effects and the streamwise variation of the temperature from inlet are neglected. When Kn in = 0 the two terms ( 56) and (57) vanish because A = 0 (see Eq. ( 50)). In this case, the power of the viscous stress at the wall is zero (no slip) and, as shown previously, the PW and VD terms compensate each other. Therefore, the temperature is equal to the wall temperature T w in the whole micro channel when A=0. On the other hand, when Kn in ≠ 0, the conduction heat flux close to the wall and the power of the viscous stress at the wall are not zero due to the slip and the temperature jump.

The other consequence of the opposite values of ( 56) and ( 57) is that the total heat flux at the wall (Eq. ( 53)) and the total Nusselt number (Eq. ( 55)) are zero: q w = 0 and Nu(X)=0

(58)