Riccati observers for position and velocity bias estimation from direction measurements

Tarek Hamel and Claude Samson

Abstract-This paper addresses the problem of estimating the position of a body moving in n (≥ 2)-dimensional Euclidean space using body velocity measurements and the measurements of direction(s) between the body and one (or several) source point(s) whose location(s) is (are) known. The proposed solutions exploit the Continuous Riccati Equation (CRE) to calculate observer gains yielding global uniform exponential stability of zero estimation errors, even when the measured body velocity is biased by an unknown constant perturbation. These results are obtained under persistent excitation (p.e.) conditions depending on the number of source points and body motion that ensure both uniform observability and good conditioning of the CRE solutions. With respect to previous contributions on the subject the proposed framework encompasses the static case, when the body is motionless and at least two source points are needed to recover its position, and the non-static case, when body motion and a single source are sufficient. Simple and explicit observability conditions under which uniform exponential stability is achieved are also worked out for each case. Simulation results illustrate the performance of the proposed observers.

I. INTRODUCTION

The general problem of estimating the position, or the complete pose (position and orientation), of a body relatively to a certain spatial frame is central for a multitude of applications. This is common knowledge. Among all sensing modalities that can be used to acquire the necessary information, source points direction (or bearing) measurements has early motivated many studies, in particular for pose estimation when body and source points are motionless in the frame of interest, a problem referred to as the Perspectiven-Point (PnP) problem in the dedicated literature [START_REF] Haralick | Review and analysis of solutions of the three point perspective pose estimation problem[END_REF]. The present paper focuses on the sole estimation of the body position. This corresponds to applications for which the body's attitude is either of lesser importance or is estimated by using other sensing modalities. In this case, iterative and recursive methods using ongoing measurements information, by opposition to non-iterative ones based on the algebraic resolution of a finite set of constraint equations, are all the more interesting that their domain of convergence can be global. The reason is that, without the compact group of rotations being involved, this simplified problem is amenable to exact linearisation and can be associated with globally convex cost functions, as shown further in the paper. Another advantage of iterative methods is that they are naturally suited to handle the non-static case, i.e. when either the body T. Hamel is with I3S UNS-CNRS, Nice-Sophia Antipolis, France, thamel@i3s.unice.fr.

C. Samson is with INRIA and I3S UNS-CNRS, Sophia Antipolis, France, claude.samson@inria.fr, csamson@i3s.unice.fr.

or the point source(s) move(s), by using on-line the extra data and information resulting from motion. In particular, the observation of a single point source may be sufficient in this case, provided that the body motion regularly grants a sufficient amount of "observability". This possibility has been studied recently in [START_REF] Batista | Globally exponentially stable filters for source localization and navigation aided by direction measurements[END_REF] where the problem is linearised by considering an augmented state vector. Another solution, not resorting on state augmentation, is proposed in [START_REF] Bras | Observer design for position and velocity bias estimation from a single direction output[END_REF]. The present paper offers a generalization of previous studies on this subject that encompasses the static and non-static cases with an arbitrary number of source points.

For five decades, Kalman filters [START_REF] Kalman | A new approach to linear filtering and prediction problems[END_REF] for linear systems, and their extensions to non-linear systems known as Extended Kalman Filters (EKF) [START_REF] Jazwinski | Stochastic process and filtering theory[END_REF], have consistently grown in popularity near engineers with various backgrounds (signal processing, artificial vision, robotics,...) to address a multitude of iterative state estimation problems involving additive "noise" upon the state and/or the measurements. The optimality of these filters in a stochastic framework under specific noise conditions and assumptions, and their direct applicability to Linear Time-Varying (LTV) systems, have undoubtedly contributed to this popularity. It is however important to keep in mind, or to recall, that the stability and robustness properties associated with them, i.e. features that supersede conditional stochastic optimality in practice, are not related to stochastic issues. They result from properties of the associated deterministic continuous-time (or discrete-time, depending on the chosen computational framework) Riccati equation that underlies a (locally) convex estimation error index (or Lyapunov function) and a way of forming recursive estimation algorithms that uniformly decrease this index exponentially (under adequate observability conditions). With this perspective, Kalman filters belong to the (slightly) larger set of Riccati observers that we intentionally derive here in a deterministic framework, knowing that a complementary stochastic interpretation may be useful to subsequently tune the Riccati equation parameters and observer gains. We also believe that, by contrast with standard Kalman filter derivations, this approach allows one to better comprehend how the system observability properties are related to the good conditioning of the Riccati equation solutions and to the observer's performance (the rate of convergence to zero of the estimation errors, in particular) via a Lyapunov analysis.

The paper is organized along five sections. Following the present introduction, Section II specifies some of the notation used throughout the paper, recalls the equations of a generic Riccati observer associated with a LTV system, as well as a few central properties of the corresponding CRE, and states complementary technical results tailored for the design and analysis of the observers here proposed. Position estimation from direction measurements in n (≥ 2)-dimensional Euclidean space is specifically treated in Sections III. A single source point is first considered, with stability and convergence of the observer relying on persistent excitation properties associated with the body motion. Solutions are then generalized to the case of multiple source points, with the augmentation of the number of these points reflecting in the gradual weakening of the body motion conditions needed to ensure uniform exponential stability. While measuring the body velocity is central to estimate the position, we show how to modify the observers via state augmentation when velocity measurement are biased by a constant vector. We also show that uniform exponential stability is preserved under the same observability conditions. Illustrative simulation results are presented in Section IV, followed by a short section V of concluding remarks.

II. RICCATI OBSERVERS

Throughout the paper the following notation is used:

• A(t)

A. Riccati observers

Consider a generic linear time-varying (LTV) system

ẋ = A(t)x + B(t)u y = C(t)x (1)
with x ∈ R n the system state vector, u ∈ R s the system input vector, and y ∈ R m the system output vector. We here call Riccati observer any observer of System (1) of the form

ẋ = A(t)x + B(t)u + K(t)(y -C(t)x) ; x(0) ∈ R n (2)
with the observer gain given by

K(t) = k(t)P (t)C (t)Q(t) ; k(t) ≥ 0.5 (3)
where P (t) is the solution to the so-called Continuous Riccati Equation (CRE)

Ṗ = A(t)P + P A (t) -P C (t)Q(t)C(t)P + V (t) (4)
with P (0) any p.d. matrix and Q(t), V (t) p.s.d. matrices that have to be specified. Note that the optimal Kalman gain in the stochastic setting where the matrices V (t) and Q -1 (t) are p.d. matrices and interpreted as covariance matrices of additive noise on the system state and output is obtained by taking k(t) = 1.

Let us now quickly recall how the stability and convergence properties of a Riccati observer is directly related to the properties of the solution P (t) to the CRE. Define the estimation error x := xx, from (1) and ( 2) one obtains the error equation

ẋ = (A(t) -K(t)C(t))x (5)
Assume (for the time being) that P (t), which is a symmetric matrix by construction, is well defined on R + and is p.d., so that its inverse is also well defined and p.d., and consider the candidate Lyapunov function

V(t) = x (t)P -1 (t)x(t).
Then, using the fact that Ṗ -1 = -P -1 Ṗ P -1 , the timederivative of P -1 satisfies the equation

Ṗ -1 = -P -1 A(t)-A (t)P -1 +C (t)Q(t)C(t)-P -1 V (t)P -1 (6)
and, using ( 3) and ( 5), one easily verifies that the timederivative of V(t) is given by

V = -x (2k(t) -1)C (t)Q(t)C(t) + P -1 V (t)P -1 x ≤ -x P -1 V (t)P -1 x ≤ - p 2 m p M vmV (≤ 0) (7)
so that V(t) ≤ V(0)exp(-

p 2 m p M v m t).
To conclude that x = 0 is globally uniformly exponentially stable it thus suffices to choose V (t) > v m I d with v m > 0 and to show that P (t) i) is always well-defined, ii) that it is p.d., and -most importantlyiii) that it is well conditioned in the sense that p m is strictly positive and p M is finite so that the ratio P M p 2 m is bounded. The central issue of boundedness and good conditioning of P (t) brings us to recall classical, and also point out less known, results concerning the CRE.

B. Properties of the Continuous Riccati Equation

The first results concerns the existence of the solutions to the CRE for t ∈ [0, +∞).

Lemma 2.1: If P (0) is p.d. and Q(t) and V (t) are p.s.d, then P (t) is p.d. and well defined on [0, +∞). The proof is given in the appendix A. Now, to ensure boundedness and good-conditioning of the solution P (t) to the CRE one has to impose other conditions upon the terms entering the equation. Sufficient conditions are pointed out in the next lemma.

Lemma 2.2: Define:

W V (t, t + δ) := 1 δ t+δ t Φ(t, s)V (s)Φ (t, s)ds (8)
and

W Q (t, t + δ) := 1 δ t+δ t Φ (s, t)C (s)Q(s)C(s)Φ(s,
such that W V (t, t + δ) ≥ µ v I d and W Q (t, t + δ) ≥ µ q I d ,
∀t, then the solution P (t) to the CRE ( 4) is bounded and well-conditioned in the sense that 0 < p m ≤ p M < ∞.

A proof of this result is, for instance, given in [START_REF] Pengov | On the boundedness of the solutions of the continuous riccati equation[END_REF] where lower and upper bounds of P (t) are also derived. In this reference the matrix Q(t) is in fact assumed p.d. because the inverse of Q is (for technical convenience) used in the proof. However, it is simple to verify that the proposed bounds for P (t) do not depend on the smallest eigenvalue of Q(t), so that these bounds are also valid when Q(t) is only p.s.d.

From now on, and by analogy with the classical observability Grammian

W (t, t + δ) := 1 δ t+δ t Φ (s, t)C (s)C(s)Φ(s, t)ds W Q is called Riccati observability Grammian. It coincides with W when Q(t) = I d . Note that if Q(t) ≥ I d > 0 and the observability Grammian W is positive then the Riccati observability Grammian W Q is also positive. This is just a consequence of that W Q (t, t + δ) ≥ λ min (Q(t))W (t, t + δ).
Also, when the output matrix C(t) varies with the the state trajectory x(t), sup(µ q ) may be interpreted as a measure of persistent excitation (p.e.) associated with this trajectory and with respect to which, given V (t) and Q(t), the lower bound of the observer's exponential convergence rate given by ( 7) varies monotically. The larger this number, the faster the observer's rate of convergence. This relation can be important for practical purposes when one has the possibility of modifying the state trajectory. Define the p.s.d. projection matrix operator Π y(t) := I dy(t)y (t) with y(t) ∈ R n and such that |y(t)| = 1 (i.e. y(t) ∈ S n-1 ). This operator plays a central role for the design of the observers proposed in the next section. In particular, when a single source point is used and the body velocity is measured with no bias, we will see that a possible modelling of the system's equations involves the matrices (A, C(t)) = (0 n×n , Π y(t) ) with y(t) denoting the unit vector along the direction between a source point and the body. Then Φ(s, t) = I d and the positivity condition upon the Riccati observability Gramian is satisfied provided that Q(t) is chosen larger than an arbitrarily small positive matrix and there exist positive numbers δ and such that ∀t :

1

δ t+δ t Π y(s) ds ≥ I d > 0 (10)
In this case it suffices to choose V (t) larger than an arbitrarily small positive matrix to ensure that the solution P (t) to the CRE is bounded and well conditioned. Note that if u(t) is differentiable and | u(t)| is bounded then (10) is satisfied provided that | ẏ(t)| is regularly larger than some positive number. Note also that the constant matrix P = I d is a solution to the CRE when choosing V (t) = Π y(t) Q(t)Π y(t) (which is only p.s.d). Although this particular solution holds independently of the persistent excitation condition (10), in view of ( 5) and ( 7) the exponential stability of the corre-sponding Riccati observer still depends on the satisfaction of this condition. The extension of these results to the biased velocity case involves the following technical lemma.

Lemma 2.3:

If 1) A = 0 n×n I d 0 n×n 0 n×n 2) C (t)Q(t)C(t) =
∆(t) 0 n×n 0 n×n 0 n×n with ∆(t) a p.s.d bounded matrix-valued function such that, for some δ > 0 and µ > 0:

1 δ t+δ t ∆(s)ds > µI d (∀t)
then there exist δ > 0 and > 0 such that

W Q (t, t+ δ) > I d (∀t).
The proof is given in the appendix B.

III. OBSERVERS FOR POSITION ESTIMATION FROM DIRECTION MEASUREMENTS

In the case of a single source point that may arbitrarily be taken as the origin of a fixed frame, the problem consists in estimating the position x of a body (or object) with respect to (w.r.t.) this frame given its velocity u and the measurement of the direction x/|x| between the source point and the body, knowing that the measured velocity may be biased by an unknown constant vector a. A situation that may, for instance, explain such a bias is when the body is a vehicle (ship, submarine, aeroplane,...) whose velocity is measured w.r.t. the ambient fluid (water or air), itself moving with a constant unknown velocity (sea-current or wind) w.r.t. an inertial frame. In practice, x will be a twodimensional vector of coordinates (in the 2D, or planar, case) or a three-dimensional vector of coordinates (in the 3D, or spatial, case). For the sake of generality, we assume here that x ∈ R n , with n ≥ 2. The corresponding modelling equations are

ẋ = u + a ȧ = 0 0 = Π y(t) x (11)
with y(t) := x(t)/|x(t)|. Let us distinguish two cases, depending on whether the velocity measurement is unbiased, i.e. a = 0, or is biased by an unknown constant vector a.

A. The unbiased velocity case

In this case the modelling equations can also be written as

ẋ = Ax + u 0 = C(t)x (12)
with A = 0 n×n -the (n × n)-dimensional null matrixand C(t) = Π y(t) . This system can be associated with the following Riccati-like observer

ẋ = Ax + u + K(t)(0 -C(t)x) (13)
with the observer gain K(t) calculated as in relation (3) from the solution to the CRE (4). The resulting observer writes as

ẋ = u -K(t)Π y(t) x (14)
with K(t) = k(t)P (t)Π y(t) Q(t) and P (t) the solution to the CRE Ṗ = -P Π y(t) Q(t)Π y(t) P + V (t)

Since ẋ = (A -K(t)C(t))x the Lyapunov analysis of section II-A applies, and global uniform exponential stability of x = 0 is obtained provided that P (t) is bounded and well-conditioned. From explanations given in the previous section such is the case if Q(t) and V (t) are chosen positive (and larger than an arbitrarily small positive matrix) or if V (t) is equal to Π y(t) Q(t)Π y(t) and, most importantly, if the p.e. condition (10) is satisfied. A loose interpretation of this condition is that the body must keep moving and not always in the direction of the source point. Note that, in the case where

V (t) = k v Π y(t) and Q(t) = k q I d , choosing the constant solution P = (k v /k q ) 0.5 I d simplifies the observer equation to ẋ = u -k(t) k q k v Π y(t)
x, so that one recovers the solution proposed in [START_REF] Bras | Observer design for position and velocity bias estimation from a single direction output[END_REF].

B. The biased velocity case

In this more difficult case the velocity bias a has to be estimated as well. The modelling equations (11) can be written in the state form as

Ẋ = AX + ū 0 = C(t)X (15)
with X := [x , a ] the 2n-dimensional extended state vector, ū := [u , 0 1×n ] , and

A = 0 n×n I d 0 n×n 0 n×n , C(t) = Π y(t) 0 n×n
Consider now the CRE

Ṗ = AP + P A -P C (t)Q(t)C(t)P + V (t) (16)
with P (0) a p.d. matrix, V (t) larger than some arbitrarily small positive matrix, and Q(t) a p.d. matrix-valued function larger than an arbitrarily small positive matrix. By application of Lemma 2.3, using the fact that

C (t)Q(t)C(t) = ∆(t) 0 n×n 0 n×n 0 n×n with ∆(t) := Π y(t) Q(t)Π y(t)
, the solution P (t) to this CRE is bounded and well-conditioned provided that the p.e. condition (10) is satisfied. Denoting the estimate of X as X and the estimation error as X := X -X, this in turn implies that the following Riccati observer

Ẋ = A X + ū + K(t)(0 -C(t) X) (17)
with X := [x , â ] and K(t) = k(t)P (t)C (t)Q(t) (k(t) ≥ 0.5), globally uniformly exponentially stabilizes X = 0 provided that the p.e. condition (10) is satisfied. This observer can also be written as

ẋ = u + â -k(t)P 11 (t)Π y(t) Q(t)x ȧ = -k(t)P 21 (t)Π y(t) Q(t)x (18)
with P ij (i, j ∈ {1, 2}) denoting the block components of P with adequate dimensions.

C. Extension to multiple direction measurements

We consider now the problem of estimating a vector x from l direction measurements y i = x-zi |x-zi| , i ∈ {1, . . . , l} with x denoting the vector of coordinates of the position of a body w.r.t. an inertial frame, and z i the known vector of coordinates of the ith source point. Setting X := [x , a ] , ū := [u , 0 1×n ] , and y = [(Π y1(t) z 1 ) , . . . , (Π y l (t) z l ) ] , one obtains the system

Ẋ = AX + ū y = C(t)X (19)
with

A = 0 n×n I d 0 n×n 0 n×n and C(t) =    Π y1(t) 0 n×n . . . . . . Π y l (t) 0 n×n   
Consider now the CRE

Ṗ = AP + P A -P C (t)Q(t)C(t)P + V (t)
with P (0) a p.d. matrix, V (t) larger than an arbitrarily small positive matrix, and

Q(t) =      Q 11 (t) 0 n×n . . . 0 n×n 0 n×n Q 22 (t) . . . 0 n×n . . . . . . . . . 0 n×n 0 n×n . . . Q ll (t)     
with Q ii (t) (i = 1, . . . , l) p.d. matrix-valued functions larger than an arbitrarily small positive matrix. The solution P (t) to this equation is bounded and well conditioned provided that the corresponding Riccati observation Grammian W Q is positive. Using the fact that

C (t)Q(t)C(t) = ∆(t) 0 n×n 0 n×n 0 n×n with ∆(t) := l i=1 Π yi(t) Q ii (t)Π yi(t)
, this condition is again satisfied by application of Lemma 2.3 provided that, for some δ > 0 and for all t > 0, 1

δ t+δ t ∆(s)ds, with ∆(t) := l i=1 Π yi(t)
, is greater than an arbitrarily small p.d. matrix. This p.e. condition clearly points out the interest of using multiple direction measurements in order to weaken, or even remove, conditions upon x and its time-variations. For instance, in the 3D case (n = 3), if l ≥ 2 this p.e. condition is satisfied provided that the body is periodically not aligned with all the source points. If three or more source points are not aligned, then this condition is automatically satisfied independently of x and its time-variations. A Riccati observer associated with this system is

Ẋ = A X + ū + K(t)(y -C(t) X) (20)
with X := [x , â ] and K(t) = k(t)P (t)C (t)Q(t) (k(t) ≥ 0.5). One easily verifies that this observer can also be written as

ẋ = u + â -k(t)P 11 (t)( l i=1 Π yi(t) Q ii (t)(x -z i )) ȧ = -k(t)P 21 (t)( l i=1 Π yi(t) Q ii (t)(x -z i )) (21)
From what precedes this observer globally exponentially stabilizes X = 0 if the previously evoked p.e. condition is satisfied.

Remarks:

• In the 3D-case, if l ≥ 2 and the matrix ∆(t) is positive, and if the body moves with a constant unknown velocity, the above observer provides also an estimation of this velocity. To this aim it suffices to set u = 0 in the algorithm. The term â is then an estimate of the body velocity that is equal to a in this case.

• In the unbiased case where a = 0 and the body velocity u is measured, the calculation of â is superfluous and the above observer reduces to

ẋ = u -k(t)P (t)( l i=1 Π yi(t) Q ii (t)(x -z i ))
with P (t) the solution to the CRE Ṗ = -P ∆(t)P + V (t)

A particular solution to this latter equation, obtained by choosing V (t) = P (0)∆(t)P (0), is any constant p.d. matrix.

IV. SIMULATIONS

We estimate the position of a possibly moving body from one or two sources. Two scenarios are considered with a minimal number of source points ensuring uniform observability. In the first scenario the body oscillates along a line segment with its position given by x(t) = (20 cos t -15, 0, 4) . In this case a single source point, here chosen at the origin of the inertial frame, suffices to grant uniform observability. In the second scenario the body is motionless and located at x = (5, 0, 4) . In this case a second source point, here chosen with coordinates (3, 3, 0) in the inertial frame, is required to grant uniform observability. For both scenarios the body velocity measurement is corrupted by the constant bias a = (0.33, 0.66, 0.99) . Initial state estimates are x(0) = (4, 6, 12) and â(0) = (0, 0, 0) . The scalar gain k(t) involved in the CRE is set to one, as for a Kalman filter, and the solution to the CRE is initialized with P (0) = 100I 6 . Simulations are first carried out with noise-free measurements, to validate theoretical exponential stability results, then with measurements corrupted by noise, to illustrate the resulting (and inevitable) slight degradation of the observers following the transient phase when the estimation errors become small but no longer converge to zero. Concerning the body velocity measurement, additive and independent zero mean Gaussian noise components with standard deviations equal to 0.1m/s are used. As for the direction measurements, they are calculated from the body position corrupted by additive and independent zero mean Gaussian noise components with standard deviations equal to 0.05m/s. The matrix V (i.e. the state noise variance in the Kalman filtering terminology) involved in the CRE is chosen equal to 0.01diag{1, 1, 1, 0, 0, 0} + v I 6 , with v = 0.001 to ensure that the matrix is positive definite. As for the matrix Q, i.e. the inverse of the output noise variance in the Kalman filtering terminology, we have used Q ii = 1.5I 3 , ∀i = 1 . . . l with l denoting the number of source points.

Figures 1(a V. CONCLUDING REMARKS Riccati observers for the estimation of a body position from direction measurements and the knowledge of the body velocity have been derived. Even when the body velocity is biased by an unknown constant vector, these observers ensure global exponential stability of zero estimation errors under uniform observability conditions that have been worked out in relation to the number of source points and the body motion. A logical prolongation of this work is the derivation of Riccati observers for the estimation of the complete body pose (position and orientation). However, due to the specific structure of the group of rotations, exact linearisation of the problem is then no longer possible and globally convex cost functions do not exist. As a consequence Riccati observers for pose estimation, and corresponding Extended Kalman Filters (EKF), have to be derived from linear approximations of the system state and output equations. This also implies that only local exponential stability of zero estimation errors can be achieved. An important complementary issue, also in the prolongation of the present work, is the characterisation of uniform observability conditions under which this latter property is granted. We foresee several other possible extensions. Let us just mention position estimation from range measurements, in relation to the recent works of Batista et al. [START_REF] Batista | Sensor-based long baseline navigation: observability analysis and filter design[END_REF], [START_REF]Tightly coupled long baseline/ultra-short baseline integrated navigation system[END_REF] for the non-static case, visionbased robotic applications involving the control of the body position from estimates provided by Riccati observers, and a deterministic approach to Simultaneous Localication and Mapping (SLAM) that could usefully complement existing studies on the subject.

VI. ACKNOWLEDGMENTS

This work was supported by the ANR-ASTRID project SCAR "Sensory Control of Unmanned Aerial Vehicles", the ANR-Equipex project "Robotex". Recall that, as long as P (t) is defined and p.d., its trace is the sum of its eigenvalues. Accordingly, since the eigenvalues of P -1 (t) are the inverse of the ones of P (t), the trace of P -1 (t) is the sum of the inverse of the eigenvalues of P (t). To prove that P (t) is well-defined for t ∈ [0, +∞) and is p.d. it suffices to show that neither the trace of P (t) nor the trace of P -1 (t), which are initially positive (since P (0) is p.d. by assumption), can tend to infinity in finite time. Indeed, this implies that none of the eigenvalues of P (t) can either reach zero or tend to infinity in finite time. To this aim, it suffices to show that neither tr(P (t) nor tr(P -1 (t)) can grow faster than exponentially, so that divergence in finite time is not possible.

Let us set x = tr(P ). In view of (4), and since tr(P (t)C (t)Q(t)C(t)P (t)) ≥ 0, one has ẋ ≤ tr(AP ) + tr(P A ) + tr(V )

Let |A(t)| denote the spectral norm of A(t). By assumption it is bounded by some positive number k a . Similarly, tr(V (t)) is bounded by a positive number v. Since P is p.s. Let us make a proof by contradiction and assume that ∀ > 0, ∀ δ > 0, ∃x ∈ S 2n-1 such that x W Q (t, t + δ)x < . Then, in view of the above expression of W Q (t, t + δ)

x W Q (t, t + δ)x = which, using the assumptions upon ∆(t), in turn implies that γ(t, δ, δ) grows to infinity when δ tends to infinity.

In particular there exists δ0 > 0 such that γ(t, δ, δ) > when δ > δ0 . This contradicts our initial assumption since x W Q (t, t + δ)x ≥ γ(t, δ, δ).

  )-1(c) and Figures 2(a)-2(c) illustrate the performance of the observer in the ideal noise-free case for scenarios 1 and 2 respectively. Sub-figures (a) show the time-evolution of the logarithm of the Lyapunov function associated with the observer. The mean slope of the curve is proportional to the rate of exponential convergence of this function to zero. Sub-figures (b) and (c) show the time-evolution of the estimation errors and their exponential convergence to zero. The last figures 1(d)-1(e) and 2(d)-2(e) show the time-evolution of these errors in the case of noisy measurements.

Fig. 1

 1 Fig. 1. Scenario 1

  d., |tr(AP )| = |tr(P A )| ≤ |A|tr(P ) and the previous inequality yields ẋ ≤ 2k a x + v This inequality in turn implies that x(t) ≤ (x(0) + v 2ka )exp(2k a t) -v 2ka , ∀t ≥ 0. Similar arguments applied to y = tr(P -1 ) yield ẏ ≤ |tr(P -1 A)| + |tr(A P -1 ) + tr(C QC) ≤ 2k a y + μq with μq denoting the supremum of tr(C (t)Q(t)C(t)).Therefore, y(t) ≤ (y(0) + μq 2ka )exp(2k a t) -μq 2ka , ∀t ≥ 0.B. Proof of lemma 2.3From the expression of the matrix A one has Φ(s, t) = exp(A(st)) = I n×n (st)I n×n 0 t)∆(s) (s -t)∆(s) (s -t) 2 ∆(s) ds

  Moreover, since ∆(s) is a p.s.d. matrix, x W Q (t, t + δ)x is larger than s 2 ( x 1 s + x2) ∆(t + s)( x 1 s + x2)dswhen δ > δ. From the latter equality we also have

			γ(t, δ, δ) := 1 δ δ = 1 δ δ-δ γ(t, δ, δ) ≥ ( δ -δ) 2 t+ δ t+ δ-δ (x1 + (s -t)x2) ∆(s)(x1 + (s -t)x2)ds δ δ δ-δ ( x 1 s + x 2 ) ∆(t + s)( x 1 s + x 2 )ds
	δ 1	t+	δ

t (x 1 + (st)x 2 ) ∆(s)(x 1 + (st)x 2 )ds < with x = [x 1 , x 2 ] . It is clear that x 2 = 0 since otherwise, the above integral is equal to 1 δ t+ δ t x 1 ∆(s)x 1 ds > µ with δ = δ.


Paul Issartel
email: paul.issartel@limsi.fr

Lonni Besançon
email: lonni.besancon@inria.fr

Tobias Isenberg
email: tobias.isenberg@inria.fr

Mehdi Ammi
email: mehdi.ammi@limsi.fr

A Tangible Volume for Portable 3D Interaction

Keywords: Tangible user interface, 3D manipulation, mixed reality, fish-tank VR H.5.1 [Information Interfaces And Presentation]: Multimedia Information Systems-Artificial, augmented, and virtual realities

: Our concept of a tangible volume consists of a fully portable and self-contained device, entirely covered with screens. A virtual scene can be seen "through" the volume of the device (left image). This volume can be directly positioned within the virtual scene (middle image), and used to grasp and manipulate virtual objects (right image). A cubic-shaped device is shown here for illustration, but other volume shapes can be used.

INTRODUCTION

Interaction with 3D data and, in particular, object manipulation [START_REF] Bowman | 3D User Interfaces: Theory and Practice[END_REF] (selecting, translating and rotating 3D virtual objects) is of major importance in many fields, such as scientific visualization, prototyping, and gaming. Until recently, these tasks were often carried out by advanced users on fixed workstations. With the development of mobile computing, however, people are now expecting to perform these tasks anywhere, with minimal set-up and learning. There is thus a need for an interface that is not only natural and efficient for 3D manipulation, but also truly portable and self-contained.

Tangible User Interfaces (TUIs) represent a promising approach for 3D manipulation. They consist of physical objects, or tangible objects, that serve as real-world representations of digital data.

In many TUIs, tangible objects are used as physical "handles" for virtual objects. These interfaces take advantage of the user's skills in interacting with physical objects [START_REF] Ishii | The tangible user interface and its evolution[END_REF], making them an attractive solution to support manipulation tasks in a natural and efficient way.

However, interacting with virtual objects also requires visual feedback. Tangible UIs for 3D manipulation often rely on an external and fixed monitor for visual output [START_REF] Hinckley | Passive realworld interface props for neurosurgical visualization[END_REF]. Hence, such interfaces cannot be considered portable. Others use mobile devices as a portable display surface [START_REF] Issartel | A portable interface for tangible exploration of volumetric data[END_REF]. Although these interfaces are portable, they still consist of multiple separate pieces, which always have to be handled and carried together-they are not self-contained. Some TUIs take a further step and use the mobile display as a physical handle [START_REF] Henrysson | Virtual object manipulation using a mobile phone[END_REF]. The device itself replaces the external tangible objects, thereby reducing the entire interface to a single portable and self-contained object. This solution, however, does not eliminate the distance between the display and the virtual objects. Since the mobile display now serves as the tangible handle, this separation creates problems during manipulation, such as a shifted center of rotation or manipulated objects leaving the field of view.

In this paper, we examine a different approach. Rather than turning a mobile device into a tangible handle, we propose to turn a tangible object into a display. We present a first, partially simulated prototype in which the surface of a tangible object appears to be covered with multiple screens. We then use fish-tank rendering to display part of the virtual scene "through" the object. In contrast to object-oriented or geometric displays [START_REF] Kawakami | Object-oriented displays: A new type of display systemsfrom immersive display to object-oriented displays[END_REF][START_REF] Stavness | pCubee: A perspective-corrected handheld cubic display[END_REF], our tangible object becomes a tangible representation of a volume of virtual space. We introduce a "grasping" metaphor that consists in positioning this volume directly onto a virtual object, and pressing the fingers to pick up the object with the tangible volume. Thus, the separation between the tangible handle and the manipulated virtual object disappears. By solving the issues caused by this separation, our solution makes it possible to preserve the advantages of tangible manipulation in a fully portable and self-contained device.

BACKGROUND

Tangible user interfaces for 3D manipulation

The main idea behind Tangible User Interfaces (TUIs) is to give physical form to digital data [START_REF] Ishii | The tangible user interface and its evolution[END_REF][START_REF] Ishii | Tangible bits: towards seamless interfaces between people, bits and atoms[END_REF]. This is accomplished through the use of real-world objects-called tangible objects-that represent the digital information. One particular type of TUI are Graspable User Interfaces [START_REF] Fitzmaurice | Graspable user interfaces[END_REF], in which tangible objects serve as physical "handles" for virtual objects. Each handle can be attached to a virtual object. Users can then manipulate a virtual object by directly moving the corresponding tangible object. By taking advantage of the user's preexisting skills in manipulating real-world objects [START_REF] Fitzmaurice | Graspable user interfaces[END_REF][START_REF] Ishii | The tangible user interface and its evolution[END_REF], this interaction mode provides a natural and immediately efficient approach to 3D manipulation.

A number of TUIs have been designed around this concept. For instance, Hinckley et al. [START_REF] Hinckley | Passive realworld interface props for neurosurgical visualization[END_REF] used passive tangible objects ("props") tracked in mid-air to position and orient 3D medical data, The result was displayed on a separate computer screen. A limitation of these interfaces is, however, that they all require an external monitor or projector. This makes them essentially fixed installations. The lack of portability creates many constraints. Users need to go to a dedicated place in order to use the system. It cannot be carried between offices or brought home. Nowadays, with the rise of mobile computing such constraints appear even more limiting. There is a need for a truly portable interface that would offer the same advantages as fixed TUIs. Our first requirement is thus that the entire interface should be portable.

Since mobile devices-such as smartphones and tactile tabletshave become widespread, they represent a readily-available solution for use as a portable display surface. Some researchers have combined mobile device with tangible objects [START_REF] Issartel | A portable interface for tangible exploration of volumetric data[END_REF], resulting in portable TUIs. However, these interfaces also consist of multiple independent pieces: the mobile device and the tangible objects, all of which need to be carried together at all times. Whenever one of the tangible objects is missing, or when the mobile device is separated from the tangible objects, the interface becomes less functional or even unusable. The portability advantage is reduced due to the inconvenience of having to carry and keep the multiple pieces together. We can thus identify another requirement: the interface should not only be portable but should also consist of a single, self-contained device.

Mobile device as tangible handle

In addition to its role as a portable display surface, the mobile device itself can be seen as a tangible object. Several researchers have proposed to use the mobile device as tangible handle [START_REF] Henrysson | Virtual object manipulation using a mobile phone[END_REF][START_REF] Mossel | 3DTouch and HOMER-S: intuitive manipulation techniques for one-handed handheld augmented reality[END_REF] to manipulate virtual objects in the space behind the device (generally in augmented reality). This eliminates the need for external tangible props, resulting in an interface that is fully portable and selfcontained in a single object: the mobile device.

Although this solution fully meets the portability requirements, it is not, however, without drawbacks. Unlike previous approaches, there is now a separation between the virtual objects (behind the device) and the tangible handle (the device itself). This separation leads to problems during manipulation. In the technique proposed by Henrysson et al. [START_REF] Henrysson | Virtual object manipulation using a mobile phone[END_REF], the manipulated object is fixed relative to the device. Since the center of rotation is now located on the device, it becomes difficult to rotate a virtual object without also translating it (Figure 2(a)). In the HOMER-S technique [START_REF] Mossel | 3DTouch and HOMER-S: intuitive manipulation techniques for one-handed handheld augmented reality[END_REF], translations and rotations are separately applied to the manipulated object, which makes it easier to rotate the object about itself. On the other hand, the object is no longer fixed relative to the mobile device and can thus leave the field of view during rotations (Figure 2(b)). The authors suggest to alter the control-display ratio to avoid this limitation, but such an indirect mapping would become less natural than direct tangible manipulation.

One way to completely eliminate the separation and its associated problems would be to position the mobile device onto the virtual object before starting the manipulation. However, a typical rendering process would clip half of the virtual object in this situation-as well as everything else in front of the screen-making it difficult to manipulate the object properly (Figure 2(c)). In addition, most current mobile devices come in the form of flat one-sided displays. Figure 2: Manipulation issues that arise when a typical mobile device is used as a tangible handle: (a) the virtual object does not rotate about its own center; (b) the virtual object leaves the field of view; (c) when trying to reduce the distance to avoid the above issues, the virtual object is clipped.

Even if clipping was not an issue, this form factor would still result in loss of visual feedback during manipulation. Therefore, we can identify two additional requirements: the portable and self-contained device should provide visual feedback on its entire surface, and should be able to display virtual objects without clipping.

Fish-tank VR and geometric displays

When rendering a 3D scene, the display is generally considered as the viewpoint on the virtual world. This causes clipping whenever a virtual object crosses the screen plane. Fish-Tank Virtual Reality (FTVR) provides a solution to address the clipping problem. FTVR turns the display into a "window" seen from the viewpoint of the user [START_REF] Ware | Fish tank virtual reality[END_REF]. This allows virtual objects to appear behind, onto, or in front of the display surface. There are two main ways to achieve this effect: stereo rendering, head-coupled perspective, or a combination of both [START_REF] Ware | Fish tank virtual reality[END_REF]. According to Ware et al. [START_REF] Ware | Fish tank virtual reality[END_REF], head coupling is a much stronger clue than stereo rendering, making it the preferred solution to implement this technique.

As explained before, a flat one-sided display is not the best shape for use as a tangible handle. A number of works have proposed volumetric objects equipped with multiple displays, capable of providing visual feedback on their entire surface. Some of these objects only support 2D rendering, such as the "Display Blocks" by Pla and Maes [START_REF] Pla | Display blocks: A set of cubic displays for tangible, multi-perspective data exploration[END_REF]. Others take advantage of FTVR to give the illusion of a 3D space inside the device. They are known as geometric displays [START_REF] Stavness | pCubee: A perspective-corrected handheld cubic display[END_REF], or object-oriented displays [START_REF] Kawakami | Object-oriented displays: A new type of display systemsfrom immersive display to object-oriented displays[END_REF]. One of the first examples is the MEDIA cube [START_REF] Inami | Media3: The virtual hologram[END_REF]: multiple LCD displays were arranged in a box shape, and combined with head coupling to give the illusion that a virtual scene was inside the box. The CoCube [START_REF] Brown | A widget framework for augmented interaction in SCAPE[END_REF] is a tangible cube that produced the same illusion when seen through a head-mounted display (HMD). Unlike the previous example, this tangible cube was freely manipulable by the user. The Cubee [START_REF] Stavness | Cubee: A cubic 3D display for physicsbased interaction[END_REF] is a cubic device that achieved the same goal without a HMD, by using integrated displays and a head tracker. The pCubee [START_REF] Stavness | pCubee: A perspective-corrected handheld cubic display[END_REF][START_REF] Tang | The new pCubee: Multi-touch perspective-corrected cubic display[END_REF] is an evolution of Cubee that was made smaller and more portable. Most of the above devices have a cuboid shape because this is the easiest way to arrange conventional rectangular displays, but other shapes are possible (ex. arbitrary polyhedra, sphere).

Geometric displays appear to meet all our requirements: they can be made small, portable, and self-contained, they provide visual feedback on all their surface, and with FTVR they can display virtual objects without clipping. However, previous work on this subject was mainly focused on the feasibility of such displays. Although significant technological contributions have been made, the potential of such devices for 3D interaction remains comparatively unknown. In addition, there seems to be a lack of examples of positioning the geometric display itself within a larger virtual scene, which is an essential part of our concept. Even though the pCubee can be used to navigate in the virtual scene, this is accomplished through an indirect velocity-based mapping. In contrast, our concept is based on a full 1-1 mapping between the real world and the virtual world, which has greater potential for direct 3D interaction. In this work, we offer a new perspective on geometric displays, by considering such a device as a portable and self-contained TUI in its own right, capable of both displaying a virtual scene and serving as a tangible handle within the scene itself.

CONCEPT: A TANGIBLE VOLUME

Based on the above considerations, we introduce the concept of a "tangible volume." A tangible volume is a single physical object, sufficiently small and lightweight to be held in the hand. The surface of this object is entirely covered in screens on which the virtual scene is displayed. The perspective of each screen is adjusted to the user's head position. As a result, part of the virtual scene appears "through" the object. The object is also tracked relative to the real world. Users can reach other parts of the virtual scene by moving the object in real space (Figure 3). At any point, the physical boundaries of the object "enclose" a corresponding part of the virtual scene. Therefore, this object is a tangible representation of a volume of virtual space that can be held in the hand and directly positioned into the virtual scene.

We use this tangible volume as an interaction device for 3D object manipulation. First, the tangible volume is positioned onto a virtual object in the scene. From there, the virtual object is attached to the volume. The virtual object then follows the tangible volume in 3D space as if it was directly held in the hand (Figure 1). Thus, there is no separation between the virtual object and the tangible handle.

Our concept integrates all input and output into a single handheld object, used both to visualize the virtual scene and to manipulate virtual objects. Unlike the alternative approach of using a mobile device as a tangible handle, our interface also eliminates the separation from the manipulated virtual objects. Our concept thus constitutes a fully portable and self-contained interface for 3D object manipulation which avoids the problems described before.

Object selection by grasping

Having an interface made of an unique tangible object has clear benefits for portability, but also raises the question of how to interact with more than one virtual object. Such an interface must provide a way to attach and detach virtual objects from the single tangible handle.

In TUIs made of multiple tangible objects, each tangible object can be linked to a different virtual object. This allows the user to interact easily with multiple virtual objects, simply by manipulating the corresponding tangible objects as desired. This is called "space multiplexing" [START_REF] Fitzmaurice | Graspable user interfaces[END_REF]. In our case, there is only one tangible object available to manipulate an arbitrary number of virtual objects. Therefore, the user has to select which virtual object should be linked to the tangible object at any given time. This is called "time multiplexing" [START_REF] Fitzmaurice | Graspable user interfaces[END_REF]. While space multiplexing is considered more desirable than time multiplexing due to the lack of an object selection step which improves efficiency and lowers cognitive load [START_REF] Fitzmaurice | Graspable user interfaces[END_REF], let us consider more closely what selection means in the specific case of tangible objects. Even though some interfaces allow the user to interact with multiple tangible objects concurrently, this is ultimately limited by the human capabilities. Tangible manipulation is typically performed with the hand(s). This means that all manipulation is accomplished through at most two effectors: the hands themselves. Any tangible object first has to be grasped with the hand in order to interact with the corresponding virtual object. Therefore, there is still an implicit selection in space-multiplexed TUIs that occurs when an object is grasped with the hand.

From that perspective, one of the main advantages of space multiplexed TUIs is not actually the lack of a selection step, but rather the fact this selection is implicit and does not require thinking. In other words, reaching for an object and grasping it with the hand constitutes a natural way to select it.

Grasping metaphor

The tangible volume provides a unique opportunity to reproduce this form of selection with virtual objects: by considering the volume as an extension of the hand. As shown above, the tangible volume can be moved in space and positioned around a virtual object. Since the volume is held in the hand, and is surrounding a virtual object, the hand is also surrounding the virtual object.

We thus designed a selection technique that consists in pressing the fingers on the tangible volume to "grasp" the virtual object located inside (Figure 4). Because the tangible volume is already held in the hand, the grasping technique is only triggered when finger pressure exceeds a given threshold. This threshold allows users to differentiate a conscious grasping action from normal manipulation of the tangible volume itself. Similarly, releasing the virtual object is done by releasing finger pressure below this threshold, while keeping hold of the tangible volume.

With our grasping metaphor, the action of picking up a virtual object becomes very similar to picking up a real-world object: placing the hand around an object, and pressing the fingers to grasp it.

Disambiguating between virtual objects

A challenge that arises from our grasping metaphor is that fingers cannot penetrate the volume. When virtual objects are close to each other, more than one may be located inside the volume bounds. Since fingers cannot be directly used to grasp the desired object, there must be a way to indicate which object will be selected among those located in the volume. One possible solution is to display an outline around the object that is closest to the center of the volume (Figure 4).

Bimanual manipulation

Space-multiplexed TUIs allow the user to manipulate different virtual objects in each hand. Since our interface is made of a single object, this form of bimanual manipulation is not directly supported. One solution could be to use two tangible volumes-one for each hand. Since our grasping metaphor makes the tangible volume an To disambiguate between nearby objects, an outline indicates which object is both inside the volume and closest to its center. This object can be grasped by pressing the fingers on the volume. The object is then attached to the volume, and can be directly moved alongside the volume in 3D space. The object is detached when finger pressure is released. If virtual gravity is enabled, it then falls to the ground.

extension of the hand, and all manipulation is accomplished through the hands, this solution would approximate space-multiplexed bimanual interaction. However, the interface would then again consist of multiple pieces, which we specifically wanted to avoid. Still, as a single object, our interface supports another form of bimanual interaction: manipulating a single virtual object with both hands. This can be accomplished by holding the tangible volume with two hands, and pressing the respective fingers on opposite sides of the volume.

Simulated physics

In his dissertation on Graspable User Interfaces, Fitzmaurice [START_REF] Fitzmaurice | Graspable user interfaces[END_REF], citing Norman [START_REF] Norman | Things that make us smart: Defending human attributes in the age of the machine[END_REF], argues that the natural laws of physics that affect tangible objects help the user during manipulation. Indeed, in graspable UIs, releasing an object from the hand makes it drop to the floor due to gravity. This familiar behavior can help understand and predict its motion.

However, in our interface, the tangible volume always remains in the hand. Releasing a virtual object simply detaches the virtual object from the volume. Without further intervention, the virtual object would remain there-floating in space. We can obtain a more realistic behavior by adding simulated physics to the virtual scene. With physics, the released virtual object falls to the virtual ground when released (Figure 4). Simulated physics also prevents virtual objects from moving through each other. This reinforces the illusion that they are solid, and thus can be grasped and manipulated directly.

It should be noted that virtual physics is not always desirable. For example, a complex manipulation task may have to be decomposed in several steps in mid-air (clutching). With virtual gravity, releasing the virtual object between each step would cause it to drop from its intermediate positions. Virtual physics enhances realism, but is not necessarily appropriate for every application.

IMPLEMENTATION

In order to study the usability of our concept, we designed a preliminary prototype. As said before, our work primarily focuses on the interaction capabilities afforded by a tangible volume, rather than on the hardware side. Therefore, we used augmented reality (AR) to simulate some hardware aspects of our concept.

One may argue that existing geometric displays could have served as a basis to conduct user studies, without requiring AR simulation. However, existing implementations described in previous work were all lacking in key aspects of our concept. Some of these devices are fixed installations, not meant to be moved by the user [START_REF] Inami | Media3: The virtual hologram[END_REF]. Others are somewhat movable, but are still tethered to a larger workstation [START_REF] Lopez-Gulliver | gCubik: realtime integral image rendering for a cubic 3D display[END_REF][START_REF] Stavness | pCubee: A perspective-corrected handheld cubic display[END_REF]. The tangible volume as intended in our concept is supposed to be fully portable and self-contained, and a tether wire could hinder manipulation [START_REF] Hinckley | Usability analysis of 3D rotation techniques[END_REF]. Finally, some of these devices are only partially covered in screens [START_REF] Stavness | pCubee: A perspective-corrected handheld cubic display[END_REF][START_REF] Stavness | Cubee: A cubic 3D display for physicsbased interaction[END_REF]. Even though this is sufficient to demonstrate the technology, for complex object manipulation (especially rotations) it is important that all sides of the volume provide visual feedback. Simulating an interaction device to conduct user studies has been done in previous work [START_REF] Baricevic | A handheld AR magic lens with user-perspective rendering[END_REF]. This approach has the advantage of providing flawless rendering and tracking, which would be difficult to achieve in a research prototype but essential for the validity of user studies.

We chose a cubic shape for the tangible volume in our prototype. Even though other shapes could have been used, we chose a cube because it was easier to build while also being easy to manipulate by users. This cube was covered with AR markers. We used a tactile tablet as an augmented reality window. When observed through the rear camera of the tablet, the faces of the cube were replaced with "virtual screens" that displayed part of the virtual scene with a fish-tank effect. A frame was added around each virtual screen, to account for the fact that real screens would not be completely borderless. Using AR provided implicit viewpoint tracking: since the cube faces were tracked by the camera, reversing this transformation produced an equivalent result. The cube was tracked relative to the real world by placing an additional AR marker in the environment. We employed the Vuforia framework 1 to track these objects. The tablet was attached to a raised stand, so that users could simply manipulate the object behind the tablet, as if they were directly looking at a cube equipped with screens.

Our grasping technique was implemented in actual hardware, by attaching six flat pressure sensors (Interlink® FSR 406) to the faces of the tangible cube. The sensors were located under the AR markers, and thus were invisible to the users. They were driven by a microcontroller2 embedded in the cube (Figure 5). It was powered by a rechargeable battery, with a charging port hidden in one corner of the cube. The microcontroller continuously transmitted the pressure values to the rendering software on the tablet, through a wireless Bluetooth connection, at a frequency of 10 Hz. Finally, we implemented physical simulation for the grasped objects with the Bullet3 physics engine.

In the rest of this paper, we use this prototype to investigate the intuitiveness of object manipulation with our grasping metaphor.

microcontroller + battery

pressure sensor

AR marker

Figure 5: View of the electronic components inside the cube. The embedded microcontroller retrieves values from the pressure sensors on the cube surface, and sends them to the rendering software on the tablet through a wireless connection.

USER STUDY: OBJECT MANIPULATION

When interacting with the real world, grasping an object by pressing fingers on it, and moving it while maintaining finger pressure is a fairly natural procedure. However, it is unknown whether the exact same procedure remains natural when interacting with virtual objects through a tangible volume. We thus conducted an experiment to evaluate the intuitiveness of object selection and manipulation in our interface.

More specifically, we wanted to see if users can understand by themselves with no prior explanations, how to grasp and move virtual objects with our interface. Of course, if some users do not succeed without help, this measure alone would not be sufficient to understand why. Therefore, we also wanted to determine at which step of manipulation those users would need guidance on their initial encounter with the interface.

Participants and apparatus

This study was conducted with 36 unpaid participants (from 20 to 52 years old, mean=29.5, sd=9.4). None of them had any prior knowledge of our interface concept or our grasping metaphor. The apparatus was our AR-based prototype described in the previous sections. We slightly raised the pressure threshold to ensure that the grasping technique would only be triggered when truly intended by the participants.

Procedure

Before starting the experiment, we first told participants that it consist in "manipulating the cube". We then explained why there was also a tablet in front of them: to "simulate screens that should have been on the faces of the cube, but were not there due to technical limitations". We followed by demonstrating how the cube turned into a cubic display when placed behind the tablet, and how a virtual scene could be seen through it. If our prototype had actual screens and head tracking, participants would likely have noticed all of this immediately. We did this short demonstration to ensure there was no confusion about our AR simulation. However, we gave them no explanation about how to interact with virtual objects.

We then introduced participants to the task: moving a virtual apple to a nearby target. The target was represented by a circle on the virtual floor (Figure 6), 6 cm to the right of the virtual object. Participants were told this task would have to be done "with the cube". They were also specifically told to ignore the tablet for manipulation. Finally, we asked them to discover how to do that "by themselves, as far as possible".

Obviously, we expected that some participants would not be able to complete the task without explanation. In order to understand how much help these participants needed, we designed a set of textual hints in increasing level of accuracy. During the task, there was a button that could be pressed to reveal a new hint. Each press on the button uncovered an additional hint on the tablet's screen, in the following order:

1. "Put the cube onto the apple" 2. "Press the cube to grab the apple" 3. "Move the cube while maintaining the pressure" These hints were specifically chosen to cover the different steps of object manipulation in our interface. Additionally, we chose to use textual hints rather than visual representations (e.g. arrows) to avoid ambiguous interpretations. We explained the role of the hint button to participants, and strongly encouraged them to use as few hints as possible.

Results and discussion

Figure 7 shows the percentage of participants according to the number of hints they needed to complete the task. A total of 19 participants (53%) successfully completed the task without requesting any hint. Among the remaining participants, all but one (45%) succeeded with the first two hints. The third hint was never used. We also report the task completion times for the participants who accomplished the task without any hint. The mean completion time was 63.5 s (SD=34.3 s).

More than half of participants discovered by themselves how to grasp and manipulate virtual objects with our tangible volume. This is an encouraging result, given that our interface is so much different from the way most people currently interact with virtual 3D objects, and more generally interact with computers. At first glance, the completion times-about one minute on average-might seem long. However, this was the time needed to discover how to use a completely unfamiliar device and accomplish a manipulation task for the first time. This, it seems that the idea of grasping a virtual object "through" a tangible volume was spontaneously considered by a majority of participants, and within a reasonable time.

For those who did not complete the task without help, the number of hints requested provides more insight into which parts of manipulation were troublesome. The first hint was designed to uncover potential difficulties in positioning the tangible volume onto the virtual object. Nearly all participants who requested hints were not helped with this first hint. Hence, positioning was likely not what prevented them from completing the task. Indeed, all participants quickly noticed that a outline appeared around the virtual object whenever it was inside the volume. Some of them, however, believed that the outline meant the object was already selected, and attempted to move it without pressing the fingers.

Among other attempted strategies, a surprising number of participants attempted to "push" the virtual object with the tangible volume. Completion time (seconds) when no hints were used This may indicate they thought the sides of the volume would be solid in the virtual scene. Many tried to tap or flick their fingers onto the surface of the volume. In some cases, it was apparently an attempt to replicate the "click" metaphor, especially when they grasped the tangible volume as if it were a mouse and tried to click (and even double-click) on the top. In other cases, it was clearly an attempt to affect the virtual object through the volume, especially when flicking a finger against one of the side faces. Finally, some participants tried to fully enclose the cube in their hands. This may have been an attempt to grasp the virtual object. However, in doing so they lost visual feedback, and it was thus impossible to see if finger pressure was sufficient to trigger the grasping technique. Nearly all participants who requested the first hint also requested the second hint to complete the task. The second hint was designed to uncover difficulties with the grasping metaphor itself. No participant ever asked for the third hint. For those who needed help, the grasping metaphor was therefore the main hurdle. The third hint was about moving the selected object by keeping it attached to the tangible volume. Since no participant requested this hint, none of them encountered any problem with this last step.

CONCLUSION

We introduced the concept of a "tangible volume": a fully portable and self-contained device for 3D interaction, made of a single tangible object entirely covered with screens. In contrast to existing object-oriented or geometric displays, this tangible object represents a volume of the virtual scene and can be positioned directly onto virtual objects. This makes 3D interaction more direct than with previous approaches. We described an object manipulation technique that consists in grasping virtual objects "through" the volume and moving them in 3D space. We created a partial prototype based on this concept, and used this prototype to investigate some aspects of its usability. In particular, we showed that the grasping metaphor was spontaneously understood by a majority of users.

Future work should focus on two aspects. On the technical side, the first step would be to create a full implementation of our concept. This would require several improvements to the current technology. In particular, achieving reliable head tracking and environment tracking in a fully portable and self-contained device remains an important challenge, which has not been addressed so far by existing prototypes (Section 2.3). While recent devices with multiple included sensors such as the Tango tablet improve the situation, they still have issues with 3D tracking and we expect that future technological developments will provide better location data. On the experimental side, a more complete implementation would allow to repeat user studies in conditions that more closely resemble a real device. For example, the influence of correct finger occlusion could be studied in a prototype with actual screens, as well as the effect of possible imperfections in tracking and rendering.

Figure 3 :

 3 Figure3: Illustration of our concept of a "tangible volume". Part of a larger virtual scene can be seen through the tangible object held by the user. On the top right, the tangible volume is observed from a different angle. On the bottom right, the tangible volume has been translated in space and now encloses a different part of the virtual scene.

Figure 4 :

 4 Figure4: Illustration of the different steps of object manipulation in our interface concept. First, the tangible volume is positioned onto a virtual object. To disambiguate between nearby objects, an outline indicates which object is both inside the volume and closest to its center. This object can be grasped by pressing the fingers on the volume. The object is then attached to the volume, and can be directly moved alongside the volume in 3D space. The object is detached when finger pressure is released. If virtual gravity is enabled, it then falls to the ground.

Figure 6 :

 6 Figure 6: Screenshot of our first experiment, as seen by participants through the tablet. Note the lack of finger occlusion, due to the AR simulation. The task was to pick up the virtual apple and move it to the red circle on the right.

Figure 7 :

 7 Figure 7: Number of hints and time needed by participants to discover how to manipulate a virtual object with our interface.

http://www.vuforia.com/

RFduino RFD22102

http://www.bulletphysics.org/


M Apreutesei

P Djemia
email: djemia@univ-paris13.fr

L Belliard

G Abadias

C Esnouf

A Billard

P Steyer

Structural-elastic relationships of Zr-TL (TL = Cu, Co, Ni) thin films metallic glasses

Keywords: Thin films, metallic glasses, ultrasonics, elasticity, mechanical properties

In this study, we investigated the structural and elastic properties of Zr-TL (late transition metal TL = Cu, Co, Ni) thin films metallic glasses (TFMG) deposited by dc magnetron co-sputtering from pure TL targets in Ar plasma discharge. The influence of the deposition parameters on the microstructure, chemical composition and elastic properties of the thin films has been explored. Advanced non-destructive techniques such as the picosecond ultrasonics and the Brillouin light scattering were employed to selectively measure the longitudinal V L and the transversal V T sound velocities, leading to the determination of the elastic constants C 11 , C 44 , respectively. From these data sets, the elastic constants of metastable amorphous single elements a-Zr, a-Cu, a-Co and a-Ni have been extrapolated. Some relationships between elastic moduli (Young's modulus E and shear modulus G = C 44 ) and the structural state either crystalline or amorphous are established. Predicted ductility/brittleness character was emphasized based on the Blackman's diagram, Pugh's ratio and the Pettifor criteria. Equivalently, the newly introduced δ parameter termed as 'the cooperation parameter' and related uniquely to the Poisson's ratio ν, confirmed the improvement of ductility with TL alloying.

Introduction

Since the metallic glasses were first discovered in 1960 [1], researchers are interested in their unique atomic packing structures [2] mechanical [3], electronic [4], and magnetic [5] properties. The initial limitation in preparing bulk metallic glasses (BMGs) by rapid quench techniques was gradually overcome by coupling appropriate elements with different atomic sizes based on the empirical rules pointed out by Inoue [6]. Many new amorphous alloys based on early transition (TE) -late transition (TL) alloying metals, such as Zr-Ni-Cu-Al [7],

Zr-Ti-Cu [8], Mg-Cu-Gd [9] and Ti-Cu-Ni-Sn [10], have been fabricated in ribbon or bulk type with diameters greater than a few mm.

Employing thin film technology routes, such as physical vapor deposition (PVD) is an alternate way to avoid the precipitation of the immiscible elements and produce thin film metallic glasses (TFMG) with over-saturated alloy concentrations. The solubility of immiscible elements and glass forming composition range using PVD processes are both much higher, compared with the liquid quenching process used for bulk or ribbon amorphous alloys. Among them, the Zr-Cu, Zr-Ni and Zr-Co systems belong to the immiscible TE-TL category which has good glass-forming ability, as previously observed in BMG [11][12][13] and TFMGs [14][15][16]. Depending on the concentration of TL elements in the matrix, homogeneous nanocrystalline and amorphous phases or mixed phases were identified.

Relationship between structural and mechanical properties remains not enough documented for amorphous metal alloys compared to conventional crystalline materials. Some recent studies have contributed to highlight the correlation between density (depending on the composition), hardness, Young's modulus and glass-forming ability (GFA) [17]. The origins of ductility/brittleness and relationship between density, elastic/plastic properties and GFA are the hottest topics in the undergoing research of amorphous alloys. Moreover, the mechanical properties are fundamental input parameters for structural design of micro-or nano-devices. In this work, we measured selectively, the two independent elastic constants, C 11 and C 44 = (C 11 -C 12 ) /2 (using Voigt notations, the C IJ 's relate the stress σ ij and the strain ε kl through the Hooke law s I = C IJ e J, with e 1 = ε 11 , e 2 = ε 22 , e 3 = ε 33 , e 4 = 2ε 23 , e 5 = 2ε 13 and e 6 = 2ε 12 ), of a series of elastically isotropic Zr-TL TFMG by the original combination of non-destructive picosecond ultrasonics (PU) [18,19] and Brillouin light scattering (BLS) [20][21] techniques.

They are well adapted for an accurate characterization of the elastic properties of Zr-TL thin films, as demonstrated previously for metastable Mo-Si [18] and nitrides [19] alloys thin films. From these measurements, the elastic moduli of the films have been calculated and ductility/brittleness behavior was predicted.

Experimental

Thin films growth and structural properties

The Zr-TL thin films were deposited on Si substrates by a direct-current co-sputtering process in a 40-liter cylinder Alcatel 450 SCM sputtering reactor. . The TL=Cu, Co and Ni targets were powered using a MDX500 generator in the range 0.1 -1.5 A at a pressure around 0.5 Pa.

More details about films deposition can be found elsewhere [15,22]. Three thin film series were produced: Zr-Cu, Zr-Co and Zr-Ni, and the content of TL atoms varied by changing the respective cathode currents. The Zr-Cu system was previously explored in the whole phase diagram, whereas in the current study, for Zr-Co and Zr-Ni thin films, we restricted the compositions in the range of < 40 -80 > at. % for the TL element.

The thickness of the thin films and chemical profiles were determined by a Zeiss Supra 55 VP scanning electron microscope (SEM) with energy dispersive X-ray microanalysis (EDX). The thickness ranged between 4 and 6 µm, so that they can be considered as a semi-infinite

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medium for BLS experiments to measure the shear elastic constant C 44 of the film. The common required structural information to extract elastic constants from PU and BLS measurements of sound velocities is the mass density, ρ. It was achieved from a fit of X-ray reflectivity (XRR) scans using an optical model [23].

Basic principles about picosecond ultrasonics and Brillouin light scattering

Only some principles will be recalled in this section while more details about both experiments can be found elsewhere [19]. The PU technique introduced by H. J. Maris in 1980 [24] derives from the well-known optical pump and probe scheme. A femtosecond laser pulse pump beam is absorbed at the sample surface. This absorption generates, by thermal expansion, a longitudinal acoustic wave that propagates through the system, perpendicularly to the free surface. For a monolithic film onto a substrate, sharp reflectivity changes are periodically detected, the so-called echoes, separated by a constant time delay (the time of flight TOF) as in a standard sonar technique.If the thickness h of the layer is known, PU gives a direct access to the longitudinal sound velocity simply defined by V L = 2h/TOF, along the normal of the film-plane, where TOF is measured with an accuracy better than 1 ps [18,19,24]. Thus, considering amorphous materials with isotropic elastic properties the result should be the same for the following elastic constants:

2 33 22 11 L V C C C ρ = = = .
In a BLS experiment, a monochromatic incident light beam probes the thermally excited acoustic waves. The BLS spectra were obtained at room temperature in air, with typical acquisition times of 1 h and 300 mW of a p-polarized incident light. For opaque films, the scattering mechanism is restricted to the scattering of light by the dynamical corrugation of the free surface by acoustic waves travelling parallel to the film plane [20,25]. For films with thicknesses well above the acoustic wavelength (~0.3 µm), we measure the Rayleigh surface wave velocity (R) of the film [25,26] closely related to its transverse velocity [27]. Also, we can observe a singularity at higher frequency that enables the measurements of V L by BLS [25].

V T =(C 44 /ρ) 1/2 M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 5
Once, V L and V T are measured, the elastic moduli (shear modulus G, Young modulus E, bulk modulus B and Poisson ratio ν) can be calculated using the following equations:

( ) ν ρ + = = - = = 1 2 2 44 12 11 2 E C C C V G T , (1) ( ) ( ) ( ) (
)

44 11 2 44 11 11 2 2 2 2 2 2 C C 2C C C 2 - - - =         - - - = T L T L L V V V V V E ρ , ( 2
)
44 11 2 2 3 4 3 4 C C V V B T L - =       - = ρ , (3) ( ) ( ) (
)

      +       - ≡ - - = - - = B B V V V V T L T L G 3 G 2 3 2 1 C C 2 2C C 2 2 44 11 44 11 2 2 2 2 ν , (4)
Other parameters such as the Pugh's ratio k = G/B [28], the Cauchy's pressure p = C 12 -C 44 [29] and the 'cooperation parameter'

( ) ( ) 2 2 2 1 65 . 12 1 ν ν δ - + =
[30] will be helpful to discuss the brittleness/ductility predictive properties of our films.

Results and discussion

The X-ray diffraction (XRD) patterns recorded in θ-2θ configuration for the three film series are reported in figure 1. They all exhibit scattering peaks in the 37 -45° 2θ region, often referred as first sharp diffraction peak or principal diffraction peak, characteristic of amorphous alloys. For all systems, a shift of the broad diffuse-scatter to higher 2θ angles is observed by increasing the content of TL atoms. This evolution suggests a continuous decrease of the average interatomic distance or equivalent first coordination sphere radius [31]. Moreover, using the Bragg's relation the equivalent lattice spacing (d) for Zr-TL films also decreased from about 2.45 Å to 2.10 Å, suggesting different bonding lengths and interatomic interactions that modifies the atomic local ordering. It has been reported by

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simulations that when mixing Zr and TL elements (TL = Cu, Ni, Co) the local ordering is organized such as polyhedral-type short and medium range ordering structures controlled by the atomic size ratio and the electronic interactions between the component elements. For instance, for a Zr 70 Cu 30 alloy the local atomic environment matched to an icosahedral-like around the Cu atoms and global short range ordering (SRO) structure [32], while for Zr 66.7 Ni 33.3 alloy around the Zr atoms a trigonal prism polyhedron dominated with medium range ordering (MRO) structure [33]. For the Zr-Co system, it has been demonstrated by molecular dynamics (MD) simulations that the local structure is quantified as short-range-order [34]. Therefore, based on the studies reported in literature it may be inferred that the local ordering of the Zr-TL (TL = Cu, Ni) thin films studied here is governed by SRO/MRO structures.

The evolution of XRD patterns of Zr-Cu films with increasing Cu in a wider range of concentration has been reported in a previous paper (see Fig. 2 of Ref. [15]). From this work, the amorphous phase region was identified for Cu content within the < 30 -85 at. % > range.

Figure 2 presents the mass density evolution as a function of films. From this figure, it can be observed that ρ increases with TL content from ρ Zr =6.0 g.cm -3 to ~8.3 g.cm -3 for the most alloyed films. In both cases the same trend implies that TL additions continuously move/reorganize the atoms at the local range. Hence, as more TL is added in the amorphous matrix, the distances between the closest neighbors is reduced and the mass density is increased, leading finally to more compact structure. Nevertheless, both extrapolated values are slightly lower than the bulk values of the bulk crystalline α-Zr (6.52 g.cm -3 ) and crystalline TL single elements (~9 g.cm -3 ) ρ remains for TL-rich concentrations below the average mass density calculated from bulk values. It means that films which lie below the Vegard's rule in Fig. 2, are not fully dense compared to the bulk, with a certain degree of porosity not investigated, here.

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7

PU measurements allow determining values of the C 11 parameters. For all samples, the C 11 elastic constant is first measured by PU and then considered as a fixed known parameter in the fitting of the BLS spectra used to evaluate the C 44 parameter. As illustrated in figure 3 within the amorphous region, a clear monotonous increase of the shear elastic constant C 44 is observed upon incorporation of a TL substituting atom, essentially due to the increase in ρ.

For the Zr-Cu system, the drop of C 11 near the 30 -35 at. % region can be attributed to the phase transition from crystalline to amorphous state, as it was similarly observed in Mo-Si system [18], and predicted in Braeckman et al. [31]. Moreover, the C 11 constant shows apparent scattering of the data in the amorphous region whereas relative uncertainty in the thickness measurements does not exceed a few percent, and seems to be much more sensitive to changes of local atomic environment related to formation of different type of SRO and different bonding energies, as a function of the Cu concentration. The scattering in C 44 data is always less pronounced as it does not depend on the thickness of these thick films, neither on the substrate. In the case of Zr-Cu alloy, the C 44 elastic constant increases linearly from 22 GPa to 35 GPa in the amorphous region. Considering ideal solution behavior [35] of the glassy state, we obtain extrapolated shear modulus for a "hypothetical" amorphous zirconium a-Zr of 17 GPa (well below the polycrystalline value of 33 GPa for hcp α-Zr) and 37 GPa for a-Cu (slightly lower than the value of 48 GPa for fcc-Cu). Considering similar ideal solution behavior [35] of glassy Zr-Co and Zr-Ni, we obtain extrapolated shear modulus of 53 GPa for a-Co and 58 GPa for a-Ni. In the case of the second constant, namely C 11 , we can obtain an extrapolated value for a "hypothetical" amorphous zirconium a-Zr of 84 GPa (well below the polycrystalline value of 141 GPa for hcp α-Zr) and 184 GPa for a-Cu, 224 GPa for a-Co and 239 GPa for a-Ni (all slightly below their crystalline phase). The large difference between the values of C 11 and C 44 for a-Zr and the crystalline Zr implies that the so-called ''rule of mixtures'', which is commonly employed in the estimation of elastic moduli of metallic

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8 glasses from those of their crystalline constituent metals, may not be applicable to a-TL-Zr alloys [35]. All elastic constants and moduli obtained for the amorphous transition metals are summarized in Table 1 and compared to their polycrystalline counterpart. The Young's modulus E can be calculated from the two independent elastic constants C 11 and C 44 using Eqn. (2). A direct correlation is found between the shear modulus G and E for amorphous alloy films investigated in the present work, yielding a linear relationship G=0.3648 E, as shown in figure 4. Note that the proportionality factor is slightly lower than what is widely found experimentally for polycrystalline metals G ~ 3/8 E [START_REF] Zwikker | Physical Properties of Solid Materials[END_REF] or even theoretically for some orthorhombic intermetallics 0.38 [START_REF] Sebehi | [END_REF] and fcc polycrystalline metals 0.39 [38]. But it is closer to 0.367 found in bcc [38]. One should notice that the slope is related to the Poisson ratio ν by Eqn. ( 1), then ν from our TFMG remains close to 0.370 in the amorphous phase. 18 -0466). Moreover, in our previous study [22] related to crystallization resistance, it has been demonstrated that at 33 at.% Cu, the first crystallized phase at temperatures above 230 ˚C was the bct-ZrCu 2 intermetallic compound. Thus, based on the HRTEM image and corroborated with the crystallization behavior [22] it can be stated that the greatest plasticity of Zr 67 Cu 33 film is due the existence of the nanocrystalline domains embedded into the amorphous matrix.

The evolution of bonding character is also reflected in figure 5b, which plots the Pugh's ratio k = G/B as a function of p. The G/B ratio has been proposed initially by Pugh [28] as an indicator to link empirically the elastic moduli to the response of materials strained outside their elastic limits. Based on an evaluation of a large experimental data set on metals, Pugh initially proposed that low G/B values, i.e., < 0.571 (for polycrystalline materials) or < 0.41 (later extended to amorphous materials), would be associated with ductility, while higher values would reveal a brittle character. According to the Pugh and Pettifor criteria, it can be seen from figure 5b that alloying enhances the ductility. In particular, Zr 67 Cu 33 alloy is expected to exhibit significant shear elastic softening (-20%) (see Fig. 4b) compared to Zr(α). Nevertheless, this film exhibits a hardness to the Young modulus ratio of H/E ~0.09 [15] which is not satisfying the condition H/E >0.1 for improved resistance to cracking [42]. This finding suggests that criteria based only on the knowledge of elastic constants, are useful indicators to predict the plastic behavior of materials. Therefore, combining BLS and PU techniques for the assessment of the elastic properties may have strong implications in the design of thin films metallic glasses with improved toughness.

The explanation of the plastic response of a metallic glass in terms of elastic moduli has not been significantly extended beyond G/B, except for showing a further link between plasticity, Poisson's ratio [43] and the fragility index m [40]. In a recent work, this issue has been clarified by introducing the cooperation parameter δ that is uniquely related to the Poisson's ratio [30]. From a thermodynamic perspective, shearing or plastic deformation is preferred in materials when fulfilling δ > 1 and cracking when δ < 1. The two modes are comparable to each other at the critical point of δ = 1 corresponding to the critical Poisson's ratio ν cri = 0.315. It is important to underline that all our Zr-TL studied films evidence δ values higher than 1, confirming their shearing behavior.

Summary and conclusions

Zr

  comparison to the enormous work done on BMGs, on TFMGs only few studies have been conducted.

Figure

  Figure 5a displays the dimensionless quantities C 12 /C 11 versus C 44 /C 11 , commonly referred as

  -TL (TL=Cu, Co and Ni) metallic glass films have been successfully deposited on Si substrate by dc co-sputtering. Glass formation ability is found in films with a TL content in the range of < 40 -80 > at. %. The two independent elastic constants C 11 and C 44 have been selectively measured by the picosecond ultrasonics and the Brillouin light scattering techniques, respectively. Extrapolated elastic constant values from the glassy state have been identified for each pure virtual amorphous elements a-Zr, a-Cu, a-Co and a-Ni. The combination of BLS and PU offers unique and useful opportunity to anticipate plastic behavior of thin films, based on simple elasticity criteria. In the specific case of glassy materials, only two independent elastic constants are needed and selectively measured by M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 11 these two techniques. Through the three series of binary films, it is clearly shown that conventional representation often used for explaining the fine mechanical behavior of crystalline materials (Pugh's ratio as a function of Cauchy's pressure, Blackman's diagram and cooperation parameter) can also be successfully involved for a better understanding of plastic/elastic behavior of TFMGs.

Figure 1 :

 1 FIGURES CAPTIONSFigure 1: Representative θ-2θ x-ray patterns of Zr-TL (TL = Cu, Ni, Co) samples as a

Figure 2 :

 2 Figure 2: Mass density of Zr-TL (TL=Cu, Co and Ni) films measured by XRR as a function

Figure 3 :

 3 Figure 3: Longitudinal C 11 and shear C 44 elastic constants as a function of TL atomic

Figure 4 :

 4 Figure 4: Shear modulus G as a function of the Young's modulus E of Zr-TL films. A linear

Figure 5 :

 5 Figure 5: (a) Blackman's diagram (C 12 /C 11 ) as a function of (C 44 /C 11 ) of the Zr-TL films. (b)

Fig 4 .

 4 Fig 4. Apreutesei et al.

TABLE CAPTION Table 1 :

 CAPTION1 Results of extrapolated elastic constants (C 11 , C 44 ) and moduli (G, E, B, ν) for pure transition metal elements a-Zr, a-Cu, a-Co and a-Ni, considering an ideal solution behavior in the glassy state of Zr-Cu, Zr-Co and Zr-Ni thin film metallic glasses. Polycrystalline values are given for comparison.

	Element C 11 (GPa)	C 44 =G (GPa)	E (GPa)	B (g/cm 3 )	ν
	a-Zr	84	17	47	61	0.373
	Zr	141	33	89	97	0.347
	a-Cu	184	37	102	135	0.374
	Cu	201	47	127	138	0.347
	a-Co	224	53	143	153	0.345
	Co	296	81	212	188	0.311
	a-Ni	239	58	155	162	0.340
	Ni	295	83	216	184	0.304

Acknowledgement

The authors would like to acknowledge the CLYM platform (Consortium Lyon Saint-Etienne de Microscopie, http://www.clym.fr) for the access to the 2010F Jeol TEM.