Raul Santiesteban

Thierry Floquet

Yuri Orlov

Samer Riachy

Jean-Pierre Richard

Second Order Sliding Mode Control of Underactuated Mechanical Systems II: Orbital Stabilization of an Inverted Pendulum with Application to Swing Up/Balancing Control

Orbital stabilization of an underactuated cart-pendulum system is under study. The quasihomogeneous control synthesis is utilized to design a second order sliding mode controller that drives the actuated cart to a periodic reference orbit in finite time, while the non-actuated pendulum produces bounded oscillations. A modified Van der Pol oscillator is introduced into the synthesis as an asymptotic generator of the periodic motion. The resulting closed-loop system is capable of moving from one orbit to another by simply changing the parameters of the Van der Pol modification. Performance issues of the proposed synthesis are illustrated in numerical and experimental studies of the swing up/balancing control problem of moving a pendulum, located on an actuated cart, from its stable downward position to the unstable inverted position and stabilizing it about the vertical.

Introduction

Motivated by applications where the natural operation mode is periodic, orbital stabilization of mechanical systems has received significant attention over the last few years (see, e.g., [START_REF] Shiriaev | Constructive tool for orbital stabilization of underactuated nonlinear systems: virtual constraints approach[END_REF] and references therein). For these systems the orbital stabilization paradigm, referred to as periodic balancing [START_REF] Canudas-De-Wit | Orbital Stabilization of Under-actuated Mechanical Systems[END_REF], differs from typical formulations of output tracking where the reference trajectory to follow is known a priori. The control objective for the periodic balancing, e.g., a walking rabbit [START_REF] Chevallereau | Rabbit: A testbed for advance control theory[END_REF] is to result in the closedloop system that generates its own periodic orbit similar to that produced by a nonlinear 1 2 This work is supported by CONACYT under grant number 45900 as well as by the regional council Nord-Pas-de-Calais and the FEDER (European Funds of Regional Development) under the projects ARCir RoboCoop and TAT31.

oscillator. Apart from this, the closed-loop system should be capable of moving from one orbit to another by simply modifying the orbit parameters such as frequency and/or amplitude.

In the present paper, a periodic balancing problem is under study for an underactuated mechanical system. A non-actuated pendulum, located on an actuated cart, appears as a special test bed. Orbital stabilization of this system captures all the essential features of the general treatment thereby forming a basis for the extension to other underactuated systems.

We demonstrate that the quasihomogeneous synthesis, developed in the first part of the paper [START_REF] Riachy | Second Order Sliding Mode Control of Underactuated Mechanical Systems I: Local Stabilization with Application to an Inverted Pendulum[END_REF], is applicable to the cart-pendulum system to design a Second Order Sliding Mode (SOSM) controller that drives an actuated part of the system to a periodic reference orbit in finite time in spite of the presence of friction forces and external disturbances with an a priori known magnitude bound. The resulting controller, that presents a contribution of the paper, exhibits an infinite number of switches on a finite time interval, however, in contrast to first order sliding mode controllers, it does not rely on the generation of sliding motions on the switching manifolds but on their intersections.

A modified Van der Pol oscillator, proposed in [START_REF] Orlov | Quasihomogeneity approach to the pendubot stabilization around periodic orbits[END_REF], is introduced into the synthesis as a reference model. The proposed modification still possesses a stable limit cycle, governed by a standard linear oscillator equation, and therefore it constitutes an asymptotic harmonic generator as opposed to a standard Van der Pol oscillator, exhibiting a non-sinusoidal response in its limit cycle.

Another example of an asymptotic harmonic generator (nearly the only one available in the literature) is the variable structure Van der Pol oscillator from [START_REF] Sira-Ramirez | Harmonic response of variable-structure-controlled Van der Pol oscillators[END_REF]. However, it is hardly possible to use that oscillator for generating a reference signal because the system response would be contaminated by high frequency oscillations (a so-called chattering effect) caused by fast switching of the structure of the Van der Pol oscillator.

In contrast to a linear oscillator, whose amplitude depends on the initial conditions, both the amplitude and frequency of the sinusoidal signal, generating by the modified Van der Pol oscillator, can readily be modified on-line by simply changing the oscillator parameters. Due to this, the modified Van der Pol oscillator is well-suited for addressing the problem in question.

Effectiveness of the orbitally stabilizing synthesis is illustrated in numerical and experimental studies of the swing up/balancing control problem for a laboratory cartpendulum system. In our studies, the pendulum, driven by a hybrid controller to be constructed, is required to move from its stable downward position to the unstable upright position and be stabilized about the vertical while the cart is stabilized about a desired endpoint.

The proposed hybrid controller is based on the orbital transfer strategy, similar to that used in [START_REF] Orlov | Model orbit robust stabilization (MORS) of Pendubot with application to swing up control[END_REF] to swing up a Pendubot to its upright position. First, a swinging controller is composed by an inner loop controller, partially linearizing the cart-pendulum system, and an orbitally stabilizing outer loop controller, that pumps into the system as much energy as required to approach a homoclinic orbit with the same energy level as that corresponding to the desired equilibrium point. Once the cart-pendulum system reaches this homoclinic orbit, the orbitally stabilizing controller is turned off and the system is therefore obliged to evolve along the homoclinic orbit.

Finally, turning on a locally stabilizing controller, when the homoclinic motion enters the attraction basin of the latter controller, completes a unified framework for the swing up/balancing control of the pendulum, located on the cart. The locally stabilizing controller from [START_REF] Riachy | Second Order Sliding Mode Control of Underactuated Mechanical Systems I: Local Stabilization with Application to an Inverted Pendulum[END_REF], which is also based on quasihomogeneous considerations, is involved into our hybrid controller design. Being verified experimentally, the proposed framework is another contribution of the paper and it presents an interesting addition to the energy-based approach from [START_REF] Astrom | Swinging up a pendulum by energy control[END_REF][START_REF] Lozano | Stabilization of the inverted pendulum around its homoclinic orbit[END_REF] to stabilization of mechanical systems with underactuation degree one.

The paper is organized as follows. The quasihomogeneous orbital stabilization of the cart-pendulum system is developed in Section 2. Numerical and experimental results on application of the orbitally stabilizing synthesis to the swing up/balancing control problem are given in Section 3. Section 4 finalizes the paper with some conclusions.

Orbitally Stabilizing Synthesis

In order to facilitate exposition, the orbitally stabilizing synthesis is developed for a laboratory cart-pendulum system from our companion paper [START_REF] Riachy | Second Order Sliding Mode Control of Underactuated Mechanical Systems I: Local Stabilization with Application to an Inverted Pendulum[END_REF].

Problem Statement

The cart-pendulum system, presented in [START_REF] Riachy | Second Order Sliding Mode Control of Underactuated Mechanical Systems I: Local Stabilization with Application to an Inverted Pendulum[END_REF], is governed by

(M + m)ẍ + ml sin θ θ2 -ml cos θ θ = τ + w 1 (t) -ψ( ẋ), (1)
4 3 ml 2 θ -ml cos θ ẍ -mgl sin θ = w 2 (t) -ϕ( θ) (2)
where x is the cart position, θ is the angular deviation of the pendulum from the vertical, M is the cart mass, m is the rod mass, l is the distance to the center of mass of the pendulum, g is the gravitational acceleration, τ is the controlled input, w 1 (t), w 2 (t) are external disturbances, ψ( ẋ) and ϕ( θ) are friction forces, affecting the cart and the pendulum, respectively. In order to describe the friction forces the classical model is utilized:

ψ( ẋ) = ψ v ẋ + ψ c sign( ẋ), ϕ( θ) = ϕ v θ + ϕ c sign( θ). (3)
The above model comes with the viscous friction coefficients ψ v , ϕ v > 0, the Coulomb friction level ψ c , ϕ c > 0, and the standard notation sign(•) for the signum function. Subject to (3) the right-hand side of the dynamic system (1)-( 2) is piece-wise continuous. Throughout, solutions of such a system are defined in the sense of Filippov [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF] as that of a certain differential inclusion with a multi-valued right-hand side.

Because the phenomenon of friction is hard to model, we have introduced the uncertain terms w 1 (t), w 2 (t) into the dynamic equations ( 1), (2) to account for destabilizing model discrepancies such as Stribeck effect and backlash. Due to dissipative properties of mechanical systems, upper bounds N i > 0, i = 1, 2 for the magnitudes of these terms can normally be estimated a priori:

|w i (t)| ≤ N i (4)
for all t.

Our objective is to design a controller that causes the actuated part of the cartpendulum system to track lim

t→∞ [z(t) + x(t)] = 0, (5)
a trajectory z(t) generated by the modified Van der Pol equation [START_REF] Orlov | Quasihomogeneity approach to the pendubot stabilization around periodic orbits[END_REF] ..

z + ε[(z 2 + ż2 ν 2 ) -ρ 2 ] ż + ν 2 z = 0, (6)
while also attenuating the effect of the friction forces (3) and external disturbances (4).

To this end, we present several arguments to consider the modified Van der Pol equation as a good choice of a reference model. The Van der Pol modification ( 6) is firstly shown [START_REF] Orlov | Quasihomogeneity approach to the pendubot stabilization around periodic orbits[END_REF] to possess a stable limit cycle, being expressible in the explicit form

z 2 + ż2 ν 2 = ρ 2 (7)
where the parameter ρ stands for the amplitude of the limit cycle and ν is for its frequency. By substituting the orbit equation ( 7) into ( 6) the limit cycle of the modified Van der Pol equation ( 6) is secondly concluded to be remarkably generated by a standard linear harmonic oscillator

.. z + ν 2 z = 0, (8)
initialized on [START_REF] Hirschorn | Generalized sliding mode control for multi-input nonlinear systems[END_REF]. This is completely opposite to the non-modified Van der Pol oscillator, whose general representation is given by the second order scalar nonlinear differential equation

.. z + ε[(z -z 0 ) 2 -ρ 2 ] ż + ν 2 (z -z 0 ) = 0 (9)
with positive parameters ε, ρ, ν, and which exhibits a nonsinusoidal periodic response in its limit cycle (see, e.g., [START_REF] Khalil | Nonlinear systems[END_REF] for details). Summarizing, the modified Van der Pol oscillator (6) constitutes a nonlinear asymptotic harmonic generator which naturally exhibits an ideal sinusoidal signal (8) in its limit cycle [START_REF] Hirschorn | Generalized sliding mode control for multi-input nonlinear systems[END_REF]. In contrast to the linear oscillator [START_REF] Isidori | Nonlinear Control Systems II[END_REF], whose amplitude depends on the initial conditions of the oscillator, the amplitude and frequency of this sinusoidal signal can be varied at will by tuning the parameters ρ and ν of the asymptotic harmonic generator (6).

Control Strategy

In order to present a control strategy that allows one to achieve the above objective let us partially linearize the cart-pendulum dynamics. For this purpose, let us rewrite the state equation ( 2) in the form

.. θ = 3 4ml 2 [ml cos θ .. x + mgl sin θ + w 2 (t) -ϕ( θ)]. (10)
Now substituting equation ( 10) into (1) yields

[(m+M )- 3 4 m cos 2 θ] .. x = τ +w 1 (t)-ψ( ẋ)-ml sin θ θ2 + 3 4 mg cos θ sin θ+ 3 4l [w 2 (t)-ϕ( θ)] cos θ. (11)
Finally, setting

J = (m + M ) -3 4 m cos 2 θ = 1 4 m + M + 3 4 m sin 2 θ and τ = Ju + ml sin θ θ2 - 3 4 mg cos θ sin θ ( 12
)
where u is the new control input, and taking into account that the relation J = 0 holds for all values of θ, the desired linearization is obtained:

..

x = u + 3 cos θ 4lJ [w2(t) -ϕ( θ)] + 1 J [w1(t) -ψ( ẋ)] (13)
..

θ = 3 4l {u cos θ + 3m cos 2 θ + 4J 4mlJ [w2(t) -ϕ( θ)] + cos θ J [w1(t) -ψ( ẋ)] + g sin θ}. ( 14
)
Since system ( 13), ( 14) describes the linearized actuated joint model it is referred to as collocated linearization [START_REF] Spong | The Swing Up Contol Problem for the Acrobot[END_REF].

The control strategy is now formalized as follows. The control input ( 12) is composed by an inner loop controller, partially linearizing the cart-pendulum, and an outer loop controller u to be constructed. Given the system output

y(t) = z(t) + x(t), (15)
that combines the actuated state x(t) of the system and the reference variable z(t) governed by the modified Van der Pol equation ( 6), the outer loop controller u is to drive the system output [START_REF] Orlov | Quasihomogeneity approach to the pendubot stabilization around periodic orbits[END_REF] to the surface y = 0 in finite time and maintain it there in spite of the friction forces ψ( ẋ), ϕ( θ) and external disturbances w 1 (t), w 2 (t), affecting the system.

SOSM Control Synthesis

Due to (6), ( 13), [START_REF] Orlov | Quasihomogeneity approach to the pendubot stabilization around periodic orbits[END_REF], the output dynamics is given by ..

y = u + 3 cos θ 4lJ [w2(t) -ϕ( θ)] + 1 J [w1(t) -ψ( ẋ)] -ε[(z 2 + ż2 ν 2 ) -ρ 2 ] ż -ν 2 z. ( 16
)
The following control law

u = 3ϕv cos θ 4lJ θ + ψv J ẋ + ε[(z 2 + ż2 ν 2 ) -ρ 2 ] ż + ν 2 z -αsign(y) -βsign( ẏ) -hy -p ẏ (17)
with the parameters such that

h, p ≥ 0, α -β > 3(ϕ c + N 2 ) 4lJ + ψ c + N 1 J (18)
is proposed. The closed-loop system (3), ( 16), ( 17) is then feedback transformed to the one

.. y = 3 cos θ 4lJ [w 2 (t) -ϕ c sign( θ)] + 1 J [w 1 (t) -ψ c sign( ẋ)] -αsign(y) -βsign( ẏ) -hy -p ẏ (19)
with piece-wise continuous right-hand side.

Relating the quasihomogeneous synthesis from [START_REF] Orlov | Finite-time stability and robust control synthesis of uncertain switched systems[END_REF], the above controller has been composed of the linear viscous friction compensator

u f = 3ϕ v cos θ 4lJ θ + ψ v J ẋ, (20)
nonlinear trajectory compensator

u c = ε[(z 2 + ż2 ν 2 ) -ρ 2 ] ż + ν 2 z, (21)
the homogeneous switching part (the so-called twisting controller from [START_REF] Fridman | Higher order sliding modes, in Sliding mode control in engineering[END_REF])

u h = -αsign(y) -βsign( ẏ), (22)
and the linear remainder

u l = -hy -p ẏ (23)
that vanishes in the origin y = ẏ = 0. By Theorem 1 from [START_REF] Riachy | Second Order Sliding Mode Control of Underactuated Mechanical Systems I: Local Stabilization with Application to an Inverted Pendulum[END_REF] the quasihomogeneous system [START_REF] Sira-Ramirez | Harmonic response of variable-structure-controlled Van der Pol oscillators[END_REF] with the parameter subordination ( 18) is finite time stable regardless of which friction forces (3) and uniformly bounded external disturbances subject to (4) affect the system. The control objective is thus achieved and the following result is obtained.

Theorem 1 Let the modified Van der Pol equation ( 6) with positive parameters ε, ν, ρ be a reference model of the cart-pendulum dynamics ( 1)-( 3) and let the system output be given by [START_REF] Orlov | Quasihomogeneity approach to the pendubot stabilization around periodic orbits[END_REF]. Then the quasihomogeneous controller ( 12), ( 17), [START_REF] Shiriaev | Constructive tool for orbital stabilization of underactuated nonlinear systems: virtual constraints approach[END_REF] drives the cart-pendulum system to the zero dynamics surface y = 0 in finite time, uniformly in friction forces [START_REF] Canudas De | A new model for control of systems with friction[END_REF] and admissible disturbances [START_REF] Chevallereau | Rabbit: A testbed for advance control theory[END_REF]. After that the actuated part x(t) follows the output -z(t) of the modified Van der Pol equation ( 6) whereas the nonactuated part θ(t) is governed by the zero dynamics equation

4 3 l .. θ = cos θν 2 x + g sin θ + 1 ml [w 2 (t) -ϕ( θ)]. ( 24
)
on any finite time interval.

Proof: By Theorem 8 of [5, p. 85] the closed-loop system (13), ( 14), ( 17) has a local solution for all initial data and admissible disturbances (4). Let us demonstrate that each solution of this system is globally continuable on the right.

Due to (4), the magnitude |W (t)| of the uncertainty

W (t) = 3 cos θ 4lJ [w 2 (t) -ϕ c sign( θ)] + 1 J [w 1 (t) -ψ v sign( ẋ)]
that appears in the right-hand side of ( 19) is upper estimated as follows:

|W (t)| ≤ 3(ϕ c + N 2 ) 4lJ + ψ c + N 1 J . ( 25
)
Since this estimate, coupled to [START_REF] Shiriaev | Constructive tool for orbital stabilization of underactuated nonlinear systems: virtual constraints approach[END_REF], ensures that

α -β > |W (t)|, (26)
Theorem 4.2 from [START_REF] Orlov | Finite-time stability and robust control synthesis of uncertain switched systems[END_REF] turns out to be applicable to the quasihomogeneous system [START_REF] Shiriaev | Constructive tool for orbital stabilization of underactuated nonlinear systems: virtual constraints approach[END_REF], [START_REF] Sira-Ramirez | Harmonic response of variable-structure-controlled Van der Pol oscillators[END_REF]. By applying this theorem, system [START_REF] Sira-Ramirez | Harmonic response of variable-structure-controlled Van der Pol oscillators[END_REF] subject to ( 18) is proved to be finite time stable, uniformly in admissible disturbances [START_REF] Chevallereau | Rabbit: A testbed for advance control theory[END_REF]. Now employing [START_REF] Orlov | Quasihomogeneity approach to the pendubot stabilization around periodic orbits[END_REF], it follows that along with a solution y(t) of ( 19), an arbitrary solution x(t) = y(t) -z(t) of ( 13) is globally continuable on the right and uniformly bounded in t.

Moreover, due to the uniform boundedness of y(t), the control signal ( 17) is uniformly bounded, too. Thus, an arbitrary solution θ(t) of ( 14) is also globally continuable on the right.

So, starting from a finite time moment the cart-pendulum system evolves in the second order sliding mode on the zero dynamics surface y = 0 and to complete the proof it remains to derive the sliding mode system dynamics. For this purpose let us apply the equivalent control method [START_REF] Utkin | Sliding modes in Electromechanical Systems[END_REF] and substitute the only solution u eq of the algebraic equation

u + 3 cos θ 4lJ [w 2 (t) -ϕ( θ)] + 1 J [w 1 (t) -ψ( ẋ)] -ε[(x 2 + ẋ2 ν 2 ) -ρ 2 ] ẋ -ν 2 x = 0
with respect to u (i.e., the equivalent control input u eq that ensures equality ÿ = 0) into [START_REF] Orlov | Finite-time stability and robust control synthesis of uncertain switched systems[END_REF]. Then, while being restricted to this surface, the system dynamics is given by

4 3 l .. θ = cos θ{ 3 cos θ 4lJ [ϕ( θ) -w2(t)] + 1 J [ψ( ẋ) -w1(t)] + ε[(x 2 + ẋ2 ν 2 ) -ρ 2 ] ẋ + ν 2 x} + 3m cos 2 θ + 4J 4mlJ [w2(t) -ϕ( θ)] + cos θ J [w1(t) -ψ( ẋ)] + g sin θ = cos θ{ε[(x 2 + ẋ2 ν 2 ) -ρ 2 ] ẋ + ν 2 x} + 1 ml [w2(t) -ϕ( θ)] + g sin θ (27)
where x(t) is the reference trajectory governed by the modified Van der Pol equation [START_REF] Fridman | Higher order sliding modes, in Sliding mode control in engineering[END_REF]. For the orbits x(t) of the Van der Pol modification (6), initialized on the limit cycle [START_REF] Hirschorn | Generalized sliding mode control for multi-input nonlinear systems[END_REF], the zero dynamics ( 27) is simplified to

4 3 l .. θ = cos θν 2 x + g sin θ + 1 ml [w 2 (t) -ϕ( θ)].
The sliding mode equation ( 27) on the surface y = 0 is thus validated for the nonactuated variable θ(t), restricted to any finite time interval. Theorem 1 is proved. Analyzing the proof of Theorem 1 one can conclude that the actuated variable x(t) remains bounded regardless of which admissible disturbances affect the closed-loop system. For practical reasons the non-actuated variable θ(t) is also required to remain bounded under arbitrary disturbances of sufficiently small magnitudes. Due to this, the zero dynamics ( 24) is required to be a bounded input -bounded state system, locally in (w 1 , w 2 ). While being beyond the scope of the paper, this item is not studied here in details, but only numerical and experimental evidences, demonstrating that this is indeed the case, are presented in the next section (for analysis of bounded inputbounded state systems see, e.g., [START_REF] Isidori | Nonlinear Control Systems II[END_REF][START_REF] Liberzon | Output-input stability and minimum phase nonlinear systems[END_REF][START_REF] Sussman | The peaking phenomenon and the global stabilization of nonlinear systems[END_REF]).

In the remainder, capabilities of the synthesis procedure, constituted by Theorem 1, are tested in numerical and experimental studies of the laboratory cart-pendulum system.

Swing up control and stabilization

In the test swing up/balancing control problem, the pendulum, located on the cart, is required to move from its stable downward position to the unstable upright position and be stabilized about the vertical while the cart is stabilized about a desired endpoint. An orbitally stabilizing controller is designed to swing up the pendulum to make it reach a homoclinic orbit of the energy level, corresponding to the cart pendulum system at the desired equilibrium point. Once the homoclinic orbit is reached at a state where the cart velocity is infinitesimal, the orbitally stabilizing controller is turned off thereby letting the system evolve along the homoclinic orbit. A locally stabilizing controller from [START_REF] Riachy | Second Order Sliding Mode Control of Underactuated Mechanical Systems I: Local Stabilization with Application to an Inverted Pendulum[END_REF] is then turned on when the homoclinic motion enters the attraction basin of the latter controller.

The hybrid control strategy, to be tested in an experimental study, is to select the amplitude ρ and the frequency ν of the model limit cycle [START_REF] Hirschorn | Generalized sliding mode control for multi-input nonlinear systems[END_REF] reasonably small and the parameter ε, controlling the speed of the limit cycle transient in the modified Van der Pol equation ( 6), reasonably large to ensure that the cart-pendulum system reaches its homoclinic orbit of the corresponding energy level. Turning off the orbitally stabilizing controller, once the system is synchronized with the homoclinic orbit, and proper switching to the stabilizing controller when the subsequent homoclinic motion enters the attraction basin of the latter controller yields the generation of a swing up motion, asymptotically stable about the vertical.

Cart-pendulum prototype

In order to observe the performance of the proposed synthesis we made simulations on Matlab using Simulink. We considered the real parameters of the laboratory cartpendulum system from [START_REF] Riachy | Second Order Sliding Mode Control of Underactuated Mechanical Systems I: Local Stabilization with Application to an Inverted Pendulum[END_REF]. These parameters are listed in Table 1.

ψ v 8.5 N • s/m ϕ v 0.0015 N • m • s/rad ψ c 6.5 N ϕ c 0.00115 N • m

Swinging controller design

By a suitable choice of the reference parameters ε, ρ, ν, the orbitally stabilizing synthesis ( 6), ( 12), ( 17) can be applied to the cart-pendulum system to swing the pendulum up from the downward position to the upright position. To successfully apply this synthesis the cumulative energy, pumped by the resulting controller into the cart-pendulum system, is to be of an appropriate level. According to the energy-based approach [START_REF] Lozano | Stabilization of the inverted pendulum around its homoclinic orbit[END_REF], such a level in the friction-free case must ensure that the total energy

E(q, q) = 1 2 (M + m) ẋ2 -ml ẋ θ cos θ + 2 3 ml 2 θ2 + mgl(cos θ -1) (28)
of the closed-loop system increases from the negative value

E 0 = -2mgl, (29)
at the initial time moment to the zero-value energy

E 0 = 0, (30)
at a time instant when the cart velocity becomes infinitesimal and ready to change the rotation direction. Thus synchronized, the unforced friction-free system generates a homoclinic orbit converging to the desired equilibrium that has the same energy level (30).

In order to take into account friction forces, resulting in an energy loss of the laboratory cart-pendulum system, the required energy level (30) has to deliberately be increased to a positive value

E 0 = δ > 0. (31)
Under a certain δ, found experimentally, the unforced friction system also generates an orbit, converging to the desired equilibrium. This orbit is further referred to as a quasihomoclinic orbit. So, tuning the energy parameter δ and the reference parameters ε, ρ, ν is crucial to a successful swing up. Appropriate values of these parameters are carried out in successive simulations. The parameter ρ, responsible for the amplitude of the cart oscillations, is simply set slightly smaller than the admissible road length of the laboratorial equipment. The parameters ε and ν are responsible for making a reasonably high convergence rate of the modified Van der Pol oscillator (6) and, respectively, for providing the desired energy level (31) of the system at a position of the cart where it possesses an infinitesimal velocity (the higher ε the faster convergence; the larger ν the greater the energy of the limit cycle and hence the greater the energy of the cart, tracking the limit cycle). These parameters are iteratively tuned to approach the quasihomoclinic orbit. Turning off the controller at the time instant, when the system attains the quasihomoclinic orbit, makes the system follow this orbit, thereby approaching the desired endpoint.

In our simulation study the controller gains were set to α = 3 N • m, β = 1 N • m, h = 0, p = 0. With these parameters, condition [START_REF] Shiriaev | Constructive tool for orbital stabilization of underactuated nonlinear systems: virtual constraints approach[END_REF] holds and the output tracking of the modified Van der Pol oscillator ( 6) is therefore guaranteed. The reference parameters were tuned to ε = 40 [rad] -2 s -1 , ρ = 0.5 rad, ν = 1 s -1 . Being found numerically for the above parameters, the energy parameter took the value δ = 0.8 N • m. Once the system energy attained this value under x = ρ, ẋ = 0, the orbitally stabilizing controller was turned off and the unforced pendulum was swang up along the quasihomoclinic orbit. Detailed experimental results, supporting the proposed swing up synthesis, are presented in Subsection 3.6.

Locally stabilizing controller design

Being developed in the companion paper [START_REF] Riachy | Second Order Sliding Mode Control of Underactuated Mechanical Systems I: Local Stabilization with Application to an Inverted Pendulum[END_REF], the locally stabilizing controller

τ = 4lJ cos θ h 3 + 8lλ 2 θ sin θ i h -µ(θ, θ) -α 1 sign(ξ) -β 1 sign( ξ) -h 1 ξ -p 1 ξi , α 1 , β 1 , h 1 , p 1 > 0 (32) subject to µ(θ, θ) = 2 tan θ cos 2 θ θ2 + 3+8lλ2 θ sin θ cos θ (M +m)g-ml cos θ θ2 D tan θ +λ 1 g + 4 3 l θ2 cos θ tan θ + λ 2 g + 4 3 l θ2 (1+sin 2 θ) cos θ θ cos 2 θ , (33)
ξ = tan θ -λ 1 η -λ 2 η, λ 1 , λ 2 > 0 (34) η = x - 4l 3 ln 1 + sin θ cos θ , |θ| < π 2 , (35)
was tested in a simulation study. Quite impressive numerical results were obtained for the local stabilization of the pendulum about the upright position with the controller parameters α 1 = 55 m/s 2 , β 1 = 35 m/s 2 , h 1 = 0, p 1 = 0, λ 1 = 0.2 1/m, λ 2 = 0.26 s/m. The results can be observed in Subsection 3.5 where the proposed controller is introduced into our hybrid synthesis of swinging the Cart-Pendulum up and balancing it about the vertical.

Hybrid Controller Design

In order to accompany swinging the pendulum up by the subsequent stabilization around the upright position the swinging controller, presented in Subsection 3.2, is turned off, once the system reaches the corresponding homoclinic orbit, and then the locally stabilizing controller from [START_REF] Riachy | Second Order Sliding Mode Control of Underactuated Mechanical Systems I: Local Stabilization with Application to an Inverted Pendulum[END_REF] is turned on once the pendulum, evolving along the quasihomoclinic orbit, enters the basin of attraction, numerically found for the latter controller. The resulting hybrid controller moves the inverted pendulum, located on the cart, from its downward position to the upright position and stabilizes it about the vertical whereas the cart is stabilized at the desired endpoint. The capability of the closed-loop system of reaching the homoclinic orbit and entering the attraction basin of the locally stabilizing controller is additionally supported by experimental results.

Numerical verification

The initial conditions of the position of the cart-pendulum system and that of the modified Van der Pol oscillator, selected for all experiments, were x(0) = 0, θ(0) = 3.14 rad, and z(0) = 0.05 rad, whereas all the velocity initial conditions were set to zero.

To begin with, we separately studied the orbitally stabilizing controller from Subsection 3.2. For demonstrating the capability of the controller to move the pendulum from one orbit to another by modifying the orbit parameters we then introduced a random time instant t 0 (it was t 0 ≈ 10s in the experiment), when the amplitude ρ and frequency ν of the model limit cycle was changed from their initial values ρ = 0.1 rad, ν = 10 s -1 to the new one ρ = 0.5 rad, ν = 1 s -1 .

Finally, the hybrid controller from Subsection 3.4 was implemented to swing the pendulum up and stabilize it about the vertical while the cart is stabilized around the initial position.

Numerical results for the resulting motion, enforced by the orbitally stabilizing controller are depicted in Figs. 1. This figure demonstrates that while being driven by the orbitally stabilizing controller, the closed-loop system, perturbed by the permanent external disturbances w 1 (t) ≡ 0.5 N, w 2 (t) ≡ 0.5 N • m, generates a bounded, quasiperiodic motion. As predicted by theory, the desired orbital transfer is achieved by simply changing the amplitude of the orbit limit cycle. Thus, good performance of the orbitally stabilizing controller is concluded from Fig. 1.

Experimental verification

The initial conditions of the position of the cart-pendulum system and that of the modified Van der Pol oscillator, selected for the experiment, were x(0) = -0.3 m, θ(0) = π rad (downward position), and z(0) = 0.3 m, whereas all the velocity initial conditions were set to zero.

The hybrid controller from Subsection 3.4 was implemented to swing the pendulum up and stabilize it about the vertical while the cart is stabilized around the desired endpoint.

Experimental results for the resulting motion, enforced by the orbitally stabilizing controller and the hybrid controller, are depicted in Fig. 2. The switching moment occurs in the interval 2-3 seconds. Notice that the control effort is zero in this interval and that the cart velocity is zero. So, as stated above, the pendulum travels on its homoclinic orbit towards the unstable equilibrium and, once it is close enough, switching to the local controller occurs. The local controller is a robust nonlinear one, having a large domain of attraction (see [START_REF] Riachy | Second Order Sliding Mode Control of Underactuated Mechanical Systems I: Local Stabilization with Application to an Inverted Pendulum[END_REF]), and switching, that occurs at 10 • from the upright position, is constrained by the road length of the laboratorial equipment.

In order to obtain a rough idea concerning the performance of the developed controller, the experimental results had been compared to the one in [START_REF] Lozano | Stabilization of the inverted pendulum around its homoclinic orbit[END_REF], where the switching from the swinging controller to a local state feedback was done after approximately 30 seconds. In the approach presented in this paper, the switching to the local quasihomogeneous controller occurs at approximately 2.5 seconds, and the stabilization is accomplished in a shorter time compared to that of [START_REF] Lozano | Stabilization of the inverted pendulum around its homoclinic orbit[END_REF]. It should also be noted that while the energy-based approach from [START_REF] Lozano | Stabilization of the inverted pendulum around its homoclinic orbit[END_REF] is limited to models with negligible friction forces, the present approach does not suffer from this drawback.

Conclusions

Orbital stabilization of a cart-pendulum system, presenting a simple underactuated (two degrees-of-freedom, one actuator) manipulator, is under study. The quasihomogeneous SOSM-based control synthesis is utilized to design a variable structure controller that drives the pendulum to a desired zero dynamics manifold in finite time and maintains it there in spite of the presence of external disturbances.

Capabilities of the quasihomogeneous synthesis and its robustness are illustrated in an experimental study of the swing up/balancing control problem of moving the pendulum from its stable downward position to the unstable inverted position and stabilizing it about the vertical.

The orbitally stabilizing synthesis, being of interest in itself and as a part of the hybrid controller design of the swing up/balancing control of the cart-pendulum system, is the main contribution of the paper.

Eliminating undesirable chattering oscillations, caused by fast switching in the implemented hybrid controller, is among other problems calling for further investigation. While being non-trivial, this problem is however well-understood from the existing literature (see, e.g., [START_REF] Hirschorn | Generalized sliding mode control for multi-input nonlinear systems[END_REF] and references therein) and hopefully general methods of chattering reduction apply here as well.

Figure 1 :

 1 Figure 1: Orbital stabilization of the cart-pendulum system: left column for the orbital recovery under the permanent disturbances, right column for the transfer from one orbit to another.

Figure 2 :

 2 Figure 2: Experimental results: left cart and pendulum position, right control input.

  Raul Santiesteban, Yuri Orlov: CICESE Research Center, P.O. Box 434944, San Diego, CA, 92143-4944. Thierry Floquet, Samer Riachy, Jean-Pierre Richard: Projet ALIEN, INRIA FUTURS and Equipe SyNeR, LAGIS UMR CNRS 8146, Ecole Centrale de Lille, P.O. Box: 48, Cité Scientifique, 59651 Villeneuve d'Ascq, France.

Table 1 :

 1 Parameters of the Cart-Pendulum.

	Notation	Value	Units
	M	3.4	kg
	m	0.147	kg
	l	0.175	m


N Cocheteau
email: cocheteau@lma.cnrs-mrs.fr

A Maurel-Pantel

F Lebon
email: lebon@lma.cnrs-mrs.fr

I Rosu
email: rosu@lma.cnrs-mrs.fr

S Aitzaid
email: sonia.aitzaid@cnes.fr

I Savin Delarclause
email: isabelle.savindelarclause@cnes.fr

On the modeling of two bonded silicon surfaces

Keywords: Direct Bonding, Bonding Energy, Double Shear Test, cleavage, wedge test, process parameters, silica glasses, and Zerodur® glasses

Direct bonding is a well-known process. However in order to use this process in spatial instrument fabrication the mechanical resistance needs to be quantified. In order to improve bonded strength, optimal parameters of the process are found by studying the influence of annealing time, temperature and roughness which are studied using three experimental methods: double shear, cleavage and wedge tests. Those parameters are chosen thanks to the appearance of time/temperature equivalence. Results brought out a predictive model of the bonding energy.

Introduction:

Direct bonding consists in joining two surfaces without the use of any adhesives [START_REF] Tong | Semiconductor wafer bonding: recent developments[END_REF][START_REF] Kendall | Molecular adhesion and its applications[END_REF]. Direct bonding is a process involved in micro-fabrication process on micro-electronics. The main applications are on silicon-on-insulator devices [START_REF] Plobl | Wafer direct bonding: tailoring adhesion between brittle materials[END_REF]. They are of particular interest to spatial instruments applications. Indeed, this is a high-precision process and assemblies obtained present a dimensional stability due to the fact that no mechanical part or glue is required and the risks of contamination associated with degassing are avoided. This kind of process has already passed with success the mechanical and thermal environment of space [START_REF] Pamplona | Three bipods slicer prototype: tests and finite element calculations[END_REF], where the constraints involved (thermal fatigue, vibrations, etc.) are very different from those encountered on Earth. However, this is necessary to quantify the bonding strength and to improve the mechanical performance of adhesive bonds without degrading optical performances of the material used.

In this paper we present the mechanical characterization of silica and Zerodur samples using double shear, cleavage and wedge tests and the influence of some parameters process on the mechanical strength of bonded interfaces in order to find the optimal parameters of the process.

Double shear tests:

Doubles shear test were performed to find optimal parameters of the process. The influence of annealing time and temperature and roughness (coupled with thermal treatment) was studied.

Samples are constituted with two cylinders of 5 mm thickness and 10 mm diameter, and one cylinder of 5 mm thickness and 15 mm diameter bonded together. Results, presents in Fig. 1a, b show a nonlinear increase of the mechanical resistance with annealing temperature and time with the appearance of a plateau for long time treatment. Moreover, a kind of equivalence between short time treatment at high temperature and longtime treatment at short temperature is observable.

Fig. 1c shows an increase of the mechanical resistance until a roughness of 0.6 nm RMS then a decrease for silica samples. For Zerodur samples, mechanical resistance can be supposed constant due to the dispersion of results. Results do not agree with literature [START_REF] Cocheteau | On the Modelling of the Direct Bonding of Two Silicon Surfaces[END_REF], but samples have undergone a thermal treatment during 120 h at 200°C for silica and 130°C for Zerodur samples. A long time treatment promotes the formation of covalent bonds [START_REF] Stengl | A model for the silicon wafer bonding process[END_REF]. The bonding energy, thus, mechanical resistance, at this range of temperature is limited by the contacted area. We are led to believe that during this annealing treatment the area of contacted zone increases with the size of roughness until reaching a maximum before decreasing again.

Despite dispersion of the double shear test, results give a good idea of optimal parameters applying to increase the mechanical resistance of bonded interfaces. Due to the time temperature equivalence and non-agreement with literature results for roughness influence combined with manufacturing constraints (high temperature are not allowed in order to not degrade optical properties of materials and coating made on assemblies) the optimal parameters chosen are for both samples: 200°C during 120h and a roughness of 0.4 nm RMS.

Cleavage tests:

Previous parameters founded using double shear tests are compared to the initial conditionno thermal treatment applied on samplesusing cleavage tests. Samples are constituted with two blades (40x40x10 mm) bonded together. Between each experiment, surfaces are cleaned and re-adhered (without being polished anew). Results presented on Fig. 2 show an increase of the mechanical resistance with optimal parameters, thus an improvement with the new parameters. Moreover, mechanical resistance slightly linearly decreased with successive adhesion/cleavage test/re-adhesion highlighting a damaging of the bonded interfaces with successive re-adhesion made in our experiments.

Results for Zerodur samples are a little worse than results for silica sample, because, samples have already undergone other treatment.

The results confirm the use of optimal parameters found to increase the mechanical resistance of samples.

Wedge test:

Important parameter to characterize direct bonding is the bonding energy. The most common method in the literature to quantify the bonding energy is the wedge test [START_REF] Maszara | Bonding of silicon wafers for silicon-oninsulator[END_REF]. Experiments are performed on silica samples constituted with two blades (80mmx10mmx500µm) bonded together. In order to complete double shear tests experiments, the same influences are studied.

Between each adhesion, surfaces are cleaned and re-adhered (without being polished anew). The same trends as double shear and cleavage tests are recovered: an increase of the bonding energy (W) with annealing time and temperature with appearance of a plateau and with roughness.

A predictive model of the bonding energy has also been identified using wedge test results: W(T, t, R) = f(R)(W (T, R) -W (R))( 1 -e ( ( ) ) ) + W (R))

Fig. 1 :

 1 Fig. 1 : Double shear test results. Influence of annealing temperature (a), annealing time (b) and roughness coupled with thermal treatment.

Fig. 2 :

 2 Fig. 2 : Cleavage tests: comparison between initial condition and optimal parameters for silica (a) and Zerodur (b) samples

Fig. 3 :

 3 Fig. 3 : results of the wedge test, influence of annealing time, temperature and roughness (solid line: adhesion 1, dashed line: adhesion 2)

With :𝑊 (𝑇, 𝑅) = 𝑊(𝑡 = 120ℎ) = 𝑔(𝑇). 𝛼(𝑅) 𝑊 (𝑅) = 𝑊(𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛) = 𝑊 . 𝛼(𝑅) 𝑎 = , = ,1 ℎ Where f(R), g(T) and 𝛼(𝑅) are found using experimental results.

Conclusion:

The three mechanical tests show the same trends results: an increase of the mechanical resistance with annealing time, temperature and with roughness (for our procedure of samples preparation). A time/temperature is observed with a phenomenon of interface damaging with successive re-adhesion. Optimal parameters of the process have been chosen and validate (200°C, 120h, 0.4 nm RMS).

A predictive model of the bonding energy has been made and need to be implemented in a finite element model in order to predict the mechanical resistance of bonded assemblies depending on process parameters.


Adrien Fonnet
email: adrien.fonnet@

Florian Melki
email: florian.melki@univ-nantes.fr

Yannick Prié

Fabien Picarougne
email: fabien.picarougne@univ-nantes.fr

Gregoire Cliquet
email: g.cliquet@lecolededesign.com

IMMERSIVE DATA EXPLORATION AND ANALYSIS

This paper belongs to the emerging immersive analytics research field. It presents a system for exploratory analysis of multi-dimensional data focusing on providing freedom of locomotion while remaining comfortable for long user session.

The solution allows to investigate temporal traces through a 3D scatterplot using a cylindrical coordinate system. Users are free to move inside the tracked area either standing up or seated in a mobile chair. A limited user study (n=6) has been conducted to investigate if the designed solution was deemed appropriate for mobility and user comfort.

INTRODUCTION

Understanding and exploring multi-dimensional datasets is a critical process for any decision maker. Immersive analytics is an emerging research field aiming at providing supportive tool for data analysis and decision making through the use of immersive environment [START_REF] Chandler | Immersive Analytics[END_REF]. Building upon the initial reflexion proposed in [START_REF] Cliquet | Towards HMD-based Immersive Analytics[END_REF] for HMD based immersive analytics, this work describes an exploratory system focusing on navigation, i.e. keeping the freedom of room-scale movement while accounting for longsession.

RELATED WORK

Even though immersive analytics terminology has only been defined in 2015, using immersive environment for multi-dimensional data analysis and exploration has been investigated since 1996 [START_REF] Symanzik | Dynamic Statistical Graphics in the CAVE Virtual Reality Environment[END_REF]. Symanzik et al. used a CAVE to display 3D scatterplot, encoding additional attributes through color and shape, followed by (Ammoura2001, Nagel2001, Nagel2003, Nagel2008). Other hardwares have also been exploited, such as augmented reality through fiducial marker [START_REF] Meiguins | Multidimensional information visualization using augmented reality[END_REF] or 3D stereo with hand-tracking [START_REF] Azzag | VRMiner: a tool for multimedia database mining with virtual reality[END_REF].

Our system focuses on the use of virtual reality head mounted display (HMD) to visualize multidimensional data like the Wizard solution [START_REF] Datey | Immersive and collaborative data visualization using virtual reality platforms[END_REF], or the works of Donalek et al. (Donalek 2014) and Filho et al. [START_REF] Filho | Immersive Analytics of Dimensionally-Reduced Data Scatterplots[END_REF]. However, those solutions are either stand-up or desk only while we combine in our system both modalities. We use a tracked mobile chair, to allow users get both the comfort of being seated for long session and the freedom of free standing movement in the tracked space. Moreover, we use the chair's tablet as a support for menu selection to give haptic feedback through a tangible surface. Tangible surfaces have been used in immersive analytics serving other purposes such as a reference map for augmented reality visualization [START_REF] Franz | More Than Blips on the Radar : Exploring Immersive Visualization for Maritime Decision Making[END_REF], or as touch table hardware [START_REF] Butscher | Clusters, Trends, and Outliers: How Immersive Technologies Can Facilitate the Collaborative Analysis of Multidimensional Data[END_REF].

IDEA IMMERSIVE ANALYTICS SYSTEM

Our immersive analytics solution has been developed with Unity3D for HTC Vive HMD rendering display and HTC controllers as the main input for interaction. SYSTEM PURPOSES Our target activity is exploratory data analysis, where users are looking for insights within trace data without any precise hypothesis to test. The trace format we use in our system is composed of event. An event has generic attributes such as event type, date, actor (e.g. who does the action depicted in the event), as well as several type specific attributes.

VISUAL MAPPING

Each tuple of the dataset is represented by a sphere. This shape's choice limits occlusion at the cost of more computer processing [START_REF] Ammoura | Immersed Visual Data Mining: Walking the Walk[END_REF]. A cylindrical coordinate system is used to place data points based on their generic attributes, see fig. 1. Vertical axis represents users, horizontal curved axis time, and depth event types. Since additional data points specific attributes depend on the event type, using another general visual mapping would not be relevant, so readability has been improved by color coding event type information. User sessions appear quite naturally in this representation as alignments of sphere. Aggregation is also used to limit occlusion, by grouping data points of the same user and same event type which are temporally close to each other. The resulting object, is a semi-transparent sphere that contains the data points spheres scaled down. This solution seems adequate to aggregate up to 10 data points.

INTERACTIONS MODALITIES

All elements for interaction have been put in the user vicinity and represents its workstation (see fig. 2). A virtual disk represents the limit of the tracked physical space. A mobile chair, equipped with a tablet, tracked via Vive tracker (see fig. 3) is modelized and positioned on top of the disk. Both Vive controllers are used, one dedicated for movement while the second is for data and menu interaction. A virtual trashcan and plank, positioned on both side of the chair, complete the setup. Up to 4 movements modalities are available for navigation. The first is physical displacement while standing up using the HTC Vive head-tracking. Second is the ability to physically move on the platform while seated, rolling the mobile chair over the disk. Third, the user can "fly" the workstation in the direction of the left controller combined with the trackpad input. Fourth, the user can teleport the workstation, by selecting direction and distance of teleportation, the point of arrival being visualized by a semi-transparent duplication of the disk + chair. Switching from flying to teleport is done via the left trigger. And last, pushing left grip button teleports the workstation back to a central position for the user to have the full data visualization in its field of view.

The user can point at a data point with a raycast from his right controller to access its full details. A window with all information appears next to the data if the sphere is closed to the user and close to the user otherwise to optimize readability of the text. The same window can be triggered by putting the right controller in contact with a data point. Upon trigger press, a selected data point can also be duplicated and kept next to the workstation.

A menu is displayed on top of the chair's tablet to choose parameters for a data selection based on event type or user ID, multi-selection is available. Upon confirmation a virtual cube is generated in front of the user at arm's length. with the selection information written on it. This "selection cube" materializes the selection. Grabbing the cube with the right controller applies a highlight, i.e. a glowing effect on the selected data points. The cube can also be released on the top of the virtual plank to filter out all the data points that are not selected. Multiple selection cubes can be on the plank keeping visible only the data points at the intersection of the selection cubes. To undo a filter, the user can simply remove a cube from the virtual plank. A temporal zoom on the time axis is also available: the user can select two markers on the curved horizontal axis using the raycast. These markers are then used to change the visualization according to the new time interval.

Last, the virtual trashcan is used to destroy any unwanted object after use by placing them in contact with it, whether it is cloned data point or a selection cube.

USER STUDY

This section describes a limited user study conducted to draw some early conclusions. The focus was on qualitative aspects of the design, aiming to get user insights on visualization, interaction and comfort of use.

DATASET DESCRIPTION

The dataset explored contains traces extracted from the coconotes platform (https://coconotes.comin-ocw.org/) which is a e-learning video platform allowing user to annotate video at specific timestamp. Traces have been monitored over 5 months, from September 2016 to January 2017, and results of over 200 000 actions from 200 users. Only 17 users were used for the visualization resulting in only 5000 data points to render ensuring acceptable framerate in our preliminary prototype. Each tuple contains a user ID, a timestamp and an event type. Additional attributes depend on the event type, e.g. play or pause events will have specific attributes such as the name of the video. EXPERIMENT A participant's session lasted one hour. The following procedure was used: first a demographic questionnaire, then the presentation of the dataset and the visualization, and some training on interactions. Once we had the confirmation that the participant was comfortable with the system the main study started with a set of tasks. A post questionnaire and an interview wrapped-up the session.

The study had two phases. In the first phase participants had to perform basic search tasks, e.g. find a video name knowing the time it was played and a user ID or determine a trend by answering questions such as "what behaviour may you observe after the completion of video reading?". In the second phase they could explore the system as they pleased, discovering new information about the dataset.

The post-questionnaire contained 4 affirmations on movement modalities, data visualization and interactions, comfort of use, and overall experience, the user had to evaluate with 7 points Likert scales.

Six male subjects from the computer science section of the University of Nantes participated to the study (mean age 20.8 ranging from 20 to 23). They had good experience in video game but few in virtual reality and data analysis. The experiment was run on a computer equipped with a CPU Intel Xeon ES-1630 v4 (3.70GHz), 16 Gb of RAM and a Nvidia GTX 1060 graphics card.

RESULTS

Movement modalities scored an average of 5 [0.58], main complaint was the slow velocity of the flying movement. However, scaling the speed up could result in increased cyber-syckness so complementary solution such as tunnelling effect should be added. The linear movement could also be replaced by dashing step.

Regarding data visualization and interactions, with a mean of 5 [1.15], the feedbacks was positive towards the cube selection object and trash can metaphor, improving the level of engagement. The details on demand feature was however unanimously disregarded due to the necessity of keeping the raycast over the point of interest, both tiring and frustrating gesture if the data is far away. Using the duplication of data point alleviated the problem during the study though this was not the intended goal of this feature.

Comfort of use was rated with a 5.33 [0.47] average and no cyber sickness was mentioned by users. Moreover, they appreciated the possibility to remain seated for analysis while having the freedom of moving when required. A full session only lasted around 30 minutes in the virtual world, so further study will be required to see if the system remains comfortable for longer period of time. It nevertheless shows encouraging results.

Last, the overall experience was rated with a mean of 5.66 [0.47], even though this can mostly be accounted for the "wow effect" of the first virtual reality and visual analytics experience based on the discussion with the subject.

CONCLUSION

This paper presented an immersive analytics system for multi-dimensional data analysis. The main contribution is the navigation solution that provides the user with the comfort of using a seated position in front of a desk while retaining the freedom of using full physical movement while being immersed. The user feedback we obtained from a limited user study confirmed the benefits of our mobile chair solution. Future work will improve the data analysis component.

Figure 1 :

 1 Figure 1: Overview of the virtual representation, each sphere representing a data point

Figure 2 :

 2 Figure 2: The workstation inside the virtual environment. The mobile chair and the tablet are modelized and tracked via Vive tracker. The virtual plank on the left is used to gather selection cube, while the trashcan on the right is used to discard cloned data point or selection cube. The disk represents the available physical area where the user can move while in the chair or standing up. "Flying" movement and teleportation move the whole workstation.

Figure 3 :

 3 Figure 3: The mobile chair setup with a user exploring the dataset in seated position.

ACKNOWLEDGMENT

This work is part of the IDEA project (Immersive Data Exploration and Analysis) funded by the Pays de la Loire French region through West Creative Industries program.