A Grochola
email: anna.grochola@fuw.edu.pl

P Kowalczyk
email: pawel.kowalczyk@fuw.edu.pl

W Jastrzebski

Rydberg states of Li 2 molecule studied by polarization labelling spectroscopy

Keywords: numbers: 33.80.Rv, 33.20.Lg, 33.20.Vq, 42.62.Fi

The polarization labelling spectroscopy technique is used to characterize three previously unknown 1 Π u electronic states in 7 Li 2 molecule. The states are identified, together with the 4 1 Π u and 6 1 Π u states reported before, as low lying members, n = 3 to 7, of a molecular Rydberg series corresponding to excitation of the npπ u electron. The dependence of their properties on the principal quantum number n is discussed.

Introduction

Rydberg states of homonuclear alkali dimers have been studied by laser spectroscopy techniques since the 1980s (see e.g the pioneering experiments on Li 2 [1], Na 2 [2] and K 2 [3]). These states attract considerable interest for several reasons. The low-lying members of the Rydberg series represent the transition from molecular valence states to Rydberg states. Since for such states quantum chemistry calculations are still feasible, experimental studies provide ground for testing reliability of computational methods. Investigation of high-lying Rydberg states allows to deduce the ionization potential and spectroscopic constants of the ground state of the molecular ion to be deduced by extrapolation, without direct observation of the ion spectra. Highly excited Rydberg states are also interesting in themselves because of several phenomena specific to them, like gradual change of coupling between angular momenta (typically from Hund's coupling case a to d), breakdown of the Born-Oppenheimer approximation (caused by changing ratio between rotation energy and the difference of the electronic energies of consecutive electronic states) and coupling of discrete levels to the electronic continuum (which leads to autoionization).

The easiest way to reach molecular Rydberg states experimentally is by stepwise excitation via some intermediate, low lying level, because two visible photons can be used then. When we start from a ground molecular state of 1 Σ + g symmetry, the selection rules allow in this case excitation of 1 Λ g states with Λ = Σ, Π, ∆. Transitions to states of ungerade symmetry require an odd number of photons. This can be realized by one photon absorption, conceptually simple but experimentally less convenient since a UV or VUV radiation is needed. An alternative way is offered by the all-optical triple resonance spectroscopy (AOTR) [4], however this method, requiring three single mode tuneable lasers, is rather extravagant and beyond capabilities of most research groups.

As a result the disparity of experimental data on gerade and ungerade Rydberg states in alkali dimers is rather large. The highest observed 1 Λ g states correspond to excitation of the Rydberg electron to a state with the principal quantum number n = 40 for Li 2 [5], n = 70 for Na 2 [6] and n = 43 for K 2 [3]. For 1 Λ u states only in diatomic sodium, where the AOTR technique has been employed, the observations reach up to high n values [4], whereas for Li 2 and K 2 the experimental data are limited to low electronic states having rather a valence than Rydberg nature.

For a few years we have been involved in investigation of excited electronic states of lithium dimer accessible in one photon transitions from the ground state, using the polarization labelling spectroscopy method [7]. The electronic states observed in our recent experiments, 5 1 Π u [8] and 6 1 Π u [9], are the highest states of singlet ungerade symmetry characterized experimentally up to now (save fragments of another 1 Π u state [10], see the Discussion section). In the current paper we present observation of three higher excited electronic states of 1 Π u symmetry in Li 2 , which apparently form, together with the previously studied 4 1 Π u [11,12] and 6 1 Π u [9] states, a Rydberg series 1 Π u (npπ u ) with n = 3 -7 (i.e. 4 1 Π u (3pπ u ), 6 1 Π u (4pπ u ), ...).

Experiment

Since the employed method and the experimental set up were identical to those used in our previous experiments on Li 2 [8,9,13], only a brief summary will be given here. Lithium dimers were formed in a heat pipe oven filled with metallic lithium (natural isotopic composition) and heated to 750 -800 • C with a background pressure of 4 Torr of helium as a buffer gas. Known levels of the ground state of Li 2 were 'labelled' with fixed frequency light of a probe laser through the

A 1 Σ + u ← X 1 Σ + g or B 1 Π u ← X 1 Σ + g
transitions, specified in Reference [13]. Transitions from the labelled (v , J ) levels to the investigated electronic states were excited in a V-type optical-optical double resonance polarization spectroscopy scheme by a tuneable pump laser. The two laser beams co-propagated at a small angle, crossing in the centre of the heat pipe oven. The intensity of the probe beam was monitored through crossed polarizers placed on either side of the oven, so that double resonance signals appeared on a black background.

As the pump laser we used an optical parametric oscillator/amplifier system provided with a frequency doubler (Continuum, Sunlite EX, pumped with an injection seeded Nd:YAG laser, Powerlite 8000). The system delivered UV pulses of 3 mJ energy, 10 ns duration and a spectral width below 0.16 cm -1 , tuneable for purposes of the present experiment between 37300 and 40900 cm -1 . Calibration of wave number scans was achieved by recording the optogalvanic spectrum of argon and transmission fringes of a Fabry-Pérot etalon 0.5 cm long (both auxiliary signals observed at the fundamental frequency of tuneable OPO/OPA radiation). The resulting accuracy in absolute determination of wave numbers was better than 0.1 cm -1 . For the probe laser we employed either a cw Ar + -Kr + laser operated on selected visible lines or a home built pulsed dye laser pumped synchronously by the same Nd:YAG laser as the OPO system and set on fixed wavelengths from the range 630 -650 nm using HighFinesse WS-5 wavemeter.

Results

With growing excitation energy, molecular spectra of lithium can be expected to become increasingly congested and intricate, according to a growing number of states accessible for excitation. In addition, perturbations between electronic states should become more frequent, adding to the complexity of the spectra. These anticipations were only partially confirmed in the course of our experiment, since the investigated spectral range was dominated by three easily recognizable band systems having the ground X 1 Σ + g state of Li 2 as the lower state. The symmetry of all three upper states is clearly 1 Π u as indicated by the presence of strong Q lines in the spectra. For the sake of clarity, in this section we shall refer to these states as α 1 Π u , β 1 Π u and γ 1 Π u . Fragments of other bands in the investigated range were too weak and too short to attempt their identification.

A visual inspection of the spectra related to the observed 1 Π u ← X 1 Σ + g systems revealed the presence of strong perturbations in the upper states. In particular, irregular patterns of P , Q and R lines originating from the same rovibrational level in the ground state were apparent and in some cases the P and R lines could not be assigned unambiguously at all. It has been easily verified that the Q lines formed regular progressions while P and R lines behaved erratically, displaying shifts from the expected positions which occasionally exceeded 1 cm -1 . We conclude that the e parity levels of the investigated 1 Π u states are strongly perturbed over a wide range of vibrational and rotational quantum numbers. This is consistent with perturbation by neighbouring 1 Σ + u states containing only levels of e parity and therefore able to influence only e levels of the 1 Π u states. However, the perturbing states could not be directly seen in the spectra, presumably because of low Franck-Condon factors for excitation from the ground electronic state of Li 2 . This made any attempt to deperturb the spectra futile and we had to confine our analysis to the observed Q lines, that is to f parity levels of the investigated 1 Π u states. This problem has been encountered in our previous studies on highly excited states in Li 2 , Na 2 and K 2 and solved in a similar way [9,13,14,15,16].

Altogether 208, 153 and 86 Q lines were identified in the α, β and γ 1 Π u states, respectively, for the main 7 Li 2 isotopologue. The vibrational numbering in each state was established by assuming that the lowest of all the observed vibrational levels in a given state corresponds to v = 0. These numberings were supported by inspection of the isotope effect for a few Q lines assigned to 6 Li 7 Li molecules. Although the Born-Oppenheimer approximation is not fully justified for lithium dimer [17], the measured isotope shifts allow confirmation of the correct vibrational numberings.

The measured positions of Q lines (limited to 7 Li 2 ) were fitted to differences of term values of the upper and lower levels, expressed by a standard Dunham expansion. Ground state Dunham coefficients were fixed at the highly accurate values given by Wang et al. [17]. Trial fits with varying numbers of fitted constants allowed us to establish statistically defined sets of coefficients describing each of the three 1 Π u states. Their values have been subjected to a rounding procedure suggested by Le Roy [18]. The Dunham coefficients listed in Table 1 reproduce the measured line positions with an accuracy consistent with the experimental uncertainty.

Discussion

In general, there are two methods of identification of excited molecular states. In one of them the state is assigned by atomic limits to which it dissociates. The method works well for the lowest electronic states where the atomic asymptotes are well separated. For higher excitation energies, a growing density of states leads to numerous avoided crossings between potential curves of Rydberg, valence and ion pair states and tracking of atomic limits becomes more involved. In such cases specification of dissociation limits of a given state usually does not provide reliable information about its properties in the region near equilibrium internuclear distance, unless a diabatic course of the potential curve is strictly followed. In another method the Rydberg states are classified according to the character of the outermost electron and labelled by its quantum numbers nlλ, Table 1. The experimentally determined Dunham coefficients for the α 1 Π u , β 1 Π u and γ 1 Π u states of 7 Li 2 (the coefficients and rms errors all in units of cm -1 ). The numbers in parentheses are 1σ uncertainties in units of the last digit. n being the principal quantum number and l, λ -quantum numbers of the electronic angular momentum and its projection on the internuclear axis, respectively.

α 1 Π u state β 1 Π u state γ 1 Π
In the absence of reliable theoretical calculations concerning higher excited states of Li 2 we can try to assign (diabatic) atomic asymptotes to the observed states only tentatively. First, we resort to a general rule that in case of transitions to highly excited molecular states the strongest ones are expected to the states corresponding to the allowed transitions between the asymptotic limits of the molecular states [1,2,19]. Since the observed transitions originate from the ground X 1 Σ + g state, which dissociates to 2s+2s lithium atoms, the three excited 1 Π u states investigated in this work correlate most likely to 2s+np asymptotes. In other words, the Rydberg electrons in the observed excited states are most likely npπ u electrons, because such orbitals display the best overlap with valence orbitals of the molecular ground state.

In a naïve approach one would assign the principal quantum number n by comparing separation of the potential wells of the observed states with energy differences between successive np atomic levels of lithium [START_REF] Radzig | Reference Data on Atoms, Molecules and Ions[END_REF], the comparison based on an assumption that the molecular dissociation energies of the Rydberg states are not significantly influenced by a distant (Rydberg) electron. For reasons explained below we shall include in further discussion two lower lying molecular states, 4 1 Π u [11,12] and 6 1 Π u [9]. The data 2 apparently suggest that the five 1 Π u states under consideration should correspond to n = 4 -8. However, the near agreement of values of ∆E and ∆T e is fortuitous. In Li 2 the order of asymptotes which are able to provide 1 Π u states is 2s+2p, 2p+2p, 2s+3p, 2s+3d, ... (see Figure 1) and, in addition, it is known that the 3 1 Π u and 4 1 Π u states display an avoided crossing [11]. Therefore the fourth state of this symmetry must correspond diabatically to the third atomic limit, 2s+3p, and not to 2s+4p, as confirmed also by theoretical calculations [START_REF] Jasik | Proceedings of the 41st EGAS Conference[END_REF]. This leads to an assignment of the discussed 1 Π u states displayed in Table 3. Determination of n is connected directly with determination of the quantum defect δ nlλ for the states under consideration, related by the formula

Π u -6 1 Π u 5p -6p 1375 1458 6 1 Π u -α 1 Π u 6p -7p 827 872 α 1 Π u -β 1 Π u 7p -8p 534 560 β 1 Π u -γ 1 Π u presented in Table
T en = T ∞ -Ry/[n -δ nlλ ] 2 , ( 1 )
where T e is the electronic energy of the given state at the potential minimum, T ∞ the energy difference between the potential minima of the Li 2 X 1 Σ + g and Li + 2 X 2 Σ + g states (41538.5 cm -1 [START_REF] Roche | [END_REF]17]) and Ry the Rydberg constant for 7 Li 2 (109733.05 cm -1 [10]). The assignment suggested in this work leads to a nearly constant value of the quantum defect for the whole series of 1 Π u states, δ nlλ ≈ -0.84 (Table 3). We note that a negative value of δ is confirmed by the only existing theoretical calculation on sufficiently highly excited electronic states of Li 2 , done by Spies [23]. Although his results apparently become less reliable with increasing excitation energy, the predicted value of δ ≈ -0.5 supports our assignment. The negative value of quantum defect shows that the Rydberg 1 Π u states are shifted upwards comparing to the hydrogenlike positions. This may be somewhat surprising in view of a bonding character of the npπ u electron. However, we must note a very low position of the doubly excited 2p+2p asymptote in Li 2 . The 2(C) 1 Π u state correlated with this asymptote pushes up all higher 1 Π u states by a "domino effect", which results in an increase in their term energies and may cause that the quantum defects assume a negative value. Interaction with the doubly excited molecular state makes the potential curves of the investigated states shallower but this effect diminishes with growing n (see the evolution of D e values in Table 3). This observation is compatible with the measured values of vibrational constants which increase systematically, as may be expected for potentials with gradually increasing depths, apparently to converge to the corresponding value of the ground state of molecular ion, Li + 2 . On the other hand, the values of rotational constants behave against the expectations, since they clearly diverge from the ionic value. Decrease of B e values, i.e. increase of equilibrium distances with growing n would mean in a simplistic picture that the molecular bond is weakened. At present we cannot explain our observation and theoretical calculation of potential curves for higher excited states in Li 2 would be highly desirable to get more insight into this problem. It must be stressed that an alternative assignment of the observed Rydberg series, as 6 1 Π u (5pπ u ), α 1 Π u (6pπ u ), ..., leading to a plausible value of quantum defect δ ≈ 0.16, would mean that a low number of the series corresponding to n = 4 has not been detected experimentally. In successive experiments done in our laboratory ( [13,12,8,9] and the present work) we have investigated excitation spectra of Li 2 in the full range of excitation energies between 29000 and 41000 cm -1 and found no 1 Π u state which might be the missing low member of the series. This experimental fact makes the alternative assignment unlikely.

We note at this juncture a 1 Π u state with T e ≈ 37260 cm -1 observed via fluorescence from some higher states and assigned as 5p 1 Π u by Ross et al. [10]. Since this state could not be seen in transitions from the ground X 1 Σ + g state in our experiment and its term value does not fit the Rydberg formula (1) established here for 1 Π u (npπ u ) states, we must conclude that the assignment by Ross et al. is incorrect and the state in question belongs to another electronic configuration. With the results of the present experiment, we may speculate about its identity. The 1 Π u state discussed in reference [10] lies between the 6 1 Π u and α 1 Π u states and therefore must correlate diabatically with atomic asymptotes placed between 2s+4p and 2s+5p. This leaves only two asymptotes: 2s+4d and 2s+4f. The 1 Π u state was seen in a transition to the 1 1 ∆ g (2p+2p) state [10]. Assuming that the states mentioned above preserve to some extent their asymptotic character even close to R e , both possible assignments would lead to a two electron process in the 1 Π u → 1 1 ∆ g transition, either (2s+4d) → (2p+2p) or (2s+4f) → (2p+2p), with the first of them clearly more probable. Therefore the assignment of the state of reference [10] as 1 Π u (2s+4d) is to be preferred. We also note that to the best of our knowledge no Table 3. Selected molecular constants for the Rydberg states of 7 Li 2 along with the corresponding constants for the ground state of the molecular ion. The units are cm -1 , except for quantum defect δ. The 4 1 Π u state is strongly perturbed by the neighbouring 3 1 Π u state [11] 263.1 0.500 10465.3 [START_REF] Roche | [END_REF] electronic state correlating to f configuration has been recognized experimentally for any of the alkali dimers. However, ab initio calculations would be necessary to resolve this question which cannot be answered definitely from spectroscopic observations alone.

Conclusion

In the range of 38300-41700 cm -1 above the minimum of the ground state of Li 2 we have observed and characterized three states of 1 Π u symmetry, which together with the 4 1 Π u and 6 1 Π u states known from other experiments form a Rydberg series corresponding to excitation of the npπ u electron. The negative value of the quantum defect for the series suggests interaction with the low lying valence C 1 Π u state correlated to doubly excited atomic configuration 2p+2p. The present work shows advantages of the polarisation labelling spectroscopy technique because simplified spectra and sensitive detection are crucial for investigation of Rydberg states.

Figure 1 .

 1 Figure1. Experimental RKR potential energy curves of the selected 1 Π u states in Li 2 with their related asymptotes. The states belonging to the Rydberg series 1 Π u (npπ u ) are depicted by thick lines. The diabatic course of the 4 1 Π u potential is shown. The 1 Π u state reported by Ross et al.[10] is marked with a dashed line and labelled with a question mark.

Figure 1 (

 1 Figure 1 (figure1.eps)

Table 2 .

 2 Comparison of atomic and molecular energy spacings in lithium. All values are in cm -1 units

	atomic pair ∆E (atomic) ∆ T e (molecular) molecular pair
	4p -5p	2546	2258	4 1

  and its potential curve is highly irregular, therefore the ω e and B e values are not given.

	state	at. asymptote	T e	δ	ω e	B e	D e
		(diabatic)
	4 1 Π u (3pπ u )	2s+3p	34605.7[12] -0.98			4942
	6 1 Π u (4pπ u )	2s+4p	36863.2[9] -0.845 253.6 0.4801	8123.0
	α 1 Π u (5pπ u )	2s+5p	38320.6	-0.840 257.5 0.4746	9211.6
	β 1 Π u (6pπ u )	2s+6p	39192.6	-0.839 259.4 0.4637	9714.8
	γ 1 Π u (7pπ u )	2s+7p	39753.4	-0.840 260.4 0.4499	9980.6
				7 Li + 2

Acknowledgments

This work has been funded in part by grant N N202 203938 from the Polish Ministry of Science and Higher Education.


S Vidal

Z P Wang

P M Dinh

P.-G Reinhard

E Suraud

Frequency dependence of level depletion in Na clusters and in C 2 H 4

Keywords: numbers: 31.15.ee, 31.70.Hq, 33.80.Eh, 36.40.Vz, 71.15.Mb

In the framework of time-dependent density-functional theory, we study electron emission from Na clusters and the C 2 H 4 molecule as induced by irradiation with an intense pulse. The collision of a charged projectile on C 2 H 4 is also explored for comparison. We look in particular at the level depletion, i.e. the electron loss in each single-electron level separately. It is found that the distribution of electron loss depends sensitively on the photon frequency. Frequencies close to visible light remove electrons exclusively from the vicinity of the Fermi surface while light in the higher UV range (up to 20 eV for Na clusters and up to 136 eV for C 2 H 4 ) depletes all levels about equally strong, down to the deepest bound valence state.

Introduction

Photons have been since long a major tool for analyzing the electronic structure of molecules [1] and clusters [START_REF] Kreibig | Optical Properties of Metal Clusters[END_REF][START_REF] Haberland | Clusters of Atoms and Molecules 1-Theory, Experiment, and Clusters of Atoms[END_REF]. In early times, one measured mostly the excitation spectrum through photo-absorption spectrocopy. In the meantime, the enormous progress in coherent light sources and detection setups has dramatically enhanced the variety of photon experiments with molecules and clusters. We mention here briefly the violent cluster dynamics induced by high-intensity infrared laser beams leading to emission of energetic electrons, ions, X-rays and even nuclear reactions, see e.g. [4,[START_REF] Springate | Electron kinetic energy measurements from laser irradiation of clusters[END_REF][START_REF] Zweiback | Detailed study of nuclear fusion from femtosecond laser-driven explosions of deuterium clusters[END_REF][START_REF] Saalmann | Mechanisms of cluster ionization in strong laser pulses[END_REF]. Pulses in the fs range have opened a wide area of fs-spectroscopy allowing the detailed tracking of ionic dynamics in molecules and clusters, see e.g [START_REF] Zewail | Femtochemistry : Atomic-Scale Dynamics of the Chemical Bond[END_REF][START_REF] Posthumus | The dynamics of small molecules in intense laser fields[END_REF][START_REF] Carley | Femtosecond lasers in gas phase chemistry[END_REF][START_REF] Hertel | Ultrafast dynamics in isolated molecules and molecular clusters[END_REF]. At the side of observables, one reveals more information by analyzing with photo-electron spectroscopy energy and angular distributions of the electrons emitted from the excited molecule [START_REF] Berkowitz | Photoabsorption, Photoionization and Photoelectron Spectroscopy[END_REF]. The experimental tools for such analysis have also much developed lately and are increasingly being used (for an overview in cluster physics, see [START_REF] Reinhard | Introduction to Cluster Dynamics[END_REF]). A rather recent achievement is the availability of coherent light beyond the visible, ranging from UV up to X-rays. A particularly versatile source is here given by the free electron lasers (FEL) [START_REF] Brau | Free-electron lasers[END_REF] which can be tuned to deliver fs pulses at high frequency and simultaneously high intensity. The new options have soon been exploited, see e.g. the pioneering experiment with short X-ray pulses on rare gas clusters [START_REF] Wabnitz | Multiple ionization of atom clusters by intense soft X-rays from a free-electron laser[END_REF], and much been used since, for a review see [START_REF] Pfeifer | Femtosecond x-ray science[END_REF]. It is now in reach to study in all detail photo-emission from deep lying electronic states in neutral or cationic molecules and clusters. Photo-electron spectra, e.g., allow to deduce to which extent each separate electron level is depleted by the photon pulse. The distribution of electron removal is of interest as such and it has consequences for the subsequent cluster dynamics [START_REF] Calvayrac | Coulomb explosion of a Na 12 cluster in a diabatic electron-ion dynamical picture[END_REF]. This motivates us to investigate the features of electron depletion in dependence of photon frequency. As a tool for this investigation, we employ time-dependent density functional theory (TDDFT) at the level of the local-density approximation (LDA). This is augmented by a self-interaction correction (SIC) to describe emission properties correctly. The scheme has been much used for the description of clusters dynamics in all regimes [START_REF] Calvayrac | Nonlinear electron dynamics in metal clusters[END_REF][START_REF] Reinhard | Introduction to Cluster Dynamics[END_REF]. We have recently extended it to the case of organic molecules [START_REF] Wang | Microscopic studies of atom-water collisions[END_REF]. In this paper, we consider as test cases cationic Na clusters and the ethylene molecule C 2 H 4 irradiated by fs laser pulses in the regime of VUV-radiation and of X-ray FEL. We address the issue of ionization mechanism, that is the extraction of electrons from least bound states or from deeper levels. We scan the level depletion as a function of laser frequency and/or intensity, in order to discriminate the ionization mechanism in each case. The paper is outlined as follows: In section 2, we introduce briefly the formal and numerical scheme. In section 3, we present and discuss the results.

Theoretical framework

The laser induced electron dynamics is described by the time-dependent local-density approximation (TDLDA) in standard manner, for details see e.g. [START_REF] Calvayrac | Nonlinear electron dynamics in metal clusters[END_REF][START_REF] Reinhard | Introduction to Cluster Dynamics[END_REF]. The calculation is restricted to valence electrons. These are the 3s electrons in Na, the 2s and 2p electrons in C, and naturally the 1s electron in H. The coupling of the ionic cores to the valence electrons is described by pseudopotentials. For the C and H atoms of the ethylene molecule, we use Goedecker-type [START_REF] Goedecker | Separable dual-space gaussian pseudopotentials[END_REF] pseudopotentials consisting in a local part and a non-local contribution, all of them employing Gaussians augmented by a polynomial. The original parameterizations of [START_REF] Goedecker | Separable dual-space gaussian pseudopotentials[END_REF] employ different Gaussian widths for the local part and for the various non-local projectors. This is a great hindrance for computations on a coordinate space grid. Thus we have refitted the pseudopotential parameters to employ the same width of 0.412 a 0 in all terms (local and non-local) and for both elements (C and H) involved in the study, see Appendix A. Metal clusters are less demanding. Much simpler local pseudopotentials suffice. We employ here the soft Gaussian potentials as given in [START_REF] Kümmel | Ionic geometries and electronic excitations of na + 9 and na + 55[END_REF]. The elimination of core electrons through pseudopotentials requires that the laser frequencies in the study stay safely below values which could ionize core electrons. For Na, the eliminated 2p state lies at -28.8 eV, while the 1s state for C is at -270.5 eV [22]. We use at most 19.4 eV for Na clusters and 136 eV for C 2 H 4 which is on the safe side in both cases.

The LDA employs the exchange-correlation energy functional from [START_REF] Perdew | Accurate and simple analytic representation of the electron-gas correlation energy[END_REF]. Pure LDA understimates grossly ionization potentials (IP), but correct IP are crucial for an appropriate dynamical description of electron emission. Therefore, we augment LDA by a SIC for which we employ actually the technically inexpensive average-density SIC (ADSIC) [START_REF] Legrand | Comparison of self-interaction-corrections for metal clusters[END_REF].

The time-dependent fields and wave functions for C 2 H 4 are represented on a 3D cartesian coordinatespace grid of dimensions 72 × 72 × 64. In the case of metal clusters, we use a cylindrically averaged pseudopotential scheme [START_REF] Montag | Small metal clusters in a cylindrically averaged pseudopotential scheme[END_REF][START_REF] Montag | Ionic structure and global deformation of axially symmetric simple metal clusters[END_REF]. The size of the corresponding cylindical box is for Na + 9 45 in radial and 91 in longitudinal directions (with a mesh size of 0.8 a 0 ), and for Na ++ 22 , 81 × 173 respectively. Electronic wave functions are propagated in time by the time-splitting method [START_REF] Feit | Solution of the schrdinger equation by a spectral method[END_REF]. We use a time step of 6 × 10 -4 fs for C 2 H 4 and 4.8 × 10 -3 fs for Na clusters. The Poisson equation is solved by a fast Fourier technique combined with separate treatment of the long-range terms [START_REF] Lauritsch | An fft solver for the coulomb problem[END_REF]. We use absorbing boundary conditions at the outskirts of the grid [START_REF] Calvayrac | Nonlinear electron dynamics in metal clusters[END_REF][START_REF] Reinhard | Introduction to Cluster Dynamics[END_REF][START_REF] Reinhard | Role of boundary conditions in dynamic studies of nuclear giant resonances[END_REF]. This absorbs all outgoing electron flow reaching the bounds of the grid and thus prevents artefacts from reflection back into the reaction zone.

We used pulse lengths of 40 fs for C 2 H 4 , and 80 fs for Na clusters. We pursued the simulations further over some tens of fs, in order to complete the direct electron emission [START_REF] Reinhard | Frequencies, times, and forces in the dynamics of na clusters[END_REF][START_REF] Calvayrac | Nonlinear electron dynamics in metal clusters[END_REF]). Although, our dynamical propagation scheme can deal with a simultaneous propagation of ionic motion [START_REF] Calvayrac | Nonlinear electron dynamics in metal clusters[END_REF][START_REF] Reinhard | Introduction to Cluster Dynamics[END_REF], we kept the ions frozen in case of C 2 H 4 to reduce computational expense. Ionic motion plays a minor role at the time scales studied here.

The observables discussed in this paper are electronic dipole moment and ionization N esc (= number of escaped electrons). The dipole moment of valence electrons with respect to ions, D(t), is evaluated as

D(t) = d 3 r (r -R cm,ion ) n(r, t) , (1)
where n(r, t) = α |ϕ α (rt)| 2 is the total electronic density and the ϕ α are the occupied electron wave functions. As mentioned above, absorbing boundary conditions are used to remove outgoing electrons.

The absorption leads to a loss of norm of each single particle state. This, in turn, allows one to compute for each state ϕ α the level depletion ν (α) , the average ionization N (α) esc out of state ϕ α and also the total average ionization N esc , by

ν (α) (t) = 1 -ϕ α (t)|ϕ α (t) , (2a) N (α) esc (t) = n α ν (α) (t), (2b)
N esc (t) = α N (α) esc (t), (2c)
where n α is the occupation number in state ϕ α . The numerical simulations are performed over a sufficiently long time to collect all ionization, i.e. up to a point where the ν (α) (t) do not change anymore.

It is to be noted that these ν (α) (t) are averaged quantities as any observable computed as an expectation value from a mean field state. And the total N esc yields the average number of emitted electrons, representing a large ensemble of measurements. One may use these numbers to recover detailed ionization probabilities for each charge state (for details see, e.g., [START_REF] Calvayrac | Nonlinear electron dynamics in metal clusters[END_REF][START_REF] Reinhard | Introduction to Cluster Dynamics[END_REF]). The present discussion is confined to average ionizations which already carry a great deal of information.

Results and discussion

In all cases considered here, the laser intensity has been adjusted so that the total ionization remains small. This means that all results stay in a perturbative regime where response amplitudes and emission yields scale with the total number of emitted electrons N esc , or intensity I respectively. We will inspect in the following the "relative level depletion" ν (α) /N esc which are in the perturbative regime independent of laser intensity or total yield.

Na clusters

We start with the case of Na clusters and first discuss Na + 9 . This cluster has three distinctive valence electron energy levels : the 1s level (filled with 2 valence electrons) at -8.6 eV, the 1p z level (with 2 electrons) at -7.4 eV, and the HOMO, nearly degenerated, 1p x,y levels (with 4 electrons in total) at -7.2 eV, which also gives the IP. As all metal clusters, the cluster shows a pronounced Mie plasmon resonance which comes here at around 2.5 eV [START_REF] Kreibig | Optical Properties of Metal Clusters[END_REF][START_REF] Haberland | Clusters of Atoms and Molecules 1-Theory, Experiment, and Clusters of Atoms[END_REF]. This cluster is irradiated with a laser pulse of total length of 80 fs and at various frequencies, ranging from 6.8 to 20.4 eV. All frequencies are far above the Mie plasmon resonance. The lowest frequency is below the IP, while the others are sufficiently large to ionize all valence levels in a one-photon process.

One may suspect that the perturbation by the external laser field spoils the levels which we want to explore. This effect was studied for Na clusters in [START_REF] Pohl | Towards single particle spectroscopy of small metal clusters[END_REF] and it was found that the amount of perturbation depends on the strength of the external field. The effect can be quantified in terms of the average number of emitted electrons. Strong perturbations are associated with about one or more emitted electrons. Staying at emission yields of a few percent means to deal with small perturbations where the electronic structure, in particular the single-particle energies, remains basically unchanged. We have taken care to tune the laser intensities properly to guarantee that one resides always in the regime of weak perturbation, see below. We have checked that the single-particle energies indeed remain well defined and quasi constant in time. To demonstrate the relevance of the emission data, we show in Fig. 1 two computed photo-electron spectra (PES) for Na + 9 irradiated by lasers whose frequencies lie in the UV domain. The PES exhibit typical couples of peaks corresponding to the 1s and the 1p states of Na + 9 . More precisely, the peaks exactly lie at an electron kinetic energy given by ε kin = ε i + n ω las where ε i denotes the energy of state i and n the number of photons involved in the process. For ω las = 6.8 eV, one can spot 2-, 3-and 4-photon processes, while for the higher frequency, 1, 2 and 3 photons are involved. Moreover, in the low ionization regime, the peaks turn out to be extremely narrow. This clearly indicates that, over the whole dynamical process, the associated single particle energies have not been modified [START_REF] Pohl | Towards single particle spectroscopy of small metal clusters[END_REF]. The sharpness of the peaks furthermore ensures that they remain well separated in energy. This figure thus demonstrates that, as soon as the total ionization remains small (in this case here, less than 0.01), no state mixing occurs and the single particle energies of the ground state can be tracked and studied within our dynamical calculations.

We now discuss more specifically the level depletion. The intensity I of the laser pulse was varied to track its effect on the detailed depletion pattern. We found that relative depletion is insensitive to intensity as long as one stays in a (perturbative) regime of low emission (less than 0.1 electron). Nonetheless, we tune the intensity to frequency such that the net yield remains roughly comparable over all frequencies in a given case. This means, e.g., that we use I = 9 × 10 12 W/cm 2 for ω las = 6.8 eV, 9 × 10 13 W/cm 2 for ω las between 10.9 and 15.0 eV, and 5 × 10 14 W/cm 2 for ω las = 19.4 and 20.4 eV. Fig. 2 shows the relative ionizations and depletions as a function of ω las . According to the top panel, the ionization yield is dominated by the 1p xy states, while the 1p z and the 1s states change relative importance with a crossing between, around 13.2 eV. Both ionizations from the latter levels stay below that from the states. Note, however, that the N (α) esc /N esc is weighted by the multiplicity of state α, see Eq. (2b), and that there are 4 electrons in the 1p xy states, while the 1s and the 1p z states have 2 electrons each. This causes the dominance of emission from 1p xy .

Probably a better comparison of ionization mechanisms is achieved by plotting the relative level depletion ν (α) /N esc , Eq. (2a), instead of the relative level ionization N (α) esc /N esc . This is done in the bottom panel of Fig. 2. For the lowest frequency, ionization still comes dominantly from the HOMO (1p xy ), but now close up with the nearly degenerate 1p z , while the 1s state contributes very little, almost one order of magnitude less. Increasing the frequency does not modify so much the depletion from the 1p xy states. But the depletion of the 1s state is almost steadily increasing while the 1p z state falls below the other states for the higher frequencies. All three states have almost equal depletion around 13 eV. Above 14 eV, the contribution from the 1p z steadily decreases, while that from the 1s level becomes more and more dominant. The trends of the level depletion show some fluctuations indicating some oscillatory behaviour of the level depletion at a fine energy scale. This point will be addressed in more detail in Fig. 4.

The results on Na + 9 indicate that low frequency laser pulses remove electrons preferentially from the Fermi surface while high frequency pulses extract electrons more equidistributed, even with a preference for the most bound state. We check that such depletion patterns are not accidental by investigating the case of Na ++ 22 . Fig. 3 presents the depletion in relation to the s.p. energies. Three cases are compared: 6.8, 10.9, and 15.0 eV. The pulse length is again 80 fs. Intensities have been adjusted in order to get in all cases N esc ∼ 0.05, so that the perturbative picture of constant s.p. energies in time remains valid in all cases. The features are much the same as in Na + 9 : for low frequencies, emission mainly comes from the HOMO whereas increasing frequency puts increasing weight on the more deeply bound states.

It is to be noted that the above statement holds for the general trends. The distribution of depletions can vary very much in detail because the emission strength depends on the spatial matrix element between bound single particle wave function and emitted continuum wave [START_REF] Faisal | Theory of Multiphoton Processes[END_REF][START_REF] Pohl | Angular distribution of electrons emitted from na clusters[END_REF]. And the latter can depend sensitively on the outgoing wave number. This detailed sensitivity to photon frequency is demonstrated in Fig. 4 showing the depletion of each occupied single electron state in Na ++ 22 as a function of photon frequency in a small window above 15 eV. The depletion can vary on a rather short frequency scale (see the deepest bound level at -10.14 eV) and there are also long range trends (see, e.g., the states at -9.25 eV and -8.04 eV) which may substantially change the relative distribution of level depletion for much different frequencies. The example shows that the detailed patterns of depletion are fluctuating. They require more systematic investigations and may, in turn, reveal more information on the spatial structure of the occupied states [START_REF] Frank | Diffraction effects in the photoionization of clusters[END_REF][START_REF] Pohl | Angular distribution of electrons emitted from na clusters[END_REF]. We will address this issue in more detail for the case of ethylene (see discussion in Sec. 3.2.3). Above all these details remains one robust effect, namely that photon frequencies below the IP deplete preferably level near the Fermi surface while frequencies which are large enough to emit the deepest bound state at once produce more equilibrated depletion from all occcupied levels.

The ethylene molecule

We now turn to the case of an organic molecule, C 2 H 4 , which will allow us to explore an even larger range of laser frequencies.

Optical spectrum

Before looking at the electronic response of this molecule after laser irradiation, we first present its optical response in Fig. 5, since the latter will help us in the choice of ω las . The optical asborbing strength is deduced from the time evolution of the electronic dipole moment, defined in Eq. (1), by means of spectral analysis [START_REF] Calvayrac | Non linear plasmon response in highly excited metallic clusters[END_REF]36,[START_REF] Calvayrac | Spectral signals from electronic dynamics in sodium clusters[END_REF]. In Fig. 5, it is averaged over the three spatial directions and it is not energy weighted. It shows strong, isolated peaks at lower energies and changes to a continuum above the IP (11.5 eV). This result is consistent with former TDLDA calculations [START_REF] Nakatsukasa | Photoabsorption spectra in the continuum of molecules and atomic clusters[END_REF]. The lowest peak is at 7.74 eV, in good agreement with previous CI calculations (7.76 eV [START_REF] Petrongolo | Nonadiabatic treatment of the intensity distribution in the v-n bands of ethylene[END_REF]). We obtain the highest peak in the optical response lying at 8.16 eV. The frequency selective laser pulses will be sensitive to this much fluctuating spectrum while a collision with fast ions will deliver a broad spectrum of frequencies and will excite all modes at once.

Irradiation by laser pulses

We now turn to the irradiation of C 2 H 4 by a laser pulse of intensity I = 10 13 W/cm 2 , total pulse length of 40 fs and polarization parallel to the C-C bond. Three typical frequencies ω las are used : below the resonant region (6.8 eV), within that region (8.16 eV) and well above (136 eV), see Fig. 5. The six occupied valence levels are successively (2a g ) 2 , (2b 1u ) 2 , (1b 2u ) 2 , (3a g ) 2 , (1b 3g ) 2 , and (1b 3u ) 2 , with energies of -11.5, -12.4, -14.4, -16.0, -18.2, and -23.4 eV respectively. They are represented in real space in Fig. 6. One can notice that the states (2a g ), (2b 1u ) and (3a g ) exhibit nodes along the C-C bond, which corresponds to the laser polarization. We also checked that no state mixing occurs in the regime of low total ionization that we explored here, as has been illustrated by the calculations of photoelectron spectra for Na + 9 , see Fig. 1. Fig. 7 presents the time evolution of N esc and of the dipole moment D(t) along the laser polarization direction. It is obvious that both observables depend sensitively on ω las . The left panels show the case ω las = 6.8 eV, i.e. below the resonant region. The electronic dipole follows nicely the envelope of the laser pulse and electronic excitation fades away as soon as the laser is switched off. Electron emission is directly connected to the dipole amplitude and thus it has a maximum when the laser pulse is at its maximum of amplitude. The total ionization remains small in this case (about 0.11 charge units). This pattern is very similar to what is observed in metal clusters when irradiated by an optical laser with frequency off the plasmon resonance [START_REF] Reinhard | Frequencies, times, and forces in the dynamics of na clusters[END_REF].

For a laser frequency highly above the resonant region (right panels), that is here ω las = 136 eV, we meet again an off-resonant process and the patterns look much similar. However, the net ionization is 48 times smaller compared with the low-frequency case. The suppression of emission is due to the fact that the laser field oscillates so fast that a large fraction of electronic density is kicked back before it can gain substantial velocity to escape. This strong suppression of frequency is well accounted for in the Keldysh parameter describing an effective laser strength [40]. One would have to increase the raw laser intensity I significantly in order to get ionization similar to the low-frequency case. At the side of the dipole moment, we also see a much smaller amplitude. Note some small and fast oscillations on top of the dipole profile. This stems from a (reduced) quiver motion.

The middle panels correspond to the case ω las = 8.16 eV, which is in resonance with the strongest peak in the optical response of C 2 H 4 [START_REF] Wang | Nonadiabatic effects in the irradiation of ethylene[END_REF]. Now the dipole signal and N esc show the typical resonant patterns which differ from the off-resonant cases. During the first 10 fs, the dipole moment still follows the envelope of the laser pulse as in the other cases. But the resonant coupling yields a much larger amplitude. As a consequence, ionization is much larger and comes up earlier. This causes a sudden reduction of the dipole oscillations. There is again a small increase of the oscillations at around 16 fs because the now charged system drives the next peak in the optical response into resonance with the laser. The amplitude of dipole moment is attenuated again due to damping by ongoing electron emission.

The attenuation is small and the dipole signal continues to oscillate on its own long after the photon pulse is switched off. Correspondingly, ionization is carrying on. It can take several ps until emission has damped the dipole signal sufficiently to terminate the growth in ionization.

Irradiation by a charged projectile

Having sorted out the dynamical regimes, we now consider the photon induced electron emission for C 2 H 4 . Two photon frequencies are considered, ω las = 6.8 eV and ω las = 136 eV, both being off-resonant. The pulse length is 40 fs to stay in a time domain where ionic motion plays no role. This leaves the electronic ground state structure the reference point throughout the whole dynamics. Additionally, we consider the case of a collision with a singly charged projectile passing by at a distance of 9 a 0 with a velocity of v = 20 a 0 /fs. Photon intensity and parameters of the projectile are tuned to obtain comparable total ionization N esc . Fig. 8 shows the results for the final depletion. The patterns are very similar to the previous test case Na ++ 22 : At low frequency, we see electron emission preferably from the HOMO and the states next to it while the high frequency (all valence levels in reach of a one-photon process) produces comparable depletion for all levels down to the deepest bound one. We now relate more precisely the level depletion to the orbital shapes (see Fig. 6), keeping in mind that the laser polarization is along the C-C bond. With ω las = 6.8 eV, the dominant effect is an energetic one (middle panel of Fig. 8). Indeed, although the HOMO (1b 3u ) has nodes along the C-C axis (visible in Fig. 6, top right), it is depleted the most because it is bound by almost 1 eV less than the HOMO-1. And comparing the depletion of the HOMO-1 and HOMO-2, the orientation of the orbitals with respect to the laser polarization now competes with the energetics : these states are separated by 2 eV but exhibit similar depletions, even higher for the HOMO-2, because it has no nodes along the C-C direction, and this is not the case for the HOMO-1 (see first row of Fig. 6). This balance between orbital energy and orbital shape is all the more predominant for the high frequency case. The least depleted states are the (2b 1u ) and (3a g ) which have nodes along the C-C axis. It is also the case for the HOMO (1b 3u ) but here, the energetics takes the lead.

The collision by a charged projectile delivers a very short Coulomb pulse to the molecule site and thus contains a broad distribution of frequencies ω which has a maximum at low ω and decays slowly with increasing ω. The depletion patterns are thus somewhat similar to the low-frequency laser case. The moderate admixture of higher frequencies allows more depletion for slightly deeper levels. There is even a small amount of emission from the deepest level. But the content of high frequency in the Coulomb pulse is obvisouly rather small. The high-frequency part could be enhanced by going to faster collisions. This, on the other hand, reduces the total emission cross section and complicates experimental detection. It is obvious that the extremely versatile (X)FEL pulses which can be tuned in pulse length, intensity, and frequency are much superior here for experimental analysis of molecular and cluster structure.

Conclusions

In this paper, we have investigated the electronic excitation of ethylene and small sodium clusters subjected to moderate external electromagnetic fields (femtosecond photon pulses or collision with a charged projectile) applying a time-dependent density-functional approach. We have analyzed the electronic dipole moments and the number of escaped electrons for C 2 H 4 . When the photon frequency is off-resonant or highly above the resonant region, the dipole signal behaves as a classical oscillator in both situations and the electron emission is rather small. In contrast, for resonant frequencies, there is a substancial electronic emission and a strong damping of the dipole moments.

We have also studied in detail the level depletion of C 2 H 4 , Na + 9 and Na 2+ 22 . For C 2 H 4 , the level depletion caused by a photon field or by a collision with a fast charged projectile are considered. Both for the organic molecule C 2 H 4 and for small sodium clusters, a low frequency photon pulse tends to remove electrons from the surface, that is from least bound states, whereas FEL frequency pulses can extract electrons from deeper bound levels. Furthermore, we find that the level depletion pattern of C 2 H 4 impacting with a fast projectile is similar to that of a low photon frequency, as long as total ionizations in both situations are comparable.

Figure 1 .

 1 photoelectron spectra (a.u., log scale)

Figure 2 .

 2 Figure 2. Relative level ionization N (α) esc /Nesc (top) and relative level depletion ν (α) /Nesc (bottom), in irradiated Na + 9 for α = 1s, 1pz and 1pxy, as a function of the laser frequency ω las . The total pulse length is 80 fs.

Figure 3 .Figure 4 .

 34 Figure 3. Depletion of single particle levels, ν (α) , in Na 22 2+ for three laser frequencies ω las and intensities I as indicated. The depletion is represented by the length of the horizontal bars while the position at the y-axis indicates the energy of the corresponding level.

4 IPFigure 5 .

 45 Figure 5. Total photo-absorption strength of C 2 H 4 . The vertical arrow corresponds to the value of the ionization potential of the molecule.

Figure 6 .Figure 7 .

 67 Figure 6. Electronic wave functions of the ground state of C 2 H 4 represented in real space. The blue balls indicate the location of the C atoms, while the H atoms are represented by green circles. Note that the spatial orientation is different for each orbital, such that the nodes are clearly visible. The polarization of the laser we used is along the C-C bond.

4 Figure 8 .

 48 Figure 8. Level depletion for C 2 H 4 irradiated by a photon pulse (bottom and middle) or by a charged projectile (top), with respective parameters as indicated. Final depletions are represented by the length of the horizontal bars while the position at the y-axis indicates the energy of the corresponding level.

ACKNOWLEDGEMENTS

This work was supported by the National Natural Science Foundation of China (Grants No. 10575012 and No. 10435020), the National Basic Research Program of China (Grant No. 2010CB832903), the Doctoral Station Foundation of Ministry of Education of China (Grant No. 200800270017), the scholarship program of China Scholarship Council and the French Agence Nationale pour la Recherche (ANR-06-BLAN-0319-02). Calculations have been performed on the French computational facilities CalMiP, CINES, IDRIS and CCRT.

Appendix A. The non-local pseudopotential for C and H atoms

The form of the Goedecker pseudopotentials at the level of refinement as we use it here is given by [START_REF] Goedecker | Separable dual-space gaussian pseudopotentials[END_REF] V PsP (r) ϕ α (r) = V loc (r) ϕ α (r)

where Z I is the charge of ion, erf is the error function, and p(r) is a radial projector in the separable non-local term. The original parameterizations employ different r loc and r nloc for each material which slows down computations. We have developed parameterizations with the same width in all terms. These pseudopotential parameters are given in Table A1. Further details will be given in a forthcoming publication.


Damien Chablat
email: damien.chablat@irccyn.ec-nantes.fr

Erika Ottaviano

Guillaume Moroz

A COMPARATIVE STUDY OF 4-CABLE PLANAR MANIPULATORS BASED ON CYLINDRICAL ALGEBRAIC DECOMPOSITION

The aim of this paper is to present a systematic method for verifying the force-closure condition for general 3-DOF fullyconstrained cable manipulators with four cables as based on the CAD (Cylindrical Algebraic Decomposition). A fundamental requirement for a cable manipulator to be fully controllable is that all its cables must be in tension at any working configurations. In other words, all the cable forces must be positive (assuming a positive cable force representing a tension and a negative cable force being a compression). Such a force feasibility problem is indeed referred to a force-closure problem (also called vector-closure problem assuming that the vectors of interest are the row vectors of the Jacobian matrix of the manipulator). The boundaries of the workspace can be obtained by the study of the Jacobian matrix of the manipulator. Therefore, this is equivalent to study the singularity conditions of four 3-RPR parallel robots. By using algebraic tools, it is possible to determine the singularity surfaces and their intersections yielding the workspace. Thus, it will be shown that the use of the CAD allows to get an implicit representation of the workspace as a set of cells. A comparative workspace analysis of three designs of mobile platforms, a line, a square and a triangle will be presented and discussed in this paper for a planar 4-cable fully-constrained robot.

INTRODUCTION

Cable-based parallel robots, in which legs are replaced by cables, are structurally similar to the classical parallel ones, and they consist of a fixed base (or frame) and mobile platform, which are connected by several cables [START_REF] Roberts | On the inverse kinematics, statics, and fault tolerance of cable-suspended manipulators[END_REF]. Unlike rigid links, the unilateral characteristic of the cables (can pull but cannot push moving platforms), and therefore the formulations and results obtained for the kinematics, workspace, trajectory planning, and dynamics of the rigid-link mechanisms cannot be directly applied. Hence, the main issue can be recognized in determining the poses whereby the moving platform can be fully constrained by the cables.

Due to the nature of cables, this type of parallel robots have in general good characteristics such as: good inertial properties, they can be modular, relatively low-cost, and easy to reconfigure. Moreover, their actuator-transmission systems can be fixed on the frame and cables are lighter and thus, they can have higher payload-to-weight ratio, which makes them attractive for a number of applications. According to their design, dynamics can be easily derived when the inertias of the cables can be neglected [START_REF] Ottaviano | A Study on the Effects of Cable Mass and Elasticity in Cable-Based Parallel Manipulators[END_REF], and this holds for many practical applications if the cable mass is negligible if compared to the combined mass of the end-effector and payload.

Cable manipulators have been classified into two basic types, the fully-constrained type and under-constrained type, based on the extent to which the end-effector is constrained by cables only or they rely on gravity [START_REF] Castelli | Analysis and simulation of a new Cartesian cable-suspended robot[END_REF], [START_REF] Verhoeven | Analysis of the Workspace of Tendon-based Stewart Platforms[END_REF]. The necessary but not sufficient condition for a mobile platform with n degrees-of-freedom to have a fully-controlled motion is considering at least m = n + 1 cables, since many cable robots can be over-determined with respect to Forward Kinematics but under-determined with respect to cable forces distribution [START_REF] Verhoeven | Analysis of the Workspace of Tendon-based Stewart Platforms[END_REF]. Therefore, a critical issue for their use is the evaluation of the cable force distributions. For cable-based parallel robots it is known that maintaining positive cable tension is critical in constraining the moving platform. Hence, the force-closure workspace is defined as the set of poses whereby resultant cable tensions can sustain an arbitrary external wrench acting on the moving platform.

A better control of the mobile platform can be obtained by increasing the number of actuated cables, which will also reduce the tension in each cable for a given payload, but the workspace will be limited by possible interferences among cables and cables with mobile platform [START_REF] Wischnitzer | Wire-driven Parallel Robot: Permitting Collisions between Wires[END_REF].

Several workspace classifications have been proposed for fully constrained cable robots, namely the controllable [START_REF] Bruckmann | Wire Robots Part I Kinematics, Analysis & Design[END_REF], wrench feasible [START_REF] Riechel | Wrench-based analysis of cable-driven robots[END_REF], dynamic [START_REF] Barrette | Determination of the dynamic workspace of cable-driven planar parallel mechanisms[END_REF] and force-closure [START_REF] Pham | Force-closure workspace analysis of cable-driven parallel mechanisms[END_REF], [START_REF] Diao | A method of verifying force-closure condition for general cable manipulators with seven cables[END_REF] workspaces. They are defined as the set of poses at which the mobile platform (or end-effector) can physically reach while all the cables have positive tension, and additional constraints are fulfilled. It is worth to note that in constraining positive cable tensions, main proposed analyses deal with the null space approach through pseudo-inverse matrices, graphical approaches, or condition expressed in terms of the convex hull which encloses the origin. Those approached have been applied to either planar and spatial cable-based parallel robots.

Many fully-constrained manipulators have been proposed for a number of possible applications, but feasible tasks are limited due to the increasing number of cables, [START_REF] Lafourcade | Design of a Parallel Wire-Driven Manipulator for Wind Tunnels[END_REF], [START_REF] Kawamura | High-speed manipulation by using parallel wire-driven robots[END_REF].

In this paper a systematic method of verifying the forceclosure condition for general 3-DOF fully-constrained cable manipulators with four cables is proposed as based on CAD (Cylindrical Algebraic Decomposition).

A comparison of the workspace of different designs is presented and discussed. The CAD algebraic tool has been successfully used for the workspace analysis and singularity determination of classical parallel manipulators as reported in [START_REF] Moroz | Cusp points in the parameter space of RPR-2PRR parallel manipulator[END_REF], [START_REF] Chablat | Uniqueness domains and non singular assembly mode changing trajectories[END_REF].

ALGEBRAIC TOOLS

To analyze the workspace of a robot, we have to determine the equations that describe the Kinematics. These equations use trigonometric functions and we have to transform them to an algebraic system to make the CAD (Cylindrical Algebraic Decomposition).

Algebraic formulation of robotics problem

There are two methods to obtain algebraic equations starting from trigonometric equations:  Change an angle  by 2 variables x  , y  and a constraint equation

          (2)
The first transformation is used for all our computation except for the rendering. In this case, we introduce a singularity representation for   .

Discussing the number of solutions of the parametric system

The workspace analysis requires the discussion of the number of solutions of the parametric system associated with the Inverse Kinematics. More precisely, we want to decompose the workspace in cells 1k ,...,C C , such that: From a general point of view, the discriminant variety can be defined for any system of polynomial equations and inequalities.

 i C is
Let 11 , ... , , ..., ml p p q q be polynomials with rational coefficients depending on the unknowns 1 , ..., n XX and on the parameters 1 , ..., = , ( ) 0,..., ( ) 0, ( ) 0, ..., ( ) 0 , ( ) 0,..., ( ) 0, ( ) 0, ..., ( ) 0

ml p p q q      v v v v v C C (3)
ml p p q q       v v v v v S C (4)
If we assume that S has a finite number of solutions over at least one real point that does not belong to

D V , then d D V  R
can be viewed as a real discriminant variety of S , with the same property: over each connected open set

d  U R such that D V    U
, C defines an analytic covering. In particular, the number of points of R over any point of U is constant. Discriminant varieties can be computed using basic and well known tools from computer algebra such as Groebner bases (see [START_REF] Cox | Ideals, Varieties, and Algorithms[END_REF], Chapter 3) and a full package computing such objects in a general framework is available in Maple software through the RootFinding[Parametric] package.

Kinematics tool for the workspace analysis

The vector of input variables q and the vector of output variables X for a n-DOF parallel manipulator are linked by a system of non linear algebraic equations as

  , F  q X 0 ( 5
)
where 0 is the n-dimensional zero vector. Differentiating Eq. ( 5) with respect to time leads to the velocity model

0  A X B q   (6)
where A and B are n  n Jacobian matrices. These matrices are functions of q and X FF    AB Xq [START_REF] Bruckmann | Wire Robots Part I Kinematics, Analysis & Design[END_REF] These matrices are useful for the determination of the singular configurations [START_REF] Chablat | Working Modes and Aspects in Fully-Parallel Manipulator[END_REF].

PLANAR 4-CABLE ROBOTS AND THEIR SUB-ASSEMBLY

Kinetostatics equations for 4-cable robots

Let us consider a planar 4-cable robot shown in Fig. 1, it is composed by a fixed frame whose dimensions are L X and L Y and end-effector, whose dimensions are referred as b and h. The manipulator has four cables to constrain the end-effector 3-DOFs in the plane of motion XY. Let us consider the attachment points at the base named as A i and those in the end-effector named as B i . The cables' lengths i l and i  angles can be evaluated as

(for i = 1…4)     00 , A R B    i i i l  PP (8)               sin cos tan cos sin iy ix iy i ix ix iy A y B B A x B B            (9) in which 0 [] P = T xy
and  represent the end-effector pose in the fixed frame and R is the rotation matrix, which relates the moving and fixed frames denoted as GX'Y' and OXY.

A fundamental requirement for a cable manipulator to be fully controllable is that all its cables must be in tension at any working configurations. In other words, all the cable forces must be positive (assuming a positive cable force representing a tension and a negative cable force being a compression). b h Such a force feasibility problem is indeed a force-closure problem (also called vector-closure problem in the mathematics literature where the vectors of interest are the row vectors of the Jacobian matrix of the manipulator).

θ 2 θ 4 A 2 A 4 O≡A 1 Y X X' Y'  l 2 l 4 cable 2 cable 4 cable 3 cable1 θ 3 θ 1 g A 3 l 3 l 1 B 2 B 4 B 1 B 3 L Y L X G
Let us consider the wrench applied at G, which is the origin of the moving frame by the i-th cable in the form [START_REF] Pham | Force-closure workspace analysis of cable-driven parallel mechanisms[END_REF] in which f i and n i are the force and moment about G produced by the i-th cable. Since the abovementioned force is parallel to its corresponding cable and its related moment is perpendicular to the plane, they can be expressed as

T T i i i n    wf
i i i t  fu and   i i i i det , n t  R B u (11)
In which u i is the unit vector along l i and t i is the tension in the i-th cable. If we arrange the wrench and tension in matrix form the static equilibrium equation for the end-effector can be expressed as G  W t w [START_REF] Lafourcade | Design of a Parallel Wire-Driven Manipulator for Wind Tunnels[END_REF] in which w G is the wrench applied to the end-effector by external forces and moments and can include gravity force, without loss of generality, and W represents the wrench matrix also called structure matrix [START_REF] Verhoeven | Analysis of the Workspace of Tendon-based Stewart Platforms[END_REF], being J the transpose of W.

A cable-based parallel manipulator is said to have a forceclosure in a particular pose if and only if any arbitrary external wrench applied at the moving platform can be sustained through appropriate tension forces in the cables [START_REF] Pham | Force-closure workspace analysis of cable-driven parallel mechanisms[END_REF].

In order to solve the force-closure problem, the linear system of equations given by Eq. ( 12) has to be solved.

From linear algebra, it is known that the vector sum of any solution of Eq. ( 12) with a vector in the null space of W is again a solution to Eq. ( 12) [START_REF] Roberts | On the inverse kinematics, statics, and fault tolerance of cable-suspended manipulators[END_REF]. Therefore, if * t is a solution of Eq. ( 12) and t  being a vector in the null space of W, then the sum

, t t t         (13)
is again a solution of Eq.( 12). Therefore, for a vector t  whose components are all strictly positive, a sufficiently large scalar multiple  of this vector can be added to any particular solution t  of Eq. ( 12) to obtain a cable-tension vector t with positive components.

Several researchers pointed out that the force-closure problem of cable manipulators is similar to that of multiple fingers grasping a frictionless rigid-body [START_REF] Ebert-Uphoff | On the connections between cable-driven manipulators, parallel manipulators and grasping[END_REF]. In the former, all the cables must be in tension while in the latter all the fingers must be in compression. Hence, the equilibrium equations with inequalities on cable tensions are similar to the equations of equilibrium for the grasped object with constraints on finger forces. Handling the force-closure problem of cable manipulators based on the similarity between cable manipulators and multi-finger hands can be found in [START_REF] Mccoll | Workspace Generation of Planar Wire-Actuated Parallel Manipulators with Antipodal Method[END_REF] in which the antipodal method is used for the workspace analysis of planar cable robots.

A graphical method was proposed in [START_REF] Azizian | On The Boundaries of the Wrench-Closure Workspace of Planar Parallel Cable-Driven Mechanisms[END_REF] to determine the types of conic sections forming the boundary of the constant orientation wrench-closure workspace of a planar cable-robot, which were firstly determined in [START_REF] Gouttefarde | On the properties and the determination of the wrench-closure workspace of planar parallel cable-driven mechanisms[END_REF], and then obtained for the spatial case in [START_REF] Azizian | On The Boundaries of the Wrench-Closure Workspace of Planar Parallel Cable-Driven Mechanisms[END_REF], when a constant orientation of the moving platform is considered. In particular, it is shown in [START_REF] Gouttefarde | Determination of the wrench-closure workspace of 6-DOF parallel cable-driven mechanisms[END_REF] that the boundaries of the constant orientation workspace of a cable robot consists of parts of cubic surfaces.

By reviewing the literature, it is worth to note that the determination of the boundaries of the wrench-closure workspace for any given planar cable-robot is still a challenging problem that has not been completely solved yet and it is worthy of investigation.

Singularity of 3-RPR parallel robot

Less work is done in developing systematic methods of judging whether the force-closure condition is satisfied for a given configuration of a cable manipulator. The proposed algorithm to determine the workspace of cable manipulator can be summarized in three steps:

1. Determine one pose where all the cables are in tension starting from the geometric center of the attached points. 2. Determine all the 3-RPR associated sub-mechanisms to get their singularity conditions. 3. Study the parametric system to have the locus where Inverse Kinematic solutions exist. As the first step has been already detailed, we will explain more deeply steps 2 and 3. We will start from the 4-cable robot used in [START_REF] Mccoll | Workspace Generation of Planar Wire-Actuated Parallel Manipulators with Antipodal Method[END_REF] and depicted in Fig. 2.

As the dimension of the wrench matrix W is four, we have four equivalent 3-RPR parallel mechanisms, as it is shown in Fig. 3. The cable robot is controllable as long as the four cables remain in positive tension. This is the case if none of the four equivalent 3-RPR manipulators cross a singular configuration. If we change the shape of the mobile platform and number of attachments k at the end-effector, then we have a great impact on the size and shape of the workspace.

In this paper, we will study the cases where k is equal to 2 (line platform), 3 (triangular) and 4 (rectangular).

A 1 A 3 A 2 A 4 B 1 P B 2 B 3 B 4 Figure 2:
An example of a planar 4-4-cable robot.

A 1 A 3 A 2 B 1 P A 1 A 2 A 4 B 1 P B 2 A 1 A 3 A 4 B 1 P B 4 A 3 A 2 A 4 B 1 P B 2 B 3 B 3 B 4 B 3 B 4 B 2

Definition of the geometric properties of the cases study and parallel singularities

For the following three examples, we fix the attachment points onto the base as [START_REF] Ebert-Uphoff | On the connections between cable-driven manipulators, parallel manipulators and grasping[END_REF]. Let  

() i Bi x Bi y Ai Bi y Bi x Ai i E q x x y x y x x y l           )
T i B i B i B x y  . Square mobile platform Let   1 1 2 B h h    ,   2 1 2 B h h  ,   3 1 2 B h h  ,   4 1 2 B h h 
in the local frame and 1 1 h  and 2 1 h  . For the four 3-RPR robots the determinants of the Jacobian matrices are

        2
                 C (15)        
                  C (16)         2
                    C (17)         2
                     C (18)
Values of i C depend on the end-effector pose only, as it is shown in Fig. 4.

Let 0 P be defined as , , , / 0 and 0 and 0 and 0

i xy E q i xy                   W C C CC (19)
Figure 5 represents the same information as Fig. 4, but the workspace is characterized by a set of cells obtained by the CAD. This result can be obtained by using the CellDecomposition function of Maple and depicted with PlotCell function for planar cases. The PlotCell3D function was implemented in Maple for this paper. This function has been extended to define the cells for spatial cases.

Figure 6 represents the workspace described by 21 cells from 5758 cells obtained with the CAD decomposition. The borders represent the parallel singularities.

                 C ( 20
) 2 : ( 5 )(1 2 2 ) 0 x x x y y y y x y x             C (21)
                    C ( 22
)
4 : ( 5)

( 1 12 2 2 6 ) 0 x x x x y y y y x y x y                 C ( 23
)
The values of i C depend on the pose of the end-effector only, as it is shown in the examples of Figs. 8 and9.

Figure 10 represents the workspace described by 21 cells from 5758 cells obtained with the CAD decomposition.

A 1 A 3 A 2 A 4 B 1 P B 2 B 3
         C (26) 4 : ( 5 )( 6 ) 0 x x x y y x y           C (27)
It is worth to note that in this case left terms of 1 C and 3 C , (resp. 2

C and 4 C ) are the same. Values of i C depend on the pose of the end-effector only, as it is shown in Fig. 12. Singularities occur whenever 1

A , 1 B , 2 B and 2 A are aligned (resp. [START_REF] Castelli | Analysis and simulation of a new Cartesian cable-suspended robot[END_REF] A , 1 B , 2 B and 4 A ). Figure 13 shows the workspace of the cable-robot with line platform when 0   and 0.04

 
. The complexity of the CAD depends on the order of the parameter in which we realize the projection. For the three cases, the simplest one is   yx  . The CAD yields 330 cells but only six represent the intervals in which  

0 0 0 P t y x 
is associated to a stable pose of the cable robot, as is shown in Fig. 14. . Moreover, this representation allows us to compute the area of each cell using the built-in integration functions of Maple. Indeed, the Maple procedure int can handle functions of the form ( ) R oot ( ( ), )

A 1 A 3 A 2 A 4 B 1 P B 2
t f t p X n 
, that maps a real t to the n th root of the polynomial () t pX whose coefficients depend on t.

A WORKSPACE COMPARISON FOR THE DESIGN OF PLANAR 4-CABLE ROBOTS

Analysis

In the following a comparison among the workspaces for the planar 4 cable robots reported in the previous Section is given. In particular, the 4-4, 3-4 and the 2-4 designs have been analyzed. As expected, increasing the number of attachment points on the mobile platform, the orientation capability of the robot decreases, as it can be clearly seen by Figs. 6, 10 and 14 and Tab. 1. Moreover, the reported case of study for the 4-4 cable robot (square platform) shows a poor orientation capability of the robot. The volume of the workspace cannot be use as a criterion to compare the cable robots. In Tab. Let us suppose we want to design a cable-based parallel robot, with requirement of the user being to obtain a regular workspace (sphere, cylinder, cube, ..) for which the angular rotation range of the moving platform is defined. This request can be given as a set of constraints. For a cylindrical shape workspace, we can use a cylinder toped by two planes or an approximation by a Lame's curve.

For example, Eq.(34) (resp. Eq.( 35)) approximates a cylinder (resp. a cube), where x c , y c and c  are the geometric centre coordinates and

                         (34)
                         (35)
These constraints can be included when we compute the workspace. If we obtain a workspace smaller than expected, it means that the cable robot has singular configurations within the requested regular shape workspace. Then we have to change the design parameters.

Fig. 17 shows the approximated cylindrical and cubic shaped workspace by Lame's curves. Figure 17 shows the maximal value of   that we can reach when

Over-constrained manipulators

Let us consider the above-mentioned planar 4-cable manipulator, which is a fully-constrained robot having 3 DOFs in the plane, if we add a 5 th cable, then we get an overconstrained manipulator. This means that we admit that the tension in one cable can have no positive value. In this case, we have to study the workspace of cable-based parallel robot taking into account four cables only. For each one, we will have four constraints (36)

CONCLUSIONS

A new method was introduced to compute workspace of the general 4-cable robot by algebraic tools. We obtain automatically the algebraic formulation of the boundary of the workspace first described in [START_REF] Gouttefarde | On the properties and the determination of the wrench-closure workspace of planar parallel cable-driven mechanisms[END_REF]. Furthermore, our approach can be generalized to higher dimension and other type of constraints. Contrary to methods based on discretization of the workspace or the interval analysis based method, we obtain an exact formulation of the boundary of the workspace. For the same location of the attachment points, we have compared the size of the workspace and the angular rotation range for several types of mobile platforms. When the shape of the mobile platform is a line, the equations representing the workspace boundary were simple and the angular rotation range was the greatest.

The proposed method provides a powerful tool to analyze fully-constrained planar or spatial cable-robots as an extension of the proposed formulation, which will be developed as future work. Furthermore, it may allow the analysis of the overconstrained manipulators, for which the number of cables is greater than the number of DOFs + 1. For the latter case, the analysis can be performed by recursively applying the described method, providing all possible combinations of sub-assembly manipulators, the workspace being the union of the computed sub-workspaces.

.

  This change of variables allows us to avoid the sine and cosine functions and keep the system algebraic. Moreover, this change of variables does not introduce any spurious solutions since the function bijective on the unit circle C .  Use the Weierstrass substitution. If we set

dUU.

  Let us consider the constructible set:

Figure 1 :

 1 Figure 1: A scheme for a planar 4-cable robot and its parameters.

Figure 3 :

 3 Figure 3: The four sub assembly for the planar 4-4-cable robot.

  pose for the cable robot, we know that the workspace W is defined up to the first singularity of at least one 3-RPR manipulator.

Figure 4 :

 4 Figure 4: The singularity curves i C for

Figure 5 : 2 Figure 6 :.

 526 Figure 5: The workspace for the square mobile platform robot in Figure 2 for 0   and

Figure 7 :

 7 Figure 7: An example of a planar 3-4-cable robot.

Figure 9 :Figure 10 :

 910 Figure 9: The workspace for

Figure 11 :Figure 12 :

 1112 Figure 11: An example of a planar 2-4-cable robot.

Figure 13 :Figure 14 :tt

 1314 Figure 13: The workspace for

  can show a meridian slice to evaluate the orientation capability, as in Fig.15for 3 x  , and as in Fig.

  c) a line mobile platform.

  its depth (resp. the sizes of a cube).

Figure 17 :

 17 Figure 17: The approximation of a cylinder and a cube by Lame's curves to approximate a regular workspace.

Figure 18 :

 18 Figure 18: The intersection of a cubic regular workspace with the workspace made by a line mobile platform for (a) 0 .7 8  

Table 1 : Comparison of the properties of the three type of mobile platform

 1

	Shape of mobile platform	Square	Triangle		Line
			m in		0.0906		0.463		0.876
			m ax	0.0906	0.463	0.876
		S	(0 )		12		7 .5		18
	S	(0 .0 4 )		7.32		7.32	17.23

ACKNOWLEDGEMENTS

The research work reported here was made possible by SiRoPa ANR Project.