Nicolas Durand

Rana Mhanna

Paul Savel

Huriye Akdas-Kiliç

Jean-Pierre Malval

Olivier Soppera

Jean-Luc Fillaut

Unexpected Disruption of the Dimensionality-driven Two-Photon Absorption Enhancement within a Multipolar Polypyridyl Ruthenium Complexes Series

The dimensionality-driven two-photon absorption (2PA) enhancement effect is investigated in a series of functionalized bipyridyl Ru-complexes. Our design strategy leads to very high 2PA responses up to ~ 1500 GM. However, we highlight that the 2PA performance vs. dimensionality correlation reaches an unexpected limit stemming from 'anti-cooperative' interchromophoric couplings.

Molecules with optimized 2PA efficiency are highly demanded in the fields of photonics and biology 1 . Indeed, 2PA materials are extensively widespread across many applications including microfabrication 2 , high density data storage 3,4 , bio-imaging 5-7 or photodynamic therapy 8 . Several critical factors determine the 2PA performances of full organic systems : the symmetry of the structure, the extent of the π-conjugation length, the occurrence of an efficient intramolecular charge transfer (ICT) both modulated by the nature and the position of electron donor and acceptor groups within the designed molecule 9 . Alternatively, coordination chemistry offers a versatile strategy for designing two-photon absorbers whose NLO properties are associated to charge-transfer processes such as intraligand charge-transfer (ILCT) and metal-ligand charge-transfer (MLCT) [10][11][12][13] . The metal center acts as a powerful template for self-organizing 2PA chromophores in a three-dimensional arrangement 9, 14 . Very high 2PA amplifications can then be observed due to the synergistic interplay and extent of electronic couplings between the organic branches 15

. In line with this strategy, we designed a series of homoleptic and heteroleptic [RuL n (bpy) With this respect, we have designed several 2,2′-bipyridinebased ligands which are 4,4'-substituted by distinctive electron-rich arylamine groups such as triphenylamine (T), carbazolyl (C) and diphenylaminofluorenyl (F) subunits through a vinylene linker (Scheme 1). These peripheral πconjugated subunits have been selected since their electron-donor ability as well as their π-conjugation length can be regulated through a planarization effect. Such a structural constraint together with the progressive increase of 3D-dimensionality will give rise to competing interactions between 'through-bond' and 'through-space' couplings 16,17 within the complexes series leading to a severe disruption in the dimensionality-induced 2PA enhancement effect. The absorption spectra of the ligands along with their corresponding Ru-complexes in tetrahydrofuran (THF) are shown in Figure 1. The absorption of Ru(bpy) 3 2+ used as a reference complex is also displayed. The spectroscopic data are summarized in Table 1. All the ligands present similar spectral features particularly with the presence of a longest wavelength absorption band located in the 330-450 nm range. This band is mainly dominated by the S 0 → S 1 electronic transition which implies a long range electronic As shown in Figures S1-S3, all the ligands are fluorescent and undergo very large amplitude Stokes shifts up to one order of magnitude on going from apolar to highly polar solvents. Based on these solvatochromic measurements, the dipole moment change between ground and relaxed excited state (∆µ 01 ) has been estimated using Lippert-Mataga formalism 21, 22 (see Figure S4 and Table S1).

Therefore, we show that all the ligands display very high ∆µ 01 with values of 13.6 D for C, 18.5 D for T, and 27.9 D for F. Note that ∆µ 01 is multiplied by a factor 2 between C and F which confirms the very strong ICT enhancement promoted by our proposed structural changes. Analogously, the coordination to Ru 2+ strongly influences the ligands absorption spectra (Figure 1). All the metal-complexes display a well distinctive band in the 270-320 nm range similarly to that observed for Ru(bpy) 3 2+ . This band presumably corresponds to the ligand-centred (LC) π → π* transitions 23 (bipyridine-centred transitions). Note that this LC band shifts to the red region and increases in intensity with the coordination number of functionalized-bipyridyl ligands (n). Similar effects can be observed when considering the lowest energy band which progressively developed in the 350-630 nm range. This very broad and structured band encompasses several electronic transitions 10,12 : the intraligand charge transfer (ILCT) transitions and the metal-to-ligand charge transfer (MLCT) ones. The ILCT band is located at the blue side of the lowest energy band and increases in intensity with n. When comparing the free ligand to its first heteroleptic complex, the ICT band clearly redshifts in concomitance with a substantial decrease in intensity. Whereas this bathochromic effect can be confidently attributed to the electron withdrawing exaltation of the branched bipyridine coordinated to Ru 2+ , the observed band decrease should be ascribed to a severe reduction of the conjugation length within the entire structure of the ligand probably due to a loss in planarity of the 2,2'-bipyridine which undergoes a trans-to-cis conformation change upon complexation. The MLCT bands which strongly overlap with the ILCT ones are typically positioned at the low energy side of the complexes absorption band. For all the heteroleptic complexes, two distinctive MLCT transitions can be observed due to the presence of two types of bipyridine ligands: i) The dπ(Ru) → π*(bpy) transitions are presumably located in the 430-460 nm range as suggested by the MLCT band of Ru(bpy) 3 2+ (see Figure 1). This band has a moderate intensity (ε MAX ~ 14 000 M -1 cm -1

) and can be distinguished on the structured absorption band of the heteroleptic complexes. ii) The dπ(Ru) → π*(bpy-substituted) transitions should correspond to the broad and intensive band at the red edge of the absorption spectrum. This red-shifted MLCT band grows incrementally with n in the same manner as for the ILCT band. The luminescence properties of the metalcomplexes in THF are presented in Figure S5 and in Table S2. All the complexes are weakly emissive with a luminescence quantum yield (Φ L ) lower than 10 . This electronic configuration implies a drift of electron density from the metal to the ligands with a dominant electronic localization on the substituted bipyridines. This assumption is both consistent with the observed luminescence spectral changes and the strong stabilization of the 3 MLCT state energy (∆E = 0.18-0.26 eV) as compared to Ru(bpy) 3 2+ . Increasing the dimensionality of the complex with functionalized bipyridines also induces a 3-fold enhancement of the complexes emissivity (see Table S2). This luminescence 'switch-on' of the 3D complexes presumably stems from a shielding effect of the peripheral donor branches from the dynamic oxygen quenching.

Increasing the dimensionality of metal-complexes also impacts the 2PA properties. Figure 2 shows 2PA spectra of the metal-complexes in THF, alongside the corresponding one-photon absorption (1PA) spectra. Figure S6 also presents the 1PA and 2PA spectra of the ligands in THF. According to our spectral resolution, the ligands display a 2PA band in the 700-900 nm range which reasonably matches their respective lowest energy 1PA band. However, the 2PA cross-sections are moderate for all ligands with δ MAX hardly reaching ~ 200 GM whatever the conjugation length or the donor group nature (see Table 1). Such modest values presumably originate from the very close centrosymmetric geometry of the ligands (i.e. D-π-Aπ-D) leading to a theoretically two-photon forbidden character of the S 0 → S 1 transition. By contrast, the complexation with Ru 2+ induces a very strong amplification of the 2PA cross-sections by more than a factor 6 between the free ligands and their respective homoleptic complexes. Moreover, increasing the number of π-functionalized bipyridines gives rise to very high δ MAX up to 1465 ± 220 GM for RuT 3 . Such a strong 2PA enhancement proceeds through a synergy effect which combines an effective increase of the intramolecular charge transfer per branch and the cooperative interbranch coupling within the entire multipolar structure 13,26 . As illustrated in Figure 2, the 2PA spectra of all the complexes feature a very intensive band which nicely coincides with the ILCT transitions and a second 2PA resonance in the lowest energy region which matches the MLCT transitions. However, the evolution of the relative intensities of these two bands all along our complexes series suggests that the MLCT resonance increases to the detriment of the ILCT one especially for the F series. Of particular interest, the ILCT-to-MLCT band ratio undergoes very strong changes which both depend on the coordination number and the nature of the ligand. For instance, this ratio is about ~ 8 for RuC 3 but progressively decreases to 3.6 and 1.9 for RuT 3 and RuF 3 respectively. Such a monotone spectral effect from C to F series illustrates the increase of dipole-dipole interchromophoric couplings 16 . However, in the present case, the magnitude and the orientation of the interbranch cross-talk within the complexes give rise to an unexpected disruptive effect of the 2PA enhancement course with dimensionality. In order to quantify such a disruption, one should first consider that δ MAX increases quadratically with the effective number of πelectrons (N eff ) contributing to the nonlinear response 27, 28 .

Therefore, δ MAX has been normalized by N eff 2 in order to access to the intrinsic 2PA cross-section which indeed accounts for the real 2PA performance of a dye. For determining N eff , we used the relevant counting method ACCEPTED MANUSCRIPT According to this methodology, it can be inferred that the triphenylamine series exhibits the optimal behaviour in term of 2PA cooperative effect. Indeed, δ/N eff 2 increases gradually with the coordination number (n) and RuT 3 exhibits the highest 2PA performance amongst its homologues. The 2PA performance of the carbazole series presents a similar behaviour but with a much weaker 2PA amplification. Finally, one of the most striking results concerns the diphenylaminofluorenyl series whose 2PA performance rapidly saturates for n = 2 and 3. Even though RuF 1 constitutes the most promising 2PA active candidate as compared to RuC 1 and RuT 1 with the highest 2PA enhancement (up to 118 %), it is clearly demonstrated that the further addition of F ligands has no effect on the 2PA performance. By contrast to the other ligands, the increase of interchromophoric interactions emerging from the close proximity of F within RuF 2 or RuF 3 paradoxically generates 'anti-cooperative' effects 16 which collapse the 2PA performance. These detrimental interactions should originate from the crowded and rigid character of F which drastically limits the fine orientation and the relative positioning of the polar branches within the complex. With this respect, there is no doubt that T appears as an optimal prototype which displays the good compromise between large dipole moment change and reasonable flexibility within a sterically constrained architecture. Such structural conditions undoubtedly establish a guideline for the future design of new multipolar metal-complexes allowing a real gain in term of 2PA performance.

Notes and references

Scheme 1 .

 1 Scheme 1. Molecular structures of the Ru-complexes.

  π-conjugated structure with a significant ICT character from the peripheral donor amino groups to the central acceptor bipyridine moiety 12, 18 . This CT band is both sensitive to the nature of the donor subunit and the conjugation length. For instance, the better donor ability of the triphenylamine moiety as compared to the carbazole one 19, 20 leads to a clear bathochromic effect on going from C to T. Moreover, such a CT process is all the more amplified when considering the dipole moment change between the ground and the excited emitting state.

Fig. 1

 1 Fig.1 Absorption spectra (full lines) of the ligands and their corresponding Ru-complexes in THF. Dashed lines: Absorption spectrum of Ru(bpy)3 2+ in acetonitrile.

  -2 in nondegassed THF. The luminescence is very sensitive to oxygen with values of Φ L enhanced by factor ~ 10 when operating in N 2 -saturated THF. This is an indication that the ACCEPTED MANUSCRIPT luminescence of the complexes arises from the triplet 3 MLCT state. It is noteworthy that the luminescence bands are strongly redshifted with respect to that of Ru(bpy) 3 2+ .This clearly underlines the electronic effect of the bipyridine functionalization. Indeed, the 4, 4'-substitution of bipyridine with electron donating groups stabilizes the

Fig. 2

 2 Fig.2 1PA (full lines) and 2PA (symbols) spectra of Ru-complexes in THF. The one-photon molar absorption is shown at the 1PA wavelength (lower horizontal scale). The 2PA cross section is shown as a function of the excitation laser wavelength (upper horizontal scale).

proposed by Kuzyk et al 27 , 16 .Fig. 3

 27163 Fig.3 Intrinsic 2PA performance of all the dyes as function of ligands π-conjugation length and complexes coordination number.

Table . 1

 . 1PA and 2PA properties of the ligands and Ru-complexes (Solvent : THF). The experimental uncertainty for δ is ± 15 %.

Université Rennes, Institut des Sciences Chimiques de Rennes CNRS UMR 6226, F-35000 Rennes.

Institut de Science des Matériaux de Mulhouse CNRS-UMR 7361, Université de Haute Alsace, 15 rue Jean Starcky, 68057, Mulhouse, France.

Yildiz Technical University, Department of Chemistry, 34220 Esenler, Istanbul, Turkey. E-Mails: jean-luc.fillaut@ univ-rennes1.fr and jean-pierre.malval@uha.fr


Mattia Giaccagli

Daniele Astolfi

Vincent Andrieu

Lorenzo Marconi

Sufficient Conditions for Output Reference Tracking for Nonlinear Systems: a Contractive Approach

This paper deals with a tracking problem for nonlinear systems. We present sufficient conditions for the state-feedback output tracking problem, in case of arbitrarily large constant references and arbitrarily large domain of attraction. We present an extension of forwarding-based control techniques applied in an incremental framework. Simulations of an academic example are presented to validate the results.

I. INTRODUCTION

This work deals with the design of control laws for nonlinear systems to track a constant reference set-point. For linear systems this problem has been completely solved in the context of the so-called internal model principle (see [START_REF] Francis | The internal model principle of control theory[END_REF]) stating that the regulator needs to include an integral action processing the regulated error. Based on this principle, many solutions have been proposed in case of nonlinear systems. We divide these solutions in two main classes.

The first one consists in employing a change of coordinate to put the system into the so-called normal form, [START_REF] Isidori | Nonlinear Control Systems[END_REF]Chapter 4.1]. Depending on the properties of the zero-dynamics, different control design have been proposed: see, among others, [START_REF] Behtash | Robust output tracking for non-linear systems[END_REF], [START_REF] Jiang | Robust nonlinear integral control[END_REF], [START_REF] Khalil | Universal integral controllers for minimum-phase nonlinear systems[END_REF], [START_REF] Marino | Output regulation for unknown stable linear systems[END_REF] for minimum-phase systems and [START_REF] Huang | Regulation of nonminimumphase nonlinear systems using slow integrators and high-gain feedback[END_REF] for non-minimum phase. In these settings, output tracking can be ensured for "large references" with a semi-global (or global) domain of attraction. However, such designs cannot be applied when the relative degree between the regulated output and the control is not well-defined, namely when the normal form doesn't exist. Furthermore, the extension to multi-input multi-output (and possibly more input than regulated output) is not trivial. See for instance, [START_REF] Bin | Output regulation by postprocessing internal models for a class of multivariable nonlinear systems[END_REF], [START_REF] Wang | Pre-processing nonlinear output regulation with non-vanishing measurements[END_REF].

The second class of approaches follows the linear paradigm: 1) extend the controlled system with an integral action processing the regulated errors; 2) stabilize the unperturbed (i.e. with zero-reference) extended system; 3) finally analyze the behaviour of the closed-loop system in presence of the reference. Such approach is in general more flexible as it's "coordinate-free" and can be easily extended to multiinput multi-output cases, but it's limited in the size of the domain of attraction and the reference, see for instance [START_REF] Khalil | Nonlinear systems[END_REF]Chapter 12.4] with a linearization-based feedback design. In order to relax such constraints, forwarding design (see, for instance, [START_REF] Kaliora | Nonlinear control of feedforward systems with bounded signals[END_REF], [START_REF] Mazenc | Adding integrations, saturated controls, and stabilization for feedforward systems[END_REF], for a review) have been proposed in [START_REF] Astolfi | Integral action in output feedback for multiinput multi-output nonlinear systems[END_REF], [START_REF] Poulain | Robust asymptotic stabilization of nonlinear systems by state feedback[END_REF] so that to achieve semi-global domain of attractions.

Nevertheless, the size of the reference still need to be "small".

The objective of this work is therefore to study sufficient conditions to design a state feedback control law for the problem of output set-point tracking for nonlinear systems with a global domain of attraction and without restrictions on the amplitude of the set-point. We aim at proposing conditions that are "coordinate-free", i.e. we don't look for any normal form. For this, we follow similar ideas to [START_REF] Pavlov | Uniform output regulation of nonlinear systems: a convergent dynamics approach[END_REF] by casting the output tracking problem into the contraction theory and we propose a new control design based on a modification of forwarding-based techniques able to preserve the desired incremental stability properties, see, for instance, [START_REF] Andrieu | Transverse exponential stability and applications[END_REF], [START_REF] Angeli | A lyapunov approach to incremental stability properties[END_REF], [START_REF] Forni | A differential lyapunov framework for contraction analysis[END_REF], [START_REF] Lohmiller | On contraction analysis for non-linear systems[END_REF] and references therein. While the extension to the incremental framework has already been done for several control techniques, such as backstepping [START_REF] Zamani | Backstepping design for incremental stability[END_REF] [37] [START_REF] Bb | Contraction theory based adaptive synchronization of chaotic systems[END_REF], LMI [START_REF] Dalto | Incremental quadratic stability[END_REF] and circle criterion [START_REF] Yakubovich | Matrix inequalities method in stability theory for nonlinear control systems: I. absolute stability of forced vibrations[END_REF] [START_REF] Waitman | Incremental stability of lure systems through piecewise-affine approximations[END_REF], to the best of authors knowledge the problem of incremental forwarding hasn't been addressed in literature. Yet, forwarding is the natural tool to stabilize systems in cascade form as the one obtained when the regulated system is extended with integral actions, see [START_REF] Astolfi | Integral action in output feedback for multiinput multi-output nonlinear systems[END_REF], [START_REF] Poulain | Robust asymptotic stabilization of nonlinear systems by state feedback[END_REF]. First we present sufficient conditions for the incremental stabilization problem via forwarding. In this context we state a non-uniform global incremental exponential stability result, which can be made uniform under more restrictive conditions. Successively, we apply the proposed design to the output set-point reference tracking problem. We establish sufficient conditions to design a stabilizing feedback law to solve such a problem for arbitrarily large references and for arbitrarily large domain of attraction. The solution is coordinate-free and does not require the knowledge of the set point in the design of the controller.

The paper is structured as follows. First, in Section II, we present the problem statement and we provide some preliminaries results in order to clarify the link between contraction theory and output tracking. Then, in Section III, we recall some results on forwarding design and we give sufficient conditions and a constructive design to enforce incremental stability properties. Finally, in Section IV, we give the main results on output tracking. An academic example is given in Section V. Conclusions are derived in Section VI. Parts of the proofs are postponed to the Appendix.

Notation: We define R >0 = (0, ∞) and R ≥0 = [0, ∞). Given a set A ⊂ R n , we denote with Cl(A) its closure. Given x ∈ R n , y ∈ R m , we denote (x, y) = (x , y ) . Given a matrix A we denote with |A| the standard induced matrix norm. We denote with I n the Identity matrix of dimension n × n; when there's no possibility of misunderstanding we drop the index and use simply I. Given two vector fields f : R n → R n , h : R n → R, we denote the Lie derivative of h along f as L f h(x) := ∂h ∂x (x)f (x).

II. PRELIMINARIES A. Problem Statement

In this paper we are interested to the output set-point tracking problem for nonlinear systems of the form

ẋ = f (x) + g(x)u, y = h(x), (1)
where x ∈ R n is the system state, u ∈ R is the control input and y ∈ R is the output. The goal is to design a state feedback control law u such that, given a reference r ∈ R, the closed loop system trajectories are bounded and, asymptotically, lim t→∞ e(t) := h(x(t)) -r = 0.

In this paper we are interested in a coordinate-free approach, namely, we look for a solution that doesn't involve the use of normal forms of (1) by working on the "original coordinates" restriction on the relative degree of the system.

In the spirit of [START_REF] Astolfi | Integral action in output feedback for multiinput multi-output nonlinear systems[END_REF], [START_REF] Poulain | Robust asymptotic stabilization of nonlinear systems by state feedback[END_REF], we extend system (1) with an integral action

η = h(x) -r , (2)
and we look for a state-feedback law of the form

u = ψ(η, x) , (3)
which leads to a closed loop system

Ẋ = ϕ(X ) + Γr, (4)
where X = (x, η) and

ϕ(X ) = f (x) + g(x)ψ(η, x) h(x) , Γ = 0 -1 . ( 5
)
This type of control law solves the tracking problem if an equilibrium X * = (x , η ) exists, is unique and is globally asymptotically stable. As a matter of fact, in this case, h(x ) -r = 0 and consequently the error e converges to zero along any solution of the closed loop system. Constructing the feedback ψ such that, for all r, an equilibrium exists and is globally asymptotically stable is not an easy task when the system is not in normal form. The results presented in [START_REF] Astolfi | Integral action in output feedback for multiinput multi-output nonlinear systems[END_REF], [START_REF] Poulain | Robust asymptotic stabilization of nonlinear systems by state feedback[END_REF] provide a solution that however poses limitations on the domain of attraction and on the amplitude of the references, in the sense that they provide a solution that is semi-global in X and only local in the size of r. To overcome this constraints, we take advantage of contraction theory, which is shortly recalled in the next section.

B. Incremental Stability and Contraction Theory

Consider a systems of the form

Ẋ = ϕ(X ), (6)
where X ∈ R is the state and ϕ is a C 1 vector field. By X (X 0 , t) we denote the trajectory of the system starting from the initial condition X 0 at time t = 0.

In the following we will also adopt the following notation. Given a 2 tensor P : R → R × and the vector field ϕ : R → R , we denote the Lie derivative of the tensor P along ϕ as L ϕ P(X ), defined as

L ϕ P(X ) := d ϕ P(X ) + P(X ) ∂ϕ ∂X (X ) + P(X ) ∂ϕ ∂X (X )
where

d ϕ P(X ) := lim h→0 P(X (X , h)) -P(X ) h .
Definition 1. We say that system (6) is Incrementally Asymptotically Stable (in short IAS) on a positively invariant subset O of R if there exists a class KL function α such that

|X (X 1 , t) -X (X 2 , t)| ≤ α(|X 1 -X 2 |, t) , (7)
for any X 1 , X 2 ∈ O, and for any t ≥ 0. If O = R , then we say that system (6) is Incrementally Globally Asymptotically Stable (in short IGAS).

IGAS property states that if we pick two different initial conditions of system (6), the two trajectories that evolve in time starting from these initial conditions, will converge with each other as time goes on. Incremental properties are useful, for instance, whenever we are not interested in the convergence of trajectories toward an equilibrium point, but rather to study the relative behaviors of trajectories. In case the function α in ( 7) has an exponential decay, we have the following definition. Definition 2. We say that system (6) is Incrementally Exponentially Stable (in short IES) on a positively invariant subset

O of R if there exist k, λ ∈ R >0 such that |X (X 1 , t) -X (X 2 , t)| ≤ k|X 1 -X 2 | exp(-λt) (8) for all X 1 , X 2 ∈ O, t ≥ 0. If O = R , then we say that (6) is Incrementally Globally Exponentially Stable (in short IGES).
It is well known that IGAS/IGES properties are related to differential properties on the vector field ϕ. We recall below an important result linking incremental stability properties and the existence of a metric which decreases along the vector field.

Theorem 1. If there exists a C 1 function P : R → R × taking positive definite symmetric values, and positive real numbers p, p, q ∈ R >0 such that

pI ≤ P(X ) ≤ pI (9) L ϕ P(X ) ≤ -qI (10)
for all X ∈ R , then system (6) is IGES. Conversely, if system (6) is IGES and ϕ is C 2 with bounded first and second derivatives, then there exist a C 1 function P and p, p, q ∈ R >0 satisfying (9), [START_REF] Bin | About robustness of internal model-based control for linear and nonlinear systems[END_REF].

Proof. The proof of the first part of the theorem can be found in [24, Theorem 1], [2, Section II-B], [START_REF] Forni | A differential lyapunov framework for contraction analysis[END_REF]. The converse can be found for instance in [2, Proposition 4].

When inequality [START_REF] Bin | About robustness of internal model-based control for linear and nonlinear systems[END_REF] holds on the (Riemannian) convex hull of a positively invariant compact set, exponential contraction is established in this one as stated in the following proposition.

Proposition 1. Assume that there exists a C 1 function P : R → R × taking positive definite symmetric values, and positive real numbers p, p, such that (9) holds for all X in R . Let C be compact invariant subset of R for the dynamical system (6) and assume that there exists q ∈ R >0 such that (10) holds for all X in O defined as

O = X ∈ R , max w∈C |X -w| < , (11)
where

> √ p 2 p max (X a ,X b )∈C 2 |X a -X b | , (12)
then system (6) is IES on C.

Proof. The proof is omitted for space reasons. It can be deduced by carefully specializing similar arguments to those employed in Theorem 1.

Finally, we recall below the notion of "Killing Vector Field" (see, for instance, [12, Appendix B]). Definition 3. Given C 1 functions P : R → R × , and γ : R → R , we say that γ is a Killing Vector Field for P if L γ P(X ) = 0 for all X ∈ R .

C. From contraction to regulation

Our motivation to study incremental stability property comes from the following result, stating that the tracking problem for system (1) is automatically solved if we are able to make the closed-loop system (4) IGES.

Theorem 2. Assume there exists a C 1 function ψ : R×R n → R such that for all r in R the control law (3) ensures that the closed loop system (4) is IGES. Then, this control law is a solution to the (global) output tracking problem stated in Section II-A i.e. lim t→∞ y(t) = r for any initial condition

X 0 = (η 0 , x 0 ) in R n+1 .
Instrumental to the proof of Theorem 2, we have the following Lemma.

Lemma 1. Suppose system (6) is IES in a closed forward invariant set C ⊂ R . Then, there exists a unique equilibrium point X ∈ C which attracts all solutions initiated from C.

Proof. See Appendix A.

Remark 1. Interestingly, a straightforward consequence of Lemma 1 is that trajectories of a IES system evolving in a closed forward invariant set cannot converge to a limit cycles, but only to an equilibrium.

Proof of Theorem 2. By applying Lemma 1 with C = R , we know the existence of unique equilibrium point denoted X = (x , η ). Note that X being an equilibrium, it implies, by definition of the η-dynamics in (2), that h(x ) = r. System (4) being IGES, there exists k, λ ∈ R >0 such that equation [START_REF] Astolfi | Francis-wonham nonlinear viewpoint in output regulation of minimum phase systems[END_REF] holds which implies that, for all initial conditions X • ∈ R n , we have

|X (X • , t) -X | ≤ k|X • -X | exp(-λt) .
Hence Theorem 2 holds.

The statement of Theorem 2 is based on the formal definition of IGES given in Definition (2). However, motivated by the sufficient conditions of Theorem 1, an equivalent result can be also stated in the metric framework, that is by asking for the existence of a feedback law ψ and metric P satisfying the properties ( 9), [START_REF] Bin | About robustness of internal model-based control for linear and nonlinear systems[END_REF] for the closed-loop system (4). Moreover, in the particular case in which the vector Γ defined in ( 4) is a Killing Vector Field for P, it follows that

L ϕ+rΓ P(X ) = L ϕ P(X ) + rL Γ P(X ) = L ϕ P(X ) .
for all r ∈ R, namely the Killing Vector property is invariant with respect to the size of r. As a consequence, based on these remarks, we can state the following result.

Corollary 1. Suppose there exist a C 1 function ψ : R × R n → R, a C 1 function P : R m → R m×m taking positive symmetric values and p, p, q ∈ R >0 such that, for the closed-loop system (4), inequalities (9), ( 10) hold. Assume, in addition, that Γ is a Killing Vector Field for P. Then, for all r ∈ R the feedback law (3) solves the global tracking problem, i.e. lim t→∞ y(t) = r.

Motivated by this analysis, in the following section, we consider the problem of designing a control law ψ ensuring the contraction property along the vector field ϕ for some P and the Killing Vector Field property with respect to the vector Γ. Furthermore, note that, in view of its structure, see [START_REF] Angeli | A lyapunov approach to incremental stability properties[END_REF], the vector field Γ acts only in the directions of η. Hence, it suffices to find a metric P independent of η to have the desired Killing Vector property satisfied.

III. INCREMENTAL FORWARDING A. Highlights on Forwarding Design

Our system (1)-( 2) is in the so-called feedforward form (see [START_REF] Mazenc | Adding integrations, saturated controls, and stabilization for feedforward systems[END_REF]). The stabilization of this class of systems has been widely studied in the last years through forwarding control techniques (see [START_REF] Poulain | Robust asymptotic stabilization of nonlinear systems by state feedback[END_REF], [START_REF] Astolfi | Integral action in output feedback for multiinput multi-output nonlinear systems[END_REF], [START_REF] Kaliora | Nonlinear control of feedforward systems with bounded signals[END_REF] and the references therein). The objective of this section is to design a forwarding-based control law ensuring the desired contractivity properties. For this, we first recall the following result.

Lemma 2. Consider system (1), ( 2) and suppose that the origin of ẋ = f (x) is globally asymptotically stable and locally exponentially stable. Then, there exists a C 1 function M : R n → R solution of

L f M (x) = h(x). ( 13
)
Proof. The proof can be found in [27, Lemma IV.2] or [19, Section 5.2].

Based on the function M defined above, we can define a stabilizing state-feedback control law for the extended system (1)-( 2) as

u = (x)L g M (x)(η -M (x)) , (14)
where : R n → R >0 is a sufficiently small function. See, for instance, [30, equation (19)] for further details. We have then the following result.

Lemma 3. Consider system (1), ( 2) and let the assumptions stated in Lemma 2 hold. Furthermore, suppose the following condition hold

|L g M (0)| > 0 . ( 15
)
Then, there exists : R n → R >0 such that the origin of system (1), ( 2) in closed-loop with ( 14) is globally asymptotically stable and locally exponentially stable for r = 0.

Proof. The proof can be found in [6, Lemma 1], [30, Assumption 4] and references therein.

Condition [START_REF] Francis | The internal model principle of control theory[END_REF] is obtained if the extended system (1), ( 2) is controllable at the origin. Such property, also known as "nonresonance condition" in regulation theory [6, Assumption 3], corresponds to the condition CA -1 B is non zero, where

A := ∂f ∂x (x) x=0 , B := g(0), C := ∂h ∂x (x) x=0 .

B. Assumptions

Our objective is to design a forwarding-based control law ensuring the desired incremental stability properties for the extended system (1), [START_REF] Andrieu | Transverse exponential stability and applications[END_REF]. To this end, we first suppose that the system (1), with u = 0, is IGES, as stated below.

Assumption 1. There exists a C 1 function P : R n → R n×n taking positive symmetric values and positive real numbers p, p, q ∈ R >0 such that, for the vector field f of system (1), inequalities (9) and (10) hold for all x ∈ R n .

Note that Assumption 1 can be also satisfied after a preliminary state-feedback, similarly to what assumed in [START_REF] Astolfi | Integral action in output feedback for multiinput multi-output nonlinear systems[END_REF]. and corresponds to a (incremental) stabilizability assumption of system (1), which is, as a matter of fact, necessary in the linear context, see [START_REF] Francis | The internal model principle of control theory[END_REF]. For instance, it can be obtained by following the design techniques in [START_REF] Pavlov | Uniform output regulation of nonlinear systems: a convergent dynamics approach[END_REF]Chapter 5], or [START_REF] Andrieu | Some results on exponential synchronization of nonlinear systems[END_REF], [START_REF] Andrieu | LMI conditions for contraction and synchronization[END_REF], [START_REF] Pavlov | Incremental passivity and output regulation[END_REF], [START_REF] Zamani | Backstepping design for incremental stability[END_REF].

Due to Lemma 1, it follows that ẋ = f (x) admits the existence of an equilibrium which we assume to be the origin (without loss of generality). Moreover, such equilibrium is globally asymptotically stable and locally exponentially stable and therefore, we can apply Lemma 2 to establish the existence of a function M solution to [START_REF] Dalto | Incremental quadratic stability[END_REF]. Similarly to the condition (15) stated in Lemma 3, we need a controllabilitylike condition for L g M (x) which, in our incremental framework, is stated as follows.

Assumption 2. Let M be solution to [START_REF] Dalto | Incremental quadratic stability[END_REF]. There exists m ∈ R >0 such that |L g M (x)| ≥ m for all x ∈ R n .

Assumption 2 can be read as a uniform controllability-like assumption for extended dynamics (1), (2) and represents the "incremental version" of condition (15) stated in Lemma 3. In particular, the new condition is assumed to hold for all x and not just for the linearized system around the origin as for [START_REF] Francis | The internal model principle of control theory[END_REF]. Finally, we have the following assumption.

Assumption 3. Let P be given by Assumption 1. Then the vector field g of system (1) is a Killing Vector Field for P , i.e., L g P (x) = 0 for all x ∈ R n . Assumption 3 guarantees that the (Riemaniann) metric induced by P is invariant along g. This means that the system preserves its contractive properties in the directions provided by g, namely, in the directions in which the control law acts. Notice that if g is constant, Assumption 3 is satisfied for every constant metric P .

C. Non-uniform Incremental Stability

In order to solve the problem, taking inspiration from the control law defined in [START_REF] Forni | A differential lyapunov framework for contraction analysis[END_REF], we focus on feedback laws of the form

u = ψ(η, x) := κ L g M (x) -1 β(η -M (x)) ( 16
)
where M is defined in ( 13), κ ∈ R >0 and β is a C 1 increasing function. We have the following result.

Theorem 3. Let Assumptions 1, 2 and 3 hold. Assume that the function M satisfying (13) is C 2 , and that there exist positive real numbers ḡ,

k 0 , k 1 ∈ R >0 satisfying |g(x)| ≤ ḡ , ∂M ∂x (x) ≤ k 0 , ∂L g M ∂x (x) ≤ k 1 , (17)
hold for all x ∈ R n . Finally, let β : R → R be any

C 1 function satisfying 1 |β(s)| ≤ 1 k 1 , 0 < β (s) ≤ 1 , ∀s ∈ R . ( 18
)
Then, there exists a positive real number κ > 0 such that, for any κ ∈ (0, κ ), the closed-loop system (4), [START_REF] Angeli | A lyapunov approach to incremental stability properties[END_REF], with ψ defined as in [START_REF] Heemels | Oblique projected dynamical systems and incremental stability under state constraints[END_REF], admits a C 1 function P : R n+1 → R (n+1)×(n+1) , a positive continuous function µ : R n+1 → R >0 and positive real number p, p ∈ R >0 satisfying (9) and

L ϕ P(X ) ≤ -µ(X )I n+1 ∀ X ∈ R n+1 . ( 19
)
Moreover, Γ is a Killing Vector Field for P.

Proof. See Appendix B

Note that we can't conclude from ( 19) that the closed loop system is IGES. Indeed, the right hand side of (19) may be not uniform. This would be the case if µ(η, x) were lower bounded. However, from the proof, we see that this is not possible since µ depends on β, and, to have µ bounded from below, β should be lower bounded by a positive real number that, in turn, would violate [START_REF] Isidori | Nonlinear Control Systems[END_REF]. Note however, that if there exists a compact set which is invariant along the solutions of the system, then continuity of µ implies that the latter can be uniformly bounded from below and uniform incremental stability holds. 1 For instance, one may select β(s) = 1 2πk 1 arctan(2πk 1 s).

D. Global Incremental Stability

In order to obtain a global contractivity, we need to suppose a stronger version of Assumption 2. In particular, we need, not only the rank of L g M to be constant, but also the function itself, as stated below. Assumption 4. There exists a real number c = 0 such that L g M (x) = c for all x ∈ R n .

Under previous assumption, we have the following result.

∈ R >0 satisfying |g(x)| ≤ ḡ , ∂M ∂x (x) ≤ k 0 . ( 20
)
for all x ∈ R n . Then, for any κ ∈ R >0 , the closed-loop system (4), ( 5) with ψ defined as

u = ψ(η, x) := κ c [η -M (x)], (21)
admits a C 1 function P : R n+1 → R (n+1)×(n+1) taking positive symmetric values and real numbers p, p, q ∈ R >0 satisfying (9), [START_REF] Bin | About robustness of internal model-based control for linear and nonlinear systems[END_REF], for all X ∈ R n+1 . Moreover, Γ is a Killing Vector Field for P.

Proof. See Section C.

IV. INCREMENTAL INTEGRAL ACTION FOR OUTPUT

TRACKING Finally, by using the feedback design proposed in Section III and the framework presented in Section II, we have the following results on output tracking, based on the design proposed in Theorem 3.

Proposition 2. Suppose all assumptions of Theorem 3 hold and consider system (1) in closed-loop with

u = ψ(η, x) , η = h(x) -r , ( 22
)
where ψ is chosen as in Theorem 3. Then, for all constant r ∈ R and all X 0 = (η 0 , x 0 ) in R n+1 such that X (X 0 , t) is bounded in positive time, the output tracking problem is solved, i.e. lim t→∞ y(t) = r for all such initial conditions.

Proof. Applying Theorem 3, picking κ ∈ (0, κ ), it yields the existence of a C 1 function P :

R n+1 → R (n+1)×(n+1) , a positive function µ : R n+1 → R >0 such that (19) holds. Let r in R and X 0 = (η 0 , x 0 ) in R n+1 such that X (X 0 , t) is bounded in positive time. Let C := Cl ∞ t≥0 X (X 0 , t) .
and note that C is a forward invariant compact set (by Birkhoff theorem [18, p. 517]). Let q = min {(η,x)∈O} µ(η, x) > 0 where O is the set defined in [START_REF] Bin | Output regulation by postprocessing internal models for a class of multivariable nonlinear systems[END_REF] for some positive real number satisfying [START_REF] Carroll | Spacetime and geometry: An introduction to general relativity[END_REF]. Keeping in mind that Γ is a Killing Vector Field for P, it yields

L ϕ+rΓ P(X ) = L ϕ P(X ) + rL Γ P(X ) ≤ -qI n+1 .
for all X ∈ O. Hence with Proposition 1, the system is IES in C. With Lemma 1, it implies the existence of a unique equilibrium X = (η * , x * ) ∈ C attracting all solution starting from C. Hence, because of the integral action, it yields h(x * ) = r.

Note that the boundedness requirement on the trajectory implies most of the time a bound for the reference r that can be tracked. The following result shows that the contractivity property always holds provided that r is sufficiently small. This is a local result in the reference signal but global in the state, (i.e. in the initial conditions). In this respect, the results extends the semi-global result obtained in [6, Proposition 3]. Proposition 3. Suppose all assumptions of Theorem 3 hold. Then there exists r ≥ 0 such that for all r such that |r| ≤ r, the feedback [START_REF] Khalil | Universal integral controllers for minimum-phase nonlinear systems[END_REF] given in Proposition 2 with ψ chosen as in Theorem 3, solves the output tracking problem for system (1), i.e. lim t→∞ y(t) = r for any |r| ≤ r and for any initial condition X 0 = (η 0 , x 0 ) in R n+1 .

Proof. Consider the closed-loop system (4), with ϕ, Γ defined as in ( 5), and ψ selected as in [START_REF] Heemels | Oblique projected dynamical systems and incremental stability under state constraints[END_REF]. For r = 0, the origin of ( 4) is an equilibrium, and because of Theorem 3, it is locally exponentially stable. This can be proved by using the linear approximation at the origin and the Lyapunov function V (X ) = X P(0)X , with P(x) satisfying [START_REF] Jankovic | Constructive lyapunov stabilization of nonlinear cascade systems[END_REF]. Hence, by [6, Lemma 5], there exists r > 0 such that, for all |r| ≤ r, system (4), admits an equilibrium X * ∈ R n+1 which is locally exponentially stable, with a domain of attraction N ⊂ R n+1 . Now, because of Theorem 3, system (4) satisfies [START_REF] Jankovic | Constructive lyapunov stabilization of nonlinear cascade systems[END_REF]. In particular, Γ is a Killing Vector and L ϕ P(X ) ≤ 0 for all X ∈ R n+1 . This implies that the (Riemannian) distance of any two trajectories of (4) is not increasing, and, in particular, the distance of any trajectory to X * is not increasing. In other words, the trajectory of ( 4) is bounded for any initial condition X 0 ∈ R n+1 . Hence, the statement of the proof follows by direct application of Proposition 2.

Finally, the last result employs Theorem 4 to obtain a global result. Proposition 4. Suppose all assumption of Theorem 4 holds. Then the control law [START_REF] Khalil | Universal integral controllers for minimum-phase nonlinear systems[END_REF] with ψ chosen in (21) solves the global output tracking problem for system (1), i.e. lim t→∞ y(t) = r for any r ∈ R and for any initial condition

X 0 = (η 0 , x 0 ) in R n+1 .
Proof. This proposition is a direct consequence of Corollary 1 and Theorem 4.

Note that in [START_REF] Astolfi | Integral action in output feedback for multiinput multi-output nonlinear systems[END_REF], [START_REF] Poulain | Robust asymptotic stabilization of nonlinear systems by state feedback[END_REF], the domain of attraction is semiglobal in X and local (i.e. small) in the references r. Differently here, we provide a result that is global (i.e. arbitrarily large) both in term of domain of attraction and reference to be tracked.

V. EXAMPLE Consider system (1) with

f (x) = -2x 1 + 3x 2 + sin(x 1 ) -x 3 1 -2x 2 , g(x) = 1 -1 ,
and h(x) = x 3 1 -sin(x 1 ) + x 2 . First, we can verify that this system verifies Assumption 1 with P = diag(1, 1 2 ), q = 1 2 , p = 1 2 , p = 1. Furthermore, the function M (x) defined in Lemma 2 can be computed as M (x) = -x 1 -2x 2 . Hence, Assumptions 2 and 4 are also verified since L g M (x) = 1. Finally, Assumption 3 is trivially satisfied since P and g are constant. Note also that the relative degree 2 is not well defined, since L g h(x) = 3x 2 1 -cos(x 1 ) + 1 is not constant. Hence, techniques based on normal forms for output tracking (see for instance [START_REF] Khalil | Universal integral controllers for minimum-phase nonlinear systems[END_REF]) doesn't apply in this case. Nevertheless, we can apply results of Theorem 4 and Proposition 4 to achieve global output tracking. Figures 1 and2 show the state x and the output y of the system in closed-loop with the control law [START_REF] Kaliora | Nonlinear control of feedforward systems with bounded signals[END_REF], with κ = 10, when r = 0 and for different initial conditions selected as X (0) = (-1, 5, 3), X (0) = (3, -6, 2), X (0) = (-2, 1, 4), X (0) = (-2.1, 3.7, -1). Figure 3 and4 shows the corresponding trajectories when r = -10. It's also interesting to notice the role of the gain κ. In particular for high-gain, the transient is faster but the error of the output tracking is higher in the transient. On the contrary, for low-gain value the transient is longer but the error of the output during is lower.

VI. CONCLUSIONS

In this paper we have studied the output set-point tracking problem for nonlinear systems by following the classical linear output regulation approach, see [START_REF] Francis | The internal model principle of control theory[END_REF]. First, we extended the system with an integral action processing the regulated output; then we designed a stabilizing feedback law for the extended system by supposing the reference to be zero; finally, we studied sufficient conditions guaranteeing that when the reference is non-zero, the tracking problem is solved. In doing so, we presented a number of new results. First, we addressed the incremental stabilization problem via forwarding by providing two different sets of sufficient conditions and constructive design in order to obtain (non-uniform) incremental exponential stability and global incremental exponential stability. Then, we studied the effect of large tracking references. In particular, we showed that, because of the contractivity properties of the system, if the trajectories of the closed-loop system are bounded in forward time, then necessarily the tracking objective is achieved. Finally, sufficient conditions to solve the global output tracking problem are given.

Future works may include a robust analysis with respect to model perturbations in the same spirit of [START_REF] Astolfi | Integral action in output feedback for multiinput multi-output nonlinear systems[END_REF], [START_REF] Bin | About robustness of internal model-based control for linear and nonlinear systems[END_REF], the design of an output feedback law [START_REF] Astolfi | Integral action in output feedback for multiinput multi-output nonlinear systems[END_REF], and the extension of the proposed technique to the case of periodic references, as done in [START_REF] Astolfi | Approximate regulation for nonlinear systems in presence of periodic disturbances[END_REF], [START_REF] Astolfi | Francis-wonham nonlinear viewpoint in output regulation of minimum phase systems[END_REF] or [START_REF] Pavlov | Incremental passivity and output regulation[END_REF]. Let τ be such that k exp(-λτ ) = γ < 1 and define T τ as the mapping that associates to any initial condition X 0 of system of (6), its corresponding solution X (X 0 , τ ) at time τ , that is T τ (X ) := X (X 0 , τ ). Since C is forward invariant, the function so defined maps points in C into C. Furthermore, by using the inequality (8) guaranteed by the IES property of system (6) on C, for any two given points X 1 , X 2 ∈ C, we have

|T τ (X 1 ) -T τ (X 2 )| ≤ ρ|X 1 -X 2 |
with ρ < 1. Recall that the Euclidean space endowed with standard Euclidean norm is a complete metric. Hence, the map T τ is a contraction and by Banach fixed point theorem (see, e.g., [START_REF] Almezel | Topics in fixed point theory[END_REF]), there exists a unique fixed point x in C such that T τ (x ) = x . Hence, for any X in C, inequality (8), implies,

|X (X , t) -X | = |X (X , t) -X (X , t)| ≤ k exp(-λt) |X -X | .
Hence, X is attracts all trajectories initiated from C. An equivalent proof of this lemma, covering also the case of periodic equilibrium, has been developed independently in [16, Section IV-B].

B. Proof of Theorem 3

In the new coordinates X = (x, z), where z = η -M (x) with M given by Lemma 2, (4) becomes

Ẋ = φ(X ) = f (x) + κg(x)L g M (x) -1 β(z) κβ(z) . (23)
The Jacobian of φ is given by

∂ φ ∂ X (X ) = J 11 J 12 0 J 22 (24)
where

J 11 = ∂f ∂x (x) -κg(x) ∂L g M ∂x (x) β(z) L g M (x) 2 , J 12 = κg(x)
β (z) L g M (x) , J 22 = -κβ (z).

Let P(x) = P(x) > 0 be defined as

P(x) = P (x) 0 0 b , (25)
for some b ∈ R >0 to be defined later. Let also R be the matrix defined as R(X ) = L φ P(X ) +

q 2 I n 0 0 κβ (z) . ( 26
)
We show that R takes only negative definite values provided κ is selected sufficiently small. To do this, note that

R(X ) = L 11 L 12 L 12 L 22 , (27)
where, employing Assumption 3, L 11 = L f P (x) + κ P (x)g(x) ∂L g M ∂x (x) β(z) L g M (x) 2 + κ P (x)g(x)

∂L g M ∂x (x) β(z) L g M (x) 2 + q 2 I n , L 12 = bκP (x)g(x) β (z) L g M (x) , L 22 = -bκβ (z).

Since, L 22 < 0, R is negative definite provided its Schur complement L 11 -

L12L 12 L22
is so. Note that, with Assumption 1, Assumption 2, inequalities [START_REF] Huang | Regulation of nonminimumphase nonlinear systems using slow integrators and high-gain feedback[END_REF] and the bound on β selected in [START_REF] Isidori | Nonlinear Control Systems[END_REF], we have

L 11 ≤ 2κ pḡ m 2 - q 2 I n . (28)
Moreover, with Assumption 2 and the fact that β (z) > 0,

|L 12 | ≤ bκ pḡ m β (z).
It implies

L 11 - L 12 L 12 L 22 ≤ 2κ pḡ m 2 - q 2 I n + bκ p2 ḡ2 m 2 β (z), (29)
namely, with the bound on β , For the last part of the theorem, since P doesn't depend on η, it yields that d Γ P(X ) = 0. Consequently, L Γ P(X ) = 0.

C. Proof of Theorem 4

The proof goes along the same lines of the one of Theorem 3. Indeed, in this case, , κ .

L
Hence, the statement of the theorem follows with p, p as in the proof of Theorem 3 and q = µ defined above. Finally, with the same arguments of the proof of Theorem 3, we have L Γ P(X ) = 0.

Theorem 4 .

 4 Let Assumptions 1, 3 and 4 hold. Assume that the function M satisfying (13) is C 2 and that there exist positive real numbers ḡ, k 0

2

  See [18, Chapter 4.1].

Fig. 1 .

 1 Fig. 1. x-dynamic in stabilization

1 =

 1 Selecting b = 1 and κ * asκ * = qm 2 2pḡ (2 + pḡ) ,it yields R negative definite and consequently, with[START_REF] Marino | Output regulation for unknown stable linear systems[END_REF],L φ P(X ) ≤ -min q 2 , κβ (|z|) I n+1 .(30)Going back into the original coordinates X = (x, η),P (x) + b ∂M ∂x (x) ∂M ∂x (x) -b ∂M ∂x (x) -b ∂M ∂x (x) b ,(31)p = max{p+bk 2 0 , b}, p = min{p+bk 2 0 , b}. Then, inequality[START_REF] Jankovic | Constructive lyapunov stabilization of nonlinear cascade systems[END_REF] holds with such P and the function µ defined asµ(η, x) = 1 (1 + k 0 ) 2 min q 2, κβ (|η -M (x)|) .[START_REF] Bb | Contraction theory based adaptive synchronization of chaotic systems[END_REF]

  11 = L f P (x) + q 2 I n ≤ -q 2 I n regardless the function β and κ, i.e. the control law has infinite gain margin [31, Definition 2.8]. By taking β(s) = s, inequality (29) modifies as

	L 11 -	L 12 L 12 L 22	≤ -	q 2	I n + bκ	p2 ḡ2 m 2 .
	For any given κ, select
		b =	qm 2 4κp 2 ḡ2 .
	This implies that equation (30) becomes
	L φ P(X ) ≤ -min		q 2	, κ I n+1 ,
	and, consequently, (32) becomes
	µ(η, x) = µ =	1 (1 + k 0 ) 2 min	q 2


Shanie A L Jayasinghe
email: shanieal@gmail.com

F R Sarlegna

R A Scheidt

R L Sainburg

The neural foundations of handedness: Insights from a rare case of deafferentation

Keywords: lateralization, proprioception, complementary dominance, motor control

The role of proprioceptive feedback on motor lateralization remains unclear. We asked whether motor lateralization is dependent on proprioceptive feedback by examining a rare case of proprioceptive deafferentation (GL). Motor lateralization is thought to arise from asymmetries in neural organization, particularly at the cortical level. For example, we have previously provided evidence that the left hemisphere mediates optimal motor control that allows execution of smooth and efficient arm trajectories, while the right hemisphere mediates impedance control that can achieve stable and accurate final arm postures. The role of proprioception in both of these processes has previously been demonstrated empirically, bringing into question whether loss of proprioception will disrupt lateralization of motor performance. In this study, we assessed whether the loss of online sensory information produces deficits in integrating specific control contributions from each hemisphere by using a reaching task to examine upper limb kinematics in GL and five age-matched controls. Behavioral findings revealed differential deficits in the control of the left and right hands in GL, and performance deficits in each of GL's hands compared to controls'. Computational simulations can explain the behavioral results as a disruption in the integration of postural and trajectory control mechanisms when no somatosensory information is available. This rare case of proprioceptive deafferentation provides insights into developing a more accurate understanding of handedness that emphasizes the role of proprioception in both predictive and feedback control mechanisms.

Introduction

Neural lateralization refers to the asymmetry in neural circuitry between analogous regions of the two sides of the central nervous system. Lateralization is a fundamental organizing feature across a large spectrum of vertebrate nervous systems. However, while lateralization of language, cognitive, emotional, and perceptual processes have been well-described [START_REF] Gazzaniga | Forty-five years of split-brain research and still going[END_REF], the neural foundations of motor lateralization that give rise to handedness in primates remains controversial [START_REF] Marcori | A historical, systematic review of handedness origins[END_REF]. Here we asked whether motor lateralization is dependent on proprioceptive feedback by examining a rare case of peripheral somatosensory deafferentation.

The paramount role of somatosensory feedback is well-established in current models of motor control, evidenced by severe deficits in movement coordination that result from somatosensory deficits [START_REF] Ghez | Roles of proprioceptive input in the programming of arm trajectories[END_REF][START_REF] Marsden | The use of peripheral feedback in the control of movement[END_REF][START_REF] Sainburg | Loss of proprioception produces deficits in interjoint coordination[END_REF], and by the fact that accuracy of the limb position estimates degrades over time when somatosensation is not available [START_REF] Medina | Contributions of efference copy to limb localization: evidence from deafferentation[END_REF]. In order to examine the role of somatosensory feedback in lateralization and in differentiating between feedforward and feedback components of motor control, it is necessary to examine behaviors in the complete absence of such feedback i.e., peripheral deafferentation. GL is an individual with chronic, permanent deafferentation acquired at age 31. While GL has a neurologically intact brain, which suggests the presence of intact motor control networks and intact interhemispheric connections, her somatic sensation of limb posture and movement are absent due to the loss of large fiber afferent projections bilaterally throughout her trunk and her upper and lower extremities. Studies in GL and a few other rare cases of deafferentation have observed profound deficits of intersegmental coordination and of limb stabilization at the end of reaching movements, suggestive of both poor optimal control of trajectory and poor impedance control of stable final postures [START_REF] Ghez | Proprioceptive control of interjoint coordination[END_REF][START_REF] Gordon | Impairments of reaching movements in patients without proprioception. I. Spatial errors[END_REF][START_REF] Messier | Influence of movement speed on accuracy and coordination of reaching movements to memorized targets in three-dimensional space in a deafferented subject[END_REF][START_REF] Sainburg | Loss of proprioception produces deficits in interjoint coordination[END_REF]). To our knowledge, differences in control strategies employed by each hand in the absence of peripheral feedback have not been determined until now.

Historically, two competing hypotheses of motor lateralization have been proposed to explain differences in how people move their hands. Global dominance hypotheses propose that the dominant hemisphere is superior in all aspects of motor control while the nondominant hemisphere is inferior in all aspects of motor control [START_REF] Annett | The control of movement in the preferred and nonpreferred hands[END_REF][START_REF] Liepmann | Die linke Hemisphäre und das Handeln[END_REF][START_REF] Volkmann | Handedness and asymmetry of hand representation in human motor cortex[END_REF][START_REF] Ziemann | Hemispheric asymmetry of ipsilateral motor cortex activation during unimanual motor tasks: further evidence for motor dominance[END_REF]. In contrast, complementary dominance hypotheses propose that each hemisphere is specialized for specific aspects of motor control [START_REF] Fisk | The effects of unilateral brain damage on visually guided reaching: hemispheric differences in the nature of the deficit[END_REF][START_REF] Haaland | Hemispheric asymmetries for kinematic and positional aspects of reaching[END_REF]). We have provided evidence that the dominant left hemisphere (in right-handed individuals) is specialized or "tuned" to control dynamic aspects of movement (e.g., movement trajectories) while the nondominant right hemisphere is specialized for stabilization of the limb about desired setpoints (i.e., posture control) during reaching tasks [START_REF] Mutha | The effects of brain lateralization on motor control and adaptation[END_REF][START_REF] Sainburg | Handedness: differential specializations for control of trajectory and position[END_REF][START_REF] Wang | The dominant and nondominant arms are specialized for stabilizing different features of task performance[END_REF]Yadav and Sainburg 2014a).

Although manual interaction with people and things requires movement control and postural stabilization in both hands, the complementary dominance hypothesis explains not only advantages in performance of dynamic movement tasks with the dominant hand and stabilizing tasks with the nondominant hand [START_REF] Sainburg | Laterality of basic motor control mechanisms: Different roles of the right and left brain hemispheres[END_REF], but also patterns of sensorimotor deficits after brain injuries, such as stroke (Haaland andHarrington 1989a, 1989b;[START_REF] Schaefer | Hemispheric specialization and functional impact of ipsilesional deficits in movement coordination and accuracy[END_REF].

According to complementary dominance hypotheses, although both hemispheres contribute different aspects of motor control to each arm, it is the contralateral, not ipsilateral, hemisphere that has a greater influence on control of each arm. We previously modelled this combination of hemisphere-specific control as a serial process, with variation in the time that the postural controller is recruited [START_REF] Scheidt | Separate adaptive mechanisms for controlling trajectory and final position in reaching[END_REF]Yadav andSainburg 2011, 2014b). These models suggest that control of the nondominant left hand shows an early shift from optimizing smooth and efficient trajectories to achieving accurate and stable final postures while the dominant right hand shows a later shift in control. However, these studies were conducted under the assumption that the two complementary (trajectory and postural) controllers received accurate somatosensory information that may be used to integrate/combine the output of each controller.

We hypothesized that motor lateralization is dependent on accurate proprioceptive feedback in order to combine control mechanisms from each hemisphere regardless of hemispheric dominance. Secondary to this hypothesis, we made predictions regarding the two popular hypotheses on handedness in a condition of complete somatosensory deafferentation. In the global dominance hypothesis, we predicted that if proprioceptive feedback were absent, the dominant hand would perform poorly in all aspects of movement as would the nondominant hand, but there would be no difference between the two hands in terms of the aspects of movement they performed poorly on. In the complementary dominance hypothesis, we predicted that the dominant hand would perform more poorly in producing accurate and stable final postures, while the nondominant hand would perform more poorly in control of trajectory dynamics. To test our hypothesis and accompanying predictions, we compared unilateral movement of each hand in a rare case of human somatosensory deafferentation (GL) as well as in age-matched controls. Here we describe how GL and a group of neurologically-intact individuals completed a unilateral reaching task on a virtual reality setup (Figure 1). Without visual feedback, control participants rely completely on somatosensation, while GL has no online peripheral feedback about her movements. We then describe how a hybrid computational model of arm movement and posture control efficiently simulates the behavior of GL and the control participants. We conclude by discussing the implications of deafferentation in developing a more accurate understanding of the neural foundations of handedness.

Materials and Methods

Participants

We recruited GL (female; age 70 years), who has been living with large-fiber sensory neuropathy for about 40 years. Biopsies of GL showed lack of large diameter afferent neurons, resulting in the loss of bilateral proprioception and tactile information from the nose down (Forget andLamarre 1987, 1995;[START_REF] Gordon | Impairments of reaching movements in patients without proprioception. I. Spatial errors[END_REF]. Nerve stimulation studies normally showed a fast and slow peak in the compound action potential; however, the fast peak did not occur in GL, indicating a loss of large diameter sensory fibers [START_REF] Forget | Rapid elbow flexion in the absence of proprioceptive and cutaneous feedback[END_REF]. GL has been confined to a wheelchair since the neuropathy onset at age 31, and performs activities of daily living (e.g., toileting, self-grooming, dressing, eating, showering) without assistance, but under constant visual guidance. She has shown no sign of recovery since the presumed viral attack about 40 years ago [START_REF] Cole | Living without touch and peripheral information about body position and movement: Studies with deafferented subjects[END_REF][START_REF] Sarlegna | Force-field adaptation without proprioception: can vision be used to model limb dynamics?[END_REF]. Prior to the acquired deafferentation, GL self-reported to have been right-hand dominant. At the time of testing, her laterality quotient was 80%, according to the 10-item version of the Edinburgh Handedness Inventory [START_REF] Oldfield | The assessment and analysis of handedness: the Edinburgh inventory[END_REF]. We also recruited 5 neurologically intact age-matched right-hand dominant individuals (4 females; mean age 67 years ± 2.91 SEM) for comparison purposes.

Exclusion criteria for the neurologically intact participants included any current or prior upper extremity motor disorder or stroke, and any cognitive disorders that would make it difficult to follow simple instructions. The Edinburgh Handedness Inventory was used to confirm hand dominance in controls, with all five participants having a 100% laterality quotient. We received written informed consent from all participants prior to study initiation (the consent form and all study procedures were translated word-for-word from English to French for GL by a native French speaker). All study procedures were approved by the Pennsylvania State University's Institutional Review Board.

Experimental setup

We used the KineReach virtual reality motion tracking system to implement all tasks (Figure 1A). Participants were seated on a height-adjustable chair in front of the system, with all arm joints distal to the elbow immobilized using an adjustable brace. Limb position and orientation of each arm were recorded at 116 Hz using two 6-degree-of-freedom magnetic sensors (trakStar ® ; Ascension Technology) placed on the hand and upper arm. We computed 10degree-of-freedom arm movements from digitized bony landmarks that allowed estimation of the wrist, elbow, and shoulder joints. Each hand was supported on an air sled that produced continuous pressurized air to reduce the mechanical effects of friction and gravity on arm movement. An inverted HD monitor displayed the task on a mirrorized screen placed at chin level, and this projection appeared in the same horizontal plane as the hand. The screen occluded direct view of the hands. A cursor represented the position of the index finger in real-time, a circle (diameter 4 cm) appeared at the beginning of each trial to signify the start position, and another circle (diameter 3.5 cm) appeared 20 cm from the start position to signify the target. A trial lasted 1 s, and consisted of a target appearing in one of three locations (90 º from start, 35 º clockwise from start, 35 º counter-clockwise from start) in a pseudorandom manner, followed by a reaching movement.

A B

Task

All participants completed a unilateral reaching task with each of the left and right hands (90 trials per hand i.e., 30 trials per target). Participants were asked to move the cursor from the start circle to the target within 1 s (Figure 1B). The target remained visible until the end of the trial. Visual feedback of the cursor was removed the moment it moved out of the start position, and visual feedback of the movement trajectory was provided at the end of each trial.

Participants were asked to make one rapid movement into the target, without any corrective movements, and to stop in the target. After a 2 s delay, the next target was displayed, and the participants were then provided cursor feedback to allow repositioning into the start position.

There was no set time between trials, with a trial beginning only when the cursor returned into the start circle, and remained there for 300 ms.

Data analysis

All kinematic data were processed and analyzed using custom programs designed on IgorPro (version 6.37; WaveMetrics). Movement onset was defined as the first minimum of tangential velocity that was less than 8% of peak velocity. Movement offset was considered to be at the first minimum of tangential velocity appearing after peak velocity that was less than 8% of peak velocity. Substantial movements produced after the initial movement were examined and classified as either drift or oscillations depending on the behavior. Error at the end of movement was measured as the distance between target center and cursor position at the end of the trial.

Initial direction error was determined to be the angle between two line segments: the line connecting the start circle and the cursor's position at peak velocity, and the line connecting the target and start circle.

We compared behaviors in each hand of GL to the corresponding hand in the control group (n=5) on each of the two measures of performance (initial direction error and error at the end of movement) using the Crawford & Howell modified t test (Crawford and Garthwaite 2012;

Crawford and Howell 1998) that has been widely used in single case-controls study designs such as ours [START_REF] Čeko | Anatomical and functional enhancements of the insula after loss of large primary somatosensory fibers[END_REF][START_REF] Cuadra | Case Studies in Neuroscience: The central and somatosensory contributions to finger interdependence and coordination: lessons from a study of a "deafferented person[END_REF][START_REF] Lafargue | Production and perception of grip force without proprioception: is there a sense of effort in deafferented subjects?[END_REF]. The modified t test calculates the t statistic using the following equation:

𝑡 = 𝑥 -𝑥 𝑠√(𝑛 + 1) 𝑛 ⁄
where 𝑥 represents GL's mean, 𝑥̅ and 𝑠 are the mean and standard deviation of the control sample, and 𝑛 is the size of the control sample. This test is designed to be robust against small control sample sizes, departures from normality, and very severe skew in the dataset, which is expected in study designs such as ours [START_REF] Crawford | Testing for a deficit in single-case studies: Effects of departures from normality[END_REF][START_REF] Crawford | Single-case research in neuropsychology: a comparison of five forms of t-test for comparing a case to controls[END_REF].

We also calculated effect sizes using zcc, which is an analog of Cohen's d that is suitable for single case-control study designs since it uses the control sample's standard deviation instead of the pooled standard deviation and is also insensitive to the size of the control sample [START_REF] Crawford | Point and interval estimates of effect sizes for the case-controls design in neuropsychology: rationale, methods, implementations, and proposed reporting standards[END_REF].

To assess the extent of within-subject differences i.e., between the left and right hand on each dependent variable, we used the revised standardized difference test (RSDT), which has been shown to be useful in comparing a single case to a small control sample [START_REF] Crawford | Point and interval estimates of effect sizes for the case-controls design in neuropsychology: rationale, methods, implementations, and proposed reporting standards[END_REF][START_REF] Crawford | Evaluation of criteria for classical dissociations in single-case studies by Monte Carlo simulation[END_REF]. This method uses Bayesian methods to standardize the difference in performance on two tasks in a case (i.e., GL) to that in a control sample, thus allowing us to judge whether significant performance differences between the left and right hand in GL are actually unusual or whether such differences are common even in controls. This also reduces the likelihood of making Type I errors [START_REF] Crawford | Point and interval estimates of effect sizes for the case-controls design in neuropsychology: rationale, methods, implementations, and proposed reporting standards[END_REF]). We calculated effect sizes for these comparisons using a different z score, zDCC, which computes the difference between the scores on the two tasks in the case (i.e., GL) by standardizing to the control sample's means and standard deviations on the two tasks and by determining the correlation between these tasks in the control sample [START_REF] Crawford | Point and interval estimates of effect sizes for the case-controls design in neuropsychology: rationale, methods, implementations, and proposed reporting standards[END_REF].

We used the Singlims_ES.exe and RSDT_ES.exe programs to implement these modified t tests and to calculate effect sizes [START_REF] Crawford | Point and interval estimates of effect sizes for the case-controls design in neuropsychology: rationale, methods, implementations, and proposed reporting standards[END_REF]). The family-wise error rate was alpha = 0.05, and we used the Bonferroni correction to set the individual test p-threshold for each family (three comparisons per dependent variable) to 0.0167. Therefore, a comparison was deemed "statistically significant" when the null hypothesis was rejected by a p-threshold of 0.0167.

Simulations

We used a two-segment model of the upper limb to perform forward dynamic simulations of goal-directed reaching movements. The limb was subject to control actions, provided by a trajectory and postural controller [START_REF] Scheidt | Separate adaptive mechanisms for controlling trajectory and final position in reaching[END_REF]. Segment lengths and centers of mass were determined using Winter's equations [START_REF] Winter | Biomechanics and motor control of human movement[END_REF]). We varied three parameters in this model: 1) joint stiffness, 2) joint damping, and 3) time of switch from trajectory to postural control. For simulating pure forms of control i.e., when the controller essentially acts alone without input from the other controller as well as typical human movement, we set joint stiffness to 2 N-m/rad. We set joint damping to 0.15 N-m-s/rad to simulate the pure postural controller in GL and 10 N-m-s/rad to simulate typical human movement. Previous studies have provided a range of values for these two parameters [START_REF] Bennett | Time-varying stiffness of human elbow joint during cyclic voluntary movement[END_REF][START_REF] Milner | Damping of the wrist joint during voluntary movement[END_REF][START_REF] Shadmehr | Adaptive representation of dynamics during learning of a motor task[END_REF]Yadav and Sainburg 2014b). We chose stiffness and damping values that were within the published range. When simulating GL's movements, we defined "high" and "low" joint stiffness as 2 N-m/rad and 0.02 N-m/rad, respectively. The corresponding joint damping was set to 0.15 N-m-s/rad and 0.0015 N-m-s/rad, respectively. We estimated these stiffness and damping values by fitting a second-order mechanical model to GL's nondominant hand's underdamped response to a step change in desired hand position. Joint damping in a deafferented individual is small but non-zero because even when the threshold for stretch reflex cannot be achieved, fractional damping effects that are inherent to muscle mechanical properties are still present [START_REF] Houk | Fractional power damping model of joint motion[END_REF]. These stiffness and damping values were consistent with antagonist muscle coactivation in the absence of large-fiber afferent feedback, and included contributions from passive muscle and connective tissue resistance, without reflex-mediated input [START_REF] Bizzi | Posture control and trajectory formation during arm movement[END_REF]. The pure trajectory controller was controlled solely by feedforward torques at the elbow and shoulder joints, while the pure postural controller involved a nearinstantaneous jump in desired limb posture. The mixing of two controllers required manipulating the switch time parameter. Here, an "early" switch in control was defined as a switch time occurring at 25% of the ideal movement's duration, thus, favoring postural stabilization. A "late" switch in control was defined as a switch time occurring at 70% of the ideal movement duration, favoring online trajectory control over postural stabilization. An assumption was made that peak velocity would be achieved at 50% of the movement trajectory. We switched to postural control at peak velocity (i.e., at 50% of the movement trajectory) or post peak velocity (i.e., at 70% of the movement trajectory) to simulate typical left and right hand behavior, respectively [START_REF] Sainburg | Interlimb differences in control of movement extent[END_REF].

Results

Without somatosensory feedback, the dominant right hand drifts and the nondominant left hand oscillates at movement's end

Figure 2 shows representative hand paths and velocity profiles from (A) a control participant and (B) a patient (GL) deprived of limb somatosensory input. Control participants made fairly straight and accurate movements toward each target, with few terminal oscillations, regardless of movement direction and hand. By contrast, GL's movements were substantially curved and showed large errors in both initial direction and at the end of movement. The most striking finding was the substantial differences between the kinematic behavior of the two hands at the end of the initial movement in GL but not in controls. While GL's left hand was far less accurate in its initial movement direction, movement termination did occur close to the target despite substantial oscillations (note the multiple peaks in her velocity profile in the left-most plot in Figure 2B). Movements made with GL's right hand showed somewhat better initial direction accuracy, but they failed to stabilize the hand near the target and instead exhibited substantial terminal drift. Crawford & Howell modified t test; statistical significance: p < 0.001 (*).

Poor accuracy at the end of movement in the dominant hand and poor trajectory control in the non-dominant hand of the deafferented patient

Figure 2 (bottom panel) shows that differences in performance observed between the hands were consistent, regardless of target location, in the deafferented patient. In Figure 2C, the mean initial direction error in each hand is shown for controls as well as for GL. The plot reveals the striking performance difference between GL's right and left hand paths compared to controls (RSDT: t(4) = 4.92, p = 0.0079, two-tailed; zDCC = 9.08); initial direction errors were higher with the nondominant left hand than with the dominant right hand. Additionally, we found marked differences between the left hand of control participants and the left hand of GL (modified t test: t(4) = 14.62, p < 0.001, two-tailed; zcc = 16.01), as well as between the right hand of controls and the right hand of GL (modified t test: t(4) = 9.35, p < 0.001, two-tailed; zcc = 10.24). These results provide evidence of suboptimal control of trajectory in the deafferented patient compared to controls, as well as significantly higher directional errors in left vs. right hand in the deafferented patient.

Figure 2D shows the mean error at the end of movement in each hand for control participants and for the deafferented patient. Here again, we found a marked difference between GL's hands in error at the end of movement compared to controls (RSDT: t(4) = 4.37, p = 0.012, two-tailed; zDCC = -5.86). While there were marked differences between the right hand of the deafferented patient and the right hand of controls (modified t test: t(4) = 9.07, p < 0.001, twotailed; zcc = 9.94), we observed no statistically significant differences between groups in the left hand with regards to average hand position error at the end of movement (modified t test: t(4) = 0.98, p = 0.4, two-tailed; zcc = 1.07). We next used a set of forward dynamic simulations to explore the potential neural basis of these limb-specific performance deficits.

Somatosensory deficits result in two poorly-tuned controllers that do not accurately modulate limb stiffness

We used a torque-activated two-segment model of the arm [START_REF] Scheidt | Separate adaptive mechanisms for controlling trajectory and final position in reaching[END_REF] to simulate reaching trajectories to the middle target in our experiments. In this simulation, two open-loop controllers were simulated: 1) a trajectory controller specified a hand trajectory optimized for straightness and smoothness; and 2) a postural controller specified positional and velocity-dependent impedance about a specified end position. We modeled the control process as a serially-engaged, dual control system that initiated movement with the trajectory controller, and terminated movement with the postural controller. We sought to determine the extent to which limb-specific deficits of initial direction accuracy and the presence of terminal oscillations and drift could result from the interaction between three parametersthe time at which the postural controller is recruited, joint stiffness values dependent on the presence or absence of muscle coactivation, and joint damping values dependent on the presence or absence of reflexes.

Figure 3 shows simulated right-hand hand paths and velocity profiles from six simulated conditions. Figure 3A shows a trajectory generated by an ideal, isolated optimal trajectory controller that has perfect knowledge of limb dynamics. Here, feedforward torques drive the hand in a straight trajectory with a bell-shaped velocity profile from perfect start to perfect end.

Since there is no postural control, movement and posture are determined solely by the feedforward torque profile at each of the two joints, which are planned prior to the start of movement. By contrast, Figure 3B depicts motions that would result from instantiating an isolated postural controller when the desired limb set-point (the set of joint angles that determine hand position) jumps from the start location to the goal. Energy from the step change in postural set-point dissipates in the form of oscillations around the desired final position. Large directional errors arise due to the inertial anisotropy of the arm [START_REF] Gordon | Accuracy of planar reaching movements. II. Systematic extent errors resulting from inertial anisotropy[END_REF] and the absence of an optimal trajectory controller concerned with the accuracy of the initial movement direction. Both of these isolated controllers likely provide unrealistic depictions of human movement, and we expect that a more realistic description of observed behaviors in both GL and in controls may result from a mixing of the two controllers. Movements are generally straight, similar to those of our control participants' (see Figure 2A).

Discussion

We explored the role that proprioceptive feedback plays in motor lateralization, and the neural foundations of handedness by examining unilateral reaching movements performed by GL, a patient with a rare case of chronic, permanent deafferentation. In the absence of concurrent visual feedback, deafferented patients have previously demonstrated deficits in the temporal coordination of sequential control actions [START_REF] Sainburg | Control of limb dynamics in normal subjects and patients without proprioception[END_REF], and in the maintenance of stable limb postures [START_REF] Rothwell | Manual motor performance in a deafferented man[END_REF][START_REF] Sanes | Motor deficits in patients with large-fiber sensory neuropathy[END_REF]. We reasoned that patterns of reach kinematics in each arm of a deafferented individual might provide insights into the role of proprioceptive feedback on the hemispheric coordination of limb movements and final stabilized postures. The most intriguing finding was the marked asymmetry in reach performance between GL's hands. Previous work in deafferented patients observed drift-like behavior at the end of single-limb reaching movements [START_REF] Rothwell | Manual motor performance in a deafferented man[END_REF][START_REF] Sanes | Motor control in humans with large-fiber sensory neuropathy[END_REF][START_REF] Sarlegna | Internally driven control of reaching movements: a study on a proprioceptively deafferented subject[END_REF]), but the issue of motor lateralization has never before been addressed in deafferentation.

In the absence of concurrent visual feedback of endpoint motion, the absence of somatosensory feedback resulted in each hand exhibiting seemingly isolated forms of either optimal trajectory control or postural control. We had previously shown, using a serial hybrid computational model of movement, that there were differences in controller switching times between the two hands in neurologically intact individuals, with the nondominant left hand switching from optimizing control of trajectory to achieving stable final postures before reaching peak velocity, and the dominant right hand switching after reaching peak velocity [START_REF] Sainburg | Interlimb differences in control of movement extent[END_REF]Yadav andSainburg 2011, 2014b). Therefore, our previous work showed that there is an unequal contribution of each controller to each limb, with the strength of contribution to the ipsilateral limb being substantially lower than that to the contralateral limb. Perturbation to the ipsilateral controller i.e., removing online peripheral input via deafferentation would, therefore, be expected to make the limb that receives less direct input from that controller, be more affected. The computer simulations suggest that the switch to postural control was late and weak in GL's right hand, resulting in an inability to stabilize at the endpoint, and instead producing the observed slow drift-like movement towards the last known set point (the start position). In contrast, GL's left hand switched to postural control very early in movement, resulting in poor online corrections of the trajectory. Therefore, the mixing of two poorly-tuned controllers can be used to describe motor behavior in the absence of somatosensory feedback.

These results provide further support for a complementary rather than a global dominance hypothesis of handedness since there was a clear difference between hands in the aspects of movement each performed more poorly on than the other.

This serial hybrid computational model of movement fills a gap in the motor control literature by explaining motor control as the coordination of two distinct control strategies interacting through limb dynamics [START_REF] Scheidt | Patterns of hypermetria and terminal cocontraction during point-to-point movements demonstrate independent action of trajectory and postural controllers[END_REF]. Current hypotheses of motor control have not succeeded in describing these differences in control between hemispheres. The equilibrium point hypothesis, a popular model of motor control, relies heavily on spinal feedback and specifies limb position via the modulation of a unique motor neuron threshold [START_REF] Asatryan | Biophysics of complex systems and mathematical models. Functional tuning of nervous system with control of movement or maintenance of a steady posture. I. Mechanographic analysis of the work of the joint on execution of a postural task[END_REF]. This hypothesis integrates both optimal trajectory and postural control mechanisms into a single mechanism, thus, not allowing for a dissociation of the two forms of control [START_REF] Sainburg | Should the Equilibrium Point Hypothesis (EPH) be considered a scientific theory?[END_REF]. Optimal feedback control is another model that has gained prominence in describing motor control as a process that involves the use of forward models to estimate limb state, and emphasizing the importance of sensory feedback [START_REF] Shadmehr | From equilibrium point to optimal control[END_REF][START_REF] Todorov | Optimal feedback control as a theory of motor coordination[END_REF]. However this approach has yet to consider the distinction between optimal trajectory and postural control mechanisms. We do not intend to conduct a lengthy review and analysis of currently-available models of motor control; however, we emphasize the current lack of a model to adequately describe laterality in motor control. Our model predicts that the activation of postural feedback mechanisms occurs via the modulation of three parametersthe time at which the postural controller is recruited, joint stiffness, and joint damping. A movement can be initiated in the absence of proprioceptive feedback, but online adjustments cannot be made without accurate estimates of the limb position and velocity.

Although our computational model of lateralized motor control provides an explanation of our empirical findings, we acknowledge that alternative mechanisms may also explain this behavior. Our model cannot, at this time, determine whether the pair of controllers (optimal trajectory and postural control) exists together in both hemispheres (thereby instantiating a superior vs. inferior form of one controller in each hemisphere) or whether each hemisphere only instantiates one controller. However, previous work in unilateral stroke patients has indicated the loss of complementary control features; left hemisphere damage produces deficits in trajectory control and right hemisphere damage produces deficits in final position control, thus, supporting the localization of each controller to each hemisphere [START_REF] Mutha | The effects of brain lateralization on motor control and adaptation[END_REF][START_REF] Schaefer | Hemispheric specialization and functional impact of ipsilesional deficits in movement coordination and accuracy[END_REF].

Another important area for future work on this model is to verify the differences in cocontraction levels between the arms predicted by our model, as well as changes in joint stiffness and damping during movement. Previous studies have examined differences in electromyographic activity between deafferented individuals and those with intact sensory afferents [START_REF] Forget | Rapid elbow flexion in the absence of proprioceptive and cutaneous feedback[END_REF][START_REF] Rothwell | Manual motor performance in a deafferented man[END_REF]; however, differences in coactivation levels between arms have not been described. We also acknowledge that the conclusions reached in this study were derived from reaching tasks and may not necessarily be valid in other types of motor tasks (for example, see [START_REF] Aoki | Handedness and index finger movements performed on a small touchscreen[END_REF]. Additionally, the results of this study were based on examining reaching in right-handed individuals and we cannot make predictions of behaviors in left-handed individuals. Left-handed individuals exhibit similar patterns of lateralization as right-handed individuals, but to a reduced extent, and there is also a large amount of variability even within their population [START_REF] Przybyla | Dynamic dominance varies with handedness: reduced interlimb asymmetries in left-handers[END_REF][START_REF] Sainburg | Laterality of basic motor control mechanisms: Different roles of the right and left brain hemispheres[END_REF].

To conclude, this rare case of deafferentation provided us with an opportunity to observe and report differences in motor behavior exhibited by each hand. We used a computational model to associate behavioral differences found in the empirical data with hypothesized differences in control mechanisms between the hemispheres. As reflected by our results, proprioceptive feedback plays a large role in mixing control mechanisms from each hemisphere in the production of human movement. This work provides key insights in developing a more accurate understanding of handedness that incorporates the vital role of proprioceptive feedback in controlling arm movement and final posture.

Figure 1 .

 1 Figure 1. Experimental setup. (A) An illustration of the KineReach setup showing a participant seated in front of a table with an inverted HD monitor that displayed the task on a mirrorized screen placed at chin level; this screen occluded direct view of the hands. Sensors placed on the hand and upper arm collected data used to compute orientation and location of the wrist, elbow, and shoulder joints. Each hand was supported on an air sled that produced continuous pressurized air to reduce the mechanical effects of friction and gravity. (B) Schematic of the unilateral reaching task showing movement of a cursor (representing the position of the hand) to the middle target (1 target presented per trial). Participants were asked to move the cursor to the target (appearing in 1 of 3 directions in a pseudorandom manner) within 1 s. The task consisted of 90 trials performed with each of the left and right hands separately. Visual feedback of the cursor was removed upon moving beyond the start circle.

Figure 2 .

 2 Figure 2. Different behaviors exhibited by each hand in the absence of somatosensation. Substantial performance differences between hands are evident in the deafferented patient's (GL) movements but not in those of neurologically-intact control participants. Hand paths and velocity profiles are shown in (A) a representative participant in the control group, and (B) in GL. Scale bars represent 2 cm hand movement. (C) Mean initial direction error and (D) mean error at movement's end is displayed for each hand of control participants and GL (female). Error bars in control data represent 1 standard deviation from the mean (n = 5, 4 females). Mean values for each control participant are plotted as purple squares (left hand) or yellow triangles (right hand).

Figure 3 .

 3 Figure 3. Serial activation of a dual-controller system describes differences in movement. Simulations of reaching movements show that two poorly-tuned controllers can predict GL's behavior. Hand paths and velocity profiles (insets) are shown for simulations of right arm movements involving (A) pure trajectory control, (B) pure postural control in a system with low joint damping, (C) activation of the postural controller at peak velocity and (E) late activation of postural control in a system with typical joint stiffness and damping values, (D) early and (F) late activation of postural control in a system with joint stiffness and damping values similar to GL's. Scale bars for trajectories represent 2 cm simulated hand movement. Scale bars for velocity profiles represent 0.5 m/s.

Figures

  Figures 3D and 3F represent systems with lower joint stiffness and damping values that are

Table Mirror HD

 Mirror

	TV
	Sensor
	Air
	sled

Acknowledgements

We would like to thank GL and the other participants for their involvement in this study. We thank Jake Schaffer for help with data collection. This work was supported by the National Institutes of Health (R01HD059783 awarded to R.L.S.) and the National Center for Scientific Research-France (PICS grant awarded to F.S.).

Disclosures

The authors declare no conflicts of interest.