Lorenzo Rosa
email: rosa@diag.uniroma1.it

Tarek Hamel ??

Robert Mahony

Claude Samson

Optical-Flow Based Strategies for Landing VTOL UAVs in Cluttered Environments

Keywords: Aerial Robotics, UAV, VTOL, visual servo control, optical flow, landing

This paper considers the question of landing an Unmanned Aerial Vehicles (UAV) using a single monocular camera as the primary exteroreceptive sensing modality. The proposed control law is based on tracking a single point feature, representing the desired landing point on a ground plane, along with optical flow computed over the full image. The bearing of the desired landing point is used as a driving term to force convergence, while the optical flow is used to provide a damping force that guarantees both obstacle avoidance as well as damping the convergence of the vehicle to the ground plane ensuring a soft touchdown. A detailed analysis of the system closed-loop dynamics is undertaken and the response of the system is verified in simulation.

INTRODUCTION

Autonomous aerial vehicles (UAVs) are a transformative technology in modern society, providing unparalleled capability to undertake difficult, dangerous and dull surveillance and monitoring tasks crucial to maintenance of infrastructure, agricultural operations and disaster recovery, to name just a handful of application domains. To perform a fully autonomous flight, an UAV requires the ability to take-off, navigate to, and accomplish a given task, and finally return to base and land. Many tasks require the vehicle to land, or at least approach and touch, its destination, and complex or long-lasting missions might require intermediate landing and taking-off operations (e.g. battery/fuel refill). Such manoeuvres generally need to be undertaken in unsurveyed, cluttered environments, and often where GPS signals are unreliable or unavailable. This is especially the case for Vertical Take-Off and Landing (VTOL) vehicles, such as helicopters and multirotor systems, that are likely to be used in tandem with ground based vehicles and other technologies, as in [START_REF] Michael | Collaborative mapping of an earthquakedamaged building via ground and aerial robots[END_REF], in extended disaster recovery or other types of missions. Classical landing control strategies require estimation of the full state of the vehicle, along with a model of the landing environment (see for example [START_REF] Flores | Trajectory generation for differentially flat systems via nurbs basis functions with obstacle avoidance[END_REF]; [START_REF] Geyer | 3d obstacle avoidance in adversarial environments for unmanned aerial vehicles[END_REF]; [START_REF] Meister | Adaptive path planning for vtol-uavs[END_REF]; [START_REF] Andert | Mapping and path planning in complex envi-ronments: An obstacle avoidance approach for an unmanned helicopte[END_REF]). If only a camera is available for state estimation then the vehicle pose must be estimated by using visual data [START_REF] Lee | Autonomous landing of a vtol uav on a moving platform using imagebased visual servoing[END_REF][START_REF] Davison | Monoslam: Real-time single camera slam[END_REF][START_REF] Kaiser | Vision-based estimation for guidance, navigation, and control of an aerial vehicle[END_REF][START_REF] Courbon | Vision-based navigation of unmanned aerial vehicles[END_REF][START_REF] Milford | Aerial slam with a single camera using visual expectation[END_REF][START_REF] Blösch | Vision based mav navigation in unknown and unstructured environments[END_REF] An alternative approach is to use the natural properties of optical flow to provide a strong visual cue for obstacle avoidance and landing regulation [START_REF] Srinivasan | How honeybees make grazing landings on flat surfaces[END_REF][START_REF] Koenderink | Facts on optic flow[END_REF][START_REF] Hérissé | Hovering flight and vertical landing control of a vtol unmanned aerial vehicle using optical flow[END_REF][START_REF] Chahl | Landing strategies in honeybees and applications to uninhabited airborne vehicles[END_REF][START_REF] Srinivasan | How honeybees make grazing landings on flat surfaces[END_REF][START_REF] Ruffier | Visually guided micro-aerial vehicle: automatic take off, terrain following, landing and wind reaction[END_REF]. Optical flow is also a powerful cue for terrain following [START_REF] Humbert | Pitch altitude control and terrain following based on bioinspired visuomotor convergence[END_REF][START_REF] Ruffier | Optic flow regulation: the key to aircraft automatic guidance[END_REF] and obstacle avoidance [START_REF] Geyer | 3d obstacle avoidance in adversarial environments for unmanned aerial vehicles[END_REF]; [START_REF] Green | Optic flow based collision avoidance[END_REF]; [START_REF] Beyeler | Vision-based control of near-obstacle flight[END_REF]. A good review of prior work in robotics applications that exploit optical flow is given by [START_REF] Mccarthy | A unified strategy for for landing and docking using spherical flow field divergence[END_REF]. A key advantage of control strategies that are based on optical flow is that they also also deal well with dynamic environments such as a moving landing surface [START_REF] Hérissé | The landing problem of a vtol unmanned aerial vehicle on a moving platform using optical flow[END_REF][START_REF] Hérissé | Landing a vtol unmanned aerial vehicle on a moving platform using optical flow[END_REF] in contrast to approaches that require modeling of the environmental motion [START_REF] Marconi | Autonomous vertical landing on an oscillating platform: an internal-model based approach[END_REF]. A disadvantage of control algorithms based on optical flow and visual data in general is the complexity of providing rigourous stability analysis of the closed loop system, especially in the case where the system has non-trivial dynamics, as is the case with flying vehicles.

In this paper, we propose a landing control algorithm for a VTOL UAV based on visual data obtained from a monocular camera coupled with the usual suite of sensors present in a standard Inertial Measurement Unit (IMU) that is common on all modern UAVs. We provide a detailed analysis of the closed-loop response of the system and prove asymptotic stability of the pose to the desired landing point, corresponding to a smooth touch down. We first consider the case where there are no obstacles and propose a reformulation of the controller formerly developed by [START_REF] Hérissé | The landing problem of a vtol unmanned aerial vehicle on a moving platform using optical flow[END_REF][START_REF] Hérissé | Landing a vtol unmanned aerial vehicle on a moving platform using optical flow[END_REF] to include a target landing point. The proposed control consists of a driving term that acts as a constant force driving the vehicle in the direction of the observed target point, based on the bearing measurement of the target, coupled with a damping proportional to optical flow measured over the whole ground plane. The natural tradeoff between velocity and distance inherent in optical flow ensures a stable soft touchdown of the closed-loop system. We then consider obstacles present in the flying space by including additional damping terms associated with divergence of the optical flow derived from the obstacle. To simplify the mathematical formulation of the problem, the ground plane is assumed to be flat and obstacles are modeled as spheres in space. The goal of this paper is to provide a detailed theoretic analysis of the system response rather than tackle some of the practical issues of implementation. To this end the visual target observation and optical flow measurements are abstracted and written in terms of the vehicle state for the purposes of the analysis. Simulations of the closed-loop dynamics are provided to demonstrate the performance of the proposed algorithm.

The content is organized as follows: Section 2 describes the landing task; the differential equations representing the motion model for small-scale VTOLs and the modelling of the environment. Section 3 proposes a control law for the landing task in obstacle-free environments and provides a detailed stability analysis. Section 4 considers the case where there are obstacles in space and provides an analysis for the proposed control. Section 5 reports on some simulation results for both obstacle-free and cluttered environments. The paper concludes with some final comments in Section 6.

SYSTEM AND ENVIRONMENT MODELLING

In this section we introduce the model used throughout the document to describe the vehicle dynamics and the environment.

Consider a Vertical-Take-Off-and-Landing (VTOL) UAV vehicle equipped with an Inertial Measurement Unit (IMU) and suitable filtering algorithms to provide reliable estimates of attitude and rotational velocities, as well as a camera, an exteroceptive passive sensor whose output is rich in information. We assume that the vehicle is equipped with vision processing capability to identify a point-feature target and compute optical flow over the full image [START_REF] Horn | Determining optical flow[END_REF].

A complete model of a VTOL aircraft includes gyroscopic and aerodynamics effects, as well as aerodynamic disturbances. In order to simplify the analysis, we neglect all second order terms [START_REF] Mahony | Robust trajectory tracking for a scale model autonomous helicopter[END_REF]. The neglected aerodynamics forces are dissipative and do not significantly effect the closed-loop response. Rotational dynamics of small-size VTOL vehicles are usually faster than translational ones, leading to a hierarchical control design methodology. The natural time-scale separation between the translational dynamics (slow time-scale) and the orientation dynamics (fast time-scale) allows one to design decoupled position and orientation controllers [START_REF] Khalil | Nonlinear Systems[END_REF][START_REF] Hérissé | Landing a vtol unmanned aerial vehicle on a moving platform using optical flow[END_REF]. The high-gain attitude control effectively dominates the attitude dynamics of the vehicle and allow one to consider a pure translational model

ξ = v m v = mge 3 -T Re 3 , (1)
in which ξ and v represent the UAV position and velocity respectively, both expressed in an inertial frame with the third direction aligned with the gravitational field, m represents the mass of the VTOL-UAV (supposed constant), g is the magnitude of the gravity force acting on the vehicle and e 3 is the third canonical basis vector. The variable T represents the thrust magnitude and R is the rotation matrix representing the orientation of the body-fixed frame with respect to the inertial frame. The high-gain attitude control assumption means that the rotation R can be viewed as a control input and we write u = -T Re 3 to represent the resulting control input to the translational dynamics of the system ξ = v m v = mge 3 + u .

(2)

In order to develop the dynamics equation of image points, in the following we suppose that the frame representing the position and orientation of the camera in the world frame is rigidly attached to the body-fixed frame, such that the linear transformation relating them degenerates to the identity matrix and their dynamics are the same. Projections of world points on the image surface are points, usually called point features or features, whose dynamics depend on the geometry of the image surface and on the physical parameters of the camera. A common choice in the aerial robotics field is to model the dynamics of the point features by using the spherical projection model for a calibrated camera as shown by [START_REF] Ma | An Invitation to 3-D vision[END_REF]. Consider a reference frame attached to the camera. If P is a generic point in the 3-D space, we denote with P the vector connecting the centre of the camera reference frame to the point P and with p the unit vector representing the direction of P , i.e. its projection on a sphere of unit radius, that represents the spherical image surface. The expression of the vector p and its derivative are

p = P |P | , ṗ = - π p |P | v , (3)
where1 π p = (I 3 -pp ) (4) is the projector in the space tangent to the spherical image surface at point p, having the following properties

p π p = π p p = 0 , (5)
π p π p = π p . (6)
We will assume that the desired landing point PL is contained locally in a planar surface whose normal η is known or can be estimated from image features (see, for instance, [START_REF] Ma | An Invitation to 3-D vision[END_REF]). In general, the normal direction cannot be easily determined unless a good knowledge of the environment is available. There are, however, certain special cases where it is relatively straightforward to extract it, such as the situation where the target lies on a flat ground. In this case η is obviously the gravitational direction e 3 and can be directly provided the IMU sensor.

A soft landing manoeuvre consists in progressively reducing the distance between the vehicle and the desired landing point, ensuring it reaches the zero value with zero speed. The environment enforces a physical constraint on the vehicle height: it has to be all time strictly positive. Indeed, the desired landing point will always lie on the boundary of the admissible region of the space, an asseveration that requires careful consideration in the stability analysis.

We approximate the cluttered environment by a collection of fixed spherical objects. For a given obstacle, let Po denote the closest point on the surface of the obstacle to the camera. Let P o be the vector from camera to Po and define d 0 = |P o | to be the distance from camera to the obstacle. The vector p o denotes the projection of P o onto the spherical image surface, p o = P o /d o . Note that p o is orthogonal to the tangent plane of the spherical obstacle. We assume that the de-rotated optic-flow Φ is computed at all points in the image. The divergence of the optic flow is a scalar field that can be computed from the optic field using either differentials of the flow or a vector convolution. Locally around p 0 in the image the divergence of the optical flow is related to the scaled rate of change of distance do we need some reference here?

divΦ(p o ) = -2 ḋo d o . (7)
In the same manner, denoting the flow over the projection of the ground plane spherical image by

D ⊂ S 2 one can show that v h = Λ D Φ(p)dp
for a matrix Λ > 0 that can be computed from knowledge of the size of D [START_REF] Hérissé | Landing a vtol unmanned aerial vehicle on a moving platform using optical flow[END_REF] and where h is the distance from the camera to the landing surface plane in the normal direction η. We will again use the expression v/h in the control analysis rather than writing the dependence on the optic flow field directly. A key aspect of the optic flow measurements is that the resulting measurements always depend inversely on distance to the observed as well as depending linearly on the camera velocity. This singularity in the velocity leads to much of the nice properties of optic flow as a cue for obstacle avoidance, but also causes significant complications in the analysis.

We will assume further that an obstacle-free path to the landing point exists and that the landing point is visible at all times during the evolution of the closed-system. Clearly this assumption will fail in highly cluttered environments and some further hybrid control laws, switching between different target points, or adding feed forward driving terms would need to be considered to overcome this in practice. Considering this issue is beyond the scope of the present paper as we concentrate on the fundamental nonlinear complexity of the problem. For the same reasons, although we recognise that real obstacles are not spherical and a practical implementation of these ideas will need to address these issues, we will not discuss this further in the present paper.

LANDING IN OBSTACLE-FREE ENVIRONMENTS

In this section, we consider the landing task in an obstaclefree environment. The proposed solution represents a vari-ant of the controllers formerly developed by [START_REF] Hérissé | The landing problem of a vtol unmanned aerial vehicle on a moving platform using optical flow[END_REF][START_REF] Hérissé | Landing a vtol unmanned aerial vehicle on a moving platform using optical flow[END_REF], that were proportional-integral nonlinear controllers, based on optical-flow data only. Here, we avoid the integration of the always noisy optical-flow measurements, by substituting it by the spherical image point of the landing point. The advantage in avoiding the integral component may become relevant in real applications where the noise in the optical flow measurement is not negligible, e.g. when the limited field of view does not allow to properly average the corrupted measurements, leading to a diverging integral term.

Note that, given any desired landing point, one can use a simple change of coordinates and rewrite the system as if the landing point was the origin of the 3-D space. Then, we assume that the desired landing point is PL = (0, 0, 0) . Theorem 1. Consider the system (2) subjected to the control input

u L = -mge 3 -mk v h + ξ |ξ| (8)
then, for any initial condition such that h(0) > 0 and for some k > 0 the following assertions hold

(1) h(t) > 0 , ∀t (implying that the dynamics of the closedloop system are well defined ∀t);

(2) h(t) and ḣ(t) converge to zero asymptotically;

(3) v(t) converges to zero asymptotically; (4) ξ(t) converges to zero asymptotically;

(5) the control law ( 8) is bounded ∀t.

The proof of the theorem is given in Appendix A.2.

It is worth noticing that, if we assume that the normal direction to the plane is known, the proposed control input can be computed by using IMU data, Optical Flow measurement and the image of the desired landing point.

LANDING AMONG OBSTACLES

Navigation and control of vehicles in cluttered environments is a central topic in the area of field robotics. Several solutions have been developed in the literature to cope with this problem. The most familiar approach is certainly the so-called artificial potential fields from [START_REF] Khatib | Real-time obstacle avoidance for manipulators and mobile robotics[END_REF] where the key idea consists in building artificial repulsive force fields that depend linearly on the inverse of the distance to the sensed obstacles. The main problems2 of the approach that arise are due to the presence of local minima that can entrap the robot, preventing it from the convergence to the desired position (a problem related to the geometry of the environment and to the incomplete knowledge of it), as discussed by [START_REF] Borenstein | Real-time obstacle avoidance for fast mobile robots[END_REF], and to the need of measuring the distance to the obstacles, since the typical sensor suite embarked consists of an inertial measurement unit (IMU), and a camera providing video data.

The key challenge here considered is to slightly modify the control strategy in order to avoid obstacles while performing the landing task using the minimal sensor suite carried by the vehicle. In particular, we exploit optical flow measurement to address the problem of a unified control task incorporating obstacle avoidance. At first glance, one can exploit the optical-flow measurements for obstacle avoidance purposes by adding to the nominal controller (8) a dissipative term

k D D i p i = k D ḋi d i p i , d i = |P i |
for each obstacle, where D i = ḋi /d i is proportional to the measured optical-flow divergence (7). Those terms act along the directions p i of the obstacles, and adding them to the landing control input one gets

u = u L + u D , where u D = k D n i=1 D i p i , K D > 0 . (9)
Since the derivative of the distance, i.e. ḋi = -v p i , is the projection of the velocity vector along the direction p i , it is negative when the vehicle approaches the obstacle. The following storage function (also exploited in the proof of Th. 1, see eq. (A.8)),

L = k|ξ| + |v| 2 2 ,
shows, via its time derivative

L = -k |v| 2 h -k D N i=1 |p i v| 2 d ≤ 0,
that the control input u D acts as a dissipative term, whose intensity can be modulated by means of the gain k D . The vehicle will slow down in the unwanted direction, while preserving its motion in the rest of the space. Being u D the sum of pure dissipative terms, it is hard to ensure by means of the control input (9) that there is no contact with the obstacles. To overcome this issue, one can suggest the design of a repulsive field around each obstacle i, derived from the following function:

φ i (d i ) = k R (d i (t) (γ i (t) -1) + d i (0)) , d i < d i (0) 0, d i ≥ d i (0) (10)
where k R is a small positive gain (k R k) and γ i is given by an integral of the term

D i γ i (t) = τ 0 ḋi (τ ) d i (τ ) dτ = ln d i (t) d i (0) (11)
Choosing the following expression of the repulsive force:

F i (d i ) = k R γ i p i . ( 12
) It follows that near the obstacle the term γ i is negative and hence the force (even small) is actually driving the vehicle away from the obstacle neighborhood. For n visible obstacles in the environment, we design the obstacleavoidance control input by summing all the contributions, generated by using eq. ( 12) for each obstacle:

u R = k R n i=1 γ i p i . ( 13
)
By defining the control action u as u = u L + u D + u R we get the following result: Theorem 2. Consider the system (2) under the control input u = u L + u D + u R .

(14) Assume the environment is populated by n (n ≥ 1) spherical obstacles and that the desired point is reachable from any initial point ξ(0) in the admissible space. Assume there is no contact between the obstacles and the landing plane and choose k R of ( 13) small with respect to k enough. Then, for any initial conditions such that h(0) > 0

(1) All solutions converge to E a ∪ E u . With E a = (0, 0) and

E u = (ξ * , 0) s.t. k ξ * |ξ * | = k R n i=1 p i (ξ * )γ i (ξ * ) ;
(2) The point E a is a locally attractive fixed point;

(3) Any point of E u is an unstable equilibrium. More precisely, for any (ξ * , 0) ∈ E u and for any neighbourhood U of (ξ * , 0), there exists ( ξ, Ṽ ) ∈ U such that the trajectories of the system starting from ( ξ, Ṽ ) converges to E a .

The proof is reported in Appendix A.3.

SIMULATION RESULTS

Here we show some simulation results obtained using the landing controller described in sec. 3, and its extension for the cluttered environment case, as described in sec. 4. All simulations are conducted including the rotational dynamics of the vehicle as well as some additive noise in the measurements of the optical flow and the bearing direction of the obstacles. To show the behaviour of the system under the control law (8), we chose a set a of different initial conditions both for position and velocity vectors.

Fig. 1 represents trajectories in the 3-D space resulting from those initial conditions. The environment is symmetric and we chose to confine the evolutions of the system in the subspace y > 0. Moreover, we chose initial conditions that are symmetric with respect to the y-h plane to visually separate trajectories resulting from different initial velocities. Indeed, initial conditions with x(0) > 0 correspond to zero initial speed and their evolutions are drawn in black (starting position are marked with a star symbol). As might be expected from the symmetry of the system, the UAV is driven along straight lines directly to the landing point.

On the other hand, red lines (and their relative starting points, marked with red circles) represent the evolutions of the system corresponding to v(0) = (-1, 0, 1) , i.e. when the initial conditions are such to drive the vehicle away from the landing point and towards the ground plane.

Consider the trajectory resulting from initial condition ξ(0) = (-5, 0, -0.2) (leftmost in Fig. 1). As h approaches zero, the control input grows and compensates the initial speed. The vertical speed approaches zero and the vehicle slowly slides in the direction of the landing point. The system shows an analogous behaviour when initial conditions are such that the initial height is close to zero, as can be seen by looking at the trajectory starting from ξ(0) = (5, 0, 0.2) (rightmost in Fig. 1). In those cases, the convergence to the landing point may require a big amount of time and the small value of the height may prevent the use of the proposed control in real systems.

The addition of the repulsive action and damping terms to the control input, (recall eq. ( 14)), modifies all the trajectories. Simulations results for this case are shown in Fig. 2. The initial conditions are the same as the previous case. The control strategy is in some sense conservative, since it modifies even those trajectories which would not collide with the obstacles in any case. This fact can be easily noticed by comparing Fig. 1 and Fig. 2.

CONCLUSIONS AND FUTURE WORKS

In this paper we addressed the problem of landing a VTOL-UAV in obstacle-free or cluttered environments. The proposed control law exploits measurements given by a camera attached to the vehicle as well as an IMU, a common sensor suite for an unmanned aerial vehicle. The simulation results shows the effectiveness of the control law in all the environments. Here we considered spherical objects, placed at a non-zero height from the ground. Considering non-spherical objects involves the estimate of the normal to their surface, and would be a useful extension towards a real-world implementation of the algorithm. We also plan to prosecute the work towards the implementation on a real small-scale UAV.

Proof. Define y : Proof. There exists a time t * such that β(t) > β 0 > 0, ∀ t > t * . Therefore, when t > t * one has

= x + p, then ẏ = -k(t)([1 + o(t)]x + p) + ṗ = -k(t)[1 + o(t)]y + k(t)o(t)p + ṗ .
t 0 α(s)β(s) ds > t * 0 α(s)β(s) ds + β 0 t t * α(s) ds ,
with the last term in the right-hand side of the inequality converging, by assumption, to infinity.

A.2 Proof of Theorem 1

Given the control (8) applied to the system (2), the equation of the closed-loop system is

ξ = -k ξ h + ξ |ξ| = -k ξ h + α ξ h , (A.4) with α(t) = h(t) |ξ(t)| , α(t) ∈ (0, 1], ∀t. (A.5)
Proof of item 1: The vehicle is initially above the target, i.e. h(0) > 0. Given an (any) initial condition (ξ(0), v(0)), as long as h(t) does not reach zero, the solution to the above equation is well defined and unique. Let us first show that h(t) cannot reach zero in finite time. By definition h is the third component of ξ. Therefore, in view of (A.4), the time-evolution of h is given by ḧ

= -k ḣ h + α . (A.6)
Integration of both sides of this equality yields

ḣ(t) = ḣ(0) -k ln h(t) h(0) -k t 0 α(s) ds . (A.7)
This relation is valid as long as h(t) = 0. Now, consider the following positive function

L = k |ξ| + |v| 2 2 , (A.8)
whose time-derivative along a solution to System (A.4) is

L = -k |v| 2 h ≤ 0 . (A.9)
One deduces from the previous two relations that |ξ| (h) and |v| (≥ | ḣ|) are uniformly bounded with respect to the initial condition, as long as the solution to the system is defined, i.e. as long as h remains positive. If we assume that h reaches zero at the finite time-instant T s > 0, then the term ln h(t) h(0) appearing in the equality (A.7) tends to infinity when t tends to T s , whereas all other terms involved in this equality remain bounded. This assumption thus yields a contradiction. Therefore h is never equal to zero, and the solution to System (A.4) is well defined and unique for t ∈ [0, ∞).

Proof of item 2: Using the following change of variables

z(t) = h(t) exp ḣ k , (A.10)
one gets ż(t) = -α(t) z(t), a first order differential equation whose solution is .11) In order to study the evolution of (h, ḣ), we can consider two different cases, depending on the sign of the initial vertical velocity.

z(t) = z(0) exp   - t 0 α(τ )dτ   . (A
Case 1: ḣ(0) < 0. Let us show that ḣ(t) < 0, ∀t. We make a proof by contradiction and assume that there exists t * such that ḣ(t * ) = 0 and ḧ(t * ) ≥ 0. Then, according to (A.6), ḧ(t * ) < 0. A clear contradiction.

Case 2: ḣ(0) ≥ 0. If ḣ(t) were always positive, or equal to zero, then, according to (A.10), z(t) would be nondecreasing. This contradicts relation (A.11) which implies that z(t) is strictly decreasing. Therefore there exists a time instant T such that ḣ(T ) < 0, and we are brought back to Case 1 with T taken as the new origin of time.

By considering these two cases, we have shown that ḣ(t) is strictly negative after a finite time. Let us now show that h(t) and ḣ(t) converge to zero. Since h(t) is positive and decreases after a finite time-instant, it converges to some limit h m which is either positive or equal to zero. Recall that the boundedness of ξ has already been proven so that ξ M := sup t∈[0,+∞) |ξ(t)| is a positive finite number. Let us assume that h m is positive, then α(t) > h m ξ M = α m > 0, ∀t. Relation (A.11) then implies z(0) e -αmt > z(t) ≥ z(0) e -t , and thus the convergence of z(t) to zero. This in turn implies the convergence of h(t) to zero, since | ḣ(t)| (≤ |v(t)|) is bounded. From this contradiction, one deduces that h m = 0 and thus that h(t) converges to zero.

Finally, using the fact that kα(t) is bounded and that k/h(t) tends to infinity, the application of Lemma 3 to the equation (A.6) establishes the convergence of ḣ(t) to zero.

Proof of item 3: We first show that the ratio |v| h is bounded. To this aim let us consider the equation governing the evolution of this ratio, as deduced from (A.4) .12) with β the angle between v and ξ, so that cos(β) = v T ξ |v||ξ| . Using the fact that k/h tends to infinity and that ḣ/k tends to zero, the boundedness of |v| h is simply obtained by applying Lemma 4 to this equation. From there, the convergence of v to zero just follows from the convergence, previously proven, of h to zero. Since h and ḣ converge to zero, one deduces from (A.7) that the integral of α tends to +∞. Therefore, by application of Lemma 6, the integral of ᾱ also tends to +∞. The convergence of ξ to zero then follows from the previous equality.

d dt |v| h = - k h 1 + ḣ k |v| h + cos(β) . ( A
Proof of item 5: The boundedness of the control u L given by ( 8) simply follows from the previously proven boundedness of v/h.

A.3 Proof of Theorem 2

Proof of item 1: Consider the following Lyapunov function:

L = k|ξ| + 1 2 |v| 2 + n i=1 φ i , (A.16) whose derivative is L = - |v| 2 2 -k D n i=1 ḋ2 i d i . (A.17)
Let's divide the analysis in two cases, related to the behaviour along the vertical axis and let define h , the value of h that characterises the height of the nearest obstacle to the ground.

Case 1: h ≥ h , ∀t. Then L = 0 implies that the velocity vector is null, and the equilibrium (ξ, v) = (ξ * , 0) = (0, 0) is a solution of the implicit equation ξ * = 0. This implies

k ξ * |ξ * | = k R n i=1 γ i p i . (A.18)
Case 2: h < h . In the hypothesis that the minimum height of the obstacles is such that their residual influence on the vehicle dynamics is negligible, then (ξ, 0) is converging to the landing point (0, 0). This case is similar to consider the neighbourhood of the landing point itself, then one can rely on the same arguments used for the next item.

Proof of item 2: To show that E a is a locally attractive point, consider the dynamics of the system in the neighborhood of the landing point i.e. when h < h . Recalling eq. ( 10), one knows that the terms γ i are negative or null. By neglecting the effect of the dissipative term due to u D it is straightforward to verify that the inner product η p i > 0 and hence the projected dynamics are approximately Recalling eq. (A.18), and taking the norm of the equilibrium point, it follows that:

n i=1 γ i p i = k k R
By choosing k R small enough, it is straight forward to verify that there exists an obstacle j ∈ {1, . . . , n} such that:

lim k R →0 n i=1 γ i p i = |γ j p j | = +∞
Since the obstacles are assumed to be scattered in the environment, the above limit is approximately valid even if k R is only chosen small enough with respect to k and consequently the problem considered is similar to the issue when only a single obstacle is present in the environment. Without loss of generality, define p = p j , γ = γ j , and d = d j and examine the linearization of the system dynamics.

Since the second element of the second row of the matrix above is negative, to prove the instability of the system in the set E u , consider the first element of the second row .21) Note that ξ * |ξ * | = ±p, according to the sign of γ (recall eq. (A.18)), and consider a rotation matrix Q, such that p = Qe 3 one gets

-k π ξ * |ξ * | -k R π p γ d + pp d . ( A
- k |ξ * | + k R γ d π e3 - k R d e 3 e 3 .
If the vehicle is approaching the obstacle, γ is negative and from (A.18), it results k R γ = -k, then one gets (A.22) whose eigenvalues are

-k 1 |ξ * | - 1 d π e3 - k R d e 3 e 3 ,
-k 1 |ξ * | - 1 d , -k 1 |ξ * | - 1 d , - k R d . (A.23)
The first two of them are positive in the neighborhood of the obstacle, since d < |ξ * |, thus the matrix is unstable.

Fig. 1 .

 1 Fig.1. Trajectories in the 3-D space (x-y-h) for different initial conditions. Black lines correspond to zero initial velocities. Red lines correspond to initial velocity V (0) = (-1, 0, -1) .

Fig. 2 .

 2 Fig. 2. Trajectories in the 3-D space (x-y-h) for different initial conditions, case with multiple obstacles. Black lines correspond to zero initial velocities. Red lines correspond to initial velocity V (0) = (-1, 0, -1) .

  and a direct application of Lemma 3 yields the desired result. Lemma 6. Consider two positive functions α(t) and β(t)

  arguments used in the proof of Th. 1 apply and one can conclude that h converges to zero.Proof of item 3:To show the instability of the set E u we compute the linearization of the error dynamics around the point (ξ * , 0) ∈ E u .Let χ = (ξ, ξ) , and recall that in this case, the full system dynamics computing the partial derivatives of ξ, ξ with respect to the components of the state vector χ, and by setting π ξ * = I 3 -ξ *

The symbol I 3 represents the identity matrix of dimension 3 by 3.

Details on drawbacks of these techniques and possible solutions have been described by[START_REF] Adeli | Path planning for mobile robots using iterative artificial potential field method[END_REF].

Appendix A

A.1 Preliminary technical lemmas

All time-dependent functions involved in the statements of these lemmas are assumed to be defined on [0, +∞) and continuous. The state variable x belongs to R n , with n a positive integer. Also, o(t) denotes any bounded vectorvalued function whose norm tends to zero when t tends to infinity. Lemma 3.


Thiomers and their potential applications in drug delivery

Keywords: Mucoadhesion, Oral delivery, Preactivated thiomers, S-nitrosothiomers, Thiomers

Introduction: Thiomers are the product of the immobilization of sulfhydryl-bearing ligands onto the polymer backbone of a conventional polymer, which results in a significant improvement in mucoadhesion, in situ gelation and efflux inhibition compared with unchanged polymers. Because of thiol groups, thiomers have more reactivity and enhanced protection against oxidation. Since the late 1990s, extensive work has been conducted on these promising polymeric excipients in the pharmaceutical field.

Areas Covered: This review covers thiomers, their classification and their different properties. Various techniques for the synthesis, purification and characterization of thiomers are described in detail. This review also encompasses their various properties such as mucoadhesion, permeation enhancement, in situ gelation and efflux inhibition, as well as different formulations based on thiomers. In addition to the use of thiomers as multifunctional excipients, this review also encompasses their use as drugs.

Expert Opinion:

The synthesis is realized by linkage of sulfhydryl-bearing ligands but reported methods give low yields. Higher degrees of modification are not necessary and would probably lead to extreme changes in properties. Nevertheless, an accurate characterization of the final product is important. The scale-up procedure for industrial manufacturing has been adapted to produce GMP materials; Lacrimera ® eye drops have already entered the European market.

• Several methods have been described to produce thiomers, but the yields of these reactions are low • An accurate characterization of the final products is important

• Thiomers have good mucoadhesive properties

• Thiomers have been used to enhance the permeation of drugs across biological barriers

• Thiomers have efflux pump inhibiting properties

• Some thiomers have in situ gel properties

• Some thiomers can be used to produce tablets, nanoparticles or microparticles • From these thiomers, S-nitrosothiomers have been synthesized and proposed as new NO donors

Introduction

Thiolated polymers, also referred to as thiomers, have encouraged the development of various fields of polymer-based medicine [START_REF] Gyarmati | Reversible disulphide formation in polymer networks: A versatile functional group from synthesis to applications[END_REF][START_REF] Bacalocostantis | Effect of Thiol Pendant Conjugates on Plasmid DNA Binding, Release, and Stability of Polymeric Delivery Vectors[END_REF][START_REF] Gyarmati | In situ oxidation-induced gelation of poly(aspartic acid) thiomers[END_REF]. Thiomers or thiolated polymers are macromolecules bearing free thiol groups on the polymer backbone. The immobilization of sulfhydryl-bearing compounds (SBC) onto the backbone of well-developed polymeric excipients such as chitosan (CS) and poly (acrylates) may lead to the generation of these thiomers [START_REF] Samuel | Thiomers: Forms, features and formulations[END_REF][START_REF] Li | A novel glutathione modified chitosan conjugate for efficient gene delivery[END_REF][START_REF] Jin | Chitosan-glutathione conjugate-coated poly(butyl cyanoacrylate) nanoparticles: Promising carriers for oral thymopentin delivery[END_REF] as illustrated in figure 1. The formation of inter-and intramolecular disulfide bonds gives strong cohesive properties to thiomers, leading to comparatively advanced stability and consequently extended disintegration times and drug release from different dosage forms such as tablets, micro particles and gels [7]. In contrast to unmodified polymers, thiomers illustrate advantages in mucoadhesion and enzyme and efflux inhibition. The use of thiolated polymers or thiomers might overcome some prominent challenges in the applications of macromolecules as therapeutic agents including deficiencies in efficiency, safety and detailed delivery approaches [START_REF] Laffleur | Thiomers: promising platform for macromolecular drug delivery[END_REF].

There has been substantial growth in the number of commercially available bioactive macromolecular therapeutic agents, such as proteins and peptides, due to current improvements in the pharmaceutical/biotechnology field. Due to their cost, these drugs are used in the treatment of major conditions such as acquired immunodeficiency syndrome (AIDS), cardiovascular disorders, hepatitis and cancer etc. [START_REF] Brannon-Peppas | Nanoparticle and targeted systems for cancer therapy[END_REF]. Macromolecules have poor oral bioavailability, and parenteral routes are necessary.

Because this latter approach is often linked with pain, fear and risks, it is of particular importance for pharmaceutical companies to make alternative routes available for drug administration. This is especially important for long-term treatments [START_REF] Laffleur | Thiomers: promising platform for macromolecular drug delivery[END_REF]. To overwhelm the targeted body diffusion barrier, drug delivery systems must possess specific properties. The bioavailability of macromolecular drugs has certain limitations due to the diffusion barriers offered by the mucosal gel layer wrapping mucosal membranes followed by the absorption barriers surrounding epithelial layers and efflux pumps, which are aided by enzymatic barriers. The polymeric excipients for drug delivery must possess certain prominent features to overwhelm these barriers, including mucoadhesive [START_REF] Balenga | Protective efficiency of dendrosomes as novel nano-sized adjuvants for DNA vaccination against birch pollen allergy[END_REF][START_REF] Kafedjiiski | Synthesis and in vitro evaluation of thiolated hyaluronic acid for mucoadhesive drug delivery[END_REF][START_REF] Shen | Mucoadhesive effect of thiolated PEG stearate and its modified NLC for ocular drug delivery[END_REF] and protective activity [13; 14], drug release control [15; 16], improved solubility [17; 18], and permeation and uptake enhancing properties [START_REF] Wong | A new polymer-lipid hybrid nanoparticle system increases cytotoxicity of doxorubicin against multidrug-resistant human breast cancer cells[END_REF][START_REF] Walsh | Oral delivery of macromolecules: Rationale underpinning Gastrointestinal Permeation Enhancement Technology (GIPET®)[END_REF][START_REF] Hauptstein | Thiomers and thiomer-based nanoparticles in protein and DNA drug delivery[END_REF].

However, thiols groups are unstable in solution (oxidized at pH ≥ 5) and some authors have demonstrated that protective substructures can be introduced to overcome this shortcoming [START_REF] Iqbal | Preactivated thiomers as mucoadhesive polymers for drug delivery[END_REF]. This idea has recently led to a second generation of thiomers ("preactivated thiomers") with improved properties (e.g., mucoadhesive properties) [START_REF] Bonengel | Thiolated and S-protected hydrophobically modified crosslinked poly(acrylic acid) -A new generation of multifunctional polymers[END_REF] (Figure 1).

Finally, thiomers can be bound to nitric oxide to act as new NO donors [24; 25] (Figure 1).

Insert Figure 1 2. Types and synthesis of thiomers

First generation

Numerous naturally occurring polymers, such as CS [26; 27], cellulose [START_REF] Laffleur | Next generation of buccadhesive excipient: Preactivated carboxymethyl cellulose[END_REF], hyaluronic acid [START_REF] Nowak | Preactivated hyaluronic acid: A potential mucoadhesive polymer for vaginal delivery[END_REF], gelatin [START_REF] Duggan | Synthesis of mucoadhesive thiolated gelatin using a twostep reaction process[END_REF], collagen [START_REF] Albrecht | Thiomers: forms, functions and applications to nanomedicine[END_REF], and synthetic polymers, such as poly(acrylic acid) (PAA) [START_REF] Grabovac | Thiomers: Influence of molecular mass and thiol group content of poly(acrylic acid) on efflux pump inhibition[END_REF], are commonly used as thiomer precursors. Thiomers are characterized as cationic, anionic and non-ionic based on the chemical structure of these parent molecules.

Cationic thiomers are prepared by immobilizing SBCs onto the 2-amino position of the glucosamines in the polymer chain. These thiomers are based on CS, such as chitosan-cysteine (CS-Cys) and chitosanthioglycolic acid (CS-TGA) etc. [START_REF] Deepak | Thiolated chitosan: modified advanced generation of mucoadhesive polymers[END_REF]. The carboxylic acid group of Cys or thioglycolic acid (TGA) can react with the -NH 2 group of CS, but the reaction has to be mediated by a carbodiimide. Other SBCs, such as N-acetylcysteine, 6-mercaptonicotinic acid and glutathione, can form amide bonds with chitosan [34; 35]. The cationic property not only comes from various groups of the SBCs (e.g., NH 3 /NH 4 + from amino acids) but also from the unreacted NH 3 groups of the glucosamine. To date, described procedures have had a low yield (e.g., 25% for chitosan -GSH = 558 µmol/g) [START_REF] Shah | Synthesis and characterization of S-nitrosoglutathioneoligosaccharide-chitosan as a nitric oxide donor[END_REF]). Amino acids can be affected by unintended reactions [START_REF] Riniker | Protection of Functional Groups in Peptide Synthesis[END_REF] that could be limited by protection of the carboxylic group.

Nevertheless, some authors have used TGA as the SBC, and because this SBC is not an amino acid, unintended reactions are not possible; however, there is also a very low yield (< 40 µmol/g) [START_REF] Kast | Thiolated polymers -thiomers: development and in vitro evaluation of chitosan-thioglycolic acid conjugates[END_REF].

Furthermore, there is a second method that is sometimes used to modify CS. With the chemical reaction between CSs and 2-iminothiolane (Traut's reagent), a cationic amidine conjugate chitosan-4thiobutylamidine (CS-TBA) is produced [38]. In another study, to achieve better stability and properties similar to those of chitosan-4-thiobutylamidine, chitosan-thioethylamidine (TEA) was synthesized by a similar mechanism [39]. All the procedures gave quite similar yields. Hence, it is possible that this issue is due to limited accessibility to the amine groups (i.e., steric hindrance).

For anionic thiomers, carboxylic acid groups are present on the polymeric substrates. Hence, the immobilization of the SBCs on these polymers can also be mediated by carbodiimides and is possible if the SBC bears a primary amine group. Examples include carboxymethylcellulose-cysteine (CMC-Cys) and alginate-cysteine (ALG-Cys) [40]. The reaction yields are also very low.

To date, these reactions are widely used to obtain thiomers and their low yields are probably not so important. It is supposed that higher degrees of modification would probably lead to extreme changes in properties.

Finally, to achieve thiolation of polysaccharides exhibiting neither carboxylic acid nor primary amino groups (e.g., nonionic polymers), other methods can be used: Insert Figure 2 1) Periodate cleavage of vicinal thiol followed by conjugation with amino acids is an advantageous technique (Figure 2). Thiomers are synthesized under fairly simple and mild conditions by covalently attaching SBCs bearing a -NH 2 to the polymer backbone. Recently, Madgulkar et al. [41; 42], synthesized a novel mucoadhesive derivative of xyloglucan via oxidation of xyloglucan followed by conjugation with the amino groups of Cys as shown in figure 2. Sodium periodate was used to oxidize the hydroxyl groups located at carbon 2 and 3, resulting in the formation of different reactive groups having more rotational autonomy along the polymer backbone. The authors found that compared with the glucose remaining on the key chain, the xylose and galactose deposits on the lateral chains were more ideal for oxidation, which may be due to the dexterity of pendant side sugar deposits compared to glucose residues on the main chain. It was also found that the vicinal -OH group of the β-(1→4) -linked Dglucopyranose backbone was in the trans state, which was kinetically less promising for periodate oxidation compared with the prime (cis) conformation detected in galactose and xylose residues.

2) A nonionic polysaccharide such as hydroxyethylcellulose (HEC) can be transformed into a positively charged thiomer after oxidation of the proximal hydroxyl groups to aldehyde followed by conjugation of Cys to the same aldehyde groups using reductive amination reactions [43]. Bromination of HEC is also used to synthesize non-ionic thiomers, and then, the bromine of bromohydroxyethylcellulose (Br-HEC) is substituted with thiol groups by treating Br-HEC with thiourea yielding 131.58 ± 11.17 µmol of thiol groups per gram of polymer [44]. The synthesized thiomers are primarily purified using dialysis followed by freeze-drying under reduced pressure. These polymers can also be precipitated using organic solvents, such as Ellman's reagent, for quantification of the immobilized thiol groups [START_REF] Ijaz | Preactivated thiomers: their role in drug delivery[END_REF].

Second generation: preactivated thiomers

The synthesis of preactivated thiomers was recent reviewed by Ijaz et al. [START_REF] Ijaz | Preactivated thiomers: their role in drug delivery[END_REF]. The protective substructures are pyridyl thiol groups derivatives: they are known to react very rapidly and quantitatively with sulfhydryl groups over a broad pH range to form disulfide bounds [45]. Since vitamin B3 (nicotinamide/nicotinic acid) has this structure and a low toxicity, 2 of its derivatives were used as protective substructures: 6-mercaptonicotinamide (6-MNA) and 2-mercaptonicotinic acid (2-MNA). For example, Iqbal et al. described a method in 2 steps: 2-MNA was first oxidized to dimer and then it was coupled to PAA-Cys via a disulfide exchange reaction to give a conjugate (PAA-Cys-MNA). This method was used in other studies [23; 46-50]. Besides, a quite similar methods was described with 6-MNA [45].

Thiomers as drugs: S-nitrosothiomers

Thiomers are not only used as drug excipients but can also be used as drugs. Because thiomers have free thiol (-SH) groups, active groups can be attached to these thiols for their therapeutic activities. Nitric oxide (NO) is an endogenous molecule and is a free radical having a very short half-life in vivo.

However, the biological half-life of NO can be increased by its chemical attachment to a free thiol group available on glutathione (GSH), and an NO-donor S-nitrosoglutathione (GSNO) is obtained with an increased half-life [START_REF] Shah | Synthesis and characterization of S-nitrosoglutathioneoligosaccharide-chitosan as a nitric oxide donor[END_REF]. Similarly, these NO-donor thiomers can be further linked to polymers for enhanced bioavailability and improved pharmacokinetics. In a study reported by Katsumi et al., a NOdonor was synthesized by the chemical attachment of SATA (N-succinimidyl-S-acetylthioacetate) to polyethylene glycol (PEG) and bovine serum albumin [START_REF] Katsumi | Development of polyethylene glycol-conjugated poly-S-nitrosated serum albumin, a novel S-Nitrosothiol for prolonged delivery of nitric oxide in the blood circulation in vivo[END_REF]. More recently, a new NO-donor was synthesized by the covalent attachment of chitosan to GSNO [START_REF] Shah | Synthesis and characterization of S-nitrosoglutathioneoligosaccharide-chitosan as a nitric oxide donor[END_REF] and alginate to GSNO [START_REF] Shah | Synthesis and characterization of S-nitrosoglutathioneoligosaccharide-chitosan as a nitric oxide donor[END_REF], with better values observed for NO-linked to the polymer backbone.

Consequently, a variety of thiomers can be modified to achieve desired properties and therapeutic effects using simple reactions with minimal steps.

Properties of thiolated polymers

Mucoadhesive properties

Various research groups have verified the mucoadhesive properties of first generation thiomers for many polymer backbones, showing a more than 100-fold enhancement of mucoadhesive properties. For instance, the mucoadhesive properties of chitosan were enhanced 140-fold by the immobilization of thiol groups [38]. Thiomers, in contrast with other polymers, are proficient at developing covalent bonds with Cys-rich glycoprotein subdomains through thiol/disulfide interchange reactions. The produced covalent linkages are reportedly much stronger than the non-covalent polymer-anionic substructure interactions of the mucous layer [START_REF] Laffleur | Thiomers: promising platform for macromolecular drug delivery[END_REF].

As depicted in table 1, in contrast with unchanged polymers, the thiomers show stronger adhesion, thus leading to increased oral bioavailability, subsequent reductions in dosing frequency and increased patient compliance [START_REF] Bernkop-Schnürch | Thiomers: potential excipients for noninvasive peptide delivery systems[END_REF].

Insert Table 1

Permeation enhancement

Different approaches have been taken to overcome absorption barriers, primarily through the use of permeation enhancers [START_REF] Bernkop-Schnürch | Biomembrane Permeability of Peptides: Strategies to Improve Their Mucosal Uptake[END_REF]. Different classes of low-molecular weight permeation enhancers, such as ethylene-diamine-tetraacetic acid (EDTA) and phospholipids, have been confirmed to be valuable in enhancing the permeation of crosswise integral epithelial membranes [START_REF] Lee | Protease inhibitors and penetration enhancers as approaches to modify peptide absorption[END_REF]. On the other hand, permeation and bioavailability also depend on a drug's physicochemical properties [START_REF] Lipinski | Experimental and computational approaches to estimate solubility and permeability in drug discovery and development settings[END_REF]. In addition to the physicochemical properties of the drug, bioavailability also depends on formulation design and GIT's physiological conditions [START_REF] Huang | Mechanistic Approaches to Predicting Oral Drug Absorption[END_REF]. Some other factors affecting bioavailability include enzymes, transit time, food material and other drugs [START_REF] Blanquet | A Dynamic Artificial Gastrointestinal System for Studying the Behavior of Orally Administered Drug Dosage Forms Under Various Physiological Conditions[END_REF]. In most circumstances, inadequate uptake from the mucosa is the main obstacle to the oral bioavailability of hydrophilic macromolecules, and this mucosal uptake can be favored by either trans-cellular or para-cellular routes of administration [START_REF] Ijaz | Preactivated thiomers: their role in drug delivery[END_REF]. Thiomers, compared with most low-molecular weight permeation enhancers, provide the benefit of not being absorbed by the mucosal membrane and thus maintain their permeation enhancement properties for long periods of time [START_REF] Samuel | Thiomers: Forms, features and formulations[END_REF]. The most probable mechanism for this enhanced permeation has been credited to claudin and occludin proteins present in tight junctions that regulate cell adhesion and ion selection [START_REF] Blanquet | A Dynamic Artificial Gastrointestinal System for Studying the Behavior of Orally Administered Drug Dosage Forms Under Various Physiological Conditions[END_REF][START_REF] Simon | Paracellin-1, a Renal Tight Junction Protein Required for Paracellular Mg2+ Resorption[END_REF][START_REF] Sreenivas | Thiolated chitosans: novel polymers for mucoadhesive drug delivery-a review[END_REF]. This mechanism further proposes that inhibition of the enzyme PTP is responsible for the dephosphorylation of the tyrosine subunits of occludin, leading to the opening and closing of tight junctions and hence a greater permeation-enhancing effect. The phosphorylation and opening of tight junctions is caused by compounds such as reduced GSH, but due to fast oxidation on the mucosal membrane, its effect is restricted [START_REF] Laffleur | Thiomers: promising platform for macromolecular drug delivery[END_REF]. The presence of thiolated polymers on the mucosal surface to prevent the oxidation of GSH is vital. Many thiomers, such as PAA-Cys [START_REF] Kast | Influence of the molecular mass on the permeation enhancing effect of different poly(acrylates). S.T[END_REF], polycarbophil-Cys (PCP-Cys) [64], CS-TGA [START_REF] Bernkop-Schnürch | Thiolated chitosans: development and in vitro evaluation of a mucoadhesive, permeation enhancing oral drug delivery system[END_REF] and CMC-Cys, have been reported to have a greater permeation-enhancing influence on some in vitro model compounds, as shown in table 2. The degree of thiolation has also been reported to have a significant impact on permeation-enhancement properties [63; 66]. We can simply assume that the higher the degree of thiolation, the higher the permeation-enhancement effect.

Insert Table 2

Efflux pump inhibiting properties

The ATP binding layer belongs to a superfamily of transmembrane-situated transporter proteins known as ABC, including P-Glycoprotein (Pgp) and multidrug-resistance protein in apical enterocytes membrane (MRP2), which are known to be efflux pumps. These pumps are the limiting factors for the oral bioavailability of many drugs and are involved in xenobiotic transport of nutrients etc. back from the serosa to the mucosa of epithelial tissue (out of the cell membrane), thus acting as a protective shield [8; 69; 70]. Cationic drugs are largely blocked by Pgp, whereas in contrast, anionic drugs are usually effluxed by MRP2. Therefore, the inhibition of these transport proteins is of high interest and has been accomplished by the use of polymers, such as thiomers, polyethylene-glycols (PEG) [START_REF] Hugger | Effects of poly(ethylene glycol) on efflux transporter activity in Caco-2 cell monolayers[END_REF] and pluronic block copolymers [START_REF] Kabanov | Pluronic® block copolymers as modulators of drug efflux transporter activity in the blood-brain barrier[END_REF], which restrict transport by the formation of disulfide linkages [START_REF] Jha | Covalent modification of cysteine 193 impairs ATPase function of nucleotide-binding domain of a Candida drug efflux pump[END_REF]. Palmberger et al. investigated the effect of thiolated anionic biopolymers in vitro and in vivo as efflux inhibitors to improve intestinal trans-cellular drug uptake. Three thiomers, pectin-cysteine (Pect-Cys), carboxymethylcellulose-cysteine (CMC-Cys) and ALG-Cys, were synthesized and tested on rat small intestine using sulforhodamine 101 (SR-101) as the MRP2 model substrate. It was found that SR-101 transport in the presence of Pect-Cys, CMC-Cys and ALG-Cys improved by 1.5, 1.8 and 3.0-fold, respectively, in vitro; unmodified alginate improved by 3.8-fold, and the AUC 0 ≥ 12 of SR-101 improved by a factor of 1.9 in vivo. It was suggested that thiolated alginate can act as a favorable auxiliary agent for different drugs by being an anionic efflux pump substrate, thereby greatly enhancing oral MRP2 substrate bioavailability [START_REF] Palmberger | In vivo evaluation of anionic thiolated polymers as oral delivery systems for efflux pump inhibition[END_REF]. Grabovac et al. investigated the effect of the molecular mass and thiol group content of poly (acrylic acid) -cysteine conjugates (PAA-Cys) on sulforhodamine and penicillin G permeation by acting as an MRP2 efflux pump inhibitor in rat small intestine. Using a 250-kDa PAA-Cys conjugate, sulforhodamine exhibited a 3.85-fold permeability improvement from 3.93 to 355.9 ± 39.5 µmol/g, and penicillin G exhibited a 1.59-fold improvement. It was proposed that thiolated PAA constrains MRP2mediated transport and that the extent of inhibition was dependent upon molecular mass and the amount of polymer thiolation [START_REF] Grabovac | Thiomers: Influence of molecular mass and thiol group content of poly(acrylic acid) on efflux pump inhibition[END_REF].

In situ gelling properties

An important limiting factor for a drug's efficacy is rapid clearance from the site of drug administration, whether ocular, nasal or vaginal mucosa. It has been generally accepted that bioavailability of drug moieties can be enhanced by increasing the drug's viscosity and subsequently limiting its clearance. In situ-based drug formulation is hence a very favorable approach to acquiring drug formulations of the desired viscosity [START_REF] Sreenivas | Thiolated chitosans: novel polymers for mucoadhesive drug delivery-a review[END_REF]. As a consequence of some physicochemical changes due to the physiological environment such as temperature [START_REF] Edsman | Rheological evaluation of poloxamer as an in situ gel for ophthalmic use[END_REF][START_REF] Bromberg | Enhanced nasal retention of hydrophobically modified polyelectrolytes[END_REF][START_REF] Wei | Thermosetting gels with modulated gelation temperature for ophthalmic use: the rheological and gamma scintigraphic studies[END_REF], pH [START_REF] Gurny | The development and use of in situ formed gels, triggered by pH[END_REF][START_REF] Chenite | Novel injectable neutral solutions of chitosan form biodegradable gels in situ[END_REF][START_REF] Srividya | Sustained ophthalmic delivery of ofloxacin from a pH triggered in situ gelling system[END_REF], electrolyte concentrations [START_REF] Deasy | Rheological evaluation of deacetylated gellan gum (Gelrite) for pharmaceutical use[END_REF][START_REF] Paulsson | Rheological studies of the gelation of deacetylated gellan gum (Gelrite®) in physiological conditions[END_REF][START_REF] Susan | Ion Gels Prepared by in Situ Radical Polymerization of Vinyl Monomers in an Ionic Liquid and Their Characterization as Polymer Electrolytes[END_REF] and the formation of inter-and/or intra-molecular disulfide bonds by oxidation, extensive crosslinking results in sol-to-gel transitions. Hintzen et al. synthesized three thiomers, i.e., pectin-cysteine (Pec-Cys), sodium carboxymethylcellulose-cysteine (NaCMC-Cys) and poly(acrylic acid)-cysteine (PAA-Cys), and measured the rheological properties of 1% (m/v) thiomer solutions with oxidizing agents and compared the oscillatory measurements over time (120 mn). It was observed that Pec-Cys and NaCMC-Cys with carbamide peroxide and hydrogen exhibited a 13,000-fold increase in dynamic viscosity, and the reason for the increase in viscosity was attributed to the formation of inter-and/or intra-molecular disulfide linkages [START_REF] Sakloetsakun | In situ gelling properties of chitosanthioglycolic acid conjugate in the presence of oxidizing agents[END_REF]. It was also found that thiomers exhibiting a polysaccharide backbone showed significant viscosity increases (p < 0.05) [START_REF] Hintzen | In situ gelling properties of anionic thiomers[END_REF]. Gyarmati et al. investigated in situ gelling poly(aspartic acid) (PASA) thiomers to determine their potential in the development of injectable preparations. It was found that time of gelation and gel strength were related to polymer concentration, but the time of gelation was controlled independently of gel strength by changing the oxidizing agent concentration. Gelation time could be altered by between 2-6 min, making thiomer PASP a good candidate for injectable drug delivery formulations [START_REF] Gyarmati | In situ oxidation-induced gelation of poly(aspartic acid) thiomers[END_REF]. In another study, Iqbal et al. measured the viscoelastic properties of poly(acrylic acid)cysteine-mercaptonicotinamide (PAA-Cys-2MNA; 100, 250 and 450 kDa), related thiomers and unmodified PAAs in pH 6.8 phosphate buffer. It was found that with an increase in molecular mass and pre-activation of thiolated PAAs, the dynamic viscosity was enhanced [START_REF] Iqbal | Preactivated thiomers as mucoadhesive polymers for drug delivery[END_REF]. The obtained results thus confirm the hypothesis that an increase in the proportionality of viscosity is associated with molecular mass and the number of sulfhydryl moieties on the polymer [START_REF] Palmberger | Thiolated polymers: Evaluation of the influence of the amount of covalently attached l-cysteine to poly(acrylic acid)[END_REF]. Table 3 depicts a comparison of some of the dynamic viscosity improvement ratios of various thiomers in the presence of mucous.

Insert Table 3 4. Different routes of drug administration using thiomers as drug carriers

Oral matrix tablets

Different routes of drug administration with thiomers as a carrier have emerged in the past few years. Among these, the oral route is considered the safest and most effective route of drug administration [START_REF] Sarti | Poly(acrylic acid)-cysteine for oral vitamin B12 delivery[END_REF].

Matrix tablets with mucoadhesive thiomers show great potential for oral, intra-oral and vaginal drug administration [40; 47; 55; 63; 89]. The swollen polymer matrix offers enhanced stability and cohesiveness in situ because of inter-and intra-molecular disulphide bond generation [START_REF] Clausen | In vitro evaluation of matrix tablets based on thiolated polycarbophil[END_REF]. Disintegration studies performed on tablets with PAA in its modified and un-modified thiolated forms to investigate overall stability revealed that tablets with un-modified polymer disintegrated within 2 h, whereas for those with PAA-Cys, no disintegration was observed after 48 h [START_REF] Bernkop-Schnürch | Development of controlled drug release systems based on thiolated polymers[END_REF]. The swelling properties of matrix tablets can be reduced by the incorporation of a hydrophobic ligand such as oxidized MNA, providing preactivated polymers [45]. This protection is very significant for prolonging mucoadhesion [35; 91] because thiolated polymers, upon excessive water uptake, attain unnecessary weight and cause the detachment of matrix tablets from the mucosal surface. In another study, the oral bioavailability of antide tablets (i.e., a peptide that acts as an LHRH antagonist), including both thiolated and S-protected thiolated chitosan, was assessed. If the antide is incorporated in chitosan-thioglycolic acid-6-mercaptonicotinic acid (CS-TGA-MNA), the peptide reaches an absolute bioavailability of 1.4%, resulting in a 421-fold increased area under the plasma concentration time curve compared with the antide solution [23; 35; 45].

In a follow-up study, the potential of a preactivated thiolated pectin derivative (Pec-Cys-MNA) to act as a mucoadhesive excipient for gastric cavity delivery was assessed. Mini-tablets were developed and evaluated for mucoadhesion, disintegration, swelling and release of rosuvastatin calcium. Adhesion increased by more than 5-fold, disintegration time was prolonged by more than 2-fold, weight gain for Pec-Cys-MNA was at least 16-fold higher, and sustained release of rosuvastatin calcium was observed over 36 h [47].

Microparticles and nanoparticles

Multiple unit dosage forms such as micro-and nanoparticles, in contrast with single unit dosage forms such as tablets, have extended residence time in small intestine [START_REF] Bonengel | Thiolated and S-protected hydrophobically modified crosslinked poly(acrylic acid) -A new generation of multifunctional polymers[END_REF]. In one study, the tablet formulation left the small intestine completely after 3 h, whereas in the case of particles, more than 50% were still present after the same time [START_REF] Coupe | Variation in Gastrointestinal Transit of Pharmaceutical Dosage Forms in Healthy Subjects[END_REF]. Barthelmes et al. revealed that thiolated nanoparticles have enhanced mucoadhesive properties compared with unmodified nanoparticles on intravesical mucosa [START_REF] Barthelmes | Thiolated particles as effective intravesical drug delivery systems for treatment of bladder-related diseases[END_REF]. Lee et al., during studies on the transfection efficiency of chitosan-4-thiobutylamidine (CS-TGA/DNA) nanoparticles, observed that higher gene expression was induced by these thiolated chitosan nanoparticles (TCS-NPs) compared with unmodified chitosan nanoparticles (CS-NPs); CS-TGA/DNA with crosslinking nanoparticles exhibited prolonged DNA release and, after transfection, uninterrupted expression for up to 60 h [START_REF] Lee | Thiolated chitosan/DNA nanocomplexes exhibit enhanced and sustained gene delivery[END_REF]. Bari et al. conducted an in vivo evaluation of thiolated chitosan nanoparticles (TCS-NPs) of buspirone hydrochloride (BUH) for delivery into the brain through the intranasal route using bovine nasal cavity tissue. It was found that TCS-NPs had 15% more binding efficiency than unmodified CS-NPs, which could be due to the low amount of CS and TCS interacting with mucin, increased particle size of BUH-loaded CS and TCS nanoparticles and absorption of drug molecules on the NP surface, leading to low NP mucosal layer penetration [START_REF] Bari | Brain delivery of buspirone hydrochloride chitosan nanoparticles for the treatment of general anxiety disorder[END_REF]. Improved TCS binding efficiency could be due to covalent attachment of TGA to CS generating a thiol group that interacts with mucin [START_REF] Kast | Thiolated polymers -thiomers: development and in vitro evaluation of chitosan-thioglycolic acid conjugates[END_REF]. After 12 h, the cumulative % of drug permeated through the nasal mucosa by CS-BUH-NPs and TCS-BUH-NPs was found to be 62.52 and 76.21%, whereas 20.97% permeated from pure drug solution, as depicted in figure 3a. The formulation (i.e. TCS-BUH-NPs) was also tested in vivo in Wistar rats: a comparatively higher brain concentration (797.46 ± 35.76 ng/ml; tmax 120 min) was achieved after intranasal administration of TCS-NPs compared with that achieved after nasal administration of the pure drug solution (417.77 ± 19.24 ng/ml; tmax 60 min), as shown in the figure 3b [START_REF] Bari | Brain delivery of buspirone hydrochloride chitosan nanoparticles for the treatment of general anxiety disorder[END_REF] .

Insert Figure 3a -Insert Figure 3b 5. Gels, liquids and other routes Thiolated liquid formulations are commonly used in ocular [START_REF] Hornof | Mucoadhesive ocular insert based on thiolated poly(acrylic acid): development and in vivo evaluation in humans[END_REF][START_REF] Wagh | Thiomers: a new generation of mucoadhesive polymers[END_REF][START_REF] Zambito | Thiolated quaternary ammonium-chitosan conjugates for enhanced precorneal retention, transcorneal permeation and intraocular absorption of dexamethasone[END_REF] and nasal [START_REF] Leitner | Nasal delivery of human growth hormone: in vitro and in vivo evaluation of a thiomer/glutathione microparticulate delivery system[END_REF][START_REF] Krauland | Thiolated chitosan microparticles: A vehicle for nasal peptide drug delivery[END_REF][100][101] preparations, as low viscosity formulations are very easy to introduce and more tempting esthetically for patients. Due to comparatively higher viscosity, hydrogels provide a sustained residence time at the site of application; therefore, lidocaine hydrogels for buccal delivery using preactivated thiolated pectin were developed, and a sustained release of lidocaine was observed [START_REF] Hauptstein | Development and in vitro evaluation of a buccal drug delivery system based on preactivated thiolated pectin[END_REF]. Laffleur et al. modified hyaluronic acid with cysteine ethyl ester (HA-Cys-EE) for the treatment of dry eye syndrome and showed a 3.81-fold increase in swelling capacity, 30.5-fold improvement in mucoadhesion and 9.72-fold higher stability [102]. Hintzen et al. synthesized S-protected thiolated pectin (Pec-Cys-MNA) resistant to oxidation in solutions above pH 5. The obtained results depicted a 1.8-fold increase in disintegration time at pH 6.8 and a 6.7-fold increase in dynamic viscosity compared with unmodified pectin. Long-term stability for a liquid formulation based on Pec-Cys-MNA was also demonstrated in a 6-month experiment [103]. In another study, thiolated polydimethylaminoethylmethacrylate (PDCys) was synthesized and exhibited higher force and work of adhesion and a 3.9-fold permeation improvement in Caco-2 cells compared with the parent polymer [104]. Similar results were found in follow-up studies [22; 85; 105]. In a study on enhanced vaginal mucoadhesion, hyaluronic acid was thiolated and then pre-activated with 6mercaptonicotinamide to develop HA-Cys-MNA. Prolonged mucoadhesion, higher stability (3.6-folds prolonged disintegration time) and no toxicity were observed [START_REF] Nowak | Preactivated hyaluronic acid: A potential mucoadhesive polymer for vaginal delivery[END_REF].

Conclusion

This review shows that there are many possibilities to produce thiomers. These thiomers are promising excipient for the development of novel formulations including tablets, microparticles and nanoparticles. Actually, thiolation dramatically increase very specific properties: in drug delivery, mucoadhesion, efflux inhibition, permeation enhancement and in situ gelation are the advantageous properties of these thiomers. Due to their properties, -SH groups can in turn link to other compounds and give rise to preactivated thiomers. Finally, thiomers have been used to produce NO-donors, which we have named S-nitrosothiomers. These polymers are potential drugs and have been proposed for the treatment of Crohn's disease.

Expert opinion

Synthesis of thiomers is realized by linkage of sulhydryl bearing compounds (SBC). Several methods have been published leading to a wide selection of polymers.

Although this review can help to compare each of these polymers, making a choice can be a complex matter. The number of-SH groups and the variations of « ratio of improvement » that have been studied are important parameters but physiochemical parameters (e.g. viscosity, hydrophilicity, ability to produces particles tablets or gels, etc.) are also critical for the final formulations. In most studies, polymers are partially modified and the final thiomers could be easily compared to the starting compound: similar formulations are usually made with the thiomer and its corresponding unmodified polymer.

To date, some toxicological data show that thiomers don't show more in vitro cytotoxicity than unmodified polymers [START_REF] Sakloetsakun | In situ gelling properties of chitosanthioglycolic acid conjugate in the presence of oxidizing agents[END_REF]. Even preactivated polymers that can release 2-MNA or 6-MNA did not show more toxicity [105].

For a further development, it is important to take in account that thiomers usually correspond to mixtures of various structures. For example, oxidation of the free thiols can lead to the formation of disulfide bounds [START_REF] Sakloetsakun | In situ gelling properties of chitosanthioglycolic acid conjugate in the presence of oxidizing agents[END_REF] and an increased viscosity . In many cases, the oxidized thiomers have been reduced by NaBH 4 [START_REF] Shah | Synthesis and characterization of S-nitrosoglutathioneoligosaccharide-chitosan as a nitric oxide donor[END_REF]. This method can give « 100%-free thiols » polymers but add another step for the production of thiomers and a necessity to purify the formulation from NaBH 4 .

Additively, if amino acids are used as SBC, other unintended reactions can occur [START_REF] Riniker | Protection of Functional Groups in Peptide Synthesis[END_REF] leading to the formation of chains of 2, 3 or more amino acids. Nevertheless this kind of reaction should be put in perspective, as yield of the reaction (i.e. SBC linkage) is usually low. At a first sight, these action are not considered to be an issue but this might lead to variations in molecular weights and physicochemical properties. The use of a protecting group (also used for the synthesis of peptides) has never been tested but could probably better control the reaction of amino acids. Traut's reactant doesn't have this shortcoming.

However, obtained polymers have very interesting properties; higher degrees of modification are probably not necessary and would lead to extreme changes in properties. After rigorous characterization, they can be considered promising tools in the development of different formulations. To overcome the challenges faced in the field of drug delivery (e.g., poor bioavailability), the use of these multifunctional polymeric excipients may be a favorable approach.

To date, a medical device (Lacrimera®: chitosan-N-acetylcysteine conjugate) has been marketed and proposed in the treatment of dry eye syndrome. PCP-Cys Penicillin G 1.09 1.30 ± 0.31 [START_REF] Grabovac | Thiomers: Influence of molecular mass and thiol group content of poly(acrylic acid) on efflux pump inhibition[END_REF] PAA 450 -Cys Penicillin G 1.17 1.41 ± 0.93 [START_REF] Grabovac | Thiomers: Influence of molecular mass and thiol group content of poly(acrylic acid) on efflux pump inhibition[END_REF]

37 * 62 * 27 * 79 LegendsFigure 1 .Figure 2 .

 3762277912 Figure 1. Thiolated polymer (Thiomers) with possibilities of further reactions on the -SH groups

Figure 3 .

 3 Figure 3. (a) Ex-vivo permeation of optimized CS-NPs and TCS-NPs. (b) Comparative BUH concentration in blood and brain at different time intervals by BUH-TCS-NP solution givenintranasally. From[START_REF] Palmberger | Thiolated chitosan: Development and in vitro evaluation of an oral delivery system for acyclovir[END_REF]

  Cys-2MNA: Cross-linked hydrophobically modified Poly (acrylic acid) Cysteine-mercaptonicotinamide; CS-TGA-MNA-660: Chitosan 660 kDa-thioglycolic acid-mercaptonicotinamide; CS-TBA-MNA: Chitosan-thiobutylamidinemercaptonicotinamide; HA-Cys-MNA: Hyaluronic acid-cysteine ethyl ester-mercaptonicotinamide; Pec-Cys-MNA: Pectincysteine-mercaptonicotinamide; PAA 450 -Cys-MNA: Poly(acrylic acid) 450 kDa-cysteine-mercaptonicotinamide; TWA: Total work of adhesion; MDF: Maximum detachment force. *: preactivated polymers

Table 1 . Ratio of improvement of thiomers with their original unmodified polymers

 1

	Polymer	Property	Ratio of Improvement of	Reference
			Polymers
			Preactivated	Thiomer
			thiomer
	AC 1030-Cys-2MNA* TWA	5.7	3.4	[23]
		Adhesion Time	18.0	4.0
		MDF	2.9	2.8
	CS-TGA-MNA-660* Adhesion Time	9.0	1.8	[45]
	CS-TBA-MNA*	TWA	79.3	14.1	[53]
		Adhesion Time	150	94.5
	HA-Cys-MNA*	TWA	18.0	2.8	[54]
		Adhesion Time	10.0	2.6
	Pec-Cys-MNA*	TWA	8.9	4.8	[89]
		MDF	1.6	1.8
	PAA 450 -Cys-MNA* TWA	3.0	1.9	[22]

Table 2 . Permeability enhancement in in vitro models and permeability difference between thiomers and their corresponding original polymers

 2

	Thiomer (Permeation	Test	R Value	Apparent permeability	Ref.
	Enhancer)	Compound	(Enhancement	Coefficient cm/s (x10 -6 )
			Ratio)
	Anionic Thiomers
	PCP-Cys	Na-Flu	1.63	11.85 ± 0.5	[63]
	PCP-Cys/GSH	Na-Flu	2.93	14.64 ± 0.93	[105]
	CMC-Cys	Na-Flu	1.8	12.92 ± 0.41	[106]
	PAA 250 -Cys-2-MNA*	Na-Flu	0.81	4.68 ± 0.80	[66]
	PAA 250 -Cys-2-MNA/GSH*	Na-Flu	1.32	7.67 ± 0.23	[66]
	PSSA-MA	Na-Flu	1.11	2.54 ± 0.17	[67]
	PC1608-2MNA*	Na-Flu	1.60	1.00 ± 0.05	[67]
	PCP-Cys	SH 101	1.27	1.60 ± 0.28	[32]
	PAA 450 -Cys	SH 101	1.95	2.41 ± 0.25	[32]

* This artcle shows the development of chitosan-thioglycolic acid conjugates

PAA-Cys-2MNA (450 kDa) 12860 ± 80 58872 ± 3227 4.5 [START_REF] Iqbal | Preactivated thiomers as mucoadhesive polymers for drug delivery[END_REF] Pec-Cys-MNA: Pectin-Cysteine-Mercaptonicotinamide; CS-TGA: Chitosan-Thioglycolic Acid; CS-TGA-MNA: Chitosan-Thioglycolic Acid-Mercaptonicotinamide; PAA-cys-2MNA: Poly(acrylic acid)-Cysteine-Mercaptonicotinamide.


The order of the MAR model should be investigated.

A source that is detected to be inactive in one of the iterations can not be reactivated in this framework (Eq (2)).

Abstract: Non-iterative two-stage approaches have been used to estimate source interactions. They first reconstruct sources and then compute the MAR model for the localized sources. They showed good results when working in high signal-to-noise ratio (SNR) settings, but fail in detecting the true interactions when working in low SNR. Our framework is based on two steps. First, we estimate sources activations for a given MAR model. Then, we estimate the MAR model. We repeat the two steps until a stopping criterion is achieved. The number of time samples, T, must be greater than the maximum number of connections per source multiplied by p.

SYNTHETIC DATA

Eq (1) is an extension to MxNE [2] in which is replaced by (a matrix that changes at each iteration)

  the MxNE and could detect true activations and functional interactions.

Data•

  http://team.inria.fr/athena/ Source activity depends on the activity of sources that are anatomically connected :-Detectable from dMRI (solid line) -Neighboring regions (dashed lines) • Sources' dynamics follow a multi-variate autoregressive model of order p whose matrix entries are constant in a time window T. B. Belaoucha et al., "Multivariate Autoregressive Model Constrained by Anatomical Connectivity to Reconstruct Focal Sources," 38th Annual International Conference of the IEEE EMBC, 2016. [2] A. Gramfort et al.,"Mixed-norm estimates for the M/EEG inverse problem using accelerated gradient methods," Physics in Medicine and Biology, vol.57, no.7, pp.1937-1961, 2012. [3] D. G. Wakeman et al., "A multi-subject, multi-model human neuroimaging dataset," Scientific Data, 2015. [4] B. Belaoucha et al., "Cortical Surface Parcellation via dMRI Using Mutual Nearest Neighbor Condition," IEEE 13th International Symposium on Biomedical Imaging (ISBI), pp. 903-906, 2016.Aknowledgments This work has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (ERC Advanced Grant agreement No 694665 : CoBCoM -Computational Brain Connectivity Mapping) and French ANR contract