Lazher Zaidi
email: lzaidi@cesi.fr

Juan Antonio Corrales
email: juan.corrales@sigma-clermont.fr

Laurent Sabourin
email: laurent.sabourin@sigma-clermont.fr

Belhassen Chedli Bouzgarrou
email: chedli.bouzgarrou@sigma-clermont.fr

Youcef Mezouar
email: youcef.mezouar@sigma-clermont.fr

Grasp Planning Pipeline for Robust Manipulation of 3D Deformable Objects with Industrial Robotic Hand + Arm Systems

Keywords: 3D deformable object, grasping, robust manipulation, robot hand

In the grasping and manipulation of 3D deformable objects by robotic hands, the physical contact constraints between the fingers and the object have to be considered in order to validate the robustness of the task. Nevertheless, previous works rarely establish contact interaction models based on these constraints that enable the precise control of forces and deformations during the grasping process. This paper considers all steps of the grasping process of deformable objects in order to implement a complete grasp planning pipeline by computing the initial contact points (pregrasp strategy), and later, the contact forces and local deformations of the contact regions while the fingers close over the grasped object (grasp strategy). The deformable object behavior is modeled using a nonlinear isotropic mass-spring system, which is able to produce potential deformation. By combining both models (the contact interaction and the object deformation) in a simulation process, a new grasp planning method is proposed in order to guarantee the stability of the 3D grasped deformable object. Experimental grasping experiments of several 3D deformable objects with a Barrett hand (3-fingered) and a 6-DOF industrial robotic arm are executed. Not only will the final stable grasp configuration of the hand + object system be obtained, but an arm + hand approaching strategy (pregrasp) will also be computed.

Introduction

Nowadays, manipulation has become an increasingly important standing research topic in robotics. Most of the related works in this field consider the grasping of rigid bodies as an extensively studied area, which is rich with theoretical analysis and implementations using different robotic hands ( [START_REF] Sahbani | An overview of 3D object grasp synthesis algorithms[END_REF][START_REF] Ramón | Finger readjustment algorithm for object manipulation based on tactile information[END_REF][START_REF] Adjigble | Model-free and learning-free grasping by local contact moment matching[END_REF][START_REF] Mavrakis | Estimation and exploitation of objects' inertial parameters in robotic grasping and manipulation: A survey[END_REF][START_REF] Wang | Feature Sensing and Robotic Grasping of Objects with Uncertain Information: A Review[END_REF][START_REF] Deng | Grasping Force Control of Multi-Fingered Robotic Hands through Tactile Sensing for Object Stabilization[END_REF][START_REF] Oravcová | Clamping Fixture Design Methodology for the Proper Workpiece Insertion[END_REF][START_REF] Velíšek | Clamping Fixtures for Intelligent Cell Manufacturing[END_REF]). Robotic grasping of deformable objects has also acquired importance recently due to several potential applications in various areas, including biomedical processing, the food processing industry, service robotics, robotized surgery, etc. ( [START_REF] Faure | Sofa: A multi-model framework for interactive physical simulation[END_REF][START_REF] Long | Force/vision control for robotic cutting of soft materials[END_REF][START_REF] Nabil | Soft material modeling for robotic task formulation and control in the muscle separation process[END_REF][START_REF] Nadon | Multi-modal sensing and robotic manipulation of non-rigid objects: A survey[END_REF][START_REF] Pirník | Integration of Inertial Sensor Data into Control of the Mobile Platform[END_REF][START_REF] Dodok | Utilization of Strategies to Generate and Optimize Machining Sequences in CAD/CAM[END_REF]). Recent surveys covering the literature related to the robotic grasping and manipulation of deformable objects can be found in [START_REF] Nadon | Multi-modal sensing and robotic manipulation of non-rigid objects: A survey[END_REF][START_REF] Sanchez | Robotic manipulation and sensing of deformable objects in domestic and industrial applications: A survey[END_REF].

Nevertheless, efficient and precise (i.e., deformation/force control) robotic grasping of 3D deformable objects remains an underdeveloped area within the robotics community due to the technical and methodological difficulties of the problem. In fact, this problem involves implementing two main steps with their own subproblems: determining the initial location of the grasp points and moving the hand + arm system towards them (i.e., pregrasping strategy), and closing the fingers of the robotic hand until a robust grasp is reached while taking into account the deformation of the object (i.e., grasping strategy).

In the case of the pregrasp strategies, two main problems arise: firstly, implementing a grasp quality measure to choose the most stable grasp configuration; secondly, applying the kinematic constraints of the hand + arm system so that the chosen grasp is reachable. Several previous surveys to quantify grasp quality were developed in [START_REF] Sahbani | An overview of 3D object grasp synthesis algorithms[END_REF][START_REF] Chinellato | Visual quality measures for characterizing planar robot grasps[END_REF][START_REF] Roa | Grasp quality measures: Review and performance[END_REF][START_REF] Pozzi | On grasp quality measures: Grasp robustness and contact force distribution in underactuated and compliant robotic hands[END_REF][START_REF] Soler | Grasp quality measures for transferring objects[END_REF]. All of these approaches have studied stable grasps and developed various stability criteria to find optimal grasps for rigid objects. Nevertheless, the development of algorithms that can efficiently synthesize grasps in 3D deformable objects is still a challenging research problem. Wakamatsu et al. [START_REF] Wakamatsu | Static analysis of deformable object grasping based on bounded force closure[END_REF] analyzed the stable grasping of deformable objects based on extending the concept of force closure of rigid objects to deformable planar objects with bounded applied forces. Mira et al. [START_REF] Mira | Study of dexterous robotic grasping for deformable objects manipulation[END_REF] presented a grasp planner that can reproduce the actions of a human hand to determine the contact points. Their algorithm combines the position information of the hand with visual and tactile data in order to achieve a hand-object configuration. This method is applied to planar objects without considering applied forces by fingers and object deformation. Xu et al. [START_REF] Xu | Minimal Work: A Grasp Quality Metric for Deformable Hollow Objects[END_REF] proposed a grasp quality metric for hollow plastic objects based on minimal work in order to combine wrench resistance and object deformation. Jorgensen et al. [START_REF] Jørgensen | An adaptive robotic system for doing pick and place operations with deformable objects[END_REF] presented a generic solution for doing pick and place operations of meat pieces with a vacuum gripper. This solution used a grasp quality metric that combines wrench resistance and object deformation. It was based on the computation of the required work to resist an external wrench and it can only be applied to planar objects, without a direct relation between applied forces and deformation. In fact, none of these works can be applied to general 3D deformable objects and they are solutions for specific types of objects.

For the second step of the grasping problem (i.e., closing of the fingers), previous grasping synthesis methods are mainly based on the generation of force closure (FC) grasps ( [START_REF] Bicchi | On the closure properties of robotic grasping[END_REF][START_REF] Lei | Object grasping by combining caging and force closure[END_REF][START_REF] Rakesh | Optimizing force closure grasps on 3D objects using a modified genetic algorithm[END_REF]). This means that the robotic hand is able to maintain the manipulated object inside the palm despite any external disturbance and evaluate the contact forces including those due to friction. FC is generally used to implement the grasping of rigid objects with a low number of frictional contacts by taking into account geometric criteria of their contact surfaces. Nevertheless, traditional FC-based approaches cannot be applied to deformable objects due the complexity of their interaction with the robotic hand's fingers: the object deforms (i.e., its size and shape are changed) and the contact surfaces evolve dynamically while contact forces are applied by the fingers. New reactive methods based on the detection and avoidance of slippage have been developed in order to solve this limitation in the grasping of deformable objects. For instance, Kaboli et al. [START_REF] Kaboli | Tactile-based manipulation of deformable objects with dynamic center of mass[END_REF] computed the tangential forces required for slippage avoidance and adjusted the fingers' relative positions and grasping forces accordingly, taking into account an estimation of the weight of the grasped object. The main drawback of this approach is that the relationship between applied forces and object deformation is not considered. Zhao et al. [START_REF] Zhao | Video-based slip sensor for multidimensional information detecting in deformable object grasp[END_REF] proposed a new slip detection sensor based on video processing in order to perceive multidimensional information, including absolute slip displacement, the deformation at the object surface, and force information. This method is only applied for 2D objects with transparent and reflective surface, such as pure glass. However, these types of materials are not common in deformable objects. In addition, the stress state of the gripper-object interface is not precisely analyzed when taking into account the properties of the object's material.

In order to solve all these limitations of reactive strategies, it is crucial to use a suitable contact model ( [START_REF] Gonthier | A regularized contact model with asymmetric damping and dwell-time dependent friction[END_REF][START_REF] Dopico | Dealing with multiple contacts in a human-in-the-loop application[END_REF][START_REF] Mohammadi | Fingertip force estimation via inertial and magnetic sensors in deformable object manipulation[END_REF][START_REF] Lu | High-gain nonlinear observer-based impedance control for deformable object cooperative teleoperation with nonlinear contact model[END_REF]) that explains the coupling between the contact forces and the object deformations. A precise contact model can be exploited to provide a set of grasping forces and torques that can be applied to maintain equilibrium before and after the deformation [START_REF] Zaidi | Model-based strategy for grasping 3D deformable objects using a multi-fingered robotic hand[END_REF]. Thereby, the grasp stability can be computed in real-time while the contact surfaces evolve during deformation. This evolution of the object surface is obtained by sampling mechanical deformation models [START_REF] Arriola-Rios | Modeling of Deformable Objects for Robotic Manipulation: A Tutorial and Review[END_REF], mainly mass-spring systems ( [START_REF] Hammer | Mass-spring model for simulation of heart valve tissue mechanical behavior[END_REF][START_REF] Jarrousse | Modified Mass-Spring System for Physically Based Deformation Modeling[END_REF]), finite element methods (FEM) ( [START_REF] Garg | Grasping and manipulation of deformable objects based on internal force requirements[END_REF][START_REF] Han | Fast deformation simulation of breasts using GPU-based dynamic explicit finite element method[END_REF]), and impulse-based methods ( [START_REF] Tagawa | Manipulation of Dynamically Deformable Object Using Impulse-Based Approach[END_REF]). The mass-spring system is a fast and interactive physical model with a solid mathematical foundation and well-understood dynamics. This model provides a more realistic force/deformation behavior in the case of large deformations, while maintaining the capability of real-time response. FEM provides higher accuracy, but it requires higher numerical computation, which is more appropriate for offline simulations. Therefore, this paper uses a deformation model based on nonlinear mass-spring elements distributed across a tetrahedral mesh: lumped masses are attached to the nodes and nonlinear springs represent the edges. The tracking of the mesh nodes positions is assured by solving a dynamic equation based on Newton's second law. The main interests of this deformation model are the dynamic predictions of the object's deformations and its realistic behavior, which results from the nonlinearity of the object. More details about this object deformation model can be found in previous works [START_REF] Zaidi | Modeling and analysis of 3D deformable object grasping[END_REF] and it will be used by the new proposed grasp synthesis algorithm.

Several previous works have taken into account the deformation of soft objects while grasping them without integrating precise contact interaction models. Berenson et al. [START_REF] Berenson | Manipulation of deformable objects without modeling and simulating deformation[END_REF] presented a vision-based deformation controller for elastic planar objects without considering the grasp properties. This method takes into account just one contact point which cannot be used with a multifingered hand for grasping deformable objects. Similarly, Nadon et al. presented a model-free algorithm in [START_REF] Nadon | Automatic Selection of Grasping Points for Shape Control of Non-Rigid Objects[END_REF] based on visual information for automatically selecting the contact points between the fingers and the object's contour in order to control its shape. However, contact forces and object stability were not considered. Grasping deformable planar objects using contact analysis was proposed in [START_REF] Jia | Grasping deformable planar objects: Squeeze, stick/slip analysis, and energy-based optimalities[END_REF]. It was based on an algorithm to track the contact regions during the squeezing process and determine the stick/slip mode in the contact area. Fingers' displacements were considered rather than forces, and object stability was not guaranteed. Similarly, Sanchez et al. [START_REF] Sanchez | Blind Manipulation of Deformable Objects Based on Force Sensing and Finite Element Modeling[END_REF] developed a manipulation pipeline that is able to estimate and control the shape of a deformable object with tactile information but the stability of the grasp is not considered. Authors in [START_REF] Lin | Picking up a soft 3D object by feeling the grip[END_REF] proposed a new approach to lift a deformable 3D object, causing small deformations of the object by employing two rigid fingers with contact friction. This approach considered small deformations within the scope of the linear elasticity during the grasp operation and did not deal with the nonlinear relationships between deformations and forces of 3D soft objects. Jorgensen et al. [START_REF] Jorgensen | Optimizing Pick and Place Operations in a Simulated Work Cell For Deformable 3D Objects[END_REF] presented a simulation framework for using machine learning techniques to grasp deformable objects. In this framework, robot motions were parameterized in terms of grasp points, robot trajectory, and robot speed. Hu et al. [START_REF] Hu | Three-Dimensional Deformable Object Manipulation Using Fast Online Gaussian Process Regression[END_REF] presented a general approach to automatically handle soft objects using a dual-arm robot. An online Gaussian process regression model was used to estimate the deformation function of the manipulated objects and low-dimension features described the object's configuration. Finally, new works (such as [START_REF] Verleysen | Simpler Learning of Robotic Manipulation of Clothing by Utilizing DIY Smart Textile Technology[END_REF][START_REF] Caporali | Pointcloud-based Identification of Optimal Grasping Poses for Cloth-like Deformable Objects[END_REF]) integrate deep-learning techniques, 3D vision, and tactile information in order to fold/unfold and pick-and-place clothes. Although all these works consider the deformation of the object while manipulating it, an initial stable grasp is supposed to be known and nonlinear force-deformation relations are not computed during the handling process.

None of these works combine all the previously explained elements that are required for the implementation of a general and complete grasp planning pipeline for 3D deformable objects: a pregrasp strategy for reaching the object that takes into account grasp quality metrics for 3D deformable objects, and a grasp strategy for closing the fingers that guarantees a stable configuration of the object by considering its deformation and the applied contact forces. This paper involves an extension of our previous grasping strategy in [START_REF] Zaidi | Model-based strategy for grasping 3D deformable objects using a multi-fingered robotic hand[END_REF] in order to implement this complete grasp planning pipeline. Firstly, we adapt the stability criteria developed for robotic grasp synthesis of rigid objects to 3D deformable objects (see Section 2 for a general description of the strategy and Section 3 for the new geometric criterion of the initial grasp). We also defined a new pregrasp strategy (Sections 4 and 5) in order to approach the deformable objects in an optimal way before manipulating them by using the previous grasping metrics. This grasping strategy is based on an interaction model developed by the authors in a previous work [START_REF] Zaidi | Modeling and analysis of 3D deformable object grasping[END_REF]. Section 6 indicates how this finger-object contact model is used in order to compute the contact forces to be applied by the fingers for a stable grasp of the deformable object and what their precise values are in the proposed experiments. We consider the problem of grasping and manipulation of a 3D isotropic deformable objects with a 3-fingered Barrett hand. This hand is installed as the end-effector of a 6-DOF Adept Viper robotic arm S1700D. Finally, a complete grasping planning strategy combining the pregrasp and grasp synthesis strategies of the hand + arm system is developed and tested with real objects (Section 7).

General Description of the Grasp Planning Pipeline

As indicated in the introduction section, by using our previously developed contact model [START_REF] Zaidi | Model-based strategy for grasping 3D deformable objects using a multi-fingered robotic hand[END_REF], we can handle highly deformable objects and give precise estimations of the contact forces generated while deforming them. Those precise estimations would guarantee the static equilibrium of the object by considering new grasps metrics (Section 3) and optimizing the pregrasp configuration of the robotic hand around the object (Sections 4 and 5).

The flowchart in Figure 1 shows all the steps that are required in order to compute and execute a robust grasp (i.e., the configuration of fingers over the surface of the object) that guarantees the stability of the object with our grasp planning pipeline. In order to execute the computed grasp, the first step determines the pregrasp configuration. In fact, the complete grasp planning strategy is divided into two phases: First, determining the configuration of the robotic arm to bring the hand close to the object so that the fingers of the hand can reach the surface of the object (Section 5). The second phase is determining the appropriate initial configuration of the fingers of the hand to grasp the object based on a new geometric criterion (Sections 3 and 4). The determination of the joint parameters of the arm and the hand for executing both phases is ensured by the resolution of their inverse kinematics (IK).

Once the pregrasp step is performed, the robot moves next to the object and the fingers reach the initial grasp. No important deformation is supposed to be generated when the fingers come into contact with the object since a very small contact force will be firstly applied in order to avoid that the grasp location changes. As soon as this initial grasp is executed precisely, the third step of the algorithm is activated and an iterative closing of the fingers begins, based on the simulation of the contact interaction model (Section 6). At each iteration of the simulation, the deformations of the object are updated and the generated contact forces are computed. The computed contact forces are used to evaluate the static equilibrium of the object + fingers system. The iterative process is repeated in simulation until the static equilibrium is reached (see all these steps in simulation on the left side of the flowchart in Figure 1).

Upon completion of the simulation process, the real handling and manipulation of the object can be performed by executing the contact forces obtained from simulation (see steps on the right side of the flowchart). Firstly, the object is installed at the workspace of the robot with the same relative configuration as at the beginning of the simulation. Then, the robotic arm moves towards the pregrasp configuration (i.e., phase 1 in reality) and the hand fingers are moved towards the initial contact points by position control (i.e., phase 2 in reality). At this stage (beginning of phase 3 in reality), the fingers are closed by force control and apply progressive squeezing of the object until the contact forces (which are measured by contact sensors installed inside each finger of the Barrett hand) become equal to the contact forces computed at the simulation step. These contact forces guarantee the equilibrium of the object-hand system (as validated by the simulation). Thereby, the object can be lifted up from the table without any risk of sliding and can be robustly manipulated. In the next sections, all these steps are described in detail and are validated with real pick-and-place experiments of deformable objects by a hand + arm robot system (Section 7).

Synthesis of the Initial Grasp Configuration

In this section, we address the characterization of the object stability for a three-fingered grasp. This implies determining a force closure configuration based on the choice of three contact points from the set of all the points representing the outer 3D surface of the object. The following assumptions are considered in this procedure:

• The use of three fingers for the grasp operation modeled as hemispheres with radius R.

• The first contacts between the fingers and the object are point contacts.

• The outer surface of the object is represented by a set of points Ω, which are described by their position vectors → P i measured with respect to a reference frame located at the center of mass ( → P co ) of the object.

In fact, a 3-fingered grasp is more reliable in terms of stability, slip avoidance, and balance of forces when it converges towards an ideal equilateral grasp [START_REF] Roa | Grasp quality measures: Review and performance[END_REF]. Thus, each three-fingered grasp can be characterized by a value which represents its similarity to an equilateral triangle. We suggest an algorithm based on geometric criteria to find this equilateral grasp. This algorithm determines, at first, the set of all possible grasping triangles by scanning the points belonging to the contact surface Ω. Then, using the Q 1 criterion presented by Roa [START_REF] Roa | Grasp quality measures: Review and performance[END_REF], our algorithm compares the angle values (alpha, beta, and lambda) of these triangles with π/3 (i.e., the angle that characterizes an equilateral triangle) in order to choose the configuration closest to an equilateral triangle (see Figure 2):

Q 1 = 3 2π α - π 3 + β - π 3 + γ - π 3 . (1)
For a target angle value of 3/2π, the minimum possible value of Q 1 is 0 for a perfect equilateral triangle, 2 for the most unfavorable case (the triangle degenerates into a segment), and 1 for an intermediate state.

For evaluating the angles of the triangle, we use a margin of error defined by the interval [0, 0.3]. Depending on the density of the points of the contact surface Ω, this algorithm can give several grasp configurations. Finally, a second criterion, referred to as Q 2 , is used in order to choose one between them. It measures the distance between the center of the mass of the object C O and the center of the grasping triangle C T (Figure 3), defined as

Q 2 = |C O -C T | .
(2) This criterion was developed to obtain stable grasps with respect to the torques generated by gravitational and inertial forces [START_REF] Roa | Grasp quality measures: Review and performance[END_REF]. The complete grasp synthesis algorithm based on the these two criteria

Pregrasp Strategy for the Robotic Hand: Placement of the Fingers

In this section, we present the pregrasp strategy for reaching the three contact points of the initial grasp computed in Section 3 with a three-fingered robotic hand (e.g., the Barrett hand [START_REF] Hasan | Modelling and Control of the Barrett Hand for Grasping[END_REF] in our experiments). In order to describe the kinematic model of the Barrett hand, we establish the joint frames of Figure 4. The movement of the fingers of the Barrett hand is controlled by defining 4 encoder values [q M1 . . . q M4 ] that are connected to the joint values ϕ i of the fingers (Figure 4) by the following coupling relationships:

              φ 98 φ 109 φ 1211 φ 1312 φ 147 φ 1514 φ 87 φ 117               =               1/125 0 0 0 1/375 0 0 0 0 1/125 0 0 0 1/375 0 0 0 0 1/125 0 0 0 1/375 0 0 0 0 2/35 0 0 0 -2/35               *      q M1 q M2 q M3 q M4      , (3)
where q Mk = 0 17, 500 for k = 1 2 3 (i.e., the range of variation of the encoders of the motors for closing/opening the three fingers), and q M4 = 0 3150 (i.e., the range of variation of the encoder of the motor 4 for separating fingers 1 and 2 around the palm).

The pregrasp strategy for the three-fingered robotic hand involves two steps: the first one aligns the orientation of the hand with the initial grasping triangle computed in Section 3; the second one adjusts this initial estimation of the grasp by taking into consideration the kinematic constraints of the hand. The first step consists of orientating the hand so that the TCP (Tool Center Point, defined along the axis perpendicular to the palm of the hand) coincides with the C T grasp triangle center (as shown in Figure 5). The second step involves the resolution of the IK of the fingers in order to estimate the joint values to reach the three grasp points (P 1 , P 2 , and P 3 ) of the initial grasp triangle. If no IK solution is found, the length of the TCP line is changed and the IK resolution is recomputed. When an IK solution is found, the motor commands q Mk are computed in order to attend to these joint values, by applying the inverse expression of (3). The implemented algorithm is shown in the diagram "Algorithm 2". In the context of this iterative solution search, we consider that the solution is validated for a value of joint angle difference for each of the fingers smaller than 2 degrees. 3. Searching for a solution for q M1 , q M2 , and q M3 by successive iteration over the length L in order to find a solution. 4. Reorientation of the hand around TCP and adjustment of the distance L to define a solution for q M4 .

Output: Definition of the values of [q M1 . . . q M4 ] and the 6 components (position + orientation) of the tool frame defining the hand's TCP.

Pregrasp Strategy for the Hand + Arm System: Reaching Trajectory by the Arm

After establishing the initial grasp points of the fingers (Section 3) and the corresponding pregrasp configuration of the hand around the object (Section 4), we should execute the latter by moving the hand with a robotic arm that carries it. Figure 6 shows the flowchart of the pregrasp strategy for the hand + arm system that uses the IK of the arm and the hand for achieving this initial grasp. For implementing this planning strategy, we have chosen a 6-DOF robotic arm (e.g., a Viper S1700D for our real experiments) so that the hand can attain any pose (position + orientation) in the space around the object. Initially, the object is resting on a table, modeled by a plane parallel to the XY plane of the arm base.

For performing this pregrasp strategy, we should represent the object pose (position [x, y, z] in mm and orientation [y, p, r] in ZYZ Euler angles) from its original WobjO frame (object mark) to the WobjU frame (user mark), known by the arm (see Figure 7). The TCP pose will also be represented firstly in the WobjO frame in order to easily define relative movements between the object and the hand. Later, it will be transformed into the arm frame (WobjU) so that final movements of the arm can be computed.

Firstly, we want to reach the relative object + hand configuration of Figure 8, which consists of putting the TCP in coincidence with C 0 (see Figure 9b). Then, we use the results from Section 3 to determine the contact points. We calculate the grasp triangle formed by these three contact points as well as its centroid C T . Finally, we define the final pregrasp configuration (defined in Figure 5) by the translation and orientation of the hand. This step consists of matching the two TCP and C T points and aligning the Z axis of the TCP coordinate system perpendicular to the grasp triangle (see Figure 9c) by computing the arm movements required to reach the tool frame obtained by Algorithm 2. When this final pregrasp configuration is reached by the hand, the fingers will be closed by applying the joints' angles obtained with Algorithm 2. An example of the main execution steps of this pregrasp strategy for a cube is shown in Figure 9.

Final Stable Grasp Execution Based on Contact Interaction Modeling

In order to be able to evaluate the equilibrium with all the forces that take place in the real experiment while the fingers of the hand are closing, a model of the contact interactions between the fingertips and the object is required for precisely determining the forces-deformations relationship during grasping. This model will receive as input the initial grasp of the object without any deformation (i.e., the initial contact points of the fingertips over the object's surface or contact polygon determined at Section 3 and the fingers' configuration of the pregrasp strategy are described in Sections 4 and 5).

After this initial grasp, the fingers should close iteratively towards the center of the object until the equilibrium of all forces over the object is reached and a stable grasp is performed. With the aim of computing the required fingertip forces for this equilibrium, the simulation of this contact model is achieved in several sequential steps. Firstly, the contact between fingers and the object is detected. Secondly, contact forces are computed (by the evaluation of the relative velocity between the fingers and the object). Finally, the static equilibrium is checked for grasping stability. At each iteration of the simulation, the dynamic model updates the overall shape and contact area deformations due to the applied forces. Contact forces are individually calculated for each contact point, which gives a realistic distribution of the contact pressure in the contact zone. The contact model takes into account the normal forces and the two modes of the tangential forces due to friction: slipping and sticking modes. As the object is modeled by a set of nonlinear spring-damper pairs, the nonlinear normal force is given by

f n j = 0 , δ j ≤ 0 max(0, (Kδ n j + C δj )) (4)
f n j = f n j n f j , (5)
where δ j is the penetration distance measured along the normal direction to the contact surface j; K and C are the contact stiffness and damping constants, respectively. Those parameters depend on Young's modulus and the Poisson ratios. Based on the work presented in the literature [START_REF] Shi | Measurement and modeling of normal contact stiffness and contact damping at the meso scale[END_REF][START_REF] Johnson | Contact Mechanics[END_REF], the stiffness constant is calculated by

K = 2Ψ √ R, (6)
where Ψ is a constant, calculated according to the mechanical properties of the two objects in contact, and it is given by

Ψ = 1 1-v 2 1 E 1 + 1-v 2 2 E 2 , (7)
where E 1 and E 2 are the Young's modulus; v 1 and v 2 are the Poisson ratios of the finger and the object, respectively. The damping constant is determined by

C = 4πRγ, (8)
where γ is a constant.

To characterize these mechanical parameters, compression tests are done for obtaining curves of stress evolution according to compressive deformation (see [START_REF] Zaidi | Model-based strategy for grasping 3D deformable objects using a multi-fingered robotic hand[END_REF] for photos and detailed results of this calibration process). Thereby, the Young's modulus (4.928 MPa) and the Poisson's ratio (0.39) of the material of the objects (i.e., foam) to be grasped during the real experiments in Section 7 are identified.

Modeling of the tangential force, acting along each contact, is required in order to prevent any slippage and to ensure grasp stability. The contact model takes into account the two modes of the tangential forces due to friction: slipping and sticking modes. In this model, a parallel spring-damper is attached to the ground at one end, via a slider element, and to the fingertip at the other end. Therefore, the contact point location dynamically changes in the slipping condition. These variables correspond to the relative displacements at the contact point due, respectively, to sticking and slipping. They are dynamically reset to zero if the contact is broken. The sliding friction force can be defined in terms of the Coulomb law as follows:

f slip = 0, V t = 0 -µ f n V t V t , (9)
where µ, f n , and V t are friction coefficient, normal force, and tangential velocity, respectively. With this model, the contact point location dynamically changes in the slipping condition. If the tangential force norm is less than the threshold of sliding, then we have a sticking mode whose tangential force is defined by

f stick = 0, v = 0 -(k t v -c t v) P c P s P c P s , v > 0 , ( 10
)
where v is the tangential deformation at the contact facet, P c is the contact point position, and P s is the contact point position at the sticking regime. The parameters k t and c t are respectively tangential stiffness and damping coefficients estimated by using Dopico's method [START_REF] Dopico | Dealing with multiple contacts in a human-in-the-loop application[END_REF].

As indicated at the beginning of this section, this contact model will be executed after the initial grasp configuration (obtained as output of the pregrasp strategy) is reached and the fingers begin to touch the surface of the object. In fact, a simulation of the grasping execution strategy is implemented in Matlab in order to determine the contact forces that should be applied to reach a robust grasp. This simulation will iteratively close the fingers towards the center of the object, evaluate the forces transmitted to the object by solving the contact model, update the state of the contact surface due to deformations, and verify the static equilibrium of the hand + object system. When this static equilibrium is obtained in the simulation, the contact forces applied by the simulated fingers will be used as references for the force control of the real robotic hand. An example of this grasping execution strategy will be shown in Section 7 in order to justify its application in our pipeline. Nevertheless, more experiments of this grasp execution strategy can be found in the previous work [START_REF] Zaidi | Model-based strategy for grasping 3D deformable objects using a multi-fingered robotic hand[END_REF] by the authors, where it was first proposed.

Experimental Validation of the Grasp Planning Pipeline for the Hand + Arm System

The different steps of our complete grasp planning pipeline are validated with simulation and real experiments. First of all, the synthesis algorithm for the first initial grasp configuration proposed in Section 3 (i.e., Algorithm 1) is validated for three different shapes: a sphere, a cylinder, and a cube (Figure 10). This algorithm has been implemented in Matlab, by integrating a discrete 3D tetrahedral representation of the shape of these objects. The number of nodes of this 3D representation and the values of the two grasping quality metrics computed by the algorithm for the three objects are shown in Table 1. The initial grasping points obtained are shown in Figure 10 and they do not consider any deformation of the objects. These initial grasp configurations are used as input for the pregrasp and grasp execution steps of the grasp planning pipeline. For validating both steps, real experiments with the Adept arm and the Barrett hand grasping a foam cube (Figure 11) and a cylinder (Figure 12) are performed. Both objects represent typical shapes found in grasping applications of industrial products: a planar-faced surface (e.g., a cube) and a revolution surface (e.g., a cylinder). After the grasping, the object is lifted and carried to another position over the table in order to validate the stability of the grasp. The first step consists of generating and executing the path of the Adept arm for the pregrasp configuration in simulation, ensuring that the object is within the Barrett hand workspace (as explained for the cube in Figure 9a-c). The trajectory of the arm and the closing of the hand are then performed in the simulation (Figure 9d) until the fingers come into contact with the object.

The second step is to model the interaction between the fingers and the object for closing the fingers against the object surface in order to get the required contact forces in simulation (as explained in Section 6). The real contact forces at each finger are obtained by strain gauges installed in the phalanxes of the Barrett hand. Thereby, the Barrett hand is iteratively closed until the contact forces obtained from the simulation of the contact model are attained (as shown in Figure 13). Thus, a stable final grasp of the deformable object is obtained and the object can be lifted without any danger of falling down. Figure 13 shows the variation in the norm of the three forces applied to the cube during the simulation of the contact model. At the initial step (t 0 ) of the simulation, the fingers move towards the object in order to reach the initial contact points. When the fingers come into contact with the object (t 1 ), the contact model starts to evaluate the applied forces. This figure also shows that the contact forces are continuous and directly related to the amount of local deformation inside the contact area. The fingers continue to apply forces until equilibrium conditions are satisfied (i.e., reaching the three fingers' contact forces thresholds, identified by horizontal dashed lines in Figure 13). Once this state is reached and the three force thresholds are obtained, the fingers should keep these contact forces in order to ensure stable handling and then the simulation can stop. The relation between these force thresholds and penetration of the fingers inside the object (i.e., the deformation of its surface) is obtained by the simulation of the contact interaction model while closing the fingertips over the object. The real relation force-penetration is very similar to the one computed in the simulation (see real validation experiments in [START_REF] Zaidi | Model-based strategy for grasping 3D deformable objects using a multi-fingered robotic hand[END_REF]). Thereby, our strategy of applying the contact forces obtained from simulation in real grasping is suitable. In fact, the mean error of the contact force obtained by our model is of 0.3N, which implies that only 15% of typical contact forces is required for grasping most common life deformable objects. In addition, the nonlinear relation force-penetration shows that simplified linear models cannot be used for grasping significantly deformable objects and justifies the necessity of the proposed model. The proposed grasp planning strategy is validated by the correct execution of two pick-and-place tasks (for a cube and a cylinder): lifting and stable transportation of both objects with the real robot without any slippage (as shown in the real photographs of Figures 11 and12).

Conclusions

This work has been devoted to the complete process of deformable object grasp planning with an industrial manipulator (robotic hand + robotic arm). First of all, this requires the definition of the grasp points over the surface of the object and thus, the determination of an initial grasp configuration. Later, we presented our approach for determining the pregrasp strategy for reaching this initial grasp by taking into account the kinematic constraints of the arm and the hand. Thereby, the grasp planning has been divided into two phases: firstly, determining the configuration of the robot arm to bring the hand close to the object; secondly, determining the appropriate configuration of the fingers of the hand to grasp the object. The determination of the joint parameters of the arm and the hand is ensured by the resolution of their inverse kinematics. In the case of the hand, the implemented strategy consists first of putting the TCP (line perpendicular to the palm of the hand) at an intersection with the center of the grasp triangle (obtained in the initial grasp synthesis) and aligning it with the normal vector of this grasp triangle. The second step is to iteratively search for a solution to the articular parameters of the three fingers of the Barrett hand in order to bring them into correspondence with the grasping points previously defined by force-closure-type stability conditions. Thereby, this final grasp solution combines the initial grasp based on a general geometric criteria for force closure (i.e., it only considers the shape of the object) with the kinematic constraints of the robotic hand by applying the proposed pregrasp strategy. Nevertheless, this grasp configuration does guarantee the robustness of the grasp only for rigid objects but not for deformable objects. Thus, a force-deformation serving scheme is activated when the fingers come into contact with the deformable object to be grasped. This scheme is based on a simulation of the object-fingers interaction (previously developed by the authors in [START_REF] Zaidi | Model-based strategy for grasping 3D deformable objects using a multi-fingered robotic hand[END_REF]), which obtains the contact force thresholds that should be applied by the fingers in order to guarantee a stable grasp while deforming the surface of the object. Finally, an iterative closing of the real fingers is performed in order to attain these contact forces with the real object.

Objects with simple geometrical shapes (sphere, cube, and cylinder) were chosen for the experiments to more efficiently simulate the interaction model described in Section 6. The proposed pregrasp/grasp pipeline is not restricted to simple shapes and can be applied to any shape of object. In fact, the mesh construction process required by the interaction model has been applied by the authors to very irregular shapes of meat pieces in a previous work [START_REF] Long | Force/vision control for robotic cutting of soft materials[END_REF]. This grasp planning pipeline is designed for nonlinear deformable objects to be manipulated by three-fingered robotic hands, but its architecture could be redesigned as a group of interchangeable modules in future works. These modules could include the following: other types of objects (e.g., changing the contact interaction model for rigid objects, granular media [START_REF] Mateo | A Manipulation Control Strategy for Granular Materials Based on a Gaussian Mixture Model[END_REF]), new sensing data (i.e., combining tactile and vision information for estimating the 3D shape of the object and the real contact forces in real-time [START_REF] Sanchez | Online Shape Estimation based on Tactile Sensing and Deformation Modeling for Robot Manipulation[END_REF]), multiple robotic fingers (e.g., dexterous robotic hands [START_REF] Aranda | Deformation-based shape control with a multirobot system[END_REF]), or even more complex manipulation tasks (e.g., in-hand manipulation [START_REF] Corrales Ramón | Multi-fingered robotic hand planner for object reconfiguration through a rolling contact evolution model[END_REF]).

Figure 1 .

 1 Figure 1. Flowchart of the proposed grasp planning pipeline (i.e., pregrasp and grasping strategies) for handling deformable objects with a hand + arm robot system.

Figure 2 .

 2 Figure 2. Force arrangement criteria.

Algorithm 1 Figure 3 .

 13 Figure 3. The distance between C O and C T .

Figure 4 .

 4 Figure 4. Barrett hand: dimensions, frames, and joints.

Figure 5 .Algorithm 2

 52 Figure 5. Intersection of the hand's Tool Center Point (TCP) with the center C T of the grasp triangle.

Figure 6 .

 6 Figure 6. Flowchart for the pregrasp planning strategy of the hand + arm system. IK-Inverse Kinematics.

Figure 7 .

 7 Figure 7. Main frames of the hand + arm system, simulated in Matlab.

Figure 8 .

 8 Figure 8. Intersection of TCP and C 0 .

Figure 9 .

 9 Figure 9. Main execution steps of the pregrasp strategy for the hand + arm system in order to grasp a cube: (a) initial robot-object configuration, (b) reaching trajectory (intersection of TCP and C O ), (c) hand orientation (intersection of TCP and C T ), (d) pregrasp configuration.

Figure 10 .

 10 Figure 10. Configuration of the initial grasping points for a sphere, a cube, and a cylinder.

Figure 11 .

 11 Figure 11. Pick-and-place of the cube by the Adept robot and the Barrett hand: (a) grasping, (b) lifting, and (c) leaving.

Figure 12 .

 12 Figure 12. Pick-and-place of a cylinder by the Adept robot and the Barrett hand: (a) grasping, (b) lifting, and (c) leaving.

Figure 13 .

 13 Figure 13. Grasping forces applied by the three fingers of the Barrett hand (F1, F2 and F3) in simulation.

Table 1 .

 1 Evaluation of the initial grasping criteria for a sphere, a cube, and a cylinder.

		Sphere	Cube	Cylinder
		(80 mm)	(80 * 80 * 80mm)	(R35mm * L140 mm)
	Number of nodes	182	162	226
	Q1	0.0491	0.2302	0.1145
	Q2 (mm)	4.8659	13.334	11.6667

Funding: This work has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement nº 869855 (Project 'SoftManBot').

Acknowledgments: L.Z. thanks the research support by the LINEACT EA 7527 laboratory at CESI Rouen.

Conflicts of Interest:

The authors declare no conflicts of interest.


Nathalie Solladié
email: nathalie.solladie@lcc-toulouse.fr

Régis Rein

Souhaila Bouatra

Sonja Merkas

Chloé Sooambar

Ivo Piantanida

Nucleosides as Organizing Agents

Keywords: porphyrins, nucleosides, supramolecular interactions, pre-organization, molecular recognition

In the present review, we discuss the results of the research we have done over the last decade on multi-porphyrinic architectures bearing a nucleotidic backbone, compare all the data, offer a novel overview and emphasize common capabilites of nucleosides appearing in various systems, providing a new insight on the preorganizing capabilities of nucleosidic backbones. The efficency of nucleosides as preorganizing agents was investigated through the synthesis and study of various bis-porphyrins bearing nucleosidic linkers, as well as through the investigation of the conformation of linear and arborescent multi-porphyrins constructed on a nucleosidic backbone. The capacity of these molecules to complex guests with a high association constant was used as a tool to evaluate their degree of pre-organization, as well as the investigation of the electronic coupling existing between their chromophores and their photo-chemical capacities. Such an overview of one decade of scientific investigations documents the fact that rigid linkers between chromophores are not necessary for their spatial pre-organization, and open new routes to the faster synthesis of flexible highly pre-organized molecular architectures avoiding the long and tedious synthesis of rigid tweezers, especially for the preparation of rigid linkers which may bring solubility and stability problems.

INTRODUCTION

Exerting control over the position and distance of the porphyrins along our specifically designed oligonucleotidic scaffold was our main goal over the past years. Elaboration of non-covalent multiporphyrinic assemblies represents an important challenge in relation to the mimicry of bacterial photosynthesis [1][2][3][4] and the design of functional molecular materials. [5] Axial coordination to the central metal ions of porphyrins by suitable ligands is an effective and widespread strategy for the assembly of non-covalently bound multi-component systems. Zn (II) derivatives have been extensively used for this purpose, [6][7][8] and examples based on Ru(II), Co(II), Rh(III) porphyrins can be found in the literature. [9][10][11][12] In the present review, we discuss the results of the research we have done over the last decade on multiporphyrinic architectures bearing a nucleotidic backbone, compare all the data, offer a novel overview and emphasize common capabilites of nucleosides appearing in various systems, providing a new insight on the pre-organizing capabilities of nucleosidic backbones. All along this review, we try to highlight how nucleosides determine both conformation and properties of various molecular architectures where porphyrins are held together by nucleosidic backbone. Tweezers as well as star-shaped and linear multi-porphyrinc arrays are discussed.

Over the years, we first focused on cofacial bis-porphyrin tweezers and host/guest interactions to investigate the possibility to obtain self-coordinated molecular systems with predictable spectral and redox characteristics. We reported some years ago the synthesis of a bis-porphyrinic tweezers bearing a tris-anthracenic spacers. [13] Specifically, a tris-anthracene spacer was chosen in order to facilitate a cofacial orientation of the chromophores while allowing free rotation around the acetylene axis, thus adjusting the cavity to a large variety of guests. Indeed, this adjustable tweezers is able to complex both small and large bidentate bases, as well as photoactive guests such as bipyridyl-porphyrins. [14] However, synthesis of rigid tweezers can be long and tedious, especially for the preparation of rigid linkers which may bring solubility and stability problems. Considering that properly functionalized nucleosides adopt a more or less blocked conformation, [15][16] we decided to synthesize three bis-porphyrins using either uridine or 2'-deoxyuridine derivatives as spacers, expecting to bring the porphyrins in a face-to-face orientation. [17] These three dimers differ by the positions where the chromophores are attached, either on the ribose or on the nucleic base. For two of these three bis-porphyrinic nucleosides, their binding studies with the bidentate base DABCO showed that the nucleoside confer sufficient preorganization to the dimers to enhance the association constants, documenting the fact that rigid dimers are not necessary for preorganization, and opening up new routes to the faster synthesis of flexible tweezers capable of complexing guests with a high association constant. Such a capacity of nucleoside to pre-organize the porpyrins attached to it was also observed in a star-shaped mutli-porphyrins with chiral and enantio-pure nucleosidic linkers we synthesized in the past. [18] If the configuration is indeed a star-shaped nanomolecule, it appears that the induced conformation is nothing as expected. The four peripheral Zn(II) porphyrins collapse over the free-base central one, what induce totally different photo-physical properties. Due to this induced conformation and despite a minor expected light energy harvesting behavior, the principal capability of this system is to quench the collected light energy.

The gain in stability and the enhancement of the association constants observed for the complexation of bidentate base such as DABCO by bis-porphyrins using either uridine or 2'-deoxyuridine derivatives as spacers can be even greater when a dinucleotide linker is used. These results document the fact that dinucleoside confer an even greater preorganization to the dimers to enhance the association constants. Our last challenge was thus to synthesize an oligonucleotide with pendant porphyrins, using a modified oligonucleotidic backbone to target a parallel conformation of the porphyrins appended to each deoxy-uridine moiety. To provide a rigid environment for the porphyrins, the uridine units were coupled in 3'-5' stepwise fashion using ether-ester type of spacer of suitable length, and porphyrins were anchored to the uridine by means of robust carbon-carbon bonds. The investigation of the conformation of the four specifically designed uridine-porphyrin conjugates (a monomer, dimer, tetramer and octamer) was realized through a comprehensive conformational analysis by NMR spectroscopy. The obtained results testify in favor of the existence of an anti glycosidic angle and suggest a slipped cofacial conformation for the porphyrin macrocycles. [19]

RESULTS AND DISCUSSION

BIS-PORPHYRINIC TWEEZERS WITH A SIMPLE NUCLEOSIDIC LINKER

As part of our studies concerning the elaboration of pre-organized systems, we synthesized some years ago three new bis-porphyrins bearing a flexible nucleosidic linker (1, 2 and 3 in Figure 1.) expected to adopt a tweezer like conformation which should facilitate the complexation of bidentate Lewis bases. These dimers differ by the attachment positions of the two porphyrins. For bis-porphyrinic dimers 1 and 2, both porphyrins are attached to the ribose, in the O- = 2.4x10 -5 M), which resulted as expected in a significant red-shift of the Soret and Q bands in the porphyrin (Figure 2). [22][23] The association constants for the formation of the complexes were calculated from UV-visible spectroscopic data. A value of 2.5x10 4 M -1 was found for tweezers 1, which is similar to the association constant of DABCO with a reference Zn(II) porphyrin 5,10,15,20-tetra-di-tert-butyl-phenyl-porphyrin A 4 (2.5x10 4 M -1 ), thus indicating as expected that no well-defined pre-organized conformation exists for bis-porphyrin 1. However, much higher binding constants were found for the formation of host/guest complexes between tweezers 2 and 3 and DABCO. Values of 6.3x10 5 M -1 and 2.0x10 6 M -1 were found. These association constants are increased by 1.5 or 2 orders of magnitude as compared to the association constants of the same bidentate ligand with the above cited reference Zn(II) porphyrin. This enhanced stability of the complex may be ascribed to a pre-organization of the bis-porphyrinic tweezers 2 and 3 forming a cavity and provides convincing evidence that the bidentate base is inserted into the cavity of both dimers via host/guest interactions. Such results were expected for tweezers 3 since it is known that the anti-conformation is blocked for uridine in the presence of bulky substituents on the nucleic base, providing more favorable conditions for the second porphyrin to come in a face-to-face orientation. However, no explanation could be found to explain such a high complexing capability of tweezers 2.

The adjustment of the cavity to bidentate guests of various size was investigated for tweezers 2 and 3 through binding studies realized in dichloromethane between these dimers and 4,4'-bipyridine. The complexations were monitored by UV-visible spectrophotometric titration of a solution of tweezers in CH 2 Cl 2 ([2] = 2.9x10 -5 M, [3] = 2.2x10 -5 M), which resulted as expected in a significant red-shift of the Soret and Q bands in the porphyrin. The association constants for the formation of the complexes were calculated from UV-visible spectroscopic data. Values of 1.3x10 5 M -1 and 3.2x10 5 M -1 were found for tweezers 2 and 3 respectively. These binding constants are increased by 1 or 1.5 order of magnitude as compared to the association constants of the same bidentate ligand with a reference Zn(II) porphyrin 5,10,15,20-tetra-di-tert-butyl-phenyl-porphyrin A 4 (7.9x10 3 M -1 ), indicating an enhanced stability of the complex, which may be ascribed to a pre-organization of the bis-porphyrinic tweezers 2 and 3 forming a cavity. These results document the insertion of the bidentate base into the cavity of the dimers via host/guest interactions, and clearly demonstrate the adjustment of the cavity to the size of the complexed guest.

STAR-SHAPED PENTA-PORPHYRIN WITH CHIRAL AND ENANTIOPURE NUCLEOSIDIC LINKERS

The study of a pentaporphyrin structure FB-Zn 4 (7) with a central free-base porphyrin and four peripheral zinc porphyrins connected via a chiral and flexible nucleosidic linker was undertaken in order to determine its potential use as an antenna system, able to collect light energy and to convey it to the energy sink in the central position. [18,24] The peripheral zinc porphyrin chromophores can extensively absorb light throughout all the visible range and convey the stored energy to the central free-base porphyrin. The system can effectively work as an antenna if the absorbed light energy is efficiently conveyed to the central porphyrin and no competitive loss of the absorbed energy occurs. The model systems ZnU (4), FBS ( 5) and the dyad FB-Zn ( 6), reported in Figure3, were synthesized to help in the elucidation of the photo-physical behavior of the pentad system FB-Zn 4 (7).

Fig. 3. The component models ZnU (4) and FBS ( 5), the dyad FB-Zn ( 6) and the pentad FB-Zn 4 (7).

The absorption spectrum of the dyad FB-Zn (6) closely matches the superposition of the two model components. This is indicative of a very modest interaction between the components, as expected from the length and the type of bridging ligand. Unlike what found for the dyad, the absorption spectrum of the pentad FB-Zn 4 (7), appears different from the simple superposition of components unit. Figure 4a reports the absorption spectrum of the pentad compared with the sum of the components. Large effects on the Soret band, namely broadening, decrease in epsilon and a minor red shift are observed and in addition, a rather important change in epsilon, a red shift and a change in spectral shape in the Q bands region are also detected. Whereas some change in the Soret band region is generally encountered whenever porphyrins are connected to other species, the effect on the Q band region is rather unusual. This fact can be taken as indication of a rather important coupling between components. Since the type and length of the bridge is not expected to couple the components to such an extent, a through space interaction can be envisaged to explain the observation. This must occur in a conformation where the components are in close contact and which can only be possible in FB-Zn 4 (7) and not in FB-Zn (6), since in the latter no such interaction was evidenced.  / nm has been stabilized by solvent interactions and accounts for the 70% of the population. A schematic illustration of the two conformers, compatible with both the experimental findings and the theoretical results, is reported on Figure 5a.

Both conformers are in a bent configuration, but whereas in isomer k, the two porphyrins are nearly overlapping, with a distance between the donor-acceptor couple of the order of 20-23 Ǻ, sufficient to promote a dipole-dipole type energy transfer process. In the conformer in the open conformation, h, the planes containing the porphyrins are displaced and the center to center distance is increased to value of ca. 32 Ǻ, distance which does not allow the energy transfer to take place effectively. In the pentad, FB-Zn 4 (7), selective excitation of the free-base component at 650 nm, indicates a decrease of the emission quantum yield of this component compared to the model. Under these strong coupling conditions, quantitative consideration on the emission quantum yield of the single components in the pentad as done for the dyad (6), is rather unreliable. This type of determination, in fact, strongly relies on the fact that the components do not substantially change their properties when bound in the arrays. This is not the case, as evident both from the ground state absorption spectrum of Figure 4a and from the free base porphyrin emission yield, which appears decreased in the pentad.

Nonetheless, a qualitative experiment comparing the overall emission quantum yield of optically matched solutions of the dyad and the pentad is reported in Figure 4b. It shows that the emission in the higher order array is strongly diminished both on the zinc porphyrin band and on the free base band, so it cannot be ascribed to an increased efficiency of the intramolecular energy transfer process, but rather to an overall increase in the non-radiative component decay involving both components of the pentad, i.e. the free base-and the Zn porphyrin. In spite of the fact that the overall yield of the luminescence of the pentad is extremely low, sensitization of the free base by the zinc component still occurs, as evidenced by the excitation spectrum registered at 720 nm, a wavelength of free-base porphyrin emission. In this excitation spectrum there is a clear contribution to the free base emission from the zinc porphyrin donor band at 550 nm, its extent is partial as in the dyad FB-Zn ( 6), and should be ascribed to the fact that some conformations in Fb-Zn 4 (7) do not contribute to energy transfer. Though, in presence of sensitization of some free-base emission, as proved by the excitation spectrum, the overall emission of the free-base porphyrin does not appear to be sensitized but rather to decrease. At the same time we determined a quenching of the zinc porphyrin, without detecting a corresponding sensitization in the free base acceptor. [25] In order to explain the observation we have to assume that a remarkable loss of emission from the zinc porphyrin component, as well as from the free-base porphyrin unit, must take place through some non-radiative path.

Some strong interacting conformer could account for the observed phenomena, namely non additivity of absorption, dramatic decrease in emission quantum yield of the pentad compared to the dyad and a new fast component (110 ps) in the quenching of the luminescence. This conformation should be precluded in the dyad, where the above effects were not observed. For the pentad a series of different conformations can be envisaged, some are reported in Figure 5b.

Whereas conformation x (fully extended) and conformation y (partially bent) are resembling and should behave similarly to the conformations of the dyad, the conformation resulting from the concomitant bending of two (or even more) Zn porphyrins over the free base component, denoted as z or z1, is unique to the pentad. Other types of conformations unique to the pentad, as the j, with the Zn porphyrins in a parallel arrangement, can be ruled out. It seems, in fact, unlikely that this type of conformation could affect the radiative parameters of the free base component which is not involved in the interaction. Type z conformers appear therefore as the most probable candidates responsible for the strong interaction detected in the pentad and this will gain further support from the data presented in the following sections. The interpretation of the data can be attempted in a rather simplified frame as follows. The pentad is assumed to take (at least) the three different types of conformations denoted as x, y and z. Whereas conformation x and y behave "regularly" i.e. similarly to the corresponding bent or extended conformations in the dyad, as confirmed by the similar lifetimes detected in the luminescence quenching of the Zn porphyrin donor, the z type conformations can be responsible for the different behavior of this array compared to the dyad. To this type of conformation is assigned the short luminescence lifetime of the zinc porphyrin component (110 ps) which, unlike the 240 ps component assigned to an energy transfer process in conformation y, has to be regarded as a fast, non-radiative decay of the excited state. In this type of conformation, an electronic, exciton type interaction involving also the freebase component can occur, and this induces a strong perturbation of the individual parameters of the components, resulting in a greatly increased non radiative decay. Since it accounts for about half of the excited population, as derived from the luminescence components, its behavior will dominate the observed phenomena. The ensemble Zn-FB-Zn in z, or Zn-Zn-FB-Zn-Zn in z1, typical of the pentad, is therefore deemed responsible for the new properties, brought about by the strong electronic coupling between components. Deactivation of the excited ensemble Zn-FB-Zn (or Zn-Zn-FB-Zn-Zn) proceeds rapidly, through non radiative paths, without producing any sensitization localized on the free base. The free-base porphyrin itself, when selectively excited, displays a decreased emission quantum yield in the pentad. This indicates that, irrespective of the excited component, the Zn-FB-Zn ensemble seems to follow non radiative decay paths.

DINUCLEOTIDE: PARTICULARLY HIGH ASSOCIATION CONSTANT FOR THE FORMATION OF A SANDWICH TYPE MOLECULAR COMPLEX.

The gain in stability and the enhancement of the association constants observed for the complexation of bidentate base such as DABCO by bis-porphyrins using either uridine or 2'-deoxyuridine derivatives as spacers can be even greater when a dinucleotide linker is used. Indeed, we recently synthesized bis-porphyrinic dimer 8 based on a di-nulceosidic linker (Figure 6) and studied its complexation capabilities by UV-Vis spectroscopy. [26] The gain in stability for the formation of sandwich type host-guest complex with DABCO can be even greater when a dinucleotide linker is used.

Such pre-organization increases the association constants by one to two orders of magnitude when compared to the association constants of the same bidentate ligands with a reference Zn(II) porphyrin.

8 Fig. 6. Rigid cofacial bis-porphrynic tweezers with di-nucleosidic spacer 8.

The ability of di-nucleotide 8 to accommodate guests was investigated through binding studies carried out in dichloromethane with DABCO as bidentate base. As reported in some of our previous publications (Solladie et al., 2008), it was assumed that the di-nucleosidic bis-porphyrin could be pre-organized and thus facilitate the coordination of this rigid bifunctional ligand. The synthesized molecular system is amenable to preorganization, because its architecture is based on combination of rigid and flexible linkers. The architecture of multi-porphyrinic systems is important because the covalent connectivity between the interacting centers commands the geometry of resulting assembly and the relative orientation of chromophores dictates the strength of the interaction with ligands. As already noted, the geometry of porphyrins in supramolecular assembly is crucial in design of artificial light-harvesting complexes. [27][28][29][30][31][32][33] Data obtained by UV-visible titrations were analyzed by using the values of the absorbance at fixed wavelengths, which method is exclusively valid for the complexes with stoichiometry 1/1, and by simultaneous fitting of the whole series of spectra collected at 1 nm intervals using the software SPECFIT. [34][35] After examining the complexation of the simple porphyrin with DABCO, we studied the binding of the same base to dimer 8, the bis-porphyrinic tweezer appended to an appropriate position of the flexible uridinic scaffold by the rigid acetylene group. The titration of the dimer with DABCO was measured by using the coordination shift of the Soret absorption as well as the Q-bands. The Soret band absorption was measured at the concentration of 1.04x10-6 M which corresponds to an absorbance of 0.95 at 422 nm. The titration was performed in two parts to be analyzed in terms of two-state equilibria. The first part of the titration was processed until 2 equivalents of DABCO were added per porphyrin. The complete titration that involved an addition of DABCO until 6000 equivalents per porphyrin was performed separately, Figure 7. The clean isosbesticity of spectra obtained in the first part of the titration demonstrates that the only colored species present in solution are free dimer 8 and one type of its complex with DABCO. The observed spectral changes are quite similar to those obtained in the titration of the monomer. However, there are some minor spectral changes, i.e. smaller red shift of the Soret peak, from 422 to 427 nm, as well as intensified and sharpened form of the titration spectrum. A shift in the Soret band of 5 nm is exactly two times smaller than the shift obtained for the 1/1 monomer/DABCO complex. Accordingly, it was reasonable to anticipate formation of the intramolecular 1/1 sandwich complex. On addition of more DABCO, intensity of the absorption band at 427 nm decreased, and a new absorption band appeared at 432 nm. As observed in the titration spectrum of the monomer, the Soret absorption at 432 nm is typical for the simple 1/1 porphyrin/DABCO complex. On the basis of the changes in absorption spectra, the possible two consecutive two-state equilibria with corresponding stepwise constants in the binding of DABCO to bisporphyrinic tweezer 8 are shown in Figure 8. The first method for calculation of the binding constant by using titration data at selected wavelengths was applied only to the first part of the titration characterized by sharp isosbestic point. This analysis could be performed because the complexation by 1/1 binding model was envisaged in the first equilibrium. The binding constant for the 1/1 dimer/DABCO complex was found to be 7.4 x 10 6 M -1 (log K 11 = 6.9). The formation of this type of a complex was verified by analysis of the whole set of spectra which correspond to the first part of the titration by using SPECFIT. The analysis resulted with only one possible 1/1 complex with corresponding binding constant log K 11 =6.42 ± 0.03. It is interesting to note that calculated log K 11 for 1/1 dimer/DABCO complex is more than one order of magnitude higher than log K 11 for 1/1 monomer/DABCO complex. Taken together, these results strongly support formation of 1/1 dimer/DABCO sandwich complex through the chelate effect. The whole spectra obtained by the complete titration was analyzed in terms of three possible colored species: free dimer 8, 1/1 sandwich complex, and 1/2 open complex (Figure 8). Fitting of the titration data revealed that 1/1 and 1/2 binding models for dimer/DABCO are only possible.

Fitting to 1/1 binding isotherm yielded the stoichiometric binding constant log K 11 = 6.65 ± 0.08, while the fit to 1/2 binding model provided log K 12 = 3.21 ± 0.10. The calculated log K 11 is almost identical to log K 11 value obtained in the analysis of the first part of the titration. Moreover, calculated log K 12 value is three orders of magnitude lower than log K 11 value. Therefore, dissociation of the more stable chelated 1/1 complex is followed by coordination of second DABCO and by that formation of the 1/2 complex in which two DABCO molecules are coordinated to two porphyrins of the dimer. Both, 1/1 and 1/2 complexes involved in these consecutive two-state equilibria are in agreement with the species represented in Figure 8.

Similar formation of sandwich complexes was evidenced by Sanders, Hunter, Anderson and co-workers. [36][37][38][39] These studies show that sandwich complex formation is in concert with the parallel orientation of two porphyrin macrocycles. In this respect, the shifts in titration spectra are caused by proximity of the porphyrins and in this line increased interporphyrinic electronic interaction. The Soret band of the free dimer 5 has full width at half maximum (fwhm) 15 nm, and the Soret band of the dimer/DABCO sandwich complex 11 nm. It is believed that this narrowing indicate a significant increase in the center-to-center distance between the two porphyrins, due to the insertion of DABCO into the bis-porphyrinic cavity. This hypothesis is in great agreement with the half width at half maximum of 13 nm for the 1/2 dimer DABCO open complex. The outer coordination of DABCO allows a shorter center-to-center distance between the porphyrins. These results thus document the fact that the choice of rigid spacers is not the only way to pre-organize bisporphyrins, and that some well-chosen nucleosidic linkers offer an interesting option for the synthesis of such devices. Furthermore, the chirality and enantio-purity of the nucleosidic linkers paves the way toward the selective complexation of enantio-pure bidentate guests and the resolution of racemates.

CONCLUSION

In the present review, we discuss the results of the research we have done over the last decade on multiporphyrinic architectures bearing a nucleotidic backbone, compare all the data, offer a novel overview and emphasize common capabilites of nucleosides appearing in various systems, providing a new insight on the pre-organizing capabilities of nucleosidic backbones. The efficency of nucleosides as preorganizing agents was investigated through the synthesis and study of various bis-porphyrins bearing nucleosidic linkers, as well as through the investigation of the conformation of linear and arborescent multi-porphyrins constructed on a nucleosidic backbone. The capacity of these molecules to complex guests with a high association constant was used as a tool to evaluate their degree of preorganization, as well as the investigation of the electronic coupling existing between their chromophores and their photo-chemical capacities. Such an overview of one decade of scientific investigations documents the fact that the choice of rigid spacers is not the only way to pre-organize bis-porphyrins, and that some well-chosen nucleosidic linkers offer an interesting option for the synthesis of such devices. Furthermore, the chirality and enantio-purity of the nucleosidic linkers paves the way toward the selective complexation of enantio-pure bidentate guests and the resolution of racemates.

  2' and O-3' positions of uridine for dimer 1 and in the O-3' and O-5' positions of 2'-deoxyuridine for dimer 2.[20-21] For dimer 2 and 3, the anchoring position of one Zn(II) porphyrin at the 0-3' position of the sugar moiety is conserved, while the second chromophore is attached either at the O-5' position of the 2'-deoxyuridine for dimer 2 or at the C-5 position of the nucleic base for dimer 3.

1 2 3 Fig. 1 .

 31 Fig. 1. Three new bis-porphyrins bearing a flexible nucleosidic linker and differing by the attachment positions of the two porphyrins.

Fig. 4 .

 4 Fig.4. a) Molar absorption coefficient of FB-Zn4(7) in toluene solution (solid line). The sum of the molar absorption coefficients of the components, one FBS (5) and four ZnU (4) units, is also reported (dotted line). b) Emission detected from optically matched (A = 0.18) toluene solutions excited at 550 nm of FB-Zn (6) (solid line) and FB-Zn 4 (7) (dashed line).

Fig. 5 .

 5 Fig. 5. Cartoon representation of the different conformations for FB-Zn (6) (a) and FB-Zn 4 (7) (b). White diamonds represent the free-base porphyrin energy acceptor and grey diamonds the Zn porphyrin energy donor. The arrows indicate exciton interactions.

Fig. 7 .

 7 Fig.7. UV-visible titration of bis-porphyrinic tweezer 8 with DABCO in CH 2 Cl 2 at rt. On the left hand side first part of the titration and on the right the complete titration.

Fig. 8 .

 8 Fig. 8. Representation of the proposed equilibria involved in the binding of DABCO to bis-porphyrinic tweezer 8. For clarity reasons tert-butyls on meso-phenyls are omitted.

Acknowledgements 12

This work was supported by the CNRS and the French Ministry of Research. We also thank the French Government for a grant to Sonja Merkas for her PhD.