Saeid Samadi

Julien Roux

Arnaud Tanguy

Stéphane Caron

Abderrahmane Kheddar

Humanoid Control Under Interchangeable Fixed and Sliding Unilateral Contacts

Keywords: Humanoid robots, multi-sliding and -fixed unilateral non coplanar contacts, Chebyshev center

In this letter, we propose a whole-body control strategy for humanoid robots in multi-contact settings that enables switching between fixed and sliding contacts under active balance. We compute, in real-time, a safe center-of-mass position and wrench distribution of the contact points based on the Chebyshev center. Our solution is formulated as a quadratic programming problem without a priori computation of balance regions. We assess our approach with experiments highlighting switches between fixed and sliding contact modes in multi-contact configurations. A humanoid robot demonstrates such contact interchanges from fully-fixed to multi-sliding and also shuffling of the foot. The scenarios illustrate the performance of our control scheme in achieving the desired forces, CoM position attractor, and planned trajectories while actively maintaining balance.

I. INTRODUCTION

M ULTI-CONTACT motion is a key technology to in- crease humanoid robots' loco-manipulation abilities in confined and narrow spaces [START_REF] Kheddar | Humanoid robots in aircraft manufacturing: The airbus use cases[END_REF]. Yet, state-of-the-art multicontact planning and control considers only creating and breaking contacts to support the motion [START_REF] Bouyarmane | Multi-contact Motion Planning and Control[END_REF]. In many situations, however, switching contacts through releasing one of the established ones is not possible. This is the case, for example, in narrow or cumbersome spaces where free space is limited. Another example is when balance cannot be kept by breaking any of the existing contacts. In such cases, using sliding contacts to support the motion and the balance is an alternative. There are other contexts where tasks require sliding to be controlled (e.g. sanding or surface smoothing). In this letter, we address whole-body humanoid multi-contact taskspace control allowing interchangeable multi-contact transitions between fixed (creating and removing) and sliding ones.

Current researches and control strategies for humanoid robots deal with multi-contact motions. Complex multi-contact motions are found in non-gaited or acyclic locomotion [START_REF] Reher | Algorithmic foundations of realizing multi-contact locomotion on the humanoid robot DURUS[END_REF], [START_REF] Kumagai | Bipedal locomotion planning for a humanoid robot supported by arm contacts based on geometrical feasibility[END_REF], ladder climbing [START_REF] Nozawa | Three-dimensional humanoid motion planning using COM feasible region and its application to ladder climbing tasks[END_REF], [START_REF] Vaillant | Multi-contact vertical ladder climbing with an HRP-2 humanoid[END_REF], grasping [START_REF] Collette | Robust balance optimization control of humanoid robots with multiple non coplanar grasps and frictional contacts[END_REF], manipulation [START_REF] Garcia-Haro | Integration of dual-Arm manipulation in a passivity based wholebody controller for torque-controlled humanoid robots[END_REF], impact generations [START_REF] Wang | Impact-aware taskspace quadratic-programming control[END_REF]... to cite a few. Such skills shall be achieved under active balance of the humanoid robot [START_REF] Kajita | Biped walking stabilization based on linear inverted pendulum tracking[END_REF], [START_REF] Caron | Stair climbing stabilization of the HRP-4 humanoid robot using whole-body admittance control[END_REF], [START_REF] Morisawa | Online 3D CoM Trajectory Generation for Multi-Contact Locomotion Synchronizing Contact[END_REF], [START_REF] Morisawa | Multi-Contact Stabilization of a Humanoid Robot for Realizing Dynamic Contact Transitions on Non-coplanar Surfaces[END_REF].

To enforce dynamic balance for multi-legged robots in multi-contact; recent methods suggest computing gravitoinertial wrench cones (GIWC) [START_REF] Caron | Leveraging cone double description for multi-contact stability of humanoids with applications to statics and dynamics[END_REF], Zero-tilting moment S. Samadi, J. Roux and A. Kheddar are with Laboratory of Informatics, Robotics and Microelectronics (LIRMM), CNRS-University of Montpellier, France. Corresponding author: saeid.samadi@lirmm.fr A. Kheddar and A. Tanguy are with CNRS-AIST Joint Robotics Laboratory (JRL), IRL, Tsukuba, Japan.

S. Caron is with ANYbotics AG, Zürich, Switzerland. This work was carried while he was with CNRS, France. points (ZMP) [START_REF] Caron | ZMP support areas for multicontact mobility under frictional constraints[END_REF], [START_REF] Harada | Dynamics and balance of a humanoid robot during manipulation tasks[END_REF] and center-of-mass (CoM) support polygons [START_REF] Audren | 3-D robust stability polyhedron in multicontact[END_REF], see also [START_REF] Bouyarmane | Multi-contact Motion Planning and Control[END_REF]. These methods end-up computing regions (e.g. polytopes, polyhedral cones...) where the CoM position or the wrenches should live. Reducing the cost of such computations is the main challenge to integrate them in control and reactive planning. Indeed, they are mostly used in planning. We propose a decent alternative with a lighter solution to tackle this shortcoming.

Sliding contacts under active balance is challenging in humanoid robots. There are however successful achievements in specific tasks such as foot shuffling [START_REF] Kojima | Shuffle motion for humanoid robot by sole load distribution and foot force control[END_REF], [START_REF] Or | Dynamic walking on slippery surfaces: Demonstrating stable bipedal gaits with planned ground slippage[END_REF]; slip-turns and maneuvers by two feet contacts [START_REF] Kojima | Rotational sliding motion generation for humanoid robot by force distribution in each contact face[END_REF], [START_REF] Miura | Slip-turn for biped robots[END_REF], [START_REF] Koeda | Shuffle turn and translation of humanoid robots[END_REF]. Sliding contacts forces must be controlled to be exactly on their friction cone [START_REF] Trinkle | On dynamic multirigid-body contact problems with coulomb friction[END_REF]. We have addressed this problem in [START_REF] Samadi | Balance of humanoid robots in a mix of fixed and sliding multi-contact scenarios[END_REF], where we considered a mix of sliding and fixed multi-contact scenarios. The CoM support area (CSA) is derived analytically when the fixed contacts are coplanar. To overcome this limitation and extend the balance criteria to be used in control, we propose a novel formulation for the computation of the CoM position and the contact wrenches using the Chebyshev center.

Multi-sliding and fixed contacts are considered as in Fig. 1. We implemented an online friction estimation to adjust the real coefficient of friction during sliding. The output reference CoM position and contact wrench distribution are achieved using task-space whole-body admittance control [START_REF] Bouyarmane | Quadratic programming for multirobot and task-space force control[END_REF].

II. BACKGROUND

Two main ingredients shall be considered for a stable dynamic balance of humanoid robots [START_REF] Featherstone | Rigid Body Dynamics Algorithms[END_REF]: (i) a good control of the robot/environment interaction forces, and (ii) a good control of the dynamic motion, governed by Newton-Euler equations, contact constraints and balancing of the external wrench. Standing or walking with a stable balance on flat terrains [START_REF] Wieber | Trajectory free linear model predictive control for stable walking in the presence of strong perturbations[END_REF], [START_REF] Brasseur | A robust linear MPC approach to online generation of 3d biped walking motion[END_REF] can be achieved by maintaining the ZMP inside its contacts support area [START_REF] Senoo | Analysis of sliding behavior of a biped robot in centroid acceleration space[END_REF]; the latter being the convex hull of all the contact points. Whereas for non-coplanar multicontact postures and motions, sustaining stable balance is less trivial to achieve.

A. Related Works and Contribution

Enforcing dynamic balance of humanoid robots in multicontact non-coplanar configurations can be addressed with one of the following inclusions:

• constraining the gravito-inertial wrench within the GIWC, e.g. [START_REF] Caron | Leveraging cone double description for multi-contact stability of humanoids with applications to statics and dynamics[END_REF]; • for a given CoM acceleration convex set and contact friction cones, build a safe region (convex polyhedron) for the CoM, e.g. [START_REF] Audren | 3-D robust stability polyhedron in multicontact[END_REF]; • for a given CoM convex set and contact friction cones, built a safe region (convex polyhedral cone) within which the CoM acceleration shall safely lie, e.g. [START_REF] Caron | Multi-contact walking pattern generation based on model preview control of 3d com accelerations[END_REF]. Efficient and fast computation of these balance equilibrium regions has been thoroughly investigated.

In [START_REF] Caron | Leveraging cone double description for multi-contact stability of humanoids with applications to statics and dynamics[END_REF] multi-contact balance is fulfilled by computing the GIWC for each stance. In this approach, there is no need to update the GIWC for fixed contact stances. However, in the case of moving or sliding contacts, GIWC needs to be recomputed at each iteration. Unfortunately, since calculating the GIWC is computationally expensive, it is not possible to use this method in closed-loop control. In a recent work, [START_REF] Abi-Farraj | Torquebased balancing for a humanoid robot performing high-force interaction tasks[END_REF] challenged this shortcoming by specifying two different set of contacts in multi-contact motion settings: interacting (hence moving) contacts, and balancing contacts that are chosen and constrained to be fixed (i.e. static contacts). Subsequently, only balancing contacts are used to compute the GIWC. Other contacts are considered as external forces induced by tasks to be balanced altogether with the robot dynamics. This approach is appropriate when the external contacts are concerned with holding a free-floating object. However, [START_REF] Abi-Farraj | Torquebased balancing for a humanoid robot performing high-force interaction tasks[END_REF] is limiting in all the other cases where task-induced contacts (including the moving ones) can be exploited for balance (e.g. pushing an object, sliding...). This is because with [START_REF] Abi-Farraj | Torquebased balancing for a humanoid robot performing high-force interaction tasks[END_REF], task contacts won't be allowed to contribute to balance and this is clearly not the way, we humans perform. By categorizing contacts this way (even if switches are possible), the multi-contact GIWC pre-calculation is made on a subset of contact (those fixed) for balance, hence relatively tractable; yet it excludes all other contacts that could eventually contribute to balance.

Computation efficiency also exists for the CoM-support regions [START_REF] Audren | 3-D robust stability polyhedron in multicontact[END_REF]. It was leveraged using 3D morphing techniques between two regions [START_REF] Audren | Stability polygons reshaping and morphing for smooth multi-contact transitions and force control of humanoid robots[END_REF] but without guarantee on the balance validity all along the morphing shape pathway between the two regions. To control a humanoid in multi-contact, [START_REF] Sentis | Compliant control of whole-body multi-contact behaviors in humanoid robots[END_REF] used the virtual-linkage model. Their method consists in constructing a multi-contact CoM area as the envelope of valid points. In our recent previous work [START_REF] Samadi | Balance of humanoid robots in a mix of fixed and sliding multi-contact scenarios[END_REF], we introduced an analytical solution to compute the CoM-support area in real-time. However, these developments apply only when the fixed contacts are coplanar.

We propose an alternative formulation that allows balance criteria to be used in closed-loop control, that applies to moving/sliding and fixed contacts. It also permits on-the-fly switching between these contact modes. Our approach do not separate contacts for balance from those of the tasks, when the latter can contribute to balance (and vice versa, i.e. balance contribute to achieving the task). Our approach distinguishes from existing work as follows:

• No need for constructing or pre-computing explicitly the multi-contact balance region (GIWC); • Fast computation performances enabling real-time closedloop control; • Calculating CoM position and wrench distribution of contacts using the Chebyshev center; • Covering all types of contact modes (e.g. multi-sliding contacts) without any of the limitations pointed in [START_REF] Samadi | Balance of humanoid robots in a mix of fixed and sliding multi-contact scenarios[END_REF]; • contrary to [START_REF] Abi-Farraj | Torquebased balancing for a humanoid robot performing high-force interaction tasks[END_REF], we do not exclude moving/tasks contacts to contribute to dynamic balance stability.

B. Centroidal Model

We consider motion scenarios without locomotion. Static equilibrium of the robot is represented by Newton-Euler equation for l limbs in contact with the environment:

w g + l i=1 w c i = 0 (1)
where w g ∈ R 6 and w c i ∈ R 6 are gravity wrench and the i th contact wrench in the world frame respectively, and w = [f τ ] T . Gravity and contact wrenches are specified in the following form:

w g = f g c × f g w c i = f c i p i × f c i + τ c i (2)
where

f g = [0 0 mg] T , c = [c x c y c z ]
T the position of the CoM, and contact points p i are given with respect to a global frame. Note that the contact wrenches are mapped from the local to the global frame by rotation matrix [R] i ∈ R 6×6 :

w c i = [R] i l w c i ( 3
)
where l • denotes the local frame of the contact point. We consider the following state variables to be computed at each iteration:

Y = [c W 1 W 2 . . . W l ] (4)
where W i is the i-th contact wrench w c i . 1) Sliding and equality bounds on contact forces: For every contact of the robot, we determine the wrench. Equations ( 2) can be re-written as:

w g = A g c -b g (5) w c i = A c i W i where i = 1, . . . , l (6)
On the other hand, the desired sliding comes with additional equality constraints as stated in [START_REF] Samadi | Balance of humanoid robots in a mix of fixed and sliding multi-contact scenarios[END_REF]. By convention, the normal contact force (f c z ) for sliding is aligned to the normal of the local surface frame. The dynamic friction coefficient (µ) of each contact, is estimated online during sliding. The force along other axes (f c

x and f c y ) of the tangent space is derived from the pre-defined velocity and trajectory of the sliding motion. Let,

l f i = f x,i f y,i f z,i T i = 1, . . . , s, (7)
be the sliding contact forces in the local frame, s being the number of sliding contacts. We can write in a matrix form

l f k =   0 0 µ x,k 0 0 µ y,k 0 0 0   l f k +   0 0 f z,k   (8) or,   1 0 -µ x,k 0 1 -µ y,k 0 0 1   l f k =   0 0 f z,k   (9)
giving the k th sliding contact velocity

v k = v x,k i + v y,k j, let α x,k = v x,k v k and α y,k = v y,k v k , from which µ x,k = µ k α x,k and µ y,k = µ k α y,k ; µ k is the k th dynamic friction coefficient.
The above equation writes in a compact form:

C l f k = K (10)
det(C) = 1, C is an invertible matrix, hence:

l f k = C -1 K (11)
Also, by using matrix transforms, we change the coordinate to the global frame:

f c k = [R k ] 3×3 C -1 K (12)
which can be written using the selection matrix [S] 3×6 = [I 3×3 0 3×3 ] as:

[S]w c k = [R k ] 3×3 C -1 K (13)
Equation ( 13) is embedded to the controller along with eqs. ( 5) and ( 6), and can be written as:

sl i W i -b sl i = 0; i = 1, . . . , s (14)
This equation is applied to the s sliding contacts w.r.t their desired sliding forces. Hence, in this equation W i refers to sliding contacts only.

2) Fixed contacts: The non-sliding condition of contacts is fulfilled when the contact force lies strictly within the friction cone. Linearized equations for non-sliding conditions were thoroughly discussed in [START_REF] Caron | Computational Foundation for Planner-in-the-Loop Multi-Contact Whole-Body Control of Humanoid Robots[END_REF]. The following equations are sufficient conditions to avoid slippage and tilting of the contacts in all R 6 coordinate of the local frame:

| f x | µf z , | f y | µf z , f min z f z f min z | τ x | σ y f z , | τ y | σ x f z , τ min z τ z τ min z (15)
where σ * are scalars defined in [START_REF] Caron | Computational Foundation for Planner-in-the-Loop Multi-Contact Whole-Body Control of Humanoid Robots[END_REF].

By considering n non-sliding contacts, the inequality constraints are in the following form:

Υ ub i W i h ub i i = 1, . . . , n (16a)
Υ lb i W i h lb i i = 1, . . . , n (16b)
where indexes ub and lb shows the upper and lower bounds. Υ i matrices and h i vectors are introduced in [START_REF] Samadi | Balance of humanoid robots in a mix of fixed and sliding multi-contact scenarios[END_REF] for both sliding and fixed contacts. Also, Ψ i enforces inequality constraints on τ z element of wrenches:

Ψ ub i W i 0 4×1 i = 1, . . . , n (17a)
Ψ lb i W i 0 4×1 i = 1, . . . , n (17b)
For the sliding contacts, the same inequality constraints are implemented only on the torque of the wrench to avoid the tilting of the sliding contacts.

III. OPTIMAL CONTROL FRAMEWORK

In this section, we propose the implementation of the Chebyshev center theorem [START_REF] Beck | Regularization in regression with bounded noise: A Chebyshev center approach[END_REF] in a two-level control framework. Inspired from [START_REF] Collette | Dynamic balance control of humanoids for multiple grasps and non coplanar frictional contacts[END_REF], [START_REF] Samadi | Balance of humanoid robots in a mix of fixed and sliding multi-contact scenarios[END_REF] such an implementation can be formulated as a first-level quadratic program (QP) that outcomes in real-time, the CoM position and contact wrenches distribution (together with the Chebyshev center and its radius). These are then integrated as tasks objectives or constraints in a running whole-body task-space controller formulated as a second-level QP [START_REF] Bouyarmane | Quadratic programming for multirobot and task-space force control[END_REF] (see later Section IV).

A. Chebyshev Center

Let U be a set of inequality constraints describing a bounded polygon. The Chebyshev center of U is the center of the largest enclosure circle [START_REF] Amir | Best Simultaneous Approximation (Chebyshev Centers)[END_REF]. Figure 2 illustrates the Chebyshev center and radius of a given polygon. To compute the Chebyshev center, we do not need to build the vertices of the polygon (i.e. no need for the v-representation of U ).

For a point, x in the interior of U , the Chebyshev center x is a point among all possible x such that [START_REF] Knitter | The informing vision of the practical: the concepts of character, virtue, vice and privation[END_REF]:

arg min x max x∈U ||x -x|| 2 (18)
This applies to any bounded non-empty convex set

U = {x | f i (x) 0; i = 1, . . . , n}, (19)
where x is a vector of a given dimension, and n the number of inequalities. These inequalities are valid for all points inside the convex set. Hence, constituent points of the Chebyshev ball enclosed in the convex set, with center x and radius r fulfill the inequalities too. The idea is to compute the largest Chebyshev ball which satisfies the convex condition [START_REF] Or | Dynamic walking on slippery surfaces: Demonstrating stable bipedal gaits with planned ground slippage[END_REF]:

max r (20a) w.r.t sup a 1 f i=1,...,n (x + ra) 0 (20b)
and the supremum form of (20b) allows accessing all points within the ball according to the variable a. For a polytope, the inequalities are in the form of {α T i xβ i 0, i = 1, . . . , n}. So the condition (20b) is modified and the optimization problem is 1 :

max r (21a) w.r.t α T i x + r α i -β i 0 i = 1, . . . , n (21b) r 0 (21c)
Figure 3 illustrates the link between the above generalities concerning the Chebyshev center and humanoid balance. The polyhedron illustrates the GIWC, that can be obtained as in e.g. [START_REF] Abi-Farraj | Torquebased balancing for a humanoid robot performing high-force interaction tasks[END_REF]. The benefit of computing the Chebyshev center, instead of the GIWC, is undeniably the speed. The price to pay for the speed is the conservativeness of the balance region as can clearly be seen in Fig. 3. 1 By replacing the inequality of the polytope into (20b), the term α T i ra appears. Supremum of this term will provide r α i * . Note that the trace norm, * , is introduced for computing the norm of the matrices. In case of vectors, one can use the simple l 1 or l 2 vector norms.

B. Chebyshev Quadratic Programming

Recall from (4) that the decision variables are noted as Y = [c W i=1,...,l ] T where Y ∈ R 3+6k . The QP formulation is structured based on the equality and inequality constraints introduced in section II-B expressed as:

AY = b (22a) GY h (22b)
whereas:

A = A g A c 0 6n×3 A sl (6+6n)×(3+6k) (23)
and the matrices A c and A sl are defined as:

A c = A c 1 . . . A c k 6×6k (24)
A sl = [diag(A sl i )]ρ sl ; i = 1, . . . , n (25)
The equation above contains a selection matrix ρ sl 6n×6k for the elements related to the sliding contacts in the A sl matrix and diag() refers to the diagonal matrix structure. The rest of the parameters of ( 22) are defined as:

b = -b g b sl 1 . . . b sl n T 1×7n (26)
G = [diag(Υ i , Ψ i )] 20k×10k ρ G ; i = 1, . . . , k (27)
h = h ub i h lb i 0 1×8k T 1×20k ; i = 1, . . . , k (28)
and ρ G 20k×3+6k is the selection matrix for the G matrix. For maximizing the radius and selecting the optimal Chebyshev center, inequality constraint (22b) is modified in the following form, based on (21b), in presence of Chebyshev radius r:

GY + rξ h ( 29
)
where ξ ∈ R 20k is a vector which consists of the norm of the rows of G matrix separately:

ξ j = G(j, :) j = 1, . . . , 20k (30)
and the operator (j, :) shows the j th row of the matrix. The vector ξ in ( 29) is equivalent to the vector α where the i-th element of the vector α(i) = α i ∀i ∈ {1, . . . , n} in (21b). This way, we aim at maximizing the Chebyshev radius by taking into account all inequalities without the need to compute the GIWC. The decision variables of this first-level QP are the position of CoM, wrench distribution, and r. To include the Chebyshev radius within the decision variables, matrices are modified as follows:

X = [Y r] T (31)
h * = [h 0] T (32)
A * = [A 0] (33)
G * = G ξ 0 -1 (34)
Maximizing the Chebyshev radius is equivalent to minimizing (-r). Other objectives of the QP can be:

• Setting the current position of the CoM as a target for the next iteration (smoothing CoM trajectory) • Minimizing the wrench distribution: sharing the load on non-constrained (contact) force.

These objectives can be defined through X des = [Y des r des ], and the optimization framework writes:

min X X -X des 2 -r ≡ 1 2 X T PX + q T X (35a) G * X h * (35b) A * X = b (35c)
where

q = [Y des -1] (36) P = 2Q (37)
We prioritize the solution of the QP by using a weight matrix Q (4+6k)×(4+6k) with chosen weights on the corresponding decision variables (the diagonal elements). However, we do not set any target for r. Therefore, we need to adopt a low computation weight for r des . This QP problem, named Chebyshev QP for the rest of the paper, provides the optimal position of CoM and wrench distribution by maximizing the Chebyshev radius.

C. Calculation of the Range of Contact Wrench and CoM Position

Chebyshev radius is the output of the minimization problem [START_REF] Beck | Regularization in regression with bounded noise: A Chebyshev center approach[END_REF]. However, this radius indicates a range that applies to all contact wrenches. We can additionally calculate the range of the gravito-inertial wrench within the GIWC based on the computed radius. Note that the contact wrench cone (CWC) is simply the opposite of the GIWC. So, the computation for range accounts for both whole-body contact and gravitoinertial wrenches. Assuming a set of contact wrenches, expressed at CoM, within their respective contact wrench cones w i ∈ C i for i = 1, . . . , l, and representation of Chebyshev method (20b), we have

w i + ra ∈ C i ( 38
)
Regarding the sum of contact wrenches and their ranges (ra) we can write

l i=1 w i + l × ra ∈ C 1 ⊕ . . . ⊕ C l ( 39
)
where ⊕ denotes the Minkowski sum of the contact wrenches cones which is the explicit definition of the CWC. So, the range for the valid whole body contact and gravito-inertial wrenches is the sphere with the radius of r w = l × r.

D. Online Estimation of the Friction Coefficient

Implementing the sliding motion of contact needs a good estimation of the friction coefficient. The latter is considered in [START_REF] Garcia-Haro | Integration of dual-Arm manipulation in a passivity based wholebody controller for torque-controlled humanoid robots[END_REF] and has a major role in matrix A sl of ( 23). Dynamic and static friction coefficients are defined as a property of a pair of surfaces in contact and can not be known intrinsically. Let l f c,tan i and l f c,nor i be the tangential and normal forces respectively in the local frame of the i-th contact surface, the norm of the Coulomb friction equation [START_REF] Greenwood | Contact of nominally flat surfaces[END_REF] leads us to the following formulation:

l f c,tan i 2 = µ mes | l f c,nor i | (40)
where

l f c,tan i = [ l f c
x,i l f c y,i ] T and µ mes is the calculated friction coefficient from the measured local forces of the force sensors:

µ mes = l f c x,i 2 + l f c y,i 2 | l f c z,i | (41)
To minimize the effect of the force measurements noise, we also apply a simple filter with 0 γ 1 and calculate the filtered friction coefficient at each time iteration t:

µ filt t = γµ mes t-1 + (1 -γ)µ mes t (42)

IV. WHOLE-BODY ADMITTANCE CONTROLLER

In this section, we present the overall task-space whole-body admittance controller as shown in Fig. 4. For each scenario, a Finite State Machine (FSM) is designed/planned. Each state of the FSM, defines a set of tasks and constraints for the controller as well as desired positions, forces, and sliding conditions for each contact, if any. The Chebyshev QP computes optimal contact wrenches and CoM position depending on the current state of the robot and FSM targets. These optimal wrenches and CoM position are then used as objectives for tasks driven by our whole-body QP (WBQP) framework. Each task is formulated as a cost function and/or eventually associated constraints that depend on joint acceleration (or torque for torque-based humanoids) and contact forces. The resulting joint acceleration is integrated twice (or torques) and sent as a target for the robot actuators' low-level controllers, see [START_REF] Bouyarmane | Quadratic programming for multirobot and task-space force control[END_REF].

The main tasks we used in our experiments are the following: (i) CoM task, to eventually track the optimal CoM position, (ii) Admittance tasks for the desired wrench which map wrench error to contact surface velocity which in turn is used as a target for an end effector trajectory task.

V. EXPERIMENTAL RESULTS

In order to assess our approach, we performed experiments with the HRP-4 humanoid robot. We investigate the capabilities of our proposed optimal control framework through a multi-contact active balance scenario that exhibits:

• combination and switching of fixed and sliding modes;

• multi-sliding contacts on non-coplanar surfaces;

• shuffling of the foot on a tilted surface. We prepared a non-coplanar multi-contact set-up consisting of a tilted fake wooden slope for the left foot, a tilted wooden board for the right hand, and a wall for the left hand. The right foot is on the experimental room ground. The three materials have different friction coefficients. The scenario starts by setting the HRP-4 in a half-sitting pose, both feet on the grounds. The motion begins with stepping the left foot up the slope and establishing a planar contact with the wall by the left hand. Then contact is established by the right hand on the tilted board (slope of 50 • ). Each contact is established while sustaining the existing ones and moving the whole body while keeping the CoM close to the suggested one by the Chebychev QP. Alike in [START_REF] Kheddar | Humanoid robots in aircraft manufacturing: The airbus use cases[END_REF] and since we use guarded motion and a calibrated environment, embedded robot vision was not used this time. At this stage, we have four contacts. Then HRP-4 gets prepared for the co-wiping motion under user-specified surface normal forces: 10 N for the right hand and 15 N for the left hand. During co-wiping, the end effectors are tracking the desired normal force and planned trajectories of circles with a radius of 10 and 8 cm for right and left hands, respectively. Notice that in a multi-contact non-coplanar setting, the range of motion of HRP-4 is limited due to intrinsic kinematics closed-chain constraints. Figure 5(a) and 6 illustrate the cowiping experiment and force tracking of the end-effectors contacts using admittance force control discussed in [START_REF] Bouyarmane | Quadratic programming for multirobot and task-space force control[END_REF]. The alignment of the end effectors with surfaces while establishing the contacts is guaranteed by zeroing the torques reference in x and y directions w.r.t the contact local frame. The trajectory tracking of end-effectors is shown in Fig. 7.

Once the co-wiping tasks are achieved, both contact switch from sliding to fixed ones, and the left foot contact switches from fixed to sliding. Now we have 3 fixed contacts and one sliding. The left foot on the slope shuffles back and forward. Figures 5(b) and 8 show the shuffling experiment and the normal force tracking for the end effectors respectively.

In the last part of the scenario HRP-4 releases both hands at a time and steps back to the floor (initial pose). During all sliding motions, online friction estimation (42) is executed. The estimation process begins with an initial guesses, as shown in Fig. 9, and computes friction only when normal forces are above a given threshold.

The experiments computations were made by a laptop computer having Intel(R) Xeon(R) E-2276M CPU at 2.80 GHz×12. With this setup, the average computation times for the Chebyshev QP and the whole framework were noted as 0.2 and 1 ms respectively. These values are largely within the implemented real-time control loop with mc_rtc framework1 that is 5 ms.

VI. CONCLUSION

In this letter, we devised a whole-body humanoid noncoplanar multi-contact motion planning and control for mixed sliding and fixed contacts that can be switched at will. Our method does not require the construction of GIWC or CoMsupport polytopes thanks to a single and fast optimization problem based on the Chebyshev center. This makes it suitable for closed-loop control. It also allows exploiting all the contacts (including moving/sliding ones) for dynamic balance when this is possible, and permit balance contacts to also contribute to force tasks. We assess our approach through complex scenarios with the HRP-4 humanoid robot: four contacts are controlled in force under dynamic balance with switches between fixed and sliding contact modes at user's will. A simple online friction estimator is implemented to update the friction coefficient of the sliding contacts. A video of the experiments 1 and the opensource code of the controller 2 are available online.

In future work, we aim at improving the force tracking based on the extension of [START_REF] Pham | Convex Controller Synthesis for Robot Contact[END_REF] that is currently limited to translation forces (i.e. non-moments). We are also considering to extend multi-contact modalities to include soft supports. That is to say, contact that combines fixed, sliding, pushing, and rolling eventually on soft supports. We then need to develop a more sophisticated friction identification models and associated filters. These modalities would then cover almost all spectrum of possible contacts encountered in the applications we are targeting. Robustness consideration, namely w.r.t. uncertain dynamic parameters shall also be accounted for (e.g. considering relative forces instead of absolute ones). Finally, we shall cover multi-contact planning that considers such multi-modal contacts in order to achieve tasks that require accessing narrow or cumbersome passages. These skills will be deployed in real use-case scenarios defined by two largescale manufacturing industrial partners, one of which is the continuation of [START_REF] Kheddar | Humanoid robots in aircraft manufacturing: The airbus use cases[END_REF] and in physical human-robot assistive robotics [START_REF] Bolotnikova | Multi-contact planning on humans for physical assistance by humanoid[END_REF].

Fig. 1 .

 1 Fig. 1. Non-coplanar (four) multi-sliding and fixed contacts and on-demand switches between them (including contact removal) with HRP-4.

Fig. 2 .

 2 Fig. 2. Chebyshev circle enclosed in a polygon.

Fig. 3 .

 3 Fig. 3. Illustration of the Chebyshev center of an enclosed Euclidean ball within a multi-contact balance GIWC.

FilterFig. 4 .

 4 Fig. 4. Schematic of the overall task-space control.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Stepping up the slope and (a) simultaneous sliding of both hands on non-coplanar wooden board and wall surfaces, and (b) shuffling of the left foot on a slope tilted by 20 • . w.r.t the ground in a multi-contact setting.

Fig. 7 .

 7 Fig. 7. Target trajectory tracking of the sliding hands. The light discrepancies are due to uncertainties of planed board and wall position w.r.t HRP-4.

Fig. 8 .

 8 Fig. 8. Normal Force tracking of hands and left foot contact during the shuffling motion. The oscillations are due to the change into the reverse direction in the shuffling motion.

Fig. 10 .

 10 Fig. 10. CoM trajectory tracking in x and y directions for co-wiping (up) and shuffling scenarios (bottom)

  Fig. 9. Estimated and filtered dynamic friction coefficient for co-wiping (up) and shuffling scenarios (bottom).

			Estimated LH Friction			Filtered RH Friction
			Filtered LH Friction				Estimated LF Friction
			Estimated RH Friction			Filtered LF Friction
		1.5
		1.0
	Friction Coefficient	0.0 0.5 0.6	40	42	44	46	48	50		52	54
		0.4
		0.2
		0.0
		40	45	50	55	60		65		70
						Time [s]
			Estimated CoM Position			Target CoM Position
	0.08
	0.06
	0.04
	Y [m]	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09
	0.10
	0.05
	0.00
	0.05
			0.01	0.02	0.03	0.04 X [m] 0.05	0.06	0.07	0.08	0.09

1 https://youtu.be/cFYd9oQueRE 2 https://github.com/SaeidSamadi/Multi-sliding-Contacts

https://github.com/jrl-umi3218/mc rtc

VII. ACKNOWLEDGEMENT

The authors would like to thank Dr. Pierre Gergondet for his technical supports on the mc_rtc framework.


Pavel A Dub

Jean-Claude Daran

Vladislava A Levina

Natalia V Belkova

Elena S Shubina

Rinaldo Poli
email: rinaldo.poli@lcc-toulouse.fr

Modeling the platinum-catalyzed intermolecular hydroamination of ethylene: The nucleophilic addition of HNEt2 to coordinated ethylene in trans

Keywords: Platinum, diethylamine complexes, olefin complexes, nucleophilic addition, hydroamination

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Introduction

A number of contributions have appeared in recent years on the intermolecular hydroamination of non-activated olefins catalysed by Pt-based systems [START_REF] Brunet | [END_REF][2][3][4][5][6][7][8]. The first step in these transformations is believed to be the nucleophilic addition of the amine to the coordinated olefin in a Pt II derivative to afford a zwitterionic complex L3Pt (-) -CH2-CH2-N (+) HR2 [5,9]. A recent computational study of the addition of aniline to ethylene catalyzed by PtBr2/Br -was found to involve the resting state complex [PtBr3(C2H4)] -and a variety of off-loop species also containing a Pt II -coordinated ethylene ligand [10]. The amine addition reaction has been very well studied experimentally for basic amines and PtCl2 derivatives such as trans-PtCl2(olefin)(am) or cis-PtCl2(C2H4)(Y) (Y = PR3, py, NH3, am, DMSO; am = amine) [11][12][13][14][15] (Scheme 1) and monochloride complexes such as [PtCl(alkene)(tmeda)] + [16] or PtCl(PyPyr)(C2H4) [17], but no evidence of addition was found when the amine is not sufficient basic (pKa < 5 for the conjugate ammonium ion). It has also been shown that basic amines add reversibly and exothermically to coordinated ethylene [15], but the system may slowly lead to irreversible transformations at room temperature, occasionally accompanied by the formation of Pt 0 [18]. For instance, the zwitterionic complexes resulting from the nucleophilic addition to the above mentioned cis-PtCl2(C2H4)(Y) may lead to the formation of platinaazacycles [13] or other ligand substitution processes, see Scheme 1 [13,16], whereas only decomposition was observed for the trans-PtCl2(C2H4)(am) isomer [19]. The cis/trans isomerisation is usually a slow process for these complexes [11,20].

Scheme 1

It is notable that no Pt-catalyzed hydroamination is reported for amines that lead to experimentally observable nucleophilic addition to Pt-coordinated ethylene, whereas no nucleophilic addition products are experimentally observable for amines that give catalytic hydroamination. Moreover, the turnover number for the PtBr2/Br --catalyzed hydroamination of C2H4 by ArNH2 is inversely proportional to the amine basicity (for instance, 22 for Ar = 4-MeOC6H4 with a pKa for the conjugated acid of 5.34 and 110 for Ar = 2-ClC6H4 with a pKa of 2.65) [START_REF] Brunet | [END_REF].

We have recently reported experimental and theoretical studies on the hydroamination of ethylene by aniline [7,8,10,21]. The catalytic system consisting of PtX2/X -shows the best activity for X = Br. The reaction between complexes trans or cis-PtBr2(C2H4)(PhNH2) and PhNH2 (pKa = 4.63) does not give observable nucleophilic addition products at temperatures up to 68°C. The present paper reports the investigation of model stoichiometric processes involving the more basic amine Et2NH (pKa of conjugated acid = 11.02) and the PtBr2/Br -catalytic system. We will compare the reactivity of complexes trans-PtBr2(C2H4)(HNEt2) [START_REF] Brunet | [END_REF] and trans-PtBr2(C2H4)(PhNH2) (1a) towards an excess of the corresponding amine, in an effort to understand the role of the amine basicity and the presence of added acids to the hydroamination catalytic activity.

Experimental part

General. All solvents were of HPLC grade and were used as received. HNEt2 (Fluka) was distilled and kept under argon in the dark. K[PtCl3(C2H4)]•H2O (Aldrich) was used as received. Ethylene (purity ≥ 99.5%) was purchased from Air Liquide.

Instrumentation. NMR investigations were carried out on Bruker DPX300 (and AV400) spectrometers operating at 300.1 (400.1) MHz ( 1 H), 75.47 (100.6) MHz ( 13 C) and 64.5 MHz ( 195 Pt). Low-temperature NMR investigations were carried out on a Bruker AV500 spectrometer, operating at 500.3 ( 1 H) and 125.8 MHz ( 13 C) and 107.2 MHz ( 195 Pt), respectively. The spectra were calibrated with the residual solvent resonance relative to TMS ( 1 H, 13 C) and Na2PtCl6 ( 195 Pt). The NMR studies were performed either in air or under argon, showing no difference. IR spectra were recorded on a Perkin-Elmer Spectrum 100 FTIR spectrometer for neat compounds (4000-600 cm -1 ; 2 cm -1 resolution) and on a Nicolet FTIR spectrometer for solutions in CaF2 cells. A Carl Zeiss Jena cryostat was used for variable temperature measurements; the accuracy of the experimental temperature adjustment was ±1 K. Elemental analyses were performed by the Microanalytical Service of the Laboratoire de Chimie de Coordination. Other characterization data (NMR in other solvents, UV-visible, IR) are given in the Supporting Information.

Synthesis of trans-PtBr2(C2H4)(HNEt2),

1. An aqueous solution (16 mL) of Zeise's salt (500 mg, 1.36 mmol) and KBr (8 g, 50 equiv) was stirred under an ethene flush overnight at room temperature. An Et2NH (140 μL, 1.36 mmol) solution in EtOH (4 ml) was then added dropwise, yielding a yellow precipitate. After stirring for 1 h at room temperature, the yellow precipitate was filtered, washed with water, dried in vacuo, redissolved in dichloromethane, filtered and again dried in vacuo (467 mg, 75 %). The compound was stored in the dark at low temperature under argon. Elem. anal. % Calcd. for C6H15Br2NPt: C, 15.80; H, 3.32; N, 3.07. % Found: C, 15.87; H, 3.28; N, 3.07. IR (neat, cm -1 ): 3198 (N-H), 1518 (δ(C2H4) + ν(C2H4)), 1249 (ν(C2H4) + δ(C2H4)). NMR (acetone-d6): 1 H (300.13 MHz, δ): 5.62 (br with Pt satellites, 1H, NH, 2 JPt-H ca. 90 Hz), 4.68 (s with Pt satellites, 2 JH-Pt = 60 Hz, 4H, C2H4), 3.35 (m, 2H, HNCH2), 3.13 (m, 2H, HNCH2), 1.52 (t, 3 JH-H = 7 Hz, 6H, HN(CH2CH3)2). 13 C{ 1 H} (75.5 MHz, δ): 70.7 (s with Pt satellites, 1 JC-Pt = 153 Hz, C2H4), 50.0 (s with Pt satellites, HNCH2, 1 JC-Pt = 14 Hz), 14.0 (s with Pt satellites, HN(CH2CH3)2, 1 JC-Pt = 8 Hz). 195 Pt (64.5 MHz, δ): -3456 (br tr, 1 JPt-N = 218 Hz).

For the NMR properties in other solvents, see SI.

Synthesis of trans-Pt2Br4(HNEt2)2, 2. Compound 1 (150 mg, 0.175 mmol) was dissolved in a mixture of CHCl3 (ca. 300 l) and petroleum ether (1.2 ml). The solution was irradiated with UV-light. After 18 h the colorless solution was decanted off from formed red-orange precipitate, which was then washed with pentane and dried in vacuo 4 JH-Pt = 62 Hz), 1.65 (t, 3 JH-H = 7 Hz, 12H, HN(CH2CH3)2); 13 Preparation of cis-PtBr2(C2H4)(HNEt2), 3. Compound 2 (100 mg) was suspended in CHCl3 (4 mL) and hexane (4 ml) and the mixture was treated with an ethylene stream in the dark. After 18 h, the 1 H NMR spectrum showed only small changes relative to the starting material. After 4 days a small amount of grey precipitate had formed while the colour of solution was slightly green. The precipitate was filtered off, washed with CHCl3 and dried (16 4 JH-Pt = 62 Hz), 1.43 (t, 3 JH-H = 7 Hz, 6H, HN(CH2CH3)2). 195 Pt (64.5 MHz, δ): -3059 (br tr, J = 213 Hz). The compound slowly decomposes in solution with the formation of black particles.

(132 mg, 94%
Computational details. All geometry optimizations were carried out on isolated molecules in the gas phase by the DFT approach with the Gaussian03 suite of programs [START_REF] Frisch | Gaussian 03, Revision C.02[END_REF] using the B3LYP functional [START_REF] Becke | [END_REF][24][25]. All molecules described in this work are diamagnetic with the paramagnetic excited states expected at much higher energy, thus the typical problem of the overestimation of spin polarization shown by hybrid functionals [26] does not negatively affect our calculations. All atoms except Pt were described by the standard 6-31+G* basis set, which includes both polarization and diffuse functions that are necessary to allow angular and radial flexibility to the highly anionic systems. The Pt atom was described by the LANL2TZ(f) basis, which is an uncontracted version of LANL2DZ and includes an f polarization function and an ECP [27]. Frequency calculations were carried out for all optimized geometries in order to verify their nature as local minima, for the calculation of thermodynamic parameters at 298.15 K under the ideal gas and harmonic approximations, and for the identification of all transition states (one imaginary frequency). Solvent effects were introduced by means of C-PCM [28,29] single point calculations on the gas-phase optimized geometries in dichloromethane ( = 8.93) and acetone ( = 20.49). Among various continuum solvent models, the C-PCM was selected because of its generally better performance [30][31][32], even though there remains an inherent error when working with charged species [30]. The solvent cavity was created by a series of overlapping spheres by the default UA0 model and all standard settings as implemented in Gaussian03 were used for the C-PCM calculations. The reaction free energy changes in solution were corrected for the change of standard state from the gas phase (1 atm) to solution (1 M) [33].

X-ray crystallography. A single crystal of each compound was mounted under inert perfluoropolyether at the tip of a glass fibre and cooled in the cryostream of an Oxford-Diffraction XCALIBUR CCD diffractometer. Data were collected using the monochromatic MoK radiation (= 0.71073). The structures were solved by direct methods (SIR97) [34] and refined by least-squares procedures on F 2 using SHELXL-97 [35]. All H atoms attached to carbon were introduced in idealised positions and treated as riding on their parent atoms in the calculations except for the H attached to the ethylene group which were refined using restraints. The drawing of the molecules was realised with the help of ORTEP3 [36]. Crystal data and refinement parameters are collected in Compound 1

Distances

Results and Discussion

(a) Synthesis and characterisation of the starting materials.

The synthetic work is summarized in Scheme 2. Complex trans-PtBr2(C2H4)(HNEt2)

(1) was synthesized in 75% yield starting from Zeise's salt by adapting the methodology previously described for the preparation of related complex trans-PtBr2(C2H4)(PhNH2), 1a [7].

Scheme 2

The trans square planar coordination geometry of 1 was confirmed by a single crystal X-ray structural study, see Figure 1, which reveals a short contact between the N-H proton and one of the Br atoms (Br1•••H-N1 = 2.71 Å). The solution 1 H NMR spectrum is consistent with the solid state geometry, with two independent resonances as complex multiplets at  3.36 and 3.13 for the diastereotopic methylene protons. One pair of these protons also shows 195 Pt satellites (J = 66 Hz, see Figure S1). The equivalent CH3 protons show a complex signal the shape of which approaches a first-order binomial triplet at  1.53 (500 MHz, Figure S1), while it is more complex at 300 MHz (Figure S2). The broad amine NH proton ( ca. 5.6) also shows coupling with the Pt nucleus (ca. 90 Hz). The thermal ellipsoids are drawn at the 50% probability level.

The 195 Pt{ 1 H} NMR resonance of 1 is peculiar, see Figure 2. It shows as a 1:1:1 triplet, the shape of which is distorted to yield approximate relative heights of 1:1.7:1 at 64.2 MHz and 1:1.25:1 at 107.2 MHz (J = 218 Hz). The observed coupling, which is attributed to the quadrupolar 14 N atom (I = 1), is not usually observed in the 195 Pt NMR spectrum of platinum amine compounds [37], but the resonance shape and relative signal intensities are well understood [38]. Other rare examples of this phenomenon have been reported for the ammonia complexes [Pt(NH3)4] 2+ , [cis-Pt(NH3)2(H2O)2] 2+ [39] and While the synthesis of 1 is relatively straightforward, that of its cis isomer 3 is more problematic. The same methodology adopted for the preparation of cis-PtBr2(C2H4)(PhNH2) [7] (ethylene addition to the aminetribromoplatinate precursor) cannot be applied here because the product cannot be kinetically stabilized by precipitation and thus led again to the thermodynamically preferred 1 as a sole product under these conditions. An alternative synthetic pathway, described for a related Cl-NHMe2 complex [13,41], involves the irradiation of trans-PtCl2(C2H4)(HNMe2) to afford trans-Pt2Cl4(HNMe2)2, from which cis-PtCl2(C2H4)(HNMe2) is obtained by addition of ethylene. This procedure indeed led to the formation of the desired complex 3, although less efficiently because the ethylene addition to the intermediate dinuclear complex 2 is extremely slow (more than 4 days vs. 2-4 h for the Cl analogue). The intermediate 2 was also obtained more slowly than the Cl-NHMe2 analogue under the same conditions (18 vs. 12 h). The molecular structure of complex 2 was also obtained, see Figure 3. x, -y, 1-z.

K[Pt(NH3)Cl3] [40]. δ -3465 -3460 -3455 -3450 -3445 (a) (b)
Like complex 1, complex 3 also shows 195 Pt- 14 N coupling ( 1 J = 213 Hz) in the 195 Pt{ 1 H} spectrum. It does not show, on the other hand, 195 Pt satellites for the broad C2H4 ligand resonance in the 1 H NMR spectrum. Since the yield of 3 is very small, we shall only concentrate on the reaction between HNEt2 and the trans isomer 1 for the model study of the nucleophilic attack to coordinated ethylene, relevant to the Pt-catalyzed ethylene hydroamination mechanism.

(b) Reaction between trans-PtBr2(C2H4)(HNEt2) and HNEt2 in dichloromethane.

The addition of NHEt2 to 1 in CD2Cl2 at 298 K was followed by 1 H, 13 C{ 1 H} and 195 Pt{ 1 H} NMR, giving evidence for the equilibrated formation, within time of mixing, of the zwitterionic complex trans-Pt (-) Br2(NHEt2)(CH2CH2N (+) HEt2) (4), according to Scheme 3. At long reaction times (2-3 days), the deposition of black particles (presumably Pt 0 ) occurred, but the solution was relatively stable over the first few hours for a full NMR investigation to be carried out. Only the spectra run in CD2Cl2 under argon are shown in the figures and discussed in detail; similar results were also obtained in CDCl3 or in air.

Scheme 3

The transformation of 1 to 4 is indicated by the decrease of the 1 significantly shift in the presence of free NHEt2 (see Figure S3), suggesting a slow NHEt2 self-exchange process (close to the slow exchange limit in NMR) for compound 1. This is much slower than for the related complex trans-PtBr2(C2H4)(PhNH2), for which a single resonance for coordinated and free PhNH2 was observed (fast exchange limit in NMR) [7]. The amine addition reaction was also monitored by IR spectroscopy. Compound 1 exhibits a NH band at 3227 cm -1 ( = 100 M -1 cm -1 ) for the coordinated Et2NH ligand in CH2Cl2. Upon addition of an increasing amount of Et2NH this band decreases in favor of a new band at 3248 cm -1 ( = 37 M -1 cm -1 ), which is attributed to the NHEt2 ligand of compound 4 (see Figure S6). The shift of this band to higher frequency on going from 1 to 4 is in agreement with predictions based on DFT calculations (vide infra). A variable temperature study on the solution containing 1 equiv of NHEt2 (Figure S7) shows an equilibrium shift toward 4 upon cooling, evidencing exothermicity for the amine addition process. Quantitative analysis of the acquired data indicated a 1:1 reaction stoichiometry (Figure S8) with thermodynamic parameters H° = -6.8 ± 0.5 kcal/mol and S° = -14.0 ± 2.0 e.u. (Figure S9). These values correspond to K298 K = 80 M -1 and G298 K = -2.6 kcal/mol. A related parameter for the addition of HNEt2 to trans-PtCl2(C2H4)(HNEt2), investigated by NMR spectroscopy in chloroform, gave values of K298 K = 12 M -1 (G298 K = -1.5 kcal/mol) [19].

Analysis of the spectra below 3000 cm -1 (Figure S10) did not reveal bands that could be clearly attributable to the ammonium NH vibration, but showed the disappearance of the bands of 1 (at 1326, 1218 cm -1 ) and Et2NH (1327, 1184, 1140, 1125 cm -1 ) and the appearance of several new bands attributed to 4 (at 1395, 1354, 1342, 1206, 1089, 1026 cm -1 ) and assigned to numerous deformational N-H vibrations of the Et2NH and -CH2CH2N (+) HEt2 ligands, mixed with CH deformations. Notably, these bands increase in a similar manner both upon the Et2NH excess increase and upon temperature decrease. There was no indication of new species generated by a higher amine excess, in agreement with the NMR results described above.

The absence of observable bands due to the ammonium NH vibration may result from strong intra-and intermolecular associations in 4 and 4•NHEt2, respectively. As generally appreciated, H-bonding has the effect of broadening the N-H vibration and shifting it toward lower frequency. In order to further confirm this hypothesis, an IR investigation in CH2Cl2 was also carried out for the protonation of Et3N by CF3COOH, yielding Et3NH + as a model of the putative -CH2CH2N (+) HEt2 unit in 4 (Figure S11). It did not reveal a NH band attributable to Et3NH + , although the formation of ionic species is evident from the presence of the CF3COO -anion  as OCO band at 1673 cm -1 [45]. Hence, this experiment justifies the lack of observation of a spectroscopic signature for the ammonium ion of 4 in the IR spectrum.

(c) Reaction between trans-PtBr2(C2H4)(HNEt2) and HNEt2 in acetone.

The same experiment described above was also carried out in acetone. The reason for this choice is that acetone is more polar than dichloromethane ( = 20.49 vs. 8.93 for CH2Cl2) and may therefore favour the formation of charge-separated species, for instance deprotonation of the zwitterion 4 by excess NHEt2 to yield a negatively charged aminoethyl Pt II complex and Et2NH2 + . The results of the NMR and IR investigations were very similar to those described above in dichloromethane within the first few hours from mixing, again in agreement with the transformations outlined in Scheme 3. Identical results

were also observed working under an argon atmosphere or in air. The relevant results of This can be easily explained by the equilibrated nature of the nucleophilic amine addition to 1 to yield 4 (Scheme 3). In a separate experiment, two equivalent solutions of 1 in acetone-d6 were treated, one with 6 equiv of Et2NH, the other one with 8 equiv of Et2NH and then immediately with 2 equiv of TfOH (aiming at the in situ generation of 6 equiv of Et2NH and 2 equiv of Et2NH2 + ). The underlying idea was to verify whether the presence of the conjugate acid would retard the decomposition of 4, thus helping to rationalize the greater catalytic activity of PtBr2/Br -in the ethylene hydroamination by aniline when a small amount of a strong acid was added. The results (shown in the supplemental figures S15 and S16) reveal that the evolution of compound 4 is essentially the same in both cases. However, an important difference is observed immediately: a new set of resonances (a triplet and a quartet) for an ethyl derivative is seen in the spectrum following the addition of TfOH. A spiking test confirmed the identity of this compound as Et3N (0.38 equiv relative to Pt, according to integration), as well as its absence in the experiment run without acid. Thus, it seems that the Pt-C bond protonolysis and the protonation of the excess amine by TfOH are competitive, even though the amine is present in much larger amount (8 equiv per Pt complex). After this initial burst of protonolysis, no additional Et3N is generated (its relative amount remains constant) while the residual 4 further decomposes, as it does in the absence of conjugate acid. This shows also that Et2NH2 + , once formed, is not sufficiently acidic to induce the release of Et3N from 4 or from any of the complexes generated by the decomposition of 4.

As mentioned above, earlier reports have already shown the stoichiometric liberation of the hydroamination products from zwitterionic compounds [46][47][48], although these studies were carried out, to the best of our knowledge, on isolated compounds and under conditions where the equilibration with the starting olefin complex is not established (addition of a mineral solution of the strong acid to the isolated solid under heterogeneous conditions). Our observations on the protonation of the equilibrated solution show that, although the free amine is protonated, shifting the equilibrium back toward the olefin complex, protonation of the alkyl ligand leading to the hydroamination product also occurs, indicating that this process is faster than the equilibration between 1 and 4 (and even competitive with the protonation of the excess free NHEt2).

The IR monitoring of the amine addition in acetone yields a spectral picture in the NH region qualitatively identical to that in dichloromethane (Figure S17), except that the NH(1) band shows widening (1/2 = 25 cm -1 in acetone vs. only 10 cm -1 in dichloromethane) and low frequency asymmetry. This is probably due to the specific solvation effects (NH•••O=CMe2 interaction). A quantitative analysis gives K298 K = 135 M -1 (G298 K = -2.9 kcal/mol) for the nucleophilic addition equilibrium constant, slightly higher than the value in dichloromethane. Careful examination of all spectral changes (including down to 1000 cm -1 ) does not show any evidence for the presence of other species; however, the bands are wide and a large portion of the spectra is masked by the solvent absorptions.

(d) Computational investigation.

In a previous contribution [7] we have shown that the reaction of trans-PtBr2(C2H4)(PhNH2) (1a) with PhNH2 does not yield an observable zwitterionic product up to ca. 68°C, the only observable process being a self-exchange of free and coordinated aniline molecules. In combination with a computational investigation [10], the lack of observation of the nucleophilic addition product was attributed to its high relative energy.

On the other hand, we have shown in this contribution that the reaction between 1 and HNEt2 gives observable equilibrium amounts of the zwitterionic product 4 and its Hbonded derivative 4•NHEt2 (Scheme 3). Hence, the reaction free energy associated to the nucleophilic amine addition must be smaller in this case. We have carried out a full QM DFT study on the conversion of 1 to 4 using the same computational level and approximations previously reported for the corresponding aniline system [10], without molecular simplification and with inclusion of the solvent effects through the C-PCM model, the only difference being the nature of the solvent (dichloromethane,  = 8.93, or acetone,  = 20.49) in the present study). It was shown for the aniline system that the most favourable nucleophilic addition pathway involves the amine attack to the olefin anti face relative to the metal, thus we restricted our calculations to this pathway also for the NHEt2 addition. Figure 5 summarizes the comparative results for the two systems. The numbering scheme used for the PhNH2 system is the same as for the NHEt2 system with an added "a" suffix. All reported energies corresponds to ∆G CPCM = ∆Ggas + ∆∆Gsolv (∆Ggas is the free energy change of the reaction in the gas phase and ∆Gsolv is the free energy of solvation for each compound), which was found to provide the best qualitative agreement with the experimental results, in spite of the inevitable uncertainty associated with the calculation of the solvation entropy [21]. The ∆G CPCM values are given relative to the common precursor [PtBr3(C2H4)] -, taking into account the values of all species that are added to or subtracted from the system. This ion was indeed shown to be thermodynamically more stable than 1a by NMR investigations in CD2Cl2 solution and computational work indicated that it is the resting state of the PtBr2/Br --catalyzed ethylene hydroamination by aniline [10].

The first notable observation from Figure 5 is that NHEt2 binds the metal atom more strongly than PhNH2, since the relative energy of 1 is 0.8 kcal/mol below that of [PtBr3(C2H4)] -and 4.2 kcal/mol below that of 1a. This result correlates at least qualitatively with the greater lability of the coordinated aniline, which undergoes selfexchange with free ligand in complex 1a considerably faster than the NHEt2 ligand in complex 1 (vide supra). The second interesting observation is that the nucleophilic addition barrier is essentially identical for both amines (20.1 kcal/mol for 1a to 4a and 20.3 kcal/mol for 1 to 4). The starting complexes have practically identical calculated C=C distances (1.398 Å for 1 and 1.396 Å for 1a), indicating an essentially identical degree of -back donation. The two transition states have also very similar geometries (C•••N distance of ca. 2.0 Å in both cases for the incipient bond) and charge distributions (Mulliken charges of -0.176 and -0.177 for the C atom in TS(1a-4a) and TS(1-4), respectively). The optimized zwitterionic structures are nearly identical, except for the presence of an intramolecular NH•••Br bond in 4 and its absence in 4a. Both complexes are calculated as less stable than the precursor olefin complex, but the difference between 4 and 1 is 6.3 kcal/mol while that between 4a and 1a is greater, 9.5 kcal/mol. The calculated free energy difference for the conversion of 1 to 4 is too large to be compatible with the observation of equilibrium between these two compounds. The computational error, given the amount of approximations related to the solvent model, inaccurate estimation of the solvation entropy, neglect of the counterion, in addition to the usual error related to the accuracy of the DFT method, are believed to be responsible for the discrepancy. However, the energy difference changes in the right direction on going from the aniline to the NHEt2 system. The calculated reaction enthalpy, including the solvation effects, is -4.9 kcal/mol, in rather good agreement with the experimentally determined value from the IR study (vide supra).

The optimized structure of the zwitterions 4 shows, like that previously reported for 4a, an anti conformation for the CH2CH2 chain linking the Pt atom and the ammonium function. We have also optimized an isomeric structure where the ammonium function establishes an intramolecular H-bond with the metal center (4', see Figure 5). Interest in this calculation stemmed from the fact that the analogue 4a' for the PhNH2 system was shown to be an intermediate for the hydroamination catalysis, delivering a proton to the Pt center in one of the key steps of the catalytic cycle. Like for the PhNH2 system, this intramolecular H-bonded isomer is found less stable than the non bonded version in terms of G C-PCM , although marginally favored in terms of H C-PCM (-0.7 kcal/mol), indicating that the enthalpic advantage of the H-bond does not compensate the increased steric tension and the increased ordering associated with the eclipsed conformation of the carbon chain. An additional calculation was also carried out on the intermolecular H-bonded adduct 4•NHEt2. This adduct is calculated as less stable on the free energy scale by 8.6 kcal/mole in dichloromethane (Figure 6), but more stable by -3.7 kcal/mol in terms of H C- PCM . Hence, the hypothesis of the formation of equilibrium amounts of 4•NHEt2 to rationalize the NMR behaviour of the excess free NHEt2 appears justified. to NHEt2 in dichloromethane and in acetone.

We have also calculated the energy cost for transferring a proton from the ammonium function of 4 to the H-bonded amine, leading from 4•NHEt2 to the aminoalkyl complex [(NHEt2)Br2Pt(CH2CH2NEt2)] -, 5, H-bonded with the corresponding counterion Et2NH2 + , and for the subsequent charge-separated ions. Since this process, proposed as responsible for the equilibration of the methylene diastereotopic protons of the ammonium moiety, occurs at different rates in CD2Cl2 and acetone-d6 (Figure 4), the calculations were carried out with the C-PCM in both solvents. The results, also reported in Figure 6, show that the proton transfer process is endoergic and requires approximately the same energy in both solvents. On the other hand, the separation of the charged species in the 5•NH2Et2 + product is quite solvent-dependent, being slightly endoergic in dichloromethane and exoergic in acetone. When considering that the entropic part of the free energy is partially quenched in solution, the last step (charge separation) becomes the rate determining step.

Thus, the more polar acetone solvent indeed seems to promote the easier equilibration of the diastereotopic methylene protons of the ammonium function via faster deprotonation and reprotonation after inversion of the pyramidal N atom.

Finally, we have also attempted to optimize a structure obtained by intramolecular proton transfer from the ammonium function to the Pt atom, namely a Pt IV hydrido complex of formula [Pt(H)Br2(NHEt2)(CH2CH2NEt2)]. A related complex was shown to be implicated as a catalytic intermediate in the ethylene hydroamination by aniline [10].

However, contrary to the aniline system a stable minimum for this complex was not obtained, leading back to the geometry of the zwitterion 4 instead, which may be rationalized by the greater basicity of the aminoalkyl ligand for the Et2NH system.

The frequency calculations have helped with the assignment of the IR spectral changes upon NHEt2 nucleophilic addition. The results are shown in the supplemental Table S1. The calculated values of the NH frequencies are ca. 200 cm -1 higher than the experimental values, but the blue shift of +29 cm -1 for the coordinated NHEt2 on going from 1 to 4 corresponds closely to the experimentally observed shift of +21 cm -1 .

According to the computation prediction, the NH frequency and intensity of the free ammonium function in 4 are similar to those of the coordinated NHEt2 ligand, but the absorption greatly red-shifts and increases in intensity when the group is H-bonded in an intra-or intermolecular fashion. Since no absorptions are experimentally observed in the predicted regions of the IR spectrum, although the intermolecular bonding interaction is suggested by the NMR data, we suspect that the compound is indeed mostly present as H-bonded adduct 4•NHEt2, the NH absorption of which is not visible given the expected broadening and the overlap with other absorptions.

Conclusions

The present study aimed at better understanding the PtBr2/Br -catalytic system, which is rather efficient for the hydroamination of ethylene [START_REF] Brunet | [END_REF]7] and higher olefins [4,5] by low-basicity amines [6], but is limited by catalyst decomposition with formation of inactive metallic platinum [START_REF] Brunet | [END_REF]8]. Greater activity was previously reported under anhydrous conditions in the presence of a small amount of a strong acid as co-catalyst [START_REF] Brunet | [END_REF]. On the basis of literature reports, the stability of Pt II (C2H4) complexes in the presence of amines seems dependent on deprotonation processes for the zwitterionic product of amine addition to the coordinated olefin to yield aminoalkyl ligands, which then may evolve in a number of ways and often lead to the deposition of metallic Pt [13,16,18]. Therefore, the role of the acid may well be that of protecting the zwitterionic intermediate against equilibrium deprotonation and subsequent decomposition.

With this concept in mind, we have investigated the addition of an amine of high basicity, NHEt2, capable of yielding observable amounts of the zwitterionic product, and used two different solvents of different polarity (dichloromethane and acetone). The choice of acetone, not frequently used in this type of investigation, was indeed guided by the consideration that its greater dielectric constant should favour the transfer of a proton to the excess of external NHEt2 and thus trigger faster decomposition. Indeed, the DFT calculations support the notion that deprotonation of the zwitterionic derivative 4 should occur more easily in acetone. Indeed, the compound decomposes faster in this solvent. No direct evidence for the formation of an aminoalkyl complex was obtained in our study. The most relevant result, suggesting an easier deprotonation in acetone, is a faster dynamic process equalizing the resonances of diastereotopic methylene protons for the ammonium function. Catalytic hydroamination, on the other hand, apparently needs intramolecular proton transfer from the ammonium function of the zwitterion to the Pt atom or directly to the Pt-bonded C atom [5,10]. Hence, conditions limiting deprotonation should lead to greater catalyst stability and hence increase the turnover number. This proposition is now subject to further testing in catalytic intermolecular hydroamination.

SYNOPSIS TOC

  C{ 1 H} (75.5 MHz, δ): 68.2 (s with Pt satellites, 1 JC-Pt = 153 Hz, C2H4), 50.0 (s with Pt satellites, HNCH2, 1 JC-Pt = 13 Hz), 13.5 (s with Pt satellites, HN(CH2CH3)2, 1 JC-Pt = 8 Hz). 195 Pt (64.5 MHz, δ): -2210 (br).

13 C{ 1

 131 H} (75.5 MHz, δ): 68.2 (s with Pt satellites, 1 JC-Pt = 185 Hz, C2H4), 50.0 (s with Pt satellites, HNCH2, 1 JC-Pt = 15 Hz), 13.5 (s with Pt satellites, HN(CH2CH3)2, 1 JC-Pt = 8 Hz).

Figure 1 .

 1 Figure 1. An ORTEP view of the molecular structure of trans-PtBr2(C2H4)(NHEt2), 1.The thermal ellipsoids are drawn at the 50% probability level.

Figure 2 .

 2 Figure 2. 195 Pt{ 1 H} NMR of 1 in acetone-d6 at 107.2 MHz (a) and 64.2 MHz (b).

Figure 3 .

 3 Figure 3. An ORTEP view of the molecular structure of trans-[PtBr2(NHEt2)]2, 2. The

  Figure S3. Integration of the residual C2H4 resonance of 1 demonstrates its persistence

Figure 4 . 6 (

 46 Figure 4. Selected region of the 1 H NMR spectra for compound 4 in the presence of 2

  Therefore, the shift and broadening of the free Et2NH resonances cannot be caused by any chemical exchange within compounds 1 and 4. We propose that it results from a rapid exchange between the excess free and H-bonded amine in a 4•HNEt2 adduct, see Scheme 3. The formation of H-bonded interactions by amines coordinated to platinum complexes is precedented.[42][43][44] The conversion of 1 to 4 is accompanied by the generation of a major resonance at  -3565 in the 195 Pt{ 1 H} NMR spectrum (FigureS4) and resonances at  -13.6 ( 1 JC-Pt = 667 Hz) and 60.7 ( 2 JC-Pt = 36 Hz) for the  and  C atoms of the CH2CH2N(+) HEt2 ligand in the 13 C{ 1 H} NMR spectrum (FigureS5). The NHEt2 13 C resonances of residual 1 are broadened by free NHEt2, whereas those of the same ligand in 4 remain sharp, confirming the 1 H NMR evidence for the amine exchange dynamics. Time evolution of these spectra shows the formation of decomposition products more clearly than the 1 H spectrum, given the reduced number of resonances and their greater spread.

the 1 H

 1 , 195 Pt{ 1 H} and 13 C{ 1 H} monitoring are shown in Figures S12-14. An interesting difference concerns the dynamic behaviour. As clearly shown in Figure 4, the diastereotropic methylene protons of the ammonium ethyl groups are collapsed into a single quartet resonance in this solvent. This phenomenon is attributed to a faster deprotonation/amine inversion/reprotonation via the 4•NHEt2 intermediate, which is favoured by the higher dielectric constant of the acetone medium. This point will be addressed again in the computational part below. The 1 H NMR spectra show the same shifting of the excess free NHEt2 resonances, confirming rapid 4/4•NHEt2 exchange, and the broadening of the coordinated NHEt2 resonance in residual 1, indicating slow selfexchange, as also observed in CD2Cl2 (vide supra). The immediate, selective and equilibrated formation of compound 4 in acetone-d6 was again confirmed by the 195 Pt{ 1 H} and 13 C{ 1 H} NMR monitoring. The former shows a singlet resonance at  -3550 while the latter reveals the characteristic Pt-C(α)-C(β) resonances with Pt satellites at  -14.7 ( 1 JC-Pt = 667 Hz) and  59.5 ( 2 JC-Pt = 36 Hz), respectively (Figures S13 and S14). These chemical shifts correspond closely to those observed in CD2Cl2 (vide supra). The decomposition occurred at a faster rate in acetone and led to different products relative to dichloromethane. The investigation of the nature of these decomposition products was not further pursued. The deposition of black solid occurred in both solvents after a few days and no formation of NEt3 was observed during the decomposition. Addition of the strong acid CF3SO3H to a partially decomposed solution did not eliminate the resonances of the decomposition products (as monitored by 13 C{ 1 H} NMR), indicating that these are not produced by equilibrium processes involving the deprotonation of the zwitterion ammonium function by the excess Et2NH. In a separate experiment, excess acid was added immediately to a solution of 4 generated in situ from 1 and 5 equiv of Et2NH. The resonances of 4 disappeared, accompanied by the reappearance of the resonances of compound 1.

9 [Figure 5 .

 95 Figure 5. G C-PCM in kcal mol -1 at 298.15 K and views of the optimized geometries for the

Figure 6 .

 6 Figure 6. G C-PCM (and H C-PCM in parentheses) in kcal mol -1 at 298.15 K relative to

  ). Elem. anal. % calc. for C8H22Br4N2Pt2, C, 11.22; H, 2.59; N, 3.27. %

	Found: C, 12.03; H, 2.63; N, 3.26. IR (neat, cm -1 ): 3183 (N-H). NMR (acetone-d6): 1 H
	(300.13 MHz, δ): 5.04 (br, 2H, NH, 2 JH-Pt = 60 Hz), 2.78 (m, 4H, HNCH2), 2.50 (m, 8H,
	HNCH2,

Table 1 ,

 1 while relevant bond distances and angles are given in Table2. Crystallographic data (excluding structure factors) have been deposited with the Cambridge Crystallographic Data Centre as supplementary publication no. CCDC 784685 & 784686. Copies of the data can be obtained free of charge on application to the Director, CCDC, 12 Union Road, Cambridge CB2 1EZ, UK (fax: (+44) 1223-336-033; e-mail: deposit@ccdc.cam.ac.uk).

Table 1 .

 1 Crystal data and structure refinement for compounds 1 and 2.

	Identification code	1	2
	Empirical formula	C6H15Br2NPt	C8H22Br4N2Pt2
	Formula weight	456.10	856.10
	Temperature,K	180(2)	180(2)
	Crystal system	Monoclinic	Monoclinic
	Space group	P 21/n	P 21/n
	a, Å	11.0458(4)	7.1779(3)
	b, Å	8.7461(3)	11.6622(4)
	c, Å	11.5095(4)	10.3715(4)
	,°	90	90.0
	,°	102.881(4)	100.113(2)
	,°	90	90.0
	V, Å 3	1083.92(7)	854.71(6)
	Z	4	2
	Dcalc, Mg/m 3	2.795	3.326

Table 2 .

 2 Relevant

bond lengths [Å] and angles [°] for compounds 1 and 2.

Acknowledgement

We are grateful to the CNRS (Centre National de la Recherche Scientifique) and the RFBR (Russian Foundation for Basic Research) for support through the GDRI (Groupe de Recherche Internationale) "Homogeneous catalysis for Sustainable Development" and a France-Russia (RFBR -CNRS) bilateral grant № 08-03-92506. The French group also thanks the ANR (Agence Nationale de la Recherche, Grant No. NT09_442499) for additional financial support, as well as the CINES (Centre Informatique National de l'Enseignement Supérieur) and the CICT (Centre Interuniversitaire de Calcul de Toulouse, project CALMIP) for granting free computational time. PD thanks the MENESR (Ministère de l'Éducation Nationale, de l'Enseignement Supérieur et de la Recherche, France) for a Ph.D. fellowship.