Swaminath Venkateswaran
email: swaminath.venkateswaran@devinci.fr

Damien Chablat
email: damien.chablat@cnrs.fr

Mapping the Tilt and Torsion Angles for a 3-SPS-U Parallel Mechanism

Keywords: parallel mechanism, tilt and torsion, singularities, control

This article presents the analysis of a parallel mechanism of type 3-SPS-U. The usual singularity approach is carried out with respect to the Euler angles of the universal joint. However, this approach is computationally expensive especially when stacked structures are analyzed. Thus, the positioning of the mobile platform for the mechanism is analyzed using the theory of Tilt and Torsion (T&T). The singularity-free workspace and the tilt limits of the mechanism are disclosed through this method. These workspaces can then be mapped to the Euler angles of the universal joint and the relation between the T&T space and the Euler space is demonstrated and validated in this study. Initially, simulations are performed using the results of singularity analysis to have a comparison between the T&T space and the Euler space. Experimental validation is then carried out on the prototype of the mechanism to perform a circular trajectory, in line with the simulations. The outcome of this study will be helpful for the integration of the mechanism for a piping inspection robot and also for the analysis of stacked architectures.

Introduction

Parallel kinematics machines (PKM) have interesting applications in industries over serial machines. PKMs offer better accuracy, lower mass/inertia properties and high structural stiffness [START_REF] Merlet | Parallel Robots[END_REF]. The performance of parallel robots varies within their workspace, which is considerably smaller when compared to serial robots. At present, many researchers work on parallel robots as they offer simple and conventional designs with ease of calibration [START_REF] Pritschow | Parallel kinematic machines (PKM)-limitations and new solutions[END_REF]. Moreover, the inverse kinematics, in general, is easier to solve for parallel robots, especially when prismatic joints are employed. With the presence of spherical joints in a parallel mechanism, the modified Euler angles, otherwise known as the "Tilt and Torsion" (T&T) angles, could be employed [START_REF] Bonev | Advantages of the modified Euler angles in the design and control of PKMs[END_REF][START_REF] Bonev | Geometric Analysis of Parallel Mechanisms[END_REF]. Bonev et al. [START_REF] Bonev | Orientation workspace analysis of 6-DOF parallel manipulators[END_REF] proposed the advantages of the T&T angles for the study of spatial parallel mechanisms in-depth. It was demonstrated that there exists a class of parallel mechanisms with zero torsion, which are referred to as zero-torsion parallel mechanisms [START_REF] Bonev | Orientation workspace analysis of 6-DOF parallel manipulators[END_REF][START_REF] Briot | Singularity analysis of zero-torsion parallel mechanisms[END_REF]. Another interesting study which exploits the T&T angles is the 3-RRR spherical parallel mechanism, which was analyzed by Tao et al. [START_REF] Tao | Interference analysis and workspace optimization of 3-RRR spherical parallel mechanism[END_REF]. This analysis helped in improving the workspace of the parallel mechanism and was applicable for architectures that have revolute joints. Generally, the applications of parallel or spherical mechanisms are found in industrial applications such as machining [START_REF] Caro | Kinematic analysis and trajectory planning of the Orthoglide 5-axis[END_REF] and nano-alignment, [START_REF] Yang | Analysis and design of a 3-DOF flexure-based zero-torsion parallel manipulator for nano-alignment applications[END_REF]. However, there is another interesting application of a spherical-type parallel mechanism for a piping-inspection robot. A bio-inspired piping-inspection robot that moves like a caterpillar was designed and developed at LS2N, France [START_REF] Venkateswaran | Numerical and experimental validation of the prototype of a bio-inspired piping inspection robot[END_REF]. This prototype was a rigid model and it was limited to straight pipe profiles. An articulation unit was proposed for this robot which addresses two main issues, namely, passive compliance and active compliance. Passive compliance is applicable for a robot working inside a pipeline having an elbow at 90 • or 45 • . In such scenarios, the articulation units usually work in passive modes. Some examples of such mechanisms that exist in the literature are the squirm-type robot of Zhang et al. [START_REF] Zhang | Design and motion analysis of a flexible squirm pipe robot[END_REF] and the caterpillar-type robot of Kwon et al. [START_REF] Kwon | Design and motion planning of a two-moduled indoor pipeline inspection robot[END_REF]. On the other hand, for the active compliance issue, the articulation unit must be actuated in order to align the robot assembly with the pipeline axis or follow a certain direction inside the pipeline. This scenario could be imagined for a robot that passes through a junction or a T-branch inside a pipeline. Some examples of active compliance include the robot of Ryew et al. [START_REF] Ryew | In-pipe inspection robot system with active steering mechanism[END_REF], which employs a dual-actuated universal joint for passing through junctions and a pantograph mechanism to ensure contact with pipeline walls during locomotion. The third criterion that must also be taken into account is the tilt limit of the articulation unit as it determines if the mechanism tilt is sufficient to overcome a bend or a junction. By taking these factors into account, a parallel mechanism that employs three tension springs and a universal joint was incorporated as an articulation unit for a bio-inspired piping inspection robot [START_REF] Venkateswaran | Numerical and experimental validation of the prototype of a bio-inspired piping inspection robot[END_REF][START_REF] Venkateswaran | A new inspection robot for pipelines with bends and junctions[END_REF]. The positioning of the end-effector with respect to the fixed base plays an important role in identifying the tilt limits of this mechanism. The conventional approach involves analyzing the end-effector for the Euler angles of the universal joint [START_REF] Venkateswaran | Singularity and Workspace Analysis of 3-SPSU and 4-SPSU Tensegrity Mechanisms[END_REF]. As the universal joint has two degrees of freedom about the x and y axis, the workspace analysis involves studying the influence of both tilt angles for each position of the end-effector within the joint limits. However, there exists an alternate approach where the end-effector can be localized using the T&T angles. When compared to the Euler space, in the T&T space the singularity analysis appears much simpler as the tilt angle remains constant while the azimuth can vary from 0 to 2π radians or vice versa. This article demonstrates a relation that exists between the T&T space and the Euler space for the 3-SPS-U parallel mechanism under study. This relation, also called the "mapping", has been demonstrated and validated for the mechanism through simulations and experiments.

The outline of this article is as follows. Section 2 presents the architecture and the geometrical equations of the mechanism using the T&T angles. In Section 3, the singularity analysis of the mechanism is presented in detail through algebraic methods. Following this, in Section 4, the mapping parameters are demonstrated for the T&T space to the Euler space. A numerical simulation is also presented to demonstrate this mapping. Section 5 presents the experimental validation of the mapping theory with the help of the prototype of the mechanism and control laws. Finally, in Section 6, closing discussions are made along with the future perspectives.

Architecture of the Parallel Mechanism

This section presents the architecture of the parallel mechanism that will be analyzed in this article. An overview of the final application of this mechanism is first presented. Followed by that, the geometrical equations that are necessary to analyze the singularities are presented.

A Flexible Mechanism for a Piping Inspection Robot

In collaboration with the enterprise AREVA in France, a rigid bio-inspired pipinginspection robot was designed and developed at LS2N, France. With the help of leg mechanisms and DC motors, the robot accomplishes the locomotion of a caterpillar to move inside a pipeline [START_REF] Venkateswaran | Numerical and experimental validation of the prototype of a bio-inspired piping inspection robot[END_REF]. A 3D model of this robot and its 2D view are shown in Figure 1. The application of this robot is limited to straight pipelines, as the model is rigid. In order to work inside a pipeline that consists of bends and junctions, a flexible version of the robot was proposed in [START_REF] Venkateswaran | A new inspection robot for pipelines with bends and junctions[END_REF]. Two similar parallel mechanisms with two degrees of freedom (DOF) were introduced between each module of the first robot. These parallel mechanisms consist of a universal joint to constrain the mobility between the base and the mobile platform and three tension springs with or without cables for the actuation. This mechanism can potentially assist the piping-inspection robot to overcome a 90 • bend or a junction. A digital model of the flexible piping-inspection robot with the parallel mechanism is shown in Figure 2.

Rear leg mechanism

Central module Front leg mechanism The primary focus of this article is to analyze the singular configurations of this parallel mechanism which will provide the tilt limits.

Geometrical Equations of the Mechanism

The parallel mechanism under study was analyzed earlier through two approaches, namely, (i) using the Tilt and Torsion (T&T) angles and (ii) singularity analysis with respect to the Euler angles. The former was employed to study the joint limits where three types of architectures were analyzed to identify an inverse pendulum configuration [START_REF] Venkateswaran | Design and analysis of a tensegrity mechanism for a bio-inspired robot[END_REF]. On the other hand, the singularity analysis was carried out for the Euler angles, which led to the conclusion that the 3-SPS-U architecture was not an ideal solution for addressing passive compliance owing to the limited tilt angles [START_REF] Venkateswaran | Singularity and Workspace Analysis of 3-SPSU and 4-SPSU Tensegrity Mechanisms[END_REF]. In this article, the singularity analysis is revisited for the 3-SPS-U mechanism by using the T&T angles. The articulation unit that consists of the tension springs and universal joints is correlated to a parallel manipulator of type 3-SPS-U [START_REF] Yigit | Design and modelling of a cable-driven parallel-series hybrid variable stiffness joint mechanism for robotics[END_REF][START_REF] Alici | Topology optimisation and singularity analysis of a 3-SPS parallel manipulator with a passive constraining spherical joint[END_REF]. The representation of the parallel mechanism under study is shown in Figure 3. In this mechanism, S indicates the spherical joint, U indicates the universal joint and P indicates the actuated prismatic joints or the springs. The mechanism also employs cables that pass through each spring from the base to the end-effector. The cable actuation is performed using external motors in the active modes and it does not resemble a classic prismatic link. Thus, under operation, the mobility of the end-effector is purely obtained through the two rotation angles of the universal joint. The home pose represented in Figure 3a is the position where the tilt angles, as well as the external forces on the mechanism, are equal to zero. The fixed coordinate frame of the base is represented by ∑ 0 . The spring mounting points on the fixed base are represented by B 1 , B 2 and B 3 and they form an imaginary equilateral triangle of the manipulator base. The vector coordinates for the base mounting points can be written as: Equation (1) provides the coordinates of B i in the fixed coordinate frame ∑ 0 . To determine the position of the end-effector, the Euler angles are usually employed [START_REF] Venkateswaran | Singularity and Workspace Analysis of 3-SPSU and 4-SPSU Tensegrity Mechanisms[END_REF], especially to study the singular configurations. In this case, the T&T angles are revisited from [START_REF] Venkateswaran | Design and analysis of a tensegrity mechanism for a bio-inspired robot[END_REF] for identifying singular configurations. It must be noted that in [START_REF] Venkateswaran | Design and analysis of a tensegrity mechanism for a bio-inspired robot[END_REF], the singularity analysis was not performed for the T&T space. As the torsion for the mechanism has been constrained, the tilt and azimuth angles are sufficient to analyze the architecture. The representation of the tilt and azimuth angles on the mechanism is revisited from [START_REF] Venkateswaran | Design and analysis of a tensegrity mechanism for a bio-inspired robot[END_REF] and it is shown below in Figure 4. This representation is correlated with the study shown by Bonev et al. [START_REF] Bonev | Advantages of the modified Euler angles in the design and control of PKMs[END_REF], where β represents the azimuth angle between the x-axis and the face of z-axis. A rotation about M causes a shift of the axis xyz to x * y * z * . The angle between z and z * represents the tilt angle, α. Here, the resultant rotation matrix R is a modified form of the equation presented in [START_REF] Bonev | Advantages of the modified Euler angles in the design and control of PKMs[END_REF] by Bonev et al. but in this case without the torsion. The matrix is given by the equation:

b 0 i =     r f cos 2π(i-1) 3 r f sin 2π(i-1) 3 -r f h     , with i = 1, 2, 3 (1)
B 2 B 1 B 3 -hr f l 1 l 2 l 3 B 1 B 2 B 3 C 1 C 2 C 3 Σ 1 Σ 0 x 1 y 1 z 1 x 0 z 0 y 0 C 1 C 2 C 3 A hr f B 1 B 2 , B 3 C 2 , C 3 C 1 r f A A (a) (b) (c)
R = R z (β)R x (α)R z (-β) (2) R =     cos(β) 2 + sin(β) 2 cos(α) cos(β) sin(β)(1 -cos(α)) sin(β) sin(α) cos(β) sin(β)(1 -cos(α)) sin(β) 2 + cos(β) 2 cos(α) -cos(β) sin(α) -sin(β) sin(α) cos(β) sin(α) cos(α)     (3)
where

R x (α) =     1 0 0 0 cos(α) -sin(α) 0 sin(α) cos(α)     , R z (-β) =     cos(β) sin(β) 0 -sin(β) cos(β) 0 0 0 1    
The moving coordinate frame of the end-effector is represented by ∑ 1 . The spring mounting points of the end-effector are represented by C 1 , C 2 and C 3 for the mechanism. In order to obtain the position of the end-effector, Equation ( 2) is multiplied by the vector coordinates of C i , which are given by:

c 1 i = R o 1 i , with i = 1, 2, 3 & o 1 i =     r f cos 2π(i-1) 3 r f sin 2π(i-1) 3 r f h     (4)
In Equation ( 4), o 1 i are the coordinates of C i in the mobile reference frame ∑ 1 of the end-effector. The length of the mechanism l i , which provides the solution for the inverse kinematic problem (IKP), can be solved by computing the distance between the vector coordinates of the base and the end-effector. The constraint equations are given by:

b 0 i -c 1 i = l i where b 0 i = [b ix , b iy , b iz ] T , c 1 i = [c ix , c iy , c iz ] T with i = 1, 2, 3 (5)
2 rf 2 (cos(β)) 2 cos(α) -2 sin(β) sin(α)h + cos(α)h 2 -(cos(β)) 2 + h 2 -cos(α) + 1 = l 2 1 (6a) -0.5 rf 2 (2 cos(β) sin(β) cos(α) √ 3 -4 √ 3 cos(β) sin(α)h + 2 (cos(β)) 2 cos(α) -2 cos(β) sin(β) √ 3 -4 sin(β) sin(α)h -4 h 2 cos(α) -2 (cos(β)) 2 -4 h 2 + cos(α) -1) = l 2 2 (6b) -0.5 rf 2 (-2 cos(β) sin(β) cos(α) √ 3 + 4 √ 3 cos(β) sin(α)h + 2 (cos(β)) 2 cos(α) +2 cos(β) sin(β) √ 3 -4 sin(β) sin(α)h -4 h 2 cos(α) -2 (cos(β)) 2 -4 h 2 + cos(α) -1) = l 2 3 (6c)
Equation (6a) to (6c) represents the solutions to the IKP. For known values of the tilt and azimuth angles as well as the design parameters r f and h, these equations can be employed to calculate the distance between the base and end-effector of the mechanism.

Singularity Analysis and Workspace of the Mechanism in the T&T Space

This section focuses on the singularity analysis of the mechanism. With the presence of the universal joint, the end-effector coordinates are identified conventionally using the Euler XY rotation angles about the origin A shown in Figure 3. In this approach, for each position of the end-effector, the influence of both rotation angles must be taken into account. However, with the help of the tilt and azimuth angles, it is still possible to analyze the presence of singularities in the mechanism. Moreover, in the T&T space, the mechanism will be simpler to analyze wherein one of the angles remains fixed or constant while the other changes. In [START_REF] Venkateswaran | Design and analysis of a tensegrity mechanism for a bio-inspired robot[END_REF], only the analysis of joint limits was carried out by setting minimum and maximum values for each prismatic link. This analysis provided the tilt angles for the joint limits; however, the direct kinematic problem (DKP), which presents the singularity-free workspace in T&T space, was not solved in [START_REF] Venkateswaran | Design and analysis of a tensegrity mechanism for a bio-inspired robot[END_REF]. The singularity-free workspace is thus identified by analyzing the mechanism in the T&T space, which provides the solution to the DKP and the tilt limits.

Singularity Equations of the Mechanism

For a parallel mechanism, the singularity equation is given by the well-known equation [START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF]:

At + B ρ = 0 ( 7
)
where t represents the angular velocity vector and ρ = [ l 1 , l 2 , l 3 ] T represents the joint velocity vector

The IKP Equations (6a) to (6c) are employed in Equation [START_REF] Tao | Interference analysis and workspace optimization of 3-RRR spherical parallel mechanism[END_REF], where their time derivatives are taken to study the singularities. In Equation [START_REF] Tao | Interference analysis and workspace optimization of 3-RRR spherical parallel mechanism[END_REF], A represents the direct-kinematics matrix or forward Jacobian matrix and B represents the inverse-kinematics matrix or inverse Jacobian matrix of the mechanism. The pose variables for the mechanism are the tilt and azimuth angles α and β. The articular variables are the lengths: l 1 , l 2 and l 3 . A parallel mechanism in general might encounter three types of singularities, namely, Type-1, Type-2 and Type-3 [START_REF] Wenger | Definition sets for the direct kinematics of parallel manipulators[END_REF][START_REF] Chablat | Working modes and aspects in fully parallel manipulators[END_REF]. When the determinant of the direct kinematics matrix A loses its rank, the determinant of this matrix becomes zero and this scenario leads to parallel singularities or Type-2 singularities. From the number of pose variables and articular variables, it could be seen that matrix A does not correspond to an n × n square matrix. For identifying the singularity equations and to construct a square matrix, the 3-SPS-U mechanism is split into three sets of 2-SPS-U architecture, which comprise the length pairs l 1l 2 , l 2l 3 and l 1l 3 . The corresponding direct kinematic matrices A are created for the three sets by differentiating Equations (6a) to (6c) with respect to the pose variables. The matrices are given by:

A 1-2 = ∂L 1 ∂α ∂L 1 ∂β ∂L 2 ∂α ∂L 2 ∂β , A 2-3 = ∂L 2 ∂α ∂L 2 ∂β ∂L 3 ∂α ∂L 3 ∂β , A 1-3 = ∂L 1 ∂α ∂L 1 ∂β ∂L 3 ∂α ∂L 3 ∂β (8)
In Equation ( 8), L 1 , L 2 and L 3 represent the left-hand sides of Equations (6a) to (6c). The parallel singularities are analyzed by calculating the determinant of matrices in Equation ( 8) and equating them to zero. The determinant values are given by:

det(A 1-2 ) , D p1 : -4 sin(β)(cos(α)) 2 (cos(β)) 2 √ 3h + 2 sin(β)(cos(α)) 2 √ 3h 3 -2 sin(α) cos(α)(cos(β)) 2 √ 3h 2 + 6 sin(β) sin(α) cos(α) cos(β)h 2 +8 sin(β) cos(α)(cos(β)) 2 √ 3h + 2 sin(α)(cos(β)) 2 √ 3h 2 -6 (cos(α)) 2 cos(β)h 3 -6 sin(β) sin(α) cos(β)h 2 -4 sin(β)(cos(β)) 2 √ 3h -2 sin(β) √ 3h 3 + sin(α) cos(α)(cos(β)) 2 √ 3 -3 sin(α) cos(α) √ 3h 2 -3 sin(β) sin(α) cos(α) cos(β) -2 sin(β) cos(α) √ 3h -sin(α)(cos(β)) 2 √ 3 -sin(α) √ 3h 2 + 3 (cos(α)) 2 cos(β)h + 6 cos(β)h 3 + 3 sin(β) sin(α) cos(β) +2 sin(β) √ 3h -sin(α) cos(α) √ 3 + sin(α) √ 3 -3 cos(β)h = 0 (9) det(A 2-3 ) , D p2 : 8 sin(β)(cos(α)) 2 (cos(β)) 2 h + 8 sin(β)(cos(α)) 2 h 3 -8 (cos(β)) 2 sin(α)h 2 cos(α) -16 sin(β) cos(α)(cos(β)) 2 h +8 sin(α)(cos(β)) 2 h 2 -6 sin(β)(cos(α)) 2 h + 8 sin(β)(cos(β)) 2 h -8 sin(β)h 3 + 4 (cos(β)) 2 sin(α) cos(α) + 12 cos(α)h 2 sin(α) +4 h sin(β) cos(α) -4 (cos(β)) 2 sin(α) -4 h 2 sin(α) + 2 sin(β)h -sin(α) cos(α) + sin(α) = 0 (10) det(A 1-3 ) , D p3 : 4 sin(β)(cos(α)) 2 (cos(β)) 2 √ 3h -2 sin(β)(cos(α)) 2 √ 3h 3 +2 sin(α) cos(α)(cos(β)) 2 √ 3h 2 + 6 sin(β) sin(α) cos(α) cos(β)h 2 -8 sin(β) cos(α)(cos(β)) 2 √ 3h -2 sin(α)(cos(β)) 2 √ 3h 2 -6 (cos(α)) 2 cos(β)h 3 -6 sin(β) sin(α) cos(β)h 2 + 4 sin(β)(cos(β)) 2 √ 3h +2 sin(β) √ 3h 3 -sin(α) cos(α)(cos(β)) 2 √ 3 + 3 sin(α) cos(α) √ 3h 2 -3 sin(β) sin(α) cos(α) cos(β) + 2 sin(β) cos(α) √ 3h + sin(α)(cos(β)) 2 √ 3 + sin(α) √ 3h 2 + 3 (cos(α)) 2 cos(β)h + 6 cos(β)h 3 + 3 sin(β) sin(α) cos(β) -2 sin(β) √ 3h + sin(α) cos(α) √ 3 -sin(α) √ 3 -3 cos(β)h = 0 ( 11
)
Using Equations ( 9) to [START_REF] Zhang | Design and motion analysis of a flexible squirm pipe robot[END_REF], it is possible to identify the pose variables and understand the presence of parallel singularities in the mechanism. On the other hand, when the determinant of the inverse kinematics matrix B loses its rank, the determinant of this matrix becomes zero and this scenario leads to serial singularities or Type-1 singularities. Similar to the parallel singularity equations, the inverse kinematics matrix B is created by differentiating the right-hand side of Equations (6a) to (6c) with respect to the articular variables. Following this, the determinant of these matrices is computed and equated to zero for verifying the serial singularities. The matrices and the determinant values are given by:

B 1-2 = ∂R 1 ∂l 1 ∂R 1 ∂l 2 ∂R 2 ∂l 1 ∂R 2 ∂l 2 , B 2-3 = ∂R 2 ∂l 2 ∂R 2 ∂l 3 ∂R 3 ∂l 2 ∂R 3 ∂l 3 , B 1-3 = ∂R 1 ∂l 1 ∂R 1 ∂l 3 ∂R 3 ∂l 1 ∂R 3 ∂l 3 (12) det(B 1-2 ) , D s1 : l 1 l 2 = 0 , det(B 2-3 ) , D s2 : l 2 l 3 = 0 , det(B 1-3 ) , D s3 : l 1 l 3 = 0 (13)
In Equation ( 12), R 1 , R 2 and R 3 represent the right-hand side of Equations (6a) to (6c). From Equation [START_REF] Ryew | In-pipe inspection robot system with active steering mechanism[END_REF], when the determinant of the inverse kinematics matrices is equated to zero, serial singularities can occur only when the length of one or two prismatic springs becomes zero. This condition is not feasible for the mechanism under study as the prismatic springs cannot have a length of 0 mm. Thus, there exist no serial singularities in the mechanism. The Type-3 singularities exist when a parallel mechanism encounters both serial and parallel singularities. Since there exist no serial singularities in the mechanism, only the parallel singularities Equations ( 9) to [START_REF] Zhang | Design and motion analysis of a flexible squirm pipe robot[END_REF] are analyzed to extract the feasible workspace of the mechanism.

Results of Singularity Analysis Using the CAD Algorithm

In order to extract the tilt limits for the mechanism from the workspace, the SIROPA library of Maple is employed [START_REF] Jha | Workspace, joint space and singularities of a family of delta-like robot[END_REF]. The vector coordinates of base from Equation (1), the endeffector coordinates from Equation ( 4) and the inverse kinematics Equations (6a) to (6c) that map the base to the end-effector are used as inputs for the CreateManipulator function of the SIROPA library. This function virtually constructs the 3-SPS-U manipulator in Maple [START_REF] Jha | Workspace, joint space and singularities of a family of delta-like robot[END_REF][START_REF] Chablat | Using Maple to analyse parallel robots[END_REF]. The tilt and azimuth angles are set as the pose variables and the lengths (l 1 , l 2 and l 3 ) are set as the articular variables for computations. Then, using the ParallelSingularities function of SIROPA, the Type-2 singularities Equations ( 9) to [START_REF] Zhang | Design and motion analysis of a flexible squirm pipe robot[END_REF] are generated. The actual prototype that will be incorporated with the piping-inspection robot has the design parameters r f as 11 mm and h = 0.6 [START_REF] Venkateswaran | An optimal design of a flexible piping inspection robot[END_REF]. As an experimental validation is carried out in the upcoming section, a scaled value of r f = 56.7 mm for h = 0.6 is chosen for the singularity analysis [START_REF] Venkateswaran | Trajectory Planning for a 3-SPS-U Tensegrity Mechanism[END_REF]. As the constraint equations require joint limits for analysis, the minimum and maximum lengths for each prismatic spring are set as l min = 40 mm and l max = 106 mm. The Cylindrical Algebraic Decomposition (CAD) algorithm is then employed to analyze the workspace of the mechanism. This approach was proposed for parallel robots by Chablat et al. [START_REF] Chablat | Uniqueness domains and non singular assembly mode changing trajectories[END_REF] wherein the workspace, as well as the joint space, are decomposed into cells from C 1 , ...., C k . The steps involved in this computation are: In Maple, the SIROPA library is exploited to use the CAD algorithm for the parallel mechanism [START_REF] Chablat | Uniqueness domains and non singular assembly mode changing trajectories[END_REF] and the analysis is carried out as follows:

• C k is
•

Computation of a subset of the workspace (resp. joint space) where the number of solutions changes: the discriminant variety

• Description of the complementary of the discriminant variety in connected cells: the Generic Cylindrical Algebraic Decomposition

•

Connecting the cells that belong to the same connected component of the complementary of the discriminant variety: interval comparisons Discriminant varieties can be computed using basic and well-known tools from computer algebra such as the Groebner bases [START_REF] Cox | Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra[END_REF]. A general framework for computing such objects is available through the RootFinding[Parametric] function of Maple. For extracting the workspace of the mechanism, the CAD algorithm combined with the parametric root-finding technique of Maple is employed to find solutions to the DKP for the tilt and azimuth angles [START_REF] Chablat | A comparative study of 4-cable planar manipulators based on cylindrical algebraic decomposition[END_REF]. This is carried out using the CellDecompositionPlus function of the SIROPA library in Maple [START_REF] Jha | Workspace, joint space and singularities of a family of delta-like robot[END_REF][START_REF] Chablat | Using Maple to analyse parallel robots[END_REF]. The parallel singularity Equations ( 9) to (11) and the joint limit Equations (6a) to (6c) are transformed as inequalities for isolating the aspects around the home pose. The results of the workspace generated by the CAD algorithm for the tilt and azimuth angles are represented in Figure 5. In Figure 5a, the blue regions indicate the feasible workspaces for the mechanism where there exists one solution to the DKP. The white regions are the zones where there exists no solution. Around the home pose, a larger workspace could be observed. Beyond the singularity zones, it is also possible to identify solutions for the direct kinematics problem, especially between -π to -2π/3 and 2π/3 to π radians. However, in order to attain these positions, the mechanism has to cross singularity boundaries, which might lead to locking of the mechanism in singular modes. Thus, the workspace around the home pose is isolated and they are represented in Figure 5b. Based on observations from Figure 5b, the maximum tilt limits that could be attained or reached by the mechanism corresponds to ±π/6 radians. These limits are represented by dotted orange lines in Figure 5b. Thus, for a maximum input tilt of α = ±π/6 radians, the mechanism is capable of following a circular trajectory around the home pose.

Mapping the T&T Angles to the Euler Angles

The singularity analysis helped in understanding the tilt limits for the 3-SPS-U parallel mechanism using the theory of T&T. However, these results must be mapped with respect to the Euler angles of the universal joint. The 3-SPS-U architecture demonstrated in Figure 3a at the home pose has two degrees of freedom about the universal joint in the center. The conventional approach to identify the vector coordinates of C i exploits the XY Euler angles about the origin A of the mechanism. In the Euler space, the two angles about the x and y axes are considered as η and ϕ. The transformation matrix E about the universal joint is given by:

E = R x (η)R y (ϕ) =     cos(ϕ) 0 sin(ϕ) sin(η) sin(ϕ) cos(η) -sin(η) cos(ϕ)
cos(η) sin(ϕ) sin(η) cos(η) cos(ϕ)

    (14)
where

R x (η) =     1 0 0 0 cos(η) -sin(η) 0 sin(η) cos(η)     , R y (ϕ) =     cos(ϕ) 0 sin(ϕ) 0 1 0 -sin(ϕ) 0 cos(ϕ)    
In Equation ( 14), E ∈ SE(3) represents the spatial transformation matrix obtained from the Euler angles of the universal joint. This matrix is used to identify the end-effector coordinates, which are given by:

c 1 i = E o 1 i , with i = 1, 2, 3 (15)
In Equation ( 15), o 1 i are the coordinates of C i in the mobile reference frame ∑ 1 whose vector equation is given in Equation ( 4). The inverse kinematic model that determines the length of the prismatic springs is computed by calculating the distance between the base and end-effector at the home pose and working conditions. The constraint equations in the Euler space are given by: 2r 2 f cos(ϕ)(h 2 cos(η) -1)h sin(ϕ)(cos(η) + 1)

+ h 2 + 1 = l 2 1 (16a) 2r 2 f ( √ 3 2 sin(η) h cos(ϕ) + h + sin(ϕ) 2 + cos(ϕ) h 2 cos(η) -1 4 + h sin(ϕ) 2 (1 + cos(η)) +h 2 - 3 cos(η) 4 + 1) = l 2 2 (16b) -2r 2 f ( √ 3 2 sin(η) h cos(ϕ) + h + sin(ϕ) 2 -cos(ϕ) h 2 cos(η) -1 4 - h sin(ϕ) 2 (1 + cos(η)) -h 2 + 3 cos(η) 4 -1) = l 2 3 (16c)
In the T&T space, the mechanism is initially tilted to its maximum limit, i.e., α = π/6 radians. Once this limit is reached, α remains a constant, whereas β changes from 0 to 2π radians, thereby achieving a complete revolution in the counter-clockwise direction. In the Euler space, the initial tilt is performed by keeping η as 0 radians and tilting ϕ = π/6 radians. Unlike the T&T space, both the angles are modified as a function of the angle β in the Euler space to achieve a complete revolution. The actuation sequence of the mechanism in both spaces is provided in Table 1. This table demostrates the relation between the T&T space and the Euler space. For each position of β between 0 to 2π, each and every value of the angles η and ϕ is computed as a function of α and β as demonstrated in Table 1 to achieve the mapping in the Euler space. A simulation of this mapping was performed in MATLAB and the video link is provided at the bottom of this page (Video link for the simulation in MATLAB: Available online: https://drive.google.com/file/d/1xXPy4oU_Q0 4Sgcu1kcQn9qt_9WIT2N5E/view?usp=share_link (accessed on: 15 January 2023) 1x Speed, Available online: https://drive.google.com/file/d/1RrIgmipxXAAhKP71hspWaot4OM5 XT4dL/view?usp=share_link (accessed on 15 January 2023) 0.4x Speed). The postures of the mechanism in both the spaces at the home pose and working conditions are represented 280 in Figure 6.

281 Table 1. The actuation angles for the mechanism in the T&T space and the Euler space.

Movement

Experimental Validation

In this section, the mapping theory is validated by performing a circular trajectory on the parallel mechanism. The prototype of the 3-SPS-U parallel mechanism is realized at LS2N, France with a universal joint of length 68 mm. As explained earlier, a value of r f = 56.7 mm was considered for h = 0.6. The base and end-effector of the mechanism are realized by rapid prototyping. Using tension springs and supports, the base and end-effector are connected. The control of this mechanism is carried out using a BeagleBone (BB) black microcomputer. Three Maxon DC motors coupled with quadrature encoders are employed to tilt the mechanism. In order to have a static model and higher torques, a planetary gearhead with a reduction of 1621:1 is coupled to each DC motor. A 36/2 servo-controller is employed for each motor to control the position. The three DC motors are assembled on a platform, which is made by rapid prototyping. As the distance between the gearbox output shaft and the mechanism was higher, an extended shaft with a flange coupling is employed to connect to the gearbox shaft. A 40 mm diameter pulley is assembled on the extended shaft and a cable of 1.5 mm diameter is wound over this pulley. The other end of the cable connects to the end-effector mounting support by passing through each spring. For eliminating the effects of radial play in the pulley shaft, flange couplings are employed. The cables are assembled in such a way that at the home pose, the tension along the cables is high. The entire experimental setup and its digital model are represented below in Figure 7. The experiments are carried out for two orientations: vertical and horizontal under no-load conditions. The vertical orientation is in line with the orientation of the piping-inspection robot inside a vertical pipeline l and this posture is represented in Figure 7a. The horizontal orientation is in-line with the orientation of the robot inside a horizontal pipeline, as shown in Figure 1.

Trajectory Generation

For the experiments, at each instance of the control loop, the tilt angles of the Euler space η and ϕ were passed as inputs to solve the IKP. This actuation is carried out through the cables that pass through the springs of the mechanism. The pose variables and the articular variables for the mechanism are given by q = [η, ϕ] and ρ = [l 1 , l 2 , l 3 ] [START_REF] Venkateswaran | Trajectory Planning for a 3-SPS-U Tensegrity Mechanism[END_REF]. The length of the springs is calculated using Equations (16a) to (16c) in the Euler space. The IKP for each spring is converted into angular displacements of the pulley, which will be the target position for each DC motor. The equation for the desired angular position of the pulley with respect to the IKP is given by:

θ di = (l i -l home ) r , with i = 1, 2, 3 (17)
In Equation ( 17), θ di is the desired angular position calculated from IKP for given input tilt angles η and ϕ. The parameter r indicates the pulley radius, which is 20 mm. At the home position, the value of l home is 68 mm. For every angular displacement of the DC motor, the encoder data are passed to the BB black ports. The angular displacement of the pulley from the encoder channel information can be calculated by:

θ mi = E i π 2CG , with i = 1, 2, 3 (18)
In Equation ( 18), E i indicates the output data of the encoder channels of Motor-i. The angle θ mi is the measured angular displacement at the output shaft of the gearbox. C indicates the counts per revolution of the encoder, which is 500, and G is the reduction ratio of the gearbox, and these values are taken from the technical datasheet of the encoder and the gearbox unit. At a given tilt angle, the mechanism tries to attain the position θ di . For the trajectory planning, the fifth-degree polynomial equation proposed by Khalil et al. [START_REF] Khalil | Modeling Identification and Control of Robots[END_REF] is employed. This equation is used to define the position s(t), velocity ṡ(t) and acceleration s(t) for the mechanism and they are given by:

s(t) = 10 t t f 3 -15 t t f 4 + 6 t t f 5 (19) ṡ(t) = 30 t 2 t 3 f -60 t 3 t 4 f + 30 t 4 t 5 f (20)
s(t) = 60 t t 3 f -180 t 2 t 4 f + 120 t 3 t 5 f (21)
An initial linear tilt is performed for the mechanism before performing the circular trajectory. For the trajectory generation, the simulation time t f is estimated and it is given by:

t f = max(γ V , √ γ A ) (22)
In Equation ( 22), γ V is the ratio of the maximum joint velocity to the speed of the motor and γ A is the ratio of the maximum joint acceleration to the acceleration of the motor [START_REF] Caro | Kinematic analysis and trajectory planning of the Orthoglide 5-axis[END_REF]. In order to demonstrate the circular trajectory, the T&T angles must be mapped to the Euler space in the control law. Using Table 1, the mapping is incorporated in the control law and it is implemented using the equation:

D(t) = η ϕ = α sin(βs(t)) α cos(βs(t)) (23)
Equation ( 23) represents the input tilt angles η and ϕ in the Euler space as a function of the tilt and azimuth angles of the T&T space. α represents the tilt angle and the azimuth value β goes from 0 to 2π radians for a constant tilt to perform a counter-clockwise circular trajectory in line with the simulation. The corresponding angular velocities and accelerations are given by:

V(t) = αβ ṡ(t) cos(βs(t))
-αβ ṡ(t) sin(βs(t)) ( 24)

A(t) = αβ s(t) cos(βs(t)) -αβ 2 ṡ(t) 2 sin(βs(t)) -αβs(t) sin(βs(t)) -αβ 2 ṡ(t) 2 cos(βs(t)) (25)
In order to calculate the Cartesian velocities of the prismatic joints, the Jacobian matrices are employed. The equation is given by:

  l 1 l 2 l 3   = J c η φ , with J c =     ∂l 1 ∂η ∂l 1 ∂ϕ ∂l 2 ∂η ∂l 2 ∂ϕ ∂l 3 ∂η ∂l 3 ∂ϕ     (26)
In Equation ( 26), J c represents the direct kinematics matrix (A). Using Equation ( 26), the joint velocities are estimated, following which the desired angular velocities of the output shaft can be estimated. The joint coordinate vector can be estimated by the equation:

q(t) = f (D(t)) (27)
In Equation ( 27), f = [l 1 , l 2 , l 3 ] T is the vector that contains the solution of the IKP for the input tilt angles. The joint velocities can be computed with the help of the Jacobian matrix J c using the equation:

q(t) = J c V(t) (28)
The accelerations in the joints can be computed using the Jacobian matrix and its time derivative, whose equation is given by: q(t) = J c A(t) + Jc V(t) (29)

PID Control Law

In [START_REF] Venkateswaran | Numerical and experimental validation of the prototype of a bio-inspired piping inspection robot[END_REF] a force control algorithm was implemented for the control of the pipinginspection robot represented in Figure 1. The same algorithm is employed here for the control of the parallel mechanism. By applying motor torques and current, the mechanism tilts to attain a certain position. Once the desired position is reached, the current and torque are cut off for attaining a static phase. For each angular position of the DC motor, the encoder position data are transmitted to the user by the BB black. However, the output data from the encoders are not directly used to calculate the solutions to the IKP as there exists a significant difference between the desired and measured positions. These differences are caused by factors such as motor inertia, frictional effects and inertial forces. In order to compensate for these factors, a closed-loop PID controller is employed, which tries to minimize the errors between the desired and measured positions. The PID control scheme as provided in [START_REF] Caro | Kinematic analysis and trajectory planning of the Orthoglide 5-axis[END_REF][START_REF] Jha | Influence of the trajectory planning on the accuracy of the orthoglide 5-axis[END_REF] for calculating the motor torques is given by: [START_REF] Jha | Influence of the trajectory planning on the accuracy of the orthoglide 5-axis[END_REF] where θ di , θ mi are the desired and measured positions θdi , θmi are the desired and measured velocities θdi is the desired acceleration

Γ = J θdi + K P (θ di -θ mi ) + K D ( θdi -θmi ) + K I t 0 (θ di -θ mi ) , with i = 1, 2, 3
In Equation [START_REF] Jha | Influence of the trajectory planning on the accuracy of the orthoglide 5-axis[END_REF], Γ represents the output torque at the gearbox after PID correction. From the technical datasheet of the gearbox unit, the inertia J is taken as 4.1 × 10 -7 kg.m 2 . The desired positions are computed using the mapping Equation [START_REF] Jha | Workspace, joint space and singularities of a family of delta-like robot[END_REF] and the IKP relations. The desired velocity and acceleration of each joint is computed with the help of Equations ( 28) and [START_REF] Khalil | Modeling Identification and Control of Robots[END_REF]. The measured positions are computed using Equation [START_REF] Yigit | Design and modelling of a cable-driven parallel-series hybrid variable stiffness joint mechanism for robotics[END_REF]. In Equation ( 30), K P is the proportionality term, which examines the error term and responds proportionally. The integral term K I tries to reduce the error at each time instant. The overshoot of value from the desired position is compensated by the difference term K D . Thus, the PID in general takes into account the past step, the present step and the future step such that the error is reduced [START_REF] Maung | DC motor angular position control using PID controller with friction compensation[END_REF]. The values for the PID terms of Equation ( 30) are given by: ω = k t k e RJ = 14 rad/s

K P = 3ω 2 = 588 , K D = 3ω = 42 , K I = ω 3 = 2744 (31)
In Equation [START_REF] Maung | DC motor angular position control using PID controller with friction compensation[END_REF], ω is a parameter that is a function of the torque constant (k t ), the speed constant (k e ), the motor resistance (R) and its inertia (J). It helps to determine the PID parameters K P , K D and K I [START_REF] Lemoine | Commande d'un Moteur éLectrique à Courant Continu Mettant en Mouvement une Charge Inertielle[END_REF] as per the relation given in Equation [START_REF] Maung | DC motor angular position control using PID controller with friction compensation[END_REF]. This approach was tested and validated on PID control of other robots at LS2N [START_REF] Caro | Kinematic analysis and trajectory planning of the Orthoglide 5-axis[END_REF][START_REF] Fasquelle | A bio-inspired 3-DOF light-weight manipulator with tensegrity X-joints[END_REF]. The closed-loop system that is incorporated for the control of the 3-SPS-U mechanism is represented below in Figure 8. In Figure 8, e and ė indicate the error and its derivative, which are computed based on the difference between the desired and measured data. Before the start of experiments, the servo-controllers are calibrated in a computer. The saturation current is used as a reference for controlling the direction of rotation of the DC motors. For safer operations, 90% and 10% duty cycles are set as the limits for the rotation of motors in either direction. The PWM calibration of each servo-controller is carried out as per Table 2. The force control algorithm is executed in the C language using the BB black microcomputer. The link for the algorithm is provided at the bottom of this page (GitHub link for the algorithm: Available online: https://github.com/stvt1991/Control-of-3-SPS-U-Tensegrity-mechanism (accessed on 20 January 2023)).

Results of Experiments

For performing the circular trajectory, it is necessary to define the tilt angle α from the T&T space. An initial tilt along one of the motors (Motor-1) is performed. A circular path in a counter-clockwise direction is then traced by the mechanism. At the end of the circular trajectory, the mechanism is pulled back to its home pose. For the circular trajectory, a tilt of α=π/10 radians within the workspace of the mechanism is chosen. The circular trajectory experiment is performed in the following sequence: Home → Motor-1: linear tilt → Circular path (counter-clockwise) → Home: linear tilt The linear tilt is performed before and after the circular trajectory for a period of 9 s. The circular path is performed for 35 s. Owing to the sleep routines in the control loop, the time taken to initiate the circular trajectory after the initial linear tilt is longer. The total time to perform the experiments is around 124 s. The plot of the joint positions at each instance of the experiments is represented in Figure 9 for both orientations of the mechanism. The initial linear tilt is performed from 0 to π/10 radians. The circular trajectory is created from this tilt position in the counter-clockwise direction. During the experimental cycle, the prismatic springs extended to a maximum length of 84.7 mm from the home position. The intermediate change in prismatic lengths can be observed in Figure 9a-f between 50 and 100 s for both orientations of the mechanism. The springs connected to each motor also reached a minimum length of 50 mm at π/10 radians. The zones between 75.9 and 84.6 mm describe the redundancy of the mechanism where one of the springs has limited effects in tilting the mechanism. The error between the measured and desired data can be calculated using the equation:

Error = (θ mi -θ di )180 π ( 32
)
The plots of the error for both orientations of the mechanism are shown in Figure 10. Due to the frequency issues with the micro-computer, noise could be observed in the final plot. From Figure 10, the error data were found to be between -0.022 • and 0.036 • in the vertical orientation and between -0.025 • and 0.025 • in the horizontal orientation. These values were found to be closer to 0, thanks to the calibration of the PID parameters. Similarly, using the current flow data from the algorithm, the output torques were calculated using the motor torque constant. The plots of the motor torques in both orientations are shown in Figure 11. The torque values were between -0.0059 N.m and 0.0053 N.m in the vertical orientation and between -0.0057 N.m and 0.0058 in the horizontal orientation. It could be observed that these values were similar in both orientations. This is due to the fact that the self-weight of the parallel mechanism had a negligible influence on the motor torques, especially for the horizontal configuration. However, with the presence of external loading, the operating torques might vary significantly in both orientations. From Figure 11 it can also be observed that the operating torques were well within the nominal torque rating of the DC motor. A video link that demonstrates the generation of the circular trajectory in the Euler space of the mechanism is provided at the bottom of this page (Video link for the circular trajectory experiments: Available online: https: //uncloud.univ-nantes.fr/index.php/s/b96nfCRTcBESzGo (accessed on 20 January 2023)).

Discussion and Perspectives

This article presented the application of the theory of Tilt and Torsion (T&T) to a spherical parallel mechanism. The estimation of the vector coordinates as well as the inverse kinematics problem is solved conventionally using the Euler angles of the universal joint. However, the singularity approach using these angles will require the analysis of the workspace by considering the presence of both tilt angles, i.e., η and ϕ [START_REF] Venkateswaran | Singularity and Workspace Analysis of 3-SPSU and 4-SPSU Tensegrity Mechanisms[END_REF]. This computation was time-consuming and it will be even more complicated to study the workspaces of the mechanism when a stacked model is analyzed as there will be four tilt angles. In order to overcome this issue, the T&T space was considered in this study. The advantage of this method is that for a constant tilt angle, the azimuth can vary between 0 and 2π radians or vice versa. The computation was much simpler and consumes less time through this approach when compared to the Euler space. The mapping was successfully implemented from the T&T space to the Euler space. The numerical simulations were first validated for mapping using MATLAB. Following this, experiments were performed on the prototype of the mechanism. The T&T space was mapped to the Euler space by using the mapping equations and the mechanism was made to perform a counter-clockwise circular trajectory in line with the simulations. At each instance of the trajectory, the equivalent X -Y Euler angles were mapped from the T&T space. The experiments also helped in calculating the error data as well as the torque generated on each motor. It was found that the operating torques were well within the nominal rating of the motor units employed. Since this parallel mechanism is proposed to be implemented in the piping-inspection robot that was presented in Figure 1, the circular trajectory can prove vital to align the axis of the robot assembly with the axis of the pipeline, especially when the robot encounters a pipe bend or a junction. This alignment can be performed using control systems that incorporate the mapping equations and thereby perform a circular path according to the profile of the pipe bend to ensure alignment.

The application of this mapping is proposed to be extended to the study of a two-stage (stacked) model of the 3-SPS-U mechanism. The tilt limits generated by the single stage were around ±π/6 radians. These limits were not sufficient to assist the piping-inspection robot to overcome a bend or a junction. With a stacked model, higher tilt limits could be obtained. The singularity analysis of the stacked structure in the Euler space will be computationally expensive due to the presence of four tilt angles. The T&T space will thus be employed and it will lead to the analysis of singularities with two tilt angles and a common azimuth. Moreover, the T&T equations facilitate an easier computational approach to calculate the value of the applied forces on the cables of the mechanism, as demonstrated in [START_REF] Venkateswaran | Design and analysis of a tensegrity mechanism for a bio-inspired robot[END_REF]. This will be very useful for the prototyping and control strategy of the stacked model in the future. Advanced industrial controllers such as the EtherCAT are proposed to be employed for the control strategy of the mechanism, as the existing BB black micro-computer has some frequency issues. This controller will also help provide an easy interfacing with the piping-inspection robot as well as the parallel mechanism and all these modules could be controlled from a central unit.

Figure 1 .Figure 2 .

 12 Figure 1. The 3D model (left) and the 2D view (right) of the rigid bio-inspired piping-inspection robot developed at LS2N, France [10].

Figure 3 .

 3 Figure 3. Representation of the (a) parallel mechanism at home pose, (b) 3D view of the correlation to a 3-SPS-U manipulator and (c) 2D view of the manipulator.

Figure 4 .

 4 Figure 4. Representation of the tilt (α) and azimuth (β) angles on the parallel mechanism under study.

  an open connected subset of the workspace • for all pose values in C k , the direct (resp. inverse) kinematics problem has a constant number of solutions • C k is maximal in the sense if C k is contained in a set E , then E does not satisfy the first or second condition.

Figure 5 .

 5 Figure 5. Representation of (a) aingularities and workspace zones for the mechanism at l min = 40 mm and (b) extraction of feasible workspace around home pose at l min = 40 mm.

Figure 6 .

 6 Figure 6. Postures of the 3-SPS-U mechanism in the T&T space and the Euler space.

Figure 7 .

 7 Figure 7. Representation of the (a) experimental setup of the mechanism in vertical orientation, (b) experimental setup of the mechanism in horizontal orientation , (c) digital model of the experimental setup and (d) one of the three ESCON 36/2 servo-controllers.

Figure 8 .

 8 Figure 8. Closed-loop PID controller employed for the mechanism [8].

Table 2 .

 2 ESCON controller calibration for the force control algorithm.

	PWM Duty Nominal Current (A) Motor Torque (mN.m)	Motor Speed (rpm)
	10%	-0.177	-6.05	3720 (Counter-clockwise)
	50% (idle)	0	0	0
	90%	0.177	6.05	3720 (Clockwise)

Acknowledgments:

We would like to thank Stéphane Jolivet, Philippe Lemoine and Denis Creusot from LS2N for their extended support in the prototyping of the mechanism and the design of the circuit for the control part.

Funding: This work has been funded by the CNRS through the "RobotVer" project within the framework of the project titled PEPS-"Ingénierie Inspirée par la Nature".

Data Availability Statement: Not applicable

Conflicts of Interest:

The authors declare no conflicts of interest.

Abbreviations

The


1 Notation used to describe the epitaxy of Pd(-Au) NPs on r-TiO 2

The epitaxial relationship between a Pd(-Au) NP and the r-TiO 2 is written as follows: Pd(-Au)(hkl)<uvw> r-TiO 2 (h'k'l') [u'v'w'] where (hkl) and (h'k'l') are respectively the Miller indices of the planes of the NP and of the support in contact. [uvw] and [u'v'w'] are two crystallographic directions of the particle and r-TiO 2 respectively that are parallel to each other and are parallel to the NP-support interface.

Dewetting of bimetallic Pd(-Au) NPs on rutile titania

Derivation of an extended Wulff-Kaishev rule

A first extension of the Wulff-Kaishev model that takes into account the effect of epitaxial stress on thermodynamic equilibrium shape (ES) of a crystal deposited on a supoport with lattice mismatch was given by P. Müller and R. Kern 1 . In particular, they showed that the expression of the elastic energy stored in a partially relaxed particle of volume V is expressed as λ Eε 2 V where Eε 2 V is the elastic energy stored in the particle before relaxation. λ is a stress relaxation factor which depends, in complex ways, on the shape of the particle, its orientation with respect to the support and the rigidity of the particle and the support 2 . It varies between 0 and 1. It is 0 for a completely relaxed particle and is 1 in the absence of elastic relaxation. E is a combination of elastic coefficients of the particle and its support and ε is the lattice parameter mismatch between the two systems. According to the theory of nucleation, ∆G, the work necessary to form a polyhedral particle of volume V from its gaseous phase on a planar support with lattice mismatch is the difference between the free energy of the system in its final and initial states plus energy terms related to epitaxial strain and particle surfaces, particle edges and particle-support interface among others 3,4 :

∆G = -n∆µ + j≠i γ j m A j + (γ i -γs)A i + λ Eε 2 V + k ρ k l k + i τ i l i (S1)
where ∆µ is the supersaturation per atom. It is the difference in chemical potential between a free atom in gaseous phase (µ 0 ) and an atom in the particle (µp), i.e., ∆µ = µ 0µp. n is the number of atoms in the particle. γ j m is the surface energy of free facet j of surface area A j , γs is the surface energy of the support and γ i is excess energy associated with the formation of the metal-support interface of surface area A i and of perimeter L i . τ i is the excess energy due the particle edge i of length l i at the particle-support-vacuum triple phase line (or, as in the main text, simply called the triple line) with ∑ i l i = L i . In comparison, ρ k is the excess energy due to free edge k of length l k .

If ν is the atomic volume, V = n.v. By setting k = λ .ν, equation (S1) can be rewritten as follows:

∆G = -V ν ∆µ + ∑ j≠i γ j m A j + (γ i -γs)A i + λ Eε 2 V + ∑ k ρ k l k + ∑ i τ i l i = λ Eε 2 -∆µ ν .V + ∑ j≠i γ j m A j + (γ i -γs)A i + ∑ k ρ k l k + ∑ i τ i l i = kEε 2 -∆µ ν .V + ∑ j≠i γ j m A j + (γ i -γs)A i + ∑ k ρ k l k + ∑ i τ i l i
In the main text, we have shown that Pd(-Au) NPs studied here were inhomogeneously strained with no clear dependence between lattice strain and size for any given composition. In the following, the relaxation factor λ will thus be considered as size-independent. We will also neglect the contribution of the particle free edge energies. Indeed, as also discussed in the main text, the free edges in Pd(-Au) NPs marginally influence their ES. Finally, the excess energy along the triple line will be considered isotropic with an average value of τ. Neglecting the anisotropy of the triple line energy leads to ∑ i τ i dl i = ∑ i τdl i = τL i . Under these assumptions and neglecting the size dependence of surface and interface energies, the elementary work associated with the nucleation of the particle, d∆G(n) is given by:

d∆G = kEε 2 -∆µ ν .dV + j≠i γ j m dA j + (γ i -γs)dA i + k ρ k dl k + τL i (S2)
For a supported particle with polyhedral morphology, its volume V can be divided into congruent pyramids of height hp and polygonal base area Ap such that :

V = 1 3 p hpAp = 1 3 ⎛ ⎝ j≠i h j A j + (h -∆h)A i ⎞ ⎠ (S3)
where (h -∆h) is the distance from the centre of the particle to the facet of the particle in contact with the support (∆h being the particle truncation due to the support). h j is the distance of free facet j (of surface area A j ) from the particle centre. The derivative of V is :

dV = 1 2 ⎛ ⎝ j≠i h j dA j + (h -∆h)dA i ⎞ ⎠ (S4)
Replacing dV in equation ( S2) by its expression in equation (S4), we obtain :

d∆G = kEε 2 -∆µ ν . ⎛ ⎝ 1 2 ( j≠i h j dA j + (h -∆h)dA i ) ⎞ ⎠ + j≠i γ j m dA j + (γ i -γs)dA i + k ρ k dl k + τL i (S5)
When the particle reaches its thermodynamic equilibrium shape, all partial derivatives of ∆G cancel out simultaneously. Thus, at equilibrium, any free facet j verifies the following equation:

∂ ∆G ∂ A j ∆µ,Ai = 0 (S6)
while at the interface with the support, we have :

∂ ∆G ∂ A i ∆µ,A j = 0 (S7)
From equations (S6) and (S7), we get:

γ i -γs + τ dLi dAi h -∆h = ∆µ -kEε 2 3ν (S8)
and

γm h = ∆µ -kEε 2 3ν (S9)
respectively. By combining equations (S8) and (S9), we obtain the extented Wulff-Kaishev rule incorporating the effect of the triple line energy on the thermodynamic ES of the particle:

∆h -h h = γs -γ i γm - τ γm dL i dA i (S10)
dLi dAi is a geometric factor which varies form one particle to another depending on the geometry of the particle-support interface. The latter is defined by the morphology of the particle and its orientation relatively to the support. In this work, this factor is expressed simply in terms of measurable distances on high resolution transmission electron microscopy (HRTEM) images of single NPs acquired parallel to the particle-support interface as shown in the following and in the main text. In the following sections, we will establish dLi dAi for a supported face-centered cubic (FCC) particles with octahedral truncated ES in two distinct orientations with respect to the support.

4 Expression of the geometric factor dL i dA i Fig. 3Sa shows an atomic model of a free truncated octahedral particle with closed-packed FCC structure in [101] zone axis. The particle is limited by ( 100) and ( 111) facets. The (100) facets have a square shape of side length a. Let W denotes the width of the particle in this projection, i.e. its lateral extension along the [10-1] direction (Fig. 3Sb).

Truncated octahedral FCC particle supported on its (010) facet

From the outline of the particle projection along the [101] direction, we show that :

• GJ being the diagonal of a (100) facet, GJ = HI = a √ 2.

• Isolating parallelogram ABCLA, AL = BC = a. By symmetry, MD = a. Thus, JI = Wa and LM = W -2a.

• ABC = ADE = DAB = DAF = θ with θ being the angle between the ( 100) and ( 111) planes. We can thus write cosθ = (111).( 100) 111 100

= 1 √ 3
and deduce that tanθ = √ 2.

When the particle is deposited on a support, the particle will present an extra truncation at the particle-support interface. Fig. 4Sa shows the outline of the projection of a supported truncated octahedral particle with one of its (100) facet in contact with the support. The particle is truncated at its base by a length ∆h. Let's first consider the length of segment PQ. From Fig. 4Sa, the length of segment PQ is expressed as :

• PQ = OP -OQ = OP -KJ = h 100 -a √ 2
. Two types of truncation are possible depending on the difference between the length of segment PQ and particle truncation ∆h. If ∆h > PQ, the interface will have an octagonal shape (Fig. 4Sb). Otherwise, the interface will be a square (Fig. 5Sb). We will now determine the expression of dLi dAi in each case. By geometrical construction on Fig. 4Sa:

∆h > PQ

SV = RX RO .AD = z+∆h z+h100 .AD = (1 + ∆h-h100 z+h100 ).AD Since, • OR = AO.tanθ = W 2 √ 2
and OR = z + h 100 , we deduce :

z + h 100 = W √ 2 (S11)
Consequently, SV can be simplified as follows :

SV = (1 + ∆h-h100 W √ 2 ).W = W + (∆h -h 100 ) √ 2 Moreover, ST = UV = QX QO .AL = ∆h-PQ QX+XO .AL = ∆h-h100+ a √ 2 a √ 2 .a = a + (∆h -h 100 ) √ 2
From these two distances, we can express the length of segments S ′ T ′ and T ′ U ′ which define the octagonal interface (Fig. 4Sb) as follows:

S ′ T ′ = S"T " = U ′ V ′ = U"V " = ST √ 2 = a √ 2 + 2(∆h -h 100 )
and

T ′ U ′ = SV -ST -UV = W + (∆h -h 100 ) √ 2) -2(a + (∆h -h 100 ) √ 2) = W -2a -(∆h -h 100 ) √ 2
Using Fig. 4Sb and by setting T ′ U ′ = a i , the expressions for the perimeter L i and the surface area A i of the octagonal particle-support interface are as follows:

L i = 4(S ′ T ′ + T ′ U ′ ) = 4(a √ 2 + 2(∆h -h 100 ) + a i ) = 4(a √ 2 + 2 W -2a-ai √ 2 + a i = 4(1 - √ 2)a i + 4 √ 2(W -a) A i = SV 2 -S ′ T ′2 = (W + (∆h -h 100 ) √ 2) 2 -(a √ 2 + 2(∆h -h 100 )) 2 = (W + W -2a-ai √ 2 √ 2) 2 -(a √ 2 + 2 W -2a-ai √ 2 ) 2 = (2W -2a -a i ) 2 -((W -a) √ 2 -a i √ 2) 2 = 2(W -a) 2 -a 2 i
Deriving the expressions of L i and A i with respect to a i (and considering the variations of L i and A i with a to be marginal), we get â Ȃć dL i = 4(1 -√ 2)da i â Ȃć dA i = -2a i da i from which we deduced the expression of the geometric factor dLi dAi for ∆h > PQ :

dL i dA i = 2( √ 2 -1) a i = 2( √ 2 -1) W -2a -(∆h -h 100 ) √ 2 (S12)
The geometric factor can be easily calculated with W , a, h 100 and ∆h known. All these dimensions are directly measurable on a HRTEM image of the particle in [101] zone axis. When the shape of the particle-support interface is a square of side length b i (Fig. 5Sb), T ′ U ′ = b i = RX RO .AD = z+∆h z+h100 .W = z+h100+∆h-h100 z+h100 .W = (1 + ∆h-h100 z+h100 ).W

∆h ≤ PQ

Fig

  Fig. 1S Dewetting of Pd 43 Au 57 NPs in epitaxial relationship Pd 43 Au 57 (111)<101> r-TiO 2 (110)[1-1-1] as a function of particle size.

Fig

  Fig. 2S Dewetting of Pd 62 Au 38 NPs in epitaxial relationship Pd 62 Au 38 (100)<101> r-TiO 2 (110)[1-10] as a function of particle size.

Fig

  Fig. 3S (a) Atomic model of a free FCC particle with truncated octahedral morphology along the [101] zone axis. Visible facets and their indices are indicated; (b) Outline of the projection of the particle in [101] zone axis. Square (100) facets of sides length a can be distinguished. O is the centre of the projection and W is the width of the particle in zone axis.

Fig

  Fig. 4S(a) Outline of the projection of a supported truncated octahedral FCC particle in [101] zone axis deposited on a (100) facet (black lines). Construction lines are shown in light blue. Point R is at the intersection of the half-lines AF and DE. P and Q are the intersections between the segment RO and segment JI and FE, respectively. We set z = PR and h 100 = OP. (b) Outline of the particle-support interface for ∆h > PQ. The interface is here of octagonal shape defined by segments S ′ T ′ and T ′ U ′ .

Fig

  Fig. 5S (a) Outline of the projection of a supported truncated octahedral FCC particle in [101] zone axis deposited on a (100) facet when ∆h ≤ PQ (b) Outline of the square particle-support interface defined by segment T ′ U ′ . Particle-support when ∆h < PQ for a truncated octahedral particle with one of its (100) facets in contact with the support. The interface has here a square shape.

Fig

  Fig. 8S (a) (top) Projection of the hexagonal facet (11-1) up and (bottom) projection of the cross section of the NP along the [101]; (b) Schematic of the equilateral triangle drawn from this facet by geometrical construction.

1-10

| 1-10

Moreover, as according to equation (S11), z + h 100 = W √ 2

, b i simplifies to:

Since the interface is a square, L i = 4b i and A i = b 2 i . By calculating the derivatives dL i and dA i with respect to b i , we obtain :

In this configuration, the determination of dLi dAi only requires the knowledge of the W , ∆h and h 100 . In our study of Pd 62 Au 38 NPs in epitaxial relationship Pd 62 Au 38 (100)<101> r-TiO 2 (110)[1-10], all NPs displayed ES satisfying ∆h > PQ. Hence, the expression of dLi dAi given by equation (S12) was used in the main text to describe the morphology of these NPs. Fig. 6Sa shows an atomic model of a free FCC particle with truncated octahedral morphology viewed along along [-121] zone axis. If the particle is supported on one of its its (111) facet, the expression of dLi dAi is, according to 5 , given by :

Truncated octahedral FCC particle supported on its (111) facet

where W is the width of the particle along the [10-1] direction with W = h 111 √ 6, ∆h the truncation of the particle at its base and ϕ the angle between the (-11-3) and (11-1) planes of the FCC lattice leading to tanϕ = 2 2 3 = 1.633 (Fig. 6Sb). In this work, all monometallic Pd and bimetallic Pd 43 Au 57 nanodecahedra in epitaxial relationship Pd(-Au)(111)<101> r-TiO 2 (110)[1-1-1] were imaged in [101] zone axis. The latter is rotated from [-121] direction by 90 around the [11-1] direction. Although the expression of dLi dAi is independent of zone axis along which a NP is viewed, neither W nor ∆h can be straightforwardly determined in [101] zone axis.

To express dLi dAi in terms of measurable distances in [101] zone axis, let's consider Fig. 7S. Fig. 7Sa shows an atomic model of a particle in [101] zone axis. The outline of the projection is shown in Fig. 7Sb.

Since, the expression of dLi dAi in [101] zone axis is the same as in equation (S14), we can write :

which simplifies to :

The geometric factor dLi dAi is here expressed as a function of ∆h and h 111 . Both distances can be determined from a projected image of a supported FCC truncated octahedron in [101] zone axis as explained in the rest of this section. Fig. 8Sa (top) shows the hexagonal shape of the (111) facet in contact with the support. The point O ′ corresponds to the perpendicular projection of the particle center O of the NP on this facet. Due to the symmetry of the particle, O ′ is at the center of the hexagon. However, O ′ does not lie in the middle of segment L ′ N ′ (L ′ and N ′ are in the middle of the segments B ′ B" and D ′ D" respectively) as the hexagon B ′ B"C"D"D ′ C ′ B ′ is not regular. It results that the position of particle centre O is also not known. Thus, the value of the truncation ∆h and h 111 are not directly measurable on the two-dimensional projection ABDFRQPA of the particle along [101] zone axis.

To determine ∆h and h 111 , we proceed as follows. The extensions of segments BD and PA meet at point K (Fig. 8Sa). Similarly, the extensions of segments B ′ C ′ , B"C" and D ′ D" delimiting the (111) facet form a triangle K ′ P ′ Q ′ (Fig. 8Sb). Note that K ′ P ′ Q ′ is an equilateral triangle of center O ′ . Hence, for an equilateral triangle, we have

3 KD. KD can be determined from the projection of the particle in [101] zone axis. Let us now express K ′ O ′ in terms of h 111 . Let's consider OKO ′′ which is a right-angled triangle with KOO ′′ the angle between the (111) and (100) planes. As K ′ O ′ = KO" and OO ′′ = h 111 , we get KO ′′ = OO ′′ .tan θ = h 111 . √ 2. Hence,

Once h 111 is known, ∆h that can be deduced from the measurement of the particle height which is equal to 2h 111 -∆h (Fig. 8Sa).

Notes and references


N Tai Nguyen

Jaysen Nelayah
email: jaysen.nelayah@univ-paris-diderot.fr

Pavel Afanasiev

Laurent Piccolo
email: laurent.piccolo@ircelyon.univ-lyon1.fr

Damien Alloyeau

Christian Ricolleau

Structural properties of catalytically-active bimetallic gold-palladium nanoparticles synthesized on rutile titania nanorods by pulsed laser deposition

Gold-palladium nanoparticles with the average size of 3.5 nm and controlled compositions were grown on rutile titania nanorods using pulsed laser deposition. The structural properties of the assynthesized nanoparticles and their interface with the oxide support were studied at the atomic scale using aberration-corrected transmission electron microscopy. High resolution TEM imaging showed that both polycrystalline Au-Pd particles with droplet morphology and single-crystalline particles with truncated octahedron morphology were stabilized on the nanorods during laser deposition depending on the degree of epitaxy between the nanoparticles and the oxide support. By analyzing the single-crystalline particles formed in the presence of a strong epitaxy, eleven epitaxial relationships between the Au-Pd nanoparticles and their supports were identified with the two dominant ones being: Au-Pd (111) [101] // r-TiO2(110) [1-1-1] and Au-Pd (100) [101] // r-TiO2 (110) [1-10]. Catalytic testing of the bimetallic titania-supported Au-Pd nanoparticles in the partial hydrogenation of 1,3-butadiene showed that they were both active and selective for this reaction. An increase in the catalytic stability with the gold content was evidenced.

By analyzing the single-crystalline particles formed in the presence of a strong epitaxy, eleven epitaxial relationships between the Au-Pd nanoparticles and their supports were identified with the two dominant ones being: Au-Pd (111) [101] // r-TiO2(110) [1-1-1] and Au-Pd (100) [101] // r-TiO2 (110) [1-10]. Catalytic testing of the bimetallic titania-supported Au-Pd nanoparticles in the partial hydrogenation of 1,3-butadiene showed that they were both active and selective for this reaction. An increase in the catalytic stability with the gold content was evidenced.

 INTRODUCTION

The combination of several metals has been long known as an important way of improving the activity, selectivity and/or stability of heterogeneous catalysts. In recent years, the renewed interest for nanoalloys, i.e. well-defined multimetallic nanoparticles, has been driven by great advances in atomic-scale characterization (e.g., aberration-corrected TEM) and structural simulation techniques. In heterogeneous catalysis, this has allowed researchers to better model and understand surface reaction processes, thereby enabling a rational design of more efficient catalysts [START_REF] Piccolo | Surface studies of catalysis by metals : nanosize and alloying effects in Nanoalloys : Synthesis, Structure and Properties[END_REF] . Au-Pd is probably the most investigated catalytic bimetallic system, due to the efficiency of Au-Pd catalysts in various reactions such as synthesis of vinyl acetate monomer [START_REF] Chen | The promotional effect of gold in catalysis by palladium-gold[END_REF] , synthesis of hydrogen peroxide [START_REF] Edwards | Direct synthesis of hydrogen peroxide from H2 and O2 using TiO2supported Au-Pd catalysts[END_REF] , oxidation of primary alcohols to aldehydes [START_REF] Enach | Solvent-free oxidation of primary alcohols to andehydes using Au-Pd/TiO2 catalysts[END_REF] , and selective hydrogenations [START_REF] Hugon | Supported gold catalysts for selective hydrogenation of 1,3-butadiene in the presence of an excess of alkenes[END_REF][START_REF] Kolli | Bimetallic Au-Pd catalysts for selective hydrogenation of butadiene: Influence of the preparation method on catalytic properties[END_REF][START_REF] Yang | High performance Pd-Au bimetallic catalyst with mesoporous silica nanoparticles as support and its catalysis of cinnamaldehyde hydrogenation[END_REF][START_REF] Kittisakmontree | The liquid-phase hydrogenation of 1-heptyne over Pd-Au/TiO2 catalysts prepared by the combination of incipient wetness impregnation and depositionprecipitation[END_REF] , among many others.

The partial hydrogenations of alkynes or alkadienes to form alkenes are applied in the chemical industry for polymer production and use palladium-based catalysts. Palladium is very selective at high concentrations of alkyne or alkadiene impurities. Nevertheless, in case of low concentrations, palladium is not selective enough to completely remove all traces of impurities. Besides, secondary polymerization of alkenes cannot be avoided and palladium is also susceptible to poisoning by various oligomers (or "green oil") during hydrogenation [START_REF] Xu | TEM study on catalysis deactivation during selective acetylene hydrogenation[END_REF] . In contrast to palladium, gold is inert in the bulk state [START_REF] Hutching | Catalysis by gold[END_REF] . However, supported gold nanoparticles recently showed interesting performance for the selective hydrogenation [START_REF] Cardénas-Lizana | The development of gold catalysts for use in hydrogenation reactions[END_REF][START_REF] Mcewan | A review of the use of gold catalysts in selective hydrogenation reactions[END_REF] . Supported gold particles with sizes between 2 and 5 nm, even if they are less active than palladium, are much more selective in the partial hydrogenation of alkynes or alkadienes [START_REF] Enach | Solvent-free oxidation of primary alcohols to andehydes using Au-Pd/TiO2 catalysts[END_REF] . Compared with palladium and other group VIII metals, the low catalytic activity of gold is mostly due to the small number of active sites able to dissociate adsorbed hydrogen molecules [START_REF] Bus | Hydrogen chemisorption on supported platinum, gold and platinum-gold alloy catalysts[END_REF] .

With the increasing use of supported Au-Pd NPs as heterogeneous catalysts, their preparation has been intensively studied in recent years. Different strategies based on both chemical and physical methods have been implemented to synthesize supported Au-Pd bimetallic catalysts, along with their monometallic counterparts. For chemical routes, two main strategies have been introduced to prepare supported Au-Pd NPs: impregnation-like and sol-immobilization techniques [START_REF] Hutchings | Strategies for the Synthesis of Supported Gold Palladium Nanoparticles with Controlled Morphology and Composition[END_REF][START_REF] Villa | New challenges in gold catalysis: bimetallic system[END_REF][START_REF] Konuspayeva | Au-Rh and Au-Pd nanocatalysts supported on rutile titania nanorods: structure and chemical stability[END_REF][START_REF] Hugon | Selective hydrogenation of 1,3butadiene in the presence of an excess of alkenes over supported bimetallic gold-palladium catalysts[END_REF] . With respect to impregnation methods, colloidal chemical reduction followed by sol immobilization enables a better tuning of particle morphology and composition, however particle size remains poorly controlled during alloy formation. Thus, developing new chemical routes to bimetallic catalysts with fine control of particle size, morphology and composition is still a key challenge in the field of catalysis. A physical route using pulsed laser deposition (PLD) in alternate mode has recently been implemented to produce monometallic Au and Pd NPs as well as bimetallic Au-Pd NPs. With this method, NPs with tuned particle size and composition can be deposited on any support. For instance, in previous works, we have succeeded in immobilizing Au-Pd NPs of various compositions on both amorphous carbon and crystalline supports (NaCl, MgO, ZrO2 and TiO2) [START_REF] Nelayah | Long-range chemical orders in Au-Pd nanoparticles revealed by aberration-corrected electron microscopy[END_REF][START_REF] Nguyen | Synthesis and study of structural, thermodynamical and catalytic properties of bimetallic Au-Pd nanocatalysts by using aberration corrected electron microscopy[END_REF] .

Here, on the basis of aberration-corrected electron microscopy analyses, we first report on the structure and the equilibrium shape of nanometer-sized monometallic Au and Pd NPs and bimetallic Au-Pd NPs fabricated by pulsed laser deposition onto TiO2-rutile nanorods exposing (110) facets. In particular, the epitaxial relationships of the Au-Pd NPs with the oxide support are studied as a function of the particle composition. We then illustrate the potential of the as-grown supported Au-Pd NPs as model catalysts for the selective hydrogenation of 1,3-butadiene. In particular, the effect of alloying gold with palladium on the catalytic activity, selectivity and stability is investigated by comparing the catalytic performances of the monometallic Pd and bimetallic Au-Pd NPs.

 EXPERIMENTAL SECTION

Gold-palladium NPs were synthesized by the pulsed laser deposition (PLD) technique in a high vacuum chamber (~10 -7 Torr). Au-Pd particles were deposited on hydrothermally obtained rutile titania (r-TiO2) nanorods [START_REF] Li | On the selective growth of titania polymorphs in acidic aqueous medium[END_REF][START_REF] Li | Hydrothermal synthesis and acidity characterization of TiO2 polymorphs[END_REF] . Before metal deposition, the nanorods were first washed two times with ammonium nitrate to remove adsorbed sulfate and then calcined in air. It results in an impurity level at the surface of the nanorods of less than 0.1 %. The r-TiO2 nanorods were dispersed on different substrates depending on the characterization method. For transmission electron microscopy (TEM), we used standard copper TEM grids (holey carbon on a 400 mesh Cu grid from Agar Scientifics). For catalytic tests, square mica sheets of 11 mm x 11 mm x 0.15 mm size were used to provide the quantity of metallic phase sufficient for detecting its catalytic activity.

The substrate holder in the vacuum chamber is designed to enable simultaneous deposition of metals on various supports having different sizes. Regardless of the substrate nature, during laser deposition the nanorods were heated at 300°C using a proportional-integral-derivative controller (Eurotherm). The deposition process was done by using a KrF laser at 248 nm with a repetition rate of 5 Hz and pulse duration of 25 ns (COMPexPro 102F, COHERENT ® ). Depending on ablation threshold of gold and palladium, the pulse energy at the exit of the laser was adjusted between 250 and 400 mJ. Bimetallic Au-Pd nanoalloys were fabricated by alternately ablating two ultrapure monometallic gold and palladium targets (99.99% purity). The deposition process was divided in sequences to form homogeneous alloy compositions. For each sequence, the total thickness of the deposited metals was fixed to 1 Å. Prior to deposition, the deposition rate of each metal was controlled by an in situ quartz balance (INFICON). This set-up enables a precise control of both the composition and the metal quantity in the deposited alloy. The nominal thickness deposited (which corresponds to the total thickness of metals deposited in a continuous thin film approximation) was fixed at 1 nm to synthesize well-separated NPs of a few nm size. The two target compositions for this study were Au57Pd43 and Au38Pd62. Pure Au and pure Pd NPs were also prepared.

The structural properties of the model catalysts were studied by high-resolution TEM (HRTEM) imaging using a Jeol ARM 200F electron microscope. This microscope combines a last-generation cold-field emission gun [START_REF] Kohno | Development of a cold field-emission gun for a 200kV atomic resolution electron microscope[END_REF] and a CEOS hexapole corrector [START_REF] Haider | Electron microscope image enhanced[END_REF] (CEOS GmbH) to compensate for the spherical aberration of the objective lens. This microscope can achieve a point resolution of 75 pm at 200 kV or 115 pm at 80 kV [START_REF] Ricolleau | Performances of an 80-200 kV microscope employing a cold-FEG and an aberration-corrected objective lens[END_REF] . To avoid structural modifications of supported nanoparticles due to the high-energy electron beam, high-resolution images were acquired at 80 kV. Sample compositions were determined using TEM energy dispersive X-ray spectroscopy (TEM-EDS) by analyzing the intensities of the Au-Mα,β (between 2.00 and 2.24 keV) and Pd-Lα,β (between 2.70 and 2.94 keV) peaks using the Cliff-Lorimer method [START_REF] Cliff | The quantitative analysis of thin specimens[END_REF] . The Cliff-Lorimer K-factor used for quantification was experimentally determined by using a thin wedge-shaped Au48Pd52 film fabricated by PLD and thinned by focused ion beam milling (PIPS TM , GATAN). Prior to thinning, the composition of the film was precisely determined by inductively coupled plasma optical emission spectrometry. Note that the NP composition corresponds to the mean value determined by TEM-EDS measurements made on large NP assemblies. The uncertainty on the atomic composition is estimated to be ± 1 at.%.

Catalytic testing in selective hydrogenation of 1,3-butadiene was carried out in a static stainlesssteel reaction cell (volume ~ 120 cm 3 ) combined with an UHV preparation/characterization setup. [START_REF] Morfin | A versatile elevated-pressure reactor combined with an ultrahigh vacuum surface setup for efficient testing of model and powder catalysts under clean gas-phase conditions[END_REF] Prior to the tests, the reactant gases (research-grade purity) were mixed in a separate UHV chamber. The catalytic tests were performed at room temperature under hydrogen-rich C4H6:H2 conditions with initial pressures of 0.5 Torr C4H6 and 5 Torr H2. 0.5 Torr argon was also present in the gas mixture for internal calibration. The reaction kinetics was continuously tracked by sampling the gases through a leak valve and analyzing them by mass spectrometry. The peak intensities for m/z = 2, 40, 54, 56 and 58 were recorded to determine the partial pressures of H2, Ar, C4H6 (1,3-butadiene), C4H8 (butenes, the three isomers could not be distinguished) and C4H10 (butane), respectively. After normalizing by the Ar signal, the data were corrected for ion fragmentation and spectrometer sensitivity to obtain partial pressures. [START_REF] Valcarcel | Alkene hydrogenation on metal surfaces: Why and when are Pd overlayers more efficient catalysts than bulk Pd?[END_REF] In order to assess the catalyst stability, repeated reaction runs separated by evacuation periods (secondary vacuum) were performed for each sample. Activation/regeneration treatments at 300 °C in O2 and H2 (10 Torr, 30 min) were also carried out in the reactor using laser heating. [START_REF] Morfin | A versatile elevated-pressure reactor combined with an ultrahigh vacuum surface setup for efficient testing of model and powder catalysts under clean gas-phase conditions[END_REF]

 RESULTS AND DISCUSSION

AuxPd1-x nanoparticles with different compositions were fabricated on r-TiO2 nanorods. Figure 1a shows a TEM bright field image of the nanorods dispersed on a holey carbon film. Those nanorods present an oblong shape (100-500nm in length and 20-40 nm in thickness) and are oriented along the [001] direction of the r-TiO2 structure. Moreover, they selectively expose 95-98% of (110) facets and their specific surface area is 60-65 m²/g 20 . The morphology of the NPs after the deposition of Au and Pd atoms by PLD is shown in Figure 1b. the supported particles display a variety of shapes. For all four samples, the average projected area diameter of particles (i.e. the diameter of a perfect circle enclosing an area equivalent to the projection area of a particle) is mostly below 10 nm and it follows a narrow distribution with a mean diameter of 3.5 nm when excluding NPs with diameter above 10 nm (see inserts of Figure 2). These large NPs result from the coalescence of smaller ones during particle growth. Since their Table 1: Epitaxial relationships and corresponding frequencies of occurrence in AuxPd1-x nanoparticles fabricated by pulsed laser deposition on r-TiO2 nanorods.

For Au, the small number of observations is due to the limited presence of truncated octahedron particles and to the difficulty of precise determination of epitaxial relationships. This weak degree of epitaxy of Au/r-TiO2 was already reported in several studies on the growth of NPs on r-TiO2

(110) single crystal [START_REF] Cosandey | Effect of substrate temperature on the epitaxial growth on TiO2 (110)[END_REF][START_REF] Okazawa | Growth mode and electronic structure of Au nano-clusters on NiO (001) and TiO2 (110)[END_REF][START_REF] Cosandey | Growth, morphology, interfacial effects and catalytic properties of Au on TiO2[END_REF][START_REF] Zhang | Initial growth and morphology of thin Au films on TiO2 (110)[END_REF] . However, when the epitaxy exists, the configuration is mostly Au (111) // r-TiO2 (110) with weak alignment between Au [101] and r-TiO2 [001] directions [START_REF] Akita | Analytical high-resolution TEM study of supported gold catalysts: orientation relationship between Au particles and TiO2 supports[END_REF][START_REF] Ichikawa | Electron holographic 3-D nano-analysis of Au-TiO2 catalyst at interface[END_REF] . Note that none of the three relationships found here have been previously reported in the literature.

Concerning Pd, the dominant epitaxial relationship, Pd ( [START_REF] Suzuki | The encapsulation of Pd by the supporting TiO2 (110) surface induced by strong metal-support interactions[END_REF] . This epitaxy was also observed by scanning tunneling microscopy in truncated octahedral Pd NPs stabilized on TiO2 (110) single crystal [START_REF] Ahmadi | Shape-selection of thermodynamically stabilized colloidal Pd and Pt nanoparticles controlled via support effects[END_REF][START_REF] Ahmadi | Size-dependent adhesion energy of shapeselected Pd and Pt nanoparticles[END_REF][START_REF] Yim | Influence of support morphology on the bonding of molecules to nanoparticles[END_REF] .

In order to determine the orientation relationship between two crystals, the calculation of coincidence of reciprocal lattice points (CRLP) is a powerful method. This method is used to determine the lattice matching between two crystals in reciprocal space [START_REF] Stemmer | Film/Substrate orientation relationship in the AlN/6H-SiC epitaxial system[END_REF] . In the case of the Au / r-TiO2 system, Akita et al. have shown that there is no preferential epitaxy between the two materials since several orientation relationships are possible [START_REF] Akita | Analytical high-resolution TEM study of supported gold catalysts: orientation relationship between Au particles and TiO2 supports[END_REF] . In addition, these unexpected results for Au and Pd can be explained by the surface roughness of r-TiO2 nanorods.

In the case of Au-Pd nanoalloys, the two dominant epitaxial relationships, Au-Pd ( 111 In bimetallic solid-solution nanoalloys, it has been reported that the lattice parameter varies linearly with the composition, according to Vegard's law [START_REF] Vegard | Die konstitution der mischkristalle und die raumfüllung der atome[END_REF] . For both previous cases, we can thus determine the NPs composition by measuring their lattice parameter using the FTs of the HRTEM images. The lattice parameters of the NPs shown in Figures 4a and5a are equal to 4.00 Å and 3.97 Å, respectively. Given the lattice parameter of pure Au (4.08 Å) and Pd (3.89 Å) bulk phases, the composition of an alloy containing x % at. Pd is given by:

xPd = 𝒂 𝒆𝒙𝒑 -𝟒.𝟎𝟕𝟖 (𝟑.𝟖𝟗𝟎-𝟒.𝟎𝟕𝟖).𝟏𝟎 -𝟐
where aexp is the alloy lattice parameter deduced from the HRTEM images. This gives xPd = 43.6% at. for the NP in Fig. 4a and xPd = 55.9 % at. for the NP in Fig. 5a. Those values are very close to the composition of the sample determined by TEM-EDS on large NP assemblies (43% at. and 62 % at. Pd, respectively).

For these two orientation relationships, the resulting morphologies exhibit the same surface indexes but the relative extension of the facets is not the same depending on a given orientation relationship. For example, in both morphologies, the total extension of the ( 111) and (100) facets is not the same. This structural difference can have an important influence on the catalytic activity of the NPs because it defines which crystallographic planes are in contact with the reacting gases.

Finally, the dominant epitaxial relationship is not the same for the Au57Pd43 and Au38Pd62 compositions, suggesting the dependence of the Au-Pd/r-TiO2 interface energy as a function of the alloy composition.

The catalytic performance of the supported NPs was evaluated in the gas-phase hydrogenation of butadiene. Three samples, Au57Pd43/r-TiO2, Au38Pd62/r-TiO2 and Pd/r-TiO2, were compared. The reaction tests were carried out at room temperature and consisted of several runs. After each run, the gases in the reactor were evacuated during fifteen minutes before injecting a new gas mixture for the next cycle. Supported Au being poorly active for hydrocarbon hydrogenation at room temperature, [START_REF] Konuspayeva | Au-Rh and Au-Pd nanocatalysts supported on rutile titania nanorods: structure and chemical stability[END_REF] Au/r-TiO2 was not tested in the reaction.

Figure 6 reports the evolution of the partial pressures of butadiene, butene and butane as a function of time over Au57Pd43/r-TiO2. The turnover frequency (TOF) is of 20 ± 2 butene molecules formed per surface metal atom per minute, which is of the order of what was previously measured under the same conditions on powder-supported Pd catalysts 42, [START_REF] Martins | Improving the catalytic performances of metal nanoparticles by combining shape control and encapsulation[END_REF] . For TOF calculation, we have used the total number of metal atoms (Pd + Au) at the surface of the NPs, which were considered as spheres [START_REF] Nguyen | Trends in the CO oxidation and PROX performances of the platinum-group metals supported on ceria (2015)[END_REF] . The spherical approximation is reasonable for FCC metals and truncated octahedra. The metal dispersion used was 1.1/d where d is the average metal particle size in nm. Surface segregation, if any, was not taken into account in this calculation. The surface composition was assumed equal to the average sample composition. Furthermore, butane is not formed during this run, i.e. the selectivity to butene is 100%. runs over Pd/r-TiO2, Au38Pd62/r-TiO2 and Au57Pd43/r-TiO2. Note that for all three catalysts, no butane was detected during the reaction, as already observed in Figure 6 for Au57Pd43/r-TiO2. After the second run, the catalysts were treated under 10 Torr O2 for 30 min then reduced under 10 Torr H2 for 30 min. This type of oxidation-reduction treatment is widely used in heterogeneous catalysis to either activate the catalysts through decomposition of the precursors followed by reduction of the metal(s), or regenerate the catalysts by removing contaminants such as atomic carbon or coke accumulated on the catalyst surface during the reaction. The latter treatment allows one to increase the catalyst lifetime [START_REF] Furimsky | Introduction of regeneration of hydroprocessing catalysts[END_REF] . As shown in Figure 7c for the Au57Pd43/r-TiO2 sample, the catalyst activity (butene formation rate) increases by a factor of ~4 after this treatment. This suggests that the carbon adsorbed at the metal surface during the sample transfer has been removed by the treatment. The decrease in initial activity as Au content increases (Fig. 7d) is due to the lesser availability of active Pd sites, Au being intrinsically poorly active in butadiene hydrogenation. The deactivation of the Pd catalyst from run 1 to run 2 is ascribed to the poisoning of active sites through gradual decomposition of hydrocarbons during the reaction. Gold appears to stabilize the NPs against poisoning, especially for the intermediate composition (Au38Pd62/r-TiO2).

The most striking observation concerns the catalyst-dependent behavior after the calcinationreduction treatment. We tentatively relate this phenomenon to the effect of Au on Pd hydride formation. Palladium is the only noble metal which, in the bulk phase, absorbs hydrogen at room temperature and atmospheric H2 pressure, and forms a stable concentrated hydride ( phase) for high H:Pd ratios [START_REF] Jewett | Diffusion of hydrogen through palladium and palladiumsilver alloys[END_REF] . The hydrogen absorption rate increases and the absorption capacity decreases upon nanosizing [START_REF] Langhammer | Size-dependent kinetics of hydriding and dehydriding of Pd nanoparticles[END_REF] . Moreover, the H absorption properties can be modulated by alloying palladium with another metal, with beneficial or detrimental effects on catalytic performance [START_REF] Kobayashi | Atomic-level Pd-Au alloying and controllable hydrogen-absorption properties in size-controlled nanoparticles synthesized by hydrogen reduction[END_REF][START_REF] Zlotea | Nanoalloying bulk-immiscible iridium and palladium inhibits hydbride formation and promotes catalytic performances[END_REF] . Hydrogen absorption may partially deactivate the Pd-rich Pd/r-TiO2 and Au38Pd62/r-TiO2 catalysts, as observed for single-crystals [START_REF] Valcarcel | Alkene hydrogenation on metal surfaces: Why and when are Pd overlayers more efficient catalysts than bulk Pd?[END_REF][START_REF] Piccolo | Pd-Au single-crystal surfaces : Segregation properties and catalytic activity in the selective hydrogenation of 1,3-butadiene[END_REF][START_REF] Piccolo | Absorption and oxidation of hydrogen at Pd and Pd-Au (1 1 1) surfaces[END_REF] . During the calcination-reduction treatment of Pd/r-TiO2 and Au38Pd62/r-TiO2 catalysts, it is likely that the deactivation induced by hydrogen absorption surpasses the beneficial effect of the catalyst surface regeneration, causing a net decrease in catalytic activity after treatment. In contrast, Au57Pd43/r-TiO2 becomes four times more active after the treatment, which might be ascribed to the vanishing solubility of hydrogen in Au-Pd for Aurich alloy compositions [START_REF] Joice | Nature and reactivity of intermediates in hydrogenation of buta-1,3-diene catalyzed by cobalt and palladium-gold alloys[END_REF][START_REF] Maeland | X-Ray and thermodynamic studies of the absorption of hydrogen by Gold-Palladium alloys[END_REF][START_REF] Luo | Thermodynamics of Hydrogen in fcc Pd-Au Alloys[END_REF] . With hydrogen absorption being negligible in Au57Pd43/r-TiO2, we expect the latter to fully benefit from the calcination-reduction treatment.

Moreover, we cannot exclude that the increase in catalytic activity of the Au57Pd43/r-TiO2 sample after treatment is due to a change in the structure of the NPs during calcination-reduction. Indeed, it has been recently shown that, besides the chemically-disordered FCC structure, the phase diagram of Au-Pd nanoalloys displays long-range chemically-ordered phases [START_REF] Nelayah | Long-range chemical orders in Au-Pd nanoparticles revealed by aberration-corrected electron microscopy[END_REF] . Around nearequiatomic composition, Au-Pd nanoalloys show L10 long-range chemical order below 600 °C. In the present study, this ordered phase could appear in Au57Pd43 under hydrogen treatment and display enhanced activity in butadiene hydrogenation as compared to the as-synthesized FCC NPs.

 CONCLUSION

In These model catalysts appear active and selective for the partial hydrogenation of butadiene. The initial activity decreases as the Au content increases. However, while a calcination-reduction treatment partially deactivates the Pd-rich samples, it strongly activates the Au-rich Au-Pd catalyst. This is tentatively attributed to an improved catalyst resistance to hydriding for high Au concentrations.

A further step will consist in optimizing the synthesis conditions to obtain only equilibrium-shaped Au-Pd particles. This will allow us to link their catalytic properties with their 3D morphology, in particular the proportions of {111} and {100} facets. Further investigations by post-reaction

Figure 1 :

 1 Figure 1 : (a) Low magnification TEM bright field images of rutile titania (r-TiO2) nanorods

Figure 2 :

 2 Figure 2 : HRTEM images of different particles supported on r-TiO2: (a) Au/r-TiO2; (b)

Figure 3 presents

 3 Figure 3 presents HRTEM images of four nanoparticles for each sample. The corresponding

Figure 3 :

 3 Figure 3 : HRTEM images and corresponding Fourier transforms (inserts) of truncated

  [101] // r-TiO2 (110) [1-1-1], has never been experimentally observed to the best of our knowledge. The only documented configuration is Pd (111) [101] // r-TiO2 (110) [1-10]. It is equivalent to the relationship Pd (111) [1-21] // r-TiO2 (110) [001], observed by Suzuki and Souda using RHEED on Pd films epitaxially formed on freshly cleaved TiO2 (110) single crystal

  ) [101] // r-TiO2 (110) [1-1-1] and Au-Pd (100) [101] // r-TiO2 (110) [1-10], are described in detail below.

Figure

  Figure4apresents a HRTEM image of an Au57Pd43 NP on r-TiO2, which shows the configuration

Figure 4 :

 4 Figure 4 : Example of epitaxial relationship Au-Pd (111) [101] // r-TiO2 (110) [1-1-1]: (a)

Figure

  Figure5apresents a HRTEM image of a supported Au-Pd particle corresponding to the

Figure 5 :

 5 Figure 5 : Example of epitaxial relationship Au-Pd (100) [101] // r-TiO2 (110) [1-10]: (a) HRTEM

Figure 6 :

 6 Figure 6 : Time evolution of the partial pressure of butadiene (blue rhombus), butene (red square) and butane (green triangle) over Au57Pd43/r-TiO2 during the 1 st test run.

Figure 7 (

 7 Figure 7 (a) to (c) shows the evolution of butene partial pressure during four consecutive catalytic

Figure 7 : 2

 72 Figure 7 : (a-c) Time evolution of the partial pressure of butenes for Pd/r-TiO2, Au38Pd62/r-TiO2,

  this work, model Au-Pd nanocatalysts supported on rutile titania nanorods were fabricated by pulsed laser deposition in vacuum at 300°C. The nanoalloys, with average diameter of 3.5 nm, present several shapes including droplet and truncated octahedron. The presence of nonequilibrium shapes is a strong indication of a weak degree of epitaxy between the Au-Pd particles and the r-TiO2 support. The HRTEM images show that most of truncated octahedral particles are FCC single crystals oriented along the [101] zone axis and delimited by {111} and {100} facets. Statistical analysis of these particles evidences various epitaxial relationships with the rutile support. Two dominant epitaxial relationships were identified: Au-Pd (111) [101] // Rutile (110) [1-1-1] and Au-Pd (100) [101] // Rutile (110) [1-10], giving rise to two types of morphology.

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript and contributed equally.

Notes

The authors declare no competing financial interest

 ACKNOWLEDGMENTS

The authors acknowledge the support of the French Agence Nationale de la Recherche (ANR) under contract reference ANR-11-BS10-009. We are also grateful to the Region Ile-de-France for convention SESAME E1845 for the support of the JEOL ARM 200F electron microscope installed at the Paris Diderot University.

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