Rohit Kalla

Latifah Nurhami

Sandipan Bandyopadhyay

Stephane Caro

Philippe Wenger

A study of Σ 2 singularities in the 3-RPS Parallel Manipulator

Keywords: Σ 2 singularity, 3-RPS manipulator, gain of degree-of-freedom, Stüdy parameterisation, spatial parallel manipulators

This paper presents some analytical results related to the determination of the singular poses of the 3-RPS parallel manipulator at which it gains two degrees of freedom. The forward kinematic univariate (FKU) of the manipulator acquires a special structure at such a pose. All such poses have been identified in the closed-form, using a Stüdy-parameter representation of SE(3), for both the operation modes of the 3-RPS. These results are novel, to the best of the knowledge of the authors, and these have been verified using the traditional method, using the criterion of loss of rank of certain Jacobian matrices. The theoretical results have been illustrated with numerical examples.

Introduction

In this paper, the problem of gain-type singularities 1 is revisited, with the goal of determining the poses in which the manipulator gains two DoF (termed hereafter as the Σ 2 singularity, following the Thom-Boardman classification of singularities 2 [START_REF] Arnold | Singularities of Differentiable Maps[END_REF]). Such poses have been mentioned in [START_REF] Basu | Singularity analysis of platform-type multi-loop spatial mechanisms[END_REF], albeit without an exhaustive coverage. The authors are not aware of any further work in this regard, except for [START_REF] Srivatsan | Analysis of the degrees-offreedom of spatial parallel manipulators in regular and singular configurations[END_REF], wherein one such pose has been analysed in detail.

The analysis performed in this paper builds upon the ideas presented in [START_REF] Schadlbauer | The 3-RPS parallel manipulator from an algebraic viewpoint[END_REF] and [START_REF] Srivatsan | Analysis of constraint equations and their singularities[END_REF]: it adopts the Stüdy-parameter representation of SE(3), (see, e.g., [START_REF] Selig | Geometric Fundamentals Of Robotics[END_REF] for the fundamentals in this regard) and the resulting identification of the operational modes from the former, while identifying the singular poses using the concept of FKU from the latter. Also, it uses some improvisations in eliminating variables from systems of polynomial equations, where more formal methods, such as the computation of the Gröbner bases (see, e.g., [START_REF] Cox | Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra[END_REF]), or even the resultants appear to be computationally intractable at 0 Rohit Kalla, Sandipan Bandyopadhyay Department of Engineering Design, Indian Institute of Technology Madras, Chennai, Tamil Nadu, E-mail: rohitkalla9@gmail.com, sandipan@iitm.ac.in. Latifah Nurhami, Stephane Caro, Philippe Wenger Institut de Recherches en Communications et Cyberntique de Nantes (IRCCyN), France, E-mail: {Latifah.Nurahmi, Stephane.Caro, Philippe.Wenger} @irccyn.ec-nantes.fr. 1 Such a singularity occurs inside the workspace of the manipulator, at which the manipulator's endeffector gains one or more degree(s) of freedom (DoF) [START_REF] Bandyopadhyay | Analysis of configuration space singularities of closed-loop mechanisms and parallel manipulators[END_REF]. This is conceptually similar to the singularity of the second kind, and are associated with the loss of rank of certain Jacobian matrices [START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF]. 2 According to the Thom-Boardman classification of singularities of a differential map f , a point x belongs to the class Σ i if the kernel of Df (x), (i.e., the differential of f at x) is of dimension i. this point. Using these techniques, the FKU is derived for the operation modes x 0 = 0 and x 3 = 0 (see [START_REF] Schadlbauer | Interdisciplinary Applications of Kinematics[END_REF] for more details on the "modes"). The required form of the FKU for Σ 2 singularities is identified, and used to derive the conditions leading to such a singularity. Finally, after extensive symbolic computations, the condition for Σ 2 singularity is obtained in closed-form, in terms of the architecture, and input variables alone. These results have been verified by reverting back to the traditional Jacobianbased analysis -first order singularities require certain Jacobian matrices to lose rank by one, while for the Σ 2 case, the loss of rank is by two, (see, e.g., [START_REF] Bandyopadhyay | Analysis of configuration space singularities of closed-loop mechanisms and parallel manipulators[END_REF][START_REF] Srivatsan | Analysis of the degrees-offreedom of spatial parallel manipulators in regular and singular configurations[END_REF]). The concepts presented in this paper are fairly generic, and can be applied to similar analysis of any other manipulator.

The rest of the paper is organised as follows: the concept of Σ 1 , Σ 2 singularities are discussed in Section 2. Section 3 describes the geometry of the 3-RPS manipulator, based on which its FKU is derived in Section 4. Section 5 presents the theoretical results on the Σ 2 singularities of the 3-RPS manipulator, which are numerically exemplified in Section 6. Finally, Section 7 presents the conclusions of the paper.

Analysis of the gain-type singularities based on the FKU

Forward kinematic analysis of a spatial parallel manipulator typically involves the following stages: (a) parametrisation of the pose of the moving platform in terms of some chosen variables; (b) construction of the loop-closure constraint equations, relating these unknown variables to the known input variables; and (c) determination of evaluation of the system of equations. The choice of the unknown variables to represent the motion of the moving platform can be the passive joint variables, as in [START_REF] Srivatsan | Analysis of constraint equations and their singularities[END_REF], or taskspace parameters, such as the Stüdy parameters, as in [START_REF] Schadlbauer | The 3-RPS parallel manipulator from an algebraic viewpoint[END_REF]. The constraint equations are then formulated based on the geometric conditions that the motion of the moving platform must satisfy. Finally, following appropriate mathematical procedures, the set of kinematic equations are reduced to a single univariate equation, termed as the FKU.

The FKU contains all the information about the manipulator's kinematics, and can be used to study its singularities, as described below.

Σ 1 singularities

The necessary and sufficient condition for Σ 1 singularity is that the FKU is of the form:

f (x) : (x -α) 2m (x -β) 2n . . . h(x) = 0, m, n ∈ Z + , α, β ∈ R. (1)
Consider, e.g., x = α. When m = 1, the implication is clear: a pair of forward kinematic branches meet at x = α, and hence the necessity and sufficiency of the condition is obvious from the statement of the inverse function theorem, which dictates that f (α) = 0 as well.

Noting that the function f has all the architecture parameters, and input variables in it, the kinematic conditions for Σ 1 singularity can be written as f (x, γ) = 0, f (x, γ) = 0 (with a slight abuse of notation), where γ represents the vector of known variables. Eliminating the sole remaining unknown, x, between these equations, one obtains the condition for Σ 1 singularity purely in terms of the architecture parameters and the input variables [START_REF] Srivatsan | Analysis of constraint equations and their singularities[END_REF].

Σ 2 singularities

The necessary condition for Σ 2 singularity, i.e., singularities corresponding to the gain of two DoF requires that two or more distinct pairs of forward kinematic branches meet in a pairwise manner at, say, x = α. This requires that in Eq. ( 1), at least one of the indices m, n,

• • • > 1.
It is interesting to note, however, that when m > 1, say, m = 2, the point x = α does not necessarily lie on a Σ 2 singularity. In general, the form of Eq. ( 1) is only necessary for Σ m singularity, and not sufficient, when m > 1. The same is true for the other points, x = β etc. This is so since the implicit function theorem does not generalise in the manner one might have expected -i.e., if m > 1, though m pairs of solutions meet at a point, the derivative does not necessarily develop a kernel of dimension m. While there are many examples of manipulators where this generalisation holds good (the 3-RPS being one of them) there are counter examples where that is not the case [START_REF] Husty | On singularities of planar 3-RPR parallel manipulators[END_REF]. This leads to the conclusion that the sufficiency conditions for the Σ 2 singularities cannot be derived based on the analysis of the FKU alone in the case of a general manipulator, and that it requires additional information, either on the structure of the FKU, or the actual kernel dimension of Df (x) computed otherwise. However, the following strategy can be used to identify all the Σ 2 singularities in any manipulator:

• Identify all the potential cases of Σ 2 singularities, using the necessary condition,

• Check all the real solutions to isolate the cases satisfying kernel(Df (x)) = 2.

Application of the above theory to the 3-RPS manipulator is detailed in Section 5.

Kinematic formulation of the 3-RPS manipulator

This section describes the geometry of the 3-RPS manipulator, and also derives the kinematics constraints the manipulator is subjected to.

Geometry of the 3-RPS manipulator

The 3-RPS parallel manipulator consists of a fixed base and a moving platform, both in the shape of equilateral triangles with circumradius 'b' and 'a', respectively. Without any loss of generality, the base radius, b, is considered to be unity. The triangular platforms are connected by three identical RPS legs, (see Fig. 1), wherein the rotational joints are attached to the base and their axes are tangent to the base circumcircle. The spherical joints are attached to the moving platform. The leg lengths can be varied by actuating the prismatic joints, thereby permitting the control of the three DoF motion of the moving-platform. The homogeneous coordi-

0 b 1 = {b, 0, 0, 1} , 0 b 2 = -b 2 , √ 3b 2 , 0, 1 , 0 b 3 = -b 2 , - √ 3b
2 , 0, 1 . The coordinates of the vertices of the top platform are first found with reference to the moving frame {1}, and then transformed to the frame {0} using the homogeneous transformation matrix 0 1 T obtained in terms of the Stüdy parameters (see, e.g., [START_REF] Schadlbauer | The 3-RPS parallel manipulator from an algebraic viewpoint[END_REF]),

x = {x 0 , x 1 , x 2 , x 3 , y 0 , y 1 , y 2 , y 3 }, i.e., 0 p i = 0 b i + 0 1 T 1 p i , i = 1, 2, 3.

Constraint equations

The 3-RPS manipulator has three active (i.e., input/known) variables, l 1 , l 2 , and l 3 , representing the prismatic actuators in the legs, and eight passive (i.e., output/unknown) variables, x i , y i , i = 0, . . . , 3, in the form of the Stüdy parameters. To obtain the eight unknown variables in terms of the known entities, eight independent equations are formulated as detailed below.

Planarity constraints: The vertex p 1 is constrained to move in the X 0 Z 0 plane, as shown in Fig. 1, i.e., the Y 0 coordinate of the vector ( 0 p 1 -0 b 1 ) must be zero:

( 0 p 1 -0 b 1 ) • e Y0 = 0
, where e Y0 = {0, 1, 0, 0} .

Similarly, the vertices p 2 and p 3 are constrained to move in their respective planes:

T Z0 -2π 3 ( 0 p 2 -0 b 2 ) • e Y0 = 0, (3)
T Z0 -4π 3 ( 0 p 3 -0 b 3 ) • e Y0 = 0, (4)
where the homogeneous transformation matrix T Z0 (θ) corresponds to a CCW rotation about Z 0 by θ. Eqs. (2-4), after some manipulations, give rise to three planarity constraints, identical to those presented in [START_REF] Schadlbauer | The 3-RPS parallel manipulator from an algebraic viewpoint[END_REF]:

f A1 : ax 2 1 -ax 2 2 -2x 1 y 0 + 2x 0 y 1 -2x 3 y 2 + 2x 2 y 3 = 0, (5)
f A2 : ax 1 x 2 -2ax 0 x 3 + x 2 y 0 -x 3 y 1 -x 0 y 2 + x 1 y 3 = 0, (6)
f A3 : x 0 x 3 = 0. ( 7
)
Leg length constraints: The respective distances between 0 p i and 0 b i yield three constraint equations, as shown below:

( 0 p i -0 b i ) • ( 0 p i -0 b i ) = l 2 i , i = 1, 2, 3. (8)
The constraints due to the legs 1, 2, and 3 are formulated as f A4 = 0, f A5 = 0, and f A6 = 0, respectively. The expressions are too big to be included here.

St üdy quadric:

In order to represent a rigid motion, the Stüdy parameters must lie on the Stüdy quadric:

f A7 : x 0 y 0 + x 1 y 1 + x 2 y 2 + x 3 y 3 = 0. (9)
Normalisation constraint: This constraint ensures that the first four of the Stüdy paramters form a unit quaternion:

f A8 : x 2 0 + x 2 1 + x 2 2 + x 2 3 -1 = 0. (10)
The set of eight constraint equations constitutes the constraint ideal

I A = f A1 , f A2 , f A3 , f A4 , f A5 , f A6 , f A7 , f A8
. Following [START_REF] Schadlbauer | The 3-RPS parallel manipulator from an algebraic viewpoint[END_REF], primary decomposition of the polynomial ideal composed of the planarity constraints and Stüdy quadric alone, i.e., f A1 , f A2 , f A3 , f A7 , is performed, to identify the operation modes of the 3-RPS, characterised by x 0 = 0 and x 3 = 0, respectively.

Derivation of the FKU

In order to obtain the FKU, the passive variables are eliminated by adapting a sequential procedure for each operation mode, as described in the following.

4.1 Operation mode corresponding to x 0 = 0, x 3 = 0

In this mode, the elimination of the unknowns are done in the following steps: 1. Consider the ideal, I A . Upon substituting x 0 = 0, the ideal I A reduces to the ideal

J A = f A1 , f A2 , f A3 , f A4 , f A5 , f A6 , f A7 .
The polynomial equations f A1 = 0, f A2 = 0 and f A6 = 0 can be solved linearly for y 1 , y 2 and y 3 . Back-substitution of the expressions of y 1 , y 2 , and

y 3 into J A results in a new ideal J B = f B1 , f B2 , f B3 , f B4 wherein, f B1 = f A3 , f B2 = f A4 , f B3 = f A5 and f B4 = f A7 .
2. The polynomials f B1 and f B2 are both quadratic in y 0 . Therefore, the remainder obtained by dividing f B1 by f B2 , while treating them as polynomials in y 0 , can be solved linearly for y 0 as shown in Eq. [START_REF] Husty | On singularities of planar 3-RPR parallel manipulators[END_REF].

y 0 = a(a + 2) 3x 2 1 + 2 √ 3x 2 x 1 -3x 2 2 + l 2 1 -l 2 2 4a 3x 1 - √ 3x 2 , a = 0. (11)
In the following analysis, it is considered that the denominator in Eq. ( 11), i.e., (3x 1 -√ 3x 2 ) is not zero, as it has been verified that vanishing of this factor finally leads only to a Σ 1 singularity4 . The denominator term a, being the circumradius of top platform, cannot be zero for a physically meaningful manipulator. Back-substituting y 0 into the ideal J B , a new ideal

J C = f C1 , f C2 , f C3 is formed, wherein, f C1 = f B2 , f C2 = f B3 , and f C3 = f B4 .
3. The polynomials f C1 and f C2 are of degree 8 in x 3 . The remainder obtained by dividing f C1 by f C2 with respect to x 3 , can be decomposed into 4 factors, given as:

f C1a : 2 √ 3x 2 x 2 1 -9a(a + 2) + l 2 1 + x 2 2 3a(a + 2) -l 2 1 -l 2 1 x 2 3 + l 2 2 3x 1 + √ 3x 2 -l 2 3 3x 1 - √ 3x 2 , ( 12
)
f C 1b : √ 3(x 2 - √ 3x 1 ), f C1c : x 2 3 , f C 1d : 4a 2 . ( 13
)
It is already stated that f C1c and f C 1d cannot be zero, and vanishing of the factor f C 1b leads to Σ 1 singularity. Therefore, only the factor f C1a will be considered for further analysis, using the new ideal

J D = f D1 , f D2 , f D3 , where f D1 = f C1a , f D2 = f C2 , and f D3 = f C3 . 4.
The polynomials f D1 and f D2 are of degrees 2 and 8 in x 3 , respectively. Upon dividing f D2 by f D1 with respect to x 3 , the remainder obtained can be decomposed into 5 factors. These factors are denoted by f D2a , . . . , f D2e . The factor f D2a is of degree 4 in both x 1 and x 2 . The expression of f D2a is large 5 , and hence is not reproduced here. The other factors are much smaller:

f D 2b : 3 x 2 - √ 3x 1 2 , f D2c : 3x 2 1 -x 2 2 2 , f D 2d : x 2 2 , f D2e : 12a 4 . (14)
As before, vanishing of the factors f D 2b , f D2c , and f D 2d lead to Σ 1 singularities, and a cannot be zero. Therefore, only f D2a is considered as the member of the new ideal

J E = f E1 , f E2 , f E3 , where f E1 = f D2a , f E2 = f D1 , f E3 = f D3 . 5.
Both the polynomials f E2 and f E3 are of degree 2 in x 3 . On dividing f E2 by f E3 with respect to x 3 , the remainder f E2a is of degree 2 in x 1 and degree 3 in x 2 . The remainder is reproduced as:

f E2a : -2 √ 3x 2 3a(a + 2) 3x 2 1 -x 2 2 + l 2 1 + l 2 2 3x 1 + √ 3x 2 + l 2 3 √ 3x 2 -3x 1 .
On dividing the polynomials f E1 by f E2a with respect to x 1 , the remainder obtained is a univariate polynomial in x 2 , which decomposes into eight factors, denoted by f E1a , . . . , f E 1h . The factor f E1a is large and of degree 8 in x 2 , while f E 1h is the integer 432. The other factors are:

f E 1b : x 4 2 , f E1c : (l 1 + l 3 ) 2 , f E 1d : (l 1 -l 3 ) 2 , (15)
f E1e : (l 1 + l 2 ) 2 , f E 1f : (l 1 -l 2 ) 2 , f E1g : (a + 2) 4 . (16)
Vanishing of the factors f E 1b , . . . , f E1g lead to Σ 1 singularities. Therefore, f E1a alone is analysed further. The polynomial f E1a is quartic in x 2 2 , is the desired FKU. It is cast in the following form:

F KU : c 0 x 4 + c 1 x 3 + c 2 x 2 + c 3 x + c 4 , where x = x 2 2 . (17)
The coefficients c i have been obtained in closed-form. However, owing to the constraint on space, only c 4 is quoted below:

c 4 = -(r 2 -r 3 ) 4 (9a 4 + 72a 3 -6a 2 (r 2 + r 3 -30) -24a(r 2 + r 3 -6) + (r 2 -r 3 ) 2 -24(r 2 + r 3 )),
where

r i = l 2 i , i = 1, 2, 3.
4.2 Operation mode corresponding to x 3 = 0, x 0 = 0

The derivation of the FKU follows the similar structure in this case, which is omitted here for the want of space. The final result is also similar:

F KU : d 0 x 4 + d 1 x 3 + d 2 x 2 + d 3 x + d 4 , where x = x 2 1 , and (18)
d 4 = -(r 2 -r 3 ) 4 (9a 4 -72a 3 -6a 2 (r 2 + r 3 -30) + 24a(r 2 + r 3 -6) + (r 2 -r 3 ) 2 -24(r 2 + r 3 )).
The other coefficients are too large to be presented here.

Σ 2 singularities of the 3-RPS manipulator

The kinematic conditions for Σ 2 singularities in the 3-RPS manipulator are derived in the following.

Structure of the FKU for Σ 2 singularities

The original FKU, as in Eq. (17) or Eq. ( 18), is converted into its monic form,

x 4 + t 1 x 3 + t 2 x 2 + t 3 x + t 4 = 0, wherein t i = c i /c 0 , or d i /d 0 , i = 1, . . . , 4. (19)
It is assumed above that c 0 = 0, d 0 = 0. In the most general case of Σ 1 singularity, Eq. ( 19) reduces to the form (see Eq. ( 1)):

(x -α) 2 (x -β) 2 = 0, x = x 2 2 , or x = x 2 1 . (20)
Comparing the equal powers of x between Eq. ( 19) and Eq. ( 20) one gets four equations in α, β. After eliminating α, β from these, the condition for Σ 1 singularities reduce to:

(4t 2 -t 2 1 ) 2 -64t 4 = 0, (21)
t 2 3 -t 2 1 t 4 = 0. ( 22
)
The Σ 2 singularities occur only when more than one pair of forward kinematic branches meet at a point. It is obvious from Eq. (20) that the (necessary) condition for the same is simply: α = β. Under this condition, Eq. (20) reduces to the form (x -α) 4 = 0. In the following, the poses satisfying the condition α = β are identified analytically.

iNaCoMM2015-176

Identification of the Σ 2 singularities

For both the operation modes, i.e., x 0 = 0 and x 3 = 0, Eq. ( 21) is of degree 12 in r 1 , degree 8 in both r 2 and r 3 . Eq. ( 22) is of degree 6 in r 1 , and degree 8 in both r 2 and r 3 . In either case, the task is to eliminate one of the variables, r i , to obtain a single equation in the other two, defining the Σ 2 singularity manifold. However, it becomes too complex to perform this last elimination step symbolically. Hence, for the mode x 0 = 0 the value a = 3 is adopted from [START_REF] Schadlbauer | The 3-RPS parallel manipulator from an algebraic viewpoint[END_REF]) and for the mode x 3 = 0 the value a = 1/2 is adopted from [START_REF] Srivatsan | Analysis of the degrees-offreedom of spatial parallel manipulators in regular and singular configurations[END_REF] for the following calculations. In x 0 = 0 mode, elimination of r 1 from Eq. ( 21) and Eq. ( 22) leads to an equation of degree 120 in r 2 and r 3 each, which decomposes into 9 factors, namely, f i , i = 1, . . . , 9. Only the factors f 7 = 0 and f 8 = 0 yields same structure as in Eq. ( 20) and lead to Σ 2 and Σ 1 singularities, respectively 6 . The factor f 7 is shown below:

f 7 : r 4 2 + 2r 3 r 3 2 -306r 3 2 + 3r 2 3 r 2 2 -576r 3 r 2 2 + 33777r 2 2 (23) + 2r 3 3 r 2 -576r 2 3 r 2 + 51678r 3 r 2 -1586304r 2 + r 4 3 -306r 3 3 + 33777r 2 3 -1586304r 3 + 26873856.
A similar set of results is obtained for the case x 3 = 0, x 0 = 0. Numerical examples of both the cases are given in Section 6 representing singular poses with a gain of two DoF.

Numerical results

In this section, numerical examples for both the operation modes are presented to demonstrate the validity of the theoretical results derived above. Case 1: Mode x 0 = 0 For the design parameter a = 3, a combination of real values7 of r 2 and r 3 satisfying Eq. ( 23) are: r 2 = 71.9999 and r 3 = 9, while a corresponding real solution of Eq. ( 22) is r 1 = 71.9999. For these numbers, Eq. (17) becomes:

362.7970x 4 -507.9158x 3 + 266.6558x 2 -62.2196x + 5.4442 = 0 ⇒(x -0.3499) 4 = 0, where x = x 2 2 .

Therefore Σ 2 singularities occur at two distinct points, x 2 = ± √ 0.3499. The corresponding singular poses are shown in Fig. 2. For the pose given by x 2 = -0.5916, (see Fig. 2(a)) the corresponding Stüdy parameters are: x = {0, -0.3415, -0.5916, -0.7302, -1.7078, -2.2135, 1.2780, 8.0540×10 -8 }. At this pose, the singular values of J ηφ are {9071.9999, 0, 0}, and the singular values of J ηX (see Appendix A for the definitions of J ηφ and J ηX ) are: {21735.7201, 1070.1392, 147.8653, 19.8730, 5.3520, 0.3835, 0, 0}. In this case too, it is verified that there are two vanishing singular values of the constraint Jacobian matrices.

Figure 1 :

 1 Figure 1: The 3-RPS parallel manipulator

  (a) Pose A: x 2 = -0.5916 (b) Pose B: x 2 = 0.5916

Figure 2 :

 2 Figure 2: Σ 2 singular poses for x 0 = 0

1 2

 1 nd international and 17 th National Conference on Machines and Mechanisms

	iNaCoMM2015-176

These coordinates are obtained by a bijective transformation from the three dimensional affine space R 3 to the projective space P 3 . In other words, these coordinates are obtained by mapping {x, y, z} to {x, y, z, w}, with the projective coordinate w = 0 set to unity, without any loss of generality.

Conditions leading to only Σ 1 singularities are discarded, as the paper focusses on Σ 2 singularities.

The size of the expression f D 2a is 134.928 KB. Here, all the symbolic computations have been performed using the commercial computer algebra software, Mathematica and "size" refers to the amount of computer memory needed to store an expression in Mathematica's internal format.

The singularities can be classified based on the values of α and β, i.e., if they are real and distinct then the manipulator undergoes Σ 1 singularities, and when they are real and equal, the manipulator undergoes Σ 2 singularities.

The numbers quoted in this section are accurate only up to the fourth digit after the decimal point. However, in actual computations, 100 digits were retained. Also, any number less than 10 -10 (in the absolute sense) is reported as zero.

ConclusionsAn analytical study of the gain-type singularities of the 3-RPS manipulator has been presented, with special focus on the indentification of the set of poses where the manipulator gains two DoF. Such a singularity is termed as the Σ 2 singularities, and these form a subset of the Σ 1 singularties, which result in the gain of a single DoF. The analytical results obtained from the study of the FKU have been verified numerically, using the standard Jacobian-based approach.

Appendix A Definition of the constraint Jacobian matrices

At a gain-type singularity, certain constraint Jacobian matrices lose rank. In this work, two such matrices have been used to validate the results obtained by the proposed new conditions based on the properties of the FKU. One such Jacobian matrix is derived from the constraint equations in the joint space [START_REF] Srivatsan | Analysis of constraint equations and their singularities[END_REF]: η(θ, φ) = 0, where θ represents the active variables, and φ the passive joint variables. The required Jacobian matrix arising out of this equation is derived in [START_REF] Srivatsan | Analysis of constraint equations and their singularities[END_REF] as: J ηφ = ∂η ∂φ . Similarly, writing the constraint equations in the task-space as: f (θ, X) = 0, one can derive the relevant Jacobian matrix in this case as: J ηX = ∂f ∂X . The latter Jacobian is the same as the one used in [START_REF] Schadlbauer | The 3-RPS parallel manipulator from an algebraic viewpoint[END_REF], and the system of equations f (θ, X) = 0 define the same ideal as I A in Section 3.


C Cordier

Pascal Dujardin

Granger

J Wu

J P Dacquin

C Dujardin

P Granger
email: pascal.granger@univ-lille.fr

Optimization of the composition of perovskite type materials for further elaboration of four-way catalysts for gasoline engine

Keywords: PGM-free TWC, Perovskite, CO oxidation, propene oxidation, La-deficient perovskite

Optimization of the composition of perovskite type materials for further elaboration of four-way catalysts

for gasoline engine

Introduction

Nowadays Three-Way-Catalysts (TWC) are considered as a mature technology suitable for the simultaneous abatement of NO x , CO and unburnt hydrocarbons. The use of precious metals and rare earths, as oxygen storage materials, is requested in significant amount to get the highest efficiency near stoichiometric conditions. More stringent standard regulations have recently arisen which account for small suspended particle emissions from gasoline engines keeping the other atmospheric pollutants at their lowest limit. Accordingly, improved technologies are expected combining particulate filter and TWC [1]. In a first approach, a particulate filter could be coated by materials currently used in three-way catalysis in a simple way taking into account a higher thermal stability than usual TWC technology. In practice, perovskite-based catalysts could be an attractive alternative as earlier demonstrated by Nishihata et al. [2] who found that nano-sized precious metal particles would be protected from thermal sintering when interacting with the perovskite. The concept of self-regenerative behavior was further debated in different environmental applications showing significant redispersion processes of precious metals at the surface according to appropriate ex situ redox thermal treatments [3][4][5][6]. Perovskites are also well-recognized for their oxidative properties and widely studied for soot combustion [7,8]. As a matter of fact, the redox properties of those varieties of mixed oxides can be carefully optimized by adjusting the extent of substitution in A-site and B-site of ABO 3 perovskite structures. Theoretical calculations predicted that the structural stability of A-site deficient cubic phase is enhanced compared to stoichiometric counterparts [9]. It was found that La-deficient materials can considerably enhance oxidation reactions and the thermal stability of those materials [10,11]. In addition, computational calculations also led to the conclusion that calcium doping can enhance oxygen conductivity in LaFeO 3 [12].

Since the discovery of Nishihata et al. [2] the debate on the use of perovskite in Three-way catalysis is still open leading to controversial statements especially if PGM-free perovskite can be reasonably envisioned for such application [13,14]. Obviously, the gap is significant because up to now most of the attempts revealed relatively low selective NO conversion to nitrogen in stoichiometric conditions. Nevertheless, the continuous evolution of the normative context, becoming more and more stringent, and taking into account the urgent need to reduce the use of critical materials of strategic importance such as Platinum Group Metals and rare earths, make that perovskite materials are always of practical interest. Parent LaFeO 3 perovskite has been considered in this study with the aim to optimize La-deficiency and subsequent substitutions of La by Ca and Fe by Cu in order to promote Oxygen Storage Capacity (OSC) and fulfill specific requirements in terms of reduction of rare earth content.

Particular attention was paid to the oxidative properties of the synthesized perovskites.

Experimental

Preparation and physicochemical characterization of perovskite based catalysts

Perovskites were prepared according to a conventional citric method earlier described [4]. The structure was obtained after calcination in air at 600°C overnight. The structural properties of freshly-prepared samples were characterized by XRD analysis performed on a Bruker AXS D8 Advance diffractometer in Bragg-Brentano geometry using Cu K radiation ( = 0.154 nm, 40 kV, 30 mA). Ex situ Mössbauer spectroscopic data were obtained at room temperature using a 57 Co (Rh) source. Data fitting was performed with Lorentzian shape peaks and hyperfine parameters (hyperfine field (HF), isomer shift (IS), quadrupole splitting (QS)) were calculated with metallic iron as reference. H 2 -Temperature-Programmed Reduction (H 2 -TPR) experiments were performed on a Micromeritics Autochem II 2920 analyzer. BET surface were calculated from N 2 physisorption measurements at -196°C on a Micrometric Flowsorb III. XPS analysis was carried out on an AXIS Ultra DLD Kratos spectrometer fitted with a monochromatized aluminium source. Binding Energy (B.E.) values were referenced to the C1s B.E. at 285 eV.

Catalytic measurements

Catalytic measurements were carried out in a fixed bed flow reactor on 200 mg of catalysts in powder form (average grain size of 150 µm) mixed with silicon carbide. Catalytic performances were evaluated from successive Temperature-Programmed Reaction (TPR), in the temperature range 100-550°C, according to the methodology described in Fig. 1 (Stoichiometry (STO1)/lean/rich/stoichiometry (STO2)). The total flow rate, adjusted to 12 Lh -1 , corresponded to a Gas Hourly Space Velocity of 60,000 mLh -1 g -1 with inlet composition reported in Table 1. The composition of the outlet gas composition was monitored by a Thermo Scientific Model 42i-HL chemiluminescence analyzer for NO x and an Agilent Technologies 490 micro GC for the detection and the quantification of N 2 , N 2 O, CO, propane, propene and CO 2 . [15]. Such changes cannot be reasonably explained by the substitution of La 3+ by Ca 2+ having a comparable ionic radius, i.e. 1.36 Å vs.

1.34 Å respectively. On the other hand, the stabilization of iron in unusual oxidation state Fe 4+ (0.585 Å) instead of Fe 3+ (0.685 Å), for preserving the electroneutrality of this system, could explain such contraction effects. The Mössbauer spectroscopy is a powerful technique to probe the oxidation state of iron stabilized in different coordination sites. As exemplified in Fig. 3(a), spectra recorded on the parent material at room temperature (RT) essentially reveals sextet environments of LaFeO 3 .

However, significant broadening is noticeable on partially substituted samples which accentuates on the stoichiometric La 0.6 Ca 0.4 Fe 0.8 Cu 0.2 O 3 perovskite (spectrum Fig. 3(f)).

Several contributions were included in spectrum fitting associated to different isomer shifts and Quadrupole Splitting which allow the identification of various coordination sites for Fe 3+ .

The main observation of a sextet on LaFeO 3 and La 0.7 FeO 3 , (figures 3 (a-b), corresponds to Fe 3+ ions in the octahedron's center surrounded by six O 2-ions on (0 0 1), (1 0 0) and (0 1 0) faces (IS = 0.29-0.57 mm/s) on whereas La 3+ ions occupy eight corners of the orthorhombic unit cell (a ≈ b < c and α = β = γ = 90°). However, the formation of Fe 3+ in distorted octahedral and tetrahedral centers (IS = 0.13-0.49 mm/s) as well as the characterization on Fe 4+ (IS = 0.07-0.15 mm/s) is observed especially for stoichiometric composition. These observations seems to agree with Pecchi et al. [15] who demonstrated that increasing the calcium substitution favors the stabilization of Fe 4+ species. On the other hand the lack of detection of Fe 4+ on La-deficient samples could suggest that the neutralization could likely involve the formation of anionic vacancies. Less structural disorder seems to occur on Ladeficient perovskites compared to La 0.6 Ca 0.4 Fe 0.8 Cu 0.2 O 3 which in a certain way could reflect a greater structural stability as reported elsewhere [10,11].

Bulk reducibility

The reducibility of La

Surface properties

Surface composition was investigated by X-ray photoelectron spectroscopy. This technique allows the quantification of surface composition on a depth of 5-10 nm which could reflect the composition of the outermost layer for bulk catalysts. In addition, the oxidation state of the different elements can be discussed on the basis of the determination of Binding Energy values for characteristic core levels. In the particular case of perovskites, the La 3d, Ca 2p, Fe 2p, Cu 2p and O 1s photopeaks have been investigated with B.E. values listed and compared in Table 3. As exemplified, they characterize essentially the stabilization of La 3+ and Ca 2+ .

The Fe 2p photopeak in Fig. 5 provides complementary information relative to the observation of a shake-up satellite structure near 719 eV which reflects the predominance of Fe 3+ . B.E. values for the Cu 2p 3/2 for reduced and oxidized copper species are currently reported near 932.6 and 934.2 eV [16,17]. Accordingly, the Cu 2p core level near 934 eV, accompanied with the shake-up satellite structure in the range 939-950 eV characterize a greater stabilization of Cu 2+ . However, some peculiar spectral features are noticeable on La 0.7 Fe 0.8 Cu 0.2 O 3 associated to a lower B.E. value and a significant attenuation of the shake-up structure which could reflect a reduction of the oxidation state of Cu 2+ with a possible stabilization of Cu + species. The origin of such formation has been already discussed [10,18] corresponding to more reducible Cu 2+ than those stabilized in octahedral geometry of the perovskite. This could correspond to extra framework Cu 2+ species as already characterized then their reduction to Cu + would proceed under irradiation during data acquisition. Let us note that the intensity of the peak satellite is progressively restored on Ca-doped samples which highlights their lower formation. The examination of the Cu LMM spectra complements XPS data to distinguish Cu(II), Cu(I) and Cu(0) species [19][20][21]. As seen in Fig. 6, the kinetic energy at 915.7 eV on La 0.7 Fe 0.8 Cu 0.2 O 3 agrees with the value earlier reported on Cu 2 O at 916.5 eV and diverges from the current values ascribed to CuO and Cu 0 respectively at 917.7 and 918.6 eV. Based on those data one can observed that the shift observed an Cadoped samples seems in relative good agreement with a preferential detection of Cu 2+ . The precise quantification of the relative amount of Cu(II) and Cu(I) species is a difficult task.

However, a significant attenuation of the intensity Auger peak is discernible on Ca-doped samples which seems consistent with a significant lessening of the concentration of more reducible extra framework copper species especially on La 0.6 Ca 0.4 Fe 0.8 Cu 0.2 O 3 . In this latter case, the weak and broad peak would likely characterize less reducible isolated Cu 2+ in octahedral position of the perovskite lattice stabilized at the surface.

The semi-quantitative analysis reveals significant changes on the surface composition. As shown in Table 3, usual surface La enrichment is noticeable on the parent stoichiometric LaFeO 3 sample which is lowered on La-deficient samples even disappearing on La 0.7 FeO 3 and calcium-doped samples. As a general trend, there is no significant deviation on the surface and bulk ratio La/Ca. On the other hand, significant fluctuations arise on the surface copper concentration, the highest surface concentration being observed on La 0.7 Fe 0.8 Cu 0. the superiority of La-deficient materials and, in a certain extent, calcium doping. It is worthwhile to note that the optimal composition for the oxidation of CO and propene differs, La 0.7 Fe 0.8 Cu 0.2 O 3 being the most active for the former reaction whereas La 0.6 Ca 0.2 Fe 0.8 Cu 0.2 O 3 exhibits the best performances for propene. A relative convergence is noticeable by examining their behavior in lean and rich conditions suggesting La 0.6 Ca 0.2 Fe 0.8 Cu 0.2 O 3 as the most suited catalytic system (see Figs. 8 and9). Important information is also provided by the comparison between STO1 and STO2 conversion curves highlighting opposite trends according the reaction considered. Interestingly, a rate enhancement is observed on La 0.6 Ca 0.2 Fe 0.8 Cu 0.2 O 3 in propene oxidation whereas no significant deviation is noticeable for CO conversion. Conversely, slight changes occur mainly at low temperature on La 0.7 Fe 0.8 Cu 0.2 O 3 for propene oxidation whereas a significant deactivation in CO conversion is observable. All those observations suggest that extensive of segregation of copper at the surface of La 0.7 Fe 0.8 Cu 0.2 O 3 could partly explain the lower of performances recorded during STO2 experiments. Ca incorporation seems to have a beneficial effect especially on La 0.6 Ca 0.2 Fe 0.8 Cu 0.2 O 3 which could correspond to a slower migration process and better dispersion of segregated copper species through a strengthening of the interaction with perovskite substrate.

Catalytic vs. physicochemical properties

This investigation shows that La-deficient La 0.7 FeO 3 materials lead to higher catalytic performance than the LaFeO 3 composition by promoting the surface enrichment of iron and large specific surface area. Subsequent substitutions of lanthanum (La 3+ ) by calcium (Ca 2+ ) and iron (Fe 3+ ) by copper (Cu 2+ ) lead to more complex surface chemistry as earlier discussed [10]. As shown in this study, the replacement of trivalent by divalent cations creates significant electronic imbalances. As a consequence, charge compensation can be provided by changes in the valence state of B-sites or oxygen vacancies. The creation of anionic vacancies is generally associated to a greater oxygen adsorption and in some extent a greater

Conclusion

This study is a first step in the frame of the development of new TWC technologies coupled with particulate filters with the aim to develop thermally stable materials. The impact of A and B-substitution on the oxidative properties of parent stoichiometric and La-deficient LaFeO 3 perovskites has been investigated. Particular attention was paid to CO and propene abatement showing that Ca-doping jointly with La-deficiency can lead to better performance monitoring the extent of copper for subsequent development as suitable substrate in Threeway Catalysis. CO and propene oxidation were investigated to probe their oxidative behavior.

It was found that the activity cannot be related neither to the specific surface area nor to the nature of oxidic iron species but more likely to intra and extra lattice of copper species as well as related defective sites.

Recherche, Région Nord -Pas de Calais and FEDER are acknowledged for supporting and funding partially this work.

Fig. 1 .

 1 Fig. 1. Protocol used for Temperature-Programmed Reaction experiments.

Fig. 3 .Fig. 2 .

 32 Fig. 3. Mössbauer spectra recorded at RT on Ca-and Cu-substituted stoichiometric and La-deficient perovskite materials: (a) LaFeO 3 ; (b) La 0.7 FeO 3 ; (c) La 0.7 Fe 0.8 Cu 0.2 O 3 ; (d) La 0.6 Ca 0.1 Fe0 .8 Cu 0.2 O 3 ; (e) La 0.6 Ca 0.2 Fe 0.8 Cu 0.2 O 3 ; (f) La 0.6 Ca 0.4 Fe 0.8 Cu 0.2 O 3 .

Fig. 4 .

 4 Fig. 4. Surface and bulk reducibility of stoichiometric and La-deficient LaFeO 3 substituted by Ca and Fe : (a) LaFeO 3 ; (b) La 0.7 FeO 3 ; (c) La 0.7 Fe 0.8 Cu 0.2 O 3 ; (d) La 0.6 Ca 0.1 Fe 0.8 Cu 0.2 O 3 ; (e) La 0.6 Ca 0.2 Fe 0.8 Cu 0.2 O 3 ; (f) La 0.6 Ca 0.4 Fe 0.8 Cu 0.2 O 3 .

Fig. 5 .

 5 Fig. 5. Fe 2p (A) and Cu 2p (B) XPS photopeak recorded on substituted LaFeO 3 and La 0.7 FeO 3 by calcium and copper. (a) LaFeO 3 ; (b) La 0.7 FeO 3 ; (c) La 0.7 Fe 0.8 Cu 0.2 O 3 ; (d) La 0.6 Ca 0.1 Fe0 .8 Cu 0.2 O 3 ; (e) La 0.6 Ca 0.2 Fe 0.8 Cu 0.2 O 3 ; (f) La 0.6 Ca 0.4 Fe 0.8 Cu 0.2 O 3 .

Fig. 6 .

 6 Fig. 6. Cu LMM Auger spectra on (a) La 0.7 Fe 0.8 Cu 0.2 O 3 ; (b) La 0.6 Ca 0.1 Fe0 .8 Cu 0.2 O 3 ; (c) La 0.6 Ca 0.2 Fe 0.8 Cu 0.2 O 3 ; (d) La 0.6 Ca 0.4 Fe 0.8 Cu 0.2 O 3 .

  Fig.7reports the CO and propene conversion profiles vs. temperature recorded in stoichiometric conditions at the initial (STO1) and final stage (STO2) as described in the experimental section. The following sequence of activity can be derived from STO1 curves for CO oxidation : LaFeO 3 < La 0.7FeO 3 < La 0.6 Ca 0.4 Fe 0.8 Cu 0.2 O 3 < La 0.6 Ca 0.1 Fe 0.8 Cu 0.2 O 3  La 0.6 Ca 0.2 Fe 0.8 Cu 0.2 O 3 < La 0.7 Fe 0.8 Cu 0.2 O 3 whereas for propene oxidation the following ranking can be established : La 0.6 Ca 0.4 Fe 0.8 Cu 0.2 O 3 < LaFeO 3 < La 0.7 FeO 3 < La 0.6 Ca 0.1 Fe 0.8 Cu 0.2 O 3 < La 0.7 Fe 0.8 Cu 0.2 O 3 < La 0.6 Ca 0.2 Fe 0.8 Cu 0.2 O 3. As a general trend, both classifications underlined

Fig. 7 .

 7 Fig. 7. TPR conversion profiles of CO (A) and propene (B) vs. temperature recorded in stoichiometric conditions on perovskites based samples : Full line (STO1), dotted line (STO2) : LaFeO 3 ( grey); La 0.7 FeO 3 ( black); La 0.7 Fe 0.8 Cu 0.2 O 3 ( green); La 0.6 Ca 0.1 Fe 0.8 Cu 0.2 O 3 ( blue); La 0.6 Ca 0.2 Fe 0.8 Cu 0.2 O 3 ( r ed); La 0.6 Ca 0.4 Fe 0.8 Cu 0.2 O 3 ( brown).

Fig. 8 .

 8 Fig. 8. TPR conversion profiles of CO in lean (A) and rich (B) conditions vs. temperature on perovskites based samples : LaFeO 3 ( grey); La 0.7 FeO 3 ( black); La 0.7 Fe 0.8 Cu 0.2 O 3 ( green); La 0.6 Ca 0.1 Fe 0.8 Cu 0.2 O 3 ( blue); La 0.6 Ca 0.2 Fe 0.8 Cu 0.2 O 3 (r ed); La 0.6 Ca 0.4 Fe 0.8 Cu 0.2 O 3 ( brown).

Fig. 9 .

 9 Fig. 9. TPR conversion profiles of propene in lean (A) and rich (B) conditions vs. temperature on perovskites based samples : LaFeO 3 ( grey); La 0.7 FeO 3 ( black); La 0.7 Fe 0.8 Cu 0.2 O 3 ( green); La 0.6 Ca 0.1 Fe 0.8 Cu 0.2 O 3 ( blue); La 0.6 Ca 0.2 Fe 0.8 Cu 0.2 O 3 ( r ed); La 0.6 Ca 0.4 Fe 0.8 Cu 0.2 O 3 ( brown).

  stabilization of supported nanoparticles of those materials. Both parameters can play an important role in determining the extent of reactivity and can reasonably explain the superiority of La 0.6 Ca 0.2 Fe 0.8 Cu 0.2 O 3 . Clearly, the segregation of Fe 2 O 3 is characterized by H 2 -TPR on La 0.7 FeO 3 , La 0.7 Fe 0.8 Cu 0.2 O 3 and La 0.6 Ca 0.1 Fe0 .8 Cu 0.2 O 3 as well as the stabilization of Fe 4+ for La 0.6 Ca 0.4 Fe 0.8 Cu 0.2 O 3 stoichiometric compositions as demonstrated by Mössbauer spectroscopic measurements. This technique also reveals significant structural constraints on Ca-doped samples with the formation of Fe 3+ in distorted octahedral and tetrahedral centers.However, these changes cannot be easily correlated to the sequences of reactivity for CO and propene oxidation.

  Additional information, provided by XRD, XPS and H 2 -TPR experiments, is related to the characterization of extra lattice copper species on La-deficient perovskites more reducible than copper species stabilized in octahedral position in the perovskite lattice. XPS analysis clearly shows the highest Cu/Fe atomic ratio on La 0.7 Fe 0.8 Cu 0.2 O 3 corresponding to a sharp increase in intensity of Auger peak corresponding to Cu(I). As explained those species could come from the in situ reduction of those extra framework Cu(II) species. As exemplified, calcium doping induces a significant lessening of the concentration of extra framework copper species which quasi disappear on the stoichiometric composition La 0.6 Ca 0.4 Fe 0.8 Cu 0.2 O 3. As a matter of fact, positive changes observed on the catalytic properties of La 0.6 Ca 0.2 Fe 0.8 Cu 0.2 O 3 could be reasonably explained by an optimized Ca-doping jointly with La-deficiency. Both, likely induce the formation cubic CuO particles in stronger interaction than on La 0.7 Fe 0.8 Cu 0.2 O 3 , in agreement with previous observations[22]. The extent of dispersion of those oxidic copper species could depend on the flux of diffusion and surface re-dispersion related to the density of anionic vacancies and their stability. Based on this, the opposite changes observed on the STO1 and STO2 curves for CO and propene oxidation on La 0.6 Ca 0.2 Fe 0.8 Cu 0.2 O 3 with respect to La 0.7 Fe 0.8 Cu 0.2 O 3 would emphasize the stabilizing effect of calcium.

Table 1 .

 1 Composition of the gas mixture during TPR experiments

	Gas (vol.%)	NO CO C 3 H 6 C 3 H 8 H 2	O 2 CO 2 H 2 O	He
	Lean	0.1 0.5 0.03 0.015 0.17 0.90 15	10 balance
	Stoichio 0.1 0.7 0.045 0.022 0.23 0.73 15	10 balance
	Rich	0.1 0.9 0.06 0.03 0.3 0.55 15	10 balance

a small segregation of cubic CuO phase with characteristic (-111) and (111) reflections. The structural parameters and the volume of the cell unit have been calculated from the Rietveld refinement method (see Table

2

). A significant lattice contraction is discernible after substitution of lanthanum by calcium which accentuates on the stoichiometric composition La 0.6 Ca 0.4 Fe 0.8 Cu 0.2 O 3 as previously observed

Table 2 .

 2 Structural and textural properties of Ca and Cu substituted stoichiometric and La-

	deficient perovskite materials
	Catalyst	a (Å)	b (Å)	c (Å)	Unit cell vol. (Å 3 )	Cryst. Size (nm)	SSA (m 2 .g -1 )	Fe 4+ (%)
	LaFeO 3	5.560(1) 7.861(0) 5.557(1)	242.880.05	29.4	15.5	4.8
	La 0.7 FeO 3	5.560(0) 7.878(1) 5.535(1)	242.440.03	21.7	35.5	-
	La 0.7 Fe 0.8 Cu 0.2 O 3	5.545(1) 7.857(1) 5.550(1)	241.80 0.05	28.6	23.2	-
	La 0.6 Ca 0.1 Fe 0.8 Cu 0.2 O 3 5.544(1) 7.837(1) 5.539(0)	240.66 0.03	32.5	22.1	-
	La 0.6 Ca 0.2 Fe 0.8 Cu 0.2 O 3 5.542(0) 7.785(0) 5.531(0)	238.63 0.03	30.4	21.3	-
	La 0.6 Ca 0.4 Fe 0.8 Cu 0.2 O 3 5.525(0) 7.810(0) 5.436(1)	234.56 0.06	24.4	32.4	17.3

  1-y Ca y Fe 1-x Cu x O 3 was investigated from H 2 -Temperature-Programmed experiments. This analysis will allow us to gain more insight into the reducibility of cations located in B-sites. The consumption profiles vs. Temperature are reported in Fig.4. They show that LaFeO 3 is unreducible in the temperature range 100-700°C. On the other hand, the reducibility of Fe 3+ is improved on La 0.7 FeO 3 with two broad signals appearing in the temperature ranges 300-400°C and 425-625°C, which can be assigned to the two-step reduction of hematite in strong interaction with the perovskite structure[3]. A prominent reduction process occurs below 275°C after Cu and Ca substitutions tentatively ascribed to the main reduction of Cu 2+ species to metallic species. Let us note that a shoulder clearly appears at 170°C on La 0.7 Fe 0.8 Cu 0.2 O 3 which attenuates on Ca-doped samples. Such asymmetry could reflect the existence of different oxidic copper species, i.e. stabilized inside the perovskite lattice and/or segregated into more reducible oxidic species. This latter assumption seems in rather good agreement with XRD analysis showing distinctly the presence of a small fraction of Cu 2+ stabilized as bulk CuO. Clearly, the fraction of extra framework copper species diminishes at increasing calcium loading especially on stoichiometric composition, i.e. on La 0.6 Ca 0.4 Fe 0.8 Cu 0.2 O 3 . This suggests that the presence of higher density of defective sites on La-deficient perovskite, especially on La 0.7 Fe 0.8 Cu 0.2 O 3 , could favor the bulk diffusion of isolated oxidic copper species in octahedral position then segregating into CuO. Calcium doping would partly prevent this process.

Table 3 .

 3 Surface analysis by XPS on calcined perovskite samples in air at 600°C Fe 0.8 Cu 0.2

	Catalyst		Binding Energy (eV)			Surface composition
		La 3d Ca 2p Fe 2p Cu 2p	O1s	La/Ca	Cu/Fe	A/B
	LaFeO 3	833.8	710.2	529.1		1.34(1.02)
	La 0.7 FeO 3	834.2	710.9	529.8		0.48(0.67)
	La 0.7

Acknowledgement

The authors acknowledge financial support by the EU-Partial-PGMs project (H2020-NMP-686086). Chevreul Institute (FR 2638), Ministère de l'Enseignement Supérieur et de la


Kostiantyn Nechay
email: kostiantyn.nechay@tuni.fi

Alexandru Mereuta

Cyril Paranthoen

Gaëlle Brévalle

Christophe Levallois

Mehdi Alouini

Nicolas Chevalier

Mathieu Perrin

Grigore Suruceanu

Andrei Caliman

* A Mereuta

M Guina

E Kapon

InAs/InP quantum dot VECSEL emitting at 1.5 µm

teaching and research institutions in France or abroad, or from public or private research centers.

demonstration of operation at 1.2 µm-1.3 µm, where InGaAs/GaAs-based QWs suffer from lattice mismatching to GaAs/AlAs distributed Bragg reflectors (DBRs). 9,10 On the other hand, the exploitation of QDs grown on InP substrates, already used in efficient edge emitting lasers, 11,12 has remained elusive for the VECSEL research community. In this letter, we introduce InP-based QD gain mirror and demonstrate a 1.5 µm emitting VECSEL operating in CW mode at room temperature. From material development point of view, the key actions have targeted modal gain increase, enabling output power as high as 2.2 W, and broad wavelength tuning.

The InAs QD structure was grown by gas source molecular beam epitaxy (MBE) on 2-inch InP(311)B substrates. SK InAs QDs were formed at 480° C after the deposition of few monolayers (ML) of InAs onto InP-lattice matched Ga0.2In0.8As0.435P0.565 alloy with a typical bandgap of 1.18 µm (addressed as Q1.18 hereinafter). This alloy is used as the absorbing layer for optical injection in VECSEL operation. When dealing with QDs as the active region of a VECSEL, the most difficult part is to get high enough gain at the operating wavelength. This fact implies a challenge of achieving high QDs density per QD layer, as well as stacking of a large number of QD layers.

Figure 1(a) reveals the QD density measured on a single uncapped QD layer as a function of the nominal thickness of InAs in MLs. By increasing the thickness of InAs from 1 to 2.5 ML, the QD density was increased from 5x10 10 cm -2 (Fig. 1(b)) up to value of 11x10 10 cm -2 (Fig. 1(c)). Such high-density QDs (HD-QD) are placed in close proximity, with typical layer-to-layer separation of 30-40 nm. The integrated photoluminescence variation as a function of the InAs thickness is presented at Fig. 1(d). No significant intensity variation is observed with InAs thickness up to 2.5 MLs. Further increase of InAs nominal thickness above 2.5 ML leads to QDs coalescence and formation of large and plastically relaxed QDs, leading to the decrease of PL efficiency. This fact evidencing formation of non-radiative recombination centers, which can prohibit VECSEL operation. Furthermore, Fig. 1(e) shows the evolution of the QD wavelength emission at room temperature, derived from PL spectra. Simultaneously with the QD density increase, the QD height increases as evidenced by the continuous redshift of the QD from 1530 nm to 1580 nm. In the case of InAs/InP QDs, independent control between QD density and wavelength emission can be achieved by applying a double-cap (DC) growth sequence. This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

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Figure 2(a) reveals the PL spectrum of a single layer of high density (HD)-QD as-grown sample, while Fig. 2(b) presents PL of the sample grown accordingly to the DC procedure. To decrease HD-QD emission wavelength, the DC growth process was applied at the same QD growth temperature. At first 2.8 nm thick cap layer of Q1.18 was deposited on the QDs and followed by a 20 seconds growth interruption under mixture of AsH3+PH3 overpressure flux. 14 Thus, HD-QDs with central wavelength at 1517 nm with full width at half maxima (FWHM) of 115 nm have been fabricated. Stacking of QD layers, separated by spacer layers (SLs), was performed in order to further increase the gain volume. Since QD formation is governed by strain effects, the thickness of the SL between two QD layers is known to play an important role on the upper QD layer. Thicker SL has been shown to decrease the upper QD layer density because of size filtering effects. Thinner SL enables keeping QD density per layer constant due to strain mediated vertical alignment of QDs, however strain accumulation still limits the number of defect-free stacked layers. 15,16 Consequently, when considering QD as active layer in photonic devices, optimal thickness of SL typically lies around 25-40 nm in order to avoid any defects, which also enables to stack up to 10 layers when necessary. When dealing with VECSELs, such thick SL limits the number of QD layers per electric field antinode to approximately three layers. In order to realize QD stacking with a reduced SL thickness, we have used a strain compensation technique, 17 thus after the growth of the first DC HD-QD, slightly tensile stressed 15-nm thick Q1.18 SL has been grown in order to mitigate strain effect for the next QD layer. This cycle has been carried out five times. Figure 2(c) shows the PL spectrum of the QD layers stack sample. The wavelength emission and linewidth of the QD stack is identical to that of single DC HD-QD sample. Furthermore, PL intensity of the stack has been increased threefold evidencing the good optical quality of these QD layers.

Following these QD growth optimizations, we have developed a VECSEL gain structure illustrated in Fig. 3(b). The structure is composed of the InP-based QD active region and a GaAsbased distributed Bragg reflector, which was grown by metalorganic vapor-phase epitaxy (MOVPE). The gain section consists of 20 QDs layers arranged into four groups of five DC HD-QD with 15 nm thick SLs. Each group was placed at the antinode of the micro-cavity electric field formed between DBR and a top-surface of the gain region. Subsequently, the InP-based QD gain structure was wafer-fused to a DBR composed of 29.5 AlAs/GaAs pairs, following the fusion process described by A. Sirbu et al. 18 This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. After the wafer fusion and selective etching of InP substrate the resulting wafer was cleaved into rectangular chips (2.7x2.7 mm 2 ), which were subsequently capillary bonded to an uncoated intra cavity diamond heat spreader, with dimensions of 3x3x0.35 mm 3 . The chip-diamond assembly was mounted onto a thermoelectrically cooled copper plate, which was kept at 15 °C throughout the measurements. The structure was tested in a V-shape resonator, which is shown in Fig. 3(a) and is composed of a highly reflective mirror (R>99.8%) with curvature radius of 200 mm and a flat output coupler with reflectivity of 99%. The distance between gain mirror and curved mirror was 150 mm, while the distance between curved mirror and output coupler was 276 mm; such resonator geometry provided a calculated mode diameter of 190 µm on the gain mirror. The VECSEL was pumped by a 980 nm fiber coupled diode laser, which collimated output was focused down to a pump spot with a diameter of ~220 µm. A birefringent filter with a thickness of 0.5 mm was used for tuning measurements and was placed at a Brewster angle inside the cavity. This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. Figure 4 shows the output power and threshold spectra (lower inset) of two exemplary VECSELs chips with different detuning (offset between micro-cavity resonance wavelength and PL peak). Upper inset shows spectra of one of the chips at different incident pump powers. The maximum output power was measured to be 2.2 W at 24 W of incident pump power, resulting in 11.7% of slope efficiency (nominal incident power was used for the fit).

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Unlike QWs, PL peak of QDs undergoes thermal red shift at a much slower rate, by this resulting in a considerably smaller detuning factor. The measured threshold spectra, recorded at 1.4 W of incident pump power, allow approximate estimation of micro cavity resonances of the chips. Thus, the optimal detuning factor was experimentally determined by testing chips across the wafer, since variation of micro cavity resonance corresponds to active region thickness fluctuation across the wafer. Although the slope curves of the two demonstrated chips are virtually identical at low pump powers (due to broad QD gain), closer to thermal roll-over the chip #1 with a larger detuning becomes less efficient, yielding lower output powers. The optimal detuning value gains significance in the context of the resonant micro cavity, where the round trip gain directly depends on overlap factor between the electric field antinodes and the QDs packages within micro cavity. 19 This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. The wavelength tuning ability is shown in Fig. 5; the tuning was achieved with the intra cavity birefringent filter. Operation in a wavelength window of about 60 nm was obtained from the single chip. The tuning curve was recorded under an incident pump power of 6 W. Inset of Fig. 5 shows exemplary beam profile measured at 22 W of incident pump power, revealing a slightly elliptical beam profile due to V-shape resonator geometry. In conclusion, implementation of DC growth sequence coupled with optimization of SL thickness by means of strain compensation allowed to demonstrate an InP-based QD gain mirror comprising dense QD layers. Such gain structure enabled demonstration of the first QD-based VECSEL emitting at 1.5 µm with output power of 2.2 W. The emission wavelength could be tuned over 60 nm using an intra cavity birefringent filter. In terms of output power the results obtained with this type of gain mirror are on par with earlier demonstrations for quantum well-based This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

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VECSELs at the same wavelength range, 20 yet opening the way for superior performance in terms of broadband tuning, temperature stability and lower threshold. Second-harmonic generation of the VECSEL with emission at 1.5 µm constitutes a viable alternative to 750 nm direct emitting VECSEL, 21,22 by offering much longer lifetime and wider wavelength tunability. In turn, lasers at 750 nm range are needed for isotope separation, 23 spectroscopy, 24 and in medicine due to the large penetration depth into tissues at the wavelengths around 750 nm. 25

13

 13

FIG. 1 .

 1 FIG. 1. QD density (a) and emission wavelength (e) variation as function of InAs thickness (in MLs). Insets (b) and (c) are AFM scans (0.5x0.5 µm²) of the uncovered QDs for InAs thickness of 1 and 2.5 MLs respectively. Dotted line represents the target wavelength at 1515 nm. (d) Integrated photoluminescence intensity of a single QD layer as function of InAs MLs.

  FIG.2. PL spectra of a single layer of as grown HD-QD (a), and DC HD-QD (b) measured at 300K and with the same input power density (500 W/cm²). PL spectrum of 5 DC HD-QD layers with 15 nm spacer layers (c) before fusion of DBR. Eye guide dashed line placed at 1510 nm.

FIG. 3 .

 3 FIG.3. Schematics of V-shaped cavity with inserted birefringent filter for tuning measurements (a) and gain mirror structure (b).

  FIG.4. Output power and threshold spectra of the free-running VECSELs, measured at 15°C. Chips were taken from across the wafer, exhibiting different micro cavity resonances.

FIG. 5 .

 5 FIG. 5. Wavelength tuning recorded as a function of birefringent filter rotation. The figure is composed from nine different spectra. Tuning obtained with the chip #2 under the incident power of 6 W.

This work has been partly supported by Agence Nationale de la Recherche (ANR) and Suisse National Science Foundation (ANR-SNSF IDYLIC project, grant ANR-15-CE24-0034-01).