Abhilash Nayak
email: abhilash.nayak@ls2n.fr

Stéphane Caro
email: stephane.caro@ls2n.frphilippe.wenger@ls2n.fr

Philippe Wenger

Comparison of 3-[PP]S Parallel Manipulators based on their Singularity Free Orientation Workspace, Parasitic Motions and Complexity

Keywords: Parallel manipulator, 3-[PP]S, MICR, parasitic motions, Pareto optimal solutions, complexity

The choice of a parallel manipulator for a particular application is a challenging task due to the existence of numerous designs that generate the same motion type. Hence, this paper attempts to compare a class of 3 degreeof-freedom parallel manipulators belonging to the 3-[PP]S family, for which each leg is confined to move in a plane with the first two joints generating a motion equivalent to two coplanar translations. Specifically, the 3-[PP]S parallel manipulators in which the planes are arranged in a Y-pattern are considered and are compared with respect to their singularity-free orientation workspace, parasitic motions within that workspace and complexity indices. Moreover, the comparison is done between the 3-[PP]S-Y family of manipulators for their different operation modes, with different actuation schemes and for different sets of design parameters. Two indices are used to quantify the singularityfree workspace and the parasitic motions of the manipulators at hand. Additionally, the non-dominated manipulator architectures with regard to their singularity-free regions and parasitic motions are highlighted thanks to a Pareto front. Finally, the Pareto-optimal solutions are ranked in ascending order of complexity.

Introduction

In recent years, Parallel Manipulators (PM) have been used in several industrial applications due to their high stiffness and high payload capacity [START_REF] Merlet | Parallel Robots[END_REF][START_REF] Hunt | Structural kinematics of in-parallel-actuated robot-arms[END_REF]3,[START_REF] Wu | Transmission quality evaluation for a class of four-limb parallel schoenflies-motion generators with articulated platforms[END_REF]. The six degree-of-freedom (dof) Gough-Stewart platform is one example with many applications like flight simulation, rehabilitation and machining. Besides, lower mobility PMs, i.e., PMs with less than six dof, have gained a lot of attention as six actuators are not required for many applications [START_REF] Li | A family of symmetrical lower-mobility parallel mechanisms with spherical and parallel subchains[END_REF]. Amongst them, the 3-dof mechanisms have been the focus of research for many interesting applications. These manipulators can be classified into different families based on their mobile-platform motion type. DELTA robot [START_REF] Clavel | Delta, a fast robot with parallel geometry[END_REF] and 3-UPU mechanism [START_REF] Tsai | Kinematics of a three-dof platform with three extensible limbs[END_REF] are some examples of 3-dof translational PMs. The Agile eye is a well known 3-dof spherical PM [START_REF] Gosselin | The optimum kinematic design of a spherical three-degree-of-freedom parallel manipulator[END_REF]. PMs with coupled motions generate both translational and rotational motions. An interesting family of 3-dof mechanisms with coupled motions is that composed of zero torsion mechanisms [START_REF] Bonev | Advantages of the modified euler angles in the design and control of PKMs[END_REF][START_REF] Bonev | Direct kinematics of zero-torsion parallel mechanisms[END_REF][START_REF] Liu | Orientation capability, error analysis, and dimensional optimization of two articulated tool heads with parallel kinematics[END_REF]. Human eye, sterno-clavicular joint and human wrist can be modelled as 3-dof zero-torsion parallel mechanisms [START_REF] Bonev | Advantages of the modified euler angles in the design and control of PKMs[END_REF]. The 3-RPS PM, introduced by Hunt [START_REF] Hunt | Structural kinematics of in-parallel-actuated robot-arms[END_REF] is one example of zero-torsion mechanism that has been studied extensively. Other examples include the 3-PRS, 3-RRS and 3-PPS PMs. We can find 3-RPS mechanisms as a part of a 6-dof robot [START_REF] Lee | Kinematic analysis of a three-degrees-of-freedom in-parallel actuated manipulator[END_REF], as a micro-manipulator wrist subsystem of the ARTISAN manipulator [START_REF] Waldron | Kinematics of a hybrid series-parallel manipulation system[END_REF] and as a reaction compensation device for space robots [START_REF] Song | A study of reactional force compensation based on three-degree-of freedom parallel platforms[END_REF]. The 3-PRS PM was first proposed for telescopic applications [START_REF] Carretero | Kinematic analysis and optimization of a new three degree-of-freedom spatial parallel manipulator[END_REF]. Another well-known application of the 3-PRS mechanism is a machining head, called the Sprint Z3, developed and patented by DS Technology [START_REF] Wahl | Articulated tool head[END_REF]. The 3-RPS PM was used as a boat simulator with different actuation modes [START_REF] López | Design of a boat simulator using two parallel manipulators[END_REF]. The 3-RRS PM was analyzed in [START_REF] Tetik | Position kinematics of a 3-RRS parallel manipulator[END_REF] to be used as a part of a 6-dof hybrid robot for pick-and-place operations. A flexure based 3-PPS PM was used for a nano imprinting tool module in [START_REF] Yang | Analysis and design of a 3-dof flexure-based zero-torsion parallel manipulator for nano-alignment applications[END_REF].

The terminology Zero torsion mechanisms arises from the fact that the rotation of the mobile platform about an axis normal to the latter is always constrained and they display a symmetric motion with respect to the constrained axis. Accordingly, the local motions of the mobile platform of such mechanisms are two non-pure rotations about horizontal axes and sometimes a pure translational motion along a direction normal to the axes of the rotational motions. A popular category of these mechanisms is the 3-[PP]S type, for which each leg is confined to move in a plane with the first two joints generating a motion equivalent to two coplanar translations. When these planes are arranged such that they intersect at 120 • in a common line, they are henceforth addressed as the 3-[PP]S-Y family. If the arrangement of those planes is based on equilateral pattern, they will be part of the so-called 3-[PP]S-∆ family. If these planes are arranged orthogonally, they are already known as the 3-[PP]S-cube family. For instance, the 3-RPS-cube manipulator analyzed in [START_REF] Nurahmi | Kinematic analysis of the 3-RPS cube parallel manipulator[END_REF] is part of the 3-[PP]S-cube family. Since the number of degree-of-freedom and motion types are the same, the 3-[PP]S PMs can be used for similar applications although their parasitic motions (horizontal displacements due to horizontal axes rotations) differ as it is shown in this paper. This paper aims at finding a common ground to compare the mechanisms belonging to the 3-[PP]S-Y family. Several techniques have already been introduced in the literature to compare PMs. A comparison study of four 3-dof translational PMs was performed by Tsai and Joshi based on their well conditioned workspace, stiffness and inertia properties [START_REF] Tsai | Comparison study of architectures of four 3 degree-of-freedom translational parallel manipulators[END_REF]. Pond and Carretero compared the 3-PRS, 3-RPS and the Tricept PMs by formulating their dimensionally homogeneous square Jacobian matrices and then calculating the condition number and singular values of the Jacobian [START_REF] Pond | Quantitative dexterous workspace comparison of parallel manipulators[END_REF]. Li et al. compared 3-PRS PMs with different limb arrangements with respect to their parasitic motions [START_REF] Li | Parasitic motion comparison of 3-PRS parallel mechanism with different limb arrangements[END_REF] and listed 1T2R PMs without parasitic motions [START_REF] Li | 1T2R parallel mechanisms without parasitic motion[END_REF]. In this paper, the comparison of mechanisms belonging to the 3-[PP]S-Y family is performed with respect to their singularity free orientation workspace and their parasitic motions. An index, named Maximum Inscribed Circle Radius (MICR), introduced in [START_REF] Nayak | Comparison of 3-RPS and 3-SPR parallel manipulators based on their maximum inscribed singularity-free circle[END_REF] is used to quantify the singularity free orientation workspace of the manipulators. The maximum parasitic motion, denoted as µ within the Maximum Inscribed Circle is used as a second performance index. Furthermore, the manipulators are compared based on their design parameters and actuation scheme. The kinematic and singularity analysis problems are tackled locally using screw theory and globally using algebraic geometry tools [START_REF] Schadlbauer | A complete kinematic analysis of the 3-RPS parallel manipulator[END_REF]. Finally, the Pareto optimal solutions obtained with respect to the two objective functions MICR and µ are ranked based on their kinematic complexities using the complexity indices defined in [START_REF] Khan | A Formulation of Complexity-Based Rules for the Preliminary Design Stage of Robotic Architectures[END_REF][START_REF] Caro | The rule-based conceptual design of the architecture of serial schnflies-motion generators[END_REF]. It should be noted that joint limits and internal collisons are not considered in this paper.

The paper is organized as follows. First, the manipulator architectures under study are described. Section 3 presents the constraint equations for the manipulators belonging to the 3-[PP]S-Y family. Sections 4 deals with the operation mode analysis of the foregoing manipulators. Section 5 presents the kinematic singularities of the manipulators at hand obtained with screw theory approaches. Some illustrative examples are given in Section 6. Finally, the manipulators belonging to the 3-[PP]S-Y family are compared based on the two performance indices introduced in this paper and the obtained Pareto-optimal solutions are highlighted and are ranked in ascending order of complexity.

Manipulators under study

Figure 1 represents six manipulators from the 3-[PP]S-Y family. The two coplanar translational motions can be obtained with the following serial kinematic chains: RP, PhR, PvR, RR, PvPh or PhPv with Ph denoting a prismatic joint of horizontal direction and Pv denoting a prismatic joint of vertical direction1 . Though there are other possible orientations of the prismatic joints [START_REF] Nurahmi | Operation modes and singularities of 3-PRS parallel manipulators with different arrangements of P-joints[END_REF], most practical applications include horizontal or vertical prismatic joints. The spherical joint centers are restricted to move along the vertical planes Π 1 , Π 2 and Π 3 shown in Fig. 2. These planes are considered in the fixed coordinate frame such that they pass through points A 1 , A 2 and A 3 , respectively. They have a common line of intersection, L along the z 0 -axis and they subtend an angle of 120 • . This geometric condition is used to derive the constraint equations for these manipulators. Table 1 gives a list of twelve manipulators belonging to the 3-[PP]S-Y family. The underlined letters denote actuated joints.

Both the base and platform are equilateral triangles with vertices A i and B i , i = 1, 2, 3 and with circumradii h 1 and h 2 , respectively. The origin O 0 of the fixed coordinate frame Σ 0 coincides with base circumcenter while origin O 1 of frame Σ 1 coincides with the platform circumcenter. The x 0 -axis of Σ 0 is along line (O 0 A 1 ) and the x 1 -axis of frame Σ 1 is along line (O 1 B 1 ). Thus, the axes y i , i = 1, 2 are parallel to lines (A 2 A 3 ) and (B 2 B 3 ), respectively and the axes z i , i = 1, 2 are normal to the base and the moving-platform, respectively.

A 2 A 1 A 3 B 2 B 3 B 1 h 2 O 0 h 1 O 1 (a) 3-RPS PM A 2 A A 3 B 2 B 3 B 1 h 2 O 1 O 0 h 1 (b) 3-PhRS PM A 2 A 1 A 3 B 2 B 3 B 1 h 2 h 1 O 1 O 0 (c) 3-RRS PM A 2 A A 3 B 2 B 3 B 1 h 2 O 1 O 0 h 1 (d) 3-PhPvS PM A 2 A 1 A 3 B 2 B 3 B 1 h 2 O 1 O 0 h 1 (e) 3-PvPhS PM A 2 A A 3 B 2 B 3 B 1 h 2 O 1 O 0 h 1 (f) 3-PvRS PM
Π 2 Π 1 Π 3 h 1 A 3 A 1 A 2 B 3 B 2 h 2 Σ 0 B 1 x 1 y 1 z 1 O 1 x 0 y 0 z 0 O 0 3 3 3 3 3

Constraint Equations

This section aims to determine the constraint equations of the manipulators under study. Those constraint equations are expressed algebraically based on the motions of the moving-platform that are constrained by the legs.

From Fig. 2, the coordinates of points A i and B i in coordinate frames Σ 0 and Σ 1 , respectively, are expressed as follows:

0 a 1 = [h 1 , 0, 0] T , 0 a 2 = [- 1 2 h 1 , √ 3 2 h 1 , 0] T , 0 a 3 = [- 1 2 h 1 , - √ 3 2 h 1 , 0] T 1 b 1 = [h 2 , 0, 0] T , 1 b 2 = [- 1 2 h 2 , √ 3 2 h 2 , 0] T , 1 b 3 = [- 1 2 h 2 , - √ 3 2 h 2 , 0] T (1)
The rotation matrix 0 R 1 from frame Σ 0 to frame Σ 1 is expressed as:

0 R 1 =               x 0 2 + x 1 2 -x 2 2 -x 3 2 -2 x 0 x 3 + 2 x 1 x 2 2 x 0 x 2 + 2 x 1 x 3 2 x 0 x 3 + 2 x 1 x 2 x 0 2 -x 1 2 + x 2 2 -x 3 2 -2 x 0 x 1 + 2 x 3 x 2 -2 x 0 x 2 + 2 x 1 x 3 2 x 0 x 1 + 2 x 3 x 2 x 0 2 -x 1 2 -x 2 2 + x 3 2               (2)
where x j , j = 0, 1, 2, 3 are the unit orientation quaternions satisfying: 2, for each leg, the vector connecting points A i and B i should always lie in the plane Π i . This geometrical constraint holds true for all manipulators belonging to the 3-[PP]S-Y family. To derive the constraint equations, we can express the normal vector s i to planes Π i , i = 1, 2, 3, in frame Σ 0 as follows:

x 2 0 + x 2 1 + x 2 2 + x 2 3 = 1. From Fig.
0 s 1 = [0, -1, 0] T , 0 s 2 = [ √ 3 2 , 1 2 , 0] T , 0 s 3 = [- √ 3 2 , 1 2 , 0] T (3)
In the fixed coordinate frame Σ 0 , the coordinates of point B i can be expressed as follows:

0 b i = 0 R 1
where 0 p 1 is the point-displacement vector from the origin O 0 of Σ 0 to the origin O 1 of Σ 1 expressed in Σ 0 as [X, Y, Z] T . The vector ---→ A i B i is perpendicular to s i , which is expressed as ( 0 b i -0 a i ) T 0 s i = 0 which after simplification yields the following three equations:

g 1 := x 0 x 3 = 0 (5) g 2 := X -h 2 x 2 1 + h 2 x 2 2 = 0 ( 6
)
g 3 := Y -4h 2 x 0 x 3 + 2h 2 x 1 x 2 = 0 (7)
Depending upon the actuated joints, three other equations, g 4 = 0, g 5 = 0 and g 6 = 0 can be derived. For instance, in case of the 3-RPS PM (Figure 1a), the Euclidean distance between A i and B i must be equal to the prismatic joint length, r i for the i-th leg of the manipulator. As a result, 0 b i -0 a i 2 = r 2 i leads to three additional equations [START_REF] Schadlbauer | A complete kinematic analysis of the 3-RPS parallel manipulator[END_REF] :

g 4 := (4 Xh 2 -4 h 1 h 2 ) x 0 2 -4 Zh 2 x 0 x 2 + 4 Yh 2 x 0 x 3 + (4 Xh 2 -4 h 1 h 2 ) x 1 2 + 4 Yh 2 x 1 x 2 + 4 Zh 2 x 1 x 3 (8) + X 2 -2 Xh 1 -2 Xh 2 + Y 2 + Z 2 + h 1 2 + 2 h 1 h 2 + h 2 2 -r 1 2 = 0 g 5 := -2 √ 3Yh 2 -2 Xh 2 -4 h 1 h 2 x 0 2 -2 √ 3Zh 2 x 0 x 1 + 2 Zh 2 x 0 x 2 + 2 √ 3Xh 2 -2 Yh 2 x 0 x 3 (9) + (-2 Xh 2 -h 1 h 2 ) x 1 2 + -2 √ 3Xh 2 -2 √ 3h 1 h 2 -2 Yh 2 x 1 x 2 -2 Zh 2 x 1 x 3 + -2 √ 3Yh 2 -3 h 1 h 2 x 2 2 -2 √ 3Zh 2 x 2 x 3 + √ 3Yh 1 + √ 3Yh 2 + X 2 + Xh 1 + Xh 2 + Y 2 + Z 2 + h 1 2 + 2 h 1 h 2 + h 2 2 -r 2 2 = 0 g 6 := 2 √ 3Yh 2 -2 Xh 2 -4 h 1 h 2 x 0 2 + 2 √ 3Zh 2 x 0 x 1 + 2 Zh 2 x 0 x 2 + -2 √ 3Xh 2 -2 Yh 2 x 0 x 3 (10)
+ (-2 Xh 2 -h 1 h 2 ) x 1 2 + 2 √ 3Xh 2 + 2 √ 3h 1 h 2 -2 Yh 2 x 1 x 2 -2 Zh 2 x 1 x 3 + 2 √ 3Yh 2 -3 h 1 h 2 x 2 2 + 2 √ 3Zh 2 x 2 x 3 - √ 3Yh 1 - √ 3Yh 2 + X 2 + Xh 1 + Xh 2 + Y 2 + Z 2 + h 1 2 + 2 h 1 h 2 + h 2 2 -r 3 2 = 0 (11)
In addition, the normalization equation of the unit quaternion is to be considered, namely,

g 7 := x 0 2 + x 1 2 + x 2 2 + x 3 2 -1 = 0 ( 12
)
It is noteworthy that Eq. ( 5) reveals that the 3-[PP]S-Y family of PMs can have at least two operation modes characterized by x 0 = 0 and x 3 = 0. The following section examines the primary decomposition of the ideal of constraint equations to identify different operation modes of the mechanisms under study.

Operation Modes

The constraint equations ( 5) to [START_REF] Tsai | Kinematics of a three-dof platform with three extensible limbs[END_REF] describe the configuration space of the manipulators. The first constraint equation is recalled:

x 0 x 3 = 0 ( 13
)
As a result, their configuration space can be split into two parts x 0 = 0 and x 3 = 0, called the operation modes [START_REF] Nurahmi | Operation modes and singularities of 3-PRS parallel manipulators with different arrangements of P-joints[END_REF][START_REF] Schadlbauer | Operation modes in lower-mobility parallel manipulators[END_REF][START_REF] Maraje | Operation modes comparison of a reconfigurable 3-PRS parallel manipulator based on kinematic performance[END_REF] separated by a constraint or C-space singularity [START_REF] Zlatanov | Constraint singularities as c-space singularities[END_REF]. To obtain other equations in each operation mode, a polynomial ideal is defined consisting of equations g 1 , g 2 , g 3 and g 7 with variables {x 0 ,

x 1 , x 2 , x 3 , X, Y, Z} over the coefficient ring C[h 1 , h 2 ] as follows: J = g 1 , g 2 , g 3 , g 7 (14)
The primary decomposition of ideal J results in two ideals J k (i = 1, 2) confirming two operation modes for all the PMs in the 3-[PP]S-Y family, no matter their actuation scheme. The analysis is completed by adding the remaining constraint equations g 4 = g 5 = g 6 = 0 to the primary ideals J 1 and J 2 , which returns two ideals K 1 and K 2 . As a consequence, the ideals K i correspond to the two operation modes and can be studied separately.

K k = J k ∪ g 4 , g 5 , g 6 k = 1, 2 (15)
Ideal K 1 -Operation mode 1, OM1 (x 0 = 0) : The moving platform is always found to be displaced about a finite screw axis by 180 degrees from the identity position where Σ 0 coincides with Σ 1 [START_REF] Schadlbauer | Operation modes in lower-mobility parallel manipulators[END_REF]. Substituting x 0 = 0 and solving for X, Y from the ideal K 1 shows that the translational motions can be parametrized by Z and the rotational motions by x 1 , x 2 and x 3 along with x 2 1 + x 2 2 + x 2 3 = 1 [START_REF] Schadlbauer | Operation modes in lower-mobility parallel manipulators[END_REF].

Ideal K 2 -Operation mode 2, OM2 (x 3 = 0) : The moving platform is displaced about a finite screw axis with a rotation angle α defined as: α = ±2 acos(x 0 ). The screw axis in this case stays parallel to the xy-plane [START_REF] Schadlbauer | Operation modes in lower-mobility parallel manipulators[END_REF]. Substituting x 3 = 0 and solving for X, Y, Z from the ideal K 2 shows that the translational motions can be parametrized by Z and the rotational motions by x 0 , x 1 and x 2 along with x 2 0 + x 2 1 + x 2 2 = 1 [START_REF] Schadlbauer | Operation modes in lower-mobility parallel manipulators[END_REF]. Therefore, PMs belonging to the 3-[PP]S-Y family exhibit two operation modes. In what remains, those manipulators are compared based on their singularity-free orientation workspace size and parasitic motions.

Singularity Analysis and Parasitic Motions

For each operation mode, the motion of the moving platform can be parametrized by Z and any two of the orientation parameters x i [START_REF] Nurahmi | Operation modes and singularities of 3-PRS parallel manipulators with different arrangements of P-joints[END_REF]. It simplifies the singularity analysis and the singular surfaces can be visualized in the three dimensional Euclidean space, E 3 . However, the rotations of the moving-platform are not pure. Indeed some undesired translational motions called as parasitic motions, arise. This section is dedicated to the evaluation of the singularity-free orientation workspace and the parasitic motions of PMs belonging to the 3-[PP]S-Y family. The forward kinematic Jacobian matrix is derived in two ways: (i) differentiating the constraint equations with respect to the output variables (ii) using screw theory. It is shown how screw theory methods simplify the determination of the Jacobian matrix for different actuation schemes. An index is defined to assess the maximum singularity-free orientation workspace of the mechanisms under study. Additionally, their parasitic motions are also assessed.

Jacobian Matrix obtained by differentiating the constraint equations

The Jacobian matrix is defined for each operation mode as a matrix composed of the first order partial derivatives of the constraint equations with respect to the parameters x 0 , x 1 , x 2 , x 3 , X, Y and Z [START_REF] Husty | Algebraic methods in mechanism analysis and synthesis[END_REF][START_REF] Husty | Algebraic geometry and kinematics[END_REF], i.e.,

J k = ∂g m ∂x i , ∂g m ∂X , ∂g m ∂Y , ∂g m ∂Z
where k = 1, 2 ; m = 1, ..., 7 ; i = 0, ..., 3

where, g m , m = 1, ..., 7 are the constraint equations for the manipulator at hand. Thus, the manipulator reaches a kinematic singularity when det(J k ) = 0. The vanishing condition of det(J k ) results in a hyper-variety for each operation mode:

S 1 : x 3 • f 1 (x 1 , x 2 , x 3 , X, Y, Z) = 0 and S 2 : x 0 • f 2 (x 0 , x 1 , x 2 , X, Y, Z) = 0 (17)
The polynomials f 1 and f 2 , whose degree depends on the PM at hand describe the singularities in each operation mode, which depend on the actuation scheme. x 0 = x 3 = 0 describes the constraint singularity that exhibits the transition between K 1 and K 2 .

It is noteworthy that this method is not always convenient to use as equations g 4 , g 5 and g 6 depend on the manipulator actuation scheme. For instance, the equations for the 3-RPS manipulator expressed in Sec. 3 depend on the actuated prismatic joint lengths. Besides, it should be noted that the derivation of equations g 4 , g 5 and g 6 as a function of the actuated revolute joint angles is usually tedious. To overcome this problem, screw theory techniques are adopted to derive the Jacobian matrices of the PMs.

Jacobian Matrix obtained based on Screw Theory

Based on the theory of reciprocal screws, the velocity model of a parallel manipulator can be expressed as

At = B ρ ( 18
)
where ρ is the actuated joint rate vector. t is the moving platform twist with respect to the fixed base i.e., it contains the angular velocity vector of the moving platform and the linear velocity vector of the MP geometric center. A is called the forward Jacobian matrix and it incorporates the actuation and constraint wrenches of the PM while B is named the inverse Jacobian matrix [START_REF] Joshi | Jacobian analysis of limited-dof parallel manipulators[END_REF]. For a non redundantly actuated and non overconstraint PM, A is a 6 × 6 square matrix whereas B is a 6 × n matrix where n is the number of actuated joints. To compare the manipulators of the 3-[PP]S-Y family, only the parallel singularities are considered such that, A is rank deficient. Since, A is square, configurations satisfying det(A) = 0 are singular. Moreover, the manipulators belonging to the 3-[PP]S family cannot meey any serial singularity corresponding to the rank deficiency of B except the 3-RRS PM.

For instance, the kinematic modeling of the 3-RPS and 3-RPS PMs following Eq. 18 can be expressed as

A 3-RPS 0 t = B 3-RPS ρ =⇒                          ( 0 R 1 1 b 1 × 0 u 1 ) T 0 u T 1 ( 0 R 1 1 b 2 × 0 u 2 ) T 0 u T 2 ( 0 R 1 1 b 3 × 0 u 3 ) T 0 u T 3 ( 0 R 1 1 b 1 × 0 s 1 ) T 0 s T 1 ( 0 R 1 1 b 2 × 0 s 2 ) T 0 s T 2 ( 0 R 1 1 b 3 × 0 s 3 ) T 0 s T 3                          0 ω 0 v = I 3×3 0 3×3           ρ1 ρ2 ρ3           (19)
A 3-RPS 0 t = B 3-RPS θ =⇒                          ( 0 R 1 1 b 1 × 0 v 1 ) T 0 v T 1 ( 0 R 1 1 b 2 × 0 v 2 ) T 0 v T 2 ( 0 R 1 1 b 3 × 0 v 3 ) T 0 v T 3 ( 0 R 1 1 b 1 × 0 s 1 ) T 0 s T 1 ( 0 R 1 1 b 2 × 0 s 2 ) T 0 s T 2 ( 0 R 1 1 b 3 × 0 s 3 ) T 0 s T 3                          0 ω 0 v = I 3×3 0 3×3           θ1 θ2 θ3           (20)
where, 0 u i is a unit vector along the prismatic joint direction and expressed in Σ 0 , 0 s i is the unit vector along the ith revolute joint axis, 0 v i is a vector normal to both 0 u i and 0 s i . Matrices A 3-RPS and A 3-RPS incorporate the actuation and constraint wrenches of the 3-RPS and 3-RPS PMs, respectively [START_REF] Joshi | Jacobian analysis of limited-dof parallel manipulators[END_REF] and they are both 6 × 6 square matrices. For the i-th leg of these manipulators, the constraint wrench, 0 ζc 0i and the actuation wrench, 0 ζa 0i are represented in Figs. 3 and4 and are expressed in frame Σ 0 as follows:

3 -RPS PM : 0 ζc 0i = 0 s i 0 R 1 1 b i × 0 s i 0 ζa 0i = 0 u i 0 R 1 1 b i × 0 u i (21)
3 -RPS PM : 0 ζc 0i =

0 s i 0 R 1 1 b i × 0 s i 0 ζa 0i = 0 v i 0 R 1 1 b i × 0 v i (22)
Thus, from Eqs. (1), ( 3) and ( 4), the determinant of the forward Jacobian matrix A 3-RPS and A 3-RPS in Eqs. ( 19) and ( 20) can be calculated in terms of x 0 , x 1 , x 2 , x 3 , X, Y, Z and the design parameters. The derivation of the forward Jacobian matrix by differentiating the constraint equations is tedious when a revolute joint is actuated whereas using screw theory, it is straightforward regardless of the actuation scheme. Therefore, the forward singularities for different actuation schemes of 3-[PP]S PMs are determined when the determinant of matrix A in Eq. ( 18) vanishes.

Singularity loci in the orientation workspace

The singularities can be expressed in the orientation workspace by parametrizing the orientation of the platform in terms of Tilt-and-Torsion (T&T) angles, φ (azimuth), θ (tilt) and σ (torsion) [START_REF] Bonev | Advantages of the modified euler angles in the design and control of PKMs[END_REF][START_REF] Bonev | Orientation workspace analysis of 6-dof parallel manipulators[END_REF] as recalled in Fig. 5 2 . As shown in [START_REF] Bonev | Advantages of the modified euler angles in the design and control of PKMs[END_REF], those angles are more suitable for 3-[PP]S manipulators, which are known to be zero torsion mechanisms (σ is always equal to 0 or π depending on the operation mode). The orientation quaternions can be expressed in terms of 2 Figure 5 is a combination of Figs. 1(a) and 1(b) in [START_REF] Bonev | Advantages of the modified euler angles in the design and control of PKMs[END_REF]. In case of ZXZ Euler angle notation, a rotation matrix is decomposed as R = R z (φ)R x (θ)R z (ψ) whereas T&T angles use only two rotations R = R a (θ)R z (σ), where θ and σ represent tilt and torsion, respectively. That is the reason why they are named as Tilt and Torsion angles. However, it should be noted that the azimuth angle is necessary to locate the axis-a and is measured from x-axis to be consistent with the ZXZ Euler angle convention. This simple yet effective change of angles makes the whole analysis much simpler and suits the behavior of any zero-torsion mechanism. [START_REF] Bonev | Advantages of the modified euler angles in the design and control of PKMs[END_REF] the T&T angles as follows 3 :

A 1 A 2 A 3 B 3 B 2 ζ c 01 ζ c 02 ζ c 03 ζ a 02 ζ a 01 ζ a 03 B 1 O 1 O 0 s 2 s 3 s 1 u 1 u 2 u 3
x 0 = cos( θ 2 )cos( σ 2
)

x 1 = sin( θ 2 )cos(φ - σ 2 )
x 2 = sin( θ 2 )sin(φ -σ 2 )

x 3 = cos( θ 2 )sin( σ 2 ) (23)
The change of parametrization results in only three parameters expressing the orientation of the moving-platform. Furthermore, for operation mode K 1 , characterized by x 0 = 0, Eq. ( 23) gives σ = π. Similarly, for operation mode K 2 , characterized by x 3 = 0, Eq. ( 23) gives σ = 0. Substituting the values of σ in the remaining equations, leads to x i expressed as functions of φ and θ angles, only.

In order to plot the singularity loci of the manipulator into the orientation workspace, the determinant of the forward Jacobian matrix of the manipulator should be expressed as a function φ and θ angles, only. All remaining variables should be eliminated from the determinant of the forward Jacobian matrix. X and Y can be eliminated using Eqs. ( 6) and ( 7) for the 3-[PP]S-Y PMs as follows:

X = h 2 x 1 2 -x 2 2 Y = -2 h 2 x 1 x 2 (24)
To this end, the determinant of the forward Jacobian matrix is a function of Z, x i , i = 0, 1, 2, 3 and of the actuated joint variables. The actuated joint variables can be eliminated using equations g 4 = g 5 = g 6 = 0 as shown in Sec. 3. Additionally, from Eq. ( 23), x i can be expressed as functions of φ and θ angles leading to the final expression of the determinant of the forward Jacobian matrix in terms of Z, θ and φ.

Maximum Inscribed Circle Radius (MICR)

From a practical point of view, it is desirable to use a PM in its singularity free region. A performance index, named Maximum Inscribed Circle Radius (MICR) and defined in [START_REF] Nayak | Comparison of 3-RPS and 3-SPR parallel manipulators based on their maximum inscribed singularity-free circle[END_REF], is used in this section to quantify the capability of the Parallel Manipulator (PM) in terms of orientation motions. For a given altitude Z, this index gives the maximum tilt angle θ that the moving-platform can reach for any azimuth angle φ without reaching any parallel singularity:

MICR = max 0≤θ≤π {θ, ∀ φ ∈ [0, π], det(A) 0} (25)
MICR values are calculated for all the mechanisms at hand in their operation modes for three ratios for

h 2 h 1 = { 1 2 , 1, 2}
and two ratios for Z h 1 = 1, 2. The ratios Z h 1 and h 2 h 1 make sure that the MICR values can be used to compare the mechanisms and their operation modes no matter the platform and base sizes. Thus, MICR is used to compare the PMS belonging to the 3-[PP]S-Y family.

Parasitic Motions

For the 3-[PP]S-Y family of PMs, the workspace is expressed in terms of Z, φ and θ. Since the torsion angle σ can only attain the values 0 or π depending on the operation mode, the constrained motions are along the x and y axes. The parasitic displacements along the x and y coordinates of the moving platform are expressed in terms of Z and x i in Equations [START_REF] Li | 1T2R parallel mechanisms without parasitic motion[END_REF].

To compare the parasitic motions of PMs belonging to the 3-[PP]S-Y family, the maximum displacement within the Maximum Inscribed Circle (MIC) [START_REF] Nayak | Comparison of 3-RPS and 3-SPR parallel manipulators based on their maximum inscribed singularity-free circle[END_REF] is considered. Here, the following index is defined to quantify the parasitic motions of the moving-platform along x 0 and y 0 axes. A parameter with units of length is defined such that

µ = max √ X 2 + Y 2 when 0 ≤ θ ≤ MICR ; -π ≤ φ ≤ π (26)
µ can be expressed in terms of h 2 , θ and φ for these manipulators.

Internal collisions

A drawback of using parallel manipulators for potential applications is the existence of internal collisions. It also applies to the PMs belonging to the 3-[PP]S-Y family listed in Table 1. For these manipulators, three kinds of internal collisions should be considered:

1. Collisions between limbs. 2. Collisions between limbs and the moving or fixed platform triangles.

Collision between the fixed and moving-platform triangles.

An accurate collision analysis requires the shape and thickness of the limbs as well as the range of motion of the spherical joints. It should be noted that this paper mainly focuses on the singularity free orientation workspace and the parasitic motions. The collision analysis is left for future work. However, it can be observed that most applications of the 3-[PP]S-Y PMs [START_REF] Carretero | Kinematic analysis and optimization of a new three degree-of-freedom spatial parallel manipulator[END_REF][START_REF] Wahl | Articulated tool head[END_REF][START_REF] Tetik | Position kinematics of a 3-RRS parallel manipulator[END_REF][START_REF] Yang | Analysis and design of a 3-dof flexure-based zero-torsion parallel manipulator for nano-alignment applications[END_REF] require them to be in the second operation mode (characterized in this paper by x 3 = 0), due to the existence of internal collisions in their first operation mode (x 0 = 0). This is partially accounted for a new complexity index introduced on the next section.

Example: The 3-RPS PM

In this section, the 3-RPS PM is considered and the forward Jacobian matrix is formulated, leading to the determination of MICR for different sizes of its fixed base and moving platforms. Furthermore, the parasitic motions are visualized within the Maximum Inscribed Circle (MIC) by plotting µ vs. φ and θ4 .

The architecture of the 3-RPS PM belonging to the family of 3-[PP]S-Y of PMs is shown in Figure 6. The determinant of A 3-RPS calculated from Eq. ( 19) turns out to be a function of x 0 , x 1 , x 2 , x 3 , X, Y, Z, h 1 and h 2 . X and Y are substituted from Eq. ( 24) and then T&T angle substitutions are done (Eq. ( 23)) resulting in the determinant of the forward Jacobian matrix to be a function of Z, h 1 , h 2 , θ, φ and σ. For operation mode 1, substituting σ = π and assigning h 1 = 1 and h 2 = 2, the implicit plot of the Jacobian can be drawn by varying Z, φ and θ as shown in Fig. 7a. For operation mode 2 with σ = 0, the implicit plot is shown in Fig. 7b. A slice of these surfaces at Z = h 1 = 1 shows the singularity loci in Fig. 8.

Along with the singularity loci, the so called intersection loci, depicting the collision between each pair of limbs A 1 B 1 , A 2 B 2 and A 3 B 3 are also plotted. They are calculated as follows:

The limbs are assumed to have no thickness and the spherical joints are assumed to have full range of motion. Thus, the problem is reduced to find all possible intersections between three pairs of line segments. For instance, to find the intersection loci of line segments A 1 B 1 and A 2 B 2 , their vector equations are written as follows:

r 1 = a 1 + (b 1 -a 1 )t 1 t 1 ∈ [0, 1] (27) r 2 = a 2 + (b 2 -a 1 )t 2 t 2 ∈ [0, 1] (28)
where r 1 and r 1 are position vectors of a point on line segments A 1 B 1 and A 2 B 2 , respectively. An intersection occurs when r 1 = r 2 . Eliminating t 1 and t 2 from this equation gives an implicit curve in terms of θ and φ where the considered line segments intersect under the condition t 1 , t 2 ∈ [0, 1]. The intersection between the other two pairs of limbs are determined in a similar manner, resulting in the intersection loci as plotted in blue in Fig. 8. It is apparent that the

x 0 y 0 z 0 x 1 y 1 z 1 O 1 O 0 h 1 A 2 A 1 A 3 B 2 B 3 r 1 r 2 s 1 s 2 s 3 r 3 B 1 h 2 Σ 0 Σ 1
h 2 h 1 = 2, Z h 1 = 1
MICR, defined in Eq. [START_REF] Nayak | Comparison of 3-RPS and 3-SPR parallel manipulators based on their maximum inscribed singularity-free circle[END_REF]. is also marked in these figures. For different ratios of h 2 to h 1 , the variations in MICR values are plotted in Figs. 9 and10. Z h 1 as a function of From Eqs. ( 23) and ( 24), the maximum displacement within the Maximum Inscribed Circle (MIC) is written as

MICR (degrees) 0 0.5 1 1.5 2 2.5 3 3.5 4 Z/h 1 h 2 /h 1 =1/4 h 2 /h 1 =1/3 h 2 /h 1 =1/2 h 2 /h 1 =1 h 2 /h 1 =3/2 h 2 /h 1 =2 h 2 /h 1 =5/2 h 2 /h 1 =3
h 2 h 1 MICR (degrees) 0 0.5 1 1.5 2 2.5 3 3.5 4 Z/h 1 h 2 /h 1 =1/4 h 2 /h 1 =1/3 h 2 /h 1 =1/2 h 2 /h 1 =1 h 2 /h 1 =3/2 h 2 /h 1 =2 h 2 /h 1 =5/2 h 2 /h 1 =3
µ = max √ X 2 + Y 2 = max h 2 2 sin θ 2 4 when 0 ≤ θ ≤ MICR ; -π ≤ φ ≤ π (29)
By varying Z from 0 to 4, θ from 0 to MICR and φ from -π to π, µ is plotted for the 3-RPS PM as shown in Figs. 11 and12. It is observed that the parasitic motion increases as the ratio h 2 h 1 increases. But, the larger the h 2 h 1 ratio, the smaller the MICR and hence, the smaller the singularity free workspace.

0 0.5 1 1.5 2 2.5 3 3.5 4 0 µ (unit of length) Z/h 1 h 2 /h 1 =1/4 h 2 /h 1 =1/3 h 2 /h 1 =1/2 h 2 /h 1 =1 h 2 /h 1 =3/2 h 2 /h 1 =2 h 2 /h 1 =5/2 h 2 /h 1 =3 0.2 0.4 0.6 0.8 1 0.1
Z/h 1 h 2 /h 1 =1/4 h 2 /h 1 =1/3 h 2 /h 1 =1/2 h 2 /h 1 =1 h 2 /h 1 =3/2 h 2 /h 1 =2 h 2 /h 1 =5/2 h 2 /h 1 =3 0.2 0.4 0.6 0.8 1 0.25

Comparison of 3-[PP]S-Y PMs based on their MICR and parasitic motions

For most applications, it is better to have a large singularity-free orientation workspace with small parasitic motions. To find out the most suitable manipulator for this purpose, MICR vs. µ is plotted for all the mechanisms listed in Table 1 in their two operation modes each for three ratios of

h 2 h 1 = { 1 2
, 1, 2} and two ratios of Z h 1 = {1, 2}. These points are plotted for the abovementioned ratios in Figures 13 to 18. From Eq. ( 24), it is noticed that parasitic motions are higher for larger singularity free regions. The abscissa is chosen to be π-MICR since the minimum of this function is preferred and the ordinate is chosen to be µ. For a PM from the 3-[PP]S-Y family, π-MICR and µ values are preferred to be smaller. Thus, plotting π-MICR vs. µ results in a Pareto front as shown in Figs. 13 to 18. Among the manipulators listed in Table 1, it is noticed that the 3-PhPvS PM behaves exactly like the 3-PvPhS PM and the 3-PhPvS PM like the 3-PvPhS PM. Hence, only the 3-PhPvS PM is considered for analysis. Due to the arrangement of prismatic joints, 3-PvRS and 3-PhPvS are inherently singular at any configuration and hence they are not considered here. Indeed, for these manipulators, the three actuation forces are coplanar and intersect at a point resulting in only two independent forces. In addition, the PMs 3-RPS and 3-RRS; 3-PhRS and 3-PhPvS; behave alike due to identical arrangement of their passive joints. It should be noted that the leg links of the 3-RRS PM are considered to be of length h 1 so that the moving platform can reach the maximum value of Z(= 2h 1 ) considered to compare the manipulators at hand. In Fig. 15, the MICR and µ values are marked for the 3-RPS PM when 2 while the latter shows the solutions that do not dominate any other solutions listed in Table 3. Their respective configurations are also displayed in these figures. Operation mode 1 is represented with crossed legs or upside down moving platform. But, this need not always be the case. In fact, the generic pose of the 3-[PP]S family of PMs in each of their operation modes is still an open problem. It is noteworthy that the ordinate is chosen to be log µ so that the smaller as well as larger values of µ are clearly visible in the same graph.

h 2 =h 1 /2 h 1 Z=h 1 Figure 13: MICR vs maximum displacement h 2 h 1 = 1 2 , Z h 1 = 1 h 2 =h 1 h 1 Z=h 1 Z h 2 Figure 14: MICR vs maximum displacement h 2 h 1 = 1, Z h 1 = 1 h 2 =2h 1 h 1 Z=h 1 MICR = 25 o µ = 0.1 MICR = 40 o µ = 0.25 Figure 15: MICR vs maximum displacement h 2 h 1 = 2, Z h 1 = 1 h 2 =h 1 /2 h 1 Z=2h 1 Figure 16: MICR vs maximum displacement h 2 h 1 = 1 2 , Z h 1 = 2 h 2 =2h 1 h 1 Z=2h 1 Figure 17: MICR vs maximum displacement h h = 1, Z h 1 = 2 h 2 =2h 1 h 1 Z=2h 1 h =2h 1 Figure 18: MICR vs maximum displacement h h = 2, Z h 1 = 2 3-[PP]S PM h 2 h 1 Z h 1 3-RPS OM2 1/2 1,2 3-PhPvS OM1 1/2 2 3-PhRS OM1 1/2 2 3-PhPvS OM2 1/2 2 3-PhRS OM2 1/2 2 3-PvRS OM1 1/2 2 3-PvRS OM2 1/2 2 3-RPS OM1 1/2 1,2 3-RRS OM1 1/2 1,2 3-RPS OM1 1/2 1,2 3-RPS OM2 1/2 1,2 3-RRS OM2 1/2 1,2 3-RRS OM1 1/2 1 3-PhRS OM1 1/2 1,2 3-PhRS OM2 1/2 2
Table 2: Pareto-optimal solutions (Fig. 19)

3-[PP]S PM

h 2 h 1 Z h 1 3-RPS OM1 2 1,2 3-RPS OM2 2 1,2 3-PhPvS OM1 2 2 3-PhRS OM1 2 2 3-PhPvS OM2 2 2 3-PhRS OM2 2 2 3-RRS OM2 2 1,2 3-PvPS OM2 2 2 3-RPS OM2 2 1,2 3-RRS OM2 2 1,2 3-PhRS OM1 2 1,2 3-RPS OM1 2 1,2 3-RRS OM1 2 1,2 3-PvRS OM1 2 2 3-RRS OM1 2 1
Table 3: Solutions that do not dominate any other solutions (Fig. 20)

Some observations that can be drawn from these curves are that all the Pareto optimal solutions consist of 3 On the contrary, the 3-RPS PM in its first operation mode with h 2 h 1 = 2, Z h 1 = 1 has the largest µ.

Ranking the Pareto optimal solutions based on complexity indices

Some complexity indices were introduced in [START_REF] Caro | The rule-based conceptual design of the architecture of serial schnflies-motion generators[END_REF][START_REF] Khan | A Formulation of Complexity-Based Rules for the Preliminary Design Stage of Robotic Architectures[END_REF] to evaluate the complexity of serial and parallel manipulators at their conceptual design stage. Those indices can be used to rank the parallel manipulators at hand. The complexity is not considered as a separate objective function along with MICR and µ. It is given a lesser priority to compare the PMs belonging to the 3-[PP]S-Y family due to the fact that the complexity indices are independent of h 2 h 1 and Z h 1 .

Moreover, due to little differences between architectures of the 3-[PP]S-Y PMs, most of the indices defined in [START_REF] Khan | A Formulation of Complexity-Based Rules for the Preliminary Design Stage of Robotic Architectures[END_REF] yield the same complexity. In order to avoid this problem and to rank the Pareto optimal solutions listed in Table 2 in order of complexity, two more indices are introduced in this paper. In total, the following six complexity indices are used to compare the Pareto-optimal solutions listed in Table 2: 8.1. Joint-number complexity K N The joint-number complexity K N is defined as

K N = 1 -exp(-q N N), (30)
where N is the number of joints in the PM and q N is the resolution parameter defined by q N = ln(0.1)/N max , for N max > 0; 0, for N max = 0.

8.5. Actuator-position complexity K P Actuator-position complexity K P is defined as:

1 m m i=1 p i -1 n i , ( 36
)
where m is the number of limbs, n i is the total number of joints in the i-th limb and p i is the location of the actuated joint i.e. p i = k if k-th joint is actuated. For 3-[PP]S-Y PMs, K P = 0 if the first joint is actuated and K P = 1/3 if the second joint is actuated. It is noteworthy that the actuator position index can be applied to any other serial kinematic chain or a PM even if the position of the actuator differs from one limb to another.

8.6. Operation mode complexity K OM Operation mode complexity K OM is defined specifically for 3-[PP]S-Y PMs. As seen in Sec. 4, a 3-[PP]S-Y PM has two different operation modes. In the first operation mode, the moving platform is rotated upside down by 180 degrees with respect to the fixed base. As a result of this complexity analysis, it is prone to have internal collisions as compared to operation mode 2, as shown for a 3-RPS PM in Sec. 6. Although a detailed collision analysis is necessary to overcome this issue, it is partially overcome in this paper by introducing a new complexity index as follows:

K OM = 1 for OM1 (37)
K OM = 0 for OM2 (38)

Total complexity

Total complexity K ∈ [0, 1] of a kinematic chain was defined as a convex combination of the different complexity indices [START_REF] Khan | A Formulation of Complexity-Based Rules for the Preliminary Design Stage of Robotic Architectures[END_REF]. In this context, for the 3-[PP]S-Y PMs, it is defined as

K = w N K N + w J K J + w L K L + w B K B + w P K P + w OM K OM (39)
where w N , w J , w L , w B , w P and w OM denote their corresponding weights, such that w N + w J + w L + w B + w P + w OM = 1

Assigning equal weights5 to all complexity indices implies

K = 1 6 (K N + K J + K L + K B + K P + K OM ) (40)
The Pareto optimal solutions listed in Table 2 are ranked in ascending order of complexity in Table 4. It is apparent that the 3-RPS, 3-PvRS and 3-PhRS PMs in OM2 dominate their counterparts with higher singularity-free orientation workspaces, lower parasitic motions and are the simplest ones, their complexity being equal to 0.4985. The 3-PhPvS PM in OM1 is the most complex Pareto optimal solution with a complexity equal to 0.7472.

Conclusion

In this paper, different operation and actuation modes of the 3-[PP]S-Y zero torsion parallel manipulators were compared based on their singularity free orientation workspace and parasitic motions. The manipulators belonging to the 3-[PP]S-Y family of PMs were considered and kinematic constraint equations were derived for their general architectures. A primary decomposition of the ideal of the constraint polynomials showed that they have two operation modes. Furthermore, their forward kinematic Jacobian matrices were derived for the mechanisms at hand. The Rank Pareto optimal solutions K N K J K L K B K P K OM K 1 3-RPS OM2 0.9 0.5078 0.9 0.6832 0 0 0.4985 3-PvRS OM2 0.9 0.5078 0.9 0.6832 0 0 0.4985 3-PhRS OM2 0.9 0.5078 0.9 0.6832 0 0 0.4985 2 3-RRS OM2 0.9 0.3489 0.9 0.6832 0.3333 0 0.5276 3 3-PhRS OM2 0.9 0.5078 0.9 0.6832 0.3333 0 0.5541 3-RPS OM2 0.9 0.5078 0.9 0.6832 0.3333 0 0.5541 4 3-PhPvS OM2 0.9 0.6667 0.9 0.6832 0.3333 0 0.5805 5 3-RRS OM1 0.9 0.3489 0.9 0.6832 0 1 0.6387 6 3-PvRS OM1 0.9 0.5078 0.9 0.6832 0 1 0.6552 3-PhRS OM1 0.9 0.5078 0.9 0.6832 0 1 0.6552 7 3-RRS OM1 0.9 0.3489 0.9 0.6832 0.3333 1 0.6942 8 3-PhRS OM1 0.9 0.5078 0.9 0.6832 0.3333 1 0.7207 3-RPS OM1 0.9 0.5078 0.9 0.6832 0.3333 1 0.7207 9 3-PhPvS OM1 0.9 0.6667 0.9 0.6832 0.3333 1 0.7472 determinant of the Jacobian matrix was expressed as a function of the orientation workspace parameters tilt, azimuth and height of the moving-platform, Z. The vanishing condition of the determinant was used to visualize the singularity loci in the orientation workspace. An index, named Maximum Inscribed Circle Radius (MICR), was used to quantify the singularity free orientation workspace of the manipulators. The maximum parasitic motion within the Maximum Inscribed Circle denoted as µ and was used as a second performance index. For

h 2 h 1 = 1 2
, 1, 2 and Z h 1 = 1, 2, MICR vs µ values were plotted for all the mechanisms in Table 1. The 3-RPS PM was taken as an example to illustrate the methodology to find the MICR and µ values. For all manipulators belonging to the 3-[PP]S-Y family, the plots of (180-MICR) vs. µ were drawn such that a minimum of these objective functions is the preferred solution. Different plots were drawn for different ratios of the circumference of the moving-platform to the circumference of the fixed base. Ultimately, (180-MICR) vs. log 10 (µ) points are plotted in the same graph for all the mechanisms for easier comparison. A Pareto front was observed in the plot and manipulator configurations were represented for the Paretooptimal solutions and the solutions that do not dominate any other solution. It was concluded that the Pareto optimal solutions only include the PMs from the 3-[PP]S-Y family with the circum-radius of the base twice as that of the platform. The solutions that do not dominate any other solutions include the PMs from the 3-[PP]S-Y family with the circum-radius of the platform twice as that of the base. Therefore, the MICR vs µ plots could be used to compare the 3-[PP]S family of PMs especially while choosing them for a particular application. Furthermore, the complexity indices of the Pareto optimal solutions were calculated. For equal weights of all complexity indices, it was shown that

Figure 1 :

 1 Figure 1: Six parallel manipulators belonging to the 3-[PP]S-Y family

Σ 1 LFigure 2 :

 12 Figure 2: Planes containing the limbs of the 3-[PP]S-Y PMs

Figure 3 : 3 Figure 4 :Figure 5 :

 3345 Figure 3: Actuation and constraint wrenches for the 3-RPS PM

Figure 6

 6 Figure 6: 3-RPS PM architecture

Figure 7 : 3 -

 73 Figure 7: 3-RPS singularity surfaces for h 2 h 1 = 2

Figure 8 : 3 -

 83 Figure 8: 3-RPS singularity loci (green), intersection loci (blue) and the maximum inscribed singularity-free circle (red), h 2 h 1 = 2, Z h 1 = 1

Figure 9 : 3 -

 93 Figure 9: 3-RPS OM1: MICR vs. Z h 1 as a function of h 2 h 1

Figure 10 : 3 -

 103 Figure 10: 3-RPS OM2: MICR vs. Z h 1 as a function of h 2 h 1

Figure 11 : 3 -

 113 Figure 11: 3-RPS OM1: µ vs. Z h 1 as a function of h 2 h 1

Figure 12 : 3 -

 123 Figure 12: 3-RPS OM2: µ vs. Z h 1 as a function of h 2 h 1

h 2 h 1 = 2 and Z h 1 = 1 .

 11 These points correspond to the MICR and µ values spotted in the MICR and µ vs Z h 1 plots in Sec. 6. To compare the 3-[PP]S-Y family of PMs, their MICR vs µ values are plotted for different ratios of h 19 and 20. The former figure shows the Pareto optimal solutions listed in Table

the 3 -

 3 RPS, 3-PvRS and 3-PhRS PMs in their second operation mode are the simplest ones amongst 3-[PP]S-Y PMs. Future work will include a comparison between 3-[PP]S-∆ and 3-[PP]S-cube PMs for different actuation schemes.Furthermore, the comparison will be done while taking into account joint limits and self collisions. Additionally, MICR, parasitic motions and proposed complexity indices can be used as a basis of comparison for other lower mobility PMs.

Table 1 :

 1 Manipulators belonging to the 3-[PP]S-Y family

Table 4 :

 4 Complexity indices of Pareto optimal 3-[PP]S-Y PMs in ascending order

Table 5 :

 5 Pareto optimal 3-[PP]S-Y parallel manipulators configurations in ascending order of complexity

It should be noted that these directions are considered following the notation defined in[START_REF] Liu | Orientation capability, error analysis, and dimensional optimization of two articulated tool heads with parallel kinematics[END_REF] and it is assumed that the fixed base is horizontal.

b i + 0 p 1 , i = 1,

2, 3 (4)

The change of parameters from quaternions to T&T angles is done for the following reasons: quaternions are used to obtain algebraic constraint equations so that the algebraic geometry techniques can be used to analyse those equations. For instance, primary decomposition of the ideal of the constraint polynomials described in Eq. (14) yields the exact number of operation modes and their characterization. However, the parametrization is changed to T&T angles to be able to visualize the singularities and maximum parasitic displacements in the orientation workspace. Moreover, they offer a more intuitive approach to deal with zero-torsion mechanisms that are compared in this paper.

Radian is used as the angular unit throughout the text but degree is used in figures for better clarity.

Figure 20: the solutions that do not dominate any other solutions.

Since K N , K L and K B are the same for all the 3-[PP]S-Y PMs, the weights w N , w L and w B do not affect the complexity ranking. However, when w L =[START_REF] Merlet | Parallel Robots[END_REF] 3 and w P = 1 12 , 3-RRS PM in OM2 becomes the simplest PM belonging to the 3-[PP]S-Y family. Eventually, the weights can be chosen according to the application.

Acknowledgments

This work was conducted with the support of both Centrale Nantes and the French National Research Agency (ANR project Kapamat #ANR-14-CE34-0008-01).

For all the 3-[PP]S-Y PMs considered here, N = N max = 9 and hence k N = 0.9.

Joint-type complexity K J

As the name suggests, the joint-type complexity K J is associated with the type of joints in the PM. The 3-[PP]S-Y PMs consist of only revolute, prismatic and spherical joints and it leads to the definition of K J as follows:

where n R , n P and n S are the numbers of revolute, prismatic and spherical joints, respectively with n = n R + n P + n S . K G|x is the geometric complexity of the pair x as introduced in [START_REF] Khan | A Formulation of Complexity-Based Rules for the Preliminary Design Stage of Robotic Architectures[END_REF]: K G|R = 0.5234, K G|P = 1 and K G|S = 0. The values of K J for the Pareto optimal solutions are listed in Table 4.

Loop complexity K L

The loop complexity of a PM is defined as:

where l is the number of kinematic loops and l m is the minimum number of loops required to produce a special displacement group or subgroup. Since each PM belonging to the 3-[PP]S-Y family has non-pure rotations about two horizontal axes and a vertical translation, the three degrees of freedom can be realized using a single kinematic chain.

Hence, for all the 3-[PP]S-Y PMs, l = 2 and l m = 0. q L is defined the same way as in Eq. ( 31):

where, L max = L = 2 for all the Pareto optimal solutions resulting in K L = 0.9.

Link diversity K B

Link diversity K B is defined to quantify the geometric constraints between neighboring joints. For a revolute joint, its axis of rotation is considered whereas for a prismatic joint, its direction. Five possible joint-constraint types between the neighboring joint axes/directions were reported in [START_REF] Khan | A Formulation of Complexity-Based Rules for the Preliminary Design Stage of Robotic Architectures[END_REF]:

1. Type B 1 : Orthogonal intersection. 2. Type B 2 : Nonorthogonal intersection. 3. Type B 3 : Parallelism. 4. Type B 4 : Orthogonal but not intersecting. 5. Type B 5 : Skew. Thus, the geometric-constraint diversity was defined as:

where B is the entropy of the joint-constraint types and B max = 2.32 [START_REF] Khan | A Formulation of Complexity-Based Rules for the Preliminary Design Stage of Robotic Architectures[END_REF]. c is the number of distinct joint-constraint types and M i is the number of instances of each type of joint-constraints.


Andrei Moroianu
email: moroianu@alum.mit.edu

Sergiu Moroianu

THE DIRAC SPECTRUM ON MANIFOLDS WITH GRADIENT CONFORMAL VECTOR FIELDS

Keywords: November 7, 2007. 2000 Mathematics Subject Classification. 58J50, 58J20 Dirac operator, hyperbolic manifolds, gradient conformal vector fields, continuous spectrum

We show that the Dirac operator on a spin manifold does not admit L 2 eigenspinors provided the metric has a certain asymptotic behaviour and is a warped product near infinity. These conditions on the metric are fulfilled in particular if the manifold is complete and carries a non-complete vector field which outside a compact set is gradient conformal and non-vanishing.

Introduction

The Dirac operator on a closed spin manifold is essentially self-adjoint as an unbounded operator in L 2 , and has purely discrete spectrum. Its eigenvalues grow at a certain speed determined by the volume of the manifold and its dimension. Hence, although determining the eigenvalues can be a daunting task, the nature of the spectrum is rather well understood.

On non-compact manifolds, the spectrum of the Dirac operator can behave in a variety of ways. For instance, the Dirac operator on R n has purely absolutely continuous spectrum, so in particular there are no L 2 eigenspinors. In contrast, Bär [START_REF] Bär | The Dirac operator on hyperbolic manifolds of finite volume[END_REF] showed that on complete spin hyperbolic manifolds of finite volume, the spectrum is purely discrete if the induced Dirac operators on the ends are invertible. In this situation even the classical Weyl law for the distribution of the eigenvalues holds [START_REF] Moroianu | Weyl laws on open manifolds[END_REF]. Otherwise, if the limiting Dirac operator is not invertible, then the essential spectrum is the whole real axis. Similar results appear in [START_REF] Golénia | The spectrum of magnetic Schrödinger operators and k-form Laplacians on conformally cusp manifolds[END_REF] for the Laplace operator on forms and for magnetic Schrödinger operators.

In this paper we show that for a class of -possibly incomplete -spin Riemannian manifolds (X, g) which includes certain hyperbolic manifolds, the Dirac operator D does not carry L 2 eigenspinors of real eigenvalue. In particular, we deduce that the L 2 index of the Dirac operator on (X, g) vanishes.

Our main result (Theorem 2.1) makes special assumptions on the metric of X. Geometrically, these assumptions imply the existence of a non-complete vector field on X which is gradient conformal on an open subset U of X. Conversely, we show in Section 4 that the existence of such vector fields implies the hypothesis of Theorem 2.1 provided that X is complete and X \U is compact. As a corollary, we obtain that on a complete spin manifold X which carries a non-complete vector field which is gradient conformal outside a compact subset of X, the Dirac operator has purely continuous spectrum (Theorem 4.1).

The main result

Let ( Xn , h) be a connected spin manifold with boundary with interior X. Assume that there exists a boundary component M so that h is a product in a neighbourhood of M . Denote by x : X → [0, ∞) the distance to M , so h = dx 2 + h M near M . Note that M inherits a spin structure from X.

Let f : X → (0, ∞) be a smooth conformal factor which depends only on x in a neighbourhood of M . Theorem 2.1. Assume that M is at infinite distance from X with respect to the conformal metric

(1) g := f 2 h = f (x) 2 (dx 2 + h M ).
Moreover assume that the Dirac operator D M on (M, h M ) is essentially selfadjoint. Then the Dirac operator of (X, g) does not have any distributional L 2 eigenspinors of real eigenvalue.

Lott [START_REF] Lott | The Dirac operator and conformal compactification[END_REF] proved that there is no L 2 harmonic spinor under somewhat similar assumptions. Namely, h could be any metric smooth up to M , and f could vary in the M directions. However, Lott assumes that f -1 extends to a locally Lipschitz function on X which eventually must be locally bounded by a multiple of x, while our hypothesis only asks that (2)

ǫ 0 f (x)dx = ∞.
In particular, unlike in [START_REF] Lott | The Dirac operator and conformal compactification[END_REF], the function f -1 may be unbounded near M .

If we assume that (X, g) is complete as in [START_REF] Lott | The Dirac operator and conformal compactification[END_REF], we deduce that (M, h M ) is complete so D M is essentially self-adjoint. With this assumption we also know that D g is essentially self-adjoint, so its spectrum is real. But we do not need to make this assumption as the statement is "local near M ". This seems to be also the case in [START_REF] Lott | The Dirac operator and conformal compactification[END_REF], although it is not claimed explicitly. However, there are many instances of incomplete manifolds whose Dirac operator is essentially self-adjoint, cf. Example 2.2.

The rest of this section is devoted to the proof of Theorem 2.1.

It is known since Hitchin [8] that the Dirac operator has a certain conformal invariance property. More precisely, if g = f 2 h, the Dirac operator D g is conjugated to f -1 D h by the Hilbert space isometry

L 2 (X, Σ, vol g ) → L 2 (X, Σ, f vol h ), ψ → f n-1 2 ψ, (3)
(see [12, Proposition 1]). Let f -n-1 2 φ ∈ L 2 (X, Σ, vol g ) be an eigenspinor of D g (in the sense of distributions) of eigenvalue l ∈ R. Then (4) (f -1 D h -l)φ = 0.
By elliptic regularity, φ is in fact a smooth spinor on X so (4) is equivalent to

(5) D h φ = lf φ.
Let c 0 denote the Clifford multiplication by the unit normal vector ∂ x with respect to h. Then D h decomposes near M as follows:

D h = c 0 (∂ x + A)
where for each x > 0, A is a differential operator on the sections of Σ over M ×{x}; moreover A is independent of x. Note that Σ M is either the spinor bundle Σ(M ) of M with respect to the induced spin structure (if n is odd), or two copies of Σ(M ) if n is even. We can describe A in terms of the Dirac operator D M on M with respect to the metric h M as follows:

(6) A =      D M for n odd, D M 0 0 -D M for n even.
In both cases, A is symmetric and elliptic. Since D M is essentially selfadjoint, so is A and we use the same symbol for its unique self-adjoint extension.

The case where A has pure-point spectrum. For the sake of clarity we make temporarily the assumption that L 2 (M, Σ, vol h M ) admits an orthonormal basis made of eigenspinors of A of real eigenvalue (equivalently, A has pure-point spectrum). Since A is essentially 1 or 2 copies of D M , this happens if M is compact, but also more generally.

Example 2.2. Suppose that either (M, h M ) is conformal to a closed cusp metric such that the induced Dirac operators on the ends are invertible (see [START_REF] Moroianu | Weyl laws on open manifolds[END_REF]), or that (M, h M ) is compact with isolated conical singularities and the Dirac eigenvalues on the cone section do not belong to (-1 2 , 1 2 ) (see e.g. [START_REF] Chou | The Dirac operator on spaces with conical singularities and positive scalar curvatures[END_REF]). Then the operator D M is essentially self-adjoint with purely discrete spectrum.

Note that since c 0 and A anti-commute, we have Aφ λ = λφ λ =⇒ Ac 0 φ λ = -λc 0 φ λ and so the spectrum of A is symmetric around 0. For each x, decompose φ onto the positive eigenspaces of A as follows:

φ = λ∈Spec A λ>0 a λ (x)φ λ + b λ (x)c 0 φ λ + φ 0 (x)
where φ 0 (x) ∈ ker A for all x > 0, λ > 0 is an eigenvalue of A, and φ λ is an eigenspinor of eigenvalue λ and norm 1 in L 2 (M, Σ, vol h M ). We compute

φ 2 L 2 (X,Σ,f vol h ) = {x>ǫ} |φ| 2 f vol h + ǫ 0 M |φ 0 (x)| 2 vol h M f (x)dx + λ∈Spec A λ>0 ǫ 0 |a λ (x)| 2 + |b λ (x)| 2 f (x)dx. In particular, φ ∈ L 2 (X, Σ, f vol h ) implies that ǫ 0 M |φ 0 (x)| 2 vol h M f (x)dx < ∞ (7) and a λ , b λ ∈ L 2 ((0, ǫ), f dx). ( 8
)
The eigenspinor equation ( 5) becomes

0 = (D h -lf )φ = c 0 (∂ x + A + c 0 lf )φ = c 0 λ∈Spec A λ>0 (a ′ λ + λa λ -lf b λ )φ λ + (b ′ λ -λb + lf a λ )c 0 φ λ + c 0 φ ′ 0 -lf φ 0 . So we get (9)
φ ′ 0 = -lf c 0 φ 0 for the part of φ which for all fixed x lives in ker A, while for λ > 0, [START_REF] Moroianu | Killing Vector Fields with Twistor Derivative[END_REF] a

′ λ = -λa λ + lf b λ b ′ λ = -lf a λ + λb λ .
First, since l is real and c 0 is skew-adjoint, Eq. ( 9) implies that the function |φ 0 (x)| 2 is constant in x. Together with (2) and ( 7), we see that

M |φ 0 (x 0 )| 2 dh M = 0 so φ 0 ≡ 0.
We show now that for all λ > 0, the system (10) does not have nonzero solutions satisfying [START_REF] Hitchin | Harmonic spinors[END_REF], i.e., in L 2 ((0, ǫ), f dx).

Remark 2.3. The Wronskian of ( 10) is constant in x, so by (2), the two fundamental solutions cannot belong simultaneously to L 2 ((0, ǫ), f dx). However, this fact alone does not stop one solution from being in

L 2 ! Fix 0 < λ ∈ Spec A and set a(x) := e λx a λ (x), b(x) := e -λx b λ (x).
Then [START_REF] Moroianu | Killing Vector Fields with Twistor Derivative[END_REF] becomes

(11) a ′ (x) = le 2λx f (x)b(x) b ′ (x) = -le -2λx f (x)a(x).
Note that the system (11) has real coefficients (here we use the hypothesis that l is real) so by splitting into real and imaginary parts, we can assume that a, b are also real. Since e ±2λx is bounded for 0 ≤ x ≤ ǫ < ∞, condition (8) implies that a, b ∈ L 2 ((0, ǫ), f dx). If l = 0 then a, b are constant functions, which by [START_REF] Bär | The Dirac operator on hyperbolic manifolds of finite volume[END_REF] do not belong to L 2 ((0, ǫ), f dx) unless they are 0. So in that case a λ and b λ vanish identically.

Let L 1 denote the space of integrable functions on (0, ǫ) with respect to the Lebesgue measure dx. Then [START_REF] Hitchin | Harmonic spinors[END_REF] implies that f ab ∈ L 1 . So from [START_REF] Moroianu | Weyl laws on open manifolds[END_REF],

(a 2 ) ′ = 2le 2λx f (x)a(x)b(x) ∈ L 1 . Hence lim x→0 a 2 (x) = a 2 (x 0 ) - x 0 0 (a 2 ) ′ (t)dt
exists, in other words a 2 extends continuously in x = 0. The same argument shows that b 2 (and so also a, b) are continuous in 0.

The case l = 0 was treated above so we can assume that l = 0. We claim that b(0) = 0. Otherwise, by continuity, b(x) = 0 for 0 ≤ x ≤ x 0 so from [START_REF] Moroianu | Weyl laws on open manifolds[END_REF], there exists C > 0 with |a ′ (x)| > Cf (x) (in particular, by continuity a ′ has constant sign) and therefore

|a(0) -a(x 0 )| = lim x→0 x 0 x |a ′ (x)|dx > C x 0 0 f (t)dt = ∞
which is a contradiction. So b(0) = 0 and similarly a(0) = 0.

We pull now our final trick. Recall that λ > 0. Consider the function

F (x) := e -4λx a 2 (x) + b 2 (x) ≥ 0.
We just showed that F is continuous in 0 and F (0) = 0. From [START_REF] Moroianu | Weyl laws on open manifolds[END_REF] we compute

F ′ (x) = -4λe -4λx a 2 (x) ≤ 0.
By collecting what we know about F , we note:

(1) F (x) ≥ 0;

(2) F (0) = 0;

(3) F ′ (x) ≤ 0 for ǫ > x > 0 Together these facts imply that F ≡ 0 on (0, ǫ). This is equivalent to saying a(x) = b(x) = 0 for all 0 ≤ x ≤ ǫ.

So we showed that φ vanishes near M . The eigensections of D have the unique continuation property, which implies that φ vanishes on X.

Remark 2.4. Recall that the spectrum of A is symmetric around 0, and a λ , b λ are the coefficients in φ of the eigensections φ λ , c 0 φ λ of eigenvalue λ, respectively -λ. It may seem that starting the decomposition using positive λ was a fortunate choice, otherwise the last argument would not hold. But in fact, the argument works for -λ by choosing a different function F (x) := a 2 (x) + e 4λx b 2 (x).

The general case. Let us remove the assumption that the spectrum of A is purely discrete. We will model the proof on the argument given above, which is now loaded with technical subtleties.

By assumption, a neighbourhood of the infinity in X is isometric to (0, ǫ)× M with the metric (1).

After the unitary transformation (3), the eigenspinor equation reads as before [START_REF] Nistor | On the kernel of the equivariant Dirac operator[END_REF] (

∂ x + A)φ = -c 0 f -1 lφ
where l is real, and φ is smooth (by elliptic regularity) and square-integrable.

To make this last condition precise, let I ǫ denote the interval (0, ǫ) with the measure f (x)dx. Denote by H the Hilbert space L 2 (M, Σ, vol h M ), then

(13) φ ∈ L 2 (X ǫ , Σ, f vol h ) = L 2 (I ǫ , H).
In particular for almost all x ∈ I ǫ , we have φ x ∈ H.

Let χ be the characteristic function of the interval [-N, N ] for some N ∈ R. Let χ(A) be the corresponding spectral projection. Since A anticommutes with c 0 and χ is even, it follows that χ(A) commutes with c 0 .

Let H 1 ⊂ H be the domain of A and H -1 ⊃ H its dual inside distributions, i.e., the space of those distributions which extend continuously to H 1 . Since χ has compact support, we deduce that χ(A) acts continuously from H to H 1 , and also from H -1 to H.

From (13) we deduce Aφ ∈ L 2 (I ǫ , H -1 ), χ(A)φ ∈ L 2 (I ǫ , H 1 ) and

χ(A)(Aφ) = A(χ(A)φ) ∈ L 2 (I ǫ , H).
Similarly,

∂ x φ ∈ H -1 loc (I ǫ , H) and χ(A)(∂ x φ) = ∂ x (χ(A)φ) ∈ H -1 loc (I ǫ , H 1
). It follows that φ := χ(A)φ satisfies (in distributions) the eigenspinor equation [START_REF] Nistor | On the kernel of the equivariant Dirac operator[END_REF]. Denote by H N the range of the projection χ(A), then φ ∈ L 2 (I ǫ , H N ). Most importantly for us, A acts as a self-adjoint bounded operator on H N . Lemma 2.5. Let H be a Hilbert space, A : H → H a bounded self-adjoint operator, and c 0 a skew-adjoint involution of H which anti-commutes with A. Then for every l ∈ R, the equation (12) does not have (distributional) solutions in L 2 (I ǫ , H) other than 0.

Proof. Let φ be a solution of [START_REF] Nistor | On the kernel of the equivariant Dirac operator[END_REF], square-integrable with respect to the measure f (x)dx on I ǫ . By elliptic regularity, φ is smooth in x. Since exp(xA)c 0 = c 0 exp(-xA), we get ∂ x (exp(xA)φ) = -lf c 0 exp(-xA)φ, hence the family of H-norms x → ∂ x (exp(xA)φ x ) is square-integrable with respect to the measure f -1 dx. Since x → φ x is L 2 with respect to f dx, and exp(xA) is uniformly bounded, by the Cauchy-Schwartz inequality we see that the function

x → d dx exp(xA)φ x 2
is integrable with respect to the Lebesgue measure dx. Thus

exp(xA)φ x 2 = exp(x 0 A)φ x 0 2 + x x 0 d dx exp(xA)φ x 2 dx
has a finite limit as x ց 0. We claim that this limit is 0. Otherwise, since lim xց0 exp(xA) = 1, we would have lim xց0 φ x 2 > 0 which, together with (2), contradicts the fact that φ is square-integrable with respect to f dx.

Thus φ x tends in norm to 0 in H as x ց 0. Let now |A| be the absolute value of A, and define

F (x) := exp(-x|A|)φ x ) 2 .
We notice that c 0 commutes with |A| since it commutes with A 2 . A direct computation shows, using that c 0 is skew-adjoint,

dF dx = -2 (A + |A|) exp(-x|A|)φ x , exp(-x|A|)φ x ≤ 0.
Hence F is decreasing, on the other hand it vanishes at x = 0 and it is non-negative, so in conclusion it vanishes identically. Since exp(-x|A|) is invertible, we conclude that φ ≡ 0.

We apply this lemma to the eigenspinor φ constructed above with H = H N . Therefore χ(A)φ = 0 for all N ∈ R. But as N → ∞ we have χ(A)φ → φ. By the uniqueness of the limit, φ must be identically zero on (0, ǫ) × M which is an open subset of X. By the unique continuation property, it follows that φ vanishes on X as claimed. This ends the proof of Theorem 2.1.

Remark 2.6. Theorem 2.1 holds for Dirac operators twisted with a Hermitian bundle E whose connection ∇ E is flat in the direction of the conformal gradient vector field ξ = ∂ x , i.e., such that the contraction of the curvature of ∇ E with the field ξ vanishes. We only need to replace in equation ( 6) the operator D M by the twisted operator D E M . The flatness condition ensures that this operator is independent of x. The rest of the proof remains unchanged.

A formal extension of Theorem 2.1

For applications, it might be useful to view the metric g given by (15) in different coordinates. We state below the most general reformulation of Theorem 2.1. Theorem 3.1. Let X be a smooth manifold with boundary, let M be a boundary component, and let [0, ǫ) × M ֒→ X be a collar neighbourhood of M . Take a smooth L 1 function ρ : (0, ǫ) → (0, ∞) and let h be a possibly incomplete Riemannian metric on X which is a warped product near M :

h = dx 2 + ρ -2 (x)h M .
Let f : X → (0, ∞) be a smooth conformal factor depending only on x near M and satisfying ǫ 0 f (x)dx = ∞. Assume that the Dirac operator D M on (M, h M ) with the induced spin structure is essentially self-adjoint. Then the Dirac operator D g of the metric g := f 2 h does not carry square-integrable eigenspinors of real eigenvalue.

Proof. The metric h is conformal to (ρ(x)dx)

2 + h M . Set t(x) := x 0 ρ(s)ds.
Since ρ is in L 1 , it follows that t is well-defined and t(0) = 0. Since ρ is positive, x → t(x) is an increasing diffeomorphism from (0, ǫ) to (0, t(ǫ)) which extends to a homeomorphism between [0, ǫ) and [0, t(ǫ)). We write x = x(t) for its inverse. Define f (t) := f (x(t)) ρ(x(t)) . Clearly, dt = ρdx so

g = f 2 (t) dt 2 + h M . Note that t(ǫ) 0 f (t)dt = ǫ 0 f (x)dx = ∞
so we can apply Theorem 2.1.

Gradient conformal vector fields

Let (X n , g) be a Riemannian manifold. A gradient conformal vector field (or GCVF) on X is a conformal vector field ξ which is at the same time the gradient of a function on X:

(14) L ξ g = αg, for some α ∈ C ∞ (X), ξ = ∇ g F,
for some F ∈ C ∞ (X).

Gradient conformal vector fields were studied intensively in the 70s (see [START_REF] Bourguignon | Transformations infinitésimales conformes fermées des variétés riemanniennes connexes complètes[END_REF] and references therein). More recently, they turned out to be a very useful tool in understanding other geometric objects, like closed twistor 2-forms on compact Riemannian manifolds [START_REF] Moroianu | Killing Vector Fields with Twistor Derivative[END_REF]. The aim of this section is to prove the following:

Theorem 4.1. Let (X, g) be a complete spin manifold. Assume that X carries a non-complete vector field which outside some compact subset is nowhere-vanishing and GCVF. Then the Dirac operator of (X, g) does not carry square-integrable eigenspinors.

This theorem is a direct consequence of Theorem 2.1 and Proposition 4.3 below.

We first recall some basic properties of GCVFs.

Lemma 4.2. Let ξ be a GCVF satisfying the system (14). Then the following assertions hold: (i) The covariant derivative of ξ depends on only the co-differential of ξ:

(15) ∇ Y ξ = φY, ∀ Y ∈ T X,
where the function φ equals -1 n δξ. (ii) Let X 0 be the set of points where ξ does not vanish. The distribution ξ ⊥ defined on X 0 is involutive and its maximal integral leaves are exactly the connected components of the level sets of F on X 0 .

(iii) The length of ξ is constant along the integral leaves of ξ ⊥ .

(iv) The integral curves of ξ are geodesics and F is strictly increasing along them.

(v) Each point p of X 0 has a neighbourhood isometric to

((-ε, ε) × V, f 2 (x)(dx 2 + h)),
where (V, h) is a local integral leaf of ξ ⊥ through p and f : (-ε, ε) → R + is some positive function. In these coordinates ξ corresponds to ∂/∂x and f (x) is the norm of ξ on the leaf {x} × V .

Proof. (i) The first equation in ( 14) is equivalent to the vanishing of the trace-free symmetric part of ∇ξ. The second equation of ( 14) implies that the skew-symmetric part of ∇ξ vanishes too. We are left with ∇ X ξ = φX for some function φ. Taking the scalar product with X and the sum over an orthonormal basis X = e i yields φ = -1 n δξ. (ii) By definition, the distribution ξ ⊥ on X 0 is exactly the kernel of the 1form dF , so it is involutive. Let M be a maximal integral leaf of ξ ⊥ . Clearly F is constant on M , which is connected, so M is a subset of some level set

F -1 (y). Moreover M is open in F -1 (y), as can be seen in local charts. Thus M is a connected component of F -1 (y). (iii) For every Y ∈ T M one can write Y (|ξ| 2 ) = 2g(∇ Y ξ, ξ) (15) = φg(Y, ξ) = 0, so |ξ| 2 is constant on M .
(iv) Let ϕ t denote the local flow of ξ and let γ t := ϕ t (p) for some p ∈ M . Taking X = ξ in (15) yields

∇ γ γ = ∇ ξ ξ = φξ = φ γ, which shows that γ t is a (non-parametrized) geodesic. Furthermore, d dt F (γ t ) = γt (F ) = ξ(F ) = |ξ 2 | > 0, so F (γ t ) is increasing. (v)
The tangent bundle of X 0 has two involutive orthogonal distributions Rξ and ξ ⊥ . The Frobenius integrability theorem shows that there exists a local coordinate system (x, y 1 , . . . , y n-1 ) around every p ∈ X 0 such that ξ = ∂ x and ∂ y i span ξ ⊥ . Let V denote the set {x = 0} in these coordinates. The metric tensor can be written

g = f 2 dx 2 + n-1 i,j=1 g ij dy i ⊗ dy j .
From (iii) we see that f only depends on x. Using the first equation in the system (14) we get

2g ij (log f ) ′ (x) = ∂g ij ∂x , ∀ i, j ≤ n -1,
which shows that g ij (x, y) = f 2 (x)h ij (y) for some metric tensor h on V .

It turns out that under some completeness assumptions, the last statement of the lemma also holds globally: Proposition 4.3. Let (X n , g) be a complete Riemannian manifold. If ξ is a non-complete vector field on X which, outside a compact subset of X, is gradient conformal and non-vanishing, then there exists an open subset of X which is isometric to ((0, c) × M, f 2 (x)(dx 2 + h)) for some complete Riemannian manifold (M n-1 , h) and smooth positive function f : (0, c) → R + with c 0 f (x)dx = ∞. Proof. Let ϕ t denote the local flow of ξ and let K be a compact subset of X such that ξ is gradient conformal and nowhere-vanishing on X \ K. By definition, ξ = ∇F for some function F defined on X \ K. Consider the open set

K ε := {p ∈ X | d(p, K) < ε}.
Since K ε is compact, there exists some δ > 0 such that ϕ t is defined on K ε for every |t| < δ. Since ξ is non-complete, there exists some p ∈ X and a ∈ R such that ϕ t (p) tends to infinity as t tends to a. By changing ξ to -ξ if necessary, we can assume that a > 0. From the definition of δ we see that

ϕ t (p) ∈ X \ K ε for all t ∈ [a -δ, a). Since lim t→a t 0 |ξ ϕs(p) |ds ≥ lim t→a d(p, ϕ t (p)) = ∞,
the norm of ξ has to be unbounded along its integral curve through p in the positive direction. Therefore one can find a point q := ϕ t 0 (p) (t 0 ∈ [0, a)) on this integral curve such that |ξ q | is larger than the supremum of the norm of ξ over K ε . Let M be the maximal leaf through q of the involutive distribution ξ ⊥ (defined on X \ K). Since the norm of ξ is constant on M , it is clear that M does not intersect K ε . We notice that M is complete with respect to the induced Riemannian metric h. This does not follow directly from the completeness of (X, g) since the distribution ξ ⊥ is only defined and involutive on X \ K. Nevertheless, since M is a connected component of some level set of F , it is closed in X, and every closed submanifold of a complete Riemannian manifold is also complete with respect to the induced Riemannian metric.

From the definition of q, it is clear that the integral curve ϕ t (q) is defined for t < a -t 0 . From Lemma 4.2, two integral curves of ξ which do not meet K, which are issued from points of the same maximal leaf, are geodesics and have the same length. Consequently, for every other point q ′ ∈ M , the integral curve of ξ in the positive direction is defined at least for all t < a -t 0 . Moreover, the map

ψ : M × (0, a -t 0 ) → X, ψ(r, t) := ϕ t (r)
is one-to-one since the vector field ξ does not have zeros on M . Finally, Lemma 4.2 (v) shows that ψ is an isometric embedding of (M × (0, a -t 0 ), f 2 (x)(dx 2 + h)) into (M, g), where f (x) denotes the length of ξ on the maximal leaf ϕ x (M ) of ξ ⊥ .

Remark 4.4. The incompleteness condition on ξ in Theorem 4.1 is necessary. Indeed, complete hyperbolic manifolds of finite volume are isometric outside a compact set to a disjoint union of cusps, i.e. cylinders (0, ∞) × T over some flat connected Riemannian manifold (T, h), with metric dt 2 + e -2t h = e -2t ((de t ) 2 + h).

The vector field e -t ∂/∂t is GCVF and complete. These manifolds are known to have purely discrete spectrum if the spin structure on each cusp is nontrivial [START_REF] Bär | The Dirac operator on hyperbolic manifolds of finite volume[END_REF], which is the case for instance in dimension 2 or 3 when there is only one cusp. The eigenvalues then obey the Weyl asymptotic law [START_REF] Moroianu | Weyl laws on open manifolds[END_REF]. On the contrary, when some cusps have non-trivial spin structures, the spectrum of D is the real line. In this case (like for the scalar Laplace operator) the existence of L 2 eigenspinors is generally unknown.

Applications

Real hyperbolic space. The Poincaré disk model of the hyperbolic space is conformally equivalent to the standard flat metric. In polar coordinates, this metric is a warped product so Theorem 3.1 shows that the Dirac operator on the hyperbolic space does not have point spectrum (the spectrum is real since H is complete). This was first studied with different methods by Bunke [START_REF] Bunke | The spectrum of the Dirac operator on the hyperbolic space[END_REF].

Hyperbolic manifolds. More generally, let (M n , h M ) be a spin hyperbolic manifold whose Dirac operator is essentially self-adjoint. Taking A = 0 in Theorem 7.2 of [START_REF] Bär | Generalized Cylinders in Semi-Riemannian and Spin Geometry[END_REF] shows that the Riemannian manifold (X a , g) := ((a, ∞) × M, dt 2 + cosh(t) 2 h M ) is a spin hyperbolic manifold of dimension n + 1 for every a ∈ R ∪ {-∞}. Setting x := e -t near t = ∞, the metric g becomes

g = x -2 dx 2 + (1 + x 2 ) 2 4 h M .
Theorem 3.1 thus shows that the Dirac operator on X a (or on any spin Riemannian manifold containing X a as an open set) does not have L 2 eigenspinors of real eigenvalue. This result was previously known when M is compact. Interesting non-compact cases are obtained when M is complete, or when M is compact with conical singularities with small angles [START_REF] Chou | The Dirac operator on spaces with conical singularities and positive scalar curvatures[END_REF].

Rotationally symmetric Riemannian manifolds. It is proved in [START_REF] Anghel | The L 2 -harmonic forms on rotationally symmetric Riemannian manifolds revisited[END_REF] that on R n with a metric which written in polar coordinates has the form ds 2 = dr 2 + ψ(r) 2 dθ 2 , there are no L 2 harmonic spinors. This metric is complete. By the change of variables

x(r) := ∞ r ds ψ(s) ,

Anghel's metric becomes a particular case of (1) with ψ(r(x)) in the rôle of f (x) from Theorem 2.1, provided that ∞ 1 dr/ψ(r) < ∞. The absence of harmonic spinors is guaranteed in this case by [START_REF] Lott | The Dirac operator and conformal compactification[END_REF] if the resulting conformal factor is Lipschitz. By Theorem 2.1 we know, even without the Lipschitz hypothesis, not only that there cannot exist L 2 harmonic spinors, but also that there are no L 2 eigenspinors at all. Both these spaces vanish by Theorem 2.1, so in particular it follows that index(D + ) = 0.

The L 2 -

 2 index of the Dirac operator. Let D + denote the chiral component of D, viewed as an unbounded operator in L 2 , acting on compactly supported smooth spinors on a spin Riemannian manifold as in Theorem 2.1. Denote by D + its closure. The L 2 -index is defined as index(D + ) := dim ker(D + ) -dim ker(D + ) * where (D + ) * is the adjoint of D + (the definition makes sense whenever both kernels are finite-dimensional, even when D + is not Fredholm). Here ker(D + ) * is precisely the distributional null-space of D -inside L 2 , while ker(D + ) is a subspace of the distributional null-space of D + inside L 2 .


Gaëlle Brévalle

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