Guanglei Wu

Stéphane Caro
email: stephane.caro@irccyn.ec-nantes.fr

Jiawei Wang

Design and Transmission Analysis of an Asymmetrical Spherical Parallel Manipulator

Keywords: Asymmetrical spherical parallel manipulator, transmission wrench screw, transmissibility, universal joint

This paper presents an asymmetrical spherical parallel manipulator and its transmissibility analysis. This manipulator contains a center shaft to both generate a decoupled unlimited-torsion motion and support the mobile platform for high positioning accuracy. This work addresses the transmission analysis and optimal design of the proposed manipulator based on its kinematic analysis. The input and output transmission indices of the manipulator are defined for its optimum design based on the virtual coefficient between the transmission wrenches and twist screws. The sets of optimal parameters are identified and the distribution of the transmission index is visualized. Moreover, a comparative study regarding to the performances with the symmetrical spherical parallel manipulators is conducted and the comparison shows the advantages of the proposed manipulator with respect to its spherical parallel manipulator counterparts.

Introduction

Three degree-of-freedom (3-DOF) spherical parallel manipulators (SPMs) are most widely used as camera-orientating device [START_REF] Gosselin | The Agile Eye: a high-performance three-degree-of-freedom camera-orienting device[END_REF], minimally invasive surgical robots [START_REF] Li | Design of spherical parallel mechanisms for application to laparoscopic surgery[END_REF] and wrist joints [START_REF] Asada | Kinematic and static characterization of wrist joints and their optimal design[END_REF] because of their large orientation workspace and high payload capacity. Since they can generate three pure rotations, another potential application is that they can work as a tool head for complicated surface machining. However, the general SPM can only produce a limited torsion motion under a certain tilt orientation, whereas an unlimited torsion is necessary in some common material processing such as milling or drilling. The co-axial input SPM reported in [START_REF] Asada | Kinematic and static characterization of wrist joints and their optimal design[END_REF] can achieve unlimited torsion, however, its unique structure leads to a complex input mechanism. Moreover, the general SPMs result in low positioning accuracy [START_REF] Wu | Mobile platform center shift in spherical parallel manipulators with flexible limbs[END_REF] without a ball-and-socket joint as the center of rotation. In this paper, an asymmetrical SPM (AsySPM) is proposed, which can generate unlimited torsion motion with enhanced positioning accuracy. This manipulator adopts a universal joint as the center of rotation supported by an input shaft at the center, which simplifies the manipulator architecture.

The design of 3-DOF SPMs can be based on many criteria, i.e., workspace [START_REF] Gosselin | The optimum kinematic design of a spherical three-degree-offreedom parallel manipulator[END_REF][START_REF] Bai | Optimum design of spherical parallel manipulator for a prescribed workspace[END_REF], dexterity [START_REF] Gosselin | A global performance index for the kinematic optimization of robotic manipulators[END_REF][START_REF] Bai | Modelling of a special class of spherical parallel manipulators with Euler parameters[END_REF][START_REF] Wu | Multiobjective optimum design of a 3-RRR spherical parallel manipulator with kinematic and dynamic dexterities[END_REF], singularity avoidance [START_REF] Bonev | Singularity loci of spherical parallel mechanisms[END_REF], stiffness [START_REF] Wu | Mobile platform center shift in spherical parallel manipulators with flexible limbs[END_REF][START_REF] Bidault | Structural optimization of a spherical parallel manipulator using a two-level approach[END_REF], dynamics [START_REF] Staicu | Recursive modelling in dynamics of Agile Wrist spherical parallel robot[END_REF][START_REF] Wu | Dynamic modeling and design optimization of a 3-DOF spherical parallel manipulator[END_REF], and so on. The prime function of mechanisms is to transmit motion/force between the input joint and the output joint. Henceforth, we will focus on the transmissibility analysis of the proposed SPM. In the design procedure, the performance index is of importance for performance evaluation of the manipulator. A number of transmission indices, such as the transmission angle, the pressure angle, and the transmission factor, have been proposed in the literature to evaluate the quality of motion/force transmission. The transmission angle was introduced by Alt [START_REF] Alt | Der Üertragungswinkel und seine bedeutung für das konstruieren periodischer getriebe[END_REF], developed by Hain [START_REF] Hain | Applied Kinematics[END_REF], and can be applied in linkage synthesis problems [START_REF] Dresner | Definition of pressure and transmission angles applicable to multi-input mechanisms[END_REF][START_REF] Bawab | Rectified synthesis of six-bar mechanisms with welldefined transmission angles for four-position motion generation[END_REF]. Takeda et al. [START_REF] Takeda | A transmission index for in-parallel wire-driven mechanisms[END_REF] proposed a transmission index (TI) for parallel mechanisms based on the minimum value of the cosine of the pressure angle between the leg and the moving platform, where all the inputs but one are fixed. Based on the virtual coefficient between the transmission wrench screw (TWS) and the output twist screw (OTS) introduced by Ball [19], Yuan et al. [START_REF] Yuan | Kinematic analysis of spatial mechanism by means of screw coordinates. part 2-analysis of spatial mechanisms[END_REF] used it as an unbounded transmission factor for spatial mechanisms. Sutherland and Roth [START_REF] Sutherland | A transmission index for spatial mechanisms[END_REF] defined the transmission index using a normalized form of the transmission factor, which depends only on the linkages' geometric properties. Chen and Angeles [START_REF] Chen | Generalized transmission index and transmission quality for spatial linkages[END_REF] proposed a generalized transmission index that is applicable to single-loop spatial linkages with fixed output and single or multiple DOFs. Wu et al. [START_REF] Wu | Optimal design of spherical 5R parallel manipulators considering the motion/force transmissibility[END_REF] introduced a frame-free index related to the motion/force transmission analysis for the optimum design of the spherical five-bar mechanism. Wang et al. [START_REF] Wang | Performance evaluation of parallel manipulators: Motion/force transmissibility and its index[END_REF] presented the transmission analysis of fully parallel manipulators based on the transmission indices defined by Sutherland, Roth [START_REF] Sutherland | A transmission index for spatial mechanisms[END_REF] and Takeda [START_REF] Takeda | A transmission index for in-parallel wire-driven mechanisms[END_REF]. Recently, some approaches to identify singularity and closeness to singularity have been reported via transmission analysis [START_REF] Liu | A new approach for singularity analysis and closeness measurement to singularities of parallel manipulators[END_REF][START_REF] Liu | A generalized approach for computing the transmission index of parallel mechanisms[END_REF]. Henceforth, the virtual coefficient based indices will be adopted in this paper for the evaluation of the transmission quality and the optimal design of the proposed manipulator. This paper presents an asymmetrical SPM and its optimum design with regard to its transmission quality. The inverse and forward kinematic problems of the AsySPM are analyzed based on the kinematic analysis of classical SPMs. By virtue of the virtual coefficient between the transmission wrench screw and output twist screw, the input and output transmission indices are defined for the optimum design, of which an optimization problem is formulated to achieve the optimal design of the proposed SPM. The performances of the proposed spherical manipulator are compared with those of its counterparts in order to highlight its advantages and drawbacks. and inner rings connected to each other with a revolute joint, the revolute joint being realized with a revolve bearing. The orientation of the outer ring is determined by two RRR1 legs and constrained in a vertical plane by a fully passive RRS leg or an alternative RPS one. Through a U-joint, the decoupled rotation of the inner ring is driven by the center shaft, which also supports the MP to improve the positioning accuracy. This manipulator can provide an unlimited rotation of the moving-platform, which can be used in milling or drilling operations and among other material processing. It can also be used as the active spherical joint, i.e., wrist or waist joint.

The coordinate system (x, y, z) is denoted in Fig. 1(b), of which the origin is located at the center of rotation, namely, point O. The ith active leg consists of three revolute joints, whose axes are parallel to unit vectors u i , v i , w i . Both of these two legs have the same architecture, defined by α 1 and α 2

angles. The design parameters of the base platform are γ and η. The design parameter of the mobile platform is β. It is noteworthy that the manipulator is symmetrical with respect to the yz plane.

Inverse Geometric Problem

Under the prescribed coordinate system, the unit vector u i is derived as:

u i = (-1) i+1 sin η sin γ cos η sin γ -cos γ T , i = 1, 2 (1)
The unit vector v i of the axis of the intermediate revolute joint in the ith leg is obtained in terms of the input joint angle θ i following the angle-axis representation [START_REF] Angeles | Fundamentals of Robotic Mechanical Systems: Theory, Methods, and Algorithms[END_REF], namely,

v i = R(u i , θ i )v * i ; R(u i , θ i ) = cos θ i I 3 + sin θ i [u i ] × + (1 -cos θ i )u i ⊗ u i ( 2
)
where I 3 is the identity matrix, [u i ] × is the cross product matrix of u i and ⊗ is the tensor product.

Moreover, v * i is the unit vector of the axis of the intermediate revolute joint in the ith leg at the original configuration:

v * i = (-1) i+1 sin η sin(γ + α 1 ) cos η sin(γ + α 1 ) -cos(γ + α 1 ) T (3)
The unit vector w i of the top revolute joint in the ith leg, is a function of the MP orientation:

w i = x i y i z i T = Qw * i ; w * i = (-1) i+1 sin β cos β 0 T (4)
where w * i is the unit vector of the axis of the top revolute joint of the ith leg when the mobile platform is located in its home configuration. Moreover, Q = R(x, φ x )R(y, φ y ) is the rotation matrix of the outer ring. Hence, the orientation of the inner ring can be described with Cardan angles (φ x , φ y , φ z ) and its output axis is denoted by:

p = Qz; z = [0, 0, 1] T (5)
According to the motion of the U-joint [START_REF] Weisbach | Mechanics of Engineering and of Machinery[END_REF], the input angle θ 3 of the center shaft is derived as:

θ 3 = tan -1 (tan φ z cos φ x cos φ y ) (6)
Referring to the inverse kinematic problem of the general SPMs, the loop-closure equation for the ith RRR leg is expressed as:

A i t 2 i + 2B i t i + C i = 0, i = 1, 2 (7)
with The input angle displacements can be solved as:

A i = (-1) i+1 x i sη + y i cη s(γ -α 1 ) -z i c(γ -α 1 ) -cα 2 (8a)
B i = (x i cη -y i sη)sα 1 (8b)
C i = (-1) i+1 x i sη + y i cη s(γ + α 1 ) -z i c(γ + α 1 ) -cα 2 (8c)
cos θ i = 1 -t 2 i 1 + t 2 i , sin θ i = 2t i 1 + t 2 i ; t i = tan θ i 2 = -B i ± B 2 i -A i C i A i (9)
The inverse geometric problem has four solutions corresponding to the four working modes characterized by the sign "-/+" of u i × v i • w i , i.e., "-+", "--", "+-" and "++" modes. Here, the "-+" working mode is selected.

Forward Geometric Problem

The forward geometric problem of the AsySPM can be obtained by searching for the angles ϕ i of a spherical four-bar linkages with the given input angles θ i , i = 1, 2, as displayed in Fig. 2, where the input/output (I/O) equation takes the form [START_REF] Yang | Application of dual-number quaternion algebra to the analysis of spatial mechanisms[END_REF][START_REF] Bai | A unified input-output analysis of four-bar linkages[END_REF]:

f (ϕ 1 , ϕ 2 ) = k 1 + k 2 cos ϕ 1 + k 3 cos ϕ 1 cos ϕ 2 -k 4 cos ϕ 2 + k 5 sin ϕ 1 sin ϕ 2 = 0 (10)
with

k 1 ≡ Cα 0 C 2 α 2 -Cβ ; k 2 = k 4 ≡ Sα 0 Sα 2 Cα 2 ; k 3 ≡ Cα 0 S 2 α 2 ; k 5 ≡ S 2 α 2 (11)
where S and C stand for the sine and cosine functions, respectively, and

α 0 = cos -1 (v 1 • v 2 ), β = 2β.
On the other hand, the motion of the unit vector e is constrained in the yz plane due to the passive leg, thus:

g(ϕ 1 , ϕ 2 ) = x 1 + x 2 = 0 ( 12
)
where the unit vector w i can be also represented with angle-axis rotation matrix, namely, Solving Eqs. ( 10) and ( 12) leads to four solutions for the angles ϕ i , i = 1, 2, i.e., the two functions have four common points in the plane z = 0 as shown in Fig. 3. Figure 4 illustrates the four assembly modes corresponding to the four solutions. Then, substituting ϕ i into Eq. ( 13), the unit vector w i and the output Euler angles φ x and φ y can be obtained, and the output angle φ z can be obtained from Eq. ( 6) accordingly.

w i = x i y i z i T = R(v i , ϕ i )R(v 0 , α 2 )v i ; v 0 = v 1 × v 2 / v 1 × v 2 ( 13
) -2 0 2 -2 0 2 -1 0 1 2 z f( 1 ,  2 ) g( 1 ,  2 ) z=0 f (φ 1 ,φ 2 ) g (φ 1 ,φ 2 ) z=0 φ 1 [rad] φ 2 [rad]

Transmission Index

The main function of the mechanism is to transmit motion from the input element to the output element. As a result, the force applied to the output element is to be transmitted to the input one. The arising internal wrench due to transmission is defined as a transmission wrench, which is characterized by the magnitude of the force and transmission wrench screw (TWS), and the latter is used to evaluate the quality of the transmission. In order to evaluate the transmission performance of the manipulator, some transmission indices (TI) should be defined.

Transmission Wrench and Twist Screw

As shown in Fig. 5(a), the instantaneous motion of a rigid body can be represented by using a twist screw defined by its Plücker coordinates:

$ T = (ω ; v) = ω $T = (L 1 , M 1 , N 1 ; P * 1 , Q * 1 , R * 1 ) ( 14
)
where ω is the amplitude of the twist screw and $T is the unit twist screw. Likewise, a wrench exerted on the rigid body can be expressed as a wrench screw defined by its Plücker coordinates as:

$ W = (f ; m) = f $W = (L 2 , M 2 , N 2 ; P * 2 , Q * 2 , R * 2 ) ( 15
)
where f is the amplitude of the wrench screw and $W is the unit wrench screw.

The reciprocal product between the two screws $ T and $ W is defined as:

$ T • $ W = f • v + m • ω = L 1 P * 2 + M 1 Q * 2 + N 1 R * 2 + L 2 P * 1 + M 2 Q * 1 + N 2 R * 1 ( 16
)
This reciprocal product amounts to the instantaneous power between the wrench and the twist. Subsequently, the transmission index is defined as a dimensionless index [START_REF] Chen | Generalized transmission index and transmission quality for spatial linkages[END_REF]:

TI = | $T • $W | | $T • $W | max (17)
where | $T • $W | max represents the potential maximal magnitude of the reciprocal product between $T and $W . The larger TI, the more important the power transmission from the wrench to the twist, namely, the better the transmission quality.

For a planar manipulator, this index corresponds to the transmission angle, which is the smallest angle between the direction of velocity of the driven link and the direction of absolute velocity vector of output link both taken at the common point [START_REF] Hartenberg | Kinematic Synthesis of Linkages[END_REF]. As illustrated in Fig. 5(b), it is the angle σ between the follower link and the coupler of a four-bar mechanism, also known as forward transmission angle.

Conversely, the angle ψ is the inverse transmission angle. Therefore, the input (λ I ) and output (λ O ) transmission can be expressed as:

λ I = | sin ψ| ; λ O = | sin σ| (18)

Input Transmission Index

The wrench applied to a SPM is usually a pure moment, thus, for a spherical RRR leg, the transmission wrench is a pure torque. As the TWS is reciprocal to all the passive joint screws in the leg, the axis of the wrench in the ith leg is perpendicular to the plane OB i C i and passes through point O, as shown in Fig. 6(a). According to Eq. ( 17), the input transmission index of the ith RRR leg is obtained as:

λ Ii = | $Ii • $W i | | $Ii • $W i | max = |u i • τ i | |u i • τ i | max , i = 1, 2 (19)
with $Ii = (u i ; 0);

$W i = (0; τ i ) = (0; v i × w i / v i × w i ) (20)
When $W i lies in the plane OA i B i , i.e., plane OA i B i being perpendicular to plane

OB i C i , |u i • τ i |
reaches its maximum value. This situation occurs when the angle between the wrench screw and the twist screw is equal to ψ 0 or 180 o -ψ 0 , namely,

|u i • τ i | max = | cos ψ 0 | = | sin α 1 | (21)
From Fig. 6(a), Eq. ( 19) is equivalent to

λ Ii = | cos ψ 1 | | cos ψ 0 | = | cos ψ 2 | = | sin ψ i | = 1 -cos 2 ψ i , i = 1, 2 (22)
where ψ i is the inverse transmission angle, i.e., the angle between planes OA i B i and OB i C i , and

cos ψ i = (v i × u i ) • (v i × w i ) v i × u i v i × w i (23)
Finally, the input transmission index of the manipulator is defined as:

λ I = min{λ Ii }, i = 1, 2 (24)

Output Transmission Index

Referring to the pressure angle at the attached point of the leg with the moving platform [START_REF] Takeda | A transmission index for in-parallel wire-driven mechanisms[END_REF], the output transmission index of a single leg can be defined by fixing the other input joints, where the parallel manipulator thus becoming a 1-DOF system. By fixing the active joint at point A 2 (point B 2 will be fixed) and keeping joint at point A 1 actuated in Fig. 6(b), the transmission wrench $ W 2 becomes a constraint wrench for the mobile platform. The instantaneous motion of the mobile platform will be a rotation about a unique vector constrained by $ W 2 and the vector e in the passive leg, namely,

s 1 = τ 2 × e τ 2 × e ; e = Qj, j = [0, 1, 0] T (25)
Henceforth, the output twist screw can be expressed as: $O1 = (s 1 ; 0). Based on Eq. ( 17), the output transmission index of the first leg is defined by

λ O1 = | $O1 • $W 1 | | $O1 • $W 1 | max = |s 1 • τ 1 | |s 1 • τ 1 | max (26)
when s 1 and τ 1 are parallel, i.e., both planes OC 1 C 2 and OB 1 C 1 being perpendicular to OB 2 C 2 , 26) is rewritten as:

|s 1 • τ 1 | is a maximum, namely, |s 1 • τ 1 | max = cos(0) = 1. Equation (
λ O1 = |s 1 • τ 1 | = |τ 12 • e|/ τ 2 × e ; τ 12 = τ 1 × τ 2 (27)
By the same token, the output transmission index of the second leg is derived as:

λ O2 = |τ 12 • e|/ τ 1 × e (28)
Similarly to Eq. ( 24), the output transmission index of the manipulator is defined as

λ O = min{λ Oi }, i = 1, 2 (29)

Transmission Efficiency of the U joint

The output of the inner ring of the mobile platform is driven by the center shaft through a universal joint, consequently, the TI of the U joint (UTI) is defined as:

λ U = |p • z| = | cos φ x cos φ y | (30)
where the vectors p and z are defined in Eq. ( 5).

Local Transmission Index

On the basis of the ITI, OTI and UTI, the local transmission index (LTI) of the manipulator under study, namely, the transmission index at a prescribed orientation, is defined as:

λ = min{λ I , λ O , λ U } (31)
The higher λ, the higher the quality of the input and output transmission. The distribution of the LTI can indicate the workspace (WS) region with a good transmissibility. Thus, this index can be used for either the evaluation of the transmission quality or the design optimization.

Optimal Design and Analysis

The optimum design of SPMs can be based on many aspects, such as workspace, dexterity, singularity, and so on. However, these criteria are usually antagonistic. In order for the proposed AsySPM to achieve a regular workspace (RWS) with a good transmission quality, the following optimization problem is formulated:

maximize f (x) = λ for θ ∈ [0, θ 0 ] ( 32
)
over x = [α 1 ; α 2 ; β; γ; η] subject to g 1 : 45 o ≤ {α 1 , α 2 } ≤ 120 o g 2 : 15 o ≤ {β, η} ≤ 60 o g 3 : 30 o ≤ γ ≤ 75 o
where θ is the tilt angle and θ 0 defines the workspace region, as shown in Fig. 7. Moreover, the lower and upper bounds of the design variables are assigned in order to avoid mechanical collisions. This problem can be solved with the optimization toolbox of the mathematical software at hand. Hereby, it is solved with the genetic algorithm (GA) toolbox in Matlab. When θ 0 = 60 o , it is found that quality transmission [START_REF] Tao | Applied Linkage Synthesis[END_REF], whence the TI is λ = sin 45 o ≈ 0.7, i.e., the manipulator at a configuration with LTI λ ≥ 0.7 has good motion/force transmission. Henceforth, a set of poses in which LTI is greater than 0.7 is identified as high-transmissibility workspace (HTW), such as the blue dashed line enveloped region displayed in Fig. 8(a). The area of HTW can be used to evaluate the manipulator performance. The larger the HTW, the better the transmission quality of the manipulator.

When the objective function in the optimization problem [START_REF] Tao | Applied Linkage Synthesis[END_REF] is replaced by f (x) = A HTW , where

A HTW is the area of HTW, the optimal parameters for θ 0 = 60 o are found as:

x = [53. geometrical parameters given in Eq. ( 34) is much larger than the HTW of the manipulator with the geometrical parameters given in Eq. (33).

To evaluate the transmissibility of the manipulator within a designated workspace, a transmission index (WTI) similar to GCI [START_REF] Gosselin | A global performance index for the kinematic optimization of robotic manipulators[END_REF] is defined over the workspace W , which is calculated through a discrete approach in practice, namely,

WTI = λdW dW or WTI = 1 W n i=1 λ i ∆W = 1 n n i=1 λ i ( 35
)
where n is the discrete number. The index obtained through the above equation is an arithmetic mean, which can be replaced with a quadratic mean for a better indication of the transmission, subsequently, WTI is redefined as

WTI = 1 n n i=1 λ 2 i ( 36
)
As a consequence, with θ ∈ [0, 60 o ], WTI is equal to 0.66 for the first design and is equal to 0.72 for the second one.

Comparison with Symmetrical SPMs

In this section, a comparative study is conducted between the asymmetrical and symmetrical SPMs.

A general SPM is shown in Fig. 9(a), which consists of three identical RRR legs connected to the base and the mobile platform. Moreover, β and γ define the geometry of two triangular pyramids on the mobile and the base platforms, respectively. A base coordinate system (x, y, z) is located at point O and the z axis is normal to the bottom surface of the base pyramid and points upwards, while the y axis is located in the plane spanned by the z-axis and vector u 1 . Figure 9(b) illustrates the Agile Wrist [START_REF] Bidault | Structural optimization of a spherical parallel manipulator using a two-level approach[END_REF] while Fig. 9(c) shows a co-axial input SPM (CoSPM) of a special case with γ = 0. Their geometrical parameters are given by Table 1.

LTI distributions

Referring to Eqs. ( 19) and ( 26), the ITI and OTI of each leg for the symmetrical SPMs can be obtained.

The difference lies in the output twist screw of Eq. ( 25) in the calculation of OTI, namely,

s i = (v j × w j ) × (v k × w k ) (v j × w j ) × (v k × w k ) ; i, j, k ∈ {1, 2, 3}, i = j = k (37)
Using the index defined in Eq. ( 31 the HTW is extremely small. When α 1 reduces to 47 o which yields better kinematic and dynamic dexterities [START_REF] Wu | Dynamic modeling and design optimization of a 3-DOF spherical parallel manipulator[END_REF], the HTW of full torsion is a spherical cap with θ ∈ [0, 30 o ].

Comparison of Overall Performances

The performance comparison of the asymmetrical SPM with the Agile Wrist and the Co-axial input spherical parallel manipulator is summarized in Table 3, which shows the advantages and drawbacks of the proposed AsySPM with respect to its symmetrical counterparts. The AsySPM can have the advantages of the general and co-axial input SPMs simultaneously except the drawback of evenly distributed power consumption.

Conclusion

This paper introduced an asymmetrical spherical parallel manipulator, whose the mobile platform is composed of an inner ring and an outer ring. The orientation of the outer ring is determined by two RRR legs as well as a fully passive leg, and the inner ring can generate a decoupled unlimited-torsion motion thanks to a center input shaft and a universal joint. Moreover, the center shaft can improve the positioning accuracy of the center of rotation for the manipulator. This manipulator can be used as a tool head for the complicated surface machining, such as milling or drilling, and can also work as As one of the most important performance, transmission analysis for the proposed manipulator was addressed. By virtue of the transmission wrench screw and the output twist screw, the input and output transmission indices are defined and are further used for the optimal design of the proposed manipulator. Two sets of optimal parameters have been identified and the isocontours of the transmission indices were traced to show the quality of the transmission. Moreover, a comparative study dealing with the mechanical design, kinematic and transmission performances was carried out between the proposed manipulator and its symmetrical counterparts, which highlights the advantages and drawbacks of the proposed manipulator with respect to its symmetrical counterparts. Besides the advantages of the general spherical parallel manipulators, such as compactness, low inertia, large regular workspace, the proposed manipulator can outperform in terms of unlimited torsion rotation, positioning accuracy and transmission quality, on the other hand, a main drawback lies in the unevenly distributed power consumption due to its asymmetrical structure.

Figure 1 :Figure 1

 11 Figure 1: The asymmetrical spherical parallel manipulator: (a) CAD model; (b) coordinate system.

Figure 2 :

 2 Figure 2: The spherical four-bar linkage.

Figure 3 :

 3 Figure 3: Graphical representation (four black points) of the four solutions to the forward geometric problem of the AsySPM.

Figure 4 :

 4 Figure 4: The four assembly modes of the AsySPM.

Figure 5 :

 5 Figure 5: Transmission wrench and twist screw: (a) the twist screw and wrench screw of a rigid body; (b) a planar four-bar linkage.

Figure 6 :

 6 Figure 6: The transmission wrench screw and transmission angle: (a) input transmission; (b) output tranmission.

Figure 7 :

 7 Figure 7: The spherical surface of a designated regular workspace.

  3 o ; 69.5 o ; 15 o ; 75 o ; 60 o ] (34) The corresponding distribution of performance index is shown in Fig. 8(b), from which it is seen that the manipulator can still reach a large RWS with θ = 75 o , whereas, the minimum LTI within θ ∈ [0, 60 o ] reduces to 0.3 compared to Fig. 8(a). In contrast, the HTW of the manipulator with the

Figure 8 :

 8 Figure 8: The isocontours of the transmission index of the asymmetrical SPM for workspace θ 0 = 60 o : (a) max(λ); (b) max(HTW).

Figure 9 :

 9 Figure 9: The symmetrical SPMs: (a) general SPM; (b) Agile Wrist; (c) co-axial input SPM.

Figure 10 :

 10 Figure 10: The LTI isocontours of the Agile Wrist: (a) φ z = 0; (b) φ z = 30 o .

Figure 11 :

 11 Figure 11: The LTI isocontours of the CoSPM with torsion φ z = 0: (a) α 1 = 60 o ; (b) α 1 = 47 o .

Table 1 :

 1 Geometrical parameters of the Agile Wrist and CoSPM.

	Agile Wrist	CoSPM
	α 1 , α deg] 90 sin -1 ( √ 6/3) 60(47) 90 90

2 [deg] β, γ [rad] α 1 [deg] α 2 [deg] β [

Table 2 :

 2 WTI for the three SPMs.

		AsySPM	Agile Wirst		CoSPM
	parameters (33) parameters (34)		α 1 = 60 o α 1 = 47 o
	θ ∈ [0, 45 o ] 0.68	0.77	0.75	0.58	0.79
	θ ∈ [0, 60 o ] 0.66	0.72	0.69	0.54

Table 3 :

 3 Performance comparison of the asymmetrical SPM with the Agile Wrist and the Co-axial input SPM.

	AsySPM Agile Wrist CoSPM

Throughout this paper, R, U, S and P stand for revolute, universal, spherical and prismatic joints, respectively, and an underlined letter indicates an actuated joint.


Lei Pan
email: panlei@ivpp.ac.cn

John Francis Thackeray

Jean Dumoncel

| Cl Ement Zanolli

Anna Oettl

Frikkie De Beer

Jakobus Hoffman

Benjamin Duployer

Christophe Tenailleau

| Jos E Braga

Intra-individual metameric variation expressed at the enameldentine junction of lower post-canine dentition of South African fossil hominins and modern humans

Objectives: The aim of this study is to compare the degree and patterning of inter-and intraindividual metameric variation in South African australopiths, early Homo and modern humans.

Metameric variation likely reflects developmental and taxonomical issues, and could also be used to infer ecological and functional adaptations. However, its patterning along the early hominin postcanine dentition, particularly among South African fossil hominins, remains unexplored.

Materials and Methods: Using microfocus X-ray computed tomography (mXCT) and geometric morphometric tools, we studied the enamel-dentine junction (EDJ) morphology and we investigated the intra-and inter-individual EDJ metameric variation among eight australopiths and two early Homo specimens from South Africa, as well as 32 modern humans.

Results: Along post-canine dentition, shape changes between metameres represented by relative positions and height of dentine horns, outlines of the EDJ occlusal table are reported in modern and fossil taxa. Comparisons of EDJ mean shapes and multivariate analyses reveal substantial variation in the direction and magnitude of metameric shape changes among taxa, but some common trends can be found. In modern humans, both the direction and magnitude of metameric shape change show increased variability in M 2 -M 3 compared to M 1 -M 2 . Fossil specimens are clustered together showing similar magnitudes of shape change. Along M 2 -M 3 , the lengths of their metameric vectors are not as variable as those of modern humans, but they display considerable variability in the direction of shape change.

Conclusion:

The distalward increase of metameric variation along the modern human molar row is consistent with the odontogenetic models of molar row structure (inhibitory cascade model).

Though much remains to be tested, the variable trends and magnitudes in metamerism in fossil hominins reported here, together with differences in the scale of shape change between modern humans and fossil hominins may provide valuable information regarding functional morphology and developmental processes in fossil species. In mammalian teeth, metameric variations observed among species may reflect differences in developmental processes [START_REF] Braga | The enamel-dentine junction in the postcanine dentition of Australopithecus africanus: Intra-individual metameric and antimeric variation[END_REF][START_REF] Evans | A simple rule governs the evolution and development of hominin tooth size[END_REF][START_REF] Morita | Exploring metameric variation in human molars: A morphological study using morphometric mapping[END_REF][START_REF] Weiss | Duplication with variation: Metameric logic in evolution from genes to morphology[END_REF]. For instance, size-related shape variations in human upper molars are consistent with odontogenetic models of molar row structure and molar crown morphology (inhibitory cascade model) (Kavanagh, [START_REF] Kavanagh | Predicting evolutionary patterns of mammalian teeth from development[END_REF]. Metameric variation could also be used to infer ecological conditions and/or functional adaptations [START_REF] Kavanagh | Predicting evolutionary patterns of mammalian teeth from development[END_REF][START_REF] Polly | Development with a bite[END_REF], as well as to distinguish symplesiomorphic traits from autamorphic traits [START_REF] Hlusko | Identifying metameric variation in extant hominoid and fossil hominid mandibular molars[END_REF]. Dental metameric studies also yield some insights into primate taxonomy (e.g., Bailey, Benazzi, andHublin, 2014, 2016;[START_REF] Braga | The enamel-dentine junction in the postcanine dentition of Australopithecus africanus: Intra-individual metameric and antimeric variation[END_REF][START_REF] Hlusko | Identifying metameric variation in extant hominoid and fossil hominid mandibular molars[END_REF][START_REF] Morita | Exploring metameric variation in human molars: A morphological study using morphometric mapping[END_REF][START_REF] Singleton | Allometric and metameric shape variation in Pan mandibular molars: A digital morphometric analysis[END_REF][START_REF] Weiss | Duplication with variation: Metameric logic in evolution from genes to morphology[END_REF]. However, few analyzes of metameric variation have yet been conducted at the intraindividual level (i.e., strictly based on teeth from the same individuals) [START_REF] Braga | The enamel-dentine junction in the postcanine dentition of Australopithecus africanus: Intra-individual metameric and antimeric variation[END_REF]: most studies were based on teeth representing different sets of individuals [START_REF] Hlusko | Identifying metameric variation in extant hominoid and fossil hominid mandibular molars[END_REF][START_REF] Olejniczak | Morphology of the enamel-dentine junction in sections of anthropoid primate maxillary molars[END_REF]Skinner, Gunz, & Wood, 2008a;[START_REF] Smith | Modern human molar enamel thickness and enamel-dentine junction shape[END_REF]. Besides the potential usefulness of metameric variation for developmental, functional/ecological and taxonomic inferences, it may also help to identify position of isolated teeth among fossil hominin assemblages.

Previous studies of metameric variation mainly focused on the outer enamel surface (OES) [START_REF] Hlusko | Identifying metameric variation in extant hominoid and fossil hominid mandibular molars[END_REF][START_REF] Singleton | Allometric and metameric shape variation in Pan mandibular molars: A digital morphometric analysis[END_REF]. Since the assessment of OES features may be obscured by occlusal wear, more recent analyses have investigated the enamel-dentine junction (EDJ). However, only a few studies have yet dealt with the relevance of the EDJ morphology for assessing metameric variation in hominin teeth [START_REF] Braga | The enamel-dentine junction in the postcanine dentition of Australopithecus africanus: Intra-individual metameric and antimeric variation[END_REF]Skinner et al., 2008a). As noted by [START_REF] Braga | The enamel-dentine junction in the postcanine dentition of Australopithecus africanus: Intra-individual metameric and antimeric variation[END_REF], modern humans exhibit stronger metameric variation between first and second molars, as compared with Au. africanus. Another study based on EDJ shape showed that Au. africanus and P. robustus display similar 3D-EDJ intra-taxon metameric trend along the molar dentition, but P. robustus preserves a marked reduction in the buccolingual breadth of the distal crown between M 2 and M 3 , and a marked interradicular extension of the enamel cap in M 1 and M 2 (Skinner et al., 2008a).

Here we assess the metameric variation within and between individuals and groups at the EDJ along the post-canine dentition in three Plio-Pleistocene hominin taxa (Australopithecus, Paranthropus and early Homo) and in modern humans. We mainly aim to test whether the intra-and inter-individual metameric patterns and scales differ between australopiths and genus Homo. In other words, is the metameric variation observed in modern humans seen in early Homo and/or australopiths? We also test if 3D-EDJ metameric variation is a useful indicator of dental position in lower postcanine teeth.

| MATERIALS A ND METHODS

| Study sample

We selected only the postcanine dentition from mandibular specimens -isolated teeth were excluded-to study the intra-individual metameric variation. Whenever the antimeres were preserved, only the teeth on the better preserved side were used in our analyses. The fossil hominin materials came from collections housed at the Ditsong Museum of Natural History (Pretoria, South Africa). Our fossil sample includes permanent lower post-canine teeth representing P. robustus (N 5 17, six individuals), Au. africanus (N 5 4, one individual) and early Pleistocene Homo specimens, attributed to Homo erectus s.l. [START_REF] Wood | Reconstructing human evolution: Achievements, challenges, and opportunities[END_REF]; but see [START_REF] Clarke | Australopithecus and early Homo in Southern Africa[END_REF][START_REF] Curnoe | Odontometric systematic assessment of the Swartkrans SK 15 mandible[END_REF]. The modern human reference material includes 88 teeth, representing 32 individuals of European, Asian and African origin (Table 1).

Because of unconformable wear stages in many modern specimens that affected the EDJ morphology in molar dentition, we used different individuals for comparisons M 1 -M 2 and M 2 -M 3 (detailed in Table 1). The degree of wear for each fossil specimen is listed in Table 1, according to tooth wear categories proposed by [START_REF] Molnar | Human tooth wear, tooth function and cultural variability[END_REF]. In the majority of cases, our specimens show no dentine horn wear on EDJs, but when necessary the dentine horn tip was reconstructed based on the morphology of the intact cusps.

The specimens were scanned using four comparable X-ray microtomographic instruments: an X-Tek (Metris) XT H225L industrial mXCT Micro-XCT image stacks were imported for semi-automated (modules "magic wand" and "threshold") and automated (module "watershed") segmentations. After the segmentation, the EDJ surface was generated using the "unconstrained smoothing" parameter.

| Analyses

In some cases, especially in fossil specimens, only a tooth from one side is preserved (details in Table 1). Any image data from the left side were flipped to obtain a homogeneous right-sided sample. For each tooth, we defined a set of main landmarks as well as a set of semilandmarks along the marginal ridge between the main dentine horns (DHs), as an approximation of the marginal ridge (Figure 1). Along each semi-landmark section, a smooth curve was interpolated using a Bspline function using the "Create Splines" module. Interpolated curves were then imported into R (R Development Core Team, 2012), and were resampled to collect semi-landmarks that were equally spaced along each section/curve, delimited by traditional landmarks. For premolars, the main landmarks were placed on the tips of the DHs (i.e., 1. protoconid, 2. metaconid), with 55 semi-landmarks in between (15 on the mesial marginal ridge, 30 on the distal marginal ridge and 10 on the essential crest that connects the two DHs). The molar landmark set included four main landmarks on the tips of the DHs (i.e., 1. protoconid, 2. metaconid, 3. entoconid, 4. hypoconid), with 60 semi-landmarks, forming a continuous line, beginning at the tip of the protoconid and moving in a counter-clockwise direction.

In order to investigate intra-taxon metameric variation, the samples were grouped into three pairs, according to tooth position (P 3 -P 4 , M 1 -M 2 , M 2 -M 3 ), and comparisons were performed within each pair.

The landmark sets were imported in R software (R Development Core Team, 2012), and statistical analyses were conducted subsequently. The study of intra-individual metameric variation in shape was completed using R packages ade4 and Morpho; each sample of landmark configurations was registered using generalized Procrustes analysis (GPA; [START_REF] Gower | Generalized procrustes analysis[END_REF], treating semi-landmarks as equally spaced points. The resulting matrix of shape coordinates was analyzed in three ways. First, to visualize the average metameric shape differences for each taxon, mean configurations of each of the tooth classes (except for early Homo and Au. africanus represented by an isolated individual) were created, and superimposed using smooth curves with regard to tooth positons.

A between-group PCA (bgPCA) was performed based on the Procrustes shape coordinates to explore the distribution of each group in shape space [START_REF] Braga | A new partial temporal bone of a juvenile hominin from the site of Kromdraai B (South Africa)[END_REF][START_REF] Gunz | The mammalian bony labyrinth reconsidered, introducing a comprehensive geometric morphometric approach[END_REF][START_REF] Mitteroecker | Linear discrimination, ordination, and the visualization of selection gradients in modern morphometrics[END_REF][START_REF] Pan | Further morphological evidence on South African earliest Homo lower postcanine dentition: Enamel thickness and enamel dentine junction[END_REF][START_REF] Ritzman | Mandibular ramus shape of Australopithecus sediba suggests a single variable species[END_REF]. The bgPCA computes a covariance matrix of the predefined group means and then projects all specimens into the space spanned by the eigenvectors of this covariance matrix. Between-group PCAs were conducted separately for each of the metameric pairs (P 3 -P 4 , M 1 -M 2 , M 2 -M 3 ). Because early Homo and Au. africanus have only one specimen, respectively, representing their groups, they were projected subsequently onto the shape space, ordinated by the modern human and P. robustus tooth means, without being assigned to groups a priori. The first two axes were plotted in order to visualize the trends and vectors of EDJ shape change between metameres of the same individual. As the metameric vectors generated by bgPCA are actually placed in a multidimensional shape space, the bgPC1-bgPC3 axes were also plotted, shown in Supporting Information Figure S1. Our interpretation of the spatial positions and metameric vectors between specimens are mainly based on the first two bgPCs, but we refer to the third bgPC as well. As a complementary method of comparing the magnitude of shape variation between metameres, hierarchical clustering (HC) as well as subsequent dendrograms were We visualized the magnitude of shape variation using dendrograms and a 0 to 25 scale. Ward's minimum variance method was applied in HC, as it aims at finding compact clusters in which individuals are grouped. We chose to use this method because it minimizes the increase of intra-group inertia, and maximizes the inter-group inertia, at each step of the algorithm.

| R E S U LTS

The within-taxon metameric variation of P 3 -P 4 , M 1 -M 2 and M 2 -M 3 mean shapes are illustrated in Figures 2 and3; bgPCA plots and dendrograms are illustrated in Figure 4, and the bgPC1-bgPC3 axes are plotted in Supporting Information Figure S1, with the max-min values along the first two bgPC axes shown as landmark series in Supporting Information Figure S2. Specimens are marked according to species and dental position.

Our analyses observed appreciable variation in the metameric relationships within and between taxa, but some common trends in shape change can be found. From P 3 to P 4 , a distalward displacement of the mesial marginal ridge is shared among taxa (Figure 2); a decrease in the height of metaconid dentine horn is presented in modern humans, P. africanus specimen Sts 52 (Figure 4A, Supporting Information Figure S1A). In the axes represented here, the shapes of their P 3 s resemble those of modern humans (note that Sts 52 P 3 is placed just between the range of modern humans and P. robustus), but their P 4 s are similar to those of P. robustus (Figure 4A, Supporting Information Figure S1A).

But as our analyses mixed interspecific and metameric variation, inferences of EDJ shape differences/similarities between isolated cases should be carefully addressed.

Almost all modern human M 2 s and many M 3 s lack a hypoconulid, and hence hypoconulid dentine horn tip was not included in the homologous landmarks (Figure 1D-F). Therefore, differences in the

| DISCUSSION

Tooth morphology is controlled by the combined effects of biochemical signaling degraded from mesial to distal direction at the tooth row level and at the individual crown level [START_REF] Jernvall | Linking development with generation of novelty in mammalian teeth[END_REF][START_REF] Weiss | Duplication with variation: Metameric logic in evolution from genes to morphology[END_REF]. As suggested by the inhibitory cascade model [START_REF] Evans | A simple rule governs the evolution and development of hominin tooth size[END_REF][START_REF] Kavanagh | Predicting evolutionary patterns of mammalian teeth from development[END_REF], the development of each deciduous and permanent molar is controlled by the balance between inhibitor molecules from mesially located tooth germs and activator molecules from the mesenchyme.

The ratio of genetic activation and inhibition during development determines the relative size of the dental elements in the dental row.

However, permanent premolars are derived independently from deciduous molars, so the tooth germs of P 3 and P 4 are not directly connected, therefore caution should be addressed when linking inhibitory cascade model to the metameric variation in permanent premolars.

Metameric differences are often subtle, and the risk of conflating metameric and taxonomic variation is a general concern [START_REF] Hlusko | Identifying metameric variation in extant hominoid and fossil hominid mandibular molars[END_REF][START_REF] Singleton | Allometric and metameric shape variation in Pan mandibular molars: A digital morphometric analysis[END_REF]. However, dental metameric variation in Plio-Pleistocene hominins remains relatively unexplored owing to difficulty in quantifying the complex and subtle shape variation in premolar and But due to the small sample size, further investigation will be needed.

In all three pairs of metameres, hierarchical clustering placed fossil specimens together, showing similar degree of shape change, but the magnitude of metameric variation is quite diversified in modern humans. Moreover, as revealed by our bivariate plots, modern human M 3 s show a large scale of shape variability. This is consistent with previous observations using conventional tools [START_REF] Garn | Third molar agenesis and size reduction of the remaining teeth[END_REF][START_REF] Townsend | Molar intercuspal dimensions: Genetic input to phenotypic variation[END_REF] or geometric morphometrics [START_REF] Morita | Exploring metameric variation in human molars: A morphological study using morphometric mapping[END_REF][START_REF] Pan | Further morphological evidence on South African earliest Homo lower postcanine dentition: Enamel thickness and enamel dentine junction[END_REF]. It has been suggested that in modern humans, the inter-individual differences are larger for the M 2 s than for the M 1 s [START_REF] Braga | The enamel-dentine junction in the postcanine dentition of Australopithecus africanus: Intra-individual metameric and antimeric variation[END_REF]. This observation is in line with the distalward molar size reduction seen in Pleistocene humans (Berm udez [START_REF] Berm Udez De Castro | Posterior dental size reduction in hominids: The Atapuerca evidence[END_REF][START_REF] Brace | Post-Pleistocene changes in the human dentition[END_REF][START_REF] Brace | Gradual change in human tooth size in the late Pleistocene and post-Pleistocene[END_REF][START_REF] Evans | A simple rule governs the evolution and development of hominin tooth size[END_REF]. Since the shape stability was considered to increase from third molar to first molar [START_REF] Dahlberg | The changing dentition of man[END_REF], it is possible that small molar size results in a more unstable shape rather than larger molar size. Just as Skinner et al. (2008a) observed a small-scale, distal expansion of the marginal ridge in M 2 -FIG URE 4 A-C: Results of bgPCA of the semi-landmark configurations. Axes represent functions of the shape variation, specimens from the same individual are marked using the same symbol (for easier visual inspection, all of the modern individuals are marked using the same symbol), and each color representing the tooth position for each taxon. Metameres are connected using lines, showing metameric vectors. D-F: Dendrograms of intra-individual metameric variation in EDJ shape yielded by Hierachical Clustering (HC), showing the magnitude of metameric variation within and between groups. HC was done between metameric pairs P 3 -P 4 (D), M 1 -M 2 (E), and M 2 -M 3 (F). Scales from 0 to 25 indicate similitude of metameric shape variation between individuals. For easier visual inspection, only fossil specimens are marked M 3 s of P. robustus, we confirm such trend in all of our P. robustus specimens and in a number of modern humans. This trend is weakly expressed in our Au. africanus and is not presented in the early Homo specimen. A previous study observed that, in australopiths there is an increase in the height of mesial dentine horns from M 2 to M 3 (Skinner et al., 2008a), in contrast, we found a reduction in relative dentine horn height from M 1 to M 2 , and from M 2 to M 3 among all the taxa examined here, similar to metameric patterns expressed in Pan [START_REF] Skinner | Discrimination of extant Pan species and subspecies using the enamel-dentine junction morphology of lower molars[END_REF]. We suggest that this is probably because of our small sample size, and the fact that our study is focused on intra-individual variation therefore only metameres from strictly the same individual were investigated. The early Homo dentition SK 15 displays similar degree of metameric variation to other fossil samples, closer to the three P. robustus specimens than to the Au. africanus specimen. In all, the three fossil groups display variable directions in the shape change. In modern humans, the direction and magnitude of metameric vectors show increased variability in metameres M 2 -M 3 , than in M 1 -M 2 . However, in P. robustus, the length and direction of metameric vectors seem more variable in M 1 -M 2 pairs, than in M 2 -M 3 pairs, which show consistent metameric shape change. Further studies including more fossil specimens will be necessary to ascertain whether the metameric patterns observed here are characteristic of these groups.

| C O NC LU S I O N S

While 2D studies based on the OES suggest the existence of a distinctive metameric pattern in modern humans compared with that found in chimpanzees and Au. africanus [START_REF] Hlusko | Identifying metameric variation in extant hominoid and fossil hominid mandibular molars[END_REF], in each comparative pair of EDJ (P 3 -P 4 , M 1 -M 2 and M 2 -M 3 ) that we investigated, we do not observe a specific metameric pattern that belongs only to extant humans, but rather a few common trends shared by groups despite a degree of inter-and intra-group variaiton. As a whole, the EDJ proves to be a reliable proxy to identify the taxonomic identity [START_REF] Pan | Further morphological evidence on South African earliest Homo lower postcanine dentition: Enamel thickness and enamel dentine junction[END_REF]Skinner et al., 2008aSkinner et al., , 2008bSkinner et al., , 2009b[START_REF] Skinner | A dental perspective on the taxonomic affinity of the Balanica mandible (BH-1)[END_REF][START_REF] Zanolli | Brief communication: Molar crown inner structural organization in Javanese Homo erectus[END_REF][START_REF] Zanolli | Brief communication: Two human fossil deciduous molars from the Sangiran Dome (Java, Indonesia): Outer and inner morphology[END_REF], but further research is needed to determine whether the metameric trends in 3D-EDJ observed here could act as one piece of evidence to identify tooth position from isolated specimens. Moreover, the underlying mechanisms remain to be answered. Along the molar dentition, based on the axes examined in this study, our results with regard to modern humans are generally in accordance with morphogenetic models of molar rows and molar crowns (inhibitory cascade model). In P. robustus specimens examined here, trends of meanshape changes from M 1 to M 2 and from M 2 to M 3 differed from each other, instead of a simple gradation, such differential expression of metamerism has been previously reported in modern human upper molars [START_REF] Morita | Exploring metameric variation in human molars: A morphological study using morphometric mapping[END_REF]. It should be noted that our study focuses only on the EDJ marginal ridges, but additional studies of the accessory ridges (e.g. protostylid), and a more global analysis of shape variation among early hominins based on the whole EDJ sur-face and diffeomorphisms [START_REF] Braga | In press. The Kromdraai hominins revisited with an updated portray of differences between Australopithecus africanus and Paranthropus robustus[END_REF] will supplement our understanding on the metameric variation in hominin dentition.

  Australopithecus africanus, early Homo, Homo sapiens, metamerism, Paranthropus robustus, tooth internal structure1 | I N T R O D U C T I O N

  system at the South African Nuclear Energy Corporation (Necsa) (Hoffman & de Beer, 2012), a Scanco Medical X-Treme micro-XCT scanner at the Institute for Space Medicine and Physiology (MEDES) of Toulouse, a Phoenix Nanotom 180 scanner from the FERMAT Federation from the Inter-university Material Research and Engineering Centre (CIRIMAT, UMR 5085 CNRS), and a 225 kV-mXCT scanner housed at the Institute of Vertebrate Paleontology and Paleoanthropology (IVPP, Chinese Academy of Sciences). Isometric voxel size ranged from 10 to 70 lm. Each specimen was segmented in Avizo 8.0 (Visualization Sciences Group, www.vsg3d.com).

  (1949); (3)[START_REF] Broom | Swartkrans ape-man, Paranthropus crassidens[END_REF]; (4)Grine and Daeglin (1993); (5)[START_REF] Grine | New hominid fossils from the Swartkrans formation (1979-1986 excavations): craniodental specimens[END_REF]; (6)Broom and Robinson (1949); (7)Rampont (1994). b Owing to different wear stages, 32 M 2 s were taken into account. 14 of them belong to the same individuals as M 1 s, they were used in the analyses M 1 -M 2 . The other 18 teeth belong to the same individuals as M 3 s, they were used in the analyses M 2 -M 3 . computed. Dendrograms were obtained from the decomposition of the total variance (inertia) in between-and within-group variance. Contrary to the bgPCA, the HC does not require any a priori classification and specimens are aggregated into clusters according to the distances recorded between them. Aimed at finding how the metameric variation between fossils compares to modern human sample in the shape space, we used Euclidean distances between metameric pairs for clustering.

  FIG URE1EDJ surface model of a lower premolar (A-C, SKX 21204 from Swartkrans) and a lower molar (D-F, SKW 5 from Swartkrans), illustrating the landmarks collected on the tips of the dentine horns (red spheres) and semi-landmarks that run between the dentine horns (orange spheres) used to capture EDJ shape. Abbreviations: buccal (B), distal (D), lingual (L) and mesial (M). Numbers on the red spheres stand for landmarks collected on the dentine horn tips, numbers next to the ridge curves stand for number of semi-landmarks. Note that relative sizes of premolars to molars are not to scale

  hypoconulid dentine horn height and relative position are not recorded in the present study. In metameres M 1 -M 2 , there is a marked reduction in the height of dentine horns of both P. robustus and modern humans, particularly in the talonid, resulting in a flattened topology in M 2 (Figure3A,B); entoconid dentine horn is more centrally placed in M 2 (a pattern that is more marked in modern humans). Unfortunaltely, materials from early Homo and Au. africanus are not avaliable for comparison. With regard to multivariate analyses, similar magnitude of shape change is seen in both species (Figure4B,E; Supporting Information FigureS1B). The bgPC1 axis is driven by the presence of the hypoconulid, and the position and height of the hypoconid dentine horn; while the bgPC2 axis is driven by the height of the lingual dentine horns (Figure4B; Supporting Information FigureS2E-H). It is worth noticing that SK 6 M 1 -M 2 show a unique trend of shape change along the bgPC1, and SK 63 has a much shorter metameric vector in bgPC1-bgPC2 plot, increasing the within-group metameric variation in P. robustus. In shape space bgPC1-bgPC3, P. robustus shows vertically-oriented metameric vectors with similar lengths contrasted with modern humans; along bgPC3, the direction of shape change in modern humans is more variable compared to bgPC1 (Supporting Information FigureS1B). Our results of hierachichal clustering reveal that, although placed separately from modern groups, the P. robustus sample displays closer affinity to a few modern individuals, indicating a similarity in intra-individual metameric distances (Figure4E).For metameres M 2 -M 3 , a slight distal expansion of the marginal ridges is observed except for the early Homo specimen, SK 15 (Figure3C-F). A reduction in the height of dentine horns on the talonid is seen in modern humans and australopiths (a pattern that is more marked in modern humans; Figure3C,D). Changes in the relative positions of the dentine horns are widely observed: for modern humans, talonid dentine horns are more centrally placed in M 2 than in M 3 buccal view); for australopiths, the hypoconid dentine horn is more centrally placed in M 3 , more markedly in P. robustus, in addition, Au. africanus individual displays a more centrally placed entoconid in M 2 (Figure3D,F; occlusal view); for the early Homo specimen SK 15, a more centrally placed protoconid and entoconid dentine horns are seen in M 3(Figure 3E; occlusal view). With regard to multivariate analyzes, the bgPC1 exhibits shape changes in the EDJ outline (oviod to elongated, Supporting Information FigureS2I,J) and changes in the

FIG URE 2

 2 FIG URE 2 Comparison of metameric variation between P 3 -P 4 based on mean shapes of the EDJ, after Procrustes superimposition. EDJ ridge curves are shown in occlusal, buccal and lingual views. Colors indicate different dental positions

FIG URE 3

 3 FIG URE 3 Comparison of metameric variation between M 1 -M 2 , and between M 2 -M 3 , based on mean shapes of the EDJ, after Procrustes superimposition. EDJ ridge curves are shown in occlusal, buccal and lingual views. Colors indicate different dental positions. A-B: M 1 (red) compared to M 2 (blue); C-F: M 2 (red) compared to M 3 (blue)

TABLE 1

 1 Composition of the study sample

										Occlusal
	Specimen	P 3	P 4	M 1	M 2	M 3	Provenance		Age	wear	Citations a
	P. robustus
	SK 6	1	1	1	1	1	Mb. 1, HR	Swartkrans	1.80 6 0.09 Ma-2.19 6 0.08	1-3	1 -2
									Ma (Gibbon et al., 2014),
									2.31-1.64 Ma (Pickering
									et al., 2011)
	SK 843				1	1	Mb. 1, HR			2-3	1
	SK 61	1	1				Mb. 1, HR			1	3
	SK 63	1	1	1	1		Mb. 1, HR			1	1
	SKW 5				1	1	Mb. 1, HR			1-late 2	4
	SKX 4446			1	1		Mb. 2		1.36 6 0.69 Ma	1-3	5
									(Balter et al., 2008)
	Au. africanus
	Sts 52	1	1		1	1	Mb. 4	Sterkfontein	3.0-2.5 Ma (White, Harris,	2-3	1
									1977; Tobias, 1978; Clarke,
									1994), 2.8-2.4 Ma (Vrba,
									1985; but see Berger et al.,
									2002), 2.1 6 0.5 Ma
									(Schwarcz et al., 1994)
	Early Homo
	SKX 21204	1	1				Mb. 1, LB	Swartkrans	1.80 6 0.09 Ma-2.19 6 0.08	1	4
									Ma (Gibbon et al., 2014),
									2.31-1.64 Ma (Pickering
									et al., 2011)
	SK 15				1	1	Mb. 2		1.36 6 0.69 Ma	2-3	6
									(Balter et al., 2008)
	Extant H. sapiens	12	12	14	14/18 b	18	South Africa/Europe/East Asia		1-late 2	7

a Citations: (1)

[START_REF] Robinson | The dentition of the Australopithecinae[END_REF]

; (2) Broom

ACKNOWLEDGMENTS

This work was supported by the Centre National de la Recherche Scientifique (CNRS), the French Ministry of Foreign Affairs, the French Embassy in South Africa through the Cultural and Cooperation Services, National Natural Science Foundation of China and the China Scholarship Council. For access to specimens we thank the following individuals and institutions: Stephany Potze (Ditsong National Museum of Natural History, Pretoria), Jean-Luc Kahn (Strasbourg University), Dr. S. Xing (Institute of Vertebrate Paleontology and Paleoanthropology, Beijing), Dr. M. Zhou (Institute of Archeology and Cultural Relics of Hubei Province, Wuhan), and Dr.

C. Thèves (UMR 5288 CNRS). We thank Dr. A. Beaudet for her technical support during the imaging processing of data and the statistical analysis. We are also grateful to the Associated Editor, and two anonymous reviewers of this manuscript, for their insightful comments and suggestions.

SUPPORTING INFORMATION

Additional Supporting Information may be found in the online version of this article at the publisher's website.