Kineto-Elasto-Static Synthesis of a 3-CRU Spherical Wrist for Miniaturized Assembly Tasks Massimo Callegari, Andrea Gabrielli Department of Mechanics, Polytechnic University of Marche, Italy E-mail: m.callegari@univpm.it; andrea.gabrielli@univpm.it Maurizio Ruggiu Department of Mechanical Engineering, University of Cagliari, Italy E-mail: ruggiu@dimeca.unica.it Abstract The article presents the preliminary studies for the prototyping of a spherical parallel robot for miniaturized assembly applications. The closed-form solution of robot’s kinematics allowed to optimise its (rotation) workspace by taking into account all singularity surfaces and the maximum strokes of the linear actuators. Then, in view of flexures design, the maximum rotations of the passive joints have been calculated and finally the design of the legs and of the whole robot has been carried out by means of a FEM software, with due regard to the non-linearities arising from large displacements. Keywords Spherical wrist, Mini-robotics, Flexible joints, Machine design, Parallel kinematics machine 1. Introduction The article describes the results obtained so far by the researches at the Polytechnic University of Marche and at the University of Cagliari within the project PRIN 2005 “Mini-robotic devices for advanced technology applications” that aimed at exploring the realisation of mini robotic devices for the orientation of parts or tools in the space. The first studies on the synthesis of a mini spherical wrist are hereby reported: the mechanism is characterized by a 3-CRU parallel kinematics architecture, Fig. 1, where revolute joints have been realised by means of flexure hinges.

The spherical motion of the machine is granted only if it works inside the singularity surfaces, which unfortunately are of different types and of complex shape. In any case it is always possible to identify a connected workspace free of singularities: an exhaustive study of the kinematics relations or even a procedure of optimization can lead to a maximization of such workspace, taking into account also the constraints of the maximum strokes of the actuated joints.

Another constraint for the operative workspace is represented by the actual strokes of the passive joints which, being realized with the technique of the flexible joints, have a maximum deformability that depends on the chosen material and on path planning. Therefore in this case joints’ design had to be integrated into the dimensioning of the legs of the manipulator and finally of the whole platform.

The first phase of flexures design is defining the geometry of the joints themselves according to the needed mobility, the geometry of the single leg and the limitations on their dimensions; then

the strain and the stress of joints and legs in each feasible configuration of the workspace has been analysed and compared with the available material performances. In fact, in order to allow the correct motion of the mobile platform, the flexible joints should guarantee localized compliance, which of course is hardly possible in reality. Therefore the kinetostatic analysis has been carried out with the aid of a FEM software and has brought to a continuous re-design of the specific parameters of the virtual hinges and of the flexible joints, such as the thickness and fillet radius.

Fig. 1 Architecture of the 3-CRU spherical wrist

2. Description of the mechanism Much research has been developed in recent years for the synthesis of spherical parallel machines (SPM’s), providing machine designers with both powerful tools, as the Lie algebra approach used by Karouia and Hervè [1], [2] and comprehensive enumerations, as the one by Kong and Gosselin [3]. Specific results have been reached also in the restricted field of design of mini- or micro-devices, as the pioneering work by Tahmasebi and Tsai [4] on a full mobility platform or the interesting study by Yi et al. [5] on the design of a parallel micro-wrists.

The device presented in the paper is based on the 3-CRU mechanism shown in Fig. 1: the concept is derived from the spherical 3-CPU mechanism studied by Callegari et al. [6] but in this case it is modified with the substitution of the prismatic pair with a revolute joint in view a miniature realization by means of flexures. It is worth noting that the 3-CRU mechanism is also mentioned by Fang and Tsai in [7].

A detailed description of the mechanism under study is provided by Callegari et al. in [8], where closed-form direct and inverse kinematic relations are worked out too: for the sake of clarity, only the main reference data are recalled in the following lines. The 3-CRU mechanism shown in Fig. 1 is composed by a mobile platform actuated in-parallel by 3 identical legs: each limb is composed by two links, joined together by a revolute pair (R), and is connected to the ground by a cylindrical joint (C) and to the mobile platform by a universal joint (U). The platform is actuated by driving the linear motion of the cylindrical pairs, i.e. by means of ground sliders powered by linear motors.

Due to machine’s geometry, the axes of the 3 base cylindrical joints are aligned along the frame O(x,y,z) fixed with the ground; in the same manner, the axes of the 3 revolute joints that

connect the limbs with the mobile platform (i.e. the “outer” revolute pairs of the 3 universal joints) are aligned along the mobile frame P(u,v,w), attached to the platform. It is noted that, with a proper assembly of the machine, the origins of the two Cartesian frames can be brought to coincide at the same fixed point P≡O.

Wrist’s kinematics shall provide the orientation of the mobile frame P(u,v,w) with respect to the global frame O(x,y,z). Such orientation can be assigned through the rotation matrix O

P R or it can be generated by three ordered rotations α, β, γ around the three body axes u, v and w respectively (Cardan angles).

Figure 2 shows more kinematic entities that will be referenced in the following sections: wji are the unit vectors of the axes of the 4 joints (j=1,2,3,4) of the 3 legs (i=1,2,3); θji are the corresponding rotations, i.e. the articular coordinates. Moreover, ai are the varying strokes of the linear actuators, while b, c and d are geometrical parameters that are fixed and have the same values for all the three legs; Table 1 summarises the values obtained after wrist design and used in the following figures.

Fig. 2 Kinematic chain of the ith limb

Table 1 Geometrical parameters of the 3-CRU Robot

Parameter Dimension [mm]

b 26.1 c 38.2 d 23.0

3. Motion range of the passive joints The assessment of the motion range of the passive joints is very important for the present study, since they have to be realised by means of the flexure hinges technique. The required strokes can be easily worked out, once the position kinematics problem has been solved in closed-form [8]: the main results are summarised in the present section, by expressing the articular coordinates as functions of wrist orientation.

The rotation of the second revolute joint i2θ of limb i is given by: 2

2

2

11

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ −+−−−=

crdb

s ijiθ

crdb

c iji

2

2

1−+−=θ with (i,j) ={(1,2), (2,3), (3,1)} (1)

where ijr is the element at the ith row and the jth column of the rotation matrix OP R that describes

the orientation of the mobile platform. The rotation of the first revolute joint i1θ of the limb i, i.e. the revolute joint of the cylindrical

pair, is:

i

jji ccb

rds

21 θ

θ⋅+

⋅=

i

kji ccb

rdc

21 θ

θ⋅+⋅−

= with (i,j,k) ={(1,2,3), (2,3,1), (3,1,2)} (2)

where the values for is 2θ and ic 2θ are provided by (1). The rotation of the third revolute joint i3θ of the limb i, i.e. the first revolute joint of the

universal pair, is: )1( 2

223 ijiiiji rscrarcsin −+−−= θθπθ if 222 bdcai −+≤ (3a)

)1( 2223 ijiiiji rscrarcsin −+= θθθ if 222 bdcai −+> (3b)

with (i,j) ={(1,2), (2,3), (3,1)}. The rotation of the fourth revolute joint i4θ of the limb i, i.e. the second revolute joint of the

universal pair, is given by:

iiii

iii sscc

rs3232

4 θθθθθ

−−

= iiii

iji sscc

rc

32324 θθθθ

θ−

= (4)

with (i,j) ={(1,3), (2,1), (3,2)}. Table 2 summarises the motion range of the passive joints when the wrist is moved throughout

its orientational workspace, as identified in section 5; for the symmetry of the 3-CRU architecture all the three limbs show the same behaviour.

Table 2 Motion range of the passive joints

i1θ i2θ i3θ i4θ ( )min

1iθ ( )max1iθ i1θΔ ( )min

2iθ ( )max2iθ i2θΔ ( )min

3iθ ( )max3iθ i3θΔ ( )min

4iθ ( )max4iθ i4θΔ

-31.5° +31.5° 63° -9.3° 0° 9.3° -22.2° 40.8° 63° -31.5° 31.5° 63°

4. Singularity surfaces It is possible to distinguish different types of singularities, which are space orientations where the wrist can not work properly: translation singularities are poses that correspond to configurations of the wrist that may cause translations in space; in inverse kinematics singularities the wrist loses one (or more) degrees of freedom while in direct kinematics singularities wrist’s mobility is increased (but it can not be controlled any more).

4.1. Translation singularities In order to prevent the 3-CRU mechanism from translating, some manufacturing and mounting conditions must be satisfied:

i. i1w and i4w incident in O ii. >⊥< iii 412 , www iii. >⊥< iii 413 ,www It can be shown that conditions (i-iii) are not verified, i.e. the wrist occurs in translation

singularities, iff the three unit vectors w3i are linearly dependent, that is: 0333231 =×⋅ www (5)

By taking the projections of such vectors in the global frame O(x,y,z): [ ]TO sc 111131 0 θθ=w [ ]TO cs 121232 0 θθ=w [ ]TO sc 0131333 θθ=w (6)

and substituting (6) into (5) it is obtained: 0131211131211333231 =+=×⋅ θθθθθθ ssscccOOO www (7)

An expression of the singularity surface (7) in the operative space can be obtained by taking into consideration (2) and after some manipulations:

0211332332211 =− rrrrrr (8) An alternative expression of the translation singularity surface can be obtained in the Cardan angles space by substituting ijr with the corresponding expressions in function of α, β, γ:

( ) ( ) ( ) 0=+⋅⋅+−⋅+−⋅ γαγβαβγαγβαβαγαγβαγβ sccssscssscccccssscc (9) that can be manipulated to obtain in the end:

( ) 022 =−− γβαγαβ sssccs (10) It is noted that such singularity surface does not depend on the geometrical parameters of the

wrist. Figure 3 maps the left hand side of (10) for a given value of α (0°, 40° and 90°): the regions where such expression approaches zero are painted black.

Fig. 3 Mapping of the expression on the left side of (10) (above) and plot of the translation singularities (below) for α equal to 0°, 40° and 90°

Of course aiming at wrist design it is very important to visualise and quantify the volume of the working space lying inside such surface: Fig. 4a plots the shape of the translation singularity surface and shows that a sphere with a radius approximately equal to 56° can be inscribed in it.

4.2. Inverse and direct kinematics singularities The differential kinematics of the 3-CRU wrist is based on the inverse Jacobian matrix I

O J and the direct Jacobian matrix D

O J , which are defined as usual for parallel mechanisms: aJωJ &⋅=⋅ I

OOD

O (11) with a& vector of joint rates and Oω angular velocity of the mobile platform. The Jacobian matrices taken on the following useful form:

( )( )

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⋅+⋅+

⋅+=

2323

2222

2121

000000

θθθθ

θθ

ccbsccbs

ccbs

IO J (12)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⋅⋅−⋅−⋅

⋅⋅−=

00

0

33113321

32133233

312231322

θθθθ

θθ

srsrsrsr

srsrdD

O J (13)

Inverse kinematics singularities are found by letting the determinant of the related Jacobian matrix vanish, therefore:

( ) ( )( )( ) 0det 232221232221 =⋅+⋅+⋅+= θθθθθθ ccbccbccbsssIJ (14) Equation (14) is satisfied either when:

02 =isθ (15) that admits the two solutions:

2

1 ⎟⎠⎞

⎜⎝⎛ −

−±=d

cbrij only if 1≤

−d

cb (16a)

2

1 ⎟⎠⎞

⎜⎝⎛ +

−±=d

cbrij only if 1≤

+d

cb (16b)

with (i,j) ={(1,2), (2,3), (3,1)} or when:

02 =⋅+ iccb θ (17) that is satisfied when:

1±=ijr (18) with (i,j) ={(1,2), (2,3), (3,1)}.

Inverse kinematics singularities are mapped in the operative space (α,β,γ) in Fig. 4b, that also shows the maximum sphere that can be inscribed in it, which has radius of 90°.

The direct kinematics singularities appear when: ( ) 0)(det 211332332211333231

6 =−= rrrrrrsssdD θθθJ (19) Equation (19) is satisfied either when:

03 =isθ (20) that admits the two solutions:

2

1 ⎟⎠⎞

⎜⎝⎛

−−=

cdbrij

only if 1≤− cdb (21a)

2

1 ⎟⎠⎞

⎜⎝⎛

+−−=

cdbrij

only if 1≤+ cdb (21b)

with (i,j) ={(1,2), (2,3), (3,1)} or when:

0211332332211 =− rrrrrr (22) that is the same as (8), already studied for the translation singularities. Figure 4c plots the direct kinematics singularity surfaces in the (α,β,γ) operative space, showing also the maximum sphere inscribed having a radius of 56°.

(a) (b) (c)

Fig. 4 Limiting surfaces of workspace: translation singularities (a); inverse kinematics singularities (b); direct kinematics singularities (c)

5. Identification of the workspace The identification of the useful workspace of the robot is very important for its performances and can be optimised by a proper design. Inside this region, taking into account the limited strokes of the actuated joints, the path of the 3-CRU wrist can be planned freely without passing through singularity surfaces. It must be noted that the device can achieve up to 8 different configurations in the operative space (α,β,γ) corresponding to the same set of joints displacements, see Innocenti and Parenti-Castelli [9].

By simple geometrical considerations it is seen that when the actuators reach the maximum stroke equal to ( ) 22

max bdca −+= the wrist gets stuck in a singular configuration characterized by the parallelism of the vectors c and d. Therefore by limiting the travel of the actuated prismatic pairs within the reasonable range:

max9.00 aa ⋅≤≤ (23) it is possible to identify the corresponding volume in the Cardan angles space: the boundary region has a quite complex shape and intersects the singularity surfaces plot in Fig. 4; the largest sphere completely inscribed in it has radius of 32° (Fig. 5a). Comparing the radiuses of all the spheres previously introduced, it turns out that, with the chosen geometrical parameters of Table 1, the safe

workspace of the wrist for motion planning lies inside the inner sphere of Fig. 5b, that is obtained by limiting the stroke of the actuators.

(a) (b)

Fig. 5 Subspace generated by the limited stroke of the actuators (a) and comparison among the limit spheres (b)

Of course, in order to plan properly the motion of the device, such inner sphere of wrist workspace should be better mapped in the space of joint actuation variables ai, i=1,2,3: by applying the inverse orientation kinematics to the boundary points, the connected and convex surface shown in Fig. 6 is obtained. It is interesting to note that such surface has a quite simple shape also in the joint space. Finally, it is observed that the scheme that has been used for representation and quantification of workspace volume is rather conservative and higher volumes would certainly be disclosed by more refined representations: on the other hand, a pretty simple criterion was needed for preliminary wrist design and related optimisation, while the following structural analysis showed that more severe limitations to wrist mobility arise for the use of flexure technique itself.

Fig. 6 Mapping of useful workspace boundary in the joint space

6. Design of the robot It has been shown that the rigid-body SPM is characterised by 3 dof's; however, when made by compliant links, the manipulator is monolithic loosing completely its mobility, Fig. 7a. In fact a simple application of the Kutzbach formula shows that the wrist is nine times over-constrained: this fact proves the difficulties in designing a manipulator capable of recovering the required three degrees of freedom, therefore needing an integrated kineto-static approach to the synthesis. Of course, such problems are well known to the designer of mini or micro-manipulators, since many interesting works have been published already on the subject, like the fundamental studies by Pernette et al. [10] or by Yi et al. [5] or the more recent realisations by Bertetto and Ruggiu [11], Kang et al. [12], Li and Xu [13-15].

As a matter of fact, the central issue in wrist design has been the proper dimensioning of flexure hinges, whose kineto-elasto-static analysis is of course complicated by the large field of displacements they undergo (see the work by Paros and Weisbord [16] providing the theoretic foundations for monolithic flexures design and the recent results by Moon et al. [17]).

According to the pseudo-rigid-body (PRB) model, see Howell [18], each leg of the manipulator with the related flexures can be thought as a proper sequence of rigid links connected each other by a traditional revolute joint coupled to a torsional spring. A schematic PRB model of the ith leg is sketched in Fig. 7b. The compliance, i.e. rotation φ with respect to applied moment M, associated to the torsion spring, is a full 3x3 matrix:

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

z

y

x

z

y

x

MMM

333231

232221

131211

δδδδδδδδδ

φφφ

(24)

A proper design of the compliant joint provides iiδ ’s much greater than ijδ ’s ( ji ≠ ) although coupling between all the elements of the matrix makes the problem pretty tough to solve. Figure 8 shows the design adopted for the three legs. It may be noted that the plane which the leg belongs to is rotated with respect to the vertical plane in order to avoid the singular poses found out in the kinematics analysis (translation singularities).

(a) (b) Fig. 7 The over-constrained structure of the manipulator (a); the PRB model of the legs (b)

Fig. 8 The design adopted for the legs

The cylindrical joint (C) is modelled by a linear pair (P) in series with a double notch flexure

hinge (R1) with constant radius r=5 mm, smallest thickness t=0.4 mm and width w=8 mm. The compliance of the flexure hinge around its sensible axis may be estimated by the formula:

5.2

5.0

1 29Ewt

rb

πδ = (25)

where E is the Young modulus of the material. Equation (25) clearly shows that the most relevant design parameter is the thickness t. 1bδ has been calculated by a separate finite element model of the flexure hinge, that was loaded by a given bending moment around the sensible axis and rigidly constrained at one end. The rotation calculated with respect to the applied moment provides the searched numerical value: it was found Nmmradb /1037.0 3

1−⋅=δ which is very close to the

theoretical value provided by (25), with an error smaller than 3%. The intermediate revolute pair (R2) and the first revolute pair of the universal joint (R3) were

modelled by single notch flexure hinges but with radius taken constant only for 11% of the entire angle span ( 6/π against 2/3π ). The theoretical formula for the single notch hinge with constant radius, different from (25) only for the values of the coefficients, was used just for addressing the design and gives values about a half of the numerical data from R2. The compliances calculated numerically are respectively Nmmradb /106.2 3

2−⋅=δ and Nmmradb /101.2 3

3−⋅=δ . The

geometrical data of the hinges are mmr 70.7= , mmw 8= , mmt 3.0= . Figure 9 shows details of the R1, R2, R3 pairs of the leg. All the revolute pairs must be stiff

with respect to moments applied around the two axes except the sensible one. In particular the stiffness of R2 and R3 influences the warping phenomena of the entire leg.

R1

R2

R3

R3

Fig. 9 Details of the flexure hinges

The second pair of the universal joint (R4) has been a crucial step of leg’s design. The pair

must work as a torsion bar along its local axis but, at the same time, it must be able to resist to the bending actions along the other two axes. For that reason a cross section was selected to be the most appropriate, see Fig. 10.

The torsion compliance δt4 of the column is given by:

pt GJ

L=4δ (26)

where G is the shear modulus of the material, L is the height of the column and Jp is the second moment of area around the torsion axis. In case of cross sections holds:

∑≅N

iiip twJ 3

31 (27)

where wi, ti may be deduced from Fig. 10. It is worth noting that the compliance strongly depends on ti. The strength of the column to bending moments does not allow to increase the height L as the formula would require. The numerical calculation of the compliance of the torsion bar provides

Nmmradt /104.2 34

−⋅=δ .

Fig. 10 The design of the torsion bar (R4 joint)

The entire wrist is shown in Fig. 11 when modelled by finite elements. A quadratic tetrahedral

element has been used for the parameterisation and the final FE mesh consists of 27 471 nodes and 14 857 elements. The material used in the calculation was the Ti-10V-2Fe-3Al alloy with E = 103 GPa, Poisson’s modulus of 0.34 and yield strength of 1 200 MPa. Several loading conditions were taken into account with proper consideration for actuators’ displacements.

Fig. 11 The parallel wrist designed with compliant links

Figure 12 shows the results of the simulation in terms either of Von Mises equivalent stresses

or displacements of some relevant points of the moving platform when a motion of 5 mm has been imposed to each one of the actuated pairs. It may be observed how the stresses are localized in the flexure hinges area; Von Mises equivalent stresses are given in Table 3 for each flexure hinge of the leg 2. The maximum stress values that results from calculations are close to the limit of the yield strength of the material: yet no effort was spent for reducing the stress peaks due to geometrical stress concentrations, i.e., change of section dimensions. Table 4 presents the maximum displacements of the vertexes A, B of the moving platform and of the (theoretical) centre of the spherical motion C.

Fig. 12 Von Mises equivalent stresses and displacements (point C free)

Table 3 Von Mises Equivalent stresses (point C free)

Name of the hinge R1 R2 R3 R4 Von Mises

equivalent stress [MPa] 1 100 1 000 900 1 000

Table 4 Displacements (point C free)

Displacement Δtot [mm] Δx [mm] Δy [mm] Δz [mm] Point A 8.00 -3.00 -5.50 5.50 Point B 11.11 -7.00 7.70 3.00 Point C 4.00 -3.95 / /

The resulting displacement of point C is quite large, i.e. about 80% of the displacement

imposed to the actuated pairs (5 mm). This result was expected since the compliant mechanisms are not able to completely restore the mobility to the manipulator. Therefore, in order to guarantee the spherical rotation of the moving platform around C, its linear displacements have been constrained. Figure 13 shows the results in this case, imposing again a displacement of 5 mm to each of the actuated pairs. The equivalent stresses reach about the maximum value of 1 100 MPa once again, as shown in Table 5.

Fig. 13 Von Mises equivalent stresses and displacements field (point C constrained)

Table 5 Von Mises Equivalent stresses (point C constrained)

Name of the hinge R1 R2 R3 R4 Von Mises equivalent

stress [MPa] 990 500 700 1 100

Main stresses (magnitudes) [MPa]

σZ=1100 MPa σX=400 MPa σxy=0 MPa σxz=30 MPa σyz=130 MPa

σZ=300 MPa σX=300 MPa σxy=120 MPa σxz=130 MPa σyz=120 MPa

σZ=400 MPa σX=200 MPa σxy=130 MPa σxz=300 MPa σyz=100 MPa

σZ=900 MPa σX=150 MPa σxy=100 MPa σxz=300 MPa σyz=250 MPa

Table 5 provides values of the components of the stress as well: they are calculated in the local

frame of the leg (xL, yL, zL) shown in Fig. 13. Stress σz assumes substantially the highest value for R1, R2, R3 according to the bending behaviour of the hinges. For the same reason σx is of the same order of magnitude for R2 and R3. The high value of σz for R4 is instead due to the stretching and

bending stresses which are basically due to the geometrical requirements imposed to the column, i.e., large slope of its axis with respect to the moving platform plane. Thus, the bending stresses are exalted whereas the torsion stresses are reduced.

Displacements of the moving platform, shown in Table 6, reach 16.6 mm as maximum and are symmetrical. The forces of the linear motor capable of providing the displacements given in the simulation have been calculated as well and are given in Table 7.

Table 6 Displacement field of platform points (point C constrained)

Displacement Δtot [mm] Δx [mm] Δy [mm] Δz [mm] Point A 16.64 / -13.99 / Point B 16.64 / 14.06 / Point C 0 0 0 0

Table 7 Actuation forces (point C constrained)

Leg Leg 1 Leg 2 Leg 3 Motor force [N] 19 21 21

Finally Fig. 14 shows the displacement field of the mobile platform when a pure torsion

behaviour is imposed to the manipulator by applying a torque of 200 Nmm: a rotation of about 0.07 rad is calculated. Roughly speaking the compliance of the entire mechanism is hence given by the ratio between the rotation calculated and the couple applied and is

Nmmradt /1035.0 34

−⋅=δ .

Fig. 14 Displacement field due to a pure moment of 200 Nmm at the moving platform From a practical point of view the theoretical centre C of the motion may be constrained by a

socket-ball joint. This solution provides very low friction, accuracy and may be miniaturized. The tension/compression force which the constraint is subjected was calculated to be of approximately 60 N and may be easily supported by such constraint.

Motors may be chosen among ultra-high-resolution (0.1 μm) linear actuators consisting in a lead-screw (ball-screw) driven by a closed-loop DC or stepper motor.

7. Conclusions

The article has described the first steps of the synthesis of a mini spherical wrist for precision manipulation: the 3-CRU mechanical architecture that has been chosen has allowed the use of flexure hinges, that are quite a standard technique in the realisation of mini or micro devices. On the other hand, their limited range of rotation constraints very much the mobility of the whole machine, therefore the design has been aimed at maximising the orientational workspace of the wrist, that in the end was rather restricted anyhow. By taking into account all the singularity surfaces that characterise the chosen kinematics and also the limited stroke of the linear actuators, a connected workspace free of singularities has been identified for the rigid body model of the wrist: it has been represented in the Cardan angles space as a sphere with radius of 32°. Finite element simulations are currently under way to establish the real workspace of the machine itself by taking into account the actual behaviour of the flexures and the properties of the selected material. It must also be noted that, due to the highly over-constrained nature of the flexible link manipulator, the (theoretical) centre of spherical motion moves around in the workspace with pretty large displacements. The solution that has been explored so far lies in the use of a ball joint to constrain the platform to the ground: such modification of the original concept seems feasible from the structural point of view but precludes the accessibility of motion centre to any possible application. An alternative opportunity would be the redesign of platform’s legs aiming at minimising this unwanted side-effect, in place of searching for maximum flexures displacements.

Acknowledgments The research has been supported by the Italian Ministry of Research and Education (MiUR) within the PRIN2005 Project “Mini-robotic devices for advanced technology applications”. The authors would like to thank Mr. Giacomo Grilli for his contribution in developing the FE model. The present work is derived from a paper previously presented by the same Authors at the 18th AIMeTA Congress of Theoretical and Applied Mechanics, Brescia, 2007.

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