Stiffness Analysis of Spherical Parallel Mechanism UP+R with 2-DOF

Yu-Lin Zhou，Long Yang，Bo Li，Da-Xing Zeng，Feng Gao

(1.School of Mechanical Engineering，Yanshan University，Qinhuangdao 006004，China;2.XCMG Xuzhou Truck-Mounted Crane Co.，Ltd.，Xuzhou 221004，China;3.State Key Laboratory of Mechanical System and Vibration，Shanghai Jiaotong University，Shanghai 200030，China)

1 Introduction

As one of the typical lower-mobility parallel mechanisms，the 2-DOF spherical parallel mechanism has broad prospects in application，such as satellite antenna［1］，spherical point positioning devices［2］and parallel rotational joints of humanoid robot.

Up to now，the research on static stiffness of parallel mechanism has got some progress.Gosselin［3］derived the mapping relationship between the operating force and the deformation of extremity of spatial mechanisms based on the principle of virtual work.Ceccarelli et al.［4］presented a formulation to deduce the stiffness matrix of the Cassino Parallel Manipulator.Tsai et al.［5］presented a comparison study of the stiffness of four 3 degree-of-freedom translational parallel manipulators considering the flexibility of active joints only.Chakarov［6］created a stiffness analytical model of redundant parallel manipulators and derived a scheme for stiffness control of them.Han et al.［7］derived the Jacobian matrix of 4-RUC parallel manipulator by screw theory and founded the stiffness matrix of it.By taking the 3-DOF parallel mechanism within the Tricept robot as an example，Wang et al.［8］presented an analytical approach which enables the stiffness of the PKM having a properly constrained passive limb to be quickly estimated within the entire workspace.By using the overall Jacobian，Wang et al.［9］presented a semi-analytical approach for the stiffness modeling of a novel hybrid robot named Bicept.Zhang et al.［10］discussed the relationship between link lengths and stiffness of the planar 2-DOF parallel manipulator with actuation redundancy and plotted corresponding atlases in the physical model of the solution space.Li et al.［11］deduced the analytical stiffness equations of the part concerning joint stiffness and configuration of 3-RRR planar parallel mechanism based on the conservative congruence transformation stiffness matrix.Based on influence coefficient method and principle of virtual work，Zhao et al.［12］established continuous stiffness nonlinear mapping general model of spatial parallel mechanism.Hu et al.［13］presented that the constrained forces/torques must be considered in analyzing the stiffness of parallel manipulators and analyzed the stiffness of a traditional parallel manipulator 3-RPS to illustrate thi s method.Zhou et al.［14］analyzed the static stiffness of spherical parallel mechanism 3-RRR with 3-DOF by means of calculating deformation and superposition principle at small strain.

However，for the spherical parallel mechanism UP+Rwith 2-DOF，it has high order overconstraints，and the bars composing the two branches are all assumed to be arc-shaped，then the calculation of its overall stiffness is partly different with general parallel mechanisms，which has not been reported so far.

In this paper，considering the elasticity of components，the deformation and displacement of each link of the UP+Rmechanism is derived sequentially.The angle displacement of the moving platform and the line displacement of the spherical center are derived based on the small strain superposition principle.Based on the relational equation between the force of the link and the external load，the analytical expression between the external load and the total deformation of the mechanism is established.And then，the static stiffness matrix of mechanism is obtained.By the means of orthogonal transformation，the static stiffness matrix is analyzed and the six principal stiffness and their direction vectors of the mechanism are derived.

2 Deformation and Displacement Analysis of the UP+R Mechanism

As shown in Fig.1，the spherical parallel mechanism UP+Rconsists of the fixed base，the moving platform and two branches connecting the fixed base and the moving platform.The first branch is composed of Link 1 and three revolute pairs，the second branch consists of Link 2－1，Link 2－2 and three revolute pairs.Two servo motors are installed on the fixed base，which axes are mutually perpendicular.The two axes of the revolute joints on the moving platform are mutually perpendicular，and the moving platform can rotate in two directions.All the axes of the revolute joints of the mechanism are concurrent in spherical center O.Then the UP+Rmechanism has two rotational degrees of freedom.

Taking the center of the moving platform as the origin of the coordinate system，and the fixed coordinate system O0-X0Y0Z0，the moving coordinate system O-XYZ and the coordinate system of each link O-XijYijZij(i=1，2;j=1，2，3 denote the j-th revolute joint of the i-th branched-chain)can be established respectively.

The X0axis coincides with the motor axis of the first branch and is along O0E pointing.The Y0axis coincides with the motor axis of the second branch and is along O0A pointing.The Y axis coincides with the support axis of the moving platform and is along OD pointing.The X axis coincides with the motor axis of first branch and is opposite to OC pointing.The Zijaxis coincides with the axis of the j-th revolute joint of the i-th branch and points to the outside of the ball.The Xijaxis is along the outer normal of the plane which is derived from the Zijaxis and the Zij+1axis.

Fig.1 Spherical parallel mechanism UP+R with 2-DOF

ui，wiand videnote the unit vector along Zi1，Zi2，Zi3(i=1，2)in the static coordinate system respectively，and their positive directions point to the outside.

For convenience of study，the deformation and gap of all rotation pairs are ignored and the following assumptions are adopted:continuity assumption，uniformity assumption，isotropy assumption，the mechanism meets the small elastic deformation conditions and the moving platform is a rigid body.

Although all the links of the mechanism are arc，the radius of the arc is larger than the size of the crosssection of the links，so these links can be regarded as straight bars to analyze their deformation.Compared with the torsion and bending deformation，the axial deformation and the shear deformation which is perpendicular to link are much smaller，so they are neglected.

For convenience of the description of the deformation of links，a new coordinate system is built as follows:the S axis is along the axis of the curved bar;the Y axis is along the curvature radius;the Z axis is perpendicular to the plane of the curved bar.

2.1 Deformation of Link 2－1

As shown in Fig.2，the bottom of Link 2－1 is subject to six constraints and can be simplified to a fixed end，while the top of Link 2－1 is subject to three forces F″22，R″22，P″22and two torques M″2r2，M″2P2.

m21(θ)，m21t(θ)and m21w(θ)denote the torques at arbitrary cross-sectionθof Link 2－1 caused by the above force respectively.Whenθ=90°，denotingθ'21z，υ'21Has the angle，deflection caused by m21(θ)，Φ21the angle caused by m21t(θ)，θ'21y，υ'21Vthe angle，deflection caused by m21w(θ).

Fig.2 Simplified model of Link 2－1

For the top of Link 2－1，the line displacement and the angle displacement which can pass to Link 2－2 are as follows:

In addition，the deformations of the two branches make the moving platform and Link 2－2 has a tiny pose adjustment.As a result，Link 2－1 has an implicated angle displacementγ2along w2relative to Link 2－2，which is shown in Fig.3.

Fig.3 Transmission of deformation of the second branch

The actual angle displacement of the top of Link 2－1 is as follows:

The implicated line displacement caused by the above angle displacement is as follows:

2.2 Deformation of Link 2－2

As shown in Fig.4，the junction between Link 2－2 and Link 2－1 is subject to five constraints，which can be simplified to a revolute pair.While the junction between Link 2－2 and the moving platform is subject to three forces F'23，R'23，P'23and two torques M'2r3，M'2p3.

m22(θ)，m22t(θ)and m21w(θ)denote the torques at arbitrary cross-sectionθof Link 2－2 caused by the above force respectively.Whenθ=90°，denotingθ'22z，υ'22Has the angle，deflection caused by m22(θ)，Φ22the angle caused by m22t(θ)，the angle and deflection caused by m21w(θ)are both zero.

Fig.4 Simplified model of Link 2－2

For the junction between Link 2－2 and the moving platform，the line displacement and the angle displacement caused by the deformation of Link 2－2 are as follows:

And then，according to the small strain superposition principle，the real line displacement and the total angular displacement are as follows:

2.3 Deformation of Link 1

As shown in Fig.5，Link 1 is split into Link 1－2 and Link 1－3 to facilitate the description of deformation.

As shown in Fig.6，the junction between Link 1－2 and the motor is subject to six constraints，which can be simplified to a fixed end.The other endpoint of Link 1－2 is subject to three forces F'1j，R'1j，P'1jand two torques M'1rj，M'1pj(j=2).Link 1－3 is just the same(j=3).

m1j(θ)，m1jt(θ)and m1jw(θ)denote the torques at arbitrary cross-sectionθof Link 1－j caused by the above force respectively.Whenθ=90°，denoting θ'1jz，υ'1jHas the angle，deflection caused by m1j(θ)，Φ1jthe angle caused by m1jt(θ)，θ'1jy，υ'1jVthe angle，deflection caused by m1jw(θ).

Fig.5 Diagram of the division of Link 1

Fig.6 Simplified model of Link 1－2 and Link 1－3

For the top endpoint of Link 1－2 and Link 1－3，the line displacement and the angle displacement which can pass to the next link are as follows respectively.

2.4 Angle Displacement of the Moving Platform and the Line Displacement of the Spherical Center

As shown in Fig.7，the projections of the line displacements of three vertices on three edges are equal.

Substitute the first part of Eqs.(5)and(6)into Eq.(8)，and thenγ2can be solved.

The angle displacement of the moving platform transmitted by Link 2，Link 1－2 and Link 1－3 are as follows:

In addition，the line displacements of three vertices of the moving platform are different，so each vertex has an additional angle displacement respectively，which are denoted asζ2，ζ12andζ13.Then the real angle displacements of three vertices are as follows:

Because the moving platform is regarded as rigid body，every point of the platform has the same angle displacement，that is

Thereforeζ2，ζ12andζ13can be derived out.

The relationship between the line displacements of three vertices and the angle displacement is shown in Fig.8.

Fig.7 Deformation of the moving platform

Fig.8 Relationship between the displacement vectors

And then the following equation can be obtained.

whereδD2and Δo2represents the angle displacement of the moving platform and the line displacement of spherical center respectively.

3 Stiffness Matrix of the UP+R Mechanism

Supposing Fg=[F M]Tis the external load of the UP+Rmechanism，δis the angle displacement of the moving platform， Δis the line displacement of spherical center，and the following equation can be derived.

where K is the stiffness matrix of the mechanism.

where C2is the 4×6 coefficient matrix derived from statics analysis，and each element of the matrix is the function of the parameters and pose of the mechanism.G2=[a2fa2pa2ra2m]is the 3×4 coefficient matrix，and

With the same method，the following equation can be derived.

where C22is the 12×6 coefficient matrix deriving from statics and each element of the matrix is the function of the parameters and pose of the mechanism，G22=［G221G222］is the 3×12 coefficient matrix，and

where af23，af13，…，c21m2p3are the functions of the pose of the mechanism.

The relationship between the total displacement of spherical point and the external load is derived as follows:

The compliance matrix of the mechanism is as follows:

IfΞis invertible，Eq.(16)can be rewritten as

Comparing with Eq.(13)，the stiffness matrix of the UP+Rmechanism can be obtained as follows:

4 Principal Stiffness of the Mechanism and Their Directions

Diagonalize the stiffness matrix by orthogonal transform method，and then the six principal stiffness，their directions of this mechanism are obtained.

For the stiffness matrix K，denote the eigenvalues asλ1～λ6，the eigenvectors x1～x6，the orthogonal vectorsβ1，β2，β3，β4，β5，β6，and the transformation matrix is P=(β1…β6)，P is a 6×6 invertible matrix.

From the above analysis，the following equation can be derived.

Given the following substitution:

where[F'M']T，[ Δ'δ']Trespectively denotes the external load and the total displacement of spherical point with respect to the new coordinate O'x'1x'2x'3x'4x'5x'6.

And then，we have whereΛdenotes the stiffness matrix and k'11，k'22，…，k'66denote the six principal stiffness respectively.

The unit vectors along the axes of the new coordinate are expressed as:

In the original coordinate system，we have X″i=β'i(i=1，2，…，6)，and then the directions of the principal stiffness isβi(i=1，2，…，6).

5 Numerical Calculation of the Stiffness of the Mechanism

For all the links，the material is 45 steel，the radius of the branch is adopted as:r1=30 mm，r2=50 mm，and the sectional dimensions are:h=10 mm，b=10 mm，β=0.141.

Three output poses of the moving platform are selected as example，and the static stiffness matrixes of the mechanism are calculated separately;these stiffness matrixes are diagonalized，and then the six principal stiffness and their directions in each output pose can be obtained separately as shown in Table 1.

Table 1 Principal stiffness and directions in three output poses of the mechanism

The results show that the principal stiffness of the mechanism is large in three certain directions，however it is small in another three directions，and the gap is huge.That is，for this mechanism，the line displacement stiffness is much smaller than the angle displacement stiffness.

The principal stiffness and directions are varied with the pose of the mechanism，and the principal stiffness is significantly increased when the output pose far away from the zero point(φx=φy=0).

6 Conclusions

Based on the small strain superposition principle，the static stiffness matrix of the spherical parallel mechanism UP+Ris obtained，and the static stiffness characteristics of this mechanism is analyzed in detail.The results show that the overall static stiffness is not only related with the parameters and pose of the mechanism，but also is affected by the overconstraint condition.The line displacement stiffness is much smaller than the angle displacement stiffness.When the pose of the mechanism is away from the zero point，the stiffness will be increased significantly，which means that the flexibility of mechanism reduces.The results will provide reference and basis for the stiffness control，parameter selection and further development of this spherical parallel mechanism.

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