Geometric error analysis of an over-constrained parallel tracking mechanism using the screw theory

Jiteng ZHANG , Binbin LIAN ,b,*, Yimin SONG

a Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education, Tianjin University, Tianjin 300072, China

b Department of Machine Design, KTH Royal Institute of Technology, Stockholm SE-10044, Sweden

KEYWORDS Error model simulation;Geometric error modeling;Over-constrained parallel mechanism;Screw theory;Sensitivity analysis

Abstract This paper deals with geometric error modeling and sensitivity analysis of an overconstrained parallel tracking mechanism. The main contribution is the consideration of overconstrained features that are usually ignored in previous research. The reciprocal property between a motion and a force is applied to tackle this problem in the framework of the screw theory. First of all, a nominal kinematic model of the parallel tracking mechanism is formulated. On this basis, the actual twist of the moving platform is computed through the superposition of the joint twist and geometric errors. The actuation and constrained wrenches of each limb are applied to exclude the joint displacement. After eliminating repeated errors brought by the multiplication of wrenches, a geometric error model of the parallel tracking mechanism is built. Furthermore,two sensitivity indices are def ined to select essential geometric errors for future kinematic calibration. Finally, the geometric error model with minimum geometric errors is verif ied by simulation with SolidWorks software. Two typical poses of the parallel tracking mechanism are selected, and the differences between simulation and calculation results are very small. The results conf irm the correctness and accuracy of the geometric error modeling method for over-constrained parallel mechanisms.

1. Introduction

Real-time target tracking systems with high precision are vastly required in industrial, medical, and military domains.Their design and control have become a research hotspot.1,2Among the main issues to be tackled, a key problem is the development of a tracking mechanism with a large workspace and a high accuracy.3,4Over the past decades,intensive efforts have been made to topological synthesis5,6and kinematic analysis and design7,8of tracking mechanisms. In this regard, we proposed a 2 rotational Degree-of-Freedom (DoF) Parallel Tracking Mechanism (PTM), as is shown in Fig. 1.9It is with a symmetrical structure, up to 90°rotational angles, and potentially high stiffness.We assume it as a promising solution for a target tracking system with high precision.10Before applying the PTM to build a tracking system, it is found that position and orientation inaccuracies of the PTM have great effects on the precision of tracking trajectory.Therefore,kinematic calibration, the technique to identify and compensate geometric errors, is inevitable to improve the accuracy of the PTM.11,12In order to implement kinematic calibration,a thorough understanding of the generation and transmission of component errors is the f irst step.Furthermore, a comprehensive analysis of the effects from these errors on mechanism accuracy is also necessary.

In the former direction, error modeling between geometric errors of parts and pose errors of the PTM is to be investigated. From the mathematical perspective, geometric errors resulted from a construction process can be described by the deviations of nominal and real kinematic models.13To study the actual kinematic features of mechanisms, scholars have applied different mathematical tools in geometric error modeling. Commonly adopted approaches are Denavit-Hartenberg(D-H)convention,the Product-of-Exponential(PoE)formula,and the screw theory.

D-H convention is a matrix-based method. The relative position and orientation of adjacent bodies are described by 4×4 homogeneous transformation matrices. Meanwhile, the nominal kinematic model of a serial limb or mechanism is the successive multiplication of these matrices. The actual kinematic model is formulated by taking small perturbation of every element in each matrix.14,15Harb and Burdekin16established a geometric error model of a spatial serial manipulator through matrix operations.For the 6-DoF Stewart mechanism, Wang and Masory17applied D-H convention to deal with the error model of each serial limb.Similarly,a geometric error analysis of some lower-mobility parallel mechanisms18,19was carried out by D-H convention. In most cases, D-H convention is chosen because (1) geometric errors of parts can be included by exhaustive differentiation,and(2)transmissions of geometric errors can be computed by multiplication and summation of matrices.However,it has been pointed out that D-H convention is not applicable for geometric error modeling of a mechanism having collinear adjacent joint axes.20

An alternative to tackle the problem is modif ied models based on D-H matrices, for instance, the six-parameter representation S-model21or the continuous and parametrically complete (CPC) model.22In these models, two additional parameters are added to allow arbitrary displacements of joint axes. Hence, the singularity problem caused by collinear adjacent joint axes is f ixed. Another option is the PoE formula. It can handle kinematic singularity by smoothly changing kinematic parameters. Moreover, different types of joints are uniformly described and modeled.23Park and Okamura24,25applied global frame representation and formulated an error model of an open-loop manipulator by using the global PoE model. Aiming at a complete, minimal, and continuous error model, Chen et al.20combined both global and local PoE formulas to analyze the errors of a serial mechanism. Although both methods are effective for the singular problem caused by collinear axes, the modeling process can be tedious when closed-loop mechanisms like the PTM are involved. This is because displacements from passive joints are included in these methods, making the geometric error model diff icult to be directly applied for kinematic calibration.

Being able to describe joint axes in a concise manner, the screw theory has been widely applied in mechanism analysis,including kinematic,stiffness,and dynamics.For the kinematics of parallel mechanisms,motions of joint axes are described by twist, and limb forces are denoted by wrench. Nominal kinematics of parallel mechanisms are computed by the accumulation of joint twist, and the actual kinematic is calculated by linear superposition of joint twist and geometric errors.26Joint displacements can be eliminated from geometric error models through the reciprocal property between twist and wrench.27,28Along this track, Charker et al.29formulated geometric error models of spatial parallel mechanisms and analyzed their position and orientation errors. Kumaraswamy et al.30proposed a screw theory-based framework for tolerance analysis of planar and spatial manipulators. By identifying joint displacements as non-compensable errors, Liu et al.31investigated a geometric error modeling method of lowermobility parallel mechanisms through the Jacobian matrix.Also relying on the generalized Jacobian matrix,Sun et al.32,33worked on the geometric error analysis of a 3-DoF parallel mechanism, which is of great help in kinematic calibrations.The effectiveness of the screw theory in geometric error modeling of parallel mechanisms has been highly recognized.However, little attention has been paid to over-constrained parallel mechanisms.

Over-constrained parallel mechanisms are those that have redundant constraints.34-37They are welcomed in practice because their stiffness and operational stability are usually better than those of non-overconstrained mechanisms. For instance,additional constrained forces are applied by two passive limbs in the PTM, and these over-constrained forces contribute to avoidance of kinematic singularity and enhancement of rigidity.10An introduction of over constraints is benef icial to the performance of parallel mechanisms, but it also brings diff iculties in geometric error modeling by the screw theory. Commonly, a 6×6 non-singular Jacobian matrix consisting actuated and constrained wrenches is applied to eliminate joint displacements in each limb. However, the constrained wrenches of over-constrained parallel mechanisms in the Jacobian matrix are usually equivalent wrenches. They are different from the actual constraints within limbs. An analysis and application of actual wrenches is necessary for accurate geometric error modeling.

What's more, due to the numerous and complicated geometric errors of parts, it remains a diff icult task for kinematic calibration. Hence, f inding out the main geometric errors is essential for increasing eff iciency of kinematic calibration.38To this end, sensitivity analysis plays an important role in assessing the effects of geometric errors on mechanism accuracy. Two approaches in terms of analytical and probabilistic methods are involved so far. Through linearization of kinematic equations, Caro et al.39worked on the sensitivity analysis of a 3-DoF parallel mechanism. Utilizing an interval analysis method, Wu and Rao40obtained sensitivity coeff icients of geometric errors from non-linear equations of a parallel mechanism. It has been pointed out that the analytical approach is computationally expensive and only suitable for parallel mechanisms with a simple structure.41On the contrary,the probabilistic approach can deal with a large amount of geometric errors in an eff icient manner. Sun et al.32and Chen et al.42assumed geometric errors following a normal distribution, and carried out sensitivity analysis of a 3-DoF parallel mechanism and an SCARA mechanism based on probabilistic models. However, Li et al.38mentioned that the probabilistic assumption is hard to guarantee in practical use. In order to eff iciently f ind out the main geometric errors for the PTM, an effective sensitivity analysis method is badly required.

In summary, kinematic calibration is of signif icance to the development of a target tracking system with high precision.Though substantial progress has been made in geometric error modeling and sensitivity analysis, methods for overconstrained parallel mechanisms are limited or even not appropriate.Taking the PTM as an example,the present study investigates generation and transmission of geometric errors,as well as the sensitivity of these geometric errors to over-constrained parallel mechanisms. Reminder of this paper is as follows.Section 2 brief ly introduces the structure of the PTM and formulates its nominal kinematic model by the screw theory.Section 3 carries out geometric error modeling of the PTM and summarizes the general error modeling procedure for over-constrained parallel mechanisms. Sensitivity analysis is implemented to select the essential geometric errors for kinematic calibration in Section 4, while verif ication of the geometric error model through simulation in Solid Works software is implemented in Section 5. Conclusions are drawn in Section 6.

2. An over-constrained parallel tracking mechanism and its nominal kinematics

As shown in Fig. 1, the main body of the PTM consists of a f ixed base, f ive limbs (four RSR limbs and one SS limb), and a moving platform. The RSR limbs are formed by a revolute(R) joint, a spherical (S) joint, and then a revolute joint. The SS limb is composed of two spherical joints. The f irst R joint in the 1st and 2nd limbs is connected to torque motors.The lengths of two links within each limb are required to be the same, and the links among limbs should be identical. The PTM has two rotational capabilities. In a physical prototype,the S joint is replaced by three R joints whose rotational axes are linearly independent. Through such an arrangement, the mobility remains to be the same while rotational angles are bigger.

Some notations and coordinate frames are assigned for the nominal kinematics of the PTM.Centers of the f ixed base and the moving platform are denoted by O and CE, respectively.They are also centers of S joints in the SS limb. Centers of joints within an RSR limb are represented by Bi, Si, and Ai(i=1,2,3,4)in sequence.Taking point O as the origin,a f ixed coordinate frame O-xyz is def ined. The x-axis points from point O to point B1, while the z-axis is perpendicular to the f ixed base. A moving coordinate frame CE-uvw is established at point CE. Its u-axis is the direction from point CEto point A1, and w-axis is normal to the plane of the moving platform.The frames satisfy the right-hand rule.Concerning the geometric constraints,the inverse position formulation of the PTM is derived by Qi et al.43. On this basis, its input-output velocity model is f irstly carried out by the screw theory in the present study.

Generally, a six-dimensional basis is associated to describe the space of an instantaneous motion (or twist). The elements of the basis are three rotations about and three translations along axes of the Cartesian frame.The basis can also be interpreted as a wrench, in which the basis vectors are pure forces along and moments about the coordinate axes. In fact,wrenches constitute the dual vector space of a twist, i.e., the standard basis of a twist and a wrench forms the basis of a six-dimensional space. The action of a wrench on a twist is the instantaneous work contributed by the wrench during the motion along the twist. This is def ined as the reciprocal screw product. If a wrench does not do work on a twist, their reciprocal product is zero, and then the wrench and the twist are described as being reciprocal. The reciprocal product can also be expressed by the generalized inner product if the twist screw is described in axis-coordinate as $t=[vT,ωT]T=[vx,vy,vz,ωx,ωy,ωz]T. Meanwhile, the coordinates of the wrench in the standard basis are denoted by $w=[fT,mT]T=[fx,fy,fz,mx,my,mz]T.

When dealing with an instantaneous motion of parts whose connecting joint is a revolute joint, a twist can be determined by the rotational axis as$t=[rr×sr,sr]T,herein sris the vector of the rotational axis and rris the vector pointing from the origin of the coordinate frame to any point on the axis.Similarly, when the connecting joint of parts is a prismatic joint, a twist would be described by the translational axis as$t=[sp,0]T, in which spis the vector of the translational axis.It is noted that joints with more than 1 DoF can always be substituted by several 1-DoF joints, and the twist of a serial linkage is derived by the superposition of the 1-DoF joint twists.For the PTM, the twist at point CEcan be expressed via RSR and SS limbs as follows:

where rCB,i, rCS,i, and rCA,iare the vectors from point CEto points Bi, Si, and Ai, respectively. sj,i(j=1,2,3,4,5,i=1,2,3,4) denotes the vector of the 1-DoF joint within the RSR limb.The twists are described in the instantaneous frame which is assigned to point CEand parallel to the f ixed frame.Similarly, the joint twist in the SS limb is derived as follows:

where r is the vector from point O to point CE, and sj,5is the vector of the 1-DoF joint within the SS limb.

Referring to the reciprocal properties of a twist and a wrench, the constraint wrench of an RSR limb is obtained by f inding out the six-dimensional vector having zero inner products with all the f ive twists, i.e.,

where sc,i=l1,i×l2,i, in which l1,i=s5,i×rSA,iand l2,i=s1,i×rBS,i.

With the same manner, the constrained wrench of the SS limb is computed as follows:

where H is the length of links SS. By locking the actuated R joint in the 1st RSR limb,the actuated wrench can be obtained from the zero products with the rest four twists and one constrained wrench. Similarly, the actuated wrench of the 2nd RSR limb is derived as

where L is the length of links RS, rSA,1are the vectors from point S1to point A1, and rSA,2are the vectors from point S2to point A2.

It is found that two actuated wrenches and f ive constrained wrenches are applied to the moving platform of the PTM. As the dimension of the Jacobian matrix is 6×6, there is one redundant constrained wrench. Usually redundant wrenches are excluded in the Jacobian matrix, as has been done by Sun et al.28. However, not only the wrenches in the full-rank Jacobian matrix but also the exact wrenches in each limb are applied for geometric error modeling in the present study.This is more accurate in the elimination of joint displacements.

3. Geometric error modeling

Geometric errors resulted from the matching and assembling process are described by the deviation of the joint axis.Transmissions of these errors are from the actual joint axis to the limb and then the whole mechanism. Therefore, geometric error modeling starts from an analysis of the actual adjacent joint axes. On this basis, the actual twist of each limb is derived, and the elimination of the passive joint displacement is implemented. Finally, a geometric model of the mechanism is formulated.

Following this procedure, body-f ixed reference frames are f irstly assigned to the actual joint axes of the PTM for the convenience of analyzing their relation. These frames are denoted by Rj,i(i=1,2,3,4, j=1,2,...,6). The zj,i-axis is the axis of each 1-DoF joint, and the xj,i-axis is perpendicular to both the zj,i-axis and the zj+1,i-axis. The yj,i-axis follows the righthand rule. As is shown in Fig. 2, point Pj,iis applied to represent the origin of the frames. Points P1,iand P2,iare the intersections of the z1,i-axis and the x1,i-axis, the z2,i-axis and the x2,i-axis,respectively.The other Pj,irepresents the intersection of the xj-1,i-axis and the zj,i-axis. The frame on the f ixed base R0,iis def ined by rotating frame O-xyz about the z-axis with an angle of (i-1)π/2. The frame on the moving platform is assigned to point CEwhose x6,i-axis is from point CEto point Aiand the z6,i-axis is colinear with the w-axis.

Similarly, body-f ixed reference frames Rj,5are established for the actual joint axis of the SS limb. The zj,5-axis denotes the axis of the j th 1-DoF joint in the SS limb, and the xj,5-axis is perpendicular to both the zj,5-axis and the zj+1,5-axis.The origin point Pj,5of frame Rj,5is the intersection of the zj,5-axis and the xj-1,5-axis, while point P1,5is the intersection of the z1,5-axis and the x1,5-axis. Frame R0,5is just the frame O-xyz, and frame R6,5is coincide with the body-f ixed frame CE-uvw of the moving platform.

Fig. 2 Actual schematic diagram of RSR and SS limbs.

With the assigned body-f ixed frames,the transformation of actual joint axis can be computed.Then the transformations of geometric errors within limbs are analyzed.It is noted that the four RSR limbs of the PTM are identical.Therefore,geometric error modeling of one RSR limb and the SS limb is demonstrated in the following sections.

3.1. Geometric error model of one RSR limb

Referring to the rotational and translational matrices between coordinate frames, the transformation of the actual joint axis within the i th RSR limb (i=1,2,3,4) can be described as follows:

where Trans(-r)is the homogeneous transformation matrix of frame O-xyz with respect to instantaneous frame CE-x′y′z′,which is parallel to frame O-xyz. Trans(x,y) is the homogeneous transformation matrix for the translation along the xaxis with a distance y. Rot(x,y) is the homogeneous transformation matrix for the rotation about the x-axis with an angle y.a denotes the radius of the moving platform.θHand θLrepresent the structural angles of the upper and lower links of an RSR limb, respectively. θj,i(j=1,2,3,4,5, i=1,2,3,4) denotes a rotational angle about the j th joint axis of the i th RSR limb,which can be solved by inverse kinematic analysis.

Generally,the number for geometric errors of any joint axis should be six, including three position errors along and three orientation errors about the axes of the coordinate frame.However, it has been proven in a previous work that there are redundant errors within a serial limb. These redundant errors are linearly dependent on the other geometric errors.28Hence, they would not affect mechanism accuracy independently. As a result, they cannot be identif ied in kinematic calibration.For this reason,redundant errors are excluded in the geometric error modeling process. For the i th RSR limb(i=1,2,3,4), all the geometric errors are listed as follows:

wherej-1δxj,i,j-1δyj,i,andj-1δzj,iare the position errors of center point Pj,iin frame Rj,iwith respect to frame Rj-1,i, whilej-1δαj,i,j-1δβj,i, andj-1δγj,iare the corresponding orentation errors. In practice,5Δ6,iis replaced by6Δ5,ibecause it is easier to evaluate the geometric errors of frame R5,iwith respect to frame R6,i.

With the transformation matrix and def ined geometric errors of each joint axis, the actual twist of an RSR limb can be expressed as

where Δθa,j,idenotes the error of θa,j,i.iAdC,j-1is the 6×6 adjoint transformation matrix of frame Rj-1,iwith respect to frame, in whichj-1rj,iis the position vector of the origin of frame Rj,iin frame Rj-1,i.

Taking the inner product on both sides of Eq.(7)with ^$wa,ito eliminate the perturbations of passive joints yields

where εae,iis the geometric errors of the i th RSR limb, Eae,iis the relative error coeff icient matrices, and

where θj,5(j=1,2,3,4,5) represents a rotational angle about the j th joint axis of the SS limb. It can also be solved through inverse kinematic analysis.17

Geometric errors of each joint in the SS limb are expressed as

herein, the expressions of each element in Eae,iare listed in Appendix A.

Similarly, taking the inner product on both sides of Eq.(7)with ^$wc,ileads to

where the geometric errors εce,iof the i th RSR limb and the error coeff icient matrices Ece,iare expressed as follows:

wherej-1δxj,5,j-1δyj,5, andj-1δzj,5are the position errors of center point Pj+1,5in frame Rj,5with respect to frame Rj-1,5.j-1δαj,5,j-1δβj,5, andj-1δγj,5are the corresponding rotational errors.

The actual twist of the SS limb can be obtained by Eqs.(11)and (12) as

herein, the expressions of each element in Ece,iare listed in Appendix A.

3.2. Geometric error model of the SS limb

Geometric error modeling of the SS limb is tackled in a similar manner to that of RSR limbs. First of all, the transformation matrices of joint axes are computed as follows:

where Δθa,j,5denotes the error of θa,j,5.5AdC,j-1is the 6×6 adjoint transformation matrix of frame Rj-1,5with respect to frame, in whichj-1rj,5is the position vector of the origin Pj,5of frame Rj,5in frame Rj-1,5.

Taking the inner product on both sides of Eq.(8)with ^$wc,5to eliminate the displacement of passive joints yields

where the geometric errors εce,5and the error coeff icient matrices Ece,5of the SS limb are expressed as follows:

3.3. Geometric error model of the PTM

During the process of applying wrenches to eliminate joint displacements from the geometric error model of limbs, it is found that some geometric errors are repeatedly included.These repeated errors contribute to the singularity of the error model and result in ambiguity in kinematic calibration.By getting rid of repeated geometric errors, the geometric errors of the PTM are written in a vector form as follows:

where

In addition,the relevant error coeff icient matrices are modif ied as follows:

The actual constrained wrenches of each limb are included in the geometric error model. There are more than six wrenches because of the over-constrained features. If we apply the full rank Jacobian as in the previous research, the

Combine the error twists of the f ive limbs,and rewrite them into a matrix form.The geometric error model of the PTM can be expressed as

where

over-constraints would not be fully considered. It is of importance to eliminate joint displacements at the limb phase for geometric error modeling of over-constrained parallel mechanisms. Multiplying the generalized inverse matrixto both sides of Eq. (19) yields

Fig. 3 Geometric error modeling procedure of over-constrained parallel mechanisms.

where Jeis the error Jacobian matrix of the PTM.There are 83 geometric errors of parts affecting the pose of the moving platform, including 2 home position errors.

3.4. Geometric error modeling of over-constrained parallel mechanisms

Taking the PTM as an example, the procedure for geometric error modeling of over-constrained parallel mechanisms can be summarized as follows (see Fig. 3).

Step 1: Replace joints within each limb with 1-DoF joints and assign body-f ixed frames to joint axes.

Step 2: Compute transformation matrices and def ine geometric errors between adjacent frames.

Step 3:Formulate the twist of a serial limb by two subsets,joint displacement and geometric errors.

Step 4: Exclude joint displacements by the actual wrenches inserted into the limb.

Step 5: Eliminate repeated geometric errors and combine the twists of all serial limbs to formulate a geometric error model.

The proposed procedure is also applicable to any types of parallel mechanism, including proper constrained and redundant parallel mechanisms.

4. Sensitivity analysis

Before employing the geometric error model in kinematic calibration, sensitivity analysis is carried out to select essential geometric errors. Mechanism errors at the end reference point include position and orientation errors. Since the units are different, it is inappropriate to simultaneously deal with these two subsets of geometric errors in the sensitivity analysis.Based on the geometric error model shown in Eq. (20), the error Jacobian matrix is divided into two sub-matrices corresponding to position and orientation errors, and two local sensitivity indices at a specif ic conf iguration are def ined as

where Je,mnis the element in the m th row and n th column of Je.ˉμrnand ˉμαndenote the position and orientation volumetric errors of the moving platform with respect to the n th geometric error (n=1,2,3, ... ,83), respectively. Herein, volumetric errors are applied to take into account the average errors along or about the x, y, z-axis.

In order to f ind out the vital geometric errors within the workspace,the global sensitivity indices are given by the average inf luence of geometric errors in the whole workspace as follows:

where V represents the workspace.

Therefore, a sensitivity analysis of the PTM can be conducted by Eq.(22).For the PTM shown in Fig.1,the circumradius of the moving platform and the f ixed base is 150 mm.The length of the link within an RSR limb is 317 mm, while that of the SS limb is 402 mm. Its orientation workspace is denoted by the azimuth angle φ and the tilt angle θ, and θ ∈[0°, 60°], φ ∈[0°, 360°]. Inspired by the statistic method, the workspace is discretized by 10°.In total,217 poses are derived.Through calculating position and orientation errors at each pose by Eq. (21) and then computing the average, the global sensitivity indices (and) are obtained, as are shown in Figs. 4 and 5, respectively.

Fig. 4 Sensitivity of geometric errors to position volumetric errors μr.

Fig. 5 Sensitivity of geometric errors to orientation volumetric errors μα.

From the sensitivity analysis of the PTM, it is found out that the home position error Δθa,1,i(i=1,2), rotational errors such as0δα1,i,0δγ1,i,4δα5,i,5δα6,i, and5δγ6,i, and translational errors such as0δx1,i,0δz1,i,1δx2,i,1δy2,i,4δz5,i,5δx6,i,5δz6,i,0δz1,5,1δx2,5,3δz4,5,4δx5,5,and6δz5,5(i=1,2,3,4)all have great effects on the position volumetric error ˉμrand the orientation volumetric error ˉμα. The number of essential geometric errors is 53.On the other hand,there are some other geometric errors having little inf luence on the accuracy of the end reference point, such as3δz4,i,6δy5,i,0δx1,5,0δy1,5,6δx5,5, and6δy5,5.

5. Verif ication of the geometric error model

In this section, the computed geometric error model of the PTM is verif ied by Solid Works software. By introducing geometric errors to a virtual prototype of the PTM, the measured coordinates from Solid Works are compared with those from the geometric error model. By giving certain values to the inputs of the PTM, the virtual prototype built by SolidWorks would drive to the expected pose without errors,which indicates that the virtual prototype is ideal. Through introducing geometric errors to the virtual prototype, the moving platform will deviate from the expected pose, and the measured pose errors will be applied to compare with the calculated pose errors under the same geometric errors.In this way, the geometric error model is verif ied if the two results are close.

Geometric errors are f irstly given in Table 1.Def ine RMand RTas the measured and theoretical orientation matrices of frame CE-uvw with respect to frame O-xyz. Assign amand atas the measured and theoretical position vectors of point P1in frame O-xyz. ‖δrM‖ and ‖δαM‖ are the position and orientation volumetric errors obtained by SolidWorks, while‖δrC‖ and ‖δαC‖ are the calculated position and orientation errors from the geometric error model.The verif ication process is implemented as is shown in Fig. 6, which is summarized as follows.

Table 1 Given values of geometric errors.

Fig. 6 Verif ication procedure of the geometric error model.

Fig. 7 Verif ication of position errors (φ=θ=0°).

1) Select one geometric error from Table 1 and assign the other errors in εeto be zero. The given error is set to be larger than its possible value for the convenience of verif ication.

Fig. 8 Verif ication of orientation errors (φ=θ=0°).

2) Choose a typical pose within the workspace(φ=θ=0°or φ=20°,θ=15°).

3) Obtain position and orientation errors of the end reference point in SolidWorks with the selected geometric errors in step 1 under the given conf iguration in step 2.Firstly, the selected geometric errors are inserted into the virtual prototype.A f ixed frame and a moving reference frame are established.Then the moving platform is driven to the given conf iguration.Moreover,the coordinates of three non-collinear points P1, P2, and P3are measured. The u-axis is decided by points P1and P2,while the w-axis is determined by points P1, P2, and P3.Hence, RM, RT, am, and atare obtained, with which‖δrM‖ and ‖δαM‖ can be formulated.

4) Derive position and orientation errors of the end reference point calculated by the geometric error model with the selected geometric errors in step 1 under the given conf iguration in step 2. The geometric error model is programed by Matlab software. ‖δrC‖ and ‖δαC‖ can be achieved by setting the selected geometric errors under the given conf iguration.

5) Compare the simulation results from step 3 with the calculation results from step 4.

6) Go back to step 1. Choose another geometric error and repeat Step 1 to Step 5.

Fig. 7 shows the position volumetric errors of the verif ication process when φ=θ=0°. The blue bars are calculated position volumetric errors ‖δrC‖, while the red bars are position volumetric errors ‖δrM‖ obtained from Solid Works. The maximum deviation between ‖δrC‖ and ‖δrM‖ is 1.4 μm.Fig.8 illustrates a comparison of orientation volumetric errors when φ=θ=0°.The blue bars are calculated orientation volumetric errors‖δαC‖,and the red bars are orientation volumetric errors ‖δαM‖ from Solid Works. The maximum deviation between ‖δαC‖ and ‖δαM‖ is 0.0054°. Tables 2 and 3 show comparisons of position and orientation errors whenφ=20°,θ=15°. The maximum deviations are 5.8 μm and-0.0063°, respectively. The differences from SolidWorks and calculation are very small.The accuracy of the geometric error model of the PTM is conf irmed, and the proposed geometric error modeling method is proven to be effective.

Table 2 Verif ication of position errors (φ=20°, θ=15°).

Table 3 Verif ication of orientation errors(φ=20°,θ=15°).

6. Conclusions

This paper deals with geometric error modeling and sensitivity analysis of an over-constrained parallel mechanism based on the screw theory. Conclusions are drawn as follows:

(1) A nominal kinematic model is established by the reciprocal property of the screw theory. Instead of formulating a full rank Jacobian matrix, the actual actuation and constraint wrenches of each limb are computed.

(2) The actual twist of the PTM is computed by each RSR limb and the SS limb. It is the superposition of joint twists and geometric errors within limbs.Body-f ixed reference frames are assigned to the actual joint axis. Geometric errors are expressed as deviations of nominal and actual joint axes, and their transmissions are conducted by transformation matrices. Then the actuation and constraint wrenches are applied to exclude joint displacements, and the repeated geometric errors brought by the exclusion are eliminated. Through this geometric error modeling process,the generation and transmission of geometric errors of over-constrained parallel mechanisms are clearly def ined and computed.

(3) In the light of an error Jacobian matrix, global sensitivity indices are def ined, and a sensitivity analysis of the PTM is carried out. 53 geometric errors are selected from the original 83 errors,which helps increase the eff iciency in future kinematic calibration. Finally, the geometric error model with minimum errors is verif ied by SolidWorks software. Results conf irm the correctness of the proposed geometric error modeling method. The proposed geometric error modeling method can also be applied to other types of parallel mechanisms.

Acknowledgments

This research work was supported by the National Natural Science Foundation of China [No. 51475321], Tianjin Research Program of Application Foundation and Advanced Technology [No. 15JCZDJC38900 and 16JCYBJC19300],and the International Postdoctoral Exchange Fellowship Program [No. 32 Document of OCPC, 2017].

Appendix A

The expressions of each element in Eae,iare shown as follows:

The expressions of each element in Ece,iare shown as follows: