Study of error modeling in kinematic calibration of parallel manipulators

Abstract

Error modeling is the foundation of a kinematic calibration which is a main approach to assure the accuracy of parallel manipulators. This article investigates the influence of error model on the kinematic calibration of parallel manipulators. Based on the coupling analysis between error parameters, an identifiability index for evaluating the error model is proposed. Taking a 3PRS parallel manipulator as an example, three error models with different values of identifiability index are given. With the same parameter identification, measurement, and compensation method, the computer simulations and prototype experiments of the kinematic calibration with each error model are performed. The simulation and experiment results show that the kinematic calibration using the error model with a bigger value of identifiability index can lead to a better accuracy of the manipulator. Then, an approach of error modeling is proposed to obtain a bigger value of identifiability index. The study of this article is useful for error modeling in kinematic calibration of other parallel manipulators.

Introduction

A parallel manipulator has become a research hot spot and attracting more and more attentions from both academia and industry after the parallel machine tool Hexapod was proposed in 1994. ¹ Compared with a serial manipulator, a parallel manipulator shows the characteristics of high stiffness–weight ratio, ² high adaptability to circumstance, ³ high-response rate, ⁴ and high-speed motions ^5,6 because of its structure feature. However, the drawback of low accuracy restricts the further development of parallel manipulators. Hence, a large number of scholars presented many solutions to improve the accuracy, mainly including accuracy design ^7,8 and kinematic calibration. ^9,10 The method of accuracy design makes the cost of assembling parallel manipulators rise a lot and it sometimes does not work well. Kinematic calibrations are studied and employed more widely in practice. This method is easy to put in practice and the cost is low. More and more researches focus on it.

However, there are still some problems in kinematic calibration. The most important one is the singular and ill-conditioning problems of an identification matrix. The problem may make the kinematic calibration badly be performed or directly failed as the identification matrix needs to be inversed when geometrical error parameters are solved in the step of parameter identification. As the error transfer matrix is the foundation of an identification matrix, the above singular and ill-conditioning problems are always solved from two parts: error modeling and parameter identification.

As to parameter identification, a lot of scholars utility numerical algorithms to overcome the problems. Least square method (LSM) is probably used most often. A plenty of scholars ^{11 –14} all adopted LSM to solve geometrical error parameters in the kinematic calibrations of different parallel manipulators. Besides, a part of scholars ^{15 –17} respectively proposed other methods such as an extended Kalman filter algorithm, an algorithm based on the regularization method and an algorithm based on QR decomposition.

To work on error modeling, many scholars investigated it from different aspects. Two general and systematic approaches of error modeling were proposed and both of them analyzed the compensatable and uncompensatable poses. ^18,19 In different situations, different error parameters in error models (EMs) were investigated, respectively, such as the joint clearance-induced errors in parallel wedge precision positioning stage parallel manipulators, ²⁰ the straightness error of guideways in prismatic and universal joint (3-PUU) parallel coordinates measuring machine, ²¹ and the constraint errors in parallel manipulators with decoupled motions. ²² Also, for some special parallel manipulators, EMs are established with special methods like the TAU parallel manipulator with the Jacobian matrix method. ²³ The investigations above focused on the result of error modeling rather than the process of error modeling. In order to improve the singular and ill-conditioning problem, it is used a lot to eliminate redundant geometrical error parameters in the process of error modeling. Two EMs of a 2-DOF (degrees of freedom) planar translational parallel manipulator were investigated. ²⁴ Both of them eliminated some error parameters for overcoming the singular and ill-conditioning problems. Some geometrical error parameters in EMs of different spatial parallel manipulators were also eliminated in a similar way. ^25,26

However, both of the two above approaches always just concentrate on the solvability of identification functions. They both ignore the integrality of the EM. The kind of integrality reflects in the choice of geometrical error parameters. Hence, the choice of geometrical error parameters must be caution. The caution means to balance in two aspects. The one is that the number of geometrical error parameters cannot be too much. If too much, an identification matrix stacked by error transfer matrixes may be singular or ill conditioning as the coupling of redundant geometrical error parameters. This kind of coupling brings a lot of trouble in kinematic calibration, which makes the identification matrix singular. If the identification matrix is singular, the kinematic calibration totally cannot be performed or badly performed. Since geometrical error parameters determine whether an identification matrix is singular, geometrical error parameters should be selected carefully in order to make the identification matrix full rank. Geometrical error parameters can be divided into identification error parameters and nonidentification error parameters. Nonidentification error parameters should be eliminated in EMs, which means that the number of geometrical error parameters cannot be too much. The other one is that the number of geometrical error parameters cannot be too lack. If too lack, it means that the real kinematic feature of the parallel manipulator is not embodied enough in the EM. The integrality of EM refers to all geometrical error parameters which can be identified. This kind of geometrical error parameters that can be identified is named as identification error parameters. It is necessary to extend the number of identification error parameters as much as possible, which leads to a better kinematic calibration. In order to achieve this aim, the error modeling is needed to study deeply.

In this article, a study of error modeling in kinematic calibration of parallel manipulators is presented as follow. In section “Definition of the identifiability index for an EM,” error parameters are investigated for determining an identifiability index. This index is used to embody the identifiability of EMs. In section “Establishing EMs with different values of the identifiability index,” a 3PRS is employed as a case to generate three different EMs. In sections “Comparisons of three EMs in calibration simulations” and “Comparisons of three EMs in practical calibration experiments,” computer simulations and prototype experiments based on the three EMs are respectively performed. In section “Discussions on simulation and experiment results,” according to the investigations of above sections, several results are discussed. Moreover, an approach of error modeling is proposed for improving the accuracy of parallel manipulators. In section “Conclusions,” conclusions are organized.

Definition of the identifiability index for an EM

In this section, three basic formats of single error vector for describing one link are proposed. Then, these three basic formats are developed on every link vector for establishing an EM and the maximum total number of identification error parameters in an EM is determined. In the end of this section, an index for evaluating the identifiability of an EM is proposed, which is called the identifiability index.

Basic formats of single error vector for one link

It is known that a closed loop vector kinematic equation is generally used to establish the kinematic model in the field of parallel manipulators. In this kinematic model, each link of the parallel manipulator is seen as a vector. In brief, error vectors in closed loop vector kinematic equations establish an EM. In other word, each error vector represents one link error in the EM. The simplest format of single error vector in Figure 1 can be written in its own coordinate

Figure 1.

The first kind of single error vector.

$\mathit{l}\prime =\mathit{l}+\Delta \mathit{l}$ 1 where $\mathit{l}\prime$ represents the vector of real link position, l represents the vector of normal link position, and $\Delta \mathit{l}$ represents single error vector of the link.

A vector can be written in different format. Hence, single error vector in Figure 2 can be rewritten

Figure 2.

The second kind of single error vector.

$\mathit{l}\prime =\mathit{l}+\Delta \mathit{l}=\left(\mathit{R}+\Delta \mathit{R}\right)\left(\mathit{w}+\Delta \mathit{w}\right)\left(\left|\mathit{l}\right|+\left|\Delta \mathit{l}\right|\right)$ 2

where R represents l ’s own coordinate and $\Delta \mathit{R}$ represents the coordinate’s error; w represents the unit vector of l ’s direction and $\Delta \mathit{w}$ represents the error of unit vector of l ’s direction; and $\left|\mathit{l}\right|$ represents the length of the link and $\left|\Delta \mathit{l}\right|$ represents the length error of the link.

Ignoring the second and higher order terms, equation (2) is rewritten

$\Delta \mathit{l}=\mathit{R}\mathit{w}\left|\Delta \mathit{l}\right|+\mathit{\text{R}}\Delta \mathit{w}\left|\mathit{l}\right|+\Delta \mathit{R}\mathit{w}\left|\mathit{l}\right|$ 3

It is easily seen that single error vector owns other formats. The right side of equation (3) consists of three parts. The first part $\mathit{R}\mathit{w}\left|\Delta \mathit{l}\right|$ represents one error parameter about length error of the link. Apparently, for single error vector, the other two parts represent pose error of the link. Then, the two parts are investigated. The two parts are multiplied by one vector e

${\mathit{e}}^{\text{T}}\mathit{\text{R}}\Delta \mathit{w}\left|\mathit{l}\right|=\left|\mathit{l}\right|{\left(\mathit{\text{R}}\Delta \mathit{w}\right)}^{\text{T}}\mathit{e}$ 4 ${\mathit{e}}^{\text{T}}\Delta \mathit{R}\mathit{w}\left|\mathit{l}\right|={\mathit{e}}^{\text{T}}\mathit{\text{Δ}}\theta \times \mathit{R}\mathit{w}\left|\mathit{l}\right|=\left|\mathit{l}\right|{\left(\mathit{\text{Δ}}\theta \times \mathit{R}\mathit{w}\right)}^{\text{T}}\mathit{e}$ 5

where $\mathit{\text{Δ}}\theta$ represents pose error of the link in angle format; $\Delta \mathit{R}$ is represented by $\mathit{\text{Δ}}\theta \times \mathit{R}$ , which means that $\mathit{\text{Δ}}\theta$ is expressed in R coordinate. It is noted that $\Delta \mathit{w}$ is also expressed in R coordinate.

It can be assumed $\mathit{\text{Δ}}\theta \times \mathit{R}\mathit{w}=\Delta \mathit{r}$ , where $\Delta \mathit{r}$ is perpendicular to $\mathit{R}\mathit{w}$ . $\Delta \mathit{r}$ is represented by $\Delta \mathit{r}\text{′}$ in R coordinate: $\Delta \mathit{r}=\mathit{\text{R}}\Delta \mathit{r}\text{′}$ . So, $\mathit{\text{R}}\Delta \mathit{r}\prime \times \mathit{R}\mathit{w}=0$ and $\Delta {\mathit{r}}^{\prime }\times \mathit{w}=0$ . At last, $\mathit{\text{R}}\Delta \mathit{r}\prime =\mathit{\text{Δ}}\theta \times \mathit{R}\mathit{w}=\mathit{\text{R}}\Delta \mathit{w}$ .

It reveals that $\Delta \mathit{R}$ and $\Delta \mathit{w}$ both represent pose error of a link. In kinematic calibration, error parameters of $\Delta \mathit{R}$ and $\Delta \mathit{w}$ are coupling. Only one of them can be stayed in EM. Hence, three basic formats of single error vector are proposed in Table 1.

Table 1.

Basic formats of single error vector.

Three basic formats of single error vector
Format 1	Format 2	Format 3
$\Delta \mathit{l}$	$\left|\Delta \mathit{l}\right|$ and $\Delta \mathit{w}$	$\left|\Delta \mathit{l}\right|$ and $\Delta \mathit{R}$ ( $\mathit{\text{Δ}}\theta$ )

It should be pointed out that $\Delta \mathit{w}$ and $\mathit{\text{Δ}}\theta$ are satisfied with the following equations

$\Delta {\mathit{\text{w}}}_x{ }^2+\Delta {\mathit{\text{w}}}_y{}^2+\Delta {\mathit{\text{w}}}_z{}^2=1$ 6 ${\text{cos}}^2\Delta {\mathit{\text{θ}}}_{\alpha }+{\text{cos}}^2\Delta {\mathit{\text{θ}}}_{\beta }+{\text{cos}}^2\Delta {\mathit{\text{θ}}}_{\gamma }=1$ 7

where $\Delta {\mathit{\text{w}}}_x$ , $\Delta {\mathit{\text{w}}}_y$ , and $\Delta {\mathit{\text{w}}}_z$ are the three components of w vector; $\Delta {\mathit{\text{θ}}}_{\alpha }$ , $\Delta {\mathit{\text{θ}}}_{\beta }$ , and $\Delta {\mathit{\text{θ}}}_{\gamma }$ are the three components of $\mathit{\text{Δ}}\theta$ vector.

Hence, both $\Delta \mathit{w}$ and $\mathit{\text{Δ}}\theta$ actually contain two parameters. Three basic formats of single error vector respectively contain three parameters.

Maximum total number of identification error parameters for an EM

In above section, basic formats of single error vector are investigated. In the following, single error vector of one link is developed on every link vectors of the parallel manipulator and the maximum total number of identification error parameters is determined.

For a general parallel manipulator in Figure 3, the normal kinematic model and the kinematic model with errors can be obtained

Figure 3.

The kinematic model of a general parallel manipulator with errors.

$\mathit{H}+{\mathit{\text{a}}}_i={\mathit{\text{b}}}_i+{\mathit{l}}_{i1}+{\mathit{l}}_{i2}+\cdots +{\mathit{l}}_{i\left(k-1\right)}+{\mathit{l}}_{\mathit{\text{i}}k}$ 8 $\mathit{H}+\Delta \mathit{H}+{\mathit{\text{a}}}_i+\Delta {\mathit{\text{a}}}_i={\mathit{\text{b}}}_i+\Delta {\mathit{\text{b}}}_i+{\mathit{l}}_{i1}+\Delta {\mathit{l}}_{i1}+{\mathit{l}}_{i2}+\Delta {\mathit{l}}_{i2}+\cdots +{\mathit{l}}_{i\left(k-1\right)}+\Delta {\mathit{l}}_{i\left(k-1\right)}+{\mathit{l}}_{\mathit{\text{i}}k}+\Delta {\mathit{l}}_{\mathit{\text{i}}k}$ 9

where H is the position vector of the moving platform; ${\mathit{\text{a}}}_i$ is the vector which the center of the moving platform directs at the joint of ith leg connecting to the moving platform; ${\mathit{\text{b}}}_i$ is the vector which the center of the base platform directs at the joint of ith leg connecting to the base platform; ${\mathit{l}}_{i1}$ , ${\mathit{l}}_{i2}$ , ${\mathit{l}}_{i\left(k-1\right)}$ , and ${\mathit{l}}_{\mathit{\text{i}}k}$ represent the link vectors on the ith leg. $k$ is the total number of the links. $\Delta \mathit{H}$ , $\Delta {\mathit{\text{a}}}_i$ $\Delta {\mathit{\text{b}}}_i$ $\Delta {\mathit{l}}_{i1}$ $\Delta {\mathit{l}}_{i2}$ $\Delta {\mathit{l}}_{i\left(k-1\right)}$ , and $\Delta {\mathit{l}}_{\mathit{\text{i}}k}$ are the corresponding error vector.

The EM is obtained by subtracting equation (8) from equation (9)

$\Delta \mathit{H}+\Delta {\mathit{\text{a}}}_i=\Delta {\mathit{\text{b}}}_i+\Delta {\mathit{l}}_{i1}+\Delta {\mathit{l}}_{i2}+\cdots +\Delta {\mathit{l}}_{i\left(k-1\right)}+\Delta {\mathit{l}}_{\mathit{\text{i}}k}$ 10

It is seen that error vectors of every links also consist of a closed loop vector kinematic equation. It should be noted that geometrical errors of a general parallel manipulator are constant. Hence, no matter how complicated the geometrical error of each link is, one and only one error vector is utilized to express the geometrical error of one link. Furthermore, it is mentioned above that a geometrical error vector contains three geometrical error parameters. Thus, for a given parallel manipulator, the maximum total number of identification error parameters is constant. The maximum total number of identification error parameters is determined

$N=3\mathit{\text{m}}j$ 11

where N is the maximum total number of identification error parameters, m represents the number of legs connecting the moving platform and the base platform, and j represents the number of all links on the ith leg.

Identifiability index for an EM

The maximum total number of identification error parameters above mentioned can be obtained if a parallel manipulator is given. However, not all error parameters in different EMs with any basic format of single error vector can be identified. Some of them may be coupling with each other, which makes the identification matrix be singular and the identification process stop. In order to figure out which basic format of single error vector is useful and evaluate whether the basic formats of single error vector adopted in an EM are suitable, an index for evaluating the identifiability of an EM with different basic formats of single error vector is proposed:

$\tau =\frac{\mathit{\text{n}}}{N}$ 12

where τ is called as an identifiability index. n represents the number of identification error parameters in a given EM. The n is determined by investigating the coupling relations of error parameters. If coefficients of the error parameters in the EM are the same (it means some error parameters are coupling with others), the coupling error parameters should be eliminated and the rest of error parameters determines the number n.

The EM with the given basic formats of single error vector has a certain n. In a given parallel manipulator, the N is constant and the n in different EMs leads to the different τ value. The different τ value can be used to evaluate the identifiability ability of a given EM. If the n approaches N (namely τ approaches 100%), it means that more error parameters can be identified and compensated, which is helpful for kinematic calibration of parallel manipulators.

Establishing EMs with different values of the identifiability index

In order to show the effectiveness of the identifiability index, a 3PRS parallel manipulator is employed as a carrier to generate three EMs with different values of the identifiability index. As the differences of the identifiability index values are determined by the combinations of error vector formats, EMs with some specified error vector formats are established in the following.

Description of a parallel manipulator and its kinematic model

The 3PRS parallel manipulator is shown in Figure 4. It is composed of a moving platform, a base platform, and three supporting legs with identical kinematic structure. Each PRS leg contains one P joint, one R joint, and one S joint. A fixed Cartesian reference coordinate system O-XYZ is fixed at the center of the base platform B ₁ B ₂ B ₃. A moving Cartesian reference coordinate system N-UVW is fixed at the center of the moving platform A ₁ A ₂ A ₃. For simplicity and without losing the generality, let the X-axis point along the direction of vector OB ₁ and the U-axis point along the vector OA ₁. B_iC_i on the base platform for i = 1, 2, and 3 represents the guide rail of each leg. The guide rail of each leg is perpendicular to the base platform. Each P joint moves along B_iC_i . P joint connects the base platform to R joint. R joint connects the P joint to the C_iA_i link. S joint (attached to the moving platform) connects the moving platform to C_iA_i link. Furthermore, all the joints attached to the base and moving platform are symmetrically distributed at vertices of the equilateral triangles.

Figure 4.

Architecture of 3PRS parallel manipulator.

T–T rotation angle is introduced to describe the rotation of the moving platform. The rotational matrix is expressed as

${\mathit{R}}_{\text{TT}}(\varphi ,\theta ,\psi )=\left[\begin{array}{ccc}\text{cos}\varphi & -\text{sin}\varphi & 0\\ \text{sin}\varphi & \text{cos}\varphi & 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}\text{cos}\theta & 0& \text{sin}\theta \\ 0& 1& 0\\ -\text{sin}\theta & 0& \text{cos}\theta \end{array}\right]$ $\left[\begin{array}{ccc}\text{cos}\varphi & \text{sin}\varphi & 0\\ -\text{sin}\varphi & \text{cos}\varphi & 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}\text{cos}\psi & -\text{sin}\psi & 0\\ \text{sin}\psi & \text{cos}\psi & 0\\ 0& 0& 1\end{array}\right]$

ψ in T–T angle represents the rotational motion of the moving platform around its normal direction. This angle is equal to zero because its accuracy is not needed to assure.

A kinematic function according to the vector loop OB_iC_iA_iN is written as

$\mathit{H}+{\mathit{R}}_{\text{TT}}{\mathit{\text{R}}}_i{\mathit{\text{a}}}_i={\mathit{\text{R}}}_i{\mathit{\text{b}}}_i+{\mathit{\text{R}}}_i{\mathit{\text{q}}}_i+{\mathit{\text{R}}}_i{\mathit{\text{l}}}_i$ 13

where, ${\mathit{\text{a}}}_i$ , ${\mathit{\text{b}}}_i$ and ${\mathit{\text{l}}}_i$ represent the position vectors of NA_i , OB_i and C_iA_i . H represents the position vector of the moving platform. ${\mathit{\text{q}}}_i$ represents the input vector of each P joint. R _i represents the rotational matrix of each leg.

Error modeling based on different error vector formats

In this section, based on different combinations of error vector formats, three EMs with different values of the identifiability index are given. In fact, 3 ⁴ = 81 EMs can be generated because four link error vectors (they are OB_i , B_iC_i , C_iA_i , and NA_i ) in 3PRS parallel manipulator are needed to be described and each link has three basic formats of single error vector proposed in Table 1. In order to prove the conclusions in this article explicitly and avoid a mass of redundant cases, three different EMs are selected as the typical cases. In these three different EMs, some error parameters of different EMs should be eliminated since they are coupling if coefficients of the error parameters in the EM are the same. Hence, identification error parameters of each EM can be obtained. In order to control the paper length, only the EM 1 is analyzed in details.

EM 1 with the error vector format 1

In this EM, based on linear perturbation method, the error vector format 1 is adopted to describe the entire geometrical error vectors. This kinematic model with errors is obtained as

$\mathit{H}+\Delta \mathit{H}+{\mathit{R}}_{\text{TT}}{\mathit{\text{R}}}_i{\mathit{\text{a}}}_i+\mathit{\text{Δ}}\alpha \times {\mathit{R}}_{\text{TT}}{\mathit{\text{R}}}_i{\mathit{\text{a}}}_i+{\mathit{R}}_{\text{TT}}{\mathit{\text{R}}}_i\Delta {\mathit{\text{a}}}_i={\mathit{\text{R}}}_i{\mathit{\text{b}}}_i+{\mathit{\text{R}}}_i\Delta {\mathit{\text{b}}}_i+{\mathit{\text{R}}}_i{\mathit{\text{q}}}_i+{\mathit{\text{R}}}_i\Delta {\mathit{\text{q}}}_i+{\mathit{\text{R}}}_i{\mathit{\text{l}}}_i+{\mathit{\text{R}}}_i\Delta {\mathit{\text{l}}}_i$ 14

where $\Delta \mathit{H}$ represents the position error vector of the output, $\mathit{\text{Δ}}\alpha$ represents the pose error vector of the output, $\Delta {\mathit{\text{b}}}_i$ represents the geometrical error vector of ${\mathit{\text{b}}}_i$ , $\Delta {\mathit{\text{q}}}_i$ represents the geometrical error vector of ${\mathit{\text{q}}}_i$ , $\Delta {\mathit{\text{a}}}_i$ represents the geometrical error vector of ${\mathit{\text{a}}}_i$ , and $\Delta {\mathit{\text{l}}}_i$ represents the geometrical error vector of ${\mathit{\text{l}}}_i$ .

The EM including output error parameters and geometrical error parameters is obtained in the following

$\Delta \mathit{H}+\mathit{\text{Δ}}\alpha \times {\mathit{R}}_{\text{TT}}{\mathit{\text{R}}}_i{\mathit{\text{a}}}_i={\mathit{\text{R}}}_i\Delta {\mathit{\text{b}}}_i+{\mathit{\text{R}}}_i\Delta {\mathit{\text{q}}}_i+{\mathit{\text{R}}}_i\Delta {\mathit{\text{l}}}_i-{\mathit{R}}_{\text{TT}}{\mathit{\text{R}}}_i\Delta {\mathit{\text{a}}}_i$ 15

This EM is expressed in scalar equations

${\left({\mathit{\text{R}}}_i{\mathit{\text{ω}}}_i\right)}^{\text{T}}\left(\Delta \mathit{H}+\mathit{\text{Δ}}\alpha \times {\mathit{R}}_{\text{TT}}{\mathit{\text{R}}}_i{\mathit{\text{a}}}_i\right)={\left({\mathit{\text{R}}}_i{\mathit{\text{ω}}}_i\right)}^{\text{T}}({\mathit{\text{R}}}_i\Delta {\mathit{\text{b}}}_i+{\mathit{\text{R}}}_i\Delta {\mathit{\text{q}}}_i+{\mathit{\text{R}}}_i\Delta {\mathit{\text{l}}}_i-{\mathit{R}}_{\text{TT}}{\mathit{\text{R}}}_i\Delta {\mathit{\text{a}}}_i)$ 16 ${\left({\mathit{\text{R}}}_i{\mathit{e}}_{\text{y}}\right)}^{\text{T}}\left(\Delta \mathit{H}+\mathit{\text{Δ}}\alpha \times {\mathit{R}}_{\text{TT}}{\mathit{\text{R}}}_i{\mathit{\text{a}}}_i\right)={\left({\mathit{\text{R}}}_i{\mathit{e}}_{\text{y}}\right)}^{\text{T}}({\mathit{\text{R}}}_i\Delta {\mathit{\text{b}}}_i+{\mathit{\text{R}}}_i\Delta {\mathit{\text{q}}}_i+{\mathit{\text{R}}}_i\Delta {\mathit{\text{l}}}_i-{\mathit{R}}_{\text{TT}}{\mathit{\text{R}}}_i\Delta {\mathit{\text{a}}}_i)$ 17

where ${\mathit{\text{ω}}}_i$ represents the unit direction vector of ${\mathit{\text{l}}}_i$ in ${\mathit{\text{R}}}_i$ coordinate.

Furthermore, equations (16) and (17) are rewritten as

${\left({\mathit{\text{R}}}_i{\mathit{\text{ω}}}_i\right)}^{\text{T}}\left(\Delta \mathit{H}+\mathit{\text{Δ}}\alpha \times {\mathit{R}}_{\text{TT}}{\mathit{\text{R}}}_i{\mathit{\text{a}}}_i\right)={\omega }_{\mathit{\text{i}}x}\Delta {\mathit{\text{b}}}_i{}_x+{\omega }_{\mathit{\text{i}}y}\Delta {\mathit{\text{b}}}_i{}_y+{\omega }_{\mathit{\text{i}}z}\Delta {\mathit{\text{b}}}_i{}_z+{\omega }_{\mathit{\text{i}}x}\Delta {\mathit{\text{q}}}_i{}_x+{\omega }_{\mathit{\text{i}}y}\Delta {\mathit{\text{q}}}_i{}_y+{\omega }_{\mathit{\text{i}}z}\Delta {\mathit{\text{q}}}_i{}_z+{\omega }_{\mathit{\text{i}}x}\Delta {\mathit{\text{l}}}_i{}_x+{\omega }_{\mathit{\text{i}}y}\Delta {\mathit{\text{l}}}_i{}_y+{\omega }_{\mathit{\text{i}}z}\Delta {\mathit{\text{l}}}_i{}_z-{\left({\mathit{\text{R}}}_i{\mathit{\text{ω}}}_i\right)}^{\text{T}}{\mathit{R}}_{\text{TT}}{\mathit{\text{R}}}_i\Delta {\mathit{\text{a}}}_i$ 18 ${\left({\mathit{\text{R}}}_i{\mathit{e}}_{\text{y}}\right)}^{\text{T}}\left(\Delta \mathit{H}+\mathit{\text{Δ}}\alpha \times {\mathit{R}}_{\text{TT}}{\mathit{\text{R}}}_i{\mathit{\text{a}}}_i\right)=\Delta {\mathit{\text{b}}}_i{}_y+\Delta {\mathit{\text{q}}}_i{}_y+\Delta {\mathit{\text{l}}}_i{}_y-{\left({\mathit{\text{R}}}_i{\mathit{e}}_{\text{y}}\right)}^{\text{T}}{\mathit{R}}_{\text{TT}}{\mathit{\text{R}}}_i\Delta {\mathit{\text{a}}}_i$ 19

where ${\mathit{\text{ω}}}_i{}_x$ , ${\mathit{\text{ω}}}_i{}_y$ , and ${\mathit{\text{ω}}}_i{}_z$ are the vector parameters of ${\mathit{\text{ω}}}_i$ ; $\Delta {\mathit{\text{b}}}_i{}_x$ , $\Delta {\mathit{\text{b}}}_i{}_y$ , and $\Delta {\mathit{\text{b}}}_i{}_z$ are the vector parameters of $\Delta {\mathit{\text{b}}}_i$ ; $\Delta {\mathit{\text{q}}}_i{}_x$ , $\Delta {\mathit{\text{q}}}_i{}_y$ , and $\Delta {\mathit{\text{q}}}_i{}_z$ are the vector parameters of $\Delta {\mathit{\text{q}}}_i$ ; $\Delta {\mathit{\text{l}}}_i{}_x$ , $\Delta {\mathit{\text{l}}}_i{}_y$ , and $\Delta {\mathit{\text{l}}}_i{}_z$ are the vector parameters of $\Delta {\mathit{\text{l}}}_i$ .

In equations (18) and (19), $\Delta {\mathit{\text{b}}}_i{}_x$ couples with $\Delta {\mathit{\text{q}}}_i{}_x$ and $\Delta {\mathit{\text{l}}}_i{}_x$ ; $\Delta {\mathit{\text{b}}}_i{}_y$ couples with $\Delta {\mathit{\text{q}}}_i{}_y$ and $\Delta {\mathit{\text{l}}}_i{}_y$ ; $\Delta {\mathit{\text{b}}}_i{}_z$ couples with $\Delta {\mathit{\text{q}}}_i{}_z$ and $\Delta {\mathit{\text{l}}}_i{}_z$ . Hence, three vector parameters of $\Delta {\mathit{\text{b}}}_i$ are chosen as identification error parameters and kept in this EM. $\Delta {\mathit{\text{q}}}_i$ and $\Delta {\mathit{\text{l}}}_i$ are eliminated.

As the existence of ${\mathit{R}}_{\text{TT}}$ , $\Delta {\mathit{\text{a}}}_i$ does not couple with any other error parameters. Three vector parameters of $\Delta {\mathit{\text{a}}}_i$ are also chosen as identification error parameters and kept in this EM.

In total, $\Delta {\mathit{\text{a}}}_i{}_x$ , $\Delta {\mathit{\text{a}}}_i{}_y$ , $\Delta {\mathit{\text{a}}}_i{}_z$ $\Delta {\mathit{\text{b}}}_i{}_x$ , $\Delta {\mathit{\text{b}}}_i{}_y$ , and $\Delta {\mathit{\text{b}}}_i{}_z$ are identification error parameters. In this EM, the number of identification error parameters is obtained as

$n_1=3\times 2\times 3=18$ 20

Moreover, the maximum total number of identification error parameters are determined

$N=3\times 4\times 3=36$ 21

According to section “Identifiability index for an EM,” the identifiability index for this EM is shown as

${\tau }_1=\frac{18}{36}=50\text{\%}$ 22

EM 2 with the error vectors format 1 and format 2

In this EM, the error vector format 1 and format 2 are adopted to describe geometrical error vector. Among them, $\Delta {\mathit{\text{b}}}_i$ $\Delta {\mathit{\text{q}}}_i$ , and $\Delta {\mathit{\text{a}}}_i$ are described by format 1. $\Delta {\mathit{\text{l}}}_i$ is described by format 2. This kinematic model with errors is obtained as

$\mathit{H}+\Delta \mathit{H}+{\mathit{R}}_{\text{TT}}{\mathit{\text{R}}}_i{\mathit{\text{a}}}_i+\mathit{\text{Δ}}\alpha \times {\mathit{R}}_{\text{TT}}{\mathit{\text{R}}}_i{\mathit{\text{a}}}_i+{\mathit{R}}_{\text{TT}}{\mathit{\text{R}}}_i\Delta {\mathit{\text{a}}}_i={\mathit{\text{R}}}_i{\mathit{\text{b}}}_i+{\mathit{\text{R}}}_i\Delta {\mathit{\text{b}}}_i+{\mathit{\text{R}}}_i{\mathit{\text{q}}}_i+{\mathit{\text{R}}}_i\Delta {\mathit{\text{q}}}_i+{\mathit{\text{R}}}_i{\mathit{\text{l}}}_i+{\mathit{\text{R}}}_i{\mathit{\text{ω}}}_i\left|\Delta {\mathit{\text{l}}}_i\right|+{\mathit{\text{R}}}_i\Delta {\mathit{\text{ω}}}_i\left|{\mathit{\text{l}}}_i\right|$ 23

where $\Delta {\mathit{\text{ω}}}_i$ represents the error of unit direction vector of ${\mathit{\text{l}}}_i$ , $\left|\Delta {\mathit{\text{l}}}_i\right|$ represents the length error of the vector ${\mathit{\text{l}}}_i$ , and $\left|{\mathit{\text{l}}}_i\right|$ represents the length of the vector ${\mathit{\text{l}}}_i$ .

After a similar analysis with EM 1, it is known that $\Delta {\mathit{\text{a}}}_i{}_x$ , $\Delta {\mathit{\text{a}}}_i{}_y$ , $\Delta {\mathit{\text{a}}}_i{}_z$ $\Delta {\mathit{\text{b}}}_i{}_x$ , $\Delta {\mathit{\text{b}}}_i{}_y$ , $\Delta {\mathit{\text{b}}}_i{}_z$ , $\Delta {\mathit{\text{ω}}}_i{}_x$ , $\Delta {\mathit{\text{ω}}}_i{}_z$ , and $\left|\Delta {\mathit{\text{l}}}_i\right|$ are identification error parameters. In this EM, the number of identification error parameters is obtained as

$n_2=9\times 3=27$ 24

Moreover, the maximum total number of identification error parameters are determined

$N=3\times 4\times 3=36$ 25

According to section “Identifiability index for an EM,” the identifiability index for this EM is shown as

${\tau }_2=\frac{27}{36}=75\text{\%}$ 26

EM 3 with the error vectors format 1,format 2, and format 3

In this EM, the error vector format 1, format 2, and format 3 are adopted to describe geometrical error vector. Among them, $\Delta {\mathit{\text{b}}}_i$ and $\Delta {\mathit{\text{a}}}_i$ are described by format 1. $\Delta {\mathit{\text{l}}}_i$ is described by format 2. $\Delta {\mathit{\text{q}}}_i$ is described by format 3. This kinematic model with errors is obtained as

$\mathit{H}+\Delta \mathit{H}+{\mathit{R}}_{\text{TT}}{\mathit{\text{R}}}_i{\mathit{\text{a}}}_i+\mathit{\text{Δ}}\alpha \times {\mathit{R}}_{\text{TT}}{\mathit{\text{R}}}_i{\mathit{\text{a}}}_i+{\mathit{R}}_{\text{TT}}{\mathit{\text{R}}}_i\Delta {\mathit{\text{a}}}_i={\mathit{\text{R}}}_i{\mathit{\text{b}}}_i+{\mathit{\text{R}}}_i\Delta {\mathit{\text{b}}}_i+{\mathit{\text{R}}}_i{\mathit{\text{q}}}_i+{\mathit{\text{R}}}_i\left|\Delta {\mathit{\text{q}}}_i\right|{\mathit{\text{e}}}_z+{\mathit{\text{R}}}_i\Delta {\mathit{\text{φ}}}_i\times {\mathit{\text{R}}}_i\left|{\mathit{\text{q}}}_i\right|{\mathit{\text{e}}}_z+{\mathit{\text{R}}}_i{\mathit{\text{l}}}_i+{\mathit{\text{R}}}_i{\mathit{\text{ω}}}_i\left|\Delta {\mathit{\text{l}}}_i\right|+{\mathit{\text{R}}}_i\Delta {\mathit{\text{ω}}}_i\left|{\mathit{\text{l}}}_i\right|$ 27

where $\left|\Delta {\mathit{\text{q}}}_i\right|$ represents the length error of the vector ${\mathit{\text{q}}}_i$ ; $\Delta {\mathit{\text{φ}}}_i$ represents pose error of the vector ${\mathit{\text{q}}}_i$ in ${\mathit{\text{R}}}_i$ coordinate. ${\mathit{\text{e}}}_z={\left[\begin{array}{ccc}0& 0& 1\end{array}\right]}^{\text{T}}$ represents the unit direction vector of ${\mathit{\text{q}}}_i$ .

In above equations, just $\Delta {\mathit{\text{b}}}_i{}_z$ couples with $\left|\Delta {\mathit{\text{q}}}_i\right|$ . $\Delta {\mathit{\text{b}}}_i{}_z$ is eliminated in this EM. Hence, $\Delta {\mathit{\text{a}}}_i{}_x$ , $\Delta {\mathit{\text{a}}}_i{}_y$ , $\Delta {\mathit{\text{a}}}_i{}_z$ $\Delta {\mathit{\text{b}}}_i{}_x$ , $\Delta {\mathit{\text{b}}}_i{}_y$ , $\left|\Delta {\mathit{\text{q}}}_i\right|$ , $\Delta {\varphi }_{\mathit{\text{i}}x}$ , $\Delta {\varphi }_{\mathit{\text{i}}y}$ , $\Delta {\mathit{\text{ω}}}_i{}_x$ , $\Delta {\mathit{\text{ω}}}_i{}_z$ , and $\left|\Delta {\mathit{\text{l}}}_i\right|$ are identification error parameters. In this EM, the number of identification error parameters is obtained as

$n_3=11\times 3=33$ 28

Moreover, the maximum total number of identification error parameters are determined

$N=3\times 4\times 3=36$ 29

According to section “Identifiability index for an EM,” the identifiability index for this EM is shown as

${\tau }_3=\frac{33}{36}=92\text{\%}$ 30

It should be noted that no matter how to change the vector format of $\Delta {\mathit{\text{a}}}_i$ , $\Delta {\mathit{\text{b}}}_i$ , $\Delta {\mathit{\text{q}}}_i$ , and $\Delta {\mathit{\text{l}}}_i$ , the number of identification error parameters of 3PRS parallel manipulator is not bigger than 33. The reason is ${\mathit{\text{q}}}_i$ perpendicular to ${\mathit{\text{b}}}_i$ , which makes $\left|\Delta {\mathit{\text{q}}}_i\right|$ couple with one of $\Delta {\mathit{\text{b}}}_i$ ’s vector parameters or $\left|\Delta {\mathit{\text{b}}}_i\right|$ couple with one of $\Delta {\mathit{\text{q}}}_i$ ’s vector parameters. Hence, ${\tau }_3$ is the biggest value of the identifiability index for this parallel manipulator.

Comparisons of three EMs in calibration simulations

In order to compare the effects of the different identifiability index values on kinematic calibration, the kinematic calibration simulations with three EMs are performed in a computer. Based on the same simulation environment, the above three EMs are respectively adopted to simulate kinematic calibrations of 3PRS parallel manipulator.

Simulation process for calibrations with three EMs

In different simulation cases, only EMs are changed. Here are the detail steps:

1. Load initial kinematic parameters ${\mathit{\text{a}}}_i$ ${\mathit{\text{b}}}_i$ , and ${\mathit{\text{l}}}_i$ ;

2. $\Delta {\mathit{r}}_{\text{real}}$ is obtained by calculating with random errors and kept in the same in each simulation;

3. Set up the normal kinematic model $\mathit{f}\left(\mathit{q}, \mathit{r}\right)=\mathit{c}$ of 3PRS parallel manipulator;

4. Calculate ${\mathit{c}}_{\text{real}}=\mathit{f}\left(\mathit{q}, \mathit{r}+\Delta {\mathit{r}}_{\text{real}}\right)$ in the given positions and poses of measurement;

5. ${\mathit{c}}_{\text{real}}+\epsilon$ is used as the data of measurement; where ε represents the measurement error;

6. An EM is given. Three different EMs are adopted respectively in this step.

7. The identification matrix W is obtained by stacking J in the given positions and poses of measurement;

8. Based on ridge estimation algorithm, geometrical error parameters are calculated by $\Delta {\mathit{r}}_{\text{cal}}={\left({\mathit{W}}^{\text{T}}\mathit{W}+\lambda \mathit{I}\right)}^{-1}{\mathit{\text{W}}}^T\Delta {\mathit{c}}_{\mathit{\text{n}}c}$ ; I is a unit matrix with the same size of ${\mathit{W}}^{\text{T}}\mathit{W}$ ; λ is called a ridge estimation parameter; $\Delta {\mathit{c}}_{\mathit{\text{n}}c}$ is stacked by ${\mathit{c}}_{\text{real}}+\epsilon -\mathit{c}$ within the measurement positions and poses;

9. Evaluate the accuracy after calibration in a given workspace;

10. The original error before calibration is obtained by $\Delta {\mathit{c}}^{*}{}_{\text{orig}}=\mathit{f}\left(\mathit{q}, \mathit{r}+\Delta {\mathit{r}}_{\text{real}}\right) -\mathit{f}\left(\mathit{q}, \mathit{r}\right)$ ;

11. The error after calibration is obtained by $\Delta {\mathit{c}}^{*}=\mathit{f}\left(\mathit{q}, \mathit{r}+\Delta {\mathit{r}}_{\text{real}}\right) -\mathit{f}\left(\mathit{q}, \mathit{r}+\Delta {\mathit{r}}_{\text{cal}}\right)$ ;

12. For a spatial parallel manipulator, the position of a tool nose point is usually adopted to evaluate the accuracy. Hence, both $\Delta {\mathit{c}}^{*}{}_{\text{orig}}$ and $\Delta {\mathit{c}}^{*}$ are transformed to the position of the tool nose point by an equation $P\_\mathit{\text{T}}\mathrm{NP}=\text{norm }\left(\Delta \mathit{H}+{\mathit{R}}_{\text{T}-\text{T}}\left(\mathit{\text{Δ}}\alpha \right)\right)$ . The $P\_\mathit{\text{T}}\mathrm{NP}$ is chosen as the output error.

Performing calibration simulations with three EMs

In order to establish a real simulation environment, another EM is proposed as the source of real errors

$\Delta \mathit{H}+\mathit{\text{Δ}}\alpha \times {\mathit{R}}_{\text{TT}}{\mathit{\text{R}}}_i{\mathit{\text{a}}}_i+{\mathit{R}}_{\text{TT}}{\mathit{\text{R}}}_i\Delta {\mathit{\text{a}}}_i={\mathit{\text{R}}}_i\Delta {\phi }_{\mathit{\text{q}}i}\times {\mathit{\text{R}}}_i{\mathit{\text{q}}}_i+{\mathit{\text{R}}}_i\left|\Delta {\mathit{\text{q}}}_i\right|{\mathit{\text{e}}}_z+{\mathit{\text{R}}}_i\Delta {\mathit{\text{b}}}_i+{\mathit{\text{R}}}_i\left|\Delta {\mathit{\text{l}}}_i\right|{\mathit{\text{ω}}}_i+{\mathit{\text{R}}}_i\Delta {\phi }_{\mathit{\text{l}}i}\times {\mathit{\text{R}}}_i{\mathit{\text{l}}}_i$ 31

where $\Delta {\phi }_{\mathit{\text{q}}i}$ represents pose error of the vector ${\mathit{\text{q}}}_i$ in ${\mathit{\text{R}}}_i$ coordinate; $\Delta {\phi }_{\mathit{\text{l}}i}$ represents pose error of the vector ${\mathit{\text{l}}}_i$ in ${\mathit{\text{R}}}_i$ coordinate.

In this section, all the simulations are based on the same following data in Tables 2 to 4. It should be noted that the number of the given real error parameters more than 33 is allowed. Because the given real error parameters influence the output of the parallel manipulator by multiplying the error transfer matrix, which process is not needed to care the coupling relationships between error parameters.

Table 2.

The normal structure parameters.

$\left|{\mathit{\text{a}}}_i\right|$ (mm)	$\left|{\mathit{\text{b}}}_i\right|$ (mm)	$\left|{\mathit{\text{l}}}_i\right|$ (mm)
201	220	487

Table 3.

The input limit for evaluating accuracy.

$q_1$ (mm)	$q_2$ (mm)	$q_3$ (mm)
92–350	92–350	92–350

Table 4.

The random limit of measurement errors.

Position measurement errors/mm	Pose measurement errors/rad
−0.01∼0.01	−0.00017∼0.00017

Based on the above simulation process and the same data, the original output error and the output error after kinematic calibration adopting EM 1, adopting EM 2, and adopting EM 3 are respectively plotted in Figures 5 to 8. The maximum output error values in each simulation are presented in Table 5.

Figure 5.

The original output error before calibration in simulation.

Figure 6.

The output error after calibration adopting EM 1 in simulation. EM: error model.

Figure 7.

The output error after calibration adopting EM 2 in simulation. EM: error model.

Figure 8.

The output error after calibration adopting EM 3 in simulation. EM: error model.

Table 5.

Maximum output error values in each simulation.

Original error/mm	EM 1 (mm)	EM 2 (mm)	EM 3 (mm)
5.903	0.155	0.120	0.021

EM: error model.

Comparisons of three EMs in practical calibration experiments

For verifying the effects of the different identifiability index values on kinematic calibration in practice, the kinematic calibration experiments with three EMs are performed on the prototype of a 5-DOF 3-P(4 R)S-XY hybrid machine tool, where the 3-P(4 R)S parallel manipulator can be simplified to a 3PRS parallel manipulator. In the same experiment environment, the above three EMs are separately adopted to perform on the kinematic calibration experiments.

Experiment process for calibrations with three EMs

Here is the process:

Locate measurement implements on the operating platform and spindle nose of the hybrid machine tool in Figure 9. A dial indicator and a spindle measuring bar clamp on the spindle nose. Let the needle of dial indicator point along the normal direction of the moving platform and point at the operating platform. A set of gauge blocks and three dial indicators are fixed on the operating platform. After locating all measurement implements, home all actuators of the machine tool. At this moment, record the output of moving platform ${\left[\begin{array}{ccc}0& 0& \begin{array}{ccc}{z}^{\text{*}}& 0& 0\end{array}\end{array}\right]}^{\text{T}}$ and the values of the three dial indicators as D . $z^{\text{*}}$ is a marker value in Z-axis directions.

Measure the data of positions by the combination of a spindle measuring bar on the spindle nose and three dial indicators on the operating platform. Drive X–Y axis and adjust the spindle measuring bar into the measurement field until let the values of the three dial indicators be D . At this moment, record values of X–Y axis as ${\left[\begin{array}{cc}{\mathit{\text{x}}}_t& {\mathit{\text{y}}}_t\end{array}\right]}^{\text{T}}$ and values of three Z-axis as ${\mathit{\text{q}}}_t$ in the numerical control system of the machine tool. This t represent t-th set of position and pose.

Measure the data of poses by a dial indicator on the spindle nose and a set of gauge blocks on the operating platform. The method of measuring poses is presented in Figure 10. The distance $\mathit{\text{Δ}}l$ along normal direction of moving platform is provided by the dial indicator on the spindle nose. Distances $\mathit{\text{Δ}}x$ , $\mathit{\text{Δ}}y$ , and $\mathit{\text{Δ}}z$ respectively are obtained in the numerical control system when the dial indicator on the spindle nose extends the distance $\mathit{\text{Δ}}l$ . Hence, the pose of moving platform is calculated by the equations $\alpha =\text{arccos}\left(\mathit{\text{Δ}}x/\mathit{\text{Δ}}l\right)$ , $\beta =\text{arccos}\left(\mathit{\text{Δ}}y/\mathit{\text{Δ}}l\right)$ , and $\gamma =\text{arccos}\left(\mathit{\text{Δ}}z/\mathit{\text{Δ}}l\right)$ . Furthermore, α, β, and γ are transform to ϕ and θ in the equations: $\text{ cos}\varphi \text{sin}\theta =\text{cos}\alpha$ , $\text{sin}\varphi \text{sin}\theta =\text{cos}\beta$ , and $\text{cos}\theta =\text{cos}\gamma$ . At this moment, record ${\left[\begin{array}{cc}{\mathit{\text{ϕ}}}_t& {\mathit{\text{θ}}}_t\end{array}\right]}^{\text{T}}$ .

Get enough measurement data. Repeat steps 2 and 3 when the moving platform performs different sets of positions and poses. Utilizing these measurement data, geometrical error parameters can be identified.

Rotation tool center point (RTCP) accuracy test: It is one of the most important accuracy indexes of five-axis machine tool. RTCP accuracy test is shown in Figure 11. In the RTCP accuracy test, three dial indicators are fixed on the operating platform and a spindle measuring bar is clamped on the spindle nose of the machine tool. The needles of three dial indicators locate on the peaks of the test ball of the spindle measuring bar, respectively, along X, Y, and Z directions. When the moving platform of the machine tool moves in a cone workspace, the reading deviations of three dial indicators are recorded as the test results.

Figure 9.

Experimental measurement implements.

Figure 10.

The method of measuring poses.

Figure 11.

RTCP accuracy test.

Performing calibration experiments with three EMs

In order to eliminate the influence of experiment operating error, the same data of measurement and RTCP accuracy test are adopted. Based on the same data of measurement, different sets of geometrical error parameter are obtained according to different EMs. Furthermore, it should be noted that RTCP accuracy test usually records the deviation along X, Y, and Z directions during maintaining the test ball motionless. In this article, the process of RTCP accuracy test is reversed. Within a given cone workspace, $\mathit{o}\mathit{u}{\mathit{t}}_{\text{M}}={\left[\begin{array}{cc}{x}_{\text{M}}& \begin{array}{cc}{y}_{\text{M}}& \begin{array}{ccc}{q}_{1\text{M}}& q_{2\text{M}}& q_{3\text{M}}\end{array}\end{array}\end{array}\right]}^{\text{T}}$ are obtained for maintaining the test ball motionless. Within the same cone workspace, $\mathit{o}\mathit{u}{\mathit{t}}_{\text{EM}}={\left[\begin{array}{cc}{x}_{\text{EM}}& \begin{array}{cc}{y}_{\text{EM}}& \begin{array}{ccc}{q}_{1\text{EM}}& q_{2\text{EM}}& q_{3\text{EM}}\end{array}\end{array}\end{array}\right]}^{\text{T}}$ based on different EMs are calculated. By comparing $\mathit{o}\mathit{u}{\mathit{t}}_{\text{M}}$ against $\mathit{o}\mathit{u}{\mathit{t}}_{\text{EM}}$ , a result of RTCP accuracy test can be obtained for each EM. The closer the values of $\mathit{o}\mathit{u}{\mathit{t}}_{\text{EM}}$ obtained from $\mathit{o}\mathit{u}{\mathit{t}}_{\text{M}}$ , the higher the accuracy is.

The original output error and the output error after kinematic calibration adopting EM 1, adopting EM 2, and adopting EM 3 are, respectively, plotted in Figures 12 to 15. The output error ranges along X, Y, and Z directions in each experiment are organized in Table 6.

Figure 12.

The original output error before calibration in experiment.

Figure 13.

The output error after calibration adopting EM 1 in experiment. EM: error model.

Figure 14.

The output error after calibration adopting EM 2 in experiment. EM: error model.

Figure 15.

The output error after calibration adopting EM 3 in experiment. EM: error model.

Table 6.

The output error ranges along X, Y, and Z directions in each experiment.

Error range	Original error (mm)	EM 1 (mm)	EM 2 (mm)	EM 3 (mm)
Along X direction	9.611	0.303	0.296	0.099
Along Y direction	8.905	0.180	0.109	0.065
Along Z direction	14.649	0.287	0.259	0.039

EM: error model.

Discussions on simulation and experiment results

In this section, the above simulation and experiment results are organized in Figures 16 and 17. The results are discussed in the following.

Figure 16.

The trend of maximum output error in simulations (Y axis value is expressed in a logarithm based on the natural constant e. “Orig” represents “Original”).

Figure 17.

The trend of output error ranges in experiments.

Firstly, different EMs of the same parallel manipulator own the different number of geometrical error parameters, especially identification error parameters. In EMs 1, 2, and 3, the numbers of geometrical error parameters are changed as adopting the different error vectors’ formats. It is needed to point out that the number of identification error parameters is also changed. The numbers of identification error parameters in different EMs, respectively, are $n_1=18$ , $n_2=27$ , and $n_3=33$ .

Secondly, kinematic calibrations based on different EMs generate different impacts on the accuracy of parallel manipulators. In Tables 5 and 6, the original output error is bigger than the output errors after any kinematic calibrations both in simulations and experiments. It proves that the kinematic calibration significantly improves the accuracy of the parallel manipulator no matter which EM is used. Furthermore, it should be noted that, for the same parallel manipulator, kinematic calibrations based on different EMs lead to the different accuracies. In simulations, the maximum output errors after adopting different EMs respectively are 0.155 mm, 0.120 mm, and 0.021 mm. Similarly, in experiments, the maximum output errors along X, Y and Z directions after adopting different EMs respectively are 0.303 mm, 0.180 mm, 0.287 mm, and 0.296 mm, 0.109 mm, 0.259 mm, and 0.099 mm, 0.065 mm, 0.039 mm.

Thirdly, a kinematic calibration adopting the EM with bigger τ value improves the accuracy of the parallel manipulator more significantly. In this article, τ value of EM 1 is ${\tau }_1=50\text{\%}$ ; τ value of EM 2 is ${\tau }_2=75\text{\%}$ ; τ value of EM 3 is ${\tau }_3=92\text{\%}$ . In Table 5, the maximum output error adopting EM 1 is bigger than adopting EM 2. Similarly, the maximum output error adopting EM 2 is bigger than adopting EM 3. Moreover, the same trend is also proved by the data of Table 6. Actually, the trend is shown more clearly in Figures 16 and 17. All above information no matter in simulations or experiments reveals that kinematic calibration adopting the EM with bigger τ value leads to the higher accuracy of the parallel manipulator. Hence, obtaining the bigger τ value is needed.

Fourthly, an approach of error modeling is proposed for obtaining a bigger τ value. First of all, three EMs with different τ values are investigated. In EM 1, error vectors $\Delta {\mathit{\text{a}}}_i$ , $\Delta {\mathit{\text{b}}}_i$ , $\Delta {\mathit{\text{q}}}_i$ , and $\Delta {\mathit{\text{l}}}_i$ adopt the same error vector format. $\Delta {\mathit{\text{b}}}_i$ couples with $\Delta {\mathit{\text{q}}}_i$ and $\Delta {\mathit{\text{l}}}_i$ . As the existence of ${\mathit{R}}_{\text{TT}}$ , $\Delta {\mathit{\text{a}}}_i$ attached on the moving platform does not couples with other error vectors. In other word, the error vector on the moving platform has the different influence on the output error from other error vectors. In general, the error vector format on the moving platform does not influence τ value. Moreover, comparing EM 1 with EM 2, just $\Delta {\mathit{\text{l}}}_i$ is replaced by another kind of error vector format. It shows that the mixed error vector formats avoid the coupling relationship between error parameters and improve τ value. In EM 3, ${\tau }_3$ is not at 100% because of the structure feature of 3PRS parallel manipulator. It shows that as the complexity of parallel manipulators’ configurations, not all parallel manipulators’ EM can achieve $\tau =100\%$ . The closer τ approaches 100%, the better performance of kinematic calibration is. At last, the approach of error modeling is organized in the following:

The vector formats of error parameters on the moving platform can be the same with others.

Error vectors are expressed in as more as possible formats.

Make τ approach 100% as closer as possible.

Conclusions

In this article, an identifiability index τ for evaluating an EM is proposed. By adopting different basic format combinations of single error vector, three typical EMs with different τ values are obtained. The computer simulations and prototype experiments of the kinematic calibration adopting these three EMs are performed on the same 3PRS parallel manipulator. The simulation and experiment results show that the maximum output error after calibrated by EM 3 with ${\tau }_3=92\text{\%}$ is smaller than the one after calibrated by EM 2 with ${\tau }_2=75\text{\%}$ , and the maximum output error after calibrated by EM 2 with ${\tau }_2=75\text{\%}$ is smaller than the one after calibrated by EM 1 with ${\tau }_1=50\text{\%}$ . One may see that a kinematic calibration adopting the EM with a bigger τ value improves the accuracy of parallel manipulators more significantly, which proves the validity of the proposed identifiability index.

Furthermore, by investigating these three EMs adopting different basic format combinations of single error vector, an approach of error modeling is proposed for obtaining a bigger τ value. This approach aims at avoiding coupling of error parameters by choosing error vector formats reasonably. The choosing principle follows three rules. The first one is that the vector formats of error parameters on the moving platform can be the same with others. The second one is that error vectors should be expressed in as more as possible formats. The third one is that make τ approach 100% as closer as possible.

The study of this article is very useful for error modeling in kinematic calibration of other parallel manipulators.