A manufacturing-oriented error modelling method for a hybrid machine tool based on the 3- P RS parallel spindle head

Abstract

Five-axis hybrid machine tools are widely used due to the advantages of high speed, high precision, and good performance in the aviation and aerospace manufacturing industry. Many scholars have studied the basic theories and key technologies along with the widespread attention of academia and industry thereon. An integrated geometric error modelling method, under the unified coordinate system framework for a general five-axis hybrid machine tool based on a 3-PRS parallel spindle head, is proposed. The kinematic inverse model of a hybrid machine tool is built using a vector chain method. On the basis of the analysis of the various error sources, an integrated geometric error model is developed based on perturbation theory. Then, the influence of each error source of a hybrid machine tool is investigated at the end-effector. Screw theory is applied to research error source separation for the 3-PRS parallel spindle head of a hybrid machine tool, and a sensitivity analysis of the error sources is undertaken using statistical methods. Based on this, the compensable error sources and non-compensable error sources of a hybrid machine tool are separated successfully. The non-compensable error sources must be strictly controlled in the process of design and manufacture, while the compensable error sources can be compensated inexpensively by software after reaching some degree of accuracy in the design and manufacture. This research provides a theoretical reference for this kind of machine tool design and its use in manufacturing.

Keywords

hybrid machine tool error model error source separation sensitivity analysis

Introduction

Parallel mechanisms have been widely applied in the manufacture of aviation and aerospace industry components with high speed, high precision, good dynamic performance, and other advantages. Some of the typical mechanisms include the Tricept series module,¹ the Exechon hybrid module,² the Sprint Z3,³ and so on. Tianjin University in China has also developed an excellent parallel spindle head A3.^4,5 The Sprint Z3 parallel spindle head has three axial linear guide rails spaced evenly around the static platform, with sliding blocks on the guide rails which can be driven back and forth by a servo motor; the sliding block and the spherical joint on the moving platform are connected by the follower lever. This topology gives the spindle head three motions, a translation along the Z-axis and two swinging motions on the A/B-axis through the movement of three branch chains along the Z-axis direction. DS-Technology developed the ECOSPEED series 5-axis high-speed machining centre by combining a Sprint Z3 with a traditional X-Y workbench. The Sprint Z3 has been widely used in the high-speed and high-precision machining of large-scale structural components in the aviation and aerospace industry.

Geometric accuracy is one of the important performance specifications of the parallel mechanism based spindle heads as the high accuracy are the essential requirements: the accuracy of a parallel mechanism is mainly improved in two ways – accuracy design and kinematics calibration. In theory, the position error at the end of a 6-degrees of freedom (DOF) parallel mechanism can be compensated by software, but for lower-mobility parallel mechanisms, only those position errors corresponding to the degrees of freedom can be compensated through kinematic calibration, the remaining errors can only be controlled by accuracy in design.^6,7 The error model is the foundation of accurate design and kinematic calibration. In recent years, many scholars made great advances in error modelling of parallel mechanisms.

One of the classical error modelling approaches is the small perturbation method of homogeneous transformation matrices based on the Denavit-Hartenberg⁸ (D-H) conventions, the approach can accurately express the physical significance and accumulation effect of the error sources and there is no tedious matrix differential operation in the process of error modelling. Yao et al.⁹ established a static position and orientation error model for a wheeled train uncoupling robot based on this method, analysed the impact of parameter error on position error, and used the input motion planning method to improve the positional accuracy of the robot. The error model for a hybrid parallel robot is proposed by Wang et al.¹⁰ using the same method, and kinematic calibration and error compensation were researched on that basis; however, the parameters become discontinuous when this method is used to research parallel adjacent axes.¹¹ To avoid this disadvantage, many scholars have proposed different improved models, such as the CPC-Model and S-Model,¹² Sung and Lu¹³ analysed a five-axis machine tool using a kind of improved D-H method.

Differential geometry is another effective method used for modelling errors in series and parallel manipulators. Chen et al.¹⁴ introduced a kinematic calibration method based on the local POE error model. The advantages of a POE method lie in the following: (1) revolute and prismatic joints can be uniformly expressed in the twist coordinates, (2) the twist coordinates of the joint axes can be set up with simple values, and (3) it needs no special treatment when two adjacent joint axes are parallel. Based on the same theory, Cheng et al.¹⁵ built error model of a five-axis NC machine tool, and analysed global sensitivity of volumetric errors using a Morris approach. This POE model can compare the error rate effectively, and describe the non-linear relationship between the errors with less computational effort. It is also precise and succinct enough to express the relationship of each of components as the Morris method is based on the elementary effect (EE).

The vector chain method is a kind of commonly used error modelling method. With this method, Ni et al.¹⁶ established an error model for a 3-PRS parallel machine tool based on perturbation principles. Cheng et al.¹⁷ developed an error model of PKM based on the vector chain method, and established the system sensitivity model using statistical methods, considering all of space error terms. Huang et al.^18,19 established an error model for a hybrid machine tool using the vector chain method, and introduced kinematic calibration based on a regularisation method; the merit of this method lies in the identification model being concise and effective, and it just requires measurement of the partial degree of freedom in the direction of movement to run. In addition to the above error modelling methods, screw theory is also a useful approach for establishing an error model. Jin and Chen²⁰ proposed an error model based on screw theory which can be used to analyse three types of constraint error problems of decoupled parallel manipulator.

On the basis of the basic modelling method, many scholars established PKM error models considering connection errors, guide errors, and clearance errors. For lower mobility parallel mechanisms such as the 3-PRS, Zhao et al.²¹ built an error model considering manufacturing and assembling tolerance errors of prismatic pairs (P). Wang et al.^22,23 established an error model including hinge joint errors of a revolute joint (R), a spherical joint (S), and zero errors of a prismatic pair, but ignored the straightness errors of the guide rail and orientation errors of the revolute joint. O’Brien et al.²⁴ analysed the same question by taking account of the straightness errors in the guide rail but ignoring the position errors of the spherical joint. For non-compensable error sources, Meng and Li²⁵ proposed a position error forecast analysis of the end-effector considering the hinge clearance errors. Chanal et al.²⁶ investigated two different inverse kinematics models (IKMs), one with 28 geometrical parameters, the other with 42 parameters, and they then proposed the IKM which best fits a given geometrical machine tool: the contribution of their work is in the analysis of the impact of the controller IKM on PKM machining behaviour and the quantification of its incidence on the assembly and machining tolerances of structural elements.

Although the above error modelling methods have many advantages, they fail to analyse the geometric error sources in the view of manufacturing and assembly. Huang et al.^27,28,29 proposed an effective error modelling method which enables the source errors affecting the compensable and non-compensable pose errors to be separated in an explicit manner, and then used it for the tolerance design of a 2T1R parallel manipulator. However, the generality of this method is quite poor. On this basis, Liu et al.³⁰ developed an error modelling method based on screw theory, this method can separate the various geometric error sources with the characteristics of twist space, wrench space, and their sub-space as a constrained body in the configuration space. It can be used to generalise lower-mobility parallel manipulators. Tian et al.³¹ used this approach to a 3-RPS parallel spindle head, but this method cannot realise the rapid and effective separation of error sources when it is used on a hybrid machine tool. Therefore, it is essential to establish a comprehensive error model which can distinguish the compensable error sources from those non-compensable error sources affecting the positional accuracy of the end-effector.

Those scholars mentioned above have conducted extensive research in the field of accuracy analysis of parallel mechanisms, but further research is needed in the area of error modelling method and the error source separation. On the basis of previous research results, this work proposes an integrated error modelling method for the general 3-PRS-XY hybrid machine tool, and analyses the various error sources affecting position errors of the end-effector from the perspective of manufacturing. This method can separate the compensable and non-compensable error sources affecting the accuracy of the end-effector, and it is critical for the further accurate design and kinematic calibration of such a hybrid machine tool.

Kinematic analysis

Structure description

The research object of this article is a five-axis hybrid machine tool consisting of a 3-PRS parallel spindle head and an X-Y workbench. The machine tool can be decomposed into two kinematic chains: one is a static platform-tool kinematic chain, which runs from the static platform via the 3-PRS parallel mechanism and moving platform to the cutting tool, the other is the static platform-workpiece kinematic chain, which runs from the static platform through the X and Y guide rails to the workpiece.

The 3-PRS parallel spindle head, which is the symmetric structure in the space, has a translation movement in the Z-direction and two rotational movements round X and Y-axes. The topology of the 3-PRS parallel spindle (Figure 1) consists of three kinematic chains, a moving platform, and a static platform. Each chain is composed of a prismatic pair (P), a revolute joint (R), and a spherical joint (S), in series, and the prismatic pair is connected with the static platform, and the spherical joint is connected to the moving platform. The spindle head is a ball screw which is directly connected to the motor, and it transfers force and movement through R and S joints, making the moving platform complete one translation and two rotation movements in the space.

Figure 1.

Structural model of the 3-PRS parallel spindle head.³

The static platform-workpiece kinematic chain, a two freedom mechanism, is composed of a vertical (Y-direction) gantry double drive moving platform and a horizontal (X-direction) moving platform (Figure 2). The five-axis linkage machining centre with its hybrid structure can not only guarantee the stiffness at the end of machine tool but also expand the workspace thus maximising the machining efficiency of aviation industry workpieces with complex shapes.

Figure 2.

Equivalent schematic diagram of the mechanism.

The 3-PRS-XY five-axis hybrid machine tool mechanism diagram is shown in Figure 2. O is the geometric centre of the static platform, $O'$ denotes the geometric centre of the moving platform, $A_{i}$ is the geometric centre of each spherical joint, $B_{i}$ is the geometric centre of each revolute joint, and $C_{i}$ is the intersection point of the static platform and each kinematic chain guide rail. So, the moving platform and static platform can be, respectively, expressed as regular triangles $Δ A_{1} A_{2} A_{3}$ and $Δ C_{1} C_{2} C_{3}$ . $O ″$ is the workpiece coordinate system origin located in the workpiece. D is the intersection of the Y-axis direction guide rail and the horizontal plane, $D'$ is the projection point of D on the X-direction guide rail, E is the intersection of the X-direction guide rail and the vertical plane, and $E'$ is the position coincidence point of E

${O}$ is the fixed coordinate system established in the static platform geometrical centre point O, where the Z-axis is perpendicular to the static platform, the X-axis is along the horizontal direction, and the Y-axis satisfies the right-hand rule. ${O_{i}}$ is the branched chain coordinate system which is established through a counter-clockwise rotation $α_{i}$ around the Z-axis, and ${C_{i}}$ is the fixed coordinate system in each $C_{i}$ , whose axis directions are the same as ${O_{i}}$ . ${B_{i}}$ is the fixed coordinate system established in the revolute joint geometric centre $B_{i}$ , the direction of the axis ${w ″}_{i}$ is same as the lever $\vec{B_{i} A_{i}}$ , ${u ″}_{i}$ is same as $x_{i}$ , ${v ″}_{i}$ meets the right-hand rule, and ${A_{i}}$ is the fixed coordinate system established at the spherical joint geometric centre $A_{i}$ , the directions of $u_{i}$ , $v_{i}$ , and $w_{i}$ are the same as ${O_{i}}$ . Then, we develop three body-fixed coordinate systems – ${C'_{i}}$ , ${B'_{i}}$ , and ${A'_{i}}$ in $C_{i}$ , $B_{i}$ , and $A_{i}$ , respectively. ${O'}$ is the body-fixed coordinate system in the geometric centre of the moving platform $O'$ , where $z'$ is perpendicular to the moving platform, the direction of $x'$ is same as vector $\vec{A_{1} A_{3}}$ , and $y'$ satisfies the right-hand rule.

${O ″}$ is the body-fixed coordinate system of the workpiece, whose axes $x ″$ , $y ″$ , and $z ″$ are same as the counterpart of x, y, and z in ${O}$ . A can only make translational movement in the x, y-direction relative to B, which lies in the direction of the X-Y guide rail. Establishing fixed coordinate systems ${D}$ and ${E}$ at points D and E, their axis directions are the same as those in ${O}$ . ${D'}$ and ${E'}$ are the body-fixed coordinate systems established in $D'$ and $E'$ . Two kinematic chains have been established: one from the coordinate system ${O}$ to ${O'}$ , that is the kinematic chain from the static platform to the tool, the other is from the coordinate system ${O}$ to ${O ″}$ that is the kinematic chain from the static platform to the workpiece.

Position inverse kinematics

A position inverse solution can establish the mapping relationship between the position and orientation of the end-effector and the kinematic chain joints variables. The 3-PRS parallel spindle head is a lower-mobility parallel mechanism and joint variables are a prismatic pair elongation $q_{i}$ . The end position vector of the mechanism can be expressed as the following formula

r = c_{i} + q_{i} z_{i} + b {w ″}_{i} - R a_{i}, i = 1, 2, 3

(1)

where, $c_{i}$ is the vector of point C under the ${O}$ , $c_{i} = R_{i} c$ , $R_{i}$ is the transformation matrix, $c = {[0, - c, 0]}^{T}$ . $z_{i}$ is the unit vector along the z-axis under coordinate system ${O_{i}}$ . b is the length of the follower lever BA, ${w ″}_{i}$ is the unit vector of lever $B_{i} A_{i}$ . $R$ is the Euler rotation matrix, $a_{i} = R_{i} a$ , $a = {[0, a, 0]}^{T}$ . a and c are the hinge centre circumscribed circle radii of the moving platform and the static platform, respectively.

Taking the dot product with $y_{i}$ on both sides of the closed-loop equation (1), the constraint equation can be written as

{(b {w ″}_{i} - r + c_{i} - R a_{i})}^{T} y_{i} = 0, i = 1, 2, 3

(2)

Similarly, taking the dot product with $x_{i}$ on both sides of the closed-loop equation (1) yields

{(b {w ″}_{i} - r + c_{i} - R a_{i})}^{T} x_{i} = 0, i = 1, 2, 3

(3)

Uniting equations (2) and (3) gives

{w ″}_{iy} = \frac{{(r + R a_{i} - c_{i})}^{T} y_{i}}{b}

{w ″}_{ix} = \frac{{(r + R a_{i} - c_{i})}^{T} x_{i}}{b}

Then

{w ″}_{iz} = \sqrt{1 - {w ″}_{iz}^{2} - {w ″}_{iz}^{2}}, i = 1, 2, 3

(4)

$q_{i}$ can be obtained by substituting ${w ″}_{i}$ into equation (1)

q_{i} = ‖ r + R a_{i} - c_{i} - b {w ″}_{i} ‖, i = 1, 2, 3

(5)

Error model

The error model for the 3-PRS parallel spindle head and the X-Y serial workbench is established based on the theory of vector chains and small perturbations: an integrated error model of a hybrid machine tool is then established.

Error model of a 3-PRS parallel spindle head

Error source analysis

The 3-PRS parallel spindle head error sources mainly come from two aspects: the position and orientation errors caused by the relative movement of the prismatic pair guide rail, the other is clearance errors caused by the clearance of the revolute joint (R) and the spherical joint (S). Ignoring the spherical joint errors, all possible geometric errors of the mechanism are considered (Table 1).

Table 1.

Geometric error sources in a 3-PRS system.

Geometric error source	Meaning
$Δ a_{i}$	Position errors of point $A_{i}$ referenced to point $O_{i}$
$Δ c_{i}$	Position errors of point $C_{i}$ referenced to point O
$Δ q_{i}$	Zero error at point $C_{i}$
$Δ b_{i}$	Length error of follower lever
$θ_{i}$	Orientation errors of ${C_{i}}$ referenced to ${O_{i}}$
$θ_{Ci}$	Orientation errors of ${{C'}_{i}}$ referenced to ${C_{i}}$
$θ_{Bi}$	Orientation errors of ${{B'}_{i}}$ referenced to ${B_{i}}$

Error model

The error model is established based on the theory of small perturbations. Taking the perturbation of the position vector equation of tool nose point in the end-effector gives

\begin{matrix} r + Δ r = R_{i} (E_{3} + θ \times) \\ [(q_{i} + Δ q_{i}) e_{3} + (E_{3} + θ_{C'_{i}} \times) (E_{3} + θ_{B'_{i}} \times) R_{B_{i}} (b + Δ b_{i})] \\ + R_{i} (c + Δ c_{i}) \\ - [E_{3} + θ \times] R R_{i} (a + Δ a_{i}) \end{matrix}

(6)

Where, $Δ r$ is the position errors of the end-effector. $θ \times$ is the skew matrix of $θ$ , $θ$ is the orientation errors of the end-effector. $E_{3} = (\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix})$ , $e_{3} = {(\begin{matrix} 0 & 0 & 1 \end{matrix})}^{T}$ . $R_{Bi}$ is the orientation transfer matrix from ${B'_{i}}$ to ${B_{i}}$ , $R_{Bi} = [\begin{matrix} c θ_{i} & 0 & s θ_{i} \\ 0 & 1 & 0 \\ - s θ_{i} & 0 & c θ_{i} \end{matrix}]$

Taking the linearization of equation (6) and subtracting it from the nominal position vector of the tool nose point and omitting the high-order items gives

\begin{matrix} Δ r = R_{i} Δ c_{i} + R_{i} Δ q_{i} e_{3} + R_{i} R_{B_{i}} Δ b_{i} e_{3} + R_{i} θ_{C_{i}} \times R_{B_{i}} b e_{3} \\ + R_{i} θ_{B_{i}} \times R_{B_{i}} b e_{3} + R_{i} θ_{i} \times q_{i} e_{3} + R_{i} θ_{i} \times R_{B_{i}} b e_{3} \\ - R R_{i} Δ a_{i} - θ \times R R_{i} a_{i} \end{matrix}

(7)

Left-multiplying equation (7) by $w_{i}^{T}$ on both sides leads to

\begin{matrix} w_{i}^{T} Δ r + {(a_{i} \times w_{i})}^{T} θ = w_{i}^{T} Δ c_{i} + w_{i}^{T} Δ q_{i} e_{3} \\ + w_{i}^{T} Δ b_{i} - w_{i}^{T} R R_{i} Δ a_{i} \end{matrix}

(8)

where $w_{i} = R_{B_{i}} R_{C_{i}} R_{i} e_{3}$

Left-multiplying equation (7) by $v_{i}^{T}$ on both sides leads to

\begin{matrix} v_{i}^{T} Δ r + {(a_{i} \times v_{i})}^{T} θ = v_{i}^{T} (R_{i} θ_{i} \times q_{i} e_{3}) + v_{i}^{T} (R_{i} θ_{i} \times R_{B_{i}} b e_{3}) \\ + v_{i}^{T} (R_{i} θ_{C_{i}} \times R_{B_{i}} b e_{3}) + v_{i}^{T} (R_{i} θ_{B_{i}} \times R_{B_{i}} b e_{3}) \\ - v_{i}^{T} R R_{i} Δ a_{i} \end{matrix}

(9)

Where, $v_{i} = R_{B_{i}} R_{C_{i}} R_{i} e_{2}$

Rewriting and simplifying equations (8) and (9) in matrix form yields

J_{o'} Δ x_{o'} = N_{o'} Δ ε_{o'}

(10)

where $Δ x_{o'} = [\begin{matrix} Δ r \\ θ \end{matrix}]$ , $J_{o'} = [\begin{matrix} J_{a} \\ J_{c} \end{matrix}]$ , $N_{o'} = [\begin{matrix} N_{a} & 0 \\ 0 & N_{c} \end{matrix}]$

Error model of an X-Y workbench

Error source analysis

The geometric error sources of an X-Y workbench mainly come from the movement of the single rail and the relative errors between the guide rails. All possible geometric errors of an X-Y workbench are shown in Table 2.

Table 2.

Error sources of an X-Y workbench.

Geometric error source	Meaning
$θ_{d}$	Orientation errors of ${D}$ referenced to ${O}$
$θ_{d'}$	Orientation errors of ${D'}$ referenced to ${D}$
$Δ q_{d}$	Zero error at point D
$θ_{e}$	Orientation errors of ${E}$ referenced to ${D}$
$θ_{e'}$	Orientation errors of ${E'}$ referenced to ${E}$
$Δ q_{e}$	Zero error at point E

Error model

A closed-loop equation for an X-Y workbench can be expressed as

r' = d y + e x + r ″

(11)

where y is the unit vector in the y-direction for the coordinate system ${D}$ and x is the unit vector in the x-direction for coordinate system ${E}$ . $r ″$ is a distance vector between coordinate systems ${O ″}$ and ${O'}$ .

Perturbation of equation (11) leads to

\begin{matrix} r + Δ r' = R_{o} (E_{3} + θ \times) R_{d} (E_{3} + θ_{d'} \times) (d + Δ q_{d}) y \\ + R_{o} (E_{3} + θ \times) R_{d} R_{e} (E_{3} + θ_{e} \times) \\ (E_{3} + θ_{e'} \times) (d + Δ q_{e}) x + [E_{3} + θ \times] (r' + Δ r ″) \end{matrix}

(12)

Simplifying equation (12) and omitting the higher-order items, subtraction from equation (11) leads to

\begin{matrix} Δ r - θ \times r' = R_{o} θ_{d} \times R_{d} y d + R_{o} θ_{d} \times R_{d} R_{e} x d \\ + R_{o} R_{d} y Δ q_{d} + R_{o} θ_{d'} \times y d + \\ + R_{o} R_{d} R_{e} x Δ d_{e} + R_{o} R_{d} R_{e} θ_{e} \times x e \\ + R_{o} R_{d} R_{e} θ_{e'} \times x e \end{matrix}

(13)

All axis directions of coordinate system ${O ″}$ are the same as those in coordinate system ${O'}$ , subtracting $Δ r'$ from $Δ r ″$ , the solution gives the end of the kinematic chain’s orientation errors $Δ r$ .

$R_{o}$ is the position transfer matrix of coordinate system ${O ″}$ referenced to ${O'}$ ; $R_{d}$ is the position transfer matrix of ${D}$ referenced to ${O}$ ; $R_{e}$ is the position transfer matrix of ${E}$ referenced to ${D}$ .

Rewriting equation (13) in matrix form yields

J_{o ″} Δ x_{o ″} = N_{o ″} Δ ε_{o ″}

(14)

where $Δ x_{o ″} = [\begin{matrix} Δ r \\ θ \end{matrix}]$ , $N_{o ″} = [\begin{matrix} {(R_{d} y d \times x)}^{T} R_{e} + {(x e \times y)}^{T} R_{o} R_{d} R_{e} \\ {(x e \times y)}^{T} R_{o} \\ y^{T} R_{o} R_{d} y \\ {(x e \times y)}^{T} R_{o} R_{d} R_{e} \\ {(x e \times y)}^{T} R_{o} R_{d} R_{e} \\ x^{T} R_{o} R_{d} R_{e} x \end{matrix}]$

Δ ε_{o ″} = [\begin{matrix} θ_{d} & θ_{d'} & Δ q_{d} & θ_{e} & θ_{e'} & Δ q_{e} \end{matrix}]

Total error model

The 3-PRS parallel spindle head error model, which corresponds to the kinematic chain from static platform to tool, and the X-Y platform error model, which corresponds to the kinematic chain from static platform to workpiece in tandem, are analysed, and then taking difference between the tool coordinate system and the workpiece coordinate system, the total error model of the five-axis hybrid machine tool can be expressed as

J_{o'} Δ x_{o'} - J_{o ″} Δ x_{o ″} = N_{o'} Δ ε_{o'} - N_{o ″} Δ ε_{o ″}

(15)

Rewriting equation (15) in matrix form leads to

J Δ x = N Δ ε

(16)

where $J = [\begin{matrix} J_{o'} \\ J_{o ″} \end{matrix}]$ , $N = [\begin{matrix} N_{o'} & 0 \\ 0 & N_{o ″} \end{matrix}]$ , $Δ x = {[\begin{matrix} Δ x_{o'} \\ Δ x_{o ″} \end{matrix}]}^{T}$ , $Δ ε = {[\begin{matrix} Δ ε_{o'} \\ Δ ε_{o ″} \end{matrix}]}^{T}$

Isolation of error sources

The topology of the machine tool is divided into the parallel and the serial components, and the parallel mechanism can be also divided into a six-DOF type and a lower-mobility type. Theoretically only the position errors at the end of the mechanism corresponding to the degree of freedom of machine tool can be compensated by an NC system.

For the lower-mobility parallel mechanisms, errors in the actuation space can be compensated, but not in the constraint space. There is no direct mapping relationship between the non-compensable error sources and the movement of the end-effector in three-dimensional space for the serial mechanism, therefore, screw theory can be employed to solve this problem, which can form a twist space and a wrench space by increasing the number of dimensions in the space. Then the error sources can be separated according to the limited relationship in that space.

The Jacobian matrix and screw theory are used to analyse the error source separation for both parallel, and serial machine tools. The various types of error sources of such hybrid machine tools are separated based on these two theories.

Error source isolation methods for the parallel mechanism

Theoretical basis

The Jacobian matrix of the parallel machine tool can be divided into a driving Jacobian matrix $J_{a}$ and the constraint Jacobian matrix $J_{c}$ . $J_{a}$ corresponds to the driving space of the machine tool in which error sources can be compensated, so the compensable error sources can be isolated by taking the dot product with the driving force direction vector. Similarly, $J_{c}$ corresponds to the constraint space in which error sources are non-compensable, so non-compensable error sources can be isolated by taking the dot product with the constraint direction vector.

Application

The theory can be used in the 3-PRS parallel mechanism. Equation (4) can be attained by left-multiplying $v_{i}^{T}$ , which is the constraint direction unit vector to equation (2), $J_{c}$ is the constraint Jacobian matrix

J_{c} = [\begin{matrix} v_{1}^{T} & {(a_{1} \times u_{1})}^{T} \\ v_{2}^{T} & {(a_{2} \times u_{2})}^{T} \\ v_{3}^{T} & {(a_{3} \times u_{3})}^{T} \end{matrix}]

Corresponding error terms can be expressed as

N_{c_{i}} = [\begin{matrix} u_{i}^{T} R_{i} & q_{i} {(w_{i} \times u_{i})}^{T} R_{i} R_{A'_{i}} & q_{i} {(w_{i} \times u_{i})}^{T} R_{i} & u_{i}^{T} R R_{i} \end{matrix}]

This comprises the non-compensable error sources. Because there is a zero term in the Jacobian matrix, the corresponding error sources have no effect on the orientation error, thus the model of error can be simplified to

N = [\begin{matrix} {N'}_{a} & 0 \\ 0 & {N'}_{c} \end{matrix}], N'_{a} = diag [{N'}_{a_{i}}], N'_{c} = diag [{N'}_{c_{i}}]

Δ ε' = {[\begin{matrix} Δ ε_{a 1}^{'} T & Δ ε_{a 2}^{'} T & Δ ε_{a 3}^{'} T & Δ ε_{c 1}^{'} T & Δ ε_{c 2}^{'} T & Δ ε_{c 3}^{'} T \end{matrix}]}^{T}

Δ ε'_{a_{i}} = [\begin{matrix} Δ c_{ix} & Δ c_{iz} & Δ q_{i} & Δ b_{i} & Δ a_{ix} & Δ a_{iy} & Δ a_{iz} \end{matrix}]

Δ ε'_{c_{i}} = [\begin{matrix} α_{i} & γ_{i} & α_{C_{i}} & γ_{C_{i}} & α_{B_{i}} & γ_{B_{i}} & Δ a_{ix} & Δ a_{iy} & Δ a_{iz} \end{matrix}]

So far, all compensable error sources and the non-compensable error sources of the 3-PRS structure are separated. The number of compensable error sources is 21, including the position error $Δ c_{ix}$ and $Δ c_{iz}$ which is the branch of the x- and z-directions, respectively, of $Δ c_{i}$ in the coordinate system ${O_{i}}$ , the length error in the follower lever $Δ b_{i}$ , zero errors $Δ q_{i}$ and position errors $Δ a_{i}$ . The number of non-compensable error sources is 27, including the orientation errors $α_{i}$ and $γ_{i}$ which are the branches of x and z with respect to $θ_{i}$ in the coordinate system ${O_{i}}$ , the orientation errors $α_{C_{i}}$ and $γ_{C_{i}}$ which are the branches of x and z with respect to $θ_{C_{i}}$ in the coordinate system ${C_{i}}$ , the orientation errors $α_{B_{i}}$ and $γ_{B_{i}}$ which are the branches of x and z with respect to $θ_{B_{i}}$ in the coordinate system ${B_{i}}$ and the position error $Δ a_{i}$ . The movements of the 3-PRS mechanism include relative motion, so the position error $Δ a_{i}$ is a common error.

Error source isolation methods for the serial mechanism

Screw theory

There is no simple mapping relationship between the non-compensable error sources and the degrees of freedom in the serial machine tool. So screw theory is needed to expand the three-dimensional space to a special six-dimensional space. The machine tool error, and the error sources, can be expressed by six-dimensional vectors, in which the error source could be separated.

Twist and wrench

Expressing the unit twist and wrench with Plücker coordinates based on a reference point on the end of the machine tool movement chain, and measuring them in the coordinate system at the end of the machine tool, the results can be expressed as

{\hat{$}}_{t} = [\begin{matrix} r \times s + h_{t} s \\ s \end{matrix}], {\hat{$}}_{w} = [\begin{matrix} s \\ r \times s + h_{w} s \end{matrix}]

(17)

where ${\hat{$}}_{t}$ is the unit of twist and ${\hat{$}}_{w}$ is the unit of wrench.

Twist space and wrench space

Within the machine tool working space, the set of error sources can be expressed in six-dimensional space, with the twist space signed with T. The force and torque acting on the end of the machine tool can be also expressed in six-dimensional space, as the wrench space signed W. T and W have a dual relationship.

On the translation axis of the machine tool, the basis of $T_{a}$ and $W_{a}$ can be expressed as

{\hat{$}}_{ta, i} = [\begin{matrix} k s_{i} \\ 0 \end{matrix}], {\hat{$}}_{wa, i} = [\begin{matrix} k s_{i} \\ r_{i} \times k s_{i} \end{matrix}]

(18)

As for the axis of rotation, the basis of $T_{a}$ and $W_{a}$ can be expressed as

{\hat{$}}_{ta, i} = [\begin{matrix} k s_{i} + r_{i} \times s_{i} \\ s_{i} \end{matrix}], {\hat{$}}_{wa, i} = [\begin{matrix} k s_{i} \\ s_{i} + r_{i} \times k s_{i} \end{matrix}]

(19)

where $s_{i}$ is unit vector of the screw axis direction and $r_{i}$ is the position vector from the reference point of the end-effector to a point along the direction of the screw axis.

Error source isolation methods

Under the permitted twist sub-space, the error screw of the machine tool can be expressed as

$_{t, T} = $_{ta} + $_{tc} = \sum_{i = 1}^{f} a_{i} {\hat{$}}_{ta, i} + \sum_{i = 1}^{6 - f} c_{i} {\hat{$}}_{tc, i}

(20)

where ${\hat{$}}_{ta, i}$ and $a_{i}$ are the unit permitted twist and its amplitude, ${\hat{$}}_{tc, k}$ and $c_{i}$ are the unit restricted twist and its amplitude. ${\hat{$}}_{wa, i}$ is the unit actuation wrench which has a dual relationship with ${\hat{$}}_{ta, i}$ , and ${\hat{$}}_{wa, i}$ is the annihilator of ${\hat{$}}_{ta, j}$ . ${\hat{$}}_{wc, k}$ is the unit constraint wrench, which is dual to ${\hat{$}}_{tc, k}$ . ${\hat{$}}_{wc, k}$ , ${\hat{$}}_{ta, i}$ and ${\hat{$}}_{tc, j}$ are a mutual annihilator.

Taking the dot product with ${\hat{$}}_{wa, i}$ and ${\hat{$}}_{wc, k}$ on both sides of equation (20) leads to

{\hat{$}}_{w a, i}^{T} $_{t} = a_{i} {\hat{$}}_{w a, i}^{T} $_{t a, i}, i = 1, 2, \dots, f

(21)

{\hat{$}}_{wc, k}^{T} $_{t} = a_{k} {\hat{$}}_{wc, k}^{T} $_{tc, k}, k = 1, 2, \dots, 6 - f

(22)

Using the same operation on the equation $$_{t} = $_{t, T} - $_{t, W} = M ε$ leads to

{\hat{$}}_{wa, i}^{T} $_{t} = {\hat{$}}_{wa, i}^{T} M ε, i = 1, 2, \dots, f

(23)

{\hat{$}}_{wc, k}^{T} $_{t} = {\hat{$}}_{wc, k}^{T} M ε, i = 1, 2, \dots, f

(24)

where $$_{t, T}$ is the error screw of the tool and $$_{t, W}$ is the error screw of the workpiece.

Combining equations (23) and (24) leads to

a_{i} = \frac{{\hat{$}}_{wa, i}^{T} M ε}{{\hat{$}}_{wa, i}^{T} {\hat{$}}_{ta, i}}, c_{k} = \frac{{\hat{$}}_{wc, k}^{T} M ε}{{\hat{$}}_{wc, k}^{T} {\hat{$}}_{tc, k}}

(25)

The machine tool error model, after isolating the compensable error sources and non-compensable error sources, can be expressed in matrix form as

J_{xa} $_{ta} = E_{a} ε_{a}

(26)

J_{xc} $_{tc} = E_{c} ε_{c}

(27)

where $$_{ta}$ is the compensable orientation error screw and $$_{tc}$ is the non-compensable orientation error screw of the end-effector of the machine tool. Therefore, $$_{ta}$ and $$_{tc}$ correspond to compensable and non-compensable error sources respectively.

Application

Concerning the serial structure X-Y workbench of the 3-PRS-XY hybrid machine tool, and depending on the type of mechanism motion pairs, the basis of its twist space can be expressed as

\begin{matrix} {\hat{$}}_{ta, 1} = [\begin{matrix} i_{x} \\ 0 \end{matrix}], {\hat{$}}_{ta, 2} = [\begin{matrix} j_{y} \\ 0 \end{matrix}], {\hat{$}}_{ta, 3} = [\begin{matrix} k_{z} \\ 0 \end{matrix}], \\ {\hat{$}}_{wc, 1} = [\begin{matrix} 0 \\ i_{o ″} \end{matrix}], {\hat{$}}_{wc, 2} = [\begin{matrix} 0 \\ j_{o ″} \end{matrix}], {\hat{$}}_{wc, 3} = [\begin{matrix} 0 \\ k_{o ″} \end{matrix}] \end{matrix}

The basis of its wrench space can be expressed as

\begin{matrix} {\hat{$}}_{wa, 1} = [\begin{matrix} i_{x} \\ {}^{o}{r_{o ″}} \times i_{x} \end{matrix}], {\hat{$}}_{wa, 2} = [\begin{matrix} j_{y} \\ {}^{o}{r_{o ″}} \times j_{y} \end{matrix}], {\hat{$}}_{wa, 3} = [\begin{matrix} k_{z} \\ {}^{o}{r_{o ″}} \times k_{z} \end{matrix}] \\ {\hat{$}}_{tc, 1} = [\begin{matrix} {}^{o}{r_{o ″}} \times i_{o ″} \\ i_{o ″} \end{matrix}], {\hat{$}}_{tc, 2} = [\begin{matrix} {}^{o}{r_{o ″}} \times j_{o ″} \\ j_{o ″} \end{matrix}], {\hat{$}}_{tc, 3} = [\begin{matrix} {}^{o}{r_{o ″}} \times k_{o ″} \\ k_{o ″} \end{matrix}] \end{matrix}

Where, $i$ , $j$ , and $k$ are the metric of the X, Y, and Z-direction’s unit vector in coordinate system ${O ″}$ , and ${}^{o}{r_{o ″}}$ is the metric of point $O ″$ in the global coordinate system ${O ″}$ .

For the static platform, the workpiece error model built in the last section is in vector form, but it can be transformed into screw form. Establishing a workpiece-XY platform error model, the transformation between the coordinate system of each axis can be established as

{}^{o}{r_{o ″}} = {}^{o}{T_{y, o}} {}^{y, o}{T_{y}} {}^{x}{T_{x, o}} {}^{x, o}{T_{x}} {}^{x}{T_{o ″}}

(28)

where ${}^{i - 1}{T_{i, o}}$ and ${}^{i, o}{T_{i}}$ are the transformation matrices between the coordinate system of the adjacent kinematic chains, respectively.

{}^{i - 1}{T_{i, o}} = [\begin{matrix} {}^{i - 1}{R_{i, o}} & {}^{i - 1}{r_{i, o}} \\ 0_{1 \times 3} & 1 \end{matrix}], {}^{i, o}{T_{i}} = [\begin{matrix} {}^{i, o}{R_{i}} & {}^{i, o}{r_{i}} \\ 0_{1 \times 3} & 1 \end{matrix}]

Taking account of the error source of each axis, and measuring each error source in the terminal coordinate system ${O ″}$ , equation (28) can be rewritten in screw form as

Δ_{o ″} = {}^{o ″}{A_{y}} Δ (q_{y}) + {}^{o ″}{A_{o}} {}^{o}{P_{y}} Δ_{y} + {}^{o ″}{A_{x}} Δ (q_{x}) + {}^{o ″}{A_{y}} {}^{y}{P_{x}} Δ_{x}

(29)

In a position on the workspace, $l_{y}$ and $l_{x}$ are the elongation of the Y and X guides, $l_{o ″ y}$ and $l_{o ″ x}$ are the distances from the ends of the Y and X guides to the workpiece, respectively. $l_{y, o}$ is the distance from the static platform centre O to the origin of the coordinate system ${O}$ . $l_{x, y}$ is the distance between the origin of the coordinate system ${E}$ and ${D'}$ . In the ideal model, $l_{y, o}$ and $l_{x, y}$ can be ignored, that is, $l_{y, o}$ and $l_{x, y}$ are zero.

In equation (29), A is the accompanying transformation matrix between adjacent coordinate systems: here, it is the unit matrix. $Δ (q_{i})$ is the six-dimensional orientation error vector, which is obtained from position related geometric error sources corresponding to i axis through isomorphic mapping. $Δ_{i}$ is the six-dimensional orientation error vector, which is obtained from position independent geometric error sources corresponding to the i-axis through isomorphic mapping. $Δ_{o ″}$ represents the six-dimensional orientation error vector corresponding to the ending error source through isomorphic mapping. This isomorphic mapping can be described by

{\hat{Δ}}_{i} = [\begin{matrix} [δ θ \times] & δ r \\ 0_{1 \times 3} & 0 \end{matrix}] ~ Δ_{i} = [\begin{matrix} δ r \\ δ θ \end{matrix}]

(30)

Equation (30) can be expressed under the permitted twist sub-space as

\begin{matrix} $_{t, W} = $_{ta} + $_{tc} \\ = a_{x} {\hat{$}}_{ta, x} + a_{y} {\hat{$}}_{ta, y} + a_{z} {\hat{$}}_{ta, z} + c_{x} {\hat{$}}_{tc, x'} + c_{y} {\hat{$}}_{tc, y'} + c_{z} {\hat{$}}_{tc, z'} \end{matrix}

(31)

The end-effector error screw of the serial kinematic chain can be expressed as a linear combination of two types of geometric error sources, each error source can be translated to a linear explicit expression with physical meaning. The error model of the static platform-workpiece can be written in matrix form as

$_{t, W} = M ε = [\begin{matrix} M_{PDGE} & M_{PIGE} \end{matrix}] [\begin{matrix} Δ_{PDGE} \\ Δ_{PIGE} \end{matrix}]

(32)

where, $M_{PDGE} = [\begin{matrix} {}^{o ″}{A_{y}} & {}^{o ″}{A_{x}} \end{matrix}]$ , $M_{PIGE} = [\begin{matrix} {}^{o ″}{A_{y}} {}^{o}{P_{y}} {}^{o ″}{A_{x}} {}^{o}{P_{x}} \end{matrix}]$

Δ_{PDGE} = {[\begin{matrix} Δ (q_{y}) & Δ (q_{x}) \end{matrix}]}^{T}, Δ_{PIGE} = {[\begin{matrix} Δ_{y} & Δ_{x} \end{matrix}]}^{T}

Combining it with the error model of the static platform-tool gives

$_{t} = $_{t, T} - $_{t, W} = M_{T} ε_{T} - M_{W} ε_{W}

(33)

Taking the dot product to equation (33) with $$_{wa, 1}$ , $$_{wa, 2}$ , $$_{wa, 3}$ , $$_{wc, 1}$ , $$_{wc, 2}$ and $$_{wc, 3}$ gives

J_{xa} $_{ta} = E_{a} ε_{a}, J_{xc} $_{tc} = E_{c} ε_{c}

(34)

Through the aforementioned error separation analysis, non-compensable error sources include the following parts: the roll angle error $ε_{x} (x)$ , the tilt angle error $ε_{y} (x)$ , and the pitch angle error $ε_{z} (x)$ on axis X; the roll angle error $ε_{y} (y)$ , the tilt angle error $ε_{x} (y)$ , and the pitch angle error $ε_{z} (y)$ on axis Y; the verticality error $ε_{yx}$ between axes X and Y, the verticality error $ε_{xz}$ between axes X and Z, and the verticality error $ε_{yz}$ between axes Y and Z.

Sensitivity analysis

The global index of sensitivity is proposed and the effects of orientation error of the end-effector, arising from each error source, are analysed. The influence of various error sources on the end-effector was confirmed. The error sources can be separated into those for which no compensation was possible, and those where it was; the former can be controlled during the manufacturing process and the latter can be improved by kinematic calibration. The use of the proposed method may assist in the design and manufacture of similar equipment.

Considering the effects on terminal position and orientation accuracy arising from the non-compensable errors statistically, a sensitivity probabilistic model can be established as

Δ x = J_{ε} Δ ε'_{c}

(35)

Where, $J_{ε} = J^{+} N_{c}$ , $Δ x$ , $Δ ε_{c}$ and $J_{ε}$ represent the tool tip point error vector, the geometric error source vector, and the error Jacobin matrix between $Δ x$ and $Δ ε_{c}$ respectively. $J_{ε}$ can be expressed with partitioning in accordance with the position and orientation errors.

J_{ε} = [\begin{matrix} J_{ε r} \\ J_{ε θ} \end{matrix}]

(36)

The global sensitivity evaluation index of volumetric error $δ_{r} = \sqrt{Δ r^{T} Δ r}$ , verticality error $δ_{α} = \sqrt{Δ α^{2} + Δ β^{2}}$ and rotating error of the tool axis $δ_{ϕ} = Δ γ$ can be established as

{\bar{μ}}_{rk} = \frac{(\int_{V} μ_{rk} dV)}{V}, μ_{rk} = \sqrt{\sum_{i = 1}^{3} \sum_{m = 1}^{3} J_{ε rimk}^{2}}

{\bar{μ}}_{α k} = \frac{(\int_{V} μ_{α k} dV)}{V}, μ_{α k} = \sqrt{\sum_{i = 1}^{3} \sum_{m = 1}^{2} J_{ε θ imk}^{2}}

{\bar{μ}}_{ϕ k} = \frac{(\int_{V} μ_{ϕ k} dV)}{V}, μ_{ϕ k} = \sqrt{\sum_{i = 1}^{3} \sum_{m = 3}^{3} J_{ε θ imk}^{2}}

where V is the volume of a design space, $J_{ε rimk}$ $(J_{ε θ imk})$ are the elements of $J_{ε r}$ $(J_{ε θ})$ , a branched chain of i, line of m, and column of k, respectively. The work space of the 3-PRS tool is shown in Table 3.

Table 3.

Nominal, and work space, parameters of the 3-PRS spindle head.

Nominal parameter				Work space parameter
$a (mm)$	$c (mm)$	$l (mm)$	$e (mm)$	$z (mm)$	$θ (°)$	$ψ (°)$
170	220	330	200	300~800	0~40	0~360

Figures 3 –5 show the sensitivity analysis of the volumetric error $δ_{r}$ , verticality error $δ_{α}$ and angular error $δ_{ϕ}$ with respect to error sources $Δ ε'_{ck}$ . From these figures, the orientation errors $α_{i}$ and $γ_{i}$ which are the x and z-direction branches of $θ_{i}$ respectively in the coordinate system ${O_{i}}$ , the x-direction branches of the orientation error $θ_{Ci}$ , $α_{Ci}$ in coordinate system ${C_{i}}$ , exert a quite larger influence on the sensitivity coefficient for volumetric error $δ_{r}$ , and angular error $δ_{ϕ}$ . The orientation errors have more significant effect on the machining accuracy than the position errors. The various error sources exert a significant influence on the cutting tool verticality errors.

Figure 3.

The sensitivity of volumetric error with respect to $Δ ε'_{ck}$ .

Figure 4.

The sensitivity of verticality error with respect to $Δ ε'_{ck}$ .

Figure 5.

The sensitivity of angular error with respect to $Δ ε'_{ck}$ .

Figures 6 –9 show the result of the sensitivity analysis of the X-Y platform, where $X \subseteq [- 600, 600]$ mm and $Y \subseteq [0, 680]$ mm. From the figures above, the effect of the error sources on each direction of the end-effector of the machine tool can be seen; for example, there are six error sources which affect the X-direction end-effector position errors, where $ε_{z} (x)$ , $ε_{z} (y)$ , $ε_{yx}$ , and $ε_{yz}$ exert the most influence. Figures 7 and 8 show that the Y- and Z-direction position errors are more sensitive to $ε_{yx}$ and $ε_{y} (y)$ , respectively, than other error sources. Meanwhile, all error sources affect the volumetric error, among which $ε_{yx}$ makes the greatest contribution. Therefore, these error sources having the most effect on machine tool accuracy must be controlled when designing the assembly tolerance.

Figure 6.

The sensitivity of the X-direction coordinate with respect to $Δ ε'$ .

Figure 7.

The sensitivity of the Y-direction coordinate with respect to $Δ ε'$ .

Figure 8.

The sensitivity of the Z-direction coordinate with respect to $Δ ε'$ .

Figure 9.

The sensitivity of volumetric error with respect to $Δ ε'$ .

After investigating the trend in each sensitivity measure to changes in the length of the follower levers, the influence of the follower lever length on machining accuracy can be ascertained. Figures 10 and 11 show the changes in the sensitivity coefficient of the volumetric error $δ_{r}$ and verticality error $δ_{α}$ with respect to error source $γ_{i}$ and $Δ a_{ix}$ , respectively.

Figure 10.

The change in volumetric error sensitivity with respect to $γ_{i}$ .

Figure 11.

The change in verticality error sensitivity with respect to $Δ a_{ix}$ .

It can be seen from these two figures that, with the change of the follower lever length from 300 mm to 360 mm, the change in sensitivity coefficient can be described as quasi-linear, and only the sign of the gradient and intercept are different. The rates of change of the volumetric error $δ_{r}$ , verticality error $δ_{α}$ , and angular error $δ_{ϕ}$ with respect to each error source are shown in Table 4. It can be concluded that the most of the rates of change are positive, except for those of the slopes of $δ_{r}$ , $δ_{α}$ , and $δ_{ϕ}$ with respect to $α_{i}$ and $γ_{i}$ , meanwhile, the absolute value of the rates of change of $α_{i}$ and $γ_{i}$ are relatively small. So, after taking all of the aforementioned information into account, an appropriate reduction in the length of the follower lever was beneficial to the reduction of machining errors.

Table 4.

Rate of change of error source sensitivities.

	$α_{i}$	$γ_{i}$	$α_{C_{i}}$	$γ_{C_{i}}$	$α_{B_{i}}$	$γ_{B_{i}}$	$Δ a_{ix}$	$Δ a_{iy}$	$Δ a_{iz}$
Volumetric error $δ_{r}$	0.1143	−0.0873	0.3205	1.0972	0.4680	1.0438	1.0972	0.4680	1.0438
Verticality error $δ_{α}$	−0.0028	−0.0026	0.0005	5.0721	0.1121	0.0270	5.0721	0.1121	0.0270
Angular error $δ_{ϕ}$	−0.0022	−0.0211	0.0000	0.0957	0.0334	0.0017	0.0957	0.0334	0.0017

Conclusion

The machine tool kinematic chains consist of the static platform–tool chain (parallel chain) and the static platform-workpiece chain (serial chain). The error model for the 3-PRS parallel spindle head and the X-Y serial workbench is established based on the theory of vector chains and small perturbations: an integrated error model of a hybrid machine tool is then established.

The error source separation methods are analysed for different topologies of machine tool. The Jacobian matrix and screw theory are used to analyse the error source separation for both parallel, and serial, machine tools. The various types of error sources of such hybrid machine tools are separated based on these two theories.

The probability model of the sensitivity of non-compensable error sources is established statistically. The global index of sensitivity is proposed and the effects of orientation error of the end-effector, arising from each error source, are analysed.

The manufacturing-oriented and integrated geometric error model proposed in this article provided a theoretical foundation for sensitivity analysis: the influence of various error sources on the end-effector was confirmed. The error sources can be separated into those for which no compensation was possible, and those where it was; the former can be controlled during the manufacturing process and the latter can be improved by kinematic calibration. The use of the proposed method may assist in the design and manufacture of similar equipment.

Footnotes

Appendix 1

Handling Editor: James Baldwin

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by a National Natural Science Foundation of China grant (No. 51575385) and a Natural Science Foundation of Tianjin grant (No. 16JCZDJC38400).

ORCID iD

Yanbing Ni

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