Error calibration of controlled rotary pairs in five-axis machining centers based on the mechanism model and kinematic invariants

International Journal of Machine Tools & Manufacture 120 (2017) 1–11

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International Journal of Machine Tools & Manufacture journal homepage: www.elsevier.com/locate/ijmactool

Error calibration of controlled rotary pairs in five-axis machining centers based on the mechanism model and kinematic invariants

MARK

⁎

Zhi Wanga, Delun Wanga, , Yu Wua, Huimin Donga, Shudong Yua,b a b

School of Mechanical Engineering, Dalian University of Technology, China Department of Mechanical & Industrial Eng., Ryerson University, Canada

A R T I C L E I N F O

A BS T RAC T

Keywords: Error calibration Motion measurement Rotary pair Ball bar Kinematic invariants Five-axis machining center

The mechanism model of ball bar testing for a two-axis rotary table of 5-axis machining center is discussed, and a new ball bar method to measure the three-dimensional motions of the rotary pairs in multi-axis machining center is developed based on the mechanism model. Then, the fixed axes and moving axes of the rotary pairs are identified by using spherical image circle fitting and striction circle fitting, according to the kinematic invariants of nominal rotation and the measured motions. The structure errors and kinematic pair errors of the rotary pairs are defined and identified by using the fixed and moving axes, and the kinematic model of the two-axis rotary table is deduced with those errors. The simultaneous two-axis motions of the rotary table are measured to verify the proposed calibration method. The experimental results show good agreements with the predicted results calculated by the calibrated kinematic model. Furthermore, the accuracy of the simultaneous multi-axis motions of the machining center is improved when the identified errors are corrected.

1. Introduction A 5-axis machining center contains two additional perpendicular rotary pairs in comparing with a 3-axis machining center composed of three orthogonal translational pairs; these provide great convenience for machining work-pieces with complicated surfaces. However, the two additional rotary pairs introduce more structure errors and kinematic pair errors [1]. The structure errors correspond to the positional and directional errors of the axis average line of a rotary pair, while the kinematic pair errors correspond to the error motion of the rotary component in relative to the axis average line. It is necessary to calibrate these errors and reduce their influences to improve the accuracy of a 5-axis machining center. In order to calibrate the errors of the rotary pairs in multi-axis machining centers, some efficient methods, based on the 3D probe-ball [2,3], 4D probe-ball [4,5], ball bar [6–9] or some other ingenious devices [10,11], are presented. In these researches, the positional and directional offsets of the rotary pairs are calibrated by the simultaneous multi-axis motions. These calibrated offsets are regarded as the structure errors, which are unrelated to the rotary angles of the rotary pairs. However, the rotary pairs also have kinematic pair errors, which are changing as the rotary angle varies. Sometimes, the ranges of the kinematic pair errors are nearly to those of the structure errors. Hence, it is beneficial to consider these two errors simultaneously during error calibration.

⁎

Corresponding author. E-mail addresses: [email protected] (D. Wang), [email protected] (S. Yu).

http://dx.doi.org/10.1016/j.ijmachtools.2017.04.011 Received 5 January 2017; Received in revised form 10 April 2017; Accepted 18 April 2017 Available online 20 April 2017 0890-6955/ © 2017 Elsevier Ltd. All rights reserved.

Generally, the structure errors and kinematic pair errors will be calibrated absolutely, if the three-dimensional motion of the rotary component is measured. In most cases, the three-dimensional motion can be described by the radial run-outs, axial run-outs and tilts of a precise artifact mounted on the rotary component, and then measured by some displacement indicators [12–16]. Unfortunately, for the controlled rotary pairs in 5-axis machining centers, such as the twoaxis rotary table, it is difficult to mount the artifact to a suitable place because of space limitation; moreover, the directional offsets and positional offsets between the rotary pairs can’t be measured by these methods. Some other precise instruments, such as the laser tracker, autocollimator, polygon, etc. [12,17,18], are also used to measure the motions of the rotary pairs. Nevertheless, these methods may timeconsuming or incomplete for three-dimensional motion measurements. As the ball bar is easy installation and efficient for error measurement of machining centers, in recently researches, the ball bar methods are used to calibrate the errors of the rotary pairs based on the threedimensional motion measurement. In these methods, the errors of the rotary pairs are defined and calibrated by using the transformation matrixes, and the mounting position errors of the ball bar are corrected before error measurement or identified from the measured results [19– 21]. However, the identified results of these methods may be different if the ball bar is mounted to different positions although the mounting

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the three-dimensional motion of the rotary table is described by three translations (xOmf, yOmf, zOmf) and three rotations (α, β, γ) of the moving frame in relative to the fixed one. The spherical pair SD moves with the rotary table, and the trajectory ΓD of the sphere-center D in the fixed frame is written as

position errors are eliminated, as the error motions of different points of the rotary component are different, and the transformation matrixes are related to the nominal installation parameters of the ball bar. Thus, is there a ball bar method independent of the mounting positions and their errors absolutely? How to identify the structure errors and kinematic pair errors of the controlled rotary pairs without considering the installation parameters? Solving these problems will improve the efficiency and accuracy of error calibration. In this paper, the error calibration of rotary pairs is divided into two steps. Firstly, a mechanism model is presented to calculate the threedimensional motions of the rotary pairs, based on the discrete data measured by the ball bar. This model is independent of the mounting position errors of the ball bar. Then, the structure errors and kinematic pair errors of the rotary pairs are identified by using the kinematic invariants of rotary error motions, in order to eliminate the influences of the installation parameters. The kinematic model of the two-axis rotary table is presented by using the identified structure errors and kinematic pair errors, and the errors of the simultaneous two-axis motions are calculated and corrected with the calibrated kinematic model. This provides a new method for error calibration of the rotary pairs in multi-axis machining centers.

ΓD: rOD = rOmf + [R (α , β , γ )] rDm

(1)

where, rOmf =[xOmf, yOmf, zOmf]T is the position vector of the origin point Om in the fixed frame; rDm denotes the position vector of point D in the moving frame; R(α, β, γ) denotes the rotational matrix, whose value is

0 ⎤ ⎡ cα − sα 0 ⎤ ⎡ cβ 0 sβ ⎤ ⎡ 1 0 R(α , β , γ ) = ⎢ sα cα 0 ⎥ ⎢⎢ 0 1 0 ⎥⎥ ⎢⎢ 0 cγ − sγ ⎥⎥ ⎢⎣ ⎥ 0 0 1 ⎦ ⎣− sβ 0 cβ ⎦ ⎣ 0 sγ cγ ⎦

(2)

The letters s and c denote sine and cosine for short. Generally, the six kinematic parameters (xOmf, yOmf, zOmf, α, β, γ) of the rotary table can be calculated by using Eqs. (1) and (2), if the j) position vectors r(Dm of three non-collinear points D(j) (j=1, 2, 3) of the rotary table in the moving frame and the corresponding position ( j) in the fixed frame are given. This can be regarded as the vectors rOD forward solution of Eq. (1) with given the position and direction of the moving frame on the rotary table. However, mounting position errors will appear during fixing the ball bar to the table, this means the values j) of the position vectors r(Dm in the moving frame have errors. In order to avoid the influences of these errors, the position and direction of the moving frame, as well as the six kinematic parameters, are inversely ( j) (j=1, 2, 3) in the fixed frame, solved with given the position vectors rOD as the moving frame can be arbitrary chosen to describe the spatial rigid motion. The origin point Om and the coordinate axes of the moving frame {Om; im, jm, km} in the fixed frame can be determined by

2. The mechanism model for ball bar to test rotary table A ball bar contains two spherical pairs and one translational pair. During ball bar testing, one spherical pair is mounted on the spindle and the other is mounted on the rotary table. The kinematic chains of the machining center and the ball bar constitute a spatial mechanism, and the displacements measured by the ball bar can be regarded as the output motion of the mechanism. Thus, the three-dimensional motion of the rotary table can be calculated by the displacement equation of the mechanism.

(1) + r(2) ⎧ rOD OD ⎪ ⎪ rOmf = 2 ⎨ (2) (1) ⎪ im = rOD − rOD ; km = ⎪ (2) − r(1) rOD ⎩ OD

2.1. The three-dimensional motion of the rotary table The controlled rotary pairs of multi-axis machining center have many different configurations, as the rotary pairs can distribute in different positions and directions of the machining center. For universality, a two-axis rotary table of 5-axis machining center is discussed, as the single rotary pair can be regarded as a special case of the two-axis table, with one of the rotary pairs locked. The ball bar testing devices of the two-axis rotary table are shown in Fig. 1(a). In order to describe the three-dimensional motion of the rotary table, a fixed frame {Of; if, jf, kf } is employed as the coordinate system of the machining center; meanwhile, a moving frame {Om; im, jm, km} is employed and attached to the rotary table, shown in Fig. 1(b). Thus,

(2) − r(1) ) × (r(3) − r(1) ) (rOD OD OD OD (2) − r(1) ) × (r(3) − r(1) ) (rOD OD OD OD

; jm = km × im

(3)

The vector rOmf corresponds to the three translations (xOmf, yOmf, zOmf) of the rotary table. The three rotations (α, β, γ) of the rotary table can be identified by using the coordinate axes, and the equations are

α = arctan(

im 2 − im 3 ); β = arctan[ ]; im1 (im1)2 + (im2 )2

γ = arctan[

jm3 ] k m3

(4)

where, imt, jmt and kmt denote the t-th elements of vectors im, jm and km.

Fig. 1. The devices of ball bar testing.

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by the ball bar. Thus, it's a feasible way to identify the kinematic parameters of the rotary table by using these equations, as the position vector rOD can be solved by Eq. (5) accurately, based on the measured data of the ball bar. 2.3. Motion identification with the displacement equations There need three independent displacement Eq. (5) to solve one undetermined position vector rOD, because the length |rED| measured by the ball bar is scalar. At any rotary position α(i), the Eq. (5) is rewritten as a discrete scalar form as (i ) (i , k ) (i , k ) ⎧ ⎪ r OD − rOE = r ED ; k = 1, 2, 3 ⎨ ⎪ (i , k ) (i , k ) (i , k ) (i , k ) ⎩ rOE = λX lX + λY lY + λZ lZ

where, the superscript i denotes the number of the discrete rotary position α(i), as the error motions of the rotary table are measured in a (i , k ) series of discrete rotary positions α(i) (i=1, 2, …, n). rOE denotes the position vector of the sphere-center E (i,k), and k=1, 2, 3 corresponds to three different positions of the sphere-center E(i,k) during measurement, as shown in Fig. 3. The Eq. (6) shows the position of the sphere-center E is changed three times by using the translational pairs of the machining center. As (i , k ) the vector rOE is known vector determined by the translational pairs, (i ) can be calculated accurately by using the the position vector rOD displacement Eq. (6), based on the measured data of the ball bar. In geometry, the sphere-center D(i) is an identical point of the moving body and the intersection point of three spheres with different radius i, k ) (i , k ) r(ED and centers rOE . The mounting position of the spherical pair SD is changed three times at the same rotary position α(i), in order to get three different points D(i, j) of the rotary table, as shown in Fig. 4. For each mounting position, the spherical pair SE moves to three different positions by using the translational pairs. In order to illustrate more clearly, the displacement equation is written as

Fig. 2. The PPP-SPS-RR mechanism.

2.2. The mechanism model and displacement equations In most cases, the cutting spindle of the machining center is omitted in kinematic analysis. Hence, the kinematic chains of the 5axis machining center and the ball bar form a closed PPP-SPS-RR mechanism, as shown in Fig. 2. The PPP-SPS-RR mechanism has 5 DOFs. The three translational pairs determine the position of the spherical pair SE, and the two rotary pairs determine the position of the spherical pair SD. In the fixed frame, the displacement equation can be written as

λX lX + λY lY + λZ lZ + rED = rOD

(6)

(5)

(i , j ) (i , j , k ) i, j , k ) rOD − rOE = r(ED ; j = 1, 2, 3; k = 1, 2, 3

where, lX, lY and lZ denote the unit direction vectors of the translational pairs PX, PY and PZ; λX, λY and λZ denote the moving distances of the translational pairs, with respect to the origin point Of. The parameters λX, λY, λZ, lX, lY and lZ are regarded as given parameters, as the error motions of the translational pairs can be calibrated by the laser interferometer and autocollimator [22,23] before measurement. rED denotes the vector from the sphere-center E to the sphere-center D, whose length |rED| is the result measured by the ball bar. rOD is the position vector of the sphere-center D, which should be determined to solve the kinematic parameters in Eqs. (3) and (4). The Eqs. (3)–(5) describe the relationship between the threedimensional motion of the rotary table and the length |rED| measured

(7)

where, the superscript j=1, 2, 3 denotes the number of the mounting positions D(i, j); i and k have the same meaning as Eq. (6). It is a worthy to notice that Eq. (7) contains nine independent equations at one rotary angle α(i), and corresponds to nine configurations of the PPP-SPS-RR mechanism. (i,1) (i,2) (i,3) Based on Eq. (7), the position vectors rOD , rOD and rOD of three non-collinear points of the rotary table will be got, and the six kinematic parameters of the rotary table will be solved by substituting these vectors into Eqs. (3) and (4). At every rotary position α(i), the six kinematic parameters can be calculated by the same way. Then, the three-dimensional motion of the rotary table is identified. In the existing researches [19–21], the measured data of the ball

(i ) Fig. 3. Measurement of the vector rOD at rotary position α(i).

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Fig. 4. Measurement of three different point D(i,

j)

at rotary position α(i).

(3) The table rotates around the axis of RA again, and the translational pairs move simultaneously to keep the ball bar parallel to the axis jf at every measuring position (k=2). (4) The table rotates around the axis of RA for the third time, and the translational pairs move simultaneously to keep the ball bar parallel to the axis kf at every measuring position (k=3). (5) The spherical pair SD is mounted to the second point D(i,2), and then repeat steps (2) to (4). (6) The spherical pair SD is mounted to the third point D(i,3), and then repeat steps (2) to (4).

bar is used to identify the errors of rotary pairs directly, based on the homogeneous transformation; thus, the position of the moving frame j) should be given and the vectors r(Dm should be identify or corrected before measurement. While in the proposed method based on the mechanism model, the measured data of the ball bar are used to identify the actual motion of the rotary table. As mentioned, the six ( j) kinematic parameters are determined by the position vectors rOD of ( j) three non-collinear points of the rotary table in the fixed frame, and rOD are calculated accurately and directly by the displacement equations of the mechanism model, based on the measured data of ball bar. Thus, j) the mounting position vectors r(Dm of the ball bar in the moving frame are unconcerned in the method proposed in this paper, and the illconditioned matrix will not appear as the transformation matrix is not used, as shown in Eqs. (3) and (4). There is no need to identify or correct the mounting position errors of the ball bar, or more exactly, the position errors of the moving frame and the errors of the vectors j) r(Dm are unconcerned. The identified six kinematic parameters are independent from the mounting position errors.

According to the above process, the rotary pair is measured nine times. The sphere-center E traces nine different arcs, whose shapes are the same as the nominal trajectories traced by the sphere-center D. This process can be realized by a multi-axis NC program, conveniently. The results measured by the ball bar are shown in Fig. 6. The horizontal axis is the nominal ration angle of the rotary pair RA, and the vertical axis is the deviations of the actual distances between the center-points D and E, in relative to the nominal value 100mm. Then, (i ) (i ) (i ) , yOmA , zOmA , αA(i ), βA(i ) , γA(i ) ) of the the three-dimensional motions (xOmA rotary table caused by rotary pair RA can be calculated by Eqs. (3), (4) and (7), based on the results measured by the ball bar.

3. Motion measurement of the two-axis rotary table The three-dimensional motion of the two-axis rotary table is influenced by the rotary pairs RA and RC. These two rotary pairs are measured independently, in order to avoid the combined errors. According to the proposed identification method, each rotary pair is measured nine times. The errors of the three translational pairs of the machining center are calibrated before measurement of rotary pairs.

3.2. Motion measurement of rotary pair RC The motion measurement of rotary pair RC is similarly to the rotary pair RA. The rotary pair RA is locked at the initial position αA=0 while RC is rotating. The configurations of the mechanism for motion measurement of RC are shown in Fig. 7. The rotary pair RC is also measured nine times, according to the proposed measuring process. The sphere-center E traces nine different circles, whose shapes are the same as the nominal trajectories of the sphere-center D. The results measured by the ball bar are shown in Fig. 8. The horizontal axis is the nominal ration angle of the rotary pair RC, and the vertical axis has the same meaning as Fig. 6. Then, the (i ) (i ) (i ) , yOmC , zOmC , αC(i ), βC(i ) , γC(i ) ) of the rotary three-dimensional motions (xOmC table caused by rotary pair RC can be calculated by Eqs. (3), (4) and (7), with the measured data. Although the measurements of the two rotary pairs are separated, the measured motions of the rotary table caused by the rotary pairs RA and RC are described in the same fixed frame, as the mounting position of spherical pair SE in relative to the spindle isn’t change during measurement. This takes some conveniences to calculate the positional and directional offsets between the rotary pairs RA and RC. However, the measured motions contain the motions caused by the structure errors of the components and the installation parameters of the ball

3.1. Motion measurement of rotary pair RA During measurement of rotary pair RA, the rotary pair RC is locked at the initial position αc=0. The configurations of the mechanism for motion measurement of RA are shown in Fig. 5. As the measurement range of the ball bar is limited, the spherical pair SE moves simultaneously by using the three translational pairs, when the rotary table is rotating. In order to improve the measuring efficiency, the three-dimensional motion measurement of the rotary pair RA is designated as the following process. (1) At the initial rotary angle, the spherical pair SD is mounted to the point D(i,1) of the rotary table and the spherical pair SE is fixed on the locked spindle with a tool holder. (2) The table rotates around the axis of RA, and the translational pairs of the machining center move simultaneously to keep the ball bar parallel to the axis if at every measuring position (k=1). 4

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Fig. 5. Motion measurement of rotary pair RA.

invariants of a ruled surface, which can be calculated by

bar, besides the error motions of the rotary pairs, as the positions of the reference points D(i, j) are chosen arbitrarily. In order to eliminate the influences of the installation parameters, the structure and kinematic pair errors of the rotary pairs are identified by using kinematic invariants of the measured motions, as discussed in the following section.

⎧ ρ(i ) = r(i ) + v(i ) l(i ) P ⎨ ⎩ l(i ) = [R (α (i ), β (i ) , γ (i ) )] lm ⎪

⎪

where, r(Pi) denotes the discrete directrix traced by a point P(xPm, yPm, zPm) on the line Lm, whose value can be calculate by Eq. (1); v(i) denotes the distance between the directrix and the striction curve; lm denotes the unit direction vector of line L in the moving frame {Om; im, jm, km}. There have three types of ruled surface ΣL traced by the line Lm of the rotary component in nominal rotation. The first is hyperboloid of one sheet, if the line Lm neither intersects to nor parallels to the rotary axis; the second is cylindrical surface, if the line Lm parallels to the rotary axis; the third is circular conical surface, if the line Lm intersects to the rotary axis; as shown in Fig. 9. These mean the spherical image curve l(i) is a spherical circle for types one and three, or a point for type two; the striction curve ρ(i) is a circle for types one and two, or a point for type three. Furthermore, the axis LQ of the rule surface is the rotary axis of the line Lm. For the measured three-dimensional motion of a rotary pair, the actual line-trajectory traced by the line Lm of the rotary table is similar to the nominal ruled surface discussed, as the error motion is much smaller than the nominal rotation. The deviations between the actual line-trajectory and the nominal one represent the errors of the rotary pair. Thus, a spherical circle is used to fit the actual discrete spherical image curve l(i) to estimate the directional errors of the line-trajectory and determine the direction of the approximate axis. The spherical circle fitting can be realized by saddle point programming, and the mathematic model is

4. Error identification based on the kinematic invariants The rotary component of an actual rotary pair has a special line, whose trajectory is most approximate to a fixed line of the fixed component. The special line of the rotary component is regarded as the approximate moving axis of the rotary pair, and the corresponding fixed line is regarded as the approximate fixed axis. These two axes represent the errors of a rotary pair [1]. Generally, the fixed axis determines the position and direction of the rotary pair; thus, the deviations between the actual and nominal fixed axis are regarded as the structure errors. The moving axis moves with the rotary component, and its motion in relative to the fixed axis are regards as the kinematic pair errors. These two axes can be identified by using the kinematic invariants of rotary motion. And then, the structure and kinematic pair errors will be calculated. 4.1. The kinematic invariants of a line-trajectory In kinematic geometry, a line Lm of the rotary component traces a ruled surface ΣL [24], whose vector equation can be written in a discrete standard form as

ΣL : r(Li ) = ρ(i ) + λ l(i ); i = 1, …, n

(9)

(8)

where, ρ(i) and l(i) denote the striction curve and spherical image curve of the ruled surface, λ denotes the line variable. i denotes the number of the rotary position. The striction curve and spherical image curve are

Fig. 6. Measured results of the rotary pair RA.

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Fig. 7. Motion measurement of rotary pair RC.

⎧ Δ = min max {g (i ) (a)} l ⎪ δ a 1≤ i ≤ n ⎪ (i ) (i ) ⎪ gl (a) = arccos(lQ⋅l ) − δ 0 ; lQ ⎨ = [ sin δQ cos θQ, sin δQ sin θQ, cos δQ]T ⎪ ⎪ s.t.δQ ∈ [0, π ), θQ ∈ [0, 2π ), δ 0 ∈ [0, π /2] ⎪ ⎪ a = (δ , θ , δ )T Q Q 0 ⎩

striction curve ρ(i) and the fitting circle at any rotary position i. For all rotary positions i=1, 2, …, n, there has a maximum value of gρ(i ) , and the maximum value to be minimum for different (xQ, yQ, zQ, rc) is the optimization goal. The striction curve ρ(i) can be calculated by Eq. (9), and the distance v(i) between the directrix and striction curve can be calculated by (10)

v (i ) =

where, Δδ is the fitting error of spherical image curve. lQ denotes the vector the center of the fitting spherical circle, which is regarded as the direction of the approximate axis. The optimization variables (δQ, θQ) are direction angles of the unit vector lQ, and δ0 is half-cone angle of the cone formed by the center Of and the fitting circle, as shown in Fig. 10(a). The optimization function gl(i ) (a) is the angular deviation between the discrete spherical image curve l(i) and the fitting circle at any rotary position i. For all discrete rotary positions i=1, 2, …, n, there has a maximum value of gl(i ) , and the maximum value to be minimum for different (δQ, θQ, δ0) is the optimization goal. Similarly, a circle is used to fit the discrete striction curve ρ(i) to estimate the positional errors of the line-trajectory and determine the position of the approximate axis. The mathematic model of saddle point programming for circle fitting is

⎧ Δ = min max {g(i ) (b)} ρ ⎪ c b 1≤ i ≤ n ⎪ ⎪ g(i ) (b) = (r (i ) − r )2 ; r (i ) = (ρ(i ) − r ) c Q ⎨ ρ ⎪ ⎪ s.t.rc ∈ (0, +∞) ⎪ b = (xQ , yQ , zQ, rc )T ⎩

l (i ) × lQ l (i ) × lQ

(rQ − r(Pi) )⋅[l (i ) − (l (i )⋅lQ)⋅lQ] 1 − (l (i )⋅lQ)2

(12)

The spherical image circle fitting and striction circle fitting determine the approximate rotary axis of line Lm, and the fitting errors Δδ and Δc are invariants errors of a line-trajectory. However, different lines will have different rotary axes, it is necessary to analyze the invariants of the line-trajectories traced by all lines of the rotary table, to find a most suitable rotary axis for the rotary pair. 4.2. The fixed axes and moving axes of the rotary pairs For all lines of the rotary component, a specific lines set with the same direction lbm , will be obtained by using the condition of minimum error Δδ defined in Eq. (10). The mathematic model is

⎧ ΔRδ = min [Δδ (lm )] ⎪ ⎨ lm = [ sin δm cos θm, sin δm sin θm, cos δm]T ⎪ s. t. δ ∈ [0, π /2], θ ∈ [0, 2π ] ⎩ m m

; rQ = [xQ , yQ , zQ]T

(13)

where, ΔRδ is the minimum spherical image curve error for all lines of the rotary component; δm and θm are direction angles of vector lm in the moving frame {Om; im, jm, km}; Δδ is the same as Eq. (10) for vector lm. The results of the Eq. (13) is the minimum spherical image curve error ΔRδ and its corresponding direction lbm , whose direction angles are denoted as (δmb, θmb ). Furthermore, a specific line with the reference position r bPm , included in the identified lines set with direction lbm , will be obtained

(11)

where, Δc is the fitting error of striction curve; rQ is the vector of the reference point Q, which is regarded as the position of the approximate axis. The optimization variables (xQ, yQ, zQ) are coordinates of vector rQ, and rc is the radius of the fitting circle, as shown in Fig. 10(b). The optimization function gρ(i ) (b) is the deviation between the discrete

Fig. 8. Measured results of the rotary pair RC.

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Fig. 9. Three types of ruled surface of nominal rotation.

μm; the horizontal axis and vertical axis are the values of xpm and ypm for different rPm. The results shows different line directions and positions have different spherical image curve errors Δδ and striction curve errors Δc. The moving axis has the minimum spherical image curve error and striction curve error, which corresponds to the minimum kinematic pair errors. The identified fixed and moving axes of the rotary pairs RA and RC are calculated by Eqs. (8)–(14) based on the measured motions, and the results are shown in Table 1. In Table 1, the parameters r bPm and lbm of the moving axes are described in the moving frame {Om; im, jm, km}, and parameters rQ and lQ of the fixed axis are described in the fixed frame {Of; if, jf, kf}. The fixed axis and moving axis of a rotary pair are similar to the axis average line and axis of rotation [12], but not the same. These two axes are determined by the characteristics of the trajectories traced by all lines of the rotary component, as shown in Eqs. (13) and (14). Thus, they are independent of the positions of the reference points D(j) measured, or the installation parameters of the ball bar. In other words, no matter where the spherical pair SD is mounted, the line of the rotary table with the minimum spherical image curve error and minimum striction curve is not change, as the properties of the spatial rigid motion are not change. The same fixed and moving axes will be obtained by searching all lines of the rotary table with Eqs. (13) and (14). The fixed and moving axes are only related to the properties of the three-dimensional motion, but unrelated to the trajectories of the measured lines or points. Furthermore, the errors ΔRδ and ΔRc are the

by using the condition of minimum error Δc defined in Eq. (11). The mathematic model is

⎧ ΔRc = min [Δc (rPm )] ⎪ ⎨ rPm = [xPm , yPm , zPm]T ⎪ s. t. x , y , z ∈ (−∞, +∞) ⎩ Pm Pm Pm

(14)

where, ΔRc is the minimum striction curve error for all lines of the rotary component; xPm, yPm and zPm are coordinates of point P on the line Lm in the moving frame {Om; im, jm, km}; Δc is the same as Eq. (11). The results of Eq. (14) are the minimum striction curve error ΔRc and its corresponding line position r bPm . The Eqs. (13) and (14) determine a special line L mb (r bPm, lbm) of the rotary component with the minimum spherical image curve error ΔRδ and the minimum striction curve error ΔRc; this line is regarded as the moving axis of the rotary pair, as its rotary error is the least and its trajectory is most approximate to an axis. Accordingly, the geometrical axis L Q (rQ, lQ) of the fitting ruled surface of the line-trajectory traced by the line L mb is regarded as the fixed axis. A part of the contour lines of Δδ for different line directions lm are shown in Fig. 11(a). The contour lines are the values of Δδ, whose dimension is μRad; the horizontal axis and vertical axis are the values of sinδm cosθm and sinδm sinθm for different lm(δm,θm). Similarly, a part of the contour lines of Δc for different line positions rPm are shown in Fig. 11(b). The contour lines are the values of Δc, whose dimension is

Fig. 10. The fittings of the spherical image curve and striction curve.

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Fig. 11. The errors Δδ and Δc for different lines of the rotary table.

least for all lines of the rotary table, and they are global invariants of rotary error motions. These invariants can be used to evaluate the accuracy of a rotary pair, as they are unrelated with the positions of the reference points. 4.3. The structure errors and kinematic pair errors The kinematic model of the two-axis table with errors is set up by using the identified fixed axes and moving axes, as shown in Fig. 12. In a the frame {Of; if, jf, kf}, r OA and laA denote the position and direction of b the fixed axis LQA of rotary pair RA, while rOA and lbA denote those of the b moving axis L mA , respectively. In order to describe the structure errors and kinematic pair errors of the rotary pair RA, a fixed frame {OAa ; iaA, jaA , k aA} is employed and attached to the fixed axis with axis k aA parallel to laA , meanwhile, a moving frame {OAb ; ibA, jbA , kbA} is employed and attached to the moving axis with axis kbA parallel to lbA . For the rotary pair RC, the parameters are denoted in the same way, with a subscript C. For convenience, the origin points OAa and OCa are located at the intersections of the fixed axes and their common vertical line, whose position vectors can be calculated by a ⎧ rOA = rQA + uA lQA ⎨ a r ⎩ OC = rQC + uC lQC

Fig. 12. The kinematic model of the two-axis table. a = [xAa , yAa , zAa]T meters of RA are represented by three translations r OA a a a a a and three Euler angles (αA , βA , γA ) of the fixed frame {OA ; i A, jaA , k aA}, in relative to the coordinate system {Of; if, jf, kf}. For the rotary pair RC, the parameters are the same with a subscript C. The results are shown in Table 2. The structure errors of the two-axis rotary table are calculated with comparing the actual structure parameters to the nominal one. Similarly, the positions and directions of the moving frame {OAb ; ibA, jbA , kbA} and {OCb ; iCb , jCb , kCb} in the frame {Om; im, jm, km} will be determined, as shown in Table 3. The kinematic pair errors of RA are described by three translations δA = [δAX , δAY , δAZ ]T and three Euler angles (αAb, βAb , γAb ) of the moving frame {OAb ; ibA, jbA , kbA} in relative to the frame {OAa ; iaA, jaA , k aA}. For the rotary pair RC, the errors are the same with a subscript C. According to these definitions, the kinematic pair errors of the rotary pairs RA and RC will be identified from the measured motions by using the parameters shown in Tables 2 and 3, because the measured motions are composed of nominal rotations and the error motions caused by the

(15)

where, rQA and rQC denote the reference positions of the fixed axes of rotary pairs RA and RC; lQA and lQC denote the unit direction vectors of the fixed axes, as shown in Table 1. The parameters uA and uC can be calculated by

⎧ ⎪ uA = ⎨ ⎪ uC = ⎩

(rQA − rQC) ⋅ [lQC (lQC ⋅ lQA) − lQA] 1 − (lQC ⋅ lQA)2 (rQA − rQC) ⋅ [lQC − (lQC ⋅ lQA) lQA] 1 − (lQC ⋅ lQA)2

(16)

The positions and directions of the fixed frames {OAa ; iaA, jaA , k aA} and {OCa ; iCa , jCa , kCa} in the frame {Of; if, jf, kf} will be determined with known origin points and coordinate axes. Then, the structure paraTable 1 The fixed axes and moving axes of the rotary pairs RA and RC.

RA RC

rQ /mm

lQ /rad

[xQ , yQ , zQ]T

δQ

[0, −243.748, −485.781]T [480.191, −243.668, 0]T

−1.73×10−5+π/2 6.30×10−5

r bPm /mm

lbm /rad

θQ

b b b T [xPm , yPm , zPm ]

δmb

θmb

1.40×10−4 −1.74×10−1

[0, −0.008, −75.097]T [0.004, −0.001, 0]T

−3.23×10−5+π/2 3.52×10−5

7.81×10−4 −7.58×10−2

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Table 2 The structure parameters of the rotary pairs RA and RC.

(xAa , yAa , zAa ) /mm

(αAa, βAa , γAa )/rad

(xCa , yCa , zCa ) /mm

(αCa, βCa , γCa ) /rad

(480.161,−243.680,−485.773)

(0,−1.73×10−4+π/2,−1.10×10−4)

(480.161,−243.663,−485.773)

(0,6.20×10−5,1.09×10−5)

identified kinematic pair errors have the minimum values as Eqs. (13) and (14) showing. For error compensation, the structure errors can be eliminated by using the inverse solution of the kinematic equation absolutely, as they are constant in the kinematic equation. The kinematic pair errors are changing with rotary angles and contain non-repeated errors. Thus, it is beneficial to reduce the kinematic pair errors in error calibration to improve the accuracy of error compensation.

structure errors and kinematic pair errors. The transformation matrix of the kinematic pair errors of the rotary pair RA is i) [TbA(i ) ] = [T aA]−1 [T(mfA ][TbmA]−1

(17)

[TbA(i ) ]

denotes the transformation matrix of the kinematic pair, where, which can be represented by

⎡ R (α b (i ), β b (i ) , γ b (i ) ) δ(i ) ⎤ A⎥ A A A [TbA(i ) ] = ⎢ ⎣ 0 1⎦

[T aA],

i) [T(mfA ]

(18) 5. Error analysis and correction of the two-axis rotary table

[TbmA]

and correspond to the structure The matrixes parameters shown in Table 2, the measured motions of rotary pair RA and the parameters shown in Table 3, respectively. The values of these matrixes are a ⎤ ⎡ R (αAa, β a , γ a ) r OA A A [T aA] = ⎢ ⎥ ⎣ 0 1 ⎦

(19)

⎡ i(i ) j(i ) k(i ) r(i ) ⎤ i) [T(mfA ] = ⎢ mA mA mA OmA ⎥ ⎣ 0 0 0 1 ⎦

(20)

⎡ R (α b , β b , γ b ) r b ⎤ mA mA mA mA [TbmA] = ⎢ ⎥ ⎣ 0 1 ⎦

The simultaneous two-axis motions of the two rotary pairs RA and RC are tested to validate the proposed calibration methods. As known from the kinematic model of the two-axis rotary table, the spherecenter D of the ball bar will traces an approximate spherical curve if the rotary pairs RA and RC rotate simultaneously. The trajectory ΓD of the sphere-center D in the fixed frame {Of; if, jf, kf} can be calculated by

⎡r ⎤ ⎡r ⎤ a ΓD: ⎢ D ⎥ = [T aA][TbA (αA)][TCA ][TCb (αC )] ⎢ Db ⎥ ⎣1⎦ ⎣ 1 ⎦

(21)

The kinematic pair errors of the rotary pair RA is calculated by Eqs. (17)–(21), and the results are shown in Fig. 13(a) and (b). It is a i) worthy to notice that the product of the matrixes [T(mfA ][TbmA]−1 b b b b represents the transformation matrix of the {OA ; i A, j A , k A} in relative to the fixed frame {Of; if, jf, kf}, whose value is i) [T(mfA ][TbmA]−1

⎡ ib (i ) jb (i ) k(i ) r b (i ) ⎤ mA OA ⎥ A =⎢A ⎣ 0 0 0 1 ⎦

(23)

where, rD and rDb denote the position vectors of point D in the frames {Of; if, jf, kf} and {OCb ; iCb , jCb , kCb}. [TbA (αA)] and [TCb (αC )] denote the transformation matrixes of the kinematic pair errors at rotary positions a ] denotes the transformation matrix from the frame αA and αC. [TCA a a a a {OC ; iC , jC , kC} to the frame {OAb ; ibA, jbA , kbA}, which can be calculated by b (0) −1 a b −1 [TCA ] = [TbA(0) ]−1 [T aA]−1 [T(0) mfC ][T mC ] [TC ]

where, [TbA(0) ]

(24)

and [TCb(0) ]

denote the matrix of the kinematic pair errors at the rotary positions αA=0 and αC=0; [T(0) mfC ] denote the matrix of the measured motions at the rotary position αC=0. The measured trajectory of the sphere-center D is calculated by Eq. (23) with the rotary pair RA rotating from −20° to 40° roundly for two times and the rotary pair RC rotating from 0° to 360°, as shown in Fig. 14. This trajectory r(Di) is measured three times with the ball bar parallel to the axes if, jf and kf. Thus, the moving distances λX, λY and λZ of the translational pairs will be calculated by Eq. (3), and the corresponded trajectories of the sphere center E of the ball bar are shown in Fig. 14(a), (b) and (c), respectively. The nominal distance between the centers D and E is 100 mm. The exact value of vector rDb, corresponds to the mounting position of spherical pair SD are identified by Eq. (6), based on the measured distance of the ball bar in the directions of if, jf and kf. Thus, based on Eq. (23), the ideal trajectory of the sphere-center D will be calculated by setting the structure and kinematic pair errors to zero, while the actual trajectory of the sphere-center D will be calculated by using the identified errors. When using the ideal trajectory to calculate the parameters λX, λY and λZ, the measured data of the ball bar are the errors of the point-trajectory r(Di) . Furthermore, when using the actual trajectory to calculate the parameters λX, λY and λZ, the measured results are the remained errors of the machining center after the structure and kinematic pair errors of the rotary pairs are corrected, as shown in Fig. 15.

(22)

b (i ) denotes the where, i denotes the number of the rotary position. rOA position vector of the origin point OAb in the fixed frame and ibA(i ), jbA(i ) i) and k(mA are the coordinate axes. As the moving axis determined by Eqs. (13) and (14) is unique of the rotary component, its corresponded frame {OAb ; ibA, jbA , kbA} is also a constant frame of the rotary component; this means the identified structure parameters and kinematic pair i) errors are both independent from the matrix [T(mfA ], or the installation parameters of the ball bar. For the rotary pair RC, the solution equations are the same, and the kinematic pair errors are shown in Fig. 13(c) and (d). In these figures, the horizontal axes αA and αC are nominal rotation angles of the rotary pairs RA and RC, the actual rotary b b and αCb = αC + δαC . angles are αAb = αA + δαA The identified structure errors and kinematic pair errors are similar to the linkage errors and volumetric geometrical errors defined in reference [19], but not the same. Actually, the linkage errors and volumetric geometrical errors can be regards as a special case of the structure errors and kinematic pair errors, if the function line or point is specified as the measured line or point. The structure and kinematic pair errors are defined by using the fixed and moving axes, which are kinematic invariants and unrelated with the mounting position errors and installation parameters of the testing devices. Furthermore, the

Table 3 The positions and directions of the frame {OAb ; ibA, jbA , kbA} and {OCb ; iCb , jCb , kCb}. b b b (xmA , ymA , zmA ) /mm

b b b (αmA , βmA , γmA ) /rad

b b b (xmC , ymC , zmC ) /mm

b b b (αmC , βmC , γmC ) /rad

(0.005, −0.008, −75.098)

(0, −3.23×10−4+π/2, −7.81×10−5)

(0.007, −0.001, −75.099)

(0, −3.51×10−5, −2.67×10−6)

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Fig. 13. The kinematic pair errors of the rotary pairs RA and RC.

pair error are nearly, and the accuracy of the machining center is improved greatly by correcting the structure errors and the kinematic pair errors of the rotary pairs.

The results shown in Fig. 15(a), (b) and (c) correspond to the measured trajectories shown in Fig. 14(a), (b) and (c), respectively. In these figures, the curve e1 denotes predicted results calculated by the calibrated kinematic model without correction; the curve e2 denotes the measured results without correction and the curve e3 denotes the measured results after correction. In order to illustrate more clearly, the ranges of the calculated errors and the measured errors are calculated and shown in Table 4. The results show the errors calculated by the kinematic model with the identified errors are good agreement with the measured results. The ranges of errors caused by the structure errors and the kinematic

6. Conclusions A ball bar method for error calibration of the controlled rotary pairs in 5-axis machining center is developed by using the mechanism model and the kinematic invariants. The main contributions are summarized as followings.

Fig. 14. The trajectories of the centers D and E.

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Fig. 15. The measured results of the ball bar before and after error compensation.

References

Table 4 The errors of the trajectory traces by the reference point D of the table. Directions

if jf kf

Calculated data by the kinematic model

Measured data by the ball bar

Structure errors

Kinematic pair errors

Total errors

No corrected

Corrected

14.7 um 28.4 um 11.9 um

14.0 um 10.5 um 19.8 um

22.3 um 36.1 um 23.2 um

22.6 um 37.5 um 36.9 um

7.7 um 10.3 um 8.9 um

[1] D.L. Wang, Z. Wang, Y. Wu, et al. Discrete kinematic geometry in testing axes of rotation of spindles, ASME 2016 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 2016. [2] W.T. Lei, Y.Y. Hsu, Accuracy test of five-axis CNC machine tool with 3D probe–ball. Part I: design and modeling, Int. J. Mach. Tools Manuf. 42 (10) (2002) 1153–1162. [3] W.T. Lei, Y.Y. Hsu, Accuracy test of five-axis CNC machine tool with 3D probe-ball. Part II: errors estimation, Int. J. Mach. Tools Manuf. 42 (10) (2002) 1163–1170. [4] B. Bringmann, J.P. Besuchet, L. Rohr, Systematic evaluation of calibration methods, CIRP Ann.-Manuf. Technol. 57 (1) (2008) 529–532. [5] S. Weikert, R-test, a new device for accuracy measurements on five axis machine tools, CIRP Ann.-Manuf. Technol. 53 (1) (2004) 429–432. [6] N. Huang, Q. Bi, Y. Wang, Identification of two different geometric error definitions for the rotary axis of the 5-axis machine tools, Int. J. Mach. Tools Manuf. 91 (2015) 109–114. [7] X. Jiang, R.J. Cripps, A method of testing position independent geometric errors in rotary axes of a five-axis machine tool usinga double ball bar, Int. J. Mach. Tools Manuf. 89 (2015) 151–158. [8] M. Tsutsumi, S. Tone, N. Kato, et al., Enhancement of geometric accuracy of fiveaxis machining centers based on identification and compensation of geometric deviations, Int. J. Mach. Tools Manuf. 68 (2013) 11–20. [9] K.I. Lee, S.H. Yang, Measurement and verification of position-independent geometric errors of a five-axis machine tool using a double ball-bar, Int. J. Mach. Tools Manuf. 70 (2013) 45–52. [10] S.H.H. Zargarbashi, J.R.R. Mayer, Single setup estimation of a five-axis machine tool eight link errors by programmed end point constraint and on the fly measurement with Capball sensor, Int. J. Mach. Tools Manuf. 49 (10) (2009) 759–766. [11] W. Jywe, T.H. Hsu, C.H. Liu, Non-bar, an optical calibration system for five-axis CNC machine tools, Int. J. Mach. Tools Manuf. 59 (2012) 16–23. [12] ASME B89.3.4-2010, Axes of Rotation: Methods for Specifying and Testing. American National Standards Institute, 2010. [13] ISO 230-7:2006. Geometric accuracy of axes of rotation. ISO, 2006. [14] R. Grejda, E. Marsh, R. Vallance, Techniques for calibrating spindles with nanometer error motion, Precis. Eng. 29 (1) (2005) 113–123. [15] X. Lu, A. Jamalian, A new method for characterizing axis of rotation radial error motion: Part 1. Two-dimensional radial error motion theory, Precis. Eng. 35 (1) (2011) 73–94. [16] P. Ma, C. Zhao, X. Lu, et al., Rotation error measurement technology and experimentation research of high-precision hydrostatic spindle, Int. J. Adv. Manuf. Technol. 73 (9–12) (2014) 1313–1320. [17] G. Zhong, C. Wang, S. Yang, et al., Position geometric error modeling, identification and compensation for large 5-axis machining center prototype, Int. J. Mach. Tools Manuf. 89 (2015) 142–150. [18] S.H. Suh, E.S. Lee, S.Y. Jung, Error modelling and measurement for the rotary table of five-axis machine tools, Int. J. Adv. Manuf. Technol. 14 (9) (1998) 656–663. [19] J. Chen, S. Lin, B. He, Geometric error measurement and identification for rotary table of multi-axis machine tool using double ballbar, Int. J. Mach. Tools Manuf. 77 (2014) 47–55. [20] S. Zhu, G. Ding, S. Qin, et al., Integrated geometric error modeling, identification and compensation of CNC machine tools, Int. J. Mach. Tools Manuf. 52 (1) (2012) 24–29. [21] J. Chen, S. Lin, X. Zhou, A comprehensive erroranalysis method for the geometric error of multi-axis machine tool, Int. J. Mach. Tools Manuf. 106 (2016) 56–66. [22] ISO 230-1, Geometric accuracy of machines operating under no-load or quasi-static conditions. ISO, 2012. [23] ISO 230-2, Determination of accuracy and repeatability of positioning numerically controlled axis. ISO, 2006. [24] D.L. Wang, W. Wang, Kinematic Differential Geometry and Saddle Synthesis of Linkages, John Wiley & Sons, 2015.

(1) The three-dimensional motions of the rotary table of a 5-axis machining center are measured by a ball bar. The six kinematic parameters are determined by the position vectors of three non( j) collinear points of the rotary table in the fixed frame, and rOD are calculated by nine displacement equations of the mechanism, formed by of the machining center and the ball bar. There is no need to correct or identify the mounting position errors by using this method, as the identified motions are not related with the mounting position errors of the ball bar. (2) The kinematic invariants of the line-trajectories, traced by all lines of the rotary component, are analyzed; and the fixed axes and moving axes of the rotary pairs are identified by using the conditions of minimum spherical image curve error and minimum striction curve error. These identified axes and errors are global invariants of rotary error motions. The moving axis is unique of the rotary component, whose trajectory has the minimum spherical image curve errors and striction curve errors. (3) The structure errors and kinematic pair errors of the rotary pairs are determined by using the fixed axes and moving axes; these errors are not related with the positions of the reference points or testing device. The identified kinematic pair errors, based on the moving axis, have the minimum values in comparing with those of the other reference lines. For the rotary table discussed, the structure and kinematic pair errors are nearly; both of them should be considered for error analysis and correction. Generally, the structure errors of the two-axis rotary table are caused by the errors of the fixed joints or components, while the kinematic pair errors are caused by the errors of the kinematical joints. The properties of these two errors are different, thus, it is more feasible to distinguish them in error testing and evaluation. Furthermore, this will provide some basis for accuracy design of the machining centers. Acknowledgments The authors acknowledge with appreciation the financial support from the National Natural Science Foundation of China (Grant No. 51275067).

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