A closed-loop error compensation method for robotic flank milling

Robotics and Computer Integrated Manufacturing 63 (2020) 101928

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A closed-loop error compensation method for robotic flank milling Gang Xiong, Zhou-Long Li, Ye Ding, LiMin Zhu

⁎

T

State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China

ARTICLE INFO

ABSTRACT

Keywords: Robot machining Flank milling Error compensation Distance function Spatial statistical analysis Moran's I Sequential linear approximation

Errors of diverse sources prevented industrial robots from being adopted into milling applications. This paper proposes a closed-loop error compensation method for robotic flank milling of complex shaped surfaces. First, the finished surface is measured in situ by a laser tracker based measuring system without unclamping the fixture. Then, the sampled points are mapped into the model reference coordinate system and a deterministic bicubic B-spline surface is fitted to extract the systematic components of the machining errors. Finally, the compensation tool path is directly generated for the mirror symmetry points of the only-systematic-error-contained sample points. The robot motion program is converted accordingly for further machining. The experiment shows that the surface accuracy is improved significantly in terms of the profile error via the proposed error compensation process, which well validates the effectiveness of the method.

1. Introduction In recent years, flank milling has been widely applied for the large and complex shaped product manufacturing in aerospace industry. Industrial robot (IR) provides a promising alternative of large-scale and high-accuracy CNC machine tool for such tasks due to its advantages of large workspace, low cost and great flexibility [1–3]. There is a trend to extend the IR application areas from traditional repetitive tasks to high accuracy milling applications [4,5]. However, despite the intensive research in academics, IRs are rarely implemented for machining tasks in industry. The primary obstacle for that is the poor machining accuracy resulting mainly from the geometric error, elastic deformation and process force induced vibration of the IR [6]. It is of great significance to reduce these errors to meet manufacturing tolerances. Generally, the vast accuracy-enhancement methods in robot machining can be summarized into four categories: machining strategy optimization, offline compensation, real-time compensation and measurement based closedloop compensation. The robot machining strategy is often improved by selecting an optimal combination of process variables via experiments or theoretical analysis. Furtado et al. [7] and Slamani et al. [8] determine the optimal process parameters, i.e., robot configuration, feedrate, spindle rotation speed etc., according to preliminary trial cut experiments. Dong et al. [9] and Ding et al. [10] optimize the posture of the machining robot based on stiffness performance indexes. Mejri et al. [11] and Cordes et al. [12] suggest to adopt cutting parameters from the stable area to avoid chatter after a sophisticated analysis of the dynamic

⁎

characteristic of the robot machining system. These methods are identified to be simple and efficient. However, the inherent drawbacks of insufficient accuracy and stiffness of the IR are not scrutinized. The improvement in the machining quality of these methods is limited. Offline compensation is a widely employed method to enhance the machining accuracy. It integrates the robot stiffness model and the process force model, predicts the end effector deviation of the robot in terms of their interaction, and then modifies the robot trajectory offline to eliminate the predicted error [13–16]. Abele et al. [17], Klimchik et al. [18] and Reinl et al. [19] develop more elaborate dynamic models of the robot to predict the motion of the tool center point, which is demonstrated to contain low frequency oscillations. Though these methods are reported valid in machining quality enhancement, they are subjected to two limits. First, it's hardly possible to establish an accurate model of the interaction system, since it involves backlash, hysteresis and other nonlinear factors in the robot structure rigidity model [15,20]. Second, the actual machining condition varies with the process, which can't stay strictly the same as the parameters used in the interaction model. The accuracy of the finished part will be confined to the deficient system model and inaccurate model parameters. The real-time error compensation method samples the process signals online and adjusts the system behavior accordingly. Schneider et al. [6,21] develop a parallel 3D-piezo compensation mechanism to directly compensate the motion error of the robot with the help of a metrological tracking system. A reasonable substitution to this costly and complex system is to measure the cutting force online and indirectly compensate the motion error via deflection prediction [22–24],

Corresponding author. E-mail addresses: [email protected] (G. Xiong), [email protected] (Z.-L. Li), [email protected] (Y. Ding), [email protected] (L. Zhu).

https://doi.org/10.1016/j.rcim.2019.101928 Received 20 May 2019; Received in revised form 18 November 2019; Accepted 13 December 2019 0736-5845/ © 2019 Elsevier Ltd. All rights reserved.

Robotics and Computer Integrated Manufacturing 63 (2020) 101928

G. Xiong, et al.

which resembles the offline compensation method. This kind of error compensation method can react immediately to the environment variation while suffers from implement difficulties or model inaccuracies, thus is not applied widely in industry. The aforementioned state-of-the-art error compensation methods for robot machining are mainly based on the understanding and offsetting errors of specific causes, while the underlying causes of dimensional inaccuracy in robotic machining are diverse and complex. Theoretically, an ideal solution would account for all possible systematic machining errors by measuring them in the part directly to compensate the finishing cut in reference to the CAD model. To this end, the so-called on-machine measurement (OMM) technique for machine tool has been widely applied [25–28]. This method features in an on-machine inspection and compensation closed loop without remounting the part. The machining quality can be guaranteed by directly compensating the systematic error of the part. Abele et al. [29] and Barnfather et al. [30] extend this concept to robotic milling via optical measurement and the experiments demonstrate the significant improvement of the finished part quality. Nevertheless, their methods are subjected to complicated dense point cloud processing and pointby-point tool path modification process. Their extension for robotic flank milling of complex shaped geometries is yet to-be-tested. This paper proposes a closed-loop error compensation method for the robotic flank milling of complex shaped surface. Within the proposed method, the surface is first measured in situ by a laser tracker based measuring system after the finishing cut. The Sampled points are then mapped into the model reference frame and fitted by a bicubic Bspline surface under the supervision of spatial statistical analysis. Finally, the compensation tool path and the corresponding robot motion program are directly generated for the mirror points of the onlysystematic-error-contained sample points. Compared with Refs. [29,30], our work suffers from less cloud point processing efforts owning to the laser tracker based measuring system and the coordinate system alignment process is greatly simplified with the help of point-tosurface distance function and its properties. Via the spatial statistical analysis based error decomposition process, it is guaranteed that the systematic machining errors are eliminated. Also, the compensation tool path is directly generated for the mirror points, which avoids the trivial point-by-point tool path correction process in traditional OMM schemes [25,28]. The closed-loop error compensation method will pave the way to wide applications of robotic flank milling of complex shaped surface in industry. The remainder of this paper is organized as follows. In Section 2, some basic concepts and conclusions used in this paper are briefly illustrated or deduced. In Section 3, the detailed error compensation procedure for robotic flank milling is presented. Section 4 gives the experiment validation of the method. Section 5 concludes the paper.

Fig. 1. Point-to-surface distance function.

Obviously, the absolute value of the signed distance function yields the unsigned distance function, i.e., |dps , S (w )|=dp, S (w ) , and the sign is usually defined to correspond to undercut (positive) and overcut (negative) of the machining error. For complex algebraic surfaces, the distance function and the corresponding foot point are usually computed numerically [33], and the special cases where there are two or more foot points are ignored here. Also, the signed distance function has the following differential property. PROPOSITION 1. . Assume that a surface has the locally parametric representation Ψ(w, u, v) and q = (w, u 0, v0) , the first-order Taylor approximant of dps , (w ) is given by

dps , (w+ w ) = dps , (w )

[nq·

q w1, …, n ·

wm]·

(1)

w

where the partial derivatives wi, i = 1, …, m are evaluated at (w, u0, v0). 3 represented PROPOSITION 2. : Given a nominal smooth surface S in its own model reference frame CW by either a parametric or an 3 measured from the actual implicit description, and a point p surface and expressed in the measurement reference frame CM, the signed point-to-surface distance function from p to S is defined as ^

g=e SE (3) q) nq , where denotes the matrix from CM to CW with v1 3 2 1 v2 SO (3) , q and nq follow the same 0 v3 2 1 0 0 0 0 definition as before. Likewise, the increment of dps , S (g ) with respect to ^ the differential rigid body motion of the surface relative to CM can be deduced as

dps , S (g ) = (g 1p transformation 0 0 ^ 3 =

2. Preliminary 2.1. Distance function and its properties Point-to-surface distance function is the basis of coordinate system alignment, surface fitting and tool path generation in the presented method. This section just briefly introduces the main achievements and conclusions in this area according to Refs. [31,32]. 3 3 and a regular surface S (w ) DEFINITION 1. Given a point p , m denotes the collection of the shape where w = [w1, …, wm]T parameters, there exists at least one closest point q ∈ S(w), termed as foot point, such that p q = min p x , where ‖ · ‖ stands for the

dps , S (g ) =

nq v

(q × nq)

where v = [v1, v2, v3 ^ coordinates of twist .

]T ,

=[

(2) 1,

2,

3

]T ,

and

=[vT ,

T ]T

are

the

2.2. Surface representation

x S (w )

Euclidean norm on 3 . The unsigned point-to-surface distance function is defined as dp, S (w ) = p q . If q is unique and lies in the interior of S(w), the signed distance function is defined as dps , S (w ) = (p q)· nq , where nq is the unit outward normal vector of surface at point q as shown in Fig. 1.

Roughly, there are two kinds of machining errors: systematic error and random error. The former one represents repeatable and reproducible error under similar operating conditions, such as the forceinduced robot deformation, tool wear and so on. While the latter one 2

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G. Xiong, et al.

Moran sI =

n i=1

n S0 n

n (e j = 1 ij i n ( e i=1 i

e¯)(ej

e¯) (6)

e¯) 2

n

where S0 = i=1 j = 1 ij , ωij is the weight representing a measure of spatial interaction of location j on location i, and it is determined by the empirical formula: ij

dij k

=

Li

i

j (7)

0i = j n

d where Li = j = 1, j i ij with dij denoting the distance between the ith location and the jth location. The value of constant k is not crucial and is set as k = 4 , which is recommended by Ref. [35]. The distribution of Moran's I is asymptotically normal with mean μM 2 2 and variance M . μM and M having the following expressions:

2 M

means the error varying with similar machining conditions, such as the error caused by the non-homogeneous workpiece material or the cutting chatter. It's obvious that only systematic error can be compensated. In this paper, a bicubic B-spline surface with m × n patches will be used as the deterministic surface to separate the systematic and random machining errors. It can be expressed as

i·

u , v ) = (1

l

As it is illustrated in Fig. 2, p (u) = i = 0 Ni, k (u) Pi and p q (u) = N (u) Qi are two B splines with Ni, k(u) the B spline basis i = 0 i, k function and Pi (i = 0, …, l), Qi (i = 0, …, p) the control points. The shape parameter wa follows the same definition as wb Eq. (3). Without loss of generality, this paper concentrates on robotic flank milling with a cylindrical flat-end milling cutter, which is mostly applied in industry. Thus the cutter swept envelope surface Ψe(we, u, v) is the offset of the tool axis trajectory surface Ψa(wa, u, v) in our case, and the offset distance is the tool radius R. As a result, we have

dps ,

e

(we) = dps ,

a

(wa )

R

2nS2 + 6S02]

1 (n

1) 2

=

S1 = n j=1

n i=1

1 2

ij ,

·i

=

n

n j=1 j=1

(

ij

+

ji , r =

)2 ,

ji m4 and m22

S2 =

m =

n

( i· + ·i ) i=1 1 n (e e¯) i=1 i n

with . To

(Moran s I )

µM

.

(10)

Whenever |Z| is greater than the specified critical value Z with the significance level, the null hypothesis that the errors are spatially statistical independence is rejected, indicating that there exists spatial autocorrelation, i.e., systematic errors. Otherwise, the null hypothesis is accepted and only random errors exist. Once a deterministic surface is fitted to the measured points, this statistic test can determine whether the surface contains all the systematic errors. PROPOSITION 3. The spatial statistical test cannot identify systematic offset errors, i.e., the test statistic Z remains unaffected even though there exists a constant bias es in each machining error. The proof is given in Appendix A. As demonstrated in Fig. 3, given a point set Ω, a surface S1 can be fitted. Let ei denote the fitting residuals, the Z-statistic is calculated as Z1 to examine whether the residuals contain random errors only. However, the Z-statistic Z2 for an offset surface S2 with ei* = ei + es the residuals is equal to Z1, though there is a systematic offset es. In this paper, we assume that the fitting residuals, i.e., random errors, have a zero mean, thus the systematic offset error can be eliminated via making the mean value of the residuals be zero.

(4)

v ) p (u) + vq (u).

nS2 + 3S02] r [n (n 1) S1 (n 1)(n 2)(n 3) S02

M

where Ni, 3(u) and Nj, 3(v) are B spline basis functions, Pij are the (m + 4) × (n + 4) control points, and the shape parameter 3(m + n + 8) × 1 wb is a collection of the coordinates of Pij. For flank milling, the tool axis trajectory surface is represented by a ruled surface as a (wa,

3n + 3) S1

examine the spatial autocorrelation of {ei}, a test statistic is chosen as

(3)

j=0

(8)

1

n [(n2

where

Z=

Ni,3 (u) Nj,3 (v ) Pij i=0

=

n

(9)

m+3 n+3

S (wb, u, v ) =

1

µM =

Fig. 2. Tool axis trajectory surface representation.

k

(5)

3. The error compensation method

for a point p on the design surface (see Fig. 2). For cases with a general cutter, the error compensation process remains unchanged except for a more sophisticated cutter geometry model [34].

3.1. Overview The whole closed-loop error compensation process is based on the

2.3. Spatial statistic analysis with Moran's I Once a deterministic surface is fitted for a set of samples, it is of great significance to determine whether or not it extracts all the systematic errors. To achieve this, a spatial statistical analysis technique is adopted in this paper. This method assumes that the sampled machining errors will have some spatial autocorrelation if there exist systematic errors. The spatial autocorrelation can be easily identified by a statistical test. Among the statistics for detecting the presence of spatial autocorrelation with interval scaled data, Moran's I is mostly used in machining applications [27,35]. Let ei (i = 1, , n) be the sampled machining error at location i and e¯ be the mean of {ei}, Moran's I is defined as

Fig. 3. Z-statistic with systematical offset errors. 3

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Fig. 4. The compensation method.

mirror symmetry principle [25,28]. As is demonstrated in Fig. 4, the method is realized via four steps. First, the finished surface is measured in situ without unclamping from the fixture by a CMM system, i.e., a laser tracker with a T-probe in this paper. Second, the sampled points are transformed into the model reference frame after a two-stage coordinate system mapping process. Third, the sampled points are fitted by a bicubic B-spline deterministic surface to extract the systematic machining error under the supervision of the spatial statistical analysis. Finally, the compensation tool path is directly generated for the mirror points of samples’ foot points on the deterministic surface and the robot motion program is generated accordingly for compensation machining. It's worth noting that the mirror symmetry approach aims at reducing systematic errors of all sources, including those induced by the deflection of the IR. As reported in the previous work [36] of our team, deflection induced machining error will converge when iteratively using the compensation strategy though the coupling relationship between the robot deflection and the depth of cut is very complex.

by solving the following nonlinear least square problem: n

min

g SE (3)

i=1

(d ps i ,S (g ) )2

(11)

where follows the definition given in PROPOSITION 2. To solve the optimality problem, the sequential linear approximation method is recommended since the closed-form differential expression of the signed distance function is derived and given in Eq. (2). That means the problem is solved by iteratively solving the linear least square problem at the kth step as

d ps i , S (g )

n

min6

i=1

[d ps i ,S (g (

k ))

+ [ niq,

qi × niq]

]2 .

(12)

This problem can be easily solved by the Levenberg–Mardquardt algorithm with ξ0 the result of the coarse mapping stage. By repeating and solving Problem (12), the precise the procedure of k + 1 = k + transformation between CM and CW can be obtained.

3.2. Measurement

3.4. Error decomposition

There're two main considerations when measuring the finished surface, i.e., the number and location of the samples. In this paper, we adopt the easy but efficient method presented in Ref. [28], which samples the surface uniformly and determines the number of samples by virtually sampling and fitting under the constraint of approximation error. Though it is proposed for CNC machine, the method can be adopted here since it's a problem of reconstructing the machined surface accurately and it's irrelevance with the machine used. The detailed process is omitted here for brevity.

This paper proposes the procedure demonstrated in Fig. 5 to extract the systematic machining errors from the sampled points. First, the sampled points are fitted by a deterministic bicubic B-spline surface expressed in Eq. (3), i.e., solving the following problem: n wb

min

3(m + n + 8)

i=1

(d ps i ,S (wb) )2

(13)

where S(wb, u, v) follows the definition given in Eq. (3). This problem has similar form with Problem (11), and can also be solved via the sequential linear approximation method by combining Eq. (1). Then, the test statistic Z is calculated according to Eqs. (6)–(10) with the sample points pi (i = 1, , n) and the fitting residuals, i.e., ei = d ps i , S (wb) , being the inputs. |Z| is compared with a critical value Zɛ, which is equal to 2.33 with the significance level = 0.01 in this paper. If |Z| ≥ Zɛ, indicating that there exist systematic errors in the residuals, the number of patches of the bicubic B-spline surface should be added to refit the sampled points. This process is repeated until |Z| ≤ Zɛ. Finally, the fitted surface should be offset slightly to cancel the bias error as implied in PROPOSITION 3. It is carried out by making 1 n µe = n i = 1 ei be equal to zero. Given the linear relationship between

3.3. Coordinate system alignment After the measurement, the measurement coordinate system CM should be aligned with the model reference coordinate system CW for error evaluation and compensation. This paper proposes a coarse-fine two-stage process for the mapping of the measurement points to the model reference frame. In the coarse stage, some benchmarks on the workblank, i.e., points, lines and planes, are measured. The transformation g from CM to CW can be roughly determined by comparing these benchmarks in measurement and CAD coordinate systems. Then the two frames are fine aligned 4

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Fig. 5. Error decomposition process.

dps , S (w ) and Δwin Eq. (1), it's easy to obtain Δwb with d ps i , S (wb) = µe . Through this step, the final deterministic surface, which contains systematic machining errors only, can be obtained. 3.5. Compensation tool path generation Once the foot points p i (i = 1, …, n) of samples pi (i = 1, …, n) on the deterministic surface are obtained, their symmetric counterparts about the design surface S(w) can be derived as

pim = p i

2d pi , S (w ) n qi

(14)

where pim (i = 1, , n) stands for the mirror points. Then the compensation tool axis trajectory surface Ψa(wa, u, v) can be directly generated with the cutter envelope surface Ψe(we, u, v) approximating pim (i = 1, , n) , i.e., solving the optimality problem

min = max d ps m ,

w

l

i

1 i n

a

(wa)

R .

(15)

By combining Eq. (1), this problem can be converted into a linear form and solved by the sequential linear approximation algorithm, too. The linear form of Problem (15) at the kth step has the following expression:

min T

[ waT ,

]

s. t . d ps m , i

d ps m , a i lk

4.2. Experiment procedure The error compensation approach is independent of the workpiece selected since the compensation tool path is only determined by the measured machining errors. As is demonstrated in Fig. 7(a), a 15mmheight blade-like ruled surface is selected as the part to be machined in this paper for simplicity. The control points and the knot vector of the surface are given in Appendix B. The actual surface finished by the robot milling system is given in Fig. 7(b). The cutting conditions for the finishing cut and the compensation cut are listed in Table 1. These cutting parameters are optimized and determined via previous trial cuts to ensure that there is no obvious machining vibration during the process since vibration is an essential factor to limit the quality of robot machining. The subsequent machining experiments are all conducted under the vibration free condition. By uniformly choosing 20×10 discrete points on the design surface, the flank milling tool path for finishing cut is generated according to the solution of Problem (15). The corresponding robot trajectory in terms of six joint angles is converted based on our previous work [10], the results are displayed in Fig. 8(a) and (b). The finished surface is sampled uniformly by the measuring system as shown in Fig. 6 with the T-probe fixed at the end effector of the robot. It is found that 175 points (25×7) are enough to reconstruct the designed surface with approximation error no more than 0.005 mm in our case. The sampled points after coordinate system transformation are displayed in Fig. 9(a). After measuring, the measurement coordinate system and the model reference coordinate system are aligned based on the two-stage process

l+1 a

(wak )

(wak )

wa

Fig. 6. Experimental setup.

R

R [nq

[nq

q w1, …, n

, …, nq

w1

uk

wl ]

wl ]

wa

wa

k

+

k

(16)

where wi is the ith component of wa and wi is the partial derivative of Ψa about wi in this case. lk and uk are the lower and up bounds for Δwa at the kth iteration. Their values are adjusted constantly in a similar method as the Levenberg-Mardquardt algorithm to guarantee the objective function to decrease, i.e., k + 1 < k . 4. Experiment validation 4.1. Experiment setup The proposed closed-loop error compensation method is validated by the experiment system shown in Fig. 6. A Motoman MH80 IR from YASKAWA with an EBS-120g electric spindle from CELL is used in this paper. The flank milling operation is carried out on an AL-6061 aluminum block using a three-fluted carbide flat end-mill with a diameter of 10 mm. The measuring system is composed of a Leica AT960 laser tracker and a T-Probe, which can obtain the three-dimensional coordinates of the center of the ruby ball head (radius: r = 3mm) with an accuracy of 0.035 mm. 5

Robotics and Computer Integrated Manufacturing 63 (2020) 101928

G. Xiong, et al.

Fig. 7. The machined surface: (a) Geometry model; (b) Finished surface.

described in Section 3.2. In the coarse mapping stage, three planes on the workblank are measured and fitted to locate the model reference frame roughly. Then the optimality Problem (11) is solved through 5 iterations of linear approximation with an accuracy of 0.005. The result of the coordinate system alignment process is presented in Appendix B. Via this step, the precise transform from CM to CW is obtained. Once the sampled points are mapped into the model reference coordinate system, they are fitted by a deterministic bicubic B-spline surface by solving Problem (13). After 10 iterations, the sequential

Table 1 Cutting conditions for finishing and compensation cut. Cutting conditions

Finishing cut

Compensation cut

Cutting Strategy Axial Depth of Cut (mm) Radial Depth of Cut (mm) Spindle Speed (r/min) Feed Speed (mm/min)

Down Milling 15~15.82 0.3 6000 600

Down Milling 15~15.82 0~0.5 6000 600

Fig. 8. Finishing and compensation cut: (a) Tool path for finish machining; (b) Robot trajectory for finish machining; (c) Tool path for compensation machining; (d) Robot trajectory for compensation machining. 6

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Fig. 9. Deterministic surface fitting: (a) Sampled points and fitted surface; (b) Distribution of fitting residuals.

linear approximation algorithm converges with an accuracy of 0.005. It is found that it's unnecessary to add extra patches since the Z-statistic is calculated as Z = 1.8503 < Z with m = n = 0 in Eq. (3), which implies that all the systematic errors are contained in the deterministic surface. The surface bias is calculated as es = 0.0006mm , and it is eliminated by the procedure proposed in Section 3.4. The final deterministic surface and its control points are displayed in Fig. 9(a), and the distribution of the fitting residuals is given in Fig. 9(b). The mirror counterparts of the foot points of the samples on the deterministic surface are calculated according to Eq. (14) and displayed in Fig. 8(c). The tool axis trajectory surface is generated for these points by solving the optimality problem (15) via 8 iterations with an accuracy of 0.005. The parameters of the tool axis trajectory surface are also given in Appendix B. Fig. 8(c) shows the tool path and the corresponding cutter envelope surface for compensation. Likewise, Fig. 8(d) lists the corresponding robot trajectory. The compensation machining is finally conducted accordingly by the robotic machining system. Seriously speaking, the compensation machining should be conducted iteratively until the machining errors converge to very small values. It is found that the machining errors are close to the measuring system's accuracy with one compensation cutting run, thus there is no need to conduct redundant compensation millings in our situation.

4.3. Experiment result The surface profile error of the two surfaces before and after compensation are measured and demonstrated in Fig. 10(a) and (b). Though the maximum overcut error increased from −0.0453 mm to −0.0737 mm at the small area of the beginning cut, it is found that the maximum error decreases 39.2% from 0.4896 mm to 0.2979 mm and the average error in terms of absolute value decreases 52.5% from 0.3555 mm to 0.1689 mm after compensation. More obviously, numbers of feature points on the design surface are selected for comparison of the profile error before and after compensation, the result is given in Table 2. Within the table, the column “(u, v)” means the u v parameters defining the feature points on the design surface, the columns eb and ea represent the profile error before and after compensation, the column “%” denotes the accuracy improvement percentage. It is seen that the accuracy improves 26.2%~77.2% at the selected feature points. Though the minimum machining errors in Fig. 10 and Table 2 are close to the accuracy limit of the laser tracker, the variation trend of the machining error before and after compensation is obvious. The surface profile accuracy is significantly improved, which validates the effectiveness of the proposed error compensation method. It should be noted that the final absolute accuracy that can be achieved is limited by

Fig. 10. Surface profile errors: (a) Before compensation; (b) After compensation.

7

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G. Xiong, et al.

Table 2 Profile error improvement at selected feature points (unit: mm). (u, v)

eb

ea

%

(u, v)

eb

ea

%

(u, v)

eb

ea

%

(0.2,0.2) (0.4,0.2) (0.6,0.2) (0.8,0.2) (1.0,0.2)

0.414 0.433 0.360 0.395 0.404

0.094 0.165 0.239 0.213 0.298

77.2 62.0 33.4 46.1 26.2

(0.2,0.5) (0.4,0.5) (0.6,0.5) (0.8,0.5) (1.0,0.5)

0.335 0.394 0.397 0.380 0.428

0.125 0.202 0.176 0.218 0.289

62.6 48.7 55.2 42.6 32.5

(0.2,0.8) (0.4,0.8) (0.6,0.8) (0.8,0.8) (1.0,0.8)

0.234 0.381 0.426 0.532 0.468

0.110 0.159 0.179 0.200 0.229

52.9 58.3 58.1 53.2 51.0

the robot repeatability, the accuracy of the measuring system and so on. The machining accuracy achieved here by the robot in our lab is suit for the machining of medium-precision part, such as vessel propeller, wind turbine blade and so on.

generation and compensation machining, the quality of the machined surface is improved significantly in terms of the profile error. By directly enhancing the final part quality of robotic flank milling, the error compensation method provides an efficient solution for the wide adoption of IR in milling in industry.

5. Conclusion

CRediT authorship contribution statement

In this paper, a mirror symmetry principle based closed-loop error compensation method for robotic flank milling is presented. The adoption of distance function and its properties in frame mapping, surface fitting and tool path generation simplifies the process. The spatial statistical analysis technique based error decomposition process guarantees that the systematic errors are separated from the measured machining errors. Also, the compensation tool path is directly generated for the mirror points by globally fitting the tool envelope surface, thus avoids the complicated process of adjusting the tool path at each cutter location. Experiment shows that via the procedure of measurement, coordinate system alignment, error decomposition, tool path

Gang Xiong: Conceptualization, Methodology, Software, Investigation, Data curation, Writing - original draft, Visualization. Zhou-Long Li: Methodology, Resources, Writing - review & editing. Ye Ding: . LiMin Zhu: Writing - review & editing, Supervision, Funding acquisition, Project administration. Acknowledgment This work is supported by the National Natural Science Foundation of China [Grant No. 91648202, 51535004, 51905345].

Appendix A. Proof of Proposition 3 Proof: Let ei = es + eri , where es and eri are the constant systematic error and the random error respectively at location i, then 1 n e¯ = es + n e = es + e¯ri , where e¯ri is the mean of {eri}. Moran's I of {ei} will be i = 1 ri n S0

Moran sI ({ei}) = =

n S0

n i=1

n i=1

n j = 1 ij (eri e¯ri )(erj n 2 i = 1 (eri e¯ri )

n j = 1 ij (ei (es + e¯ri ))(ej n 2 i = 1 (ei (es + e¯ri ))

(es + e¯rj ))

e¯rj )

= Moran sI ({eri}). Similarly, µM ({ei}) = µM ({eri}) ,

2 M ({ei})

=

2 M ({eri}) .

Thus

Z ({ei}) = Z ({eri}). This implies that the test statistic Z of the sampled machining error {ei} is equal to that of the random error {eri} even though there exists systematical error es at each sampling point. The spatial statistical test cannot identify the systematical offset error. Appendix B. Results of Coordinate System Alignment and Tool Path Generation Let gcand gf denote the transformation from CM to CW after coarse and fine alignment respectively, in our case, they are identified as

gc =

0.0967 0.9953 0.0064 0

0.9953 0.0967 0.0016 0

0.0010 0.0065 1.0000 0

491.7342 338.7784 , gf = 767.4933 1.0000

0.9994 0.0179 0.0312 0

0.0182 0.0310 0.9998 0.0116 0.0110 0.9995 0 0

0.5730 0.3562 , 1.0110 1.0000

The design and the compensation tool axis trajectory surface have the same expression as following: 3

(w, u, v ) = (1

v)

3

Ni,3 (u) Pi + v i=0

Ni,3 (u) Qi . i=0

The knot vector is [0, 0, 0, 0, 1, 1, 1, 1]T for both surfaces. The control points for the design and the tool axis trajectory surface are listed as wd and wa as below in Table B.1 and Table B.2:

8

Robotics and Computer Integrated Manufacturing 63 (2020) 101928

G. Xiong, et al.

Table B.1 Parameters of wd. Pi(mm) xpi

ypi

zpi

Qi(mm) xqi

yqi

zqi

−65 −53 −30 −15

55 60 55 50

−2 −2 −2 −2

−65 −53 −30 −15

55 65 60 55

−17 −17 −17 −17

Table B.2 Parameters of wa. Pi(mm) xpi

ypi

zpi

Qi(mm) xqi

yqi

zqi

−67.04 −51.89 −23.87 −14.57

60.46 66.77 60.09 56.19

−2.01 −1.99 −2.04 −2.03

−68.50 −52.14 −19.00 −11.97

59.66 72.31 62.50 58.91

−17.10 −16.94 −17.00 −17.02

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