Neural Network Based Sensor Fusion Strategy for Multi-Axis Machine Accuracy Monitoring and Control Abderrazak El Ouafi* and Abdellah Bedrouni** *

Mathematics, Computer and Engineering Department, University of Quebec at Rimouski, Canada.

**

Mechanical Engineering Department Laval University, Quebec, Canada.

ABSTRACT. This paper describes the initial phase of a major research work devoted to the investigation of a number of factors influencing the accuracy of machine tools. The investigation is indeed conducted so as to develop methods capable of providing practical solutions to the well-known machine tool loss of accuracy phenomenon. Based on a multi-sensor monitoring system, a novel software error-compensation approach is proposed to provide the ability to predict geometric, thermal and dynamic errors and, hence, offer the possibility to improve the accuracy of multi-axis machines. The proposed method can be divided into the following steps: (i) design of a geometric model describing a multi-axis machine, (ii) measurement of path-dependent rigid-body errors according to the model, (iii) fusion of sensors and the measured parametric errors via an artificial neural network model (ANN), (iv) one-line prediction and integration of the individual errors to produce a correction vector, (v) compensation of imbedded errors. Implemented on a turning centre, the approach led to a substantial improvement of the machining accuracy characterised by a reduction of the maximum error from 70 µm to less than 4 µm.

KEYWORDS: Multi-axis machines, Error compensation, Sensor fusion, Neural network. 1. Introduction Recent developments in production technology display the ever-increasing need for machine tools capable of achieving higher throughput and extreme accurate machining. In response to this on-going challenge facing the manufacturing industry, research efforts are directed towards the development and implementation of new cost-effective and practical methods that allow substantial improvements in the accuracy, speed, and repeatability of machine tools. These efforts have generally focused on enhancing the positioning accuracy of CNC machine tools through process monitoring and on-line software error prediction and correction. The concept of error correction is based on monitoring the parameters of the machine tool and using the real-time data to maintain the desired machining process accuracy. The concept requires the development and

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implementation of a sensor fusion technique combined to a machine tool model that represents the interactions of the parameters being monitored. Indeed, the performance of a machine tool is assessed according to its capability to accurately position the cutting tool. Obviously, each element involved in positioning the cutting tool with respect to the workpiece contributes to the resultant accuracy of the complete system. Resulting from various disturbances, errors in the positioning of the cutting tool affect the metal-removal process and introduce unacceptable deviations. These errors are generally classified according to their source and behaviour in the time domain [ HOC 77]. Considered as slowly varying in time, quasi-static errors associated to the machine tool structure are due to imperfect geometry and kinematics of moving components, static deflections, and thermal distortions. On the other hand, dynamic errors are related to tool wear, tool chatter, spindle run-out, machine self-induced and forced vibrations, and other disturbances associated to the machining process. Although the dynamic errors are also important, the quasi-static errors are considered to be responsible for a very large proportion of the machine inaccuracy, contributing as much as 70% of the total positioning error [ RAG 85]. Specific efforts in the area of machine tool metrology have focused on developing error modelling and prediction software [ ZHA 85], [ DON 86]and supporting the implementation of standard performance evaluation tests [ BRY 82]. Several approaches to enhancing machining accuracy have been proposed. The conventional approach is based rigid body kinematics for modelling the errors in machine elements ([ ZHA 85]and [ FER 86]). Other alternative methods consist in using empirical models [ BEL 87]or homogeneous transformation matrices [ DON 86]to represent the final observed volumetric error in the workspace of a multiaxes machine. Early research in the area of machine tool metrology has concentrated on applying methods based on analytic [ POR 80], trigonometric [ LOV 73], vectorial [ SCH 77]and [ HOC 77]and matrix ([ ZHA 85]and [ FER 86]) error representations of multi-axis machines. More recent attempts propose the use of artificial intelligence concepts to estimate errors induced in a multi-axes machine[ SRI 92]. The approach proposed in this paper is based on an accuracy-monitoring scheme designed to improve multi-axis machine performance by compensating for geometric, thermal, load-induced, and inertial errors. The essential feature of the monitoring system consists of measuring and modelling the individual errors through an ANN based multi-sensor fusion technique combined to a time-variant and spatial-variant position error kinematic integration model. 2. The proposed compensation system 2.1. On-line error compensation system

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As noted above, the accuracy of a multi-axis machine is adversely affected by various error sources such as geometric imperfections, thermal deformations, load effects, and dynamic disturbances. The implementation of the concept of error compensation as a way to achieve accuracy improvements of a machine tool requires the need to ensure on-line evaluation or prediction of individual error components. Attempts to conduct on-line measurements of the parametric errors uncovered the difficulties and problems involved with integrating and using optical instruments in a hostile machining environment. Hence, in the absence of reliable, accurate, and hardened sensors and measuring devices, indirect methods provide an effective and continuous real-time error correction task that prevents defective parts from being machined. The application of artificial intelligence methods to a machine tool error compensation system offers, in addition, the opportunity to allow integration and faster processing of multiple sensor signals and enhance error prediction accuracy. As illustrated in Figure 1, the proposed compensation scheme is designed to provide the ability to monitor the machine tool parameters (tool nominal position, cutting forces, temperature at various positions on the structure, speed and feedrate) and use the real-time data to control the accuracy of the machining process. The error-correction technique is implemented on a 285 x 1090 Mori Seiki SL 25 SE turning center. The sensors monitor the machine parameters and provide continuously the data that a multi-layer feedforward ANN uses to generate the individual error components. These errors are synthesized through a kinematic model to provide a correction vector that is fed into the servo loop of the machine to adjust the machining process accuracy. S8 S5

S6

Dynamometer Accelerometer

WORKSTATION

Turret CNC CONTROL

Tool

S3

S1

S4

S2

Speed feed and position control

INTERFACES ACQUISITION

An. Input Interface

PROCESSING Data Acquisition and Control Board

wo rkpiece ANN Models

Thermistor S7

CONTROL

Dig. Output Interface Kinematic model error prediction

Figure 1. Schematic diagram of accuracy monitoring system

2.2. Error integration model In a typical machine tool, error is the difference between the actual and the anticipated response of the machine to a command issued according to the machine's accepted protocol [HOC 77]. This error results in a deviation of the

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cutting tool tip from the desired trajectory. The components of the positioning error of a two-axis turning center are shown in Figure 2. Assuming that the structural components of the machine tool are rigid bodies, the resultant error at the tool tip can also be described by a combination of individual displacement and rotational path-dependant errors. A first step in the development of a compensation scheme consists, as stated above, in establishing a model in order to estimate the resultant error and hence derive the correction vector from the interaction in space of the individual error components. For this purpose, a general model providing the ability to estimate the correction vector has been developed from four sub-models: the basic geometric model, the coordinate system thermal drift model, the spindle error model, and the dynamic model. Then, the general model is used to define the necessary task of measuring the error components. ∆x εx X εx Spindle

Workpiece

∆x

εy

∆z εz Z

∆y

Nonsensitive direction

Cutting tool

Figure 2. The resultant error component at the tip of the cutting tool Basic geometric model The individual errors are defined with respect to a single reference point representing the zero of the machine tool. The error characteristics are obtained by moving the carriage in the Z-axis and the cross slide in the X-axis. Displacement of the cutting edge from the machine original position [Xo, Zo] to a nominal position [Xn, Zn] introduces the geometric errors [ ∆x, ∆z]. As illustrated in Figure 3, these position-dependent errors resulting from the displacement of the machine components are obtained through a combination of the individual errors representing the linear, the straightness, the angular and the non-orthogonal errors induced along a machine axes. Coordinate system thermal drift model The second part of the general model intends to take into account the problem of the coordinate system drift due to the thermal disturbances. Assuming linear x y z T effects, the error vector ∆ d = δd , δd, δd is defined as the thermal drift. Obtained at

ANN Based Sensor Fusion Strategy for Multi-Axis Machine Accuracy Monitoring and Control

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various machine thermal conditions, the thermal drift vector is added to the error vector derived from the geometric model.

Figure 3. Schematic diagram of a two-axis turning center Spindle error model The third model is established to allow the derivation of the spindle thermal drift. In a turning center, three components associated to the spindle thermal drift are z

critical to the machine accuracy. The first component is the axial thermal drift δ s responsible for a displacement along the Z-axis. The second represents the radial x

thermal drift δ s acting in a direction perpendicular to the Z-axis. The third y component is associated to the tilt thermal drift εs representing the angular deviation of the spindle axis in the X-Z plane. Dynamic error model The fourth model is developed to allow the computation of the dynamic effects on the machine tool accuracy. The dynamic effects include herein two categories of x y z T error sources. The error vector ∆ f = δf , δf , δf defines the first category representing x y z T the cutting force effects. The vector ∆ i = δi , δi , δi represents the second category associated to the inertial effects. As already mentioned, the components required to build the sub-models and consequently the general model are determined at various machine tool thermal states. The error components are obtained in terms of the machine tool measured parameters and operating conditions. As can be noticed, the absence of time as a variable is created by the need to simplify the modelling procedure. The use of time as a variable is susceptible of unnecessarily complicating the modeling procedure. From this point of view, the resultant error is, as illustrated in Figure 2, synthesized as follows: ∆x = δdx + δsx − δxx − δxz − δfx − δix − Z o (ε xy + ε zy ) + Z p εsy

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∆z = δdz + δsz − δzz − δzf − δiz − δzx + Xε zy + X o (ε xy + ε yz ) − X p εsy

The algorithm relative to the implementation of this model is schematically illustrated in Figure 4. Process sensing devices monitoring the nominal positions of the machine slides, temperature at various positions on the machine structure, cutting forces, spindle speed, and feed-rate generate signals that are scanned at a constant sampling rate. At every sample, each individual error is predicted using the appropriate model. Vxi

Fi

x

δi

Inertial effects X-axis

x

δf

Cutting forces effectsX-axis

X-axis X-axis res ultant ∆x error co mponent

x

Thermal d rift error X-axis Xn

Ti

N

δx x y X n + δd Geometric errors ε x

X-axis

z

x δb

x

Axial thermal d rift

δb

Axial thermal d rift

ε by z

δb

δx ε by

Spindle Thermal errors

Axial thermal d rift Zn

Thermal d rift error Z-axis

z

δb ε by z

δz

x

Z n + δd

y

εz

Geometric errors x δz Z-axis z

Fi

Cutting forces effectsZ-axis

Vxi

Inertial effects Z-axis

δf

X-axis res ultant ∆z error co mponent

X-axis

z

δi

Figure 4. The bloc diagram of the integration scheme 2.3. The proposed sensor fusion method Prediction of the individual errors enables the evaluation of the resultant error components at any location within the machine tool working space using the multiple variable models. These models are developed to include all factors contributing to the deviation of the cutting tool from the desired trajectory. Since error sources exhibit highly non-linear interactions with the machine conditions, a precise quantitative prediction of the individual errors is difficult to achieve using theoretical analysis. Indeed, on-line individual error evaluation through a kind of multiple-input/output empirical model allows for faster

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processing and enhanced prediction accuracy. The implementation of this point of view comes up against the following difficulties: (i) choice of the modeling technique and (ii) selection of the predictor variables. Model building analysis is often conducted using a large set of variables. From these candidate predictor variables, only an optimal subset is indeed useful for predicting the response. The identification of important input variables is crucial to the success of any empirical model. In this study, a systematic procedure designed for model building is presented. Compared to the GMDH and the Least-Squares Regression Technique, neural networks provide a more effective modeling capability for predicting the error components. This is particularly true when the relationship between the sensorbased information and the actual error is non-linear. Based on various selection criteria, the selection of the candidate predictor variables can be achieved using some statistical techniques. Neural network analysis Artificial neural network models are used to express the geometric path-dependant errors as a function of the cutting tool nominal position, temperature, cutting forces, speed and feed-rate. This approach offers the ability to model and generalise without overfitting highly non-linear relationships. The use of the ANN model can further significantly reduce measurement and off-line calibration efforts. The ANN model used is a multilayered Perceptron involving a collection of simple interconnected non-linear processing elements. The elements or neural nodes are arranged in patterns and operate in parallel. Using the well-known backpropagation technique and the generalised delta rule [ MCC 88], the multilayer Perceptron is trained to find an acceptable weight solution. Variables selection procedure The idea behind the sensor selection using statistical techniques is based on the comparison of a complete model containing all predictor variables and a model with reduced number of input variables. This procedure can be implemented as follows: i) Train a sufficient number of fusion models. Each model should be designed with a subset of input sensors selected randomly. ii) Compute the overall performance of each model under development according to the selection criterion. iii) Determine the contribution of each sensor to the overall performance using appropriate statistical tools. Many statistical criteria can be used to assess whether a reduced model is an adequate representation of the relationship between model response and predictor variables. The performance evaluation of fitted models is based on the principle of

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reduction in some statistical criteria such as error sum of squares with training data (SSEt), error sum of squares with checking data (SSEc), variance of the residual error (Vr), variance of the modeling error (Vm), variance of the error transmitted from input to output (Vt), and total variance (Vtot). The traditional selection procedure involves the comparison of the selection criteria for all possible subsets of predictor variables. Three alternative selection methods are used when a large number of candidate predictors are available: the forward, backward and stepwise variable selection procedures. The forward selection procedure starts with no input variables in the model so as to allow the addition of one variable at a time. This process is halted when a satisfactory fit is achieved or when all input variables have been added. The backward selection method begins with all predictor variables in the model and applies a process of elimination. As a result, variables are deleted one at a time until an unsatisfactory fit is encountered. Finally, the stepwise procedure combines features of both the forward and the backward selection procedures. While variables can be added one at a time, the procedure also allows the elimination of variables. Though the F-statistics method is generally applied, the decision to halt the process of adding or eliminating variables can be made using any of the selection criteria mentioned above. The classical selection methods offer the possibility to isolate one reduced model. However, the main drawback associated to these methods lies in their inability to identify alternative candidate subsets of the same size or a model considered being optimal according to the above selection criteria. Hence, the traditional selection procedures could lead to poor results and often to different subsets because of their inability to consider any interaction between sensors. Thus, the basic condition for a successful implementation of a sensor fusion method requires a simultaneous application of the selection criteria so as to ensure the independent selection of sensors. The proposed sensor fusion method involves the use of experimental data relative to the individual error components with respect to different factors such as processinput parameters and operating conditions. Accordingly, the use of an efficient test strategy appears to be most appealing. Though not recommended for a prohibitively large number of predictor variables, the factorial design allows the greatest scrutiny of alternative candidates. The use of orthogonal arrays (OAs) would reduce significantly the number of fitted models. The OA-based model building procedure can be summarized in the following steps: q

Collect the training and checking data. The most important factors believed to influence the investigated features must be identified and considered in the measurement tests.

q

Select the modeling technique and optimize the training performance.

q

Select the OA for the design of models including all potential predictor variables. Every column in the OA corresponds to a variable, Vi, with two

ANN Based Sensor Fusion Strategy for Multi-Axis Machine Accuracy Monitoring and Control

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levels indicating if a sensor information is input to the fitted model (1: included) or not (0: not included). q

Train the models generated in the OA and compute their performance index (SSEt, SSEt, Vr, Vm, Vt and Vtot).

q

Determine the effect of each variable on every performance index. These effects can be considered as the rate of reduction in the sum squared errors SSEi and variances Vi when a sensor is or is not input to the fitted model. For each criterion, the effect of each predictor variable can be estimated by taking the arithmetic difference between the two average values corresponding to the two variable levels (0 and 1). Based on these results, sensors contributing to the reduction of the criteria values are selected.

q

Obtain the final fusion model. The final fusion model is built once the sensors providing the best information on the error sources behavior are determined.

3. Models building and simulation To build the compensation ANN models, the error components were classified into five groups having similar characteristics and requiring the same measurement procedures and instrumentation. These groups are the co-ordinate system thermal drift, the geometric errors, the cutting force induced errors, the inertial errors and the spindle thermal drift. 3.1. Measurement of error components Using a 5528A-laser interferometer system, measurement of the individual error components was conducted along lines parallel to the machine tool axes. The displacement intervals for recording the error values amounted to 10 mm along the X-axis and 20 mm along the Z-axis. In addition, the transitional drifts and the angular inclination of the spindle axis were also measured using two capacitance sensors. The behavior of the average thermal drift error of the coordinate system at the zero of the machine tool observed over the machine warm-up is illustrated in Figure 5. It can be seen that the thermal expansion of the machine frame resulting from a 12 hrs run introduced an error of about 25 µm. As shown in Figure 6, the linear displacement error along the X-axis is 15 µm at a nominal position Xn=380 mm. The linear displacement error along the Z-axis reaches at a nominal position Zn=840 mm a maximum value of 70 µm, exceeding the specified machine tool accuracy (± 20 µm). Evaluated at a given machine thermal state, the maximum straightness error reaches 10 µm along the X-axis and 12 µm along the Z-axis. Furthermore, the yaw errors measured along the X and Z axes are similarly presented in Figure 6 and 7. Finally, the behaviour of spindle thermal drift error observed under various running conditions is also illustrated in Figure 8.

II International Seminar on Improving Machine Tool Performance Coordinate system thermal drift [micron]

10

30

20

10 0 0

180

360

540

720

Time [min]

Figure 5. Coordinate system thermal drift error X_axis displacement error

10

Straightness error [µm]

5

Yaw error [arc sec]

0 -5 Linear displacement error [µm]

-10 -15 0

90

180

270

340

Nominal X position [mm]

Z_axis displacement error

Figure 6. The X displacement error measurement (20 °C) 10 0 -10 -20 -30 -40 -50 -60 -70

Yaw error [arc sec]

Straightness error [µm]

Linear displacement error [µm] 0

190

380

570

760

Nominal Z position [mm]

Spindle thermal drift errors

Figure 7. The Z displacement error measurement (20 °C) 12 10

Axial thermal drift [µm] Radial thermal drift [µm] Spindle tilt drift error [arc sec]

8 6 4 2 0 -2 0

250

500

750

1000

Time [min]

Figure 8. Spindle thermal drift error measurement taken under continuous running conditions

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3.2. Error modelling To provide quasi-real time tool position correction, the static, thermal and dynamic errors are predicted using data from an array of sensors. These sensors constantly monitor the machine tool parameters related to the cutting tool nominal position, the temperature at various locations, the forces along three perpendicular directions, the speed, and the feed-rate. Data from the sensing devices is fed into the ANNs in order to estimate the individual error components. The axial thermal drift error of the spindle δzb is considered to illustrate the procedure designed to build the ANN based error model. Initial investigation suggests a strong relationship between δzb and the spindle speed and the rise of the average temperature along the spindle and the bed location. As a result, measurement data used to train the ANNs was collected at three spindle rotational speeds of 1000, 2000, and 3000 rpm using temperature sensors S1, S2, S3, and S4 (Figure1). Before measurement task, the turning center was run at a constant spindle speed for a 12-hrs warm-up period. Then, the axial thermal drift and the temperature history were continuously registered. Before selecting the variables and training the neural networks, it is important to establish the size of the hidden layer and optimise the training performance. The number of inputs in the models under evaluation is not constant. The idea is to approximate the relationship between the size of the hidden layer and the complexity of each parameter to estimate. For this evaluation 5 nets have been studied {[IxI/2x 1], [IxIx1], [Ix2Ix1], [IxIx2Ix1] and [Ix2Ix3Ix1]}, where I is the number of inputs. The backward error propagation using the generalised delta involves setting the gain and momentum, such that accurate results can be achieved within the shortest time. For all trained models, an adequate knowledge representation with an average error of less than 1% was observed, irrespective of the hidden layer size. Consequently, to avoid long training and overfitting that could disturb its accuracy, the [I x I+1 x 1] network structure was selected. To select the predictor variables, the procedure consists in establishing the OA for the design of models including all potential predictor variables. As illustrated in table 1, the OA that fits in this procedure is the L8 representing a total of 8 models to be designed. A relatively accurate relationship for δzb prediction is obtained using [I x I+1 x 1] three layer ANNs models. Results relative to each model for both training and checking operations are also presented in table 1. Deviations of the model's estimates are presented as a function of six selection criteria. All models fitted the training data relatively well as indicated by the SSEt and the Vr. As shown in table 1, the results obtained from the checking data were less accurate. Using these results, the effect of each predictor variable on the selection criteria was evaluated. As illustrated in Figure 9, the average effect of each input variable on the six criteria is represented by its contribution to each model accuracy

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improvement. These results reveal that only the temperature sensors S2 and S4 and the spindle speed N have positive effects on the models. Curve fitting results of the spindle axial thermal drift error obtained from training the ANN model with the selected subset are shown in Figure 10. Examination of these results demonstrates that the ANN model can fit the thermal drift very well. Generally, the sensor fusion procedure reveals that each individual error component depends strongly on the temperature sensors distributed along its generative axis. Table 1. Subsets variables evaluation Predictor variables Model # 1 2 3 4 5 6 7 8

N 1 1 1 1 0 0 0 0

Selection criteria

S1 S2 S3 S4 SSEt 1 1 1 1 8.92 1 1 0 0 12.98 0 0 1 1 10.34 0 0 0 0 89.34 1 0 1 0 22.27 1 0 0 1 10.83 0 1 1 0 26.21 0 1 0 1 28.23

SSEc 17.92 33.38 20.17 256.3 44.33 22.49 53.51 44.02

Vr 0.335 0.402 0.358 1.047 0.524 0.366 0.569 0.59

Vm 0.072 0.124 0.09 1.108 0.244 0.096 0.295 0.322

Vt 0.106 0.139 0.094 0.198 0.245 0.105 0.174 0.134

V tot 0.846 1.118 0.894 3.603 1.534 0.938 1.61 1.565

Predictor variable contribution %

60 S1

40

S2 S3 S4

20

N Er

0 SSEt SSEc Vr Vm

Vt Vtot

Selection c riteria

Figure 9. Average contribution of the predictor variables in reduction of statistical criteria

Axial spindle error [micron]

25

N=3000 rpm N=2000 rpm

20 15

N=1000 rpm

10 5

Measured Fitted

0 -5 0

250

500

750

1000

Time [min]

Figure 10. The measured and the modelled spindle axial thermal drift as the machine warm-up.

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The sensor fusion procedure described above and used to model the spindle axial thermal drift error has also been similarly implemented to establish specific optimal models for the rest of the individual errors. The results of these ANN sensor fusion models are summarized in Table 2. The line entries correspond to the monitored machine parameters or variables. The column entries are the measured individual errors associated to various error sources. For each error, the sign "x" is used to identify the optimal combination of the monitored parameters. Obviously, all of the variables could have been used as inputs to each model. Consequently, the modelling procedure could have been simplified to the detriment of the error prediction accuracy. Table 2. Selected optimal subsets variables for models building V A R

Position

Temp. sensors

Force

Speed

Px Pz S1 S2 S3 S4 S5 S6 S7 S8 Sm Fx Fy Fz Vx Vy N

Geometric errors

Thermal drift

X axis

Z axis

x δ d

z δ d

x δ x x

z δ x x

εxy x

x

x

x

x x x x

x x x x

x x

x x

x x x x x

x

z δ z

x δ z

εzy

x x

x

x

x x

x

x x

Spindle errors

x x

x x x x

x z δ δ sp sp

εpy

x

x

x

x

x

x

Load induced errors

Inertial errors

x δ f

z δ f

x δ i

z δ i

x

x

x

x

x x

x x

x x x

x

x x x

x

x

Indeed, the accuracy of the models presented in Table 2 has been thoroughly investigated. The results of this evaluation are shown in Figures 11-13. It can be seen that the maximum residual error that is the difference between the observed quantity and the response of the model is less than 1µm. Compared to the maximum linear displacement error observed along the X-axis (15µm) and the Zaxis (70µm), it can be concluded that the proposed sensor fusion technique offers the ability to predict the individual errors with sufficient accuracy.

Residual errors microns / Arc sec

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εy z

0.125

z

εy

z

δx

x

δz

x

δz

x

δx

0.100 0.075 0.050 0.025

#1

#2

#3

#6

#4 #5 Measuring cycle #

#7

#8

Residual errors microns / Arc sec

Figure 11. Maximum residual prediction errors of geometrical deviations under various thermal conditions 0.4

ε by

z

δb

x

δb

0.3 0.2 0.1

3000 2000 1000 Sp indle speed (Rpm)

Residual errors microns / Arc sec

Figure 13. Maximum residual prediction errors of spindle deviations 1.25

z

z

δf

δi

δi

#4 #5 #6 Measuring cycle #

#7

#8

x

δf

x

1.00 0.75 0.50 0.25

#1

#2

#3

Figure 12. Maximum residual prediction errors of load-induced and inertial errors under various thermal conditions

ANN Based Sensor Fusion Strategy for Multi-Axis Machine Accuracy Monitoring and Control

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3.3. Simulation Once the various individual error models were established, the compensation values ∆x and ∆z are synthesized using the algorithm illustrated in Figure 4. Simulation tests conducted using various machine tool conditions display the effectiveness of the proposed compensation approach. Figures 14-16 visualize the spatial-variant error components at an arbitrary temperature (average temperature of 24.3 °C) and show a comparison of measured and predicted errors in the X-Z plane. Maximum errors without compensation are 30 µm in the X-direction and 65 µm in the Z-direction. As illustrated in Figure 16, residual errors estimated after compensation are within a 2 µm range. The aptitude of the model to identify thermal effects has also been verified. The models were tested under different machine thermal conditions. Thermal effects were predicted with an average error of less than 2 µm. The approach provided good prediction capabilities, particularly when the machine tool average temperature varies within a range of 18 °C.

Figure 14. Measured error surfaces in X and Z directions

Figure 15. Predicted error surfaces in X and Z directions

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Figure 16. Residual error surfaces in X and Z directions 4. Conclusion In response to the increasing demand for higher quality of machined parts, accuracy improvement of multi-axis CNC machines through software error compensation has become increasingly important in modern manufacturing. The success of such approach depends on the degree of accuracy, robustness and reliability of the model to estimate on-line the resultant error components at any location within the machine workspace. The neural network-based multi-sensor fusion approach proposed in this paper is built to satisfy these requirements. Applicable to multi-axis machines of divers configuration, the proposed general methodology has been developed to formulate the mathematical model that relates quasi-static and dynamic errors to the machine tool operating conditions. Combined to the selection of optimal subset sensors, the use of the ANN technique provides a powerful modelling tool of the individual error components. The procedure expresses in a rational clear manner the contribution of the machine conditions to the individual error components. Indeed, the performance evaluation of the error compensation approach led to a significant reduction of the machine error to an average error of 2 microns. In addition to its applications for improving point-to-point positioning accuracy, the methodology reported herein can help designers and users evaluate machine tools performance for acceptance, testing and identifying the machine's optimal working region. 5. Acknowledgements The authors are grateful to the Natural Sciences and Engineering Research Council of Canada for their financial support through individual grant RGPIN# 217395. 6. References [ BEL 87] Belfore, G., Bona, B., Canuto, E., Donati, F., Feraris, F., Gorini, I., Morei, S., Peisino, M., and Sartori, S., "Coordinate Measuring Machine And Machine Tool Selfcalibration", Annals of the CIRP Vol. 36, No. 1, 1987.

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[ BRY 82] Bryan, J. B., "A Simple Method for Testing Measuring Machines and Machine Tools, Part I", Precision Engineering, Vol. 4, No. 2, 1982. [ BRY 82] Bryan, J. B., "A Simple Method for Testing Measuring Machines and Machine Tools, Part II", Precision Engineering, Vol. 4, No. 3, 1982. [ DON 86] Donmez, M. A., Blomquist, D. S., Hoccken, R. J., Liu, C. R. and Barash, M. M., "A General Methodology for Machine Tool Accuracy Enhancement by Error Compensation", Precision Engineering, Vol. 8, No. 4, 1986. [ FER 86] Ferreira, P. M. and Liu, C. R., "A Contribution to the Analysis and Compensation of the Geometric Error of Machining Center ", Annals of the CIRP, Vol. 35, No. 1, 1986. [ HOC 77] Hocken, R. J., Simpson, J. A., Borchardt, B., Lazar, J., Reeve, C. and Stein, P., "Three Dimensional Metrology" Annals of the CIRP, Vol. 26, No. 2, 1977. [ LOV 73] Love, W. J. and Scarr, A. J., "The Determination of the Volumetric Accuracy of Multi-Axis Machines", MTDR Proc. Conf., Vol. 14, 1973. [ MCC 88] McClelland L. and Rumelhart D. E., "Explorations in Parallel Distributed Processing", MIT Press, 1988. [ POR 80] Portman, V. T., "Error Summation in the Analytical Calculation of Lathe Accuracy", Machines & Tooling, Vol. 51, No. 1, 1980. [ RAG 85] Ragunath, V., "Thermal Effects on the Accuracy of Numerically controlled Machine Tools" Ph. D dissertation, Perdue University, West La fayette, Indiana, 1985. [ SCH 77] Schultschick, R., "The components of volumetric Accuracy", Annals of the CIRP, Vol. 25, No. 1, 1977. [ ZHA 85] Zhang, G., Veale, R., Charlton, T., Borchardt, B. and Hocken, R., "Error Compensation Of Co-ordinate Measuring Machines", Annals of the CIRP Vol. 34, No. 1, 1985.