Neural network based surface shape modelingof stressed lap optical polishing
Min-you Chen,1,* Yong-tao Feng,1 Yong-jian Wan,2 Yang Li,2 and Bin Fan2
1School of Electrical Engineering, State Key Laboratory of Power Transmission Equipment and SystemSecurity and New Technology, Chongqing University, Chongqing 400030, China
2Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu 610209, China
Large-scale aspheric mirrors have been widely usedin various areas, such as astronomical object obser-vation, space communication, and laser systems.Although aspheric mirrors have the advantages ofbeing lightweight, having small volume, and highimaging quality in comparison with sphericalmirrors, they are difficult to produce. Computer-controlled stressed lap polishing is a relatively newoptical burnishing technology for large aspheric mir-rors [1] that can actively deform the lap surface to anoff-axis aspheric surface depending on different lappositions on the mirror surface and different anglesof the lap [2]. In large optical component processing,computer-controlled actively stressed lap polishinghas a number of advantages, such as high processingefficiency, natural smoothness [3,4], andproduction ofa smooth surface with sharp edges.Although computer-controlled stressed lap proces-
sing is advantageous for fabrication of large aspheric
surfaces, there are some difficult problems, such asmodeling uncertainty of the stressed lap, high non-linearity, and a high performance requirement of op-tical processing, especially for deep off-axis asphericsurfaces. In the computer-controlled polishing pro-cess, it is crucially important to build an accuratestressed lap surface model for shape control, so thatthe lap shape can match the ideal parabolic mirrorsurface at all times, while the lifting actuators canbe used to vary the pressure in proportion to the sur-face error and to balance forces when the lap extendsover the edge of the mirror. To address the problem,we emphasize stressed lap surface modeling. First,we introduce a computer control system for thestressed lap polishing process. The shape controlsystem structure and the surface figure measure-ment method are discussed. Then a stressed lapsurface model based on a radial basis function(RBF) network is proposed to represent the dynamicdeformation of a stressed lap surface. This addressesissues such as shape accuracy and deformationhysteresis. Simulation experiments for stressed lapmodeling were performed with different actuatordriving forces. The experimental results show that
the RBF network model fitted the lap surface accu-rately. In comparison with traditional Zernikepolynomial shape representation [5,6], the proposedneural network model is more effective and accurate.
2. Stressed Lap Surface Control and Measurement
Stressed lap polishing was developed to polish largeasphericmirrors [4,5]. It consists of analuminumdiskthat changes shape dynamically under the effect ofsome symmetrically distributed moment-generatingactuators. These computer-controlled actuators areprogrammed to produce the shape changes necessaryto make the lap fit the mirror surface as it movesacross the surface and rotates. The general structureof the control system for lap surface is shown in Fig. 1,where P represents the given ideal surface, T repre-sents the corresponding surface data, the circledenotes a comparison unit,E ¼ T −D is themodel er-ror, F represents the control forces, and D representsthe measured lap surface data. The task of the con-troller is to output the appropriate driving forces tomake the lap surface match the given ideal surfaceaccurately. It is important to build an efficient andaccurate lap surface model to facilitate lap surfacecontrol. We describe the principles and design ofthe surface deformation control and a measurementmethod of the stressed lap and develop an effectivemodel relating the actuator forces and the shapechanges of the stressed lap surface.
A. Stressed Lap Surface Shape Control
As shown in Fig. 2, under the effect of 12 variabletorques generated by installed drive and draw-bardevices around the lap, the stressed lap can be de-formed according to the requirements of the polish-ing process. The 12 actuators are composed of fourgroups of drives; each consisting of three symmetri-cally positioned drives connected by a tension band.The lap, a 60 cm diameter aluminum plate, is visiblebelow the bending actuators. The plate is faced withpitch to form the polishing surface. The twelve actua-tors apply lifting forces normal to the plate to controlpolishing pressure and pressure gradients. Thebending actuators are programmed to make the lapshape match the ideal parabolic mirror surface,while the lifting actuators can be used to vary thepressure in proportion to the surface error and to bal-ance forces when the lap extends over the edge of the
mirror. The resultant force of the four groups ofactuators produce the required bending and twistingtorque to change the surface shape of the lap. Thedeveloped computer-controlled stressed lap, asshown in Fig. 3, has been used for large asphericpolishing.
B. Measurement Method of the Surface Displacement
To measure the surface changes of the lap, wedesigned a displacement measuring plate that com-prised 60 displacement sensors uniformly distribu-ted on the plate [6]. The lap is mounted on an arrayof displacement sensors that measure its changes inshape for testing and calibration, as shown in Fig. 4.Any distortion of the lap surface will be detected bythe sensor array. The surface displacement sensorsare symmetrically and uniformly distributed overthe circular plate, as shown in Fig. 5, which can beused to detect any changes on the lap surface. Themeasured surface change data are then interpolatedto represent the surface shape of the stressed lap.
3. Neural Network Based Stressed Lap Modeling
It is important to establish the relationship betweendriving forces and surface changes of the lap for com-puter-controlled stressed lap polishing. Since thechanges of the lap surface are usually continuousand smooth in a circular area, one can representthe lap surface bya set of independent basis functions.
Fig. 1. Basic structure of the stressed lap surface control system.
Fig. 2. Top view of the actuators positioned on the stressed lap.
In optical processing, the Zernike polynomial is com-monly used to represent the surface shape of mirrors[7,8]. First, the measured surface data are mappedonto a unit circle. Then the Zernike basis functionswould be selected and the coefficients of the Zernikepolynomial are usually determined by an orthogonalleast-squares algorithm [7] or by neural networktraining [8]. However, such a polynomial model,which is used for optical surface description, is diffi-cult to use for surface change predictions and thestressed lap control system because of its poor gener-alization performance. To facilitate the stressed lapsurface control system and improve the model accu-racy, we proposed a RBF network lap surface modeldirectly obtained from detected surface data.A RBF network comprises a single hidden layer of
nonlinear neurons followed by a linear output layer,as shown in Fig. 6. Xu et al. [9] derived a set of the-oretical results that commend the use of the RBF asuniversal function approximators. That is, for anygiven model error ε > 0, a 60 cm RBF network modelf ðc; σ;w; Þ exists with optimal numbers of hiddenunits and optimal parameters ðc; σ;w; Þ such thatthe error functionE ¼ ‖y − f‖ satisfies the inequalityEðc; σ;w; Þ < ε. For a stressed lap surface model, theinput variables are control forces F from the actua-tors, and the outputs are surface displacement D.
The acquired RBF network model can be presentedas
D ¼ f ðF1;F2;…;FmÞ; ð1Þ
Dj ¼XSi¼1
wijhiðF1;F2;…;FmÞ; ð2Þ
where j ¼ 1; 2;…;n and RBFs hi are defined as
hi ¼ expð−‖F − ci‖2=σ2i Þ=Mi; ð3Þ
Mi ¼XSi¼1
expð−‖F − ci‖2=σ2i Þ; ð4Þ
where wij are output weights, ci is the center of theith radial unit, and each can be considered to be aprototype vector that represents a region of inputspace; σi is the unit width that determines over whatdistance in the input space the unit will have a sig-nificant influence. The prototype vectors ci can be de-termined in different ways; the commonly used waysinclude the K-means algorithm, the Kohonen self-organizing map, and fuzzy clustering [10–12]. Forthe stressed lap surface model, the input variablesare control forces F from the actuators that form auniform distribution. Usually, it is difficult to decidethe number of hidden units in a RBF network. Thisissue could be addressed by an empirical approach ordata partition validation [11]. Here, the number ofradial basis units (hidden units) is selected as thesame number of force actuators by the trial-and-errormethod that is due to its good balance between modelaccuracy and complexity. The initial values of ci andwij are generated by a fuzzy c-means algorithm [12].σi is the average distance that is produced by thenear field within the scope of the m RBFs, namely,
σi ¼0@Xm
j¼1
‖ci − cj‖=m
1A
12
: ð5Þ
To enhance the approximation accuracy of the RBFnetworkmodel, all the network parameters ci, σi, and
Fig. 4. Displacement sensor array for measuring the shapechanges of the lap surface.
Fig. 5. (Color online) Distribution of the stressed lap surfacedisplacement detecting sensors.
wij are trained by the backpropagation algorithmwith the error function
E ¼ 12
XN
Xnj¼1
ðDj − djÞ2; ð6Þ
where dj is the jth desired output, Dj is the jth modeloutput, N denotes the number of training data, andthe parameter learning algorithms are derived as
ΔwijðtÞ ¼ βhiej þ γΔwijðt − 1Þ; ð7Þ
ΔcikðtÞ ¼ βhi
Xnj¼1
ejðFk − cikÞ
σ2ikðwij −DjÞ þ γΔcikðt − 1Þ;
ð8Þ
ΔσikðtÞ ¼ βhi
Xnj¼1
ejðFk − cikÞ
σ2ikðwij −DjÞ þ γΔσikðt − 1Þ;
ð9Þwhere i ¼ 1; 2;…; s; j ¼ 1; 2;…;n; k ¼ 1; 2;…;m: β isthe learning rate, γ is the momentum rate, k refers tothe iteration number, and ej ¼ ðDj − djÞ. To increasethe convergence speed and improve the learningefficiency of the backpropagation algorithm, we in-troduced an adaptive tuning algorithm proposed in[11] to self-adjust both the learning rate β and themomentum rate γ, both of which help to obtain theoptimal parameters of the RBF network model.
4. Simulation Experiments
To verify the effectiveness of the proposed neural net-work model, various simulation experiments havebeen conducted. The simulation was based on setsof data consisting of 12 actuator driving forces andthe corresponding lap surface changes obtained fromthe 60 displacement sensors. Various data sets wereobtained by measuring the displacements driven by
different combinations of actuator forces. The RBFnetwork models were trained by 1000 experimentaldata including the lap surface changes under a oneunit to a three unit force driven by different actua-tors. The measured driving forces were used as inputsamples, and the measured data from the microdis-placement sensor array were used as output samplesto train the RBF neural networks. We used 70% ofthe data for model training and 30% of the datafor validation. Thirteen sets of measured data con-sisting of a one unit actuator force from different ac-tuators and corresponding surface data were used forthe neural network model testing.
After training and cross validation, we obtainedthe RBF network based lap surface model. Figure 7displays the lap surface shape produced by the ob-tained RBF network model with a one unit drivingforce from actuator 2. Figure 8 displays the lap sur-face shape produced by the trained RBF networkmodel with a one unit driving force from actuators2 and 6. The corresponding measured lap surfacedata were used to test the model accuracy. The aver-age value of the mean-square error of the modelpredictions is less than 10−2 μm. The residual fitting
Fig. 7. (Color online) Surface shape of the stressed lap with a oneunit force produced by actuator 2.
Fig. 8. (Color online) Surface shape of the stressed lap with a oneunit force produced by actuators 2 and 6.
error distribution is shown in Fig. 9. The modelaccuracy for other surface shapes (as shown in thefollowing figures) is similar and consistent. The sur-face shape of the stressed lap with a one unit forceproduced by actuators 1, 5, and 9 is shown in Fig. 10.Different surface shapes obtained through the RBFnetwork model reveal the effect of different actuatorforces on the lap surface changes, which is helpful forinvestigating the relationship between actuatorforces and surface changes.Figure 11 displays the surface shape with all 12
actuators producing a one unit force at the sametime. It can be seen that the lap surface became aparaboloid because of the uniform force distribution.Again, the obtained neural network model providedaccurate lap surface changes. The simulation experi-ments show that the obtained RBF network modelcan be used to represent the stressed lap surface ef-fectively. The link between actuator forces and lapsurface changes has been established with theRBF network model that can be used to facilitatethe stressed lap control system design.
5. Conclusions
We have introduced a stressed lap control systemand established a radial basis function networkbased on a multiple input–multiple output modelto represent a stressed lap surface shape. In compar-ison with the conventional Zernike polynomialmodel, the RBF network model is simple, effective,and easy to achieve. Simulation experiments showthat the proposed neural network model not only re-presents the lap surface accurately but also revealsthe effects of different actuator forces on the lap sur-face change. It is seen that the proposed RBF net-work model can be used as an alternative modelfor a stressed lap control system design. Therefore,the RBF network based surface shape modeling ofa stressed lap in a large mirror optical polishing pro-cess provides a method with the potential to be usedwith ease in practice.
The authors are grateful for the support from theNational 111 Project of China. We thank D. S. Holderfrom the University College London, UK, and R. F.Harrison from the University of Sheffield, UK, fortheir helpful discussions and comments.
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Fig. 10. (Color online) Surface shape of the stressed lap with aone unit force produced by actuators 1, 5, and 9.
Fig. 11. (Color online) Surface shape of the stressed lap with aone unit force produced by all 12 actuators.
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