光学曲面确定性抛光的面型精度控制研究
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摘要
光学曲面零件是光学系统的关键元器件。近年来，随着光学曲面零件需求量的日趋增加和光学系统对其光学性能要求的不断提高，光学曲面零件在加工精度、轻量化程度、生产成本和生产效率等方面的要求越来越高。高精度光学曲面零件的生产效率和成本已经成为反映一个国家光学工业乃至整个制造业现代化水平的重要标志。抛光工艺常常作为光学曲面制造的最后一道工序，对光学曲面的表面质量、面型精度和使用寿命等起到至关重要的作用。本文结合国家重点基础研究发展计划（973计划）“光学自由曲面制造的基础研究”的子课题二“光学自由曲面成形过程的物理解析与精度控制”（课题编号：2011CB706702），以实现确定性抛光面型精度控制为主题，深入探讨了抛光过程的局部和整体材料去除理论、抛光力控制和抛光面型工艺控制策略等关键技术。
目前，确定性抛光工艺研究的核心问题包括单位时间材料去除函数建模、驻留时间计算、抛光路径规划和材料去除量预测等。经典的抛光理论为，整体材料去除量等于单位时间材料去除函数与抛光轨迹上各抛光点的驻留时间的卷积。上述经典理论将抛光过程假设成一个线性移不变系统，即假设抛光工具单位时间内的材料去除量不随位置的移动而变化。对于低陡度的球面和非球面而言，运用该理论能够对抛光过程有很好的控制。但是，对于形状比较复杂的非球面和自由曲面，曲面曲率、抛光轨迹和抛光姿态的复杂性使得该理论的适用性降低。本文研究了抛光工艺参数（包括抛光力、主轴转速、进给速度、抛光姿态角等）、抛光工艺条件（包括抛光液和抛光垫条件等）、抛光轨迹、曲面几何/物理特性（曲率和弹性模量等）等一系列因素对抛光材料去除影响，构建了综合考虑以上各因素影响的抛光表面去除模型，揭示了工艺参数和条件变化对抛光材料去除的影响规律，实现了确定性材料去除的抛光工艺控制。在此基础上，构建了面向整体曲面面型改善的确定性抛光面型误差补偿模型，通过面型误差测量，余量分布提取以及进给速度优化，提高了确定性抛光整体材料去除的可控性。
本文的研究工作主要包括以下几部分：
1.传统的定点抛光材料去除模型只能预测定点抛光时的去除深度。针对以上不足，本文建立了移动抛光的材料去除廓形模型。将材料去除指数定义为单位轨迹长度的材料去除深度，根据Preston经验方程，接触区域内一点的材料去除指数与该点的接触压强，相对线速度和进给速度相关。根据抛光工具与工件曲面的局部接触和姿态的描述，对抛光区域的接触压强和相对线速度分布进行建模。在此基础上，利用对材料去除指数在接触区域内沿着抛光轨迹积分的方法，分别对球形工具和盘形工具的材料去除廓形进行理论推导，建立了综合考虑抛光力、抛光轴转速、抛光姿态、进给速度、工件/工具几何性质的材料去除廓形模型。实验结果显示，该模型具有很高的预测精度。
2.抛光轨迹在抛光工艺规划中占有重要地位，为了增加轨迹覆盖曲面的均匀性，各种复杂抛光轨迹应运而生。根据材料去除指数的定义和去除廓形计算的基本方法，进一步研究复杂轨迹抛光的材料去除，从理论上揭示了抛光轨迹相关参数对材料去除廓形的影响。本文着重研究抛光轨迹大曲率过渡区域和螺旋线轨迹的材料去除廓形的理论，对实际抛光过程中的轨迹规划和工艺参数设计有重要的指导意义。其它复杂抛光轨迹均可运用类似方法对其材料去除进行建模与分析，优化抛光工艺。
3.抛光时的抛光垫和抛光液性质对抛光质量和材料去除的影响是至关重要的，而利用Preston方程所建立的模型忽略了这两个因素。本文从抛光垫、自由磨粒和工件三者间的微观接触出发，分析了单个磨粒的材料去除体积、磨粒发生材料去除的临界条件和有效磨粒的数目，建立了基于自由磨粒材料去除机理的材料去除指数模型。对该材料去除指数在抛光区域进行积分，求得抛光后的材料去除深度和廓形。该预测模型不但包含了抛光工艺参数对材料去除的影响，并且创新性地将抛光垫拓扑参数、磨粒大小、磨粒物理性质和抛光液体积比浓度等参数对材料去除的影响也考虑在内，增强了抛光过程的可预测性。实验结果显示，所述的理论结果与实验结果有较好的一致性，验证了该理论的有效性。根据所述模型和实验结果，自由磨粒抛光时的材料去除深度正比于(法向抛光力)0.65、(抛光液体积比浓度)2/3和自由磨粒平均半径，反比于(抛光垫微观高度分布标准差)0.3。
4.根据柔顺性抛光原理，设计和研制了基于气缸和压力比例阀的抛光力伺服控制系统。理论上推导了力控制系统的系统传递函数，并利用PRBS信号和最小二乘系统辨识方法对系统的模型进行辨识。分别利用PID、RST和ISMC控制算法对工具系统的抛光力输出进行控制。基于极点配置理论，设计了RST三相控制器；由于气动系统辨识模型的不确定性，RST控制器能够维持系统的稳定，但是稳态误差较大。在RST控制器设计的基础上，推导了非最小相位系统的ISMC控制器的控制结构，并利用根轨迹法确定了滑模控制参数。通过ISMC对工具系统的控制实验发现，由于加入了滑模趋近分量，ISMC能够抑制系统模型的不确定性和系统的扰动，稳态误差和超调量小。当抛光系统加入抛光轴转速后，ISMC控制器显示出比PID控制更好的抑制干扰的能力，输入力波动范围小，从而验证了利用ISMC控制器的有效性和实用性。
5.通过对廓形测量方向与抛光轨迹方向关系的分析，建立了沿着廓形测量方向的局部材料去除模型。将沿着测量方向的各驻留点上的局部材料去除深度在廓形控制点上进行局部累加，提出了二维和三维整体材料去除的离散化矩阵预测模型。该模型中，整体材料去除深度矩阵等于材料去除影响矩阵与进给速度矩阵的乘积。本文利用非负最小二乘最优化方法，对模型中的进给速度进行求解，构建了确定性抛光面型误差补偿的策略。以此为指导，对一平面零件进行误差补偿抛光，经过一次抛光后的表面面型Wv值从抛光前的1.8989μm提高到了0.4251μm，而Wt值从28.4896μm提高到了9.3354μm。对一球面零件进行均匀性抛光，每个抛光周期的总体材料去除深度为3μm，经过抛光后的表面粗糙度从135.8nm提高到14.7nm，且整个曲面的表面质量具有较好的一致性。
Optical part surface is the key component for an optical system. In recent years, withthe improving requirement of the optical properties, the requests of accuracy, relativeaperture, weight reduction, production efficiency and cost of the optical part surfaces aregetting higher and higher. Nowadays, the production cost and efficiency of the optical partsurface has become one of the signs for a country reflecting the modern level of the opticalindustry and even the whole manufacturing ability. The polishing process is usually used asthe final step to fabricate the optical part surface, which has large effect on the accuracy andduration of the surface. This dissertation is supported by the Chinese National Program onKey Basic Research Project (973Program)[Grant number2011CB706702],“BasicResearch on the Fabrication of the Optical Freeform Surface—Project2: Physical Analysisand Accuracy Control Strategy for the Generation of the Optical Freeform Surface”. Thisstudy focused on the technical architecture of the deterministic polishing process and the keytechnologies of the deterministic polishing. The prediction of the local and glocal polishedprofiles, polishing force control, control strategy of the surface form, et al. are explored indepth both theoretically and experimentally.
The core issues in the deterministic polishing process include modelling of the unitmaterial removal function, dwell time calculation, planning of the polishing path, predictionof the material removal depth, et al. For the fundamental principle of the polishing processcorrecting the surface form, the global removal depth is modelled as the convolution of theunit material removal function and the dwell time. It is assumed that the polishing process istime-invariant, which means the unit material removal function does not vary with themovement of the polishing tool. For the spherical optical surfaces and aspherical surfaceswith low gradient, this assumption is reasonable. However, for the high gradient asphericalsurfaces and the freeform surfaces, the surface curvature, tool path and polishing posture arevarying during the polishing, thus the unit material removal function is time variant. Thisdissertation addresses the problem of material removal in the polishing process. The effectsof some polishing conditions upon the material removal are analyzed, including not only theprocess parameters, which refer to the normal force, angular spindle velocity and angularfeed rate, but also the abrasive grain size, polishing slurry properties, topographicalparameters of the sub-aperture pad, as well as tool path curvature. Based on the analysis, anew control strategy to improve the surface form accuracy is proposed. The main researchcontents in this dissertation are as follows:
(1) A new material removal model for the velocity-dwell-mode polishing process isproposed. The material removal index, which means the material removal depth at unitlength of polishing path, is defined and derived based on the Preston equation. Thedistribution of material remval index in the polishing contact is affected by the contact pressure and relative sliding velocity between the tool and the surface. The material removalprofile during the polishing can be obtained by intergrating the material removal index alongthe polishing path in the contact region. Based on the fundamental theory, the materialremoval models of the spherical tool and sub-aperture tool are proposed in this dissertation.Accordign to the models and experimental results, the polishing material removal is affectedby the polishing normal force, spindle velocity, polishing attitude, geometrical/physicalpropertities of the tool/surface, et al.
(2) The effect of polishing path on the polishing material removal is modelled andanalyzed. Especially, the polished profiles as the sub-aperture polishes at the path corner andalong the trochoidal path were modeled by integrating the material removal index along thetool path in the contact region. The effects of the parameters of the polishing path on thepolished profile were considered in different cases. The theory in this paper is potentiallyuseful for the planning of tool path and processing parameters in the deterministic polishingprocess.
(3) A novel mathematical model of the material removal profile for the free abrasivepolishing (FAP) is developed, which successfully explains the effects of the properties ofabrasive grains/slurry and topographical parameters of sub-aperture pad on the profile. Byanalyzing the interations among polishing pad, abrasive grain and workpiece surface in themicro level, the material removal for a single abrasive and the critical condition of materialremoval for the abrasives are modelled. On the basis of above, the material removal indexconsidering the removal mechanism of FAP is proposed to facilitate more accurate polishing.According to the simulation and experimental results, the removal depth is proportional to(normal polishing force)0.65,(volume concentration)2/3and the radius of abrasive grains, andinversely proportional to (deviation of pad asperity height)0.3.
(4) The control system of polishing force composed by the pneumatic cylinder andpressure propotional valve is developed. The software platform of the control system isbased on the NI PCIe6321DAQ card and the real time module. The transfer function of thepolishing force control system is derived theoretically. Pseudo Random Binary Sequence(PRBS) signals were applied at the control input, and the output date was caputured. Withthe input and output data, the model of the system is identification by least square method.An integral sliding mode controller (ISMC) is proposed to control the polishing force inorder to improve the performace. Root locus is used to find an appropriate value of ki, theintergral control coefficient, to ensure the closed-loop stability. By compared with the RSTand PID controller, the robustness of the ISMC to the disturbance and model uncertainties isensured by the implementation of an integral sliding control action. The force controlexperiments show that the proposed algorithm is effective in improving the control accuracyof the polishing force.
(5) By extending the theory of removal profile orthogonal to the tool path, the localpolished profile, which is defined as the polished depth in the measuring direction, ismodeled. According to the model, the local polished profile is determined not only by the process parameters, the tool attitude, the geometrical/mechanical properties of workpieceand tool, but also the measuring angle. The linear algebraic expression of the2D/3D globalpolished profile is derived by convoluting the local polished depth at each dwell point of thepolishing process. In this model, the matrix of the global polished depth equals to theproduct of influential matrix and feed rate matrix. On the basis of above, the errorcompensation method for the polishing process is developed as an optimition problem of thefeed velocity in polishing. The nonnegative least square method is used to solve this problem.The polishing experiments composed by polihsing of a flat surface and a spherical surfaceare conducted to verify the proposed model. After polishing for a cycle, the Wvvalue of theflat surface was improved from1.8989μm to0.4251, and Wtvalue from28.4896μm to9.3351μm respectively. The spherical surface was polished with the uniform materialremoval control method. For every polishing cycle, the global polished depth is3μm. Afterpolishing, the roughness of the spherical surface reduced from135.8nm to14.7nm, and theuniformity of the whole surface was guaranteed.
目前，确定性抛光工艺研究的核心问题包括单位时间材料去除函数建模、驻留时间计算、抛光路径规划和材料去除量预测等。经典的抛光理论为，整体材料去除量等于单位时间材料去除函数与抛光轨迹上各抛光点的驻留时间的卷积。上述经典理论将抛光过程假设成一个线性移不变系统，即假设抛光工具单位时间内的材料去除量不随位置的移动而变化。对于低陡度的球面和非球面而言，运用该理论能够对抛光过程有很好的控制。但是，对于形状比较复杂的非球面和自由曲面，曲面曲率、抛光轨迹和抛光姿态的复杂性使得该理论的适用性降低。本文研究了抛光工艺参数（包括抛光力、主轴转速、进给速度、抛光姿态角等）、抛光工艺条件（包括抛光液和抛光垫条件等）、抛光轨迹、曲面几何/物理特性（曲率和弹性模量等）等一系列因素对抛光材料去除影响，构建了综合考虑以上各因素影响的抛光表面去除模型，揭示了工艺参数和条件变化对抛光材料去除的影响规律，实现了确定性材料去除的抛光工艺控制。在此基础上，构建了面向整体曲面面型改善的确定性抛光面型误差补偿模型，通过面型误差测量，余量分布提取以及进给速度优化，提高了确定性抛光整体材料去除的可控性。
本文的研究工作主要包括以下几部分：
1.传统的定点抛光材料去除模型只能预测定点抛光时的去除深度。针对以上不足，本文建立了移动抛光的材料去除廓形模型。将材料去除指数定义为单位轨迹长度的材料去除深度，根据Preston经验方程，接触区域内一点的材料去除指数与该点的接触压强，相对线速度和进给速度相关。根据抛光工具与工件曲面的局部接触和姿态的描述，对抛光区域的接触压强和相对线速度分布进行建模。在此基础上，利用对材料去除指数在接触区域内沿着抛光轨迹积分的方法，分别对球形工具和盘形工具的材料去除廓形进行理论推导，建立了综合考虑抛光力、抛光轴转速、抛光姿态、进给速度、工件/工具几何性质的材料去除廓形模型。实验结果显示，该模型具有很高的预测精度。
2.抛光轨迹在抛光工艺规划中占有重要地位，为了增加轨迹覆盖曲面的均匀性，各种复杂抛光轨迹应运而生。根据材料去除指数的定义和去除廓形计算的基本方法，进一步研究复杂轨迹抛光的材料去除，从理论上揭示了抛光轨迹相关参数对材料去除廓形的影响。本文着重研究抛光轨迹大曲率过渡区域和螺旋线轨迹的材料去除廓形的理论，对实际抛光过程中的轨迹规划和工艺参数设计有重要的指导意义。其它复杂抛光轨迹均可运用类似方法对其材料去除进行建模与分析，优化抛光工艺。
3.抛光时的抛光垫和抛光液性质对抛光质量和材料去除的影响是至关重要的，而利用Preston方程所建立的模型忽略了这两个因素。本文从抛光垫、自由磨粒和工件三者间的微观接触出发，分析了单个磨粒的材料去除体积、磨粒发生材料去除的临界条件和有效磨粒的数目，建立了基于自由磨粒材料去除机理的材料去除指数模型。对该材料去除指数在抛光区域进行积分，求得抛光后的材料去除深度和廓形。该预测模型不但包含了抛光工艺参数对材料去除的影响，并且创新性地将抛光垫拓扑参数、磨粒大小、磨粒物理性质和抛光液体积比浓度等参数对材料去除的影响也考虑在内，增强了抛光过程的可预测性。实验结果显示，所述的理论结果与实验结果有较好的一致性，验证了该理论的有效性。根据所述模型和实验结果，自由磨粒抛光时的材料去除深度正比于(法向抛光力)0.65、(抛光液体积比浓度)2/3和自由磨粒平均半径，反比于(抛光垫微观高度分布标准差)0.3。
4.根据柔顺性抛光原理，设计和研制了基于气缸和压力比例阀的抛光力伺服控制系统。理论上推导了力控制系统的系统传递函数，并利用PRBS信号和最小二乘系统辨识方法对系统的模型进行辨识。分别利用PID、RST和ISMC控制算法对工具系统的抛光力输出进行控制。基于极点配置理论，设计了RST三相控制器；由于气动系统辨识模型的不确定性，RST控制器能够维持系统的稳定，但是稳态误差较大。在RST控制器设计的基础上，推导了非最小相位系统的ISMC控制器的控制结构，并利用根轨迹法确定了滑模控制参数。通过ISMC对工具系统的控制实验发现，由于加入了滑模趋近分量，ISMC能够抑制系统模型的不确定性和系统的扰动，稳态误差和超调量小。当抛光系统加入抛光轴转速后，ISMC控制器显示出比PID控制更好的抑制干扰的能力，输入力波动范围小，从而验证了利用ISMC控制器的有效性和实用性。
5.通过对廓形测量方向与抛光轨迹方向关系的分析，建立了沿着廓形测量方向的局部材料去除模型。将沿着测量方向的各驻留点上的局部材料去除深度在廓形控制点上进行局部累加，提出了二维和三维整体材料去除的离散化矩阵预测模型。该模型中，整体材料去除深度矩阵等于材料去除影响矩阵与进给速度矩阵的乘积。本文利用非负最小二乘最优化方法，对模型中的进给速度进行求解，构建了确定性抛光面型误差补偿的策略。以此为指导，对一平面零件进行误差补偿抛光，经过一次抛光后的表面面型Wv值从抛光前的1.8989μm提高到了0.4251μm，而Wt值从28.4896μm提高到了9.3354μm。对一球面零件进行均匀性抛光，每个抛光周期的总体材料去除深度为3μm，经过抛光后的表面粗糙度从135.8nm提高到14.7nm，且整个曲面的表面质量具有较好的一致性。
Optical part surface is the key component for an optical system. In recent years, withthe improving requirement of the optical properties, the requests of accuracy, relativeaperture, weight reduction, production efficiency and cost of the optical part surfaces aregetting higher and higher. Nowadays, the production cost and efficiency of the optical partsurface has become one of the signs for a country reflecting the modern level of the opticalindustry and even the whole manufacturing ability. The polishing process is usually used asthe final step to fabricate the optical part surface, which has large effect on the accuracy andduration of the surface. This dissertation is supported by the Chinese National Program onKey Basic Research Project (973Program)[Grant number2011CB706702],“BasicResearch on the Fabrication of the Optical Freeform Surface—Project2: Physical Analysisand Accuracy Control Strategy for the Generation of the Optical Freeform Surface”. Thisstudy focused on the technical architecture of the deterministic polishing process and the keytechnologies of the deterministic polishing. The prediction of the local and glocal polishedprofiles, polishing force control, control strategy of the surface form, et al. are explored indepth both theoretically and experimentally.
The core issues in the deterministic polishing process include modelling of the unitmaterial removal function, dwell time calculation, planning of the polishing path, predictionof the material removal depth, et al. For the fundamental principle of the polishing processcorrecting the surface form, the global removal depth is modelled as the convolution of theunit material removal function and the dwell time. It is assumed that the polishing process istime-invariant, which means the unit material removal function does not vary with themovement of the polishing tool. For the spherical optical surfaces and aspherical surfaceswith low gradient, this assumption is reasonable. However, for the high gradient asphericalsurfaces and the freeform surfaces, the surface curvature, tool path and polishing posture arevarying during the polishing, thus the unit material removal function is time variant. Thisdissertation addresses the problem of material removal in the polishing process. The effectsof some polishing conditions upon the material removal are analyzed, including not only theprocess parameters, which refer to the normal force, angular spindle velocity and angularfeed rate, but also the abrasive grain size, polishing slurry properties, topographicalparameters of the sub-aperture pad, as well as tool path curvature. Based on the analysis, anew control strategy to improve the surface form accuracy is proposed. The main researchcontents in this dissertation are as follows:
(1) A new material removal model for the velocity-dwell-mode polishing process isproposed. The material removal index, which means the material removal depth at unitlength of polishing path, is defined and derived based on the Preston equation. Thedistribution of material remval index in the polishing contact is affected by the contact pressure and relative sliding velocity between the tool and the surface. The material removalprofile during the polishing can be obtained by intergrating the material removal index alongthe polishing path in the contact region. Based on the fundamental theory, the materialremoval models of the spherical tool and sub-aperture tool are proposed in this dissertation.Accordign to the models and experimental results, the polishing material removal is affectedby the polishing normal force, spindle velocity, polishing attitude, geometrical/physicalpropertities of the tool/surface, et al.
(2) The effect of polishing path on the polishing material removal is modelled andanalyzed. Especially, the polished profiles as the sub-aperture polishes at the path corner andalong the trochoidal path were modeled by integrating the material removal index along thetool path in the contact region. The effects of the parameters of the polishing path on thepolished profile were considered in different cases. The theory in this paper is potentiallyuseful for the planning of tool path and processing parameters in the deterministic polishingprocess.
(3) A novel mathematical model of the material removal profile for the free abrasivepolishing (FAP) is developed, which successfully explains the effects of the properties ofabrasive grains/slurry and topographical parameters of sub-aperture pad on the profile. Byanalyzing the interations among polishing pad, abrasive grain and workpiece surface in themicro level, the material removal for a single abrasive and the critical condition of materialremoval for the abrasives are modelled. On the basis of above, the material removal indexconsidering the removal mechanism of FAP is proposed to facilitate more accurate polishing.According to the simulation and experimental results, the removal depth is proportional to(normal polishing force)0.65,(volume concentration)2/3and the radius of abrasive grains, andinversely proportional to (deviation of pad asperity height)0.3.
(4) The control system of polishing force composed by the pneumatic cylinder andpressure propotional valve is developed. The software platform of the control system isbased on the NI PCIe6321DAQ card and the real time module. The transfer function of thepolishing force control system is derived theoretically. Pseudo Random Binary Sequence(PRBS) signals were applied at the control input, and the output date was caputured. Withthe input and output data, the model of the system is identification by least square method.An integral sliding mode controller (ISMC) is proposed to control the polishing force inorder to improve the performace. Root locus is used to find an appropriate value of ki, theintergral control coefficient, to ensure the closed-loop stability. By compared with the RSTand PID controller, the robustness of the ISMC to the disturbance and model uncertainties isensured by the implementation of an integral sliding control action. The force controlexperiments show that the proposed algorithm is effective in improving the control accuracyof the polishing force.
(5) By extending the theory of removal profile orthogonal to the tool path, the localpolished profile, which is defined as the polished depth in the measuring direction, ismodeled. According to the model, the local polished profile is determined not only by the process parameters, the tool attitude, the geometrical/mechanical properties of workpieceand tool, but also the measuring angle. The linear algebraic expression of the2D/3D globalpolished profile is derived by convoluting the local polished depth at each dwell point of thepolishing process. In this model, the matrix of the global polished depth equals to theproduct of influential matrix and feed rate matrix. On the basis of above, the errorcompensation method for the polishing process is developed as an optimition problem of thefeed velocity in polishing. The nonnegative least square method is used to solve this problem.The polishing experiments composed by polihsing of a flat surface and a spherical surfaceare conducted to verify the proposed model. After polishing for a cycle, the Wvvalue of theflat surface was improved from1.8989μm to0.4251, and Wtvalue from28.4896μm to9.3351μm respectively. The spherical surface was polished with the uniform materialremoval control method. For every polishing cycle, the global polished depth is3μm. Afterpolishing, the roughness of the spherical surface reduced from135.8nm to14.7nm, and theuniformity of the whole surface was guaranteed.
引文
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