摘要
位移测量是最基本、最普遍的测量。从宇航飞行卫星探测到超大规模集成电路生产，从物质结构研究到纳米技术的探索，无一不需要高精度位移测量。在精密测量中用得最广泛的是以光栅为代表的栅式位移传感器，但在我国，高端位移传感器目前绝大部分依赖进口，不仅价格高，而且进口某些高精度传感器常常受限，这些不利因素直接制约着我国制造业和国防工业的发展。
作者所在课题组自1995年开始从事精密位移传感器及相关技术的研究，研制出了一种具有自主知识产权全新测量原理的精密位移传感器——时栅。根据“用时间作为空间位移测量基准”的思想，使得时钟脉冲具有唯一的空间当量，从而实现“采用时间测量位移”的新方法。目前场式圆时栅的检定精度已达到±0.8″，直线时栅的检定精度达到±0.5μm/m，而分辨力分别达到0.1″和0.1μm。
由于时栅是采用时间测量空间，按时间均分的等时采样，因此时栅属于绝对式静态位移传感器。而在动态测量和实际生产应用中，有许多场合需要按空间均分的等空间间隔采样，这就需要将时栅的原始绝对角位移转化为空间均分的增量式脉冲信号。在三项国家自然基金的支持下，本课题引入预测理论实现时栅位移传感器的增量式动态测量和普通栅式位移传感器的细分方法设计，并由此进行相关预测理论、算法和实验的研究，主要研究和创新成果如下：
（1）从空间和时间的角度研究两种位移测量模型。由此展开对普通栅式位移传感器和时栅位移传感器测量模型和测量原理的深入分析和讨论。从深层次上阐述栅式位移传感器和时栅位移传感器在测量原理及物理意义上的相关性和差异性。
（2）提出将预测理论用于精密位移测量。从数学角度对经典的预测理论进行分析和对比，寻求用于精密位移测量的最佳预测理论，从理论上证明了预测理论用于精密位移测量的有效性和可行性。
（3）提出了按时间序列生成连续空间位置信号的新全闭环控制方案。结合预测理论和时栅的测量原理，从数学和运动学的角度，采用时序理论对预测回归模型进行辨识、建模、检验和优化，其中还包含对预测回归模型的定阶和参数估计。为了获得最佳预测效果，提出了自适应时序预测模型，实现了预测模型参数的时变性。其目的是通过时栅测量得到的原始绝对式位移信号实时、有效地预测下个测量周期的位移值，并将此预测增量值通过脉宽调制方式转换成连续空间均分的预测脉冲信号，完成原本静态测量式时栅用于全闭环数控转台的动态位置反馈，解决了时栅动态位置的反馈误差问题和数控系统接口兼容性问题。
（4）研究预测技术用于滚齿机床传动链误差检测。将时栅用作滚齿机床传动链传动误差的检测元件，采用预测技术设计了一套用于传动链高速端时栅和低速端时栅测量的软同步技术。解决了测量过程中出现的时-时、时-空和空-空不同步问题，实现了传动误差的同步精确测量。
（5）提出了按空间序列生成连续反映空间位置信号的时间序列新方案。采用时空对偶方式研究利用时间序列理论构建空间序列理论，建立状态时间模型和空间序列模型，以完成对时间量的精确预测。从而提出一种基于时空转换技术的栅式位移传感器信号细分新技术。这种细分方法突破传统细分方法的思维限制，从原理上创新，是一种与栅式位移传感器输出信号的正交性和等幅性无关的新技术。
（6）研究基于测量基准时空变换技术具有空间意义的时间脉冲产生机理。利用时间脉冲实现对时栅空间脉冲的实时细分和对普通栅式位移传感器脉冲信号的细分方法研究，以及相关的细分误差实时修正技术的研究。
（7）研究了动态和静态标定实验中的误差补偿。提出了基于多位置测头法和傅立叶级数谐波修正法的静态测量误差修正和基于离散标准量插入的动态测量误差修正法，并通过数学和运动学角度建立起一系列基于预测理论的精密位移测量的误差修正理论与方法。
在上述研究工作中，最成功的理论与技术成果有：
（1）建立了用于精密位移测量自适应回归预测理论。结合精密位移测量和预测理论重点研究了应用最多、最广的多元统计回归、时间序列和支持向量机，并在系统分析了三种预测方式的优点和缺点的基础上，提出了自适应回归预测理论，为预测理论用于精密位移测量提供有力的理论依据。
（2）研究了动态和静态标定实验中的误差补偿技术。采用多位置测头误差分离与傅立叶级数谐波修正技术，把时栅传感器静态误差修正到1″之内，实现了无需高精度机械制造完成高精度测量的目的。在动态预测测量过程中，采用标准量插入法在预测当前测量值后减去上次预测误差，目的是保证预测精度，消除预测误差累计。
（3）实现基于预测理论的精密角位移动态测量实验研究及相关产品研制。①实现了基于时间序列理论的时栅全闭环数控转台研制。采用自适应时序预测模型实时、有效地预测下个测量周期的位移值，并通过嵌入式技术将此预测增量值转换成连续空间均分的预测脉冲信号，实现时栅用于全闭环数控转台的动态位置反馈，目前预测精度能达到±2″。②实现了基于预测技术的时栅测量传动误差的软同步技术。采用预测技术设计了一套用于传动链高速端时栅和低速端时栅测量的软同步技术。解决了测量过程中出现的时-时、时-空和空-空不同步问题，实现了时栅对传动误差的同步精确测量。③实现了基于空间序列理论的传统栅式位移传感器新型细分卡研制。提出空间序列的概念，并采用时空对偶方式研究利用空间序列理论建立空间状态时间模型，实现对栅距运动时间量的精确预测，可实现最大细分倍数400，角位移细分精度优于信号周期的±4％，细分误差达到±1.3″。
Displacement measurement is one of the most basic and common physical measurement.Precision displacement measurement cover many areas such as space navigation, satellite sounding,super-large-scale integration production, material structure study and nanometer technology. Gratingtyped displacement sensors, such as optical grating sensors, are most widely applied in precisiondisplacement measurement. However, such high precision displacement sensors mainly depend onimport in our country. These high precision displacement sensors cost too much and are usuallyrestricted to import. These negative factors directly influence the development of our nationalmanufacturing industry and national defense industry.
Our research team have been involving in precision displacement study since1995, and a novelprecision displacement sensors named time grating sensors with proprietary intellectual propertyrights have been invented. Time grating displacement sensors adopt time quantity as measurementstandard to measurement spatial value. Because time pulses in time grating sensors have a specialspatial equivalent, a new measurement principle, special measurement using time as measurementstandard, can be realized. The measurement precision of existing field typed circular time gratingsensors can reach±0.8″, and that of linear time grating sensors can reach±0.5μm/m. And theresolution of time grating sensors can reach0.1″and0.1μm, respectively.
Time grating sensors adopt time to measure spatial value, i.e. time grating sensors measurespatial value every equal time interval. Therefore, time grating sensors are mainly used in staticalmeasurement state. However, sampling equal spatial interval is needed in many applications. Inorder to adopt time grating sensors as position detectors in dynamic measurement for suchapplications, original absolute time grating signal should be transformed into incremental pulseswith equal spatial value. Funded by three National Natural Science Foundation of China (NSFC),this study adopts forecast theory to realize the dynamic measurement of time grating sensors and thesignal subdivision of traditional grating typed displacement sensors. Furthermore, forecast theory,algorithm and related experiments are also studied. The main research content and innovations are asfollows:
（1）Two displacement measurement models are analyzed from spatial quantity and timequantity. After analyzing the measurement models and principles of grating typed displacementsensors and time grating sensors deeply, the correlation and variation in physical meaning betweenthem are discussed.
（2）Forecast theory is presented for precision displacement measurement. Some classicalforecast theory are analyzed and discussed with mathematics, and the optimal forecast theory isobtained, which validates the effectiveness and availability of forecast theory for precisiondisplacement measurement.
（3）The principle of continuous spatial position signal generation is presented in time order fora new principle of full closed loop control. Combing with forecast theory and the measurementprinciples of time grating, the principles of identification, checkout, optimization of forecastregression model are presented, as well as the model ordering and parameter estimation withmathematics and kinematics. In order to obtain optimal forecast results, adaptive forecast model withtime order is presented. As a result, the time variation of model parameters can be realized. Thepurpose is to forecast the displacement of time grating sensors during the next measurement periodbased on series of past measured absolute displacement values. The forecast incrementaldisplacement values are transformed into continuous equal spatial forecast pulses with pulse widthmodulation (PWM). Therefore, time grating sensors applied for statical measurement can be adoptedas position dectors for dynamic position feedback of a full closed loop numerical control system.And the problems of feedback errors and compatible interface of numerical control system can besolved.
（4）Forecast technology are pesented for transmission error measurement of gear-hobbingmachine. Time grating sensors are adopted as position detectors for transmission error measurementof gear-hobbing machine. A soft synchronization technology is presented based on forecasttechnology for time grating sensors equipped in the high speed terminal and low speed terminal oftransmission chain. In this way, the problems of time-time asynchronization, time-spaceasynchronization and space-space asynchronization can be solved, and precision transmission errorscan be measured synchronously.
（5）The principles of time series signal generation which represent spatial position values arepresented in spatial order. Spatial series theory is established using time series with a time-spacecorresponding method. State-time model and space series model will be established to forecastprecise time quantity. Therefore, a novel subdivision technology based on time space transformationmethod for grating typed sensors is proposed. This subdivision technology broaden out thetranditional think model of subdivision technology and bring some innovations for the signalsubdivision of grating typed sensors. This method has nothing to do with the orthogonal and equalamplitude of signal outputted from grating typed sensors.
（6）The principle of time pulse generation with spatial value based on time-space transformation of measurement standard is presented. The real time subdivision technology arepresented for spatial pulses of time gratings and grating typed displacement sensors, and real timeerror correction technology are also presented for calibrating subdivision errors.
（7） Error correction methods are presented for statical measurement and dynamicmeasurement. The methods of multi-probe and harmonic correction based on fourier series arepresented for statical measurement, and discrete standard quantity interpolation method is presentedfor dynamic error correction. Therefore, a set of error correction theory and methods are establishedbased on mathematics and kinematics.
Among the mentioned research works, the most successful achievements of theory andtechnology are as follows:
（1）A self-adaptive regression forecast theory is established for precision displacementmeasurement. Combining with precision displacement measurement and forecast theory, themulti-statistical regression model, time theory and support vector regression machine which areadopted most widely are discussed. And a self-adaptive regression forecast theory is presented basedon the deep analysis on the binifits and drawbacks of three classic forecast theory, which provides apowerful theoretical foundation for precision displacement measurement.
（2）Error correction technology are established for statical and dynamic measurement. Staticalerrors of the time gating sensor are calibricated within1″with a multi-probe method and a harmoniccorrection technique based on fourier series, which realizes precision measurement without precisionmechnical fabrication. In addition, the next future displacement value is forecasted with standardquantity interpolation method. In order to ensure forecasting precision and eliminate accumulativeforecasting errors, the forecasting errors during the last measurement period are deducted from theoriginal forecasting value.
（3）The experiment study of dynamic precision angular displacement measurement arerealized, and related products are designed.①A full closed loop numerical control rotary tablewith time grating sensors as position detectors based on time series theory is designed. A self-adaptive regression forecast model is adopted to forcast the effective displacement value of nextmeasurement period in real time. And the forecasting incremental angular displacement istransformed into continuous forecasting pulses with equal spatial value using embedded technology.Therefore, time grating sensors can be used as a position feedback sensors for full closed loopnumerical control rotary table. The forecasting precision can reach±2″.②A soft synchronizationtechnology is realized based on forecast technology for transmission error measurement oftransmission chain. A soft synchronization technology is presented based on forecast technology for time grating sensors equipped in the high speed terminal and low speed terminal of transmissionchain. In this way, the problems of time-time asynchronization, time-space asynchronization andspace-space asynchronization can be solved, and precision transmission errors can be measuredsynchronously, and the synchronized transmission errors can be measured with time grating sensors.③The novel signal subdivision devices of traditional grating typed displacement sensors aredesigned based on space series theory. Space series theory is presented. And state-time model andspace series model will be established to forecast precise moving time quantity of line-space ofgrating using time series with a time-space corresponding method. And maximum subdivisionnumber can achieve400, and the subdivision accuracy of angular displacement measurement ishigher than±4％of the original signal period. And forecasting errors are limited within±1.3″.
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