5.1 Ultra-precision Machining: Nanometric Displacement Sensors E. Brinksmeier, Universität Bremen, Bremen, Germany Sensors used for ultra-precision machines include position and velocity sensors, which are mostly part of a control loop, and for some special applications accelera- tion sensors for active vibration control, and thermal sensors for thermal error compensation [1]. An essential component of each positional feedback loop in a machine tool is the displacement measurement system for detecting the actual position of moving machine parts. It is the performance of such measurement systems that limits the accuracy of machine tools and hence directly affects the quality of the ma- chined part. In ultra-precision machining, accuracies of interest are of one part in 10 6 or even 10 7 with a measurement range of up to 1 m. Adequate resolutions are in the range of nanometers [2]. Such demands can only be fulfilled by laser inter- ferometers, optical scales, and linear voltage differential transducers, which will be covered in detail. Although capacitive and piezoresistive sensors are able to of- fer resolutions in the sub-nanometer range, their measurement range is rather small, which limits their application to measuring macroscopic workpieces [1]. 5.1.1 Optical Scales Optical scales are based on a repetitive pattern of reflective or transmissive material. On scanning along the scale, a periodic signal is obtained which is a direct measure for the traveled path. Conventional optical encoders, also called photoelectric encoders, are based on counting Moiré lines by means of a light source and a photodiode. Beside optical scales, the patterns might also be made of magnetic or conductive material. The principle is then based on magnetic, capacitive, or inductive measurement [2, 3]. The optical set-up for an incremental photoelectric device is shown in Fig- ure 5.1-1. The scale is divided into alternating optically opaque and transparent sectors. A mask with the same grating period and a phase shift of 908 with re- 343 5 Developments in Manufacturing and Their Influence on Sensors Sensors in Manufacturing. Edited by H.K. Tönshoff, I. Inasaki Copyright © 2001 Wiley-VCH Verlag GmbH ISBNs: 3-527-29558-5 (Hardcover); 3-527-60002-7 (Electronic) spect to each other is located between the light source and the scale. The images of the sectors are projected on to the scale by the collimated light from the light source. As the scale moves, four photodiodes on the other side of the mask re- ceive light pulses which pass through the transparent slits of the mask and the scale. Thus, four square sine signals with a relative phase shift of 908 are gener- ated. The determination of the displacement x from these signals is depicted in Figure 5.1-2. Two of the signals with a phase shift of 1808 are subtracted so that two signals, S 1 and S 2 , which are symmetrical with respect to the x-axis and have a phase shift of 908, are generated. These signals are triggered and finally a resolution of ¼ of the grating period g is achieved. By counting the intersections of both functions with the x-axis, the traveled path can be determined. A further increase in resolu- tion can be achieved by means of electronic interpolation. In Figure 5.1-2, the in- tensities of S 1 and S 2 can be expressed as sine and cosine functions depending on the traveled path: S 1 $ sin 2px g S 2 $ cos 2px g : Dividing S 1 by S 2 , the path x can be derived by extracting the arctangents from the coefficient: 5 Developments in Manufacturing and Their Influence on Sensors344 Fig. 5.1-1 Photoelectric scale. Source: Ernst [3] x g 2p arctan S 1 S 2 : One serious drawback of incremental encoders is that the information about the position is lost when the system is switched off. This disadvantage can be avoided with absolute encoders. Today, most absolute linear encoders are based on the photoelectric principle combined with a coded scale. The linear encoder gives an output which is a binary number that is directly proportional to the distance of the encoder from a fixed point. The simplest code is the binary code consisting of alternating opaque and transparent blocks, as shown in Figure 5.1-3. The number of strips determines the number of binary digits and the total number of incre- ments that can be detected. For example, using 16 bits (and thus strips), the num- ber of resolvable positions is 65 536. The position is measured with a linear array of photodiodes mounted above each strip. One problem that arises with binary coded scales of the above-mentioned type is that reading errors of the photodiodes result in large errors in position. Therefore, another code is used where only a change in one digit occurs when moving from one position to the next. This code is termed the gray code. Transformation from the gray code to the binary code and vice versa can be done using simple logical operations. However, a general drawback of both the gray code and the binary code is the huge number of strips required for high resolutions and large measurement ranges which requires a lot of space within a machine tool. Absolute coding with only a few strips can be 5.1 Ultra-precision Machining: Nanometric Displacement Sensors 345 Fig. 5.1-2 Interpolation techniques for optical scales. Source: Ernst [3] achieved with the random code. Here, the strip is coded in a random fashion over the entire length so that a sector of a certain critical length never repeats. If a sen- sor array with equal critical length scans the coded strip, the information obtained from the array is unique and can be compared with the code stored in the mem- ory. Thus, the absolute position is obtained. Theoretically, one strip is sufficient for absolute encoding, available random coded encoders have three strips for in- creased resolution [3]. Conventional optical photometric encoders are limited in resolution owing to diffraction effects. On the other hand, so-called interferential scales are based on diffraction [5]. The optical set-up of the interferential encoders (by courtesy of Hei- denhain Co.) consist of a movable reflective scale (grating), a transmissive grating with the same grating period as the scale, a light-emitting diode (LED) as light source, a condenser and three photodiodes for scanning the signals. The scale grating consists of 4 lm wide, rectangular plateaus of gold with a height of 0.2 lm, which are located at equal intervals of 8 lm on a gold-plated substrate. 5 Developments in Manufacturing and Their Influence on Sensors346 Fig. 5.1-3 Binary and gray code. Source: Sinclair [4] Fig. 5.1-4 Principle of interfero- metric scale. Source: Spies [5] Figure 5.1-4 shows the operating scheme of the Heidenhain linear encoder. On passing through the transmissive grating, a first-order diffraction pattern is created, with three partial waves of nearly the same intensity. The grating is de- signed in such a way that the zero-order wave is subject to a phase shift w, with respect to the first-order diffraction waves [5]. At the reflective scale grating the light is diffracted so that the first-order diffrac- tion has the highest intensity while the zero-order diffraction is negligible. Be- sides being diffracted, the three waves are also subject to another phase shift X at the scale grating. Passing through the transmissive phase grating again, three new interfering waves with different phases are generated. These waves are colli- mated by the condenser at three different spots to be analyzed by the photo- diodes. The phase shift of the three new waves is given as by D 1 2X 2w D 2 2X D 3 2X À 2w : However, the phase shift X is not a constant like w but depends on the move- ment of the scale grating. When the scale is moving by a distance x, the phase shift is X 2px g where g denotes the grating period. Hence the phase shifts D 1 , D 2 , and D 3 can be expressed as a variable depending on the constant w and the displacement x. The three photodiodes in the focal plane of the condenser generate three periodic sig- nals with twice the period of the grating and having a phase shift of 2w each: D 1 2 2px g 2w D 2 2 2px g D 3 2 2px g À2w : Common to all different interference scales is that the interference phenomenon is used to generate sinusoidal signals which are a function of linear displacement. These signals can further be processed electronically to increase the resolution. Generally, the displacement x can be expressed as [2, 3, 5]: x i kf g where i is the number of counts, g the grating period, f the optical multiplication factor, and k the electronic interpolation factor. With electronic interpolation, an 5.1 Ultra-precision Machining: Nanometric Displacement Sensors 347 increase in resolution by up to the factor f = 400 is possible, so that nanometric resolution can be achieved. Since scales are made from glass or steel, they are subject to thermal expansion when the ambient temperature changes. To relate linear displacement measure- ment to the internationally agreed reference temperature of 20 8C, the thermal coefficient of expansion a and the temperature of the scale must be known. While the temperature can be measured with an accuracy of better than 0.05 8C, careful attention has to be paid to local temperature gradients [2]. 5.1.2 Laser Interferometers The basis of nearly all laser interferometric displacement sensors is the Michelson interferometer and its variations. Modern interferometers for measuring displace- ments in the nanometer range are based on the Doppler effect, in which two waves of slightly different frequency are made to interfere. As depicted in Figure 5.1-5, a heterodyne He-Ne laser emits two mutually perpendicular polarized beams with the frequencies f 1 and f 2 with a frequency difference Df=f 2 –f 1 = 20 MHz, which serves as a reference signal [2, 6]. The reference signal can be interpreted as a single wave with a beat frequency of Df = f 2 –f 1 . This reference signal is detected in the photodiode P1, after a certain amount of the wave emitted by the laser has been split in the half-reflecting mirror HRM. Next, the wave passing through the half-reflecting mirror is split in a polariz- ing beamsplitter PBS. The wave with frequency f 2 is led directly to the photodiode P2, whereas the wave f 1 is reflected by the moving reflector MR. Both waves are again made to interfere in the photodiode P2, and this signal serves as measure- ment channel. If the moving reflector does not move, this P2 likewise detects an amplitude modulation of 20 MHz. However, a moving reflector with velocity v causes a frequency alteration of Df 1 to the wave f 1 so that the resulting beat fre- 5 Developments in Manufacturing and Their Influence on Sensors348 Fig. 5.1-5 Laser interferometer. Source: Sommargren [6] quency detected at P2 is (f 1 ± Df 1 )–f 2 . This wave contains information about the op- tical path change of the moving reflector in the form of a phase change between the reference signal and the measurement signal. The phase difference between the two signals is measured every cycle and any phase changes are digitally accumu- lated. Figure 5.1-6 depicts how the phase shift and the optical path change are gen- erated. Both the sinusoidal reference and measurement signal are each converted into square waves. The reference signal is further integrated to produce a triangular wave. At each positive transition of the measurement square wave, the triangular wave is sampled and digitized. Its value is then compared with the previous reading and any difference is added to the memory. Furthermore, the polarity of the refer- ence square wave and the number of transitions of the triangular wave is monitored for correction of the phase measurement. An optical path change of k, which is equivalent to a movement of k/2 of the moving reflector, causes the measurement signal to shift by one cycle with respect to the reference signal. A complete cycle is equivalent to a change of 2 m –1 levels of digitization, where m is the number of available bits in the analog to digital converter. Thus, the optical path change can be resolved with a resolution of k/(2 m –1). This measurement technique achieves both a high resolution and a high rate of optical path change [6]. The accuracy of laser interferometers is generally limited by the stability of the wavelength, which is the measurement reference and is a function of the refrac- tive index of air. This index varies with temperature, humidity, pressure, CO 2 con- tent, and contamination by other gases. Beside these errors, which can be com- pensated when the properties of the air are monitored permanently, another source of errors is arising from non-linearities in the interpolation process. Mis- alignment and imperfections of optics increase the systematic errors of laser inter- ferometers [2]. 5.1 Ultra-precision Machining: Nanometric Displacement Sensors 349 Fig. 5.1-6 Interpolation technique for laser interferometer. Source: Sommargren [6] To overcome the problems arising from the changing properties of the air, the ultimate solution is to encapsulate the laser interferometer in vacuum; however, this requires huge efforts regarding the design and implementation in a machine structure and is only used for very high demanding applications, eg, calibration for unencapsulated interferometers. In general, it can be stated that for ultimate accuracy in dimensional metrology, as the first choice vacuum laser interferometers or laser interferometers under ex- tremely stable environmental conditions with very sensitive and calibrated tem- perature sensors have to be considered. For high accuracy in the range 10 –7 –10 –6 / 10–100 nm, scales and laser interferometers are more or less equivalent. The same applies to accuracies in the range 10 –6 –10 –5 but scales are mostly preferred as they are easier to integrate into a machine tool or coordinate measurement ma- chine [2]. A recent application of the heterodyne laser interferometer is the laser ball bar (LBB) for machine tool metrology, whose principle is depicted in Figure 5.1-7 [7]. It is designed to measure volumetric errors rapidly by directly measuring the spa- tial coordinates of the tool. The LBB consists of a displacement-measuring laser interferometer whose axis is aligned between the centers of two precision spheres as depicted in Figure 5.1-7. Since the LBB is a linear displacement measurement device, it can be used to locate the actual position of a tool relative to a base frame when combined with the technique of triangulation. The tool point posi- 5 Developments in Manufacturing and Their Influence on Sensors350 Fig. 5.1-7 Laser ball bar. Source: Ziegert and Mize [7], Srinivasa and Ziegert [8] tioning error can be derived from the difference between the measured position and the position indicated by the machine. Sequential triangulation leads to a sys- tematic error mapping of the machine tool [8]. Conventionally, error mapping of the volumetric errors of a machine tool, which is the first step in error compensation of a machine tool, is done by mea- suring parametric error functions for each axis. These parametric functions are combined through rigid body modeling to obtain the volumetric errors of the ma- chine. However, this procedure is difficult and time consuming, eg, a three-axis machine requires a total of 21 error measurements (three translatory and three ro- tational for each axis and three orthogonality errors for the axes). A useful alterna- tive is to measure directly the spatial displacements of the tool and thus obtain the error map of the machine tool. Here, the LBB serves as a convenient measure- ment device. Srinivasa and Ziegert [8] applied LBB triangulation to determine the spindle thermal drift of a two-axis turning center with respect to the machine frame. Thermal errors in machine tools arise from thermal deformations of the machine tools caused by a complex temperature field within the machine. Existing heat sources within the machine tool are the leadscrew bearings and nut, axis drive motors, spindle bearings and drive, heat caused by any friction within the ma- chine, and the cutting process itself. Thermal errors in machine tools, in particu- lar spindle thermal drift, are considered to be the main contributors to overall ma- chine accuracy and are thought to be the main reason for dimensional and geo- metric errors in workpieces produced on machine tools [8]. 5.1.3 Photoelectric Transducers Both laser interferometers and scales require photoelectric transducers in order to transform the optical signals into electrical information. The most commonly used photoelectric transducers are photodiodes, which are based on the photovolt- aic principle. In photovoltaic action, a voltage is generated when light is incident on the photosensitive material. Since photovoltaic cells have an efficiency of only 10–20%, attention has to be paid to ensure that the dissipated thermal power does not cause unacceptable thermal errors to ultra-precision machines. The types of available photodiodes include monolithic and hybrid devices which can produce digital or analog output. Their spectral sensitivity may range from infrared to ultraviolet. Monolithic analog output photodiodes are commonly used in conjunc- tion with LEDs in optical encoders and proximity sensors, and in conjunction with lasers in interferometers. Two-dimensional arrays of photodiodes form the imaging device of most video cameras [9]. For optical position sensors, monolithic photodiodes are most often used which provide an analog voltage output. When photodiodes are used to measure the in- tensity of incident light, eg, from laser interferometers or optical scales, a band- pass filter, eg, a 630–635 nm filter for the He-Ne laser, is used for excluding ambi- ent light [9]. 5.1 Ultra-precision Machining: Nanometric Displacement Sensors 351 5.1.4 Inductive Sensors Inductive sensors are, unlike scales or interferometers, analog position sensors. The most commonly used sensor for measuring displacements in the millimeters and centimeters range is the linear variable differential transducer (LVDT) [1, 10]. The LVDT, depicted in Figure 5.1-8, consists of three coils. An AC voltage, typi- cally 5 kHz, is applied to the primary coil, which is inductively coupled to the sec- ondary coils. For small displacements of the core about the central position, the amplitude of the output voltage across the counterwired secondary windings will be linearly proportional to the displacement. The direction of the core is deter- mined by analyzing the phase of the signal with respect to the reference phase. As the name suggests, the output from the phase-sensitive detector will be linear- ly proportional to the distance, as resulting non-linearities are subtracted due to the use of two counterwired coils [10]. Short-stroke LVDTs have useful linear range of movement of a few millimeters only. The long-stroke type can provide a displacement range of as large as ± 60 mm [4]. 5.1.5 Autocollimators Autocollimators are devices for the precise measurement of small horizontal and/ or vertical angular displacements. They are usually not an integral part of a machine sensor system, but are used as inspection and calibration devices. In environmentally controlled laboratories, autocollimators provide a fast and simple method to measure flatness of a machined surface and straightness and orthogon- ality of moving axes. Both manual and electronic autocollimators are available [9]. 5 Developments in Manufacturing and Their Influence on Sensors352 Fig. 5.1-8 Linear variable differential transducer (LVDT). Source: Smith and Chetwynd [10]