Estimating the Resolution of Nanopositioning Systems from Frequency Domain Data

Transcription

1 01 IEEE International Conerence on Robotics and Automation RiverCentre, Saint Paul, Minnesota, USA May 14-18, 01 Estimating the Resolution o Nanopositioning Systems rom Frequency Domain Data Andrew J. Fleming School o Electrical Eng. and Computer Science The University o Newcastle NSW 300, Australia andrew.leming@newcastle.edu.au Abstract Mechanical and electrical noise in nanopositioning systems is unavoidable and dictates the maximum positioning resolution. The proper speciication o resolution is critical or deining the smallest possible dimensions in a manuacturing processes or the smallest measurable eatures in an imaging application. This article deines a standard or the reporting o resolution and demonstrates how this parameter can be measured and predicted rom requency domain data. I. INTRODUCTION A nanopositioning system is an electromechanical device or maneuvering an object in three or more degrees o reedom. A typical nanopositioner consists o base, a moving platorm, actuators, position sensors, and a control system [1]. These devices are commonly used in scanning probe microscopes [] to develop displacements o between one and one-hundred micrometers with a resolution on the order o one nanometer or less. Other applications o nanopositioning systems include nanoabrication [3], data storage [4], cell surgery, beam pointing, and precision optical alignment. A key perormance speciication o a nanopositioner, or indeed many other controlled systems, is the resolution. The resolution is essentially the amount o random variation that remains at the output, even when the system is at rest. The resolution is critical or deining the smallest possible dimensions in a manuacturing processes or the smallest measurable eatures in an imaging application. Although the resolution is a key perormance criteria in many applications, there is unortunately no strict deinition available in the literature. There are also no published industrial standards or the measurement or reporting o positioning resolution. Predictably, this has led to a wide variety o ragmented techniques used throughout both academia and industry. As a result, it is extremely diicult to compare the perormance o dierent control strategies or commercial products. The most reliable method or the measurement o resolution is to utilize an auxiliary sensor that is not involved in the eedback loop. However, this requires a sensor with less additive noise and greater bandwidth than the displacement to be measured. Due to these strict requirements, the direct measurement approach is oten impractical or impossible. Instead, the closed-loop positioning noise is usually predicted rom measurements o known noise sources such as the sensor noise. In industrial and commercial applications, the methods used to measure and report closed-loop resolution are widely varied. Unortunately, many o these techniques do not provide complete inormation and may even be misleading. For example, the RMS noise and resolution is commonly reported without mention o the closed-loop or measurement bandwidth. In the academic literature, the practices or reporting noise and resolution also vary. The most common approach is to predict the closed-loop noise rom measurements o the sensor noise [4], [5]. However, this approach can underestimate the true noise since the inluence o the high-voltage ampliier is neglected. In the hard drive industry, the standard perormance metric or resolution is the track pitch and the standard deviation o the measurement [6], [7]. However, the main sources o error in a disk drive are due to aeroelastic eects and track eccentricities which are not present in a nanopositioning system. In this article, the resolution is deined as the minimum distance between two points that can be uniquely identiied. Although the ocus is on nanopositioning applications, the background theory and measurement techniques are applicable to any control system where resolution is a actor. II. RESOLUTION AND NOISE When a nanopositioner has settled to a commanded location, a small amount o random motion remains due to sensor noise, ampliier noise, and external disturbances. The residual random motion means that two adjacent commanded locations may actually overlap, which can cause manuacturing aults or imaging arteacts. To avoid these eventualities, it is critical to know the minimum distance between two adjacent points that can be uniquely identiied. Since the noise sources that contribute to random position errors can have a potentially large dispersion, it is impractically conservative to speciy a resolution where adjacent points never overlap. Instead, it is preerable to state the probability that the actual position is within a certain error bound. Consider the example o random positioning errors plotted in Figure 1(a). Observe that the peak-to-peak amplitude o random motion is bounded by and δ y, however this range is occasionally exceeded. I the random position variation is assumed to be Gaussian distributed, the probability density unctions o three adjacent points, spaced by, are plotted in Figure 1(b). In this example, is equal to ±3σ x or 6σ x which means that 99.7% o the samples all within the range speciied by. Restated, there is a 0.3% chance that the position is exceeding and straying into a neighboring area, this probability is shaded in grey. For many applications, a 99.7% probability that the position alls within = 6σ x is an appropriate deinition or the resolution. To be precise, this deinition should be reerred to as the 6σ-resolution and speciies the minimum spacing between two adjacent points that do not overlap 99.7% o the time. Although there is no international standard or the measurement or reporting o resolution in a positioning system, /1/$ IEEE 4786

2 y S ns () 1/ δ y A s nc (a) Two dimensional random motion x Fig.. Power spectral density o a baseband sensor (solid line) and a modulated sensor (dashed line). A s is the noise density and nc is the 1/ noise corner requency. 0.4 x approximate the power spectrum o physical processes such as licker noise in resistors and current noise in transistor junctions. The power spectral density o a baseband sensor S ns () can be written σ m x 3σ (b) The probability density unctions o three adjacent points on the x-axis Fig. 1. The random motion o a two-dimensional nanopositioner. The random motion in the x and y-axis is bounded by and δ y. In the x- axis, the standard deviation and mean are σ x and m x respectively. The shaded areas represent the probability o the position being outside the range speciied by. the ISO 575 Standard on Accuracy (Trueness and Precision) o Measurement Methods and Results [8] deines precision as the standard deviation (RMS Value) o a measurement. Thus, the 6σ-resolution is equivalent to six times the ISO deinition or precision. III. SOURCES OF NANOPOSITIONING NOISE The three major sources o noise in a nanopositioning systems are the sensor noise, external noise, and the ampliier output voltage noise. The power spectral density o each source is derived in the ollowing to allow the estimation o closed-loop position noise. A. Sensor noise The noise characteristics o a position sensor depend mainly on the physical method used or detection. Although there are a vast range o sensing techniques available, or the purpose o noise analysis, these can be grouped into two categories: baseband sensors, and modulated sensors. Baseband sensors involve a direct measurement o position rom a physical variable that is sensitive to displacement. Examples include resistive strain sensors, piezoelectric strain sensors and optical triangulation sensors [9], [10]. The power spectral density o a baseband sensor is typically described by the sum o white noise and 1/ noise, where 1/ noise has a power spectral density that is inversely proportional to requency [11], [1]. 1/ noise is used to x 4787 S ns () = A s nc +A s, (1) where A s is the mid-band density, expressed in units /Hz and nc is the 1/ corner requency. This unction is plotted in Figure. In contrast to baseband sensors, modulated sensors use a high-requency excitation to detect position. Examples include capacitive sensors, eddy-current sensors, and Linear Variable Displacement Transormers (LVDTs) [9]. Although these sensors require a demodulation process that inevitably adds noise, this disadvantage is usually outweighed by the removal o 1/ noise. The power spectral density S ns () o a modulated sensor can generally be approximated by S ns () = A s, () where A s is the noise density, expressed in units /Hz. The power spectral density o a modulated sensor is compared to a baseband sensor in Figure. In nanopositioning applications, modulated sensors can be preerable to baseband sensors as they do not exhibit 1/ noise. Thus, in the ollowing, the ocus is on modulated sensors with an approximately constant noise spectral density. B. External noise The external orce noise exerted on a nanopositioner is highly dependent on the ambient environmental conditions and can not be generalized. Typically, the power spectral density consists o broad spectrum background vibration with a number o narrow band spikes at harmonic requencies o the mains power source and any local rotating machinery. Although the external orce noise must be measured in-situ, or the purposes o simulation, it is useul to assume a white power spectral density A w, that is S w () = A w. (3) Clearly a white power spectral density does not provide an accurate estimate o externally induced position noise. However, it does illustrate the response o the control system to noise rom this source. That is, it reveals whether the control system attenuates or ampliies external noise and over what requency regions. A constant power spectral density o A w is used or this purpose in the ollowing sections.

3 V o Va w V in C(s) V o R V n I n C(s) V R V o R r C(s) P(s) d n s V R1 Fig. 5. C. A single axis eedback control loop with a plant P and controller (a) Voltage ampliier (b) Equivalent noise circuit Fig. 3. The simpliied schematic o a voltage ampliier and its equivalent noise circuit. The noise sources V n and I n represent the equivalent input voltage noise and current noise o the ampliier. V R1 and V R are the thermal noise o the eedback resistors. S Vn () A V 1/ nc (a) Input voltage noise V n S Vo () A V β nc V (b) Output voltage noise V o Fig. 4. Power spectral density o the input and output voltage noise o a high-voltage ampliier. nc is the noise corner requency. C. Ampliier noise The high-voltage ampliier is a key component o any piezoelectric actuated system. It ampliies the control signal rom a ew volts up to the hundreds o volts required to obtain ull stroke rom the actuator. For the purpose o noise analysis, the simpliied schematic diagram o a noninverting ampliier in shown in Figure 3(a). This model is suicient to represent the characteristics o interest. The opamp represents the dierential gain stage and output stage o the ampliier. As high-voltage ampliiers are oten stabilized by a dominant pole, the open-loop dynamics can be approximated by a high-gain integrator C(s) = α ol /s, where α ol is the open-loop DC gain. With this approximation, the closed-loop transer unction is where β is the eedback gain gain and bandwidth are: V o = 1 α ol β V in β s+α ol β, (4) R +. The closed-loop DC DC Gain = 1 β = R +, (5) Bandwidth = α ol β = α ol R + rad/s. The input voltage noise o a practical high-voltage ampliier can be approximated by the sum o a white noise process and 1/ noise, that is, the power spectral density can be written S Vn () = A V nc +A V. (6) where nc is the noise corner requency and A V is the mid-band density, expressed in V /Hz. The power spectral density o the ampliier output voltage is then approximately S Vo () = A V β ( nc +1 ) V +V, (7) where V = α ol β/π is the closed-loop bandwidth o the ampliier (in Hz) and 1/β is the DC gain. The power spectral density o the output voltage noise is plotted in Figure 4(b). IV. CLOSED-LOOP POSITION NOISE A. Noise sensitivity unctions To derive the closed-loop position noise, the response o the closed-loop system to each noise source must be considered. In particular, we need to speciy the location where each source enters the eedback loop. The ampliier noise V o appears at the plant input. In contrast, the external noise w acts at the plant output, and the sensor noise n s disturbs the measurement. A single axis eedback loop with additive noise sources is illustrated in Figure 5. For the sake o simplicity, the voltage ampliier is considered to be part o the controller. The transer unction rom the ampliier voltage noise V o to the position d is the input sensitivity unction, V o (s) = P(s) 1+C(s)P(s). (8) Likewise, the transer unction rom the external noise w to the position d is the sensitivity unction, w(s) = 1 1+C(s)P(s). (9) Finally, the transer unction rom the sensor noise n s to the position d is the negated complementary sensitivity unction, n s (s) = C(s)P(s) 1+C(s)P(s) (10) B. Closed-loop position noise spectral density With knowledge o the sensitivity unctions and the noise power spectral densities, the power spectral density o the position noise due to each source can be derived. The position noise power spectral density due to the ampliier output voltage noise S dvo () is S dvo () = A ( ) V nc β +1 V d(jπ) +V V o (jπ). (11) 4788

4 Similarly, the position noise power spectral density due to the external orce noise S dw () is d(jπ) S dw () = A w w(jπ). (1) Finally, the position noise power spectral density due to the sensor noise S dns () is S dns () = A s. d(jπ) n s (jπ). (13) The total position noise power spectral density S d () is the sum o the three individual sources, S d () = S dvo ()+S dw ()+S dns (). (14) The position noise variance can also be ound rom the Wiener Khinchin relation E [ d ] = 0 S d () d, (15) which is best evaluated numerically. I the noise is assumed to be Gaussian distributed, the 6σ-resolution o the nanopositioner is 6σ-resolution = 6 E[d ] (16) C. Closed-loop noise approximations with integral control I a simple integral controller is used, C(s) = α/s, the transer unctions rom the ampliier and external noise to displacement can be approximated by V o (s) = sp(0) s+αp(0), w(s) = s s+αp(0), (17) where P(0) is the DC-Gain o the plant. Likewise, the complimentary sensitivity unction can be approximated by n s (s) = αp(0) s+αp(0). (18) With the above approximations o the sensitivity unctions, the closed-loop position noise power spectral density can be derived. From (11) and (17) the position noise density due to the ampliier voltage noise S dvo () is ) V S dvo () A VP(0) β ( nc +1 + V +cl, (19) where cl = αp(0) π is the closed-loop bandwidth. As illustrated in Figure 6(a), the position noise due to the ampliier has a bandpass characteristic with a mid-band density o A V P(0) /β. From (1) and (18) the position noise density due to the external noise S dw () is S dw () A w +cl, (0) which has a high-pass characteristic as illustrated in Figure 6(b) with a corner requency equal to the closed-loop bandwidth. The closed-loop position noise due to the sensor S dns () can be derived rom (13) and (18), and is cl S dns () A s +cl, (1) S dvo () A V P(0) β 4789 nc cl V S dw () (a) The position noise power spectral (b) The position noise power spectral density due to ampliier voltage noise density due to external noise S dw () S dvo () S dns () A s cl A w cl (c) The position noise power spectral density due to sensor noise S dns () Fig. 6. The position noise power spectral density due to the ampliier voltage noise (a), external disturbance (b) and sensor noise (c). which has a low-pass characteristic with a density o A s and a corner requency equal to the closed-loop bandwidth, as illustrated in Figure 6(c). Although the expression or variance (15) is generally evaluated numerically, in some cases it is straightorward and useul to derive analytic expressions. One such case is the position noise variance due to sensor noise (E [ d ] due to n s ) when integral control is applied. As demonstrated in the orthcoming examples, sensor noise is typically the dominant noise process in a eedback controlled nanopositioning system. As a result, other noise sources can sometimes by neglected. As the sensor noise density is approximately constant and the sensitivity unction (18) is approximately irst-order, the resulting position noise can be determined rom E[d ] due to n s = A s 1.57cl, () The corresponding 6σ-resolution is 6σ-resolution = 6 A s 1.57cl. (3) This expression can be used to determine the minimum resolution o a nanopositioning system given only the sensor noise density and closed-loop bandwidth. It can also be rearranged to reveal the maximum closed-loop bandwidth achievable given the sensor noise density and the required resolution. ( ) 6σ-resolution maximum bandwidth (Hz) = (4) A s For example, consider a nanopositioner with integral eedback control and a capacitive sensor with a noise density o 30 pm/ Hz. The maximum bandwidth with a resolution o

5 Parameter Value Closed-loop bandwidth cl 50 Hz Controller gain α 314 Ampliier bandwidth V khz Ampliier gain 1/β 50 Ampliier input voltage noise A V 100 nv/ Hz Ampliier output voltage noise 5 µv/ Hz Ampliier noise corner requency nc 100 Hz Sensor noise A s 0 pm/ Hz Position range 100 µm Sensitivity P(0) 500 nm/v Resonance requency ω r π 10 3 r/s Damping ratio ζ r 0.05 TABLE I SPECIFICATIONS OF AN EXAMPLE NANOPOSITIONING SYSTEM 10 Sd SdVo Sdns (Hz) (a) 50 Hz Closed-loop bandwidth 10 Sd 1 nm is maximum bandwidth = ( = 11 Hz ) Sdns SdVo V. SIMULATION EXAMPLES A. Integral controller noise simulation In this section an example nanopositioner is considered with a range o 100 µm at 00 V and a resonance requency o 1 khz. The system model is P(s) = 500 nm V s +ω r ζ r s+ωr, (5) where ω r = π1000 and ζ r = The system includes a capacitive position sensor and voltage ampliier with the ollowing speciications. The capacitive position sensor has a noise density o 0 pm/ Hz. The voltage ampliier has a gain o 0, a bandwidth o khz, an input voltage noise density o 100 nv/ Hz, and a noise corner requency o 100 Hz. The eedback controller in this example is a simple integral controller with compensation or the sensitivity o the plant, that is 1 α C(s) = 500 nm/v s, (6) whereαis the gain o the controller and also the approximate bandwidth (in rad/s) o the closed-loop system. All o the system parameters are summarized in Table I. The total density o the position noise can now be calculated rom equation (14). The total spectral density and its components are plotted in Figure 7(a). Clearly, the sensor noise is the dominant noise process. This is the case in most nanopositioning systems with closed-loop position eedback. The variance o the position noise can be determined by solving the integral or variance numerically, The result is σ = E [ d ] = 0 ω r σ = 0.4 nm, or σ = 0.49 nm, which implies a 6σ-resolution o.9 nm. S d () d (7) (Hz) (b) 1 Hz Closed-loop bandwidth Fig. 7. The spectral density o the total position noise S d () and its two components, the ampliier output voltage noise S dvo () and sensor noise S dns () (all in pm/ Hz). In systems with lower closed-loop bandwidth, the 1/ noise o the ampliier can become dominant. For example, i the closed-loop bandwidth o the previous example is reduced to 1 Hz, the new power spectral density, plotted Figure 7(b), diers signiicantly. The resulting variance and standard deviation are σ = nm, or σ = 0.30 nm, which implies a 6σ-resolution o 1.8 nm. Not a signiicant reduction considering that the closed-loop bandwidth has been reduced to % o its previous value. More generally, the resolution can be plotted against a range o closed-loop bandwidths to reveal the trend. In Figure 8, the 6σ-resolution is plotted against a range o closed-loop bandwidths rom 100 mhz to 60 Hz. The curve has a minima o 1.8 nm at 0.4 Hz. Below this requency, ampliier noise is the major contributor, while at higher requencies, sensor noise is more signiicant. B. Noise simulation with inverse model controller In the previous example, the integral controller does not permit a closed-loop bandwidth greater than 100 Hz. Many other model-based controllers can achieve much better perormance. One simple controller that demonstrates the noise characteristics o a model based controller is the combination o an integrator and notch ilter, or direct inverse controller. 4790

6 Resolution (nm) Integral Control Inverse Control Closed-loop Bandwidth (Hz) Fig. 8. Resolution o the example nanopositioning system with integral control (solid line) and inverse control (dashed). The transer unction is an integrator combined with an inverse model o the plant, C(s) = α s 1 s +ω r ζ r s+ωr. (8) 500 nm/v ω r The resulting loop-gain C(s)P(s) is an integrator, so stability is guaranteed and the closed-loop bandwidth is α rad/s. With such a controller it is now possible to examine the noise perormance o eedback systems with wide bandwidth. Aside rom improved bandwidth, the inverse controller also eliminates the resonance peak in the sensor induced noise spectrum. This beneit also occurs with controllers designed to damp the resonance peak [13]. Ater ollowing the same procedure described in the previous section, the resulting variance or a closed-loop bandwidth o 500 Hz is σ = 0.37 nm, or σ = 0.61 nm, which implies a 6σ-resolution o 3.7 nm. This is not signiicantly greater than the 50 Hz controller bandwidth in the previous example, which resulted in a.9 nm resolution. When the closed-loop bandwidth o the inverse controller is reduced to 50 Hz, the resolution is.1 nm, which is slightly better than the previous example. The dierence is due to the absence o the resonance peak in the sensor induced noise. The resolution o the inverse controller is plotted or a wide range o bandwidths in Figure 8. The minimum resolution is 1.8 nm at 1 Hz. Ater approximately 100 Hz, the position noise is due predominantly to the sensor-noise which is proportional to the square-root o closed-loop bandwidth, as described in equation (3). C. Feedback versus eedorward control A commonly discussed advantage o eedorward control systems is the absence o sensor induced noise. However, this view does not take into account the presence o1/ ampliier noise that can result in signiicant peak-to-peak amplitude. It is not necessary to derive equations or the noise perormance o eedorward systems as this is a special case o the eedback examples already discussed. The positioning noise o a eedorward control system is equivalent to a eedback control system when C(s) = 0 or equivalently, when the closed-loop bandwidth is zero. Thus, the plots o resolution versus bandwidth in Figure 8 are also valid or eedorward control. The eedorward controller resolution is the DC resolution o these plots, which in both cases is.60 nm. It is interesting to note that both the integral and inverse controller can achieve slightly less positioning noise than a eedorward control system when the closed-loop bandwidth is very low. This is because the ampliier noise density is greater than the sensor noise density at low requencies. In the examples considered, the optimal noise perormance could be achieved with a eedback controller o around 1-Hz bandwidth. To increase the positioning bandwidth, a eedorward input would be required [14]. VI. CONCLUSIONS In this article, a requency domain approach was used to quantiy noise sources and predict the closed-loop resolution o a nanopositioning system. The oremost noise sources were identiied as the ampliier voltage noise and the displacement sensor noise. Simulation examples demonstrate that the minimum positioning noise usually occurs in openloop or with very low closed-loop bandwidth. This implies that combined eedback and eedorward control can achieve the best positioning resolution. ACKNOWLEDGEMENTS This research was supported by an Australian Research Council Discovery Project (DP ). REFERENCES [1] S. Devasia, E. Eletheriou, and S. O. R. Moheimani, A survey o control issues in nanopositioning, IEEE Transactions on Control Systems Technology, vol. 15, no. 5, pp , September 007. [] S. M. Salapaka and M. V. 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