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Abstract

This article develops a long-stroke nano-positioning stage and discusses the impacts of sensor layouts on the positioning performance. The stage consists of a piezoelectric-transducer stage and a motor stage. First, we obtain the transfer functions of the two stages by experiments and design robust loop-shaping controllers to improve their performance. Second, we integrate the two stages and designed two sensor layouts, a local sensor layout using encoders and a global sensor layout using laser interferometers, to achieve precision positioning for large travels. Finally, we implement the designed controllers and sensor layouts for experimental verification. Based on the results, the proposed combined stage is deemed effective in accomplishing nano-positioning for long displacements. In addition, the local sensor layout can achieve high precision but with global misalignments, while the global sensor layout can eliminate the global misalignments but suffer large sensor noises. Therefore, we further constructed a modified sensor layout that can combine the merits and achieve precision positioning with a root-mean-square error of 5 nm and a misalignment error of 16 nm for a 10-cm travel.

Keywords

Lead zirconate titanate precision positioning robust control loop shaping stage sensor

Introduction

Precision positioning control is increasingly important as technologies advance. For example, Gan et al.¹ used a linear switched reluctance motor with H-infinity loop-shaping design to reach a steady-state error (SSE) of 3.5 µm. Park² constructed a 3 degree-of-freedom hybrid stage by three linear motors and four voice coil motors and applied proportional–integral control to acquire travels up to 300 mm and a minimum resolution of 10 nm. Chen et al.³ approximated electromagnetic nonlinearity by cubic polynomials and proposed adaptive robust control to achieve tracking performance for industrial applications. Yao et al.⁴ applied a robust integral with adaptive control to a hydraulic rotary actuator and verified the high-accuracy tracking performance by experiment. Piezoelectric transducer (lead zirconate titanate (PZT)) actuators are frequently considered in precision positioning because of high-driven forces and fast responses. However, PZT nonlinearities, such as hysteresis and creep, might degrade system performance. Therefore, several nonlinear models were proposed, including the Bouc-Wen model,^5,6 the Preisach model,^7,8 and the Prandtl-Ishlinskii inverse model.^9,10 Liu et al.¹¹ identified the hysteresis and creep dynamics of piezoelectric actuators and applied model-inversion control to get an error of less than 4.32% at 600 Hz. Many advanced controls have been applied to improve the performance of PZT stages. For example, Kenton and Leang¹² applied repetitive control to achieve a root-mean-square error (RMSE) of 0.6% at 100 Hz. Helfrich et al.¹³ combined iterative learning control and robust control to obtain an RMSE of less than 40 nm on three axes. Wang et al.¹⁴ applied robust loop-shaping techniques to PZT to achieve an RMSE of 5.4 nm for a travel of 1000 nm. Other robust control techniques can also be considered. For instance, Sun et al.¹⁵ applied finite frequency H_∞ control and constrained adaptive robust control technology¹⁶ to vehicle active suspension systems. Yao et al.¹⁷ proposed output feedback nonlinear robust control with an extended state observer to a hydraulic system.

Because the working ranges of PZTs are limited, many studies tried to extend their travel ranges. For instance, Chu and Fan¹⁸ designed mechanisms for a PZT stage to achieve a maximum stroke of 10 mm with an error of less than 50 nm. Peng et al.¹⁹ applied adaptive sliding mode control to PZT and electromagnetic actuators and reached a positioning field of 1 mm² with an RMSE of 30 nm. Wu et al.²⁰ built a one-axial combined stage and obtained an RMSE of 5 nm with a working range of 50 mm. This article extends these ideas to integrate a two-axial PZT stage with a motor stage for achieving nano-positioning over long travels and further investigates the influences of displacement measurements on stage performance. The frequently used measuring instruments include linear variable differential transformers, encoders, capacitive sensors, charge sensors, and laser interferometers.²¹ The former four sensors detect local displacements while the last one provides global positions. However, there has been no profound discussion on how different sensor layouts might influence the positioning performance of precision stages. Therefore, in this article, we apply encoders and interferometers to build a local sensor layout and a global sensor layout, respectively, and discuss the pros and cons of these layouts based on experimental results.

This article is organized as follows: section “System description and identification” describes the PZT and motor stages and derives their dynamic models by experiments; section “Controller design and implementation” designs and implements robust loop-shaping controllers for the stages and demonstrates the experimental responses; section “Stage integration and control by different sensor layouts” integrates the two stages for nano-positioning during long travels and develops two sensor layouts for performance comparison. The results indicate that the local sensor layout can achieve better precision but suffers global misalignments, while the global sensor layout can eliminate the global misalignments but endures larger noises. Therefore, we further build a modified sensor layout to combine the advantages of the aforementioned two layouts and to achieve an RMSE of about 5 nm and a misalignment error of about 16 nm for a 10-cm travel. Finally, we draw conclusions in section “Concluding remarks.”

System description and identification

The combined stage consists of a PZT stage and a motor stage, as shown in Figure 1(a). The PZT stage was driven by two PZTs to manipulate 15 µm × 15 µm stokes on two axes and was equipped with linear encoders with a resolution of 1.22 nm on both axes for measuring local displacements. The motor stage was steered by two stepper motors to provide 100 mm × 100 mm displacements on two axes and was equipped with two encoders with a resolution of 0.1 µm for measurement and control. Therefore, we can integrate the PZT (see Figure 1(b)) and the motor stages (see Figure 1(c)) to accomplish precision positioning over long travels. In addition, we applied laser interferometers (see Figure 1(d)) to measure the global displacements of the combined stage. The specifications of the equipment are illustrated in Table 1. The control structure is shown in Figure 1(e), where we applied the data-acquisition (DAQ) cards to receive signals from the PZT stage and stepper motors. The control inputs were calculated by the computer with the software of Visual Studio 2010 C++ and transmitted to the stages through the DAQ cards.

Figure 1.

The proposed system: (a) the combined stage, (b) the PZT stage, (c) the motor stage, (d) the laser interferometer, and (e) the control structure.

Table 1.

Specifications of the experimental instruments.

The PZT stage
Linear encoder resolution	1.22 nm/count
Travel	15 µm
Voltage range	−20 to 120 V
Resonance frequency	50 kHz
Stiffness	60 N/µm
The motor stage
Linear encoder resolution	0.1 µm/count
Travel	100 mm
Maximum speed	8 mm/s
Mass	16 kg
Maximum load	40 kgf
The laser interferometer
Power	281 µW
Split frequency	3.76 MHz
Sensor resolution	0.155 nm/count

Figure 2 illustrates the hardware layouts of the combined stage. We used two DAQ boards, PCIe-6323 and PCI-6221, to control the stepper motors by pulse-width modulation signals and to feedback the encoder signals. In addition, we applied DAQ PCI-6229 to transmit the control signals and encoder signals of the PZT stage. Furthermore, the PZT control signals were amplified 10 times by the Amplifier E-663 to drive the PZTs. Finally, the global stage positions were recorded by an axis laser board N1231B.

Figure 2.

Hardware layouts of the combined stage.

Identification of the PZT stage

The models of the PZT stage were derived by experiments. First, we applied a swept sinusoidal signal, $v_{P} = v_{P, x} = v_{P, y}$ , with a frequency range of 0.01–100 Hz and a decreasing magnitude of 5–0 V, to drive the PZT. Second, we measured the corresponding PZT displacements $x_{P}$ and $y_{P}$ by the encoders and $x_{I}$ and $y_{I}$ by the interferometers. Finally, we applied the numerical subspace state space system identification (N4SID) methods²² to derive the PZT models. Considering system variation, we repeated the experiments five times for both axes and defined the transfer functions as

\begin{matrix} G_{P}^{x_{i}} (s) = T_{v_{P, x} \to x_{P}} (s), G_{P}^{y_{i}} (s) = T_{v_{P, y} \to y_{P}} (s) \\ {\bar{G}}_{P}^{x_{i}} (s) = T_{v_{P, x} \to x_{I}} (s), {\bar{G}}_{P}^{y_{i}} (s) = T_{v_{P, y} \to y_{I}} (s) \end{matrix}

for $i = 1, \dots, 5$ . The Bode plots of these transfer functions are illustrated in Figure 3.

Figure 3.

Bode plots of the PZT stage: (a) $G_{P}^{x_{i}} (s)$ and ${\bar{G}}_{P}^{x_{i}} (s)$ and (b) bode $G_{P}^{y_{i}} (s)$ and ${\bar{G}}_{P}^{y_{i}} (s)$ .

Identification of the motor stage

Similarly, for the motor stage, we applied a swept sinusoidal input signal, $f_{S} = f_{S, x} = f_{S, y}$ , with a frequency range of 0.001–20 Hz and a magnitude of ±20,000 pulses to make the motors move within a velocity of ±2 mm/s, and recorded the corresponding displacements ( $x_{S}$ and $y_{S}$ ) of the motor stage by encoders. We took system variation into account by repeating the identification 10 times and applied the N4SID method to derive the following transfer functions

\begin{matrix} G_{S}^{x_{i}} (s) = T_{f_{S, x} \to x_{S}} (s) \\ G_{S}^{y_{i}} (s) = T_{f_{S, y} \to y_{S}} (s) \end{matrix}

for $i = 1, \dots, 10$ , with the Bode plots of Figure 4. These transfer functions will be used for controller design and simulations in the next sections.

Figure 4.

Bode plots of the motor stage: (a) bode plots of $G_{S}^{x_{i}} (s)$ and (b) bode plots of $G_{S}^{y_{i}} (s)$ .

Controller design and implementation

This section demonstrates the design and implementation of robust controllers for the PZT and motor stages and compared the simulation and experimental results.

Controller design for the PZT stage

From Figure 3, we can note the variation of the PZT models. Therefore, we can regard the systems as linear transfer functions with uncertainties and disturbances and apply robust control techniques to improve system performance while guaranteeing stability. Assume a nominal plant $G_{0}$ has the following normalized left coprime factorization

G_{0} = {\tilde{M}}^{- 1} \tilde{N}

(1)

where $\tilde{M}, \tilde{N} \in R H_{\infty}$ and $\tilde{M} {\tilde{M}}^{*} + \tilde{N} {\tilde{N}}^{*} = I$ . Note that $R H_{\infty}$ represents the set of real rational stable transfer function matrices. Suppose a perturbed plant $G_{Δ}$ can be expressed as $G_{Δ} = (\tilde{M} + Δ_{\tilde{M}})^{- 1} (\tilde{N} + Δ_{\tilde{N}})$ with $Δ_{\tilde{M}}, Δ_{\tilde{N}} \in R H_{\infty}$ . Because coprime factorization of a system is not unique, we can define the gap between the nominal plant $G_{0}$ and the perturbed plant $G_{Δ}$ as:²³The smallest value of $‖ [\begin{matrix} Δ_{\tilde{N}} & Δ_{\tilde{M}} \end{matrix}] ‖_{\infty} < ε$ , which perturbs $G_{0}$ into $G_{Δ}$ , is called the gap between $G_{0}$ and $G_{Δ}$ , and is denoted as $δ (G_{0}, G_{Δ})$ . Note that $‖ P ‖_{\infty}$ denotes the infinity norm of P.

Therefore, we can select the nominal plants $G_{0}$ from Figure 3, with the principle of minimizing the maximum gaps between $G_{0}$ and other plants, as follows

G_{0} = \arg {min_{G_{0}} max_{G_{i}} δ (G_{0}, G_{i})}, \forall G_{i}

(2)

The gaps of the identified PZT models are illustrated in Appendix 1, where the nominal plants are selected as in the following

\begin{matrix} G_{0, P}^{x} = G_{P}^{x_{4}} = \frac{- 6.55 \times 10^{5} s^{4} - 1.26 \times 10^{8} s^{3} - 3.33 \times 10^{12} s^{2} + 8.27 \times 10^{14} s + 2.649 \times 10^{17}}{s^{5} + 996.4 s^{4} + 7.61 \times 10^{7} s^{3} + 3.04 \times 10^{10} s^{2} + 1.15 \times 10^{15} s + 1.935 \times 10^{17}} \\ G_{0, P}^{y} = G_{P}^{y_{4}} = \frac{- 1.67 \times 10^{6} s^{4} + 6.30 \times 10^{8} s^{3} - 5.10 \times 10^{12} s^{2} + 7.47 \times 10^{14} s + 4.81 \times 10^{16}}{s^{5} + 1135 s^{4} + 7.523 \times 10^{7} s^{3} + 2.06 \times 10^{10} s^{2} + 7.903 \times 10^{14} s + 3.24 \times 10^{16}} \end{matrix}

which gives $δ (G_{0, P}^{x}, G_{P}^{x_{i}} \cup {\bar{G}}_{P}^{x_{i}}) \leq 0.164$ and $δ (G_{0, P}^{y}, G_{P}^{y_{i}} \cup {\bar{G}}_{P}^{y_{i}}) \leq 0.108$ for the x- and y-axis, respectively, where $α \cup β$ indicates the union sets of $α$ and $β$ . These gaps can be regarded as the maximum system variations due to system uncertainties and operating conditions.

A closed-loop system with a perturbed plant $G_{Δ}$ and a controller K can be represented as Figure 5(a) and rearranged as Figure 5(b). Therefore, from the Small Gain Theorem,²⁴ the system is internally stable for all perturbations with $‖ [\begin{matrix} Δ_{\tilde{N}} & Δ_{\tilde{M}} \end{matrix}] ‖_{\infty} \leq ε$ if and only if

‖ [\begin{matrix} K \\ I \end{matrix}] {(I - G_{0} K)}^{- 1} {\tilde{M}}^{- 1} ‖_{\infty} = ‖ [\begin{matrix} K \\ I \end{matrix}] {(I - G_{0} K)}^{- 1} [\begin{matrix} I & G_{0} \end{matrix}] ‖_{\infty} < \frac{1}{ε}

Hence, we can further define the system’s stability margin $b (G_{0}, K)$ as

b (G_{0}, K) = {‖ [\begin{matrix} K \\ I \end{matrix}] {(I - G_{0} K)}^{- 1} {\tilde{M}}^{- 1} [{\begin{matrix} I & G \end{matrix}}_{0}] ‖_{\infty}}^{- 1}

(3)

so that the closed-loop system is internally stable for all uncertainties with $‖ [\begin{matrix} Δ_{\tilde{N}} & Δ_{\tilde{M}} \end{matrix}] ‖_{\infty} \leq ε$ if and only if its stability margin $b (G_{0}, K) > ε$ . Therefore, the principle of robust control is to design a controller K for the nominal plant $G_{0}$ such that $b (G_{0}, K)$ is greater than the system gaps.

Figure 5.

Robust stability analysis: (a) the original closed-loop system and (b) rearrangement.

We applied H_∞ loop-shaping techniques,¹² as shown in Figure 6(a), to improve the system performance. The weighting functions for the PZT stage were adjusted by experiments and selected as follows

W_{P}^{x} = \frac{1.2 \times 10^{6}}{s (s + 100) (s + 300)}, W_{P}^{y} = \frac{9 \times 10^{5}}{s (s + 150) (s + 200)}

Figure 6.

Loop-shaping design for the PZT stage: (a) loop-shaping technique, (b) bode plot for x-axis, and (c) bode plot for y-axis.

The Bode plots of the original plant $G_{0}$ and the weighted plant $W_{P}^{i} G_{0, P}^{i}$ are shown in Figure 6(b) and (c). Note that the gains of the weighted plants were increased at the low-frequency range to suppress system disturbances and decreased at the high-frequency range for attenuating system noises. Using these weighting functions, we designed the following controllers

K_{\infty}^{x} = \frac{1.556 s^{4} + 632.6 s^{3} + 3.246 \times 10^{7} s^{2} + 1.287 \times 10^{10} s + 9.626 \times 10^{11}}{s^{4} + 485 s^{3} + 2.095 \times 10^{7} s^{2} + 9.891 \times 10^{9} s + 1.502 \times 10^{12}}

K_{\infty}^{y} = \frac{1.483 s^{4} + 673.2 s^{3} + 1.884 \times 10^{7} s^{2} + 7.116 \times 10^{9} s + 5.431 \times 10^{11}}{s^{4} + 507.5 s^{3} + 1.278 \times 10^{7} s^{2} + 5.466 \times 10^{9} s + 7.864 \times 10^{11}}

which gave a stability margin of $b (G_{0, P}^{x}, K_{\infty}^{x}) = 0.541$ and $b (G_{0, P}^{y}, K_{\infty}^{y}) = 0.559$ . Because the stability margins are much greater than the system gaps (0.164 for the x-axis and 0.108 for the y-axis), system stability can be assured during operations.

We implemented the weighted controllers, $W_{P}^{x} K_{\infty}^{x}$ and $W_{P}^{y} K_{\infty}^{y}$ , to the PZT stage to track step commands of 1 and 5 µm. The results are illustrated in Figure 7 and Table 2. First, the overshoots of the responses were 1%–2% in both simulation and experiments. Second, we defined the settling time as the time when the output responses remain within an error of 2% of the inputs and found that the simulation and experiments achieved similar settling time of 0.06–0.08 s. Third, we considered the RMSE, the maximum absolute error (MAE), and the SSE of the responses in the last second as follows

SSE = \int_{T - 1}^{T} e (t) dt

(4)

RMSE = {(\int_{T - 1}^{T} {| e (t) |}^{2} dt)}^{1 / 2}

(5)

MAE = max | e (t) |, (T - 1) \leq t \leq T

(6)

where e indicates the error between the input command and the output response, and T is the final time in simulation or experiments. It is noted from Table 2 that the theoretical errors (SSE, RMSE, and MAE) were zero while the experimental errors were small but not exactly zero because of system disturbances and sensor noises. Based on these comparisons, the designed robust controllers can provide satisfactory responses.

Figure 7.

Step responses of the PZT stage: (a) 1 µm x-axis and (b) 5 µm y-axis.

Table 2.

Response analyses of the PZT stage.

Command			1 µm	5 µm
Simulation	Overshoot (%)	x-axis	1.12	1.10
	Overshoot (%)	y-axis	1.13	1.12
	Settling time (s)	x-axis	0.07	0.07
	Settling time (s)	y-axis	0.08	0.08
	SSE (nm)	x-axis	0.00	0.00
	SSE (nm)	y-axis	0.00	0.00
	RMSE (nm)	x-axis	0.00	0.00
	RMSE (nm)	y-axis	0.00	0.00
	MAE (nm)	x-axis	0.00	0.00
	MAE (nm)	y-axis	0.00	0.00
Experiment	Overshoot (%)	x-axis	1.52	2.23
	Overshoot (%)	y-axis	1.49	1.64
	Settling time (s)	x-axis	0.06	0.08
	Settling time (s)	y-axis	0.07	0.06
	SSE (nm)	x-axis	0.06	0.23
	SSE (nm)	y-axis	0.09	0.31
	RMSE (nm)	x-axis	1.89	1.93
	RMSE (nm)	y-axis	2.27	2.05
	MAE (nm)	x-axis	7.32	7.32
	MAE (nm)	y-axis	8.54	8.54

PZT: lead zirconate titanate; SSE: steady-state error; RMSE: root-mean-square error; MAE: maximum absolute error.

Controller design for the motor stage

For the motor stage, we considered the models of Figure 4 and illustrated their gaps in Appendix 1. Similarly, we applied (2) to select the nominal plants $G_{S}^{x_{4}}$ and $G_{S}^{y_{4}}$ for the x- and y-axis, respectively, and calculated their gaps by (3) as $δ (G_{S}^{x_{8}}, G_{S}^{x_{i}}) \leq 0.042$ and $δ (G_{S}^{y_{6}}, G_{S}^{y_{i}}) \leq 0.060$ for $i = 1, \dots, 10$ .

We further noted that the Bode plots of Figure 4 can be approximated by an integral. Therefore, we can simplify the control design by directly deriving the mathematical model of the motor stage. The stepper motor is controlled by pulse signals and makes one complete revolution with 20,000 pulses, while the pitch of the screw is 2 mm/rev. In other words, given an input signal of $f_{S} (pulse / s)$ , the motor is rotated with an angular velocity of $f_{S} / 20, 000 (rev / s)$ and the stage is driven with a linear velocity of $f_{S} / 10, 000 (mm / s)$ . Hence, the mathematical model of the steeper stage can be represented as in the following

G_{S} = T_{f_{S} \to x_{S}} = T_{f_{S} \to y_{S}} = \frac{0.1}{s} (μ m \times s / pulse)

(7)

We calculated the gaps between $G_{S}$ and the identified models, $G_{S}^{x_{i}} and G_{S}^{y_{i}}$ , as $δ (G_{S}, G_{S}^{x_{i}}) \leq 0.087$ and $δ (G_{S}, G_{S}^{y_{i}}) \leq 0.073$ for $i = 1, \dots, 10$ . Because these gaps are small, we can use the mathematical model of (7) to simplify the robust control design for both axes with the principle of finding a controller whose stability margin is greater than 0.087.

Applying the loop-shaping design, we adjusted and selected the following weighting function

W_{S} = 200

to increase the system gain at the low-frequency range to suppress disturbances. Because the mechanical friction of the motor stage can effectively isolate high-frequency disturbances and noises, we tried higher-order weightings to suppress the high-frequency gains but obtained similar results. Therefore, we can use the constant weighting to derive a lower-order H_∞ controller that can simplify controller implementation. The robust controller using $W_{S} = 200$ was designed as follows

K_{\infty} = \frac{0.9507 s + 10.54}{s + 10.02}

which gave a stability margin of $b (G_{S}, K_{\infty}) = 0.707$ . Because the stability margin is much greater than the system gap, 0.087, system stability can be guaranteed during operations. We further considered the velocity limitation (8 mm/s; see Table 1) of the motor stage and adjusted the input command to avoid motor stall.²⁵ That is, if the command velocity was greater than 8 mm/s, we set it as 8 mm/s and adjusted the command accordingly.

We implemented the designed controller $W_{S} K_{\infty}$ on both axes of the motor stage to track step commands of 100 µm, 1000 µm, 10 mm, and 100 mm. The simulation and experimental results are shown in Figure 8 and Table 3. First, the simulation had overshoots of about 0.45% while the experiments showed no overshoot because of the mechanical frictions. Second, the experimental settling time was much larger than that of the simulation because of the mechanical friction and the velocity limitation on 8 mm/s. Finally, the simulation errors by (4–6) were zero, while the experimental errors were small but not exactly zero because the encoders had a resolution of 0.1 µm.

Figure 8.

Step responses of the motor stage: (a) 100 µm, (b) 1000 µm, (c) 10 mm, and (d) 100 mm.

Table 3.

Responses analyses of the motor stage.

Command			100 µm	1000 µm	10 mm	100 mm
Simulation	Overshoot (%)		0.45	0.45	0.45	0.45
	Settling time (s)		0.17	0.17	0.17	0.17
	SSE (µm)		0.00	0.00	0.00	0.00
	RMSE (µm)		0.00	0.00	0.00	0.00
	MAE (µm)		0.00	0.00	0.00	0.00
Experiment	Overshoot (%)	x-axis	0.00	0.00	0.00	0.00
	Overshoot (%)	y-axis	0.00	0.00	0.00	0.00
	Settling time (s)	x-axis	0.19	0.23	1.25	12.24
	Settling time (s)	y-axis	0.18	0.22	1.24	12.24
	SSE (µm)	x-axis	0.00	0.00	0.00	0.00
	SSE (µm)	y-axis	0.00	0.00	0.00	0.00
	RMSE (µm)	x-axis	0.07	0.06	0.07	0.07
	RMSE (µm)	y-axis	0.05	0.03	0.04	0.06
	MAE (µm)	x-axis	0.10	0.10	0.10	0.10
	MAE (µm)	y-axis	0.10	0.10	0.10	0.10

SSE: steady-state error; RMSE: root-mean-square error; MAE: maximum absolute error.

Stage integration and control by different sensor layouts

We integrated the PZT and the motor stages, as shown in Figure 1(a), to achieve precision positioning for long travels. We applied encoders and interferometers to construct a local sensor layout (see Figure 9(a)) and a global sensor layout (see Figure 9(b)), respectively, and compared their impacts on positioning performance. The local sensor layout used four encoders to measure the local positions of the PZT stage, $z_{P} = [\begin{matrix} x_{P} & y_{P} \end{matrix}]^{T}$ , and the motor stage, $z_{S} = [\begin{matrix} x_{S} & y_{S} \end{matrix}]^{T}$ , and calculated the displacement of the combined stage as $z = z_{P} + z_{S}$ . On the other hand, the global sensor layout applied two laser interferometers to directly measure the displacements of the combined stage, $z_{I} = [\begin{matrix} x_{I} & y_{I} \end{matrix}]^{T}$ . In both layouts, the motor stage was controlled by a feedback error, $e_{S} = \bar{r} - z_{S}$ or $e_{S} = \bar{r} - z_{I}$ , to provide long travels. Then, when the position error $(e)$ was within a threshold, the PZT stage was activated. Because the PZT stage had faster responses and much smaller moving ranges (15 µm) than the motor stage, we designed the following anti-locking function to prevent the PZT stage from locking at its dead-end

e_{P} = {\begin{matrix} e, if | e | \leq 5 μ m \\ 0, otherwise \end{matrix}

(8)

as shown in Figure 9(c), so that the PZT stage was activated when the error $(e)$ was less than 5 µm.

Figure 9.

Control block diagram: (a) the local sensor layout with four encoders, (b) the global sensor layout with two interferometers, and (c) the anti-locking function.

We further applied multi-thread control²⁰ for the PZT and the motor stages for synchronization because the sampling rates of the PZT and the motor stage were 5 and 1 kHz, respectively. If we used single-thread control with a unified sampling frequency for both PZT and motor stages, the PZT responses would be slowed down and the system performance degraded. Therefore, we controlled the two stages simultaneously with the following multi-thread control: the PZT and the motor stage was sampled and controlled five and one times, respectively, in 1 ms.

Stage control by the local sensor layout

We first implemented the designed controllers with the local sensor layout to track step commands of 100 µm, 1000 µm, 10 mm, and 100 mm on both axes. The experimental results are shown in Figure 10(a) and (b) and Table 4. Note that Figure 10 only illustrates the x-axis responses because the y-axis responses are similar. First, the overshoots were about 1–3 µm for all experiments so that the percentage overshoots decreased as the displacements increased. Second, the settling time was slightly smaller than that of the motor stage (see Table 3) because the motor stage made long travels and the PZT stage provided faster responses at the final 5 µm travel by (8). Third, the SSEs were nearly zero because of the integral terms in the loop transfer functions, while the RMSEs and MAEs were about 5 and 15 nm, respectively, by the designed robust controllers. Finally, we compared the global positions measured by the interferometers and calculated the misalignment errors, which were caused by the inclined angles of the axes. As illustrated in Figure 10(c), when the encoders were not placed exactly on the same axis as the interferometers, the local sensor layout resulted in global positioning errors. We measured the global displacements $z_{I}$ and the system output z and calculated the misalignment errors defined as follows

r - \int_{T - 1}^{T} z_{I} (t) dt

(9)

where T is the final time in experiments. As shown in Table 4, the misalignment errors were 80 nm, 820 nm, 6.78 µm, and 66.45 µm for the step commands of 100 µm, 1000 µm, 10 mm, and 100 mm, respectively. This resulted in about a 0.07% global positioning error and an inclined angle of $θ_{S} \approx 2.2 ° .$ . Similarly, we estimated the inclined angles of the PZT stage as $θ_{P} \approx 0.5 ° .$ . The corresponding misalignment errors caused by the PZT, however, were less than 0.2 nm and can be neglected because the maximum PZT displacements were 5 and 0.1 µm at the transient and steady states, respectively.

Figure 10.

Combined stage control with the local sensor layout: (a) x-axis responses: 100 and 1000 µm, (b) x-axis responses: 10 and 100 mm, and (c) misalignment errors caused by sensor coordinates.

Table 4.

Response analyses using the local layout.

Command		100 µm	1000 µm	10 mm	100 mm
Overshoot (%)	x-axis	1.32	0.23	0.01	0.00
Overshoot (%)	y-axis	1.21	0.29	0.01	0.00
Settling time (s)	x-axis	0.16	0.22	1.23	12.15
Settling time (s)	y-axis	0.15	0.22	1.21	12.14
SSE (nm)	x-axis	−0.01	−0.07	−0.10	−0.21
SSE (nm)	y-axis	0.01	−0.04	−0.09	−0.17
RMSE (nm)	x-axis	4.77	4.89	5.04	5.12
RMSE (nm)	y-axis	4.63	4.82	4.99	5.07
MAE (nm)	x-axis	15.02	14.98	15.11	15.38
MAE (nm)	y-axis	14.46	14.79	15.13	15.20
Misalignment error (µm)	x-axis	0.08	0.82	6.78	66.45
Misalignment error (µm)	y-axis	0.08	0.84	7.23	70.29
Calculated angle $θ_{S}$ (°)	x-axis	2.286	2.320	2.113	2.094
Calculated angle $θ_{S}$ (°)	y-axis	2.045	2.229	1.982	2.166

SSE: steady-state error; RMSE: root-mean-square error; MAE: maximum absolute error.

Stage control by the global sensor layout

Similarly, we implemented the designed controllers with the global sensor layout to track step commands of 100 µm, 1000 µm, 10 mm, and 100 mm. The experimental results are illustrated in Figure 11 and Table 5. First, the percentage overshoots were less than 2% and decreased as the travels increased. Second, the settling time was smaller than that of the motor stage because both stages worked together to reach the final states. Finally, the SSEs were nearly zero because of the integral terms in the loop transfer functions. The errors (SSE, RMSE, and MAE), however, were much larger than those by the local sensor layout (see Table 4) because, though they can achieve global positioning, the interferometers suffered much greater noises than the encoders. That is, the local sensor layout can accomplish smaller RMSE and MAE but suffers misalignment errors. On the other hand, the global sensor layout can achieve global positioning but bears greater RMSE and MAE. Therefore, we need to consider a new layout to combine the merits of these two sensor layouts.

Figure 11.

Step responses of the combined stage with the global sensor layout: (a) x-axis responses: 100 and 1000 µm and (b) x-axis responses: 10 and 100 mm.

Table 5.

Response analyses using the global sensor layout.

Command		100 µm	1000 µm	10 mm	100 mm
Overshoot (%)	x-axis	1.83	0.71	0.02	0.00
Overshoot (%)	y-axis	1.78	0.72	0.01	0.00
Settling time (s)	x-axis	0.17	0.27	1.23	12.14
Settling time (s)	y-axis	0.17	0.26	1.22	12.14
SSE (nm)	x-axis	0.17	0.44	0.72	1.21
SSE (nm)	y-axis	−0.15	−0.50	−0.85	−1.38
RMSE (nm)	x-axis	26.82	31.25	30.80	34.49
RMSE (nm)	y-axis	28.93	32.47	33.79	32.88
MAE (nm)	x-axis	48.27	50.32	47.76	49.02
MAE (nm)	y-axis	46.66	48.42	48.81	51.28

SSE: steady-state error; RMSE: root-mean-square error; MAE: maximum absolute error.

Modified sensor layout for the combined stage

We constructed a modified control structure to combine the advantages of the local and global sensor layouts. The modified structure is shown in Figure 12(a), which is similar to the local sensor layout but with shaped input commands to eliminate the misalignment errors. In Figure 12(c), the global position can be seen as $z_{I} = z_{S} \cos θ_{S} + z_{P} \cos θ_{P}$ , which can be approximated as $z_{I} \approx z_{S} \cos θ_{S}$ because $| z_{P} |$ is less than 5 and 0.1 µm at the transient and steady responses, respectively. Because the inclined angle is caused by hardware deployment and does not vary significantly in different experiments, we can calculate the inclined angle $(θ_{s})$ using both interferometers and encoders in a test run. Therefore, as shown in Figure 12(a), the input command can be shaped as

r^{*} = \frac{r}{\cos θ_{S}}

(10)

where we adjusted the command and allowed the motor stage to travel a longer distance to compensate the misalignment errors. We set the inclined angle as $θ_{s} = 2.09 °,, 2.17 °$ for the x- and y-axis, respectively, from the 100 mm test (see Table 4) because their signal noise ratios were the largest. The x-axis step responses to a 100 µm command with and without input shaping are shown in Figure 12(b) and (c), where the combined stage was successfully adjusted to the correct position using the modified sensor layout and (10). Table 6 illustrates the results of the system responses to step commands of 100 µm, 1000 µm, 10 mm, and 100 mm. First, the overshoots, settling time, and errors were similar to the local sensor layout (see Table 4) for all experiments. Second, we used the interferometers and (9) to double check the misalignment errors, which were about 16 nm for a 100-mm travel and can be neglected. That is, the modified layout possessed the merits of the local and global sensor layouts.

Figure 12.

Effects of the modified sensor layout: (a) the modified sensor layout, (b) without input shaping, and (c) with input shaping.

Table 6.

Response analyses using the modified sensor layout.

Command		100 µm	1000 µm	10 mm	100 mm
Overshoot (%)	x-axis	1.22	0.24	0.01	0.00
Overshoot (%)	y-axis	1.22	0.29	0.01	0.00
Settling time (s)	x-axis	0.15	0.23	1.20	12.13
Settling time (s)	y-axis	0.15	0.22	1.22	12.12
SSE (nm)	x-axis	0.02	0.06	0.15	0.20
SSE (nm)	y-axis	0.01	0.05	0.09	0.19
RMSE (nm)	x-axis	4.67	4.82	4.98	5.34
RMSE (nm)	y-axis	4.36	4.77	4.89	5.22
MAE (nm)	x-axis	15.36	15.62	15.71	15.98
MAE (nm)	y-axis	14.57	15.79	15.86	15.50
Misalignment error (nm)	x-axis	0.46	1.12	2.84	15.67
Misalignment error (nm)	y-axis	0.33	0.94	3.47	16.32

SSE: steady-state error; RMSE: root-mean-square error; MAE: maximum absolute error.

Concluding remarks

This article has integrated a PZT stage and a motor stage to achieve long-stroke precision positioning and demonstrated the impacts of sensor layouts on system performance. First, we designed robust controllers to reach an RMSE of about 2 nm for the PZT stage and a travel of 10 cm for the motor stage. Second, we assembled the two stages and designed two sensor layouts to accomplish nano-positioning for long travels. Third, we discussed the pros and cons of the local and global sensor layouts: the local sensor layout acquired better precision but induced misalignment errors, while the global sensor layout eliminated the misalignment error but suffered larger sensor noises. Therefore, we further proposed a modified sensor structure that combined the merits of the local and global sensor layouts. The experimental results showed that the proposed stage and control can achieve an RMSE of about 5 nm and a misalignment error of about 16 nm for a 10-cm travel. In the future, we would like to apply the combined stage to micro-structure fabrication.

Footnotes

Appendix 1

Academic Editor: Jianyong Yao

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was financially supported, in part, by the Ministry of Science and Technology of Taiwan with Grants 98-2221-E-002-167-MY3 and 102-2221-E-002-150.

References

Gan

Cheung

Qiu

Position control of linear switched reluctance motors for high-precision applications. IEEE T Ind Appl 2003; 39: 1350–1362.

Park

Precision motion control of a three degrees-of-freedom hybrid stage with dual actuators. IET Control Theory Appl 2008; 2: 392–401.

Chen

Yao

Wang

QF.

Accurate motion control of linear motors with adaptive robust compensation of nonlinear electromagnetic field effect. IEEE/ASME T Mech 2013; 18: 1122–1129.

Yao

Jiao

. High-accuracy tracking control of hydraulic rotary actuators with modeling uncertainties. IEEE/ASME T Mech 2014; 19: 633–641.

Song

Kiureghian

AD.

Generalized Bouc–Wen model for highly asymmetric hysteresis. J Eng Mech: ASCE 2006; 132: 610–618.

Fung

Hsu

Huang

MS.

System identification of a dual-stage XY precision positioning table. Precis Eng 2009; 33: 71–80.

Song

Zhao

Zhou

. Tracking control of a piezoceramic actuator with hysteresis compensation using inverse Preisach model. IEEE/ASME T Mech 2005; 10: 198–209.

Ting

Jar

CC.

Measurement and calibration for Stewart micromanipulation system. Precis Eng 2007; 31: 226–233.

Shen

Jywe

Chiang

. Precision tracking control of a piezoelectric-actuated system. Precis Eng 2008; 32: 71–78.

10.

Dong

Tan

YH.

A modified Prandtl–Ishlinskii modeling method for hysteresis. Physica B 2009; 404: 1336–1342.

11.

Liu

Tan

Teo

. Development of an approach toward comprehensive identification of hysteretic dynamics in piezoelectric actuators. IEEE T Contr Syst T 2013; 21: 1834–1845.

12.

Kenton

Leang

KK.

Design and control of a three-axis serial-kinematic high-bandwidth nanopositioner. IEEE/ASME T Mech 2012; 17: 356–369.

13.

Helfrich

Lee

Bristow

. Combined H-infinity-feedback control and iterative learning control design with application to nanopositioning systems. IEEE T Contr Syst T 2010; 18: 336–351.

14.

Wang

Chen

Tsai

. Robust loop-shaping control for a nanopositioning stage. J Vib Control 2014; 20: 885–900.

15.

Sun

Gao

Kaynak

Finite frequency H_∞ control for vehicle active suspension systems. IEEE T Contr Syst T 2011; 19: 416–422.

16.

Sun

Gao

Kaynak

Vibration isolation for active suspensions with performance constraints and actuator saturation. IEEE/ASME T Mech 2015; 20: 675–683.

17.

Yao

Jiao

Extended-state-observer-based output feedback nonlinear robust control of hydraulic systems with backstepping. IEEE T Ind Electron 2014; 61: 6285–6293.

18.

Chu

Fan

SH.

A novel long-travel piezoelectric-driven linear nanopositioning stage. Precis Eng 2006; 30: 85–95.

19.

Peng

Huang

. Design and implementation of an atomic force microscope with adaptive sliding mode controller for large image scanning. In: Proceedings of the 50th IEEE CDC and ECC conference, Orlando, FL, 12–15 December 2011, pp.5577–5582. New York: IEEE.

20.

Tsai

Wang

. Design and control of a long-stroke nanopositioning stage. In: Proceedings of the 2013 IEEE/SICE international symposium on system integration, Kobe, Japan, 15–17 December 2013, pp.138–143. New York: IEEE.

21.

Prelle

Lamarque

Revel

Reflective optical sensor for long-range and high-resolution displacements. Sensor Actuat A: Phys 2006; 127: 139–146.

22.

Van Overschee

De Moor

. N4SID: subspace algorithms for the identification of combined deterministic-stochastic systems. Automatica 1994; 30: 75–93.

23.

Georgiou

Smith

MC.

Optimal robustness in the gap metric. IEEE T Automat Contr 1990; 35: 673–686.

24.

Zhou

Doyle

JC.

Essentials of robust control. Upper Saddle River, NJ: Prentice Hall, 1998.

25.

Tounsi

Bailey

Elbestawi

MA.

Identification of acceleration deceleration profile of feed drive systems in CNC machines. Int J Mach Tool Manu 2003; 43: 441–451.

Impacts of sensor layouts on the performance of a long-stroke nano-positioning stage

Abstract

Keywords

Introduction

System description and identification

Identification of the PZT stage

Identification of the motor stage

Controller design and implementation

Controller design for the PZT stage

Controller design for the motor stage

Stage integration and control by different sensor layouts

Stage control by the local sensor layout

Stage control by the global sensor layout

Modified sensor layout for the combined stage

Concluding remarks

Footnotes

Appendix 1

Declaration of conflicting interests

Funding

References