26 KiB
+++ title = "Design, modeling and control of nanopositioning systems" author = ["Thomas Dehaeze"] draft = false +++
Tags :
- Reference
- (Fleming and Leang 2014)
- Author(s)
- Fleming, A. J., & Leang, K. K.
- Year
- 2014
1 Introduction
1.1 Introduction to Nanotechnology
1.2 Introduction to Nanopositioning
1.3 Scanning Probe Microscopy
1.4 Challenges with Nanopositioning Systems
1.4.1 Hysteresis
1.4.2 Creep
1.4.3 Thermal Drift
1.4.4 Mechanical Resonance
1.5 Control of Nanopositioning Systems
1.5.1 Feedback Control
1.5.2 Feedforward Control
1.6 Book Summary
1.6.1 Assumed Knowledge
1.6.2 Content Summary
References
2 Piezoelectric Transducers
2.1 The Piezoelectric Effect
2.2 Piezoelectric Compositions
2.3 Manufacturing Piezoelectric Ceramics
2.4 Piezoelectric Transducers
2.5 Application Considerations
2.5.1 Mounting
2.5.2 Stroke Versus Force
2.5.3 Preload and Flexures
2.5.4 Electrical Considerations
2.5.5 Self-Heating Considerations
2.6 Response of Piezoelectric Actuators
2.6.1 Hysteresis
2.6.2 Creep
2.6.3 Temperature Dependence
2.6.4 Vibrational Dynamics
2.6.5 Electrical Bandwidth
2.7 Modeling Creep and Vibration in Piezoelectric Actuators
2.8 Chapter Summary
References
3 Types of Nanopositioners
3.1 Piezoelectric Tube Nanopositioners
3.1.1 63mm Piezoelectric Tube
3.1.2 40mm Piezoelectric Tube Nanopositioner
3.2 Piezoelectric Stack Nanopositioners
3.2.1 Phyisk Instrumente P-734 Nanopositioner
3.2.2 Phyisk Instrumente P-733.3DD Nanopositioner
3.2.3 Vertical Nanopositioners
3.2.4 Rotational Nanopositioners
3.2.5 Low Temperature and UHV Nanopositioners
3.2.6 Tilting Nanopositioners
3.2.7 Optical Objective Nanopositioners
References
4 Mechanical Design: Flexure-Based Nanopositioners
4.1 Introduction
4.2 Operating Environment
4.3 Methods for Actuation
4.4 Flexure Hinges
4.4.1 Introduction
4.4.2 Types of Flexures
4.4.3 Flexure Hinge Compliance Equations
4.4.4 Stiff Out-of-Plane Flexure Designs
4.4.5 Failure Considerations
4.4.6 Finite Element Approach for Flexure Design
4.5 Material Considerations
4.5.1 Materials for Flexure and Platform Design
4.5.2 Thermal Stability of Materials
4.6 Manufacturing Techniques
4.7 Design Example: A High-Speed Serial-Kinematic Nanopositioner
4.7.1 State-of-the-Art Designs
4.7.2 Tradeoffs and Limitations in Speed
4.7.3 Serial- Versus Parallel-Kinematic Configurations
4.7.4 Piezoactuator Considerations
4.7.5 Preloading Piezo-Stack Actuators
4.7.6 Flexure Design for Lateral Positioning
4.7.7 Design of Vertical Stage
4.7.8 Fabrication and Assembly
4.7.9 Drive Electronics
4.7.10 Experimental Results
4.8 Chapter Summary
References
5 Position Sensors
5.1 Introduction
5.2 Sensor Characteristics
5.2.1 Calibration and Nonlinearity
5.2.2 Drift and Stability
5.2.3 Bandwidth
5.2.4 Noise
5.2.5 Resolution
5.2.6 Combining Errors
5.2.7 Metrological Traceability
5.3 Nanometer Position Sensors
5.3.1 Resistive Strain Sensors
5.3.2 Piezoresistive Strain Sensors
5.3.3 Piezoelectric Strain Sensors
5.3.4 Capacitive Sensors
5.3.5 MEMs Capacitive and Thermal Sensors
5.3.6 Eddy-Current Sensors
5.3.7 Linear Variable Displacement Transformers
5.3.8 Laser Interferometers
5.3.9 Linear Encoders
5.4 Comparison and Summary
5.5 Outlook and Future Requirements
References
6 Shunt Control
6.1 Introduction
6.2 Shunt Circuit Modeling
6.2.1 Open-Loop
6.2.2 Shunt Damping
6.3 Implementation
6.4 Experimental Results
6.4.1 Tube Dynamics
6.4.2 Amplifier Performance
6.4.3 Shunt Damping Performance
6.5 Chapter Summary
References
7 Feedback Control
7.1 Introduction
7.2 Experimental Setup
7.3 PI Control
7.4 PI Control with Notch Filters
7.5 PI Control with IRC Damping
7.6 Performance Comparison
7.7 Noise and Resolution
7.8 Analog Implementation
7.9 Application to AFM Imaging
7.10 Repetitive Control
7.10.1 Introduction
7.10.2 Repetitive Control Concept and Stability Considerations
7.10.3 Dual-Stage Repetitive Control
7.10.4 Handling Hysteresis
7.10.5 Design and Implementation
7.10.6 Experimental Results and Discussion
7.11 Summary
References
8 Force Feedback Control
8.1 Introduction
8.2 Modeling
8.2.1 Actuator Dynamics
8.2.2 Sensor Dynamics
8.2.3 Sensor Noise
8.2.4 Mechanical Dynamics
8.2.5 System Properties
8.2.6 Example System
8.3 Damping Control
8.4 Tracking Control
8.4.1 Relationship Between Force and Displacement
8.4.2 Integral Displacement Feedback
8.4.3 Direct Tracking Control
8.4.4 Dual Sensor Feedback
8.4.5 Low Frequency Bypass
8.4.6 Feedforward Inputs
8.4.7 Higher-Order Modes
8.5 Experimental Results
8.5.1 Experimental Nanopositioner
8.5.2 Actuators and Force Sensors
8.5.3 Control Design
8.5.4 Noise Performance
8.6 Chapter Summary
References
9 Feedforward Control
9.1 Why Feedforward?
9.2 Modeling for Feedforward Control
9.3 Feedforward Control of Dynamics and Hysteresis
9.3.1 Simple DC-Gain Feedforward Control
9.3.2 An Inversion-Based Feedforward Approach for Linear Dynamics
9.3.3 Frequency-Weighted Inversion: The Optimal Inverse
9.3.4 Application to AFM Imaging
9.4 Feedforward and Feedback Control
9.4.1 Application to AFM Imaging
9.5 Iterative Feedforward Control
9.5.1 The ILC Problem
9.5.2 Model-Based ILC
9.5.3 Nonlinear ILC
9.5.4 Conclusions
References
10 Command Shaping
10.1 Introduction
10.1.1 Background
10.1.2 The Optimal Periodic Input
10.2 Signal Optimization
10.3 Frequency Domain Cost Functions
10.3.1 Background: Discrete Fourier Series
10.3.2 Minimizing Signal Power
10.3.3 Minimizing Frequency Weighted Power
10.3.4 Minimizing Velocity and Acceleration
10.3.5 Single-Sided Frequency Domain Calculations
10.4 Time Domain Cost Function
10.4.1 Minimum Velocity
10.4.2 Minimum Acceleration
10.4.3 Frequency Weighted Objectives
10.5 Application to Scan Generation
10.5.1 Choosing β and K
10.5.2 Improving Feedback and Feedforward Controllers
10.6 Comparison to Other Techniques
10.7 Experimental Application
10.8 Chapter Summary
References
11 Hysteresis Modeling and Control
11.1 Introduction
11.2 Modeling Hysteresis
11.2.1 Simple Polynomial Model
11.2.2 Maxwell Slip Model
11.2.3 Duhem Model
11.2.4 Preisach Model
11.2.5 Classical Prandlt-Ishlinksii Model
11.3 Feedforward Hysteresis Compensation
11.3.1 Feedforward Control Using the Presiach Model
11.3.2 Feedforward Control Using the Prandlt-Ishlinksii Model
11.4 Chapter Summary
References
12 Charge Drives
12.1 Introduction
12.2 Charge Drives
12.3 Application to Piezoelectric Stack Nanopositioners
12.4 Application to Piezoelectric Tube Nanopositioners
12.5 Alternative Electrode Configurations
12.5.1 Grounded Internal Electrode
12.5.2 Quartered Internal Electrode
12.6 Charge Versus Voltage
12.6.1 Advantages
12.6.2 Disadvantages
12.7 Impact on Closed-Loop Control
12.8 Chapter Summary
References
13 Noise in Nanopositioning Systems
13.1 Introduction
13.2 Review of Random Processes
13.2.1 Probability Distributions
13.2.2 Expected Value, Moments, Variance, and RMS
13.2.3 Gaussian Random Variables
13.2.4 Continuous Random Processes
13.2.5 Joint Density Functions and Stationarity
13.2.6 Correlation Functions
13.2.7 Gaussian Random Processes
13.2.8 Power Spectral Density
13.2.9 Filtered Random Processes
13.2.10 White Noise
13.2.11 Spectral Density in V/sqrtHz
13.2.12 Single- and Double-Sided Spectra
13.3 Resolution and Noise
13.4 Sources of Nanopositioning Noise
13.4.1 Sensor Noise
13.4.2 External Noise
13.4.3 Amplifier Noise
13.5 Closed-Loop Position Noise
13.5.1 Noise Sensitivity Functions
13.5.2 Closed-Loop Position Noise Spectral Density
13.5.3 Closed-Loop Noise Approximations with Integral Control
13.5.4 Closed-Loop Position Noise Variance
13.5.5 A Note on Units
13.6 Simulation Examples
13.6.1 Integral Controller Noise Simulation
13.6.2 Noise Simulation with Inverse Model Controller
13.6.3 Feedback Versus Feedforward Control
13.7 Practical Frequency Domain Noise Measurements
13.7.1 Preamplification
13.7.2 Spectrum Estimation
13.7.3 Direct Measurement of Position Noise
13.7.4 Measurement of the External Disturbance
13.8 Experimental Demonstration
13.9 Time-Domain Noise Measurements
13.9.1 Total Integrated Noise
13.9.2 Estimating the Position Noise
13.9.3 Practical Considerations
13.9.4 Experimental Demonstration
13.10 A Simple Method for Measuring the Resolution of Nanopositioning Systems
13.11 Techniques for Improving Resolution
13.12 Chapter Summary
References
Electrical Considerations
Amplifier and Piezo electrical models
{{< figure src="/ox-hugo/fleming14_amplifier_model.png" caption="Figure 1: A voltage source \(V_s\) driving a piezoelectric load. The actuator is modeled by a capacitance \(C_p\) and strain-dependent voltage source \(V_p\). The resistance \(R_s\) is the output impedance and \(L\) the cable inductance." >}}
Consider the electrical circuit shown in Figure 1 where a voltage source is connected to a piezoelectric actuator. The actuator is modeled as a capacitance \(C_p\) in series with a strain-dependent voltage source \(V_p\). The resistance \(R_s\) and inductance \(L\) are the source impedance and the cable inductance respectively.
Typical inductance of standard RG-58 coaxial cable is \(250 nH/m\). Typical value of \(R_s\) is between \(10\) and \(100 \Omega\).
When considering the effects of both output impedance and cable inductance, the transfer function from source voltage \(V_s\) to load voltage \(V_L\) is second-order low pass filter:
\begin{equation} \frac{V_L(s)}{V_s(s)} = \frac{1}{\frac{s^2}{\omega_r^2} + 2 \xi \frac{s}{\omega_r} + 1} \end{equation}
with:
- \(\omega_r = \frac{1}{\sqrt{L C_p}}\)
- \(\xi = \frac{R_s \sqrt{L C_p}}{2 L}\)
Amplifier small-signal Bandwidth
The most obvious bandwidth limitation is the small-signal bandwidth of the amplifier.
If the inductance \(L\) is neglected, the transfer function from source voltage \(V_s\) to load voltage \(V_L\) forms a first order filter with a cut-off frequency
\begin{equation} \omega_c = \frac{1}{R_s C_p} \end{equation}
This is thus highly dependent of the load.
The high capacitive impedance nature of piezoelectric loads introduces phase-lag into the feedback path. A rule of thumb is that closed-loop bandwidth cannot exceed one-tenth the cut-off frequency of the pole formed by the amplifier output impedance \(R_s\) and load capacitance \(C_p\) (see Table 1 for values).
| Cp = 100 nF | Cp = 1 uF | Cp = 10 uF | |
|---|---|---|---|
| Rs = 1 Ohm | 1.6 MHz | 160 kHz | 16 kHz |
| Rs = 10 Ohm | 160 kHz | 16 kHz | 1.6 kHz |
| Rs = 100 Ohm | 16 kHz | 1.6 kHz | 160 Hz |
The inductance \(L\) does also play a role in the amplifier bandwidth as it changes the resonance frequency. Ideally, low inductance cables should be used. It is however usually quite high compare to \(\omega_c\) as shown in Table 2.
| Cp = 100 nF | Cp = 1 uF | Cp = 10 uF | |
|---|---|---|---|
| L = 25 nH | 3.2 MHz | 1 MHz | 320 kHz |
| L = 250 nH | 1 MHz | 320 kHz | 100 kHz |
| L = 2500 nH | 320 kHz | 100 kHz | 32 kHz |
Amplifier maximum slew rate
Further bandwidth restrictions are imposed by the maximum slew rate of the amplifier. This is the maximum rate at which the output voltage can change and is usually expressed in \(V/\mu s\).
For sinusoidal signals, the amplifiers slew rate must exceed: \[ SR_{\text{sin}} > V_{p-p} \pi f \] where \(V_{p-p}\) is the peak to peak voltage and \(f\) is the frequency.
If a 300kHz sine wave is to be reproduced with an amplitude of 10V, the required slew rate is \(\approx 20 V/\mu s\).
When dealing with capacitive loads, the current limit is usually exceed well before the slew rate limit.
Current and Power Limitations
When driving the actuator off-resonance, the current delivered to a piezoelectric actuator is approximately: \[ I_L(s) = V_L(s) C_p s \]
For sinusoidal signals, the maximum positive and negative current is equal to: \[ I_L^\text{max} = V_{p-p} \pi f C_p \]
| Cp = 100 nF | Cp = 1 uF | Cp = 10 uF | |
|---|---|---|---|
| f = 30 Hz | 0.19 mA | 1.9 mA | 19 mA |
| f = 3 kHz | 19 mA | 190 mA | 1.9 A |
| f = 300 kHz | 1.9 A | 19 A | 190 A |
Chapter Summary
The bandwidth limitations of standard piezoelectric drives were identified as:
- High output impedance
- The presence of a ple in the voltage-feedback loop due to output impedance and load capacitance
- Insufficient current capacity due to power dissipation
- High cable and connector inductance
References
Bibliography
Fleming, Andrew J., and Kam K. Leang. 2014. Design, Modeling and Control of Nanopositioning Systems. Advances in Industrial Control. Springer International Publishing. https://doi.org/10.1007/978-3-319-06617-2.