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+++ title = "Design, modeling and control of nanopositioning systems" author = ["Dehaeze Thomas"] description = "Talks about various topics related to nano-positioning systems." keywords = ["Control", "Metrology", "Flexible Joints"] draft = false +++
- Tags
- [Piezoelectric Actuators]({{< relref "piezoelectric_actuators.md" >}}), [Flexible Joints]({{< relref "flexible_joints.md" >}})
- Reference
- (Fleming and Leang 2014)
- Author(s)
- Fleming, A. J., & Leang, K. K.
- Year
- 2014
Introduction to Nanotechnology
Introduction to Nanopositioning
Scanning Probe Microscopy
Challenges with Nanopositioning Systems
Hysteresis
Creep
Thermal Drift
Mechanical Resonance
Control of Nanopositioning Systems
Feedback Control
Feedforward Control
Book Summary
Assumed Knowledge
Content Summary
References
The Piezoelectric Effect
Piezoelectric Compositions
Manufacturing Piezoelectric Ceramics
Piezoelectric Transducers
Application Considerations
Response of Piezoelectric Actuators
Modeling Creep and Vibration in Piezoelectric Actuators
Chapter Summary
References
Piezoelectric Tube Nanopositioners
63mm Piezoelectric Tube
40mm Piezoelectric Tube Nanopositioner
Piezoelectric Stack Nanopositioners
Phyisk Instrumente P-734 Nanopositioner
Phyisk Instrumente P-733.3DD Nanopositioner
Vertical Nanopositioners
Rotational Nanopositioners
Low Temperature and UHV Nanopositioners
Tilting Nanopositioners
Optical Objective Nanopositioners
References
Introduction
Operating Environment
Methods for Actuation
Flexure Hinges
Introduction
Types of Flexures
Flexure Hinge Compliance Equations
Stiff Out-of-Plane Flexure Designs
Failure Considerations
Finite Element Approach for Flexure Design
Material Considerations
Materials for Flexure and Platform Design
Thermal Stability of Materials
Manufacturing Techniques
Design Example: A High-Speed Serial-Kinematic Nanopositioner
State-of-the-Art Designs
Tradeoffs and Limitations in Speed
Serial- Versus Parallel-Kinematic Configurations
Piezoactuator Considerations
Preloading Piezo-Stack Actuators
Flexure Design for Lateral Positioning
Design of Vertical Stage
Fabrication and Assembly
Drive Electronics
****0 Experimental Results
Chapter Summary
References
Introduction
Sensor Characteristics
Calibration and Nonlinearity
Drift and Stability
Bandwidth
Noise
Resolution
Combining Errors
Metrological Traceability
Nanometer Position Sensors
Resistive Strain Sensors
Piezoresistive Strain Sensors
Piezoelectric Strain Sensors
Capacitive Sensors
MEMs Capacitive and Thermal Sensors
Eddy-Current Sensors
Linear Variable Displacement Transformers
Laser Interferometers
Linear Encoders
Comparison and Summary
Outlook and Future Requirements
References
Introduction
Shunt Circuit Modeling
Open-Loop
Shunt Damping
Implementation
Experimental Results
Tube Dynamics
Amplifier Performance
Shunt Damping Performance
Chapter Summary
References
Introduction
Experimental Setup
PI Control
PI Control with Notch Filters
PI Control with IRC Damping
Performance Comparison
Noise and Resolution
Analog Implementation
Application to AFM Imaging
References
Introduction
Modeling
Actuator Dynamics
Sensor Dynamics
Sensor Noise
Mechanical Dynamics
System Properties
Example System
Damping Control
Tracking Control
Relationship Between Force and Displacement
Integral Displacement Feedback
Direct Tracking Control
Dual Sensor Feedback
Low Frequency Bypass
Feedforward Inputs
Higher-Order Modes
Experimental Results
Experimental Nanopositioner
Actuators and Force Sensors
Control Design
Noise Performance
Chapter Summary
References
Why Feedforward?
Modeling for Feedforward Control
Feedforward Control of Dynamics and Hysteresis
Simple DC-Gain Feedforward Control
An Inversion-Based Feedforward Approach for Linear Dynamics
Frequency-Weighted Inversion: The Optimal Inverse
Application to AFM Imaging
Feedforward and Feedback Control
Application to AFM Imaging
Iterative Feedforward Control
The ILC Problem
Model-Based ILC
Nonlinear ILC
Conclusions
References
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
11.1 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
12.1 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
13.1 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="<span class="figure-number">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