Integrated Design and Control of Flexure-Based Nanopositioning Systems — Part I: Methodology

TL;DR: In this article, the authors present an integrated design and control method for implementing flexure-based nanopositioning systems, where an automation engine generates a set of flexurebased design topologies and also controllers of varying order in the optimization.

About: This article is published in IFAC Proceedings Volumes.The article was published on 2011-01-01 and is currently open access. It has received 1 citations till now. The article focuses on the topics: Integrated design & Automation.

Summary

1. INTRODUCTION

• Precision positioning applications built around conventional bearings (such as sliding contact or rolling contact bearings) are often hindered by friction, backlash, hysteresis, and other motion non-linearities.
• In Part I of this paper, a novel methodology integrating design and control considerations was presented.
• The topology generation is aimed as a valuable addition to the design toolkit, facilitating novel designs that could not have been conceived otherwise.
• A piezoelectric stack actuator with a lever amplification mechanism is suggested for generating a large displacement range on the order of 100 μm required for the gap z.

2.1 Varying Design and/or Control Parameters

• In what follows, the authors first review works reported on optimizing a or controller so that a desired performance metric ( or control) is met under physical constraints, and state/output and control constraints.
• Different approaches for integrated design and control have been studied from an optimization theory standpoint in [17] and [15].
• External damping such as squeeze film damping and foamdamping have been suggested and explored for flexures in the past.
• The design and control performance space in terms of performance requirements, such as (i) the positioning error and (ii) control bandwidth of the drive and (iii) the maximum acceleration of the carriage, were captured for the entire range of geometry, material, and other parameters.
• Preprint submitted to 18th IFAC World Congress.

2.2 Varying Design Topology

• Unlike most of the methods reported above, few references address changing the design structure or configuration (referred to as the topology) itself, so that control performance is enhanced.
• The authors examine here two specific cases from the literature that illustrate the importance of selecting an appropriate design topology before deploying any optimization routine.
• Since the actuator and the sensor are not at the same location in space, i.e. the system is non-collocated.
• In order to avoid the occurrence of the non-minimum phase zero, the actuation point shown in Fig. 1 (a) can be moved away from the motor closer to the end-point, as shown in Fig. 1(b).
• The presence of the bearing translation mode is undesirable for two reasons: (i) the translation shows up in the displacement at the read head and (ii) the transfer function between the applied force and the measured displacement at read head can be non-minimum phase under certain 2.

3. PROPOSED METHOD

• Based on the examples of integrated design and control described in Section 2, the authors identify the four possible cases for integrated design and control in Table 1.
• In Case II, for a fixed design topology, the controller is allowed to vary.
• The primitives are then subjected to these operations generate building blocks that meet the desired performance requirement.
• Step 5: Optimization: Given that the nominal design and the nominal controller have passed the screening test, the authors now feed them to an optimization procedure.
• If the performance requirements are not met at the end of the optimization procedure and the maximum number of iterations has not been reached, the nominal design topology is revised.

5. SUMMARY

• The authors presented a method for iterating on design topologies and controller order to achieve a desired closed-loop system specification.
• Instead, the authors need to iterate over design topologies and controller order.
• An automated topology generation engine is discussed.
• Further, a novel controller parameterization is used to vary the controller order while directly tuning the sensitivity function to a desired form.
• The first author is thankful to Xerox Foundation and MIT Dean School of Engineering for fellowship support.

2.2 Problem Statement

• The problem statement for applying the proposed integrated design and control methodology to the example of the positioning system of Fig. 2 is as follows: Given a lever amplification mechanism of Fig. 2 with the following parameters: (ii) output displacement yout measured at a distance Ls = 2.
• The authors approach of integrated design and control is implemented for achieving this feature.
• In the example of disk drive actuator system given in [4], altering geometry of the given topology eliminates nonminimum phase zeros.
• In a actual multi-DOF system, given many constraints on geometry, and design requirements, both (i) varying parameters within a topology and (ii) varying the topology (and parameters within each topology) should be explored.
• Preprint submitted to 18th IFAC World Congress.

3. IMPLEMENTATION OF METHODOLOGY

• Given the above parameters for the lever and the piezoelectric stack actuator, the authors examine the topology, shape-size optimization and control performance of the system when a flexure-based mechanism is used as a pivot for the lever.
• While the notch flexure joint in Fig. 5(a) has a localized compliance around its neck, the beam flexure of Fig. 5(b) has a compliance distributed over its length.
• The rest of the design topologies shown in Figs. 5(c)(j) are obtained as follows.
• The details of the controller optimization are as follows: Control Parameter.

6. RESULTS AND DISCUSSION

• An optimal solution was found for the case of flexurebased pivots of Fig. 5(g)-(j) for both the design and control optimization problems.
• The design topologies of Fig. 5(c)-(f) turn out to be infeasible, the reason for which is discussed as follows.
• Hence, the flexure-based pivots of Fig. 5(c)-(f) need to be discarded in their integrated design and control methodology.
• The nominal sensitivity transfer function resulting from a nominal controller C0(s) = 1000s has a low bandwidth, while the desired sensitivity has a bandwidth of 1000 Hz, a rollon of 40db/dec.
• Avoiding the non-minimum phase zero may require reconsidering where to measure relative to where the authors actuate the system.

7. SUMMARY

• The authors presented a flow chart for iterating on design and controller to achieve a desired closedloop system specification.
• An example of a flexure-based 1-DOF positioning system was worked out to show the integrated design and control methodology.
• The methodology was worked out step-by-step to cover (i) generation of design topologies (ii) screening of topologies for obvious design choices that cannot work for the given application, (iii) optimization formulation in terms of design parameters, cost functions, and equality and inequality constraints, and (iv) controller generation based on model-matching of a sensitivity transfer function.
• White JR, “The nanogate: nanoscale flow control,” Ph.D. dissertation, Cambridge, MA: Massachusetts Institute of Technology, Department of Mechanical Engineering, June 2003. [2].
• Shilpiekandula V, “Progress through Mechanics: Small-scale Gaps,” Mechanics (Publication of the American Academy of Mechanics), vol. 35, no.

Figures

Table 1. All possible cases for integrated design and control.

Fig. 2. Design for control example from [35] modifying actuator design to eliminate translational loading in voice-coil motors. (a) In disk drives, the actuator for the read/write head is a typical lorentz-force voice coil motor that produces a force at an offset. The force also excites the translational mode of the bearing. (b) A novel design for the voice coil motor, based on a magnetic array called as Hallbach array, which is commonly found in linear motors, is used in a rotary configuration to produce a unidirectional magnetic field in the hub. (c) The resulting actuator is a pure torque motor that minimizes the effect of the translational mode of the bearing.

Fig. 1. Schematic diagram showing a positioning system example. The goal is to vary the gap z over a large range of motion and control bandwidth.

Fig. 2. Design for control example from [35] modifying actuator design to eliminate translational loading in voice-coil motors. (a) In disk drives, the actuator for the read/write head is a typical lorentz-force voice coil motor that produces a force at an offset. The force also excites the translational mode of the bearing. (b) A novel design for the voice coil motor, based on a magnetic array called as Hallbach array, which is commonly found in linear motors, is used in a rotary configuration to produce a unidirectional magnetic field in the hub. (c) The resulting actuator is a pure torque motor that minimizes the effect of the translational mode of the bearing.

Fig. 10. Lumped parameter model for depicting dynamic behavior of topology concepts using flexure-based mechanisms of Fig. 5(g)-(j) as pivots in the 1-DOF positioning system.

Fig. 4. Integrated design and control methodology for meeting performance requirements.

Fig. 3. A performance-driven design library shown as constructed from building blocks prepared by performance-driven operations on a set of primitives. Novel designs synthesized with this method are schematically shown in the Design Library block.

Fig. 8. Lumped parameter model for depicting dynamic behavior of topology concepts using flexure-based mechanisms of Fig. 5(c)-(f) as pivots in the 1-DOF positioning system.

Fig. 12. Comparison of magnitude response of desired and achieved sensitivity transfer function designed with a model-matching matching procedure. A corresponding response obtained for the case of mixed-sensitivity design is also shown.

Fig. 6. Primitive beam flexure shown in (a) is enhanced in its load-capacity by adding a redundant constraint in (b) to produce a double-sided beam flexure primitive.

Fig. 7. Double-sided notched flexure primitive. This primitive is the notched equivalent of the double-sided beam flexure of Fig. 6 (b). Unlike the beam flexure, which has a distributed compliance, the notch flexure has a localized compliance at the thin necks of the notch.

Fig. 4. Integrated design and control methodology for meeting performance requirements.

Fig. 5. A library of candidate design topologies for a flexural pivot.

Fig. 3. A performance-driven design library shown as constructed from building blocks prepared by performance-driven operations on a set of primitives. Novel designs synthesized with this method are schematically shown in the Design Library block.

Fig. 1. Schematic diagram showing a positioning system example. The goal is to vary the gap z over a large range of motion and control bandwidth.

Table 2. Results of Optimization of Design and Control for the case of topologies of Fig. 5(g)(j) used as flexure-based pivot.

Fig. 11. Pole-zero plot of open-loop plant corresponding to design topologies using flexure-based mechanisms of Fig. 5(c)-(f) as pivots in the 1-DOF positioning system. The zeros of the system are on the real axis and symmetric about the imaginary axis, hence resulting in a non-minimum phase behavior.

Frequently asked questions

Q1. What have the authors contributed in "Integrated design and control of flexure-based nanopositioning systems - part i: methodology" ?

In this paper, the authors present an integrated design and control method for implementing flexurebased nanopositioning systems. The authors discuss the need for varying design topology and order of a controller in design and control optimization. This work may not be copied or reproduced in whole or in part for any commercial purpose. Permission to copy in whole or in part without payment of fee is granted for nonprofit educational and research purposes provided that all such whole or partial copies include the following: a notice that such copying is by permission of Mitsubishi Electric Research Laboratories, Inc. ; an acknowledgment of the authors and individual contributions to the work ; and all applicable portions of the copyright notice. 

Q2. What are the future works in "Integrated design and control of flexure-based nanopositioning systems - part i: methodology" ?

The details of the controller parameterization are not covered here and will be part of a future paper from their group. 

Q3. What is the main reason for the characterization of the design and control parameters?

Since unmodeled dynamics in the control bandwidth can adversely affect the performance, it is necessary to account for model-truncation errors in the design and control optimization. 

Q4. What is the way to generate a library of design topologies?

Once the building blocks are generated, a library of design topologies can be generated by using the building block as an implementation of the constraints (following a constraint-based synthesis approach [29]) for satisfying the necessary kinematics. 

Q5. What are the advantages of flexure-based mechanisms?

Flexure-based mechanisms are composed of slender beam-like spring elements in their mechanical design; they are close to being ideal motion bearings with minimal friction, backlash, and other uncertainties. 

Q6. What is the common type of damping in flexures?

Physical damping is low in flexures made from metals such as aluminium (used in development stages of the design process for ease of machining), or titanium (used in the implementation and testing phase because of its high fatigue strength and other material properties). 

Q7. What is the way to achieve the desired closed-loop control performance?

A common systems-based methodology can facilitate developing valuable synthesis tools for achieving the desired closed-loop control performance. 

Q8. What is the way to solve the varying design topology problem?

since the number of possible design configurations in typical nanopositioning system applications are finite, the varying design topology problem can be broken down into a number of fixed design (each tested with a controller of varying order) problems. 

Q9. What is the effect of moving the actuation point closer to the sensor?

With the actuation location moved closer to the end-point, the portion of the link from the new actuation point to the sensor location is shorter, and hence stiffer. 

Q10. What are some of the applications for nanopositioning?

Many applications for nanopositioning systems have emerged over the past few decades in various contexts, such as semiconductor manufacturing, metrology, x-ray crystallography, and biological imaging. 

Q11. What are the possible operations that can be used to generate a design library?

These operations could be, for example, a parallel or serial replication, or a geometrical transformation, or adding a redundant constraint that imparts symmetry. 

Q12. What is the effect of the change in the topology?

It is shown in [36] that, under certain geometry conditions, this topology change results in moving the zeros from the real-axis on to the imaginary axis, making the system minimum-phase. 

Q13. How can the actuation point be moved away from the motor?

In order to avoid the occurrence of the non-minimum phase zero, the actuation point shown in Fig. 1 (a) can be moved away from the motor closer to the end-point, as shown in Fig. 1(b). 

Q14. What is the simplest way to change the topology of a robot?

Without this topology change, with the actuator just as the motor and sensor at the end-point, the system would be non-minimum phase and pose critical control challenges.