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Tilt Mirror for raster scan applications
8th IFAC Symposium on Mechatronic Systems 8th IFAC Symposium on Mechatronic Systems 8th IFAC IFACAustria, Symposium on Systems Vienna, Sept. on 4-6,Mechatronic 2019 8th Symposium Systems Available online at www.sciencedirect.com Vienna, Austria, Sept. 4-6,Mechatronic 2019 8th IFACAustria, Symposium Systems Vienna, Sept. 4-6, 2019 Vienna, Austria, Sept. on 4-6,Mechatronic 2019 Vienna, Austria, Sept. 4-6, 2019
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IFAC PapersOnLine 52-15 (2019) 289–294
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Piezo Actuated Tip/Tilt Mirror Piezo Actuated Tip/Tilt Mirror Piezo Actuated Tip/Tilt Mirror raster scan applications Piezo Actuated Tip/Tilt Mirror raster scan applications raster scan applications raster scan applications E. Csencsics, B. Sitz, G. Schitter
for for for for
E. Csencsics, B. Sitz, G. Schitter E. Csencsics, B. Sitz, G. Schitter E.Laboratory Csencsics, Sitz, G.Engineering Schitter for Automated Christian Doppler forB.Precision Precision Christian Doppler Laboratory for Engineering for Automated Christian Doppler Laboratory for Precision Engineering for Automated In-line Metrology, Metrology, Automation Automation and and Control Control Institute Institute (ACIN), (ACIN), Vienna In-line Vienna Christian Doppler of Laboratory for Precision Engineering for Automated University Technology, 1040 Vienna, Austria (e-mail: In-lineUniversity Metrology, Automation and Control Institute (ACIN), Vienna of Technology, 1040 Vienna, Austria (e-mail: In-lineUniversity Metrology,ofAutomation Control Institute (ACIN), [email protected]). Technology,and 1040 Vienna, Austria (e-mail:Vienna University of [email protected]). Technology, 1040 Vienna, Austria (e-mail: [email protected]). [email protected]). Abstract: Abstract: This This paper paper presents presents the the system system and and control control design design of of aa novel novel high high performance performance piezopiezoAbstract: This paper mirror presents(FSM) the system andspeed control designscanning of a novelwhich high performance piezoactuated fast steering for high optical is centered centered around around actuated fast steering mirror (FSM) scanning which for high speed optical is Abstract: This paper presents the system and control design of a novel high performance piezoactuated fast steering mirror (FSM) for high speed optical scanning which is centered around an easy-to-manufacture membrane-like mechanical flexure and uses optical system for an easy-to-manufacture membrane-like mechanical flexure and uses an anwhich opticalissensor sensor system for actuated fast steering mirror (FSM) formechanical high speedflexure optical scanning centered around position measurement. With the membrane flexure the first fundamental resonance mode of the the an easy-to-manufacture membrane-like and uses an optical sensor system for position measurement. With the membrane flexure the first fundamental resonance mode of an easy-to-manufacture membrane-like mechanical flexure and uses an optical sensor system for fundamental resonance mode of the actuator is as while enabling an angular range ±4.8 position measurement. With thekHz, membrane flexure the first actuator is placed placed as as high high as 6.7 6.7 kHz, while still still enabling an optical optical angular range of of mode ±4.8 mrad. mrad. position measurement. With the membrane flexure the first fundamental resonance of the actuator is placed as high as 6.7 kHz, while still enabling an optical angular range of ±4.8 mrad. Using a controller together with three notch filters in loop approach for Using a PI PI controller together three notch filters designed designed in aaangular loop shaping shaping for actuator is placed as high as 6.7with kHz, while still enabling an optical range ofapproach ±4.8 mrad. Using a PI controller together with three notch filters designed in araster loop trajectories, shaping approach for maximizing the system bandwidth and tracking of high resolution aa closedmaximizing the system bandwidth and tracking of highdesigned resolution raster trajectories, closedUsing a PI controller together with three notch filters in a loop shaping approach for loop bandwidth as high as 2.7 kHz and a positioning uncertainty of 3.8 µrad rms is obtained. maximizing the system bandwidth and tracking of high resolution raster trajectories, a closedloop bandwidth as highbandwidth as 2.7 kHzand andtracking a positioning uncertainty of 3.8 µrad rms is obtained. maximizing the system of high resolution raster trajectories, a closedThe tracking aa 250 raster aa scan loop bandwidth as higherror as 2.7when kHz tracking and a positioning of 3.8trajectory µrad rms with is obtained. The obtained obtained tracking when 250 Hz Hzuncertainty raster scanning scanning scan loop bandwidth as highiserror as 2.7 kHzastracking and a µrad positioning uncertainty of 3.8trajectory µrad rms with is obtained. The obtained tracking error when tracking a 250 Hz raster scanning trajectory with a scan amplitude of 2.2 mrad as small 73.8 rms (3.6% of scan amplitude). amplitude of 2.2 mrad is as small as 73.8 µrad rms (3.6% of scan amplitude). The obtained tracking whenastracking a 250 raster scanning trajectory with a scan amplitude of 2.2 mrad iserror as small 73.8 µrad rms Hz (3.6% of scan amplitude). © 2019, IFAC (International Federation Automatic Control) Hosting by amplitude). Elsevier Ltd. All rights reserved. amplitude of 2.2 mrad is as small asof73.8 µrad rms (3.6% of scan Keywords: Keywords: Fast Fast steering steering mirror, mirror, System System analysis analysis and and design, design, Motion Motion control, control, Piezo Piezo actuation, actuation, Keywords: Fast steering mirror, System analysis and design, Motion control, Piezo actuation, Optomechatronics Optomechatronics Keywords: Fast steering mirror, System analysis and design, Motion control, Piezo actuation, Optomechatronics Optomechatronics 1. INTRODUCTION abling aa high high control control bandwidth bandwidth for for good tracking tracking and and disdis1. abling 1. INTRODUCTION INTRODUCTION abling a high control bandwidth [Kluk for good good tracking and disturbance rejection performance et al. (2012)]. Piezo turbance rejection performance [Kluk et (2012)]. Piezo 1. INTRODUCTION abling a high control bandwidth for good tracking and disturbance rejection et al. al. (2012)]. Piezo actuated FSMs andperformance other high [Kluk stiffness systems, such as actuated FSMs and other high stiffness systems, such as turbance rejection performance [Kluk et al. (2012)]. Piezo Fast steering mirror (FSM) systems [Hedding and Lewis actuated FSMs and other high stiffness systems, such as AFMs [Schitter et al. (2007)], are usually controlled by PI Fast steering mirror (FSM) systems [Hedding and Lewis AFMs [Schitter et al. (2007)], are usually controlled by PI actuated FSMs and systems, such as Fast steering mirror tip/tilt (FSM) mirrors, systemsare [Hedding and Lewis AFMs (1990)], also termed opto-mechatronic [Schitter et al.other (2007)], arestiffness usually controlled by PI controllers, which place the high crossover frequency of the the loop (1990)], also mirrors, are opto-mechatronic controllers, which place the crossover frequency of loop Fast steering mirror tip/tilt (FSM) systems [Hedding and Lewis AFMs [Schitter et al. (2007)], are usually controlled by PI (1990)], also termed termed tip/tilt mirrors, are opto-mechatronic devices that enable tip and tilt motion of a mirror for controllers, which place the crossover frequency of the loop gain below below the the first first fundamental fundamental resonance resonance mode mode of of the the devices that enable tip and tilt motion of mirror for (1990)],pointing also termed are opto-mechatronic controllers, which place the crossover frequency of the devices that enable tip andmirrors, tilt motion ofina a various mirror scifor gain optical andtip/tilt scanning applications gain below theare first fundamental resonance mode ofloop the actuator and well suited for tracking conventionally optical pointing and scanning applications in various sciand are well suited for conventionally devicesand thatcommercial enable and tilt motion ofinafor mirror for actuator gain below the first resonance mode of the optical pointing and tip scanning applications various scientific systems. Applications pointing actuator and are wellfundamental suited for tracking tracking conventionally used raster trajectories [Csencsics et al. al. (2016)]. (2016)]. Their entific and commercial systems. Applications for pointing used raster trajectories [Csencsics et Their optical pointing and scanning applications in various sciactuator and are well suited for tracking conventionally entific and commercial systems. Applications for pointing operations include include beam beam stabilization stabilization in in optical optical systems systems used raster trajectories [Csencsics et limited al. (2016)]. control bandwidth is thereby typically by the theTheir first operations bandwidth is typically limited by first entific et and systems. Applications for pointing used raster trajectories [Csencsics et al.system (2016)]. operations include stabilization in optical systems [Kluk al.commercial (2012)],beam tracking of objects objects and acquisition of control control bandwidth is thereby thereby typically limited by [Ito theTheir first or second mechanical resonance of the and [Kluk et al. (2012)], tracking of and acquisition of second mechanical resonance of system and operations beam in optical systems control bandwidth is thereby typically limited by [Ito the first [Kluk etsignals al.include (2012)], tracking objectsimage and acquisition of or optical [Guelman etstabilization al.of(2004)], (2004)], stabilization or second mechanical resonance of the the system [Ito and Schitter (2016)]. optical signals [Guelman et al. image stabilization (2016)]. [Kluk etsignals al. (2012)], tracking and acquisition of Schitter or second mechanical resonance of the system [Ito and optical [Guelman et al.of(2004)], image stabilization in stare imaging systems [Sun etobjects al. (2015)] (2015)] and adaptive Schitter (2016)]. in stare imaging systems [Sun et al. and adaptive optical signals [Guelman et al. (2004)], image stabilization Various piezo actuated FSMs FSMs are are reported in in literature literature Schitter (2016)]. in stare imaging systems [Sun et al. (2015)] and adaptive optics in in telescopes telescopes [Janssen [Janssen et et al. (2010)]. (2010)]. Applications Applications Various piezo actuated optics piezovalues actuated FSMs range are reported reported in resonance literature in stareinimaging systems [Sun al. (2010)]. (2015)]from and adaptive Various with typical for angular and main optics telescopes [Janssen etet al. al. Applications requiring good scanning properties range scanning with typical values for range and resonance Various actuated FSMs are [Mokbel reported literature requiring good scanning properties range from scanning typical for angular angular range and main main resonance opticssensors in telescopes [Janssen et al. over (2010)]. Applications mode, ofpiezo e.g.values ±1 mrad and 3 kHz kHz etin al. al. (2012)] requiring good scanning properties rangescanning from scanning laser [Schlarp et al. al. (2018)] optical with mode, of e.g. ±1 mrad and 3 [Mokbel et (2012)] with typical values for angular range and main resonance laser sensors [Schlarp et (2018)] over scanning optical mode, of e.g. ±1 mrad and 3 kHz [Mokbel et al. (2012)] requiring good scanning properties range from scanning or ±1.6 mrad and 5.3 kHz [Park et al. (2012)]. Recent relaser sensors [Schlarp et al. (2018)] over scanning optical lithography [Zhou [Zhou Q., Q., et et al. al. (2008)] (2008)] to to confocal confocal microscopy microscopy or ±1.6 and 5.3 kHz [Park et al. (2012)]. Recent remode, ofmrad e.g. were ±1 mrad andon 3extending kHz [Mokbel et al. (2012)] lithography or ±1.6 mrad and 5.3 kHz [Park et al. (2012)]. Recent relaser sensors [Schlarp et al. (2018)] over scanning optical search efforts focused the range of piezo lithography [Zhou Q., et al. (2008)] to confocal microscopy [Yoo et et al. al. (2014)] and and material processing processing [Hedding [Hedding and and search were focused extending the range of piezo or ±1.6efforts mrad and 5.3 kHz on [Park et al. (2012)]. Recent re[Yoo efforts were focused on extending the(2018)] range of piezo lithography [Zhou Q., al. (2008)] to confocal microscopy actuated FSMs up to [Shao or even [Yoo et(1990)]. al. (2014)] (2014)] andetmaterial material processing [Hedding and search Lewis actuated FSMs up focused to ±7 ±7 mrad mrad [Shao et et al. al. (2018)] orpiezo even search efforts were on extending the range of Lewis (1990)]. actuated FSMs up to ±7 mrad [Shao et al. (2018)] or even [Yoo et al. (2014)] and material processing [Hedding and ±17.5 mrad [Wei et al. (2014)] by employing sophisticated Lewis (1990)]. mrad [Wei (2014)] by employing sophisticated actuated FSMs mrad [Shao et al. flexures. (2018)] orThese even FSMs are most most commonly commonly actuated actuated by by either either using using piezopiezo- ±17.5 ±17.5 mrad [Weiupet ettoal. al.±7 (2014)] by employing sophisticated Lewis (1990)]. mechanical amplification structures and FSMs are mechanical amplification structures and flexures. These ±17.5 mrad [Wei et al. (2014)] by employing sophisticated FSMs are mostetcommonly actuated by either using[Csencpiezo- mechanical electric [Tapos al. (2005)] or voice coil actuators amplification structures and flexures. These range extensions extensions however however come come on the the cost cost of of the first first electric [Tapos al. or voice coil actuators [CsencFSMs are mostet actuated by piezo- range mechanical amplification and flexures. electric [Tapos etcommonly al. (2005)] (2005)] ormore voicerecently coileither actuators [Csencsics and Schitter (2017)], and byusing reluctance range extensions however structures come on on the cost of the the1These first resonance modes occuring significantly lower than kHz sics and Schitter (2017)], and more recently by reluctance resonance modes occuring significantly lower than 1 kHz electric et (2017)], al. (2005)] voice coil actuators [Csenc- resonance range et extensions however come on thewhich cost reduces of the1 first sics and[Tapos Schitter andor more recently bycomparably reluctance actuators, which are rather rare due to their modes occuring significantly lower than kHz [Shao al. (2018); Wei et al. (2014)] the actuators, which are rare due to comparably [Shao et al. (2018); Wei et al. (2014)] which reduces the sics and Schitter and more recently by reluctance resonance modes occuring significantly lower than 1 kHz actuators, which (2017)], areetrather rather rare due to their their comparably large size [Csencsics al. (2018)]. Voice coil actuators are [Shao et al. (2018); Wei et al. (2014)] which reduces the achievable closed-loop bandwidth and thus the scan speed. large size [Csencsics et al. (2018)]. Voice coil are [Shao et al.closed-loop bandwidth and the reduces scan actuators, which are rareadue to scan their comparably (2018); Wei et al. (2014)] which the large sizesystems [Csencsics etrather al.require (2018)]. Voice coil actuators actuators used for which large range andareaa achievable achievable closed-loop bandwidth and thus thus scan speed. speed. Key components, components, commonly highlighted in the reported piezo used for systems which require a large scan range and Key commonly highlighted in reported piezo large size [Csencsics et al. (2018)]. Voice coil actuators are achievable closed-loop bandwidth and thus the scanforspeed. used for systems which require a mrad large and scan350 range and a Key moderate bandwidth, such as ±87 Hz [Xiang components, commonly highlighted in reported piezo FSM systems, are the rather complex flexures susmoderate bandwidth, such as ±87 and Hz systems, are the rather complex flexures for susused systems require large scan350 and a FSM Key components, highlighted reported moderate bandwidth, such ±87a mrad mrad and 350 Hz [Xiang [Xiang et al. for (2009)], and which which areastypically typically larger inrange size. PiezoFSM systems, arecommonly the pre-stressing rather complex flexures for suspending the mover mover and the in actuators aspiezo well et al. (2009)], and which are larger in size. Piezopending the and pre-stressing the actuators as well moderate bandwidth, such ±87 mrad and 350 Hzsize [Xiang FSM systems, are the pre-stressing rather structures. complex flexures for suset al. (2009)], andadvantage which areasof typically larger in size. Piezobased FSMs take the small actuator and pending the mover and the actuators as well as the employed amplification They typically based FSMs take advantage of the small actuator size and the employed amplification structures. They typically et al. (2009)], andadvantage which are of typically larger in size. Piezopending the mover and pre-stressing the actuators as well based FSMs take the small actuator size and as result in comparably compact systems that enable a higher as the employed amplification structures. They typically require aa considerable considerable design design effort, effort, are are most commonly commonly result comparably compact that enable a higher based in FSMs takescan advantage ofsystems the small actuator and require as the cut employed structures. They typically result in comparably compact systems that enable higher bandwidth and speed at at a rather rather small scanasize range, require a considerable design effort, are most most EDM and in in amplification general rather challenging tocommonly optimize bandwidth and scan speed a small scan range, EDM cut and general rather challenging to optimize result in comparably compact systems that enable a higher require a considerable design effort, are most commonly bandwidth and scan speed at a rather small scan range, such that they appear suited for the integration in high EDM cut and in general rather challenging to optimize and manufacture [Jing et al. (2015)]. such that they appear suited the integration inrange, high et (2015)]. bandwidth and systems scan speed at for a rather small scan EDMmanufacture cut and in [Jing general rather challenging to optimize such that they appear suited for high and speed scanning [LeLetty etthe al.integration (2004)]. Toinobtain obtain and manufacture [Jing et al. al. (2015)]. speed scanning systems [LeLetty et al. (2004)]. To such that they appear suited for the integration in high This paper presents the system and control control design design of of aa and manufacture [Jing et system al. (2015)]. speed scanning systems [LeLetty et al. (2004)]. To obtain This the required levels of precision, FSM systems, indepenpaper presents the and the required levels of precision, FSM systems, indepenThis paper presents the system and control design of a speed scanning systems [LeLetty et al. (2004)]. To obtain novel high performance piezo-actuated FSM system with the required levels of precision, FSM systems, independent of of the the actuation actuation principle principle and and the the application application class, class, novel high performance FSM system with This paper presents thepiezo-actuated system and control design et ofal. a dent novel high performance piezo-actuated FSM system with the required levels of precision, FSM systems, indepenpiezo actuators in a push-pull configuration [Mokbel dent of the actuation principle and the application class, are most most commonly commonly controlled controlled by by feedback feedback controllers controllers enen- piezo in aa push-pull configuration al. novel actuators high performance piezo-actuated FSM[Mokbel system et with are piezo actuators in push-pull configuration [Mokbel et al. dent of the actuation principle and the application class, are most commonly controlled by feedback controllers enactuators in a push-pull configuration [Mokbel et al. are most © commonly byFederation feedbackofcontrollers en- piezo 2405-8963 2019, IFACcontrolled (International Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Copyright © 2019 IFAC 814 Copyright © under 2019 IFAC 814 Control. Peer review responsibility of International Federation of Automatic Copyright © 814 Copyright © 2019 2019 IFAC IFAC 814 10.1016/j.ifacol.2019.11.689 Copyright © 2019 IFAC 814
2019 IFAC MECHATRONICS 290 Vienna, Austria, Sept. 4-6, 2019
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(2012)], which is centered around an easy-to-manufacture membrane-like flexure to ease the mechanical design challenges, and features an optical sensor system for position measurement. It investigates the resulting system performance when the FSM is operated with a feedback position controller as well as the error when it is tracking a designed high speed raster scanning trajectory. Section 2 describes the system design and elaborates on the various system components and design choices. The system identification is shown in Section 3, followed by the model-based controller design in Section 4. Section 5 evaluates the system performance along a classical raster scan case. Section 6 concludes the paper. 2. FSM SYSTEM DESIGN 2.1 System Overview The proposed FSM system relies on two pairs of piezo stack actuators in push-pull configuration that are aligned along the two system axes and actuate the mover in tip and tilt direction. An overview of the proposed FSM system design is shown in Fig. 1. A suspension membrane is mounted to the top of the system body, suspending the mover and carrying a mirror on its top side. This membrane also provides the preloading of the piezos, which is required for dynamic operation. The actuators are mounted to a piezo mounting plate and are pushing against the backside of the mover at specific contact points. A reflective element is mounted on the backside of the mover to reflect a laser beam of the optical sensor system, which enters the system through the base plate and propagates through the center of the cylindrical system. The reflected beam hits a PSD that measures the position of the laser spot in two dimensions. Mirror Suspension mount
Mover
Mounting screws Suspension
Balls for force transmission Body
Piezo mounting plate
Piezo-Stack actuators
Base plate
Fig. 1. Cross section view of the piezo FSM showing the system components. The mirror and the mover are attached to the suspending flexure membrane and are actuated with two stack actuators per axis. The mover, which is a solid ø 5.5 x 20 mm aluminum cylinder, is designed to ensure sufficient stiffness of the moving part in order to shift structural modes to high frequencies and avoid deformations of the mirror surface during dynamic operation. To provide a force interface 815
from each actuator to the rotary mover that is not able to transmit tensile and shear forces, steel balls are bonded to the bottom of the mover providing a single contact point to the steel plates that are bonded to the top of each piezo actuator enabling the intended rotational motion of the mover. For actuation four PICMA stack actuators (Model P885.91, Physik Instrumente GmbH & Co. KG, Germany) with a length of 36 mm (max. displacement 38 µm) are used and placed in a distance of 6 mm from the system center. Each actuator pair is operated in a push-pull configuration to obtain a pure rotational motion of the mover. In the zero position the actuators are already elongated by half their range, in order to enable bi-directional motion of each actuator [Schitter et al. (2007)] and thus bidirectional rotational movement of the mirror around the zero position. This denotes that at the maximum rotational displacement, one actuator has its minimal, while the other actuator has its maximum elongation. As the piezo actuators can not be used to exert pulling forces on the mover, they have to be pre-stressed in order to maintain contact to the mover when retracting and thus enable dynamic operation. The actuators are bonded to a thin piezo mounting plate (cf. Fig. 1), which is attached to the base plate. This mounting plate restricts lateral movement but enables individual vertical adjustment of the stack actuator to fine align them to the flexure by means of set screws in the base plate. 2.2 Suspension System The suspension system is designed as a metallic membrane which carries the mirror and the mover centrically bonded to its front and backside, respectively. It is clamped onto the body of FSM system with two mounting rings, connecting the static and moving part of the system. It is further used for pre-stressing the piezo actuators for dynamic operation. When increasing the stiffness of the flexure (by making it thicker), the range of the actuators in the pre-stressed case is reduced accordingly [MunnigSchmidt et al. (2014)] kA ∆LA ≈ L0 1 − (1) kA + kS with kA the stiffness of the actuator, kS the stiffness of the flexure, and L0 being the range of the stack actuator in the stress-free case. Making the flexure stiffer on the other hand also increases the frequencies of its structural modes, which is beneficial from a controls perspective. This results in two competing design aspects: (i) the flexure needs to be sufficiently stiff to push the main resonance mode to a high high frequency and (ii) it needs to be compliant enough to not significantly reduce the range of the piezo stacks. After evaluating several membrane-material-configurations on the basis of a static mechanical finite element analysis performed in ANSYS (Ansys Inc., PA, USA) a 0.8 mm brass flexure is chosen for the FSM design, as it provides a good trade-off between range loss and main resonance mode. A modal analysis is used throughout the flexure design to obtain the structural modes of the flexure. The designed flexure shows a piston mode at 3.8 kHz, the
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desired tilt mode at 6.5 kHz, which is depicted in Fig. 2, and a first higher strucutral mode at 28.4 kHz.
Fig. 2. Modal analysis of the flexure membrane. The tilt mode of the flexure occurs at 6.5 kHz. 2.3 System Setup
291
mode two more modes at 3 kHz and 3.6 kHz can be observed. The mode at 3.6 kHz can be matched to the piston mode of the modal analysis, which might be excited due to unsynchronized piezo motion caused by hysteresis or imperfect alignment. The mode at 3 kHz does not appear in the modal analysis but may be explained by the finite stiffness of the upright supporting structure Schitter et al. (2007). Above the main resonance mode the system shows fourth order lowpass characteristics, which reduces the excitation of higher system modes and is due to the piezo amplifier bandwidth. For modeling the measured frequency responses of both axes an 8th order plant model 2 2 2 s + 2ζzi ωzi s + ωzi · e−sT , (2) G(s) = K · i=1 4 2 2 s + 2ζpi ωpi s + ωpi i=1
The entire experimental system setup is depicted in Fig. 3a, showing the FSM system with the mirror mount and the components of the optical sensor system, including the custom made readout electronics. The optical sensor system is located behind the mirror unit and comprises a laser source, a beam splitter and a position sensitive device (PSD; S5990-01, Hamamatsu Photonics, Japan) for detecting displacements of the reflected laser spot in x- and y-direction. A front view of the system with the 10 mm mirror bonded to the flexure membrane is shown in Fig. 3b, while Fig. 3c depicts the backside of the flexure with the mounted mover, the steel balls and the sensor mirror for the optical sensor system, which reflects the incoming laser beam. The piezo actuators are operated in a push-pull configuration and are driven by a piezo amplifier (Techproject EMC GmbH, Austria; output voltage 0-150V) with a measured bandwidth of 10 kHz and a second order lowpass characteristic at higher frequencies. The comparably low amplifier bandwidth is chosen deliberately as it can be directly employed to significantly reduce the excitation of higher modes of the mechanical system. For deriving input signals for the amplifier, acquiring the position signals, and controller implementation a real-time data acquisition system (Type: DS1202, dSPACE GmbH, Germany) is used. The optical sensor system is calibrated by using an external reference metrology system (Confocal sensor confocalDT 2451, Micro-Epsilon Messtechnik GmbH & Co. KG, Germany), which measures the mirror position from the front side and uses a small angle approximation to obtain the angular displacement. 3. SYSTEM IDENTIFICATION For identifying the dynamics of the FSM prototype system a system analyzer (3562A, Hewlett-Packard, Palo Alto, CA, USA) is used. The inputs of the piezo amplifier and the output signals of the optical sensor system are considered as system inputs and outputs, respectively. Figure 4 shows the measured frequency response data of the x- (blue) and y-axis (red) of the system. It can be seen that both system axes show almost identical dynamics, with only slight deviations resulting most likely from mounting tolerances. The main resonance mode occurs at 6.7 kHz and shows good agreement with the predicted value from the flexure design in Section 2.2. Below this 816
with gain K = 5.82e16, time delay T = 16.7 µs (sampling delay of the dSpace system), and parameters according to Table 1 is obtained. Crosstalk measurements (data not Table 1. Coefficients of the FSM system model G(s). Index z1 z2 p1 p2 p3 p4
ωIndex [rad/s] 2.05e4 2.41e4 1.93e4 2.29e4 4.26e4 5.03e4
ζIndex 0.03 0.02 0.05 0.02 0.05 0.44
shown) revealed that the crosstalk between the axes at DC is more than 30 dB smaller than the magnitudes of the individual system axes, which justifies the application of one single-input-single-output (SISO) feedback controller per axis. The mechanical system range and hysteresis of the stack actuators are measured by applying a 1 Hz sinusoidal signal with 10 V amplitude to the piezo amplifier input of one system axis at a time. The measurement results (data not shown) reveal a mechanical range of ±2.4 mrad (±4.8 mrad optical) in tip and tilt direction, as well as a maximum hysteresis of about 15% for both system axes. 4. CONTROLLER DESIGN The design of the feedback controller is done with the aim of maximizing the closed-loop system bandwidth for high speed scanning operation. 4.1 PI Controller with Notch Filters (PI+ Controller) As most commercial piezo actuated FSM systems use feedback bandwidth controllers for position control of the mirror, a PI based controller design is taken as starting point for both axes. PI controllers with a crossover frequency below the first fundamental resonance mode of the actuator are also applied in many other piezo actuated high stiffness systems, such as AFMs [Schitter et al. (2007)], for tracking raster trajectories with their various frequency components [Fleming and Wills (2009)]. The P-gain is used to vertically shift the loop gain and adds artificial stiffness
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BNC connectors
FSM
PSD board Steel balls
Beam splitter
Sensor mirror
Flexure
Mirror mount Mirror
Laser source
(a)
(b)
(c)
Magnitude [dB]
Fig. 3. Experimental FSM system setup. (a) depicts an overview of the entire setup with the FSM, its mounting plate and the optical position sensor system with laser source, beam splitter and PSD board. (b) shows a front view of the FSM, with the mirror, the flexure membrane and the BNC connectors for the actuators observable. (c) shows the backside of the mover with the mounted mover, the steel balls and the sensor mirror. 3 2 2 s + 2d w ω s + ω n n n n 0 (s + ω4 ) i=1 · (4) C(s) = K · 3 2 −20 s s + 2wn ωn s + ωn2 −40 −60 −80
i=1
with K = 50 and parameters according to Table 2, is designed for a crossover frequency of 1.4 kHz. The
X−axis Y−axis Model
Table 2. Coefficients of the designed PI+ controller.
−100
Phase [°]
0 −100 −200 −300 −400 −500 1 10
2
10
3
10 Frequency [Hz]
4
10
Fig. 4. Frequency response of the FSM x- (solid blue) and y-axis (solid red) measured with the optical sensor system. The fitted system model (dashed black) models the first three system resonances at 3, 3.6 and 6.7 kHz. to the system. The integrator increases the loop gain at low frequencies, enforcing high tracking performance and zero steady state error. Looking at the mode at 6.7 kHz and its peak value of -16.9 dB the achievable crossover frequency would be limited to about 75 Hz when using solely a PI controller. To extend the bandwidth three tuned notch filters [Munnig-Schmidt et al. (2014)] (also see Fig. 5) of the form N (s) =
s2 + 2dn wn ωn s + ωn2 , s2 + 2wn ωn s + ωn2
(3)
with dn the depth, wn the width of the notch and ωn the frequency at which the notch is placed, are designed and added to the controller to cancel the first three resonance peaks identified in Section 3. In a loop shaping approach, aiming to maximize the system bandwidth, the PI+ controller 817
Index
ωIndex [rad/s]
dIndex
wIndex
1 2 3 4
1.94e4 2.29e4 4.26e4 1e4
0.5 0.5 0.025 -
0.2 0.1 2 -
simulated open loop response of the designed controller and the derived plant model shows a gain margin of 6.4 dB and a phase margin of 63◦ . The simulated closed-loop bandwidth is 3 kHz. As the dynamics of both system axes are similar, the same controller can be applied to both system axes. 4.2 Controller Implementation For implementation on the dSpace system the controllers are discretized using Pole-Zero-Matching for a sampling frequency of fs = 60 kHz [Franklin et al. (1997)]. The poles and zeros are directly transformed to the discrete time domain by using the relation z = es/fs to guarantee that the notch filter are located exactly at the desired frequencies. Fig. 5 shows the frequency responses of the implemented PI+ controller together with the frequency responses of its PI and notch filter components. The PI component (dashed blue) shows integrating behavior at low frequencies and constant gain at high frequencies. The first two notch filters N1 (s) and N2 (s) (dotted and dashed black) have the same depth value d1 = d2 = 0.5 (w1 = 0.2, w2 = 0.1), are located at 3 and 3.6 kHz, respectively, and are used to cancel the two small resonance peaks at these frequencies. The third notch filter N3 (s) (dashdotted black) is placed at 6.7 kHz and is significantly deeper and wider (d3 = 0.025, w3 = 2) in order to notch out the main resonance mode of the FSM dynamics. The
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Fig. 5. Measured frequency response of the implemented PI+ controller (solid red). It is designed for a crossover frequency of 1.4 kHz, comprising a PI component (dashed blue), as well as notch filters (black) at 3 (N1 (s)), 3.6 (N2 (s)) and 6.7 kHz (N3 (s)). 5. EVALUATION OF SYSTEM PERFORMANCE For evaluation of the system performance, the closedloop system dynamics, the tracking performance, and the positioning uncertainty are investigated. For testing the tracking performance a raster trajectory with high spatial and temporal resolution, which has its first 11 harmonics covered by the closed-loop bandwidth, is obtained by selecting f1R = 250 Hz and f2R = 0.5 Hz for the fast and slow scan axis, respectively. The resulting spatial resolution is 4 µrad/mrad at a frame rate of 1 frame/s. The measured complimentary sensitivity function of the FSM system with the PI+ controller is shown in Fig. 6, together with the modeled closed-loop system dynamics. The measured frequency response shows good agreement with the modeled response. The resulting system bandwidth 2.7 kHz matches well with the 3 kHz predicted by the model. The measured dynamics show a small gain peaking of 2.5 dB at 1.9 kHz, which is due to the relatively large phase margin. To evaluate the tracking performance of the closed-loop system, the designed raster trajectory is applied with a scan amplitude of 2.2 mrad and tracked with the PI+ controllers. The measured signals are post-processed by low-pass filters with a bandwidth of 20 kHz for reducing high frequency noise. The results of the fast scan axis are depicted in Fig. 7. The tracking error for the entire trajectory results to 73.8µrad rms (3.6% of the scan 818
Fig. 6. Modeled (solid blue) and measured (dashed red) complimentary sensitivity function the FSM with the PI+ controller. The measured dynamics shows a control bandwidth of 2.7 kHz. amplitude) and 0.61 mrad peak-to-valley error, with a controller output of 71.9 V rms. Disregarding the data around the turning points of the fast axis (80% threshold; see Fig. 7) where the tracking error is largest, the tracking error in the region of interest results to 41.4µrad rms (1.9% of the scan amplitude) and 0.48 mrad peak-to-valley. The resulting positioning uncertainty of the FSM system controlled by the PI+ controller is determined at zero reference applied to both axes and is as small as 3.8 µrad rms, which is 0.08% of the full scale range. In summary it is successfully demonstrated that the proposed piezo actuated FSM with the novel membrane flexure, optical position sensor system and ±2.4 mrad mechanical angular range achieves a closed-loop bandwidth of 2.7 kHz with the PI+ controller, and shows good tracking performance as well as a low positioning uncertainty. 6. CONCLUSION This paper presents a novel piezo-actuated FSM system design with an optical position sensor system for high speed scanning applications. The design is centered around an easy-to-manufacture membrane-like metallic flexure and two pairs of stack actuators that are operated in a push-pull configuration. It provides an mechanical angular range of ±2.4 mrad (±4.8 mrad optically) in both axes, while achieving a first fundamental resonance mode of the actuator as high as 6.7 kHz. After a detailed system description and analysis, a model-based PI+ controller for maximizing the system bandwidth and tracking of raster trajectories is designed. The resulting closed-loop system has a bandwidth as high as 2.7 kHz and a small positioning uncertainty of 3.8 µrad rms. ACKNOWLEDGEMENTS The financial support by the Christian Doppler Research Association, the Austrian Federal Ministry for Digital and
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Fig. 7. Measured raster scan trajectory. The reference (dashed blue), the measured position signal (solid red) and the tracking error (solid black) of the fast scan axis tracking a 250 Hz triangular signal are shown. Economic Affairs, and the National Foundation for Research, Technology and Development, as well as MICROEPSILON MESSTECHNIK GmbH & Co. KG and ATENSOR Engineering and Technology Systems GmbH is gratefully acknowledged. REFERENCES Csencsics, E., Saathof, R., and Schitter, G. (2016). Design of a dual-tone controller for lissajous-based scanning of fast steering mirrors. American Control Conference, Boston, MA, USA. Csencsics, E. and Schitter, G. (2017). System design and control of a resonant fast steering mirror for lissajousbased scanning. IEEE TMECH, 22(5), 1963–1972. Csencsics, E., Schlarp, J., and Schitter, G. (2018). Highperformance hybrid-reluctance-force-based tip/tilt system: Design, control, and evaluation. IEEE TMECH, 23(5), 2494–2502. Fleming, A.J. and Wills, A.G. (2009). Optimal periodic trajectzories for band limited systems. IEEE TCST, 17(3), 552–562. Franklin, G.F., Powell, D.J., and Workman, M.L. (1997). Digital Control of Dynamic Systems. Prentice Hall. Guelman, M., Kogan, A., Livne, A., Orenstein, M., and Michalik, H. (2004). Acquisition and pointing control for inter-satellite laser communications. IEEE Transactions on Aerospace and Electronic Systems, 40(4), 1239. Hedding, L.R. and Lewis, R.A. (1990). Fast steering mirror design and performance for stabilization and single axis scanning. SPIE 340, 1304, 14–24. Ito, S. and Schitter, G. (2016). Comparison and classification of high-precision actuators based on stiffness influencing vibration isolation. IEEE TMECH, 21(2), 1169–1179. 819
Janssen, H., Teuwen, M., Navarro, R., Tromp, N., Elswijk, E., and Hanenburg, H. (2010). Design and prototype performance of an innovative cryogenic tip–tilt mirror. Modern Technologies in Space and Ground-based Telescopes, Proceedings of the SPIE, 77394A. Jing, Z., Xu, M., and Feng, B. (2015). Modeling and optimization of a novel twoaxis mirror-scanning mechanism driven by piezoelectric actuators. Smart Materials and Structures, 24(2), 025002. Kluk, D.J., Boulet, M.T., and Trumper, D.L. (2012). A high-bandwidth, high-precision, two-axis steering mirror with moving iron actuator. Mechatronics, 22(3), 257. LeLetty, R., Barillot, F., Fabbro, H., Claeyssen, F., Guay, P., and Cadiergues, L. (2004). Miniature piezo mechanisms for optical and space applications. 9th International Conference on New Actuators, Germany. Mokbel, H.F., Yuan, W., Ying, L.Q., Hua, C.G., and Roshdy, A.A. (2012). Research on the mechanical design of two-axis fast steering mirror for optical beam guidance. International Conference on Mechanical Engineering and Material Science. Munnig-Schmidt, R., Schitter, G., Rankers, A., and van Eijk, J. (2014). The Design of High Performance Mechatronics. Delft University Press, 2nd edition. Park, J.H., Lee, H.S., Lee, J.H., Yun, S.N., Ham, Y.B., and Yun, D.W. (2012). Design of a piezoelectric-driven tilt mirror for a fast laser scanner. Japanese Journal of Applied Physics, 51(9S2), 09MD14. Schitter, G., Astrom, K.J., DeMartini, B.E., Thurner, P.J., Turner, K.L., and Hansma, P.K. (2007). Design and modeling of a high-speed afm-scanner. IEEE TCST, 15(5), 906–915. Schlarp, J., Csencsics, E., and Schitter, G. (2018). Optical scanning of laser line sensors for 3d imaging. Applied Optics, 57(18), 5242–5248. Shao, S., Tian, Z., Song, S., and Xu, M. (2018). Twodegrees-of-freedom piezo-driven fast steering mirror with cross-axis decoupling capability. Review of Scientific Instruments, 89(5), 055003. Sun, C., Ding, Y., Wang, D., and Tian, D. (2015). Backscanning step and stare imagingsystem with high frame rate and wide coverage. Applied Optics, 54(16), 4960–4965. Tapos, F.M., Edinger, D.J., Hilby, T.R., Ni, M.S., Holmes, B.C., and Stubbs, D.M. (2005). High bandwidth fast steering mirror. Proceedings of SPIE, 5877. Wei, C., Sihai, C., Xin, W., and Dong, L. (2014). A new two-dimensional fast steering mirror based on piezoelectric actuators. 4th IEEE International Conference on Information Science and Technology, 308–311. Xiang, S., Wang, P., Chen, S., Wu, X., Xiao, D., and Zheng, X. (2009). The research of a novel single mirror 2d laser scanner. Proc. of SPIE, 7382. Yoo, H., van Royen, M.E., van Cappellen, W.A., Houtsmuller, A.B., Verhaegen, M., and Schitter, G. (2014). Automated spherical aberration correction in scanning confocal microscopy. Review of Scientific Instruments, 85, 123706. Zhou Q., et al. (2008). Design of fast and steering mirror and systems for precision and laser beams and steering. IEEE Workshop on Robotic and Sensors Environments.
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A A A A
Fast Fast Fast Fast
Piezo Actuated Tip/Tilt Mirror Piezo Actuated Tip/Tilt Mirror Piezo Actuated Tip/Tilt Mirror raster scan applications Piezo Actuated Tip/Tilt Mirror raster scan applications raster scan applications raster scan applications E. Csencsics, B. Sitz, G. Schitter
for for for for
E. Csencsics, B. Sitz, G. Schitter E. Csencsics, B. Sitz, G. Schitter E.Laboratory Csencsics, Sitz, G.Engineering Schitter for Automated Christian Doppler forB.Precision Precision Christian Doppler Laboratory for Engineering for Automated Christian Doppler Laboratory for Precision Engineering for Automated In-line Metrology, Metrology, Automation Automation and and Control Control Institute Institute (ACIN), (ACIN), Vienna In-line Vienna Christian Doppler of Laboratory for Precision Engineering for Automated University Technology, 1040 Vienna, Austria (e-mail: In-lineUniversity Metrology, Automation and Control Institute (ACIN), Vienna of Technology, 1040 Vienna, Austria (e-mail: In-lineUniversity Metrology,ofAutomation Control Institute (ACIN), [email protected]). Technology,and 1040 Vienna, Austria (e-mail:Vienna University of [email protected]). Technology, 1040 Vienna, Austria (e-mail: [email protected]). [email protected]). Abstract: Abstract: This This paper paper presents presents the the system system and and control control design design of of aa novel novel high high performance performance piezopiezoAbstract: This paper mirror presents(FSM) the system andspeed control designscanning of a novelwhich high performance piezoactuated fast steering for high optical is centered centered around around actuated fast steering mirror (FSM) scanning which for high speed optical is Abstract: This paper presents the system and control design of a novel high performance piezoactuated fast steering mirror (FSM) for high speed optical scanning which is centered around an easy-to-manufacture membrane-like mechanical flexure and uses optical system for an easy-to-manufacture membrane-like mechanical flexure and uses an anwhich opticalissensor sensor system for actuated fast steering mirror (FSM) formechanical high speedflexure optical scanning centered around position measurement. With the membrane flexure the first fundamental resonance mode of the the an easy-to-manufacture membrane-like and uses an optical sensor system for position measurement. With the membrane flexure the first fundamental resonance mode of an easy-to-manufacture membrane-like mechanical flexure and uses an optical sensor system for fundamental resonance mode of the actuator is as while enabling an angular range ±4.8 position measurement. With thekHz, membrane flexure the first actuator is placed placed as as high high as 6.7 6.7 kHz, while still still enabling an optical optical angular range of of mode ±4.8 mrad. mrad. position measurement. With the membrane flexure the first fundamental resonance of the actuator is placed as high as 6.7 kHz, while still enabling an optical angular range of ±4.8 mrad. Using a controller together with three notch filters in loop approach for Using a PI PI controller together three notch filters designed designed in aaangular loop shaping shaping for actuator is placed as high as 6.7with kHz, while still enabling an optical range ofapproach ±4.8 mrad. Using a PI controller together with three notch filters designed in araster loop trajectories, shaping approach for maximizing the system bandwidth and tracking of high resolution aa closedmaximizing the system bandwidth and tracking of highdesigned resolution raster trajectories, closedUsing a PI controller together with three notch filters in a loop shaping approach for loop bandwidth as high as 2.7 kHz and a positioning uncertainty of 3.8 µrad rms is obtained. maximizing the system bandwidth and tracking of high resolution raster trajectories, a closedloop bandwidth as highbandwidth as 2.7 kHzand andtracking a positioning uncertainty of 3.8 µrad rms is obtained. maximizing the system of high resolution raster trajectories, a closedThe tracking aa 250 raster aa scan loop bandwidth as higherror as 2.7when kHz tracking and a positioning of 3.8trajectory µrad rms with is obtained. The obtained obtained tracking when 250 Hz Hzuncertainty raster scanning scanning scan loop bandwidth as highiserror as 2.7 kHzastracking and a µrad positioning uncertainty of 3.8trajectory µrad rms with is obtained. The obtained tracking error when tracking a 250 Hz raster scanning trajectory with a scan amplitude of 2.2 mrad as small 73.8 rms (3.6% of scan amplitude). amplitude of 2.2 mrad is as small as 73.8 µrad rms (3.6% of scan amplitude). The obtained tracking whenastracking a 250 raster scanning trajectory with a scan amplitude of 2.2 mrad iserror as small 73.8 µrad rms Hz (3.6% of scan amplitude). © 2019, IFAC (International Federation Automatic Control) Hosting by amplitude). Elsevier Ltd. All rights reserved. amplitude of 2.2 mrad is as small asof73.8 µrad rms (3.6% of scan Keywords: Keywords: Fast Fast steering steering mirror, mirror, System System analysis analysis and and design, design, Motion Motion control, control, Piezo Piezo actuation, actuation, Keywords: Fast steering mirror, System analysis and design, Motion control, Piezo actuation, Optomechatronics Optomechatronics Keywords: Fast steering mirror, System analysis and design, Motion control, Piezo actuation, Optomechatronics Optomechatronics 1. INTRODUCTION abling aa high high control control bandwidth bandwidth for for good tracking tracking and and disdis1. abling 1. INTRODUCTION INTRODUCTION abling a high control bandwidth [Kluk for good good tracking and disturbance rejection performance et al. (2012)]. Piezo turbance rejection performance [Kluk et (2012)]. Piezo 1. INTRODUCTION abling a high control bandwidth for good tracking and disturbance rejection et al. al. (2012)]. Piezo actuated FSMs andperformance other high [Kluk stiffness systems, such as actuated FSMs and other high stiffness systems, such as turbance rejection performance [Kluk et al. (2012)]. Piezo Fast steering mirror (FSM) systems [Hedding and Lewis actuated FSMs and other high stiffness systems, such as AFMs [Schitter et al. (2007)], are usually controlled by PI Fast steering mirror (FSM) systems [Hedding and Lewis AFMs [Schitter et al. (2007)], are usually controlled by PI actuated FSMs and systems, such as Fast steering mirror tip/tilt (FSM) mirrors, systemsare [Hedding and Lewis AFMs (1990)], also termed opto-mechatronic [Schitter et al.other (2007)], arestiffness usually controlled by PI controllers, which place the high crossover frequency of the the loop (1990)], also mirrors, are opto-mechatronic controllers, which place the crossover frequency of loop Fast steering mirror tip/tilt (FSM) systems [Hedding and Lewis AFMs [Schitter et al. (2007)], are usually controlled by PI (1990)], also termed termed tip/tilt mirrors, are opto-mechatronic devices that enable tip and tilt motion of a mirror for controllers, which place the crossover frequency of the loop gain below below the the first first fundamental fundamental resonance resonance mode mode of of the the devices that enable tip and tilt motion of mirror for (1990)],pointing also termed are opto-mechatronic controllers, which place the crossover frequency of the devices that enable tip andmirrors, tilt motion ofina a various mirror scifor gain optical andtip/tilt scanning applications gain below theare first fundamental resonance mode ofloop the actuator and well suited for tracking conventionally optical pointing and scanning applications in various sciand are well suited for conventionally devicesand thatcommercial enable and tilt motion ofinafor mirror for actuator gain below the first resonance mode of the optical pointing and tip scanning applications various scientific systems. Applications pointing actuator and are wellfundamental suited for tracking tracking conventionally used raster trajectories [Csencsics et al. al. (2016)]. (2016)]. Their entific and commercial systems. Applications for pointing used raster trajectories [Csencsics et Their optical pointing and scanning applications in various sciactuator and are well suited for tracking conventionally entific and commercial systems. Applications for pointing operations include include beam beam stabilization stabilization in in optical optical systems systems used raster trajectories [Csencsics et limited al. (2016)]. control bandwidth is thereby typically by the theTheir first operations bandwidth is typically limited by first entific et and systems. Applications for pointing used raster trajectories [Csencsics et al.system (2016)]. operations include stabilization in optical systems [Kluk al.commercial (2012)],beam tracking of objects objects and acquisition of control control bandwidth is thereby thereby typically limited by [Ito theTheir first or second mechanical resonance of the and [Kluk et al. (2012)], tracking of and acquisition of second mechanical resonance of system and operations beam in optical systems control bandwidth is thereby typically limited by [Ito the first [Kluk etsignals al.include (2012)], tracking objectsimage and acquisition of or optical [Guelman etstabilization al.of(2004)], (2004)], stabilization or second mechanical resonance of the the system [Ito and Schitter (2016)]. optical signals [Guelman et al. image stabilization (2016)]. [Kluk etsignals al. (2012)], tracking and acquisition of Schitter or second mechanical resonance of the system [Ito and optical [Guelman et al.of(2004)], image stabilization in stare imaging systems [Sun etobjects al. (2015)] (2015)] and adaptive Schitter (2016)]. in stare imaging systems [Sun et al. and adaptive optical signals [Guelman et al. (2004)], image stabilization Various piezo actuated FSMs FSMs are are reported in in literature literature Schitter (2016)]. in stare imaging systems [Sun et al. (2015)] and adaptive optics in in telescopes telescopes [Janssen [Janssen et et al. (2010)]. (2010)]. Applications Applications Various piezo actuated optics piezovalues actuated FSMs range are reported reported in resonance literature in stareinimaging systems [Sun al. (2010)]. (2015)]from and adaptive Various with typical for angular and main optics telescopes [Janssen etet al. al. Applications requiring good scanning properties range scanning with typical values for range and resonance Various actuated FSMs are [Mokbel reported literature requiring good scanning properties range from scanning typical for angular angular range and main main resonance opticssensors in telescopes [Janssen et al. over (2010)]. Applications mode, ofpiezo e.g.values ±1 mrad and 3 kHz kHz etin al. al. (2012)] requiring good scanning properties rangescanning from scanning laser [Schlarp et al. al. (2018)] optical with mode, of e.g. ±1 mrad and 3 [Mokbel et (2012)] with typical values for angular range and main resonance laser sensors [Schlarp et (2018)] over scanning optical mode, of e.g. ±1 mrad and 3 kHz [Mokbel et al. (2012)] requiring good scanning properties range from scanning or ±1.6 mrad and 5.3 kHz [Park et al. (2012)]. Recent relaser sensors [Schlarp et al. (2018)] over scanning optical lithography [Zhou [Zhou Q., Q., et et al. al. (2008)] (2008)] to to confocal confocal microscopy microscopy or ±1.6 and 5.3 kHz [Park et al. (2012)]. Recent remode, ofmrad e.g. were ±1 mrad andon 3extending kHz [Mokbel et al. (2012)] lithography or ±1.6 mrad and 5.3 kHz [Park et al. (2012)]. Recent relaser sensors [Schlarp et al. (2018)] over scanning optical search efforts focused the range of piezo lithography [Zhou Q., et al. (2008)] to confocal microscopy [Yoo et et al. al. (2014)] and and material processing processing [Hedding [Hedding and and search were focused extending the range of piezo or ±1.6efforts mrad and 5.3 kHz on [Park et al. (2012)]. Recent re[Yoo efforts were focused on extending the(2018)] range of piezo lithography [Zhou Q., al. (2008)] to confocal microscopy actuated FSMs up to [Shao or even [Yoo et(1990)]. al. (2014)] (2014)] andetmaterial material processing [Hedding and search Lewis actuated FSMs up focused to ±7 ±7 mrad mrad [Shao et et al. al. (2018)] orpiezo even search efforts were on extending the range of Lewis (1990)]. actuated FSMs up to ±7 mrad [Shao et al. (2018)] or even [Yoo et al. (2014)] and material processing [Hedding and ±17.5 mrad [Wei et al. (2014)] by employing sophisticated Lewis (1990)]. mrad [Wei (2014)] by employing sophisticated actuated FSMs mrad [Shao et al. flexures. (2018)] orThese even FSMs are most most commonly commonly actuated actuated by by either either using using piezopiezo- ±17.5 ±17.5 mrad [Weiupet ettoal. al.±7 (2014)] by employing sophisticated Lewis (1990)]. mechanical amplification structures and FSMs are mechanical amplification structures and flexures. These ±17.5 mrad [Wei et al. (2014)] by employing sophisticated FSMs are mostetcommonly actuated by either using[Csencpiezo- mechanical electric [Tapos al. (2005)] or voice coil actuators amplification structures and flexures. These range extensions extensions however however come come on the the cost cost of of the first first electric [Tapos al. or voice coil actuators [CsencFSMs are mostet actuated by piezo- range mechanical amplification and flexures. electric [Tapos etcommonly al. (2005)] (2005)] ormore voicerecently coileither actuators [Csencsics and Schitter (2017)], and byusing reluctance range extensions however structures come on on the cost of the the1These first resonance modes occuring significantly lower than kHz sics and Schitter (2017)], and more recently by reluctance resonance modes occuring significantly lower than 1 kHz electric et (2017)], al. (2005)] voice coil actuators [Csenc- resonance range et extensions however come on thewhich cost reduces of the1 first sics and[Tapos Schitter andor more recently bycomparably reluctance actuators, which are rather rare due to their modes occuring significantly lower than kHz [Shao al. (2018); Wei et al. (2014)] the actuators, which are rare due to comparably [Shao et al. (2018); Wei et al. (2014)] which reduces the sics and Schitter and more recently by reluctance resonance modes occuring significantly lower than 1 kHz actuators, which (2017)], areetrather rather rare due to their their comparably large size [Csencsics al. (2018)]. Voice coil actuators are [Shao et al. (2018); Wei et al. (2014)] which reduces the achievable closed-loop bandwidth and thus the scan speed. large size [Csencsics et al. (2018)]. Voice coil are [Shao et al.closed-loop bandwidth and the reduces scan actuators, which are rareadue to scan their comparably (2018); Wei et al. (2014)] which the large sizesystems [Csencsics etrather al.require (2018)]. Voice coil actuators actuators used for which large range andareaa achievable achievable closed-loop bandwidth and thus thus scan speed. speed. Key components, components, commonly highlighted in the reported piezo used for systems which require a large scan range and Key commonly highlighted in reported piezo large size [Csencsics et al. (2018)]. Voice coil actuators are achievable closed-loop bandwidth and thus the scanforspeed. used for systems which require a mrad large and scan350 range and a Key moderate bandwidth, such as ±87 Hz [Xiang components, commonly highlighted in reported piezo FSM systems, are the rather complex flexures susmoderate bandwidth, such as ±87 and Hz systems, are the rather complex flexures for susused systems require large scan350 and a FSM Key components, highlighted reported moderate bandwidth, such ±87a mrad mrad and 350 Hz [Xiang [Xiang et al. for (2009)], and which which areastypically typically larger inrange size. PiezoFSM systems, arecommonly the pre-stressing rather complex flexures for suspending the mover mover and the in actuators aspiezo well et al. (2009)], and which are larger in size. Piezopending the and pre-stressing the actuators as well moderate bandwidth, such ±87 mrad and 350 Hzsize [Xiang FSM systems, are the pre-stressing rather structures. complex flexures for suset al. (2009)], andadvantage which areasof typically larger in size. Piezobased FSMs take the small actuator and pending the mover and the actuators as well as the employed amplification They typically based FSMs take advantage of the small actuator size and the employed amplification structures. They typically et al. (2009)], andadvantage which are of typically larger in size. Piezopending the mover and pre-stressing the actuators as well based FSMs take the small actuator size and as result in comparably compact systems that enable a higher as the employed amplification structures. They typically require aa considerable considerable design design effort, effort, are are most commonly commonly result comparably compact that enable a higher based in FSMs takescan advantage ofsystems the small actuator and require as the cut employed structures. They typically result in comparably compact systems that enable higher bandwidth and speed at at a rather rather small scanasize range, require a considerable design effort, are most most EDM and in in amplification general rather challenging tocommonly optimize bandwidth and scan speed a small scan range, EDM cut and general rather challenging to optimize result in comparably compact systems that enable a higher require a considerable design effort, are most commonly bandwidth and scan speed at a rather small scan range, such that they appear suited for the integration in high EDM cut and in general rather challenging to optimize and manufacture [Jing et al. (2015)]. such that they appear suited the integration inrange, high et (2015)]. bandwidth and systems scan speed at for a rather small scan EDMmanufacture cut and in [Jing general rather challenging to optimize such that they appear suited for high and speed scanning [LeLetty etthe al.integration (2004)]. Toinobtain obtain and manufacture [Jing et al. al. (2015)]. speed scanning systems [LeLetty et al. (2004)]. To such that they appear suited for the integration in high This paper presents the system and control control design design of of aa and manufacture [Jing et system al. (2015)]. speed scanning systems [LeLetty et al. (2004)]. To obtain This the required levels of precision, FSM systems, indepenpaper presents the and the required levels of precision, FSM systems, indepenThis paper presents the system and control design of a speed scanning systems [LeLetty et al. (2004)]. To obtain novel high performance piezo-actuated FSM system with the required levels of precision, FSM systems, independent of of the the actuation actuation principle principle and and the the application application class, class, novel high performance FSM system with This paper presents thepiezo-actuated system and control design et ofal. a dent novel high performance piezo-actuated FSM system with the required levels of precision, FSM systems, indepenpiezo actuators in a push-pull configuration [Mokbel dent of the actuation principle and the application class, are most most commonly commonly controlled controlled by by feedback feedback controllers controllers enen- piezo in aa push-pull configuration al. novel actuators high performance piezo-actuated FSM[Mokbel system et with are piezo actuators in push-pull configuration [Mokbel et al. dent of the actuation principle and the application class, are most commonly controlled by feedback controllers enactuators in a push-pull configuration [Mokbel et al. are most © commonly byFederation feedbackofcontrollers en- piezo 2405-8963 2019, IFACcontrolled (International Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Copyright © 2019 IFAC 814 Copyright © under 2019 IFAC 814 Control. Peer review responsibility of International Federation of Automatic Copyright © 814 Copyright © 2019 2019 IFAC IFAC 814 10.1016/j.ifacol.2019.11.689 Copyright © 2019 IFAC 814
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(2012)], which is centered around an easy-to-manufacture membrane-like flexure to ease the mechanical design challenges, and features an optical sensor system for position measurement. It investigates the resulting system performance when the FSM is operated with a feedback position controller as well as the error when it is tracking a designed high speed raster scanning trajectory. Section 2 describes the system design and elaborates on the various system components and design choices. The system identification is shown in Section 3, followed by the model-based controller design in Section 4. Section 5 evaluates the system performance along a classical raster scan case. Section 6 concludes the paper. 2. FSM SYSTEM DESIGN 2.1 System Overview The proposed FSM system relies on two pairs of piezo stack actuators in push-pull configuration that are aligned along the two system axes and actuate the mover in tip and tilt direction. An overview of the proposed FSM system design is shown in Fig. 1. A suspension membrane is mounted to the top of the system body, suspending the mover and carrying a mirror on its top side. This membrane also provides the preloading of the piezos, which is required for dynamic operation. The actuators are mounted to a piezo mounting plate and are pushing against the backside of the mover at specific contact points. A reflective element is mounted on the backside of the mover to reflect a laser beam of the optical sensor system, which enters the system through the base plate and propagates through the center of the cylindrical system. The reflected beam hits a PSD that measures the position of the laser spot in two dimensions. Mirror Suspension mount
Mover
Mounting screws Suspension
Balls for force transmission Body
Piezo mounting plate
Piezo-Stack actuators
Base plate
Fig. 1. Cross section view of the piezo FSM showing the system components. The mirror and the mover are attached to the suspending flexure membrane and are actuated with two stack actuators per axis. The mover, which is a solid ø 5.5 x 20 mm aluminum cylinder, is designed to ensure sufficient stiffness of the moving part in order to shift structural modes to high frequencies and avoid deformations of the mirror surface during dynamic operation. To provide a force interface 815
from each actuator to the rotary mover that is not able to transmit tensile and shear forces, steel balls are bonded to the bottom of the mover providing a single contact point to the steel plates that are bonded to the top of each piezo actuator enabling the intended rotational motion of the mover. For actuation four PICMA stack actuators (Model P885.91, Physik Instrumente GmbH & Co. KG, Germany) with a length of 36 mm (max. displacement 38 µm) are used and placed in a distance of 6 mm from the system center. Each actuator pair is operated in a push-pull configuration to obtain a pure rotational motion of the mover. In the zero position the actuators are already elongated by half their range, in order to enable bi-directional motion of each actuator [Schitter et al. (2007)] and thus bidirectional rotational movement of the mirror around the zero position. This denotes that at the maximum rotational displacement, one actuator has its minimal, while the other actuator has its maximum elongation. As the piezo actuators can not be used to exert pulling forces on the mover, they have to be pre-stressed in order to maintain contact to the mover when retracting and thus enable dynamic operation. The actuators are bonded to a thin piezo mounting plate (cf. Fig. 1), which is attached to the base plate. This mounting plate restricts lateral movement but enables individual vertical adjustment of the stack actuator to fine align them to the flexure by means of set screws in the base plate. 2.2 Suspension System The suspension system is designed as a metallic membrane which carries the mirror and the mover centrically bonded to its front and backside, respectively. It is clamped onto the body of FSM system with two mounting rings, connecting the static and moving part of the system. It is further used for pre-stressing the piezo actuators for dynamic operation. When increasing the stiffness of the flexure (by making it thicker), the range of the actuators in the pre-stressed case is reduced accordingly [MunnigSchmidt et al. (2014)] kA ∆LA ≈ L0 1 − (1) kA + kS with kA the stiffness of the actuator, kS the stiffness of the flexure, and L0 being the range of the stack actuator in the stress-free case. Making the flexure stiffer on the other hand also increases the frequencies of its structural modes, which is beneficial from a controls perspective. This results in two competing design aspects: (i) the flexure needs to be sufficiently stiff to push the main resonance mode to a high high frequency and (ii) it needs to be compliant enough to not significantly reduce the range of the piezo stacks. After evaluating several membrane-material-configurations on the basis of a static mechanical finite element analysis performed in ANSYS (Ansys Inc., PA, USA) a 0.8 mm brass flexure is chosen for the FSM design, as it provides a good trade-off between range loss and main resonance mode. A modal analysis is used throughout the flexure design to obtain the structural modes of the flexure. The designed flexure shows a piston mode at 3.8 kHz, the
2019 IFAC MECHATRONICS Vienna, Austria, Sept. 4-6, 2019
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desired tilt mode at 6.5 kHz, which is depicted in Fig. 2, and a first higher strucutral mode at 28.4 kHz.
Fig. 2. Modal analysis of the flexure membrane. The tilt mode of the flexure occurs at 6.5 kHz. 2.3 System Setup
291
mode two more modes at 3 kHz and 3.6 kHz can be observed. The mode at 3.6 kHz can be matched to the piston mode of the modal analysis, which might be excited due to unsynchronized piezo motion caused by hysteresis or imperfect alignment. The mode at 3 kHz does not appear in the modal analysis but may be explained by the finite stiffness of the upright supporting structure Schitter et al. (2007). Above the main resonance mode the system shows fourth order lowpass characteristics, which reduces the excitation of higher system modes and is due to the piezo amplifier bandwidth. For modeling the measured frequency responses of both axes an 8th order plant model 2 2 2 s + 2ζzi ωzi s + ωzi · e−sT , (2) G(s) = K · i=1 4 2 2 s + 2ζpi ωpi s + ωpi i=1
The entire experimental system setup is depicted in Fig. 3a, showing the FSM system with the mirror mount and the components of the optical sensor system, including the custom made readout electronics. The optical sensor system is located behind the mirror unit and comprises a laser source, a beam splitter and a position sensitive device (PSD; S5990-01, Hamamatsu Photonics, Japan) for detecting displacements of the reflected laser spot in x- and y-direction. A front view of the system with the 10 mm mirror bonded to the flexure membrane is shown in Fig. 3b, while Fig. 3c depicts the backside of the flexure with the mounted mover, the steel balls and the sensor mirror for the optical sensor system, which reflects the incoming laser beam. The piezo actuators are operated in a push-pull configuration and are driven by a piezo amplifier (Techproject EMC GmbH, Austria; output voltage 0-150V) with a measured bandwidth of 10 kHz and a second order lowpass characteristic at higher frequencies. The comparably low amplifier bandwidth is chosen deliberately as it can be directly employed to significantly reduce the excitation of higher modes of the mechanical system. For deriving input signals for the amplifier, acquiring the position signals, and controller implementation a real-time data acquisition system (Type: DS1202, dSPACE GmbH, Germany) is used. The optical sensor system is calibrated by using an external reference metrology system (Confocal sensor confocalDT 2451, Micro-Epsilon Messtechnik GmbH & Co. KG, Germany), which measures the mirror position from the front side and uses a small angle approximation to obtain the angular displacement. 3. SYSTEM IDENTIFICATION For identifying the dynamics of the FSM prototype system a system analyzer (3562A, Hewlett-Packard, Palo Alto, CA, USA) is used. The inputs of the piezo amplifier and the output signals of the optical sensor system are considered as system inputs and outputs, respectively. Figure 4 shows the measured frequency response data of the x- (blue) and y-axis (red) of the system. It can be seen that both system axes show almost identical dynamics, with only slight deviations resulting most likely from mounting tolerances. The main resonance mode occurs at 6.7 kHz and shows good agreement with the predicted value from the flexure design in Section 2.2. Below this 816
with gain K = 5.82e16, time delay T = 16.7 µs (sampling delay of the dSpace system), and parameters according to Table 1 is obtained. Crosstalk measurements (data not Table 1. Coefficients of the FSM system model G(s). Index z1 z2 p1 p2 p3 p4
ωIndex [rad/s] 2.05e4 2.41e4 1.93e4 2.29e4 4.26e4 5.03e4
ζIndex 0.03 0.02 0.05 0.02 0.05 0.44
shown) revealed that the crosstalk between the axes at DC is more than 30 dB smaller than the magnitudes of the individual system axes, which justifies the application of one single-input-single-output (SISO) feedback controller per axis. The mechanical system range and hysteresis of the stack actuators are measured by applying a 1 Hz sinusoidal signal with 10 V amplitude to the piezo amplifier input of one system axis at a time. The measurement results (data not shown) reveal a mechanical range of ±2.4 mrad (±4.8 mrad optical) in tip and tilt direction, as well as a maximum hysteresis of about 15% for both system axes. 4. CONTROLLER DESIGN The design of the feedback controller is done with the aim of maximizing the closed-loop system bandwidth for high speed scanning operation. 4.1 PI Controller with Notch Filters (PI+ Controller) As most commercial piezo actuated FSM systems use feedback bandwidth controllers for position control of the mirror, a PI based controller design is taken as starting point for both axes. PI controllers with a crossover frequency below the first fundamental resonance mode of the actuator are also applied in many other piezo actuated high stiffness systems, such as AFMs [Schitter et al. (2007)], for tracking raster trajectories with their various frequency components [Fleming and Wills (2009)]. The P-gain is used to vertically shift the loop gain and adds artificial stiffness
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BNC connectors
FSM
PSD board Steel balls
Beam splitter
Sensor mirror
Flexure
Mirror mount Mirror
Laser source
(a)
(b)
(c)
Magnitude [dB]
Fig. 3. Experimental FSM system setup. (a) depicts an overview of the entire setup with the FSM, its mounting plate and the optical position sensor system with laser source, beam splitter and PSD board. (b) shows a front view of the FSM, with the mirror, the flexure membrane and the BNC connectors for the actuators observable. (c) shows the backside of the mover with the mounted mover, the steel balls and the sensor mirror. 3 2 2 s + 2d w ω s + ω n n n n 0 (s + ω4 ) i=1 · (4) C(s) = K · 3 2 −20 s s + 2wn ωn s + ωn2 −40 −60 −80
i=1
with K = 50 and parameters according to Table 2, is designed for a crossover frequency of 1.4 kHz. The
X−axis Y−axis Model
Table 2. Coefficients of the designed PI+ controller.
−100
Phase [°]
0 −100 −200 −300 −400 −500 1 10
2
10
3
10 Frequency [Hz]
4
10
Fig. 4. Frequency response of the FSM x- (solid blue) and y-axis (solid red) measured with the optical sensor system. The fitted system model (dashed black) models the first three system resonances at 3, 3.6 and 6.7 kHz. to the system. The integrator increases the loop gain at low frequencies, enforcing high tracking performance and zero steady state error. Looking at the mode at 6.7 kHz and its peak value of -16.9 dB the achievable crossover frequency would be limited to about 75 Hz when using solely a PI controller. To extend the bandwidth three tuned notch filters [Munnig-Schmidt et al. (2014)] (also see Fig. 5) of the form N (s) =
s2 + 2dn wn ωn s + ωn2 , s2 + 2wn ωn s + ωn2
(3)
with dn the depth, wn the width of the notch and ωn the frequency at which the notch is placed, are designed and added to the controller to cancel the first three resonance peaks identified in Section 3. In a loop shaping approach, aiming to maximize the system bandwidth, the PI+ controller 817
Index
ωIndex [rad/s]
dIndex
wIndex
1 2 3 4
1.94e4 2.29e4 4.26e4 1e4
0.5 0.5 0.025 -
0.2 0.1 2 -
simulated open loop response of the designed controller and the derived plant model shows a gain margin of 6.4 dB and a phase margin of 63◦ . The simulated closed-loop bandwidth is 3 kHz. As the dynamics of both system axes are similar, the same controller can be applied to both system axes. 4.2 Controller Implementation For implementation on the dSpace system the controllers are discretized using Pole-Zero-Matching for a sampling frequency of fs = 60 kHz [Franklin et al. (1997)]. The poles and zeros are directly transformed to the discrete time domain by using the relation z = es/fs to guarantee that the notch filter are located exactly at the desired frequencies. Fig. 5 shows the frequency responses of the implemented PI+ controller together with the frequency responses of its PI and notch filter components. The PI component (dashed blue) shows integrating behavior at low frequencies and constant gain at high frequencies. The first two notch filters N1 (s) and N2 (s) (dotted and dashed black) have the same depth value d1 = d2 = 0.5 (w1 = 0.2, w2 = 0.1), are located at 3 and 3.6 kHz, respectively, and are used to cancel the two small resonance peaks at these frequencies. The third notch filter N3 (s) (dashdotted black) is placed at 6.7 kHz and is significantly deeper and wider (d3 = 0.025, w3 = 2) in order to notch out the main resonance mode of the FSM dynamics. The
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0
N2 -10
N1
-20
Ng
-10 -20 -30 101
103
102
103
104
102
103
104
104
50
0
Phase [°]
Magnitude [dB]
0
N3
-30 100
10
Magnitude [dB]
Magnitude [dB]
resulting transfer function of all notch filters Ng (s) (dashed cyan) is also shown.
293
0 -50 101
102
103
-100 -200
104
Modeled Measured
-300 101
Phase [°]
100
PI+ controller PI controller N1(s)
Frequency [Hz]
N2(s) N3(s) Ng(s)
0 -100 101
102
103
104
Frequency [Hz]
Fig. 5. Measured frequency response of the implemented PI+ controller (solid red). It is designed for a crossover frequency of 1.4 kHz, comprising a PI component (dashed blue), as well as notch filters (black) at 3 (N1 (s)), 3.6 (N2 (s)) and 6.7 kHz (N3 (s)). 5. EVALUATION OF SYSTEM PERFORMANCE For evaluation of the system performance, the closedloop system dynamics, the tracking performance, and the positioning uncertainty are investigated. For testing the tracking performance a raster trajectory with high spatial and temporal resolution, which has its first 11 harmonics covered by the closed-loop bandwidth, is obtained by selecting f1R = 250 Hz and f2R = 0.5 Hz for the fast and slow scan axis, respectively. The resulting spatial resolution is 4 µrad/mrad at a frame rate of 1 frame/s. The measured complimentary sensitivity function of the FSM system with the PI+ controller is shown in Fig. 6, together with the modeled closed-loop system dynamics. The measured frequency response shows good agreement with the modeled response. The resulting system bandwidth 2.7 kHz matches well with the 3 kHz predicted by the model. The measured dynamics show a small gain peaking of 2.5 dB at 1.9 kHz, which is due to the relatively large phase margin. To evaluate the tracking performance of the closed-loop system, the designed raster trajectory is applied with a scan amplitude of 2.2 mrad and tracked with the PI+ controllers. The measured signals are post-processed by low-pass filters with a bandwidth of 20 kHz for reducing high frequency noise. The results of the fast scan axis are depicted in Fig. 7. The tracking error for the entire trajectory results to 73.8µrad rms (3.6% of the scan 818
Fig. 6. Modeled (solid blue) and measured (dashed red) complimentary sensitivity function the FSM with the PI+ controller. The measured dynamics shows a control bandwidth of 2.7 kHz. amplitude) and 0.61 mrad peak-to-valley error, with a controller output of 71.9 V rms. Disregarding the data around the turning points of the fast axis (80% threshold; see Fig. 7) where the tracking error is largest, the tracking error in the region of interest results to 41.4µrad rms (1.9% of the scan amplitude) and 0.48 mrad peak-to-valley. The resulting positioning uncertainty of the FSM system controlled by the PI+ controller is determined at zero reference applied to both axes and is as small as 3.8 µrad rms, which is 0.08% of the full scale range. In summary it is successfully demonstrated that the proposed piezo actuated FSM with the novel membrane flexure, optical position sensor system and ±2.4 mrad mechanical angular range achieves a closed-loop bandwidth of 2.7 kHz with the PI+ controller, and shows good tracking performance as well as a low positioning uncertainty. 6. CONCLUSION This paper presents a novel piezo-actuated FSM system design with an optical position sensor system for high speed scanning applications. The design is centered around an easy-to-manufacture membrane-like metallic flexure and two pairs of stack actuators that are operated in a push-pull configuration. It provides an mechanical angular range of ±2.4 mrad (±4.8 mrad optically) in both axes, while achieving a first fundamental resonance mode of the actuator as high as 6.7 kHz. After a detailed system description and analysis, a model-based PI+ controller for maximizing the system bandwidth and tracking of raster trajectories is designed. The resulting closed-loop system has a bandwidth as high as 2.7 kHz and a small positioning uncertainty of 3.8 µrad rms. ACKNOWLEDGEMENTS The financial support by the Christian Doppler Research Association, the Austrian Federal Ministry for Digital and
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2.5 2
80%
1.5
Angle [mrad]
1 0.5 0 -0.5 -1 -1.5
80%
-2
Measured Reference
Error [mrad]
-2.5 0
0.005
0.01 Time [s]
0.015
0.02
0
0.005
0.01 Time [s]
0.015
0.02
0.5 0 -0.5
Fig. 7. Measured raster scan trajectory. The reference (dashed blue), the measured position signal (solid red) and the tracking error (solid black) of the fast scan axis tracking a 250 Hz triangular signal are shown. Economic Affairs, and the National Foundation for Research, Technology and Development, as well as MICROEPSILON MESSTECHNIK GmbH & Co. KG and ATENSOR Engineering and Technology Systems GmbH is gratefully acknowledged. REFERENCES Csencsics, E., Saathof, R., and Schitter, G. (2016). Design of a dual-tone controller for lissajous-based scanning of fast steering mirrors. American Control Conference, Boston, MA, USA. Csencsics, E. and Schitter, G. (2017). System design and control of a resonant fast steering mirror for lissajousbased scanning. IEEE TMECH, 22(5), 1963–1972. Csencsics, E., Schlarp, J., and Schitter, G. (2018). Highperformance hybrid-reluctance-force-based tip/tilt system: Design, control, and evaluation. IEEE TMECH, 23(5), 2494–2502. Fleming, A.J. and Wills, A.G. (2009). Optimal periodic trajectzories for band limited systems. IEEE TCST, 17(3), 552–562. Franklin, G.F., Powell, D.J., and Workman, M.L. (1997). Digital Control of Dynamic Systems. Prentice Hall. Guelman, M., Kogan, A., Livne, A., Orenstein, M., and Michalik, H. (2004). Acquisition and pointing control for inter-satellite laser communications. IEEE Transactions on Aerospace and Electronic Systems, 40(4), 1239. Hedding, L.R. and Lewis, R.A. (1990). Fast steering mirror design and performance for stabilization and single axis scanning. SPIE 340, 1304, 14–24. Ito, S. and Schitter, G. (2016). Comparison and classification of high-precision actuators based on stiffness influencing vibration isolation. IEEE TMECH, 21(2), 1169–1179. 819
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