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7th IFAC Symposium on Mechatronic Systems 7th IFAC Symposium on MechatronicUniversity, Systems UK September 5-8, 2016. Loughborough 7th IFAC IFAC Symposium Symposium on Mechatronic Mechatronic Systems Systems 7th on September 5-8, 2016. Loughborough University,online UK at www.sciencedirect.com Available September 5-8, 2016. Loughborough University, September 5-8, 2016. Loughborough University, UK UK

ScienceDirect IFAC-PapersOnLine 49-21 (2016) 133–140

Mechatronic Design of an Active Two-body Mechatronic Design of an Active Two-body Mechatronic Design of an Active Two-body Vibration Isolation System Vibration Isolation System Vibration Isolation System E. Csencsics, M. Thier, P. Siegl, G. Schitter E. Csencsics, M. Thier, P. Siegl, G. Schitter E. E. Csencsics, Csencsics, M. M. Thier, Thier, P. P. Siegl, Siegl, G. G. Schitter Schitter Automation and Control Institute, Vienna University of Technology, Automation and Control27-29, Institute, University of Technology, Automation and Institute, Vienna University of Gusshausstrasse 1040Vienna Vienna, Austria (e-mail: Automation and Control Control27-29, Institute, Vienna University of Technology, Technology, Gusshausstrasse 1040 Vienna, Austria (e-mail: Gusshausstrasse 27-29, 1040 Vienna, Austria (e-mail: [email protected]). Gusshausstrasse 27-29, 1040 Vienna, Austria (e-mail: [email protected]). [email protected]). [email protected]). Abstract: Abstract: Abstract:modes as for example decoupling of a mechanical subsystem are in general unwanted Structural Abstract: Structural modes as for example decoupling aa mechanical subsystem are in generaldecoupling unwanted Structural modes as decoupling of subsystem are in unwanted effects in high precision positioning systems.of paper proposes a well Structural modes as for for example example decoupling of This a mechanical mechanical subsystem are designed in general generaldecoupling unwanted effects in high precision positioning systems. This paper proposes a well designed effects in high precision positioning systems. This paper proposes a well designed decoupling mechanism as a design choice to improve the energy efficiency of an active vibration isolation effects in high precision positioning systems. This paper proposes a well designed decoupling mechanism aa design choice the energy an active vibration isolation mechanism asneeds design choice to toa improve improve thecomprising energy efficiency efficiency ofand an low active vibration isolation system thatas to position structure a high of precision subsystem. mechanism as a design choice to improve the energy efficiency of an active vibration isolation system structure comprising higha and and low subsystem. system that that needs needs toofposition position aDoF structure comprising high low precision precision subsystem. Experimental setupsto a single aa system comprising with a rigidaaa and decoupling mechanical system system that needs to position structure high and low precision subsystem. Experimental setups of aaand single DoF system aa rigid and aa decoupling mechanical system Experimental setups of single DoF system with rigid and decoupling mechanical system structure are developed analyzed. PD andwith a PID controllers are designed for the rigid and Experimental setups of a single DoF system with a rigid and a decoupling mechanical system structure are developed and analyzed. PD and a PID controllers are designed for the rigid and structure are developed and analyzed. PD and a PID controllers are designed for the rigid and the equivalent decoupling structure, respectively, resulting in the same disturbance rejection structure are developed andstructure, analyzed.respectively, PD and a PID controllersthe are designed for the rigid and the resulting rejection the equivalent equivalentfordecoupling decoupling structure,subsystem. respectively, resulting in in the same same disturbance disturbance rejection performance the high precision Experiments demonstrate that disturbances of the equivalent decoupling structure, respectively, resulting in the same disturbance rejection performance for the high precision subsystem. Experiments demonstrate that disturbances of performance the high precision Experiments demonstrate that disturbances of 12.4 µm rms for amplitude are reducedsubsystem. to below 118 nm rms for both systems and that the rms performance for the high precision subsystem. Experiments demonstrate that disturbances of 12.4 µmconsumption rms amplitude are reduced to structure below 118can nm rms for both that the 12.4 rms are reduced below nm rms for systems and that rms energy of the bysystems 68% as and compared to rms the 12.4 µm µmconsumption rms amplitude amplitude are decoupling reduced to to structure below 118 118can nm be rmsreduced for both both systems and that the the rms energy of the decoupling be reduced by 68% as compared to the energy consumption of the decoupling structure can be reduced by 68% as compared to rigid system structure. energy consumption of the decoupling structure can be reduced by 68% as compared to the the rigid system structure. rigid system system structure. structure. rigid © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Vibration isolation, Systems design, Mechanical decoupling, PID controllers, Keywords: Vibration isolation, Systems design, Mechanical decoupling, PID controllers, Keywords: Vibration isolation, Reduced energy consumption Keywords:energy Vibration isolation, Systems Systems design, design, Mechanical Mechanical decoupling, decoupling, PID PID controllers, controllers, Reduced consumption Reduced Reduced energy energy consumption consumption 1. INTRODUCTION In contrast to active concepts structural modes, such as 1. INTRODUCTION In to structural as 1. INTRODUCTION In contrast contrast sub-systems, to active active concepts concepts structural modes, modes, such as decoupling are intentionally used in such passive 1. INTRODUCTION In contrast to active concepts structural modes, such as decoupling sub-systems, are intentionally used in passive decouplingisolation sub-systems, are intentionally used in passive vibration systems. Applications proposed in Gidecoupling sub-systems, are intentionally used in passive systems. proposed in High precision positioning systems in production and vibration vibration isolation systems. Applications proposed in GiGiaime et al.isolation (1996) and PirroApplications (2006) use passive vibration vibration isolation systems. Applications proposed in GiHigh precision positioning systems in production and aime et al. (1996) and Pirro (2006) use passive vibration High precision positioning systems in production and metrology often require high control bandwidths to ensure aime et al. (1996) and Pirro (2006) use passive vibration isolation stacks to reduce the transmissibilty of the reHigh precision positioning systemsbandwidths in production and aime et al. (1996) and Pirrothe (2006) use passiveofvibration metrology often require high to stacks to reduce transmissibilty the remetrology often require high control control bandwidths to ensure ensure the required levels of precision (Munnig Schmidt et al. isolation isolation stacks to reduce the transmissibilty of the resulting structure and to isolate sensitive equipment from metrology often require high control bandwidths to ensure isolationstructure stacks to reduce the transmissibilty of thefrom rethe levels of precision Schmidt et al. sulting isolate equipment the required required levels of precision (Munnig Schmidt et al. (2014)). External vibrations are (Munnig a common problem for sulting structure structure and to isolate sensitive sensitive equipment from ground vibrations.and In to Csencsics et al. (2016) it is shown the required levels of precision (Munnig Schmidt et al. sulting and to isolate sensitive equipment from (2014)). External are aa common problem for ground vibrations. In Csencsics et al. (2016) it is shown (2014)). External vibrations are problem these tasks, as theyvibrations are in general sensitive to disturbances ground vibrations. In decoupling Csencsics et etmechanism al. (2016) (2016) can it is isalso shown that a well designed be (2014)). External vibrations are sensitive a common common problem for for that ground vibrations. In Csencsics al. it shown these tasks, as they are in general to disturbances aa well designed decoupling mechanism can also be these tasks, as they are in general sensitive to disturbances (Amick et al. (2005)). In literature thus numerous passive that well designed decoupling mechanism can also be introduced in an active vibration isolation system to imthese tasks, as they areIn in literature general sensitive to disturbances that a well indesigned decoupling mechanism can also be (Amick thus passive an active vibration isolation system to im(Amick et etetal. al.al.(2005)). (2005)). Inand literature thus numerous numerous passive (Carrella (2007))In active vibration isolation sys- introduced introduced in an active vibration isolation system to improve versatility, without impairing the controllability of (Amick et al. (2005)). literature thus numerous passive introduced in an active vibration isolation system to im(Carrella and active vibration isolation sysprove versatility, (Carrella etetal. al.al.(2007)) (2007)) and active vibration isolation systems (Kimet (2009)), equipped with sensors and actuprove versatility, without without impairing impairing the the controllability controllability of of the system. (Carrella et al. (2007)) and active vibration isolation sysprove versatility, without impairing the controllability of tems et equipped with and tems (Kim (Kim et al. al. (2009)), (2009)), equipped with sensors sensors and actuactu- the ators to actively reject external vibrations, are commonly the system. system. tems (Kim et al. (2009)), equipped with sensors and actuthe system. ators reject vibrations, are This paper presents the advantage of a designed decouators to to actively actively reject external external vibrations, are commonly commonly proposed countermeasures in these fields. Active concepts This presents the of ators to actively reject external vibrations, are commonly proposed countermeasures in these fields. Active concepts This paper paper presents the advantage advantage of aaa designed designed decoupling mechanism introduced in an active vibrationdecouisolaThis paper presents the advantage of designed decouproposed countermeasures in these thesebetween fields. Active Active concepts that maintain constant proximity a probe and a pling mechanism introduced in an active vibration proposed countermeasures in fields. concepts that maintain constant proximity between aaalso probe and aa tion plingsystem mechanism introduceda in in an active active vibrationaisolaisolathat positions structure comprising high pling mechanism introduced an vibration that maintain constant proximity between probe and sample by means of closed loop control are reported system that positions aa structure comprising aaisolahigh that maintain constant proximity between aalso probe and a tion sample by means of closed loop control are reported tion system that positions structure comprising high and a low precision subsystem in constant distance to a reftion asystem that positions a structure comprising a ahigh sample et byal. means of Ito closed loop control are are also also reported reported and (Thier (2015), et al. (2015)). low precision subsystem in constant distance to refsample by means of closed loop control (Thier et al. (2015), Ito et al. (2015)). and a low precision subsystem in constant distance to a reference. The proposed approach offers the design freedom and a low precision subsystem in constant distance to a ref(Thier et et al. al. (2015), (2015), Ito et et al. al. (2015)). (2015)). The approach offers freedom (Thier Structural modes of Ito the positioned structure represent a erence. erence. The proposed approach offers the theofdesign design freedom to reduce theproposed overall energy consumption the system for erence. The proposed approach offers the design freedom Structural modes the the energy of for Structuralfor modes of the positioned positioned structure represent challenge such of active closed loopstructure controlledrepresent concepts,aaa to to reduce reduce the overall overall energy consumption consumption of the the system system for high bandwidth positioning of the high precision subsysStructural modes of the positioned structure represent to reduce the overall energy consumption of the system for challenge such closed loop controlled concepts, bandwidth positioning of the high precision subsyschallenge for such active active closedthe loop controlled concepts, as they infor general may limit achievable closed loop high high bandwidth positioning of the high precision subsystem. To demonstrate this targeted property the system dechallenge for such active closed loop controlled concepts, high To bandwidth positioning of the high precision subsysas may limit the closed loop this the system deas they they in in general general maymodes limit originate the achievable achievable closedmodes loop tem. bandwidth. Structural either from tem. and To demonstrate demonstrate this targeted targeted property the design the experimental setup ofproperty a rigid system concept as they in general may limit the achievable closed loop tem. To demonstrate this targeted property the system system debandwidth. Structural modes originate either from modes sign and the experimental setup of a rigid system concept bandwidth. Structural modes originate either from modes of the individual components of the positioned structure sign and the experimental setup of a rigid system concept with a high and a low precision sub-metrology-system is bandwidth. Structural modes originate either from modes sign and the experimental setup of a rigid system concept of components the positioned structure and aa low is of the the individual components of the the case positioned structure with or theirindividual interconnections, as inof of a decoupling with aaa high high and low precision precision sub-metrology-system is presented and analyzed in Sectionsub-metrology-system 2. Section 3 introduces of the individual components of the positioned structure with high and a low precision sub-metrology-system is or as the case aa decoupling and analyzed in Section 2. Section 33 introduces or their their interconnections, interconnections, as etin in al. the(2014)). case of of Almost decoupling sub-mass (Munnig Schmidt every presented presented and analyzed in Section 2. Section introduces the developed single axis prototype of a two-body system or their interconnections, as in the case of a decoupling presented and single analyzed inprototype Section 2. Section 3 introduces sub-mass et every aa two-body sub-mass (Munnig (Munnig Schmidt et al. al. (2014)). (2014)). Almost every the structure that can Schmidt be considered rigid at Almost low frequenthe developed developed single axis axismechanism prototype of of two-body system design with decoupling and provides asystem thorsub-mass (Munnig Schmidt et al. (2014)). Almost every the developed single axis prototype of a two-body structure that can be considered rigid at low frequendesign with decoupling mechanism and provides aasystem thorstructure that can be considered rigid at low frequencies showsthat internal structural modes, meaning additional designsystem with decoupling decoupling mechanism and provides providesdata thorough analysis. Based on the identification PD structure can be considered rigid at low frequendesign with mechanism and a thorcies shows internal structural modes, meaning additional ough system analysis. Based on the identification data PD cies shows internal structural modes, meaning additional system dynamics, at higher frequencies, depending on its ough system analysis. Based on the identification data PD and PID controllers are designed in Section 4 for the rigid cies shows internal structural modes, meaning additional oughPID system analysis.are Based on the identification datarigid PD system dynamics, higher depending on controllers designed in Section 44Section for the systemand dynamics, at higher frequencies, depending on its its and shape density. at There arefrequencies, different strategies proposed PID controllers are designed in Section for the rigid and the decoupling system, respectively. In 5 it is system dynamics, at higher frequencies, depending on its and PID controllers system, are designed in Section Section 4 for the5rigid shape and There different strategies proposed respectively. shape and density. density. There are differentmodes strategies proposed in literature to cope withare structural ranging from and and the the decoupling decoupling system, respectively. In Section 5 it it is is demonstrated that the control effort for In high bandwidth shape and density. There are different strategies proposed and the decoupling system, respectively. In Section 5 it is in to cope modes ranging from that the effort bandwidth in literature literature to (Babakhani cope with with structural structural modes ranging from demonstrated active damping and De Vries (2010)) and overdemonstrated thatprecision the control control effort for for high bandwidth control of the high subsystem canhigh be significantly in literature to cope with structural modes ranging from demonstrated that the control effort for high bandwidth active (Babakhani and Vries overprecision be significantly active damping damping (Babakhani and De De Falangas Vries (2010)) (2010)) and overcontrol actuation (Schneiders et al. (2003), et al.and (1994)), control of ofinthe the high precision subsystem can be reduced thehigh system with subsystem decouplingcan mechanism, while active damping (Babakhani and De Vries (2010)) and overcontrol of the high precision subsystem can be significantly significantly actuation (Schneiders et al. (2003), Falangas et al. (1994)), reduced in the system with decoupling mechanism, while actuation (Schneiders et al. al. (2003), (2003), Falangas et al. al. (1994)), (1994)), to the appropriate placement of actuators (Nestorovic and maintaining reduced in the system with decoupling mechanism, while an equal disturbance rejection performance as actuation (Schneiders et Falangas et reduced in the systemdisturbance with decoupling mechanism, while to of (Nestorovic and an rejection as to the the appropriate appropriate placement of actuators actuators (Nestorovic and maintaining Trajkov (2013)). Allplacement of them, however, require an increased maintaining an equal equalSection disturbance rejection performance as in the rigid system. 6 concludes theperformance paper. to the appropriate placement of actuators (Nestorovic and maintaining an equal disturbance rejection performance as Trajkov (2013)). require an increased in Trajkovcomplexity (2013)). All Allorof ofathem, them, however, require an increased system high however, system analysis effort. in the the rigid rigid system. system. Section Section 666 concludes concludes the the paper. paper. Trajkov (2013)). All of them, however, require an increased in the rigid system. Section concludes the paper. system a high system analysis effort. system complexity complexity or or system complexity or aa high high system system analysis analysis effort. effort. Copyright © 2016, 2016 IFAC 133 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016 IFAC 133 Copyright © 2016 IFAC 133 Peer review under responsibility of International Federation of Automatic Copyright © 2016 IFAC 133Control. 10.1016/j.ifacol.2016.10.532

2016 IFAC MECHATRONICS 134 E. Csencsics et al. / IFAC-PapersOnLine 49-21 (2016) 133–140 September 5-8, 2016. Loughborough University, UK

2. RIGID SYSTEM DESIGN AND IDENTIFICATION

2.2 Experimental setup

2.1 System description and modeling

The rigid system structure in the experimental setup is composed of a solid aluminium block (m1,2 =5.7 kg). It is placed directly on the mover of a voice coil actuator (Shaker S51110, TIRA GmbH, Germany) that is placed on mechanical ground and used for vertical actuation. The actuator is driven by a custom made current amplifier (Amplifier type MP38CL, Apex Microtechnology, Tucson, AZ, USA). The amplifier is controlled by a current controller with a bandwidth of 10 kHz, implemented on the FPGA of a dSpace-platform (Type: DS1005, dSPACE GmbH, Germany). The controller implementation is done on the processor of the dSpace-platform running at a sampling frequency of 20 kHz.

In Fig. 1 the concept of a high precision metrology system which is positioned in constant distance ∆z to a measurement sample is shown. It comprises a lightweight sub-metrology-system MS1 which requires high precision and bandwidth and a heavy sub-metrology-system MS2 requiring only low precision and bandwdith. Both submetrology-systems are mounted on a rigid metrology platform with an entire mass m1,2 (platform plus submetrology-systems). The platform is connected to mechanical ground via the actuator suspension (k1 and d1 ) and actuated via the force F . It is subject to external ground vibrations, which disturb the constant distance ∆z to the sample. The coordinate z1,2 represents the vertical position of the platform body.

sample

MS2

z1,2

2.3 System identification

m1,2 k1

F

d1

floor vibrations

Fig. 1. Mechanical model of the rigid system design concept of a high precision metrology system that is positioned in constant distance to a sample. The high precision subsystem MS1 and the low precision subsystem MS2 are stiffly mounted to the metrology platform (rigid body). m1,2 represents the mass of the entire metrology platform and is connected to mechanical ground via the actuator suspension (k1 and d1 ) and actuated by force F . To model the dynamics of the rigid system along a single degree of freedom the lumped mass models in Fig. 1 is considered. The differential equation, describing the motion of the body of the rigid system, is m1,2 z¨1,2 (t) = F − k1 z1,2 (t) − d1 z˙1,2 (t). (1) The transfer function (TF) from the applied force F to the vertical position z1,2 can be obtained by reordering (1) and applying the Laplace transformation. This results in the second order TF: 1 Z1,2 (s) GR (s) = = . (2) F (s) m1,2 s2 + d1 s + k1 To fulfill the requirements on disturbance rejection for both subsystems in closed loop control, the entire platform mass m1,2 needs to be actuated over the entire frequency range, that is required to reach the high precision requirements of MS1. Especially for platforms with large mass and target cross over frequencies of several hundred Hertz this might quickly exceed the capabilities of the power amplifier or the actuator in terms of maximum rms current. This can impose stringent requirements of the platform weight, require unnecessary large actuators or high current power amplifiers, or even trouble the feasibility of the targeted system bandwidth and precision requirements. 134

The frequency response is measured by applying a sine sweep and measuring magnitude and phase response by the Lock-In principle with the dSPACE-platfrom (Masciotti et al. (2008)). The measured frequency responses of the rigid system is depicted in Fig. 2. The 2nd order mass-

Magnitude [dB]

MS1

0 −50 Measured RS Modeled RS

−100 0

10

0

10

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1

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2

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1

10 Frequency [Hz]

3

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3

10

0 Phase [°]

Δz

For measuring the position z1,2 of the mass m1,2 an eddy current sensor SE (eddyNCDT DT3702-U1-A-C3, MicroEpsilon GmbH, Germany) is used. The input of the system is the input of the current amplifier, which via the motor constant exerts a force on the moving mass. The signal of the position sensor is considered as the system output.

−200

−400 10

4

Fig. 2. Measured and modeled frequency response of the rigid system (RS). The frequency response shows mass-spring dynamics resulting from the rigid mass and the actuator suspension with a suspension mode at 11 Hz. spring characteristic with a suspension mode at 11 Hz and a -40 dB slope after this frequency is clearly visible. Fig. 2 additionally shows the system model GRS (s) = K · GR (s), (3) with K=1.686e4, GR (s) according to (2) and coefficients according to Table 1, which is fitted to the measured data.

2016 IFAC MECHATRONICS September 5-8, 2016. Loughborough University, E. Csencsics et al. / IFAC-PapersOnLine 49-21 (2016) 133–140 UK

The sampling delay of Ts =50µs of the digital system is also included in the system model. It can be seen that for target cross over frequencies between 200 Hz and 2 kHz the controller already needs to lift the mass line of the system between 50 and 100 dB.

Value 5.7 25e3 13

Δz

MS1

MS2

z2

m2 k2

d2 z1 m1

Unit kg N/m N·s/m

k1

3. DECOUPLING SYSTEM DESIGN AND IDENTIFICATION 3.1 System description and modeling Considering that the entire metrology system is composed of two functionally decoupled sub-metrology-systems, one with high (MS1) and one with low (MS2) precision requirements, a mechanical decoupling mechanism is introduced between the two subsystems. Due to this decoupling mechanism the heavy low precision subsystem MS2 decouples at a designed frequency from the lightweight high precision subsystem MS1, which is directly mounted to the actuator. The decoupling frequency is chosen such that the precision requirements on MS2 are met. Above the decoupling frequency only the mass of the subsystem with MS1 remains for positioning, due to the decoupling of the subsystem with MS2. This makes a targeted high cross over frequency, to fulfill the high precision requirements of MS1, even for heavy platforms feasible, without running into current limitations. A needed change of the actuator, the power amplifier or the entire platform weight and design can thus be avoided. Fig. 3 shows the concept of the metrology system with decoupling mechanism. For the decoupling system the mass of the entire platform m1,2 of the rigid system is distributed over the high precision inner body with m1 and the low precision outer body with m2 (m1,2 =m1 +m2 ). The inner body is directly attached to the actuator and actuated by the force F . It is connected to mechanical ground via the actuator suspension k1 and d1 . Both subsystems are connected via spring k2 and damper d2 , representing the dynamics of the designed decoupling mechanism (grey area in Fig. 3). The coordinate z1 represents the vertical position of the inner body. In contrast to the rigid system only the body with m1 needs to fulfill the precision and bandwidth requirements of MS1. For modeling the dynamic behavior of the decoupling system along a single degree of freedom the lumped mass model in Fig. 3 is considered. The differential equations, describing the motion of the inner body with m1 and the outer body with m2 , are m1 z¨1 (t) = F −k1 z1 (t) − k2 (z1 (t) − z2 (t)) (4) −d1 z˙1 (t) − d2 (z˙1 (t) − z˙2 (t)) , and m2 z¨2 (t) = k2 (z1 (t) − z2 (t)) + d2 (z˙1 (t) − z˙2 (t)) .

designed decoupling mechanism

sample

Table 1. Coefficients for the system model of the rigid system. Parameter m1,2 k1 d1

135

(5) 135

F

d1

floor vibrations

Fig. 3. Mechanical model of the dual body mass-springdamper system of a design concept of a high precision metrology system with decoupling mechanism. The former rigid metrology platform is distributed over two coupled parts. The high precision subsystem MS1 with mass m1 is connected to mechanical ground via the actuator suspension (k1 and d1 ) and actuated by force F . The low precision subsystem MS2 with mass m2 is connected to MS1 via the designed decoupling mechanism comprising k2 and d2 . The TF from the applied force F to the vertical positions z1 is obtained by combining (4) and (5), and by applying the Laplace transformation, resulting in m 2 s 2 + d2 s + k2 Z1 (s) = GD (s) = , F (s) m 1 m2 s 4 + D 3 s 3 + D 2 s 2 + D 1 s + k 1 k 2 (6) with D1 = d 1 k 2 + d 2 k 1 , (7) (8) D2 = d1 d2 + k1 m2 + k2 m1 + k2 m2 , (9) D3 = d1 m2 + d2 m1 + d2 m2 . 3.2 Experimental setup The experimental setup of the decoupling system structure is shown in Fig. 4. It is composed of an inner solid aluminium block (m1 =1.4 kg) and an outer aluminium frame (m2 =4.3 kg) connected by a designed spring-damper decoupling mechanism (Csencsics et al. (2016)). The inner block of the mechanical structure is mounted on the actuator mover. The input of the system is the input of the current amplifier. The position sensor is used to measure the position z1 of the inner body with mass m1 , which is the considered as system output. 3.3 System Identification The measured frequency response of the decoupling system is depicted in Fig. 5. It shows the same dynamics as the rigid system up to 60 Hz, with a suspension mode at 11 Hz and a -40 dB slope above this frequency. It further shows a damped anti-resonance and resonance at fa =78 Hz and fr =170 Hz, due to the decoupling mass m2 . Above the decoupling only the mass m1 of the inner body remains for positioning. To demonstrate good accordance with theory, Fig. 5 additionally shows the system model GDS (s) = K · GD (s), (10) with K=1.686e4, GD (s) according to (6) and coefficients according to Table 2, which is fitted to the measured data.


Table 2. Coefficients for the system model of the decoupling system. Parameter m1 m2 k1 k2 d1 d2

SE

m1

k2

A

Magnitude [dB]

Fig. 4. Experimental lab setup of the decoupling system. The inner body with m1 and the outer body with m2 are depicted. A is the actuator, SE is the eddy current sensor that measures the position z1 and k2 are the leaf springs of the decoupling mechanism.

−50 Measured DS Modeled DS 0

10

1

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2

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3

10

12dB

−50 Measured RS Measured DS

−100 0

10

1

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3

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4

10

4

10

4. CONTROLLER DESIGN

0 Phase [°]

0

Fig. 6. Comparison of the magnitude plots of the measured frequency responses of the rigid (RS) and the decoupling system (DS). The magnitude plot of the decoupling system (grey) shows that at fr =170 Hz the mass m2 decouples, leaving only the mass m1 for high bandwidth positioning. The reduced mass leads to a lift of the mass line of 12 dB with respect to the magnitude plot of the rigid system (black).

0

−100

Unit kg kg N/m N/m N·s/m N·s/m

is vertically lifted. Due to the ratio of entire mass m1 + m2 to the mass m1 the mass line of the decoupling system is in this area 12 dB higher than the one of rigid system. This suggests that for the decoupling system the same cross over frequency of the controlled inner high precision body can be achieved with less control effort (see Section 4.1).

m2

Magnitude [dB]

k2

Value 1.4 4.3 25e3 13e5 13 400

To allow a comparison of the consumed energy an equally good disturbance rejection performance for the body with m1,2 of the rigid system and the high precision part (inner body) with m1 of the decoupling system is targeted. Feedback controllers are thus designed for both systems. The target open loop bandwidth of the systems is set to 600 Hz in order to achieve good disturbance rejection and to avoid an increased controller order in order to deal with the anti-resonance of the decoupling system.

−200

−400 0

10

1

10

2

10 Frequency [Hz]

3

10

4

10

Fig. 5. Measured and modeled frequency response of the decoupling system (DS). The frequency response shows the same mass-spring dynamics as the rigid system up to 60 Hz with a suspension mode at 11 Hz. It further shows a well damped decoupling anti-resonance resonance combination at fa =78 Hz and fr =170 Hz, respectively. The sampling delay (Ts =50µs) of the digital system is again included. Fig. 6 shows a comparison of the magnitude plots of the measured frequency responses of the rigid and the decoupling system. It can be seen that due to the decoupling of the mass m2 the mass line after the decoupling frequency 136

As the suspension mode of the rigid system lies at 11 Hz and the dynamics are dominated by the mass line above this frequency, a tamed PD controller is designed according to Munnig Schmidt et al. (2014). The P-gain shifts the intersection of the mass- and 0 dB-line to the targeted cross over frequency, while the D-gain is chosen to maximize the phase lead at the cross over frequency, resulting in a phase margin of 36◦ . A realization term stops the differential action at a frequency of 1.8 kHz to limit the control effort at higher frequencies. This results in a first order controller of the form s + ωz,RS CRS = KRS (11) s + ωp,RS with KRS =1.07e3, ωz,RS =1.26e3 and ωp,RS =1.131e4.


A test signal complying to the BBN VC-A criterion, see Ungar and Gordon (1983), is used to experimentally evaluate the disturbance rejection performance. Thus the equality of the sensitivity functions of both system is verified in simulation up front using the power spectral density (PSD) defined by the criterion.

Magnitude [dB]

80

60

40

Measured CRS Measured C

DS

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50 Phase [°]

The targeted cross over bandwidth and the decoupling resonance are well separated, such that the CRS with an adjusted DC-gain, to account for the lifted mass line, is taken as starting point. To ensure an equal disturbance rejection performance at low frequencies, where both plant TFs are equal, a tamed integrating action between 50 Hz and 200 Hz is added, that results in an equal DC level for both controllers. This results in a tamed PID controller of the form (s + ωz,DS )2 CDS = KDS (12) (s + ωp1,DS )(s + ωp2,DS ) with KDS =275.12, ωz,DS =1.26e3, ωp1,DS =1.131e4 and ωp2,DS =307.87.

137

0 −50 1

10

10 Frequency [Hz]

4

4.1 Controller implementation

It shows that at frequencies below 30 Hz the gains of both controllers are equal, to achieve an equal disturbance rejection performance in this range where both plant TFs are still identical. At 50 Hz the integrator of CDS starts and lowers the gain of CDS compared to CRS . The Daction starts for both controllers at 200 Hz resulting in a 14◦ reduced phase lead of the CDS at the targeted cross over frequency. Above the decoupling frequency fr only the mass of the inner body is left for positioning, such that the gain of CDS can be 12 dB smaller than the gain of CRS , resulting in an reduced control effort.

Fig. 7. Measured frequency response of the rigid and decoupling system controller CRS and CDS . CRS is a tamed PD controller designed for an open loop cross over frequency of 600 Hz with the rigid system. CDS is a tamed PID controller, tuned to result in an equal bandwidth and disturbance rejection performance with the decoupling system.

80 Magnitude [dB]

Both controllers are discretized using Pole-Zero-Matching, see Franklin et al. (1997), for a sampling frequency fs =20 kHz, where poles and zeros are directly transformed by z = es/fs . The measured controller TFs of both controllers are depicted in Fig. 7.

60 40 20 0 −20 −40

5. EXPERIMENTAL RESULTS

Open Loop RS Open Loop DS 1

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The measured open loop TFs of the rigid and the decoupling system with the related controllers are depicted in Fig. 8. Both TFs show a cross over frequency of 650 Hz with phase margins of 36◦ and 25◦ and gain margins of 8.8 dB and 6.8 dB, respectively. The measured sensitivity functions of the rigid and the decoupling system are shown in Fig. 9. At frequencies below 20 Hz both sensitivity functions are equal, while the decoupling system lies above the rigid system between 20 Hz and 100 Hz and below it between 100 Hz and the cross over frequency. This behavior results from the decreased and increased loop gain around the decoupling anti-resonance and resonance, respectively. Both sensitivity functions cross the 0 dB line around 400 Hz and attain peaks of 12.5 dB (RS) and 9.7 dB (DS), due to the Waterbed effect (Preumont (2012)). Between 1 kHz and 3 kHz the sensitivity function of the rigid system lies below the sensitivity function of the decoupling system. The evaluation of the disturbance rejection performance in the time domain and the energy consumption is done 137

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Fig. 8. Measured open loop TFs of the rigid (black) and the decoupling system (grey) with the related controllers. Both show cross over frequencies around 650 Hz and phase margins of 36◦ (RS) and 25◦ (DS). by applying a disturbance profile that satisfies the spectral BBN VC-A criterion, see Ungar and Gordon (1983), as reference, while the error signal is considered as output. This relation results in an equal TF as the output-disturbance to output relation. Fig. 10 shows the results for both systems. The positioning error of the rigid system shows a peak-to-peak value of 1.26 µm and a rms value of 116 nm with an rms current value of 513 mA. The decoupling

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Fig. 10. Disturbance rejection and current evaluation of the closed loop rigid (RS) and the decoupling system (DS). The disturbance with 62.8µm peak-to-peak and 12.4µm rms value is applied and the error is evaluated. (a) shows the error of the rigid system with 1.26µm peak-to-peak and 116 nm rms value. The rms current is 513 mA. (b) shows the error of the decoupling system with 1.17µm peak-to-peak and 118 nm rms value. The rms current is 292 mA.

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Fig. 9. Measured sensitivity functions of the closed loop rigid (black) and decoupling system (grey). At frequencies below 20 Hz both sensitivity functions are equal. Between 20 Hz and the cross over the sensitivity function of the decoupling system lies first above and then below the one of the rigid system, respectively. Both cross the 0 dB line around 400 Hz. system offers an equal disturbance rejection performance with a peak-to-peak error value of 1.17 µm and a rms value of 118 nm. The rms current value is 292 mA and thus 43% smaller than in the case of the rigid system. This denotes a reduction of the energy consumption by 68%. Additionally providing a frequency domain analysis of the time signals, Fig. 10 and Fig. 11 show the power spectral densities (PSDs) of the disturbance signal and the tracking error of the rigid and the decoupling system. At frequencies below 10 Hz both disturbance PSDs are 138

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Fig. 11. Power spectral density (PSD) of the disturbance and the error signal of the rigid (RS) and the decoupling system (DS). The error components are equal except around the decoupling aantiresonance/resonance and match the shape of the sensitivity function. attenuated by six orders of magnitude, which corresponds to the −60 dB of the sensitivity functions. Around the decoupling anti-resonance/resonance a difference in the error spectra comparable to the different shapes of the sensitivity functions is observable. After a peak of the decoupling system at about 70 Hz and a valley just


below 200 Hz both error spectra approach the disturbance spectrum at frequencies above 1 kHz, as disturbances at these frequencies can not be rejected (compare Fig. 9). The PSDs of the current signals are shown in Fig. 12. The spectrum corresponds to the spectrum of the disturbance weighted with the input sensitivity function U (s) = C(s)/(1 + G(s)C(s)) of the respective system. It can be seen that at frequencies above the decoupling resonance the current components of the decoupling system are significantly smaller than the current components of the rigid system.

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Fig. 12. Power spectral density (PSD) of the current signal at the controller output of the rigid (RS, black) and decoupling system (DS, grey). The current components above the decoupling resonance of 190 Hz are significantly smaller in the case of the DS signal. In summary it is shown that by introduction of a mechanical decoupling mechanism the energy consumption in the closed loop controlled system can be reduced by a factor of 68% as compared to a rigid structure, while maintaining an equal disturbance rejection performance of the high precision subsystem. 6. CONCLUSION In this paper a mechanically decoupling mechanism is proposed as a design choice to reduce the energy consumption in an active vibration isolation system composed of a lightweight subsystem with high precision requirements and a heavy subsystem with low precision requirements. It is shown that by introducing a well-damped decoupling mechanism (in this case at 170 Hz) into the initially rigid system strucutre and applying the designed PID controller the rms current can be reduced by 43% as compared to the original rigid system with the designed PD controller. This equals an reduced energy consumption of 68% and can be obtained while still maintaining the same disturbance rejection performance for the high precision subsystem. As introduced this principle can be used in an energy efficient design of metrology platforms that carry two metrology systems with different requirements. Future work focuses on the implementation of the decoupling mechanism in systems with 3 degrees of freedom. 139

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ACKNOWLEDGEMENTS The authors acknowledge funding from the EU commission under the FP7 NMP Program, project title aim4np, grant number 309558. REFERENCES Amick, H., Gendreau, M., Busch, T., and Gordon, C. (2005). Evolving criteria for research facilities: vibration. Optics & Photonics 2005, 593303. Babakhani, B. and De Vries, T.J. (2010). Active damping of the 1d rocking mode. International Conference on Mechatronics and Automation (ICMA), IEEE, 1370– 1375. Carrella, A., Brennan, M., and Waters, T. (2007). Static analysis of a passive vibration isolator with quasi-zerostiffness characteristic. Journal of Sound and Vibration, 301(3), 678–689. Csencsics, E., Thier, M., Hainisch, R., and Schitter, G. (2016). System and control design of a voice coil actuated mechanically decoupling two-body vibration isolation system. IEEE Transactions on Mechatronics, submitted. Falangas, E.T., Dworak, J.A., and Koshigoe, S. (1994). Controlling plate vibrations using piezoelectric actuators. Control System IEEE, 14(4), 34–41. Franklin, G., Powell, D., and Workman, M. (1997). Digital Control of Dynamic Systems. Prentice Hall. Giaime, J., Saha, P., Shoemaker, D., and Sievers, L. (1996). A passive vibration isolation stack for ligo: design, modeling, and testing. Review of scientific instruments, 67(1), 208–214. Ito, S., Neyer, D., Pirker, S., Steininger, J., and Schitter, G. (2015). Atomic force microscopy using voice coil actuators for vibration isolation. 2015 IEEE International Conference on Advanced Intelligent Mechatronics (AIM), 470–475. Kim, Y., Kim, S., and Park, K. (2009). Magnetic force driven six degree-of-freedom active vibration isolation system using a phase compensated velocity sensor. Review of Scientific Instruments, 80(4). Masciotti, J.M., Lasker, J.M., and Hielscher, A.H. (2008). Digital lock-in detection for discriminating multiple modulation frequencies with high accuracy and computational efficiency. IEEE Transactions on Instrumentation and Measurement, 57(1), 182–189. Munnig Schmidt, R., Schitter, G., Rankers, A., and van Eijk, J. (2014). The Design of High Performance Mechatronics. Delft University Press, 2nd edition. Nestorovic, T. and Trajkov, M. (2013). Optimal actuator and sensor placement based on balanced reduced models. Mechanical Systems and Signal Processing, 36(2), 271–289. Pirro, S. (2006). Further developments in mechanical decoupling of large thermal detectors. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 559(2), 672–674. Preumont, A. (2012). Vibration control of active structures: an introduction, volume 50. Springer Science & Business Media. Schneiders, M.G.E., van de Molengraft, M.J.G., and Steinbuch, M. (2003). Introduction to an integrated design


for motion systems using over-actuation. Proceedings of the European Control Conference. Thier, M., Saathof, R., Csencsics, E., Hainisch, R., Sinn, A., and Schitter, G. (2015). Design and control of a positioning system for robot-based nanometrology. atAutomatisierungstechnik, 63(9), 727–738. Ungar, E. and Gordon, C. (1983). Vibration criteria for microelectronics manufacturing equipment. Proceedings of International Conference on Noise Control Engineering, Edinburgh, Scotland, 487–490.

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