Self-tuning of position feedback and velocity feedback of active vibration isolation system with 6 DOFs

Vibration isolation systems are widely used in high-precision motion systems such as IC stepper and high-precision measurement devices such as electron microscope [1-3, 4]. Ground vibration and direct disturbance force such as motion stages lead to payload vibration. Traditionally, passive isolators, such as air springs and Euler springs, are employed to isolate the vibrations above 10 Hz [5]. However, the passive isolator amplifies the vibrations near the resonance frequency which is normally in range of 2 ~ 6 Hz and greatly degrades the performance of precision equipment. Unlike passive isolator, Active Vibration Isolation System (AVIS) can isolate both base vibration and direct disturbance without amplifying vibrations near resonance frequency. For a typical 6-DOF AVIS, position loop and velocity loop are both employed to achieve high antivibration performance and maintain position stability [1, 6-8]. Instead of velocity feedbacks, force feedbacks or acceleration feedbacks are used in pneumatic vibration isolation system and in aerospace vibration isolation system [9-11], in order to realize sky-hook damping [12-13]. Generally, loop-shaping tuning method is used to determine parameters of the sky-hook damping loop and the position feedback loop in sequence. However, there are 12 feedback loops, and at least 18 parameters in a 6-DOF AVIS. So the loop-shaping tuning method costs much time and greatly depends on the operator’s experience. In this paper, a fast self-tuning procedure is presented to get optimal damping for the 6-DOF AVIS with sky-hook damping. Firstly, the dynamic model of the 6-DOF AVIS is built. Secondly, we propose the self-tuning procedure and simulate the performance. At last, the selftuning controller is validated with an experiment.


Introduction
Vibration isolation systems are widely used in high-precision motion systems such as IC stepper and high-precision measurement devices such as electron microscope [1][2][3]4].Ground vibration and direct disturbance force such as motion stages lead to payload vibration.Traditionally, passive isolators, such as air springs and Euler springs, are employed to isolate the vibrations above 10 Hz [5].However, the passive isolator amplifies the vibrations near the resonance frequency which is normally in range of 2 ~ 6 Hz and greatly degrades the performance of precision equipment.
Unlike passive isolator, Active Vibration Isolation System (AVIS) can isolate both base vibration and direct disturbance without amplifying vibrations near resonance frequency.For a typical 6-DOF AVIS, position loop and velocity loop are both employed to achieve high antivibration performance and maintain position stability [1,[6][7][8].Instead of velocity feedbacks, force feedbacks or acceleration feedbacks are used in pneumatic vibration isolation system and in aerospace vibration isolation system [9][10][11], in order to realize sky-hook damping [12][13].Generally, loop-shaping tuning method is used to determine parameters of the sky-hook damping loop and the position feedback loop in sequence.However, there are 12 feedback loops, and at least 18 parameters in a 6-DOF AVIS.So the loop-shaping tuning method costs much time and greatly depends on the operator's experience.
In this paper, a fast self-tuning procedure is presented to get optimal damping for the 6-DOF AVIS with sky-hook damping.Firstly, the dynamic model of the 6-DOF AVIS is built.Secondly, we propose the self-tuning procedure and simulate the performance.At last, the selftuning controller is validated with an experiment.

Structure dynamics
A typical 6-DOF AVIS includes 3 vibration isolators, as shown in Fig. 1.The payload is supported by 3 vibration isolators displaced at the vertices of equilateral triangle on the base.We assume that the centre of equilateral triangle is the origin of the coordinate system OXYZ, the Z axis is in the gravity direction, and the X axis in the direction perpendicular to the connection line of Isolator 1 and Isolator 3. As is shown in Fig. 2, the displacement between each two isolators is L, and the three isolators distribute on the circle with the radius equates R. The system geometry is depicted in Fig. 2. It shows both the sensor and actuator locations together with the centre of gravity, indicated with m, with respect to the virtual reference point indicated with O.In the coordinates system OXYZ, the Cartesian coordinates of the system is defined as Six geophone sensors and six eddy sensors are used to measure the absolute velocity and the relative position of the payload.So the measurement results is written as [ ] where V indicates the vertical direction and H indicates the horizontal direction, A Hi and A Vi (i = 1, 2, 3) denotes the horizontal and vertical velocity (or displacement) signals http://dx.doi.org/10.5755/j01.mech.17.6.1006 for i-th isolator.
Because the angles x ϕ , y ϕ , and z ϕ are very small [1], the sensor output in terms of generalized position or velocity is given by where matrix T S is used to transform the measurement results of geophones and eddy sensors from physical axis into logical axis.

Fig. 3 Model of structure dynamics
Based on the principle of structure dynamics, each vibration isolator is equivalent to three mutually orthogonal springs and dampers, as shown in Fig. 3 [1,14].By using the Euler-Lagrange formalism, the load dynamic equation is expressed as M Mθ + Cθ + Kθ = F (1) where M represents a symmetric mass matrix K is a symmetric stiffness matrix and F M is the matrix of the control force in logical axisθ The actuator output L F in terms of physical coordinates is given by

Control strategies
As shown in Fig. 4, Six single input, single output (SISO) controllers with a velocity feedback and a position feedback are used to compensate the vibrations in six directions.The decoupling matrix T S is calculated from geometric relations.Sensor matrix T S is used to transform the measurement results of geophones and eddy sensors from physical axis into logical axis.Actuator decoupling matrix T A is used to transform the control signal from logical axis into actuator axis.With the coupling matrix and decoupling matrix, the 6×6 multiple input, multiple output (MIMO) controllers of AVIS are reduced to six SISO controllers in six logical axes.A velocity proportionalintegral-derivative (PID) controller is used to increase the damping, and a position PID controller to maintain the position stability.As the structure coupling is weak in six orthogonal coordinates, six SISO feedback controllers can be selftuned separately.For each logical axis, the control schematic of SISO feedback is simplified as shown in the Fig. 5.Only 3 parameters of a typical SISO controller, velocity proportional gain k v , position proportional gain k p and integral gain k i , must be determined with self-tuning procedures.Besides these 3 parameters, SISO controller includes notch filters and lowpass filters.The notch filters, which are used to depress the structure modes, depend on the actual systems and their parameters are easily determined.The lowpass filters are used to depress high frequency noise, and the cross frequency for velocity loop is commonly set to 50 Hz while the cross frequency for the position loop is about 10 Hz.To simplify the question, the self-tuning controller only determines the 3-PID parameters.

Self-tuning method for SISO feedback
The payload vibration is caused by both base vi-bration and direct force disturbance.As shown in Fig. 4 Normalizing Eq. 2 and Eq. 3 with respect to n ω , n s , 1 ξ , ξ , m and η , we can get ( ) ( ) with ( ) It should be noted that the parameters k p , k i and k v do not appear in Eq. 4 and Eq. 5, and the mechanical parameters n ω and m only appear as gains.These imply that only the parameter 1 ξ is needed to determine the optimum control parameters for the feedback control parameters ξ and η .The parameter 1 ξ must be very small to achieve high vibration isolation performance in high-frequency band.Its typical value is set to 0.02.As the parameters ξ and η are determined, the parameter n ω can be optimized by minimizing the vibration of the payload.
So we can follow the following procedure: first determine the damping rate ξ and integration rate η, and then determine the n ω .
3.1.Damping rate ξ and integration rate η a) Stability Fig. 6 is root locus with different integration rate η.The damping rate ξ and integration rate η are limited into the stable area shown in Fig. 7.
Fig. 6 Root locus with different integration rate η  The position response to step disturbance force on the payload is shown in Fig. 9, where P 0 and T 0 are the position response amplitude and period without active control separately.Fig. 9 Step response of ( ) To limit the setting time into T 0 , η and ξ are restricted into the area of 'Setting time 1T 0 ' area, as shown in Fig. 10.
Above all, the optimal η and ξ should set into 1T 0 range.Damping ξ = 0.7 and η = 0.36 are recommend to achieve maximum damping rate.( ) is the disturbance force PSD.By minimization of J in Eq. 7, the optimal resonance frequency n ω can be calculated.Follow the procedure of chapter 3, we can calculate the feedback control parameters.There are two primary vibrations, vibration from base floor and direct disturbance acoustics, the acceleration PSD of base vibration is depicted in Fig. 11 while the random disturbance force caused by acoustics noise below the level of 0.01 N.    The metrology vibration with or without using self-tuning controller are shown in Figs. 12 and 13.With active control, the resonant PSD is dropped down to below 10 -6 (m/s 2 ) 2 /Hz, and the horizontal acceleration and vertical acceleration are depressed to 10 -8 (m/s 2 ) 2 /Hz in the frequency range above 3 Hz and above 4 Hz separately.

Experiment and verification
To verify the vibration isolation performance of self-tuning controller, an experiment is executed.The AVIS test platform is shown in Fig. 14 Compared with the base vibration, the effects of acoustics noise can be ignored.The self-tuning controller of AVIS is turned on with the parameters listed in Table .Both the base and payload vibration are measured by accelerometers, which are shown in Figs. 15 and 16.   16 shows that the PSD level of payload acceleration is satisfied and Fig. 16 shows that the acceleration PSD value is below the level of 10 -10 (m/s 2 ) 2 /Hz in the frequency range of 10 Hz to 100 Hz.These imply that the self-tuning controller can achieve high vibration isolation performance and shorten the tuning time.

Conclusion
A self-tuning procedure is proposed for 6-DOF AVIS with position feedback and velocity feedback, which aims at optimal damping of AVIS with Sky-hook.As the structure coupling is weak in six orthogonal coordinates, self-tuning is realized in three steps: calculation of the decoupling matrix, identification of the structure parameters based on the dynamic model of the AVIS, optimization of the feedback controller by minimizing the vibration of the payload.An experiment is implemented to verify the selftuning method and the performance of AVIS with a metrology test platform.The acceleration PSD value is below the level of 10 -10 (m/s 2 ) 2 /Hz in range of 10 Hz and 100 Hz by using the self-tuning controller.Compared with the traditional loop shaping method, this self-tuning procedure can quickly and efficient determine the feedback control parameters and doesn't depend on operator's experience.Active Vibration Isolation Systems (AVIS), which are composed of position feedback loop and velocity feedback loop, are widely used in IC equipments and high precision metrology devices to achieve high vibration isolation performance and maintain position stability.However, the performance of AVIS is always affected by both position feedback loop and velocity feedback loop.Traditionally, the control parameters of the velocity loop and feedback loop are determined in sequence.But theses tuning procedures are time-consuming and the performance of AVIS greatly depends on operator's experience.Further more, AVIS controlled with these parameters can not attain the best performance.In this paper, we propose a self-tuning procedure for the AVIS with velocity feedback and position feedback, which aims at optimal damping of AVIS with Sky-hook.As the structure coupling is weak in six orthogonal coordinates, the self-tuning procedure is realized in three steps: calculation of the decoupling matrix, identification of the structure parameters based on the dynamic model of the AVIS, optimization of the feedback controller by minimizing the vibration of payload.A simulation and an experiment are implemented to verify the self-tuning method and the performance of AVIS.

Fig. 5
Fig. 4 Control block of the AVIS

F
to payload x m can be deprived and normalized as

Fig. 7
Fig. 7 Stable area b) Base vibration isolation Fig. 8 denotes the vibration transmissibility from base at resonance frequency according to the integration rate η and the damping rate ξ.In order to limit the vibration transmissibility resonance into 5 dB, η and ξ are limited into the area of "Resonance < 5 dB" as shown in Fig. 8.

Fig. 8
Fig. 8 Resonance vibration transmissibility contour c) Setting-time With step disturbance on the payload, the settingtime is restricted by the η and ξ at the same n ω .

4. 1 .
Self-tuning procedure for AVIS: a) calculation of the sensor decoupling matrix and actuator decoupling matrix, based on the geometry relations; b) estimation of system parameters in six logical axes, such as relative damping c, stiffness k, m.This parameter estimation approaches are divided into indirect and direct techniques who work together to use online [2, 15, 16].c) Determination of the optimal controller parameters.

Fig. 11
Fig. 11 Base acceleration PSD level The resonance frequencies n ω for each SISO loop are optimized based on the Eq. 7. All three parameters are then calculated based on the Eq. 8.The results are depicted in the Table.

Fig. 12 Fig. 13
Fig. 12 Horizontal acceleration PSD comparison with the use of self-tuning controller . The payload is supported by three vibration isolators, and all the parameters are described in Chapter 4. The vibration source includes base vibration disturbance and acoustics noise disturbance.The vibration level is measured using the selftuning controller.Lightweight piezoelectric accelerometers (PCB 352A10 ICP accelerometers, frequency range 0.003 -10 kHz, sensitivity ~ 1.052 mV/m/s 2 , 0.7 gr) are used to measure the base vibration and metrology frame vibration at two locations at the same direction.The measured signals are conditioned and subsequently sampled by a SigLab data acquisition module (featuring four 20-bit simultaneously sampled A/D channels, two 16-bit D/A channels, and analog 4-th order quasielliptic anti-aliasing filter).

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. H. Liao, X. P. Li, Z. Y. Yuan SELF-TUNING OF POSITION FEEDBACK AND VELOCITY FEEDBACK OF ACTIVE VIBRATION ISOLATION SYSTEM WITH 6 DOFS S u m m a r y