Active vibration iolation for a 6 degree of freedom cale model of a high preciion machine W.B.A. Boomma Supervior Report nr : Prof. Dr. Ir. M. Steinbuch : DCT 8. Eindhoven Univerity of Technology Department of Mechanical Engineering Control Sytem Technology Eindhoven, February 8
Table of content Introduction... 3 Chapter : General introduction... 4. Sytem decription... 4. Active Vibration iolation... 6.3 Dynamic of the active vibration iolation ytem... 9 Chapter : Analyzing and controlling D.... Sytem analyi.... loop-haping....3 verifying the controlled ytem... 4 Chapter 3: Analyzing and controlling 6 DOF... 5 3. Sytem analyi... 5 3. loop-haping... 6 3.3 verifying the controlled ytem... 9 3.4 Horizontal geophone output... Chapter 4: Concluion and recommendation... 4. Concluion... 4. Recommendation...
Introduction The main goal of thi project i the intallation of a 6 degree of freedom vibration iolation device. Beide the intallation, the econdary goal i to deign and intall a controller for active damping. Thi approach will prove that the machine function properly and that the machine i ready for the implementation and evaluation of more advanced control trategie. For thi project a cale model of a high preciion machine i granted by Philip. The cale model can roughly be decribed a a platform hanged by three pring on a bae frame. Furthermore there are 6 actuator intalled which able the platform to move in 6 direction. The relative poition of the platform to the bae frame i meaured in all direction by 6 poition enor, 6 velocity enor meaure the abolute peed of the platform. For the control purpoe, firt a dynamical model i derived, afterward the dynamic for degree of freedom i analyzed and a controller i tuned by mean of loop haping. In chapter, only degree of freedom i analyzed and controlled. In chapter 3, all 6 degree of freedom are analyzed and controlled in a imilar way a in chapter. Thi i done for each degree of freedom eparately. Beide aforementioned, thi report alo function a an intruction guide for operating the cale model of the high preciion machine. 3
Chapter : General introduction. Sytem decription The ytem i already briefly decribed in the introduction. The ytem i baically a platform which i upported by 3 pring. Thi enable the platform to move in 6 degree of freedom. 6 actuator are intalled for the motion of the platform. Thee actuator are each accompanied by enor: poition enor and geophone (velocity). The ignal of the enor i guided through an electronic board and end to the dspace d4 ytem. The Architecture of the experimental etup i chematically repreented in figure.. Mechanical tructure enor Actuator Electronic board Amplifier dspace d4 Figure. Architecture of the experimental etup In order to get a better overview of the mechanical tructure of the ytem and the meaured propertie, degree of freedom i chematically repreented in figure.. The relative poition of the platform to the bae frame i called x relative, the abolute velocity of the platform i called ẋ frame. So in thi cae xrelative i meaured by the poition enor and the ẋ frame i meaured by the velocity enor. 4
ẋ frame x relative Figure.: Model of DOF of the ytem To be able to do meaurement and for the implementation of a controller a computer equipped with a dspace d4 controller board i available. The controller board i upplied with a panel with 8 ADC input and 8 DAC output. The Lorentz actuator are Technotion coil type UC3, which are ued a a phae coil. The velocity i meaured by geophone, the 4.5 Hz variant of the GS-D produced by Geopace. The geophone add an important characteritic for the ue on thi model, becaue they have a high-pa character for the meaurement of the velocity. Thi will be explained later on in thi chapter. For implification the horizontal actuator en enor will be indicated with y, x and x 3. The poition of y i near z, the poition of x i near z and the poition of x 3 i near z 3. In the following part of thi chapter, active vibration iolation for active iolation will be dicued. For implicity the concept i explained uing a one degree of freedom ytem. 5
. Active Vibration iolation For implification, at firt only the firt vertical motion, z, i analyzed. Looking at that particular degree of freedom, the uncontrolled ytem ha a tranfer function of a tandard ma-pring ytem with negligible damping. In figure. a Bode plot of uch a ytem with arbitrary choen parameter i depicted. In thi cae the input of the ytem i the poition of the bae frame and the output i the poition of the platform. H original k m + k (..) Figure.3: Bode plot ma pring ytem with poition a output The figure how, in the magnitude part, for frequencie below rad/ a lope due to the pring and for frequencie above rad/ a lope of - caued by the ma. Furthermore a reonance peak i located between the lope. Thi reonance peak i undeirable for vibration iolation and need to be eliminated. There are different method available to anwer to thi need. Virtual damping can be added uing the derivative of the poition meaurement or the meaered in a controller. Another method make ue of meaured velocity of the platform. Figure.4 how a ytem where the controller make ue of the derivative of the poition meaurement. Thi mean, in thi cae there i no need for any velocity enor. In figure.4 x relative i the ymbol for the poition of the platform in relation to the poition of the bae frame. 6
ref u P x relative ẋ frame ẋ relative K D Figure.4: Schematic repreentation of the active damped ytem uing the poition enor For the ytem depicted in figure.4, the tranfer function with frame-poition a input and platform poition a output i given by: H damped d + k m + d + k (..) The Bode plot of thi controlled ytem i hown in figure.5. The Bode plot how the reonance peak ha vanihed. Conequently the high frequency lope i reduced to the lope - due to the introduced virtual damping. Thi effect i unwanted, becaue the vibration iolation at high frequencie i now reduced. Figure.5: Bode plot of the active damped ytem uing the poition enor Therefore another way of controlling the ytem i needed. In the following cae the damping of the ytem i achieved uing the abolute velocity of the frame. Thi i in contrat to the derivative of the poition, which i actually the relative velocity of the platform with repect to the bae frame. 7
ref u P x relative ẋ frame ẋ frame K D Figure.6 Schematic repreentation of the active damped ytem uing a velocity enor Uing the abolute frame velocity a input for the controller, the tranfer function in equation (..) change to: H damped k m + d + k (..3) The correponding Bode plot i given in figure.7. In the figure can be een that the reonance peak ha diappeared without lifting the high frequency lope to -. Figure.7: Bode plot active damped ytem uing a velocity enor 8
.3 Dynamic of the active vibration iolation ytem A mentioned in de introduction, the ytem i equipped with a poition enor a well a a velocity enor. Thi implie the tranfer function can alo be derived from the poition of the bae frame to the velocity of the platform. The tranfer function of the uncontrolled ytem uing the velocity a the output and the poition of the bae frame i imilar to function (..) but derived one time. Thi reult in the following tranfer function: H velocity k m + k (..4) The correponding Bode plot i depicted in figure.8. Figure.8: Bode plot ma pring ytem with velocity a output Unfortunately the geophone, ued to meaure the velocity, have dynamic of there own. The dynamic of the geophone mut be accounted for in the controller to prevent intable behavior of the ytem. A implified way to decribe thee dynamic i of a econd order low pa filter. H geophone m m d k g g + g + g (..5) The Bode plot of the dynamic of the geophone i depicted in figure.9. 9
Figure.9: Bode plot dynamic geophone. The Bode plot of the dynamic of the complete ytem, in thi cae the combination of function (..4) and (..5), i depicted in figure.. Figure.: Bode plot complete ytem
Chapter : Analyzing and controlling D.. Sytem analyi Before analyzing the entire ytem, firt the ytem i analyzed uing actuator and it accompanying enor. In thi cae the firt vertical motion i of interet, which i labeled z on the model and will be mentioned by that label in thi report. The motion of z i meaured by enor, enor for the velocity and for the poition. Firt the relative poition of z i analyzed. White noie i generated by Matlab and injected at the input of the actuator of z. The output of the poition enor i filtered by a low pa filter to avoid aliaing and the filtered ignal i recorded. The noie ignal i filtered in a imilar way and i recorded a well. Afterward the frequency repone data i calculated and repreented in figure.. The tranfer function i calculated uing a tranfer function etimate routine, which i a tandard implemented in Matlab. 6 4 - - -.5 - Figure.: Frequency repone of force to poition of firt vertical motion The figure how a typical ma-pring-ytem behavior. The lope of the frequencie below Hz i. The lope of the frequencie above.5 Hz i -. The frequency repone data uing the output of the geophone i important, becaue the geophone will be ued for the control of the ytem. So again a noie ignal i repreented to the actuator of z. The noie ignal i filtered and recorded. The output ignal of the geophone i filtered by the ame filter and recorded a well. The frequency repone data of the meaurement i depicted in figure..
6 4 - -4 - - -.5 - Figure.: Frequency repone of force to geophone of firt vertical motion In the figure, the +3 lope at low frequencie with a correponding phae of 7 can be recognized, a well a a hort + lope and - lope at higher frequencie. Thi i correponding with the expected dynamic depicted in figure.6. Furthermore at 3 Hz an anti reonance can be een, a well a a reonance peak at 4 Hz. Thee reonance are caued by dynamic of the velocity enor in combination with the ytem.. loop-haping. For the control of the ytem a controller i deigned by mean of loop haping. In thi cae the controller conit of 5 part. - high pa filter - lag filter - notch filter - low pa filter - Proportional gain A high-pa filter i ued to filter low frequency noie from the geophone ignal. The filter i of the econd order and ha a cut off frequency of. Hz and a damping factor of.7. Thi reult in the tranfer function: C high pa ( *pi*.) ( *pi*.) *.7* + + *pi*. (..) The +3 lope of a part of the tranfer function introduce tability iue. A a olution for thee problem a econd order lag filter i ued. Thi filter reduce a part of the +3 lope to a + lope. The damping factor i choen.7. The pole lie at. and.5 Hz.
C lag..5 ( *pi*.5) ( *pi*.) *.7* + + *pi*.5 *.7* + + *pi*. (..) The reonance peak at 4 Hz i neutralized uing a notch filter. The notch filter enure afe level for the gain and phae margin of the controlled ytem. C notch + + pi*4* + ( *pi*4) ( *pi*4) (..3) Furthermore a firt order low-pa filter i ued. The filter enure table behavior at higher frequencie and reduce the influence of meaurement noie. The cutoff frequency of the filter lie at 3 Hz. C low pa + *pi*3 (..4) Finally a proportional gain i added, it gain i. C. (..5) prop The tranfer function of the controller combined with the tranfer function of the plant i depicted in figure.3. Furthermore the open loop tranfer function of the controlled ytem, calculated by multiplying both aforementioned tranfer function, i tated in the figure. The frequency repone function of the plant i given in red, the calculated open loop tranfer i depicted in blue and the controller function i depicted in black. 4 - -4-6 - - - - Figure.3: Frequency repone of force to geophone of firt vertical motion of the uncontrolled (red) and the calculated controlled (blue) ytem 3
.3 verifying the controlled ytem Finally the open loop tranfer function of the controlled ytem i verified. Thi i done by adding noie at the output ignal of the controller and determining the tranfer from ignal W to ignal U. Thi i chematically depicted in figure.4 W ref C U P ẋ Figure.4: Schematically repreented etup for verifying the controlled ytem The calculated tranfer function i the enitivity, in hort: S + CH So the open loop tranfer function can be calculated uing the enitivity: CH S S Thi reult in the tranfer function given in figure.5. The figure how the new calculated tranfer function in red and the one already given in figure.3 in blue. The figure how roughly the ame pattern, but with more noie at higher frequencie. Alo the low frequency gain margin i a bit le. The magnitude pike through the db line, but thi i only caued by the uncertainty of the meaurement. Thi ha no influence on the tability of the controlled ytem. 5-5 - - - - -.5 - - Figure.5: Frequency repone of force to geophone of firt vertical motion of the calculated controlled (blue) and meaured controlled ytem (red). 4
Chapter 3: Analyzing and controlling 6 DOF 3. Sytem analyi In the previou chapter the firt vertical motion i analyzed and controlled. In thi chapter the entire ytem will be analyzed and controlled. The ytem i analyzed one dimenion at the time. So for all degree of freedom the tranfer function are eparately determined. The tranfer function are determined in a imilar way a in chapter. In thi cae, only the output of the geophone i ued, becaue that i the only output which i going to be ued by the controller at thi time. The tranfer function are determined in the ame way a the tranfer of ż decribed in paragraph.. Frequency repone plant zdot 6 4 - -4 - - -.5 - Figure 3.: Frequency repone of force to poition of firt vertical motion Frequency repone plant zdot 6 4 - -4 - - -.5 - Figure 3.: Frequency repone of force to poition of econd vertical motion Frequency repone plant z3dot 6 4 - -4 - - -.5 - Figure 3.3: Frequency repone of force to poition of third vertical motion Frequency repone plant ydot 6 4 - -4 - - -.5 - Figure 3.4: Frequency repone of force to poition of firt horizontal (y ) motion 5
Frequency repone plant xdot 6 4 - -4 - - -.5 - Figure 3.5: Frequency repone of force to poition of econd horizontal (x ) motion Frequency repone plant x3dot 6 4 - -4 - - -.5 - Figure 3.6: Frequency repone of force to poition of third horizontal (x 3 ) motion Figure 3.4 to 3.6 how an intereting phenomenon. Below ±.6 Hz the frequency repone plot how an unexpected hape. In tead of howing the +3 lope a decribed in chapter, the figure how a lope of +. Thi i probably caued by the dependency of the output of the geophone on the tilt of the platform. Beide thi, the of the tranfer function i unatifactory at thee frequencie. Thi make the interpretation of the data difficult. The phenomenon will be further explained in chapter 3.3. Thi phenomenon will eriouly jeopardize the control of the platform in the horizontal plane. 3. loop-haping For the control of the entire ytem 6 controller are eparately tuned by mean of loop haping. In thi cae the controller for z and z conit of 5 part, for the other controller the notch filter i not neceary, o only 4 part are preent. The controller are tuned eparately. Thi mean the ytem i analyzed one degree of freedom at the time without applying a controller on any other degree of freedom. After the 6 controller are tuned, the controller are all applied on the ytem, and the tuning of the control loop i checked. The controller will conit of 5 part: - high pa filter - lag filter - notch filter(only z and z ) - low pa filter - Proportional gain The function of the controller part are tated in the following equation A high-pa filter i ued to filter low frequency noie from the geophone ignal. The filter i of the econd order and ha the tranfer function: 6
C high pa ( *pi*f ) ( *pi*f ) HP HP *.7* + + *pi*f HP (3..) The +3 lope of a part of the tranfer function introduce tability iue. A a olution for thee problem a econd order lag filter i ued. Thi filter reduce a part of the +3 lope to a + lope. C lag ( *pi*flag ) f lag lag f lag *.7* + + *pi*flag ( *pi*flag ) *.7* + + *pi*f (3..) The reonance peak at 4 Hz i neutralized uing a notch filter. The notch filter enure afe level for the gain and phae margin of the controlled ytem. C notch + ( *pi*f notch ) ( ) + pi*f * + *pi*f notch notch (3..3) Furthermore a firt order low-pa filter i ued. The filter enure table behavior at higher frequencie and reduce the influence of meaurement noie. C low pa *pi*f LP + (3..4) Finally a proportional gain i added, it gain i determined for each actuator eparately, for example C Y3. C. (3..5) prop The controller i tuned in a imilar way a the controller decribed in paragraph.. Figure 3.7 to 3. how the open loop tranfer function of the controlled motion combined with the deigned controller and the open loop tranfer function of the controlled ytem. The figure how each degree of freedom eparately. 7
open loop tranfer zdot open loop tranfer zdot 4 4 - -4-6 - -4-6 - - - - - - - - Figure 3.7: Frequency repone of firt vertical motion of the uncontrolled (red) and the calculated controlled (blue) ytem Figure 3.8: Frequency repone of econd vertical motion of the uncontrolled (red) and the calculated controlled (blue) ytem open loop tranfer z3dot open loop tranfer ydot 4 4 - -4-6 - -4-6 - - - - - - - - Figure 3.9: Frequency repone of third vertical motion of the uncontrolled (red) and the calculated controlled (blue) ytem Figure 3.: Frequency repone of firt horizontal motion (y ) of the uncontrolled (red) and the calculated controlled (blue) ytem open loop tranfer xdot 4 - -4-6 - open loop tranfer x3dot 4 - -4-6 - - - - - - - Figure 3.: Frequency repone of econd horizontal motion (x ) of the uncontrolled (red) and the calculated controlled (blue) ytem Figure 3.:Frequency repone of third horizontal motion (x 3 ) of the uncontrolled (red) and the calculated controlled (blue) ytem 8
The parameter of the controller hown in aforementioned figured i repreented in Table 3.. Z Z Z 3 Y X X 3 Controller gain...5.5.. High Pa frequency....3..3 Firt lag filter frequency..3..3.3.3 Second lag filter.5.5.5.8.6.6 frequency Notch frequency 4.5 4.5 N.A. N.A. N.A. N.A. Low Pa frequency 3 3 3 3 3 3 Table 3. Controller parameter obtained by loop haping 3.3 verifying the controlled ytem According to the open loop tranfer function of the controlled ytem (figure 3.7 to 3.), the ytem i uppoed to be table. Appling the controller on the cale model, the platform i uppoed to hang till. However thi i not the cae, the platform make a mall motion at a low frequency. Thi motion and to verify the open loop tranfer of the controlled ytem, the tranfer function of the ytem are meaured in the imilar way a decribed in chapter.3. 5-5 open loop tranfer zdot - - - 5-5 open loop tranfer zdot - - -.5 -.5 - Figure 3.3: Frequency repone of firt Figure 3.3: Frequency repone of econd vertical motion of the calculated controlled (red) vertical motion of the calculated controlled (red) and the meaured controlled (blue) ytem and the meaured controlled (blue) ytem 9
5-5 Frequency repone plant z3dot - - -.5 - Figure 3.5: Frequency repone of third vertical motion of the calculated controlled (red) and the meaured controlled (blue) ytem Gain [db] 5-5 open loop tranfer ydot - - -.5 - Figure 3.6: Frequency repone of firt horizontal (y ) motion of the calculated controlled (red) and the meaured controlled (blue) ytem 5-5 open loop tranfer xdot - - - 5-5 open loop tranfer x3dot - - -.5 -.5 - Figure 3.5: Frequency repone of econd Figure 3.6: Frequency repone of third horizontal motion (x ) of the calculated controlled horizontal (x 3 ) motion of the calculated controlled (red) and the meaured controlled (blue) ytem (red) and the meaured controlled (blue) ytem The horizontal motion of the platform i caued by the controller of y, x and x 3. Diabling thee actuator will top the motion. The only olution to the problem i to lower the controller gain. Thi reult in a lower damping of the ytem for the horizontal motion. Thi olution reult in altered controller parameter. Z Z Z 3 Y X X 3 Controller gain...5..5. High Pa frequency....3.3.3 Firt lag filter frequency..3..3.3.3 Second lag filter.5.5.5.8.6.6 frequency Notch frequency 4.5 4.5 N.A. N.A. N.A. N.A. Low Pa frequency 3 3 3 3 3 3 Table 3.: Final controller parameter
3.4 Horizontal geophone output A already tated in the previou chapter, the output of the horizontal geophone are influenced by the angle the platform make to the horizontal plane. A olution might be to correct the output of the geophone for the influence of thi angle. Aumed a perfect model can be derived, the corrected geophone output will only repreent the velocity of the platform in the horizontal direction. The tranfer function calculated uing the corrected geophone output will how a different picture than in figure 3. to 3.. Thi will give the opportunity to retune the controller to a higher bandwidth To get a better inight of the phenomenon the tranfer function i determined of the tilt of the platform to the geophone output. The meaurement i done by adding white noie to z, while fixating z. The tranfer function i determined of z to x. Thi way of working i caued by the anti-aliaing filter, which ha only input and output. The meaurement actually need 3 input for a proper meaurement. Figure 3.7: Frequency repone of the tilt to the econd horizontal geophone (x ) The frequency repone diagram (figure 3.7) how a + lope with a correponding +9 degree phae. However, the quality of the meaurement i poor. Thi require further invetigation with additional equipment.
Chapter 4: Concluion and recommendation 4. Concluion The cale model of the ytem i intalled and made operational. The tranfer function are determined to get a better inight of the dynamic of the ytem. Furthermore the controller are deigned and tuned by mean loop haping uing thee tranfer function. The controller make only ue of the output of the geophone, a required. Afterward the controller are verified by determining the open loop tranfer function of the controlled ytem. 4. Recommendation The difference of the calculated open loop tranfer of the controlled ytem and the meaured verion of z i unexpected and need further invetigation. Furthermore, due to the nonlinearity in the ytem and particularly in the horizontal plane, the output of the enor need to be corrected. Thi will require further tudy of the ytem en particularly the influence of the angle of the platform to the output of the horizontal geophone. In addition, the controller decribed in the report hould be able to handle a lightly higher gain than the gain ued. However, thi reult in untable behaviour of the ytem. It will be ueful to check the meaurement of the tranfer function of the uncontrolled and controlled ytem.