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Modelling, instrumentation and control of a 6

1. Table B 1 Dimensions of magnetic actuator 67 Four different surfaces can be distinct through which the flux paths pass Si dj hg So dysha Sa d ba Sa dj bg For each part of the actuator the reluctance can be computed using Equation 4 9 R TE ES amp Ra a TTE A Rs fais EN Ra u Raz Fi m The reluctances Ry Ro and Ra ate constant while R and R are dependent on z the mag nitude of the air gap Since eventually only very small rotations of the I with respect to the E are considered it is assumed that the air gaps at all of the 3 legs of the E are equal Using the well known laws for electrical circuits an equivalent reluctance Ry for the total circuit can be derived A distinction is made between the constant reluctance Rc and the term depending on air gap Rx Rr Ro Rx Ro 3Ri 3R24 R3 Rx Rar Raz 68 Appendix C Code C 1 C code for reading parallel laser include periph h include stdio h include lt math h gt define AO 24 define Ai 25 define A2 26 define A3 27 define A4 28 define RESET 0x1D define IGE OxiE define SCLK1 Ox1F Globals volatile int module 0 unsigned int errors volatile unsigned int readback fi Prototypes void test_dio void test_base_dio void read _laser void pause JJ nt di main Henu of test choices dd ISIS re main 1 enable_monitor e
2. 4 2 1 Experimental setup 4 22 Results on de an Shi Sh Is 4 2 3 Modelfit 11 Conclusion to sen we mere ed Aeg eee eee ee eee eee ee eee ee ee ee eo ee ee ee eo ee eee ee ae ee ee ee ee ee ee ee a en ee ee eee eee ee ee eo eo eo eo ee ee iii 5 Control of 1 DOF system with magnetic actuator 53 System description menr tk sagene la ee hd Gol ende See oe 5 2 near Controle mre mek tari E Pa sms K Linearization jy A ay rae abs ORGS ROR ek EA WK A 5 2 2 System identification ee 5 2 3 Simulations ooo doncs ae a ee 5 24 Real time results zer an nertsen edn Al len EE p se bedt 5 25 1CORCLUSIOM une se sa ge etende ar kast Sak B Hede ge tee ete t 53 Nonlinear control naat ene ens ad de GR Rwa eee 5 3 1 Feedback linearization o e e e 5 3 2 Sliding mode control of feedback linearized system o 5 3 3 Sliding mode control of non affine system with algebraic input invertibility 53 4 535 Real time results ee Gonclusion ae bo Seed gn Seek Sek eho ee eB aes EEL ed 6 Modelling and control of the 6 DOF system 61 Systemmodel es se dei 4 ka eerde sara ee GS ae OW 6 1 1 6 1 2 6 1 3 6 1 4 6 1 5 Assumptions and definitions i o ee aa Sete Equations of motion ee ee Master slave sa ares Decoupling a io w w p es tees Aae be ete BO wide je ance U Measured variables vs performance variables 6
3. Ez Eet li Fa Ls La 0 0 0 Fa Se ae This relation cannot be inverted directly When the control actions Fagg Tecos are computed the approach proposed by 6 is used to determine the force each pair of coils has to deliver Use Fin Frcoc RO ER 6 23 Fa Tzooe With M Mo My M i IME Meg gt mz With M the first row of M etc such that MM is a diagonal matrix with ones on the diagonal when the tray is centered 51 So after the control actions with respect to the COG are computed they are translated to a force for every pair of opposed coils After this the previously used nonlinear transformation is used to compute the input current to each coil Since each coil can only exert attractive forces on any moment only one coil of an opposite pair is active depending on the sign of the required force For example for the horizontal coils in magnet box 1 F Tha 914 Fn and Iyg 0 for Fy 20 6 24 F fis iiz and Lj 0 for Fi lt 0 6 25 6 2 2 Simulations A typical application of this system is high precision positioning in photo lithography or the produc tion of Micro Electro Mechanical Systems MEMS Some specifications with respect to maximum positioning errors are given in 10 but no information is available with respect to typical reference trajectories In simulations smooth 3rd order profile steps were chosen as setpoint for each DOF The amplitude of the steps is limited
4. Elil Compensate force in z direction for gravity Fcl2 Fc 2 mt gc Map control action to actuators for i 0 i lt 4 i Horizontal A Lil Ms i 0 Fc 0 Ms i 1 Fo 1 Ms i 2 Fc 2 Ms i 3 Fo 3 M5 i 4 Fe 4 1 Ms 33 5 Fc 5 for i 4 zirt H Vertical Prec i 4 Ms i 0 Fe 0 Ms i 1 Fe 1 Ms i 2 Fc 2 Ms i 3 Fc 3 Ms i 4 Fc 4 Ms i 5 Fc 5 Gap measurements from capacitive sensors gap 01 0 U 18 gid 86 gap 1 0 0 19 H g21 gapf21 1 0 20 g32 gap 3 1 U 21 1f g42 gap 0 3 0 22 gi4 gapli 3 U 23 H g24 gap 2 3 U 24 g34 gap 3 3 U 25 LI gal Compute complementary gaps gap 0 1 2 gnx gap 0 0 gapf0 2 2 gnz gap 01 3 gap 1 1 2 gnx gap 1 0 gap 1 2 2 gnz gapfilis gapi21 01 2 gnx gap 2 i gap 2 2 2 gnz gap 2 3 gap 3 0 2 gnx gap 3 i gap 3 2 2 gaz gap 3 3 Input to linear motors DONT SENT TO LINEAR MOTORS BUT USE FOR FEED FORWARD yl03 0 yti 0 ti yt2 Fc 0 Fx y 2j yi3 kscale KD2 U 26 KP2 U 0 U 6 KID U 32 Fe i Fy Translate actuator forces to input currents using inverted non linearity non robust for i 031 lt 4 i Horizontal coils 1 if FHact i gt 0 1 yl4 4 i gaplil Ol sgrt FHactli kc giS 4 i 0 3 else 1 yl4 4xi 0 yl5 4 i
5. gap i 1 sqrt FHact i xc r for i 0 i lt 4 i fi Vertical coils i if FYact i gt 0 y 6 4 i gaplilf2 sgrt FVactli kc yl7 4 i 0 else 1 y 6 4x3 0 y 7 4x1 gap i 3 sqrt FVact i kc I r Th SSeS EES SS Somat eee Terminate me re cs p etn static void mdiTerminate SimStruct 45 i I tifdef MATLAB MEX FILE Is this file being compiled as a MEX file include simulink c MEX file interface mechanism Helse include cg sfun h Code generation registration function tendif 87 C 5 Demo program The following code has been used to demonstrate some basic functionalities of the linear motors and Unidex software It contains initialization basic motions like translations and circular motion of the middle point of the tray independent operation or master slave operation of the linear motors and some examples of WHILE and FOR loops in G code For more details and advanced options refer to 3 PROGRAM METRIC INCREMENTS 3Use metric system WAIT AL sWait until all previous commands are executed ENABLE XY ZU Enable all axes MESSAGE DISPLAY Axis X Y Z and U enabled HOME XYZU Home all axes GEAR 3 1 1 1 Link axis Z as slave to axis X at 1 1 ratio GEAR 4 2 1 1 L nk axis U as slave to axis Y at 1 1 ratio 3i 0 WHILE i lt 10 iWhile loop example not used Enter commands here i i 1 ENDWHILE Gi x10 yi0 600 s
6. Magnetic forces real T FL 4 Forces driving linear motors real T Li L2 L3 L4 Arm length ml int_T i Index real T KC 161 Force constant per coil real T sumFZ Sum of forces in z direction static real_T wkefi6 41 1 1 1 1 1 1 1 1 1 1 1 1 1 1 175 for i 0 i lt 16 i 1 KC i kc wkc i Multiplicative perturbation of coil force constants Compute arm lenghts Li LO x 51 Loty L2 LO x 4 LOtx L3 LO x 5 L y 14 LO x 4 LO x Compute gaps and store in matrix GIO gaz x 4 x 0 Li tan x 9 Gli 2 gnx G 0 GI23 gz x 6 Li tan x 7 G 3 2 gnz GI2 G 4 gox xl5 xL1 L2 tan x 9 GI5 2xgnx 614 GI6 gaz xL6 L2rtan x 81 G 7 2 gnz GI6 GIS gnx x 4 x 2 L3 tan x 9 GI9 gnx G 8 G 10 gaz x 6 L3 tan x 7 GI11 2gnz G 10 G 12 gnx x 5 x 3 La tan x 9 G 13 2 gnx G 121 G 14 gnz xf6 L4 tan x 8 1 G 15 2 gnz G 14 Show error message if any gap is smaller than 0 for i 0 i lt 16 i 1 if G i lt le 6 I ssSetErrorStatus Gap 0 MECHANICAL CONTACT Compute magnetic forces for i 0 i lt 16 i i FL KC i I i I i GLi G i 83 Compute forces driving linear motors for i 0 i lt 4 i 1 FL i klin U i fi Compute ist derivatives for i 0 i lt 12 i 1 dxfi x
7. gi zi deanta and d 5 40 JK 5 40 il o min 37 Equivalent control The equivalent control is the control when the system is in sliding mode 68 68 68 681 fle 0 M It follows that 6S Meg a 2Af1 x fala p x Solving for Ieg gives Ia 92 01 4 JD and Nhy 0 if plz gt 0 ha T ri y ES and L 0 if ple lt 0 Stability In the theory of 12 which is applied here it is assumed that the equivalent control and switching term can be added directly This can lead to the case where both actuators are active at the same moment which does not seem logical considering energy consumption for example Stability can be proved for all possible situations I Is Leg 5 41 Stability can be proved for all possible situations 1 S gt 0andp gt 0 2 S gt Oandp lt 0 3 S lt 0andp gt 0 4 S lt 0andp lt 0 For case 1 Jy 0 and Ig ga x1 Espia ga x1 SEAS Stability is guaranteed if pla h so 5 42 Substituting J and fo in h and simplifying gives Zy Ko Jzep Jeen z Kan z PE q ELGEN Y A 5 43 ii Jrz yKe YK in Koran If Equation 5 43 is substituted in Equation 5 42 the term p x cancels out Since all parameters are positive both resulting terms are negative and stability is guaranteed A Jzzp 2 J zn z Kan z lt 0 5 44 yKo YR an Koen For case 2 f gi xi des E and Js go z1 aS Substituting J and Ig in h and simplifying gives ipa gp 2 5 45
8. pause test_dio Test all digital 1 0 bits id test_dio int attr int i volatile unsigned int shadow 0 volatile unsigned int rtn volatile unsigned int err 0 volatile unsigned int reps 0 int col row cirscr printf Testing OMNIBUS DIG digital I 0 Cable Loopback Tests H printf n nInstall Loopback Cable on Digital Connector getchar cursor OFF 72 attr get_attribute for i 0 i lt 2 i 4 int highState lowState err 0 reps 0 cirser printf n nPress any key to terminate test 1 initialize Half for input half output if i 0 i highState DI0_DIR_OUTPUT lowState DIO_DIR_INPUT printf n nLoopback High word to Low word wherexy col raw else 1 highState DIO_DIR_INPUT lowState DI0 DIR OUTPUT printf n nLoopback Low word to High word wherexy amp col amp raw y DIO_dig_dir module highState highState lowState lowState 1 Loop doing writes to Digital 1 0 while kbd_hit i int output shadow unsigned 1 Complement shadow register output shadow OxAS5A5A5A5 Ox5A5A5A5A DI0 write dig module output if shadow set_attribute RGB 255 0 0 else set_attribute RGB 0 0 255 5 rtn DIO read dig module gotoxy col row 1 printi 410x 10x rtn output if rtn output errtt repst if reps4100 0 1 gotoxy col row printf id Errors in d tries
9. ssSetNumInputPorts S 6 return 4 inputs x r rddot and gaps ssSetInputPortWidth S 0 6 Measured states ssSetInputPortWidth S 1 6 Reference ssSetInputPortWidth S 2 6 Second derivative of reference ssSetInputPortWidth S 3 8 Gaps gii g21 g32 g42 g14 g24 g34 g44 ssSetInputPortWidth S 4 6 Derivative of errors ssSetInputPortWidth S 5 6 Integral of errors ssSetInputPortDirectFeedThrough S 0 1 if ssSetNum0utputPorts S 2 return ssSetOutputPortWidth S 0 4 Output to linear motors ssSetOutputPortWidth S 1 16 Current to coils ssSetNumSampleTimes S 1 ssSetNumRWork S GLOBALREALS set number of global reals ssSetNumIWork S GLOBALINTS set number of global ints ssSetNumPWork S 0 set number of pointer work vector elements ssSetNumModes S 0 set number of operating modes for the block ssSetNumNonsampledZCs S 0 set number of nonsampled zero crossings that black detects F static void mdi nitializeSanpleTimes SimStruct S 85 1 ssSetSampleTime S 0 CONTINUOUS SAMPLE TIME ssSetoffsetTime 5 0 0 0 static void mdlOutputs SimStruct S int_T tid InputRealPtrsType uPtrs ssGetInputPortRealSignalPtrs S 0 nn real T ay ssGetOutputPortRea1Signal S 0 Parameters sliding mode algorithm real T LABDA 6 25 25 10 15 15 20 Labda for x y z theta_x theta_y theta_z real_T PHI 6 fie 2 1e 2 ie 2 7e 3 Te 3 1e 2 Ph
10. 1 write_dig_bit A1 0 write_dig bit A2 0 write_dig_bit A3 1 write_dig_bit A4 0 ns 20 write dig bit SCLKL 1 ns 2Q write dig bit IOE 0 ne 120 rtn read_dig read data from laser pos rtny0x1000000 if first_time diff 0 first_time 0 2 else diff pos old_pos old_pos pos if abs diff gt pow 2 23 I overflow if diff gt 0 counter else counter 70 I position counter pow 2 24 pos position position 1 235977184 position position 1e6 write dig bit SCLK1 0 write dig bit IDE 1 gotoxy 0 row 1 printf Current value 10x Current displacement d mm pos position R ms 1 write dig bit A0 0 write dig bit Al 1 write dig bit A2 0 write dig bit A3 1 write dig bit A4 0 ns 20 write dig bit SCLK1 1 ns 20 write dig bit IDE 0 ns 120 rtn read dig read data from laser velocity rtn 1 235977184 velocity velocity 1000 write_dig_bit SCLKi 0 vrite_dig_bit 10E 1 gotoxy 0 row 2 printf Current value 10x Current velocity d um s rtn velocity void test_base_dio void t 44 int attr int i BaseboardDio dio BaseboardDio amp Periph gt Dio volatile unsigned int shadow 0 volatile unsigned int rtn volatile unsigned int err 0 volatile unsigned int reps 0 int col row clrscr printf nTesting M x baseboard digital n n printf n nInst
11. 8 05 s 1 5 i mi i i i i 2 i i i i i 0 5 1 1 2 2 5 3 0 0 5 1 2 2 5 3 5 Time s Figure 6 14 0 error 55 1 5 Time s Figure 6 15 0 error Position Imi E Y x107 s x107 Position Reference Y o I 7 Position m 1 N T L T I a T i 1 5 1 5 Time s Time s Figure 6 16 xg error Figure 6 17 yg error 56 Chapter 7 Conclusion and recommendations 7 1 Conclusion During this project several steps towards the design and implementation of MIMO control for the 6 DOF positioning stage were made For control of the linear motors hardware was assembled and electronics were connected Preparing the laser system for position feedback involved replacement and re design of several components addition of kinematic mounts and debugging In addition to alignment of the laser and setting up the control software this took several months Once the system was operational system identification of the linear motors was straightforward A linear controller was designed and implemented The closed loop bandwidth is over 50 Hz while the steady state error is limited to 50 nanometer To gain understanding about the behavior and control of the magnetic actuators a 1 DOF setup was made in which one crossmember is actuated by one magnetic actuator A theoretical force model for the magnetic actuators was derived and experiments
12. That is the flux paths only pass through the laminations and the air gaps between E en I laminations e The air gap a is small compared to the dimensions dj hg and ha see Figure B 1 e The metal core of the actuator does not saturate This means the relation between B and H is linear in the operating region used e There are no Eddy current losses 17 4 12 Model E As explained in Chapter 2 a magnetic field is generated when a current is passed through the coil around the center leg of the E shaped laminations The work exerted by the magnetic force on the P laminations equals the change of magnetic energy stored in the air gap This force is always directed so that it tends to reduce the air gap Fdl dW 4 1 In the linear operating range no saturation the flux density B is given by the permeability of the material times the field intensity H B uH 4 2 The energy density is given by 1 1 B B wm 5 Fg 4 3 Integrating over the volume gives the total energy 2 W funav 8 wal 55 fa 4 4 Now substitution of Equation 4 4 in Equation 4 1 gives 1 B Fat as J dj 4 5 This can be reduced to 1 B 4 6 The total flux through a surface is given by the integral of the flux density B over the surface Ur J Bas 4 7 The flux generated by a current through the coil is given by NI Wp 4 8 Here the reluctance R is defined as pe 49 pS Where I represe
13. amp id_struct printf n nIDROM coefficients used for gain and offset correction 15 int init gain and scale factor values to IDROM contents all channels for i 0 i lt 16 i SERVO16_set_adc_gain module i SERVO16 from fixed id struct adc gain coeff Li SERVO16 set adc offset module i id struct adc offset coefflil SERV016_set_dac_gain module i SERVO16 from fixed id struct dac gain coeff GJ SERVO16 set dac offset module i id struct dac offset coeffi 7 debo Start DDS timebase DDS_TIMER CLOCK_RATE CLK_FACTOR DDS_CLOCK Allow DDS to settle at new rate ns ie7 1 7 7 Initialize Servol module Ha am SS SS SS Set decimation mode DECIMATION points are thrown out for every one point stored in the A D FIFO SERV016_set_adc_decimation module DECIMATION Set additional delay ns between A D and DAC conversion This delay should be at least 10800ns the maximum calculation time SERVO16_set dac_delay module DAC DELAY Toggle reset SERV016_reset module Enable required channel pairs SERVO16 enable dac pair module PAIR_MASK_OUT SERVO16 enable adc pair module PAIR MASK IN Set FIFO level threshold so interrupt fires when there is only one sample per channel in the FIFO SERVO16 set adc fifo threshold module ActivePairsIn 1 Ulea nnn ian aa Initialize Interrupts VS es ClearInterrupt
14. ko gt ms El R co o a I sa gt o kane NS L Gl a o man o Figure A 2 Aluminum mounting block for optical components axis 3 amp 4 62 Mirror amp Beamsplitter Mount P N 6191 0445 0X Description Used to mount optical components with 1 inch square cells Dimensions See Figure Weight 86 grams 3 0 oz Materials 431 Stainless Steel Tit Clamping Screws 33 Allen head capscrews 1254 Beam Height Anguiar Adjustment Yaw 8 degrees Tilt 4 degrees vhen used with 3 clamping screws 8 degrees when used with 2 clamping screws in center slots of Rocker Plate 440 Threads 4 ito mount Fold Marar Yaw Adjustment Sicis 2 r clearance for 4 or 3 mm Dimensions shown in inches and millimeters Figure A 3 Kinematic mount for mirror and beam splitter 24 i Description Used to mount optical components with 1 1 2 inch cells Dimensions See Figure Weight 137 grams 5 0 oz interferometer Mount P N 6191 0446 0X Materials 431 Stairiass Steel Angular Adjustment Yaw 8 degrees THE 5 degrees 1 00 Nominal 84 40 Threads 4 75 4 Beam Height to mount nierferometa Tilt Clamping Screws 3 slotted hex head screws Yaw Adjustment Slots 2 clearance for 4 or 3 mm screws Dimensions shown in inches and millimeters Figure A 4 Kinematic mount for interferometer 24 63 A 3 Alig
15. min So _ Kan z pla 1 E 5 46 Since Ka Ka and n x are positive the sliding condition is met For case 3 and 4 stability can be guaranteed in a similar way As mentioned before this approach has not been implemented real time due to time constraints In future research this approach could be investigated further since it has lead to good results on a teeing ta the nne 1705 Ad SG PATNA setup similar to the one used in this project 12 5 3 4 Real time results Results of feedback linearization in combination with sliding mode control first approach can be seen in Figure 5 18 through Figure 5 27 x10 Position Position m Position m SD ed Reference bee Position 0 5 0 6 0 7 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 1 0 2 0 3 0 4 Time fs Time sj Figure 5 18 Smooth step position and reference Figure 5 19 Smooth step error Step resporise Error Position fm Sensor voltage M i 0 4 0 45 0 5 0 55 0 6 0 65 0 7 0 1 0 15 Time sj Time 5 Figure 5 20 Smooth step error zoomed Figure 5 21 Steps of 50 pm 39 Reference Position 5 1 5 1 0 Position fm Position m a older 3 15 Time s Time s Figure 5 22 1 Hz reference amplitude 250 um Figure
16. 2 Control of non linear system LL Lu eee nia 6 2 1 6 2 2 Mapping a veelen aon he A od W ah eat ee Simulations sie da dere Sa dead done ede n BS 633 CONClUSIO 60 Book Bede ewe Bae WSE e hdd ne oe rd 7 Conclusion and recommendations YE Conclusi n sake mod aoe bake kr ee CR Re EP tee Gad a 7 2 Recommendations oo oo o sa References A Instrumentation for linear motors AGT er PINUS E a V t ES A IA AS ve e ted A a Mounting of laser system ro A3 Aligning laser system ooo A 3 1 A 3 2 A 3 3 A34 A 3 5 Fold Mirrors and Beam Splitters i nna nana o Interterometers vats wet n min Sadki A glede de h Target MUITO Lik noteerden aen ane A Gap a Seated Fiber Optic Pickup Gw 24 4 wha oan ende eden AG Re ae Feedback and dip switch settings a ee ee ee ee ee B Actuator dimensions and equivalent circuit C Code C 1 C code for reading parallel laser data Lae aaa ee C 2 C code for control of 1 DOF system l kritt Ho Beeta Sie t t G 3 6 DOF model S function aa aa eeen OCZNE C 4 MIMO controller S function 2 anna ee C5 Demo program oe eda Mal de Bred E a i a me ae se el ai D M files D 1 Fit model to measurement data D 2 Closed loop system ID see ee Chapter 1 Introduction As nowadays technology tends towards the development of smaller components the high precision manufacturing industry is continuously pursuing higher ac
17. 7 0 295 Fo ee RE 0 315 i 4 031 i i 0 305 z 5 ost 5 3 8 0 295 0 29 i benemen a z a 0285 003 i i 4 SH 0 5 1 15 2 025862 04 06 68 1 12 14 16 18 2 Time s Time s Figure 5 11 Control signal simulated Figure 5 12 Control signal real time 32 x107 Step response simulation 0 5 7 E k i 4 Step response E os _ i i 0 oF 0 5 4 05 4 a z 45 4 E 5 5 1 5F ere 1 B al a 25H sl 3 is J RAL 3 5 io Reference 3 5 24 Si Reference i i i Position Position 4 i i i i i i y 7 4 K i i i i i 7 0 0 2 0 4 0 6 0 8 1 1 2 14 1 6 0 0 2 0 4 0 6 0 8 1 1 2 14 1 6 Time s Time s Figure 5 13 Reference and position simulated Figure 5 14 Reference and position real time Error simulation Position m 2 02 04 06 08 i 12 14 1 6 Time s Figure 5 15 Error simulated Figure 5 16 Error real time x10 Error Position m 1 1 1 1 2 13 14 15 16 Time s Figure 5 17 Steady state error 33 5 3 Non linear control The linear controller described in the previous section is only suited to control the system in a small range around the operating point Literature 21 16 17 and 15 shows that consistent perfor mance over the whole operating range can be obtained by feedback linearization of the system This is d
18. Figure 3 8 shows that the steady state error ess lt 50 nm 14 Phase degrees Magnitude dB o Phase degrees o tok 200 100 100 Frequency Hz Figure 3 6 Axis 1 Closed loop measurement At Magnitude dB 200 100 100 200 10 Frequency Hz Figure 3 7 Axis 1 Sensitivity measurement 15 Vibration in steady state position closed loop 150 T T T T T Position nm ms EE SE 150 i i i 1 1 0 0 2 0 4 0 6 0 8 1 12 Time s Figure 3 8 Steady state error c osed loop measurement 3 7 Conclusion Hardware and software for the linear motor have been set up This was a time consuming process connecting all electronics redesigning and aligning the laser system ordering and replacing several devices debugging software etc The implementation of a linear controller shows that this process has been completed successfully Maximum achieved bandwidth is 5o Hz while steady state error is limited to 50 nm Severa limitations have been encountered especially considering the restricted freedom in con troller design To improve performance it will be necessary to implement a controller with more complex structure For example it is desired to implement a feed forward term Such a controller could be implemented on a DSP and the control signal could be sent to the linear motors bypassing the preprogrammed con
19. Gap 12 m 2 Gap 13 m 3 Gap 14 m 4 Gap 21 ml 14 Gap 43 m 15 Gap 44 ml No of input ports 0 Torque command linear motors V i Coil currents A Input port 0 width 0 3 Torque command linear motor 1 4 Input port 1 width 0 Current to coil ii 1 Current to coil 12 2 Current to coil 13 3 Current to coil 14 4 Current to coil 21 14 Current to coil 14 15 Current to coil 15 No of output ports 0 Translations linear motors m 1 6 DOF s of tray m deg 2 Physically sensed gaps m 3 Short stokes xS and yS m Output port 0 width 0 3 Translations linear motors 1 4 Output port 1 width 0 5 Tray DOF s x y z theta_x theta_y theta_z Dutput port 2 width Gap 11 Gap 21 Gap 32 Gap 42 Gap 14 Gap 24 Gap 34 Gap 44 NOD UP WHE O Gutput port 3 width 0 xS 80 HU 1 ys define U element uPtrsO element Pointer to Input Porto Torque command lin motors define I element uPtrsifelement Pointer to Input Porti Coil currents System parameters A Tray def ine Jxx 0 5364 di kgx m 2 Inertia tray define Jyy 0 536 17 Ukg m 2 Inertia tray define Jzz 1 0708 Ikgam21 Inertia tray define LO 0 55 ml Nominal arm length define mt 10 9 ti kg Mass tray iidefine ge 9 81 m s 2 Gravity constant Linear motors define klin 18 96 EN V Force constant linear motor 1 4 define mlin 3 8 kg Mass of linear motor 1 4
20. considering setpoints and system specifications is required Some steps towards implementation have been made such as ordering equipment and writing code to read parallel laser data Many other issues will have to be addressed before a implementation is possible see Section 7 2 Considering the results so far it seems very hard to meet the design specifications Although the object of 10 was design of a ultra high position system as used in photolithography the current system seems more suited as a high precision gantry system as used for flat panel inspection or electronic assembly 57 7 2 Recommendations A lot of work has to be done before the system performance even gets close to the design specifications First of all assembly of the hardware has to be completed Design of the passive magnetic suspension has to be finished so it can be manufactured A second laser system is necessary to measure the position of the tray with high accuracy This also requires a high precision mirror at the bottom of the tray Other hardware necessary to get closer to the specified accuracy includes a faster DSP with better noise properties Fluctuations of temperature and humidity in the lab should be prevented In order to finish the design and assembly of the hardware some design principles should be investigated further During the past year new ideas considering the metrology frame were born These ideas include a different setup for the laser s
21. edt 0 With e 21 Tia and A 0 0 0 0 0 0 0 0 0 0 AJO 9 8 0 0 0 io o 0 22 00 0 0 0 0 As 0 0 0 0 0 0 Ag Define Ay an 24011 A and As a22 2Aaz2 Ayre Agres u ug Substitution of a11 a12 a23 and azz gives An A n 2A 50 Now u ug AW tg Ameri n 62 With ug an array containing the desired inputs ug na And us and array containing the switching terms Us G arctan S With Gy an array containing the gain of the switching term for each DOF Gug Ska Iks Ska Iks kel And a diagonal matrix containing the boundary layer thickness for each DOF 0 0 0 0 0 0 s 0 0 0 0 5 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 s 6 2 1 Mapping The computed control actions u represent forces and moments with respect to the COG To translate these control actions to actuators inputs a mapping will be used Since a pair of opposed coils is needed to be able to produce attractive and repulsive forces for this moment one pair of opposed coils is considered as one actuator The first index is the number of the axis the second index tells if the horizontally h or vertically v oriented pair of coils is considered It is easy to see that the resultant forces and moments with respect to the tray s COG equal Fi Fscoc 1 0 16 0 0 O 0 Fa Piene 0 1 0 dy 0 0 0 0 Fa Frcoa MTI 0 0 0 0 1 1 1 1 Fa 6 22 bo 0 0 0 0 L 0 Lg 0 Fi Roe 0 0 0 0 0 k 0
22. frequencies Frequency Hz Magnitude dB Eel The mean ratio 2 equals 4 96 N Vm According to 1 and 4 the motor constant equals 18 96 N V resulting in a mass of 3 82 kg for one moving unit This is close to the estimated mass of 4 0 kg confirming that the result is correct Since the moving units on all 4 axes are identical it is assumed this mass model accurately describes them ali 3 5 Control As mentioned before the linear motors are controlled using the Unidex 500 CNC card and accessory MMI software A fixed control structure is implemented in the software so the user can only set the control gains The structure of the controller is given in 3 page 6 2 but this schedule contains sev eral errors The correct structure of the controller can be seen in Figure 3 5 The sample frequency F is set to the maximum of 4000 Hz The user can tune the gains Kos Ko Ki and Ars The velocity feed forward gain Vy is only used if a secondary feedback device is used to measure velocity In this case velocity is derived from the position data and V7 should be set to 256 This results in a PID controller with acceleration feed forward Rewriting the scheme in Figure 3 5 shows how the gains of this PID controller relate to Kyos Ky and Ky 13 Aff 256 F s Fs VI2SB Ci r counts Kpos 4096 10 Kp Torque command v 83886085 Figure 3 5 Controller as implemented in the MMI soft
23. i T Y mi i 5 i is z k Hi E 4 i i in t y i y X 4 E i E Q p 4 i i i i a 2 ok i 2 i i i i 4 S I t t 5 i A 4 2 NR FE GENE 4 w AJ i f f il i o a E j a tr k nii id 0 1 0 2 0 3 0 4 0 6 0 7 0 8 0 8 1 0 5 Time s Figure 5 6 Tracking of a 5 Hz sinusoidal over large range simulation 30 5 2 4 Real time results The implementation of a linear controller on this system is difficult First problem is the fact that a controller with good performance around the equilibrium position is not suited for the whole vertical range Since the crossmember initially rests on the lower coil it is far away from the operating point for which the controller is designed the middle of the vertical range Second problem is that in the initial position the lower actuator exerts a theoretically infinite force on the crossmember as long as Jo is non zero This means the crossmember is lifted when the control current AJ equals the bias current 15 so Iz 0 At this moment J Je AI Itis not hard to see that this situation does not match the linearized system equations Because of these problems implementation of the controller designed in simulation leads to an unstable system A controller with lower gains was designed The closed loop system can perform a smooth step acceleration is a linear function of time so position is a third order curve towards 0 0 However the performance of this controll
24. in the middle of the vertical range dre Adr BAI 5 8 y C r DAI Taking the appropriate Jacobian and substituting the equilibrium point leads to the following lin earized system 0 1 A 2y Ki T 2y Ko La 0 5 9 Jez Gz Jeti 0 B Zy Kali ZyKal PP mer Celi 0 5 11 D 0 5 12 Assuming both actuators are equal Ky K2 the poles of the linearized system are 22K I T A Ei En wi 5 13 EEINZ 27 The bias current I can be chosen arbitrarily by the control designer Typically the bias currents equal half of the maximum current 7 Since the current drive amplifiers saturate at 3 0 A Ip 1 5 A 7 follows from the value 7 and the condition that resultant force in the operating point is o IK In mal To gA LS h 5 14 BOB Ko Inz yKo With a bias of I 1 5 A To 1 02 A and the poles are 283 2 rad s Actuator saturation The amplifiers for the magnetic actuators are designed to deliver up to 4 0 A but only for short experiments They will become extremely hot and will fail if maximum current is delivered for longer periods Much heat development in the coils is also undesired The amplifier gain was adjusted so that currents are limited to 3 0 A and the amplifiers can run continuously for long periods without problems 5 2 2 System identification To verify the linearization frequency response of the system can be measured in the operating
25. linear model from closed loop system ID close all clear all clc load CLIDsim2 mat Contains u v w and test_e test_u fs 4000 Npoints 2 12 4096 Noverlap round 0 9 Npoints SensCoh F cohere w u Npoints fs hanning Npoints Noverlap SensFRF F tfe w u Npoints fs hanning Npoints Noverlap CLCoh F cohere r thx Npoints fs hanning Npoints Noverlap CLFRF F tfe r thx Npoints fs hanning Npoints Noverlap figure subplot 2 1 1 semilogx F 20 logi0 abs SensFRF k grid ylabel Mag dB gt fontsize 16 title Sensitivity fontsize 16 subplot 2 1 2 semilogx F angle SensFRF pi 180 k grid ylabel Phase deg fontsize 16 xlabel Frequency Hz1 fontsize 16 figure semilogx F SensCoh k grid xlabel Coherence fontsize 16 xlabel Frequency Hz fontsize 16 t tle Coherence of sensitivity measurement fontsize 16 figure subplot 2 1 1 semilogx F 20 10g10 abs CLFRF k grid ylabel Mag dB fontsize 16 title Closed loop Simulated fontsize 16 subplot 2 1 2 semilogx F angle CLFRF pi 180 k grid ylabel Phase deg fontsize 16 xlabel Frequency Hz fontsize 16 figure semilogx F CLCoh k7 grid xlabel Coherence fontsize 16 xlabel Frequency Hz fontsize 16 title Coherence of closed loop measurement fontsize 1
26. mi Ya za site i I n4 I O l I x o poe x aa Figure 6 2 Geometry ET 1 Sa i 233 I gt i gl4 234 gt lt LI L3 ty g23 gt i g43 g24 Figure 6 3 Gaps 45 ee The physical system has 8 capacitive sensors measuring the air gaps 911 921 932 942 914 924 934 and gas The complementary gaps are computed using the geometry of the system For example Gji 912 29nx Here 911 912 gna is the nominal gap when the tray is in centered position all states o e The DOF s of the center of gravity COG are measured with respect to a reference frame that is fixed to ground The origin of this frame coincides with the COG of the tray when it is in centered position poy Lye 4 a near e In z and y direction distinction is made between long range motion zr yr of the linear motors and short range motion xs ys of the tray with respect to the linear motors The overall translations of the tray are ar 11 zs and yr yL ys ee The linear motors move in pairs It is assumed that travel of master and slave is equal 1 2 Er and yj ya yz so linear motors do not influence the angle 9 ee The origin of the long range motions zz and yy lies in the middle of their range e While computing moments acting on the tray the influence of the DOF s zg ys 2 Oz Oy and 9 on the arm lengths is neglected This is valid since the tray has only very small strokes order 107 and 1074 in t
27. n err reps getchar eat the character gotoxy col row set_attribute attr printf 4d Errors in d tries err reps gotoxy 0 20 getchar set_attribute attr cursor ON clrscr printf n nRemove Loopback Cable from Digital Connector getchar AJ zw nnn nnn te M pause wait for a key to be hit Ml oi SS SS SS SS void pause void printf n nPress any key to continue getchar 73 C 2 C code for control of 1 DOF system Feedback linearization in combination with SMC E E Visser Uf nr nn en ne rn rn a SS SS en mm SS include periph h include stdio h include dsp h finclude math h Servo parameters define CLK_FACTOR 24 This number should not be changed define CLOCK_RATE 100000 0 A D conversion rate define DECIMATION 39 Number of samples thrown out for each one kept define DAC_DELAY 100000 At least 10800ns max calc time define IN SCALE 10 0 32767 0 Input scaling 10 to 10 V 2716 define DUT SCALE 3276 7 Idem for output fidefine GAIN SCALE 3 3333 Amplifier gain 10 V input 3 output fdefine ts 4 0e 4 Sample time define PAIR MASK OUT Oxff 16 output channels 8 pairs Oxff define PAIR MASKIN Oxf 8 input channels 4 pairs Oxf System parameters fdefine Jxx 0 15 ti kg m2 define yy 0 3353 Im define ys 0 275 ti Im define gni 5 81e 4 ti Im 9 50 50e 6 yy
28. non affine system with algebraic input invertibility l Here the theory of 12 and 13 is applied to the system The system equations defined earlier in Equation 5 6 can be written as f x Bh h 2 x SA With PE 5 31 Bie _ mgl get 5 32 pis 0 5 33 1 yK I yK B hll h z FA 5 34 Hi 12 2 dya gi 21 Jaz 92 1 Switching In 12 switching term and equivalent control are derived separately To determine the switching term first the sliding surface is defined t S x t Det xi edt with e x r 5 35 0 Asymptotic stability is guaranteed if the following condition is met dS s s s TD sar OP or S i S Cf t 37 BRL 12 2 zi lt 5 36 In case S gt 0 the sliding condition is ue lt 0 Define s s pla ft z Now the sliding condition is p x h h l 1 lt 0 Choose n x such that it is a upper boundary for p z 0 lt p z lt nfz 5 38 5 37 Least control effort is needed when only 1 coil current is used to meet the sliding condition The following switching control term always meets the sliding condition for S gt 0 a Jran z Ba l 92 x1 Eas and h 0 5 39 With Ko a lower boundary for Ko Note that the sliding condition is already met when S gt 0 and p x lt 0 but the input proposed above will be given anyway Later more about this In a similar way a switching control action for S lt 0 can be derived J zn z h
29. point System identification for this system can only be performed closed loop Once a linear controller is designed that stabilizes the system in the operating point 6 0 sensitivity can be measured From the sensitivity measurement open loop and subsequently plant can be derived See Appendix D 2 for the M file used Main challenge in this process is to design a stabilizing controller before the system identification is performed In practice this is an iterative process The plant derived from the sensitivity measurement can be seen in Figure 5 3 Figure 5 2 shows a Bode plot of the linearized system equations Note that there is a gain difference of roughly 80 dB between the two This can be explained by the fact that in the sensitivity measurement the sensor sensitivity 50 1079 m V and arm y 7355 m are included accounting for a 76 dB difference Both plants are very similar considering gain and poles at 45 Hz In the real system a resonance can be seen around 120 Hz Note that the phase difference of 180 degrees represents a minus sign The real time measurement shows additional phase delay at high frequencies which is caused by the sampling of the white noise before it is injected to the system as explained before in Chapter 3 Plant from sensitivity Bode Diagram 30 Magnitude dB 19 e 10 1 10 Frequency Hz Frequency Hz Figure 5 2 Bode plo
30. rddot rvddot 0 Qther near double gi Zegnl Initial upper gap m near double g2 0 JI initial lower gap m near double Y_INIT 9 0 di Initial sensor voltage V near double 2 Sensor 2 V vertical displacement Current 8 Time s near double 11 12 near double t near volatile int ClockCount It ooo We we we near double Ki K2 4 55e 6 near double P1 near double P2 near double P3 3 55671335314818 0 00855707663713 T 363349702380939e 007 near int odd 0 For fine tuning of servo timing main 1 EnableCache Initialize DSP Enable interrupt on Servo16 allow data acquisition to start SERV016_start module Infinite loop abort when user hits any key while kbd_hit 0 stopcondition 1 print n nSTOP mm mm a a A vennen Initialize_DSP Ph a nennen void Initialize_DSP 1 int i SERVO16_ID id_struct ActivePairsin active_pairs_in ActivePairsOut active_pairs_out EnableInterrupts enable monitor gotoxy 30 0 bold printf 7Servo 16 applicationin normal Get module site printf n nServo 16 on module 4d module cursor OFF use A D FIFO level interrupt xrpt module EINT2_INTERRUPT EINTO_INTERRUPT xrpt_source module EINT2_INT_SRC EINTO_INT_SRC disable interrupt on module disable FIFO data acquisition SERV016_stop module SERV016_read_idrom module
31. setup For example in the 1 DOF setup position is measured with capacitive sensors Their resolution is significantly lower than the resolution of the laser system Also the DSP used at this moment limits sample frequency of the controller and the amplifiers for the magnetic actuators have limited bandwidth In this stage of the project the focus lies on finding a controller that is stable for the whole ver tical stroke of the crossmember Obviously high bandwidth and small static error are desired If a controller meeting these requirements is implemented with succes it can be used as starting point for an iterative process to improve system performance This proces will eventually have to lead to a competitive design However the necessary modifications of the system position measurement with laser magnetic suspension faster DSP will cost time and money and will not be performed before the end of this project More information about the implementation can be found in 22 25 5 1 System description The test set up consists of a crossmember with I rotational DOF which is actuated by a pair of opposite placed magnetic actuators A schematic view of this system is shown in Figure 5 1 Using the force model derived in Chapter 4 the following equation of motion can be derived Tun h h A yKo mgl cos 61 aha g0 1 A raj Cad ine description of the gaps 91 01 gnz ytam 6 92 01 gn ytan 0 W
32. with e 0 01 0 5 21 Writing B 0 H with H e R leads to 1 411 1 ai2 2 5 22 s 42181 a22 2 H u ug 5 23 A sliding surface S with integral term is chosen t S r Ber x edt 5 24 9 The derivative of S is a 246 Aer 5 25 35 Substitution of 5 22 and 5 23 gives aqq 211 Me a22 2Xajz es H u ua 5 26 Define Ay a2 24011 X and Aoi a22 24a12 Aue Anes H u ua 5 27 Now the following control law is chosen Or bo 00 lt Al i u uq H us 41167 Agie2 It is easy to see that substitution leads to u An appropriate choice for input us like us gr sign S will force S to o This can be verified by analyzing the Lyapunov function V 182 Since the sign term leads to a discontinuous control action chattering it is replaced by the continuous term us gy atan 18 Implementation of this control action will force S to stay within a boundary layer with width 26 For the feedback linearized system ai 421 072 b 0 and aja baj 1 Accordingly H 1 Ay A and Ag 24 The control law simplifies to u ug us Me 24e2 5 29 Note that input u equals the desired angular acceleration v of the crossmember The input currents I and Jo are computed using the nonlinear transformation described in Equation 5 17 36 5 3 3 Sliding mode control of
33. xrpt ActivateInterrupt xrpt InterruptSource xrpt xrpt_source InstallIntVector analog_isr xrpt ClearInterrupt xrpt EnableInterrupt zrpt printf n n Servo 16 initialized active_pairs_in volatile int pairs 0 volatile unsigned int mask PAIR_MASK_IN while mask i if mask amp 1 pairst mask gt gt 1 return pairs active_pairs_out volatile int pairs 0 volatile unsigned int mask PAIR MASK BUT 76 while mask 1 if mask amp 1 pairst mask gt gt 1 I return pairs gt Md A A M m di analog_isr Analog 170 interrupt routine ff zzz A a void analog_isr t Get pointer to base address of Servol module volatile SERVO16 servol6 SERVO16 pModule module dma copy samples from A D FIFO to input buffer Blocking SERV016_bleed_adc_fifo module uint32 Input ActivePairsIn Servo calculations DoServoCalc dma copy samples from output buffer to DAC FIFO non Blocking dma_mem_to_port 0 int Output int kservol6 gt fifo both ActivePairsOut 0 ack interrupt on the module required SERV016_ack_adc_interrupt module A SS III mm mm A SS SS SS SS SS SS DoServoCalc Servo calculations VE Ks void DoServoCalc 4 int k Update time ClockCount t ClockCount ts Read input channels PWM board introduces a sign on inputs S2 IN SCALE double Input 0 half low Sensor 2 V ve
34. zs is myt Fr The resultant force Fr required to maintain zg 0 can be computed using the linear controller designed in Chapter 3 R POLAR A 6 20 Mp Now the required force delivered by the linear motor Fijn can be determined easily m tmi p Fun magn FR 6 21 m T 48 6 1 5 Measured variables vs performance variables Goal is to control the DOF s 24 Yr Z Bz y and 0 of the center of gravity of the tray At this moment these DOF s cannot be measured directly The necessary high precision mirror and laser system will only be purchased once the machine design is permanent and budget allows These DOF s have to be extracted from the available data using knowledge of the system geometry In the physical system linear motor displacements 21 12 yi yo are measured with a laser system and the gaps 911 921 932 942 914 924 934 and gas are measured using capacitive sensors Neglecting rotational cross terms width and height of the arms the measured gaps can be described as a function of the 6 DOF s of the tray see also Figure 6 3 911 gna Er 21 In tan 0 921 Iny Yr Y1 La tan 0 932 Gna Lr 22 La tan 0 942 Iny Yr Ya La tan 8 914 gne 2 Ly tan 8 924 Inz Z Lo tan y 934 nz Z La tan 6z 944 Gnz Z La tan 0 So 911 Ina T1 l 0 0 0 o Li 921 Iny TY 0 1 0 0 0 Lo Ly 932 Inz T2 1 0 0 0 0 L
35. 1 half high int GAIN_SCALE 0 0UT_SCALE Output 3 pin 29 Output 2 half low int 10 Ss DUT_SCALE Output 4 pin 23 Output 2 half high int rv 5500 0 D0UT SCALE Output 5 pin 21 f Output pulse freq should match controller sample freq if odd 1 Output 2 half low int 1 0UT_SCALE odd 0 else Output 2 half low int O DUT_SCALE odd 1 Ha nn re rn nc cnn nan ncn ene cnn SS Write O ta all output channels if key has been pressed pnw nnn nnn nn nnn nnn nnn nnn w nan else t for k 0 k lt ActivePairsQut k Output k both 0 3 3 N ti Reference Smooth step N void Reference near static double TO 0 001 Start time of step reference s near static double Tstep 1 0 Step duration s near static double Amp 1 0 Step amplitude corresponding to sensor voltage near double dt Tstep 1 0 near double Ti TO dt near double T2 T0 3 0 dt near double Tend TO Tstep near double jerk hmp 2 dt dt dt near double a02 jerk dt near double a03 a02 near double v02 jerk adt dt 2 0 18 near double v03 v02 near double x02 jerk dt dt dt 6 0 near double x03 jerkxdt dt dt 11 0 6 0 if t lt TO 1 rddot 0 V_INIT 9 0 VLINIT 3 else if t gt TO k t lt TL 1 rddot jerk T0 r jerk t TO t TO t T0 6 V_INIT E else if 4 gt T1 d lt T2
36. 15 0 15 20 5 10107 KO pis root 5 Te ma DOF ET 10 YT DOF ar r ii Y Table 6 2 Controller gains fja D gt x The linear controllers have the gains as used in Chapter 3 6 3 Conclusion system model was made and a controller based on temporary results was designed Simulated dis _ tribution of the control action for 6 DOF s to 20 coils works properly when linear motors are centered This indicates that the approach using a mapping is suited When the tray moves off center the quality of the distribution decreases and cross terms are introduced This implies that the control action for a certain DOF also affects other DOF s In principle the robustness of the controller is high enough to deal with this disturbances However the vertical forces are large since they have to carry the weight of the tray The disturbing effect of the vertical forces on the DOF s 0 and 6 is large and limits the maximum speed of the tray in X and Y direction This problem is solved when a magnetic suspension becomes available The force feed forward leads to a good decoupling between the long stroke and short stroke mo tion since xg and ys stay very small 53 Position m Position m Position m Yr 0 035 r 9 025 0 02F 0 015 0 035 r T r Position Reference i i ej 0 025 0 03 4 0 02 0 015 Positi
37. 3 Up 942 Iny Ya 0 10 0 0 La z 914 nz 0 01 0 0 tan 0 924 Inz 0 01 0 La 0 tan 0 934 Inz 0 0 1 La 0 0 tan 0 944 Inz 0 0 1 0 a 0 8 There are 4 relations for the horizontal DOF s z Y and tan 8 and 4 relations for the vertical DOF s 2 tan 0 and tan 0 They are dependent the matrix has rank 6 so 6 independent rela tions can be established for 6 DOF s Solving them gives r Y z tan 0 tan 0 tan 8 7 and subsequently 9 6 and 9 can be derived 49 6 2 Control of non linear system Due to delayed delivery of the current drive amplifiers no thorough comparison between different control strategies could be made yet on the 1 DOF system Based on temporary results the following approach using sliding mode control of the feedback linearized system was chosen The 6 DOF system can be described as follows m ler yr z 6 dy 9 F IT Ka jer yr 0 Oy A FE a a Je Ta a21 Q22 T2 ba With ar Ol eze a2 aa Oleze a22 O 626 ooo SES momo om ooo S kose bu Olcezo ba OOOO 8 e ooo oO yyt o ooo oi o o oo lko o o okoo oo Sy Gl x NB With u Pesse Fycoe essa m9 Posa Toda Tacoc the resultant forces and mo ments with respect to the center of gravity Using forces and moments as input the sliding mode algorithm is straightforward Array S contains sliding surfaces for all 6 DOF s t s e 2ne A
38. 5 23 1 Hz reference amplitude 250 um x 103 Position at Liege i TA 3 AVE 4 dt E i 2p 2 E E E oj E 8 gr L 1H a 3 2 4 3t 6 Ar zak n en ANC E LONG tery gt 3 A E 1 105 1 1 1 15 12 1 25 13 1 35 14 145 15 1 1 05 1 1 1 15 12 125 1 3 1 35 14 1 45 15 Time s Time fs Figure 5 24 10 Hz reference amplitude 40 um Figure 5 25 10 Hz reference amplitude 40 pm x 105 Position Position m Position m i 1 2 1 25 13 a E 1 1 05 14 145 12 1 25 13 1 1 05 14 1 15 Time s Time Is Figure 5 26 20 Hz reference amplitude 40 um Figure 5 27 20 Hz reference amplitude 40 m 5 3 5 Conclusion From the real time results it can be concluded that a sliding mode controller in combination with feedback linearization has a larger operating range than a linear controller Figure 5 21 shows steps of 50 um each over the whole vertical range Clearly oscillations occur when the crossmember comes close to the upper actuator If a setpoint closer than approximately 75 um from the upper actuator is given the crossmember sticks to this actuator Apparently the force model of the actuator is not exact for this region which falls within the region where no force measurement could be performed see Chapter 4 As Figure 5 22 and Figure 5 23 show the system can track a 1 Hz sinusoidal with amplitude 250 um which corresponds to ha
39. 6 Controller used in closed loop 0 09478 0 9 605e 005 1 L 9 605e 005 1 633e 008 6 828e 009 1 48e 009 I 7 5e 005 _PIDdisc ss a b c d 2 5e 4 C_PIDi freqresp C_PIDdisc 2 pi F for i 1 2049 C_PID i 1 C_PID1 4 3 US nan do end 94 Derive open loop and plant from sensitivity OpenLoop 1 SensFRF 1 Plant OpenLoop C_PID Linearization of system model Jxx 0 3897 kg m 2 ma 1 7327 kg g 9 81 m kg s 2 1 0 2456 Im yy 0 3353 Im Ki 4 1603e 6 N m72 A 2 K2 4 1603e 6 N m 2 A 21 guz 6 3500e 4 m Ini 1 0 4 A In 1 0 gnz sqrt K1 K2 Ini 2 gnz 2 A Ts 1 fs s A 0 1 5 2 yy 2 K1 Ini 2 gnz 3 2 yy 2 K2 In2 2 gnz 3 Jxx 04 B 0 2 yy Ki Ini gnz 2 2 yy K2 In2 gnz 2 Jxx 1 C 1 01 D 0 LinSys ss A B C D figure MagLinSys PhLinSys bode LinSys 2 pi F Plot plant and compare to linearization of model figure subplot 2 1 1 semilogz F 20 ogiO abs Plant k gt hold on grid semilogx F 20 1ogi0 abs MagLinSys gt r ylabel Mag dB fontsize 16 title Plant fontsize 16 subplot 2 1 2 semilogx F angle Plant pi 180 k hold on grid ylabel Phase deg fontsize 16 xlabel Frequency Hz gt fontsize 16 95
40. 70 335 If arm sticks force measurement is not valid so measurement value is 0 4 V This leads to high gt 80 N actuator force Remove from data for i 1 length I for m 1 14 if Fact m i gt 80 Fact m i 0 end end end Plot measurements figure mesh Xtot Itot Fact hold on title Force measurements xlabel Air gap Emi ylabel Current LAJ zlabel Force N Fit theoretical model to measurements by adjusting gain and additional gap length Compute absolute error for different additional gap lenghts and gains for gi 1 100 gain gi 0 5 0 01 gi for add_x 0 1 300 x0 add_x 1e 6 for r 1 14 for ix 1 length Xtot Fmodel r ix gain gi 0 5 N 2 mu0 4 1 mu0 A 0 5 R1 0 5 R2 R3 x0 Xtot r ix 72 Itot r ix 2 if Fact r ix 0 Fmodel r ix 0 end err r ix Fact r ix Fmodel r ix err_crit2 r ix sqrt err r ix 2 end end GO gi add xti XO gi add_x 1 E2 gi add xti end gain gi x0 sum sum err_crit2 end Plot error as function of gain and x0 figure mesh X0 G0 E2 grid on xlabel Additional air gap imi ylabel Gain J zlabel Force N title Sum of absolute error as function of x 0 and gain Idem zoom For better visualization don t plot values larger than 225 for gi 1 100 for add_x 0 1 300 if E2 gi add_x i gt 225 E2 gi add x 1 225 end end end figure mesh X0 G0 E2 gr
41. Coils define gnx 3 8ie 4 3 5e 4 ti fm Nominal gap size horizontal gny gnx def ine gnz 6 35e 4 6 0e 4 ti m Nominal gap size vertical def ine ke 1 00e 5 1 5e 5 Nx m 2 472 Force constant of magnetic actuator define z_INIT 0 6 33e 4 ti Im Initial vertical position tray rests on lower coils H Maintain gap of 2e 6 m otherwise dividing by 0 int_T initpos 1 ff Boolean variable tray is in initial position TRUE FALSE f FZZDDDDZDDZZ O seas a IIS St SSS SSS Initialize ned static void mdlInitializeSizes SimStruct S ssSetNumSFcnParams S 0 Number of expected parameters if ssGetNumSFcnParams S ssGetSFcnParamsCount S 4 return Parameter mismatch will be reported by Simulink ssSetNumContStates S CONT_ST ssSetNumDiscStates S DISC ST if ssSetNuminputPorts S NIP return ssSetInputPortWidth S 0 IPWO ssSetInputPortWidth S 1 IPW1 ssSet InputPortDirectFeedThrough S 0 1 if ssSetNumOutputPorts S NOP return ssSetOutputPortWidth S 0 OPWO ssSetU0utputPortWidth S 1 OPW1 ssSetOutputPortWidth S 2 OPW2 ssSetOutputPortWidth S 3 OPW3 ssSetNumSampleTimes S 1 ssSetNumRWork S GLOBALREALS ssSetNumIWork S GLOBALINTS ssSetNumPWork S 0 ssSetNumModes S 0 ssSetNumNonsampledZCs S 0 Take care when specifying exception free code see sfuntmpl doc c ssSetOptions S SS OPTION EXCEPTION FREE CODE static void mdlinitializeSampleTimes SimStruct S 1
42. Disable masters Manually move masters and slaves will track MESSAGE DISPLAY Axis X Y Z and U disabled EXIT 89 90 Appendix D M files D 1 Fit model to measurement data close all clear all cic load all measured forces load meas2308conditioned Geometry a 1 753 25 4e 3 ml b 0 455 25 4e 3 Im c 0 631 25 4e 3 Im d 0 433 25 4e 3 Im e 2 628 25 4e 3 Cm 0 648x25 4e 3 4 m g 0 325 25 4e 3 m h 0 5 25 4e 3 Im i 0 5 e 0 5 d 25 4e 3 m a 0 5 b 25 4e 3 Em k 0 272 25 4e 3 lm Si c f m 2 S2 def m21 S3 bef m 2 S4 heft im 2 Constants N 258 mo 4 pixie 7 mur 47000 From Thesis L Fevre p 17 mu mu0 mur Ri k mu 1 i mu S3 j mu S2 R2 i mu S4 R3 g mu 1 A S1 2x52 2 81xS2 Sensor values for different air gaps x 8 5 Compute air gap for izi length 1 for m 1 14 Ttot m 1 1 i Xtot m i 9 50 x m 50e 6x 0 335 0 275 end end Force measurements VJ Fsens F8 F7 F6 F5 F4 F3 F2 Fi FO Fplusi Fplus2 Fplus3 Fplus4 Fplus5 Force N Fmeas Fsens 10 1 Equilibrium of moments 0 335 Fact Fmeas 0 405 1 69 9 81 0 245 8 0382 9 81 0 4 used load arm is 0 37 This is a calculated value to get the balance of moments correct at zero current 91 Fact 1 69 9 8140 245 8 0382 9 81 0 375 Fmeas 0 405
43. Modelling instrumentation and control of a 6 DOF positioning stage E E Visser DCT 2004 112 Master s thesis Coach es Prof Dr Ir M Steinbuch Supervisor H Gutierrez B Sc M Eng Ph D Technische Universiteit Eindhoven Department Mechanical Engineering Dynamics and Control Technology Group Eindhoven November 2004 Preface This thesis describes the research I have done at the Department of Mechanical amp Aerospace Engineer ing of the Florida Institute of Technology Melbourne FL I performed this project from September 2003 until September 2004 under direct supervision of Hector Gutierrez FIT and with feedback from Maarten Steinbuch It was a challenging year and although results are different from my expec tations one year ago it has been a very valuable experience I would like to thank some of the people that contributed to my graduation project First of all I would like to thank Hector Gutierrez for his supervision during the past year Hector motivated me in his own unique style He was always late or in a hurry but very enthusiastic and positive The fact that he overlooked many problems sometimes drove me crazy but also taught me some valuable lessons I really appreciate the opportunity Hector gave me to gather this experience abroad Furthermore I would like to thank Rahul Agarwal for his work on the laser system electronics and particularly the software I want to thank Lionel Fevre for his help manu
44. Move 10 mm away from home position feedrate 600 mm s REPEAT LOOP 5 Loop 5 times forward and back over 180 mm Gi x180 y180 30000 G1 x 180 y 180 30000 NEXT GEAR 3 0 Decouple Z from X GEAR 4 0 sDecouple U from Y G1 y180 z180 30000 Move Y and Z to opposite end of range DWELL 1000 Dwell 1000 milliseconds Gi x180 25000 Motors chase each other Gi u180 25000 Gi z 180 25000 Gi y 180 25000 Gi y180 25000 Gi 2180 25000 Gi u 180 25000 Gi x 180 25000 DWELL 1000 LOOP 1 Motors travel full range in 15 small steps forward and back LOOP 15 G1 x10 y 10 z 10 u10 20000 DWELL 300 NEXT LOOP 15 Gi x 10 y10 210 u 10 20000 DWELL 300 NEXT NEXT DWELL 1000 Gi y 180 z 180 20000 Bring Y and Z back to same positions as X and U resp GEAR 3 1 1 1 Link slaves to masters again GEAR 4 2 1 1 Gi x100 y10 20000 DWELL 1000 CW XO YO 10 375 Clockwise cirkel start point 0 0 mm and middle point 0 75 iw r t current position of middle point of all axes 88 DWELL 1000 CCW XO YO 10 J75 DWELL 1000 Gi x 100 y 10 20000 DWELL 2000 GOTO REPEAT SDISABLE X Y MESSAGE DISPLAY DWELL 5 SEC sDWELL 5000 GEAR 3 0 GEAR 4 0 DISABLE U Z DIXYZU Counter clockwise cirkel 60 to REPEAT infinitely loop this demo Commend out to use following code z utotuning command generates sinusoidal reference with increasing frequency and decreasing amplitude
45. all Loopback Cable on Digital Connector pause cursor OFF attr get_attribute H set software triggering set_dig_trigger DI0_TRIGGER_SOFT for i 0 i lt 2 i int highState lowState err 0 reps 0 clrser printf n nPress any key to terminate test H initialize Half for input half output if i 0 highState DI0 DIR OUTPUT lowState DI0_DIR_INPUT printf n nLoopback High word to Low word wherexy amp col amp row I 71 va i else 1 highState DID_DIR_INPUT lowState DIG DIR OUTPUT printf n nLoopback Low word to High word wherexy amp col amp row 3 dig dir highState highState lowState lowState 11 Loop doing writes to Digital 1 0 while kbd_hit i int output Complement shadow register shadow Oxffffffff output shadow 70zA5A5A5A5 OxSA5A5A5A write_dig output if shadow set_attribute RGB 255 0 0 5 else set_attribute RGB 0 0 255 5 rtn read_dig gotoxy col row 1 printf 10x 1i0x rtn output if rtn output err reps if reps 100 0 1 gotoxy col raw printf d Errors in d tries n err reps 2 gt getchar eat the character gotoxy col row set attribute attr printf d Errors in Ad tries err reps gotoxy 0 20 pause y set attribute attr cursor ON clrscr printf n nRemove Loopback Cable from Digital Connector
46. anslations of the slaves motor 3 and motor 4 equal the translations of the masters motor 1 and motor 2 respectively imposes the following condition on the forces acting on the slaves Em Fig Fu N a Faa F31 6 13 Ema mal m Em F F Fn ml Ca zi D at Ha lt f m 4 Fag Fay 6 1 4 Decoupling In z and y direction the total translation of the tray is the sum of the short stroke translation xg ys of the tray with respect to the linear motors and the long stroke translation zr yz of the linear motors with respect to the ground TT TL s 6 15 YT TYL tys 6 16 Since the equations of motion for both the tray and the linear motors are known the 2nd derivative of the short stroke follows F Ey 6 17 T Flin SE Fmagn so mom 1 L m 6 18 Fun 1 Fmagn _ Fein Fmogn _ Mr m JARE TE 6 19 My m mrm m Translations zg and ys of the tray with respect to the linear motors are undesired These translations reduce the air gaps thus limiting the maximum stroke of rotation 0 The same translation of the tray can be achieved by translating the linear motors and keeping the crossmembers centered in the magnet boxes So ideally zs 0 This can be achieved as follows Translations x7 and yr are controlled using the magnetic actua tors which will lead to known reaction forces on the linear motors The equation of motion for the short stroke
47. ase should be o or 180 degrees for a mass but Figure 3 4 shows phase increases with frequency This can be explained by the way this measurement was performed The white noise is injected to the BBsor board so it is sampled first The delay of this A D conversion shows in the phase measurement Also position is measured with the laser system A program running on the DSP converts the parallel output of the laser to a voltage which is used for the frequency response measurement The limited sample frequency of the program running on the DSP also introduces delay Theoretically the first delay could be eliminated by measuring the frequency response between the echoed white noise white noise is sampled and sent to output channel thus undergoing the delay of the A D and D A conversions and the output voltage In practice this is not possible because the BBsor board has no output channels available The second delay can not be eliminated at all since there is no other way to obtain a position signal using this setup Since a number of scaling factors is used in the MMI software a simple check is done to verify the results The equation of motion for a single mass kmu ME 3 1 Here z m is motor translation u V is torque command m kg is the mass of the linear motor and km N V is the motor constant The ratio between mass and motor constant can be determined from Figure 3 4 5 20 Table 3 1 Ratio Em for several
48. ch receives position measurement from the laser system feedback for commutation status of limit and home switches etc A schematic overview of the system can be found in Figure 3 2 The structure of a PID controller is preprogrammed in the MMI software After system identification and linear control design the control gains can be set More about this in Section 3 5 If the software settings are correct motion of the linear motors can be controlled with the CNC card An example program demonstrating some basic functionalities can be found in Appendix C 5 This program is written in so called G code 3 3 Laser measurement system 3 3 1 Operating principle Translation of the moving units is measured using a Zygo ZMI 501 laser This laser has a vacuum wavelength A of 632 992 nm The laser beam is split into two signals a reference beam and a measurement beam The measurement beam is reflected by a target mirror on the moving unit Based on the phase difference between reference and reflected measurement beam the displacement of the target is computed The laser has a maximum resolution of 1 24 nm 35 The controller in the MMI software receives position feedback from the ZMI501 measurement board This is serial data with a maximum rate of 23 04 MHz which limits the target velocity to 28 48 mm s When a lower laser resolution is used a higher target velocity is possible High accuracy also limits measurement range but using the high
49. curacy levels Wafer positioning stages in photolithography are certainly among the most accurate and demanding high precision systems Strong research efforts are dedicated to reduce positioning errors to nanometer level According to 10 high precision positioning stages can be classified as high precision gantry sys tems or as photolithography wafer stages Using two examples ro describes how gantry systems achieve long travels over 1 m in X and Y direction with an average position accuracy of 3 0 micron This type of machine is typically used for flat panel inspection or electronic assembly Ultra high posi tion lithography systems achieve position accuracy in the order of 30 nm but with smaller range of motion in X and Y 0 35 m and very small range in vertical direction The ultimate goal of this multiple year project is development of a positioning stage that can achieve both high resolution and large travel In ro the design of this machine is described It provides long ranges of motion with high accuracy and can be customized to fit different applications such as assembly of optical components and MEMS Micro Electro Mechanical Systems manufactur ing Production costs of the proposed machine will be significantly lower than existing positioning systems since standard machining tolerances apply for the manufacturing of the machine s compo nents Main goal for the project described in this report is design of a control syste
50. e current to the magnetic actuators coils and the air supply for the air bearing Pin outs and other details about the electronic connections can be found in Appendix AT a Front view b Side view Figure 3 1 One moving unit OT Host Unidex 500 CNC card M67 DSP DIG 32 module Servo 16 module BB501 PCB Interface board ELI CCW limit Switch signals A 20 40 A Amplifier Serial data Motor feedback Motor power Home BLMC 92 A BLMC 92 A Linear motor gt Linear motor A Parallel data NI 6024E Data aqcuisition card Analog input output Data aqcuisition box custom build de PWM board ZMI 501 ZMI 501 Reference signal Laser head ke Measurement board Position interferometer measurement inget e urrent man Amp 2 Coil 2 Amp 3 el Coil 3 Amp4__ Sensor signal DMT 22 Driver module Figure 3 2 Schematic overview of one axis Coil 4 Capacitive sensor h Capacitive sensor v 3 2 Software The position of the moving units is controlled by a Unidex 500 CNC card and accessory MMI software The Usoo card gets input from a controller board BB501 whi
51. e of the vertical range not at the lower actuator However for this project it is not useful to modify the setup First of all the required time is not available More important is the fact a controller is pur sued that can position the crossmember accurately over the whole vertical range Initially positioning the system in the equilibrium point might show that linear control works for small regions around that point but the closed loop system will still not be stable for the whole vertical range Final reason is that the 6 DOF system will experience the same problem but in this case it cannot be solved with hinges 31 x10 4 Hz reference simulation 6 1 Hz reference 6 x10 Reference i Reference 4 Position i 4 Position al i i ol E Bo E 3 2H 4 E i i i i i i i i 6 i i i i i i i i i 0 0 2 04 06 08 1 12 14 16 18 2 0 02 04 0 6 0 8 1 12 14 16 18 2 Time fs Time s Figure 5 7 Reference and position simulation Figure 5 8 Reference and position real time x103 Error simulation 6 Error Position m Position m o i i i i gi NE FE NE ODER BR UE BE OE A 0 02 04 06 08 1 12 14 16 18 2 0 02 04 06 08 1 12 14 16 18 2 Time fs Time s i Figure 5 9 Error simulation Figure 5 10 Error real time Control signal simulation Control signal 0 32 E H
52. eam splitters fold mirrors and interferometers is a time consuming proces that has to be performed accurately For future reference a step by step instruction is given in Appendix A 3 11 den Doe Be s I ry At PE 5 Straight trough interferometer R Right angle interferometer B 50 beam splitter F Fold mirror Figure 3 3 Laser setup 3 4 System identification An external signal can be added to the control action of the preprogrammed controller on the Usoo If all control gains are set to zero the frequency response of a linear motor can thus be measured open loop by externally supplying white noise The frequency response of the linear motor on axis 1 can be seen in Figure 3 4 White noise up to 400 Hz was injected Axis 1 Plant NE Td T e E 50 g o 2 8 8 100 150 10 200 g 100 8 e gt D 2 op a 2 E amp too 200 Frequency Hz Figure 3 4 Frequency response of linear motor on axis 1 12 From this figure it can be concluded that the linear motor behaves as a mass This was to be expected since it Hoats on air bearings and experiences minimal friction Only the stiffness of the E chain carrying all cables to the motor and coils resists the translation of the motor but this can be neglected The ph
53. ection should also be present on the alignment card which can be moved by changing the pitch and yaw of the target mirror The two reflected beams should overlap each other completely for a good signal see Figure A g Move the axis from end to end and observe the reflections and adjust the target mirror so that reflections do not drift Two spots should be observed on the target mirror horizontally separated by about 25 inch They should not move with respect to each other or w r t the mirror for the entire trajectory of the axis see Figure A 8 Target mirror Mounting holes for fibre optic pickup Figure A 7 Position of beam on input beam Figure A 8 Reflections on target mirror aperture 65 A 3 4 Fiber Optic Pickup The final step is to align the fiber optic pickup to the interferometer Connect the pickup to the fiber optic cable and interferometer loosely to allow some adjustments Check the output of the fiber optic cable on the other end by directing it to a piece of white paper as mentioned in 24 page 7 11 The beam should be a solid spot and not a ring or a donut Connect the optic cable to the ZMI 501 enclosure and measure the voltage on the test point for the channel Adjust the pickup to get a voltage in excess of 2 0 V for the entire range of the axis Note that the 2 4 V mentioned in the manual is not correct As long as the error light on the enclosure is off the signal is fine The voltage should be a
54. eel van de slag en kan een laagfrequent referentie signaal volgen De statische positie fout ligt ruim beneden 1 micrometer Vanwege de beperkte beschikbaarheid van de versterkers kon een grondige analyse van de regeling van deze testopstelling niet worden voltooid Dit is een essentieel onderdeel van het project en vraagt om meer aandacht in de toekomst Ondertussen werd een model voor het 6 vrijheidsgraden systeem opgesteld Op basis van voor lopige resultaten van de regeling van de testopstelling werd een MIMO regelaar ontworpen en ges imuleerd Gezien de resultaten tot dusver lijkt het erg moeilijk om aan de ontwerp specificaties te voldoen vi Contents Preface Abstract Samenvatting 1 Introduction 2 Previous research 2 1 2 2 Technical Background Machine Design 2 2 1 General assembly 2 2 2 Moving units 2 2 3 Magnetboxes 2 2 4 Trayandarms 2 2 5 Passive magnetic suspension 2 2 6 Position measurement 2 2 7 Metrology mapping 3 Linear motors 3 1 3 2 33 3 4 3 5 3 6 3 7 Laser measurement system 3 3 1 Operating principle 3 3 2 Layout laser system 3 3 3 Alignment of optical components System identification Control Real time results Conclusion 4 Force model of magnetic actuator 41 43 Theoretical model 4 1 1 Assumptions 412 Model 2112 Empirical model
55. eersceereerseensecereeneeneoecereeeneceereen Vf E Visser June 2004 H Sliding mode control of tray f snaeemm eeezzenen Ens sene E tn define S_FUNCTION NAME longstrokeSMC def ine S_FUNCTION LEVEL 2 include simstruc h include math h define Ulelement uPtrs element Pointer to Input Port0 def ine CONT_ST o 41 No of continuous states define DISC_ST 0 ti No of discrete states define GLOBALINTS 0 H No of global integers def ine GLOBALREALS 0 VER No of global reals Tray def ine Jxx 0 5364 lkg m 2 Inertia tray def ine Jyy 0 536 kg m 21 Inertia tray define Jzz 1 0708 4 1 Dkgem 2 Inertia tray def ine LO 0 55 ti Im Nominal arm length define mt 10 9 ti kg Mass tray define gc 9 81 di m s 2 Gravity constant Coils define gnx 3 81e 4 17 Im Nominal gap size horizontal gny gnx iidefine gnz 6 35e 4 Im Nominal gap size vertical def ine ke 1 00e 5 W m 2 4 2 Force constant of magnetic actuator Other def ine Ts 2 5e 4 s Sample time def ine pi 3 14159265358979 fae sss sae SS SS SSS SSS Z SSS SS SSS SSS SSSR ZZ Initialize H static void mdlInitializeSizes SimStruct S i ssSetNumSFcnParams S 0 if ssGetNumSFcnParams S ssGetSFonParamsCount S 1 return Parameter mismatch will be reported by Simulink ssSetNumContStates S CONT ST set number of continnous states ssSetNumDiscStates S DISC ST set number of discrete states if
56. ent of the maps origin with the origin of the metrology frame is suggested in ro The proposed approach has been subject to discussion and criticism Using feedback from 11 modifications to the design will be made before it is implemented In the meanwhile a temporary measurement system will be used for the design of a control system This limits the maximum achiev able accuracy but in this stage of the project this is hardly a restriction since nanometer level accuracy is not to be expected yet Chapter 3 Linear motors 3 1 Hardware In this chapter the term linear motor describes a so called moving unit ro One moving unit con sists of two BLMC92 A linear motors see Figure 3 1 The motors slide through two parallel magnet tracks and actuate a platform that floats on an air bearing This platform carries the magnetic actua tors that levitate the tray and the capacitive sensors to measure position of the tray with respect to the moving unit The two linear motors are connected in parallel to one amplifier Within the magnet track each motor has only 1 translational DOF Since the motors are mounted rigidly to the same platform it is assumed their translation is equal Feedback from only one motor is used for commutation The position of the platform is measured with a laser system All cables to the moving unit are guided by an E chain This E chain also carries the capacitive sensor wires the cables that provid
57. ents in Z 6 and 0 is proposed in 10 This is a novel approach that is still topic of research and discussion If the principle is feasible the implementation will require special attention calibration implementation of large look up table Another issue are the specifications with respect to speed and accuracy in 10 These specifications will not be used as target since it is not realistic to think design and implementation of a control system meeting all design specifications can be realized within a year Instead control systems will be designed that achieve maximum performance in terms of accuracy and bandwidth on the 1 DOF setups Only when MIMO control of the 6 DOF system is implemented it will become clear if the specifications come close to those specified in ro From that point iterations will have to lead to improved performance in order to eventually meet the design specifications Chapter 2 Previous research This chapter contains a summary of Design and construction of a 6 DOF positioning system with nanometer accuracy master s thesis of L Fevre ro At the end of Fevre s project the machine described in this thesis was partially manufactured and assembled 2 1 Technical Background ro shortly describes the applications of high precision positioning devices and gives an indication of the performance of state of the art devices in terms of accuracy stroke etc The performance of this particular design i
58. er is really poor as can be seen in Figure 5 7 through Figure 5 17 While the single DOF of the system is a rotation both reference position and error are converted a displacement m at the position of the actuator Position error in m at this point gives a good indication of the accuracy of the system The system can only track a low frequency sinusoidal with an amplitude of a few micron A good closed loop frequency response measurement could not be performed 5 2 5 Conclusion The results obtained so far with linear control are poor The system is highly nonlinear so the lin earized system equations only describe the system accurately c ose to the operating point in which it was linearized As the crossmember moves away from the operating point the quality of the lin earization decreases and the performance of the system degrades Since no magnetic suspension is available yet the crossmember initially rests on the lower actuator In this position it is far from the operating point A linear controller with good performance as tuned in simulation could not be implemented The system could only be stabilized with a linear controller with lower gains but with very poor results The steady state position error is approximately I micron In literature for example 21 similar experimental setups can be found in which the effect of gravity is cancelled out using hinges The equilibrium position of the crossmember will be in the mid dl
59. es for this system are less strict than for existing high precision positioning sys tems Only manufacturing of the granite base interferometer mirrors and reference surface requires extreme accuracy and has to be performed by professional companies There are 3 possible operating modes Depending on sensor sensitivity settings a trade off is made between stroke and accuracy The table shown is based on sensor settings as used during this entire project Figure 2 1 Overview of the complete positioning table 10 Figure 2 2 Magnet box 10 Figure 2 3 Crossmember with integrated lami nations 10 Stack of E laminations Figure 2 4 Nominal position of crossmember in Figure 2 5 Magnetic flux paths magnet box 10 Parameter Axis Specification Travel X 203 mm Y 203 mm Z 1 mm Pitch Or 20 arc sec Oy 20 arc sec Yaw 6 20 arc sec Orthogonality X to Y 0 5 arc sec Z to XY p ane 0 02 arc sec Resolution X and Y 1 25 nm Z 60 01 nm 04 and Oy 0 02 arc sec 6 0 009 arc sec Repeatability X and Y 1 25 nm Z 4 01 nm 6 and 6 0 051 arc sec 0 0 084 arc sec Accuracy X 3 3 nm S 3 3 nm Z 64 0 nm Velocity X and Y 500 mm sec Acceleration X and Y 8 92 m sec z 12 m sec Total mass N A 445 kg Table 2 1 Capabilities and perf
60. escribed in the following subsection Theoretically the feedback linearized system behaves as a double integrator and can be controlled with a standard linear controller Drawback of the feedback linearization method is that it is not robust In practice it will only work if the system model is exactly correct Any model errors or unmodelled phenomena that introduce a mismatch between model and physical system will lead to deviations Literature 21 17 shows feedback linearization has successfully been implemented on similar magnetic suspension systems before Implementation of feedback linearization in combination with a linear controller leads to poor re sults Oscillations occur when the crossmember is positioned close to the upper actuator Although the force model derived in Chapter 4 closely matches empirical results it can be concluded that this model is not 100 accurate Differences between model and actual system can be the result of cali bration errors sensor drift unmodelled friction etc Since feedback linearized system could not be controlled satisfactory with a linear controller due to limited model accuracy a different control strategy was used One option was the use of a robust controller However in general robust control leads to high or der controllers compromising fast implementation of the control algorithm Even a simple controller like a PID could only be implemented at a maximum sample frequency of 4000 Hz A h
61. est resolution a position range of approximately 1 33 m is possible The maximum stroke per axis is approximately 0 208 m so this is no limitation The ZMIsor board also has parallel ports which can be read using the digital I O of the DSP This functionality can be valuable later when control loops for the linear motors and magnetic actuators have to be integrated for control of the 6 DOF system An additional advantage is that now maximum resolution is possible without limitations on the target speed Appendix C 1 contains C code used for collecting parallel laser data with the DSP 3 3 2 Layout laser system Using several optical components the laser signal can be split and directed to multiple target mirrors The layout used to measure displacement on 4 axes can be seen in figure 3 3 This layout was designed in ro but is not the most efficient Displacement on 4 axis could also be measured using less optical components In the original design of ro no kinematic mounts for the optical components were included These mounts can be used to secure the components and provide freedom for small adjustments These kinematic mounts are an absolute requisite for a good alignment In order to include kinematic mounts in the laser setup the aluminum mounting blocks were redesigned Drawings of the mounts and the modified aluminum blocks can be found in Appendix A 2 3 3 3 Alignment of optical components The alignment of the laser head b
62. etup could not be finished This is an essential part of the design and requires attention in further research In the meanwhile a model of the 6 DOF system was derived Based on temporary results obtained on the 1 DOF setup a MIMO controller was designed and simulated Preliminary simulation results were obtained Considering the results so far it seems very hard to meet the design specifications iii Samenvatting Het uiteindelijke doel van dit meerjarige project is ontwikkeling van een positioneertafel met hoge nauwkeurigheid en groot werkgebied Het ontwerp van deze machine wordt beschreven in ro Lin eaire motoren zorgen voor een grote slag in X en Y richting terwijl magnetische actuators zorgen voor kleine slag in 6 vrijheidsgraden De machine is geschikt voor verschillende applicaties zoals de assernblage van optische componenten en de productie van MEMS Micro Electro Mechanical Sys tems Doel van het hier beschreven project is het ontwerp van een regelsysteem voor de 6 vrijheidsgraden positioneertafel beschreven in ro Ontwerp en implementatie van een regelsysteem voor deze ma chine is een complex en tijdrovend proces Aangezien de ontwikkeling van een prototype nog in een vroeg stadium verkeerd omvat dit proces niet alleen het modelleren en regelen van het systeern maar ook instrumentatie assemblage en her ontwerp van componenten Vanwege de beperkte beschikbare tijd is dit project gericht op een aantal essenti le
63. facturing the 1 DOF setup and Dennis Miller for the time he voluntarily spent to help me getting the laser system and DSP up and running Many other people have supported me during my education and were particulary important to me during the 20 months I spent abroad I want to thank Olaf Miriam and my high school friends for their loyalty and my roommates for their interest support and especially for the necessary distraction from my work Finally I would like to thank my parents for their support financially but foremost mentally They made it possible for me to fully enjoy my student days and stimulated me to seize all the opportunities I got u Abstract The ultimate goal of this multiple year project is development of a positioning stage that can achieve both high resolution and large travel The design of this machine is described in to Linear motors provide long range motion in X and Y direction while magnetic actuators provide small strokes in 6 DOF s with high accuracy The system can be customized to fit different applications such as assembly of optical components and MEMS Micro Electro Mechanical Systems manufacturing Main goal for the project described in this report is design of a control system for the 6 DOF positioning stage described in 10 Design and implementation of MIMO control for this machine is a complex and time consuming process Since development of a prototype is in an early stage this process also i
64. he tray provides a repelling force Main requirements to the repelling force are that it is constant over the vertical range of motion 1 mm and over the X Y operating range Peaks in the magnetic field should be avoided because this will induce vibration in vertical direction when the tray travels in X and or Y direction The design of this magnetic suspension has not been finished A scale model has been made and currently efforts are made to fit both a FEA model and analytical model to measurement data 9 Once a good model is available design can be finished and the suspension will be manufactured 2 2 6 Position measurement Two measurement frames are used to determine the position and orientation of the tray A laser interferometer system will be used to measure the horizontal translations X and Y and the rotation around the vertical axis 0 For this purpose a high precision L shaped mirror is attached to the bottom of the tray see Figure 2 7 For the measurement of Z 0 and 0 a set of three capacitive sensors is used These sensors are mounted to the bottom of the tray and accurately measure distance based on the capacitance between two plates One plate is the sensor surface the other plate is the target which has to be a metallic surface A target made of Zerodur a high stiffness type of glass with a metallic surface is recommended ro This target has to be manufactured with very precise flatness The resolution of this ty
65. hese DOF s Only translations of the linear motors strokes limited to plus or minus 0 104 m have significant influence on the arm lengths In centered position all magnetic actuators have an arm of 0 55 m w r t the COG of the tray 6 1 2 Equations of motion The system model consists of 5 rigid body s 4 linear motors and the tray The linear motors have one translational DOF each From Figure 6 1 it is easy to see that the equations of motion for these DOF s are A Fm Fr Fi Ti mi z Fma Foz Fx y ma Fina F32 Fai Ta ma Fma Fag Fa Ya ER en The center of gravity of the tray has 6 DOF s with respect to ground Fu Fia F31 F32 T My j Fa Fa Pa Faz T Mo ga Fis Fu Paz Fx Fag F34 Faz Faa Mg My lt L Fis Fia La Fas F34 6 3 Am eN pa Nee 6 5 6 6 6 7 6 8 6 9 6 10 6 11 6 12 a I j La F23 Fra L F z Faa y MB SS A Jyy i La Fy Foi Lal Fai Fa Ls Fsi Fan La Fai Fa2 e Jez I Here L1 L4 represent the arm between each magnetic actuator and the COG The arms can be approximated by L Lo yr Lo Lo zr La Lo yr La Lo zr Lo is the arm length when the tray is centered La La L3 La Lo 47 6 1 3 Master slave The assumption that the tr
66. i real_T GK 6 I 10 10 15 10 10 15 Gain of switching term real_T KI2 4 3656e 4 1 24e 9 real T KP2 3 1549e 5 1 24e 9 real T KD2 1 1921e 7 1 24e 9 real T klin 18 96 real T mlin 3 8 real T kscale mt klin mlin For mapping of Fcog to Fact see Compute M m static real T Ms 81 6 110 5 0 0 0 gt 0 0 4545 40 0 5 0 0 0 0 4545 0 5 0 0 0 0 gt 0 4545 0 0 5 0 0 gt 0 0 4545 0 0 0 25 0 9091 0 0 F 0 0 0 25 O 0 9091 0 10 0 0 25 0 9091 0 gt 0 gt 40 0 0 25 0 0 9091 O Non zero entries diagonal elements of B matrix static real T H 6 0 0917 0 0917 0 0917 1 8643 1 8643 0 9339 i mt 1 mt 1 mt 1 Jxx 1 Jyy 1 Jzz real_T Ss 6 Sliding surface for 6 DOF s real_T Us 6 Switching terms real_T Fc 6 1 Control action force torque w r t COG real_T FHact 4 Control action mapped to horizontal actuators real_T FVact 4 Control action mapped to vertical actuators real T gap 4 4 Gaps int T i Gompute control action for 6 DOF s for i 0 i lt 6 i 1 Compute Sliding surface Ss i U i 26 2 0 LABDA i U i U i 6 LABDA i LABDA i U i 32 Compute switching term Us i GK i 2 0 pi atan Ss i PRI i Compute control signal force torque w r t COG Fcfil U i 12 LABDAfi LABDA i U i U i 6 2 0 LABDA i U i 26 Us i
67. i 12 gt Compute 2nd derivatives Translations of linear motors dx 12 U 0 U 2 mlin FL O F 1 FLO mlin dxL13 0 U 1 mlin FLI1 F 5 F 4 mlin dx 14 dx 10 Slave to motor 1 FL 2 F 9 F 8 mlin dx 15 dx 11 Slave to motor 2 L 3 F 13 F 12 mlin Translations of tray dx 16 U 2 mt F 01 F 1 F 8 F 9 mt dx 17 0 0 3 mt I F 4 F 5 F 12 F 13 mt sumFZ F 2 F 3 F 6 F 7 F 10 FL11 F 14 F 15 mt gc if sumFZ lt 0 4 x 6 lt 6 3e 4 2 initpos 1 1 dx 18 0 3 else 1 dx 18 sumFZ mt initpos 0 KJA dx 18 0 Rotations of tray dx 19 0 Li F 2 FE3 3 F 10 F 11 Jxx dx 20 0 L2 F 6 F 7 LA F 14 F 15 Jyy dx 21 0 Li F 0 F 1 L2 F 4 F 5 L3 F 8 F 9 L4 F 12 F 13 1 Jzz Short strokes l x 22 dx 14 dx 10 x _ddot dx 22 U 0 mlin mlin mt mlin mt U 2 dx 23 dx 15 dx 11 LI yS ddot 3 Nje SS t Terminate Nje is see SS ae se ee SSS SSS mm static void mdlTerminate SimStruct 45 1 UNUSED_ARG S unused input argument E fifdef MATLAB MEX FILE Is this file being compiled as a MEX file fincinde simulink c MEX file interface mechanism felse include cg_sfun h Code generation registration function fendif C 4 MIMO controller S function SMC6DOF eeeeeeneeemeeernene
68. id on axis 0 300e 6 0 5 1 5 205 2251 xlabel Additional air gap m gt ylabel Gain 1 zlabel Force N title Sum of absolute error as function of x 0 and gain ZDOMED Find minimum value Minimum value for every column Ecolmin Row index min E2 Minimum value of row Y Emin Col index min Ecolmin Check E2 Row index Col index Col index 92 Return optimum values gain GO Row_index Col_index Col index x0 XO Row_index Col_index Col index Compute model with optimum parameter values and plot clear Fmodel Calculate model with optimum values for gain and x0 for ix 1 14 for ii i length Itot Fmodel ix ii gain 0 5 N 2 mu0 A 1 mu0 A O 5 R1 0 5 R2 R3 z0 Xtot ix ii 2 Itot ix ii 12 end end where actuator force was not measurable and made 0 make model force O as well for i i length I for m 1 14 if Fact m i 0 Fmodel m i 0 end end end Plot measurement and model figure mesh Xtot Itot Fact hold on mesh Xtot Itot Fmodel title Force measurement and model xlabel Air gap m ylabel Current 4 zlabel Force N Plot difference between model and measurements figure mesh Xtot Itot Fact Fmodel hold on title Force measurement minus model xiabei Air gap m ylabel Current IAj zlabel Force N gt 93 D 2 Closed loop system ID System ID for arm with 1 DOF vertical Derive
69. igh order robust controller would be very slow especially for the 6 DOF system Sliding mode control is an other control strategy that provides robustness against modelling er rors and unmodelled phenomena Due to its relatively simple algorithm it can be implemented with a high sample frequency Sliding mode control is often used for this kind of suspension systems 7 but typically using voltage as input Choosing voltage as input will make the system affine but is not an option for this project The amplifiers have been designed to drive current Two approaches to the sliding mode control of the system have been considered In the first one the system is made affine by applying feedback linearization Sliding mode control of a linear system is applied according the theory of 14 20 and 8 using force as input Theory of this approach is described in subsection 5 3 2 Subsection 5 3 3 describes the second approach where the system is considered non affine using currents as input and the theory of 12 and 13 is applied Both approaches were compared in simulation with very similar results Since very limited ex perimental time was left only the first approach has been implemented and optimally tuned Further investigation of the second approach deserves attention 34 5 3 1 Feedback linearization To obtain consistent performance over the whole operating range feedback linearization was applied Here a nonlinear state transf
70. in the 1 DOF setup showed that this model closely matches the empirical results Linear and nonlinear control strategies for control of the 1 DOF setup were simulated and imple mented This part of the project was delayed significantly since the originally designed current drive amplifiers were not suited due to low bandwidth and failure Design and manufacturing of new am plifiers took much time Debugging and programming of a DSP for control implementation was time consuming too Due to limited availability of amplifiers a thorough analysis of control of the 1 DOF setup which was one of the main goals of this project could not be finished A linear controller was implemented with very poor results Best results were obtained using a sliding mode controller in combination with feedback linearization of the system The closed loop system is stable for a large operating range only close to the upper actuator it is not The system can track a low frequency reference signal but improvement is definitely required The steady state positioning error is well below 1 micron A start was made with the control of the 6 DOF system A model was made using the results from system identification of the linear motors and the magnetic actuators Long stroke and short stroke were decoupled and the issue of controlling 6 DOF s with 20 actuators was addressed using a mapping to distribute the control signals Some simulation results were shown but more information
71. ing experiments 4 3 Conclusion The model fit to the measurement data is close to the theoretical model Due to practical limitations no measurements could be done over the whole vertical range Possibly saturation of the core material occurs This could not be determined with the available tools but should be investigated further Force N Force N Force measurement and model Current A sh nde Figure 4 3 Force measurements and fit Force measurement minus model Current A Air gap m Figure 4 4 Difference between force measurements and fit 23 24 Chapter 5 Control of 1 DOF system with magnetic actuator Before addressing the problem of controlling the 6 DOF system a good understanding of the control of a magnetic actuator is necessary To achieve this a setup with a 1 DOF crossmember actuated by I magnetic actuator is used Linear and non linear controllers will be designed implemented and compared using this system Once a good control strategy is found this can be extended for use in the more complex 6 DOF system In ro several specifications are mentioned which the 6 DOF system eventually has to meet This concerns specifications like a maximum static error and the stroke for all DOF s No explicit control specifications are defined in terms of bandwidth Since not all hardware is available these specifications cannot be translated to specifications for the control of the 1 DOF
72. is no horizontal drift in the input beam Tighten the screw to secure the mount completely to the block Now the vertical drift has to be removed Loosen the screws to adjust the pitch of the fold mirror Tighten them back when there is no vertical drift in the input beam Repeat the process till the input beam is stationary on the alignment target for the entire range of motion of the axis Next replace the prism with the second fold mirror and repeat the process to align the second mirror To align the first beam splitter put the penta prism on the right angle output of the splitter see Figure A 6 Repeat the above procedure by adjusting the pitch and yaw of the beam splitter to stabilize both output beams on the target Next replace the prism with a fold mirror and repeat the procedure Continue the process to align all the mirrors and splitters on the aluminum block A 3 2 Interferometers The next step is to align the interferometers The input beam aperture on the interferometers is rather small after mounting the fiber optic pickup Hence it needs to be carefully aligned Remove the fiber optic pickup from the interferometer and align its input side so that the input beam is on the opposite side to where the pickup gets mounted see Figure A 7 A part of the beam may overflow the aperture Tighten the screw on the input side to avoid translations Now follow the previous instructions to align the yaw and pitch of the right angled interfer
73. ith J Tx 0 Il u H Moment of inertia around x axis Rotation around x axis Mass of crossmember Gravity constant Distance from CM to point of rotation Distance from magnetic actuators to point of rotation Force constant Currents trough coil 1 and coil 2 Nominal gap when 6 0 gif y Ly 92 Fo 5 1 kg m degrees kg m s m m IN m A A m Figure 5 1 Schematic view of 1 DOF arm with one pair of actuators 26 5 2 Linear control 5 2 1 Linearization In order to make the system single input a premagnetization is introduced by applying bias currents to both coils The variational current AJ is the single control input hash Al 5 2 hahaa 5 3 Here I and Jp represent a bias current on coil 1 and coil 2 respectively The control current has to meet the requirement AJ lt Ip l to prevent the occurrence of negative currents in both coils Currents flowing in opposite direction will also generate tractive forces The state vector z consists of angle and angular velocity 01 c 54 6 A ee 5 5 Z 5 5 T2 a yKy b AD Ko ba AD mgl cos a 5 6 RE EG a OD mgheos ra J gi x1 Jaz 95 1 Jez In order to linearize the system define 11 ZZ T 65 La Ta dra Where dx denotes small perturbations with respect to the operating point 7 Now the system is lin earized around the operating point 7 0 Z2 0
74. lf of the vertical range Tracking error rapidly increases when when frequency is increased see Figure 5 24 through Figure 5 27 which indicates that the relation between ait gap current and actuator force is frequency dependent Figure 5 18 through Figure 5 20 show tracking of a smooth step from initial position to half of the vertical range The steady state error lies well below 1 um Al 42 Chapter 6 Modelling and control of the 6 DOF system Due to the previously mentioned problems with the amplifiers the project was considerably delayed In the meanwhile some preliminary work was done on the modelling and control of the 6 DOF system The work presented in this chapter should be seen as a starting point for future research A 6 DOF model is derived based on results of system identification of the linear motors and the force model derived in Chapter 4 An approach is suggested to distribute the control action for 6 DOF s over 20 actuators Preliminary simulation results are shown 6 1 System model This section describes the modelling of the MIMO system Definitions can be found in Figure 6 1 Figure 6 2 and Figure 6 3 6 1 1 Assumptions and definitions The following assumptions were made while modelling the system Only rigid body modes are considered The lowest eigenfrequency is a flexible mode of the tray in vertical direction and lies at approximately 74 Hz For more details see ro e The linear motors only ha
75. linear input system application to a magnetically levitated fast tool servo IEEE transactions on Industrial Electronics vol 45 no 6 December 1998 14 Jager A de Advanced control lecture sheets TU e 2001 15 Levine J Lottin J and Ponsart J A nonlinear approach to the control of magnetic bearings IEEE transactions on CST vol 5 no 4 September 1996 16 Lindlau J D and Knospe C R Feedback linearization of an active magnetic bearing with voltage control IEEE transactions on CST vol 10 no 1 January 2002 17 Ludwick Stephen J Trumper D L Holmes M L Modeling and control of a six DOF mag netic fluidic motion control stage IEEE transactions on CST vol 4 no 5 September 1996 59 18 Sadiku Matthew N O Elements of Electromagnetics 3rd edition Oxford University Press New York 2001 19 Scherpen J M A Kerk B van der Klaassens J B Lazeroms M and Kan S Y Nonlinear control for magnetic bearings in deployment test rigs Simulation and experimental results Proc of the 37th IEEE conference on Decision amp Control Tampa FL USA December 1998 20 Slotine J and Li W Applied nonlinear control Prentice Hall London 1991 21 Trumper D L Olson S M and Subrahmanyan P K Linearizing control of magnetic suspension systems IEEE transactions on CST vol 5 no 4 July 97 22 Visser E E Modelling and linear control of a magnetically suspended arm Traineeship rep
76. llers is limited so fast acceleration of the tray can leads to mechanical contact between tray an linear motors which is obviously undesired Simulations show that the mapping proposed in the previous section is suited to distribute the control actions over the 20 coils When the tray is centered MM is a diagonal matrix with 1 s at the diagonal However if the tray moves away from the center non zero terms occur outside the diagonal of MM This basically means that the control action for a certain DOF influences other DOF s as well These cross terms can be seen as disturbance Since no passive magnetic suspension is available the vertical actuators have to provide a large force to carry the weight of the tray This force is an order 100 higher than the other forces When the tray moves off center motion in xy and yr after 1 second the cross terms between the vertical force and the DOF s 6 and 6 increase from o to 0 3 The effect of this disturbance on 6 and 6 can clearly be seen in Figure 6 7 and Figure 6 8 This severely limits the maximum vertical speed in zr and yr 52 If the tray moves faster in the horizontal plane the disturbance on 6 and 0 is larger and leads to instability Note that this problem is solved when a passive magnetic suspension carries the weight of the tray 201073 6 2 0 Vuo SU AV IN 2 0 0 3 0 107 O 2 0 0 05 4 010 7 Table 6 1 Setpoints Labda 15 10 10 15
77. lmost constant for the entire range of the motion If the voltage varies more than roomV this indicates misalignment Once a constant voltage is achieved for the entire range screw in the pickup completely This step is difficult and requires patience Sometimes using 1 screw instead of 2 is better to maintain satisfactory output voltage _ Input beam Alignment card Interferometer Figure A 9 Overlapping beams for whole axis A 3 5 Feedback and dip switch settings Finally check the feedback in the Usoo software The axis should move for about 208 mm and should be constant for every move The right angled interferometer gives opposite output as compared to the straight through Hence the right angled interferometer gives increasing feedback when the target mirror is moving closer to it As explained in 24 page 3 3 switch S8 can be used to swap the direction of the quadrature serial output going to the control software The motor expects an increasing feedback in the clockwise direction of the motion the side where the wires of the motor come out Therefore adjust switch 8 for all the axes accordingly An incorrect feedback will make the axis go unstable by just enabling it Also note that the position of switch S8 does not affect the parallel output 66 Appendix B Actuator dimensions and equivalent circuit Mere x RI Rat Figure B 1 Actuator dimensions Figure B 2 Equivalent circuit
78. m for the 6 DOF positioning stage described in 10 The design and implementation of a control system for this type of machine is a complex and time consuming process Since development of the prototype is in an early stage this process does not only include modelling and control but also assembly instrumentation and re design of components In the limited time available the project will focus on several essential steps that must form a basis for the control system that eventually will be implemented As described in Chapter 2 the system contains linear motors for long range motion in X and Y direction and magnetic actuators fot small adjustments in all 6 DOF s Chapter 3 describes system identification and control of the linear motors In Chapter 4 a theoretical force model for the magnetic actuators is derived This model is verified with experiments on a 1 DOF setup In Chapter 5 linear and nonlinear control strategies for this 1 DOF system are described Due to problems with the hardware a good controller of the 1 DOF setup has not been obtained yet Finally Chapter 6 describes a 6 DOF model of the system Based on temporary results of the 1 DOF control a MIMO controller was designed and simulated Several issues will not be addressed in this report that are certainly non trivial They will require attention in future research One of those issues is the feasibility of several design principles For example a metrology mapping for measurem
79. minations with a coil around the center leg are placed that can only exert tractive forces For the following experiments only the upper coil is used A force sensor carries the weight of the crossmember and a known preload generated by adding extra mass to the crossmember Increasing the current through the upper coil results in an increasing actuator force in upward direction while the resultant force is still directed downward Since mass preload and all arms with respect to the point of rotation are known accurately the actuator force can be derived from equilibrium of moments The air gap is measured with a capacitive sensor By adjusting the height of the force sensor the air gap between crossmember and actuator can be varied For each air gap value the current is increased stepwise until the actuator force exceeds the preload For several reasons the magnitude of the applied preload is limited It has to approximate a point force and the force beam should not bend significantly under the preload Since the preload is limited only forces up to approximately 70 N can be measured Higher actuator forces will pull the arm including weights up Figure 4 1 Arm with 1 rotational DOF in 1 magnet box 20 4 2 2 Results Figure 4 2 shows the measured force as function of current and air gap For good visualization force has been plotted with a magnitude of o N for the region where the actuators force exceeds the preload so
80. n x0 7 GE11 gnz GI10 G 12 gax x0 5 x0 3 LO4 tan x0 9 6113 2 gnx 61121 G 14 gnz x0 6 LO4 tan x0 8 G 15 2 gnz 6114 Define outputs static void mdlQutputs SimStruct S int_T tid real_T TLM ssGet0utputPortRealSignal 5 0 real_T DOF ssGetQutputPortRealSignal S 1 real_T GAP ssGetOutputPortRealSignal S 2 real_T SHS ssGetQutputPortRealSignal 5 3 real_T x ssGetContStates S real_T xG ssGetRWork S int_T Ls UNUSED_ARG tid Hot used in single tasking mode for i 0 i lt 4 i Output port 0 translations of linear motors 1 4 1 TIME x i for i 0 i lt 6 i Output port 1 6 DOF s of tray 1 DOF i x i 4 Output port 2 8 gaps that are physically sensed GAP 0 GEO Gap 11 82 GAP 1 G 4 Gap 21 GAP 2 GE9 Gap 32 GAP 3 G 13 Gap 42 GAP 4 G 3 Gap 14 GAP 5 GIT H Gap 24 GAP 6 G 11 Gap 34 GAP 7 G 15 Gap 44 SBS OJ x 10 H x8 SBS 1 x i 1 ys 3 J seme cee ee Se ee SS ESS Compute derivatives of continuous states Vj me ss ss define MDL DERIVATIVES static void mdiDerivatives SimStruct S t InputRealPtrsType uPtrso InputRealPtrsType vPtrsi ssGet InputPortRealSignalPtrs 5 0 ssGetInputPortRealSignalPtrs S 1 NA bow a real T dx ssGetdX S real T xx ssGetContStates S real_T G ssGetRWork S real_T F 16
81. nable_clock enable_interrupts start cirsct gotoxy 35 0 bold printf 7Laser Data testin normal printf n nCurrent module 4d module printf n n Main Menu n printf 1 Read Laser Data n printf 2 BaseBoard Digital I 0 n printf 3 Module Digital 1 0 n printf 4 Select Module to Testln printf 5 Quit n printf mSelect test to run switch getchar 1 case 1 read_laser break case 2 69 data test_base_dio break case 3 test_dio break case 747 printf VoModule Site 0 2 scanf d amp module printf ln break default printf n nTesting Complete monitor break printf n nPress a key to return to main menu getchar goto start void read_laser volatile unsigned int rtn int col row volatile int counter 0 old pos pos position diff first_time 1 set software triggering set_dig_trigger DIO_TRIGGER_SOFT clrscr printf Reading Laser telemetry data cursor OFF cirscr 0 printf n nPress any key to terminate test wherexy amp col amp row initialize all baseboard pins for output control signal for laser dig_dir DIO_DIR_OUTPUT DIO_DIR_INPUT DIO DIR INPUT DIO DIR INPUT write dig 0 write dig bit RESET 1 lactive low write dig bit IOE 0 Hactive low Loop doing writes to Digital 1 0 while kbd_hit 1 write dig bit A0
82. ning laser system This Appendix describes how to align all optical components of the laser system in a correct manner Alignment Tools Penta prism alignment card alignment target with grid pattern 1 Penta prism reflects light at go degrees irrespective of the relative angle between the prism and the input beam An alienment card has an adjustable hole for the in put light to pass and can be used to see the All diigiiliitiit Cait Lid DIC LUV 101 uit inp Mt sin LO pp ass and can USC position of the reflected beam from the interferometer without blocking the input beam N 3 The alignment target is a piece of paper or cardboard with fine grid pattern used to track the target beam drift on the target mirror A 3 1 Fold Mirrors and Beam Splitters Mount all the optical components on their kinematic mounts so that each component s rocking plate rotates around the input bearn Mount the laser head and the first fold mirror Adjust the screw on the kinematic mount towards the input laser beam so that the beam is centered in the input aperture of the fold mirror Tighten the screw almost completely to avoid any further translations of the mount Now put the penta prism in the path of the beam to fold the beam Adjust the position of the prism so that the light hits the alignment target on Axis 2 see Figure A5 Now move the axis across its entire range and note the horizontal drift in the input beam Adjust the fold mirror until there
83. nts the magnetic path length In Appendix B the total reluctance Ar of the actuator is derived Distinction is made between the constant reluctance Rg of the actuator core and the variable reluctance Rx of the variable air gap z Rr Ro Rx 4 10 Substituting Rr in Equation 4 8 gives the total generated flux This flux passes through the cross section of actuator s center leg 51 see Figure B r so that Equation 4 7 equals Ur BS 4 11 Assuming no flux leakage occurs the same flux returns through the two outer legs with cross section Sa Ur 2B S2 4 12 18 The total tractive force on the T is the sum of forces at all three legs of the E The force in the center leg is defined as Fj the forces at the two outer legs as Fo they are equal due to symmetry Using Equation 4 6 the total force can be described by Fr F 2F is Big 4 13 2 Lg Ho This can be written as m Nu T A Fy AE 4 14 With a Si 293 25152 The numerical values for these parameters are A 6 543 10 r m Ro 857 8 A Wb N 258 H uo 4r 107 H m Hr 47000 E 19 4 2 Empirical model 4 2 1 Experimental set up Experimental data was gathered using the setup in which one arm has 1 rotational DOF see Figure 4 1 and Figure 5 1 In the notation of ro this arm is called a crossmember The T laminations are integrated in the crossmember as described in Chapter 2 At both sides of the arm E shaped la
84. nvolves assembly instrumentation and rejdesign of components In the limited time available this project focussed on several essential steps Hardware and software for the control of the linear motors have been set up This includes a laser system for position feedback which was redesigned debugged and aligned Once the linear motors were operational system identification was performed and a linear controller was designed and implemented The closed loop bandwidth is over 50 Hz while the steady state error is limited to 50 nanometer A 1 DOF setup was made in which an arm with 1 rotational DOF is actuated by one magnetic actuator A theoretical force model for the magnetic actuator was derived and experiments on the 1 DOF setup show that this model closely matches the empirical results Linear and nonlinear control strategies for the 1 DOF setup were simulated and implemented Several months were lost by delayed manufacturing of new amplifiers and by many flaws of the DSP that was used for implementation of the controller A linear controller was implemented with poor results Best results were obtained using a sliding mode controller in combination with feedback linearization of the system The closed loop system is stable for a large operating range and able to track a low frequency reference signal Steady state positioning error is well below 1 micron Due to limited availability of amplifiers a thorough analysis of control of the 1 DOF s
85. ometer For a good pickup signal from the straight through interferometer it is important to adjust pitch and yaw even though the main output beam is unaffected by yaw A very faint secondary spot is visible on the target plane right of the main beam The secondary beam is affected by yaw and pitch of the interferometer and is used for alignment following the same procedure previously described The secondary spot should not drift for the entire range of the motion of the axis A 3 3 Target Mirror Next step is alignment of the target mirror Remove the alignment target from the target mirror Adjust the optical table on which the laser head is mounted so that the input beam hits the target mirror on the left side at center height Adjust the mirror tilt and rotation so that output beam goes back into the aperture of the interferometer Axis 2 Axis 2 Target Target mirror mirror LEE peey Jese KO x Y Ne E Penta Fold old i mirror mirror Fold e prism mirror 1 50 beamsplitter pesuy Jose Figure A 5 Alignment of fold mirror Figure A 6 Alignment of beam splitter Now place the alignment card before the interferometer and adjust the position so that none of the input beam going to the interferometer is blocked An almost circular reflection should be visible on the alignment card to the right of the input hole when looking from target mirror A second refl
86. on m i Position Reference 15 Time s 25 Figure 6 4 ge position and reference 0 5 Figure 6 6 1 5 Time s Position Reference 15 Time 5 2 25 yr position and reference z position and reference Position m 5 Time fs i 2 2 5 Figure 6 7 0 rotation and reference Rotation Reference 3 i Position m X 0 57 Rotation Rotation Reference i Reference 0 5 i i 0 5 1 5 Time sl 2 5 Figure 6 8 0 rotation and reference 15 Time s Figure 6 9 0 rotat 54 inn a 1041 2 2 5 and reference 2r rror m E I EN T 6r 8 Error m ans 4 i i i E 7 6 3 8 i 3 i i i A 40 i i a i i 0 5 1 1 5 2 25 3 o 0 5 2 25 3 Time s Figure 6 10 x7 error 15 Time s Figure 6 11 yr error Error degrees 15 Time s Figure 6 12 z error 5 ie x10 0 5 1 5 Time s Figure 6 13 0 error 1 57 Error degrees nm 2 D 0 5 e der 0 5 de Peng i E i i 8 2 gt i g
87. operation amp tech nical manual V1 3a Pittsburgh PA USA November 17 2000 4 AEROTECH Inc U channel linear motors user s manual V1 8 Pittsburgh PA USA May 29 2002 5 AEROTECH Inc Personal communication mr Ratin and mr Fry December 2003 May 2004 6 Boeij J de Electrodynamic magnetic levitation NASA s vision for the future Traineeship re port Eindhoven University of Technology CST 2003 55 2003 7 Charara A De Miras J and Caron B Nonlinear control of a magnetic levitation system without premagnetization IEEE transactions on CST vol 4 no 5 September 1996 8 DeCarlo R A Zak S H and Drakunov S V Variable structure sliding mode controller design from The control handbook page 941 951 edited by W S Levine CRC Press 1996 9 Dehmelt Klaus Personal communication September 2003 September 2004 10 Fevre L J P Design and construction of a 6 DOF positioning system with nanometer accuracy M Sc Thesis Florida Institute of Technology December 2003 11 Gutierrez H M and Fevre L J P Design and construction of a 6 DOF positioning system with long range motion in XY and nanometer resolution usign magnetic servo levitation presented at MIT April 2004 12 Gutierrez H M and Ro P I Magnetic servo levitation by sliding mode control of non affine systems with algebraic input invertibility 13 Gutierrez H M and Ro P I Sliding mode control of a non
88. ormance 10 2 2 2 Moving units Each moving unit see Figure 3 1 consists of a pair of linear servo motors carrying a platform with a magnet box The platform floats on an air bearing while the linear motors move in a magnet track without making mechanical contact Theoretically the position resolution is only limited by the resolution of the measurement device 2 2 3 Magnet boxes Each electromagnetic actuator or magnet box see Figure 2 2 consists of two pairs of opposed copper coils that are each wound around the center leg of a stack of E shaped iron laminations I shaped laminations are embedded in each supporting arm of the tray see Figure 2 3 When current is passed through the coil this induces magnetic flux paths that form two loops through the outer legs of the E shaped laminations and the I shaped laminations see Figure 2 5 Each coil can on y exert attractive force to the I shaped laminations so two opposite coils are needed to control one DOF Consequently the vertical coil pairs control the DOF s Z 8 and 9 of the tray while the horizontal pairs control X Y and 0 Based on the magnetic laws governing the system the relations for induction and force were es tablished Magnetic flux BA A 2 1 4 2 Hrho pod Induction _ NG N L 257 2 2 HrkoA poA Magnetic force NIu so F Y uoA 2 Tog 0 2 3 With Hr Relative permeability of metallic core uo Permeability of ai
89. ormation is described that cancels nonlinearities so that the system dynamics are reduced to a linear form double integrator Goal of the feedback linearization is to obtain the simple relationship t T2 za 5 15 In this equation v represents the angular acceleration to be applied to the crossmember From Equa tion 5 6 it follows that K 12 yKa 13 mgl y ME o CA cos z 8 16 J z gi z1 J z 92 x1 Jaz The following transformation linearizes the system in terms of the new input v h g r gee and LO for Jesu mgl gt 0 5 17 T go 11 gl esn and h 0 for J gv mgl lt 0 Note that using this transformation at any instant current is flowing in only one of the two coils Desired resultant forces can also be achieved with two actuators active at the same time but the above result is simple and most efficient considering energy consumption 5 3 2 Sliding mode control of feedback linearized system The system including feedback linearization can be written in the standard state space formulation t Az Bu 61 di 012 0 bi z 5 th 9 0 0 5 18 la fe Al T ba HO 8 With x e R the u e R and rank B m Now the desired state vector is defined as 0 2 D 5 19 ca do 5 19 Assume that the desired state can be realized with a bounded input signal so that for t gt 0 a Ara Bua 5 20 The tracking error is defined as e 6 64 so Ae Bl u ug
90. ort Eindhoven University of Technology CST 2003 120 2003 23 Woodson H H and J R Melcher Jr Electromechanical Dynamics Part 1 New York Wiley 1968 24 Zygo Corporation ZMI500 series Interferometer system with 501 measurement board Mid dlefield CT USA 2004 60 Appendix A Instrumentation for linear motors A 1 Pinouts This appendix gives pinouts of custom made connections red Phase B white Phase C green yellow Common L nc 3 ne ne 9 me ae Table A 1 Motor feedback Table A 2 Switches Table A 3 Motor power A PCB was designed to connect the parallel output of both enclosures to the DSP s baseboard dig ital I O and the DIG32 module s digital I O The data from enclosure 1 laser axis 1 and 2 is read using the baseboard DIO the data from enclosure 2 laser axis 3 and 4 is read using the DIG32 mod ule DIO Both enclosure outputs are 44 pins with identical pin out the two digital VO s are both 50 pins and only have different grounds The PCB is used to route the signals from the enclosure output to the right pins of the DIO s The pin out of connectors on the PCB is as follows Enclosure parallel output DSP Digital I O DIG32 module Table A 4 Pin outs w r t parallel laser data 61 A 2 Mounting of laser system o o o i D G ka I 2 o P o 1 Ei 24
91. pe of sensor depends on its capacitance A low capacitance provides a very accurate measurement but limits the operating range of the sensor so a trade off has to be made The sensors used in this project can be used in two sensitivity settings 50 um V over a 1 0 mm range or 5 im V over a 0 1 mm range The resolution is Go nm or 4 nm respectively Additionally 2 capacitive sensors are used in each magnet box to measure the gaps in vertical and horizontal direction These gaps are used in the computations for feedback control 2 2 7 Metrology mapping According to 10 the resolution of the position measurements can be improved by mapping of the sensor targets These targets the L shaped mirror for the laser system and a coated Zerodur surface for the three capacitive sensors can be mapped with an accuracy of up to 0 1 nm A lookup table should be used to compensate measurements for imperfections of the targets Two major problems arise using metrology mapping First of all the operating conditions must be equal to the mapping conditions This means that both surfaces have to be mounted using kine matic mounts and cannot be disassembled or changed once they are mapped The environmental conditions must be controlled accurately Second problem is that each surface map has a coordinate system but this is not linked to the mapped object the mapping is done with respect to an external reference frame A procedure for alignm
92. r H m N Number of coil windings I Coil current A A Cross section of the coil m Ga Nominal gap size m i Magnetic path length m The magnetic actuators are designed to provide forces high enough to keep the tray and arms in position while the linear motors provide maximum force This means the arms will not make contact with the magnetic actuators during maximum acceleration of the moving units The required force was computed for a worst case situation when the tray is as far off center as possible and the air gap is at its maximum Additional mass for example of an extra device to hold the work piece was taken into account too Based on this computation the magnetic actuators are designed to provide a force of at least 200 N The maximum coil current is limited to 3 0 A in order to avoid high heat dissipation in the coils Furthermore the inductance of the coils is kept to a minimum The electromagnetic actuators in vertical direction are equal to those in horizontal direction even while lower forces are required in vertical direction a passive magnetic suspension carries the weight of the stage The nominal air gaps in vertical direction see Figure 2 4 are 500 am making the vertical range of motion 1 mm in total Additional room is added to allow for the rotations 0 and 94 The nominal air gaps in horizontal direction are 250 um each and allow for small rotations 6 2 2 4 Tray and arms The tray is made as
93. rtical P S vi vdes pen s n ae dp mana roo Compute reference trajectory and 2nd derivative f ammmm cere nr mm mm tm mm mm SS SS m Reference rv r rvddot rddot hewn SS SS tm mm rnc tm mm VERTICAL Compute errors fame nt ms mm mm mm M x atan S2 50e 6 ys Rotation around x axis eold err gi gni yy atan x Upper gap g2 gn2 yy atan x Lower gap err X TV edot err eold freq eint 0 5 err eold freq Pf o a 1 o i A ea VERTICAL Sliding mode control Dl e nM edot labdaitlabda2 err labdaixlabda2teint Us gki 2 0 pi atan Ss phii 1 0 rvddot labdaltlabda2 tedot labdal labda2terr Us H 77 Tx m kg l i 0 Compensate gravity if Tx gt 0 1 I1 g1 1 0 70 0e 6 sgrt Tx 0 335 7 0e 6 12 0 7 else i Ti 03 12 g2ksqrt Tx 7 0e 5 fren nnn ne nnn nn ree ener nnn STISI SST NO NEGATIVE CURRENT COMMANDS SHOULD BE PASSED Al nnn nnn nnn nnn nnn nnn me nnn nnn a m if 11 lt 0 T1 033 di i 12 lt 0 12 03 P ISIS SS SIS SST SS SIS SS SS Update output channels if no key has been pressed anam EST SIS SS SS SIT TITISTE SST if stopcondition 0 1 Output 0 half low int GAIN_SCALE I1 OUT_SCALE Output O pin 48 Output 0 half high int GAIN_SCALE I2 OUT_SCALE Output 1 pin 46 Dutput 1 half low int GAIN_SCALE 0 0UT SCALE Dutput 2 pin 27 Output
94. s summarized in Table 2 1 Note I arc sec 4 8345 prad 2 2 Machine Design 2 2 1 General assembly An overview of the machine can be seen in Figure 2 1 The work piece is carried by a support surface tray This tray has a long range of motion in the horizontal plane XY 250 x 250 mm and can be accurately positioned in 6 degrees of freedom The four arms carrying the tray are actuated by a combination of linear motors on air bearings for the long range motion in X and Y direction and novel non contact electromagnetic actuators for small adjustments in all DOF s A passive magnetic suspension carries the weight of the tray DOF s X Y and 6 are measured using a laser interferometer systern A set of three capacitive sensors is used to measure Z 0 and 0 Similar capacitive sensors are used to measure the relative motion of the arms with respect to the magnetic actuators The maximum achievable accuracy of the machine is limited by the resolution of the measurement devices This resolution can be improved by using a mapping of the sensor target surfaces that will be used to compensate measurements for imperfections on these surfaces According to ro the machine can be rescaled to achieve larger travels without affecting the perfor mance Only the manufacturing and mapping of sensor target surfaces will be more time consuming and expensive Most machine components can be manufactured by a standard machine shop since machining toleranc
95. ssSetSampleTime S 0 CONTINUGUS_SAMPLE_TIME ssSetO0ffsetTime S 0 0 0 E ame PO AA PAS Initialize continuous states amp global real variables gg s His seas ass define MDL INITIALIZE CONDITIONS static void mdlinitializeConditions SimStruct S 81 real T x0 ssGetContStates S real T G ssGetRWork 5 real T LO1 L02 L03 L04 Initial arm lengths ml int_T i mat xolo 0 Translation motor 1 x 1 x0 11 0 Translation motor 2 y_1 x0 2 0 Translation motor 3 x_2 x013 0 ti Translation motor 4 y_2 x0 4 0 Translation tray z x0 5 0 Translation tray y x016 z_INIT Translation tray 2 x0 7 0 Rotation tray theta_x x0 8 0 Rotation tray theta_y x0 9 0 Rotation tray theta_z x0 10 0 fi Short stroke xS x0 11 0 Short stroke yS for i 0 1 lt 12 i i x0 12 i 0 Derivatives of state 0 11 3 Compute initial arm lenghts ignore short strokes L01 LO x0 51 LO2 LO x0 4 103 LO x0 5 LO4 LO x0 4 Compute initial gap sizes GEO gax x0 4 x0 0 LOl tan x0 91 Gli 2 gnx GLO GI gnz x0 61 LOi tan x0 7 1 GI3 gnz G 2 GI4 gax x0 5 xO 1 LO tan x0 9 GI5 2 gnx GI4 GI6 gaz x0 6 LO2 tan x0 8 GI7 gnz G 6 GI8 gox x0 4 x0f2 LO3wtan x0 9 G 9 2 gnx G 8 G 10 goz x0 6 LO3 ta
96. stappen die uiteindelijk moeten leiden tot het ontwerp en de implementatie van een MIMO regelsysteem voor het 6 vrijheidsgraden positioneer systeem Hardware en software voor de regeling van de lineaire motoren zijn opgezet Dit proces omvatte onder andere het herontwerpen en opzetten van een laser systeem voor positie feedback Zodra alle apparatuur operationeel was kon systeem identificatie en regeling van de lineaire motoren worden uitgevoerd Een lineaire regelaar werd ontworpen en ge mplementeerd De closed loop bandbreedte is ruim so Hz en de statische positie fout is maximaal so nm Een opstelling waarin een arm met 1 vrijheidsgraad rotatie wordt gedragen door een magnetische actuator werd gebruikt voor experimenten Een theoretisch kracht model voor de magnetische actua tor werd afgeleid en vervolgens met experimenten in deze opstelling geverifieerd De experimentele resultaten komen goed overeen met het theoretische model Lineaire en niet lineaire regel strategie n voor deze testopstelling zijn gesimuleerd en geimple menteerd Enkele maanden vertraging ontstonden bij de fabricage van nieuwe versterkers en van wege de vele gebreken van de DSP waarop de regelaar ge mplementeerd werd Een lineaire regeling werd ge mplementeerd met slecht resultaat Het beste resultaat tot nu toe werd behaald met een sliding mode regelaar in combinatie met een feedback linearisatie van het systeem Het closed loop systeem is stabiel over een groot d
97. stiff and light as possible to keep its natural frequencies high and to allow a competitive acceleration of the system The main stage consists of two plates sandwiching four center beams to which the arms are attached see Figure 2 6 To save material and weight the I shaped laminations are integrated in the arm structure Three stacks of these laminations are assembled using two aluminum C shaped channels see Figure 2 3 and Figure 2 4 Apart from obvious advantages like low density low price and easy machining the aluminum also has a relative magnetic permeability of 1 so it is not affected by the magnetic field in the magnetic boxes A kinematic mounting system will be used to attach a L shaped mirror under the tray see Fig ure 2 7 This mirror is used with the interferometer system to measure the DOF s X Y and 9 More about this in Subsection 2 2 6 Figure 2 6 Tray assembly top view 10 Figure 2 7 Tray assembly bottom view 10 2 2 5 Passive magnetic suspension A passive magnetic suspension provides a lifting force to the tray to compensate gravity Consequently no bias current through the upper coils of the magnetic actuators is needed to carry the tray A bias current leads to heat generation An array of equally oriented magnets generates a uniform magnetic field at the base of the ma chine covering the entire operating range of the tray An array of magnets in opposed orientation mounted to the bottom of t
98. t rddot jerk t Ti 202 r jerk t T1 t T1 t T1 6 a02 t T1 t T1 2 v02 t T1 202 V_INIT else if t gt T2 amp amp t lt Tend 1 rddot jerk t T2 203 r jerk t T2 x 1 12 t T2 6 a03 t T2 t T2 2 vO3 t T2 x03 V_INIT else if t gt Tend g t lt T02 gt i rddot 0 r V_INIT Amp I Step defined in V for more practical use LI now compute corresponding angle degrees rddot rddot 5500 0 50e 6 ys V s 2 to degrees s 2 r atan r 50e 6 ys LN to degrees J 19 C 3 6 DOF model S function define S_FUNCTION NAME longstrokeMIMO fdefine S FUNCTION LEVEL 2 E Visser June 2004 6 DOF model of tray Includes motion of linear motors include simstruc h include math h define define define define define define define define define define define define CONT_ST DISC_ST GLOBALINTS GLOBALREALS NIP IPHO IPW1 NOP OPWO OPW1 OPW2 OPW3 24 0 0 16 16 No of continous states 0 3 Translation linear motors 1 4 in m 4 9 Tray DOF s x y z in m theta_x theta y theta_z in deg 10 11 Short range translations xS and yS m 12 23 Derivatives of state 0 11 STATES ARE DEPENDENT x0 x2 xL xi x3 yL x4 x0 x10 xL xS x5 zi x11 yL y5 No of discrete states Ho of global integers No of global reals 0 Gap 11 m 1
99. t of linearized system Figure 5 3 Plant derived from sensitivity mea surement 28 5 2 3 Simulations A linear controller for the linearized system was designed with DIET and analyzed in simulations The simulated closed loop and sensitivity can be seen in Figure 5 4 and Figure 5 5 The bandwidth is approximately 55 Hz Sensitivity stays below 6 dB for all frequencies Closed loop Simulated 29 TT Te yr Mag dB 5 D 5 8 oj c a amp 200 A b SE 10 10 10 10 10 10 Frequency Hz Figure 5 4 Linear control closed loop simulation Sensitivity Mag dB o 3 o 8 Of ao bu Ka a sna 200 3 k gt Re RS ROMA 40 10 10 10 10 10 Frequency Hz Figure 5 5 Linear control sensitivity simulation 29 Since the linearized system equations are strictly valid for small variations around the equilibrium point it is interesting to see how the system behaves for large strokes Figure 5 6 shows the closed loop system can track a 5 Hz sinusoidal with amplitude 1 2 107 degrees which is roughly 80 of the whole vertical range ies 5 Hz sinusoidal T T T m T Reference A E Rotation R A NR mara ik I is pode J fer i i i i i Hi NSE AE I N SQ H E i y ish to by o 05H Yi if He if hi s ti if i oj E ae pa A O I i a
100. the arm sticks to the upper actuator Force measurements Force N Current A Or Air gap m Figure 4 2 Force measurements 4 2 3 Model fit In literature 21 7 the following model is used to describe the tractive force in this type of magnetic suspension Note that the structure is equal to 4 14 I F Kgx zo z With K a constant and xo an additional gap length to model the finite reluctance of the actuator s metal core A Matlab script see Appendix D 1 was written to fit K and zo by minimizing the error criterium i J E o Be reas la 7 c Emoa t J 1 1 With i and j the number of variations in current and position The best fit is achieved with the following values K To 7 5951 10 6 Nm A 73 1075 m 21 In Figure 4 3 the fitted model and measured data are plotted together For a good visualization the model is only plotted for those points where actual measurement were available Figure 4 4 shows the difference between measurements and fit The theoretical model gives a gain K of 6 3824 1079 IN m A which is close to the empirically determined gain The additional gap length found during measurements 73 micron is much larger than the theoretical value 1 4 micron The difference can be explained by the way the sensor was calibrated The zero level actually included an air gap caused by a protecting lip preventing the cross member from hitting actuator or sensor dur
101. to half of the maximum stroke for each DOF to make sure no mechanical contact occurs between the crossmembers and the magnet boxes Settings e Since the amplifiers for the magnetic actuators have limited bandwidth they are modelled as a first order low pass filter with a pole at 100 Hz This is a conservative estimate The actuators saturate at 3 0 A input current e Setpoints used are smooth steps which means acceleration is a linear function of time The tray has an initial z displacement of 6 35 107 m since it initially rests at the lower coils Step commands are given after 1 0 s so the tray can be lifted first and settle at z 0 m e Better results are achieved when the steps in x7 yr don t coincide with the step in 6 Reason is that with 6 o all gaps have nominal width which gives room for overshoot in z and y If the cross is simultaneously rotated half of the air gaps become smaller than the nominal distance gna leaving less room for overshoot So it seems logic first to perform the translations in x and y and to adjust the other DOF s after that Results The sliding mode controllers for the 6 DOF s of the tray have been tuned such that the closed loop system has short settling time and low residual error Gains were increased to improve performance until the system started to chatter and than reduced a little The step time for zr and yr can not be taken too short Stiffness of the sliding mode contro
102. troller in the MMI software This controller could used parallel data instead of the serial data used now At this moment such a solution is not possible since the available DSP has 16 analogue output channels that are all needed to control the magnetic actuators 16 Chapter 4 Force model of magnetic actuator The magnetic actuator has been described previously in Chapter 2 In this chapter a theoretical force model will be derived for the actuator The model has been verified with measurements on an experi mental setup This model will be used for nonlinear control of a rotating arm actuated by one magnetic actuator and in the 6 DOF model of the total system In the electromagnetic theory in this chapter the following definitions are used B Flux density Wb m F Force N H Magnetic field intensity A m 1 Current A N Number of coil windings H R Reluctance A Wb Wm Energy density U m5 Wm Energy pi pu Permeability of a material u pro H m Ho Permeability of free space 47 1077 H m pr Relative permeability of a material Fi y Flux Wb 4 1 Theoretical model 4 1 1 Assumptions The magnetic actuators in this machine behave like tractive type magnetic suspensions A force model is derived using the electromagnetical theory as presented in 23 and especially 18 The following assumptions are made e There are two closed flux paths see Figure 2 5 Flux leakage is assumed negligible
103. ve 1 translational DOF each z and zz for motors 1 and 3 and y and Ya for motors 2 and 4 Their air bearings provide infinite stiffness in z direction The motors behave as a pure mass Height and thickness of the arms are neglected in the computation of gaps moments and forces There are two types of actuators in the system linear motors and magnetic actuators Forces generated by the linear motors are described by Fm kiV 6 1 with k the motor constant and V the torque command in V Index i 1 4 gives the number of the linear motor The forces generated by the magnetic actuators are modelled as kac T AE 6 2 go Jac with parameter values as derived in Chapter 4 Here I is the current input A k is a force constant Y m and g is the air gap m The magnet box on each axis index a 1 4 consists A of 4 coils index c 1 4 See Figure 6 2 for definitions ac 43 F32 1 F31 4F21 i F22 F21 F12 Fil i xl y Fml AA z x S Fi2 Fil Figure 6 1 Free body diagram Fala Fm4 F42 F41 E ni 1 41 ie cos e gt s22 12 J LA Tee 122 Al Li I i y i 13 I I I 1 I g13 z uz 74 i Coil 12 ne Amaja pf Col gl2 EH f I NE gil
104. ware Kpos 10K 10K mag o P 7668358668 s388608 V count pos 10K F l 4096 8388668 V s count Kp D Ke s V count This allows linear control design using DIET after which the correct settings for Kpos Kp and K can be retrieved and implemented Normally setpoint and position are measured in m in which case these gains have to be divided by the laser resolution m count The MM I software has an option to automatically tune the controller gains but manual tuning using DIET leads to better results A second order filter is preprogrammed as well which can be used either as a notch filter or as a low pass filter When used as a notch the resonance around 150 Hz can be reduced at the cost of some phase loss leading to lower bandwidth It is preferred to implement a low pass filter to filter out high frequency noise This filter has a double pole at 300 Hz Implementation of the feed forward term led to strange results Consulting Aerotech Inc 5 learned that the feed forward implementation in the MMI software contains errors and that feed forward should not be used until the next software upgrade becomes available Consequently the results shown here were obtained without feed forward Af zo 3 6 Real time results Figure 3 6 shows that the closed loop bandwidth 45 degrees phase lies between 50 and 55 Hz The sensitivity Figure 3 7 stays below 6 dB which is a requirement for good disturbance rejection
105. ys fdefine gn2 5 50e 4 Im 9 00 50e 6 yy ys define m 1 694 1 kg define g 9 81 ti m s 2 define 1 0 2456 Um define H 6 67 1 Jxx Hdefine pi 3 14159265358979 Prototypes pragma CODE SECTION analog isr onchip pragma INTERRUPT analog_isr void analog_isr pragma CODE SECTION DoServoCalc onchip void DoServoCalc pragma CODE SECTION Initialize DSP offchip void Initialize DSP pragma CODE SECTION Reference onchip void Reference pragma CODE SECTION active pairs out offchip int active pairs out pragma CODE SECTION active pairsin offchip int active pairs in General near volatile AIO PAIR Input 8 near volatile AIO PAIR Output 8 near volatile AID PAIR ZeroOut 8 unsigned int xrpt unsigned int xrpt_source near int ActivePairsOut near int ActivePairsIn near int module 1 Servo16 module on site 1 of M67 baseboard near int stopcondition 0 Emergency stop near double freq 1 ts Sample freq controller Parameters for SMC near double labda 70 0 near double phi 15e 3 near double gk 4 0 near double labdai 125 0 near double labda2 80 0 near double phil 50 0e 3 near double gki 7 0 near double Ss 0 Sliding surface near double Us 0 Switching term 74 near double Tx 0 H Torque around x axis Error and reference near double x eold err edot eint 0 near double r rv
106. ystem and abandon the principle of metrology mapping These principles were not within the scope of this project but definitely require attention From a control point of view a lot of interesting work has to be done Control of the magnetic actuators is far from perfect Due to delayed delivery of equipment this could not be investigated thoroughly yet Now the hardware is all operational this step can be performed Another important issue is the integration of the controllers for linear motors and magnetic actua tors At this moment control for the linear motors runs on a CNC card while the magnetic actuators are controlled with a DSP Real time communication between a host program and the CNC card has been established but communication between the host program and DSP is not possible yet This is an absolute requirement for successful implementation of one control system that controls both types of actuators If these steps are performed a control system for the 6 DOF system can be implemented Only at that moment performance of the complete 6 DOF system can be measured in terms of accuracy and speed From there performance should be improved iteratively 58 Bibliography 1 AEROTECH Inc BAL linear amplifier series users manual V1 5 Pittsburgh PA USA May 23 2002 2 AEROTECH Inc BBso1 interface board manual V1 5 Pittsburgh PA USA May 2 2001 3 AEROTECH Inc The UNIDEX 500 motion controller and windows software

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