Gemini. The errors from the servo system are considered as the superposition of three things:

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1 Gemini Mount Control Sytem Report Prediction Of Servo Error Uing Simulink Model Gemini 9 July 1996 MCSJDW (Iue 3) - Decribe the proce of etimating the performance of the main axi ervo uing the non-linear Simulink model a developed by Mike Burn. Thee error are then compared with the error pecification a agreed with the IGPO. 1. INTRODUCTION The error from the ervo ytem are conidered a the uperpoition of three thing: i. Linear Servo Error, E. Error due to the bandwidth (or lack of it) of the ervo. Thi can be found by performing a frequency repone of ervo error with repect to input. ii. Encoder Error, E e. The difference between the actual poition and the poition indicated by the encoder. Thi ha been found by experimentation - ee MCSJDW11. iii. Telecope Error, E t. Error that i caued by non-linear effect that are outide the bandwidth of the telecope. Thi i found by performing an FFT upon the ervo error ignal from a non-linear imulation. Figure 1 how where thee error occur within the telecope ervo loop. E Et CONTROLLER TELESCOPE ENCODER Figure 1 - Location Of The Variou Error Signal To predict the error from the telecope the reduced verion of Mike Burn non-linear model ha been ued. Thi conit of two model, one for each axi. See Appendix C of the MCS CSDD for more detail. Note that thee model do not include the effect of the telecope pier - thi allow the elevation axi model to achieve a much larger bandwidth than i poible in reality. A linear verion of each model, which had all the non-linear effect removed, wa alo ued. A model of the PMAC card ha been ued to act a the poition controller. Thi i hown in Figure. Ee TN-C-G4 1 of

2 T.1 T.1 AA/^7 VA/^7 1 Po. Dem. - Po. Error I/^3 P/^19 n() d() 1/376.8 DAC 1 Volt Out Po. F/B T.1 D/^7 - Figure - Linear, Continuou Time PMAC Model. AZIMUTH AXIS The controller wa optimied to give a large bandwidth. The following PMAC parameter, with reference to Figure, were choen: P = I = 64 D = 3 VA = AA =.448 Thee give a bandwidth of about 6.3 rad/ec (1 Hz)..1 Linear Servo Error Taken from the linearied model uing the linmod and bode function in MATLAB. The repone i a hown in Figure 3. 3 rad/ec Gain (db) Slope = -4 db/decade Frequency (Rad/Sec) Figure 3 - Azimuth Linear Error Bandwidth of TN-C-G4

3 Figure 3 how that the linear ervo error i dependant upon frequency. In fact, if we approximate the error bandwidth to a cut-off frequency (BW in rad/ec) and a lope (Sl in db/decade), for a given inuoid (P eq in W eq t), the error will be given by: Error = P eq BW Sl log W eq Thi give the azimuth axi error, the error upon the ky i modified by a projection factor which i the coine of elevation (E), thu ky error i given by: SkyError = Error co E The parameter of the inuoid (P eq & W eq ) will be dependant upon the azimuth velocity and acceleration, thu: P eq = W eq = V A A V For a given elevation, the value of V max and A max can be found uing the equation a decribed by Pat Wallace in Geometrical Tranformation for the Gemini Telecope (TCS/PTW/1.5). Uing all thi, we can calculate the ky error for tellar tracking over a range of elevation. Figure 4 how the reult of doing thi for latitude = 19.8 (Mauna Kea), BW = 3 rad/ec and Sl = -4 db/decade (See Figure 3). max max max max, 3.5 Azimuth Sky Error (milli-arcecond) Elevation (Degree) Figure 4 - Sky Error Due To Azimuth Linear Servo Error From thi, we can ee that the maximum error ( 3.3 milli-arcecond) occur at approximately 88.9 elevation. The W eq correponding to thi i 1.4x -3 rad/ec - o mall it can be conidered a DC error. TN-C-G4 3 of

4 Note that the equivalent to thi maximum error on the azimuth axi i 17 milli-arcecond. Thi i very large, but the error only occur at uch magnitude at a very few value of elevation, at mot elevation (ay 8 and below) the azimuth error i le than 3 milli-arcecond.. Encoder Error From encoder tet carried out in the lab, ee MCSJDW11, the encoder error i expected to be around milli-arcecond RMS at a velocity dependant frequency. The profile of thi diturbance then, i given by the cloed loop ervo bandwidth, i.e. at frequencie of 1 Hz and lower the error will be milliarcecond. At higher frequencie the error will be attenuated according to the roll-off of the frequency repone. 5 Wore Cae Error (milli-arcecond) Frequency (Rad/Sec) Figure 5 - Expected Encoder Error With Repect To Frequency Note the naty pike in the error at 41 and 97 rad/ec (6.5 and 15.4 Hz repectively), which will be due to tructural reonance. The maximum encoder error would occur when the axi velocity caued the error frequency to be 1.6 rad/ec (.55 Hz). The velocity at which thi occur will depend upon which tape encoder i choen (.44 arcecond/econd for Heidenhain and.95 arcminute/econd for Inductoyn). The amplitude of thi error i 1.7 milli-arcecond..3 Telecope Error The non-linear model wa imulated for econd with the variou input lited below. An FFT wa then performed upon the ervo error ignal. Only the lat ten econd worth of data wa ued for the FFT to ignore the ettling part of the repone. The input ued were: 1 Encoder Count (E.C.) Step 6 E.C. Step (3 arc) 7 E.C. Step (1 ) 3 E.C./ec Ramp (15 arcec/ec) 36 E.C./ec Ramp (.5 /ec) There wa alway an error at 6.6 Hz, though at different amplitude depending on the input. An error at 9.3 Hz appeared in the 36 E.C./ec ramp cae. An error at 15.1 Hz appeared in the 1 E.C. tep cae. The ummary of the wort cae reult i: 6.6 Hz 13 E.C. (65 milli-arcecond) peak. 4 of TN-C-G4

5 9.3 Hz 5 E.C. (5 milli-arcecond) peak Hz 3 E.C. (15 milli-arcecond) peak. The 6.6 and 15.1 Hz error will be due to excitation of tructural reonance. The 9.3 Hz error, doe not correpond to a tructural reonance o mut be due to ome non-linear proce..4 Total Error To uper impoe the three type of error dicued above, the wore cae frequency component mut be combined in a root um quare (RSS). Thi i hown in Table 1. Frequency (Hz) RMS Amplitude (milli-arcecond) Caue D.C. 13* Linear Servo Error.55 Hz 1.7 Encoder Error 6.6 Hz Hz 17.7 Telecope Error 15.1 Hz.6 TOTAL (RSS) 55 / 134* Table 1 - Wore Cae Azimuth Error, Without Tip/Tilt Correction * Note that it i unfair to ue the large value for linear ervo error ince it only occur at a mall number of elevation, at which the azimuth error contribute little to the overall ky error anyway. At mot elevation, the azimuth linear ervo error i negligible..5 Effect Of Tip/Tilt Correction The effect of the active tip/tilt econdary loop can be modelled by a econd order ervo loop, with a bandwidth of 5 Hz and a damping factor of.77. The tranfer function of uch a loop i: G( ) = Thi loop ue an actual tar for poition feedback, therefore all the error dicued previouly will be act a diturbance to the tip/tilt loop. The diturbance rejection repone of the tip/tilt loop i given by the error tranfer function: E( ) = 1 G( ) = 4649 The frequency repone of thi tranfer function i the diturbance rejection repone and it i hown in Figure 6. The effect of the azimuth encoder error and telecope error after tip/tilt correction hown in Figure 7. The effect of the linear ervo error, becaue of it low frequency, i negligible after tip/tilt correction. TN-C-G4 5 of

6 Gain (db) Frequency (Rad/Sec) Figure 6 - Diturbance Rejection Repone Of Secondary Tip/Tilt Loop 7 Encoder Error Telecope Error 6 Error (milli-arcecond) Error (milli-arcecond) Frequency (Rad/Sec) Frequency (Rad/Sec) Figure 7 - Effect Of Tip/Tilt Upon Azimuth Error Note that the maximum encoder error now occur at 6.6 Hz. Thi correpond to an axi velocity of arcecond/econd for the Heidenhain tape or 4.73 arcminute/econd for the Inductoyn tape. Table how the combined effect of all azimuth error after tip/tilt correction. Frequency (Hz) RMS Amplitude (milli-arcecond) Caue 6.6 Hz 6.9 Encoder Error 6.6 Hz Hz 9.6 Telecope Error 15.1 Hz 9.6 TOTAL (RSS) 3 Table - Wore Cae Azimuth Error, After Tip/Tilt Correction 3. ELEVATION AXIS The controller wa optimied to give a bandwidth of about 1 Hz. The following PMAC parameter, with reference to Figure, were choen: P = 9 6 of TN-C-G4

7 I = 76 D = 3 VA = AA = Thee give a bandwidth of about 6 rad/ec ( 1 Hz). 3.1 Linear Servo Error Taken from the linearied model uing the linmod and bode function in MATLAB. The repone i a hown in Figure 8..3 rad/ec - Gain (db) - -3 Slope = -4 db/decade Frequency (Rad/Sec) Figure 8 - Elevation Linear Error Bandwidth Unlike azimuth, there i no ky projection factor to be taken into account for elevation. The maximum ky error, i therefore equal to the maximum axi error. Uing Pat Wallace equation, it can be hown that the maximum velocity and acceleration will occur at the cloet zenith point (zenith ditance =.5 ). The value for Mauna Kea are: V max = 3.9x -3 /ec A max = 3.1x -5 /ec Uing the equation derived above for the azimuth axi, thi value give a maximum equivalent inuoid.5 in (8x -3 )t. Uing BW =.3 rad/ec and Sl = -4 db/decade (See Figure 8), we get a maximum error of milli-arcecond at a frequency of 1.3 mhz (effectively D.C.). 3. Encoder Error A with the Azimuth axi, the error due to the encoder i expected to be around milli-arcecond RMS at a velocity dependant frequency or frequencie. Figure 9 how thi applied to the elevation cloed loop frequency repone. TN-C-G4 7 of

8 5 Wore Cae Error (milli-arcecond) Frequency (Rad/Sec) Figure 9 - Expected Encoder Error With Repect To Frequency Note the abence of any enitive reonance uch a thoe een with the azimuth axi. Thi i probably becaue the natural mode of the telecope tube are at higher frequencie than thoe of the azimuth axi and are completely damped out by the ervo bandwidth. The maximum encoder error would occur when the axi velocity caued the error frequency to be 1.5 rad/ec (.4 Hz). Thi correpond to an axi velocity of.5 arcecond/econd for Heidenhain and 1.6 arcecond/econd for Inductoyn). The amplitude of thi error i.9 milli-arcecond. 3.3 Telecope Error The ame tet were applied to the elevation axi a the azimuth axi. There were no ignificant error and again, thi i put down to the high (a compared to ervo bandwidth) frequency mode of the elevation axi. 3.4 Total Error A in Section.4, to uper impoe the error dicued above, the wore cae frequency component mut be combined in a root um quare (RSS). Thi i hown in Table 3. Frequency (Hz) RMS Amplitude (milli-arcecond) Caue D.C Linear Servo Error.4 Hz.9 Encoder Error TOTAL (RSS) 7.7 Table 3 - Wore Cae Elevation Error, Without Tip/Tilt Correction 3.5 Effect Of Tip/Tilt Correction The elevation error, can be applied to the diturbance rejection repone a hown in Figure 6. The effect of the encoder error after tip/tilt correction i hown in Figure. The effect of the linear ervo error, becaue of it low frequency, i negligible after tip/tilt correction. 8 of TN-C-G4

9 .5 Error (milli-arcecond) Frequency (Rad/Sec) Figure - Effect Of Tip/Tilt Upon Elevation Encoder Error It can be een that, after tip/tilt, the combined effect of all elevation error i negligible, i.e. le than.5 milli-arcecond RMS. 4. SERVO SPECIFICATION The error budget agreed with the IGPO i milli-arcecond RMS, on axi, at zenith. Thi increae with zenith ditance, Z, according to the following equation: E =. 1 in Where E i the permitted error, on axi, in arcecond. ( Z) The on axi error i calculated from individual axi error by uing the following equation: Where: E T i total on axi error. E A i total azimuth error. E E i total elevation error. [ in ( )] T E A E = E E Z Table 4 ummarie the reult of the previou ection and how maximum RMS error expected from each axi. RMS milli-arcecond Without Tip/Tilt With Tip/Tilt Azimuth 55 3 Elevation Table 4 - Summary Of Error Prediction Figure 11 how how the predicted on axi error, with and without tip/tilt, compare to the current ervo pecification. TN-C-G4 9 of

10 7 6 Error Without Tip/Tilt 5 Milli-arcecond 4 3 Error Budget Error With Tip/Tilt Zenith Ditance (Degree) Figure 11 - On Axi Error For The Range Of Zenith Ditance of TN-C-G4