J. Kissel

I use data from recent aLOGs LHO:60446 and LHO:60447 to calibrate the standard HXDS sliders in YAW, and compare that against the standard method, which is to use the from-first-principles suspension dynamical model, and dead-reckoned model values for electronics components of the actuator chain.

The Answer:
       Unit       Value       Abs. Unc.  (Rel. Unc)     Source  
YAW    [urad/ct]  7.856e-1    5.5091e-2  (7.0%)         Weighted Mean of (ZM5 to ZM6) measurement, (ZM4 to ZM5) measurement, and first-principles model

Not the uncertainty is quite small, mostly because of the precision of the ZM4 to ZM5 results that Betsy quotes of "13-15 [mm] of motion" over the nominal ZM4 to ZM5 distance lever arm of 1.6812 [m].
Let's get these kind of measurements all the time! 

For Pitch, I don't have a displacement over lever measurement (yet??), so we must rely on the from-first-principles suspension dynamical model, and dead-reckoned model values for electronics components of the actuator chain like we've used for many other suspension types -- which -- the above data proves is quite imprecise.

       Unit       Value       Abs. Unc.  (Rel. Unc)     Source     
PITCH  [urad/ct]  4.498e-01   4.498e-01  (100.0%)       first principles model alone

The Thought Process & Calculation:

Using ZM5 to ZM6 distance as YAW calibration.
                Unit       Value       Abs. Unc.  (Rel. Unc)   Source          
arc             [m]        0.02000     0.00500    (25%)        LHO:60446, guess at uncertainty of +/-5 mm.

lever           [m]        4.6900      0.07620    (1.6%)       D1900436, 184.64 [inches] in [meters], rounded to nearest [mm], 
                                                               wild guess at +/- 3 [inch] = 7.62 [cm] uncertainty between HR surface quoted 
                                                               in D1900436 and where surface of "ZM6 baffle" lie.

angle span      [rad]      4.265e-3    1.069e-03  (25%)        angle value = (arc value / lever value) and (+/-) abs. unc = (angle value) * sqrt((arc unc / arc value)^2 + (lever unc / lever value)^2).
   
slider span     [ct]       4700.0      0.00000                 LHO:60446, no uncertainty.


calibration     [rad/ct]   9.075e-07   2.274e-07  (25%)        calibration value = (angle / slider span) and (+/-) abs. unc = (calibration value) * (angle rel. uncertainty).

YAW calib.      [urad/ct]  9.075e-01   2.274e-01  (25%)     [rad/ct] value * 1e6 [urad/rad].


Using ZM4 to ZM5 distance as YAW calibration.
                Unit       Value       Abs. Unc.  (Rel. Unc)   Source     
arc             [m]        0.01400     0.00100    (7.1%)       LHO:60447, "13-15 mm of motion"

lever           [m]        1.6812      0.02540    (1.5%)       D1900436, 66.19 [inches] in [meters], rounded to the nearest [mm],
                                                               wild guess at 1 [inch] uncertainty between HR surface to HR surface, 
                                                               from design to in-situ value.

angle span      [rad]      8.328e-3    6.081e-04  (7.3%)       angle span value = (arc value / lever value) and (+/-) abs. unc = (angle value) * sqrt((arc unc / arc value)^2 + (lever unc / lever value)^2).

slider span     [ct]       10600.0     0.00000                 LHO:60447, no uncertainty.

calibration     [rad/ct]   7.857e-07   5.736e-08  (7.3%)       calibration value = (angle / slider span) and (+/-) abs. unc = (calibration value) * (angle rel. uncertainty).

YAW calib.      [urad/ct]  7.857e-01   5.736e-02  (7.3%)    [rad/ct] value * 1e6 [urad/rad].

Using "From first Principles" (boldly claiming no uncertainty) knowledge of the M1 actuation chain (but checks out against modeled vs. measured M1 drive to M1 response transfer functions [which should be equivalent to the M1 drive to M2 response at low frequency] within a factor of 2.0):
                Unit       Value       Abs. Unc.  (Rel. Unc)   Source     
slider span     [ct]       10600.0     0.00000                 LHO:60447, no uncertainty.

   EULER2OSEM   [N/(N.m)]  (cancels w/ nActs & leverarm)       

   DAC gain     [V/ct]     20/2^16                             "This is known" Top, M1 masses of HXDSs, like HTTSs, use 16-bit DACs, with 20 V peak-to-peak range.

   Coil Driver  
   Transconduct.[A/V]      0.988e-3                            T1200264 HAM-A Driver design study, with a BOSEM coil.

   nActs        []         (cancels w/ EUL2OSEM)

   Magnet 
   Strength     [N/A]      0.963                               T1000164 "fmax" thoery, from Table 2 for 10 mm Diam x 5 mm Thick NdFeB magnet

   lever arm    [(N.m)/M]  (cancels w/ EUL2OSEM)

   YAW
   M1 drive to 
   M2 optic DC 
    compliance  [rad/N.m]  1.382                               hxdsopt_doublep.m parameters using ssmake2MBf.m model

angle span      [rad]      4.2535e-03  4.2535e-03 (100%)       angle span value = (slider span) * (DAC gain) * (CD TC) * (manget strength) * (Yaw M1 to M2 compliance), with uncertainty = +/- value

calibration     [rad/ct]   4.0127e-07  4.0127e-07 (100%)       calibration value = (angle span) / (slider span)
              
YAW calib.      [urad/ct]  4.0127e-01  4.0127e-01 (100%)    [rad/ct] value * 1e6 [urad/rad].

From which, we can derive the weighted mean, and weighted standard error on the mean, (in the Standard, 1960's, frequentist, blinding assuming bounds are gaussian sigma without correcting, low-number statistics don't matter, and assuming no systematics, manor -- i.e. really crude uncertainty combination that is certainly is drastically underestimating or misrepresenting the uncertainty),

                Unit       Value       Abs. Unc.  (Rel. Unc)   Source     
YAW
calib.
weighted mean   [urad/ct]  7.856e-1    5.5091e-2 (7.0%)      (weighted mean) = sum(x./sigma.^2) / sum(1./sigma.^2), (standard error on the mean) = sqrt( 1 ./ sum(sigma.^(-2)))
where "x" and "sigma" are the [urad/ct] value and absolute uncertainty from each of the three methods, respectively. 


Just in case we don't get the opportunity to measure similar values for the in-situ pitch slider, we can still repeat the (wildly imprecise, as shown indicated above) first principle's calculation from above:
    (DAC gain) * (CD TC) * (manget strength) * (Pitch M1 to M2 compliance) * ([um/m]) 
    = (20/2^16) * (0.988e-3) * (0.963) * (1.549) * (1e6) 
    = 4.4976e-01
where the pitch compliance comes from the same SUS matlab model as in Yaw, and again, comparing M1 to M1 transfer function measurements against this model a low frequency reveal that this is good to about a factor of 2.


                Unit       Value       Abs. Unc.  (Rel. Unc)   Source     
PITCH           [urad/ct]  4.498e-01   4.498e-01  (100.0%)     the above quick calculation above, only augmented by the M1 to M2 Pitch compliance, 
                                                               rather than the yaw compliance (thanks to the already-installed EUL2OSEM matrix 
                                                               which accounts for the lever arm difference).