[mmaimcal]Re: [Almasci] Antenna Acceleration
Mark Holdaway
mholdawa at nrao.edu
Tue Nov 18 11:43:05 EST 2003
Richard,
As a linux-residing person, I am Spreadsheet Illiterate.
However, I am sending you an ASCII file that has an *approximate*
profile for a turnaround from the maximum JERK method, at .48 msec
resolution (ie, 100 samples in 48 msec). Looking at this, it is
apparent that the acceleration profile isn't exactly parabolic
(that would have a jerk with a constant slope), so I did something
more elaporate. (Still, the desired motion is approximately cubic
on timescales of 48 msec.)
The turnaround from +1 deg/s (ie, 0.5deg/s on the sky in AZ, at 60 deg
elevation) to -1 deg/s takes just over 0.3 seconds to achieve -- I don't
have any clue about how realistic this would be for the ALMA antennas,
given that I heard a rumer that it took the Vertex antenna longer to turn
around.
I am thinking that it would not be difficult for you to repeat your
spreadsheet analysis on this motion profile. If it IS difficult,
maybe it is time for me to start running spreadsheets.
Take care,
-Mark
> "The commanded position and rate apply at the second timing event after the
> command is received. The ALMA computer will guarantee that the command is
> available no later than 10 ms in advance of the next timing event. If no
> position command is received between two timing events, the position and rate
> applicable at the next timing event shall be derived by constant velocity
> extrapolation of the most recently received command. In this way, there is a
> commanded position and rate that applies at each timing event. At any time
> between timing events, the commanded trajectory shall be determined by a fixed
> interpolation rule chosen by the Contractor to produce smooth motion, taking
> into account the response of the servo. The position and velocity command is
> the primary control interface.
>
> The servo shall attempt to make the actual trajectory of the antenna follow
> the commanded trajectory. If the latter contains first or second derivatives
> exceeding the maximum angular velocity or acceleration (see 3.4.4), the servo
> shall drive the antenna so as to converge to the commanded trajectory as
> quickly as possible while meeting the settling time requirements (3.4.4)."
>
> I interpret this as saying that during each 48msec interval the ACU has 4
> numbers to play with - the postion and velocity at the beginning of the
> interval and the position and velocity at the end. Although the contractors
> are free to do as they choose, it seems to me that the only sensible thing to
> do with these is to fit a cubic, e.g. x = a + b*t + c*t^2 + d*t^3. Here d is
> the jerk that Mark talked about. So long as the contractor does this, and the
> motion that is demanded can also be described by a cubic, then the
> interpolated trajectory should be exactly that demanded.
>
> If the higher level software is perverse and tries to change things part way
> through an interval (e.g. by reversing the jerk) then there will be an error.
> It is clear however that any such error will be small. Remember that the rate
> of the commands is one per 48msec, i.e. 20.8Hz, so that the position data is
> fully sampled up to ~10Hz, whic is probably more than the servo bandwidth, and
> we are giving it the velocity information as well.
>
> I have done a little spreedsheet (attached) to make sure about this. The plot
> shows the required position and the interpolated position (using a cubic to
> interpolate the position, given the position and velocity at the beginning and
> end of each 48msec interval). This is an extreme case with the jerk going
> from +100 to -100 deg/sec^3. The initial velocity and acceleration are chosen
> to keep the excursions small.
>
> For the first two intervals the jerk is fixed for the duration of the
> intervals and only swithces at the boundary. It is seen that the
> interpolation is good (the error that it is just visible is probably due to
> the crude numberical integration).
>
> For the second interval the jerk is changed and the mid-point, and one does
> see some error, but it is not significant, espcially as this would presumably
> not be a moment when one was expecting the full accuracy. I imagine that the
> limited servo bandwidth and the inertial forces in the structure would cause
> much greater errors under these circumstances.
>
> I hope this helps
> Best Richard
-------------- next part --------------
# time[s] x[deg] v[deg/s] acc[deg/s^2] jerk[deg/s^3]
#
0.00000 0.500119633 0.999992573 -0.0599627933 -99.9961872
0.00048 0.500599629 0.999963791 -0.107960963 -99.9910061
0.00096 0.501079612 0.99991197 -0.155956646 -99.9835181
0.00144 0.501559569 0.999837111 -0.203948735 -99.9737292
0.00192 0.502039491 0.999739215 -0.251936125 -99.9616373
0.0024 0.502519366 0.999618286 -0.299917711 -99.9472405
0.00288 0.502999183 0.999474325 -0.347892386 -99.9305408
0.00336 0.503478931 0.999307337 -0.395859046 -99.9115432
0.00384 0.503958598 0.999117325 -0.443816586 -99.8902356
0.00432 0.504438174 0.998904293 -0.4917639 -99.8666321
0.0048 0.504917648 0.998668246 -0.539699883 -99.8407297
0.00528 0.505397009 0.99840919 -0.587623433 -99.8125204
0.00576 0.505876246 0.998127131 -0.635533443 -99.7820151
0.00624 0.506355347 0.997822075 -0.68342881 -99.749212
0.00672 0.506834301 0.997494029 -0.731308432 -99.7141089
0.0072 0.507313098 0.997143001 -0.779171204 -99.676708
0.00768 0.507791727 0.996768999 -0.827016024 -99.6370121
0.00816 0.508270176 0.996372031 -0.87484179 -99.5950194
0.00864 0.508748435 0.995952107 -0.922647399 -99.5507327
0.00912 0.509226492 0.995509236 -0.970431751 -99.5041512
0.0096 0.509704336 0.995043429 -1.01819374 -99.4552788
0.01008 0.510181957 0.994554696 -1.06593228 -99.4041155
0.01056 0.510659343 0.994043049 -1.11364625 -99.3506583
0.01104 0.511136484 0.993508498 -1.16133457 -99.2949132
0.01152 0.511613368 0.992951058 -1.20899613 -99.2368863
0.012 0.512089984 0.99237074 -1.25662983 -99.1765635
0.01248 0.512566322 0.991767557 -1.30423458 -99.1139629
0.01296 0.513042371 0.991141525 -1.35180929 -99.0490754
0.01344 0.513518119 0.990492656 -1.39935284 -98.9819071
0.01392 0.513993555 0.989820967 -1.44686416 -98.9124559
0.0144 0.514468669 0.989126472 -1.49434214 -98.8407299
0.01488 0.51494345 0.988409188 -1.54178569 -98.766724
0.01536 0.515417886 0.987669131 -1.58919371 -98.6904414
0.01584 0.515891968 0.986906318 -1.63656513 -98.6118879
0.01632 0.516365683 0.986120766 -1.68389883 -98.5310597
0.0168 0.516839021 0.985312495 -1.73119374 -98.4479636
0.01728 0.517311971 0.984481522 -1.77844876 -98.3625977
0.01776 0.517784522 0.983627867 -1.82566281 -98.2749671
0.01824 0.518256663 0.982751548 -1.87283479 -98.1850687
0.01872 0.518728384 0.981852588 -1.91996363 -98.0929125
0.0192 0.519199673 0.980931005 -1.96704822 -97.9984955
0.01968 0.51967052 0.979986822 -2.0140875 -97.9018168
0.02016 0.520140914 0.97902006 -2.06108037 -97.8028864
0.02064 0.520610843 0.978030742 -2.10802576 -97.7017012
0.02112 0.521080298 0.977018889 -2.15492258 -97.5982653
0.0216 0.521549267 0.975984526 -2.20176974 -97.4925817
0.02208 0.52201774 0.974927677 -2.24856618 -97.3846474
0.02256 0.522485705 0.973848365 -2.29531081 -97.2744764
0.02304 0.522953152 0.972746616 -2.34200256 -97.1620597
0.02352 0.523420071 0.971622455 -2.38864035 -97.0474023
0.024 0.523886449 0.970475907 -2.4352231 -96.9305152
0.02448 0.524352278 0.969307 -2.48174975 -96.8113895
0.02496 0.524817545 0.96811576 -2.52821922 -96.6900362
0.02544 0.525282241 0.966902215 -2.57463044 -96.5664542
0.02592 0.525746354 0.965666392 -2.62098233 -96.4406466
0.0264 0.526209874 0.964408321 -2.66727384 -96.3126184
0.02688 0.52667279 0.96312803 -2.7135039 -96.1823696
0.02736 0.527135091 0.961825548 -2.75967144 -96.0499061
0.02784 0.527596767 0.960500905 -2.80577539 -95.9152302
0.02832 0.528057808 0.959154133 -2.8518147 -95.7783406
0.0288 0.528518202 0.957785262 -2.89778831 -95.6392495
0.02928 0.528977939 0.956394324 -2.94369515 -95.4979519
0.02976 0.529437008 0.95498135 -2.98953416 -95.3544557
0.03024 0.529895399 0.953546374 -3.0353043 -95.208757
0.03072 0.530353101 0.952089428 -3.08100451 -95.0608728
0.0312 0.530810104 0.950610545 -3.12663373 -94.9107952
0.03168 0.531266397 0.949109761 -3.17219091 -94.758529
0.03216 0.53172197 0.94758711 -3.217675 -94.6040794
0.03264 0.532176812 0.946042626 -3.26308496 -94.4474543
0.03312 0.532630912 0.944476345 -3.30841974 -94.2886488
0.0336 0.533084261 0.942888303 -3.35367829 -94.1276739
0.03408 0.533536847 0.941278538 -3.39885957 -93.9645266
0.03456 0.533988661 0.939647085 -3.44396254 -93.7992199
0.03504 0.534439692 0.937993983 -3.48898617 -93.6317458
0.03552 0.534889929 0.93631927 -3.53392941 -93.4621194
0.036 0.535339362 0.934622984 -3.57879123 -93.2903376
0.03648 0.535787981 0.932905164 -3.62357059 -93.1164054
0.03696 0.536235775 0.93116585 -3.66826646 -92.940331
0.03744 0.536682735 0.929405082 -3.71287782 -92.7621112
0.03792 0.537128849 0.927622901 -3.75740363 -92.5817572
0.0384 0.537574108 0.925819347 -3.80184288 -92.3992689
0.03888 0.538018502 0.923994462 -3.84619453 -92.2146514
0.03936 0.538462019 0.922148289 -3.89045756 -92.0279096
0.03984 0.53890465 0.920280869 -3.93463096 -91.8390496
0.04032 0.539346385 0.918392247 -3.9787137 -91.6480704
0.0408 0.539787213 0.916482464 -4.02270477 -91.45498
0.04128 0.540227125 0.914551566 -4.06660316 -91.2597845
0.04176 0.54066611 0.912599596 -4.11040786 -91.0624868
0.04224 0.541104158 0.9106266 -4.15411785 -90.863086
0.04272 0.541541258 0.908632624 -4.19773214 -90.661597
0.0432 0.541977402 0.906617712 -4.2412497 -90.458019
0.04368 0.542412578 0.904581913 -4.28466955 -90.2523539
0.04416 0.542846778 0.902525271 -4.32799068 -90.0446097
0.04464 0.54327999 0.900447836 -4.37121209 -89.8347925
0.04512 0.543712205 0.898349654 -4.41433279 -89.6229063
0.0456 0.544143413 0.896230774 -4.45735179 -89.408954
0.04608 0.544573603 0.894091245 -4.50026809 -89.1929378
0.04656 0.545002767 0.891931117 -4.5430807 -88.9748746
0.04704 0.545430894 0.889750438 -4.58578864 -88.7547555
0.04752 0.545857974 0.887549259 -4.62839092 -88.5325924
0.048 0.546283998 0.885327632 -4.67088656 -88.3083914
0.04848 0.546708955 0.883085606 -4.71327459 -88.0821546
0.04896 0.547132836 0.880823234 -4.75555403 -87.8538889
0.04944 0.547555632 0.878540568 -4.79772389 -87.6235963
0.04992 0.547977331 0.876237661 -4.83978322 -87.3912889
0.0504 0.548397925 0.873914565 -4.88173104 -87.1569677
0.05088 0.548817404 0.871571334 -4.92356638 -86.9206327
0.05136 0.549235758 0.869208022 -4.96528829 -86.682305
0.05184 0.549652978 0.866824684 -5.00689579 -86.4419685
0.05232 0.550069054 0.864421374 -5.04838794 -86.1996513
0.0528 0.550483976 0.861998148 -5.08976377 -85.9553394
0.05328 0.550897735 0.859555061 -5.13102233 -85.7090539
0.05376 0.551310322 0.85709217 -5.17216268 -85.4607886
0.05424 0.551721726 0.854609532 -5.21318386 -85.2105568
0.05472 0.552131939 0.852107204 -5.25408492 -84.9583603
0.0552 0.55254095 0.849585243 -5.29486494 -84.7042103
0.05568 0.552948751 0.847043708 -5.33552296 -84.4481006
0.05616 0.553355332 0.844482657 -5.37605805 -84.1900565
0.05664 0.553760684 0.841902149 -5.41646927 -83.9300608
0.05712 0.554164797 0.839302244 -5.4567557 -83.6681407
0.0576 0.554567662 0.836683001 -5.49691641 -83.404287
0.05808 0.55496927 0.834044481 -5.53695047 -83.1385159
0.05856 0.555369611 0.831386745 -5.57685696 -82.8708254
0.05904 0.555768677 0.828709854 -5.61663495 -82.6012295
0.05952 0.556166457 0.826013869 -5.65628354 -82.3297262
0.06 0.556562944 0.823298853 -5.69580181 -82.0563286
0.06048 0.556958127 0.820564868 -5.73518885 -81.7810426
0.06096 0.557351999 0.817811977 -5.77444375 -81.5038663
0.06144 0.557744548 0.815040244 -5.8135656 -81.2248188
0.06192 0.558135768 0.812249733 -5.85255352 -80.943894
0.0624 0.558525648 0.809440507 -5.89140659 -80.6611079
0.06288 0.558914179 0.806612632 -5.93012392 -80.3764657
0.06336 0.559301353 0.803766172 -5.96870462 -80.0899623
0.06384 0.559687161 0.800901194 -6.0071478 -79.8016247
0.06432 0.560071593 0.798017763 -6.04545258 -79.511442
0.0648 0.560454642 0.795115946 -6.08361808 -79.2194292
0.06528 0.560836298 0.792195809 -6.1216434 -78.9255894
0.06576 0.561216552 0.789257421 -6.15952769 -78.6299345
0.06624 0.561595395 0.786300847 -6.19727005 -78.3324645
0.06672 0.56197282 0.783326158 -6.23486964 -78.0331936
0.0672 0.562348816 0.78033342 -6.27232557 -77.7321197
0.06768 0.562723376 0.777322704 -6.30963699 -77.4292599
0.06816 0.563096491 0.774294078 -6.34680303 -77.1246131
0.06864 0.563468152 0.771247613 -6.38382285 -76.8181905
0.06912 0.563838351 0.768183378 -6.42069558 -76.509997
0.0696 0.564207079 0.765101444 -6.45742038 -76.2000416
0.07008 0.564574328 0.762001882 -6.4939964 -75.8883325
0.07056 0.564940089 0.758884764 -6.5304228 -75.5748686
0.07104 0.565304353 0.755750161 -6.56669873 -75.2596719
0.07152 0.565667113 0.752598145 -6.60282338 -74.9427335
0.072 0.566028361 0.74942879 -6.63879589 -74.6240755
0.07248 0.566388086 0.746242168 -6.67461544 -74.3036907
0.07296 0.566746283 0.743038353 -6.71028122 -73.9816004
0.07344 0.567102941 0.739817418 -6.74579238 -73.6578034
0.07392 0.567458053 0.736579438 -6.78114813 -73.3323058
0.0744 0.567811611 0.733324486 -6.81634764 -73.0051247
0.07488 0.568163607 0.73005264 -6.8513901 -72.6762591
0.07536 0.568514033 0.726763972 -6.8862747 -72.345715
0.07584 0.568862879 0.72345856 -6.92100064 -72.0135104
0.07632 0.569210139 0.72013648 -6.95556713 -71.6796444
0.0768 0.569555805 0.716797808 -6.98997336 -71.344124
0.07728 0.569899868 0.713442621 -7.02421854 -71.0069622
0.07776 0.57024232 0.710070996 -7.05830188 -70.6681681
0.07824 0.570583154 0.706683011 -7.0922226 -70.3277366
0.07872 0.570922362 0.703278744 -7.12597991 -69.985695
0.0792 0.571259936 0.699858274 -7.15957305 -69.642033
0.07968 0.571595868 0.696421679 -7.19300122 -69.2967718
0.08016 0.57193015 0.692969038 -7.22626367 -68.9499094
0.08064 0.572262775 0.689500431 -7.25935963 -68.6014629
0.08112 0.572593736 0.686015939 -7.29228833 -68.2514323
0.0816 0.572923023 0.68251564 -7.32504902 -67.8998325
0.08208 0.573250631 0.678999617 -7.35764094 -67.5466657
0.08256 0.573576551 0.675467949 -7.39006334 -67.1919399
0.08304 0.573900775 0.671920719 -7.42231547 -66.8356761
0.08352 0.574223297 0.668358007 -7.45439659 -66.4778602
0.084 0.574544109 0.664779897 -7.48630597 -66.1185195
0.08448 0.574863203 0.66118647 -7.51804286 -65.7576558
0.08496 0.575180573 0.65757781 -7.54960653 -65.3952743
0.08544 0.57549621 0.653953999 -7.58099626 -65.0313859
0.08592 0.575810108 0.65031512 -7.61221133 -64.6660028
0.0864 0.576122259 0.646661259 -7.64325101 -64.2991268
0.08688 0.576432657 0.642992498 -7.67411459 -63.9307692
0.08736 0.576741293 0.639308923 -7.70480136 -63.5609388
0.08784 0.577048162 0.635610619 -7.73531061 -63.1896467
0.08832 0.577353255 0.63189767 -7.76564164 -62.816896
0.0888 0.577656566 0.628170162 -7.79579375 -62.4426967
0.08928 0.577958087 0.624428181 -7.82576625 -62.0670638
0.08976 0.578257813 0.620671813 -7.85555844 -61.6899943
0.09024 0.578555735 0.616901145 -7.88516963 -61.3115104
0.09072 0.578851848 0.613116263 -7.91459916 -60.9316069
0.0912 0.579146144 0.609317256 -7.94384633 -60.5503071
0.09168 0.579438616 0.60550421 -7.97291048 -60.1676028
0.09216 0.579729258 0.601677213 -8.00179093 -59.7835232
0.09264 0.580018063 0.597836353 -8.03048702 -59.3980552
0.09312 0.580305024 0.593981719 -8.05899808 -59.0112289
0.0936 0.580590136 0.5901134 -8.08732347 -58.6230353
0.09408 0.58087339 0.586231485 -8.11546253 -58.2334935
0.09456 0.581154781 0.582336063 -8.14341461 -57.8426126
0.09504 0.581434302 0.578427224 -8.17117906 -57.4503944
0.09552 0.581711947 0.574505058 -8.19875525 -57.0568571
0.096 0.58198771 0.570569655 -8.22614254 -56.6620007
0.09648 0.582261583 0.566621107 -8.2533403 -56.2658433
0.09696 0.582533561 0.562659503 -8.28034791 -55.8683868
0.09744 0.582803638 0.558684937 -8.30716473 -55.4696413
0.09792 0.583071807 0.554697497 -8.33379016 -55.069624
0.0984 0.583338062 0.550697278 -8.36022358 -54.6683306
0.09888 0.583602396 0.546684371 -8.38646438 -54.2657834
0.09936 0.583864805 0.542658868 -8.41251195 -53.8619834
0.09984 0.584125281 0.538620862 -8.43836571 -53.4569415
0.10032 0.584383819 0.534570447 -8.46402504 -53.0506688
0.1008 0.584640413 0.530507715 -8.48948936 -52.6431765
0.10128 0.584895057 0.52643276 -8.51475808 -52.2344654
0.10176 0.585147744 0.522345676 -8.53983063 -51.8245577
0.10224 0.58539847 0.518246557 -8.56470642 -51.4134513
0.10272 0.585647229 0.514135498 -8.58938487 -51.0011603
0.1032 0.585894014 0.510012593 -8.61386543 -50.5876968
0.10368 0.58613882 0.505877938 -8.63814752 -50.1730678
0.10416 0.586381641 0.501731627 -8.6622306 -49.7572783
0.10464 0.586622472 0.497573756 -8.68611409 -49.3403483
0.10512 0.586861308 0.493404422 -8.70979746 -48.9222779
0.1056 0.587098142 0.489223719 -8.73328015 -48.5030812
0.10608 0.587332969 0.485031744 -8.75656163 -48.0827672
0.10656 0.587565784 0.480828595 -8.77964136 -47.6613458
0.10704 0.587796582 0.476614367 -8.8025188 -47.2388232
0.10752 0.588025357 0.472389158 -8.82519344 -46.8152194
0.108 0.588252104 0.468153065 -8.84766474 -46.3905273
0.10848 0.588476817 0.463906186 -8.8699322 -45.9647752
0.10896 0.588699492 0.459648619 -8.89199529 -45.5379579
0.10944 0.588920124 0.455380461 -8.91385351 -45.1100936
0.10992 0.589138706 0.451101811 -8.93550635 -44.6811923
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