Apa maksud dari np1 matriks gauss

Page 2

Description: Subroutine SOLVE employs Gauss' pivotal or elimination method to first

triangularize the matrix B and then solve for the elements of the solution vector C in reverse order. The product of the diagonal terms of the triangularized matrix is the determinant, the value of which is printed to indicate a proper range of amplitudes of numbers involved. The matrix B consists of A except for the right-hand column which is Y. Both matrix A and vector C use the vector format described earlier.

SUBROUTINE SOLVEIA, C) VECTOR FORMAT

DIMENSION All), C(1)
101 FORMAT(E15.7)

N = All) - .99 NMI : N - 1 NPI = N + 1 NP3 = N 3 DO 6 K 1, NM1

AKKI 1./ A(K*NFT + K - NP3)

KIL = KI #NPI + L NP3 6 A(KILI = A(KILO AK IKEA (K*NPI + L - NP3)

DETA : A(5)
CIN+4) = A (NP1 *NPI NP 3) / A(N*NPI + 3) DO 16 I = 2,N NIT = N I + 1

CINIT + 4) = A(NII+NPI + 4)

DO 8 J = NI2, N 8 CINI 1+4) = CINII+4) CO J+4) *AINII*NPI + J - NP3)

CINII+41 = CINII+4)/A (NIT*NPI + NIT - NP3)
16 DETA - DETA*A(INPI + 1 - NP3)

PRINT 101, DETA C(1) = 1.0

C12) = A(1) 1.0

Page 3

NDA 1 2 2

99999. 1.0 NDA

6 2

1.0 NOR 1 2 2 2

99999. 1.0 NDR

6 2 2

1.0 1 2 2 3

99999. 1.0 6 2 3

1.0 1 1 3

99999. 1.0 5 3

1.0 1 1 3 3 99999.

1.0 VB

6 3 3

1.0 VO 1 2 3 3

99999. 1.0 VO 3

1.0 END -1 0. O.

1.0000E+00 8.9372E-05 -4.1952E-05 2.2927E-05 2.2944E-06 1.6415E-03 -8.0941E-04 4.7933E-04 5.0000E-02 0.

1.0000E+00 1.6384E-04 -8.0879E-05 4.7857E-05 8.6888E-06 1.3345E-03 -7.4711E-04 5.1709E-04 1.0000E-01 0.

1.0000E+00 2.2256E-04 -1.1661E-04 7.4556E-05 1.8416E-85 1.0118E-03 -6.8167E-04 5.5005E-04 1.5000E-01 0 0.

1.0000E+00 2.6481E-04 -1.4900E-04 1.0278-04 3.0670E-05 6.7648E-04 -6.1363E-04 5.7803-04 2.0000E-01 0. O.

1.0000E+00 2.9005E-04 -1.7794 E-04 1.3227E-04 4,4614E-05 3.3153E-04 -5.4357E-04 6.0087E-04 2.5000E-01 0. O.

1.00005+00 2.9786E-04 -2.0334 E-04 1.6278E-04 5.9385E-n5 -1.9951E-05 -4.7208E-04 6.1847E-04 3.0000E-01 O 0.

1.0000E+00 2.8800E-04 -2.2513E-04 1.9403E-04 704105E-05 -3.7489E-04 -3.9974 E-04 6.30766-04 3.5000E-01 0. 0.

1.0000E+00 2.6036E-04 -2.4331E-04 2.2577E-04 8.7888E-05 -7.3023E-04 -3.2714E-04 6.3773E-04 4.0000E-01 O.

1.0000E+00 2.1502E-04 --2.5785E-04 2.5772E-04 9.9846E-n5 -1.0829E-03 -2.5484E-04 6.3940E-04 4.5000E-01 O O.

1.0000E+00 1.5216E-04 -2.6880E-04 2.8962E-04 1.0910E-04 -1.4301E-03 -1.8342E-04 6.3581E-04 5.0000E-01 0. O.

1.0000E+00 7.2150E-05 -2.7622E-04 3.2121E-04 1.1478E-n4 -1.76886-03 -1•13425-04 6.27085-04 5.5000E-01 0.

1. 0000E+00 -2.4530E-05 -2.8018E-04 3.5224E-04 1.1604E-94 -2.0963E-03 -4.5389E-05 6.1334E-04 6.0000E-01 0.

1.0000E+00 -1.3725E-04 -2.8080E-04 3.8247E-04 101206E-14 -2.4 100E-03

2.0174E-05 5.9477E-04 6.5000E-01 0

1•0000E+00 -2.6526E-04 -2.7821E-04 4.1164E-04 1.0206E-04 -2.7074E-03

8.2782E-05 5.7158E-04 7.0000E-01 0.

1.0000E+00 -4.0768E-04 -2.7258E-04 4,3955E-04 8.5290E-05 -2.9861E-03

1.41985-04 5.4401 E-04 7.5000E-01 0.

Page 4

-9.6934E+00 1.2847E+01 1.5004E+00 -5.3840E+00 2.7500E+00

1.0000E-01 -1.3000E+00 1.0000E+00 -1.4551E+01 3.7727E+00 -1.4118E +00 -1.0439E+01 1.5719E+01 9.0566E-01 -5.3590E+00 2.8000E+00

1.0000E-ni-1.3000E+00 1.0000E+00 -1.3695E+01 3.8034 E+00 -1.6782E+00 -1.1146E+01 1.8524E+01 3. 2229E-01 -5.2904E+00 2.8500E+00

1.0000E-01 -1.3000E+DO 1.0000E+00 -1.2701E+1 3.8052E+00 -1.9401E +00 -1.1807E+01 2.1237E+01 -2.4515E-01 -5.1794E+00 2.9000E+00

1.0000E-01 -1.3000E+00 1.0000E+00 -1.1573E+01 3.7792 E+00 -2.1954E+00 -1.2414E+ni 2.3837E+01 -7.9231E-01 -5.0272E+00 2.9500E+00

1.0000E-ni -1.3000E+00 1.0000E+00 -1.0319E+01 3.7264 E+00 -2.4422E+00 -1.2962E+01 2.6303E+01 -1.3151E+00 -4.8359E+00 3.0000E+00

1.0000E-01 -1.3000E+00 1.0000E+00 -8.9454E+00 3.6482E+00 -2.6784E+00 -1.3444E+01 2.8615E+01 -1.8096E+00 -4.6072E+00 3.0500E+00

1.0000E-ni-1.3000E+00 1.0000E+00 -7.4605E+00 3.5460E+00 -2.9023E+00 -1.3855E+01 3.0754E+01 -2.2723E+00 -4.3437E+00 3.1000E+00

1.0000E-01 -1.3000E+00 1.0000E+00 -5.8732E+00 3.4215E+00 -3.1122E+00 -1.4188E+01 3.2705E+01 -2.6998E+00 -4.0477E+00 3.1500E+00

1.0000E-01 -1.3000E+00 1.0000E+00 -4.1934 E+00 3.2766E+00 -3.3066E +00 -1.4440E+01 3.4451E+01 -3.0892E+00 -3.7220E+00 3.2000E+00

1.0000E-01 -1.3000E+00 1.0000E+00 -2.4317E+00 3.1133E+00 -3.4840E+00 -1.4606E+01 3.5981E+01 -3.4379E+00 -3.3696E+00 3.2500E+00

1.0000E-01 -1.3000E+00 1.0000E+00 -5.9913E-01 2.9336E+00 -3.6432E+00 -1.4682E+01 3.7282E+01 -3.74 35E+00 -2.9937E+00 3.3000E+00

1.0000E-01 -1.3000E+00 1.0000E+00 1.2926E+00 2.7397E+00 -3.7830E+00 -1.4665E+01 3.8346E+01 -4.0042E+00 -2.5974E+00 3.3500E+00

1.0000E-01 -1.3000E+00 1.0000E+00 3.2314E+00 2.5339E+00 -3.9026E +00 -1.4552E+01 3.9165E+01 -4.2183E+00 -2.1841 E+00 3.4000E+00

1.0000E-ni -1.3000E+00 1.0000E+00 5.2049E+00 2.3186E+00 -4.0012E+00 -1.434 1 E+01 3.9735E+01 -4.3849E+00 -1.7574E+00 3.4500E+00

1.0000E-01 -1.3000E+00 1.0000E+00 7.2006E+00 2.0962E+00 -4.0782E+ -1.4031E+01 4.0052E+1 -4.5030E+00 -1.3206E+00 3.5000E+00

1.0000E-01 -1.3000E+00 1.0000E+00 9.2059E+00 1.8692E+00 -4.1332E+00 -1.3621 E+01 4.0115E+01 -4.5724E+00 -8.7740E-01 3.5500E+00

1.0000E-01 -1.3000E+00 1.0000E+00 1.1208E+01 1.6398E+00 -4.16596 +00 -1.3111E+01 3.9925E+01 -4.5931E+00 -4.3131E-01 3.6000E+00

1.0000E-01 -1.3000E+00 1.0000E+00 1.3194E+01 1.4106E+00 -4.1763E +00 -1.2501 E+01 3.9485E+01 -4.5654E+00 1.41389-02 3.6500E+00

1.0000E-01 -1.3000E+00 1.0000E+00 1.5152E+01 1.1841 E+00 -4.1645E +00 -1.1792E+ni 3.8801E+01 -4.4902E+00 4.5547E-01 3.7000E+00

1.0000E-ni -1.3000E+00 1.0000E+00 1.7070E+01 9.6240E-01 -4.1309E+00 -1.0986E+ni 3.7878E+01 -4.3685E+00 8.8929E-01 3.7500E+00

1.0000E-ni -1,3000E+00 1.0000E+00 1.8936E+01 7.4796E-01 -4.0758E+00 -1.0086E+01 3.6725E +01 -4.2019E+00 1.3123E+CO 3.8000E+00

1.0000E-01 -1.3000E+00 1.0000E+00 2.0739E+01 5.4293E-01 -3.9999€ +00 -9.0935E+00 3.5353E+01 -3.9922E+00 1.7213E+00 3.8500E+00

1.0000E-01 -1.3000E+00 1.0000E+00 2.2468E+01 3.4943E-01 -3.9039E+00 -8.01 30E+00 3.3773E+01 -3.7414E+00 2.1134E+00 3.9000E+00

1.0000E-ni -1.3000E+00 1.0000E+00 2.4113E+01 1.6943E-01 -3.7889E+00 -6.8481E+00 3.2000E+01 -3.4521E+00 2.4856E+00 3.9500E+00

1.0000E-01 -1.3000E+00 1.0000E+00 2.5665E+01 4.8128E-03 -3.6557E +00 -5.6033E+00 3.0047E+01 -3.1270E+00 2.8353E+00 4.0000 E+CO

1.0000E-01 -1.3000E+00 1.0000E+00 2.7115E+01 -1.4271E-01 -3.5058E+00 -4.2833E+00 2.7930E+01 -2.7689E+00 3.1601E+00 4.0500E+00

1.0000E-ni -1.3000E+00 1.0000E+00 2.8456E+01 -2.7158E-01 -3.3402E+00 -2.8936E+00 2.5667E+01 -2.3810E+00 3.4577E+00 4,1000E+00

Page 5

4.00 1.0000E-01 -1.3000E+00 1.0000E+00 2.7115E+01 -1.42718-01 -3.5058E+00 -4.2833E+00 2.1930E+01 -2.7689€:00

1.0000E-01 -1.3000E+00 1.0000E+00 3.07815+01 -4.6788E-01 -2.9681E+00 1.2358E-02 2.0776E+01 -1.5297E+00 3.9638E+00

-1.80961.00 -4.6072E 00 1.0000E+00 -7.4605E00 3.10 1.0000E-01

3.0754E01 -2.27236.00 -4.34375.00 -1.3000E00

1.0000E00 -5.8732E+00 3.4215.00 -3.1122E.00 -1.4188E-01 3.2705E+01 -2.6998E+00 -4.0477600 3.20 1.0000E-01

3.2766E.00 -3.3066E 00 -1.4440E+01 3.4451E01 -3.0692E.00 -3.7220E+00 3.25 1.0000E-01 -1.3000E+00

3.5981E01 -3.4379E+00 -3.3696E+00
1.0000E-01 -1.3000E00

-3.74 355.00 -2.9937E+00 2.7397E+00 -3.7830E+00 -1.4665E +01

3.8346E+01 -4.0042E.00 -2.5974 E +00 1.0000E-01 -1.3000E+00

3.9165E+01 -4.2183E+00 -2.1841E00 1.0000E+00

3.9735E+01 -4.38496.00 -1.7574E+00 7.2006E+00 1.0000E-01 -1.3000E+00 1.0000E+00

4.0052E+01 -4.5030E + 00 -1.3206E+00

4.0115€+01 -4.5724E+00 -8.77406-01 1.0000E+00

3.9925E+01 -4.5931E+00 -4.3131E-01 3.70

1.4138E-02 1.7070E+01 1.0000E-01 -1.3000E+00

3.8801E+01 -4.4902E+00 4.5547E-01 3.80 1.0000E+00 1.8936E+01

3.7878E+01 -4.3685E+00 1.0000E-01 -1.3000E+00 1.0000E+00 2.07396+01 1.4796E-01 -4.0758E+00 -1.0086E+01 3.6725E+01 -4.2019E+00

8.8929E-01 3.85 1.0000E-01 -1.3000E+00 1.0000E+00

1.3123E+00 2.2468E+01

5.4293E-01 -3.9999E+00 -9.0935E+00 3.5353E+01 -3.9922E+00 1.7213E+00 3.90 1.0000E-01 -1.3000E+00 1.0000E+00

3.4943E-01 -3.9039E+00 -8.0130E+00 3.3773E+01 -3.7414E+00 2.1134E+00 3.95 1.0000E-01 -1.3000E+00

2.41138+01 1.6943E-01 -3.7889E+00 -6.8481E+00 3.2000E+01 -3.4521E+00 2.4856E+00 1.0000E+00 2.5665€+01 4.81286-03 -3.6557E+00 -5.6033E+00 3.0047E+01 -3.1270E+00

2.8353E+00

3.1601E+00 4.05 1.0000E-01 -1.3000E+00

1.0000E+00 2.8456E+01 -2.7158E-01 -3.3402E+00 -2.8936E+00 2.5667E+01 -2.3810E+00 3.4577E+00 4.10 1.0000E-01 -1.3000E+00 1.0000E+00 2.9680E+01 -3.80386-01 -3.1605E+00 -1.4397€ +00 2.3276E+01 -1.9668E+00 3.7262E+00 4.15

1.0000E+00 3.1756E+01 -5.3302E-01 -2.7646E+00 1.6363E+00 1.81858+01 -1.0733E+00 4.1692E+00 4.20 1.0000E-01 -1.3000E+00

3.25998+01 -5.7495E-01 - 2.5517E+00 3.2458E+00 1.55 25E+01 -6.0151E-01 4.3410E+00 4.25 1.0000E-01 -1.3000E+00 1.0000E+00

4.4 784E+00 4.35 1.0000E-01 -1.3000E+00 1.0000E+00 3.3880E+01 -5.8663E-01 -2.1045E+00 6.574 2E+00 1.0074E+01 3.7315E-01 4.5807E+00

3.431 5E+01 -5.5560E-01 -1.8736E+00 8.2797E+00 7.3254E+00 8.6823E-01 4.6476 E+00 4.45 1.0000E-01 -1.3000E+00 1.0000E+00 3.461 3E+01 -4.9981E-01 -1.6403E+00 1.0003E+01 4.5875E00 1.36 336.00 4.6790E+00

4.6752E+00 4.55 1.0000E-01 -1.3000E+00 1.0000E+00 3.4801E+01 -3.1447E-01 -1.1734E+00 1.3479E+01 -7.7582E-01 2.3383E+00 4.63656 +00

2.8108E+00 4.5537E+00 1.0000E-01 -1.3000E+00

3.4467E+01 -3.3644 E-02 -7.1755E-01 1.6946E+01 -5.8629E+00 3.2685E+00 4.4579E+00 4.65 1.0000E-01 -1.3000E+00 1.0000E+00 4.70 1.0000E-01 -1.3000E+00 1.0000E+00 3.411 3E+01

1.8661E+01 -8.2584E+00 3.7083E00 4.3202E+00

4.1268E+00 4.1521E+00 1.0000E+00 3.306 2E+01 5.5313E-01 -8.3234E-02 2.2024E+01 -1.2673E+01 4.5211E+00 4.80 1.0000E-01 -1.3000E+00

3.9553E+00 4.85 1.0000E-01 -1.3000E+00 1.0000E+00 3.2378E+01 7.8849E-01 1.0905E-01 2. 3660E+01 -1.4663E+01 4.8885E +00 3.731 7E+00

3.1598 +01 1.0415E +00 2.8953E-01 2.5260E+01 -1.6491E+01 5.22656+00 3.48 34E+00 1.0000E-01 -1.3000E+00 4.95 1.0000E+00 3.0732E+01 1.3106E +00 4.5702E-01 2.6818E+01 -1.8146E+01

3.2126E+00

1.6637E+00 6.0836E-01 2.8322E+01 -3.9592E+01 8.5789E+00 2.8105E +00 5.05 -4.8000E+00 -1.30 COE+00 1.0000E+00 2.6783E+01 2.16 70E +00 7.3364E-01 2.9727E+01 -6.0442E+01 1.1536E+01 2.1709E+00 5.10 -7.2500E+00 -1.3000E+00 1.0000E+00 2.32557+01 2.8153E+00 8.2115E-01 3.0981E+01 -8.0572E+01 1.4379E+01 1.3010E+00

1.8740E+01 3.6025E+00 8.5959E-01 3.2034E+01 -9.9863E+01 1.7086E+01 2.1011E-01 5.20 -9.5000E+00 -9.0000E-01 1.0000E+00 1.3868E+01 4.4086E+00 8.4121E-01 3.2849E+01 -9.4956E+01 1.5140E+01 -9.3039E-01 5.25 -9.3000E+00 -5.00COE-01 1.0000E+00 9.2538E+00 5.1149E+00 7.6876E-01 3.3426E+01 -8.9510E+01 1.3098E+01 -1.9513E+00 5.30 -7.9000E+00 -5.0000E-01 1.0000E+00 5.1280E+00 5.7205€ +00 6.4863E-01

1.1110E+01 -2.8332E+00

3.378 3E+01 -7.5464E+01 5.35 -6.5000E+00 -5.0000E-01 1.0000E+00 1.7113E+00 6.2248E +00 4.8833E-01 3.395 2E+01 -6.1161E+01 9.0563E+00 -3.5574E+00 5.40 -5.1000E+00 -5.0000E-01 1.0000E+00 -9.85755-01 6.6253E+00 2.9585E-01 3.3967E+01 -4.66 95E+01 6.955 8E+00 -4.1199E+00 5.45 -3.7000E+00 -5.0000E-01 1.0000E+00 -2.9574E+00 6.9200E+00 7.9348E-02 3.3866E+01 -3.2160E+01 4.8261E+00 -4.5179E+00 5.50 -2.3000E+00 -5.0000E-01 1.0000E+00 -4.2025E+00 7.1078E.00 -1.5292E-01 3.3685E+01 -1.7652E+01 2.6853E+00 -4.7503E+00 5.55 -9.0000E-01 -5.0000E-01 1.0000E+00 -4.72496.00 7.1887E+00 -3.9265E-01 3. 3459E+01 -3.2624E+00 5.5161E-01 -6.8168E+00

5.60 5.0000E-01 -5.00005-01 1.0000E+00 -.53263 +00 7.1634E+00 -6.3158E-01 3.3225E+01 1.0915E+01 -1.55 74E+00 -4.7187E+00

5.1300E+01 -7.5649E+00 -3.4686E+00

6.3710E+01 -9.4072E+00 -2.7496E+00

3.2904E+01 5.85 7.5000E+00 -5.0000E-01 1.0000E+00 6.54 75E+00 5.5328E+00 -1.535 15 +00 3.3142E+01 7.5618E+01 -1.1144E+01 -1.8904E+00 5.90 8.9000E+00 -5.0000E-01 1.0000E+00 1.0610E+01 4.9346E+00 -1:6053E+00 3.3570E+01 6.6775E+01 -1.2763E+01 -8.9924E-01

1.52123+01 4.2588E+00 -1.6227E+00 3.421 4E+01 9.7179€+01 -1.4250E+01 2.1490E-01 1.0400E+01 -5.00C0E-01 1.0000E+00 6.00

2.0049E+01 3.5489E+00 -1.5827E+00 3.5095E+01 9.6176E+01 -1.4123E+01 1.3832E+00

2.4819E+01 2.84 85€ +00 -1.4846E+00 3.6218E+01 9.4529E+01 -1.3872E+01 2.5371E+00

Page 6

.25 0.

1.0000E+00 3.5258E-04 -2.1986E-04 1.4863E-04 6.5468E-05 2.6441E-04 -5.5047E-04 5.7563E-04 .30 0.

0.

1.0000E+00 3.5741E-04 -2.4567E-04 1.7789E-04 8.3288E-05 -7.2146E-05 -4.81596-04 5.94026-04 . 35 0. 0.

1.0000E+00 3.4530E-04 -2.68016-04 2.0795E-04 1.0093E-04 -4.128 6E-04 -4.11746-04 6.0728 E-04 . 40 0.

0.

1.0000E+00 3.1610E-04 -2.8684E-04 2.3854E-04 1.1753E-04 -7.54 76E-04 -3.4148E-04 6.1537E-04 0. 0.

1.0000E+00 2.6985E-04 -3.0216E-04 2.6940E-04 1.3225E-04 -1.0949E-03 -2.71 38 E-04 6.1829E-04 . 50 0. 0.

1.0000E+00 2.000 96-04 -3.13996-04 3.0028E-04 1.44 24E-04 -1.4304E-03 -2.01996-04 6.1610E-04 .55 0. 0.

1.0000E+00 1.2693E-04 -3.2238E-04 3. 3092E-04 1.5264E-04 -1.7584E-03 -1.3384E-04 6.0888 E-04 0. 0.

1.0000E+00 3.10162-05 -3.2740E-04 3.6109E-04 1.5666E-04 -2.0763E-03 -6.7472E-05 5.9676E-04 .65 0.

0

1.0000E+00 -8.04882-05 -3.2916E-04 3.9052E-04 1.5549E-04 -2.3815 E-03 -3.37376-06 5.79906-04 .70 0. 0.

1.0000E+00 -2.0688E-04 -3.2779E-04 4.1900E-04 1.4836E-04 - 2.67146-03 5.79756-05 5.5850E-04 0. 0.

1.0000E+00 -3.47348-04 -3.2342E-04 4.46 30E-04 1.3456 E-04 -2.9439E-03 1.1612E-04 5.3279E-04 .80 0.

5.0000E-01 1. C000E+00 5.1043E-02 -4.7042E-02 1.7455E-03 1.4086E-03 2.05 38E+00 -1.868 0E+00 5.1208 E-02 0.

1.0000E+00 1.0000E+00 2.0323E-01 -1.8645E-01 8.0880E-03 7.7910E-03 4.0197E+00 -3.7057 +00 2.0175E-01 4. 9000E+00 1.0000E+00 1.0000E+00 1.3980E+00 -5.0881E-01 2.8672E-02 4.7942E-02 4.3731E+01 -9.1814E+00 6.1996E-01 .95 9.8000E+00 1.0000E+00 1.0000E+00 4.5563:00 -1.1019E+00 8.1014E-02 1.9711E-01 8.25 25E+01 -1.45 296 +01 1.4705E+00 1.00 9.5500E+00 1.0000E+00 1 1.0000E+00 8.5764E+00 -1.8125E+00 1.806 36-01 5.2590E-01 7.8188E01 -1.3875E01 2.5093E+00 1.05 9.3000E+00 1.0000E+00 1.0000E+00 1.2367E+01 -2.4877E+00 3.3084E-01 1.0501E+00 7.3338E+01 -1.3118E01 3.4932E+00 1.10 2.5000E+00 1.00COE+00 1.0000E+00 1.4567E+01 -2.9373E+00 5.21296-01 1.7240E+00 1.4618E+01 -4.8522.00 4.11856.00 1.15 -4.3000E+00 1.0000E+00 1.0000E+00 1.3846E+01 -2.9746E+00 7.2658E-01 2.434 8E+00 -4.3444E+01 3.3563E+00 4.0875€ +00

1.20 - 4.3333E+00 1.0000E+00 1.0000E+00 1.1626E+01 -2.7961E+00 9.2172 E-01 3.0719E+00 -4.532 3E+01 3.7777E+00 3.7135600

2.25 1.0000E-01 -1.3000E+00 1.0000E+00 -1.5266E+01 1.8500E+00 1.0145E+00 -2.4017E+00 -1.1873E+01 6.5714E+00 - 3.6257E+00

6.1636E-01 -3.9798E+00 -7.0352E+00 5.5794E+00 -4.3123E+00
2:40 1.0000E-01 -1.3000E+00 1.0000E+00 -1.6503E+01 2.7242E+00

3.9348E-01 -4.7984E+00 -4.3991E+00 5.0401E+00 -4.5961E+00 1.0000E+00 -1.6654E+01 2.9623E+00 1.5746E-01 -5.6279E+00 -1.6518€* 00

4.4782E+00 -4.8378E+00
1.0000E+00 -1.6666E+01 3.1718E+00 -8.9578E-02 -6.4615E+00

1.1810E+00 3.8985600 - 5.0363E+00 2.45 1.0000E-01 -1.3000E+00

1.0000E+00 -1.6535E+01 3.3519E+00 - 3.4543E-01 -7.2921E+00

4.0733E+00 3.30596.00 -5.1905E00 2.50 1.0000E-01 -1.3000E+00 2.55 1.0000E-01 -1.3000E+00

3.5022E+00 -6.0788E-01 -8.1126E+00

6.9990E+00 2.70556.00 -5.2999E+00 2.60 1.0000E-01 -1.3000E+00 1.0000E+00 -1.6259E+01

3.6224E+00 -8.7467€ -01 -8.91556700

9.9321E+00 2.1020E+00 -5.3643E+00 2.65 1.0000E-01 -1.3000E+00 1.0000E+00 -1.58358+01

3.7125€ +00 -1.1436E+00 -9.6937E+00

1.2847E+01 1.50056.00 -5.3838E+00 2.70 1.0000E-01 -1.300000 1.0000E+00 -1.5266E+01

3.7726E+00 - 1.4123E+00 -1.0440E+01

1.5719E+01 9.0576E-01 -5.3588E+00 2.75 1.0000E-01 -1.3000E+00 1.0000E+00 -1.4551E+01

3.8032E+00 -1.6787E+00 -1.1146E+01

1.85 24E+01 3.2238E-01 -5.2903E+00 2.80 1.0000E-01 -1.3000E+00 1.0000E+00 -1.3695E+01

3.80518.00 -1.9406E+00 -1.1807E+01

2.1237€+01 -2.4506E-01 -5.1792E+00 2.85 1.0000E-01 -1.3000E+00 1.0000E+00 -1.27003+01

3.7791E+00 -2.1960E+00 -1.2414E+01 2.3837E+01 -7.9224E-01 -5.0271E+00 2.90 1.0000E-01 -1.30000.00 1.0000E+00 -1.1573E-01

3.7263E+00 -2.4427E+00 -1.2962E+01

2.6303E+01 -1.3150E+00 -4.8358E+00 2.95 1.0000E-01 -1.3000E+00 1.0000E+00 -1.0319+01

3.6480E+00 -2.6789E+00 -1.3444E+01 2.8615E+01 -1.8096E+00 -4.6071E+00 3.00 1.0000E-01 -1.3000E+00 1.0000E+00 -8.9453E+00

3.5458E+00 -2.9028E+00 -1.3855E+01 3.0755E+01 -2.2723E+00 -4.3436E+00 3.05 1.0000 E-01 -1.3000E+00 1.0000E+00 -7.46036+00

3.4214E+00 -3.1127E+00 -1.4180E+01 3.2705E+01 -2.6998E+00 -4.0476E+00 3.10 1.0000E-01 -1.3000E00 1.0000E+00 -5.8730E+00

3.2765E+00 -3.30716.00 -1.4441E+01 3.445 16.01 -3.0892E.00 -3.7220E+00 3.15 1.0000E-01 -1.3000E+00 1.0000E+00 -4.1932 +00

3.11316.00 -3.48456.00 -1.4606E+01 3.59 81+ 01 -3.379E •00 - 3.3696E •00 3. 20 1.0000E-01 -1.3000E+00 1.0000E+00 -2.4315.00

2.9334E+00 -3.6437E+00 -1.4682E +01 3.7282E+01 -3.74 355 00 ,2.9937600 3.25 1.0000E-01 -1.3000E+00 1.0000E+00 -5.99946-01

2.7395E+00 -3.78356.00 -1.4665€ +01 3.8346E01 -4.0041E.00 -2.59746.00 3. 30 1.0000E-01 -1.3000E+00 1.0000E+00 1.2928E00

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Page 7

9.35 -5.4545E-02 -1.0909E-01 1.0000E+00 -3.9758E+01 1.2483E.00 -7.57815.00 -3.9113E01 9.64626.01 -1.6285E01 -5.2471E+00 9.40 -5.0000E-02 -1.0000E-01 1.0000 6.00 -3.4898E+01 4.2431E-01 -7.81176.00 -4.0979E+01 9.7817E+01 -1.6654E01 -4.0935E+00 9.45 -4.54556-02 -9.0909E-02 1.0000E+00 -2.9988E+01 -4.1489E-01 -7.9871E+00 -4.2602E+01 9.84936+01 -1.6893E+01 -2.91956.00 9.50 -4.0909E-02 -8.1818E-02 1.0000E+00 -2.5060E+01 -1.2628 E.00 -8.1034E+00 -4.3978 E +01 9.8492E+01 -1.7002E+01 -1.7345E+00 9.55 -3.63646-02 -7.2727E-02 1.0000E+00 -2.0150E+01 -2.1129E+00 -8.1605E+00 -4.5108E+01 9.7819E+01 -1.6981E+01 -5.4806E-01 9.60 -3.1818E-02 -6.3636E-02 1.0000E+00 -1.5289E+01 -2.95 88E+00 -8.1584E+00 -4.5993E+01 9.6485E+01 -1.6834E+01 6.3068E-01 9.65 -2.7273E-02 -5.4545E-02 1.0000E+00 -1.0512E+01 -3.7942E+00 -8.0977E+00 -4.6638E+01 9.4507E+01 -1.6561E+01 1.7926E+00 9.70 -2.2727E-02 -4.5455E-02 1.0000E+00 -5.8489E+00 -4.6129E+00 -7.9795E+00 -4.7047E+01 9.1906E+01 -1.6168E+01 2.9289E+00 9.75 -1.8182E-02 -3.6 364E-02 1.0000E+00 -1.3311E+00 -5.4090E+00 -7.8053E+00 -4.7225E+01 8.8708E+01 -1.5657E+01 4.0310E+00 9.80 -1.3636E-02 -2.7273E-02 1.0000E+00 3.012 5E+00 -6.1768E+00 -7.5771E+00 -4.71826+01 8.4943E+01 -1.5036E+01 5.0908€+00 9.85 -9.0909E-03 -1.6182E-02 1.0000E+00 1.154 3E+00 -6.9108E+00 - 7.2971E+00 -4.6927E+01 8.0645E+01 -1.4310E-01 6.1005E+00 9.90 -4.54556-03 -9.0909E-03 1.0000E+00 1.1069E+01 -7.6061E+00 -6.9680E+00 -4.6471E+01 7.5853E+01 -1.34 85E +01 7.0531 5+00

9.95 -5.6932E-13 -1.1386E-12 1.0000E+00 1.4732E01 -8.2578E+00 -6.5928E+00 -4.5825E+01 7.0606E+01 -1.25 70E+01 7.9417E+00 THE VALUE OF THE FIT ERROR IS 2.08800E-05 FOR THE ITERATICN NUMBER 9

MATRIX EE*D HAS I ROWS AND 7 COLUMNS

1.4898E-06 3.8844E-08 1.46 90E-06 1.7081E-05 5.6953E-07 4.7047E-08 1.8428E-07 MATRIX RMS HAS 1 ROWS AND 7 COLUMNS 1.22066-03 1.9709E-04 1.2120E-C3 4.13306-03 7.54676-04 2.1690E-04 4.2928E-04 MATRIX GRAD HAS I ROWS AND 16 COLUMNS

9.13976-05 -9.2442E-06 1.4474E-C5 -1.98 78 E-05 9.30 30E-06 -9.6792E-06 4.4500E-04 9.2840E-05 -5.0273E-05 7.7592E-05 -1.25876-05 4.5126E-05 -1.4111E-03 -7.63910-07 3.0171E-C7

2.0880E-05 WOULD YOU BELIEVE.

8.34966136-05 MATRIX DC HAS 1 ROWS AND 15 COLUMNS -1.91496-06 -1.8729E-06 -1.74 36 E-C5 9.9519E-08 -7.6805E-06 -1.2673E-04 4.9877E-07 2.9333E-06 4.5277E-00 6.6766E-07

2.6512 E-06 1.8164E-05 -3.6874E-C7 1.88 26E-06 -1.1702E-05 MATRIX A HAS 4 ROWS AND 4 COLUMNS -2.73 96 E-CI 6.3028E-02 -1.1400E+C1 -0.

1.3100E-03 -2.0814 E-01 2.1701E.CO -0. 8.2997E-02 -1.0000E+00 -3.1494E-01 7.8900E-02 1.0000E+00 -0.

-0. MATRIX B HAS 4 ROWS AND 3 COLUMNS

8.2232E+00 4.1772E+00 1.9426 E-C3 -1.1406E+00 -3.7614E+00 -9.0107 E-04 -0. -0.

3.5644E-C4 -0.

-0. MATRIX CZT HAS 1 ROWS AND 7 COLUMNS 0. 0. 0. 0.

0. 0.

0.
MATRIX CZN HAS 1 ROWS AND 7 COLUMNS 0. 0. 0. 0.

0. 0.

0. MATRIX F HAS 3 ROWS AND 4 COLUMNS -2.73 96E-01 6.3028E-02 -1.14005 +01 0. 1.3100E-03 -2.0814E-01 2.17CIE + CO 0. 8.2957E-02 -1.0000E+00 -3.1494E-CI 7.8900E-02 MATRIX G HAS 3 ROWS AND 3 COLUMNS

8.2232E+00 4.1772E+00 1.94 26 E-C3 -1.1406E+00 -3.7614E+00 -9.01C7E-C4 0. 0.

3.5644 E-04

8.34 96613E-05 MATRIX S-IS HAS 15 ROWS AND 15 COLUMNS

1.0000E+00 -1.705 3E-13 -S.38 145-15 8.8818E-15 1.0214E-14 -1.27906-13 -3.7517E-12 -1.0658E-12 3.6948 E-13 -2.4025E-13

3.0909E-13 -7.9581E-13 2.4727E-12 1.0658E-14 1.91856-13 -2.7569E-12 1.0000E+00 -2.984 3E-13 8.1712E-14 3.3751E-14

Page 8

A PROCEDURE FOR ESTIMATING STABILITY AND CONTROL PARAMETERS FROM FLIGHT TEST DATA BY USING

MAXIMUM LIKELIHOOD METHODS EMPLOYING andbruks A REAL-TIME DIGITAL SYSTEM

amunican activitas roceeding

3. Recipient's Catalog No.

5. Report Date

May 1972 6. Performing Organization Code

1. Report No.

2. Government Accession No. NASA TN D-6735 4. Title and Subtitle

A PROCEDURE FOR ESTIMATING STABILITY AND CONTROL PARAMETERS FROM FLIGHT TEST DATA BY USING MAXI- MUM LIKELIHOOD METHODS EMPLOYING A REAL-TIME

DIGITAL SYSTEM 7. Author(s)

Randall D. Grove; Roland L. Bowles; and

Stanley C. Mayhew, Electronic Associates, Inc. 9. Performing Organization Name and Address

& Performing Organization Report No.

L-8178

10. Work Unit No.

136-62-01-03

11. Contract or Grant No.

NASA Langley Research Center Hampton, Va. 23365

13. Type of Report and Period Covered

Technical Note

12. Sponsoring Agency Name and Address

National Aeronautics and Space Administration Washington, D.C. 20546

14. Sponsoring Agency Code

16. Abstract

A maximum likelihood parameter estimation procedure and program were developed for the extraction of the stability and control derivatives of aircraft from flight test data. Nonlinear six-degree-of-freedom equations describe the aircraft dynamics and are used to derive the sensitivity equations for the method of quasilinearization. The maximum likeli hood function with quasilinearization was used to derive the parameter change equations, the covariance matrices for the parameters and measurement noise, and the performance index function. The maximum likelihood estimator was mechanized into an iterative estimation procedure utilizing a real-time digital computer and graphic display system. This program was developed for 8 measured state variables and 40 parameters. Test cases were conducted with pseudo or simulated data for validation of the estimation procedure and program. This program has been applied to a V/STOL tilt-wing aircraft, a military fighter airplane, and a light single-engine airplane. The appendixes describe in detail the particular nonlinear equations of motion, derivation of the sensitivity equations, addition of accelerations into the algorithm, operational features of the real-time digital system, and test cases.

17. Key Words (Suggested by Author(s) )

18. Distribution Statement Maximum likelihood estimation procedure

Unclassified Sensitivity equations Extraction of aerodynamic parameters

from flight test data 19. Security Classif. (of this report)

20. Security Classif. (of this page) Unclassified

Unclassified

'For sale by the National Technical Information Service, Springfield, Virginia 22151

Page 9

A maximum likelihood parameter estimation procedure and program were developed for the extraction of the stability and control derivatives of aircraft from flight test data. Nonlinear six-degree-of-freedom equations describe the aircraft dynamics and are used to derive the sensitivity equations for the method of quasilinearization. The maximum likelihood function with quasilinearization was used to derive the parameter change equations, the covariance matrices for the parameters and measurement noise, and the performance index function.

The maximum likelihood estimator was mechanized into an iterative estimation procedure utilizing a real-time digital computer and graphic display system. This program was developed for 8 measured state variables and 40 parameters. Test cases were conducted with pseudo or simulated data for validation of the estimation procedure and program. This program has been applied to a V/STOL tilt-wing aircraft, a military fighter airplane, and a light single-engine airplane.

The appendixes describe in detail the particular nonlinear equations of motion, derivation of the sensitivity equations, addition of accelerations into the algorithm, operational features of the real-time digital system, and test cases.

The problem of estimating stability and control parameters of an aircraft from flight test data dates from the early days of flight. The results of early investigations were frequently limited, however, due primarily to insufficient estimation technology and restricted computational resources. Since 1960 there has been significant progress in

Electronic Associates, Inc.

correcting these deficiencies; therefore, it seems appropriate to reevaluate, in a more general setting, the problem of estimating aerodynamic coefficients from flight data.

Parameter estimation is the process of determining the parameters in a mathematical model after having been supplied measured values for the variables of state and the input to the dynamic system. The accuracy of the resulting estimate is degraded by a combination of measurement, modeling, and numerical errors. Obtaining this estimate is a problem in inverse computation and the matter of existence and uniqueness of solution must be resolved at least to some relative degree. The fact that in the defining of the discipline it is necessary to refer to modeling errors and the existence and uniqueness of solution suggests that the expression "parameter estimation" is not adequate to describe the task, the task being a study undertaken by an analyst using the parameter estimation program. A meaningful consideration must include studies of the model definition in relation to the engineering application, the accuracy and precision to which the parameters are computed, and some indication of the uniqueness of solution. It is suggested that the expression "system identification" better represents the task being considered and more accurately implies the interrelated disciplines the study requires. A survey of the general problem of identifying a dynamic system from input-output measurements is given in reference 1.

The objectives of this paper are twofold:

(1) To present the development of an estimation procedure, based on maximum

likelihood statistics, suitable for extracting stability and control param

eters of an aircraft from flight test data for realistic aircraft models (2) To develop a computer program and provide operational features of the estima

tion procedure when implemented on the Langley real-time simulation system The general approach adopted in this paper was based on a maximum likelihood output error method. The case where process noise is present is not considered. The assumed mathematical model for the aircraft was based on a standard six-degree-of-freedom rigid body description with linearized aerodynamic forces and moments. The interactive role of the analyst is discussed as well as various program options which are available. Also included in the paper is a demonstration of the performance of the estimation procedure and the computer program using pseudo flight data. The program developed has been successfully applied to the analysis of flight data for generically different aircraft (refs. 2 and 3).

longitudinal, lateral, and vertical accelerations at center of gravity

longitudinal, lateral, and vertical accelerations at instrument

location

rolling-moment and yawing-moment coefficients at B = oa = Op = 0

pitching-moment coefficient at Qa = de = 0

longitudinal-, lateral-, and normal-force coefficients

longitudinal-force and normal-force coefficients at Ca = e = 0

бе

longitudinal, lateral, and vertical velocity components

Page 10

Early methods for identifying aircraft stability and control parameters are typically characterized as equation error methods. They were basically least-squares estimators which minimize the equation error and are known to give biased estimates in the presence of measurement noise. Details of these various methods can be found in references 4 to 7. Since the unknown parameters enter the equations of motion in a linear fashion, equation error methods are characterized computationally as single step processes and, therefore, are simple to deal with.

More recent parameter identification methods are generally classified as output error techniques. These methods minimized the output error (measurement noise) between the measurements and the true states of the dynamic system and are often used to modify the initial estimates obtained by equation error methods. Unlike equation error methods, the identification problem using output error methods is nonlinear and this requires an iterative solution. Standard output error methods include the NewtonRaphson iteration method (ref. 8), modified Newton-Raphson method (ref. 9), quasilineari zation method (refs. 10 to 14), and various forms of gradient-dependent methods (ref. 15). The quasilinearization and modified Newton-Raphson methods can be shown to be identi cal. The Kalman filter, a sequential estimation method, can be shown for certain restrictive cases to be equivalent to the two techniques just mentioned and, therefore, can be considered an output error method. Output error methods are known to produce unbiased parameter estimates under realistic conditions on the measurement noise. However, if a significant amount of process noise exists, that is, gusts and modeling errors, then the validity of estimates obtained by using output error methods is subject to serious question,

PROBLEM STATEMENT AND ASSUMPTIONS

Assume that the equations of motion of an aircraft are in the form

The essential problem is to estimate the parameter vector a when given the equations of motion of the dynamic system and the measurements M(t) and 5M(t).

The solution of the problem as posed in the preceding paragraph is probably not practical at the present time. Various approaches to the general problem have been attempted (for example, refs. 16 to 18). The results of these studies indicate that for dynamic systems as complicated as aircraft, a solution of the estimation problem requires substantial computational effort. In order to proceed with a solution of the estimation problem which is computationally feasible and for which theoretical techni ques have been developed, the following assumptions are made:

(1) Rigid body aircraft (six degrees of freedom) (2) Only measurement noise (ū(t) = 7 (t) = 0) (3) The measurement noise ñ(ti) is a sequence of independent Gaussian random

where the matrix R1 is unknown.

The foregoing assumptions imply the following conditions:

(1) The aircraft maneuvers do not exceed a dynamic range consistent with linearization of the aerodynamic forces and moments.

(2) Wind gusts, modeling errors, and inaccuracies of measuring physical movement of a control surface are considered sufficiently small to warrant the use of an output error method.

(3) Alinement and location of rate gyros and angle of attack and sideslip instrumentation are of sufficient quality to preclude their inclusion in the measurement model. Anomalies introduced by accelerometer location are included in the measurement model.

PARAMETER ESTIMATION PROCEDURE

The parameter estimation procedure, using the method of maximum likelihood with quasilinearization, is diagramed in figure 1. The basic components are the (1) equations of motion, (2) sensitivity equations, (3) maximum likelihood estimation equations, (4) performance index, and (5) flight test data. These components are described next and the procedure in figure 1 is explained in a subsequent section.

The equations of motion are for a six-degree-of-freedom rigid body aircraft and are stated in detail in appendix A. The equations of motion are written in a general vector form for simplicity in formulating the parameter estimation algorithm.

Page 11

The state vector is defined as

x = x (@,t) = [X1, X2, ..., xn] where xlā,t) is the solution of the equations of motion and xão,t) = xo(t) is the

°

The parameter vector, the components of which are the system parameters, is

T [Q1, Q2, ..., Qp]" Pp

(3) where p' is the total number of system parameters and ã o is the nominal parameter vector. These parameters are the aerodynamic coefficients (stability and control derivatives) and the initial conditions of the state. These parameters must be initialized for the first iteration of the algorithm. The coefficients are initialized by a prior estimate (wind-tunnel data or analysis) and the state is initialized from the flight test data.

The input to the system is

7 = õ(t) = [01, 02, ..

(4) where ő is the control deflection vector with dimension m'. The control vector ő

o is set equal to the measured control input to the aircraft

The sensitivity equations form a basis for the method of quasilinearization and are derived by formally differentiating the equations of motion with respect to the parameters. The sensitivity equations are integrated in parallel with the equations of motion to yield the sensitivity coefficients, which reflect the sensitivity of the nominal solution with respect to each parameter. The method of quasilinearization uses these coefficients to linearize the change in the solution of the nonlinear equations of motion due to a change in the system parameters. Reference 19 describes the use of these parameter sensitivity coefficients in dynamic systems.

Sensitivity coefficients.- Used in the method of quasilinearization is a linear approximation expressing the change in the state vector as a linear function of the changes in the parameters. The expansion of the nominal solution about the nominal parameter vector, neglecting second and higher order terms, is

where each partial derivative (sensitivity coefficient vector) is evaluated along the nominal solution.

ha = [-Q7, AQ,,.

Δα = 2 This matrix equation expresses the change in the nominal solution as a linear function of the parameter changes and the sensitivity coefficients. This equation is used in the expansion of the maximum likelihood function about the nominal parameter vector.

Derivation.- The sensitivity equations are derived from the equations of motion by taking the partial derivative of each equation with respect to each parameter. The sensitivity equations corresponding to the equations of motion in appendix A are derived in detail in appendix B.

By assuming that the derivatives are continuous in ã and t (ref. 20),

that is, the order of differentiation can be interchanged. This result is used to derive the sensitivity equation

This system of equations represents p' sets of n simultaneous first-order linear differential equations with time varying coefficients. The solutions of this system are the sensitivity coefficients exk/adi where i = 1, 2, . . .,p' and

əxk

... k = 1,2,..., n. These coefficients are the elements of the matrix A(t) used in the linear approximation of the change in the nominal solution.

The initial conditions of the sensitivity coefficients corresponding to the aerodynamic parameters are

and the initial conditions of the sensitivity coefficients corresponding to the initial conditions of the state (parameters) are

The maximum likelihood method is used to estimate the stability and control parameters of the aircraft from flight test data. This method has the asymptotic properties of unbiased and minimum variance estimates (ref. 18). Maximum likelihood estimation requires initial parameter values to start the algorithm, but it assumes that the covariance matrix of the measurement noise is unknown.

Maximization of the likelihood function yields the parameter change equations and the covariance matrix for the parameters. These equations yield the changes in the nominal parameters to improve the fit between the measured and calculated variables of state. The covariance matrix gives the variances (or standard deviations) of the parameters, or in an inverse sense the sensitivity of the parameters in the equations of motion. This matrix also indicates the dependency or correlation among the parameters.

Maximization of the likelihood function also yields the covariance matrix for the measurement noise based on the current nominal solution. This matrix gives the vari ances (or standard deviations) of the difference of the measured state and the nominal solution. The inverse of this matrix is used in the parameter change equations as a weighting matrix. The performance index function is derived by substituting the covariince matrix of the measurement noise into the likelihood function (ref. 21). The performance index function derived is the determinant of the covariance matrix of the measurement noise which is defined as the criterion for the maximum likelihood method.

Measurement noise. - Let the measurement noise be

ñ(ti) = M(ti) - xlāti) = [21, 72,.... ., "

(i = 1, 2, ...,N)

(11) where N is the number of data points, x M(ti) is the measured data, šlā,ti) is the calculated solution for the true ā values, and ñ(ti) are independent Gaussian vari ables with zero mean. The noise is assumed to have the statistical properties of E[(

(12) e [bi(t) **(!;)] R10ij

(13) where E denotes the mathematical expectation and R1 is the unknown covariance matrix of the measurement noise.

Maximum likelihood function.- The maximum likelihood estimation of Ad and R1 is obtained by maximizing the likelihood function Lão+A2, Rı) with respect to da and R1, respectively. Although the term Ad is not explicit in the function Lã,Ra), its use is evident in the maximization procedure.

where |R1| denotes the determinant, R;? denotes the inverse of the symmetric covariance matrix Rį, and || |

= Tr;'s.

Fr

Parameter change estimation.- The maximum likelihood function for the nominal solution is

N 1

(15) i=1

and R1 is estimated from the nominal solution. The likelihood function is expressed for a change in the nominal solution by using equation (7) as

#Mti) - (ao+aa, tj) = xM(ti) - toti) - A(ti) Aa

(

To maximize this equation, the partial derivatives of L(@o+Ãa, Rı) with respect to each parameter change Adi are set to zero; that is,

Page 12

For p'> n, there exists more unknowns than equations; the system is overdetermined by calculating the nominal solution and sensitivity coefficients at N data points. The solution of AR is given by

-1 Δα = B В

(21)

where the new nominal parameter vector is incremented by AQ. A necessary condition for this to be the estimate to maximize the likelihood function is that the matrix BTW1B must be positive definite.

Maximum likelihood methods (ref. 18) give asymptotically unbiased estimates and

is the error covariance matrix for the following estimated

the inverse of BTW,B

[] E[aa]

The error covariance matrix for the estimated parameters is (ref. 22)

Estimation of measurement noise statistics.- The likelihood function of equation (17) for the change in the nominal solution is

where Ri denotes the unknown covariance matrix. The estimated value of the covariance matrix R1 is denoted by R (N).

In a similar maximization procedure as for Aa, the likelihood function is maxi mized with respect to R1. The derivative with respect to a matrix is defined as follows:

(') The derivative of llao+Ãa, Rı) with respect to R1 is set to zero. But (ref. 21)

hy (aot#, Ry) = 0

ha

where R (N) is the predicted estimate of the covariance matrix for N data points due to the change in the nominal solution.

The equation for calculating the estimate of the covariance matrix used in this algorithm is

R (N) 4 Estimate of Ry

The schemes for calculating the estimate of R1 are discussed later.

The matrix R (N) is written as

Performance index. - The performance index or index function evaluation gives a measure of performance for the iterative estimation procedure. Selection of the index function for the maximum likelihood estimator is an important condition as to whether the estimation procedure converges to the true parameter values. (See ref. 21.)

The index function is derived from the likelihood function (ref. 21). Substituting

the covariance matrix estimate R (N) (eq. (30)) into the likelihood function 1(@°,R1)

Thus maximization of the likelihood function is equivalent to the minimization of

which is defined as the index function for the maximum likelihood estimation procedure. The minimization of R (N) with respect to ad yields equivalent parameter change equations as in equation (19).

The flight test data are composed of the onboard instrument measurements of the aircraft behavior and are assumed to be the output of the aircraft mathematical model superimposed with instrument noise. These data contain many individual aircraft maneuvers stored on one magnetic tape, with each maneuver easily accessible to the central memory of the computer. These data are used for comparison with the mathematical model output and for initialization of and control input to the equations of motion. The

-M , X

) i = 1, N, control deflections corresponding to the equations of motion.

measurements, Mti) and 5M(ti) for i = 1, 2, ..., N, are known for all states and

The steps in the maximum likelihood estimation procedure, corresponding to figure 1, are as follows:

(1) Initialize the system parameters, where ão denotes the nominal or current values of the parameters.

(2) Integrate the equations of motion and the sensitivity equations to obtain the nominal solution and the sensitivity coefficient matrix, respectively.

(3) Form the comparisons of the flight test data and nominal solution for each data point time ti, where i = 1, 2,.. N and t1 = 0 and tn = T.

(4) Form the maximum likelihood estimation equations from the comparisons in step (3) and the sensitivity coefficient matrix in step (2) (dash lines indicate accumulation of information over the flight test time period T).

) (5) Calculate the performance index Julão).

(6) Calculate the parameter changes AQ and the statistical information matrix R (N).

(7) Update the nominal parameter values in step (1) to start the next iteration procedure and repeat steps until convergence.

Each iteration of the procedure extends over the flight test time period T and results in the update of the parameters. Evaluations within the period are at specified data point times tị, where the intervals ti+1 - ti are integer multiples of the integration step size. The integration step size is made compatible with the flight data intervals and the problem dynamics.

Acceleration measurements and equations were added later to improve the estima tion procedure and the addition is presented in appendix C.

COMPUTATIONAL CONSIDERATIONS

The flight tests do not always necessitate the use of all the state variables and parameters in the estimation algorithm (n state variables and p' parameters) for specific cases. These cases involve only a specific portion of the program, as with an excitation of only the longitudinal motion of the aircraft. The total number of differential equations in the algorithm for evaluation and integration is n + np'; in appendixes A and B, n = 8 and p' = 40.

The parameter estimation algorithm was programed with a variable dimensioning capability with respect to the number of state variables and parameters necessary for any specific case. The analyst through the operational control features (appendix D) could select any subsets of the state variables and parameters to be active in the parameter estimation algorithm. Computer program parameters were activated for each state variable and each parameter desired. This operation generated two sequences of numbers specifying the state variables and parameters which were active in the algorithm and neglected the remaining ones.

This variable dimensioning of the algorithm allowed flexibility in the parameter identification study in that the analyst could alter the program easily for each specific aircraft maneuver or computer run. In addition, the number of integrations was reduced.

Page 13

In the calculation of Ad the matrix equations (eq. (20)) were

If the matrix B and the vector ħ were calculated for N data points, this would result in the storage of an nN Xp' matrix and an nN vector. The storage would increase as the number of data points N increased. To eliminate the storage being dependent on N, equation (19), which is restated, was used:

In these equations the matrix products are formed as a function of time and the dimensions of the matrix products were not a function of the number of data points N. In fact, the dimensions depended only on p' and n, the number of parameters and state variables.

In the calculation of the measurement noise covariance matrix, three computational schemes can be used. The matrix can be updated (1) on the same iteration as AQ, (2) one iteration behind Ād, and (3) with a two step procedure. Scheme (1) uses equation (29): N

T N

Δα N

i=1

In this equation the matrix Asti) and the vector ñ(ti) must be stored for each increment of time until AQ is calculated. Scheme (2) is easy to incorporate into the program by using equation (30):

R (N) 4 Estimate of R1

In this equation the covariance matrix is calculated from the nominal solution and not with the predicted change in the nominal solution. The matrix is one iteration behind in the algorithm but the effect is negligible when the change in the solution is small, that is, when convergence is achieved.

Scheme (3) uses a two step process for each parameter update. The first step is to calculate Ad in the usual manner. The second step is to update the parameters and integrate only the equations of motion. Thus, the covariance matrix is N

T -M

-M X α+Δα

(36)

Scheme (3) is similar to scheme (1) in that scheme (1) uses the predicted change, whereas scheme (3) uses the calculated change in the nominal solution. Scheme (3) approaches scheme (2) when AQ approaches zero.

Scheme (2) was used in the parameter estimation program with the option of using scheme (3). In test cases using schemes (2) and (3), the indication was that the difference is not significant.

The testing procedure used pseudo flight data in checking the maximum likelihood estimation algorithm. The data were generated by integrating the equations of motion and then adding measurement noise, all states assumed being measured. The measurement noise was sequences of pseudorandom numbers (random within the capability of a digital computer) having the normal (Gaussian) distribution with zero mean and known standard deviation. These data were assumed to be the flight test data or measured data for specified parameter values. The parameter values were then offset to become the nominal parameter values for the parameter estimation algorithm.

Test cases using the pseudo flight data were conducted for the longitudinal motion of the aircraft and are presented in appendix E. The maximum likelihood algorithm computed the standard deviation of the measurement noise and the parameter values and their standard deviations; no statistical information was assumed concerning the noise. Results obtained from the test cases indicated that the calculated parameter values and standard deviations of the noise were converging to the true values.

A maximum likelihood parameter estimation procedure and program have been developed and validated for the extraction of the stability and control derivatives of

aircraft from flight test data. A nonlinear six-degree-of-freedom aircraft mathematical model was used in the derivation of the sensitivity equations. Instrument measurement noise was accounted for by the maximum likelihood estimator. The program was developed for 8 measured state variables and 40 parameters, from which subsets could be selected for program operation. Real-time digital simulation and graphic display provided the analyst with interactive control and display capabilities during the study. The program has been applied to a V/STOL tilt-wing aircraft, a military fighter airplane, and a light single-engine airplane.

Langley Research Center, National Aeronautics and Space Administration,

Hampton, Va., April 6, 1972.

The equations of motion are stated for reference and in particular for the derivation of the sensitivity equations in appendix B. The following equations of motion are for a V/STOL tilt-wing aircraft:

Тх

qc + +

(A1)

ů = -qw + rv - g sin 0 + 1 v2s(Cx,0

lo cx+

morbo) 1x - Ly , xZco - gr) +

1 v2sb(C1,0 + Cngß + Cn; 2V i-po) ,

Cap po la

Mz

cos dow) * Men 1 Vss

Trim conditions of the aircraft are added by substituting, respectively, for da, da, Oe, and or in equations (A1) to (A6) the following terms:

where the subscript t denotes the trim conditions (values) of the aircraft.

The equations of motion are altered in two ways: (1) solve for i (eq. (A2)) explicitly, and (2) decouple the p (eq. (A4)) and · (eq. (A6)) equations. The equations of motion are then written in simplified notation for use in the derivation of the sensitivity equations.

The equations of motion are

= F (3,2,0,0,0,0,3,3)

Page 14

v=7(3,2,5,v,ca)

Tx g sin + a, v2 (Cx1 + Cx2)

Tx aq v2 (Cx,0 + Cxoa da) + a2 V (Cxq4) + m v = F (5,7,6,v,8)

Ty - ru + pw + g cos o sin 0 + aj v2 (Cy1+ Cy2) + - azCy;

Тү - ru + pw + g cos e sin $ + aqv2(CY,0 + Cypß + CYoroz! + ag V (Cypp + Cyrt) +

+

=as_b2qr + Ixzpq + a_v?(C21 + G12) + aqvs?(C23) + Mx ag + b [bgpq - Ixzar + a, v2 (€n1 + 6n2) – 24Vs?(Cn3) + M2]

- a + Ixzpq + a 4v2(Cl2B + C267°r) + aş v(CE8 + Cap® + Ctr")

Cepp + 24Vs? (Cona) + Mx] + b [13] - Ixzar + axv?(C1,0 + Cng8 + Cn6402)

bzpq ,

Cnor agv (Cn38 + Cnpp + Cnpt) - aqvs ? (Cn62@a) + Mz

DETAILS OF DERIVATION OF SENSITIVITY EQUATIONS

The sensitivity equations for the method of quasilinearization are derived in detail for the equations of motion in appendix A.

' t

(i, k = 1, 2, ...,

., 8) The functions F2 and Fz do not contain dg or B. Thus,

F3

da

,

The following equations are used in the derivation of the sensitivity equations:

The numbers in the parentheses above each term indicate the derivations in which they are used; this is in reference to the equations of motion used in the derivation of sensitivity equations.

(1) Sensitivity equations derived from i equation (eq. (A14)): 1

. (:

Page 15

(2) Sensitivity equations derived from y equation (eq. (A15)):

a, v2.cy1 + Cy2 - Cyyv (u2 + w251/3+a,v2Cy7(u2 + w29-1/2

азсүз + a, w2cyl + Cya - Cyju (u 2 + m2)-1/2

(3) Sensitivity equations derived from w equation (eq. (A16)):

(4) Sensitivity equations derived from pequation (eq. (А17)):

p= F(x,8,5,7,В,в) = aer (я, а,б,у,В,в) + b, F(x, а,б,у,В,в) x

V

Page 16

(5) Sensitivity equations derived from ģ equation (eq. (A18)):

(6) Sensitivity equations derived from equation (eq. (A19)):

where all the terms have been defined in the derivation of the sensitivity equations for P equation.

(7) Sensitivity equations derived from ė equation (eq. (A20)):

(8) Sensitivity equations derived from o equation (eq. (A21)):

ADDITION OF ACCELERATIONS INTO ALGORITHM

The acceleration measurements and equations were included in the parameter estimation algorithm to improve the extraction process. They are used with or can replace the linear velocities u, v, and w. The acceleration equations were transformed to the instrument location from the center of gravity.

The equations are transformed to the instrument location (ref. 23), that is,

where xa,Ya,za are the center-of-gravity offsets of the accelerometer measurements.

The acceleration sensitivity equations were derived in terms of the sensitivity equations and coefficients stated in appendix B and need only to be evaluated and not integrated.

The maximum likelihood function was modified to include the accelerations in the algorithm. The likelihood function is

Page 17

and R2 is measurement noise covariance matrix with accelerations included; that is,

The maximization procedure is similar to the previous developments and similar esti mation equations can be derived.

The computer program was written in FORTRAN IV language (75 000 octal locations) and run on the Control Data series 6000 digital computer complex, a major application being real-time simulation (RTS) (ref. 24). Incorporated in the RTS system are the cathode ray tube (CRT) graphic display units.

The computer program was mechanized into an iterative estimation procedure with manual interactive control through the utilization of the RTS system. The operational diagram of the RTS system is shown in figure 2, the main components being the computer complex, control console, and CRT. The remaining equipment is for output of information and monitoring the program. Figure 3(a) shows a photograph of the program control station and figure 3(b) shows a closeup of the control console.

The maximum likelihood estimation program resides in the central memory of the computer. The analyst investigating the stability and control derivatives of the aircraft has direct control of the computer program through the control console. The control console has mode control switches for program operation, a data entry keyboard for inputing program parameters and logic controls, logical switches for program options, and indicator lights for program status. The digital decimal display was used to monitor continuously any selected parameter or variable in the program, particularly the performance index function.

The CRT displayed the flight test maneuver at the start of each iteration. The response of the equations of motion was plotted simultaneously as it was computed in the digital program and was plotted with the flight test maneuver for direct comparison. This display permitted quick analysis of each flight test case on an iteration to iteration basis. Figure 4 shows three CRT displays; they are a portion of the dynamic check. (Note that symbols on CRT display in figure 4 are not the standard symbols defined in the Symbols section.) Permanent pictures of the CRT displays were obtained directly from the hard copy unit in the facility or from postprocessing of the plotting routine in the computer program. The plotting routine generated figure 4 by plotting the CRT display and adding the additional labeling on the right.

The information output consisted mainly of calculated data preselected by the analyst and routed to the high-speed printer. The information could be printed for any iteration by activating a logical control switch. The printer is located in the proximity of the program control station and easily accessible to the computer operator. The output consisted of the following information:

(2) covariance matrix of the measurement noise (Rs(N)), its determinant, and its

(3) variables of state and parameters active in algorithm (4) nominal parameter values a; (i = 1, 2, ...,

., p') (5) calculated changes in the nominal values Adi (i = 1, 2,...,p') (6) covariance matrix for the parameters in a modified form more readable to the

analyst

in that the standard derivations and correlation coefficients are expressed explicitly.

The integration scheme that was used for the parameter estimation procedure was second-order Adams-Bashforth, a 1-pass integration scheme. The real-time system provides the option of four integration schemes: (1) second-order Runge-Kutta (2 pass), (2) fourth-order Runge-Kutta (4 pass), (3) second-order Adams-Moulton (2 pass), and (4) fourth-order Adams-Moulton (2 pass). The Adams-Bashforth scheme was obtained by program logic limiting scheme (3) to a 1-pass operation; this thus reduced the computation time for the integration of the equations of motion and sensitivity equations. The dynamic check was run by using the Adams-Bashforth scheme and scheme (2); the indication was that the Adams-Bashforth scheme was adequate for the parameter estimation procedure.

The test cases were for the longitudinal motion of the aircraft and included the effects of measurement noise on the pseudo data.

The test cases were for different noise levels of 1, 2, 5, and 10 percent on the variables u, w, q, and e. Table I shows the known and calculated standard deviations

e of the noise for each percent level. The calculated standard deviations agreed closely with the known input, with an error of less than 1 percent. Table II shows the true and calculated parameter values and their standard deviations at each noise level. The calculated parameter values indicate convergence to within one standard deviation of the true values based on a fixed number of iterations. Figure 5 shows the CRT display of the converged solution and the pseudo flight data. (Note that symbols on CRT display in figure 5

5 are not the standard symbols defined in the Symbols section.)

1. Åström, K. J.; and Eykhoff, P.: System Identification, A Survey. Identification and

Process Parameter Estimation, Pt. I, 0.1, Int. Fed. Automat. Contr., June 1970.

2. Steinmetz, George G.; Parrish, Russell V.; and Bowles, Roland L.: Longitudinal

Stability and Control Derivatives of a Jet Fighter Airplane Extracted From Flight Test Data by Utilizing Maximum Likelihood Estimation. NASA TN D-6532, 1972.

3. Suit, William T.: Aerodynamic Parameters of the Navion Airplane Extracted From

Flight Data. NASA TN D-6643, 1972.

4. Gerlach, O. H.: Determination of Performance and Stability Parameters Fror Nonsteady Flight Test Maneuvers. Preprint] 700236, Soc. Automot. Eng., Mar. 1970.

( 5. Greenberg, Harry: A Survey of Methods for Determining Stability Parameters of an

Airplane From Dynamic Flight Measurements. NACA TN 2340, 1951.

6. Milliken, William F., Jr.: Progress in Dynamic Stability and Control Research. J.

Aeronaut. Sci., vol. 14, no. 9, Sept. 1947, pp. 493-519.

7. Shinbrot, Marvin: On the Analysis of Linear and Nonlinear Dynamical Systems From

Transient-Response Data. NACA TN 3288, 1954.

8. Ortega, J. M.; and Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, Inc., 1970.

,

9. Taylor, Lawrence W., Jr.; Iliff, Kenneth W.; and Powers, Bruce G.: A Comparison of

Newton-Raphson and Other Methods for Determining Stability Derivatives From Flight Data. AIAA Paper No. 69-315, Mar. 1969.

10. Kumar, K. S. Prasanna; and Sridhar, R.: On the Identification of Control Systems by

the Quasi-Linearization Method. IEEE Trans. Automat. Contr., vol. AC-9, no. 2, Apr. 1964, pp. 151-154.

11. Larson, Duane B.: Identification of Parameters by the Method of Quasilinearization.

CAL Rep. No. 164, Cornell Aeronautical Lab., Inc., May 14, 1968.

12. Denery, Dallas G.: An Identification Algorithm Which Is Insensitive to Initial Parame

ter Estimates. AIAA Paper No. 70-34, Jan. 1970.

13. Young, Peter C.: Process Parameter Estimation and Self Adaptive Control. Theory

of Self-Adaptive Control Systems, P. H. Hammond, ed., Plenum Press, 1966, pp. 118-140.

14. Dolbin, Benjamin H., Jr.: A Differential Correction Method for the Identification of

Airplane Parameters From Flight Test Data. Proceedings of the National Electronics Conference, Vol. XXV, 1969, pp. 90-94.

Page 18

15. Bard, Yonathan: Comparison of Gradient Methods for the Solution of Nonlinear

Parameter Estimation Problems. SIAM J. Numerical Anal., vol. 7, no. 1, Mar. 1970, pp. 157-186.

16. Kerr, R. B.: Bayesian Identification of System Parameters. J. Eng. Math., vol. 4,

no. 3, July 1970, pp. 273-281.

17. Tyler, James S.; Powell, J. David; and Mehra, Raman K.: The Use of Smoothing and

Other Advanced Techniques for VTOL Aircraft Parameter Identification. Contract No. N00019-69-C-0534, Systems Control Inc., June 30, 1970.

18. Mehra, Raman K.: Maximum Likelihood Identification of Aircraft Parameters.

Eleventh Joint Automatic Control Conference of the American Automatic Control Council, Amer. Soc. Mech. Eng., c.1970, pp. 442-444.

19. Meissinger, Hans F.: The Use of Parameter Influence Coefficients in Computer

Analysis of Dynamic Systems. Simulation, vol. 3, no. 2, Aug. 1964, pp. 53-63.

20. Bekey, George A.; Optimization of Multiparameter Systems by Hybrid Computer

Techniques. Pt. I. Simulation, vol. 2, no. 2, Feb. 1964, pp. 19-32. 21. Kashyap, R. L.: Maximum Likelihood Identification of Stochastic Linear Systems.

IEEE Trans. Automat. Contr., vol. AC-15, no. 1, Feb. 1970, pp. 25-34.

22. Compton, Harold R.: A Study of the Accuracy of Estimating the Orbital Elements of

a Lunar Satellite by Using Range and Range-Rate Measurements. NASA TN D-3140, 1966.

23. Wagner, William E.; and Serold, Arno C.; Formulation on Statistical Trajectory

Estimation Programs. NASA CR-1482, 1970.

24. White, Ellis: Eastern Simulation Council Meeting. Simulation, vol. 12, no. 2,

Feb. 1969, pp. 53-56.

TABLE II.- PARAMETER VALUES AND STANDARD DEVIATIONS USING MAXIMUM LIKELIHOOD ESTIMATION

Calculated Standard Calculated Standard Calculated Standard

value deviation value deviation value deviation

Figure 2.- Operational diagram of Langley real-time simulation system

for parameter estimation.

Page 19

Page 20

15. Supplementary Notes

The MSC Director waived the use of the International System of Units (SI) for this Apollo Experience Report, because, in his judgment, use of SI Units would impair the usefulness

of the report or result in excessive cost. 16. Abstract

Because of the increased complexity of the Apollo manned spacecraft, corresponding increases in the capability, and thus complexity, of the ground checkout equipment were required. In this report, the acceptance checkout equipment for the Apollo spacecraft is described, and the history of the major equipment modifications that were required to meet the Apollo Program checkout requirements is traced. Some major problem areas are outlined, and a discussion of future checkout methods is included. The concept of the future checkout methods presented in this report provides for an increase in test-equipment standardization among NASA programs and among all testing phases within a program. The capability for increased automation and reduction in the test-equipment inventory is provided in the proposed concept.

ACCEPTANCE CHECKOUT EQUIPMENT FOR THE APOLLO SPACECRAFT

By 1. J. Burtzlaff Manned Spacecraft Center

The acceptance checkout equipment for spacecraft (ACE-SC) system is a computerized checkout system which provides for centralized, programed control of spacecraft checkout operations in the Apollo Program. Most of the changes in the evolution of the ACE-SC system capability occurred in the computer programs. The initial design of the ACE-SC was based on a concept of simple, computer-controlled, commandand-response checkout of the Apollo spacecraft systems. Because of the developmental modifications to the Apollo spacecraft, changes were required in the ACE-SC system to keep pace with the Apollo Program test requirements.

The first major update was in the checkout-system software and was required to meet the test requirements for the guidance, navigation, and control system of the spacecraft. The second major change was implemented to optimize the use of ACE-SC memory. Because of the problems associated with accomplishing integrated testing of the Apollo spacecraft, approval was given in February 1968 for the addition of more memory capacity in the checkout system. Further system improvements included the capability for automatic data compression and automatic test sequencing during spacecraft checkout.

The ACE-SC system proved to be an effective tool; however, several undesirable conditions arose which should be corrected in the checkout systems and procedures used for future programs. These conditions can be summarized as follows.

1. Total ground operations were not integrated within the ground checkout system.

2. Different checkout systems were used for the booster and the spacecraft.

3. Extensive use of special test equipment increased program costs.

Page 21

The unified-test-equipment concept, which has been proposed and is presently under development at the NASA Manned Spacecraft Center, significantly reduces these conditions in the following manner.

1. General-purpose test equipment, rather than costly special test equipment that cannot be reused in subsequent test phases, is to be used during the vendor testing and the subsystem buildup.

2. The test equipment is to be modular and expandable to meet the integratedacceptance-test requirements during prelaunch operations.

3. The total test-equipment complex is to be designed to use significantly less ground-test equipment for an integrated test than was used in the Apollo Program.

4. Upon completion of factory testing and integrated acceptance testing, the test equipment is to be used for operational spacecraft testing and the costly development of new test equipment thus avoided.

5. The test equipment is to be designed to adapt easily to other NASA program checkout requirements.

The experience gained with the ACE-SC used for the Apollo spacecraft is expected to lead to the development of checkout systems which will reduce significantly the cost of testing and checkout activities in future programs.

The effectiveness of the testing and checkout phases of the Apollo missions has played a significant role in the accomplishment of a successful lunar- landing mission. The acceptance checkout equipment for spacecraft (ACE-SC) system has supported all Apollo spacecraft factory-acceptance testing, environmental-simulation testing, and prelaunch testing at the NASA John F. Kennedy Space Center (KSC). No major delays have been attributed to a failure of the ACE-SC system. Changes in testing and checkout requirements were important in stimulating the evolution of the ACE-SC system. Significant problems were involved in implementing major system changes and testphilosophy changes in the middle of a program as dynamic as the Apollo Program has been. This report provides a basic description of the ACE-SC system, a statement of the original goals and requirements for the ACE-SC system, a discussion of the ACE-SC system evolution and the problem areas encountered, and a definition of proposed checkout-system concepts that will resolve some of the undesirable conditions encountered in the use of the ACE-SC system during the Apollo Program.

The information used in this report was accumulated through working experience with the ACE-SC system over a period of several years. The NASA and contractor personnel who have contributed to the success of the ACE-SC program (and hence to the information presented in this report) are acknowledged for the support and significant contributions they have provided.

DESCRIPTION OF THE ACE-SC SYSTEM

The ACE-SC system is a computerized system that provides for centralized, programed control of spacecraft checkout operations. The ACE-SC system provides for manual, semiautomatic, and automatic operational modes to accommodate subsystem testing, integrated-system testing, and launch support. For the purpose of this discussion, the ACE-SC system (fig. 1) is considered in two parts, which are (1) the command subsystem and (2) the displayand-recording subsystem, also referred to as up-link and down-link subsystems, respectively.

Figure 1. - Simplified data-flow diagram

of the ACE-SC.

The testing of spacecraft subsystems is controlled from selection-to-activaterandom-testing (START) modules that are located in associated system consoles (fig. 2). Spacecraft subsystems that are tested by the various START modules include the environmental-control subsystem, the fuel-cell and cryogenics subsystem, the power and sequential subsystem, the guidance-and-navigation subsystem, the stabilization-andcontrol subsystem, the propulsion subsystem, the biomedical subsystem, the instrumentation subsystem, and the communications subsystem. The START modules facilitate the input of the appropriate command selections, computer-subroutine selections, or spacecraft-guidance-computer information to the spacecraft.

Checkout-subsystem consoles can operate simultaneously with and independently of other subsystem consoles. Each console has a variety of test-command capabilities which are necessary for the testing and checkout of a particular spacecraft subsystem. The up-link computer (fig. 3) interprets and reacts to the commands initiated from the subsystem console. A specific command may instruct the computer either to initiate an automatic test or to transmit a single command to the spacecraft. The signals generated by the ACE-SC ground station are transmitted by hardlines to the spacecraft vicinity, where the command signals are distributed by the digital test command system (DTCS). Redundant transmission checks and proper transmission verification tests are accomplished to ensure maximum confidence in proper command transmission.

Figure 2. - Typical ACE-SC control room.

Test data to be processed by the downlink equipment are obtained from sensors in the spacecraft, from the carryon equipment, and from the ground-support equipment (GSE). The test data are transmitted as serial pulse-code-modulation (PCM) data to the recording and display equipment which receives, records, and displays the spacecraft performance data, as required for the particular test procedure being conducted. The digital acquisition and decommutation equipment synchronizes on the incoming serial PCM bit stream, decommutates the data, and routes the data for appropriate processing or display (or both).

The down-link computer conducts the required processing, which includes such functions as predetermined limit checks, engineering unit conversions, data compression, and a variety of special processing for each spacecraft subsystem. Display information is transferred to a symbol generator and storage unit, which generates alphanumeric-character display signals for display on the appropriate subsystem console. The display can be specific parameters that will blink if the tolerance is exceeded, unique outputs based on special processing requirements, or status information for automatic test sequences.

INITIAL APOLLO PROGRAM REQUIREMENTS FOR THE ACE-SC SYSTEM

It was recognized early in the Apollo Program that the magnitude of the Apollo command and service module (CSM) and lunar module (LM) checkout requirements precluded the use of manual test equipment because of the excessive checkout time that would have been required. The ACE-SC system underwent a significant evolution from the initial Apollo Program requirements to the final capability that was used in the Apollo 11 lunar- landing mission. A major part of ACE-SC system evolution took place in the ACE-SC computer-programing system, which had to be modified to meet the evolving test requirements as the missions became more complex.

The initial requirements placed on the ACE-SC system for the early Apollo Program test phases were based on a concept of providing simple, computer-controlled

command-and-response checkout of the Apollo spacecraft. The capability to test several spacecraft subsystems simultaneously from one ACE-SC ground station also was included in the initial requirements. Basically, the requirements included the capability to perform the following functions.

1. To send a wide variety of test stimuli to the spacecraft from any ACE-SC subsystem console

To monitor and display several hundred spacecraft parameters in real time

3. To provide special processing and formatting of approximately 400 spacecraft parameters for real-time display

4. To provide complete documentation of the tests by recording all commands, recording digitally all out-of-limit parameters, recording raw PCM data, and recording selected parameters on strip-chart recorders

EVOLUTION OF THE ACE-SC SYSTEM

Because of the evolutionary development of the Apollo spacecraft, changes were required in the ACE-SC system to keep pace with Apollo Program test requirements. The number of spacecraft parameters to be processed increased substantially throughout the program. Special computer programs had to be written to accomplish Apollo guidance-and-navigation-subsystem testing, to achieve automatic control for static firing of the service-propulsion-subsystem engine, to control simulated altitude tests, to provide emergency detanking and spacecraft-system safing, and to meet many other requirements which caused the evolution of the ACE-SC hardware and software. The major advances which were accomplished during the ACE-SC system evolution are discussed in the following paragraphs.

Increased requirements. - The first major changes to the ACE-SC system capability were accomplished by modification of the system software. These changes were necessary to meet the guidance-and-navigation-subsystem test requirements. The modifications facilitated the interpretation by the ACE-SC system of guidance-computer digital outputs, the loading of guidance-and-navigation tests from the ACE-SC system, and the recording of special guidance-computer data. In addition, the ACE-SC system software was expanded for additional spacecraft parameters and some special real-time data-processing functions.

Memory limitations. - The next major system software change was made because of the ACE-SC system memory limitations. The addition of numerous parameters on the spacecraft, the addition of special processing requirements, and the addition of new GSE and launch-support requirements at the KSC caused the ACE-SC system memory capacity to be exceeded in 1966. The memory capacity of the ACE-SC system could not be increased at the time; therefore, a major effort was initiated to rewrite the system software, with the emphasis on conserving memory space. The system software

was altered to provide the capability to call alternate test loads, which caused serialization of testing and constrained the amount of integrated testing that could be accomplished. Control programs and subroutines were integrated, and common subroutines were used where possible to conserve computer memory. This method of resolving the immediate problem caused the computer processing capability to be exceeded when new programs were added to meet increasing test requirements. The trade-offs between memory capacity and computer processing time must be considered carefully for each checkout requirement to ensure that the optimum system software efficiency is obtained. Checkout systems for future programs must be designed for modular adaptation (of both hardware and software) to different and increasing testing and checkout requirements.

Expansion of the ACE-SC System Memory

As the testing and checkout activities evolved, it became apparent that additional memory capacity for the ACE-SC system would be required to provide the type of integrated testing necessary for the Apollo Program. Serialization of testing at the LM prime-contractor facility and at the KSC, as a result of alternate computer-program loads, would have had an intolerable impact on the schedule if the additional memory capacity had not been provided. The addition of 24 000 words of memory capacity was approved in February 1968, and modifications to the system software were made to allow use of the additional memory capacity.

Data-Compression Techniques

The amount of testing and checkout data that had to be processed for the Apollo Program became such an overwhelming problem that a requirement was generated by the Apollo Spacecraft Program Office to provide the capability to reduce significantly the amount of test data. This capability was provided in the ACE-SC system by initiating a decommutator design change which enabled the decommutator to perform fixedlimit checks and dynamic-limit checks (on a change greater than 1.2 percent or on any data change) in conjunction with a computer program that filed only significant changes on digital tapes. In addition, this modification relieved the computer from some of its routine limit-checking functions and provided for the transmission of data to the computer in a direct-access mode. By providing for the transmission of data to the computer in a direct-access mode, a significant step was taken toward achieving the capability for automatic test sequencing during the testing and checkout of the Apollo spacecraft.

One of the most significant steps in the evolution of the testing and checkout capa bility for the Apollo Program was the implementation of a computer program, which was referred to as the Adaptive Intercommunications Routine (ADAP). The ADAP provided the ACE-SC with the capability for closed-loop automatic test sequencing. "Adaptive' refers to the capability of the program to adapt to new test requirements and to add new test sequences without affecting other computer-program functions. The term

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"Intercommunications" refers to the closed-loop aspects of the programs, which provide the capability to transmit up-link commands to the spacecraft, to receive the downlink response, and to initiate the appropriate action in a completely automatic mode. The status of the automatic operations is displayed in the ACE-SC control room, and appropriate override and restart-recycle capability is provided. Test sequences are called from digital tape, as required to meet the particular test-procedure requirements.

The next generation of the ADAP system is being designed and will be implemented for Apollo telescope mount testing at the NASA Marshall Space Flight Center (MSFC). Of major significance is the capability of the ADAP program to generate test sequences in a higher level, engineering-oriented language to help bridge the communications gap between engineering and programing personnel and to reduce the time required for definition of computer-programing requirements.

Prior to the implementation of the ADAP program, there was no capability to provide special processing of checkout data without stopping the test or going to another available ACE-SC station. This problem has been reduced somewhat by using the ADAP test-sequence memory area and loading special data reduction programs to support "quick-look" requirements.

Present Checkout-System Problems

The ACE-SC system used on the Apollo missions does not adequately support all phases of ground testing that are desired in checkout systems for future programs. Vendor-acceptance and preinstallation testing are now delegated to special bench-test and acceptance-test equipment. The present ACE-SC system role begins after the subsystems are installed in the spacecraft. A complete ACE-SC station (control-room and computer-complex combination) is configured for a CSM or an LM, regardless of the magnitude of testing required (that is, subsystem testing or integrated testing). A complete ACE-SC station is required for each vehicle (CSM and LM) during combined CSM and LM testing; consequently, redundancy occurs in the system hardware elements. Station reconfiguration and site activation are lengthy processes that involve many test and operations personnel.

Lack of adequate control and monitoring capability in spacecraft systems and GSE prevents the automation of the entire checkout operation by using the present ACE-SC system. The entire subsystem diagnostic capability is vested in the ACE-SC computer system, with no provisions for a built-in self-test in the spacecraft subsystems. The lack of provisions for a built-in self-test is a major constraint on the simplification of ground checkout systems. Future spacecraft subsystems should have built-in test logic to ensure adequate and efficient checkout operations.

Digital-Test Command-System Problems

One of the most significant problems encountered with the ACE-SC system was in the spacecraft-to-ACE-SC interface equipment. This equipment, which is referred to as the DTCS, is the means by which commands are decoded and routed to the appropriate spacecraft test point.

On December 14, 1968, spacecraft 104 (the Apollo 11 command module) was inadvertently powered down as a result of the erroneous resetting of two latching relays within the DTCS located on Mobile Launcher no. 2. The cause of the problem was be

. lieved to have been a transient voltage generated by some external source. There were subsequently 15 additional occurrences of the DTCS problem.

A concentrated effort to solve the problem was initiated by the NASA Manned Spacecraft Center (MSC), the KSC, and all contractors that were associated with the use or manufacture of DTCS equipment. Because of the random nature of the problem, the source of the transients has never been definitely determined.

Several significant points should be made for the purpose of improving GSE and facilities for future space programs. Improvements that need to be made can be summarized as follows.

1. Susceptibility of critical GSE to transient voltage levels

2. Complexity of GSE cabling and patching arrangements

3. Adequacy of launch facility grounding systems, which require careful attention during facility design and subsequent configuration control procedures that are comparable to those used for spacecraft systems

An attempt to improve these areas in the early stages of future programs will not only reduce hazardous situations but should help reduce the time required to perform prelaunch operations.

Mechanical and Radio-Frequency System Interface Problem

One of the problems that impeded the advancement of automated checkout operations was the lack of digital interfacing of mechanical, hydraulic, and radio-frequency (rf) test-support equipment with computerized checkout systems. Much of the time involved in checkout operations is consumed because voice communications are required in order to coordinate the manual operation of valves, solenoids, and rf equipment. This equipment should be automated for future space programs. Automation will provide a reduction in the test time as well as in the number of personnel required to perform the more menial, repetitive operations. Automation of these areas will free technical personnel to apply their efforts to the analysis and evaluation of critical system areas and to the troubleshooting of system problem areas, which would provide for the more efficient use of manpower.

Facility design and consideration of potential problems associated with power supply, equipment grounding, and equipment access can have considerable impact upon spacecraft testing if these areas are not given proper attention during the early stages of system development. Examples of problems which were encountered on the ACE-SC system during the Apollo Program include the following.

1. The ACE-SC ground stations at the KSC underwent a significant number of power transients during spacecraft testing. In most cases, the ground-station computers required a program reinitialization or reload (or both) before the test activity could be continued. Power failures and transients also occurred at the contractor ACE-SC stations at Downey, California, and Bethpage, New York. After a total power failure at the prime-contractor site, a high incidence of ACE-SC hardware failures occurred. Normally, the failures were detected by the preoperational test programs and the diagnostic programs before spacecraft testing was continued.

2. Air-conditioning deficiencies and failures created various ACE-SC system problems. At the contractor facility, a broken water main to the air-conditioning system caused the shutdown of all ACE-SC stations. After the air-conditioning system had been repaired, the stations were powered up. Several ACE-SC system problems were detected and repaired before the spacecraft testing was continued. At the MSC, a deficiency in the air-conditioning system caused corrosion in the data-entry units in both ACE-SC control rooms. After several failures that were related to humidity and component corrosion had been identified, the affected elements were coated to reduce the effect of humidity on the system. The connections continued to corrode, however, and the problem was finally resolved by modifying the air-conditioning system to correct the temperature and humidity problems.

3. Isolated instances of poor equipment layout caused minor impact to spacecraft testing. The contractor recorded one incident in which two spacecraft being tested were interrupted as access to a unit in one control room had to be made through the control room of the other station. Another problem associated with poor equipment layout was the lack of isolated convenience outlets for the FR1400 recorders. Because the outlets were not isolated, there were instances of erroneous transients being recorded with the test data.

2. To have a modular subset of the standard system that will be capable of supporting preinstallation and bench-maintenance testing to reduce the requirement for special-purpose test equipment

3. To have a built-in test capability both in subsystems and GSE to minimize ground diagnostic requirements

4. To have universal control and display consoles that are designed to allow maximum automation, thus reducing the number of required test personnel

5. To have a GSE control and monitor capability from the checkout station

6. To have a basic system that will be capable of modular adaptation to the checkout requirements throughout the program evolution

7. To have a central data facility for spacecraft system test history recording and data-processing support

8. To have a serial digital interface with the spacecraft and GSE

FUTURE CHECKOUT-SYSTEM CONCEPTS

The next-generation checkout-system concept is described in the following section under the two basic categories which encompass the more significant features of the system: equipment configuration and system interfaces. A basic diagram of the system is shown in figure 5.

to the spacecraft or the GSE. Responses would be received by the acquisition module and routed to the control and display modules for processing, display, and recording. Test history data would be recorded on the storage and retrieval module for subsequent use. The test history data would be transmitted to the central data facility on a noninterference basis. One console would be required for subsystem preinstallation tests.

The system functional interfaces have three basic elements: (1) the space-station and space-shuttle hardware and the GSE, (2) the centralized computer facility, and (3) the operator interface. Each of these functional interfaces must be connected by a serial digital interface to provide the required information or data-acquisition capability.

The complexity of the ground-system cabling presently used for the Apollo spacecraft checkout will not satisfy future space-program requirements for simplicity, reliability, and fast turnaround time. The system under consideration would use serial digital up-link commands to control the spacecraft and PCM data for response monitoring. The interface between the checkout station and the GSE would be in the form of a

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coaxial cable that carries information and data at a frequency of 1 to 5 megahertz. Because the central data facility would be performing a non-real-time function, standard remote-terminal computer techniques would be used for test-area-to-centralfacility communications.

One of the major goals of the new system is to minimize the number of personnel involved in the on-line, real-time conduct of spacecraft tests, which would thereby minimize the complexity of the man and machine interface. This goal can be accomplished by using the standard test console, by integrating the GSE with the checkout system, by providing specialist backup to the test conductors through standard television links, and by modular sizing of the control and display system to meet program phases.

CONCLUSIONS AND RECOMMENDATIONS

The ACE-SC system has performed all test and support functions in an outstanding manner. Major advances have been made in the automation of selected test sequences, and confidence has been established in computerized systems for real-time checkout and evaluation of manned-spacecraft systems. However, major areas for improvement still remain that will contribute significantly to the efficiency of future testing and checkout programs. Areas for future improvements include the following.

1. In the past, spacecraft systems and ground-support equipment have not always been designed with testing and checkout in mind. Future spacecraft and groundsupport-equipment systems must have the designed-in capability for automatic checkout.

2. The present checkout system does not provide the capability to support predelivery and preinstallation testing of spacecraft subsystems. Future checkout systems must be capable of modular adaptability to all phases of test activity as well as to changing test requirements. This will reduce the amount of special-purpose test equipment which has been required in the past as well as the modifications to the checkout equipment.

3. The present ACE-SC system does not adequately provide for "quick-look" data processing with hard-copy data. Future checkout systems should be interfaced with a central computer facility to provide this capability as well as data-recording and trend-analysis capabilities.

4. The communication of test requirements from engineers to the computer programer is a problem that can be reduced by the implementation of engineering-testlanguage compilers. Therefore, a test engineer with a minimum of training is allowed to submit requirements for test programs in engineering terms with which he is familiar. This procedure has been implemented with the Adaptive Intercommunication

. Routine, which is presently being used on the ACE-SC system; however, this computer program is highly oriented toward the ACE-SC system. A higher level test-language compiler is presently in development and will be made available for future testing and checkout activities.

The experience gained in the use of the ACE-SC system has been invaluable. It is anticipated that the benefit of this experience will contribute to the design of even more efficient and successful checkout systems for future programs.

Manned Spacecraft Center National Aeronautics and Space Administration Houston, Texas, December 7, 1971

914-50-17-08-72

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15. Supplementary Notes

The MSC Director waived the use of the International System of Units (ST) for this Apollo Experience Report, because, in his judgment, use of SI Units would impair the usefulness of the report or result in excessive cost.

A description of the construction and use of crew provisions and equipment subsystem items for the Apollo Program is presented. The subsystem is composed principally of survival equipment, bioinstrumentation devices, medical components and accessories, water- and wastemanagement equipment, personal-hygiene articles, docking aids, flight garments (excluding the pressure garment assembly), and various other crew-related accessories. Particular attention is given to items and assemblies that presented design, development, or performance problems: the crew optical alinement sight system, the metering water dispenser, and the wastemanagement system. Changes made in design and materials to improve the fire safety of the hardware are discussed.

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CREW PROVISIONS AND EQUIPMENT SUBSYSTEM

By Fred A. McAllister Manned Spacecraft Center

A description of equipment and experience gained during development of the crew provisions and equipment subsystem items for the Apollo Program is presented in this document. Details and understanding about the crew-related systems used on the Apollo 11 mission and about the individual equipment and equipment-related problems are presented. The rationale for selection of materials and design philosophy is discussed. Also, several recommendations are presented for future improvement of spacecraft hardware.

This report is a discussion of the Apollo crew equipment items used on the command module (CM) and lunar module (LM). These crew equipment items include the restraint systems, docking aids, water-management systems, waste-management systems, crew accessories, medical components, bioinstrumentation, survival equipment, stowage, and flight garments. System changes have been made for Apollo flights subSequent to the Apollo 11 mission, but only a few are referenced in this document. An alphabetical listing of the equipment discussed in this document is presented in appendix A. Acronyms used are listed in appendix B.

Valuable contributions to this document have been made by Maxwell W. Lippitt, Jers; William L. Burton, Jr., James H. Barnett, Ralph J. Marak, Thomas F. Gallagher, William F. Reveley, and Richard S. Serpas of the NASA Manned Spacecraft center; and Kevin J. Gravois, Elizabeth W. Gauldin, and Robert C. Hill of the General Electric Company, Houston, Texas.

The problems associated with the development of the spacecraft (SC) crew provisions and equipment items were discovered from use and comments by crewmen. The initial design of items often is not discernible in the evolved product. Prior to the approval of a design for flight, the items were subjected to hardware design reviews,

bench evaluations, mockup evaluations, zero-gravity water tests, high-fidelity fit and function tests, and finally manned-chamber evaluation under simulated altitude conditions. During the early crew-interface tests, the design remained fluid and changed, as required, with each review.

As experience from tests and mockup reviews increased, changes to the equipment decreased. Designers were better able to anticipate the requirements of the Apollo missions. Eventually, a point of minimum change and maximum efficiency was attained, this being a fine blend of design intuition and crewman participation in the development effort (ref. 1). This same general philosophy of development was applied to the LM restraint system, the docking aids, and the more personal equipment and provision items, such as the waste-management systems.

Crew equipment engineers learned to remain closely involved with the equipment from the time of initial design concept until completion of the postflight analysis. After the Apollo fire, it became mandatory to make SC cabin materials less flammable. This new emphasis completely changed the design philosophy of the crew equipment. The design process (as described) began with new ground rules and new restrictions that required the use of nonflammable materials. Yet, few major crew equipment changes were made after the Apollo 7, 8, and 9 missions. This fact is a credit to the success and efficiency of the design effort.

There have been continuous reevaluations in the categories of waste management and bioinstrumentation. These groups of personal equipment and provision items are used by the crewmen, who offer suggestions for improvement.

CREWMAN-RESTRAINT SYSTEMS

Crewman restraints for all mission modes are provided in Apollo vehicles. The primary systems are the CM couch-harness assembly and the LM "standup" restraint hardware. These systems are designed to provide stability and safety during phases of earth and lunar launch and landing. Handholds, Velcro attachments, and sleep restraints are used also for inflight tasks and general mobility. The sleeprestraint assembly provides the crewmen with a comfortable sleeping enclosure for use in zero-gravity environment. These items, general accessory items, and special equipment designed to allow unsuited entry are discussed in this section.

The headrest pad, which is attached to the couch headrest in the Apollo CM, is used to create stability and acts as a buffer for the crewman's head during unsuited entry mission modes. The assembly must mate with and enclose the couch headrest. Thus, firm cushioning is provided for the unsuited crewman's head during entry vibration and shock. To satisfy SC flammability requirements, the headrest-pad assembly is made of a firm Fluorel outer case filled with an inner Fluorel-foam cushioning section. Fluorel is used because of its strength, wear, and flammability characteristics. The outer Fluorel case is molded to fit the contours of the couch headrest.

The heel restraint is used to secure the crewman's feet in a fixed position in the CM during entry. The restraint is designed to be strapped over the coverall boot assembly and is used only during the portion of entry in which the pressure garment assembly (PGA) is not worn by the crewman. A simple design is necessary to allow easy attachment to the foot before entry. The heel must be constructed of a durable material that will mate in a locked position with the SC heel-locking device.

The assembly consists of two subassemblies: an aluminum heel and a strap assembly. The aluminum heel is constructed with three slot openings to permit strap passage. The straps are constructed of polybenzimidazole (PBI) webbing and pass through the aluminum heel and around the ankle. The PBI is used because it possesses desirable structural-strength and flammability characteristics. Velcro strips on the straps fasten around the crewman's ankle to hold the heel firmly in place. The metal heel is slotted to fit directly into the SC heel-locking device (as the PGA boot does). Heel-restraint assemblies have been used successfully on each Apollo flight.

The "sleeping bag'' sleep-restraint enclosure provided in the CM restrains the crewman in the sleep station during zero-gravity environment. The sleep restraints also provide warmth during sleep, and perforated cloth construction provides aeration.

The three sleep restraints (fig. 1) consist of 64-inch-long by approximately 21-inch-wide bags equipped with longitudinal-axis zippers. Each bag has a neck opening and is constructed of perforated Teflon-coated Beta cloth. "Dog leashes" are used to attach one bag under the right couch and another under the left couch. The right couch harness constrains the other bag, which is on top of the right couch. Crewmen enter the sleep restraints through the zipper openings.

-Ventilation holes 0.060-in. diameter

Teflon-coated Beta cloth was chosen for the sleeping bags because this material has good abrasion resistance and meets the fire-retardation requirements of the potentially dangerous cabin atmosphere.

Several controlled environment tests were evaluated by crewmen and contractor personnel to determine the number and size of the perforations needed for maximum crew comfort. These tests indicated that perforations which have a 0.060-inch

Figure 1.- Sleep-station restraints in

diameter and are on 2-inch centers provide maximum crewman comfort. Blankets, rather than sleeping bags, were considered but never used. Stiffeners were added, according to crewman recommendations, in the lower-torso area to provide leg support while in the sleep restraints.

Handholds are provided to assist crewman ingress and egress from the CM side hatch and for periods of gravity loads. Two aluminum handholds are provided to aid crewmen in the CM. These handholds are located by the side windows, near the main display console (MDC). A handbar is provided on the MDC near the side hatch as an aid to ingress and egress. The handbar can be stowed or extended. Five handstraps, located behind the MDC, plus another handstrap over the environmental control system (ECS) access panel, are maneuverability aids. These handstraps are constructed of Viton material with metal-reinforced interiors.

Velcro (H549 hook with P537 pile) provides a simple way to prevent floating of small objects in a zero-gravity environment and is used extensively throughout the crew compartment of the CM for temporary inflight stowage of loose crew equipment items. Patches of Velcro hooks are bonded at convenient locations on the SC structure, and Velcro pile is fastened to most of the loose crew equipment items. Items may be attached, at any location in the SC where Velcro is available, by using hand pressure.

An interface control document (ICD) was originated to establish the Velcro locations within the SC. The ICD is reviewed and updated periodically because crewman preference may dictate many changes in Velcro location.

Lunar Module Crewman-Restraint System

The LM crewman-restraint system restrains the crewmen during powered flight, zero gravity, and the shock of lunar landing. The system must function without seriously reducing the mobility, visibility, or dexterity of the crewmen. The system must also maintain proper pilot orientation, with respect to instruments and controls, during powered flight and zero-gravity environment. During landing, the system must prevent the crewmen from striking adjacent structures and must facilitate absorption of the impact shock.

The development of the restraint system involved three phases: zero-gravity tests using aircraft, ground-based tests using acceleration rigs, and manned drop tests of the lunar test article (LTA-3) vehicle.

The initial landing-acceleration limits at the LM crew station were formulated during October 1965. The landing conditions considered were within 10 ft/sec vertical at a 0-ft/sec horizontal velocity and within 7 ft/sec vertical at a 4-ft/sec horizontal velocity. Landing conditions were limited by the kinematic capability of the landing gear. The landing gear was considered to be elastic and the LM, rigid.

Zero-gravity test planning was conducted (1) to determine the effectiveness of proposed foot restraints and of devices to stabilize the crewmen at the crew station and (2) to determine the best locations and configurations for handholds and handgrips. A KC-135 jet aircraft was used in the zero-gravity tests. The landing-shock test program involved the use of manned shock rigs to simulate the acceleration envelope.

The restraint system that evolved (fig. 2) includes (1) Velcro on the boot soles and the LM floor, (2) restraint cables (from a constant-force reel) attached to the crewman to produce a constant 15-pound downward force on each side of the crewman, and (3) a set of armrests to absorb loads from the crewman's upper torso. The restraint armrests are equipped with hydraulic dampers for energy absorption.

Handgrip Downward forces from the middle and lower torso are intentionally absorbed by

assembly the crewman's legs. The panel handholds

force and the armrests are designed to supply

Pulley

spring system

tension lateral support. The constant-force reel is equipped with a cable lock to restrain the upward motion of the crewman during the lunar landing.

Figure 2. - Crewman-restraint system Difficulties were encountered with

in the LM. the armrest hydraulic damper during component design. The magnesium damper assembly that was developed in a weight-reduction program did not operate properly; the magnesium did not provide adequate piston-chamber surface hardness to protect the sliding seal on the piston. Various corrective approaches, including coating techniques, were tried. Schedule difficulties and lack of success resulted in changing the design to aluminum, and no further difficulties were encountered.

The constant-force reel involved unique design requirements: the unit had to be mobile, lightweight, and lockable. Three negator springs are used to roll and unroll a central (takeup) drum that drives one of two restraint cables operated by the reel.

Changes in material types occurred during hardware development. Flammable plastic parts were replaced with flame-resistant Teflon or metal parts. The syntheticfiber rope assembly was replaced with a Teflon-coated, braided steel cable. The Velcro was replaced with a better, flame-resistant variety.

The adequacy of the LM restraint design is demonstrated by the flight program. Crewmen indicated that it would be desirable to make the restraint-reel force variable, thereby allowing adjustment for the various mission phases; however, this change is not critical.

Equipment was developed to enable the crewmen to meet the Apollo Program requirement that the LM and the CM rendezvous and dock in earth and lunar orbits.

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Tracking and running lights are on both vehicles; however, the prime docking aid is the crew optical alinement sight (COAS) system (fig. 3) and the respective COAS system targets for the LM (fig. 4) and CM. The COAS system is versatile enough to supply

range, range-rate, and attitude information as navigation aids and as a supplement to docking information. The process of

rendezvous and docking is a relatively glass

simple series of maneuvers with the use of this system.

design concept was modified to meet this requirement. The result, the COAS system, was installed in the CM as a docking aid. Basically, the COAS system is a collimator device consisting of an intensity control, a reticle, a barrel-shaped housing, a combiner assembly, a power receptacle, a clip-on filter, and a mount.

An active LM mode was required during rendezvous; however, a COAS system was not required for the LM because, in this mode, use of the LM docking port as the forward hatch was sufficient. Thus, the crewmen can directly observe and control the docking operation. No auxiliary devices are needed. In the process of LM development, the docking port was changed to become the overhead hatch, not directly visible to the LM pilot. A device similar to the CM COAS system was obviously needed in the LM; thus, the CM COAS system was modified to be compatible with LM design requirements. The device modified for the LM provides range, range-rate, and attitude information to the LM pilot during docking. A second function of the LM COAS system is to provide the crewmen with a fixed line-of-sight attitude-reference image which, when viewed through the combiner lens, appears to be superimposed on a lighted target

located in the CM right rendezvous window. This image is boresighted parallel to the X-axes of the LM and the CM (fig. 5). A similar but larger target is located on the exterior of the LM, adjacent to the overhead hatch, for use with the CM COAS system during the CM active docking maneuver.

During the LM-3 crew-compartment Figure 5. - Predocking condition of the fit and function (CCFF) test, an erratic in

COAS and target. tensity control of the flight COAS system was discovered. Failure analysis indicated that a circuit transistor caused the difficulty. The failure was not restricted to a single unit. The faulty transistor was not compatible with the design of the intensity control; therefore, all COAS system units that were equipped with this transistor in the intensity control were returned to the manufacturer for rework. The defective transistors were replaced with a more satisfactory type, and erratic intensity controls ceased to be a problem.

To establish the appropriate range of reticle brightness, the crewmen visited the manufacturer to view a fifth-magnitude star under laboratory conditions. An internal neutral-density filter was added to the COAS system to reduce the brightness of the reticle. However, severe difficulty was encountered during the Apollo 9 docking phase; high ambient lighting conditions caused a washout of the reticle image on the combiner glass. This problem subsequently resulted in removal of the internal neutral-density filter from the COAS system, allowing the original high-reticle-brightness capability of the COAS system to be used. The ability to sight on a fifth-magnitude star was retained through addition of an external clip-on filter. The Apollo 9 docking difficulty also resulted in the decision that the CM would be the active docking vehicle. Subsequent Apollo missions were flown without docking difficulties. Use of the COAS system in both the LM and the CM has been nominal, and no further design changes are considered necessary or desirable.

The metering water dispenser (MWD) and the water dispenser/fire extinguisher (WD/FE) are both pistol-shaped devices that dispense potable water for drinking, food reconstitution, and fire extinguishing.

The initial dispenser was a modified Gemini water metering device (fig. 6). Modification made the device compatible with the Apollo SC requirements. The device metered 15 milliliters (+10 percent) of water for each trigger cycle and recorded the number of trigger cycles on an integral mechanical counter. The MWD (initial configuration) was cycled 20 000 times more than necessary to meet Apollo qualification requirements. Because an anomaly occurred during the Apollo 7 mission (SC-101), a series of tests was performed to determine the compatibility of the MWD O-rings with the excessive chlorine content of the CM potable water. These tests, in which chlorine concentrations of 2 to 5000 ppm and durations up to 16 days were involved, revealed that a change was required in the material of the forward plunger or metering O-ring of the MWD. This O-ring was changed from neoprene to ethylene propylene, resulting in the 05 configuration

Figure 6. - Gemini water metering MWD, which was used successfully on the

device. Apollo 8 and 9 missions (SC-103 and SC-104).

The Apollo MWD was intended for use in both the CM and the LM. However, before the first manned Apollo flight, a requirement was included that the water dispenser used in the LM should have the additional capability of dispensing a continuous cone-shaped spray of water for firefighting purposes. To meet this requirement, an early Gemini continuousflow water dispenser was redesigned. The modified WD/FE unit was selected for LM use on the Apollo 9 (LM-3) and subsequent missions, and for CM use on the Apollo 10 (SC-106) and subsequent missions. Operation of the water/gas separation equipment was simplified significantly by using the modified WD/FE in the CM (fig. 7). The WD/FE also provided the crewmen with drinking water that was relatively free of excess hydrogen.

A bacterial filter has been designed that can be mated to the WD/FE by means of the existing WD/FE water-inlet quick-disconnect fitting. This filter ensures that the drinking and food-reconstitution water dispensed by the unit is free of bacteria. A water-gas separator can be attached to the WD/FE outlet (in the CM) to ensure that the drinking and food-reconstitution water dispensed by the unit is free of entrained gas.

To remove and dispose of crewman waste matter, various waste-management systems were developed for the Apollo Program. Separate systems, for use in both the CM and the LM, were designed for management of feces and urine. The CM wastemanagement system is shown in figure 8. The LM waste-management system is similar to that of the CM.

The Apollo fecal-collection system consists of the fecal-collection assembly (FCA) on the CM and the defecation-collection device (DCD) on the LM. The design and operation of the DCD are similar to the design and operation of the FCA. The FCA provides a method of collecting, inactivating, and stowing feces for 14-day missions with a minimum of crewman effort. The FCA consists of an inner fecal/emesis bag, a germicide pouch, an outer fecal bag, and a wrapper. A waste compartment with an overboard vent system for odor removal is provided in the SC cabin for stowage of the used fecal

bags. The outer fecal bag and the inner fecal/emesis bag are constructed of a heatsealed laminate film. The germicide is added to the feces to prevent or reduce gas and bacteria.

To use the FCA, the crewman attaches the outer fecal bag properly and proceeds with fecal elimination. Upon completion of the action and subsequent sanitary cleansing, the tissues and refuse are placed in the inner fecal/emesis bag. The crewman then removes the germicide pouch, cuts the outer protective seal, and places it in the inner bag. Finally, all items are placed into the outer fecal bag, the bag is sealed, the germicide pouch is ruptured by hand pressure, the bag is kneaded, and the contents are stowed in the waste-stowage compartment.

Although the Apollo fecal-collection system is the same as that used in the Gemini Program, many new concepts and designs were investigated and tested. Various types of canisters, with and without air blowers, were developed with some success. In all cases, the primary problem has been the separation, in a weightless environment, of the fecal wastes from the crewmen. Nothing has proved more effective than the current system, which has proved adequate for all flights, although the crewmen have expressed dislike for it. Other methods are being investigated, and experiments will be conducted on future missions. A better method of collecting fecal wastes must be found for future flights, particularly those of longer duration. Many promising designs are being investigated and may be incorporated into future space vehicles.