guidance, flight mechanics and trajectory optimization

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Ii _ 6 5 9 3 I 8 I I The cost integral is represented by 9 /O // +qF i 5G A x, instead of the continuous form in Equation 2.2.6. The cost of each transition is shown, <strong>and</strong> the optimal value of the cost of each point is encircled. possible transitions that must be considered for each of the.above points The are shown below. 25 I
z I-M J J-M E, E-I E-J E-K E 'F-I F-J F-K G, G-I G-J G-K .235 .2 .166 + .235 .235 + .2 0372 .282 + .235 .2 + .2 .282 .559 + .235 .352 + .2 -25 = 0 .235 = 0 .200 = .401 = 0 .435 = .372 = .517 = 94 = 0 ,282 = .794 = .552 = 0 .250 L D-E D-F D-G D-H c C-E C-F C-G C-H B B-E B-F B-G B-H A A-E A-F A-G A-H .166 + 372 .235 + .282 .372 + .25 .527 .282 + .372 .2 + .282 ,282 + .25 l 447 .559 + .372 .353 + .282 .25 + .25 .353 1.05 f .372 .745 + .282 .471 -t .25 933 = .538 0 = .517 = .622 = .527 = .654 = 0 .a!+82 = .532 = ,447 = .931 = .635 = .50 0 = ,353 = 1.422 = 1.027 = .721 = 0 .33 This process continues until the optimal path can be found by following the decisions that were made by starting at the origin <strong>and</strong> progressing to the right. 26
Page 1 and 2: NASA OI 0 0 P; U 4 cd 4 z . . _ -.
Page 4: c FOREWORD This report was prepared
Page 7 and 8: SECTION PAGE 2.5 Dynamic Programmin
Page 10 and 11: 2.0 STATE OF THE ART 2.1 Developmen
Page 12 and 13: There are several ways to accomplis
Page 14 and 15: 2.2 Fundamental Concepts and Applic
Page 16 and 17: II. Although the previous problem w
Page 18 and 19: I,. Citg B. C optimum cost Path for
Page 20 and 21: The optimum path can be found by st
Page 22 and 23: The integral in Equation 2.2.1 can
Page 24 and 25: 2.2.2.1 Shortest Distance Between T
Page 26 and 27: The cost of the allowable transitio
Page 28 and 29: In a,completely analogous manner th
Page 30 and 31: 2.2.2.2 Variational Problem with Mo
Page 34: 2.2.2.3 Simple Guidance Problem As
Page 37 and 38: This process continues in the same
Page 39 and 40: The algorithm for solving this prob
Page 41 and 42: The reader, no doubt, has a reasona
Page 43 and 44: are applied to Min (fl), the range
Page 45 and 46: Note that each diagonal corresponds
Page 47 and 48: The value of X can be found by empl
Page 49 and 50: The initial condition is: K.&P : P(
Page 51 and 52: it is seen that (for a two stage pr
Page 53 and 54: The optimal policy can now be found
Page 55 and 56: 2.3 COMPUTATIONAL CONSIDERATIONS So
Page 57 and 58: If classical techniques were to be
Page 59 and 60: 2.3.3.1 The Curse of Dimensionalitv
Page 61 and 62: Next, fl is evaluated for all allow
Page 63 and 64: % Xl + 5 = 2 (A2 = 2) 5+X2=3 (A2=3.
Page 65 and 66: that minimizes f. This interchange
Page 67 and 68: 2.3.3.2 Stability and Sensitivity I
Page 69 and 70: Let the lower limit of integration
Page 71 and 72: But (2.4.10) since the function on
Page 73 and 74: Now,. noting that the second MIN op
Page 75 and 76: where v 'is determined from and wit
Page 77 and 78: NOW, combining these two expression
Page 79 and 80: The situation is pictured to the ri
Page 81 and 82: Hence, the boundary condition OAI f
Page 83 and 84:
The boundary condition to be satisf
Page 85 and 86:
2.4.7. Discussion of the Probltsi o
Page 87 and 88:
characteristics associated with Eqs
Page 89 and 90:
2.4.8 The Problem of Bolza The prec
Page 91 and 92:
The initial position, velocity and
Page 93 and 94:
where I is the moment of inertia, F
Page 95 and 96:
or, as . -... .-. . .._-_--- to ind
Page 97 and 98:
Since R(t, x(t) ) is the minimum va
Page 99 and 100:
2.4.10 Ljnear Problem with Quadrati
Page 101 and 102:
where S(t) is some n x n symmetric
Page 103 and 104:
The governing equations for the att
Page 105 and 106:
y 2.4.11 Dynamic Programming and th
Page 107 and 108:
M Since 3t does not depend on u exp
Page 109 and 110:
with The P vector for the system is
Page 111 and 112:
where With the control known as a f
Page 113 and 114:
with and with the boundary conditio
Page 115 and 116:
2.5 DYNAMIC PROGRAMMING AND THE OPT
Page 117 and 118:
2.5.2 Problem Statement Let the sys
Page 119 and 120:
Now, introducing the variables qs2,
Page 121 and 122:
Thus, the performance index takes t
Page 123 and 124:
#R where tr denotes the trace of th
Page 125 and 126:
The minimum value of the performanc
Page 127 and 128:
variance characterizing Ilt) can be
Page 129 and 130:
The solution takes the form with s
Page 131 and 132:
Since V is positive definite for f
Page 133 and 134:
where the first the second with exp
Page 135 and 136:
Taking the limit and using the expr
Page 137 and 138:
Since V is positive definite for t
Page 139 and 140:
2.5.3 The Treatment of Terminal Con
Page 141 and 142:
F while Equation (2.5.87B) requires
Page 143 and 144:
._ __. ..- . . - which will equal t
Page 145 and 146:
is determined. Let p(z,t') be given
Page 147 and 148:
of the p and $ equations (i.e., Eqs
Page 149 and 150:
(2.5.I.21) Thus, the control is to
Page 151 and 152:
Note, as in.Section (2.5.2.3), the
Page 153 and 154:
.6 t -(h.(S~tj~fitr~f-‘~~~~n~) =
Page 155 and 156:
3.0 RECOMMENDED PROCEDURES - ._ .-_
Page 157 and 158:
4 Section 2.5 (2.5.1) (2.5.2) (2.5.
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