guidance, flight mechanics and trajectory optimization

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x1=0,x2=0 x1 = 0, x2 = 1 xl =1,x,=0 y1 + y2 = B2 yl y2 f2 f2 f2 0 0 0 @ 0 0 1 0 1 0 0 2 1 0 cjj 3 2 0 12 5 5 / 2 ljl ; 0 0 3 2 0 4 5 5 0 3 9 10 10 12 0 @ @ 21 0 @ @ 3 0 9 10 10 0 indicates optimum value for stage (2 indicates redundant optimal values for different policies at the same stage
% Xl + 5 = 2 (A2 = 2) 5+X2=3 (A2=3.) x1=0, x2=2 x1=1, x2=1 x1=2, x2=0 xp, *=3 x1=1, x2=2 x1=2, x24 x1=3, x2"o Yl + 92 =B2 ~1 ~2 f2 f2 *2 f2 *2 f2 *2 0 01 0 0 4 5 0 4 5 10 9 0 @ I:$ :b:\ 1: 1 10 5 I:?! 5 10 $! :z+j 10 0 2 8 6 8 13 9 9 13 2 11 6 @* 6 11 @ $? 11 2 0 8 6 8 13 9 9 13 i 0 3; 13 11 13 18 14 14 18 21 / 1 9 0 9 14 0 1 $3 14 3 2 1 9 0 9 14 @ 53 14 3 0, 13 11 13 18 14 14 18 / Q optimal value for problem
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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 32 and 33: Ii _ 6 5 9 3 I 8 I I The cost integ
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: Next, fl is evaluated for all allow
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
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2.5.2 Problem Statement Let the sys
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Now, introducing the variables qs2,
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Thus, the performance index takes t
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#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
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Since V is positive definite for f
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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
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F while Equation (2.5.87B) requires
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._ __. ..- . . - which will equal t
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is determined. Let p(z,t') be given
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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
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.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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