guidance, flight mechanics and trajectory optimization

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2.4.6 N-Dimensional Lagrange Problem The concepts of Sections (2.4.1) to (2.4.5) which have been developed in connection with minimizing the integral X,9 fi J =f f(x, y, y”dy %9)1. where y is a SCalar (l-dimensiona variable, can be extended to the case in which y is an n dimensional vector (2.4.56) (2.4.58) Then, following a procedure identical to that employed in Eqs. (2.4.20) to (2.4.24) but with the scalar t( replaced by the vector y as indicated in Eq. (2.4.56), it can be shown that'RcX,y) satisfies the equation (2.4.60) where superscript T denotes the transpose and a,R and ' are the vectors du if 7.5 (2.4.61)
The boundary condition to be satisfied by R(x,yj will in all cases take the form R(X/, Q-0 (2.4.62A) whether the Ix, the point ) space. (%+-,g ) is fixed or allowed to vary on some surface in In tile latter case, however, Eq. (2.4.62A) has several alterna f e representations similar to those developed for the l-dimensional problem "e.g. Eqs. (2.4.46) to (2.4.52) . For example, if the terminal Point straint (xr 9gf 1 is required equations to lie in the surface specified by the icon- (2.4.62~) the boundary condition ofp as given in (2.4.62A) can also be mitten as R(z,@=O OA/ G(z,yI=03jg,.(~,y)=~ ; i=l,mj (2.4.6yi) or analogous to Eq. (2.4.52), as where// is the m dimensional vector (2.4.63~) (2.4.64) Thus, by combining (2.4.63B) with (2.4.60) and using the optimization condition inherent in (2.4.60), that -+- af JR = 0 a$( afi i L’= 1, n (2.4.65) the transversality conditions of the Calculus of Variations, corresponding to the terminal constraints of Eq. (2.4.62B), can be developed. These 76
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NASA OI 0 0 P; U 4 cd 4 z . . _ -.
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c FOREWORD This report was prepared
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SECTION PAGE 2.5 Dynamic Programmin
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2.0 STATE OF THE ART 2.1 Developmen
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There are several ways to accomplis
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2.2 Fundamental Concepts and Applic
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II. Although the previous problem w
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I,. Citg B. C optimum cost Path for
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The optimum path can be found by st
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The integral in Equation 2.2.1 can
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2.2.2.1 Shortest Distance Between T
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The cost of the allowable transitio
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In a,completely analogous manner th
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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 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: Hence, the boundary condition OAI f
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
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where the first the second with exp
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Taking the limit and using the expr
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Since V is positive definite for t
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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
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(2.5.I.21) Thus, the control is to
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Note, as in.Section (2.5.2.3), the
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.6 t -(h.(S~tj~fitr~f-‘~~~~n~) =
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3.0 RECOMMENDED PROCEDURES - ._ .-_
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4 Section 2.5 (2.5.1) (2.5.2) (2.5.
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