guidance, flight mechanics and trajectory optimization

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The integral in Equation 2.2.1 can now be written in its discrete form as i=l (2.2.2) The evaluation of the ith term can be seen for a typical transition in the above sketch. The choiz of yt. can be thought of as being the decision parameter. The similarities to'the previous examples should now be evident. Each transition in the space has an associated "cost" just as in the previous travel problem. The problem is to find the optimum path from (x0, y ) to (XfJ Yf) such that J, or the total cost, is minimized. Obviously, i? a fairly accurate solution is desired, it is not advantageous to choose big increments when dividing t@e space. It must be kept in mind, however, that the amount of computationinvolved increase quite rapidly as the number of increments increases. A trade-off must be determined by the user in order to reach a balance between accuracy and computation time. The problem of Mayer can be shown to be equivalent to the problem of Lagrange (see Ref. 2.1). This problem will be included in this discussion because it is the form in which guidance, control and trajectory optimization problems usually appear. The general form of the equations for a problem of the Mayer type can be written as 2 =f (%, u) (2.2.3) 15
where x is an n dimentional state vector and u is a r dimensional control vector. It is desired to minimize a function of the terminal state and terminal time, i.e., subject to the terminal constraints y=LxI), $’ j=l,m (A more detailed statement of the problem of Nayer can be found in Section 2.4.8 or Reference 2.1). The approach that is used to solve this problem with Dynamic Programming is quite similar to the Lagrange formulation. The state space component is divided into many increments. The ltcostlV of all the allowable transitions is then computed. Each diffierent path eminating of the same point in the state'space corresponds to a different control, which can be thought of as being analogous to the decision at that point. With these preliminary remarks in mind, some illustrative examples will now be presented. 16
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 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 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
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The boundary condition to be satisf
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2.4.7. Discussion of the Probltsi o
Page 87 and 88:
characteristics associated with Eqs
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2.4.8 The Problem of Bolza The prec
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The initial position, velocity and
Page 93 and 94:
where I is the moment of inertia, F
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or, as . -... .-. . .._-_--- to ind
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Since R(t, x(t) ) is the minimum va
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2.4.10 Ljnear Problem with Quadrati
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where S(t) is some n x n symmetric
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The governing equations for the att
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y 2.4.11 Dynamic Programming and th
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M Since 3t does not depend on u exp
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with The P vector for the system is
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where With the control known as a f
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with and with the boundary conditio
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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
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The minimum value of the performanc
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variance characterizing Ilt) can be
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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
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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 - ._ .-_
Page 157 and 158:
4 Section 2.5 (2.5.1) (2.5.2) (2.5.
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