guidance, flight mechanics and trajectory optimization

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The Hsmiltonian for the problem is: H = P’i =/+Art&) But, the Pontryagin Msximum Principle states that the function e(t) which minimizes the coast, #(t> , is that function which causes the Hamiltonian be a maximum at each instant of time. Therefore, this value of 8 can be to found by setting the derivative d/4 dH/dB equal to zero: or dB =-$aodjnB tGao cbo~ =o The Pi are the co-state variables and obey the differential equations: or q-0, Since, the boundary condition on Pl is: or And since P, = pczp=o 0, the solution for Pl (t) is: /9 (t) = 0 Solving the differential equations for P3 and P4 with P1 = 0 gives and the steering law becomes: or letting tLwe = c, f c- t 77 (4.38)
This equation gives the optimum steering schedule as a function of time. It remains to evaluate Cl and C2 from the initial condition. This objective is realized by first writing Equation (4.38) in terms of J& 8 and ~0. These values can now be used in the expression for X and Y velocity. Performing the indicated integration gives: (Recall that the coordinate system was chosen so that the initial velocity vector is along the X axis therefore ,$, = v . - 0 9 j% -0 . Applying the terminal conditions reduces the y equation to: Substitutions of this in the 2 equation gives Integrating the second equation in (4.39) to get. Y(t$) and applying the boundary conditions gives (4.4oj This equation and Squation (4.40) are adequ,ate to determine the constants Cl and C2 from the initial condition. The complete solution is summarized below: 78
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NASA CONTRA REPORT CTOR GUIDANCE, F
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_- .-- --- I
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110 2.1 2.1.1 2.1.2 2.1.3 2.1.3.1 2
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. 0 i time difference between a ref
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only line-of-sight thrusting is use
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2.0 STATE-OF-THE-ART The significan
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Thus in cylindrical coordinates, th
Page 19 and 20:
In this form the equations are vali
Page 21 and 22:
For all the equations given to this
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of inertial directions which are ta
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which form the columns of F must be
Page 27 and 28:
Thus, the solution is written The t
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This matrix is simply written as G(
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The fundamental matrix for A can be
Page 33 and 34:
2.1.5 Approximate Second Order Solu
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where the superscript p can be eith
Page 39 and 40: w” 2 E IOOJ- -100 - 200 AV = 400
Page 41 and 42: cause a significant effect for rend
Page 43 and 44: Figure 2.1 Polar Coordinate System
Page 45 and 46: . will attain, z , can be calculate
Page 47 and 48: This equation may now be solved for
Page 49 and 50: If the matrix 121 model discussed'&
Page 51 and 52: One method which can be employed (a
Page 53 and 54: If a collision is to occur at time
Page 55 and 56: The solution to these equations are
Page 57 and 58: Fixed Range: Range-Rate Schedule Ra
Page 59 and 60: The velocity correction, SVtO) repr
Page 61 and 62: (2.17) The solutions of all of thes
Page 63 and 64: I(7) I COMPUTE c (7) =- MISS g-r -1
Page 65 and 66: 0 MOO0 60000 il 20& 40&o hod00 Y (f
Page 67 and 68: Manipulation of the equations invol
Page 69 and 70: 2.4.1 Optimal Stepwise Thrusting As
Page 71 and 72: Note that if it, can occur, this sc
Page 73 and 74: where F (Q) is the matrix given by
Page 75 and 76: where Since 0 and Eq. 4.23 shows th
Page 77 and 78: In contrast, the fuel optimal probl
Page 79 and 80: significantly reducing the out-of-p
Page 81 and 82: Figure 4.3 Velocity Increments to R
Page 83 and 84: If it is assumed that 4~ is constan
Page 85 and 86: Examples of the effectiveness of th
Page 87: performed using the Pontryagin Maxi
Page 91 and 92: where 2((t) is given by the first e
Page 93 and 94: Cd&t- d&J dt I where Xc9 #II 2 % 9
Page 95 and 96: 3.0 RECOMMENDED PROCEDURES In the p
Page 97 and 98: For the guidance technique which nu
Page 99 and 100: 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13
Page 101: 4.20 Bakalyar, G., Continuous Thrus
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