guidance, flight mechanics and trajectory optimization

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2.2.4 Impulsive Rendezvons Techniques -.. - ---I--.-----. Tn thi.s section, the development of guidance equations in a linear gravity field will be restricted to determining an impulsive velocity correction such that a rendezvous till take place at some time in the future. The techniques of this secti.on can be thought of as defining a reference trajectory along which the rendezvous vehicle is to travel. Once this reference trajectory has been established, the actual steering equatinns (i.e., orientation of the thrust vector as a function of time) can be determined by the methods discussed in a previous monnqraph of this series on boost-guidance equations (Reference 2.16). The section below gives a discussion of two simp3.e techniques for utilizing cal cu!.ated velocity impulses. Impulsive velocity changes have been considered in detail in the monograph on mi.dcourse guidance (Reference 2.15) and this reference is recommended for a more com@ete formulation including the effects of errors in data and optimization of mu!.ti.ple i.rnpulse schemes. 2.2.4.1 Approximating Velocity Impulses The use of a model having a linear variation in gravity generates the necessity of neglecting the effects of finite bnrning time in order that closed form solutions be obtainable. Al though, th,e use of imp111 sT.ve ve3 ocity changes is essential to the simoV fication of the equ,ati ons, their physical realjzation can be only anproximate. If a true impulse could he achieved, then the guidance mechanization won!d simply be to orient the thrustor along the direction of the required vel.ocity chanece and apply an impulse equal to the magnitude of the required chance. Since th.e rocket motor must bllrn for a finite time (the length of t?me depends upon the thrust capability in relation to the magnitude of the velocity increment) the orientation of the required veloci.tg increment will., generally, nnt be the same at the initiation and termination of the thrusting period. Thus, if the rocket v !ENDEZVOUS VEHICLE ORBIT Figure 2.4 Di.rection of Velo&tp Correction TRAJECTORY Rocket motors were oriented along the required AV vector and maintained in that orientation throughout the thrusting period, the resulti.ng velocity increment wnnld have the correct magnitude, but it would be directly incorrectly. 39
One method which can be employed (and has been employed in the Gemini missions (Reference 2.3))t o compensate for the finite thrusting time is to center the thrusting period about the time at which the velocity impulse is to be applied. For example, if it is calculated, that the velocity change to be applied at t = to to achieve rendezvous is AV, and the acceleration produced by the rocket motor is T/M then the length of time the motor is required to burn is The thrust period'is then centered about time to by initiating the thrust at &=
Page 1 and 2: NASA CONTRA REPORT CTOR GUIDANCE, F
Page 3 and 4: - ..a --.
Page 5 and 6: _- .-- --- I
Page 8: 110 2.1 2.1.1 2.1.2 2.1.3 2.1.3.1 2
Page 11 and 12: . 0 i time difference between a ref
Page 13 and 14: only line-of-sight thrusting is use
Page 15 and 16: 2.0 STATE-OF-THE-ART The significan
Page 17 and 18: 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
Page 23 and 24: of inertial directions which are ta
Page 25 and 26: which form the columns of F must be
Page 27 and 28: Thus, the solution is written The t
Page 29 and 30: This matrix is simply written as G(
Page 31 and 32: The fundamental matrix for A can be
Page 33 and 34: 2.1.5 Approximate Second Order Solu
Page 35: 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: If the matrix 121 model discussed'&
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 and 88: performed using the Pontryagin Maxi
Page 89 and 90: This equation gives the optimum ste
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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