guidance, flight mechanics and trajectory optimization

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ii 2 I 2 4!m 4000 3500 3000 250C I- rood I-- )- I- I- ‘5 0 CUAORATIC THEORY P LlhEAR THEORY - - AV AV -827.52 FPS Figure 1.5 Extreme Case Comparison Figures 1.5 illustrate a rather extreme case of velocity difference in which even the quadratic theory breaks down rapidly. However, even if this case for relative distance is less than 2500 miles, the predictions are quite accurate. 2.1.6 The Effects of Perturbations on Rendezvous Satellite orbits are perturbed from pure conic sections by forces due to the earth's atmosphere, the non-sphericity of the earth, the attraction of the moon or sun, and the radiation pressure of solar radiation. As was indicated earlier, the term A g really represents the difference between the accele- rations of the two objects due to all forces. The earth's atmosphere could 29 3n
cause a significant effect for rendezvous maneuver sufficiently low, but the assumption iz this study is that the maneuver is as a suf.ficiently high altitude.( >150 Km) that air drag is negligible. The largest of the remaining forces in th's region near the earth is that due to the oblateness of the earth. The magnitude of this force is about 10-3 times that of the inverse square force and while its gradient coefficient is twice that of the ma-in term, it is clear that the magnitude of the contribution to Ag is not more than l/500 that of the main term which is itself expressed only approxi- mately. Consequently,.all perturbative differences can be safely neglected in studying rendezvous in orbits near single attracting centers that are no more oblate than the earth. Rendezvous in earth-moon space is thus considered in the monograph only if near enough to ona of the attract?ng centers, that the perturbative force due to the other is substantially smaller than the main gravity turn. 2.2 GTJIDAVCE F,QUATIONS In this section the methods that have been proposed for mechanizing the rendezvous are presented. 19 several cases , guidance schemes are chosen without regard to the degree of optimization;-however, when the a preach is taken, tlie considerations will be detailed L see Section (2.L) 4 -, 2.2.1 Fixed Inertial Line of SFuht &-CC] - Free Space ~ --__ - _____C_.__._ -_- ..a--.. . ..--.---1_ It has been shown in a previcus section that the Vatural.'t maneuver of thrusting f-n the direction of the T,c)S w-X.1 not, in general, produce rendezvous. However , for the special case with the relative velocity vector aligned parallel to the LOS, then thrust thrusting along the LOS is the optimum method of achieving rendezvous (in free space). This fact is demonstrated in Section 2.4 where optimization is discussed. This solution suggests that a mar.euver which first orients the relative velocity vector along the LOS could be advantageous. Aligning the velocity vector in this way is equi- valent to nu13ing the angular rate of the LOS in inertial space'and can be acccmplished by appldg thrust normal to t!?e YE. The overall maneuver, +.'lus, retains a degree of naturalness in that an astronaut performing a man-la: rendezvous can easily determine the directions in whFch the thrust i 9 t 0 be applied. The discussion Fresented here is limited to its application as a manual hack up guidance technique for Gemini as presented in papers by Chamberline and Rose, and Burton and Ha.yes (References 2.2 and 2.3), and to an extension cf the technique by Steffan (Reference 2.4) which separates the guidance and navigation tasks. Because of the approximation that there is no relative acceleration due to gravity, the range of initial ccndjtjons for which this t.ec~njql~e has acceytab1.p accuracy is limited. A larger set of initial conditions can he handled if a number of fVmidcourse 1' correctinns are made. nowever, fr-m a-? eff; ciency st.andpoin+., undesirab!e. The nri.ncina! I advantaee a natural basis for manual gui.dance. these midcourse corrections are of th5.s method lies in its use as 30
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: w” 2 E IOOJ- -100 - 200 AV = 400
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 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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