guidance, flight mechanics and trajectory optimization

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When these expressions are substituted into the,control law, the result may be interpreted as a sampled data feedback law where the last measurement occurred at to. Thus, Another result may be obtained by imagining that the position and velocity are continuously sensed and immediately used to correct the thrust. In this event, a A09 go, and to become ,& , g , and t and a continuous feedback guidance law is obtained, which is y =- b u/r -- (7-tP (7-t) To express this la-~r in terms of uR (toyCard the target) and u1 (normal to the line of sight), note that 4 = - R&, L/ = -RI, - R E. Next adopting the notation of expressing a small displacement from the nominal path as a small angle, Yields TARGET f VEHICI.JZ Figure 4.5 - Geometry of Rendezvous 73 NOMINAL PATH
Examples of the effectiveness of this form of proportional guidance for chemical engines used in earth orbital rendezvous are given by Goldstein, et.al, (Reference 4.9) and an illustration of their results is given in Section 2.4.1. 74
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NASA CONTRA REPORT CTOR GUIDANCE, F
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- ..a --.
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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
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
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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 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: If it is assumed that 4~ is constan
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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