guidance, flight mechanics and trajectory optimization

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and assume an elliptical reference orbit. Note that the reference system is centered at the target. Now, using the derivative of UJ and xl as in Section 2.1.3.1 the equations are found to be: Note that moving the origin to the target vehicle has the effect of causing a K term to occur in all three equations. The second-order form of these equations as used by Anthony and Sasaki (Reference 1.2) is obtained by introducing changes of scale for both distance and time. In.this reference, the semi-major axis, aT' of the elliptical thus, x = aTX19 y = apl$ to mean anomaly, M, and reference is used as a normalizing variable; z = aTzlt and = aTq. The time is then changed d/dM is represente 'a with the open dot, O, as before. Including the second-order terms on the right, the equations become: For circular orbits the equations simplify to equations of exactly the same form as Equations 1.20 thus 'indicating the equivalence of the two origins (either active or target) for rendezvous. Thus, one finds 2-3n”x -2nj =a, Y +Zfl;r: =a Y it + n2t =a * For the final two forms of the equations, consider that 2 is expressed in a set centered at the target in an elliptical orbit and oriented in a fixed set 11 1.25
of inertial directions which are taken to be those of perigee (P), Q = WxP and W (binormal). First choose a rectangular set (Figure 1.2)-wheFe -- The result is: Here, of course, the terms involving p are the first linear terms in the series expansion of Ag. Finally, using a cylindrical set of coordinates (R, 'I', 2) where kZ= R&s the equations are seen to be From sets 1.26 and 1.27 the usual expression for circular reference orbits is obtained by substituting nz '//fl,3 Figure 1.2 Target Centered Coordinate Systems 12 1.26 1.27
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: For all the equations given to this
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 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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