guidance, flight mechanics and trajectory optimization

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Goldstein et al, (Reference 4.9) utilizes linear equations for the in- plane motion only and describes a procedure for obtaining the optimum thrusting and steering program. The problem is formulated as a Mayer problem in the calculus of variations and the switching function is developed. A sequence of thrusting programs is then constructed and developed in such a way as to approach that one which satisfies the maximizing conditions. Computational' results for a series of cases are given and these will be summarized below. The analysis to be presented is similar to that of Paiewonsky and Woodrow. In this case, the full set of equations for the circular reference orbit is used because the function to be minimized involves all three thr,ust directions, that is, Equation 4.1 is used. In addition, both the time optimal and fuel optimal rendezvous problems are developed. Much of the analysis is common to the two procedures; therefore, these discussions will be carried together until it finally becomes necessary to distinguish one problem from the other. These two problems are in fact very similar to the time and fuel optimal orbit transfer problems which are described by McIntyre (Ref. 4.12, Section 2.3.4, pages 61-68) as illustrations of the Pontryagin Maximum Principle. The differences between rendezvous and transfer problems are: first, that for the three dimensional transfer problem only five variables corresponding to some set of five distinct orbital elements have to be matched at the end, whereas for rendezvous six variables are needed so that the final position on the target orbit matches a given phase situation; and second, that the transfer problem is extremely non-linear while the differential equations for the present problem are linear with constant coefficients. In order to attack either of these two problems in three dimensions, it is necessary to combine the differential equations of motion and one for a measure of fuel consumption into a set of seven linear differential equations. The equations of motion were integrated in section 2.1.4.1 as follows: to begin with, the differential equations of motion are: (Eq. 1.21). where the els are distances expressed in units of the radius of the circular reference orbit; the independent variable, 8, can be taken as the true anomaly or the mean anomaly in radians (time in units of the time for the reference particle to move one radian around its orbit); and ui are the forces/unit mass in units of the acceleration of gravity at the reference orbit. The variables (51; 32, r; si 53, s; >= 'rT were transformed to WT according to Eq. 1.59 and thi sol&on was obtained as (Eq. 1.64) as 4.10 w = F (Q) (w 0 - z) 4.l-l 61
where F (Q) is the matrix given by Eq. 1.60. Then, the six integrals (z) of the components of the forcing variables ul, u2, u3 are given in Eq. 1.63. Now the acceleration produced by the engine is described by three control variables: b, LL , ,& where the magnitude of the acceleration will be specified by, 6 . (This variable will have only the values u. or 0 as called for by the switching function) and the components are described by two angles, Q!,,& , similar to latitude and longitude in the rotating coordinate system as shown in Figure &,2. Thus, the forcing functions are specified by the controls according to .Ul = b cosa cos p u2 = b coso( sin / “3 = bsinoc where the permissible values are 6 = 0 C?? 0, ONLY ACTIVE SPACECRAFT Figure 4, 2 Thrust Direction at Active Craft 62 4.12
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 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: Note that if it, can occur, this sc
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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