guidance, flight mechanics and trajectory optimization

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where A= B= The characteristic equation for A is A4 + A2 = 0 from which it is seen that the four independent functions out of which solutions are formed are 1, 0, sin 6, and cos 8. The fundamental matrix may be'taken to be: ( F(0)= 0 1 -30 -3 2 0 cos Sin 2 -2 cos Sin 6 8 0 8 cos -2 Sin cos Sin 8 6 8 1.50 The determinant of F is equal to unity,and the inverse is 1 -2 38 0 0 1 -3 case Sin6 0 -Sin@ case -2 cos Sin 0 8 > Thus, making the substitution z = F-‘(e)[ allows the differential equation for Z to be written as or Z' = ( -Sin z’ = F-‘(e) B(;) 2 cos 0 8 -2 1 38 cos Sin e 8 ) The state transition matrix for the variables y is thus Fce,F-?q = G(8-8,) 17 1.51 1.52 1.53 1.54
This matrix is simply written as G(8) (i.e., this notation is used to avoid writing) 4-3 cos 8 0 Sin 6 2(1 - cos e) 6(Sin 0 - '3) 1 ~(COS 8 -1) 4 Sin e - 3 G(8) = 3 Sin 0 0 cos 0 2 Sin 8 6(Cos 8 - 1) 0 -2 Sin 6 4 cos 0 - 3 This matrix, when combined with that of Equation 1-38 and both expressed in terms of 8 = nT (and n added as needed to give dimensions correctly), are seen to be exactly that of page 145 of SID 65-1200-5 (Reference 1.1). In the case of the representation of state transition matrix for locally level inertial systems (Table 2.4.2, page 147 of Reference l.l), the coordinate transformation required is only that between inertial and rotating systems at the moment they are aligned. Reverting to 6 = nT, the rotation rate is one of the angular velocity, n, about the z or third axis and the transformation matrix, T, for 6 =TX is with T = [ ‘& ---d- I ' 0 1 0 -n! n o!I The state transition matrix for inertial locally level (at both times) may thus be obtained from G(nT) and it is Q! '1 = T'lG(nT)T = : 2-Cos nT sin nT l/n Sin nT 2/n(l-Cos nT) 2 Sin nT-3nT 2 Cos nT-1 2/n(Cos nT-1) l/n(4 Sin n(3nT Sin nT) n(l-Cos nT) 2-Cos nT 3nT-2 Sin nT 1.58 n(Cos MT-l) -n(Sin nT) -Sin nT 2 Cos nT - 1 The development of the system by Tschauner and Hampel (Reference 1.4) involves a substitution to simplify the matrix of coefficients. In addition, the out-of-plane motion will be included and the set of six equations solved with matrix notation for later reference. It is simplest to add the out-of- plane coordinates to the set of four in-plane variables of Equation 1.48 as 18 1.55 1.56 1.57
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: Thus, the solution is written The t
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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