guidance, flight mechanics and trajectory optimization

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when a = semi major axis Q = one of Herrick's universal variables A = matrix A similar relation can be written for the velocity and variation in the orbit parameters The "correctl' velocity required for rendezvous is then given by 2.3 PROPORTIONAL GUIDANCE xr = 2 +si 4. = ‘gi ‘&&)&A? i’ Proportional guidance refers to a class of guidance laws which exhibits smooth thrust functions chosen to be proportional to some function of range and range rate. The mechanization of these laws requires throttable motors, this requirement may be undesirable when compared with the relative simplicity and reliability of constant thrust motors. Also, these methods are ex- pected to be less optimum than those using constant-thrust motors since optimization of maneuvers indicates that thrusting should occur in periods of either full or zero thrust (see Section 2.4). The computation of the thrust vector, however, is quite simple since the gravity model is not introduced into the equations. An example of proportional guidance is presented by Sears and Fellman (Reference 2.10). In this reference, the rocket thrust is determined from the equations c = s,[p- p I-s,+ St) c = s3[p-$a+s, (p (ys) where Sl, S2, S3, S4, S , and S6 are constants and LS is the angular rate of the line of sight. 'i The subscripts r, c and z refer io the radial, circumferential and out of plane direction.) If the constants Sl and S3 are made equal and S2 and Sk are set to zero, the thrust direction will be along the line of sight and the thrust magnitude will be determined by the difference between the range rate and the square root of the range. For this case, the trajectory as seen by an observer on the target satellite will appear as in Figure 2.13. 53
0 MOO0 60000 il 20& 40&o hod00 Y (feet) Figure 2.13 Rendezvous Trajectory with Thrust Along the Line of Sight If all four of the guidance constants Sl, S2, S3 and Sk-are made equal, a rendezvous will occur and the trajectory will appear as in Figure 2.14 - s V E x 40000 - 60000 - Y (feet) 20000 40000 60000 80000 Figure 2.14 Rendezvous Trajectory with All Sensitivities Made Equal 54
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: I(7) I COMPUTE c (7) =- MISS g-r -1
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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