guidance, flight mechanics and trajectory optimization

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This updating can be accomplished, however, at a much slower rate than is necessary for steering and cut-off calculations. In the reference, Gunckel proposes to update one element of the matrix each computation cycle; thus, updating the complete matrix in nine cycles. Since the quantities under the integral sign are taken to be constant during the time period, the above equation can be written as 2.2.4.2 Two Impulse Rendezvous The first linear impulsive guidance scheme to be considered was pre- pared by Clohessy and Wiltshire (Reference 2.1) and is the basis of the Gemini automatic rendezvous guidance (Reference 2.2 and 2.3). Two major velocity impulses are used through several smaller mid-course corrections calculated from the same equations and based on intermediate measurements of the state attained by the first change in velocity may also be required. The first velocity impulse is applied at the beginning of the rendezvous maneuver and establishes the target and rendezvous vehicle on a collision course. The second impulse is required to null the range rate at the time of closure. Consider a target vehicle in a circular orbit and a coordinate system fixed to the target such that the Y axis is always along the radius from the center of attraction, the X axis is circumferential inthe direction of the motion of the target, and the Z axis completes the right handed XYZ set. If the rendezvous vehicle is in an orbit which closely approximates that of the target vehicle, then the transition matrix of Equation (1.58) can be used to describe VEHICU the relative motion. In this Linear Rendezvous Coordinate System I scheme, the only concern is the Figure 2.5 reduction of the range to zero; thus, the only part of the matrix which is needed is rewritten below. 41
If a collision is to occur at time r then the components on the left of this equation must be zero. If this substitution is made and the matrix multiplication indicated on the right performedi the three resulting equations are assuming that the position at t = 0 is fixed, the velocity necessary to achieve a collision at time 'f can be determined by solving the above sets of equations for X0, Y,, and Z,. 2d = nZo coYnr -4 Thus, if the rendezvous vehicle is at a position (X0, Y,, Zo) with velocity lVXo' vY ' V z, ) and the desired time to rendezvous velocity'increment necessary is given by. is -r then the impulsive "$=4-v, AVy=i -‘I, Av,=r, =s 0 The velocity which must be nulled by the braking impulse can now be determined by again app$ying the transition matrix to the initial conditions x 0’ Yo' Zo' X0' yo, Zo' =3n&nr& %- +won7-ia +ZAnZk;~ A = 6n(mnr-i).Zd - Zdinnr20 f fC(CdOnT-3)fi kr = -nknrr, +c&-onT2* Note that this method requires knowledge of the time at which rendezvous is to occur. However, there are several factors which will indicate a suitable choice for 5 . For example, in the Gemini rendezvous, it was found that the velocity requirements for a rendezvous maneuver which caused interceptions in three-fourths of an orbit were less than for other fractions, therefore, r is chosen to cause such an interception (Reference 2.2). Another criteria would be the adjustment of the closing rate to be compatible 42
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: One method which can be employed (a
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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