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p The kinetic energy of the system is Thus, Lagranges equations of motion for periods of normal thrusting are seen to be where a,, is the acceleration due to the normal thrust. Now substituting relation for p as a function of time from Equation (2.2) in the second equation above gives or This equation can be considered a first order differential equation in the variable 4 , i.e., But this equation has as a general solution Equation (2.4) is the solution for the angular rate of the LOS (e ) during periods of normal thrusting. This equation, however, is not the one employed to determine the length of time that the normal thrust is to be applied, but rather the simpler equation which assumes ,d '= 0 is used. The time of normal thrusting is calculated as l That is, if T0 is the threshold to zero the length of time that A? ?3 (2.5) tn = - an . value of t and it is desired to drive 8 normal thrusting should be applied is calculated from (2.5). If the motor used for normal control is actually operated for this time, 8 will not be driven to zero. Rather, the value that it 33 (2.4)
. will attain, z , can be calculated by substituting the value of t, from Equation (2.5) into Equation (2.4). This substitution gives Thus, assuming l&j L4 Ia,1 (this approximation becomes better as r-0 as will be seen when motion along the LOS is discussed) the ex- pression becomes The overall behavior of 6 can thus be determined from this equation and from the one given earlier for a no-thrust condition L Equation (2.3)]. This overall behavior will be a limit cycle between the values f0 and x as characterized in Figure (2.4). The time between applications of normal thrust ( tc) and fl can be calculated from Equations (2.3) and (2.6). Letting z= -$ (the time until rendezvous occurs assuming no radial thrust is applied) quation (2.3) results in a'Limit Cycle Figure 2.2 Finally, substituting for Z$ from Equation (2.6) gives A certain minimum time will be necessary to take and process the data to determine the current position and velocity. Therefore, it will be necessary to require that the time between corrections, t,, be longer,than the minimum data taking time. Since p is to be driven to zero and F, is small, it can be seen from Equation,(2.7) that a minimum t, will be maintained ifaminimum 1 is maintained. In turn, a constant7 ( or more practically a constant range of values for T ) is obtained by a proper choice of thrusting periods for the LOS thruster. A discussion of motion along the LOS is now called for so that the behavior of 2 during periods of LOS thrusting 34 (2.6) (2.7)
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: Figure 2.1 Polar Coordinate System
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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