guidance, flight mechanics and trajectory optimization

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Again, a test of the second derivatives will indicate that only the plus sign for D is allowed. The quadrant determined by the sign of C. for a (- 3 4 or L YZ ) is thus The portion of the Hamiltonian containing the control is H, = b (D + q) = b (D + p7) Thus, it is clear that to maximi ze the Hamiltonian when D + P7 '0, the value of b should be uo, whereas when D + P7 < 0, b should be 0. The case when D + P7 = 0 over an appreciable time is called a singular arc and is discussed by McIntyre (p 67 of Reference 4.12). Such arcs are not excluded and have not been found to occur for these problems. Note that since D is either positive or zero, the value of P7 must be negative for coasting arcs to occur. The function k = D + P is called the switching function since it controls the value of the thrus z . Thus, to summarize If k Lo, fhen 6-O a coasting arc, or IT k 70, &nb=U, a full power thrusting arc and the thrust components are ul = b A/D u2 = b B/D u3 = b D/C. The distinction between the time optimal problem and the fuel optimal problem lies in the function to be optimized, the constraints, the boundary conditions on the adjoint variables, and the Hamiltonians. The time optimal problem is the determination of the controls such that 4.27 4.28 4.29 J = Qf isaminimum 4.30 under the constraint that w7 ( Qf ) f AV, the total allowab:e velocity increment. It is necessary to include the constraint w (Q,) - A V in some way since otherwise P = 0 and the engine will be 7. urned on all the time. (That is, the time ?I o rendezvous would be minimized if there were no coast arc; that is,the vehicle should accelerate thus increasing the closing rate until it is necessary to reverse the thrust to be able to stop at the target). If the fuel is limited, this result, suggests that the optimal time will be attained by using all of the fuel and hence, w7 (Qf) = AV is the boundary condition. Thus, the seventh constraint is taken to be Y7 = “7 @,I - A V = 0 for the time optimal problem. 65
In contrast, the fuel optimal problem is the determination of the controls such that J =W7(Qf) is a minimum (4.31) under the constraint that 0f L T. Suppose that a local optimum has been found which produces rendezvous as the result of a one, two, or three burn maneuver. This solution is the program sought and represents a lower limit to the possible value of T which might be imposed in view of the thrusting limitations. Hence, the final time 8 will be left open for the fuel optimal case; that is, only locally optimal solutions will be sought in the neighbor- hood of one, two,, or three burn maneuvers, The boundary conditions on the adjoint variables and the Hamiltonian are given by (Equation 2.3.36, P. 55 Reference 4, 12) where 7J. w7. J i= / 9 **-, 7 (Q c ) = 0 are the boundary conditions for the variables w l>""' (4.32) (4.33) In summary, for the time optimal problem, the results are (at 8 = Qf) P +p =o (for the first six) P7 +/$=o (4.34) H = 1 and for the fuel optimal problem (time open) P +p= 0 (for the first six) p7 +1= 0 (p7 = -1) (4.35) H = 0 The computational problem involved in finding the optimal maneuver is one of determining the six components of ,A4 ( or of p,) from the six conditions w(Q,) = 0 (and /u 7 from w (Qf) = AV for the problem has been reduce 2 to a two point the time oPtimaI problem). Thus, boundary value problem (as may be expected, this problem is one of considerable sensitivity). Paiewonsky and 66
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NASA CONTRA REPORT CTOR GUIDANCE, F
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- ..a --.
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_- .-- --- I
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110 2.1 2.1.1 2.1.2 2.1.3 2.1.3.1 2
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. 0 i time difference between a ref
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only line-of-sight thrusting is use
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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 and 72: Note that if it, can occur, this sc
Page 73 and 74: where F (Q) is the matrix given by
Page 75: where Since 0 and Eq. 4.23 shows th
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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