Space Trajectory Optimization and L1Optimal Control Problems

Department of Mechanical and Astronautical Engineering, Naval Postgraduate School, Monterey, CA 93943 6.1 Introduction 6.2 Geometry and the mass flow equations 158 6.3 Cost functions and Lebesgue norms 160 6.4 Double integrator 6.5 Issues in solving nonlinear -optimal control problems 170 6.6 Solving nonlinear -optimal control problems 175 6.7 -Formulation of the minimum-fuel orbit transfer problem 179 6.8 A simple extension to distributed space systems 181 6.9 Conclusions The engineering...

Info

*(C(i), i) - V9C j (2.43) is the convective term. Since we chose the constants Cj to make canonical pairs (Q, P) obeying Eq. (2.39-2.40), then insertion of Eq. (2.39) into Eq. (2.43) will result in * E Qn(i) + E (2.44) So canonicity is incompatible with osculation when Aft depends on p. Our desire to keep the perturbed equations (2.39) canonical makes the orbital elements Q, P non-osculating in a particular manner prescribed by Eq. (2.44). This breaking of gauge invariance reveals that the...

Index

Abraham, R., 57, 70 Advanced Composition Explorer (ACE), 203 Albedo, 14 Andoyer elements, 23 Arnold, V.I., 57, 60 Artificial three-body equilibria ideal solar sail, 199-202 realistic solar sail, 202-203 Atmospheric density models, 7-12 Ballistic capture transfers, 111-16 method of determining, 116-20 properties, 116-20 Barker, 6 Battin, R.H., 2, 55, 66 Bekey, I., 212, 213 Belbruno, E., 125 Beletsky, V.V., 218-19, 220, 221 Betts, J.T., 171, 175, 177 Boundary value problem (BVP), Brumberg,...

X1

Fig. 5.4. (a) Approximation of the relative global attractor for the Henon mapping after 18 subdivision steps (b) attractor of the Henon mapping computed by direct simulation. In Figure 5.4(a) we show the rectangles covering the relative global attractor after 18 subdivision steps. Note that a direct simulation would not yield a similar result. In Figure 5.4(b) we illustrate this fact by showing a trajectory, neglecting the transient behavior. This difference is due to the fact that the...

T

X (Ml,M2,M3)T ( p,hp sin 7p,hp cos 7p) (2.125) 1. Efroimsky, Michael, and Peter Goldreich. (2003). Gauge symmetry of the N-body problem in the Hamilton-Jacobi approach. Journal of Mathematical Physics, 44, pp. 5958-5977 astro-ph 0305344. 2. Efroimsky, Michael, and Peter Goldreich (2004). Gauge freedom in the N-body problem of celestial mechanics. Astronomy & Astrophysics, 415, pp. 1187-1199 astro-ph 0307130. 3. Euler, L. (1748). Recherches sur la question des inegalites du mouvement de...

Value Problems Using Generating Functions Theory and Applications to Astrodynamics

Scheeres * MBDA France f University of Michigan, Ann Arbor, Michigan Contents 3.1 Introduction 3.2 Solving two-point boundary value 3.3 Hamilton's principal 3.4 Local solutions of the Hamilton-Jacobi 3.6 Conclusions Appendix A. The Hamilton-Jacobi equation at higher orders 99 Appendix B. The Hill three-body problem 102 Two-point boundary value problems have a central place in the field of astrodynamics. In general, most of the hard problems in this field...

Iit N

Where we have ignored the fact that this equation is an approximation for p to. Note that p is now a design option (i.e., gimbaled single engine or multiple ungimbaled engines). The unified rocket equation holds in other situations as well. For example, in cases when it is inconvenient to use spacecraft body axes, Eq. (6.2) can be used if (Tx,Ty,Tz) are any orthogonal components of T. In such cases, steering must be interpreted to be Fig. 6.2. Space vehicle thruster configurations (a) l2, (b)...

U

-----2x x r - x r - x x (x x r), (2.67) dots standing for time derivatives in the co-precessing frame, and being the coordinate system's angular velocity relative to an inertial frame. Formula (2.125) in the Appendix gives the expression for in terms of the longitude of the node and the inclination of the equator of date relative to that of epoch. The physical (i.e., not associated with inertial forces) potential U(r) consists of the (reduced) two-body part Uo(r) -GMr r3 and a term AU(r)...

Ta

For a required characteristic acceleration, Eqs. (7.3) and (7.4) may now be used to size a solar sail while imposing constraints on the total mass of the solar sail to satisfy the capacity of the launch vehicle. A typical design space is shown in Figure 7.2 for a characteristic acceleration of 0.25 mms-2, which is representative of the level of performance required for science missions such as a Mercury sample return. Both the payload mass and the total launch mass are shown. It is clear that...

Perturbed Motion

1.1 Basic 1.4 Drag 1.6 Solar radiation 1.10 Propagating the 1.13 Semianalytical 1.14 Variation of 1.15 Lagrangian VOP conservative 1.16 Gaussian VOP nonconservative 1.17 Effect on 1.18 J2 1.19 Comparative force model Perturbations are deviations from a normal, idealized, or undisturbed motion. The actual motion will vary from an ideal undisturbed path (two-body) due to perturbations caused by other bodies (such as the Sun and Moon) and additional forces not considered in Keplerian motion (such...

AyAay AtyAayy Aayy A7

Where n is the dimension of the configuration space. The Hamilton-Jacobi equation becomes jy yj + A j + + h ,jx xj + h ,j,*x xjx* + 0 . (A.8) Replacing X by y in Eq. (A.8) using Eq. (A.7), and keeping only terms of order less than 3 yields 0 jy yj+Aj yj + ha,by yb + ha,b,cyaybyc + (ha,n+s + + hn+a,n+b(A,ty* + A,ay* + A, Uy*y + Aa,tyty + *, ,ay*y ) X (A ,mym + m,bym + b,m,pymyp + p,b,mymyp + m,p,bymyp) + (hn+a,b,c + h c,n+a,b + hb ,c,n+a )ybyc ( a, ,a + hn+a,n+b,n+c( a,*y* + A,ayt)(A, y + Aby...

Gauge Freedom in Astrodynamics

2.1 Introduction 2.2 Gauge freedom in the theory of 2.3 A practical example on gauges a satellite orbiting a precessing oblate 2.4 Conclusions how we benefit from the gauge freedom 48 Appendix 1. Mathematical formalities Orbital dynamics in the normal form of Cauchy Appendix 2. Precession of the equator of date relative to the equator of epoch . . 50 Both orbital and attitude dynamics employ the method of variation of parameters. In a non-perturbed setting, the coordinates (or the Euler angles)...