Manifold structures also exist for fixed points along periodic orbits and are com- puted in a manner similar to the manifolds associated with equilibrium points. These manifolds asymptotically approach or depart a periodic orbit, therefore, such struc- tures are useful for designing transfers to or from periodic orbits. However, the natural flow reflected in the evolution of manifolds can also be leveraged for other spacecraft destinations.
Computation of periodic orbit manifolds follows from the discussion of periodic orbit stability in Section 3.6.3. The eigenvalues in that analysis signal stable, unstable, or marginally stable behaviors depending upon the modulus of the eigenvalues. The number of each eigenvalue type is once again assigned to three different categories and sum to n, the total number, i.e.,
n = nS+ nU + nC nS |λj| < 1 nU |λj| > 1 nC |λj| = 1 (3.39)
The three eigenvalue types correspond to stable, unstable, and center subspaces, each
with dimensions indicated by nS, nU, and nC, respectively. Analogs of the stable and
center manifold theorems exist for fixed points along periodic orbits. These theorems
support the existence of local stable WS
loc(Γ), unstable WlocU(Γ), and center WlocC(Γ)
manifolds associated with a periodic orbit Γ. These local manifolds are then tangent
to the stable, unstable, and center eigenspaces, ES(Γ), EU(Γ), and EC(Γ). The rate
at which manifolds approach or depart a periodic orbit is determined by the stability of the orbit. When an orbit is highly stable the unstable manifold, defined by a real eigenvalue with magnitude greater than one, requires a longer time to depart the orbit while the stable manifold takes longer to approach it. The opposite is true for highly unstable periodic orbits where the unstable and stable manifolds, respectively, depart and approach the periodic orbit more quickly.
The computation of invariant manifolds are initiated from any fixed point along a periodic orbit and, because the theory is largely analogous, the computation of periodic orbit manifolds generally follows the same steps as in Section 3.8.1. The eigenvalues of the monodromy matrix are independent of the fixed point along a pe- riodic orbit, however, the eigenvectors evolve from one fixed point to the next. These eigenvectors are computed at each fixed point from the monodromy matrix. Recall that the monodromy matrix is defined as the STM after one complete revolution. A new monodromy matrix at each fixed point could be calculated by propagating over a complete revolution at each fixed point, but such an approach is inefficient. Rather, a similarity transformation is employed to generate the eigenvectors at any point “downstream” from the initial fixed point on the periodic orbit through the following relationship,
νi(t1) = φ(t1, t0)ν(t0) (3.40)
In Equation (3.40), the STM from t0 to t1 advances the eigenvectors from t0 to
t1. The eigenvectors are then nondimensionalized to calculate the directions of the
eigenspaces. ˆ ν(t1)S = νS(t1) |νS(t1)| (3.41) ˆ νU(t1) = νU(t1) |νU(t1)| (3.42) Recall the 6-D eigenvectors are nondimensionalized using their magnitude. When the directions of the eigenspaces are constructed, an initial point on the manifold surface is determined by stepping off of the periodic orbit in the direction of one of the eigenspaces. The value of the step size is sufficiently small such that the first-order approximation remains valid, but also large enough that the numerical integration time is reasonable, i.e.,
xS(t1) = xeq(t1) ± dˆνS(t1) (3.43)
Moon
𝑊𝑈−
𝑊𝑆+
(a) Global Manifolds of L1 Halo Orbit
0.75 0.8 0.85 0.9 0.95 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 X [nondimensional] Y [nondimensional]
(b) Region about L1 Periodic Orbit
Figure 3.8.: Stable and Unstable Manifolds Associated with an L1 Halo Orbit as
Viewed in Configuration Space; the Manifold Structures are Represented by Trajec- tories Along the Surface
The initial conditions for the stable and unstable manifolds approaching and departing
an L1 halo orbit are constructed and then propagated for t = 5 nondimensional time
units. The resulting 6D paths are projected onto configuration space and the two- dimensional x-y position history is viewed in Figure 3.8. The invariant manifolds in position space actually form a three-dimensional invariant surface with a tube- like structure. The three dimensional velocity space can also be represented but are not plotted here. The manifolds corresponding to other periodic orbits form similar surfaces that suggest low-cost structures that may be exploited to transfer between regions of space in the CR3BP.
4. INDIRECT OPTIMIZATION
Mission design for spacecraft equipped with low-thrust engines typically requires the use of optimization techniques; and indirect methods are an approach that has yielded useful results and significant insights. Indirect optimization methods are generally based on analytical techniques derived from the calculus of variations and have been developed since the conception of this field in 1696. Indirect optimization employs the necessary conditions from the calculus of variations to formulate a two-point bound- ary value problem (TPBVP) that renders an optimal solution. These techniques reflect the classical approach to optimization having been employed long before the ubiquity of computers. However, computational power and numerical methods have enabled the evaluation of more complex TPBVPs, broadening the scope of problems for application via indirect optimization techniques.
The theory behind indirect optimization is introduced, and applied to the general low-thrust variable specific impulse (VSI) problem. One of the primary disadvan- tages of an indirect approach is the non-physical nature of the costate variables. This challenge is alleviated by introducing an adjoint control transformation, that is, a scheme that relates the costates to physical values and, thus, enables a simpler ini- tial guess. Presentations of indirect optimization techniques and the adjoint control transformation are found in numerous textbooks, however this content is reviewed to set up several sample problems that demonstrate the strengths and weaknesses of an indirect optimization method.