Multiple Gravity-Assist Low-Thrust Using a Sundman Transformation

In the original MGALT model, a numerical optimizer selects the phase flight time variable (∆tp), and a

Kepler solver then iteratively determines the corresponding change in eccentric anomaly (∆Ep) by solving

Kepler’s equation. Analogously, a Kepler solver using a universal formulation determines the change in generalized anomaly (∆χp) given a particular flight time. The generalized anomaly arises from applying a

Sundman transformation to the Kepler solver’s independent time variable (Eq. (4.3)).

For the original MGALT transcription, the propagation time input to the Kepler propagator ∆t is simply determined by dividing the phase flight time by N , the number of discretization segments. The goal of this new algorithm is to instead place the control nodes at equal intervals of anomaly around the orbit, instead of positions separated by equal propagation times. The optimizer will now select a total propagation anomaly variable for the phase ∆χp, that will be divided into N equal segments



∆χ =∆χp

N



. It is not intuitive for a mission designer to place bounds on this new independent angle variable. It is much more natural to think in terms of flight times, therefore, the MGALT flight time constraints are retained in this new method for the mission designer to specify, with the task of selecting the total displacement in generalized anomaly being left to the NLP solver. The universal Kepler solver is used in an inverse sense to compute the time-of-flight that results from the angular displacement in each low-thrust control segment, in addition to propagating the state without the need for iteration. An epoch constraint is then imposed at the match point ensuring that the accumulated propagation time arcsPN

k=1∆tk



sum to the total phase flight time variable ∆tp.

The procedure for evaluating a trajectory phase, and for obtaining all the necessary partials to compute match point derivatives, is as follows (definitions for the symbols below are provided in the Appendix):

1. ∆χpis selected by the NLP solver and is the total change in generalized anomaly required to propagate

one trajectory phase. The size of the propagation segment, between each control node, is ∆χ = ∆χp

N

2. Starting at the left boundary, propagate inwards to the phase center for N₂ segments of generalized anomaly

3. For each propagation segment in the forward half-phase:

(a) beginning with some post-maneuver state X+_k−1=r+_k−1 v+_k−1T, compute α (Eq. (3.11)), the reciprocal of the semimajor axis, to determine if the current conic orbit is elliptic, hyperbolic or parabolic (this affects the computation of the universal functions {Un} and their derivatives)

(b) compute the universal functions Un n = 0, 1, 2, 3 (Eq. (3.15) - (3.17)).

(c) compute ∂Un

∂α and ∂Un

∂χ (Eq. (4.25)) for n = 0, 1, 2, 3 (for use in computing step g).

(d) compute the segment propagation time ∆tk (Eq. (4.7)) resulting from a propagation through ∆χ

of generalized anomaly as well as ∂∆tk

∂χ (Eq. (4.22))

(e) compute the Lagrange coefficients and their derivatives F,∂F_∂χ, ˙F ,∂ ˙_∂χF, G,∂G_∂χ, ˙G and∂ ˙_∂χG (Eq. (3.18) - (4.30))

(f) compute the propagated state X−_k =r−_k v−_kT (Eq. (3.1)) (g) compute ∂X

− k

∂X+_k−1

1_and ∂∆tk

∂X+_k−1 (Eqs. (4.23) and (4.24)), which form the basis for computing match

point derivatives

(h) compute the maximum available thrust Tmaxk using Eq. (2.15)

(i) compute ∆vmaxk with Eq. (3.2)

(j) compute the throttle magnitude constraint kukk < 1

(k) apply the impulsive maneuver v+_k = v−_k + uk∆vmaxk

(l) compute the spacecraft’s mass after the applied impulse m+_k+1with Eq. (3.6)

4. Locate the right hand phase boundary using the phase flight time NLP decision variable ∆tp

5. Repeat step 3 for the backwards half-phase

6. compute the match point constraint cmp and its partials ∂cmp ∂p cmp=          rB_{− r}F vB_{− v}F mB− mF tB− tF          = 0 (4.4)

where the F and B superscripts denote quantities at the end of the forward and backward propagated half-phases respectively. The epochs on either side of the match point are computed as follows, with t0referring to the epoch at the beginning of the phase, and tf referring to the epoch at the end of the

phase:

1_{This matrix is calculated using Eqs. (9.84)-(9.87) in Battin [160] or by differentiating Eq. (3.1) and using the additional}

tF = t0+ N/2 X k=1 ∆tk (4.5) tB= tf − N X k=N/2+1 ∆tk (4.6)

In the procedure above, the - and + superscripts indicate a quantity immediately prior to and after an applied impulse respectively. The 3x1 vector uk contains the control parameters of the kth control node.

The scalar quantity ∆vmax represents the maximum ∆v achievable by the spacecraft were it to operate its

thruster continuously over the course of the propagation segment, and is computed using Eq. (3.2). The spacecraft’s mass across the kth_{bounded impulse is computed using Eq. (3.6).}

One peculiarity of this method is that the maximum size of a maneuver ∆vmaxk can only be determined

after the propagation time ∆t has been computed. This is in contrast to using an equivalent time spacing of the nodes (MGALT) where segment propagation time is known prior to propagation and hence the trajectory may be computed in half-segments. For this reason, maneuvers must occur at the the right-hand boundary of a segment instead of in the center as they would for the original MGALT transcription. This is immaterial, as the location of the impulse is more or less arbitrary for an impulsive approximation of continuous-thrust, save for the case of the last segment in any half-phase where placing the maneuver on the right-hand boundary results in it occurring exactly at the match point. When both half-phases are considered, this results in the burn at the match point having a maximum potential size that is twice as large as the others. This is not a major concern, however, as the purpose of bounded-impulse methods is to produce an accurate estimation of the total mass requirement of a mission, and not necessarily the exact topology of its trajectory, and this method does not violate the total possible ∆v that the low-thrust engine is capable of imparting. This issue also reduces asymptotically with the use of more control nodes.