Lambert Modified Sims-Flanagan Low-Thrust Model

Figure 7.1 illustrates the low-thrust trajectory model first described by Sims and Flanagan [89]. For each leg of the mission the low thrust trajectory arcs are modeled as a series of small impulsive ∆V maneuvers connected by conic 2-body arcs. Each leg of the trajectory is broken into N segments, with an impulse ∆V applied at the mid-point of each section.

The ∆V for each segment is not allowed to exceed a maximum magnitude, denoted by

∆Vmax−i. This maximum impulsive ∆V is determined so that it cannot exceed the capability of the two Busek BHT-20K solar electric thrusters at full power. The maximum ∆V for each

segment can then be determined as a function of the maximum thrust and mass flow rate of the thrusters as follows:

∆Vmax,i= Tmax

˙

m ln mi

mi− ˙m∆t (7.1)

where the maximum time for each segment, ∆t, is defined as

∆t = t_leg

N (7.2)

In low-thrust optimization problems, the objective is often to maximize the final spacecraft mass. It is therefore necessary to update the spacecraft mass after each ∆V is applied. For this purpose, Tsiolkovsky’s rocket equation is used to determine updated masses and is expressed as

mi+1= mie

−−∆V_i

g₀I_sp (7.3)

For each leg of the mission, the trajectory is propagated, via solutions to Kepler’s problem, forward from the starting point and backward from the ending point to a match point. The forward and backward propagated half legs are required to meet at the match point, which is typically ensured with six non-linear equality constraints. The match point is defined as follows:

Z_{f w}− Z_bw= [∆r_x, ∆r_y, ∆r_z, ∆V_x, ∆V_y, ∆V_z]^T (7.4) The standard Sims-Flanagan transcription can be modified, in order to remove the need for the six equality constraints, by utilizing a solution to Lambert’s problem to eliminate the match point equality conditions. Solutions to Lambert’s problem can be used to determine the trajectory between two radius vectors and a given time-of-flight [7, 9, 10, 12, 13]. An illustration of the modified Sims-Flanagan transcription is shown in Fig. 7.2.

With this new formulation a velocity discontinuity will occur at both end points of the Lam-bert arc segment. These velocity discontinuities are then subject to two inequality constraints, in order to ensure the two velocities discontinuities don’t exceed the capabilities of the solar

Lambert Arc Boundary Burn Point

Segment Boundary

Figure 7.2 An Impulsive ∆V Low-Thrust Trajectory Model by Sims and Flanagan.

electric thrusters. The advantage to this modification is that six equality constraints, which can be difficult for NLP solvers to enforce, are replace with only two inequality constraints.

The two inequality constraints are represented as:

g1 = ∆Vf w− ∆V_max (7.5)

g₂ = ∆V_bw− ∆V_max (7.6)

The Earth departure velocity is determined as a function of the Earth escape C3, provided by the lunar gravity-assist, and the departure ascension and declination angles. These two angles provide the direction of the Earth-system departure. The final Earth departure velocity is given as

V~_dep=√ c₃h

cos α cos β ~I + sin α cos β ~J + sin β ~Ki

(7.7) Sphere point picking is then used to determine the direction of each impulse maneuver.

The direction of the ∆V is determined by the two spherical pointing angles, θ and φ. The

clock angle, θ has a range from 0 to 360 degrees, while the cone angle, φ, ranged from 0 to 180 degrees.

A throttle parameter, , which ranges from 0 to 1, is used to determined the magnitude of the ∆V applied at each burn point. This allows a ∆V range from 0 up to the maximum possible ∆V for each segment and is defined as

∆Vi= ∆Vmax,i (7.8)

The final ∆V vector is calculated from the two spherical angles and ∆V magnitude as follows:

∆~Vi = ∆Vi

h

cos θ sin φ~I + sin θ sin φ ~J + cos φ ~K i

(7.9) For the low-thrust transcription method used in this paper, using the spherical coordinates pose some problems. If the ∆V magnitude for any of the individual burns is 0, the two angle, θ and φ become meaningless. If this happens, the optimizer may make large changes to these angle which have no effect on the solution. When the ∆V is turned back on it could potentially be pointed in the wrong direction. This problem is minimized by limiting the minimum throttling parameter to a small non zero value. In this algorithm a minimum of 1e-5 is used, which allows the individual ∆V values (and the actual thrust) to be close to, but not exactly zero.

Another problem with the standard spherical coordinates occurs when the two pointing angles are near their bounds. If the cone angle, φ, is exactly 0 or 180 degrees θ becomes meaningless. As with the zero ∆V problem the optimizer will make large unnecessary changes to θ. This problem is alleviated by limiting φ to a range of 2 to 178 degrees. Problems with the clock angle, θ can also occur when the optimizer approaches a value close to the lower angle bound of 0 degrees. The optimal solution may actually be a negative angle, which is the same as a large position angle. If this happens the optimizer will be unable to make the jump towards the upper limit of 360 degree. This problem is solved by changing the clock angle limits form 0 to 360 degrees to -360 to 360 degrees. The variable limits for the spherical Sims-Flanagan transcription used with the GNLP optimization algorithm are shown in Table 7.2.

Table 7.2 Variable limits for the spherical Sims-Flanagan transcription model.

α β θ_i φ_i _i

Ub 360 89 360 178 1

L_b -360 -89 -360 2 1.E-05