Optimal Flight Time Estimation

In order to estimate the optimal flight time of a vehicle, we consider a simple kinematic model of the spacecraft with a fixed mass throughout the whole flight. We also assume that the spacecraft will have a vanishing velocity when arriving at the target, which is true for the planned maneuvers with EAGLE. In a 1-D example, the vehicle tries to reach the point send starting from s0. In the time-optimal case, the thrusters will always run on maximum

power, which provides a maximum acceleration for the spacecraft.

Assume a spacecraft is at rest at t0. At the beginning of the maneuver, it will apply full thrust

accelerating the vehicle towards the target. At a certain point t1, which in this example is

right in the middle of the trajectory due to the vanishing velocities at start and end, the ve- hicle has a velocity of v2 as a result of the previous acceleration. It will then turn around at

this point and provide full thrust in the opposite direction for deceleration while the move- ment towards the target continues. Because of the basic laws of kinematics, it will have the desired final velocity of vend= 0at the target. This example shows the fundamental concept

on which the proposed flight time estimation is based on and is illustrated in image 5.1.

FIGURE5.1: Sketch of the heuristic defining the optimal time of flight in one

dimension

As in the example above, one can divide the movement of the vehicle into two segments of uniform acceleration: send= s1+ s2+ s0 = 1 2a1t 2 1+ v1t1+ 1 2a2t 2 2+ v2t2+ s0. (5.1)

In this formulation, the index 1 always denotes the segment, in which the vehicle is acceler- ating with a1 towardsthe target point, while 2 denotes the segment where the acceleration

is pointing away from the target. If gravity is not acting along the considered dimension, then a1 = −a2. The velocity v1 is the initial velocity of the spacecraft v1 = v(t0), while v2 is

the velocity at the end of the segment 1. The entire optimal flight time can then be calculated from the flight times for both segments with to= t1+ t2. Hence, the goal is to write equation

5.1 as a function only containing one of the two times for the segments together with only known parameters like initial and final conditions, such that it can be solved and the flight times can be obtained.

Depending on the initial velocity, there exist possible cases where the segment 1 is not exist- ing, since the vehicle will have to break for the entire time and in even further cases it might overshoot over the desired target send. We therefore start with the calculations related to

segment 2, since there will always be a segment involved in which the vehicle will have to break in order to stop at the final point.

The time to reach vend= 0starting with the velocity v2is given by

t2 = vend− v2 a2 = −v2 a2 (5.2) ⇔ v₂ = vend− t2a2= −t2a2. (5.3)

Note, that the segment 2 has been defined to be the segment where a2is pointing away from

the target. Therefore, if in the 1-D case se< s0, then a2 > 0and v2 < 0. In contrast, if se> s0,

then a2 < 0and v2 > 0, such that we will always obtain a positive time for the braaking

segment. Similar, the flight time t1 for the first segment is the time to accelerate the vehicle

from the initial velocity v1to the velocity v2 at the switching point

t1 = v2− v1 a1 ⇔ t₁ = vend− t2a2− v1 a1 = −t2a2− v1 a1 , (5.4)

where equation 5.3 has been inserted. This flight time also considers the case where there is no first segment, meaning that t1= 0if v1= −t2a2 = v2. The switching point of starting the

braking then falls onto the initial point of the maneuver.

Replacing t1in equation 5.1 by equation 5.4 and a reformulation results in a quadratic equa-

tion on t2that can be solved:

send= 1 2a2  1 −a2 a1  t2₂− 1 2 v₁2 a1 + s0 (5.5) ⇔ t₂ = v u u u t send− s0+1₂v 2 1 a1 1 2a2  1 −a2 a1  . (5.6)

The corresponding flight time t1 of the first segment can be obtained by inserting t2 into

equation 5.4, and therefore the estimated optimal flight time is to = t1+ t2. In the following,

we will refer to this procedure as flight time algorithm 1. The above equations only result in a reasonable flight time, if the vehicle’s initial velocity is small enough such that it does not overshoot the desired target point.

If the vehicle actually overshoots the target point, one can consider the trajectory separated into two parts: first, the spacecraft decelerates during overshooting, reaching a velocity of v₁0 = 0at the point s00. Afterwards, it accelerates towards the target point in one segment of

the trajectory and decelerates again when approaching the target point. Such a maneuver is sketched in figure 5.2. The second part can be solved by the flight time algorithm 1 using s0₀ as initial position and the initial velocity v0₁ = 0. In order to include the overshooting for an

FIGURE5.2: Sketch for a maneuver with overshooting

arbitrary maneuver, the following procedure can be applied: First, the time tdecto decelerate

the vehicle to zero velocity can be calculated as tdec = v_a1₃. The corresponding final point s00

is given by

s0₀= 1 2a3t

2

dec+ v1tdec+ s0. (5.7)

In this case, a3is the acceleration pointing against the velocity vector, either being equal to

a1 or to a2. Then the flight time algorithm 1 can be applied with the inital position s00 and

velocity v0₀ to calculate t1 and t2 for the second part of the trajectory, resulting in the total

flight time of to= tdec+ t1+ t2.

The complete proposed and adapted method for estimating the optimal flight time t0 is

therefore given by Calculate tdecand s00

if s0₀ further away from s0than sendthen

Calculate t1and t2with s00and v00 as initial conditions with flight algorithm 1

to = tdec+ t1+ t2

else

Calculate t1and t2directly from s0 and v0with flight algorithm 1

to = t1+ t2