This document discusses using artificial potential functions (APF) to design spacecraft trajectories. It presents examples of using an APF method to plan coplanar and inclined orbital transfers between Earth orbits. For the coplanar transfer, the APF method produced a trajectory with a total delta-V of 1.25 km/s, matching the optimal solution. For the inclined transfer, the APF trajectory had a delta-V of 3.46 km/s compared to 6.11 km/s for the optimal Lambert solution. The document also uses the APF method to plan a lunar return trajectory, calculating a delta-V of 1.9483 km/s to match arrival conditions. It concludes the APF method is

1. Introduction
Artificial Potential Function Trajectory Design
Examples
Conclusions
Applications of Artificial Potential Function
Methods to Autonomous Space Flight
Sara K. Scarritt∗
and Belinda G. Marchand†
AAS/AIAA Astrodynamics Specialist Conference
July 31 - Aug. 4 2011
Girdwood, Alaska
∗Graduate Student, Aerospace Engineering and Engineering Mechanics, The
University of Texas at Austin, 210 E. 24th St., Austin, TX 78712.
†Assistant Professor, Aerospace Engineering and Engineering Mechanics, The
University of Texas at Austin, 210 E. 24th St., Austin, TX 78712.
Sara K. Scarritt and Belinda G. Marchand 1

2. Introduction
Artificial Potential Function Trajectory Design
Examples
Conclusions
Background
Artificial Potential Function (APF) Methods
◮ Extensive use in path planning applications
◮ Global minimum at goal, peaks at constraints
◮ Vehicle follows steepest descent of potential
◮ Extendable to more general trajectory planning
Sara K. Scarritt and Belinda G. Marchand 2

3. Introduction
Artificial Potential Function Trajectory Design
Examples
Conclusions
Background
Modification for General Trajectory Design
◮ Discrete control parameter (∆v) vs. continuous control
◮ Formation flight problem - time derivative of Φ to define
switching time
◮ Minimum of Φ → lowest available maneuver cost
◮ Potential as a function of velocity error 1
(i.e. ∆v)
◮ Dynamical model to calculate desired velocity field
1Neubauer, J., Controlling Swarms of Micro-Utility Spacecraft, Ph.D.
dissertation, Washington University in St. Louis, August 2002
Sara K. Scarritt and Belinda G. Marchand 3

4. Introduction
Artificial Potential Function Trajectory Design
Examples
Conclusions
Maneuver Planning
Potential Function Construction
◮ Desired state/desired orbit (transfer time constraint)
◮ Two distinct cases: (1) intersecting orbits, and (2)
non-intersecting orbits
◮ Initial orbit intersects target orbit → maneuver at intersection
point:
Φint = (rintersect − r0)T
(rintersect − r0),
◮ For non-intersecting orbits, Φ = Φ(∆v):
Φvel = ∆v2
,
Sara K. Scarritt and Belinda G. Marchand 4

5. Introduction
Artificial Potential Function Trajectory Design
Examples
Conclusions
Maneuver Planning
APF Maneuver Planning
Sara K. Scarritt and Belinda G. Marchand 5

6. Introduction
Artificial Potential Function Trajectory Design
Examples
Conclusions
Desired Velocity Field
Coplanar Transfer
Lunar Example
The Desired Velocity Field
◮ Desired velocity depends on target point rf along final orbit
◮ Target point assumed to be
◮ apoapsis of the transfer orbit, if transferring to higher altitude
◮ periapsis of the transfer orbit, if decreasing altitude
◮ Gives eccentricity vector direction, leads to desired velocity
vector:
rf → ˆet → et → vt
Sara K. Scarritt and Belinda G. Marchand 6

7. Introduction
Artificial Potential Function Trajectory Design
Examples
Conclusions
Desired Velocity Field
Coplanar Transfer
Lunar Example
Coplanar Transfer: Desired Velocity Field
◮ Target point: 180◦
from current position
ˆrf = −
r0
r0
.
−8000 −6000 −4000 −2000 0 2000 4000 6000 8000
−6000
−4000
−2000
0
2000
4000
6000
x (km)
y(km)
(a) Velocity Field (b) Resulting Potential
Figure: Coplanar Velocity Field and Potential Function
Sara K. Scarritt and Belinda G. Marchand 7

8. Introduction
Artificial Potential Function Trajectory Design
Examples
Conclusions
Desired Velocity Field
Coplanar Transfer
Lunar Example
Coplanar Transfer
◮ Total ∆v: 1.25 km/s
◮ Doubly cotangential 180◦
transfer - matches analytical optimal
result
Table: Coplanar Transfer
Initial and Target States
Parameter Initial Target
x (km) -6478.145 0.000
y (km) 0.000 12587.983
z (km) 0.000 0.0000
vx (km/s) 0.000 -4.708
vy (km/s) -7.844 0.000
vz (km/s) 0.000 0.000
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
x (×104
)
y(×104
)
Target Orbit
Initial Orbit
Transfer Orbit
Maneuvers
Initial Position
Figure: Coplanar Transfer
Sara K. Scarritt and Belinda G. Marchand 8

9. Introduction
Artificial Potential Function Trajectory Design
Examples
Conclusions
Desired Velocity Field
Coplanar Transfer
Lunar Example
The Desired Velocity Field: Inclined Transfer
◮ Target point: intersection of initial and desired orbit planes
◮ Higher altitude solution to minimize cost of the plane change
−8000 −6000 −4000 −2000 0 2000 4000 6000 8000
−6000
−4000
−2000
0
2000
4000
6000
x (km)
y(km)
(a) Velocity Field (b) Resulting Potential
Figure: Non-Coplanar Velocity Field and Potential Function
Sara K. Scarritt and Belinda G. Marchand 9

10. Introduction
Artificial Potential Function Trajectory Design
Examples
Conclusions
Desired Velocity Field
Coplanar Transfer
Lunar Example
Inclined Transfer
◮ Total ∆v: 3.46 km/s → Compare to 6.11 km/s for Lambert
solution
◮ Consider three-maneuver sequence to reduce plane change
Table: Inclined Transfer
Initial and Target States
Parameter Initial Target
x (km) -5610.238 190.512
y (km) 0.000 12396.744
z (km) 3239.073 2177.562
vx (km/s) 0.000 -4.690
vy (km/s) -7.844 0.000
vz (km/s) 0.000 0.410
−0.5
0
0.5
−0.5
0
0.5
1
−0.2
0
0.2
x (×104
)
y (×104
)
z(×104
)
Desired Orbit
Initial Orbit
Transfer Orbit
Maneuvers
Initial Position
Figure: Non-Coplanar
Transfer
Sara K. Scarritt and Belinda G. Marchand 10

11. Introduction
Artificial Potential Function Trajectory Design
Examples
Conclusions
Desired Velocity Field
Coplanar Transfer
Lunar Example
Lunar Example (1/3)
◮ More complex test of APF algorithm
◮ Specific time needed at target state; not guaranteed by APF
method as-is
◮ Offset targeting to do timing match, typically converges in 3-4
iterations
Table: Initial Conditions
Epoch 2-Aug-2018 17:16:06 TDT
x (km) -1834.7155
y (km) -66.2361
z (km) -73.9653
vx (km/s) -0.0864
vy (km/s) 0.8139
vz (km/s) 1.4136
Table: Estimated Arrival Conditions
Epoch 7-Aug-2018 00:52:08 TDT
Geocentric Altitude (km) 121.92
Longitude (deg) -134.5456
Geocentric Latitude -19.20410
Geocentric Azimuth (deg) 13.9960
Geocentric Flight Path Angle (deg) -5.8600
Sara K. Scarritt and Belinda G. Marchand 11

12. Introduction
Artificial Potential Function Trajectory Design
Examples
Conclusions
Desired Velocity Field
Coplanar Transfer
Lunar Example
Lunar Example (2/3)
◮ Total ∆v: 1.9483 km/s
Figure: APF Lunar Return, 1.9483 km/s
Sara K. Scarritt and Belinda G. Marchand 12

13. Introduction
Artificial Potential Function Trajectory Design
Examples
Conclusions
Desired Velocity Field
Coplanar Transfer
Lunar Example
Lunar Example (3/3)
◮ Investigate phasing effects on total cost
◮ Shift departure/arrival epoch by n revolutions
−15 −10 −5 0 5 10 15
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
# of Revolutions
∆v(km/s)
Figure: ∆v of Time-Shifted Transfers vs. # of Revolutions
Sara K. Scarritt and Belinda G. Marchand 13

14. Introduction
Artificial Potential Function Trajectory Design
Examples
Conclusions
Conclusions
◮ Preliminary exploration of artificial potential function methods
as a design tool for generating startup arcs
◮ Candidate potential function construction presented
◮ Method for calculating a desired velocity field developed based on
two-body analysis
◮ APF trajectory design algorithm developed and tested
◮ APF method is promising, but room for improvement in design
of potential field
◮ Future work will focus on constructing more complex potentials
for use in multi-body regimes
Sara K. Scarritt and Belinda G. Marchand 14