Information about Applications of Artificial Potential Function Methods to Autonomous...

Talk charts from AAS/AIAA Astrodynamics Specialist Conference, held July 31 - Aug. 4 2011 in

Girdwood, Alaska

Girdwood, Alaska

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

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

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

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

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

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

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

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

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

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

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

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

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

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