An Autonomous Onboard Targeting Algorithm using Finite Thrust

Maneuvers

Sara K. Scarritt, Belinda G. Marchand, Michael W. Weeks

1

IntroductionIntroduction Onboard guidance for Orion lunar return Two-level targeting algorithm Based on linear system theory Designed for impulsi e maneu ers Designed for impulsive maneuvers

In a main engine failure scenario, impulsive approximation invalid

Adapt two-level targeter to incorporate finite burns while retaining its simplicity

22

Classical Impulsive Level I ProcessGoal: Position Continuity Only Control Variables: ΔV’s

BEFORE LEVEL IBEFORE LEVEL I

AFTER LEVEL I

Classical Level II Process:Goal: Meet Specified Constraints (e.g. Velocity Continuity), Control Variables: Time & Position of Patch States

BEFORE LEVEL II

IMPLEMENTATION IN THE N/L SYSTEM

LEVEL II:LINEAR CORRECTION

Level 1: Impulsive vs Finite Burn

kδr

Level 1: Impulsive vs. Finite Burn

IMPULSIVE FINITE BURNδ kδr

kkδr

k

TTΔv

rv

6 1×

=

rx

v1k −1k−Δv

1k −

11 1

g

mm

×

=

x

u

6 1×

1

Constraint:

Control Variables: ,k

k Tt

δδ+

=r 0u1

Constraint: Control Variables:

k

k

δ

−

=Δ

r 0v

11 1×

5

1,k T−1k−

Variational Equations: Impulsive vs Finite BurnImpulsive vs. Finite Burn

IMPULSIVE

, 1 , 1 1 1 1

, 1 , 1 1 1 1

k k k kk k k k k k

k k k kk kk k k k

A Bt tC Dt t

δ δ δ δδ δ δ δ

− +− − − − −

− − + +− − − − −

− −= − −

r v r vv a v a

IMPULSIVE

1 1 1 1 1k k k kk k k k k kA Bt tδ δ δ δ++ + +

−+ + − −r v r v

FINITE BURN

, 1 , 1 1 1 1

, 1 , 1 1 1 1

k k k kk k k k k k

k k k kk kk k k k

t tC Dt t

δ δ δ δδ δ δ δ

+ + + + ++ + −

+−

+ + + +

= − −

r v r vv a v a

, 1 , 1 , 1 , 1 , 1

, 1 , 1 , 1 , 1 , 1

, 1 , 1 , 1 , 1 , 1T

T k T k T k T k T kT T T

T k T k T k T k T kT T T

T k T k T k T k T kT g T

A B E F GtC D H I JtK L M N Om m t

δ δδ δ

δ δ

−− − − − −

− −− − − − −

− −− − − − −

− − + =

r vv a

1

1 1 1

1 1 1

1 1k

k k k

k k k

k g k

tt

m m t

δ δδ δ

δ δ−

+− − −

+ +− − −

+ +− −

− − +

r vv a

, 1 , 1 , 1 , 1 , 1T T

T

T k T k T k T k T kg g T

T g T

P Q R S Tm m tt

δ δδ δ

− −− − − − −

− −

− − u u

1 1 1

, 1 , 1 , 1 , 1 , 1 1 1 1

k kg g k

T k T k T k T k T k k k k

m m tU V W X Y t

δ δδ δ

− −

+ +−

+ +− − − − − − − −

− − u u

1 1 1 1 1k k k kk k k k k kA Bt tδ δ δ δ++ +

− − −r v r vk T k TA Bt tδ δ δ δ− + − −r v r v

6

, 1 , 1 1 1 1

, 1 , 1 1 1 1

k k k kk k k k k k

k k k kk kk k k k

A Bt tC Dt t

δ δ δ δδ δ δ δ

+ + + + ++ + −

+−

+ + + +

= − −

r v r vv a v a

, ,

, ,

k T k Tk k k T T T

k T k Tk k T Tk T

A Bt tC Dt t

δ δ δ δδ δ δ δ− − + +

= − −

r v r vv a v a

Level 1 TargetingLevel 1 Targeting Direct from TEI-3 to Earth entry Entry targets:

Geodetic Altitude (km) 121.92 Longitude (deg) 175.6365g ( g) Geocentric Azimuth (deg) 49.3291 Geocentric Flight Path Angle (deg) -5.86

7

Level II Algorithm: Impulsive vs Finite BurnImpulsive vs. Finite Burn

[ ]1 2 1Constraints: , , , , , , , , , , vjn TEIh λ δ γ χ−

Δ Δ Δ Δ Δ

V = v v v A =

[ ]0 0 1 1Control Var , ,iables: , , ,,j

n nt t tδ δ δδ δ δ

b = r r r

k+v

k

1k +k+v

1k +IMPULSIVE FINITE BURN

k−v

1k −

k

k−v

1k −

kT

( ) 1 T TM MM

δ δδ δ

−

∂Δ Δ Δ ∂= → ∂

=

VV VbA A

bbA

8

( )

M

δ δ∂ ∂

A AbA

Variational Equations: Impulsive vs Finite BurnImpulsive vs. Finite Burn

IMPULSIVE

, 1 , 1 1 1 1

, 1 , 1 1 1 1

k k k kk k k k k k

k k k kk kk k k k

A Bt tC Dt t

δ δ δ δδ δ δ δ

− +− − − − −

− − + +− − − − −

− −= − −

r v r vv a v a

IMPULSIVE

1 1 1 1 1k k k kk k k k k kA Bt tδ δ δ δ++ + +

−+ + − −r v r v

FINITE BURN

, 1 , 1 1 1 1

, 1 , 1 1 1 1

k k k kk k k k k k

k k k kk kk k k k

t tC Dt t

δ δ δ δδ δ δ δ

+ + + + ++ + −

+−

+ + + +

= − −

r v r vv a v a

, 1 , 1 , 1 , 1 , 1

, 1 , 1 , 1 , 1 , 1

, 1 , 1 , 1 , 1 , 1T

T k T k T k T k T kT T T

T k T k T k T k T kT T T

T k T k T k T k T kT g T

A B E F GtC D H I JtK L M N Om m t

δ δδ δ

δ δ

−− − − − −

− −− − − − −

− −− − − − −

− − + =

r vv a

1

1 1 1

1 1 1

1 1k

k k k

k k k

k g k

tt

m m t

δ δδ δ

δ δ−

+− − −

+ +− − −

+ +− −

− − +

r vv a

, 1 , 1 , 1 , 1 , 1T T

T

T k T k T k T k T kg g T

T g T

P Q R S Tm m tt

δ δδ δ

− −− − − − −

− −

− − u u

1 1 1

, 1 , 1 , 1 , 1 , 1 1 1 1

k kg g k

T k T k T k T k T k k k k

m m tU V W X Y t

δ δδ δ

− −

+ +−

+ +− − − − − − − −

− − u u

1 1 1 1 1k k k kk k k k k kA Bt tδ δ δ δ++ +

− − −r v r vk T k TA Bt tδ δ δ δ− + − −r v r v

9

, 1 , 1 1 1 1

, 1 , 1 1 1 1

k k k kk k k k k k

k k k kk kk k k k

A Bt tC Dt t

δ δ δ δδ δ δ δ

+ + + + ++ + −

+−

+ + + +

= − −

r v r vv a v a

, ,

, ,

k T k Tk k k T T T

k T k Tk k T Tk T

A Bt tC Dt t

δ δ δ δδ δ δ δ− − + +

= − −

r v r vv a v a

Total Cost Constraint:Impulsive vs Finite BurnImpulsive vs. Finite Burn

v | |k k k+ −Δ = −v v ( )1 1, , ,k k k k k kt t− −

− −=v v r rIMPULSIVE

( )1 1, , ,k k k k k kt t+ ++ +=v v r r

( )T km t t− ( )v , ,k k T kf t t mΔ =

FINITE BURN

( )0v ln 1 kg T k

k spk

m t tI g

m

Δ = − −

( ), ,k k T kf1

01

[ ]n

k g burn jj

m m m t−

=

= − Δ

10

Main Engine SimulationMain Engine Simulation Initial guess data

E h 4 A 2024 15 30 00TDT Entry constraints:

G d ti Altit d (k ) 121 92 Epoch: 4-Apr-2024 15:30:00 TDT Initial mass: 20339.9 kg (total fuel =

8063.65 kg) Main EngineThrust: 33 361 6621 N

Geodetic Altitude (km): 121.92 Longitude (deg): 175.6365 Geocentric Azimuth (deg): 49.3291 Geocentric Flight Path Angle (deg) Main Engine Thrust: 33,361.6621 N

Main Engine Isp: 326 sec State (J2000 Moon-centered inertial

frame):

Geocentric Flight Path Angle (deg): -5.86

) X: -1236.7970783385588 km Y: 1268.1142350088496 km Z: 468.38317094160635 km Vx: 0.0329108058365355 km/sec Vy: 0.589269803607714 km/sec Vz -1.528058717568413 km/sec

11

Results (1/2)Results (1/2)

Moon

Earth

12 MCI Frame Perspective

Results (2/2)Results (2/2) Comparison of finite burn and impulsive algorithms:

13

Auxiliary Engine SimulationAuxiliary Engine Simulation Same initial guess data and constraints Assume main engine failure after TEI-1 TEI-2 and TEI-3 performed using auxiliary engines: Auxiliary Engine Thrust: 4,448.0 N Auxiliary Engine Isp: 309 sec

1414

ResultsResults Maneuver and final constraint data:

1515

Lunar Cycle SimulationsLunar Cycle Simulations Simulations run for 10 different days spanning February 2024 Patch points from converged impulsive runs

Initial lunar orbit of 100 km, targeting altitude (121.9 km) d fli ht th l ( 5 86o)and flight path angle (-5.86o)

Auxiliary engines used for TEI-2 and TEI-3

ResultsResults

Delayed Patch PointsDelayed Patch Points Patch points associated with specific epoch Targeter must converge even if the patch points are not

currentU F b 1 f l f l l Using February 1 input file from previous example, initial epoch delayed for (a) 3 hours and (b) 12 hours

ResultsResults

Conclusions and Future WorkConclusions and Future Work Two-level targeting algorithm developed for finite burn

maneuvers Algorithm successfully targets lunar return trajectory U i i i Using main engines Using auxiliary engines following simulated failure of main

engines after TEI-1g

Future work Implementing thruster steering law Automated patch point selection

20