Minimum Energy Trajectories for Techsat 21
Earth Orbiting Clusters
Edmund M. C. Kong
SSL Graduate Research Assistant
Prof David W. Miller
Director, MIT Space Systems Lab
Space 2001 Conference & Exposition
Albuquerque
August 28-30, 2001
Objective and Outline
Objective
: To determine the optimal trajectories to reorient a cluster of spacecraft
Motivation : To maximize the full potential of a cluster of
spacecraft with minimal resources
Presentation Outline
•
•
Techsat 21 Overview
Optimal Control Formulation
– Equations of Motions
(Dynamics)
– Propulsion System (Cost)
– LQ Formulation
– Terminal Constraints
Space Systems Laboratory
•
•
•
Results
– Tolerance setting
– Cluster Initialization
– Cluster Re-sizing
(Geolocation)
Future Work
Conclusions
Massachusetts Institute of Technology
Techsat 21
• To explore the technologies required
to enable a Distributed Satellite
System
• Sparse Aperture Space Based Radar
• Full operational system of 35 clusters
of 8 satellites to provide global
coverage
• 2003 Flight experiment with 3
spacecraft
• Spacecraft will be equipped with Hall
Thrusters
– 2 large thrusters for orbit raising
and de-orbit
– 10 micro-thrusters for full threeaxis control
* Figure courtesy of AFOSR Techsat21
Research Review (29 Feb - 1 Mar 2000)
Space Systems Laboratory
Techsat 21 Flight Experiment
Number of Spacecraft
Spacecraft Mass
Cluster Size
Orbital Altitude
Orbital Period
Geo-location size
:3
: 129.4 kg
: 500 m
: 600 km
: 84 mins
: 5000 m
Massachusetts Institute of Technology
Equations of Motions
•
•
First order perturbation about
natural circular Keplerian orbit
Modified Hill’s Equations:
a x = &&
x − ( 5c 2 − 2 ) n 2 x − 2 ( nc ) y&
a y = &&
y + 2 ( nc ) x&
a z = &&
z + k2z
where
3J 2 Re2
⎡1 + 3cos ( 2iref ) ⎤
c = 1+ s
⎦
8rref2 ⎣
2
3nJ 2 Re2
⎡cos ( iref ) ⎤
k = n 1+ s +
2
⎦
2rref ⎣
s=
Possible trajectory for Techsat 21:
x = Ao cos( nt 1 − s )
2 1+ s
y=−
Ao sin( nt 1 − s )
1− s
z=−
2 1+ s
Ao cos( kt )
1− s
Space Systems Laboratory
Zenith-Nadir
•
400
2x1 Ellipse
200
Elliptical Trajectory
0
-200
-400
400
Projected Circle
200
0
-200
Cross Axis
-400
400
200
0
-200
Velocity Vector
Massachusetts Institute of Technology
-400
Propulsion Subsystem (Hall Thrusters)
• High specific impulse
– low propellant expenditure
• Electrical power required:
m 2u 2
Pe =
2m& η
200 W Hall
100 - 200 W
where
Thruster *
Hall Thruster *
m - mass of spacecraft (129.4 kg)
u - spacecraft acceleration (m/s)
BHT-200-X2B Hall Thruster
m& - mass flow rate of propellant (kg/s) Specific Impulse
: 1530 s
η - thruster efficiency (%)
Thrust
: 10.5 mN
Mass flow rate
: 0.74 mg/s
• Objective is to minimize electrical
Typical Efficiency
: 42%
energy required:
Power Input
: 200 W
t
J = ∫ Pe dt
f
to
Space Systems Laboratory
* Figures courtesy of AFOSR Techsat21
Research Review (29 Feb - 1 Mar 2000)
Massachusetts Institute of Technology
Optimal Control Theory
•
Linear Dynamics
•
tf
x& = Ax + Bu
•
Augmented Cost
(Method of Lagrange)
tf
J a ( u) = ∫ { 12 uT Ru
Quadratic Cost
J = ∫ Pe dt
to
•
First order variation
+ pT [ Ax + Bu − x& ]}dt
Boundary Conditions
x*(to) = xo
x*(tf) = xf
x*(to) = xo
p*(tf) = 0
3. x(tf) on the surface x*(to) = xo
k
∂m
*
m(x(t)) = 0
− p (t f ) = ∑ d i [ i ( x* (t f ))]
∂x
i =1
*
m(x (tf)) = 0
Space Systems Laboratory
tf
δJ a ( u) = − p (t f )δx f + ∫ {[ p*T A + p& *T ]δx
to
1. x(tf) = xf specified
terminal state
2. x(tf) free
1 tf T
J ( u) = ∫ u Rudt
2 to
T
to
+ [u*T R + p*T B]δu
+ [ Ax * + Bu*-x& * ]δp}dt = 0
Linear Quadratic Controller
(to to tf)
x& * = Ax * + Bu*
p& * = − A T p*
u* = − R -1BT p*
Massachusetts Institute of Technology
Terminal Conditions (Multi-Spacecraft)
For each spacecraft (Ro projection on
y-z plane):
• Position Conditions
2
2
⎡ y ⎤ ⎡ x sin γ + z cos γ ⎤
m1 = ⎢ ⎥ + ⎢
⎥ −1
(
)
R
5
/
2
R
sin
γ
o
⎣ o⎦ ⎣
⎦
m2 = x cos γ − z sin γ
• Velocity Conditions
2
2
⎡ y& ⎤ ⎡ x& sin γ + z& cos γ ⎤
m3 = ⎢
⎥ + ⎢ (5 / 2 )nR sin γ ⎥ − 1
nR
o
⎣ o⎦ ⎣
⎦
m4 = x& cos γ − z& sin γ
• Tying Condition
m5 = y& ( x sin γ + z cos γ )
5
− y ( x& sin γ + z& cos γ ) + nRo2 sin γ
2
Phasing Condition (Cluster):
N
⎡ y⎤ ⎡ y⎤
m5 N +i = ∑ ⎢ ⎥ − ⎢ ⎥ − C i
j =i ⎣ z ⎦ i
⎣z⎦j
Space Systems Laboratory
where
N
2 Ro ∑ 1 − cos θ i , j
Ci =
for i = 1, 2, …, N-1
j =i
4 spacecraft example:
C1 = 4.35
C2 = 3.42
C3 = 1.67
N-th Condition (Total of 6N conditions)
− p (t f ) =
*
6 N −1
∑ di [
i =1
∂mi *
(x (t f ))]
∂x
Massachusetts Institute of Technology
Multiple Shooting Method
Solving two point boundary value problems
x
x
(t3,s3)
(tm,sm)
(tm-1,sm-1) (tm,sm)
(t1,s1)
(t1,s1)
(t2,s2)
t1
t2
t3
tm-1
tm
t
Simple shooting method
• Guess the missing states at to
and compare the integrated
states at tf with terminal
constraints
• Numerically unstable - errors
are amplified due to integration
Space Systems Laboratory
t1
t2
t3
tm-1
tm
t
Multiple shooting method
• Guess states at tk and
compare the integrated states
at tk+1 with states at tk+1
• Numerically more stable
• Computationally expensive
Massachusetts Institute of Technology
Tolerance Setting
(a)
(c)
Tolerance Set at 10-3
Space Systems Laboratory
Massachusetts Institute of Technology
Tolerance Setting
(a) (e)
(c)
Tolerance Set at 10-3
Space Systems Laboratory
Massachusetts Institute of Technology
Cluster Initialization (1)
• Cluster initialization from Hill’s origin to Ro = 250 m
•
•
•
In general, average energy required are similar for different N spacecraft
clusters
Slight differences in energy requirements are due to the more stringent
constrains placed on phasing the array (eg. E2sc < E3sc)
Average energy required decay rapidly as a function of initialization time
Space Systems Laboratory
Massachusetts Institute of Technology
Cluster Initialization (2)
200 W
Power History for cluster of 3 S/C
• Peak power required is below Techsat 21
maximum (200 W) for initialization periods
greater than 0.2 period
• ∆V required asymptotes to ~ 0.45 m/s
• Recommend initialization time of 1 period
due to significant ∆V savings (67%)
Space Systems Laboratory
Massachusetts Institute of Technology
Cluster Re-sizing (1)
5km
500m
Objective of Techsat 21 Geolocation mission is to provide
10-50 m geo-location accuracy
•
Geo-location accuracy is
inversely proportional to size
of cluster
•
Re-size cluster to an elliptical
trajectory of 2.5 km to achieve
approximately 10 m ground
resolution
•
Example application is to
quickly locate a lost pilot
(Time critical mission)
Reconfigure
Radar Mode
Geolocation Mode
* Figure courtesy of AFOSR Techsat21
Research Review (29 Feb - 1 Mar 2000)
1500
Zenith-Nadir (m)
•
750
0
-750
-1500
3000
1500
0
-1500
Cross Axis (m)
-3000
3000
1500
0
-1500
-3000
Velocity vector (m)
Optimal Cluster Re-sizing
Space Systems Laboratory
Massachusetts Institute of Technology
Cluster Re-sizing (2)
Corresponding ∆V Required
Maximum Power Required For Geo-location
200 W
•
•
Minimum re-sizing time of 0.5
periods (48 mins) is required for
Techsat 21 geo-location
Maximum size of 1250 m can be
attained if re-sizing time of 30
minutes is allowed
Space Systems Laboratory
•
•
Minimum ∆V of 10 m/s is required to
perform Techsat 21 geo-location
operation (25% of total ∆V budgeted)
Significant ∆V savings can be
achieved by increasing re-sizing
time to at least 1 period (97 mins)
Massachusetts Institute of Technology
Future Considerations
Solutions obtained are only
guaranteed to be local minimum not global optimum
Must check for minimum energy
trajectories
•
1500
Zenith-Nadir (m)
•
750
0
-750
-1500
3000
1500
0
-1500
Zenith-Nadir (m)
1500
Cross Axis (m)
-3000
3000
1500
0
-1500
Velocity vector (m)
750
0
•
-750
•
-1500
3000
1500
0
-1500
Cross Axis (m)
Space Systems Laboratory
-3000
3000
1500
0
-1500
Velocity vector (m)
-3000
•
Plume contamination due to
thruster firings
Penalize thruster firings at
other spacecraft
Penalize firings in the plane of
elliptical trajectory
Massachusetts Institute of Technology
-3000
Conclusions
Developed a tool to
– determine minimum energy trajectories
– evaluate minimum resources required
for cluster re-configuration
– size power subsystem for propulsion
1500
Zenith-Nadir (m)
•
750
0
-750
-1500
3000
1500
0
-1500
Cross Axis (m)
•
•
3000
1500
-1500
-3000
Velocity vector (m)
Techsat 21 Cluster initialization
– achievable even with a short initialization time
– recommend an initialization time of at least 1
period due to significant ∆V savings
Techsat 21 Geo-location problem
– a minimum re-orientation time
of at least 1 period
– extremely high ∆V expenditure
operation
Space Systems Laboratory
-3000
0
5km
500m
Reconfigure
Radar Mode
Geolocation Mode
Massachusetts Institute of Technology