Thruster failure recovery strategies
for libration point missions
Maksim Shirobokov
Keldysh Institute of Applied Mathematics
Moscow Institute of Physics and Technology
Sergey Trofimov
Keldysh Institute of Applied Mathematics
Moscow Institute of Physics and Technology
Contents
• Motivation
• Problem statement
• Theory background
• Test case: Sun-Earth L2, planar periodic orbits
• Conclusion
2/21
Missions to libration points
• Successfully accomplished missions:
– ISEE-3, WIND, SOHO, ACE, Genesis
• Promising near-future projects
– Deep Space Climate Observatory (NASA)
– LISA Pathfinder (ESA/NASA)
– Spektr-RG (Roscosmos/ESA)
3/21
Features related to periodic motion
around collinear libration points
High instability of motion requires the application of
station-keeping techniques and their essential components:
• Accurate trajectory determination
• Regular control-law updates
In average, 2-12 m/s per year is required
Thus, any possible thruster (or communication) failure1
threatens a mission and can lead to a significant deviation
of the spacecraft from the nominal periodic orbit
1The
largest percentage of all fail occurrences relating to the control system falls on
thruster failure, see Tafazoli [2009] “A Study of On-Orbit Spacecraft Failures”, Acta
4/21
Astronautica
Thsuter failure issue
If a thruster fails, the control is allocated to a redundant set
of thrusters:
• attitude control thrusters
• a backup orbital thruster
Most of publications are related only to collision avoidance
during rendezvous and docking. The problem of libration
point mission recovery has not been deeply studied yet
5/21
Problem statement
Basic assumptions:
• the main orbit control thruster failed and produces no thrust
• the planned correction maneuver is not performed on time
• with some delay, a redundant set of thrusters is used
Transfer to the nominal periodic orbit may appear to be too
expensive:
• unstable environment leads to fast orbit decay
• redundant thruster has usually less fuel than the main one
Therefore, not enough fuel is left to perform station-keeping
maneuvers during the planned mission lifetime
6/21
Thruster failure recovery strategies
Two strategies are considered:
• periodic orbit targeting (POT)
• stable manifold targeting (SMT)
In both cases, the aim is the same—to find the “cheapestto-get” periodic orbit for different values of correction
maneuver delay (the time passed since the moment of
unsuccessful correction maneuver)
7/21
Circular restricted three-body problem
The planar circular restricted three-body problem (CR3BP)
is studied:
• a spacecraft of negligible mass moves under the
gravitational influence of two masses m 1 and m 2
• the spacecraft is supposed to move in the orbital plane
of the primaries
Note: the proposed recovery strategies can be applied to
the spatial case (for example, for halo orbits)
8/21
Reference frame
Mass parameter
  m2
 m1  m 2 
Non-dimensional units:
m1  1  
xm1   
m2  
xm 2  1  
0  1
For the Sun-(Earth+Moon) system   3.03939  10  6
9/21
Equations of motion
In rotating frame
x  2 y  U x ,
y  2 x  U y
where
U  x, y   
x  y
2
2
2

1 
r1


r2

 1   
,
2
is the so called effective potential; U x and U y are the partial
derivatives of with respect to the position variables. The
distances between the spacecraft and the primaries equal
r1 
x  
2
 y
2
r2 
x 1   y
2
2
10/21
Libration points
Equilibrium (libration) points can be found from the equations
Ux Uy  0
Collinear libration points
x L  1  rH 
1
x L  1  rH 
2
xL  1 
3
5
12
rH    3 
1
3
1
3
2
rH

rH 
2

26
9
28
3
rH
x L  0.989987
3
x L  1.010074
12
1
rH
9
23  49
4
Sun-(Earth+Moon) system

2
3
x L   1.000001
3
1 3
11/21
Richardson’s third-order
approximation of periodic orbits
The third-order approximation of periodic orbits in normalized
variables x   x  x L   and y  y  expressed as follows:
2
2
3
x  a 2 1 A x  A x c os  1  a 2 3 A x cos 2 1  a 3 1 A x cos 3 1  Aexp u x ,st exp    t 
2
3
y  kA x sin  1  b 21 A x sin 2 1  b31 A x sin 3 1  Aexp u y ,st exp    t 
where
k   p  2 c2  1
2
p 
2  c
2

 2 p   0
9 c  8c2
2
2
a 21 , a 23 , a 31 , b21 , b31 , s1 

2
 1   p t  
c2   x L  1  
  1  s1 A x
2
3
 1    x L  
some constants
12/21
3
Periodic orbit targeting strategy
J
 y 
 v 1   v 2  m in
y   A x , T1 , T 2 , 
Reference
periodic
orbit

Backup
periodic
orbit
13/21
Gain in delta-v for POT strategy
t d , reference orbit periods
14/21
Change in amplitude for POT strategy
t d , reference orbit periods
15/21
Stable manifold targeting strategy
J
Reference
periodic
orbit
Stable
manifold
 y 
 v 1   v 2  m in
y   A x , T1 , T 2 ,  , t 
16/21
Gain in delta-v for SMT strategy
t d , reference orbit periods
17/21
Change in amplitude for SMT strategy
t d , reference orbit periods
18/21
Conclusion
• Two recovery strategies in case of possible
thruster failure—periodic orbit targeting and
stable manifold targeting—are proposed for
collinear libration point missions
• The proposed approach reduces delta-v spent
by the redundant set of thrusters and
increases the lifetime of the spacecraft
19/21
Future work
• Targeting to periodic orbits with larger amplitudes
requires higher-order approximations of these orbits
• Different positions of the unsuccessful correction
maneuver may bring to different results
20/21
Thank you!