Sub-Optimal Solutions for Satellite Orbital
Transfers
6th International Workshop and Advanced
School
"Spaceflight Dynamics and Control"
Antonio Fernando Bertachini de Almeida Prado
Division of Space Mechanics and Control – INPE
C.P. 515, 12227-310 São José dos Campos - SP, Brasil
prado@dem.inpe.br
INTRODUCTION
The idea of the present research is to study low thrust maneuvers
for a satellite that needs to escape from a collision with a cloud of
particles.
The cloud of particles is assumed to come from a close approach
with a celestial body, that causes a dispersion of particles.
So, we first study the orbital behavior of the cloud to define an
orbit that is safe and than we optimize transfers to achieve this
safe orbit.
Sub-optimal control will be used to complete this task.
INTRODUCTION
To study the orbit of the cloud we use analytical equations for the
variations of velocity, energy, angular momentum and inclination
for a spacecraft that passes close to a celestial body.
This passage is assumed to be performed around the secondary
body of the system. In 2D:
V+


ΔV  ΔV  2 V sin δ  2V sin δ 


B
Vp
P M3
rp
V2
Rumo a M1

E  2V2V sen  sen 


X
M2
A
V
SWING-BY OF A SINGLE PARTICLE
It is assumed that the system has three bodies: a primary (M1) and a
secondary (M2) bodies with finite masses that are in circular
orbits around their common center of mass and a third body with
negligible mass. In 3D:
z
B

PM
p
1-
y
M1
M2
A
3
x
EXTENSION TO A CLOUD OF PARTICLES
It is assumed that all the particles that belong to the cloud
have similar orbital elements before the close approach,
with a small difference from each member of the cloud,
that will be varied in a short interval to represent the group
of particles.
We then study the change of the relative orbital elements of
the orbits of this cloud of particles after the close approach
with the planet, as well as the variations in velocity, energy
and angular momentum.
VARIABLES THAT DEFINE AN ORBIT
The following five variables are used to define one trajectory:
1)
Vp, the velocity of the spacecraft at periapsis;
2)
Two angles ( and ), that specify the direction of the
periapsis;
3)
rp the periapsis distance;
4
, the angle between the velocity vector at periapsis and
the intersection between the horizontal plane that passes
by the periapsis and the plane perpendicular to the
periapsis that holds .
ANALYTICAL EQUATIONS FOR FHE SWING-BY
IN 3 DIMENSIONS
First, it is calculated the initial conditions with respect to M2 at the periapsis:
Position:
xi  rp cos  cos 
yi  rp cos  sin 
z i  r p sin 
Velocity: V xi  V p sin  sin  cos   V p cos  sin 
V yi  V p sin  sin  sin   V p cos  cos 
Vzi  Vp cos sin

V  A

rp
rp
B

Vp
Vp
ANALYTICAL EQUATIONS FOR FHE SWING-BY
IN 3 DIMENSIONS

V  A (cos  cos , cos  sin , sin ) 
 B( sin  sin  cos   cos  sin , sin  sin  sin  
 cos  cos , cos  sin  )







r
V
p
p 
V  Vp   A
B
 Vp  BVp  V Vp cos 

rp
Vp 





Vi  V  V2  V sin(cos  cos , cos  sin, sin) 
 V cos ( sin sin cos   cos  sin,  sin sin sin 
 cos  cos , cos  sin )  (0, V2 ,0)



V0  V  V2  V sin(cos  cos , cos  sin, sin) 
 V cos (sin sin cos   cos  sin,  sin sin sin 
 cos  cos , cos  sin )  (0, V2 ,0)
VARIATIONS

V  V  2Vsin
E 


1 2
V0  Vi2  2V2 V cos sinsin
2


C  2dVsin cos 2 sin 2  sin 2
C
Cosii   iZ 
Ci
C
Cosi o   oZ 
Co

1
2
1




sin

sin


cos

cos

sin


1 
 V2


cos

cos

cos


cos

sin

sin


cos

sin

sin

sin



V
 

2
1




sin

sin


cos

cos

sin


1 
 V2


cos

cos

cos


cos

sin

sin


cos

sin

sin

sin



 V

2
RESULTS (PLANAR CASE)
Some results, for the case a = e = 0.001, rp = 1.4 RJ
RESULTS (PLANAR CASE)
Some results, for the case a = e = 0.001, rp = 1.4 RJ
RESULTS (3D)
Initial conditions (rp, vp, , , ) are varied and the effects of the
close approach are studied.
A satellite exploded when passing by the periapsis.
This position is given by  = 30,  = 45.
The reference value for the direction of the velocity is  = 60.
Two values were used for the velocity at periapsis (vp = 4.0 and
vp = 4.5) and for the periapsis distance (rp = 1.5 rJ, rp = = 5.0 rJ).
The vertical axis shows the difference between the value of
every single particle and a reference value, assumed to be the
value of the particle that remains with the nominal values of .
The horizontal axis shows the value of , in radians.
0.15
0.004
0.1
0.003
0.05
0.002
1.45
0.001
1.5
1.55
-0.05
1.45
1.5
1.55
-0.1
-0.001
-0.15
Fig. 1 – Variation in Inclination
(rad) for rp = 1.5 rJ and vp = 4.0.
0.1
Fig. 2 – Variation in Velocity for
rp = 1.5 rJ and vp = 4.0.
0.04
0.05
0.02
1.45
1.5
1.45
1.55
-0.05
1.5
1.55
-0.02
-0.04
-0.1
Fig. 3 – Variation in Angular
momentum for rp = 1.5 rJ and vp = 4.0.
Fig. 4 – Variation in Energy for
rp = 1.5 rJ and vp = 4.0.
0.1
0.0075
0.005
0.05
0.0025
2.45
2.55
2.6
2.45
2.55
2.6
-0.0025
-0.05
-0.005
-0.1
-0.0075
Fig. 5 – Variation in Inclination
(rad) for rp = 1.5 rJ and vp = 4.5.
Fig. 6 – Variation in Velocity for
rp = 1.5 rJ and vp = 4.5.
0.075
0.03
0.05
0.02
0.025
0.01
2.45
2.55
2.6
-0.025
2.45
2.55
2.6
-0.01
-0.02
-0.05
-0.03
-0.075
Fig. 7 – Variation in Angular
momentum for rp = 1.5 rJ and vp = 4.5.
Fig. 8 – Variation in Energy for
rp = 1.5 rJ and vp = 4.5.
0.03
0.002
0.02
0.001
3.35
0.01
3.45
3.5
3.35
3.45
-0.001
-0.01
-0.002
-0.02
3.5
-0.03
Fig. 9 – Variation in Inclination
(rad) for rp = 5.0 rJ and vp = 4.0.
Fig. 10 – Variation in Velocity for
rp = 5.0 rJ and vp = 4.0.
0.01
0.02
0.005
0.01
3.35
3.45
3.5
-0.01
-0.02
Fig. 11 – Variation in Angular momentum
for rp = 5.0 rJ and vp = 4.0.
3.35
3.45
3.5
-0.005
-0.01
Fig. 12 – Variation in Energy for
rp = 5.0 rJ and vp = 4.0.
0.02
0.0015
0.001
0.01
0.0005
3.95
4.05
4.1
3.95
4.05
4.1
-0.0005
-0.01
-0.001
-0.0015
-0.02
Fig. 13 – Variation in Inclination
(rad) for rp = 5.0 rJ and vp = 4.5.
Fig. 14 – Variation in Velocity for
rp = 5.0 rJ and vp = 4.5.
0.0075
0.015
0.005
0.01
0.0025
0.005
3.95
4.05
4.1
-0.005
-0.01
-0.015
Fig. 15 – Variation in Angular
momentum for rp = 5.0 rJ and vp = 4.5.
3.95
4.05
4.1
-0.0025
-0.005
-0.0075
Fig. 16 – Variation in Energy for
rp = 5.0 rJ and vp = 4.5.
INTRODUCTION TO MANEUVERS
The problem of spacecraft orbit control with minimum fuel
consumption is considered, using low thrust maneuvers.
The goal is to go to a safe orbit, obtained from the orbital evolution
of the cloud of particles.
A numerical suboptimal solution is tried, where the direction of the
thrust is assumed to be a quadratic form function of the position of
the satellite in the orbit.
The main idea is to verify the effects of using a new propulsion
system that is under development in Brazil (Phall 1) for a mission
of this type.
ASSUMPTIONS
The spacecraft is supposed to be in Keplerian motion controlled
only by the thrusts, whenever they are active.
This means that there are two types of motion:
i) A Keplerian orbit. This motion occurs when the thrusts are not
firing;
ii) The motion governed by two forces: the Earth's gravity field
and the force delivered by the thrusts. This motion occurs during
the time the thrusts are firing.
CHARACTERISTICS OF THE THRUSTS
i)
Fixed magnitude;
ii) Constant Ejection Velocity;
iii) Constrained angular motion: the direction of the thrust is
assumed to be a quadratic form function of the position of the
satellite in the orbit;
iv) Operation in on-off mode.
DEFINITION OF THE PROBLEM
The problem is to find how to transfer the spacecraft between two
orbits in such way that the fuel consumed is minimum.
There is no time restriction involved here and the spacecraft can
leave from any point in the initial orbit.
The maneuver is performed with the use of an engine that is able to
deliver a thrust with constant magnitude and constrained variable
direction.
This is a typical optimal control problem:
Objective Function:
Mf,
where Mf is the final mass of the vehicle and it has to be
maximized with respect to the control u(.);
Subject to:
Equations of motion, constraints in the state
(initial and final orbit) and control (quadratic form, limits in the
angles of "pitch" and "yaw", forbidden region of thrusting and
others);
And given:
All parameters (gravitational force field,
initial values of the satellite and others).
THE NON-SINGULAR STATE
X1 = [a(1-e2)/]1/2
X2 = ecos(-)
X3 = esin(-)
X4 = (Fuel consumed)/m0
X5 = t = time
X6 = cos(i/2)cos((+)/2)
X7 = sin(i/2)cos((-)/2)
X8 = sin(i/2)sin((-)/2)
X9 = cos(i/2)sin((+)/2)
 =  + - s, and s is the range angle of the spacecraft
EQUATIONS OF MOTION
dX1/ds = f1 = SiX1F1
dX2/ds = f2 = Si{[(Ga+1)cos(s)+X2]F1+F2sin(s)}
dX3/ds = f3 = Si{[(Ga+1)sin(s)+X3]F1-F2cos(s)}
dX4/ds = f4 = SiF(1-X4)/(X1W)
dX5/ds = f5 = Si(1-X4)m0/X1
dX6/ds=f6 = - SiF3[X7cos(s)+X8sin(s)]/2
dX7/ds = f7 = SiF3[X6cos(s)-X9sin(s)]/2
dX8/ds = f8 = SiF3[X9cos(s)+X6sin(s)]/2
dX9/ds = f9 = SiF3[X7sin(s)-X8cos(s)]/2
EQUATIONS OF MOTION
where:
Ga = 1 + X2cos(s) + X3sin(s)
Si = ( X14)/[Ga3m0(1-X4)]
F1  F cos() cos()
F2  Fsin () cos()
F3  Fsin ()
and F is the magnitude of the thrust, W is the velocity of the
gases when leaving the engine,  is the true anomaly of the
spacecraft.
THE CONTROL
A quadratic parametrization is used as an approximation for the
control law (angles of pitch () and yaw ()):
 = 0 + ' * ( s – s0 ) + '' * ( s – s0 )2
 = 0 + ' * ( s – s0 ) + '' * ( s – s0 )2
where 0, 0, ', ', '', '' are parameters to be found, s is the
instantaneous range angle and s0 is the range angle when the motor
is turned-on.
THE CONTROL
Considering these assumptions, there is a set of eight variables to be
optimized for each burning arc:
-Start and end of thrusting and;
- The parameters 0, 0, ', ', '', ‘’ for each "burning arc" in the
maneuver.
Note that this number of arcs is given "a priori" and it is not an
"output" of the algorithm.
By using parametric optimization, this problem is reduced to one of
nonlinear programming, which can be solved by several standard
methods.
NUMERICAL METHOD
To solve the nonlinear programming problem, the gradient
projection method was used.
It means that at the end of the numerical integration, in each
iteration, two steps are taken:
i) Force the system to satisfy the constraints by updating the control
function according to:
u i 1  u i  f . f. f
T
T 1
f
where f is the vector formed by the active constraints;
NUMERICAL METHOD
After the constraints are satisfied, try to minimize the fuel
consumed. This is done by making a step given by:
u i 1
d
 ui  
d
where:
J (u )
 
J (u ).d


d   I  f T f.f T

1

f .J(u )
where I is the identity matrix, d is the search direction, J is the
function to be minimized (fuel consumed) and  is a parameter
determined by a trial and error technique.
SOLUTIONS
The solution is given in terms of the constants that specifies the control to be
applied and the fuel consumed.
Several numbers of "thrusting arcs" (arcs with the thrusts active) can be used
for each maneuver.
Instead of time, the "range angle" (the angle between the radius vector of the
spacecraft and an arbitrary reference line in the orbital plane) is used as
the independent variable.
PHALL 1
Institute of the Brasilia University (UNB) is developing a propellant
that uses a plasma propulsion system based on Stationary Plasma
Thrusters (SPT).
They use permanent magnets that generates the magnetic field,
reducing the electricity consumption.
The characteristics of the propulsion system are:
2.
specific impulse Is = 1607s;
3.
specific energy  = 0.06;
4.
T = 2126 mN;
RESULTS
Several maneuvers are used to test the method and to predict
capabilities of the thrusters.
MANEUVER 1:
Initial orbit: Semi-major axis: 99000 km, eccentricity: 0.7,
inclination: 10 deg, longitude of the ascending node: 55 deg,
argument of periapsis: 105 deg.
Initial data of the spacecraft: Total mass: 300 kg, Thrust magnitude:
1.0 N, Initial position: 0, True anomaly: -105 deg.
MANEUVER 1
Condition imposed in the final orbit: Semi-major axis = 104000 km.
Propulsion: 1 arc.
Solution obtained: s0 = 80.3 deg, sf = 134.5 deg, 0 = -3.2, 0 =
0.0,  = 0.443,  = 0.0,  = 0.041,  = 0.00
Fuel consumed = 2.35 kg, Duration of burn = 6088.4 s.
Final orbit obtained: Semi-major axis: 104000.71 km, eccentricity:
0.712, inclination: 10 deg, longitude of the ascending node: 55 deg,
argument of periapsis: 105 deg, True Anomaly = 30.1 deg.
MANEUVER 1
Considering a linear approximation:
s0 = 78.0 deg, sf = 132.5 deg, 0 = -8.8, 0 = 0.0,  = 0.469,  =
0.0
Fuel consumed = 2.39 kg (was 2.35 kg) , Duration of burn = 6111.6
s (was 6088.4 s).
Final orbit obtained: Semi-major axis: 104000.73 km, eccentricity:
0.714, inclination: 10 deg, longitude of the ascending node: 55 deg,
argument of periapsis: 105 deg, True Anomaly = 28.2 deg.
MANEUVER 2
Initial orbit: Semi-major axis: 99000 km, eccentricity: 0.7,
inclination: 10 deg, longitude of the ascending node: 55 deg,
argument of periapsis: 105 deg.
Initial data of the spacecraft: Total mass: 300 kg, Thrust magnitude:
1.0 N, Initial position: 0, True anomaly: -105 deg.
Condition imposed in the final orbit: Semi-major axis = 104000
km.
Propulsion: 1 arc, with restriction in applying thrust between the
true anomalies of 120.0 deg and 180.0 deg (difference from 1).
MANEUVER 2
Solution obtained: s0 = 23.1 deg, sf = 63.1 deg, 0 = -15.2, 0 = 0.0,
 = 0.098,  = 0.0,  = 0.034,  = 0.00,
Fuel consumed = 2.76 kg, Duration of burn = 7001.1 s.
Final orbit obtained: Semi-major axis: 104000.03 km, eccentricity:
0.711, inclination: 10 deg, longitude of the ascending node: 55 deg,
argument of periapsis: 100.3 deg, True Anomaly = 318.2 deg.
MANEUVER 2
Considering a linear approximation:
s0 = 25.2 deg, sf = 65.0 deg, 0 = -26.3, 0 = 0.0,  = 0.179,  =
0.0
Fuel consumed = 2.80 kg (was 2.76 kg), Duration of burn = 7037.2 s
(was 7001.1 s).
Final orbit obtained: Semi-major axis: 104000.06 km, eccentricity:
0.713, inclination: 10 deg, longitude of the ascending node: 55 deg,
argument of periapsis: 102.1 deg, True Anomaly = 320.1 deg.
Maneuver 2 considers the same situation with the extra constraint in
the propulsion phase.
MANEUVER 3
Initial orbit: Semi-major axis: 9900 km, eccentricity: 0.2, inclination:
10 deg, longitude of the ascending node: 0 deg, argument of
periapsis: 25 deg.
Initial data of the spacecraft: Total mass: 300 kg, Thrust magnitude:
2.0 N, Initial position: 0, True anomaly: -10 deg.
Condition imposed in the final orbit: Semi-major axis = 10000 km.
Propulsion: 1 arc.
Smaller amplitude from previous ones.
MANEUVER 3
Solution obtained: s0 = 0.0 deg, sf = 178.1 deg, 0 = 0.1, 0 = 0.0, 
= 0.032,  = 0.0,  = 0.022,  = 0.00
Fuel consumed = 3.68 kg, Duration of burn = 4601.1 s.
Final orbit obtained: Semi-major axis: 10000.00 km, eccentricity: 0.2,
inclination: 10 deg, longitude of the ascending node: 55 deg,
argument of periapsis: 22.9 deg, True Anomaly = 163.2 deg.
MANEUVER 3
Considering a linear approximation: s0 = 0.0 deg, sf = 179.3 deg, 0
= 2.1, 0 = 0.0,  = 0.058,  = 0.0
Fuel consumed = 3.73 kg (was 3.68 kg), Duration of burn = 4675.3 s
(was 4601.1 s).
Final orbit obtained: Semi-major axis: 10000.00 km, eccentricity: 0.2,
inclination: 10 deg, longitude of the ascending node: 55 deg,
argument of periapsis: 23.2 deg, True Anomaly = 165.9 deg.
Note that the constraint s0  0.0 is active.
MANEUVER 4
Initial orbit: Semi-major axis: 9900 km, eccentricity: 0.2, inclination:
10 deg, longitude of the ascending node: 0 deg, argument of
periapsis: 25 deg.
Initial data of the spacecraft: Total mass: 300 kg, Thrust magnitude:
2.0 N, Initial position: 0, True anomaly: -10 deg.
Condition imposed in the final orbit: Semi-major axis = 10000 km.
Propulsion: 2 arcs (the new fact, rest is similar to the previous one).
MANEUVER 4
Solution obtained:
First arc: s0 = 0.0 deg, sf = 89.7 deg, 0 = 1.0, 0 = 0.0,  = 0.172,
 = 0.0;
Second arc: s0 = 299.1 deg, sf = 417.6 deg, 0 = -8.1, 0 = 0.0,  =
0.091,  = 0.0
Fuel consumed = 3.21 kg, Duration of burn = 4009.9 s.
Final orbit obtained: Semi-major axis: 10000.02 km, eccentricity:
0.207, inclination: 10 deg, longitude of the ascending node: 0.0 deg,
argument of periapsis: 21.2 deg, True Anomaly = 45.1 deg.
This maneuver shows that the use two arcs reduces the fuel
consumption, from 3.73 kg to 3.21 kg in this case.
MANEUVER 5
Initial orbit: Semi-major axis: 4500 km, eccentricity: 0.5,
inclination: 8 deg, longitude of the ascending node: -145 deg,
argument of periapsis: -20 deg.
Initial data of the spacecraft: Total mass: 11300 kg, Thrust
magnitude: 60000 N, Initial position: 0, True anomaly: 170 deg.
Condition imposed in the final orbit: Semi-major axis = 10000 km,
eccentricity = 0.122, Inclination = 2.29 deg (3 constraints!!).
Propulsion: 1 arc and the burn must be completed before the true
anomaly of 35.0 deg.
MANEUVER 5
Solution obtained: s0 = 6.6 deg, sf = 27.8 deg, 0 = 0.8, 0 = 16.5,
 = -0.033,  = -0.069
Fuel consumed = 5249.9 kg, Duration of burn = 377.4 s.
Final orbit obtained: Semi-major axis: 7435.00 km, eccentricity:
0.122, inclination: 2.290 deg, longitude of the ascending node: 255.2
deg, argument of periapsis: 169.0 deg, True Anomaly = 324.6 deg.
This maneuver considers the case where the thrust is large and that
there are three keplerian elements to be changed.
CONCLUSIONS
In this paper, analytical equations based in the patched conics
approximation were used to calculate the variation in velocity,
angular momentum, energy and inclination of a cloud of
particles that performs a swing-by maneuver.
The results show the distribution of those quantities for each
particle of the cloud. Those results can be used to estimate the
position of each individual particle in the future.
From this information, it is possible to find a safe orbit for a
spacecraft that will pass close to this cloud.
CONCLUSIONS
Then, suboptimal control was explored to generate algorithms to
obtain solutions for the minimum fuel maneuvers for a spacecraft,
to allow an escape.
Then we could test the physical parameters of a propulsion system
that is under development in Brazil at UNB, called Phall 1.
The results showed that the method applied here, as well as the
propulsion system can be used to solve the proposed problem.