Title: Dr' Samuel Schweighart
1Dr. Samuel Schweighart
Propellantless Formation Flight Operations in LEO
AIAA SpaceOps 2006
2Motivation
- Satellites that fly in formation require station
keeping in order to maintain their relative
position. - Typically this is achieved by using propulsive
thrusters. - Can control the formation center of mass.
- Can individually control each satellites
position. - Optics can become coated from expelled
propellants. - Can require large amounts of propellant mass.
- Hot propellants can blind some optics.
3(No Transcript)
4Overview of Presentation
- Model the magnetic forces and torques.
- Describe the equations of motion (EOM).
- Solve the EOM for the magnetic dipole solutions.
- Solve for all points along a trajectory.
- Manage the angular momentum buildup on the
reaction wheels. - Examine and overcome challenges of operating in
low Earth orbit.
5The Magnetic Dipole
- The basic building block of electromagnetism is
the magnetic dipole, µ. - A dipole is an approximation of a magnetic field
source at a distance. - The far field approximation
- A dipole can be approximated by
- A bar magnet
- A loop of current
- A solenoid
- For a loop of current, the dipole is given by
N
S
6Modeling Magnetic Field
- Far Field Model
- Created by linearizing the Near Field Model
- Models the coils as dipoles (bar magnets)
- Equations can be solved for the dipoles
- Three rings on one vehicle can be combined to
form one dipole - Accurate when the dipoles are 6-8 coil radii
apart
7The Force Equations of Motion
- At every instant in time, it is assumed that EMFF
will be called upon to provide a specific force
to each satellite in the formation.
- The magnetic forces are described by polynomial
equations of motion. - 23N-6 possible solutions
- Can be solved using
- Analytic Methods
- Only possible for 2 satellite formations.
- Newtons Method
- Only provides one solution per initial value.
- Reduction Methods
- Not practical for large systems.
- Continuation Methods
- Finds every possible solution.
- Relatively slow
8Newtons 3rd Law and theFree Dipole
For every action, there is an equal and opposite
reaction.
- All the inter-satellite forces are internal to
the formation. - There are no external forces.
- EMFF cannot affect the formation center of mass
- The sum of the magnetic forces on the satellites
must be zero - Three of the equations of motion are dependent.
- There are 3N-3 equations and 3N variables (dipole
components) - The 3 extra dipole components can be chosen at
will. - The extra dipole components are called the Free
Dipole - The force equations of motion can be solved for
any force distribution for (almost) any choice of
the free dipole.
92D Example
- The five satellites initially start at rest in a
line along the axis. - The arrows represent the dipoles at different
times. - The arrows point in the N direction.
- As the formation spins up, the dipoles align to
provide the additional radial forces necessary to
counteract the centripetal forces.
103D Example
- Three satellites are in an equilateral triangle
configuration rotating about their center of
mass. - The plane that the three satellites lie in also
rotates about the y axis. - The free dipole is on satellite 3.
y
x
z
113D Example
- Using the continuation method, all possible
dipole solutions can be found. - The choice of the free dipole affects the other
dipoles and the number of possible solutions - The inter-satellite forces are the same for each
dipole solution
y
3
40,000Am2
x
z
2
1
123D Example
- Using the continuation method, all possible
dipole solutions can be found. - The choice of the free dipole affects the other
dipoles and the number of possible solutions - The inter-satellite forces are the same for each
dipole solution
y
3
50,000Am2
x
z
2
1
133D Example
- Using the continuation method, all possible
dipole solutions can be found. - The choice of the free dipole affects the other
dipoles and the number of possible solutions - The inter-satellite forces are the same for each
dipole solution
y
60,000Am2
3
x
z
2
1
14Angular Momentum Management
- Whenever shear forces are applied, torques are
generated on the satellites. - Reaction wheels are used to counteract the
torques and store the angular momentum. - Ideally, the angular momentum would be evenly
distributed among the satellites. - The torque distribution is affected by the free
dipole. - How is the free dipole chosenso that the desired
torque distribution is achieved?
Free Dipole Chosen at Will
Dependent Dipole Solved the EOM
15Angular Momentum Distribution
- The choice of the free dipole affects the angular
momentum distribution. - The plots below show the angular momentum stored
in each vehicle. - Ideally, the angular momentum would be evenly
distributed among the satellites. - The total angular momentum stored is equal and
opposite to the formations angular momentum. - EMFF cannot change the total angular momentum.
16Adjusting the Dipole Solution
How does one choose the free dipole so that the
torque/angular momentum distribution is favorable?
- Directly Solve the Equations
- 3N-3 Force Equations 3N-3 Torque Equations
- 3N Dipoles
- Not enough variables to solve for a desired force
profile AND a desired torque profile. - There are only 3 degrees of freedom, or
essentially, the torque can be specified for only
one vehicle. - Nullspace Method
- Approach 1 Find the best torque distribution.
- Approach 2 Specify the angular momentum
distribution at a point in time. - Approach 3 Minimize the angular momentum
distribution at all points in time.
17The Nullspace Method
- Linearize the Force and Torque Equations of
motion at a point in time. - The change in Force/Torque due to a change in the
dipoles - There can be NO change in the force profile.
- The nullspace of A gives the allowable change in
the dipoles. - There are three directions that the dipole
distribution can change. - The allowable change in torque distribution is a
function of the s. - The alphas are chosen to so that the best torque
distribution is achieved.
18Approach 1 Specifying the Torque Distribution
at one Point in Time
- Due to the lack of degrees of freedom, the
desired torque distribution cannot be achieved. - Goal is to find the closest possible solution.
- Using the nullspace method, the local minimum is
found. - Due to the non-linearity of the Equations of
motion, multiple local minima are present.
- In order to find the global minimum, different
choices for the free dipoles are used.
19Approach 1 Specifying the Torque Distribution
at one Point in Time
- Many different choices for the initial free
dipole are used. - Shown as blue points.
- In this example, seven different local minima are
found. - Shown as black points.
- The points in the plots on the right are given by
the following equation.
20Approach 3 Specifying the Angular Momentum
Distribution at all Points in Time
- Goal is to evenly distribute the angular momentum
among the satellites. - Since we are minimizing at every point in time,
there are 2(3N-3)k equations, and only 3k degrees
of freedom ( ). - Can be accomplished by
- Can be solved by using gradient projection method
with the nullspace method and the conjugate
gradient method.
21Approach 3 Specifying the Angular Momentum
Distribution at all Points in Time
Initial Angular Momentum Distribution (Free
dipole 60,000Am2)
Adjusted Angular Momentum Distribution
- The goal is to evenly distribute the angular
momentum at all times.
22Approach 3 Resulting Change in Dipole Solution
Initial Dipole Solution (Free dipole 60,000Am2)
Adjusted Dipole Solution
- The dipole solution does not change significantly.
23Operating in the Earths Gravity
- Formations that operate in Earth orbit are
subject to the Earths gravitational field. - Formations must either be in Keplarian orbits or
fight the gravitational forces. - Fighting gravity typically requires shear forces,
and thus angular momentum is transferred to the
satellite formation.
- Earth pointing
- Space pointing
- Rotating formations
- Earth looking
- Space looking
24Operating in the Earths Gravity
Two satellites (300 kg) separated by 8m
- To maintain this formation, EMFF will apply
radial and shear forces. - Torque will also be applied to the formation.
- This will cause angular momentum to build-up on
the formation
- The angular momentum build-up can be removed by
- Rotating the formation to an opposite orientation
- Using the Earths magnetic field to remove the
angular momentum.
25Modeling the Earths Magnetic Field
- The Earths magnetic field to first order can be
approximated as a large magnetic dipole - A typical EMFF dipole is
- If the Earth is modeled as a dipole, it can be
easily included into the current model. - If the Earth is treated as another satellite, we
can distribute the angular momentum onto the
Earth. - Essentially, remove the angular momentum from the
formation and transfer it to the Earth.
26The Disturbance Forces
- The disturbance forces are a function of
- Because of the large distance separating the
Earth and the satellite, the 1/r4 dominates. - The resulting disturbance force is very small and
can be neglected. - A method for incorporating the disturbance forces
without loosing the ability to set the free
dipole has been created.
d15m
d10m
d5m
d1m
27The Disturbance Torques
- The disturbance torques are a function of
- Unlike the disturbance forces, the disturbance
torques cannot be neglected. - Depending on the separation distance, the
disturbance torques can larger than the
inter-satellite torques. - The plots show the inter-satellite torques
(multi-colors) compared to the disturbance torque
(orange).
d1m
d5m
d8m
d10m
d15m
28Disturbance Torques
Angular Momentum Distribution without the Earths
magnetic field
Adjusted Angular Momentum Distribution with the
Earths magnetic field
- The Earths magnetic field produces a sizable
disturbance in the angular momentum.
29Angular Momentum Management Overview
- The algorithm to manage angular momentum in the
Earths field has two modes Normal and Momentum
Reduction. - The algorithm typically operates in the normal
mode until one satellites angular momentum
crosses a threshold (40 Nms). - Then the algorithm reduces the angular momentum
of that satellites to a lower threshold (20 Nms)
and then returns to the normal mode. - Normal Mode
- Arbitrarily set the free dipole.
- Minimize the angular momentum transferred to the
formation. - Set the torque on one satellite to zero.
- Angular Momentum Reduction Mode
- Specifically target a satellite to have its
angular momentum reduced. - Minimize the angular momentum across the
formation.
30Angular Momentum Management
Angular Momentum without Angular Momentum
Management
Angular Momentum Management Arbitrary Free
Dipole in Normal Mode
- The algorithm successfully manages angular
momentum buildup. - This includes the angular momentum from the
Earths gravitational field.
31Angular Momentum Management
Angular Momentum without Angular Momentum
Management
Angular Momentum Management Minimizing the
Torque in Normal Mode
- By minimizing the angular momentum build-up
during the normal mode, the momentum distribution
is much smoother.
32Summary
- Thee different models were created to describe
the magnetic forces and torques. - Near-Field, Far-Field, Mid-Field
- Dipole solutions can be found for any desired
force profile. - The continuation method found all possible dipole
solution profiles - The dipole solutions were not always continuous.
- EMFF cannot control the formation center of mass.
- This allows for three extra degrees of freedom,
and allows for the presence of the free dipole. - The free dipole enables the ability to manage the
angular momentum distribution.
33Summary
- Due to the lack of degrees of freedom, the
torque distribution cannot be directly specified
for each satellite. - Found the best possible torque distribution.
- Minimized the angular momentum distribution.
- The Earths gravity field.
- In order to maintain their shape, most formation
designs require EMFF to continuously apply
corrective forces. - Angular momentum can build up due to the
corrective forces. - The Earths Magnetic Field
- Disturbance forces are small and can be
neglected. - Disturbance torques are significant, but can be
exploited to remove angular momentum from the
formation.
34Questions
35BackupSlides
36J2 Geopotential
- The J2 geopotential refers to the fact that the
Earth is not a perfect sphere. - The J2 geopotential force causes satellites in
formation to Separate in the cross-track
direction. - The disturbance force is on the order of ?N.
- The resulting torques on the formation are also
very small and nearly periodic.
- The angular momentum build is on the order of
?Nms per orbit. - Any method used to manage the angular momentum
when operating in LEO can easily handle the
effects of the J2 geopotential.
37Specifying the Torque on One Satellite
- The presence of an external magnetic field allows
for the ability to directly specify the torque on
one satellite. - Without the Earths magnetic field, it is not
always possible to set the torque on one
satellite even though there are enough degrees of
freedom.
- This allows for the possibility of having a
zero-torque satellite. - Setting the torque on one satellite to zero
prevents any other angular momentum management
scheme.
38Approach 2 Specifying the Angular Momentum
Distribution at a Point in Time
- The angular momentum is the sum of torques at the
previous time steps. - is a linear equation in
- At each time step, the torque can change in three
directions. - The degrees of freedom is now 3k. (There are 3k
) - The equations are now under- constrained, and
multiple solutions can be found.
- Solving a simple quadratic program with linear
constraints - is a (3k x 1) vector
39Approach 2 Specifying the Angular Momentum
Distribution at a Point in Time
Initial Angular Momentum Distribution (Free
dipole 40,000Am2)
Adjusted Angular Momentum Distribution
- The goal is to evenly distribute the angular
momentum at t3600.
40Incorporating the Disturbance Forces
- Some mission designers may wish to incorporate
these disturbance forces. - Treat the Earth as another dipole and incorporate
it into the equations of motion. - Because the Earths dipole cant change, it must
become the free dipole. - This locks the dipole solution and prevents
angular momentum management. - Solution is to allow for the force applied to the
formation center of mass to be unspecified. - Treat the Earth as an external magnetic field.
- This allows for the relative disturbance force
from the Earth to be incorporated, but does not
allow for control of the formations center of
mass - However, due to the extremely small disturbance
forces, this force is negligible.
41Comparing the Models
A
B
r
- If the models are considered valid at 10 error,
then - The Far Field Model is valid at 6-8 coil radii
- The Mid Field Model is valid at 3-5 coil radii