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Dr' Samuel Schweighart

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Optics can become coated from expelled propellants. ... Some mission designers may wish to incorporate these disturbance forces. ... – PowerPoint PPT presentation

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Title: Dr' Samuel Schweighart


1
Dr. Samuel Schweighart
Propellantless Formation Flight Operations in LEO
  • June 20th, 2006

AIAA SpaceOps 2006
2
Motivation
  • 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)
4
Overview 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.

5
The 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
6
Modeling Magnetic Field
  • Far Field Model
  • 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

7
The 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

8
Newtons 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.

9
2D 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.

10
3D 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
11
3D 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
12
3D 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
13
3D 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
14
Angular 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
15
Angular 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.

16
Adjusting 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.

17
The 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.

18
Approach 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.

19
Approach 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.

20
Approach 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.
  • G and d are given by

21
Approach 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.

22
Approach 3 Resulting Change in Dipole Solution
Initial Dipole Solution (Free dipole 60,000Am2)
Adjusted Dipole Solution
  • The dipole solution does not change significantly.

23
Operating 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

24
Operating 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.

25
Modeling 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.

26
The 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
27
The 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
28
Disturbance 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.

29
Angular 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.

30
Angular 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.

31
Angular 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.

32
Summary
  • 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.

33
Summary
  • 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.

34
Questions
35
BackupSlides
36
J2 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.

37
Specifying 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.

38
Approach 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

39
Approach 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.

40
Incorporating 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.

41
Comparing 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
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