Research Background
Nonlinear Control of Electro

Mechanical Systems
Design
Analysis
Experimental
Validation
Robot Manipulators

Constrained robots

Electrically driven
robots

Flexible joint robots
Magnetic
Bearings
Flexible Mechanical
Systems (PDE)
Aerospace
Systems

Formation Flying

Attitude Control

VTOL
Nonlinear Control of Multiple
Spacecraft Flying in
Formation
Dr. Marcio S. de Queiroz
Outline
Multiple Spacecraft Formation Flying (MSFF)
Concept
Dynamic Model
Nonlinear Control
Simulation Results
Fuel Consumption Issue
Ongoing and Future Research
MSFF Concept
Distribute the functionality of a large spacecraft
among an array of highly

coordinated, autonomous
micro

spacecraft
(“Virtual spacecraft”)
Large, specialized
spacecraft
Virtual spacecraft
MSFF Concept
Virtual spacecraft
Mission
hardware
function spread across micro

spacecraft
Coordination via
software
Analogous to network of PCs vs. mainframe
MSFF Concept
Why?
Micro

spacecraft are less expensive
Mass production
Low weight/volume for launch
Increases the baseline of scientific instruments
Widens coverage area of satellites
Reduces ground support
Micro

spacecraft are
autonomous
MEMS are an enabling technology
Micro

instruments, micro

propulsion
MSFF Concept
Why?
(cont.)
Flexible architecture
Robustness, redundancy, and
reconfigurability
Minimizes effects of failure
Multi

mission capability
Reduces mission cost and increases performance
MSFF Concept
Current related applications
Spacecraft rendezvous
Satellite recovery and servicing
Potential future applications
Surveillance
Earth surface mapping
Space

based communication system
Interferometer
MSFF Concept
Facts
MSFF idea was first proposed in 1984
Has not yet been flight

tested
NASA’s New Millennium Interferometer (NMI)
Formation of 3 spacecraft for long baseline optical stellar
interferometry
NASA’s Earth Orbiter

1 (EO

1)
Formation of 2 spacecraft with the Landsat 7 satellite for stereo
imaging
MSFF Concept
More facts
AFOSR’s TechSat 21
Several application missions to demonstrate MSFF paradigm
Micro

satellite dimensions: 2

7 meters, weight: < 100 kg
AFOSR/DARPA University Nanosatellite Program
NMSU/ASU/UC 3 Corner SAT
Researchers are exploring methods
to … use
midget
spacecraft

some
weighing less than a pound and
hardly larger than a pack of cards

that could be used alone to perform
simple tasks or
flown in formations
to
execute more complex ones.
… next month … the Air Force
launches a fleet of tiny … satellites
made of miniature components

diminutive machines that could
…
work together in groups to replace
or supplement larger spacecraft
.
“We’re talking about fully integrated
satellites that could be
mass
produced cheaply
by the hundreds
and sent into space to
perform a
of variety tasks
.”
If one or several of the machines in
a formation
fails
, others in the group
could
redistribute themselves
and
the continue performing the same
task ...
Peter Panetta of NASA’s Goddard Space
Flight, agrees, saying there is a growing
interest in increasingly smaller …
spacecraft.
“This isn’t just a fad. A lot
of people see this as the future …”
MSFF Concept
Guidance and control challenges
Reliable onboard sensing to determine relative
position/attitude
Global positioning system (GPS)
High

level control
Fleet path planning, navigation strategy
Fault

tolerance schemes
Centralized vs. decentralized control
Low

level control
Accurate
control of the relative position/attitude
(NMI mission:
order of a
centimeter
; EO

1 mission: order of
10

20 meters
)
Should be fuel

efficient
Sensing
Low

level
control
High

level
control
MSFF Concept
Two

phase, low

level control operation
Formation reconfiguration
Spacecraft are commanded to their respective positions and
orientations in the formation
Requires
propulsion
Formation

keeping
Once in formation, spacecraft move in their
respective
natural
orbits about the Earth
Maintained by orbital mechanics and propulsion
Dynamic Model
Spacecraft is a point

mass
Only position dynamics
MSFF fleet composed of a
leader

follower
pair
Leader
provides a reference motion trajectory
Follower
navigates in neighborhood of leader according to
a desired, relative trajectory
Navigation strategy motivated by marching bands
Designated band leaders provide basic reference path
Band members navigate by tracking certain leaders
Dynamic Model
Schematic representation of the MSFF system
R
(
t
)
: Position vector of leader from Earth center
r
(
t
)
: Position vector of follower
relative
to leader
Dynamic Model
Newton’s law of gravitation
Two bodies attract each other with a force acting along the
line that joins them
G
:
Universal gravitational constant
Dynamic Model
Dynamics of
leader
spacecraft
Dynamics of
follower
spacecraft
M
:
Earth mass
m
l
, m
f
:
Spacecraft masses
F
dl
(
t
)
, F
df
(
t
)
:
Disturbance force vectors (3x1)
u
l
(
t
)
, u
f
(
t
)
:
Control input vectors (3x1)
Dynamic Model
Nonlinear, relative position dynamics
F
d
(
t
)
:
Composite disturbance force
Dynamic Model
Dynamics are given w.r.t. inertial coordinate frame
Spacecraft masses vary
slowly
in time due to fuel
consumption and payload variations
m
l
and
m
f
are constant parameters
Disturbance forces result from solar radiation,
aerodynamics, and magnetic field; hence, vary
slowly
in time
F
d
is a constant vector
Nonlinear Control
Common practice:
Linearize
relative position dynamics
Hill’s or Clohessy

Wiltshire equations
Design standard, linear controllers
Assumptions
for all time
Leader in circular orbit around the Earth
Reasonable approach for formation

keeping
Nonlinear Control
Problems with linearized dynamics
Initial position of follower relative to leader may be
large
During formation reconfiguration maneuvers, leader
will not
be in
circular orbit
Control system will need to download a
new
linear controller
Control design based on nonlinear model
Same controller valid for formation reconfiguration
and
formation

keeping
Extrapolates “valid” operating range
Nonlinear Control
Significant contributions can be made to advance
MSFF technology by exploiting
nonlinear control
Several issues tailored for nonlinear control
Dynamic model is nonlinear
Higher performance under broader operating conditions
Trajectory tracking problem
Reconfiguration maneuvers, collision avoidance, minimize fuel
Uncertainties in system model
Mass, inertia, disturbance, drag
Expensive sensor technology (GPS) may limit state info
Actuator saturation
Physical limit or need to minimize fuel
Nonlinear Control
Goal:
Design a new class of MSFF controllers that
addresses these issues
Theoretical tools:
Lyapunov

based
control design
Easily handle nonlinearities
Flexible
Tracking or setpoint problems
Adaptive or robust controllers for uncertainties
Output feedback controllers for lack of full

state feedback
Bounded controllers for actuator saturation
Guaranteed stability properties
Implementation tools: Low

cost and computational
power of microprocessors
Nonlinear Control
(Design)
Adaptive tracking control
objective
Given the nonlinear MSFF dynamics
and a
desired
position trajectory of follower w.r.t. leader,
r
d
(
t
)
. Design
u
f
(
t
)
such that
Assumption:
Spacecraft masses and disturbance forces
are not known precisely
Nonlinear Control
(Design)
Property:
Dynamics can be
parameterized
Known
matrix:
Unknown
, constant parameter vector:
Nonlinear Control
(Design)
Quantify control objective
Position tracking error:
Control objective is then
Parameter estimation error:
is a dynamic, parameter estimate
Filtered tracking error:
L >
0
is a constant, diagonal, control gain matrix
Allows 2nd

order dynamic equation to be written as a 1st

order
If
r
(
t
)
0
then
e
(
t
)
0
Nonlinear Control
(Design)
Write dynamics in terms of
r
(
t
)
Substitute for
using dynamic equation, and
apply
parameterization property
Nonlinear Control
(Design)
Adaptive control law
(standard, “robot” adaptive controller)
K > 0
is constant, diagonal, control gain matrix
G
> 0
is constant, diagonal, adaptation gain matrix
Closed

loop system dynamics
Linear feedback stabilizing term
Helps “cancel”
Attempts to “cancel”
Nonlinear Control
(Stability Analysis)
Lyapunov Stability Analysis
If system’s total energy is continuously dissipating
†
獹獴emevenua汬yge猠anequ楬楢r極mp楮
Determination of system’s stability properties
Construct a scalar,
energy

like
function (
V
(
t
)
0
)
Examine function’s time variation
却慢汥
r湳慢汥
Nonlinear Control
(Stability Analysis)
Define the non

negative function
Differentiate
V
along closed

loop dynamics
Apply Barbalat’s lemma
Position tracking error is
asymptotically stable
Simulation Results
System parameters
Leader spacecraft in
natural
orbit around the Earth
Radius:
4.224
x
10
7
m
Angular velocity:
w
= 7.272
x
10

5
rad/s
(orbit period = 24 h)
No control required (
u
l
=
0
)
Simulation Results
(Unnatural Trajectory)
Initial position and velocity of follower relative to
leader
Desired relative trajectory
Follower is commanded to move
around leader
in a
circular orbit of radius
100
m
with angular velocity
4
w
Parameter estimates initialized to 50% of actual
parameter values
Simulation Results
(Unnatural Trajectory)
Relative Trajectory
(‘
*
’ denotes leader spacecraft)
Simulation Results
(Unnatural Trajectory)
Position Tracking Errors
Parameter Estimates
Simulation Results
(Unnatural Trajectory)
Control Forces
Maximum magnitude =
0.02 N
Simulation Results
(Natural Trajectory)
Follower commanded to move in natural,
elliptical
orbit around the Earth with orbit period = 24 h
Typical of formation

keeping
Elliptical orbit for
r
d
(t)
obtained by integrating
Relative dynamics with
u
l
= u
f
= 0
and
F
d
= 0
Proper initial conditions must be selected
Parameter estimates initialized to zero
Simulation Results
(Natural Trajectory)
Position Tracking Errors
Parameter Estimates
Disturbance estimates
converge to actual values
Simulation Results
(Natural Trajectory)
Control Forces
Maximum magnitude =
4 x 10

5
N
Fuel Consumption Issue
Continuous
thruster
Ideal scenario
Control amplitude can be
continuously
modulated but
maximum amplitude is limited
Nonlinear saturation control results apparently can be
applied with guaranteed closed

loop stability
On/off type
thruster
Currently, a more realistic scenario
Control amplitude can be modulated only for certain
periods of time
Not clear how to rigorously address closed

loop stability
under a pulse

type, nonlinear control law
Fuel Consumption Issue
Formation

keeping
On/off thrusters may suffice
When “off”, orbital mechanics maintain natural orbit
Formation reconfiguration
Demanding maneuvers will require significant control
effort
When on/off thrusters are used, obvious trade

off between
performance and fuel consumption
Reconfiguration may last for only short periods of time
Fuel Consumption Issue
Simple, ad

hoc solution to reduce fuel consumption
Let
q
d
(
t
) = [
x
d
(
t
),
y
d
(
t
),
z
d
(
t
)]
T
be a
desired
spacecraft
trajectory
Define a ball centered at
{
x
d
(
t
),
y
d
(
t
),
z
d
(
t
)}
with radius
e
Fuel Consumption Issue
Gradient at a point
q
*
= {
x
*
,
y
*
,
z
*
}
on the ball surface:
Fuel Consumption Issue
On/Off Type Control Algorithm
Goal:
Control spacecraft position such that it never
leaves the ball
q
s
(
t
) = [
x
s
(
t
),
y
s
(
t
),
z
s
(
t
)]
T
:
spacecraft position
1.
If
2.
If
Control off
Control on
Fuel Consumption Issue
On/Off Type Control Algorithm
(cont.)
3.
If
If
Else
Control off
Control on
Fuel Consumption Issue
Control on
means:
Control is set to the designed nonlinear control
Left on for some finite time interval
T
Algorithm is resumed only after
T
has expired
Case 2 (
spacecraft outside ball
) may occur during
initialization of formation reconfiguration
Trade

off between tracking performance and fuel
consumption
Asymptotic tracking vs. bounded tracking with less fuel
Ongoing and Future Research
Account for spacecraft attitude dynamics
MSFF position/attitude tracking controller
4

parameter kinematic representation (quaternions)
Account for higher

order gravitational perturbations
(J2 effect) and atmospheric drag
Output feedback controller
Only GPS position measurements
No GPS “estimation” architecture for velocity
Formation control of autonomous vehicles
Aircraft, ships, underwater vehicles, mobile robots
Ongoing and Future Research
Testbed for preliminary experiments
3 DOF
DC motor

propeller pairs provide actuation
Optical encoders sense the 3 angular positions
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