Boundary Control of the Timoshenko Beam with Free-End ...

loutsyrianΜηχανική

30 Οκτ 2013 (πριν από 3 χρόνια και 7 μήνες)

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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emevenua汬yge猠anequ楬楢r極mp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