Research Background Nonlinear Control of Electro-Mechanical Systems DesignAnalysis Experimental Validation Robot Manipulators - Constrained robots - Electrically

Research Background

Nonlinear Control of Electro-Mechanical Systems

Design AnalysisExperimental

Validation

Robot Manipulators

- Constrained robots- Electrically driven robots- Flexible joint robots

MagneticBearings

Flexible MechanicalSystems (PDE)

AerospaceSystems

- 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

Earth

MSFF Concept

Virtual spacecraft Mission hardware function spread across micro-spacecraft Coordination via software

Analogous to network of PCs vs. mainframe

Hardware function 1

Virtual Spacecraft

Hardware function 2Hardware function 3

Software

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 performanceReduces mission cost and increases performance

E arth

Com municationm ission

Radar m ission

Fleet augm entation and/orreplacement

Radiom etry m ission

Spacecraft failure

MSFF Concept

Current related applications Spacecraft rendezvous

Satellite recovery and servicing

Potential future applications Surveillance

Earth surface mapping

Space-based communication system

Interferometer

U -V P lane

Light Pa th

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 SpaceFlight, 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-levelcontrol

High-levelcontrol

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 propulsionEarth

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

(t): Position vector of follower relative to leader

Y

X

Z

Leader Spacecraft

Follow er Spacecraft

R

O

Earth

Inertia l CoordinateSystem

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

r

r

r

mmGF

221

21

Y

X

Z

m 1

m 2

r

F 21

Dynamic Model Dynamics of leader spacecraft

Dynamics of follower spacecraft

M: Earth mass ml, mf : Spacecraft masses

Fdl (t), Fdf (t): Disturbance force vectors (3x1)

ul(t), uf (t): Control input vectors (3x1)

Dynamic Model

Nonlinear, relative position dynamics

Fd(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 ml and mf are constant parameters

Disturbance forces result from solar radiation, aerodynamics, and magnetic field; hence, vary slowly in time Fd 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

)()( tt R

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 rangeExtrapolates “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,

d(t). Design uf(t) such that

Assumption: Spacecraft masses and disturbance forces are not known precisely

)()(lim tt dt


Property: Dynamics can be parameterized

Known matrix:

Unknown, constant parameter vector:


Quantify control objective Position tracking error:

Control objective is then

Parameter estimation error:

is a dynamic, parameter estimate

Filtered tracking error:

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

0)(lim

tet


Write dynamics in terms of r(t)

Substitute for using dynamic equation, and apply parameterization property


Adaptive control law (standard, “robot” adaptive controller)

K > 0 is constant, diagonal, control gain matrix > 0 is constant, diagonal, adaptation gain matrix

Closed-loop system dynamics

)(W

Linear feedback stabilizing term

Helps “cancel” )(WAttempts to “cancel”

+

Nonlinear Control (Stability Analysis)

Lyapunov Stability Analysis

If system’s total energy is continuously dissipating system eventually goes to an equilibrium point

Determination of system’s stability properties Construct a scalar, energy-like function (V(t) 0)

Examine function’s time variation

Stable

Unstable

V t 0

V t 0

t t= 0

U nstab le

S tab le

Nonlinear Control (Stability Analysis)

Define the non-negative function

Differentiate V along closed-loop dynamics

Apply Barbalat’s lemma0)(lim 0)(lim

tetr

tt

Position tracking error is asymptotically stablePosition tracking error is asymptotically stable

Simulation Results

System parameters

Leader spacecraft in natural orbit around the Earth Radius: 4.224 x 107 m Angular velocity: = 7.272 x 10-5 rad/s (orbit period = 24 h) No control required (ul = 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

Parameter estimates initialized to 50% of actual parameter values


Relative Trajectory (‘*’ denotes leader spacecraft)

-100

-50

0

50

100

-150-100

-50

050

100150

-200

-150

-100

-50

0

50

100

x [m]

Relative Trajectory

y [m]

z [m

]


Position Tracking Errors

Parameter Estimates

0 5 10 15 20 25 30 35 40 45 50-50

0

50

[m]

Error (x-direction)

0 5 10 15 20 25 30 35 40 45 50

0

50

100

[m]

Error (y-direction)

0 5 10 15 20 25 30 35 40 45 50-100

0

100

200

Time [hours]

[m]

Error (z-direction)

0 10 20 30 40 50-4000

-3000

-2000

-1000

0

1000

2000

3000Theta-hat(1)

0 10 20 30 40 50-4

-2

0

2

4

6

8

10x 10

4 Theta-hat(4)

0 10 20 30 40 50

-8

-6

-4

-2

0

2

4x 10

4

Time [hours]

Theta-hat(5)

0 10 20 30 40 50-4

-2

0

2

4

6

8x 10

4

Time [hours]

Theta-hat(6)


Control Forces

Maximum magnitude = 0.02 N

0 5 10 15 20 25 30 35 40 45 50

-0.02

0

0.02

[N]

Control Force (x-direction)

0 5 10 15 20 25 30 35 40 45 50

-0.02

0

0.02

[N]

Control Force (y-direction)

0 5 10 15 20 25 30 35 40 45 50

-5

0

5

x 10-3

Time [hours]

[N]

Control Force (z-direction)

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 d(t)obtained by integrating

Relative dynamics with ul = uf = 0 and Fd = 0

Proper initial conditions must be selected

Parameter estimates initialized to zero

0)0( ̂


Position Tracking Errors

Parameter Estimates

Disturbance estimatesconverge to actual values

0 10 20 30 40 50 60 70-0.2

-0.1

0

0.1

0.2

[m]

Error (x-direction)

0 10 20 30 40 50 60 70-0.2

-0.1

0

0.1

0.2

[m]

Error (y-direction)

0 10 20 30 40 50 60 70-0.1

-0.05

0

0.05

0.1

Time [hours]

[m]

Error (z-direction)

0 10 20 30 40 50 60 700

100

200

300

400Theta-hat(1)

0 10 20 30 40 50 60 700

1

2

3x 10

-5 Theta-hat(4)

0 10 20 30 40 50 60 70-3

-2.5

-2

-1.5

-1

-0.5

0x 10

-5 Theta-hat(5)

Time [hours]0 10 20 30 40 50 60 70

-2.5

-2

-1.5

-1

-0.5

0x 10

-5 Theta-hat(6)

Time [hours]


Control Forces

Maximum magnitude = 4 x 10-5 N

0 10 20 30 40 50 60 700

1

2

3x 10

-5

[N]

Control Force (x-direction)

0 10 20 30 40 50 60 70-4

-3

-2

-1

0x 10

-5

[N]

Control Force (y-direction)

0 10 20 30 40 50 60 70-3

-2.5

-2

-1.5

-1

-0.5

0x 10

-5

Time [hours]

[N]

Control Force (z-direction)


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


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


Simple, ad-hoc solution to reduce fuel consumption

Let qd(t) = [xd(t), yd(t), zd(t)]T be a desired spacecraft

trajectory

Define a ball centered at {xd(t), yd(t), zd(t)} with radius

2222 )()()()()()( tztztytytxtx ddd

q (t)d


Gradient at a point q* = {x*, y*, z*} on the ball surface:

T

ddd zzyyxxqgrad

)(2 )(2 )(2)( **** ,,

q ( t)d

q *

g ra d q *( )


On/Off Type Control Algorithm

Goal: Control spacecraft position such that it never leaves the ball

qs(t) = [xs(t), ys(t), zs(t)]T: spacecraft position

1. If

2. If

2222s dsdsd zzyyxx Control offControl off

2222s dsdsd zzyyxx Control onControl on


On/Off Type Control Algorithm (cont.)

3. If

If

Else

2222s dsdsd zzyyxx

0 ss qgradq Control offControl off

g ra d q( )s

q s

.

90

Control onControl ong rad q( )s

q s

.90


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

Yaw

Roll

P itch

DC Motor

Propeller

Base

Leader"Spacecraft"

Follower"Spacecraft"

Documents

Research Background Nonlinear Control of Electro-Mechanical Systems DesignAnalysis Experimental Validation Robot Manipulators - Constrained robots - Electrically