On-Orbit Assembly of Flexible Space Structures with SWARM

Jacob Katz, Swati Mohan, and David W. Miler

MIT Space Systems Laboratory

AIAA Infotech@Aerospace 2010

April 22, 2010

1

Autonomous On-Orbit Assembly

Enabling technology for Large telescopes Orbiting solar arrays Interplanetary spacecraft

Challenges– Flexible structures (solar panels,

lightweight materials)– Multiple payloads with uncertain

parameters

2

Self-assembling Wireless Autonomous Reconfigurable Modules (SWARM) Testbed

docking port 2007-2009 (Phase II) SBIR sponsored by MSFC

2D flat floor demonstration

Goals: maneuvering and docking with flexibility

Hardware: SPHERES on propulsion

module Flexible segmented beam Docking ports

propulsion module

flexible beam element

SPHERES satellite

3

Key Challenges Requirements for assembly

Follow trajectories for positioning and docking

Minimize vibrational disturbances Desired

Handle parameter uncertainty for unknown payloads

Fewer actuators than degrees of freedom: underactuated control

This talk: Ideas for adaptive control Initial hardware testing

4

Incremental Test Plan

5

Test 1: Beam Control

6

SWARM as a Robot Manipulator

7

miδ1

δ2

δ3

0y

ki

x

SWARM DynamicsBeam joints modeled as torsional springs

δ1

δ2

δ3

0y

Fy

Fx

x

3

2

1

0

y

x

q

8

Inertia Matrix Coriolis Matrix

BuG(q)q)qC(q,qM(q) Potential Terms

0

Inertia Matrix Coriolis Matrix

YaG(q)q)qC(q,qM(q) Potential Terms

“Linear in the parameters”

00

00

u

a

u

a

uuua

auaa

u

a

uuua

auaa

q

q

Kq

q

CC

CC

q

q

MM

MM

0

y

x

F

F

SWARM Dynamics

9

0

3

2

1

0

y

x

q

Beam joints modeled as torsional springs

δ1

δ2

δ3

0y

Fy

Fx

x

underactuated

Simplified Dynamic Model

Most important measurement for docking is tip deflection Reduces complexity of dynamic model for control and

estimation

10

δf

0y

x

k1

f

y

x

q

0

00

00

f

a

ff

a

fffa

afaa

u

a

fffa

afaa

q

q

Kq

q

CC

CC

q

q

MM

MM

Nonlinear Adaptive Control for Robot Manipulators

11

sPYa T̂

P

K

a

Y

q

s

weighted tracking error

tracking time constant

kinematic regressor

parameter vector

control vector

state vector

PD gains

adaptation gains

)( dd qqqqs

adaptive feed-forward PD term

Tracking Error

Control Law

Adaptation Law

KsaY ˆ

dim(τ) = dim(q), how do we apply this to underactuated control?

Underactuated Adaptive Control

12

qJy

yy

qhy

sqa

fa

)(

Main idea: perform tracking in a lower dimensional task space y

)dim()(

)dim(

qJrank

p

y

sq

p

subject to

f

y

x

y

0

For example:

weighted combination of beam deflection and base rotation

Underactuated Adaptive Control

13

qJy

yy

qhy

sqa

fa

)(

Main idea: perform tracking in a lower dimensional task space y

)(

1

1

1

dd

sqaaa

sqT

sq

sqT

sq

yyyys

JKyyCyM

CJJC

MJJM

Underactuated Adaptive Control

14

qJy

yy

qhy

sqa

fa

)(

Important to note inherent sacrifice in underactuated control Lose guarantee of tracking convergence for arbitrary state trajectories Best we can do is achieve tracking in the output space Need to show zero output error implies convergence of internal states

Main idea: perform tracking in a lower dimensional task space y

)(

1

1

1

dd

sqaaa

sqT

sq

sqT

sq

yyyys

JKyyCyM

CJJC

MJJM

0ˆ

01 KsaYJ sq

Beam State Estimation Overview

Requirement Provide an estimate of beam

state variablesDesign Camera mounted to SPHERES

body frame Observe infrared LED on beam

end Calculate beam deflection using

LED position State estimate relative to

SPHERES body frame

DSP

Image

EstimatorLED(X,Y)

State Estimate

f

Side View

Image Processing Demonstration

Threshold

Centroid

X

Y

pixelspixels

Time (s)

EstimatorDSP

16

Beam Estimator

f

u

Z ≈ Beam Len

X

Image Plane

IR LEDSchematic View

Z

X

f

u

Perspective Projection

f

f

u

f

uf

1tan

Measure beam angle directly using perspective projection

Differentiate δf using LQE

DSP Estimator

ttft

ttftftf

vy

wt

1

17

Beam Simulation Full nonlinear model built in

Simulink/SimMechanics Simulation of SWARM

thrusters, camera, and control/estimation system

Autocoding capability for rapid deployment and testing

18

Test 1: Beam Maneuvering Test

19

Toward Assembly: Tests 3, 4, 6

20

Typical Assembly Sequence1. Docking2. Beam Maneuvering3. Beam Docking

21

Typical Assembly Sequence1. Docking2. Beam Maneuvering3. Beam Docking

22

Typical Assembly Sequence1. Docking2. Beam Maneuvering3. Beam Docking

23

Typical Assembly Sequence1. Docking2. Beam Maneuvering3. Beam Docking

24

Test 6: Hardware Assembly Test

25

Trajectory Tracking Performance

26

Test 3: Beam Docking

27

Trajectory Tracking Performance

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0 20 40 60 80 100 120 140 160 180 200

-0.6

-0.4

-0.2

X (

m)

0 20 40 60 80 100 120 140 160 180 200-0.6

-0.4

-0.2

0

0.2

Y (

m)

0 20 40 60 80 100 120 140 160 180 200

1.4

1.6

1.8

2

Time (s)

(r

ad)

TargetState

Alignment Approach Docking

Conclusions and Future Work

Conclusions Robot manipulator analogy is a

useful tool for analyzing flexible assembly problem

Adaptive control with a simple dynamic model looks promising but further testing will be required to compare it to other methods

Future Work Adaptive control in hardware

testing Look into better trajectories for

beam vibration control 6DOF extensions and on-orbit

assembly testing with SPHERES

29

Acknowledgments:

This work was performed under NASA SBIR Contract No. NNM07AA22C Self-Assembling Wireless Autonomous Reconfigurable Modules.

Backup Slides

30

Perpendicular Docking

31

Stability for Fully Actuated Adaptive

32

0

~ˆ~

~ˆ

~ˆ2

1

~~

~~2

1

2

1

1

1

1

1

1

KssV

aPaaYKss

aPaKqqCqMs

aPasMsKqCsqCqMs

aPasMssMsV

aPaMssV

T

TT

Trr

T

TTrr

T

TTT

TT

symmetric skew)2(

)(

CM

qCqMKqCssM

qqqq

qqs

rr

dd

r

Flexible Structure Dynamics

33

Shahravi, 2005

Docking Drives Control Approach1. Move

2. Damp

3. Dock

(+) Trajectory specified for satellite end (collocated)

(-) Requires accurate pointing and low vibration

(+) Relative metrology to guide beam end into docking port

(-) Trajectory specified for docking end (non-collocated)

34

Start simple: collocated trajectory with beam damping

Dynamics Derivation

Kinetic Energy: 3

0

1 1

2 2i i i i i ii

T m v v I

Potential Energy:3

2

0

1

2 i ii

U K

m1,I1

m2,I2

m3,I3

m4,I4

Q1

Q2

Q3

00

00

u

a

u

a

uuua

auaa

u

a

uuua

auaa

q

q

Kq

q

CC

CC

q

q

MM

MM

Inertia Matrix Coriolis Matrix

BuG(q)q)qC(q,qM(q) Potential Terms

35