Instrumental aspects of stellar interferometry : p 302-313

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Dynamics and
and Controls
Controls Modeling
Modelingand
and
Analysis Toolbox
Toolbox (DOCS)
(DOCS) for
for
Space-Based Observatories
Observatories
Interferometry Summer
Interferometry
SummerSchool
School
18-22 September
September 2000,
2000,Leiden,NL
Leiden,NL
Olivier de
Olivier
de Weck,
Weck, Prof.
Prof. David
DavidMiller
Miller
Space Systems
Systems Laboratory
Laboratory
Massachusetts Institute
Massachusetts
Institute of
ofTechnology
Technology
deweck@mit.edu
deweck@mit.edu
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Abstract
The DOCS (Dynamics-Optics-Controls-Structures) framework presented here
is a powerful toolbox for the modeling and analysis of precision optomechanical systems such as interferometers. Within the MATLAB
environment a model of the system can be created, which simulates the
dynamic behavior of the structure, the optical train, the control systems and the
expected disturbance sources in an integrated fashion. This modeling is critical
in order to identify important modal and physical parameters of the system
that drive the opto-mechanical performance. Other uses are the development
and tuning of attitude and optical controllers, uncertainty analysis, model
updating with test data, the development of error budgets and the flowdown of
subsystem requirements. This presentation outlines the technical challenges
faced by precision telescopes and gives an overview of how the DOCS
framework can assist in solving observatory design problems. Two specific
analysis examples are presented. First the derivation of an error budget for
NGST with two performance metrics and three error sources is shown.
Secondly the effect of reaction wheel disturbances and OPD control bandwidth
on the transmissivity function of TPF is presented. Experimental validations of
the toolbox are carried out with telescope testbeds in 1g. Preliminary versions
of the framework have been successfully applied to conceptual designs of
SIM, NGST, TPF and Nexus.
1
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Technological Challenges
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HST - 1990
Science
Requirements
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Engineering
Requirements
D&C System
Requirements
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Sample Requirements
Flowdown for SIM
TPF-2011
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The Hubble Space Telescope (HST) has celebrated it’s 10th year of on-orbit
operation in 2000. Hubble has broken technological ground for space based
astronomy as a multi-purpose UV/visual and IR observatory. The main science
objectives for Hubble are multi-purpose astrophysical imaging and
spectroscopy. The next generation of space based observatories, including
interferometers is being designed at this time and is expected to provide
significant improvements in angular resolution, spectral resolution and
sensitivity. In spaceborne interferometry SIM will provide precision
astrometry for faint stars and TPF (and DARWIN) will work as a nulling
(Bracewell) interferometer for direct plant detection in the IR. In designing
these ambitious missions a requirements flowdown is taking place from the
science requirements, to the engineering requirements to sub-discipline
requirements such as dynamics and controls (D&C). Thus phasing and
pointing requirements for the metrology-, guide- and science-light can be
postulated in terms of wavefront error (WFE), optical pathlength difference
(OPD), wavefront tilt (WFT), line-of-sight(LOS) jitter, beam shear (BS) and
others. The primary objective of DOCS is to address these dynamic system
requirements. This can be done for the various telescope modes such as
science light integration, tracking, retargeting and slewing.
2
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Research Motivation - Problem Statement
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z
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z,1=RMS
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Performances
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The present work is motivated by the need to simulate the dynamic behavior of
the future generation of space interferometers during the conceptual and
preliminary design phases. In order for these space or ground-based telescopes
to meet their stringent phasing (e.g. OPD) and pointing requirements (e.g. LOS
jitter), the path from disturbances to the performance metrics of interest ,z ,
must be modeled in detail. It is assumed that the systems are linear and timeinvariant (LTI). The premise is that a number of disturbances sources (reaction
wheel assembly (RWA), cryocooler, guide star noise etc.) are acting during the
various operational modes of interest as zero-mean random stochastic
processes. Their effect is captured with the help of state space shaping filters
[A d,Bd,Cd,Dd], such that the input to the appended system dynamics
[A zd,Bzd,Czd,Dzd] is assumed to be unit-intensity white noise d, which is
uncorrelated for each disturbance source. The shaped disturbances w are
propagated through the opto-structural plant dynamics [Ap,Bp,Cp,Dp], which
include the structural dynamics of the spacecraft and the linear optics matrices.
A compensator [A c,Bc,Cc,Dc] is often present in order to stabilize the
observable rigid body modes (ACS) and to improve the disturbance rejection
capability (ODL, FSM). The sensor outputs y and actuator inputs u might be
subject to noise. One of the goals of the DOCS framework is to accurately
predict the root-mean-square (RMS) values and sensitivities of the
performances z under the above assumptions.
Reaction Wheel Picture: http://www.ithaco.com/T-Wheel.html, FSM picture:
http://www.lefthand.com/prod_fsm.html, SIM Picture: http://sim.jpl.nasa.gov
3
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DOCS Toolbox Structure
Modeling
Hall How
Feron
System
Control
Strategy
Model Prep
Moore Skelton
Zhou
Model
Model
Reduction
Reduction&&
Conditioning
Conditioning
Baseline
Baseline
Control
Control
Balmes
Design
Structure:IMOS
Structure:IMOS
Crawley
Data
Model
Model
Assembly
Assembly
van Schoor
DYNAMOD
DYNAMOD
Jacques
Hasselman
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Uncertainty
Uncertainty
Database
Database
Control
Control
Tuning
Tuning
Disturbance
Disturbance
Analysis
Analysis
Gutierrez
Model
Model
Updating
Updating
System
Requirements
Optimization
Optimization
Feron
Sensitivity
Sensitivity
DE
Gutierrez
Margins
Isoperformance
Isoperformance
Examples
Blaurock
Campbell
Bourgault
Miller
Masters
Crawley
Haftka
Blaurock
Masterson
Disturbance
Disturbance
Sources
Sources
Mallory
Sensor
Sensor&&
Actuator
Actuator
Topologies
Topologies
ControlForge
ControlForge
JPL
Optics:MACOS
Optics:MACOS
Design
Analysis
Uncertainty
Uncertainty
Analysis
Analysis
- . / 0 1 2 0 1 35460 / 1 78:9546460 1:8; < 3=3=> ?@ 0A B50.6C@
Subsystem
Requirements
Error
Budgets
This block diagram shows an overview of the DOCS framework. The existing
toolboxes are compatible with IMOS (version 4.0), MSC/NASTRAN as well
as DynaMod® and ControlForge®. Once an initial model has been created and
numerically conditioned, the root-mean-square (RMS) values of scientific and
opto-mechanical performance metrics of the system
(e.g. pathlength
difference, pointing jitter, fringe visibility, null depth) can be predicted. The
exact sensitivities of the RMS with respect to modal or physical design
parameters can be computed. These sensitivities are essential for conducting
gradient-based optimization, redesign or uncertainty analyses. The goal of the
uncertainty analysis is to associate error bars with the predicted RMS values,
which are based on an uncertainty database resulting from past ground and
flight experience. The actuator-sensor topology of the system can be analyzed
numerically to ensure that the control system uses the actuator-sensor pairs
that will ensure maximum disturbance rejection or tracking performance. Once
a design has been found that meets all requirements with sufficient margins, an
isoperformance analysis can be conducted. Treating the performance as a
constraint the expected error sources (error budgeting) or key design
parameters (subsystems requirements definition) can be traded with respect to
each other. If hardware exists, the experimental transfer functions can be used
to update the structural, avionics and uncertainty models throughout the life of
the program to achieve a convergent design that will achieve mission success.
4
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Example 1: Error Budgeting (I)
(1) Why is error budgeting important ?
(2) How is it done today?
(3) How can DOCS/ isoperformance
help error budgeting ?
Establishes feasibility of dynamic system
performance given noise source assumptions.
Ad-Hoc error budgeting, RSS error tree,
limited physical understanding of interactions.
Leverages sensitivity analysis and integrated
modeling. Creates link to physical parameters.
Goal:
Balance anticipated error sources, which are given by physical process limits
and imperfections of hardware in a predictable and physically realizable
manner. Example: balancing of sensor vs. process noise.
NGST Example : Assume 3 Main Error Sources
Error Source 1: CRYO
Error Source 2: RWA
Error Source 3: GS Noise
3x 10
5
0.03
RWA Force Fx
4
Axial Force [N]
2
0.02
3
2
0.01
x
1
0
0
0
-0.01
-1
-0.02
-2
-3
-2
Centroid Pos
-4
0
0.5
Qc: Amplification Factor [-]
0.005 <= Qc <= 0.05
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1
[sec]
-0.04
-0.04
-5
0
1
2
3
4
5
6
7
Time [sec]
Us: Static Imbalance [gcm]
1.0 <= U s <= 30.0
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-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
8
Tint: Integration Time [sec]
0.020 <= Tint <= 0.100
The first example for the usefulness of the DOCS toolbox is error budgeting.
Error budgeting is the process of capturing and allocating allowable
contributions to all potential dynamic error sources of the opto-mechanical
system. Typically error budgeting is done in an RSS fashion and in a tree
structure, where the total error for the system, e.g. 10 nm OPD RMS is
subdivided among the expected error sources. The problem with this
traditional approach is that an allocation of a subportion of the allowable error
to a particular error source is not in the same units as the modal or physical
parameters, which directly describe the error source. (E.g. How does a 5 nm
OPD RMS error allocation to the reaction wheels relate to allowable static
imbalances Us of the wheels ?). The Isoperformance module of the DOCS
framework can assist in the error budgeting process by leveraging the
integrated model and sensitivity analysis to yield an error budget (capability),
which is as close as possible to the desired budget (allocation), but takes into
account the limitations/feasibility on the physical parameters, which describe
the error sources. In this example for NGST we assume three main error
sources. The cryocooler is located in the integrated science instrument module
(ISIM) and produces tonal mechanical vibrations to to a linear compressorexpander (Physical parameter: Qc). The reaction wheel assembly in the
spacecraft support module (SSM) produces attitude command torques and
superimposed disturbance forces and torques due to static and dynamic
imbalances (Physical parameter: Us). Guide star noise is introduced as a NEA
(noise equivalent angle) due to photon noise in the detector (Physical
parameter: Tint). The module assumes a 16.5 Mag guide star in the K-band.
5
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Example 1: Error Budgeting (II)
ERROR SOURCE 1
x' = Ax+Bu
y = Cx+Du
White
Noise
Cryo
Cryocooler
Disturbance
Mux
K
x' = Ax+Bu
y = Cx+Du
Mux
Demux
x' = Ax+Bu
y = Cx+Du
Hexapod
Isolator
Spacecraft
Structural
Dynamics
x' = Ax+Bu
y = Cx+Du
K
K
-K-
ps
Demux
ps-1
x' = Ax+Bu
y = Cx+Du
XY Graph
FSM Controller
FGS x' = Ax+Bu
Noise y = Cx+Du
Dynamics and Controls
Block Diagram for NGST
486 states
Disturbance Parameters
(Inhomogeneous Dynamics)
(variable)
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Plant Parameters
(Homogeneous Dynamics)
(fixed)
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t_wfe
PERFORMANCE 1
Dot Product2
Math
sqrt
Function1
ERROR SOURCE 3
ERROR SOURCE 2
Scope
Kfsm
-K-
White
Noise
RWA
Math
Function
Dynamic WFE
ps
K
Attitude
Control System
sqrt
nm Dot Productmean
Klos
Kacs
Reaction
x' = Ax+Bu
Wheel Assembly
y = Cx+Du
Disturbance
nm
Kwfe
Mux
Mux
White
Noise
Ch X
White
Noise
Ch Y
LOS Jitter
t_los
PERFORMANCE 2
Control Parameters
(Homogeneous Dynamics)
(fixed)
This slide presents the dynamics block diagram (Simulink®) for NGST used in
the error budgeting example. The dynamics of the disturbance sources (CRYO,
RWA, FGS) are shown in magenta. The assumption is that these disturbances
act as random, zero-mean processes which can be represented as colored
(filtered) white noise such that the PSD’s of experimentally measured noise
processes can be reproduced. These processes represent the inhomogeneous
dynamics of the system. There are two performance metrics z1 (Dynamic WFE
RMMS) and z2 (LOS Jitter). The requirements that have to be met by the
system are 55nm at λ=2.2 mm (λ/40) and σLOS=5 mas respectively. The
cryocooler disturbance acts directly onto the structure of the ISIM. The RWA
forces and torques are low-pass filtered by a mechanical hexapod isolator. The
displacements and rotations of the attachment points of optical elements
(primary mirror, secondary mirror, detector etc…) on the structure are
converted to optical quantities via linear sensitivity matrices. There are two
independent control loops in the model. The Attitude Control System (ACS)
uses measured attitude angles (star tracker) and rates (gyros) to command
torques in order to stabilize the rotational rigid body modes of the observatory
(bandwidth ~ 0.02 Hz). The LOS stabilization loop senses the guide star
position with the science camera with a frequency of 1/T int and commands
two angles of the fast steering mirror (FSM) gimbal. The plant and controller
parameters are assumed to be fixed during the error budgeting computations.
These are the homogenous dynamics of the system.
6
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Example 1: Error Budgeting (III)
Find
Error Source
Contributions
var_contr
Evaluate
Error Contribution
Sphere
Req:
55.00
z2: LOS Jitter [asec]
VAR%
Capability VAR % Allocation
0.72
46.6467
0.40
0.003162
0.28
29.1543
0.40
0.003162
0.00
0.0000
0.20
0.002236
55.0081
0.005000
Req:
0.005
VAR% Capability
0.31
0.002823
0.53
0.0037016
0.17
0.0020829
0.0051
WFE Budget
LOS Budget
Capability
0.8
GS Noise
Isoperformance
solution set
z1: WFE RMS [nm]
Error Source VAR % Allocation
Cryocooler
0.49
38.50
RWA
0.49
38.50
GStar Noise
0.02
7.78
Total
55.00
Capability =
Closest Feasible Error Budget
Isoperformance
Engine
Example: NGST Error Budget (Excel)
Allocated Budget
LTI System , DE z,req,
p_bounds, p_nom
Error
Contribution
Sphere
0.6
0.4
0.2
0
0
0.2
0.2
0.4
FGH GIJK JH LNMONP QRS RTUVWX QYZS RUV[N\ ] ^ QRS RZR_ RWA 0.6
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0.4
0.6
0.8
1.0
0.8
CRYO
This slide presents the results of the error budgeting exercise. The Excel
spreadsheet in the upper right corner shows the error allocation (blue) for the
three error sources as a percentage of the total variance for both performances.
This allocation is done apriori based on empirical experience. This desired
error budget is fed into the isoperformance engine of DOCS together with a
LTI model of the system dynamics, the system requirements and upper and
lower bounds on the physical parameters, which describe the error sources. An
isoperformance solution set is computed. This set contains all combinations of
solutions, which produce the required performance levels and do not violate
any constraints. For this set of solutions the error contributions are computed
and plotted on an error contribution sphere. The error budget, which comes
closest to the desired budget (allocation) on this sphere is computed. This is
called the capability budget, which is then returned to the error budgeting
spreadsheet (yellow columns). It is also possible to assign a weighting factor to
the performances, while finding the capability error budget. Another useful
result are the physical (error source) parameters, which correspond to the
capability error budget. In this case these parameters are Qc=0.029, I.e. the
cryocooler disturbances have to be reduced to 2.9% of the (uncompensated)
experimental levels, Us=14.09 gcm, i.e. the static imbalance of the reaction
wheels should not exceed 14.09 gcm and the integration time Tint of the fine
guidance sensor should be set to 40 msec. We see that the WFE is dominated
by the cryocooler disturbance, whereas the pointing performance (LOS jitter)
is mainly determined by the RWA imbalances.
7
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Example 2: RWA Noise and Nulling Performance
Normalized Intensity
Interferometer Transmissivity Function
0.5 AU
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TPF
Nominal Test data : Scale factor =1.0
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The second example shows a performance prediction analysis conducted for
the TPF mission. This chart shows the effect that reaction wheel imbalances
can have on the transmissivity function of the nulling interferometer and
ultimately the signal to noise ratio. The reaction wheel disturbance data was
obtained from a test of the ITHACO E-Wheel conducted at NASA GSFC in
1998. The wheel speed distribution was assumed to be uniform between 0 and
2000 RPM. The combined effect of 4 wheels in a pyramidal configuration is
taken into account. The left subplot shows the effect of the reaction wheel
imbalances that where obtained from the test without any modification to the
test data. We see that the transmissivity for a linear symmetric array with four
apertures (1-2-2-1) has four symmetric lobes (areas of peak intensity) and that
the suppression of starlight meets the specification (upper left plot) of 10-6 out
to the star diameter. The right subplot however demonstrates the effect if the
wheel imbalances are scaled up by a factor of 10. This could occur if the
wheels are poorly balanced or if a ball bearing failure occurs during
operations. The effect on the transmissivity is dramatic. Firstly we notice that a
pair of lobes is now being washed out by the vibrations, secondly the nulling
no longer meets requirements. In the nominal case the σOPD (average) is 76
nm, where it is 762 nm in the second case, which corresponds to roughly λ/16.
At 12 microns. For non-interferometric systems such a wavefront error might
be acceptable. Thorough analysis and testing of reaction wheel imbalances
before launch is paramount.
8
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Example 2: Effect of Optical Control on Nulling
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Interferometer Transmissivity Function
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4 Apertures, SCI,
linear symmetric @ 1AU
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values shown
are average for
all apertures
10
This chart shows the level to which the dynamics and controls can be
captured in a systems analysis (such as TPF) with the DOCS toolbox. The
effect of optical control on the system is modeled using a high-pass filter
approach, where each OPD channel is attenuated by the optical control at low
frequencies but not at high frequencies due to the limited sensor and actuator
bandwidths. The sensor is the internal and external metrology system and the
actuators are the multi-stage optical delay lines (ODL’s). The chart shows the
effect of changing the optical control bandwidth on the transmissivity
function of TPF assuming λ=12 µm. If the optical control bandwidth is too
low, the optical pathlength differences between the apertures creates a timevarying phase difference φi between the light beams at the combiner. This
phase shift disturbs the +/- 180 degree phase shift required for perfect nulling.
A simplifying assumption is that the OPD’s which are the square roots of the
variance of a stochastic random signal are added to the phase shift used to
compute the transmissivity function as if they were deterministic. Thus the
perturbations from the perfect transmissivity shown above are to be
understood in a 1 sigma sense. The preliminary results indicate that the
science requirements for TPF cannot be met with an optical bandwidth of 5
Hz, but increasing the bandwidth to 100 Hz leads to sufficient suppression of
the dynamic onboard disturbance sources. In this sense the DOCS toolbox
can be used to design and size important components (controllers, sensors,
actuators) of the observatory prior to manufacturing, integration and test.
9
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0
x
Experimental Validation (1)
Reaction Wheel Disturbance Testbed
Goal: Validate predictive capability
of DOCS toolbox on a test article
Performance Prediction (6 state model)
Displacement RMS [ m]
EF
20
15
Demo
Upper Stage: RWA
10
5
Video Clip
0
3000
200
150
2000
Wheel Speed [RPM]
100
1000
50
0
0
Mass m 1 [lbs]
Weight Bed
Kistler Accelerometer Results
Lower Stage
Displacement RMS [ m]
CD
20
15
10
Axial Stabilization System
5
0
3000
200
2000
1000
Wheel Speed [RPM]
! " # $
% & ' ! & () & * * +* $ $ $
50
0
0
150
100
Mass m 1 [lbs]
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Test Article allows to trade:
imbalance Us, mass m1, Wheel Speed
Ro, Suspension Spring Stiffness k1
This chart shows the Reaction Wheel Disturbance Testbed (RWDTB) at MIT’s
Space Systems Laboratory. This testbed is being designed with two purposes
in mind. First it serves as a platform to investigate coupling and impedance
effects, which occur when a dynamic disturbance source such as a reaction
wheel drives a flexible mechanical structure. Secondly it enables an
experimental verification of other elements of the DOCS toolbox such as the
isoperformance module. This is achieved by allowing variable parameters
(wheel speed, imbalance, spacecraft mass, base stiffness) and several
operational modes. The right side shows the testbed, which is comprised of an
upper stage (reaction wheel, DC motor, load cell, coupling plate, optics truss),
a lower stage (spacecraft truss, weight bed) and an axial stabilization system.
The instrumentation consists of three accelerometers, a laser displacement
sensor, an inductive proximitor, a six axis load cell and an analog tachometer.
The left side shows a comparison between a numerical prediction with a
simple 6 state DOCS-model (top) and initial experimental measurements
(bottom). It can be seen that the general trends are captured such as an
increase in accelerometer response with wheel speed and the occurrence of
two resonances including the suspension mode. The maximum predicted and
measured displacement RMS of the lower stage is 15 microns. A possible
extension to this testbed is to add a Michelson interferometer in the upper
stage and to predict and measure the impact of reaction wheel imbalances on
fringe visibility.
10
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0 2
Experimental Validation (2)
ORIGINS Telescope Testbed
Testbed Transfer Functions
2
1
Actuators
! " # #
$ % & ! % '( % ) ) *) + + +
RWA
VC
PZT
FSM
Reaction Wheel Assembly
Voice Coil Mirror
Piezo Mirror
Fast Steering Mirror
Sensors
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ENC
RGA
DPL
QC
Digital Angle Encoder
Rate Gyro Assembly
Differential Path Laser
Quad Cell Pointing
This chart shows the second ground based testbed, which is being used for
experimental validation of the DOCS toolbox. The ORIGINS Telescope
Testbed is able to slew by +/- 40 degrees about one axis and simulates the
main operational modes of a spaceborne imaging or interferometric telescope
such as slewing, target acquisition and tracking. It contains reaction wheels, a
gimbal connecting it to the floor structure, a voice coil and piezo stage for
pathlength control and a fast steering mirror for pointing control as actuators.
The sensors are a digital angle encoder and rate gyro for attitude
determination, a heterodyne laser measuring differential pathlength with a
resolution of 10 nm and a CCD or quad cell for pointing. The left side of the
chart shows the experimentally determined transfer function matrix from all
actuators to all sensors. It can be seen that the first flexible mode of the testbed
occurs at 2 Hz and corresponds to a flexible appendage simulating solar
panels. A non-dimensionalization and scaling analysis ensures traceability of
the results to full-size observatories. The testbed has been used to validate
several modules of the DOCS toolset such as DynaMod (obtaining
measurement models from test data), Sensor-Actuator topologies and
Controller tuning.
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0 0
Conclusions
•
•
•
•
Integrated modeling and simulation are critical for space
and ground-based interferometry before committing to a
particular system architecture
A MATLAB based analysis toolbox has been developed
and is integrated into the IMOS and MACOS environments.
Can work with numerical models or component test data
Experimental validation using laboratory testbeds in 1g
has been conducted
Supports dynamics and controls analysis: performance
prediction, uncertainty analysis, error budgeting, subsystem
requirements definition, controller development
Further information contact: deweck@mit.edu
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% & ' ! & () & $ $ *$ + + +
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The DOCS toolset is in a continuous flux of development. With each new
program (SIM, NGST, TPF, Nexus) the tools are becoming more robust
and user-friendly. The components MACOS and IMOS are available for
academic licensing from JPL. The components DynaMod and
ControlForge are available from Mide Technology Corporation
(www.mide.com) for commercial licensing. Other components might be
available upon request from MIT or will be transitioned to commercial
products in the future.
12