NGAO
Hard Real Time Controller
Algorithms
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Table of Contents
1. Overview .................................................................................................................................... 1
2. Algorithms (Mathematical Descriptions) ................................................................................ 2
2.1
2.2
2.3
2.4
2.5
2.6
Centroiding....................................................................................................................................... 2
Wave Front Reconstruction .............................................................................................................. 2
Tomography ..................................................................................................................................... 2
DM Command Generation ............................................................................................................... 2
Pseudo Open Loop Control .............................................................................................................. 2
Split Tomography............................................................................................................................. 3
3. Algorithm Interpretation ......................................................................................................... 4
3.1 Spatial Domain Interpretation .......................................................................................................... 4
3.2 Fourier Domain Interpretation ......................................................................................................... 4
4. Requirements Flow Down ........................................................................................................ 7
4.1 Speed of Calculations ....................................................................................................................... 7
4.2 Accuracy of Calculations ................................................................................................................. 7
4.3 Interface Data ................................................................................................................................... 7
4.3.1
4.3.2
4.3.3
4.3.4
Cameras ................................................................................................................................................. 7
DMs....................................................................................................................................................... 7
Diagnostic Data ..................................................................................................................................... 7
Telemetry Data Capture ........................................................................................................................ 7
4.4 Calibration Requirements ................................................................................................................. 8
5. Algorithms ................................................................................................................................. 9
5.1 Wave Front Sensor Processing ......................................................................................................... 9
5.1.1
5.1.2
5.1.3
5.1.4
5.1.5
5.1.6
Raw Pixel Processing ............................................................................................................................ 9
Centroiding............................................................................................................................................ 9
Wave Front Reconstruction ................................................................................................................... 9
Calibration ............................................................................................................................................. 9
Geometric Correction .......................................................................................................................... 10
Tip/Tilt processing .............................................................................................................................. 10
5.2 Tomography ....................................................................................................................................11
5.2.1
5.2.2
5.2.3
5.2.4
5.2.5
Forward Propagation ........................................................................................................................... 11
Reapplying the Aperture and Calculating the Errors ........................................................................... 13
Testing for Completion of the Iterations ............................................................................................. 15
Back Propagation ................................................................................................................................ 16
Calculating a New Atmospheric Estimate ........................................................................................... 17
5.3 DM Command Generation ..............................................................................................................18
5.3.1
5.3.2
5.3.3
5.3.4
5.3.5
5.3.6
5.3.7
5.3.8
5.3.9
Split Tomography................................................................................................................................ 18
Forward propagation (collapsing the layers) ....................................................................................... 18
Distortion Correction........................................................................................................................... 18
Woofer DM ......................................................................................................................................... 18
Science Object DMs (Tweeters) .......................................................................................................... 19
Point and shoot DMs (Tweeters) ......................................................................................................... 19
DM Phase to Volts Computation ......................................................................................................... 19
Saturation Correction .......................................................................................................................... 19
Unpowered Shape and Other Calibration ............................................................................................ 19
5.4 Tip/Tilt and Split Tomography Processing .....................................................................................19
5.4.1
Recombining Tip/Tilt using Split Tomography................................................................................... 20
6. Control Loop ........................................................................................................................... 21
7. Calibration ............................................................................................................................... 22
Table of Figures
Figure 1
Figure 1
Figure 2
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Split Tomography ....................................................................................................................... 3
Basic Loop.................................................................................................................................11
Forward Propagation .................................................................................................................12
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Figure 3
Figure 4
Figure 5
Calculating the error ..................................................................................................................14
Back Propagation ......................................................................................................................16
Split Tomography ......................................................................................................................18
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1. Overview
We divide our time line into frames (~500 µsec). At the start of each frame, we send the
DM commands that we have calculated in the previous frame to the DMs and Tip/Tilt
mirrors. Simultaneously, we read Shack-Hartmann images and Tip/Tilt/Truth
information from our WFS cameras to use in preparing an estimate of the atmosphere
used for the DM command generation for the next frame.
The tip/tilt information from the Tip/Tilt/Truth sensors is processed and sent to the
Tip/Tilt mirror and also used later in the generation of the DM commands.
We process the S-H images to handle dark current, pixel sensitivity, etc.
To prepare DM commands, we use the processed S-H images and generate tip/tiltremoved wave fronts for the tomography engine, which will create a tomographic volume
estimate of the atmosphere.
We then extract the low order information from the tomography and place it on a woofer,
which will handle the generally larger compensations. The woofer is conjugate to a low
altitude.
In MOAO, to correct the light for each science object, we forward project from each one
through the estimated atmosphere and send the resultant wave fronts to the individual
DMs for each science object.
To generate the actual DM commands we must recombine these wave fronts with the
tip/tilt/truth information (split-tomography) to establish actual compensation for each
segment of each DM. We then process the displacements to generate the actual voltages
needed by each segment.
These voltages are the actual commands used for the DMs.
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2. Algorithms (Mathematical Descriptions)
2.1 Centroiding
2.2 Wave Front Reconstruction
2.3 Tomography
2.4 DM Command Generation
2.5 Pseudo Open Loop Control
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2.6 Split Tomography
Split Tomography
From
High Order
Cameras
HOWFS
From
Low Order
Cameras
LOWFS
+
WFS
Data
Science
Object
Data
Value
Adj.
Tomography
-
Blind Modes
Low Pass
Filter
To
Tip/Tilt
Mirror
Tip/Tilt
Data
-
To DM
Command
To DM
Command
To DM
Command
Tip/Tilt
Filter
To
Woofer
Woofer
Data
Ver 1.2
Figure 1
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Split Tomography
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3. Algorithm Interpretation
3.1 Spatial Domain Interpretation
We divide the atmosphere into layers and then divide each layer into small regions called
voxels. The result is a volume model of the atmosphere and we want to obtain an
estimate for the effect of each voxel on the light passing through it.
To obtain this estimate we need to solve Ax=y for x: where A is a matrix that describes
the effect of the atmosphere on the light passing through it, x is a vector of the estimated
volume of the atmosphere that we are modeling, and y is a vector of the measurements
from our WFSs through the real atmosphere.
We do this iteratively starting from an estimate of x, usually the last estimate we made in
the previous frame (~500 µsec old), and the measurements of the WFSs, y (also from the
previous frame).
We forward propagate our guide stars through our estimated atmosphere to obtain an
estimated wavefront for each guide star. Forward propagation is just adding up the
values in each voxel through which a light beam passes on its way from the guide star to
the pixel on the WFS.
Then we subtract the resultant estimated wave front from that measured by the WFSs to
establish the resultant errors and compare the total error to a reference to determine
whether we need to continue iterating. If the total error is less than a predetermined value,
we are done iterating.
If we are not done iterating, we pre-condition our errors and back propagate them through
the atmosphere along the same paths by which we generated them, distributing them to
each voxel in proportion to the strength ( Cn2 ) of its layer. We then average the back
propagated errors for each voxel, apply a Kolmogorov filter to the errors to condition
them to condition them based on our understanding of atmospheric statistics. In the
spatial domain, filtering is accomplished by performing a convolution, a computationally
costly operation.
The averaged and filtered errors are added to the previous estimate for each voxel, and
we start the cycle again, forward propagating, etc. until the error is below our minimum.
We now have an estimate of the atmosphere in the spatial domain within the accuracy
requirements of our specifications.
3.2 Fourier Domain Interpretation
We maintain the estimated atmosphere and the measured values from the WFSs in the
Fourier domain.
The algorithm starts by projecting through the atmosphere from the guide stars to their
WFSs. We must account for the angular shift to each guide star and in the Fourier
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domain; these shifts are simply a complex multiply of the coefficient and the shift
variable.
After accounting for the guide star shift, we must account for the height of the laser guide
stars. For laser guide stars, the light from the guide star to the telescope forms a cone.
As a result, high spatial frequencies in the upper part of the cones, translate to lower
frequencies at the WFSs, which are at the ground level. We correct for this in both
backward and forward propagation using a bi-linear interpolation.
After we have accounted for the shift for each guide star and the height of each layer
relative to the laser guide star, we add the adjusted values for each guide star for each
layer.
This gives us an initial estimated value for our WFSs. However, after forward
propagating in the Fourier, if we analyze the data in the spatial domain, we find non-zero
values in the areas that correspond to the area outside the aperture due to spatial aliasing.
These values should be zero since there is no light there in our measurements (zero
values) and our estimate should be consistent and produce zeros there as well. We
correct for these incorrect values by converting to the spatial domain, applying an
aperture and converting back to the Fourier domain.
In the Fourier domain, applying an aperture function requires convolving the forward
propagated WFS images with the aperture image, a computationally intensive operation.
In the spatial domain, we can simply multiply the forward projection images by an
aperture image (consisting of 1’s inside the aperture and 0’s outside), a trivial operation.
Consequentially, we must perform an inverse Fourier transform on our forward
propagated images, multiply the resulting spatial images by the aperture image and
transform the results back into the Fourier domain. This does require additional
calculations, but far less than performing convolutions.
As a note, zeroing out the forward propagated spatial values, outside the aperture, does
not have the same effect as zeroing frequency bins, which would cause artifacts in the
spatial domain. In this case, we are simply duplicating, in our forward propagation, what
the telescope aperture does to the light in our WFS measurements.
Now, we subtract each of our forward propagated WFS values from the measured WFSs
(both in the Fourier domain) and obtain error values. If our total error value is less than a
predetermined value, the tomography calculations are done.
If we are not done, we must back propagate the errors to obtain a new estimate of the
atmospheric volume. Before back propagating, however, we must precondition the errors.
Preconditioning in the Fourier domain is accomplished by a vector matrix multiply of our
error vector (size W where W equals the number of WFSs) by a WxW matrix.
Preconditioning in the spatial domain requires ______.
In back propagation, we must propagate the errors back along the path by which they
were forward propagated. As in forward propagation, this is accomplished with a
complex multiply of the error frequency bins.
After the errors have been shifted to accommodate the guide stars position, they must be
scaled, stretching the low frequencies into higher frequencies using linear interpolation.
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The errors for each frequency bin for each layer that have been shifted and scaled are
averaged.
A Kolmogorov filter is now applied to the errors to condition them to be consistent with
atmospheric statistics. This is accomplished by performing a simple complex multiply.
The averaged and filtered errors are now added to the previous estimate of the
atmosphere and we start the cycle again, forward propagating, etc. until the error is below
our minimum.
We now have a Fourier domain estimate of the atmosphere within the accuracy
requirements of our specifications.
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4. Requirements Flow Down
4.1 Speed of Calculations
4.2 Accuracy of Calculations
4.3 Interface Data
4.3.1 Cameras
4.3.2 DMs
4.3.3 Diagnostic Data
4.3.4 Telemetry Data Capture
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4.4 Calibration Requirements
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5. Algorithms
5.1 Wave Front Sensor Processing
5.1.1 Raw Pixel Processing
The camera transfers raw camera data to the WFS processor. This defines the start of the
camera frame and the WFS processing.
The camera image is adjusted to compensate for the dark current values of the camera
and a threshold is applied to insure the subapertures are above certain intensity.
5.1.2 Centroiding
The algorithm to be used will be either center-of-mass, quad-cell or ______. Centroid
values will be compensated by stored reference centroids and centroid offset data.
Centroid values may also be linearized if necessary.
5.1.3 Wave Front Reconstruction
Reconstruction of the wave fronts from the centroids can be accomplished by using one
of the following algorithms: Fourier domain iterative, ________
5.1.4 Calibration
The RTC will provide capabilities to calibrate the opto-mechanical systems. These will
include
Saving the following:
 Reference dark currents for each camera
 Thresholds for each camera
 Current centroids as reference centroids for each WFS
 Last loaded centroid offsets as the offsets for each WFS
Setting or loading the following:



Set current reference centroids for each WFS to zero
Set current centroid offsets for each WFS to zero
Set current centroid linearization data to ones (no linearization) for each WFS
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 Load current reference centroid data to use for each WFS
 Load current centroid offset data to use for each WFS
 Load current centroid linearization data to use for each WFS
Also, see the DM command generation section (5.3.9 Unpowered Shape and Other
Calibration)
5.1.5 Geometric Correction
This compensates for pupil rotation.
5.1.6 Tip/Tilt processing
We compensate for field rotation here.
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Basic Loop
— 1-Nov-06 —
5.2 Tomography
Start
Calculate
New
Estimate
Back
Propagate
Forward
Propagate
Error > Limit
Calculate
Error
Error <= Limit
Measured
Value From
WFS
Done
Error Limit
Figure 2
Basic Loop
5.2.1 Forward Propagation
In preparation for the forward propagation for each WFS, the layer information is
adjusted for its shifted position in the sky based on angle to its guide star and height. For
laser guide stars, an adjustment is also made to account for the cone effect.
The estimated WFS value for each path from a guide star though the estimated values of
the layers to its WFSs is obtained by adding the adjusted values above in each layer.
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Forward Propagate
— 1-Nov-06 —
Start
For each sub aperture, sum all the estimated
Voxel values, along the path of a ray to each
Guide Star.
Calculate New
Estimate
Back Propagate
In the Fourier domain, you must account for the
spatial shift that occurs in the spatial domain.
This will be the complex conjugate of the shift
value used in the back propagation.
Forward Propagate
Error > Limit
Calculate Error
Measured Value From
WFS
Error Limit
Error <= Limit
Done
New Voxel
Estimate,
Layer(0)
Adjust for
Shift for
Layer(0)
New Voxel
Estimate,
Layer(N)
Adjust for
Shift for
Layer(N)
Sum All
Adjusted
Values in the
Ray Path
Calculate
Error Guide
Star (N)
Measured
Value From
WFS(0)
Figure 3
Forward Propagation
5.2.1.1 Shifting
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5.2.1.2 Scaling
Interpolation
Cone Beams
5.2.2 Reapplying the Aperture and Calculating the Errors
5.2.2.1 Inverse Fourier Transform
5.2.2.2 Applying the aperture
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5.2.2.3 Calculating the error
Calculate Error
— 1-Nov-06 —
Start
Calculate New
Estimate
Back Propagate
For each sub aperture, for each ray from a
Guide Star, calculate the error between the
forward propagated value and the
measured value from the WFS for that
Guide Star.
Forward Propagate
Error > Limit
Calculate Error
Measured Value From
WFS
Error Limit
If you have reached the limit stop, else
continue with another loop
Error <= Limit
Done
Back
Propagate
Forward
Propagate
Error > Limit
Subtract
Forward
Propagated
Value from
Measured
Value for
each WFS
Compare
Error to Limit
Error <= Limit
Done
Measured
Value From
WFS(0)
Error Limit
Figure 4
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Calculating the error
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5.2.2.4 Fourier Transform
5.2.3 Testing for Completion of the Iterations
There are two methods of stopping the tomography iterations during normal operations:
reaching a preset number of iterations or falling below a preset combined RMS wave
front error. When either condition happens, we stop the iterations and use our estimated
atmosphere to generate DM commands.
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5.2.4 Back Propagation
Back Propagate
— 10-Apr-09 —
Start
Calculate New
Estimate
For each guide star, for each sub aperture, back propagate
the error for a guide star to all the Voxels along the ray
back to the Guide Star (These should be the same Voxels
used in calculating the Forward Propagated value.)
Adjust the error for each Voxel for the Cn2 value for its layer
Back Propagate
Forward Propagate
In the Fourier domain, you must account for the spatial shift
that occurs in the spatial domain. This will be the complex
conjugate of the shift value used in the Forward
propagation.
Error > Limit
Calculate Error
Measured Value From
WFS
Error Limit
Error <= Limit
Calculate
New
Estimate for
this Voxel
Done
Adjust
Error(0) for
Shift for this
Layer
(Multiply)
Adjust
Error(N) for
Shift
(Multiply)
Adjust
Error(0) For
Gain and Cn2
(Multiply)
Adjust
Error(N) For
Gain and Cn2
(Multiply)
Calculate
Error Guide
Star (N)
Calculate
Error Guide
Star (0)
Forward
Propagate
Guide Star
(0)
Forward
Propagate
Guide Star
(N)
Measured Value
From WFS(N)
Measured Value
From WFS(0)
Error or Iteration
Limit
Figure 5
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Back Propagation
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5.2.4.1 Pre conditioning the errors for back propagation
Faster convergence and stability.
5.2.4.2 Weighting the errors
C n2
5.2.4.3 Shifting
Guide star position
5.2.4.4 Scaling
Laser guide star
5.2.4.5 Averaging the Errors
Multiple paths through voxels
5.2.4.6 Post Conditioning (Applying a Kolmogorov Filter)
Better estimate
5.2.5 Calculating a New Atmospheric Estimate
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5.3 DM Command Generation
We forward propagate from the MOAO science objects and use the wave fronts for DM
command generation. For MCAO, we would use the layer information to generate
commands for the conjugate DMs.
5.3.1 Split Tomography
Split Tomography
From
High Order
Cameras
HOWFS
From
Low Order
Cameras
LOWFS
+
WFS
Data
Science
Object
Data
Value
Adj.
Tomography
-
Blind Modes
Low Pass
Filter
To
Tip/Tilt
Mirror
Tip/Tilt
Data
-
To DM
Command
To DM
Command
To DM
Command
Tip/Tilt
Filter
To
Woofer
Woofer
Data
Ver 1.2
Figure 6
Split Tomography
5.3.2 Forward propagation (collapsing the layers)
Propagate from science object based on its position in the sky
Shifting but no cone effect
5.3.3 Distortion Correction
Camera, WFS, Optical, DM distortion correction
5.3.4 Woofer DM
The woofer operates in pseudo closed loop.
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Woofer processing
5.3.5 Science Object DMs (Tweeters)
The science object DMs operate in open loop.
Woofer displacement is removed.
5.3.6 Point and shoot DMs (Tweeters)
These DMs run operate in open loop
Woofer displacement removed?
5.3.7 DM Phase to Volts Computation
Open Loop control requirements
Phase to Force
Phase/Force to Volts
5.3.8 Saturation Correction
If we are saturating the DMs, we need to correct more in the Woofer.
5.3.9 Unpowered Shape and Other Calibration
The RTC uses unpowered shape data to compensate for the non-planar nature of the DM
when unpowered when generating DM commands.
The RTC can apply both internally generated and arbitrary DM data saved as files to
provide various patterns on the DMs for use in calibrating and characterizing the AO
system.
5.4 Tip/Tilt and Split Tomography Processing
Tip/Tilt measurement and processing
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5.4.1 Recombining Tip/Tilt using Split Tomography
Pre and post tomography processing
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6. Control Loop
The Woofer operates in closed loop.
All(?) WFSs see the light after the Woofer.
The science DMs see the light after the Woofer.
The science DMs are operated open loop.
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7. Calibration
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