1   Introduction¶

The LUTs of the various Active Optics System (AOS) components are of vital importance to the image quality of the Rubin Observatory. It is expected that most of the on-sky commissioning time will be spent calibrating these LUTs. And the improvements to the LUTs will be a long-term process lasting well into operations. We will use the engineering data from initial science observations for trend analyses so that the various LUT components can be characterized more and more accurately over time.

The following subsystems need LUTs to offset the impacts of changing operational parameters,

• M1M3
• M2
• M2 hexapod
• Camera hexapod

Each subsystem LUT has various components, depending on the input variables,

• Elevation
• Static component
• Actuator internal weight
• Temperature (bulk, gradients)
• Azimuth
• Camera rotator angle

The primary input variable to all the LUTs is the elevation angle. The only exception is the z-axis displacement of the hexapods which is also significantly affected by bulk temperature changes.

The elevation components of the M1M3 and M2 LUTs are required for safe operation of the optical systems. They are the only components that should not have a disabling switch. All other LUT components need to have switchs to allow users to turn them off when needed. For example, the thermal LUTs are often hard to determine with good accuracy, especially for the mirrors. This functionality will enable the users to easily determine whether the thermal LUTs are helping with the performance or not.

The toggle switches for the LUT components must be implemented in both the engineering user interfaces (EUIs) and as Software Abstraction Layer (SAL) commands.

2   Objective¶

The implementation details we cover in this tech note are mostly about conventions, i.e., if we make different choices, it probably won’t matter much to the performance of the AOS. The main objective of this tech note is to make the implementation of the LUTs as consistent as possible across the components, while accommodating the unique design and functionalities of each. This is crucial for avoiding confusions later, both during commissioning of the AOS and in operations.

3   Elevation angle vs zenith angle¶

We use elevation angle for elevation, instead of zenith angle. The elevation angle is zero when the telescope points at horizon, and 90 deg when the telescope points at zenith.

It is OK if a subsystem or an analysis notebook uses zenith angle internally, if the variable name is self-explanatory. The input variable to the gravity component of the LUTs is the elevation angle. The output of data analysis, which would be passed to software developers to implement, also uses the elevation angle.

The gravity components of the LUTs have different elevation ranges,

• [-8.5, 98.5] deg for M1M3. Occasionally M1M3 may go out of the [0, 90] deg range, and we want to make sure the glass mirror can still be supported properly. Hence the 8.5 deg margin on both sides.
• [-270, 90] deg for M2. In normal operations, M2 elevation will never go beyond the [0, 90] deg range. But M2 has a cart. For testing and engineering purposes, it needs to be able to be rotated by 360 deg. Beyond the [0, 90] deg range, glass safety is still the concern, but not image quality.
• [0, 90] deg for the M2 and camera hexapods. The hexapods may go out of the [0, 90] deg range. But we won’t be imaging outside of that range. Unlike the mirrors, where outside of [0, 90] deg proper LUT forces are still needed to counter gravity, for hexapods, we do not have to compensate position offset due to gravity. If the elevation angle goes smaller than 0 deg, we will use LUT values for 0 deg. If it goes larger than 90 deg we will use LUT values for 90 deg.

The M1M3 and M2 use their onboard inclinometers to determine the elevation angle. If due to some reason the elevation angle is not available from the onboard instruments, the LUTs use the elevation angle from the TMA telemetry. There are no inclinometers on the hexapods. The hexapods can only get the elevation angle by subscribing to the TMA telemetry.

4   Functional Form of the gravity components¶

We use 5th order standard polynomials of the elevation angle (:math:`theta_e`) for all the gravity components of the LUTs.

To avoid confusion, when the coefficients are output to a configuration file, they always follow the order of

Other functional form for the gravity component which we also considered include:

• Nth order standard polynomials of the cosine of the elevation angle (\(\cos(\theta_e)\)),
• Fourier series up to N cycles in the [0, 90] deg range, and
• piecewise linear interpolation with input data at 5 deg increment.

In choosing which functional form to use, our main considerations are:

1. Consistency between the LUTs. If the needs of each component is drastically different, it is OK to use different functional forms for them. However, unless there are good reasons for doing so, we prefer to keep the functional form consistent between the LUTs. This will make future work for fine-tuning the LUTs using real measurements and updating the LUTs much more straightforward and less error-prone.
2. Simplicity. There may be many functional forms that can meet our accuracy requirements. Here the criteria is to make sure the error due to finite number of coefficients is much less than the non-repeating error of the corresponding degree of freedom (DOF). Document-36395 [9] shows an analysis that demonstrates that the 5th order standard polynomials are accurate enough in all cases. And it is the simplest form among the above.
3. Various requirement documents (for example, LTS-88 [4] and LTS-206 [2][1]) specify that 5th order standard polynomials be used. We do not want to go through the change request process unless it is necessary.

5   Actuator internal weight components¶

For M1M3 and M2, the actuators have to supply the forces that can support their own internal weights first. Additional forces are then used to support and shape the glass. The LUTs therefore have actuator internal weight components. These forces can be calculated analytically. As a result of its two internal pneumatic cylinders angled at 45 degrees, the effects of the internal weight of the M1M3 actuators are more complex than would be expected.

The calculations for M1M3 actuators (both single-axis and dual-axis actuators) are given in Document-32192 [6]. Measurements were also performed for standalone M1M3 actuators on the test bench in Tucson [1]. The results were consistent with the analytical calculations within measurement errors. We therefore use the results from the calculations in the final actuator internal weight component of the M1M3 LUT.

The M2 LUT actuator internal weight component was supplied by the M2 vendor Harris. The Rubin team plans to recalculate these forces to crosscheck the Harris results. We will also measure these when the mirror is removed. But note that the measurements will include the effects of the load cell zero offsets. Therefore the measurements will be used as a crosscheck. For the M2 LUT actuator internal weight component we will use results based on the engineering model.

The M2 and the camera hexapods do not have actuator internal weight components in their LUTs because the output of the hexapod LUTs are positions instead of forces.

6   Static components¶

The static component of the LUT doesn’t vary with external conditions. For the mirrors, these are the forces that are needed to bend out the low spatial frequency factory figuring error. These were supplied by the vendors during factory acceptance testings. We will not change these components during commissioning and operations, unless somehow it can be proven that the figuring errors are different from what were determined at the factories.

As for the hexapods, the \(C_0\) defined in Sec. 4   Functional Form of the gravity components is the static component. For now, all six coefficients for the 5th order standard polynomial for each hexapod have been determined using results from FEA analyses. Once we have the hexapods mounted on the telescope mount assembly (TMA), we will use Laser Trackers (LTs) to calibrate the LUTs for both hexapods. It is expected that the calibrated values of \(C_0\) will be quite different from the FEA values. The \(C_0\) represents the variation of the hexapod locations from the theoretically perfect TMA and optical system. This variation is primarily the result of fabrication and assembly tolerances.

The static components \(C_0\), especially those for the hexapods, are defined at a reference temperature (\(T_{\rm ref}\)). We use

for all the LUTs. Per LTS-54 [3] the operational temperature range is -3 to 19C, and the mean temperature is expected to be 11.5C.

7   Thermal components¶

For now, the thermal LUTs only use the bulk temperature as the input variable. There is no plan to utilize the thermal gradients.[2] The functional form of the thermal compensations will be the 5th order standard polynomials, to comply with LTS-88 [4] and LTS-206 [2]). All the thermal coefficients are set to zeros before we have good measurements of the thermal commpensations.

The degradation in image quality resulting from thermal variations will occur slowly relative to the cadence of the telescope and the AOS response. Consequently, the system will already compensate for this degradation.

8   Azimuth angle¶

All the LUTs have azimuth components where the azimuth angle of the telescope is the input variable, to comply with LTS-88 [4] and LTS-206 [2]. The functional form of the azimuth angle dependence also uses a 5th order standard polynomial. It is understood that a Fourier series will have the advantage of being continuous at 0/360 deg. However, it is expected that the azimuth corrections will be small and not worth the complexity.

9   Rotator angle¶

Only the camera hexapod LUT has a rotator angle component. This is due to a small angle of tilt between the rotator’s rotational axis and the camera’s optical axis, and the asymmetry in the camera mass distribution around the optical axis.

10   Load cell zero offsets¶

The M1M3 and M2 control systems need to have LUTs to account for the load cell zero offsets.

These can be configured individually for the x, y, and z-components of each actuator. Initially these are all set at zero. After we obtain the offset values through measurements, these tables will be populated. They represent an important contributor to the overall commanded forces.

Like the static components, the load cell zero offsets are not dependent on any of the other variables discussed in this document. For conceptual clarity and easier maintanence, we choose to keep them separate from the static forces.

11   Units, DOF names, and CS¶

The outputs of the M1M3 and M2 LUTs are forces for individual actuators. The units are Newtons. The CSs are M1M3 CS and M2 CS, respectively. See SITCOMTN-003 [8] for definitions of these CSs.

The outputs of the hexapod LUTs are x, y, and z displacements and rotations around the x, y, and z axes. The rotations follow the right-hand rule. The units shoulld be microns for displacements and arcseconds for rotations. Even though the Data Management (DM) standard for rotations is to use degrees, we decide to make an exception here because we will be dealing with small angle all the time.[3]

In the XML interface, we need to keep uniform naming to avoid confusions. The x, y, and z displacements have parameter names of dx, dy, and dz. The rotations around the x, y, and z axes have parameter names of rx, ry, and rz. The parameter names having “d” and “r” in them indiciate that these are offset commands, not the new positions for the commanded components. For clarity, these are required even when the topic name already indicates it is an offset command, since they are not much longer than x, y, z, u, v, and w. The offsets are always in M2 CS [8] for the M2 hexapod, and CCS [8] for the camera hexapod.

We realize that the parameter names for mirror positions are not as consistent as one may wish. Right now the M1M3 positions use units of meters and degrees, while M2 positions uses microns and arcseconds. The parameter names are xPosition, yPosition, zPosition, xRotation, yRotation, and zRotation for M1M3. For M2 they are x, y, z, xRot, yRot, and zRot. Since these are only used in engineering modes, and not controlled by the AOS, they are less likely to cause confusions. To reduce the amount of work for the developers we choose not to change these.

Note that for the hexapods, rz, the rotation about the optical axis, does not affect the optical system and is not used by the AOS system. It is only used for engineering/diagnostic purposes.

12   Future work¶

Things we need to do before the next round of testing:

• finish FEA analysis on M1M3 gravity LUT, and make sure we account for the weights of all the interface plates and cups correctly; also revise Document-34898 [7] accordingly;
• measure the M1M3 actuator load cell zero offsets before they are attached to the glass mirror on the summit;
• perform FEA analysis on M2 gravity LUT to verify Harris values;
• determine M2 actuator internal weight component and compare against Harris results;
• measure the M2 actuator load cell zero offsets when the mirror is removed;
• perform analysis to determine if M2 static forces from Harris make sense;
• change M2 gravity functional form to 5th order polynomial;
• add Harris M2 LUT dependence on thermal gradients;
• change the M2 thermal LUT reference temperature from 21C to 11.5C;
• Provide software switches to disable separate LUT components;
• change all thermal components in the form of 5th order polynomial;
• check and ensure that we use the following everywhere in the XML
  • elevation angle instead of zenith angle (also revise SITCOMTN-003 [8] accordingly);
  • parameter names for offset commands: dx, dy, dz, rx, ry, rz;
  • units for hexapod offsets: microns and arcseconds.

Future milestones for LUT updates:

• M3 summit testing;
• Updates of the M2 and camera hexapods LUTs using laser tracker measurements on the TMA;
• Initial Optical Testing Assembly (IOTA) (if we eventually do get a time window);
• Commissioning Camera (ComCam);
• LSSTCam Full-Array Mode (FAM);
• LSSTCam normal operation mode (using four corner wavefront sensors).

13   Appendix A - Useful links¶