Jitter Characterization of the HyTI Satellite

The concept behind this optical, low-cost metrology setup is to measure the jitter of HyTI by using a detector to measure and record small deflections of stationary laser beam off of a mirror that is attached to the satellite. To achieve low cost and replicability, commercial off-the-shelf (COTS) components were used to develop the metrology system, each superscript in the text is linked to a node or connection in Figure 3.

The S1FC635PM fiber-coupled laser source¹ from Thorlabs is used with a single-mode fiber optic cable ² to feed a laser collimation package³ that is secured to a kinematic v-mount. A standard two-adjuster kinematic mirror mount holds the mirror⁴ and is rigidly mounted to the supported jig with a 3-D printed clamp. The Thorlabs Lateral Effect Position Sensor⁵ measures the displacement of the incident beam relative to the calibrated center of the detector. This sensor was chosen due to its high accuracy and ability to provide positional information of any spot within the detector region, independent of beam shape, size, and power distribution. This obviates the need for optical gain calibration from other wavefront sensors or an expensive, high-speed, pixelated detector. At the maximum incident power level, the sensor has a resolution of 0.75 micrometers limited by electronics. A Kinesis K-Cube position sensing detector and auto aligner⁶ was used in open loop mode to measure beam position. The K-Cube generates a left-minus-right (X DIFF) signal, a top-minus-bottom (Y DIFF) signal, and a sum (SUM) signal. The LabJack T7-Pro⁷ was used to convert the analog signal from the K-Cube into reported $x$ and $y$ values (in the dimension of length) for the center of the incident beam with a Nyquist frequency of 500 Hz (half the 1 kHz sampling rate). These digital position values are then stored in a text file for analysis in Python. To convert the position values into an angular displacement in units of arcseconds, the centroid values reported by the setup were divided out by the distance between the mirror and the lateral effect position sensor, $l$. Figure 4 shows the setup with HyTI on the air bearing along and the throw distance labeled.

This setup can provide highly accurate measurements of laser beam deflection. Electronic noise limits the lateral effect position sensor to a resolution of $\delta$= 0.75 $\mu$m in both the x and y directions and given $l=1~{}\mathrm{m}$ (the approximate throw distance between trials), this corresponds to a minimum measurable angular displacement of 0.154”, shown in Equation 1. This system is also scalable to measure jitter simultaneously in three degrees of freedom; one would just need to duplicate the mirror to the position sensor path orthogonally to the existing path.

The metrology system was set up alongside the ADCS test facility in the 10,000-class HSFL Cleanroom at the University of Hawai ‘ i at Mānoa. The test facility, built by Astrofein, can provide highly accurate and complete end-to-end verification of ADCS systems for small satellites using a rotary air-bearing platform (to which HyTI was attached) by Physik Instrumente (PI) and a Helmholtz cage to simulate Earth’s magnetic field in orbit. The acceptance test from PI for our specific air bearing reports an axial error motion of 0.036 $\mu$m, a radial error motion of 0.058 $\mu$m, and a tilt error of 0.1 arcsec. Using $\delta_{ax}=0.036~{}\mu\mathrm{m}$ in Equation 1 for an $l=1~{}\mathrm{m}$ this corresponds to a d$\theta$ = 0.0074”, much less than the minimum angular displacement measurable at this throw distance calculated in the previous section and thus, we do not expect the axial error to contribute any systematic error to our measurements. The radial error does not contribute to measurements made since that motion occurs orthogonally to the sensor and does not result in any displacement of the incident beam in the directions measured. The tilt error, which is the combined effect of the axial and radial errors, is on the order of the fundamental precision of our metrology setup.

A 3-D printed plastic clamp secures the kinematic mount holding the mirror to the 6U support jig. This is shown in Figure 5, where calipers were used to place the mirror as accurately as possible to the calculated center of mass on the jig. Errors in this case come from the misalignment between the mirror face and the edge of the calipers which is no more than 0.05 mm. In this case and shown in Figure 4, the x-axis of the sensor corresponds to yaw and the y-axis corresponds to pitch.

Damping effects from the 3-D printed plastic clamp are assumed to be negligible and approximated as a rigid body at these frequencies for this work. In orbit and deployment, the jitter in the x- and y-directions will correspond to the jitter in the in-track and across-track dimensions respectively. This is pictured in Figure 6.

Air bearing imbalance introduced challenges in the form of undesired oscillatory motion, where the incident beam would occasionally drift off of the sensor area leading to a lack of high signal-to-noise readouts from the K-cube.

The process used to evaluate the jitter contribution from various sources involves incrementally adding vibrational sources (or active components) and investigating the impacts of the presence (or absence) of certain sources on the detected vibratory modal frequencies through the comparison of their power spectral density (PSD) plots and calculated jitter values.

The most relevant vibration influences are the ADCS reaction wheels, cryocooler for the payload, and the air bearing (which will isolate the vibratory motion of HyTI from the ADCS test facility when active). Other components include the satellite bus electronics and the ADCS electronics. These electronics have no moving parts so we do not expect to see any change in the PSDs or calculated jitter values when these components are active. Testing with these configurations allows us to gain confidence in the results and indicate the presence of any systematic errors (if any). By incrementally adding these sources, we can compare PSD plots and the calculated jitter values and attribute differences to the addition of a known source.

Configuration 1 trials were measured with the air bearing off and the satellite bus electronics turned on. All sources of vibration are off and observed vibration responses present in this trial are representative of the ambient environment in the cleanroom. In Configuration 2 the cryocooler and camera were on, actively taking images while the air bearing remained off. Then, any observed changes in the PSD plots between these two configurations would be from the vibrations of the cryocooler. Jitter values found using Equation 5 are not representative of the response of the satellite when the air bearing is off since it turns the satellite and bearing structure into a rigid body. We instead these values and compare them between different configurations with the air bearing off to help identify any potential systematic errors with the system. In Configuration 3, the air bearing was on while the bus electronics remained on and all other components were turned off. In a similar manner to the first configuration, this allows for a baseline for subsequent results. In Configuration 4, the ADCS electronics were turned on. Since this is not an active vibration source, any changes between this and the previous configuration will be due to dynamics from the air bearing. In Configuration 5, the x-direction reaction wheel was spun at a constant rate of 3000 rpm. This angular velocity was chosen to try and detect a fundamental modal frequency at 50 Hz and any generated harmonics. In Configuration 6, x-direction reaction wheel was then turned off and the z-direction reaction wheel was spun at 3000 rpm. Finally, in Configuration 7, both reaction wheels were spun at 3000 rpm to investigate their combined effects. This information is summarized in Table 1 and the following subsections describe how data was collected and analyzed for these configurations.

One permutation from Table 1 is selected. Over a three-minute period, the lateral effect position sensor detects and reports the measured power center of the incident laser spot. The data is sampled at a rate of 1kHz. This results in two time series: i) a plot of the laser center as a function of time in the x-direction as shown in Figure 4, and ii) a plot of the laser center in the y-direction as a function of time. The number of trials conducted per configuration is detailed in Table 1.

To investigate the frequency content of the system dynamics, the Fourier transform for each time series was generated. From the Fourier transform, a PSD was calculated using Equation 2, where the $G(T,f)$ is the PSD of some zero-mean random signal $x(t)$ and $X(T,f)$ is the Fourier transform of $x(t)$ over the interval $[0,T]$. The resolution of the PSD, given by the inverse of the duration of the time series, is 5.55$\times$10^-3 Hz. All PSD plots per configuration were averaged to obtain a more representative sample of the data. No binning or interpolation was required between trials since the resolution of the PSD plots was equal. From the PSD plots, our primary interest lies in quantifying the amount of jitter introduced by system dynamics in the form of pointing errors at two frequencies relevant to the HyTI mission: i) the integration time frequency $f_{int}$ = 2000 Hz (0.5 ms), ii) the frame rate frequency $f_{frame}$= 139 Hz (7.2 ms).

Because HyTI’s payload is a spatially modulated imaging Fourier Transform spectrometer, the orbital motion of the spacecraft moves scene elements through fixed optical path differences created by the interferometer. Thus, pointing errors in the integration frequency will result in sampling light from a larger area on Earth’s surface and the resulting image will appear blurred. Pointing errors in the frame rate frequency will affect the rate at which the interference pattern moves over the ground. This will affect how scene elements move through successive optical path differences and is equivalent to speeding up or slowing down the spacecraft, changing the effective spectral sampling rate. Oversampling at 139 Hz helps to mitigate this problem slightly for HyTI specifically, but the results will be useful to inform future mission design and planning.

From [13], the variance is the cumulative power from the frequency of interest to the highest measurable frequency, $f$ shown in Equation 3. Because variance measurements made in each axis are independent, the aggregate variance is defined by Equation 4, where $\sigma^{2}_{x}$ is the result of Equation 3 on the in-direction PSD and $\sigma^{2}_{y}$ is the result of Equation 3 on the across-track PSD. After the aggregate variance is calculated, we calculate the $n\sigma$ pointing jitter for our frequency of interest by Equation 5. Because the Nyquist frequency for the system is 500 Hz, ($f_{max}=500$), we will not be able to compute the jitter via Equation 3 for the integration time frequency since $f_{int}$ is less than $f_{max}$.

As an additional objective, we are interested in identifying any fundamental or resonance modal frequencies that originate from individual vibrational sources. An algorithm found peaks and fit Lorentzian profiles to give each modal frequency [Hz], the amplitude [arcsec²/Hz], the full width at half maximum (FWHM) [Hz], and their respective variances. By comparing the differences in the amplitude and the FWHM of common frequencies between configurations, we are able to disambiguate the contributions of different sources at the same frequency. While these values are not useful for calculating the pointing error due to jitter directly, they help accomplish our second objective to better understand system dynamics and inform mission planning and structure and assembly design. The modal frequencies that were identified are compared to those present in the previous configuration. The plot of the peak identification is shown in Figure 7.

Configuration 1 inspected the ambient environment of the cleanroom and the air bearing and all active sources were turned off. The variance for the frame rate jitter was 0.002 arcsec², which yields 1$\sigma$, 2$\sigma$, and 3$\sigma$ values of 0.014 arcsec, 0.028 arcsec, and 0.043 arcsec respectively.

Table 2 lists all the modal frequencies that were identified in Configuration 1. The difference in the number of frequencies in the in-track and across-track dimensions suggests that the ambient environment in the cleanroom may have more seismic activity normal to the ground. In the across-track dimension, aside from those 35.02 Hz, and 120.10 Hz and their resonances at 70.46 Hz and 240.12 Hz, the rest of the observed frequencies appear to occur at irregular intervals. By subtracting out the modal frequencies present here from any tests run with the bearing off, we expect that the difference in the PSD plots represents only contributions from the satellite’s vibrational sources.

Configuration 2 added the contribution from the cryocooler. The variance for the frame rate jitter was 0.0037 arcsec², which yields 1$\sigma$, 2$\sigma$, and 3$\sigma$ values of 0.06 arcsec, 0.12 arcsec, and 0.18 arcsec respectively.

The identified modal frequencies found for this configuration are listed in Table 3 where frequencies that were still present from the previous configuration but have different sources are highlighted in orange and new frequencies that were not present in the previous configuration are highlighted in green. In this configuration, they were determined to be contributions from the cryocooler by comparing the amplitude and FWHM values in Tables 10 and 11 in the Appendix. Because the cryocooler was operated at 60 Hz, the fundamental modal frequency as well as the resonance frequencies occurred in multiples of 60 in both the in-track and across-track dimensions. There was also a new modal frequency at 16 Hz in the in-track dimension as well as modal frequencies that are not multiples of 60 Hz, including 35 Hz, 93 Hz, and 460 Hz in the across-track dimension, and 340 Hz and 460 Hz in the in-track dimension.

To inspect the dynamics of the air bearing itself, measurements were made with no active vibrational sources from HyTI turned on. Similar to Configuration 1, this configuration offers baseline values for comparison of vibratory modal frequencies. Due to oscillations caused by air bearing imbalance, the noise floor generated is high - approximately 10^-2 arcsec²/Hz, as shown in Figure 11 in the Appendix. The resulting frame rate variance was 7.68 arcsec² which yields 1$\sigma$, 2$\sigma$, and 3$\sigma$ values of 2.77 arcsec, 5.54 arcsec, and 8.31 arcsec respectively. While these values are not relevant from the mission perspective as there are no active sources of jitter, we will compare them to the values from the next configuration to gain insight into the consistency of the metrology system.

There was only one modal frequency identified by the algorithm in Table 4. However, this modal frequency is shown to be noise from sensor electronics for measurements recorded by the K-Cube when the laser drifts off of the sensor area of the detector. This peak occurs at 120 Hz because of the flicker from the fluorescent lighting in the cleanroom and is highlighted in orange to indicate that while it was present in Configuration 2, the source is not the same.

In this configuration, the ADCS electronics were turned on, but the reaction wheels, cryocooler, and payload remained inactive. The resulting frame rate variance was 1.85 arcsec² which yields 1$\sigma$, 2$\sigma$, and 3$\sigma$ values of 1.36 arcsec, 2.72 arcsec, and 4.08 arcsec respectively. These values were a slight decrease from the results of the previous configuration and are the result of the air bearing settling. This is also visible as a slight decrease in the baseline noise level in the PSD (Figure 12) as this configuration was tested immediately after the previous one. The only modal frequency that was identified in this configuration shown in Table 5 was the one generated by noise as in Configuration 3. Both have a comparable amplitude and FWHM as shown in Table 13 in the Appendix.

The air bearing remained on while the x-direction reaction wheel was spun at a constant 3000 rpm and the cryocooler and payload remained off. The variance for the frame rate jitter is 0.35 arcsec², which yields 1$\sigma$, 2$\sigma$, and 3$\sigma$ values of 0.60 arcsec, 1.19 arcsec, and 1.79 arcsec respectively. The 3$\sigma$ frame rate jitter value is smaller than the HyTI mission requirement for the integration time jitter.

The expected vibratory modal frequency in the in-track direction near 50 Hz that would be generated by the x-direction reaction wheel is not present and could be buried in the baseline nose.

The air bearing remained on, the x-direction reaction wheel was turned off and the only z-direction reaction wheel was spun at a constant 3000 rpm. The variance for the frame rate jitter is 0.021 arcsec², which yields 1$\sigma$, 2$\sigma$, and 3$\sigma$ values of 0.14 arcsec, 0.29 arcsec, 0.44 arcsec respectively, significantly less than previous trials and may suggest that the addition of angular momentum in the along the optical axis helps to stabilize the system more than the jitter it generates. The identified modal frequencies are listed in Table 7. There is no peak from the noise at 120 Hz in both the in-track and across-track dimensions. This is because the oscillations of the air bearing were constrained to the sensor area of the detector. From the PSD in Figure 14, we observe a further decrease in the baseline values.

In the in-track dimension, there was only one modal frequency identified at 0.03 Hz due to the low-frequency air-bearing drift. In the across-track dimension, there were two modal frequencies were identified, one at 11 Hz and another at 58 Hz. We attribute the modal frequency at 58 Hz to the spinning of the reaction wheel but would suggest that the wheel is not accurately spinning at 3000 rpm, which would be 50 Hz from the datasheet. Future trials hope to compare these results with rotor encoder data and compare the values of the reaction wheel angular rates.

The air bearing remained on and the x-direction wheel was turned on to spin at the same time as the z-direction reaction wheel at a constant 3000 rpm while the cryocooler and payload remained off. The variance for the frame rate jitter is 0.019 arcsec², which yields 1$\sigma$, 2$\sigma$, and 3$\sigma$ values of 0.14 arcsec, 0.28 arcsec, 0.42 arcsec respectively - similar to Configuration 6. Identified modal frequencies are presented in Table 8.

In the in-track dimension, another modal frequency was identified at 0.02 Hz and can reasonably be considered the same modal frequency as one identified in Configuration 6 from the low-frequency drift of the air bearing. The modal frequencies at 10 Hz and 58 Hz remain present from the previous trial and we observed the addition of two higher frequency modal frequencies at 208 and 368 that were not present in Configuration 5 or Configuration 6. As these values are not multiples of 58 Hz (or 50 Hz), this demonstrates the benefits of investigating system dynamics with a fully integrated system.

The values of the frame rate jitter calculated for all configurations are summarized in Table 9. The frame rate jitter values from configurations 5 - 7 are within the integration time jitter requirement for HyTI at the 3$\sigma$ level.