1.1

Fine Pointing Requirement

The communications link budget (see Table 2) allots 3 dB loss due to pointing error, which corresponds to pointing within the full-width, half-max (FWHM) divergence angle of 14.6 arcsec (full-cone). Assuming symmetric pointing error, which is reasonable as discussed in Section 2, the resulting single-axis pointing requirement is ±5.18 arcsec. As discussed in Long [9], the contributions to this error include the residual pointing error from the FPS control system as well as error from optical, mechanical, and environmental sources. Contributors to the FPS control error include the sensor noise, spacecraft jitter, and the inertial coarse pointing error. Following the methodology in Long [9], the baseline FPS control error is estimated as ±2.27 arcsec. The mechanical and environmental errors include launch-induced shift, thermoelastic deformation, and residual misalignment following mechanical assembly and calibration. This error is estimated as ±4.50 arcsec [9]. The optical source of error, chromatic shift, is estimated as ±1.06 arcsec [9]. The total estimated error via RMS is ±5.15 arcsec, which results in a narrow margin of 0.6% to meet the 3 dB pointing loss requirement. The primary goal of this work is to improve this margin by further analysis and design refinement of the flight-level FPS control system. Improvements to mechanical, thermal, and chromatic shift error are not considered in this paper but are ongoing parallel efforts. The estimated FPS control error incorporates measurements from preliminary prototype experiments [10]. As a first prototype, the hardware used did not include flight-level electrical and optical components or use the flight laser wavelength. The control law used was a digital integrator tuned using in-place calibration. This paper examines the FPS control error using a simulation created in MATLAB/Simulink®. This is used to rapidly test different control laws and tune parameters without risk to the actual hardware. It is also used to verify system requirements for the selected flight hardware before prototyping. Last, it allows analysis of the primary sources of error to better focus efforts for the refinement of the design. This paper presents the results of the first iteration of the simulation software, focusing on the development of the model and the control law development.

Table 2:

Example intersatellite laser communications link budget for the CLICK mission.

2.1

Review of the Coarse Pointing System

One of the more common PAT architectures consists of a large gimbal that the optical bench is mounted to [16,17,18]. In this type of design, the optical bench includes the fine pointing and tracking system, and the gimbal is used for coarse pointing, acquisition, and tracking. However, in order to meet the low-SWaP requirements of a CubeSat, the CLICK payload does not use a coarse pointing gimbal. Instead, it directly relies on the ADCS of the spacecraft, which is aided by a CMOS camera and wide angle (2700 arcseconds full-cone) beacon laser. This approach was initially developed by Yoon [10], and the modeling software has been maintained for continued design iteration. The inertial pointing error data used in this work was generated with Yoon’s software.

2.2

Residual Inertial Pointing Error

The largest disturbance in the fine pointing system is due to the residual inertial pointing error from the attitude control system, which serves as the coarse pointing actuator for the PAT system. This error signal is generated from the coarse pointing model of the satellite, the details of which are described in Yoon [10].

The satellites are each equipped with a GPS unit and an RF crosslink for relaying GPS ephemerides between them. The payload contains a beacon system that utilizes a 10 degree FOV CMOS camera to measure the beacon signal (see Fig. 2). The coarse pointing system uses these two measurements of the relative state of the two satellites in order to estimate the relative pointing direction between them, which is the unit vector along the relative displacement vector. These measurements are synthesized in a specialized extended Kalman filter derived in Yoon [10]. The coarse pointing simulation also contains a model of the spacecraft dynamics including environmental disturbances as well as the attitude determination and control system. The environmental disturbances modeled so far include the gravity gradient torque and the magnetic dipole torque. Additional disturbances are currently modeled by Gaussian white noise. The results in this paper were derived from a coarse pointing simulation that was run for 1 hour of spacecraft time. The data was then transferred to the fine pointing system model, where a random time interval of 1 min, excluding the ADCS settling time, was selected and used as the residual inertial pointing error model. The field angle dynamics are shown in Figs. 4 and 5.

Figure 4:

X-Y field angle from inertial pointing error.

Figure 5:

Timeseries of inertial pointing error.

The total pointing error is the RMS of the X and Y component errors and is therefore modeled effectively by a Rician distribution as shown in Fig. 6. The noncentrality parameter of this distribution is 2.55 with a 95% confidence interval of (0.00, 9.24). The X and Y distributions are approximately Gaussian. The X fit gives a mean pointing error of -4.00 arcsec within (-4.21, -3.78) with 95% confidence and a standard deviation of 20.94 arcsec within (20.79, 21.10) with 95% confidence. The Y fit gives a mean pointing error of 9.83 arcsec within (9.47, 10.19) with 95% confidence and a standard deviation of 34.74 arcsec within (34.74, 35.00) with 95% confidence.

Figure 6:

Rician distribution of inertial pointing error.

2.3

Optical Models

The optical modeling of the payload is complicated by the fact that the system operates at different wavelengths. As described in Long [9], the telescope design is optimized for the communications wavelengths of 1537 nm and 1563 nm, which means that the 976 nm beacon signal is not collimated at the exit pupil of the resizing telescope. In order to accurately model the beacon spot location on the quadcell as a function of the FSM tilt angle and the input field angle, a Zemax® model (see Fig. 7) is utilized. For this iteration of the model, decoupled X and Y axis dynamics were assumed, which is supported by Yoon [10] and the manufacturer [12,13]. The Y axis dynamics were used for this iteration of the model. The optical transfer functions to quadcell from the telescope aperture and from the FSM were estimated independently for the range of angles, that covered the quadcell’s 1.13 mm sensor diameter. The transfer functions models are affine for the small range of maximum angles of the input disturbance: 134 arcsec in Y and 72.5 arcsec in X. The affine transfer function from the field angle to the quadcell Y axis centroid is -1.057 um/arcsec with a bias of -0.021 um and an R2 value of 1. The affine transfer function from the FSM tilt angle to the quadcell Y axis centroid is 0.186 um/arcsec with a bias of 0.001 um and an R2 value of 1.

Figure 7:

Zemax® Model of CLICK Payload Optics.

2.4

Electrical Models

Two simplifying assumptions were made about the electronics for this iteration of the system model. Future model iterations will include higher fidelity electronic models to anticipate sources of error due to digital sampling rates and software delays. The sampling rate on the ADC was assumed to be significantly larger than the inertial error dynamics, which is reasonable since the inertial pointing dynamics are estimated via power spectral density as being less than 3 Hz. Therefore, a continuous time version of the controller was used. The second simplification was to replace the quadcell sensor dynamics with an inverse system model that converts the quadcell position to a voltage in the range that is useable by the Raspberry Pi. The sensor dynamics, amplification, and filtering electronics will be included in future model iterations. The FSM is designed to minimize X-Y coupling [12,13], so the system is modeled at this stage by treating these as independent degrees of freedom. The open-loop FSM dynamics were modeled using a simple-harmonicoscillator (SHO) for the FSM in series with the 6th order Bessel low-pass filter. The FSM transfer function is given in Eqn. 1.

Control design using the SHO model has been verified on the hardware by the manufacturer [12]. The parameters were taken from the specification sheet for part number S7105, which uses A1B1.4 actuators and a 3.6 mm diameter mirror. The DC gain K = 0.0252 deg/V. The natural frequency is 3.10 x 103 rad/s, and the quality factor Q = 20. Specifications for all Mirrorcle FSMs can be found at their website [14]. The 6th order Bessel low pass filter transfer function is shown in Eqn. 2.

The defining parameter is the 200 Hz cutoff frequency, and the purpose of the filter is to eliminate the resonance modes associated with the FSM, which can easily damage the MEMS assembly. The effect of the filter is shown in Fig. 8. The use of this filter has been verified on the hardware by the manufacturer [12] as well as in previous experiments [10,15]. The frequency responses of each of these transfer functions, as well as the overall open-loop system, are shown in Fig. 9. The gain margin is 17.70 dB, and the phase margin is infinite, which shows that the baseline open-loop pointing system is stable and relatively robust to sources of error. The optics are designed to align the Rx and Tx beams when the beacon beam is centered on the quadcell. The closed-loop system design therefore focuses on rejecting disturbances like inertial pointing error and jitter.

Figure 8.

FSM & Low-Pass Filter (LPF) frequency responses

Figure 9.

Quadcell Y-Axis Centroid Control Result

2.5

Jitter Model

The last element of the forward path is reaction wheel jitter, which is modeled as a low-frequency noise signal generated by passing Gaussian white noise of intensity 0.10 arcsec through a 2nd order, 1 Hz cutoff low-pass filter. This is an approximation of the model of a BCT 15 to payload frequency response given in Shields et al. [19]. This simplification of the model does not include potential resonance effects. Further structural analyses and experiments are ongoing to develop a high-fidelity jitter model for the CLICK spacecraft.