M. E. Pittelkau and W. G. McKinley, "Optical Transfer Functions, Weighting Functions, and Metrics for Images with Two-Dimensional Line-of-Sight Motion", Optical Engineering, SPIE, Volume 55, Issue 6, June 2016. 18 pages.
DOI: http://dx.doi.org/10.1117/1.OE.55.6.063108 [Open Access]
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We define the displacement, smear, and jitter components of image motion and derive the two-dimensional statistical Image Motion Optical Transfer Function (OTF) corresponding to each component. These statistical OTFs are parameterized by means and covariances, which are computed most conveniently from a weighted power spectrum of the line-of-sight motion. Another feature of these results is the realization that all temporal and spatial frequencies contribute to each statistical OTF and that one can determine the frequencies that contribute most significantly to each OTF. Additionally, optical system design is typically based upon the properties of an individual image. In a comprehensive optical system design, the statistical properties of an ensemble of images should also be considered. For individual images subject to a constant but possibly unknown smear length, the OTF is a sinc function. This is called a deterministic smear OTF because it does not describe the smear statistically. The statistical smear OTF describes the average smear OTF for an ensemble of images.
"Cascaded and Decoupled RIMU Calibration Filters", Journal of the Astronautical Sciences, Vol. 54, Nos. 3 & 4, July–December 2006, pp. 449-466. (Special Edition for the Malcolm D. Shuster Astronautics Symposium)
A model for a redundant inertial measurement unit (RIMU) and a Kalman filter for estimating calibration parameters is reviewed. We then derive a transformation of the physical calibration parameters that reduces the model to a simpler form. A calibration filter based on this simpler form executes faster and therefore is of benefit to real-time in-flight operation and to ground processing. Not only does the simpler form save on computation, the calibration filter naturally breaks into cascaded and decoupled forms, depending on the linearization chosen. Simulation results are provided to demonstrate the filter’s performance and to compare it to previous results based on the model with physical parameters.
"Calibration and Attitude Determination with Redundant Inertial Measurement Units", AIAA Journal of Guidance, Control, and Dynamics, Vol. 28, No. 4, July–August 2005, pp. 743–752.
A calibration filter is developed for redundant inertial measurement units (RIMUs), which are IMUs that have more than three sense axes. It is shown that for a Kalman filter based on an attitude measurement model and an attitude kinematics model in the "model replacement mode", a linear combination of the calibration parameters is not observable and therefore cannot be estimated. This observability problem is not related to dynamic observability, which requires calibration maneuvers. A null-space measurement equation, together with the attitude measurement and kinematics models, provides complete observability so that all calibration parameters can be estimated. Estimator performance without and with the null-space measurement update is demonstrated via simulation results.
"An Analysis of the Quaternion Attitude Determination Filter", Journal of the Astronautical Sciences, Vol. 51, No. 1, 2003, pp. 103–120.
The full-quaternion attitude determination filter is analyzed to address questions of covariance singularity and quaternion normalization. It is shown how nonsingularity of the covariance in the extended Kalman filter depends on the initial covariance, the process noise matrix, and implementation details of the filter. The covariance of a normalized quaternion estimate and the various means to achieve normalization are examined. The effect of a quaternion measurement update on the covariance and on the norm of the estimated quaternion is analyzed. It is also shown that the multiplicative and additive quaternion updates are equivalent. These are distinguished from an update called "rotational," which was proven elsewhere to be the constrained maximum-likelihood optimal update. It is demonstrated that the reduced-order body-referenced attitude determination filter is embedded in the full-quaternion filter.
"Rotation Vector in Attitude Estimation", AIAA Journal of Guidance, Control, and Dynamics, Vol. 26, No. 6, 2003, pp. 855–860.
An alternative derivation of the spacecraft attitude determination filter is developed to avoid questions of quaternion normalization or attitude matrix orthogonality constraints, quaternion covariance, and subterfuges used to circumvent these problems. This derivation is based on the Bortz equation for the rotation vector. Because the rotation vector is an unconstrained representation of attitude, the aforementioned questions do not arise. Singularities in the state dynamics equation are avoided by maintaining the predicted body attitude as the inertial reference for the filter. A simple discrete solution to the Bortz equation provides accurate attitude propagation for highly maneuverable spacecraft and also in the presence of jitter.
"Composite Estimate of Spacecraft Sensor Alignment Calibrations", AIAA Journal of Guidance, Control, and Dynamics, Vol. 26, No. 2, Mar–Apr 2003, pp. 371–374.
This paper has been superceded by later developments.
"Everything is Relative in System Alignment Calibration", AIAA Journal of Spacecraft and Rockets, Vol. 39, No. 3, May–June 2002, pp. 460–466.
The concepts of absolute and relative alignment calibration of spacecraft attitude sensors are examined. It is known that three degrees of freedom of attitude associated with absolute alignment calibration are unobservable unless payloaddata are processed. It is shown that an absolute alignment model is equivalent to a relative alignment model when the payload is regarded as an attitude sensor. Then it is shown that the three unobservable degrees of freedom are eliminated by defining the gyro as the body reference frame, attributing only three nonorthogonal misalignment parameters to the gyro. The payload misalignment can then be parameterized and calibrated in the same manner as any attitude sensor, or this can be left strictly to the payload data processing, thus creating a well-defined boundary between attitude control system calibration and payload calibration. The new parameterization introduced is illustrated via simulation results.
"Kalman Filtering for Spacecraft System Alignment Calibration", AIAA Journal of Guidance, Control, and Dynamics, Vol. 24, No. 6, Nov–Dec 2001, pp. 1187–1195.
The problem of attitude sensor and payload alignment calibration for spacecraft systems is addressed. An absolute alignment error model and a gyro error model are derived and implemented in a Kalman filter using UD factorization. Advantage is taken of the structure of the model to efficiently compute the factors of the process noise matrix. Simulation results illustrate the ill-effects of misalignment on attitude estimation and illustrate the performance of the absolute alignment calibration filter. The presence of unobservable degrees-of-freedom associated with the absolute alignment error model is demonstrated in these results. We also offer a simple modification of the typical onboard real-time attitude estimator that is useful to mitigate the effects of minor miscalibration (misalignment and gyro scale factor error) without requiring estimation of calibration parameters.
"Optimal Periodic Control for Spacecraft Pointing and Attitude Determination", AIAA Journal Guidance, Control, and Dynamics, Vol. 16, No. 6, Nov–Dec 1993, pp. 1078–1084.
A new approach to autonomous magnetic roll/yaw control of polar-orbiting, nadir-pointing momentum bias spacecraft is considered as the baseline attitude control system for the next TIROS series. It is shown that the roll/yaw dynamics with magnetic control are periodically time varying. An optimal periodic control law is then developed. The control design features a state estimator that estimates attitude, attitude rate, and environmental torque disturbances from Earth sensor and sun sensor measurements; no gyros are needed. The state estimator doubles as a dynamic attitude determination and prediction function. In addition to improved performance, the optimal controller allows a much smaller momentum bias than would otherwise be necessary. Simulation results are given.