"Attitude Determination Kalman Filter with a 1/f Flicker Noise Gyro Model", Institute of Navigation, GNSS Conference (ION GNSS 2013), 16–20 September 2013, Nashville, Tennessee. Abstract Angle random walk, rate random walk, and bias instability are key noise parameters in gyro performance for spacecraft attitude determination and navigation system performance. The bias instability is due to flicker noise, which generally determines the noise floor of gyros. Conventional attitude determination filters model angle random walk and model the bias instability as a rate random walk, and therefore are suboptimal. An overview of gyro noise characteristics, methods of noise characterization, and various flicker noise models is given. Attitude determination Kalman filters that incorporate models of flicker noise are developed and their performance is analyzed via simulation and compared with the performance of the conventional two-state Kalman filter in the presence of flicker noise. The flicker noise Kalman filters are expected to be particularly beneficial to low-cost gyros with high bias instability, and will also benefit high-accuracy and strategic systems, particularly in GPS-denied environments. An extension of the flicker noise Kalman filter to accelerometers, gyros, and clocks in navigation systems is straightforward.
Pittelkau, M. E.; O'Shaughnessy, D. O.; "Gyro Misalignment Decomposition Applied to MESSENGER Calibration", Paper No. AAS 09-203, AAS/AIAA Spaceflight Mechanics Meeting, Savannah, Georgia, 8–12 February 2009, in Advances in the Astronautical Sciences, Vol. 134, Part I, 2009, pp. 1527–1539. [CD, Book] Abstract In attitude sensor misalignment estimation, a rotational misalignment vector, or a linear combination of rotational misalignments, must be constrained to zero for full observability. For this reason, one attitude sensor is generally designated the body reference sensor. Alternatively, the Inertial Measurement Unit (IMU) can be the body reference sensor. A method for removal of a rotational misalignment from a Redundant IMU (RIMU), which has more than three sense axes, was developed recently. We demonstrate the method using telemetry from the MESSENGER spacecraft. Results are compared with earlier results where one star tracker is the body reference sensor.
“General Definition of Relative Misalignment”, Paper No. AAS 08-289, F. Landis Markley Astronautics Symposium, Cambridge, Maryland, 29 June – 2 July 2008, in Advances in the Astronautical Sciences, Vol. 132, 14 pp. Abstract It is well known that if misalignments are modeled at all attitude sensors on a spacecraft, which may include the gyro, a linear combination of misalignments is unobservable. Typically one sensor (called the primary, master, or body reference sensor) is not parameterized with misalignments so that the remaining misalignments are fully observable. Another parameterization is to model the misalignment of two sensors as being equal and opposite in direction. In this paper we examine various misalignment models used to calibrate the alignment of spacecraft attitude sensors and gyros, and we discuss the pros and cons of each. We present a general formula for eliminating the unobservable misalignment.
“Survey of Calibration Algorithms for Spacecraft Attitude Sensors and Gyros”, Paper No. AAS 07-295, AAS/AIAA Astrodynamics Specialists Conference, Mackinac Island, MI, 19–23 August 2007, in Advances in the Astronautical Sciences, Vol. 129, 2008, pp. 651–705. An outline of the survey paper and access to the paper and presentation are on the Survey Paper page. Abstract This paper is an historical survey of algorithms for ground-based and on-orbit calibration of attitude sensors and gyros on spacecraft. These algorithms include attitude-independent alignment calibration algorithms, the classical Davenport gyro calibration algorithm, and various Extended Kalman Filters. The calibration algorithms are outlined and compared, and an assessment of the algorithms is offered. Attitude maneuvers required for calibration are also discussed.
O’Shaughnessy, D.; Pittelkau, M. E., “Attitude Sensor and Gyro Calibration for MESSENGER”, (revised) 20th International Symposium on Space Flight Dynamics, Annapolis, MD, 24–28 September 2007, NASA Conference Publication NASA/CP-2007-214158, Document ID 20080012629. Abstract The Redundant Inertial Measurement Unit Attitude Determination/Calibration (RADICAL) filter was used to estimate star tracker and gyro calibration parameters using MESSENGER telemetry data from three calibration events. We present an overview of the MESSENGER attitude sensors and their configuration is given, the calibration maneuvers are described, the results are compared with previous calibrations, and variations and trends in the estimated calibration parameters are examined. The warm restart and covariance bump features of the RADICAL filter were used to estimate calibration parameters from two disjoint telemetry streams. Results show that the calibration parameters converge faster with much less transient variation during convergence than when the filter is cold-started at the start of each telemetry stream.
“Autonomous On-Board Calibration of Attitude Sensors and Gyros”, (revised) 20th International Symposium on Space Flight Dynamics, Annapolis, MD, 24–28 September 2007, NASA Conference Publication NASA/CP-2007-214158, Document ID 20080012629. Abstract This paper presents the state of the art and future prospects for autonomous real-time on-orbit calibration of gyros and attitude sensors. The current practice in ground-based calibration is presented briefly to contrast it with on-orbit calibration. The technical and economic benefits of on-orbit calibration are discussed. Various algorithms for on-orbit calibration are evaluated, including some that are already operating on board spacecraft. Because Redundant Inertial Measurement Units (RIMUs, which are IMUs that have more than three sense axes) are almost ubiquitous on spacecraft, special attention will be given to calibration of RIMUs. In addition, we discuss autonomous on board calibration and how it may be implemented.
"Advances in Attitude Determination with Redundant Inertial Measurement Units", Paper No. AAS 06-110, AAS/AIAA Spaceflight Mechanics Meeting, Tampa, Florida, 22–26 January 2006, in Advances in the Astronautical Sciences, Vol. 124, Part I, 2006, pp. 163–178. [CD, Book] Abstract Key results of several recent papers on attitude determination and calibration with Redundant Inertial Measurement Units (RIMU) are summarized, and some subtleties are illuminated. The RIMU Attitude Determination/Calibration (RADICAL) filter has been implemented in commercial off-the-shelf software. A RIMU, which has more than three angular rate sense axes, offers enhanced system performance in terms of availability, reliability, redundancy, and accuracy. We discuss how to exploit the analytical redundancy of the RIMU to maximize the potential performance of a spacecraft system. Some of the observability properties that result from analytical redundancy are demonstrated via simulation. It is shown that calibration accuracy can be partially maintained even in the absence of attitude measurements.
"Recent Advances in Calibration of Redundant Inertial Measurement Units", Flight Mechanics Symposium, NASA Goddard Space Flight Center, NASA/CP-2005-212789, 18–20 October 2005. [CD] Abstract Key results of several recent papers on calibration and attitude determination with Redundant Inertial Measurement Units (RIMU) are summarized in this paper. A RIMU has redundant (> 3) sense axes, which is of benefit in several ways in addition to functional redundancy. This redundancy also presents additional properties that must be exploited to estimate RIMU calibration parameters as well as attitude, and other properties that can be exploited to improve calibration performance or to improve computational efficiency. A general RIMU measurement model is given and the approach to developing a RIMU calibration filter is outlined. A minimal parameterization of RIMU misalignments is presented and observability properties of the calibration parameters are discussed. Reduced-order, cascaded, and decoupled RIMU calibration filters are then presented.
"Cascaded and Decoupled RIMU Calibration Filters", Paper No. AAS 05-466, Malcom D. Shuster Astronautics Symposium, Grand Island, NY, 12–15 June 2005. In Advances in the Astronautical Sciences, Vol. 122, Part I, 2006, pp. 273–288. [CD, Book] Abstract 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.
"Observability and Calibration of a Redundant Inertial Measurement Unit (RIMU)", Paper No. AAS 05-105, AAS/AIAA Space Flight Mechanics Meeting, Copper Mountain, CO, 23–27 January 2005. In Advances in the Astronautical Sciences, Vol. 120, Part I, 2005, pp. 71–84. [CD, Book] Abstract A calibration model for a redundant inertial measurement unit (RIMU) comprises a bias, a symmetric and an asymmetric scale factor, and two misalignment angles for each of *n* sense axes (*n* > 3) for a total of 5*n* parameters. A linear combination of these parameters is observable through the attitude kinematics model and attitude measurements; an orthogonal linear combination is observable through null space angular rate measurements. Transformations are derived to separate the 5*n* calibration parameters into two independent reduced-order parameter sets, from which the observability of the parameters is characterized. These parameters can be estimated by a pair of cascaded or decoupled Kalman filters. Sensitivity matrices needed to construct these filters are derived.
"RIMU Misalignment Vector Decomposition", AAS/AIAA Astrodynamics Specialist Conference, Paper No. AIAA-2004-4856, Providence, RI, 16–19 August 2004. [CD, On-line] Abstract A general misalignment parameter vector for a redundant inertial measurement unit (RIMU) comprises two misalignment angles for each of *n* sense axes (*n* > 3). A transformation is derived to separate 3 rotational misalignments and 2*n* – 3 nonorthogonal misalignments in the parameter vector. The model order can be reduced by explicitly eliminating the three rotational misalignments. It is shown that the model order can also be implicitly reduced by appropriately initializing the covariance with the aid of the transformation. The equivalence of rotational misalignments modeled at either the RIMU or at the attitude sensors is shown empirically via simulation. Results apply also to nonredundant IMUs (*n* = 3).
"Attitude Determination and Calibration with Redundant Inertial Measurement Units", Paper No. AAS 04-116, AAS/AIAA Space Flight Mechanics Meeting, 8–12 February 2004, Maui, HI. In Advances in the Astronautical Sciences, Vol. 119, Part I, pp. 229–248, 2005. [CD, Book] Abstract A calibration filter is developed for redundant inertial measurement units (RIMUs), which have more than three sense axes. It is shown that a linear combination of the calibration parameters is not observable in attitude and therefore cannot be estimated by updating a Kalman filter with only attitude measurements. This is a geometric observability problem and is not related to dynamic observability. Calibration maneuvers are required for dynamic observability but do not affect the geometric observability. A null-space measurement equation, together with an attitude measurement model, 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.
"Pointing Error Definitions, Metrics, and Algorithms", Paper No. AAS 03-559, AAS/AIAA Astrodynamics Specialists Conference, Big Sky, MT, 3–7 August 2003. In Advances in the Astronautical Sciences, Vol. 116, Part II, pp. 901–920, 2003. [CD, Book] See the Errata sheet. Abstract This paper introduces pointing error definitions and metrics that are mathematically well-founded and are meaningful to image quality and to other types of measurements. The definitions and metrics are accuracy, displacement, jitter, stability, and windowed stability. The metrics are time-domain and equivalent frequency-domain formulas. The frequency-domain formulas are particularly easy to apply. Useful metrics for displacement and jitter were introduced a decade ago but have not been utilized in requirements documents, and so they are reintroduced here. New definitions and metrics for stability and windowed stability are introduced in this paper.
"Definitions, Metrics, and Algorithms for Displacement, Jitter, and Stability", Flight Mechanics Symposium, NASA Goddard Space Flight Center, NASA/CP-2003-212246, 28–30 October 2003. [CD] See the Errata sheet. Abstract This paper introduces pointing error definitions and metrics that are mathematically well-founded and are meaningful to image quality and to other types of measurements. The definitions and metrics are accuracy, displacement, jitter, stability, and windowed stability. The metrics are time-domain and equivalent frequency-domain formulas. The frequency-domain formulas are particularly easy to apply. Useful metrics for displacement and jitter were introduced a decade ago but have not been utilized in requirements documents, and so they are reintroduced here. New definitions and metrics for stability and windowed stability are introduced in this paper.
Note: This paper contains a little more information than AAS 03-559.
"Measurement Sensitivity Equations in Attitude Determination", Flight Mechanics Symposium, NASA Goddard Space Flight Center, NASA/CP-2003-212246, 28–30 October 2003. Abstract This paper focuses on constrained and unconstrained partial derivatives of the measurement equations used in spacecraft attitude determination. Six different approaches to these calculations are examined. Although all of these are found in the literature, there are many missing details and unexplained subtleties. This paper presents these details and explains the subtleties.
Note: The following paper should have been cited: SHUSTER, M. D., "The Quaternion in Kalman Filtering", Paper No. AAS-93-553, Proceedings of the AAS/AIAA Astrodynamics Specialists Conference, Victoria, British Columbia, Canada, August 16–19, 1993; in Advances in the Astronautical Sciences, Vol. 85, 1993, pp. 25–37.
"An Analysis of the Quaternion Attitude Determination Filter", AAS/AIAA Space Flight Mechanics Meeting, Ponce, PR, 9–13 February 2003, Paper No. AAS 03-194, in Advances in the Astronautical Sciences, Vol. 114, Part II, pp. 1337–1352, 2003. (See the journal publication.) Abstract 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.
Note: The following paper should have been cited: SHUSTER, M. D., "The Quaternion in Kalman Filtering", Paper No. AAS-93-553, Proceedings of the AAS/AIAA Astrodynamics Specialists Conference, Victoria, British Columbia, Canada, August 16–19, 1993; in Advances in the Astronautical Sciences, Vol. 85, 1993, pp. 25–37.
M. Pittelkau and W. Dellinger, "Attitude Sensor Alignment and Calibration for the TIMED Spacecraft", Paper No. AAS 03-153, AAS/AIAA Space Flight Mechanics Meeting, Ponce, Puerto Rico, in Advances in the Astronautical Sciences, Vol. 114, Part II, pp. 821–833, 9–13 February 2003. Abstract This paper presents the attitude sensor alignment and gyro calibration for the Thermosphere-Ionosphere-Mesosphere Energetics and Dynamics (TIMED) and demonstrates the effectiveness of an Alignment Kalman Filter and a composite calibration algorithm. The on-orbit attitude determination performance prior to and after calibration is presented. The optical alignment performed prior to launch is compared to the estimated alignment parameters. Various means of validating the alignment calibration are discussed.
"Square Root Quaternion Estimation", AIAA Paper No. 2002-4914, AIAA Guidance, Navigation, and Control Conference, Monterey, CA, August 2002.
"Attitude Determination Using the Bortz Equation", Paper No. AAS 01-310, AIAA/AAS Astrodynamics Specialist Conference, Quebec City, Quebec, Canada, 30 July – 2 August 2001, in Advances in the Astronautical Sciences, Vol. 1, Part 1, 2001, pp. 153–165. Abstract This paper didn't meet its objective and has been superceded by the journal paper "Rotation Vector in Attitude Estimation". Look in the journal paper section.
"A Factorization and Least-Squares Method for Multi-Pass Sensor Alignment Calibration", Paper No. AAS 01-151, AAS/AIAA Space Flight Mechanics Meeting, Santa Barbara, CA, 11–15 February 2001. Abstract This paper has been superceded by later developments.
"Everything is Relative in System Alignment Calibration", Paper No. AIAA-2000-4246, AIAA/AAS Astrodynamics Specialist Conference, August 2000. [CD, On-line] Abstract Much of the literature on alignment calibration is based on the concepts of absolute and relative alignment calibration. Three degrees of freedom of attitude associated with absolute alignment calibration are unobservable unless payload data is processed. We show in this paper that an absolute alignment model is equivalent to a relative alignment model when the payload is regarded as an attitude sensor. We show that the extra degrees of freedom are eliminated by defining the gyro as the body reference frame, attributing only three non-orthogonal misalignment parameters to the gyro. The payload misalignment can then be parameterized and calibrated in exactly 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 in this paper is illustrated via simulation results
"A Kalman Filter Approach to System Alignment Calibration", Paper No. AAS 00-127, AAS/AIAA Space Flight Mechanics Meeting, January 2000. Abstract Many spacecraft missions require precise pointing of their payload boresight and precise knowledge of the payload boresight attitude from ground processing of attitude sensor, gyro, and payload telemetry. Such precision is obtained at reasonable expense in part by system alignment calibration. This paper treats specifically the calibration of a spacecraft system comprising two star trackers, a gyro, and an imaging instrument via an Alignment Kalman Filter. This approach is not restricted to those specific sensors. An overview of the misalignment processes from assembly to launch to on-orbit operation is given and the concepts of relative and absolute alignment are defined and illustrated. Misalignment models and gyro error models are derived and their implementation in the Kalman filter is presented. Results demonstrate convergence and performance characteristics. The effect of miscalibration on a six-state on-orbit attitude estimation filter is also presented. A simple modification of the six-state filter that can be used in real-time to squash the effects of miscalibration is presented.
L. Haas and M. Pittelkau, "Real-Time High Accuracy GPS Onboard Orbit Determination for Use on Remote Sensing Satellites", ION GPS '99, Proceedings of the 12th International Technical Meeting of the Satellite Division of the Institute of Navigation, Nashville Convention Center, Nashville, TN, 14–17 September 1999, pp. 829–836.
A. Garber, L. Haas, M. Pittelkau, "Performance Evaluation of the GPS Onboard Orbit Determination Software to Ensure Improved Positioning Accuracy", Flight Mechanics Symposium, NASA/GSFC, NASA/CP-1999-209235, 18–20 May 1999, pp. 141–154.
"Comparison of Four Torque Distribution Methods for Attitude Control", Flight Mechanics Symposium, NASA/GSFC, NASA/CP-1999-209235, May 1999, pp. 253–267.
"Distortion and Alignment Calibration of a CCD-Based Fine Sun Sensor", Flight Mechanics and Estimation Symposium, NASA/GSFC, May 1997.
"Frequency Weighted LQG Control of Spacecraft Attitude", IEEE Conference on Control Applications, 13–16 September 1992, Vol. 1, pp. 336–341. Abstract A frequency-weighted Linear Quadratic Gaussian approach to control of pitch attitude and pitch wheel momentum of an Earth-orbiting spacecraft is presented. Attitude is measured by an Earth horizon sensor and cotnrol torque is produced by magnetic torquer bars and a pitch momentum or reqction wheel. Environmental disturbances are estimated for disturbance rejection. The design features control weighting so that low frequency control is effected by the torquer bars and high frequency control is effected by the pitch wheel. Control weighting is also used to provide a steep roll-off of the open-loop gain to achieve a wide bandwidth while avoiding control-structural interaction. A frequency response analysis and simulation results are provided.
"Parameter Adaptive Gun Fire Control", Proceedings of the Ninth Meeting of the Coordinating Group on Modern Control Theory, Army Mathematics Steering Committee, ARDEC, Picatinny Arsenal, Picatinny, NJ, 12 October 1989.
"Adaptive Load-Sharing Force Control for Two-Arm Manipulators", 1988 International Conference on Robotics and Automation, 24-29 April 1988, Vol. 1, pp. 498–503. (Philips Prize paper for best paper submitted by a graduate student, awarded by the North American Philips Corporation.)
"Reflex-Action Position Control of Robots: Comparison with Impedance Control", IEEE SouthEastCon '88, 11-13 April 1988, pp. 408–412.
"Decoupled Simulation Results for Two-Arm Robotic Systems", 1987 IEEE Southeastern Symposium on System Theory.
Scheeres, D.J.; Pittelkau, M.E.; Proulx, R.J.; Cangahuala, L.A.; (Editors) Spaceflight Mechanics 2003, Vol. 114, Parts I–III, Advances in the Astronautical Sciences, American Astronautical Society, 2003, 2500 pages. [Book, CD]
Segerman, A. M.; Lai, P. C.; Wilkins, M. P.; Pittelkau, M. E.; (Editors) Spaceflight Mechanics 2009, Vol 134, Parts I--III, Advances in the Astronautical Sciences, American Astronautical Society, 2009, 2287 pages. [Book, CD]
A frequency-weighted Linear Quadratic Gaussian approach to control of pitch attitude and pitch wheel momentum of an Earth-orbiting spacecraft is presented. Attitude is measured by an Earth horizon sensor and cotnrol torque is produced by magnetic torquer bars and a pitch momentum or reqction wheel. Environmental disturbances are estimated for disturbance rejection. The design features control weighting so that low frequency control is effected by the torquer bars and high frequency control is effected by the pitch wheel. Control weighting is also used to provide a steep roll-off of the open-loop gain to achieve a wide bandwidth while avoiding control-structural interaction. A frequency response analysis and simulation results are provided |