Attitude Determination Kalman Filter with a 1/f Flicker Noise Gyro Model

M.E. Pittelkau

Abstract:	BACKGROUND Flicker noise processes have a power spectral density (PSD) that decreases as 1/f^gamma, where f is frequency and 0 < gamma < 2. The PSD thus has constant slope -gamma. Flicker noise processes exhibit infinitely long memory as well as short-term fluctuations. Flicker noise is stationary for gamma < 1 and nonstationary for gamma >= 1. For convenience we will consider gamma = 1 (pink noise). In this case, the power spectral density, in units of power per Hz, is inversely proportional to frequency and there is equal energy or power in each decade of frequency. The Allan Variance of 1/f noise is constant with integration time. Flicker noise generally sets the noise floor of gyros and accelerometers, and therefore is a limiting factor in the performance of attitude determination and navigation systems. In reference to gyros, rate flicker noise is also known as rate bias instability. (We have to be careful that we don't confuse bias with bias instability, which are both called drift.) Angle flicker noise (gamma = -1), if present, can be regarded as integrated rate flicker noise. To date, attitude determination filters and navigation filters have used a rate random walk or a first-order correlated (Gauss-Markov) noise model for the bias instability and therefore are suboptimal. In addition, gyro noise is typically simulated with only quantization, angle random walk, rate random walk or a first-order correlated (Gauss-Markov) noise, and a fixed bias. Attitude determination and navigation system performance determined via simulation can be optimistic if flicker noise is omitted from the filter and sensor models. This is particularly true during long intervals when an attitude update is not available, which occurs on geosynchronous spacecraft. Several space-qualified gyros are available with a range of gyro drift stability performance. The Northrop-Grumman SIRU is a high-performance space-qualified IMU containing four Hemispherically Resonating Gyros with a bias instability of 0.001 deg/hr. Medium-performance space-qualified gyros include the LN200S (about 0.015 deg/hr bias stability), the 2.5" Honeywell IFOG (0.005 deg/hr bias stability), and the European SAGEM DS HRG (0.01 deg/hr navigation grade, 0.1 to 1.0 deg/hr tactical grade). MEMS gyros are attractive for their very low cost, mass, volume, and power. MEMS gyros are almost ubiquitous in a variety of low-performance applications including nanosats, cubesats, missiles, piloted aircraft, UAVs, munitions, automobiles, robotics, personal navigation, hobby aircraft, and toys. MEMS gyros are currently being used on nanosats and cubesats simply because there is no other gyro that has the necessary low mass and power. The bias instability of MEMS gyros is large (ranging from 0.5 to 35 deg/hr), but it may be several years before MEMS gyros with bias instability less than 0.1 deg/hr become available. Most small spacecraft require bias stability less than 0.1 deg/hr. Bias stability in the range of 0.01 to 0.001 deg/hr is needed to achieve medium to high performance attitude determination and navigation. OBJECTIVES A conventional attitude determination filter models gyro drift as a random walk, and performance analysis via simulation also uses a random walk model. However, gyro drift is typically dominated by flicker noise. We investigate the feasibility and benefit of using a flicker noise model in an attitude determination Kalman filter. We will examine, in particular, the estimability of the flicker noise states. METHODOLOGY An overview of flicker noise characteristics, modeling, and simulation is given, and two linear time-invariant models that approximate flicker noise are provided. One of the flicker noise models has been investigated for use in estimation of time in ensemble atomic clocks, where the clock drift is modeled as flicker noise. A detailed gyro error simulator also incorporates a flicker noise model Allan variance plots show the validity of the gyro noise model. This more realistic gyro simulator is used to evaluate performance of the attitude determination filter augmented with the flicker noise model. The augmented filter is compared with an attitude determination filter that is based on the random walk drift model. ANTICIPATED AND ACTUAL RESULTS An attitude determination Kalman filter that incorporates a model of flicker noise is expected to yield greater performance than a conventional attitude determination filter, particularly when a low-cost gyro with poor drift stability is used, as occurs when using a MEMS gyro. Preliminary filter covariance results show that the higher-frequency states in the flicker model are not estimable. This is in accordance with an estimability criterion defined by Baram & Kailath (IEEE TAC-33-12 1988) and Baram & Shaked (IEEE TAC-30-5 1985). The effect of attitude sensor accuracy and update rate on estimability will be investigated. The estimation error for the augmented filter will be compared to that of the conventional filter to assess the suboptimality fo the conventional filter as well as the error in performance analyses that use only a random walk drift model. CONCLUSION AND SIGNIFICANCE The use of a flicker noise model in an attitude determination filter could lead to improved performance compared to a conventional filter that uses only a random walk drift model. Similarly, the use of a flicker noise model in a gyro simulator yields a more accurate performance assessment. This paper will lay the groundwork for investigating the use of a flicker noise model in accelerometers as well as gyros in an airborne, underwater, or terrestrial navigation system, particularly when position fixes (GPS updates or other) are intermittent. Applications also include orbit determination for spacecraft whose orbits extend well beyond the receiving range of GPS signals. There are also implications of estimability of flicker states in ensemble time estimates based on time measurements from atomic clocks.
Published in:	Proceedings of the 26th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2013) September 16 - 20, 2013 Nashville Convention Center, Nashville, Tennessee Nashville, TN
Pages:	2143 - 2160
Cite this article:	Pittelkau, M.E., "Attitude Determination Kalman Filter with a 1/f Flicker Noise Gyro Model," Proceedings of the 26th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2013), Nashville, TN, September 2013, pp. 2143-2160.
Full Paper:	ION Members/Non-Members: 1 Download Credit Sign In