A Unifying Approach to Quaternion Adaptive Filtering: Addressing the Gradient and Convergence

A novel framework for a unifying treatment of quaternion valued adaptive filtering algorithms is introduced. This is achieved based on a rigorous account of quaternion differentiability, the proposed I-gradient, and the use of augmented quaternion statistics to account for real world data with noncircular probability distributions. We first provide an elegant solution for the calculation of the gradient of real functions of quaternion variables (typical cost function), an issue that has so far prevented systematic development of quaternion adaptive filters. This makes it possible to unify the class of existing and proposed quaternion least mean square (QLMS) algorithms, and to illuminate their structural similarity. Next, in order to cater for both circular and noncircular data, the class of widely linear QLMS (WL-QLMS) algorithms is introduced and the subsequent convergence analysis unifies the treatment of strictly linear and widely linear filters, for both proper and improper sources. It is also shown that the proposed class of HR gradients allows us to resolve the uncertainty owing to the noncommutativity of quaternion products, while the involution gradient (I-gradient) provides generic extensions of the corresponding real- and complex-valued adaptive algorithms, at a reduced computational cost. Simulations in both the strictly linear and widely linear setting support the approach

A Sounding Rocket Attitude Determination Algorithm Suitable For Implementation Using Low Cost Sensors

Thesis (Ph.D.) University of Alaska Fairbanks, 2003The development of low-cost sensors has generated a corresponding movement to integrate them into many different applications. One such application is determining the rotational attitude of an object. Since many of these low-cost sensors are less accurate than their more expensive counterparts, their noisy measurements must be filtered to obtain optimum results. This work describes the development, testing, and evaluation of four filtering algorithms for the nonlinear sounding rocket attitude determination problem. Sun sensor, magnetometer, and rate sensor measurements are simulated. A quatenion formulation is used to avoid singularity problems associated with Euler angles and other three-parameter approaches. Prior to filtering, Gauss-Newton error minimization is used to reduce the six reference vector components to four quaternion components that minimize a quadratic error function. Two of the algorithms are based on the traditional extended Kalman filter (EKF) and two are based on the recently developed unscented Kalman filter (UKF). One of each incorporates rate measurements, while the others rely on differencing quaternions. All incorporate a simplified process model for state propagation allowing the algorithms to be applied to rockets with different physical characteristics, or even to other platforms. Simulated data are used to develop and test the algorithms, and each successfully estimates the attitude motion of the rocket, to varying degrees of accuracy. The UKF-based filter that incorporates rate sensor measurements demonstrates a clear performance advantage over both EKFs and the UKF without rate measurements. This is due to its superior mean and covariance propagation characteristics and the fact that differencing generates noisier rates than measuring. For one sample case, the "pointing accuracy" of the rocket spin axis is improved by approximately 39 percent over the EKF that uses rate measurements and by 40 percent over the UKF without rates. The performance of this UKF-based algorithm is evaluated under other-than-nominal conditions and proves robust with respect to data dropouts, motion other than predicted and over a wide range of sensor accuracies. This UKF-based algorithm provides a viable low cost alternative to the expensive attitude determination systems currently employed on sounding rockets

Adaptive filtering algorithms for quaternion-valued signals

Advances in sensor technology have made possible the recoding of three and four-dimensional signals which afford a better representation of our actual three-dimensional world than the ``flat view'' one and two-dimensional approaches. Although it is straightforward to model such signals as real-valued vectors, many applications require unambiguous modeling of orientation and rotation, where the division algebra of quaternions provides crucial advantages over real-valued vector approaches. The focus of this thesis is on the use of recent advances in quaternion-valued signal processing, such as the quaternion augmented statistics, widely-linear modeling, and the HR-calculus, in order to develop practical adaptive signal processing algorithms in the quaternion domain which deal with the notion of phase and frequency in a compact and physically meaningful way. To this end, first a real-time tracker of quaternion impropriety is developed, which allows for choosing between strictly linear and widely-linear quaternion-valued signal processing algorithms in real-time, in order to reduce computational complexity where appropriate. This is followed by the strictly linear and widely-linear quaternion least mean phase algorithms that are developed for phase-only estimation in the quaternion domain, which is accompanied by both quantitative performance assessment and physical interpretation of operations. Next, the practical application of state space modeling of three-phase power signals in smart grid management and control systems is considered, and a robust complex-valued state space model for frequency estimation in three-phase systems is presented. Its advantages over other available estimators are demonstrated both in an analytical sense and through simulations. The concept is then expanded to the quaternion setting in order to make possible the simultaneous estimation of the system frequency and its voltage phasors. Furthermore, a distributed quaternion Kalman filtering algorithm is developed for frequency estimation over power distribution networks and collaborative target tracking. Finally, statistics of stable quaternion-valued random variables, that include quaternion-valued Gaussian random variables as a special case, is investigated in order to develop a framework for the modeling and processing of heavy-tailed quaternion-valued signals.Open Acces