HIS report reviews the methods of Kalman filtering in attitude estimation and their development over the last two decades. This review is not intended to be complete but is limited to algorithms suitable for spacecraft equipped with three-axis gyros as well as attitude sensors. These are the systems to which we feel that Kalman filtering is most ap- plicable. The Kalman filter uses a dynamical model for the time development of the system and a model of the sensor measurements to obtain the most accurate estimate possible of the system state using a linear estimator based on present and past measurements. It is, thus, ideally suited to both ground-based and on-board attitude determination. However, the applicability of the Kalman filtering technique rests on the availability of an accurate dynamical model. The dynamic equations for the spacecraft attitude pose many difficulties in the filter modeling. In particular, the external torques and the distribution of momentum internally due to the use of rotating or rastering instruments lead to significant uncertainties in the modeling. For autonomous spacecraft the use of inertial reference units as a model replacement permits the circumvention of these problems. In this representation the angular velocity of the spacecraft is obtained from the gyro data. The kinematic equations are used to obtain the attitude state and this is augmented by means of additional state-vector components for the gyro biases. Thus, gyro data are not treated as observations and the gyro noise appears as state noise rather than as observation noise. It is theoretically possible that a spacecraft is three-axis stabilized with such rigidity that the time development of the system can be described accurately without gyro information, or that it is one-axis stabilized so that only a single gyro is needed to provide information on the time history of the system. The modification of the algorithms presented here in order to apply to those cases is slight. However, this is of little practical importance because a control system capable of such

Kalman Filtering for Spacecraft Attitude Estimation

Adetailed survey ispresentedoftheliteratureonattituderepresentationdatingfromtheearlyworkofEulerand Hamilton to recent publications in e elds such as navigation and control. The scope is limited to the development of the aircraft kinematic transformation equations in terms of four different attitude representations, including the well-known Euler angles, the Euler-axis rotation parameters, the direction cosines, and the Euler ‐Rodrigues quaternion.Theemphasisisdirectedattheapplicationofthequaternionformulationtoaircraftkinematics.Results are presented that reinforce observations that the quaternion formulation, typically implemented to eliminate singularities associated with the Euler angle formulation, is far superior to the other commonly used formulations based on computational efe ciency alone. A development of quaternion constraints necessary to independently constrain roll, pitch, yaw, bank angle, elevation angle, and/or azimuth angle is presented. For verie cation of simulation codes, a general closed-form solution to the quaternion formulation, for the case of constant rotation, is also presented. Additionally, a discussion is provided of numerical integration methods and numerical errors for the quaternion formulation. This discussion is especially important for simulations that may still utilize a common error reduction scheme originally developed for analog computers.

Review of Attitude Representations Used for Aircraft Kinematics

Several schemes in current use for sequential estimation of spacecraft attitude using Kalman filters are examined. These differ according to their treatment of the attitude error, namely: using the complete four-component quaternion; using a truncated quaternion in which one of the components has been eliminated; or using a quaternion referred to approximate body-fixed axes. These schemes are examined for the case of a spacecraft carrying line-of-sight attitude sensors and three-axis gyros whose measurements are corrupted by noise on both the drift rate and the drift-rate ramp. The analysis of the covariance is carried out in detail. The historical development of Kalman filtering of attitude is reviewed.

Kalman filtering for spacecraft attitude estimation

The Delft Parallel Processor is a multiple instructions/multiple data stream (MIMD) parallel computer. The system consists of eight processing elements, which can operate in parallel. The Delft Parallel Processor has been designed and built in a pure university environment (Delft University of Technology/Department of Applied Physics) and became operational in the spring of 1981. Since that time the parallel processor is used for different purposes, such as research with respect to parallel algorithms, parallel computation in problems solving and interactive parallel systems simulation. In this paper the Delft Parallel Processor is considered from a simulation point of view. The simulation — oriented set-up of the computer and the facilities for interactive modelling and experimentation on models are briefly discussed. Several aspects of simulation applications are dealt with, such as simulation of continuous systems, studying coupled excitable units, experimentation on sets of models and simulation of discrete event systems. As to continuous systems simulation, the relevant topic to be treated is the parallel implementation of differential equations; the attention is focussed on the equations segmentation method and its implementation in the Delft Parallel Processor. Simulation studies, that include the coupling of excitable units or the experimentation on sets of different models, have urged us to consider respectively the linking and retrieval of preprogrammed (sub) models. With respect to the simulation of discrete event systems we discuss the use of parallel random number generators, which have been incorporated in the Delft Parallel Processor. time. Vertically, respectively the number of ships in the system, the contents in hectoliters of an arriving oil-tanker, the contents in hectoliters of the oil-tank and the consumption in hectoliters of the oil refining plant are plotted.

The delft parallel processor in a simulation environment

Large-scale and complex simulation studies require the use of advanced information processing (AIP) techniques, such as number crunching facilities and database-management capabilities. Recently there is an increasing interest to incorporate another AIP-technique in simulation: artificial intelligence (in particular its applied branch “knowledge engineering”). Without having the pretention to be complete, in this paper we try to survey AIP in simulation.

Advanced simulation: Advanced data/knowledge-processing

The objective of this paper is to develop a new and more accurate local linearization algorithm for numerically solving sets of linear time-varying differential equations. Of special interest is the application of this algorithm to the quaternion rate equations. The results are compared, both analytically and experimentally, with previous results using local linearization methods. The new algorithm requires approximately one-third more calculations per step than the previously developed local linearization algorithm; however, this disadvantage could be reduced by using parallel implementation. For some cases the new algorithm yields significant improvement in accuracy, even with an enlarged sampling interval. The reverse is true in other cases. The errors depend on the values of angular velocity, angular acceleration, and integration step size. One important result is that for the worst case the new algorithm can guarantee eigenvalues nearer the region of stability than can the previously developed algorithm.

Improved Local Linearization Algorithm for Solving the Quaternion Equations

Algorithms are presented which reduce the computation time per integration step compared to the more common methods. This is accomplished by using only algorithms which permit the simultaneous calculation of x and f(x) where x = f(x). These results are important not only in the field of digital simulation, but also in realizing dynamic controllers/estimators digitally, e.g., realizing a nonlinear observer.

Kenneth Yen

Papers

Improved Local Linearization Algorithm for Solving the Quaternion Equations

Digital Simulation Algorithms Using Parallel Processing