F. Landis Markley

Survey of Nonlinear Attitude Estimation Methods

This paper provides a survey of modern nonlinear filtering methods for attitude estimation. Early applications relied mostly on the extended Kalman filter for attitude estimation. Since these applications, several new approaches have been developed that have proven to be superior to the extended Kalman filter. Several of these approaches maintain the basic structure of the extended Kalman filter, but employ various modifications in order to provide better convergence or improve other performance characteristics. Examples of such approaches include: filter QUEST, extended QUEST and the backwards-smoothing extended Kalman filter. Filters that propagate and update a discrete set of sigma points rather than using linearized equations for the mean and covariance are also reviewed. A twostep approach is discussed with a first-step state that linearizes the measurement model and an iterative second step to recover the desired attitude states. These approaches are all based on the Gaussian assumption that the probability density function is adequately specified by its mean and covariance. Other approaches that do not require this assumption are reviewed, Associate Professor, Department of Mechanical & Aerospace Engineering. Email: johnc@eng.buffalo.edu. Associate Fellow AIAA. Aerospace Engineer, Guidance, Navigation and Control Systems Engineering Branch. Email: Landis.Markley@nasa.gov. Fellow AIAA. Postdoctoral Research Fellow, Department of Mechanical & Aerospace Engineering. Email: cheng3@eng.buffalo.edu. Member AIAA.

Unscented Filtering for Spacecraft Attitude Estimation

A new spacecraft attitude estimation approach based on the unscented filter is derived. For nonlinear systems the unscented filter uses a carefully selected set of sample points to map the probability distribution more accurately than the linearization of the standard extended Kalman filter, leading to faster convergence from inaccurate initial conditions in attitude estimation problems. The filter formulation is based on standard attitude-vector measurements using a gyro-based model for attitude propagation. The global attitude parameterization is given by a quaternion, whereas a generalized three-dimensional attitude representation is used to define the local attitude error. A multiplicative quaternion-error approach is derived from the local attitude error, which guarantees that quaternion normalization is maintained in the filter. Simulation results indicate that the unscented filter is more robust than the extended Kalman filter under realistic initial attitude-error conditions.

Attitude Error Representations for Kalman Filtering

The quaternion has the lowest dimensionality possible for a globally nonsingular attitude representation. The quaternionmustobeyaunitnormconstraint,though,whichhasledtothedevelopmentofanextendedKalmane lter usingaquaternionfortheglobalattitudeestimateandathree-componentrepresentationforattitudeerrors.Various attitude error representations are considered for this multiplicative extended Kalman e lter, which incorporates a nonlinear, norm preserving quaternion reset operation. Second-order bias corrections are computed in this framework.

Predictive Filtering for Attitude Estimation Without Rate Sensors

A real-time predictive e lter is derived for spacecraft attitude estimation without the utilization of angular rate measurements from gyros. The formulation is shown using only attitude sensors (three-axis magnetometers, sun sensors,startrackers, etc. ).Thenew real-timenonlineare lter predicts the requiredtorquemodeling errorinput in ordertopropagatethespacecraftdynamicmodel,sothatmodelresponsesmatchthemeasuredvectorobservations. The real-time predictive e lter is used to estimate the attitude of the Solar, Anomalous, Magnetospheric Particle Explorer (SAMPEX) spacecraft. Results using this new algorithm indicate that the real-time predictive e lter accurately estimates the attitude of an actual spacecraft with the sole use of magnetometer sensor measurements.

Nonlinear Attitude Filtering Methods

This paper provides a survey of modern nonlinear filtering methods for attitude estimation. Early applications relied mostly on the extended Kalman filter for attitude estimation. Since these applications, several new approaches have been developed that have proven to be superior to the extended Kalman filter. Several of these approaches maintain the basic structure of the extended Kalman filter, but employ various modifications in order to provide better convergence or improve other performance characteristics. Examples of such approaches include: filter QUEST, extended QUEST, the backwards-smoothing extended Kalman filter, and the interlaced extended Kalman filter. Filters that propagate and update a discrete set of sigma points rather than using linearized equations for the mean and covariance are also reviewed. A two-step approach is discussed with a first-step state that linearizes the measurement model and an iterative second step to recover the desired attitude states. These approaches are all based on the Gaussian assumption that the probability density function is adequately specified by its mean and covariance. Other approaches that do not require this assumption are reviewed, including particle filters and a Bayesian filter based on a non-Gaussian, finite-parameter probability density function on SO(3). Finally, the predictive filter, nonlinear observers and adaptive approaches are shown. The strengths and weaknesses of the various approaches are discussed.

Optimal Variable-Structure Control Tracking of Spacecraft Maneuvers

An optimal control approach using variable-structure (sliding-mode) tracking for large angle spacecraft maneuvers is presented. The approach expands upon a previously derived regulation result using a quaternion parameterization for the kinematic equations of motion. This parameterization is used since it is free of singularities. The main contribution of this paper is the utilization of a simple term in the control law that produces a maneuver to the reference attitude trajectory in the shortest distance. Also, a multiplicative error quaternion between the desired and actual attitude is used to derive the control law. Sliding-mode switching surfaces are derived using an optimal-control analysis. Control laws are given using either external torque commands or reaction wheel commands. Global asymptotic stability is shown for both cases using a Lyapunov analysis. Simulation results are shown which use the new control strategy to stabilize the motion of the Microwave Anisotropy Probe spacecraft.

Minimum Model Error Approach for Attitude Estimation

In this paper, an optimal batch estimator and smoother based on the Minimum Model Error (MME) approach is developed for three-axis stabilized spacecraft. The formulation described in this paper is shown using only attitude sensors (e.g., three-axis magnetometers, sun sensors, star trackers, etc). This algorithm accurately estimates the attitude of a spacecraft, and substantially smoothes noise associated with attitude sensor measurements. The general functional form of the optimal estimation approach involves the solution of a nonlinear two-point-boundary-valueproblem that can only be solved using computational intense methods. A linearized solution is also shown that is computationally more efficient than methods which solve the general form. The linearized solution is useful when an a priori estimate of the angular velocity is already known, which may be obtained from a finite difference of a determined attitude, or from propagation of a dynamics model. Results using this new algorithm indicate that an MME-based approach accurately estimates the attitude of an actual spacecraft with the sole use of magnetometer sensor measurements. 1 Assistant Professor, Catholic University of America, Dept. of Mech. Eng., Washington, D.C. 20064. Member AIAA. 2 Staff Engineer, Goddard Space Flight Center, Code 712, Greenbelt, MD 20771. Associate Fellow AIAA

Efficient and optimal attitude determination using recursive global positioning system signal operations

In this paper, a new and efficient algorithm is developed for attitude determination from Global Positioning System signals. The new algorithm is derived from a generalized nonlinear predictive filter for nonlinear systems. This uses a one time-step ahead approach to propagate a simple kinematics model for attitude determination. The advantages of the new algorithm over previously developed methods include: it provides optimal attitudes even for coplanar baseline configurations; it guarantees convergence even for poor initial conditions; it is a non-iterative algorithm; and it is computationally efficient. These advantages clearly make the new algorithm well suited to on-board applications. The performance of the new algorithm is tested on a dynamic hardware simulator. Results indicate that the new algorithm accurately estimates the attitude of a moving vehicle, and provides robust attitude estimates even when other methods, such as a linearized least-squares approach, fail due to poor initial starting conditions.

Global Positioning System Integer Ambiguity Resolution Without Attitude Knowledge

Inthispaper,anewmotion-basedalgorithmforglobalpositioningsystemintegerambiguityresolutionisderived. Thealgorithmrepresentstheglobalpositioningsystem sightlinevectorsinthebodyframeasthesumoftwovectors, onedependingonthephasemeasurementsandtheotherontheunknownintegers.Thevectorcontainingtheinteger phases is found using a procedure developed to solve for magnetometer biases. In addition to a batch solution, this paper also provides a sequential estimate, so that a suitable stopping condition can be found during the vehicle motion. The newalgorithm has several advantages: it doesnot requirean a prioriestimateofthevehicle’ s attitude; it provides an inherent integrity check using a covariance-type expression; and it can sequentially estimate the ambiguitiesduring thevehiclemotion. Itsonly disadvantageisthatit requiresatleastthreenoncoplanar baselines. The performance of the new algorithm is tested on a dynamic hardware simulator.

Attitude estimation for large field-of-view sensors

The QUEST measurement noise model for unit vector observations has been widely used in spacecraft attitude estimation for more than twenty years. It was derived under the approximation that the noise lies in the tangent plane of the respective unit vector and is axially symmetrically distributed about the vector. For large field-of-view sensors, however, this approximation may be poor, especially when the measurement falls near the edge of the field of view. In this paper a new measurement noise model is derived based on a realistic noise distribution in the focal plane of a large field-of-view sensor, which shows significant differences from the QUEST model for unit vector observations far away from the sensor boresight. An extended Kalman filter for attitude estimation is then designed with the new measurement noise model. Simulation results show that with the new measurement model the extended Kalman filter achieves better estimation performance using large field-of-view sensor observations.