Dual Quaternions for Gravity Recovery Missions

A dual quaternion-based modeling, state estimation and control approach is introduced as a better alternative to the traditional methods which are currently utilized for gravity recovery missions. The proposed modeling and control approach was verified against and compared to the tangent bundle to Special Euclidean Group 3 through MATLAB simulations. The dual quaternion-based approach shows superior performance over traditional linearized and uncoupled methodologies, in both modeling accuracy of spacecraft translational position, and the ability to control the pose of a test mass relative to its host spacecraft. Utilizing data products from the Gravity Recovery and Climate Experiment Follow-On mission, a comparison of modeling methods is presented which demonstrates the advantage of the proposed dual quaternion-based modeling approach. In addition, a dual quaternion-based PD-type controller is designed and proven to control a test mass within specified gravitational reference sensor development requirements, and outperforms the traditional linear control technique. A dual quaternion-based extended Kalman-Bucy filter which remains in dual quaternion space was derived to filter the relative pose measurements and estimate the relative dual velocity between the spacecraft and test mass, which were utilized for the PD-type controller. Additional fidelity was added to the system by modeling the coupling between the spacecraft and the test mass, and voltage-to-force and-torque equations of the actuators with appropriate white noise added to the stepped commanded and measured signals. This work also presents two novel variations of the Kalman-Bucy filter, incorporating sensor measurements into the covariance matrix dynamics. The first formulation is an attitude quaternion-based formulation which provides quasi-optimal estimates of the relative attitude between two frames. The second formulation builds on the attitude quaternion-based filter by extending the results to dual quaternions, which allows the filter to provide quasi-optimal relative pose estimations. The utility of this filter is attributed to the fact the filter will estimate the relative attitude or pose between two frames provided the measurements of any two quantities as long as the quantities are measured in both frames. This filter is intended to improve the pose estimations of the formation flying spacecraft utilized in future gravity recovery missions. Furthermore, both formulations remain in quaternion and dual quaternion space which retains the computational efficiency of quaternion and dual quaternion algebra. Another major innovation of this research is a novel investigation into the use of newly-available relative angular acceleration measurements between a spacecraft utilized for gravity recovery missions and an internally located test mass. The gravity-gradient torque equation for spherical harmonic order $n$ is formulated for a known gravitational potential field, and through simulations it is proven that the Simplified-Gravitational Reference Sensor will be sensitive to the gravity-gradient induced torques acting on its test mass. This research then demonstrates how the presented gravity-gradient torque equation and the measured relative angular acceleration between the spacecraft and test mass will improve the accuracy of the gravity field models acquired by future gravity recovery missions by directly measuring the drag acting on the spacecraft with a single accelerometer. This novel research ultimately lays the groundwork and proves the need for dual quaternions on future gravity recovery missions.