# Rotation Identification in Geometric Algebra: Theory and Application to the Navigation of Underwater Robots in the Field

### Abstract

We report the derivation and experimental evaluation of a stable adaptive identifier to estimate rigid body rotations using rotors in Geometric Algebra (GA). This work is motivated by the need for in situ estimation of the alignment between sensors commonly used in underwater vehicle navigation. Here we derive an adaptive identifier using a geometric interpretation of the error to drive first-order rotor kinematics. We prove that it is Lyapunov stable, and we show that it is asymptotically stable in the presence of persistent excitation. We use the identifier to estimate the alignment between the Doppler velocity log sonar and the fiber optic gyrocompass used by underwater vehicles for dead reckoning (DR). We evaluate this method in the laboratory with a remotely operated vehicle (ROV), and then with an autonomous underwater vehicle (AUV) operating in the field at 1,200 m depth. Our results show that this technique reduces dead reckoning navigation errors on these platforms and provides comparable performance to previously reported SO(3) constrained Linear Algebra (LA) approaches. The rotor identifier has a number of advantages over these previously reported methods, including a more straightforward derivation, simpler gain tuning, increased computational efficiency, and reduced data manipulation.

Type
Publication
Journal of Field Robotics 32(5)
Date

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$y = R u \tilde{R}$