Quaternions and Dual Quaternions for State and Parameter Estimation
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This thesis presents new results on the use of quaternions and dual quaternions for the estimation of position and orientation using cameras and inertial sensors. The unit quaternion is a four-parameter representation of orientation, while the unit dual quaternion is a dual extension of the unit quaternion which represents pose with eight parameters, where pose is the combined description of position and orientation. Previously published results on pose estimation with dual quaternions are extended in three different solutions for state estimation. In the first solution, pose measurements are obtained by finding the rigid displacement between consecutive point clouds from a 3D camera. The pose measurements are converted to unit dual quaternions, which are filtered with the system dynamics using a multiplicative extended Kalman filter. In the second solution, a moving horizon estimator for attitude estimation with quaternions is extended to a moving horizon estimator for pose estimation with dual quaternions. In the third solution, a particle filter for pose estimation with dual quaternions is proposed. Dual modified Rodrigues parameters are used as a local representation of pose with six parameters, whereas unit dual quaternions are used for global representation. Experimental results are presented, where the motion of a swinging crane payload was estimated with six degrees-of-freedom using a 3D camera. Parameter estimation with unit quaternions is addressed for the estimation of static orientations in dynamic systems. To this end, a Riemannian gradient descent algorithm formulated with unit quaternions as a Lie group is proposed, and gradients with respect to orientation for quaternion functions are derived. The methods are demonstrated for a crane payload system, where the motion of the payload is estimated using a quaternion particle filter. A sensor system consisting of an inertial measurement unit and a 2D camera with a fiducial marker is considered, where the extrinsic orientation and position of the sensors are required in the sensor models. The extrinsic orientations are estimated simultaneously with the system states using the proposed Riemannian gradient descent method. The final contribution of this thesis addresses the discrete integration of the quaternion kinematics, which is used for the discrete implementation of quaternion state estimators and computer simulations. The Lie group integrator of Runge-Kutta-Munthe-Kaas is formulated for the quaternion kinematics. This integrator is more computationally efficient than the previously proposed Crouch-Grossman integrator, and preserves the unit constraint of the quaternion by construction, as opposed to standard Runge-Kutta methods.