Lie Group Approach to Dynamic-Model-Aided Navigation of Multirotor Unmanned Aerial Vehicles

Sensor Fusion for Underwater Vehicle Navigation Compensating Misalignment Using Lie Theory

“…In Equation (13), p(t) = x(t) y(t) z(t) T represents the position of the vehicle which belongs in three-dimensional real vector space. R(q(t)) is the rotation matrix corresponding to the unit quaternion q(t).…”

“…The arithmetic operations on the attitude and misalignment, including differentiation, integration, and covariance calculations required to implement KF methodology are not trivial since attitude does not belong in a linear space [ 9 , 10 , 11 ]. Attitude and misalignment belong to nonlinear spaces, such as the 3-dimensional special orthogonal group and 3-sphere space [ 12 , 13 ]. The research [ 14 , 15 ] used usual operations on attitude assuming that linear approximation does not cause a big problem.…”

Three-Dimensional Dead-Reckoning Based on Lie Theory for Overcoming Approximation Errors

“…Commonly used Lie groups in robotics include the n-dimensional special orthogonal group (SO(n)) [28], which represents rotation transformations, the n-dimensional special unitary group (SU(n)), which plays an important role in quantum mechanics, and the special Euclidean group (SE(3)), which represents rotations and translations in three-dimensional space [29][30][31]. In addition, there is SE n (3) extended from SE(3) for rigid body transformations related to transformations or kinematic changes [26,[32][33][34].…”

Enhanced Autonomous Vehicle Positioning Using a Loosely Coupled INS/GNSS-Based Invariant-EKF Integration

“…The invariant extended Kalman filter (IEKF) is based on the symmetry-preserving observer theory which claims that the estimation error is invariant under the action of a Lie group matrix [23][24][25][26]. The main advantage of IEKF over EKF is that the state linearization and measurement models are independent of the current estimation of the state leading to state-independent Jacobians at any linearization point [27,28].…”