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Classical mathematical techniques such as discrete integration, gradientdescent optimization, and state estimation (exemplified by the Runge-Kuttamethod, Gauss-Newton minimization, and extended Kalman filter or EKF,respectively), rely on linear algebra and hence are only applicable to statevectors belonging to Euclidean spaces when implemented as described in theliterature. This document discusses how to modify these methods so they can beapplied to non-Euclidean state vectors, such as those containing rotations andfull motions of rigid bodies. To do so, this document provides an in-depthreview of the concept of manifolds or Lie groups, together with their tangentspaces or Lie algebras, their exponential and logarithmic maps, the analysis ofperturbations, the treatment of uncertainty and covariance, and in particularthe definitions of the Jacobians required to employ the previously mentionedcalculus methods. These concepts are particularized to the specific cases ofthe SO(3) and SE(3) Lie groups, known as the special orthogonal and specialEuclidean groups of R3, which represent the rigid body rotations and motions,describing their various possible parameterizations as well as their advantagesand disadvantages.

The SO(3) and SE(3) Lie Algebras of Rigid Body Rotations and Motions and  their Application to Discrete Integration, Gradient Descent Optimization, and  State Estimation