Continuous-time filtering in Lie groups: estimation via the Fr{\'e}chet mean of solutions to stochastic differential equations

We compute the Fr\'echet mean $\mathscr{E}_t$ of the solution $X_{t}$ to a continuous-time stochastic differential equation in a Lie group. It provides an estimator with minimal variance of $X_{t}$. We use it in the context of Kalman filtering and more precisely to infer rotation matrices. In this paper, we focus on the prediction step between two consecutive observations. Compared to state-of-the-art approaches, our assumptions on the model are minimal.