  Simultaneous Localization and Mapping (SLAM) is one of the key robotics tasks
as it tackles simultaneous mapping of the unknown environment defined by
multiple landmark positions and localization of the unknown pose (i.e.,
attitude and position) of the robot in three-dimensional (3D) space. The true
SLAM problem is modeled on the Lie group of $\mathbb{SLAM}_{n}\left(3\right)$,
and its true dynamics rely on angular and translational velocities. This paper
proposes a novel geometric nonlinear stochastic estimator algorithm for SLAM on
$\mathbb{SLAM}_{n}\left(3\right)$ that precisely mimics the nonlinear motion
dynamics of the true SLAM problem. Unlike existing solutions, the proposed
stochastic filter takes into account unknown constant bias and noise attached
to the velocity measurements. The proposed nonlinear stochastic estimator on
manifold is guaranteed to produce good results provided with the measurements
of angular velocities, translational velocities, landmarks, and inertial
measurement unit (IMU). Simulation and experimental results reflect the ability
of the proposed filter to successfully estimate the six-degrees-of-freedom (6
DoF) robot's pose and landmark positions. Keywords: Simultaneous Localization
and Mapping, nonlinear stochastic observer for SLAM, stochastic differential
equations, pose estimator, position, attitude, Brownian motion process,
inertial measurement unit, landmarks, features, SDE, SO(3), SE(3), SLAM.
