A Geometric Nonlinear Stochastic Filter for Simultaneous Localization and Mapping (2102.06150v1)

Abstract: 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.

• The paper introduces a geometric nonlinear stochastic filter that operates on a Lie group framework to simultaneously localize and map environments.
• It compensates for constant sensor biases and stochastic noise using a nonlinear estimation method based on stochastic differential equations.
• Simulation results demonstrate global asymptotic stability and SGUUB performance, underscoring its potential for autonomous navigation and terrain mapping.

Overview of Geometric Nonlinear Stochastic Filter for SLAM

The paper introduces a novel geometric nonlinear stochastic filter aimed at addressing the Simultaneous Localization and Mapping (SLAM) problem within robotics. SLAM entails mapping an unknown environment while simultaneously localizing a robot within it. This dual problem is fundamentally modeled on the Lie group of $\mathbb{SLAM}_{n}(3)$, which incorporates both angular and translational dynamics of the robot and the spatial distribution of landmark features. Notably, the proposed solution compensates for previously unaddressed challenges related to constant biases and stochastic noise in velocity measurements—a common issue in existing SLAM solutions.

The SLAM problem remains relevant due to the unreliability of absolute positioning systems, particularly in occluded environments, making it indispensable for applications like autonomous navigation, terrain mapping, and exploration. Most traditional filtering approaches, such as Gaussian and deterministic filters, fall short by either inadequately handling nonlinearity or lacking robustness against noise and biases present in sensor data.

Key Findings and Contributions

1. Nonlinear Stochastic Filter on Manifold: The algorithm designed operates directly on the geometric manifold of $\mathbb{SLAM}_{n}(3)$, effectively tracking both the pose and mapping environment under stochastic conditions. It ensures robust performance through incorporation of sensor measurements, including angular velocities, translational velocities, landmark data, and inertial measurement units (IMU).
2. Accounting for Biases and Noise: The filter innovatively compensates for constant biases and noise in velocity sensor measurements, elements typically neglected or oversimplified in previous SLAM solutions. The algorithm uses a structurally sound approach involving stochastic differential equations to enhance the robustness of localizing and mapping capabilities.
3. Performance Guarantee: Through simulation studies, the algorithm demonstrates the capacity for achieving global asymptotic stability and semi-global uniform ultimate boundedness (SGUUB) in mean square signal estimations. This robust convergence was verified despite initial errors and the stochastic nature of real-world environments.
4. Stochastic Nonlinear Estimator Design: The filter adapts a nonlinear stochastic structure for the first time in SLAM research, paving the way for potential extensions in not just landmark estimation but also comprehensive robot state estimation.

Implications and Future Directions

The integration of stochastic elements into SLAM is a forward step in addressing the unpredictability inherent in real-world robotic applications. The proposed method’s capability to handle noise and biases heightens both theoretical insights into manifold-based nonlinear estimation and practical outcomes in robotics and autonomous navigation.

Speculatively, future research may leverage this innovative approach to expand the use of Lie group techniques in broader AI systems, potentially enhancing machine perception, navigation, and interaction within dynamic environments. There is also room for exploring hybrid methods that fuse the benefits of deterministic and stochastic approaches for contexts where rapid response times are critical.

The findings in this paper contribute to the refinement of SLAM techniques by injecting stochastic realism into filter designs, thereby setting a precedent for continued exploration of advanced geometric and probabilistic methods in robotics.