navlie: A Python Package for State Estimation on Lie Groups (2310.15774v1)

Abstract: The ability to rapidly test a variety of algorithms for an arbitrary state estimation task is valuable in the prototyping phase of navigation systems. Lie group theory is now mainstream in the robotics community, and hence estimation prototyping tools should allow state definitions that belong to manifolds. A new package, called navlie, provides a framework that allows a user to model a large class of problems by implementing a set of classes complying with a generic interface. Once accomplished, navlie provides a variety of on-manifold estimation algorithms that can run directly on these classes. The package also provides a built-in library of common models, as well as many useful utilities. The open-source project can be found at https://github.com/decargroup/navlie.

• The paper demonstrates that navlie simplifies robotic state estimation by efficiently integrating Lie group theory with practical algorithm frameworks.
• The package supports a variety of estimation algorithms, including EKF variants, Unscented Kalman Filter, batch estimation, and interacting multiple model filters.
• Its utility features, such as numerical differentiation and interpolation tools, enable rapid prototyping and seamless integration with complex sensor data.

A Technical Exposition of the navlie Python Package for On-Manifold State Estimation

The paper meticulously delineates the functionality and structure of navlie, a Python package developed to facilitate state estimation using Lie groups. The focus of this research work has been to provide an accessible, flexible, and profound tool for rapid algorithm prototyping in robotic navigation systems, specifically accommodating the manifold-based representations routinely encountered in such applications.

Core Functionality and Implementation

navlie is designed to enable both seasoned and novice researchers to implement manifold state estimation frameworks efficiently. It incorporates two fundamental components: support for a wide spectrum of on-manifold estimation algorithms and a comprehensive library of utilities to aid in their deployment. This approach integrates recent advancements in Lie group theory application within robotics, contributing to the ongoing transition of traditional state space representations to more complex manifold structures like $SO(3)$ and $SE(3)$.

Estimation Algorithms

The paper highlights several prominent on-manifold state estimation algorithms consolidated within navlie. These include:

• Extended Kalman Filter (EKF) variants, such as Iterated and Invariant EKF, which are tailored to handle nonlinear state-space models where system states reside on manifold structures rather than Euclidean space.
• Sigma-Point Filters, which offer non-linear transformations without assuming linear approximations—a prominent example being the Unscented Kalman Filter.
• Batch Estimation frameworks for smoothing applications, providing maximum a posteriori (MAP) estimates over trajectories.
• Interacting Multiple-Model Filter techniques, designed to manage systems with discrete mode changes by modeling these modes as a Markov process.

These implementations are made accessible via Python, distinguishing navlie from other libraries primarily implemented in C++, which, although optimized for real-time applications, pose barriers to algorithmic experimentation and prototyping.

On-Group Mathematical Structures

The theoretical framework employed within navlie capitalizes on the mathematical structures of Lie groups and algebras. The package supports various standard groups like $SO(n)$, $SE(n)$, and more intricate constructs such as $SE_2(3)$, facilitating operations including generalized addition and subtraction, essential for consistent handling of states on manifold structures. The paper also details the Jacobian calculation methods tailored for these operations—critical for executing filters like the EKF that necessitate such derivatives.

Utility Integration

The paper articulates the utility features of navlie that enhance its usability and application breadth:

• Numerical Differentiation Tools: Finite-difference and complex-step differentiation methodologies are integrated to derive accurate Jacobian matrices without requiring explicit analytical forms.
• Interpolation and Preintegration Methods: These tools support bespoke implementations such as interpolation on manifold states and preintegration for common robotic process models, including IMU and odometry systems, leveraging intrinsic $SE_2(3)$ representations.
• Composite State Handling: navlie allows for the composition of multiple state representations into a unified framework, simplifying the management of complex systems with multifaceted state dimensions.

Practical and Theoretical Implications

The navlie package heralds significant implications for both practical robotics applications and theoretical advancements in state estimation. Practically, it lowers the barrier for researchers to trial various estimation techniques in diverse contexts—from sensor fusion to complex SLAM problems—warming the transition of these methods from theoretical to real-world applications. Theoretically, navlie encourages exploration into novel estimation strategies employing Lie theory underpinnings, fostering innovation in robust state estimation against manifold representations.

Speculations on Future Directions

As on-manifold estimation solutions become increasingly pivotal in addressing the complex spatial challenges posed by autonomous systems, navlie could catalyze advancements in multi-sensor fusion and real-time dynamic systems control. Future developments may see extensions into more sophisticated dynamic modeling, seamless integration with machine learning for adaptive state estimation strategies, and parallel processing support to enhance real-time applicability.

In conclusion, navlie is presented as a pivotal advancement for state estimation on Lie groups, offering a robust, adaptable framework for addressing diverse algorithmic requirements in modern robotics, steadfastly supported by detailed mathematical and software engineering principles. The adoption and further development of such frameworks promise to significantly impact the efficiency and effectiveness of robotic state estimation methodologies in academia and industry alike.