This letter extends the exactly sparse Gaussian variational inference (ESGVI) algorithm for state estimation in two complementary directions. First, ESGVI is generalized to operate on matrix Lie groups, enabling the estimation of states with orientation components while respecting the underlying group structure. Second, factors are introduced to accommodate heavy-tailed and skewed noise distributions, as commonly encountered in ultra-wideband (UWB) localization due to non-line-of-sight (NLOS) and multipath effects. Both extensions are shown to integrate naturally within the ESGVI framework while preserving its sparse and derivative-free structure. The proposed approach is validated in a UWB localization experiment with NLOS-rich measurements, demonstrating improved accuracy and comparable consistency. Finally, a Python implementation within a factor-graph-based estimation framework is made open-source (https://github.com/decargroup/gvi_ws) to support broader research use.

Neural Information Processing Systems http://nips.cc/

We present a geometric neural network-based tracking controller for systems evolving on matrix Lie groups under unknown dynamics, actuator faults, and bounded disturbances. Leveraging the left-invariance of the tangent bundle of matrix Lie groups, viewed as an embedded submanifold of the vector space $\R^{N\times N}$, we propose a set of learning rules for neural network weights that are intrinsically compatible with the Lie group structure and do not require explicit parameterization. Exploiting the geometric properties of Lie groups, this approach circumvents parameterization singularities and enables a global search for optimal weights. The ultimate boundedness of all error signals -- including the neural network weights, the coordinate-free configuration error function, and the tracking velocity error -- is established using Lyapunov's direct method. To validate the effectiveness of the proposed method, we provide illustrative simulation results for decentralized formation control of multi-agent systems on the Special Euclidean group.

Consistent localization of cooperative multi-robot systems during navigation presents substantial challenges. This paper proposes a fault-tolerant, multi-modal localization framework for multi-robot systems on matrix Lie groups. We introduce novel stochastic operations to perform composition, differencing, inversion, averaging, and fusion of correlated and non-correlated estimates on Lie groups, enabling pseudo-pose construction for filter updates. The method integrates a combination of proprioceptive and exteroceptive measurements from inertial, velocity, and pose (pseudo-pose) sensors on each robot in an Extended Kalman Filter (EKF) framework. The prediction step is conducted on the Lie group $\mathbb{SE}_2(3) \times \mathbb{R}^3 \times \mathbb{R}^3$, where each robot's pose, velocity, and inertial measurement biases are propagated. The proposed framework uses body velocity, relative pose measurements from fiducial markers, and inter-robot communication to provide scalable EKF update across the network on the Lie group $\mathbb{SE}(3) \times \mathbb{R}^3$. A fault detection module is implemented, allowing the integration of only reliable pseudo-pose measurements from fiducial markers. We demonstrate the effectiveness of the method through experiments with a network of wheeled mobile robots equipped with inertial measurement units, wheel odometry, and ArUco markers. The comparison results highlight the proposed method's real-time performance, superior efficiency, reliability, and scalability in multi-robot localization, making it well-suited for large-scale robotic systems.

Classical adversarial attacks are phrased as a constrained optimisation problem. Despite the efficacy of a constrained optimisation approach to adversarial attacks, one cannot trace how an adversarial point was generated. In this work, we propose an algebraic approach to adversarial attacks and study the conditions under which one can generate adversarial examples for post-hoc explainability models. Phrasing neural networks in the framework of geometric deep learning, algebraic adversarial attacks are constructed through analysis of the symmetry groups of neural networks. Algebraic adversarial examples provide a mathematically tractable approach to adversarial examples. We validate our approach of algebraic adversarial examples on two well-known and one real-world dataset.

Matrix Lie groups provide a language for describing motion in such fields as robotics, computer vision, and graphics. When using these tools, we are often faced with turning infinite-series expressions into more compact finite series (e.g., the Euler-Rodriques formula), which can sometimes be onerous. In this paper, we identify some useful integral forms in matrix Lie group expressions that offer a more streamlined pathway for computing compact analytic results. Moreover, we present some recursive structures in these integral forms that show many of these expressions are interrelated. Key to our approach is that we are able to apply the minimal polynomial for a Lie algebra quite early in the process to keep expressions compact throughout the derivations. With the series approach, the minimal polynomial is usually applied at the end, making it hard to recognize common analytic expressions in the result. We show that our integral method can reproduce several series-derived results from the literature.

Neural Information Processing Systems http://nips.cc/

This paper presents an invariant Rauch-Tung- Striebel (IRTS) smoother applicable to systems with states that are an element of a matrix Lie group. In particular, the extended Rauch-Tung-Striebel (RTS) smoother is adapted to work within a matrix Lie group framework. The main advantage of the invariant RTS (IRTS) smoother is that the linearization of the process and measurement models is independent of the state estimate resulting in state-estimate-independent Jacobians when certain technical requirements are met. A sample problem is considered that involves estimation of the three dimensional pose of a rigid body on SE(3), along with sensor biases. The multiplicative RTS (MRTS) smoother is also reviewed and is used as a direct comparison to the proposed IRTS smoother using experimental data. Both smoothing methods are also compared to invariant and multiplicative versions of the Gauss-Newton approach to solving the batch state estimation problem.

This paper presents a solution for the state estimation and control problems for a class of unconventional vertical takeoff and landing (VTOL) UAVs operating in forward-flight conditions. A tightly-coupled state estimation approach is used to estimate the aircraft navigation states, sensor biases, and the wind velocity. State estimation is done within a matrix Lie group framework using the Invariant Extended Kalman Filter (IEKF), which offers several advantages compared to standard multiplicative EKFs traditionally used in aerospace and robotics problems. An SO(3)- based attitude controller is employed, leading to a single attitude control law without a separate sideslip control loop. A control allocator is used to determine how to use multiple, possibly redundant, actuators to produce the desired control moments. The wind velocity estimates are used in the attitude controller and the control allocator to improve performance. A numerical example is considered using a sample VTOL tailsitter-type UAV with four control surfaces. Monte-Carlo simulations demonstrate robustness of the proposed control and estimation scheme to various initial conditions, noise levels, and flight trajectories.

This work presents a novel approach for the optimization of dynamic systems on finite-dimensional Lie groups. We rephrase dynamic systems as so-called neural ordinary differential equations (neural ODEs), and formulate the optimization problem on Lie groups. A gradient descent optimization algorithm is presented to tackle the optimization numerically. Our algorithm is scalable, and applicable to any finite dimensional Lie group, including matrix Lie groups. By representing the system at the Lie algebra level, we reduce the computational cost of the gradient computation. In an extensive example, optimal potential energy shaping for control of a rigid body is treated. The optimal control problem is phrased as an optimization of a neural ODE on the Lie group SE(3), and the controller is iteratively optimized. The final controller is validated on a state-regulation task.