Geometric Fault-Tolerant Neural Network Tracking Control of Unknown Systems on Matrix Lie Groups

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.

Fault-Tolerant Multi-Modal Localization of Multi-Robots on Matrix Lie Groups

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.

Fast and Modular Whole-Body Lagrangian Dynamics of Legged Robots with Changing Morphology

Fast and modular modeling of multi-legged robots (MLRs) is essential for resilient control, particularly under significant morphological changes caused by mechanical damage. Conventional fixed-structure models, often developed with simplifying assumptions for nominal gaits, lack the flexibility to adapt to such scenarios. To address this, we propose a fast modular whole-body modeling framework using Boltzmann-Hamel equations and screw theory, in which each leg's dynamics is modeled independently and assembled based on the current robot morphology. This singularity-free, closed-form formulation enables efficient design of model-based controllers and damage identification algorithms. Its modularity allows autonomous adaptation to various damage configurations without manual re-derivation or retraining of neural networks. We validate the proposed framework using a custom simulation engine that integrates contact dynamics, a gait generator, and local leg control. Comparative simulations against hardware tests on a hexapod robot with multiple leg damage confirm the model's accuracy and adaptability. Additionally, runtime analyses reveal that the proposed model is approximately three times faster than real-time, making it suitable for real-time applications in damage identification and recovery.