Geometric Fault-Tolerant Neural Network Tracking Control of Unknown Systems on Matrix Lie Groups
This work addresses fault-tolerant control for robotic and multi-agent systems on Lie groups, offering a novel geometric approach that is incremental in combining neural networks with existing geometric control frameworks.
The authors tackled the problem of tracking control for systems on matrix Lie groups with unknown dynamics, actuator faults, and disturbances by developing a geometric neural network-based controller that leverages Lie group properties to avoid singularities and enable global optimization. They demonstrated the method's effectiveness through simulation results for multi-agent formation control on the Special Euclidean group, establishing ultimate boundedness of all error signals using Lyapunov analysis.
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.