On the analytical mechanics methods in mathematical modeling the dynamics systems with geometric constraints

Aleksandr Ya  Krasinskiy

doi:10.59400/mea2512

On the analytical mechanics methods in mathematical modeling the dynamics systems with geometric constraints

Aleksandr Ya Krasinskiy Department of Computer Modeling and Probability Theory, Moscow Aviation Institute, National Research University, 125993 Moscow, Russia; Information and Computing Center, Rosbiotech, 125080 Moscow, Russia

Article ID: 2512

DOI: https://doi.org/10.59400/mea2512

Keywords: mathematical dynamics model; geometric constraints; stabilization

Abstract

A necessary condition for the most effective application of the mathematical control theory results to the modern automatic devices dynamics сonsideration is the presence of an adequate nonlinear mathematical model obtained by strict general methods. Methods for reducing the dimensions of dynamic models of systems with geometric constraints by analytical mechanics methods for non-free systems are considered due to the transition to equations in redundant coordinates free of constraint multipliers. A detailed algorithm for this procedure and its justification is given. Using the theory of critical cases, a complete solution is given to the stabilizing problem of a given configuration of systems with geometric constraints.

Published

2025-05-08

How to Cite

Krasinskiy, A. Y. (2025). On the analytical mechanics methods in mathematical modeling the dynamics systems with geometric constraints. Mechanical Engineering Advances, 3(2), 2512. https://doi.org/10.59400/mea2512

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Vol. 3 No. 2 (2025)

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This work is licensed under a Creative Commons Attribution 4.0 International License.

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This manuscript presents a sophisticated deep neural networks (DNNs)-driven adaptive control paradigm for concurrently regulating the attitude and suppressing structural oscillations of a flexible spacecraft in a fully three-dimensional domain. By leveraging Hamilton’s principle, the spacecraft’s motion is formulated as an infinite-dimensional dynamic model described by partial differential equations, capturing the subtle interactions between rigid-body rotational maneuvers and flexible panel deformations. In contrast to traditional schemes, the proposed control methodology integrates a DNNs module to compensate for uncertain actuator anomalies and external input disturbances in real time, thereby ensuring fault tolerance under arbitrary, potentially unbounded actuator malfunctions. A rigorously constructed Lyapunov-based stability analysis corroborates that the system’s energy, angular rates, and transverse deflections remain uniformly bounded and asymptotically converge to zero, even in the face of multiple actuator failures. This theoretical guarantee stems from the synergistic interplay between the network’s representational power and the adaptive control law’s robust learning capabilities. Extensive computational experiments demonstrate the efficacy of the developed framework in orchestrating high-precision attitude stabilization while simultaneously mitigating detrimental vibrations, showcasing the superior performance and resiliency of the proposed DNNs-infused control architecture.

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