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Research Article Volume 11 Issue 1
On the analytical mechanics and nonlinear stability theory methods in mathematical modeling of the controlled dynamics parallel manipulators by incomplete state information
Aleksandr Ya Krasinskiy, Andrey A Yuldashev
Received: February 24, 2025 | Published: April 21, 2025
Citation: Krasinskiy AY, Yuldashev AA. On the analytical mechanics and nonlinear stability theory methods in mathematical modeling of the controlled dynamics parallel manipulators by incomplete state information . Int Rob Auto J. 2025;11(1):13-17. DOI: 10.15406/iratj.2025.11.00294
Abstract
Despite the fact that parallel kinematic manipulators are becoming more and more widespread due to their advantageous operational characteristics, modern methods of mathematical control theory have not yet found application in their control systems. A necessary condition for the these methods effective use is the availability of an adequate mathematical model in analytical form, obtained by rigorous methods and including the all components automatic system. Despite intensive research, there are currently no general methods for obtaining such models. Almost all known results are based on the inverse kinematics problem consideration, the which solution in general analytical form, as is known, is impossible. In the studies published to date on modeling the specific technical devices dynamics with parallel kinematics, modern information technologies for processing symbolic information are used to one degree or another. In this article, we develop the such rigorous methods application of theoretical non-free systems mechanics and nonlinear stability theory for nonlinear mathematical dynamics modeling of systems with geometric constraints, which in the general case eliminate the need to consider the inverse kinematics problem. The application of vector-matrix equations obtained in analytical form in redundant coordinates in the general stabilization problem provides the possibility of reducing the volume of measurement information by reducing the dimensionality of the controlled subsystem from which the stabilizing control is determined. The controlled subsystem does not contain dependent coordinates, therefore, to form a stabilizing control, only information on independent coordinates and their velocities is sufficient. The analytical form model of made it possible to specify in the work the observability conditions that create the possibility for further reducing the measurement information volume by forming a stabilizing control using an phase state estimate of the controlled subsystem obtained using the evaluation system constructed in the article. The coefficients of the stabilizing control and the parameters for the state estimate system are determined from the corresponding linear-quadratic problems solution by Krasovskii’s method.
Keywords: mathematical dynamics model, geometric constraints, measurement information, state estimate linear-quadratic problem
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