employee from 01.01.1997 to 01.01.2026
Irkutsk, Irkutsk region, Russian Federation
This article examines an applied model in the field of robotics, which is mathematically formalized as a degenerate optimal control problem with discontinuous trajectories. The analysis relies on first-order optimality conditions in the form of the generalized maximum principle and the Kelly conditions. The aim of the study is to analyze a degenerate optimal control problem for the motion of a two-link robotic manipulator with three degrees of freedom. The problem is characterized by a linear dependence of the robot’s dynamics on control inputs, which precludes the application of the classical Pontryagin maximum principle and requires a transition to impulse control theory. The object of study is a two-link manipulator robot, one of whose links is of variable length (telescopic). Its kinetic energy is described by a degenerate quadratic form of velocities. The equations of motion are derived from the Lagrangian equations and reduced to an affine system of differential equations with respect to control. The latter constitutes a so-called undercontrolled system, since the number of degrees of freedom in it exceeds the number of control inputs. For a qualitative analysis of the problem, methods of optimal impulse control theory are applied in the form of the maximum principle for generalized impulse processes and trajectories of limited variation. In the course of the study of the model, a synthesis of a nonlinear impulse control law for a robotic manipulator was obtained, taking energy conservation into account. Furthermore, it is proven that the structure of this control is characterized by the presence of two impulses. The solution obtained for the problem in the class of impulse processes demonstrates that, to achieve maximum speed in systems with a control deficit, a step-like change in velocities is characteristic. The results of this work can be applied in the design of energy-efficient control algorithms for high-speed industrial robots and medical manipulators.
ROBOTIC MANIPULATOR, OPTIMAL IMPULSE CONTROL, DISCONTINUOUS TRAJECTORIES, NECESSARY OPTIMALITY CONDITIONS, GENERALIZED MAXIMUM PRINCIPLE, PERFORMANCE OPTIMIZATION
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