Математичне моделювання побудови траєкторій руху захватного пристрою колаборативного робота

The object of the study is the process of constructing and analyzing the trajectories of the gripping device of a collaborative robot-manipulator under spatial constraints and the presence of obstacles in a dynamic environment. The subject of the study is mathematical models, algorithmic and software for modeling the optimal motion of the manipulator end effector taking into account kinematic, dynamic and energy constraints. The aim of the research is to construct trajectories of the collaborative robot's gripping device, taking into account constraints and optimal control actions in continuous time, which ensure the construction of trajectories with minimal energy consumption, compliance with given spatial constraints, and avoidance of collisions with obstacles. The research methodology is based on the application of the Pontryagin maximum principle to form the conditions for optimal control and the construction of a system of differential equations with boundary conditions. A special cost functional has been developed to quantify energy consumption and take into account penalties for approaching prohibited zones. The numerical solution of the problem was implemented using the Euler method, and the optimization of the trajectory parameters with fixed final effector coordinates was implemented using the least squares method with constraints. The Python programming language and the Matplotlib library were used to visualize the results. As a result of the study, optimal trajectories of the gripping device were obtained, which ensure collision avoidance, compliance with spatial constraints, and reduced energy consumption when reaching the specified final effector positions. The simulation confirmed the effectiveness of the developed method and its resistance to changes in environmental parameters. The conclusions of the study indicate that the proposed approach allows for a comprehensive solution to the problem of planning the movement of collaborative robots in the optimal control mode taking into account constraints. The results obtained can be applied in Industry 5.0 production systems, robotic service complexes, automated warehouse systems, and robots that interact with humans in a limited space.