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Application of numerical approximation methods in control systems which describe movement of autonomous vehicles in lane free roads

Tzitzikopoulos Nikolaos-Marinos

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Summary

Automobiles have changed people's everyday life and have become essential for the personal transportation of millions of people. Advancements in technology are growing and enhancing the driver's experience, from automatic headlights to automatic emergency braking and autonomous driving, nowadays. Autonomous driving on lane-free roads is a complex system where the vehicles have to be “connected” with each other and “cooperate” to perform their movement with safety. Usually, this kind of problems consists of non-linear or stiff differential equations which cannot be solved analytically, thus in this thesis we utilize numerical approximation methods to investigate a proposed system of cooperative autonomous vehicles in lane-free roads [ ] to observe the simulation's results along with the system's control functions. However, such complex systems describing autonomous vehicles driving on lane-free roads tend to be a challenge for numerical approximation methods, where high order Runge-Kutta methods may not be applicable while low order Runge-Kutta methods may present numerical instability if the initial step size is not sufficiently small. To achieve our goals, we analyze the system's characteristics and utilize a variety of numerical approximation methods to observe the vehicle's behavior on lane-free roads, as also their results, and make comparisons between them and their errors. Furthermore, we utilize an adaptive step size control in order to maintain the numerical solutions inside a defined open set, as also have the advantage of increasing and decreasing the step size, depending on the behavior of the system at any given moment. Following, we will use a Lyapunov function of the system, which represents the energy developed between the vehicles as the step size evaluator at an adaptive numerical method, in contrast to the regular adaptive methods which utilize the vehicle's characteristics, their lateral and longitudinal positions, their velocities and their wheel orientations. Lastly, we will try to investigate certain repulsive potential functions of the system, bound to keep the integrity of the vehicles hoping for smoother and more desired trajectories developed.

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