# Feasible Trajectories Generation for Autonomous Driving Vehicles
###### Abstract  
This study presents smooth and fast feasible trajectory generation for autonomous driving vehicles subject to the vehicle physical constraints on the vehicle power, speed, acceleration as well as the hard limitations of the vehicle steering angle and the steering angular speed. This is due to the fact the vehicle speed and the vehicle steering angle are always in a strict relationship for safety purposes, depending on the real vehicle driving constraints, the environmental conditions, and the surrounding obstacles. Three different methods of the position quintic polynomial, speed quartic polynomial, and symmetric polynomial function for generating the vehicle trajectories are presented and illustrated with simulations. The optimal trajectory is selected according to three criteria: Smoother curve, smaller tracking error, and shorter distance. The outcomes of this paper can be used for generating online trajectories for autonomous driving vehicles and auto-parking systems. autonomous driving vehicle; steering angle; vehicle velocity; optimal trajectory; vehicle physical constraints +
Footnote †: journal: Applied  
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The body of literature for recent autonomous vehicle paths and trajectory generation is enormous. This paper reviews some update information on the model predictive controls, optimal iterative calculations for real-time control of autonomous ground vehicles, auto-driving, and auto-parking systems. Basic mathematic functions to determine feasible trajectories were presented in [2], in which the authors introduced different mathematical methods for generating optimal paths. The vehicle was controlled and simulated moving on its feasible paths, including forward and reverse velocity, into and out of parking places. However, Authors in [2] lacked the generation of continuing trajectories and online calculations for feasible and optimal trajectories subject to the vehicle's dynamic constraints. Vehicle steering dynamic control and simulation are referred to in [3]. In [3], the author provided analyses and simulations of sideslips and the steering angle depending on the vehicle velocity and the environmental limitations. The vehicle steering angle in forward and reverse speeds were also analyzed and simulated. Fundamental parameters of vehicles regarding the position of the gravity center, the stiffness of front and rear wheels, under-steering and over-steering conditions, as well as the sideslip dynamics were also included in the analyses and calculations. Therefore, data from [3] is useful for generating the vehicle trajectories subject to physical constraints. Regarding the model predictive control for vehicles' tracking setpoints, reference [4] is considered. The authors in [4] presented a fundamental new robust model of predictive control for uncertain and constrained systems. The tracking setpoints of the robust model predictive control for input-saturated and softened-state constraints were introduced. The softened-state constraints showed higher superiority since, in many cases, the autonomous vehicle might be allowed to violate certain constraints while being subject to heavy penalty conditions. After this, the control system will find it easier to determine the optimal solution and minimize the tracking error. Recent studies on vehicle trajectory generation and tracking control are referred as following: Reference [5] introduced the problem of tracking trajectories under certain traffic conditions. The design of feasible paths was performed step by step based on the traffic environments. The closed-loop control was determined by a Proportional Integral Derivative (PID) controller. The trajectory generation and the tracking path controller were feasible but lacked optimal calculation and were not subject to vehicle constraints. A new study on vehicle trajectory generation for autonomous systems subject to emergency situations was referred to in [6].