# Simultaneous Trajectory and Speed Planning for Autonomous Vehicles Considering Maneuver Variants
## 1 Introduction  
Modern developments in the domain of autonomous vehicle (AV) motion planning are distinguished by a wide range of proposed techniques and methods having advantages and drawbacks depending on the forecasting goals under current conditions. The main planning challenge concerns the need for simultaneously satisfying multiple requirements that guarantee both the motion feasibility and its safety, including the controllability and stability within the subsequent tracking process. The basic planning task consists of generating a forecast of trajectory and speed mode. On the one hand, it must conform to the principles of simplicity, smoothness, and unambiguity. For these purposes, analytical functions of polynomial and trigonometric types are often used [1], which change shape depending on parameters. The advantage of this approach implies that, with known parameters, any trajectory's point may be directly computed. However, there are also problems associated with the fact that trajectories represented by polynomials of small degrees do not provide high differentiability. For large degrees, the polynomials may be unstable, and the sensitivity of DOFs corresponding to higher derivatives increases. This complicates the optimization procedure since the determined parameters, in this case, may differ by an order of magnitude. On the other hand, the trajectory curve must meet the requirements of continuity and differentiability. Since the forecast optimality criteria themselves depend on the trajectory curvature's derivatives, it is also necessary to ensure the equality of these derivatives at the point of changing planning cycles. In this case, most often it is impossible to ensure the continuity of higher speed derivatives, as well as kinematic parameters based on them such as yaw rate, angular acceleration, and jerks. In most of the existing techniques, these aspects are neglected, which affects the irregular and intermittent nature [2] of the listed parameters at transient points. However, these parameters are associated with vehicle control systems and actuators. Therefore, it is desirable to ensure the continuity of their functions and derivatives for transient processes. To guide the AV on a roadway, two approaches may be used. The first one involves the formation of roadway boundaries and motion zones in the Cartesian coordinate system [3]. In this case, a space available for predicting the AV movement may be obtained by limiting motion zones for other traffic participants in such a way that a minimum safe distance is guaranteed. However, most often in this case, there is a need for separating the planning stage into sequential phases of determining the trajectory and distributing the speed mode. Another approach allows for the consideration of the formation of AV motion in Frenet coordinates [4; 5]. In this case, usually, the midline of a road lane is in some way expressed as a reference path function. Thus, instead of Cartesian coordinates, two Frenet coordinates arise, such as the lane reference curve's length and the lateral displacement perpendicular to the reference curve's tangent. In this case, it is much easier to work with spatial constraints by imposing them on the lateral displacement within the boundaries of a road lane. However, we still need to make the transition to the AV's local Cartesian system to decompose the components of kinematic parameters into longitudinal and transversal ones. Due to curvature, the full representation of the motion model in curvilinear coordinates becomes more complicated. Therefore, in this work, we will use a mixed approach in which there is a reference path curve in the Frenet coordinates, and the trajectory is predicted in the Cartesian coordinates.