# Simultaneous Trajectory and Speed Planning for Autonomous Vehicles Considering Maneuver Variants
## 1 Introduction
Forecasting the speed mode [6] involves obtaining several kinematic parameters (linear and angular speeds, linear and angular accelerations, linear jerks) characterizing the AV dynamics and providing direct ties with vehicle sensors' data (accelerometers, gyroscopes, radars), which allows for the monitoring and correcting of the AV control at the tracking stage. These same parameters may be used as criteria for optimizing the AV motion. The main task of speed planning aims at compromising the vehicle's maximum performance, safety, and motion smoothness. To implement the above, two different approaches may be used. The first one is represented by discrete modeling [7], where a system of equations based on vehicle dynamics (or kinematics) allows simultaneous optimization [8] of the trajectory and speed related differentially to each other. However, to ensure good convergence of the numerical solution, it is necessary to reduce the integration step, which increases iteration time. Another approach proceeds from representing the trajectory and speed by piecewise analytical functions [9], forms of which are known in advance and only the parameters that determine the nature of these functions are optimized. In this case, there are no problems with step control, but the motion can only be represented by a vehicle kinematic model. Constraints are a separate issue. There are quite a number of restrictions stipulated by both the parameter nature and the possibility of its assessment. Thus, geometric restrictions [10] imply a non-violation of the roadway boundaries (permissible zone) and keep safe positions relative to other traffic participants. Kinematic constraints concern parameters based on speed [11] and its derivatives. Dynamic restrictions are determined by limits imposed on the vehicle's powertrain system to realize its traction potential. The vehicle acceleration naturally decreases with speed, which must be reflected in the driving mode being planned. Physical limitations are based on the tire's maximum adhesion property and the potential for motion stability. All these restrictions may be represented as hard and/or soft constraints. In the first case, satisfying limiting conditions is mandatory and formed by additional linear or nonlinear equalities and inequalities. In the second case, the restrictions themselves are included in the objective function [12], which determines the influence of the criteria but allows for violations of the limits for exceptional reasons. For example, this approach is appropriate in cases when a safe distance between vehicles in the same lane is not ensured. Obstacles: In real conditions, the AV must maneuver among moving and stationary obstacles [13]. One of the main tasks on AV's motion-planning strategy is to prevent collisions and maintain safe distances [14]. The issue is complicated by the fact that it is difficult to accurately estimate the motion directions and speeds of other participants for a relatively long time. In this regard, it may be assumed that during [PERSON]'s maneuvering, other vehicles barely change the nature of their movement. Thus, the task is to maximize the distances between vehicles while maneuvering, considering vehicles' safety contours. This study continues a series of our research on developing algorithms for AV motion planning. Initially, in paper [15], we outlined the main approaches to modeling trajectory and speed modes based on the vehicle kinematics model. At the same time, the sequential approach to optimizing the trajectory and speed was used. In the next research [16], we stipulated an advantage of using integral forms of the equality constraints instead of point ones. In articles [17], we tested various numerical integration schemes in optimization problems of motion planning. Based on the above, we shall consider a motion-planning technique that involves the simultaneous prediction of trajectory and speed mode within a single finite element (FE) of the variable reference length.