Modelling and Simulation in Engineering

Volume 2018, Article ID 9348907, 19 pages

https://doi.org/10.1155/2018/9348907

## Time-Optimal Trajectory Planning along Parametric Polynomial Lane-Change Curves with Bounded Velocity and Acceleration: Simulations for a Unicycle Based on Numerical Integration

^{1}Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan^{2}Institute of Information Science, Academia Sinica, Taipei 115, Taiwan

Correspondence should be addressed to Jing-Sin Liu; wt.ude.acinis.sii@uil

Received 15 June 2018; Accepted 27 August 2018; Published 19 November 2018

Academic Editor: Mohamed B. Trabia

Copyright © 2018 Chien-Sheng Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

lane-change path imposes symmetric conditions on the path geometric properties. This paper presents the comparative study of time-optimal velocities to minimize the time needed for traversal of three planar symmetric parametric polynomial lane-change paths followed by an autonomous vehicle, assuming that the neighboring lane is free. A simulated model based on unicycle that accounts for the acceleration and velocity bounds and is particularly simple for generating the time-optimal path parameterization of each lane-change path is adopted. We base the time-optimal trajectory simulations on numerical integration on a path basis under two different end conditions representing sufficient and restricted steering spaces with remarkable difference in allowable maximum curvature. The rest-to-rest lane-change maneuvering simulations highlight the effect of the most relevant path geometric properties on minimal travel time: a faster lane-change curve such as a quintic Bezier curve followed by a unicycle tends to be shorter in route length and lower in maximum curvature to have achievable highest speed at the maximum curvature points. The results have implications to path selection for parallel parking and allow the design of continuous acceleration profile via time scaling for smooth, faster motion along a given path. This could provide a reference for on-road lane-change trajectory planning along a given path other than parametric polynomials for significantly more complex, complete higher-dimensional highly nonlinear dynamic model of autonomous ground vehicle considering aerodynamic forces, tire and friction forces of tire-ground interaction, and terrain topology in real-world.

#### 1. Introduction

As an essential part of the active safety system of autonomous driving or human-driven cars [1–3], a lane-change maneuver performed by a vehicle on a terrain [4, 5] is a path following or trajectory tracking task for avoiding vehicle-to-vehicle collisions. It involves decision, sensing of traffic flow and environmental conditions, planning, and control subject to the constraints such as path constraints, kinodynamic constraints, environmental constraints, and real-time requirements. The planning task of lane change requires the generation of path and velocity, or trajectory for the vehicle to follow. It is commonly used for testing autonomous or human-driving vehicle performance such as critical speed for no sideslip, or for path and trajectory design [6]. Besides, it is considered as a method for measuring vehicle-handling performance such as safety and completing time of lane-changing maneuver, especially at high speed or within restricted steering space.

Curves which produce a transition between parallel lanes (from current lane to a neighboring target lane) in the same direction are called lane-change curves [5]. The lane-change curve connects symmetric interpolating boundary configurations, that is, the same tangent angle and curvature. Following a lane-change curve, the vehicle will be traveling in the same direction at the end of the maneuver as it was traveling at the start; that is, there is no change of heading between the start and the final configurations. Lane-change trajectory consists of a lane-change path and a velocity profile along the path. For lane changing from current lane to a neighboring lane, the allowable longitudinal displacement for longitudinal trajectory planning and travel time for lateral trajectory planning [1, 7] are used to determine the feasibility of performing a lane change. Various forms of lane-change trajectories are proposed (e.g., [1–3, 5, 8–10]), which serve as reference trajectories for the vehicle controller (e.g., PID or model predictive control) to follow in high-speed autonomous driving. The approach of intervehicle traffic gap and time instance [1] was proposed to generate safe and smooth lane-change (longitudinal and lateral) trajectories for a discrete-time vehicle model of double integrator. Bai et al. [2] and Altch’e et al. [11] proposed a 5-degree polynomial trajectory of time function as the vehicle trajectory. The jerk, as a measure of comfort, is a quadratic polynomial of time. Geng et al. [3] proposed a 6-degree polynomial trajectory of the form , where *x* and *y* denote the forward/longitudinal and lateral position, respectively. Based on field test vehicle-driver integration data, Wang et al. [9] proposed a lane-change trajectory represented by a combination of linear and sinusoidal functions in terms of lane-change ratio. McNally [10] used a cubic polynomial to blend a target lane-change curve. Under the assumption that the vehicle keeps its longitudinal vehicle constant throughout the lane-change maneuver, comparative simulations of candidate curves were performed in terms of path length, minimal yaw transients, and jerk but no time efficiency in an earlier work [8].

In addition to smooth shorter motion, time-efficient or faster motion is of major concern for trajectory optimization. Due to fuel economy, smooth (or continuous curvature) motion generation and motion optimization along the given path or for state-to-state transfer [12–16] subject to the velocity or acceleration and other constraints are desirable to achieve time or energy efficiency and safety. In [15], quintic B-splines are used to blend linear segments to enhance the smoothness of cornering motion and cycle time of CNC machine tools. Model predictive control is applied by Mahdi Ghazaei Ardakani [16] to deal with fixed-time trajectory-generation problem with a minimum-jerk cost functional under velocity and acceleration constraints. Aspects of time-efficient state-to-state motions for different mobility platforms like mobile robots, humanoid robots, robotic manipulators, or autonomous-driving vehicles have been studied for decoupled motion planning approach of trajectory optimization. This path-velocity decomposition approach gains its popularity due to the reduced complexity at the cost of losing generality. In the decoupled approach, two subproblems of the geometric path planning between two configurations and time scaling (or velocity planning) along the planned path subject to kinodynamic constraints are treated independently. Algorithms and properties of a time-optimal trajectory along prespecified paths parameterized by arc length, that is, the computation of switch points and optimal input subject to state and input constraints, were developed [17–25]. The solution is based on necessary conditions derived from Pontryagin maximum principle (PMP) or dynamic programming principle (DPP) for the optimality of trajectory and control. There are different numerical algorithms based on Pontryagin maximum principle or receding horizon techniques to generate the control input and state trajectory to move the systems from an initial state to a goal state as fast as possible, while respecting the equality or inequality constraints imposed on the state and input of systems [20, 21, 22, 26].

Velocity planning along a path not only depends on vehicle dynamics, its kinodynamic constraints such as the velocity and acceleration constraints, and all other constraints on motion [17, 19] but also on the characteristics of the path such as the curvature defining the ratio of angular over linear velocity (see, e.g., [23, 24]). There exist a number of time-optimal velocity planning algorithms along a prespecified path whose study of motion optimization is to minimize the travel time required to move along the whole path subject to vehicle system dynamics, any other constraints such as path constraints, torque/acceleration constraints, and velocity constraints for different systems (see, e.g., [18, 20]). The adopted vehicle model can be varied for different studies [2, 11] where the bounding geometry of the vehicle shape could be circle, ellipse, or rectangular. Unicycle is a popular model for control and trajectory planning of nonholonomic wheeled mobile robots in that it offers very good compromise between accuracy and computational efficiency for simulation and prediction of nonholonomic autonomous vehicle motion (e.g., [12, 25, 27–29]), that is, zero lateral velocity for emulating the vehicle performance. It is used in this paper as a simulated vehicle model for lane-change trajectories.

As mentioned in [3], aggressive lane change is completed within the shortest possible time. This paper takes the overall travel time along the path as the cost to be minimized over the velocity for safety-critical concern of lane-change maneuvering by a unicycle subject to kinodynamic constraints on velocity and acceleration. More specifically, we focus on three types of symmetric parametric polynomial curves with closed-form position expressions that are very popular for robotic applications as candidate lane-change paths for autonomous driving. The aim of this work is to study via simulation-based evaluation the path geometric characteristic related to time-optimal lane-change path constrained trajectory planning. Among the methods to solve the time-optimal velocity planning or time-optimal path parameterization along the fixed lane-change path, numerical integration (NI) [20, 26] is directly employed as the main simulation tool to make a number of runs to obtain the time-optimal velocity profile on each path in different simulation conditions without violating the bounds on velocity and acceleration. In particular, the acceleration and velocity constraints for a unicycle can be expressed as linear in squared path velocity ( in (12)), which allows the solution to TOPP for each curve particularly simple. The major contribution of this paper based on time-optimal velocity simulation-based evaluation of parametric polynomials for lane-change maneuvering as a path following task by a unicycle in constrained maneuver space is as follows:(i)Simulation results based on unicycle provides a good understanding of velocity and acceleration characteristics and the switching structure defined by the velocity limit curve caused by all the constraints of time-optimal solution along parametric polynomial lane-change curves. It provides a supplementary comparative study of assessment of least amount of travel time along parametric polynomial lane-change curves, as compared to earlier work [8]. After NI simulation-based evaluation of the minimum travel time along all candidate curves, in order to perform a lane-change maneuver faster, the selection of lane-change path based on path geometric characteristics is proposed. The results such as feasibility of parametric polynomials for a lane-change maneuver in a wider range of situations could be a reference for simulating more refined and complete dynamic model and lane-change curves other than the families of parametric polynomials. The minimal lane-changing time by a unicycle provides an ideal lower bound estimate for further optimization of lane-change maneuver along a given path, such as the data-driven design of continuous acceleration profile for smooth, faster motion. In addition, the highest speed allowed is also concerned for comfort and safety in autonomous driving.(ii)Velocity along a given path is curvature dependent, which is determined by available constrained maneuver space, for example, as in parallel parking [30]. The authors in [27] among others pointed out that the travel time of mobile robot navigation depends on the path shape and the velocity profile, in which the admissible velocity is affected by the curvature. These observations are based on piecewise constant longitudinal accelerations defined over distance along the curve. Among all the evaluated curves, our simulations confirm how the minimum travel time required for a unicycle is affected by the path characteristics such as path length and maximum curvature coupled with the bounds on velocity and acceleration in a more concrete setting of simulating lane-change curves in different scenarios.(iii)The comparison results of lane-change paths have implications for curve selection of parallel parking in constrained steering space studied in [30], since parallel parking maneuver bears similarity to lane-change maneuver in terms of interpolating boundary conditions (position, heading, and curvature at the endpoints).

This paper is an extension of our conference paper [36]. The rest of the paper is organized as follows. In Section 2, kinematic unicycle model, symmetric planar curve, time-optimal velocity planning with velocity and acceleration bounds, and numerical integration to compute the maximum velocity profile on the path are introduced. In Section 3, an offered set of three symmetric parametric polynomial curves for lane change is derived. In Section 4, the comparative simulation-based evaluation of each parametric polynomial curve is presented for two end configurations (loose and hard curvature conditions) to highlight the effect of length and maximum curvature along the curve on the minimum travel time in distinct situations. Conclusion is made in the last section.

#### 2. Background

##### 2.1. Kinematic Model of Unicycle

Let *O*-*x*-*y* be a global coordinate frame with *x*-axis being the forward/longitudinal direction. The configuration of a unicycle, depicted in Figure 1, moving in a planar workspace along a parametric path parameterized by the path parameter is given by its Cartesian position coordinates and orientation denoting the heading of unicycle. For unicycle, is equal to the angle between the *x*-axis and the tangent of the path . The kinematic equations of motion are described as follows:where the symbol “” denotes the derivative with respect to the argument (here ), and are the inputs of linear and angular speed, respectively. The nonholonomic nature (1) is caused by the fact that wheels can only roll but not slip, which satisfies the following equation: