Informed sampling-based trajectory planner for automated driving in dynamic urban environments Robin Smit1 Chris van der Ploeg12 Arjan Teerhuis1 Emilia Silvas13

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Informed sampling-based trajectory planner for automated driving in

dynamic urban environments

Robin Smit1, Chris van der Ploeg1,2, Arjan Teerhuis1, Emilia Silvas1,3

Abstract— The urban environment is amongst the most

difﬁcult domains for autonomous vehicles. The vehicle must

be able to plan a safe route on challenging road layouts, in

the presence of various dynamic trafﬁc participants such as

vehicles, cyclists and pedestrians and in various environmental

conditions. The challenge remains to have motion planners that

are computationally fast and that account for future movements

of other road users proactively. This paper describes an compu-

tationally efﬁcient sampling-based trajectory planner for safe

and comfortable driving in urban environments. The planner

improves the Stable-Sparse-RRT algorithm by adding initial

exploration branches to the search tree based on road layout

information and reiterating the previous solution. Furthermore,

the trajectory planner accounts for the predicted motion of

other trafﬁc participants to allow for safe driving in urban

trafﬁc. Simulation studies show that the planner is capable

of planning collision-free, comfortable trajectories in several

urban trafﬁc scenarios. Adding the domain-knowledge-based

exploration branches increases the efﬁciency of exploration of

highly interesting areas, thereby increasing the overall planning

performance.

I. INTRODUCTION

Since dense trafﬁc and cluttered environments bring uncer-

tainties and interaction challenges with other road users, one

of the most challenging tasks in automated driving is safe

and comfortable urban driving. Amongst others, trajectory

planning is a far from trivial task due to the varying road

layouts, different (types of) road users, and highly dynamic

scenarios. The task of planning a trajectory for the automated

vehicle (AV) to follow has gained increasing attention and

focus over the last years [1][2].

To compute fast (near-)optimal trajectories in terms of

safety and comfort, several types of algorithms exist,

falling in one of these categories: optimization-based,path

primitive-based and sampling-based. The ﬁrst approach for-

mulates the environment, such as the road layout and other

trafﬁc participants, as artiﬁcial potential ﬁelds or hard con-

straints and tries to solve the optimization problem analyt-

ically. For instance, [3] tries to ﬁnd an optimal trajectory

by formulating a Model Predictive Control (MPC) problem.

However, usually such methods are limited in ﬂexibility of

environment representations due to the analytical formula-

tion, limiting their applicability in real-world autonomous

1Netherlands Organisation for Applied Scientiﬁc Research, Integrated

Vehicle Safety Group, 5700 AT Helmond, The Netherlands.

2Eindhoven University of Technology, Dynamics and Control Group,

Mechanical Engineering Dept., P.O. Box 513, 5600 MB, Eindhoven, The

Netherlands.

3Eindhoven University of Technology, Control Systems Technology

Group, Mechanical Engineering Dept., P.O. Box 513, 5600 MB, Eindhoven,

The Netherlands.

driving systems. The second approach generates a set of

path primitives and calculates an optimal velocity and/or

acceleration proﬁle per path primitive, after which the most

optimal trajectory in terms of safety and comfort is selected.

In [4], a real-time motion planner is developed based on

sampling path primitives as 5th-order Bezier curves and

calculating suitable velocity proﬁles based on the motion pre-

diction of dynamic objects. However, the ﬁxed mathematical

description of the path primitives can limit the ﬂexibility of

the solution, especially in complex road layouts and highly

cluttered environments. Moreover, the decoupling of the

velocity proﬁle from the path primitives causes problems for

online decision making such as overtaking a slower vehicle

versus cruising behind it.

To overcome the limited ﬂexibility of the aforementioned

approaches, the trajectory planning problem can be solved

by using a sampling-based approach, which has shown great

potential in several robotics applications [5][6]. In [7], an al-

gorithm called Stable-Sparse-RRT (SST) is introduced where

piecewise-constant control samples are generated following

a stochastic distribution and propagated through a kinematic

model. Sampling and propagating control inputs guarantee

that the solution adheres to the vehicles non-holonomic

kinematic constraints. Furthermore, by applying forward

propagation of the kinematic system in time, a time space can

be mapped on the system states. A time-bounded collection

of vehicle states will be henceforth referred to as a trajectory.

Mapping calculated vehicle states in time enables the planner

to take time-bounded motion prediction of other road users

into account. However, due to the inherently exploring nature

of such sampling based algorithms, convergence to high

quality trajectories is slow and (sub-)optimality is often not

guaranteed [8]. To overcome these limitations and enable

SST solutions to be applied for AV trajectory planning, initial

state space exploration of high potential regions such as the

lane center or around a previously found solution is enforced,

resulting in higher quality solutions while maintaining the

same planning update rate.

The main contributions of this work are as follows:

1) We apply the generic algorithm introduced in [7] in a

real-time automated driving trajectory generation appli-

cation

2) We present a novel extension to the trajectory planner,

improving its efﬁciency, by providing an initial explo-

ration method of high potential regions such as the lane

center and previous solutions.

The proposed trajectory planner is implemented in a real-

arXiv:2210.02335v1 [math.OC] 5 Oct 2022

istic simulation environment and validated based on several

urban-driving scenarios, showing its capability of providing

safe and comfortable trajectories in the case of complex road

layouts and other static and dynamic trafﬁc participants.

This article is outlined as follows. In Section II, the

problem statement and preliminaries are deﬁned. Section

III focuses on the method used for solving the trajectory

planning problem. This method is proven in a simulation

study in Section IV and future work is presented in Section

II. PRELIMINARIES

To improve on-road safety in extended operational design

domains, novel motion planning algorithms are developed in

the EU Horizon Safe-up Project. Based on prediction models

of other trafﬁc participants’ behaviour and advanced risk

estimation, motion planners and vehicle control strategies

need to show improved safety and capabilities in various

use cases, such as bad weather, faulty sensors or dynamic

urban environments including pedestrians or cyclists.

The focus of this work is to ﬁnd safe, comfortable and

feasible vehicle trajectories in an urban environment. Due to

the highly dynamic nature of this environment, the trajectory

planner should be able to ﬁnd suitable solutions in real-

time with a sufﬁcient update rate. Furthermore, the solution

should take into account the kinodynamic constraints of the

vehicle, the road layout and the predicted motion of other

trafﬁc participants.

A. Problem statement

To deﬁne the constraints and goals of the planner, ﬁrst

consider the state of the vehicle at time kdenoted by

xk= (xk, yk, θk, vk), where xand yrepresent the vehicle

ground-plane position, θis the vehicle planar orientation

and vis the vehicle longitudinal velocity. To allow for safe

and comfortable driving in urban environments, the vehicle

needs to proactively account for the predicted motion of other

trafﬁc participants. Often these predictions are neglected

which can lead to reactive and thus more aggressive vehicle

behavior. To account for predicted motions, assume a set

of Oobjects, where the measured state of an object at

time kis given by its planar coordinates denoted by oo

(xo

k, yo

k, θo

k),∀o∈O. Furthermore, assume the predicted

trajectories of these dynamic objects to be available with no

noise or inaccuracies. The challenge is then to include these

into the trajectory planning problem in a (near) optimal way

by balancing progress along the road and minimize driving

risks. In the absence of static or dynamic obstacles, the

vehicle is expected to follow the lane centers of the road

network. To model that, the road infrastructure is described

by a set of Llanes, where each lane is described as a

collection of lane center points, together with a lane width.

A lane is denoted by li= (r1, r2, . . . , rR, ω),∀i∈[1 . . . L],

where rj∀j∈Ris a lane center point described by its planar

position, (xr, yr)and ωis the lane width. To lead the vehicle

along a global route, the trajectory planner receives global

goals to plan towards. Here, the goal is represented by a

goal space G⊂R4, and it is computed such that its position

components (x, y)span over all available lanes at a ﬁxed

distance gdalong the vehicles position, with longitudinal

threshold gt. Furthermore, the orientation component θand

velocity component vare unbounded, meaning that a speciﬁc

orientation and velocity are not required for the goal to

be reached. The schematic overview of the goal space

is depicted in Figure 1. Considering the aforementioned

Goal area

𝑔𝑑

𝑔𝑡

Fig. 1: Ground plane visualization of goal space

requirements for safe and comfortable driving, the trajectory

planning problem at time krequires ﬁnding a trajectory ξk=

x0x1. . . xf, which satisﬁes the kinodynamic constraints

of the vehicle, is collision-free given the predicted object

states and brings the vehicle to the goal space such that

xf∈G. Besides these constraints, the planner should also

minimize the risk of collision for the vehicle as well as

other trafﬁc participants by maintaining sufﬁcient distance

to others, follow the appropriate lane center for a given road

network whenever possible and maximize comfort.

B. State and control space deﬁnition

Most sampling-based planners generate search trees by

sampling new states in a selected state space. Such state

spaces can be R2, Dubins or Reeds-Shepp [9]. However,

planning in such spaces either puts a limit on the movement

of the car (e.g., the constant curve radius in Dubins), does

not satisfy the differential and/or non-holonomic kinematic

constraints of the vehicles motion model (e.g., instantaneous

discrete change of vehicle heading) or require a speciﬁc

steering function for the system. To facilitate the integration

of kinematic and differential constraints, while maintaining

ﬂexibility on the reachable space of the vehicle, control

space sampling poses an alternative. Here, control inputs are

sampled and propagated in the system over a speciﬁed time.

Given a ﬁxed initial state (sample), this results in a new state

space sample that satisﬁes the kinematic constraints. As often

used in motion planning designs and also in this work, a

non-linear kinematic bicycle model as shown in Figure 2 is

used to allow for real-time trajectory planning for larger time

horizons, i.e., 4-10 seconds [1]. The model is described by its

discretized kinematic equations, using Euler discretization,

as:

xk+1 =xk+tsvkcos θk,

yk+1 =yk+tsvksin θk,

θk+1 =θk+tsvk

Lwtan δk,

vk+1 =vk+tsak,

(1)

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Informedsampling-basedtrajectoryplannerforautomateddrivingindynamicurbanenvironmentsRobinSmit1,ChrisvanderPloeg1;2,ArjanTeerhuis1,EmiliaSilvas1;3AbstractTheurbanenvironmentisamongstthemostdifcultdomainsforautonomousvehicles.Thevehiclemustbeabletoplanasaferouteonchallengingroadlayouts,inthepresence...

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Informed sampling-based trajectory planner for automated driving in dynamic urban environments Robin Smit1 Chris van der Ploeg12 Arjan Teerhuis1 Emilia Silvas13

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