Optimization-based Motion Planning for Autonomous Parking Considering Dynamic Obstacle A Hierarchical Framework Xuemin Chi1 Zhitao Liu1 Jihao Huang1 Feng Hong1 Hongye Su1

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Optimization-based Motion Planning for Autonomous Parking

Considering Dynamic Obstacle: A Hierarchical Framework

Xuemin Chi1, Zhitao Liu1, Jihao Huang1, Feng Hong1, Hongye Su1

1. State Key Laboratory of Industrial, Control Technology, Zhejiang University, Hangzhou, 110004

E-mail: chixuemin@zju.edu.cn, ztliu@zju.edu.cn, jihaoh@zju.edu.cn, hogfeg@zju.edu.cn, hysu@iipc.zju.edu.cn

Abstract: This paper introduces a hierarchical framework that integrates graph search algorithms and model predictive

control to facilitate efﬁcient parking maneuvers for Autonomous Vehicles (AVs) in constrained environments. In the high-

level planning phase, the framework incorporates scenario-based hybrid A* (SHA*), an optimized variant of traditional

Hybrid A*, to generate an initial path while considering static obstacles. This global path serves as an initial guess

for the low-level NLP problem. In the low-level optimizing phase, a nonlinear model predictive control (NMPC)-based

framework is deployed to circumvent dynamic obstacles. The performance of SHA* is empirically validated through 148

simulation scenarios, and the efﬁcacy of the proposed hierarchical framework is demonstrated via a real-time parallel

parking simulation.

Key Words: autonomous parking, trajectory planning, model predictive control, dynamic obstacles

1 INTRODUCTION

Autonomous parking systems are a critical component of

autonomous driving technologies. These systems comprise

several interrelated modules such as sensing, localization,

decision-making, planning, and control [1]. This paper nar-

rows its focus to address challenges in the planning module

of autonomous parking systems.

In an autonomous parking system, the planning module

is primarily responsible for generating a viable trajectory

that allows the vehicle to park in a designated space with-

out colliding with any obstacles. This trajectory is subse-

quently executed by a lower-level control module [2]. The

task of trajectory planning encompasses two major con-

cerns: comfort and collision avoidance. An optimal trajec-

tory minimizes time, while also considering factors such

as passenger comfort and vehicular stability. In this con-

text, we propose a hierarchical framework that integrates

graph search algorithms with nonlinear model predictive

control methods to generate safe and efﬁcient parking tra-

jectories, even in constrained environments populated by

dynamic obstacles.

1.1 Graph Search-based methods

Graph search methods are a popular choice in the realm of

path planning due to their computational efﬁciency relative

to generic optimization techniques. In this approach, the

environment is discretized into a grid, within which nodes

are sampled based on speciﬁc rules. Subsequently, an algo-

rithm searches for the optimal nodes that form the desired

This work was partially supported by National Key R&D Pro-

gram of China (Grant NO. 2021YFB3301000); Science Fund for Cre-

ative Research Group of the National Natural Science Foundation of

China (Grant NO.61621002), National Natural Science Foundation of

China (NSFC:62173297), Zhejiang Key R&D Program (Grant NO.

2021C01198,2022C01035).

path. Although these methods are typically computation-

ally efﬁcient in low-dimensional spaces, their performance

deteriorates in higher dimensions. Furthermore, the result-

ing paths are often sub-optimal.

Conventional deterministic graph search algorithms, such

as A* and Hybrid A [3], rely on ﬁxed motion primitives for

sampling. The motion primitives in Hybrid A* are particu-

larly well-suited to accommodate the non-holonomic con-

straints of vehicles. Stochastic graph search techniques like

Rapidly-exploring Random Trees (RRT) [4] and its vari-

ants [5, 6] employ random sampling within grids. While

these methods can be more ﬂexible, they often produce

paths with curvature discontinuities, making them unsuit-

able for immediate use without further reﬁnement.

1.2 Related work

Two primary approaches dominate the landscape of path

and trajectory planning for autonomous parking: search-

based methods and optimization-based methods. While the

former offers computational efﬁciency, the latter allows for

a more nuanced consideration of a vehicle’s dynamic or

kinematic characteristics through model-based algorithms.

Optimization-based methods also have the capability to

handle complex constraints such as comfort and stability

through mathematical modeling.

Zhang et al. [7] proposed the optimization-based collision

avoidance (OBCA) algorithm, which reformulates colli-

sion avoidance as smooth constraints using duality and

Slater’s condition. Despite these advances, their method

is not suitable for real-time applications and struggles with

dynamic obstacles.

Model predictive control has been extensively employed in

trajectory planning and tracking due to its ability to manage

multiple constraints effectively [8]. Soloperto et al. [9] ap-

plied OBCA within a tube-based robust MPC framework

arXiv:2210.13112v3 [cs.RO] 14 Nov 2023

Figure 1: The AV executes a parking maneuver from

start(solid black) to end(solid blue) while avoiding a dy-

namic obstacle(solid red)

for car overtaking simulations. However, the robustness

of real-time applicability in structured roads is not well-

established. Br¨

udigam et al. [10] implemented a stochastic

MPC scheme for overtaking maneuvers but did not address

the complexities of parking scenarios that involve both for-

ward and backward driving.

Contributions: This paper presents a novel hierarchical

framework designed to perform real-time parking maneu-

vers. The contributions of our work are summarized as fol-

lows:

• We introduce a hierarchical framework capable of

accommodating general parking scenarios in con-

strained environments.

• We propose a faster variant of the traditional Hybrid

A*, termed scenario-based hybrid A* (SHA*), whose

computational advantages can be leveraged in various

contexts.

• Our framework explicitly accounts for dynamic ob-

stacles, thus allowing for real-time implementation of

autonomous parking maneuvers.

2 PROBLEM DESCRIPTION

This section presents the parking scenario under consider-

ation and outlines the problem formulation.

We focus on a constrained parking scenario, as depicted in

Fig. 1. In this environment, the AV aims to transition from

a given start conﬁguration to a predetermined goal conﬁg-

uration. A signiﬁcant challenge arises from the presence of

a dynamic obstacle (DO) within the parking lot. The AV

must execute a parking maneuver that not only fulﬁlls the

objective but also avoids collision with the DO. Notably,

the entire scenario is modeled in a 2-dimensional space.

2.1 Vehicle Description

In the considered scenario, parking maneuvers occur at low

speeds. To model the vehicle’s dynamics, we employ the

kinematic bicycle model, expressed as:

Figure 2: The proposed hierarchical parking framework

˙z=





˙x

˙y

˙v







=





vcos(ϕ)

vsin(ϕ)

Ltan(δ)







, u =δ

a,(1)

where the system state zt∈ Z ⊆ R4,xand yare states cor-

responding to the center position of the rear axle in global

coordinates. State ϕis the yaw angle related to the x-

axis and vis the velocity for the center of the rear axle.

The front steering angle δfand acceleration aare input

ut∈ U ⊆ R2. L is the length between the rear axle and

front axle shown in Fig. 2.

Remark 1. We assume that the AV parking at a low

speed(i.e. less than 5m/s), therefore the tire slip angle and

inertial effects can be ignored.

In a tight-parking lot, a full-dimensional model allows the

AV to navigate less conservative. we modeled the con-

trolled AV as a full-dimensional object as follows

E(zt) := R(ϕt)B0+T(xt, yt),(2)

where zt∈ Z ⊆ R4is the state of the AV at time t, E(zt)is

the ”space” occupied by the AV, R(·)is rotation matrix, and

T(·)is translation operation, B0is a rectangle represented

by a convex polyhedron

B0:= y∈R2|Gy ≤g,(3)

where Gand gare related to the car shape, G=

[1,0; 0,1; −1,0; 0,−1] and g= [L/2, W/2, L/2, W/2],

the motion of the AV can be regarded as a polyhedron

through rotation and translation.

2.2 Obstacle Description

In this paper, we consider M∈N, M ≥0dynamic obsta-

cles in the parking lot that the AV must avoid. DO can be

described as compact polyhedrons, i.e.

t:= y∈R2|Am

ty≤bm

t, m = 1, . . . , M, (4)

where Am

t∈Rhm×2and bm

t∈Rhmare matrices with

respect to obstacles at time tand hmis the number of hy-

perplanes which formulates the m-th obstacle.

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Optimization-basedMotionPlanningforAutonomousParkingConsideringDynamicObstacle:AHierarchicalFrameworkXueminChi1,ZhitaoLiu1,JihaoHuang1,FengHong1,HongyeSu11.StateKeyLaboratoryofIndustrial,ControlTechnology,ZhejiangUniversity,Hangzhou,110004E-mail:chixuemin@zju.edu.cn,ztliu@zju.edu.cn,jihaoh@zju.edu.c...

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Optimization-based Motion Planning for Autonomous Parking Considering Dynamic Obstacle A Hierarchical Framework Xuemin Chi1 Zhitao Liu1 Jihao Huang1 Feng Hong1 Hongye Su1

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