Received October 22, 2018, accepted November 11, 2018, date of publication November 26, 2018,
date of current version January 7, 2019.
Digital Object Identifier 10.1109/ACCESS.2018.2883405
Terminal Guidance Based on Bézier Curve for
Climb-and-Dive Maneuvering Trajectory With
Impact Angle Constraint
ZHEN QIN , XIAOYE QI, AND YONGLING FU
School of Mechanical Engineering and Automation, Beihang University, Beijing 10083, China
Corresponding author: Yongling Fu (fuyongling@buaa.edu.cn)
This work was supported by the National Natural Science Foundation of China under Grant 61327807.
ABSTRACT To improve the penetration capability and the damage effectiveness of cruise missiles, maneu-
vering trajectory with specific impact angle is applied in terminal guidance. In this paper, an innovative online
method for terminal guidance is presented, which can guide the cruise missile to a stationary target along a
climb-and-dive maneuvering trajectory while meeting the expected impact angle. The basic principles of the
proposed guidance method are the Bézier curve theory and the missile inverse dynamics. The maneuvering
trajectory is shaped by a third-order Bézier curve based on the missile current state, the target position,
and the impact angle constraint. The real-time guidance command represented by attack angle α is obtained
based on the missile inverse dynamics. The shape of flight trajectory can be modified by regulating the Bézier
parameters b
1
and b
2
to satisfy different tactical requirements without changing the boundary conditions.
Simulation results demonstrate the effectiveness of the proposed method applied in cruise missile terminal
guidance.
INDEX TERMS Cruise missile, terminal guidance, maneuvering trajectory, impact angle, Bézier curve.
I. INTRODUCTION
Cruise missiles play an important role in modern warfare due
to their advantages of small size, low cost, high precision, and
high maneuverability. However, the survival of cruise mis-
siles has become increasingly severe with the developments
of anti-missile systems [1]. Planning climb-and-dive maneu-
vering trajectory, especially at the terminal guidance phase,
can improve the penetration capability of the missile and
reduce the interception probability. In addition, some tactical
tasks require the missile to attack the target with a specific
impact angle to improve the damage effectiveness [2], [3].
Previous researchers conducted a series of work on
missile terminal guidance. To achieve the climb-and-dive
maneuvering trajectory, methods such as two-phase propor-
tional navigation guidance law [4]–[6] and optimal guidance
law [7]–[10] are applied before. The two-phase proportional
navigation guidance assumes a transition point within the
flight trajectory which divides the entire trajectory into two
phases. At the first phase, the missile climbs from the start
point until it reaches the transition point. Then at the second
phase, the missile turns down and dives until it hits the
target. To meet the expected impact angle, methods such
as biased proportional navigation guidance law [11]–[14],
optimal guidance law [15]–[17], model predictive static pro-
gramming technique [18], [19], and sliding mode control
theory [20]–[22] are applied before. But most of these meth-
ods only focus on solving one of the maneuvering trajectory
or impact angle problems.
Thus, an innovative online method for terminal guidance
is proposed here, which can guide the cruise missile to a sta-
tionary target along a climb-and-dive maneuvering trajectory
while meeting the expected impact angle. The maneuvering
trajectory is shaped by a third-order Bézier curve accord-
ing to the missile current state, the target position, and the
impact angle constraint. The real-time guidance command
represented by attack angle α is obtained based on the missile
inverse dynamics.
The rest of this paper is organized as follows: section II
describes the guidance problem, section III introduces the
guidance method, section IV shows the simulation results,
and section V demonstrates the conclusions.
II. PROBLEM STATEMENT
A. GUIDANCE PROBLEM
The purpose of this paper is to present a terminal guid-
ance method for cruise missile attacking a stationary target.
VOLUME 7, 2019
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