Planar Smooth Path Guidance Law for a Small Unmanned Aerial

Planar Smooth Path Guidance Law for a Small Unmanned Aerial
Hindawi
Journal of Control Science and Engineering
Volume 2017, Article ID 6712602, 11 pages
https://doi.org/10.1155/2017/6712602
Research Article
Planar Smooth Path Guidance Law for a Small
Unmanned Aerial Vehicle with Parameter Tuned by Fuzzy Logic
Yang Chen,1,2 Jianhong Liang,2 Chaolei Wang,3 Yicheng Zhang,2
Tianmiao Wang,2 and Chenghao Xue2
1
School of Mechanical and Electrical Engineering, Longyan University, Fujian 364000, China
Robotics Institute, Beihang University, Beijing 100191, China
3
Science and Technology on Special System Simulation Laboratory, Beijing Simulation Center, Beijing 100854, China
2
Correspondence should be addressed to Yang Chen; [email protected]
Received 16 November 2016; Accepted 21 February 2017; Published 26 March 2017
Academic Editor: William MacKunis
Copyright © 2017 Yang Chen 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.
A guidance law has been designed to guide the small unmanned aerial vehicle towards the predefined horizontal smooth path. The
guidance law only needs the mathematical expression for the predefined path, the positions, and the velocities of the vehicle in the
horizontal inertial frame. The stability of the guidance law has been demonstrated by the Lyapunov stability arguments. In order
to improve the path following performance, one of the parameters of the guidance law is tuned by using the fuzzy logic which will
still keep its stability. The simulation experiments in the Matlab/Simulink environment to realize the square-, circular-, and the
athletics track-style paths following are given to verify the effectiveness of the proposed method. The simulation results show that
the path following performance will be improved with smaller overshoot and oscillation amplitude and shorter arrival time with
the parameter tuned.
1. Introduction
In the past decades, the small unmanned aerial vehicles
(UAVs) have been widely applied in the military and civilian
domains with their ability of automation, such as fire detection, surveillance, aerial photography over the disaster area,
and forbidden area monitoring [1]. To realize these functions,
it is required that the UAVs have the capability of tracking or
following the predefined path or trajectory accurately.
During recent years, the approaches to realize the path
following or trajectory tracking control of a small unmanned
aerial vehicle appear on the linear and nonlinear control
methods. The linear techniques include the PID and linear
quadratic regulator (LQR), while the nonlinear methods
mostly appear on the backstepping technique, the fuzzy
logic based control, the sliding mode control, and the model
predictive control (MPC).
While applying linear methods, in [2], the outer-loop
design for translational motion is based on the PD controller together with the disturbance accommodating control
method to estimate the aerodynamic force. Compared to the
traditional PID technique, Luo et al. applied the directional
fractional order (PI)𝛼 method to the lateral control of a
small fixed-wing unmanned aerial vehicle with its parameters
tuned based on the mathematical model [3]. However, the
PID controller has little robustness to the disturbance, and
it is mostly used in the lateral/longitudinal maneuvers by
reducing the lateral deviation from a desired flight path.
The genetic algorithm has also been used to optimize the
control parameters of the trajectory tracking control law
which will improve the tracking performance [4]. But it
needs more calculation which makes it limited to be applied
on the low-cost autopilot. In [5], the linear quadratic regulator (LQR) with an adaptive running cost was applied
to generate the lateral acceleration command to guide the
vehicle towards the path. Lee et al. used the LQR technique
to reach the path following by finding the optimal state
feedback law where its state is the combination of the lateral
attitudes, the cross-track, and heading errors [6]. However,
the heading and cross-track errors should be saturated to
2
make the system stable due to the limitation of the linear
region.
Alternative approaches to realize the path following
are based on the nonlinear techniques, which include the
intelligent control, the Lyapunov stability based control, and
model predictive control. For using the intelligent method,
an adaptive neuron-fuzzy inference system based flight controller has been designed to realize the autonomous flight
[7]. Cancemi et al. applied the fuzzy logic to generate the
heading reference which will guide the UAV towards the
predefined waypoint trajectory [8]. The fuzzy based method
was implemented in the hardware-in-the-loop simulation.
The fuzzy logic method can reach better track performance
compared to the traditional PID controller.
Based on the Lyapunov stability theory, the sliding mode
and backstepping techniques are often applied. In [9, 10], the
sliding mode control technique together with the vector field
is used to guide the UAV towards the desired path. A spatial
sliding mode controller [10] defined as a stable differential
operator with respect to a defined spatial variable is presented
to track the predefined path. Liang et al. applied the vector
field for the vehicle to reach the planar path following by
using the sliding mode control [11]. In [12, 13], Ren applied the
backstepping techniques to obtain the velocity and roll angle
control laws to track the predefined path. Brezoescu obtained
the direction control of the UAV with the help of the adaptive
backstepping method in the presence of the crosswind, which
has been tested in simulation [14]. For real flight tests, the
backstepping technique is usually used as the inner loop to
stabilize the dynamics of the UAV [15]. Based on the MPC
technique, a nonlinear model predictive controller is used to
design the high level controller based on the kinematic model
of the UAV to follow the line path [16]. In [17], an adaptive
NMPC that varies the control horizon is used for design as the
tracking layer to follow the predefined path by minimizing
the mean and maximum errors between the reference path
and the UAV. For normal MPC tracking controller, the error
can converge fast with respect to its predictive ability, but
there exist some static state error and some wave control due
to the computational accuracy, limited predictive horizon,
and weight matrixes selection.
Motivated by the above discussions, it can be found that
the guidance control laws generated by the Lyapunov-stable
based methods are usually simple and useful. In our preview
works [18], a Lyapunov-stable based nonlinear control law
has been designed to guide the UAV towards the predefined
line path. In this paper, we have expanded the result to a
class of smooth curve-type path in the horizontal plane. The
proposed guidance law generated for the smooth path still
has one of its parameters tuned by the fuzzy logic, which
will make the vehicles fly towards the predefined path more
directly and smoothly.
In the outer loop, with the altitude held by a simple PI
controller, the concept of vector field which has been applied
in fixed-wing UAV to generate the heading rate command
was adopted [19]. However, the authors only considered the
straight line and circular-type path for the fixed-wing UAV. In
this study, we have expanded the result to a class of smooth
curve-type path in the 2D plane. While applying the vector
Journal of Control Science and Engineering
field to generate the heading rate command, the necessary
information including the ground speed, position, and heading angle is measured by a GPS/INS integrated navigation
system. Additionally, the asymptotic approximation to the
desired path is demonstrated by use of the sliding mode
approach and Lyapunov stability arguments.
The organization of this paper is as follows. In the next
section, the problem description is presented. In Section 3, a
Lyapunov-stable nonlinear controller is designed to generate
desired course rate to guide the UAV towards the planar
smooth path together with a key parameter of the controller
tuned by the fuzzy logic technique. The simulation results
in Matlab/Simulink environment to realize the square-,
circular-, and the athletics track-style paths are given in
Section 4. Finally, some concluding remarks are summarized
in Section 5.
2. Problem Description
Small fixed-wing UAVs equipped with low-level flight controller, which can provide altitude- and airspeed-hold functions, are assumed to be used for the path following. The
following kinematics of the UAV in inertial horizontal frame
can be used for the controller design:
π‘₯Μ‡ = π‘‰π‘Ž cos (πœ“) + π‘Šπ‘₯ ,
𝑦̇ = π‘‰π‘Ž sin (πœ“) + π‘Šπ‘¦ ,
(1)
πœ“Μ‡ = 𝑒0 ,
where (π‘₯, 𝑦) is the inertial position of the UAV, π‘Šπ‘₯ and π‘Šπ‘¦ are
the corresponding components of the wind velocity, π‘‰π‘Ž is the
airspeed, πœ“ is the heading, and 𝑒0 is the commanded heading
rate. By introducing the ground speed 𝑉𝑔 and the course πœ’
shown in Figure 1, the kinematics of the UAV (see (1)) can be
expressed as
π‘₯Μ‡ = 𝑉𝑔 cos (πœ’) ,
𝑦̇ = 𝑉𝑔 sin (πœ’) .
(2)
Following [9, 10], with another assumption that the
autopilot can provide the course-hold function adopted, by
introducing the course rate control πœ’Μ‡ = 𝑒cmd , the system
shown as (2) becomes as follows:
π‘₯Μ‡ = 𝑉𝑔 cos (πœ’) ,
𝑦̇ = 𝑉𝑔 sin (πœ’) ,
(3)
πœ’Μ‡ = 𝑒cmd .
The relation between the heading rate and course rate is
given by [11]
𝑒cmd = 𝐿 (πœ“) 𝑒0
=
π‘‰π‘Ž2 + π‘‰π‘Ž (π‘Šπ‘₯ cos (πœ“) + π‘Šπ‘¦ sin (πœ“))
π‘‰π‘Ž2 + π‘Šπ‘₯2 + π‘Šπ‘¦2 + 2π‘‰π‘Ž (π‘Šπ‘₯ cos (πœ“) + π‘Šπ‘¦ sin (πœ“))
β‹… 𝑒0 .
(4)
Journal of Control Science and Engineering
3
point (π‘₯, 𝑦) lies on the curve 𝑓(π‘₯, 𝑦) = 0. In order to avoid
the vehicle being trapped in the center point of the closed
curve (such as the center of a circle), an interesting domain
is defined as 𝐷int = {(π‘₯, 𝑦) | π‘₯, 𝑦 ∈ 𝑅, β€–βˆ‡π‘“β€– β‰₯ πœ†}, where
πœ† is a positive constant. When the vehicle’s initial position is
within 𝐷int , the following control law shown as (7) will guide
the UAV towards the desired path in finite time.
y
f(x, y) = 0
Desired
Wy
path
Vg
Va
πœ’
W
Wx
𝑑
σ΅„© σ΅„©
𝑒cmd = βˆ’π‘˜1 σ΅„©σ΅„©σ΅„©βˆ‡π‘“σ΅„©σ΅„©σ΅„© 𝑉𝑔 𝑓sat (𝑑 (π‘₯, 𝑦)) βˆ’ π‘˜2 𝑉𝑔 𝑑 (π‘₯, 𝑦)
𝑑𝑑
πœ“
+
𝑓
𝑑
tanβˆ’1 (βˆ’ π‘₯ ) ,
𝑑𝑑
𝑓𝑦
(7)
(0, 0)
x
Figure 1: Position of the UAV with respect to the desired path and
the relationship among the ground speed, airspeed, and the wind
speed.
For flying a UAV, it has the fact that the airspeed π‘‰π‘Ž
is larger than the wind speed π‘Š. When there is no wind,
𝐿(πœ“) = 1, which means that the course of the UAV equals
its heading angle. So it can be easily derived that 𝐿(πœ“) ∈
(1/2, 1]. In the following, we control the UAV’s course instead
of its heading. The command design is based on model (3).
Therefore, the motions of the UAV can be expressed in terms
of the ground speed and the course angle. Since the ground
speed contains the information of wind velocity, the wind
disturbance rejection will be naturally considered by using
the inertial measurements [9, 18].
3. Planar Smooth Path Following
3.1. Normal Planar Smooth Path Following Controller. Consider a generic desired smooth path in the horizontal plane
which can be described as a continuously differentiable
implicit function 𝑓 : 𝑅2 β†’ 𝑅 with
𝑓 (π‘₯, 𝑦) = 0.
(5)
In this paper, the desired path shown as (5) is a generic
curve. Since it is not trivial to express the distance from
a generic curve in closed form, the curve function 𝑓(π‘₯, 𝑦)
instead of the Euclidean distance from the curve is used as the
distance function [20]; that is, 𝑑(π‘₯, 𝑦) = 𝑓(π‘₯, 𝑦). The partial
derivatives of 𝑑(π‘₯, 𝑦) with respect to π‘₯ and 𝑦 are
𝑓π‘₯ =
πœ•π‘‘ (π‘₯, 𝑦) πœ•π‘“ (π‘₯, 𝑦)
=
,
πœ•π‘₯
πœ•π‘₯
πœ•π‘‘ (π‘₯, 𝑦) πœ•π‘“ (π‘₯, 𝑦)
=
.
𝑓𝑦 =
πœ•π‘¦
πœ•π‘¦
where π‘˜1 is a positive constant. π‘˜2 β‰₯ π‘˜2𝑝 ; π‘˜2𝑝 is an arbitrary
chosen positive value, and the saturation function 𝑓sat is
defined as
𝑉𝑔 ,
π‘₯ > 𝑉𝑔
{
{
{
{
𝑓sat (π‘₯) = {π‘₯,
βˆ’π‘‰π‘” ≀ π‘₯ ≀ 𝑉𝑔
{
{
{
{βˆ’π‘‰π‘” , π‘₯ < βˆ’π‘‰π‘” .
(8)
Here, the saturation function is used to limit distance, just
because if the initial distance from the path is sufficiently high
and without the saturation function, the UAV will perform
unnecessary endless loops away from the desired path. It
should be noted that the term (𝑑/𝑑𝑑)tanβˆ’1 (βˆ’π‘“π‘₯ /𝑓𝑦 ) in (7) can
guarantee that the vehicle will fly along the given path when
the distance to the curve is zero.
For analyzing the stability of the path tracking control
law shown as (7), two other following variables π‘₯1 and πœ’1 are
introduced as
π‘₯1 = 𝑑 (π‘₯, 𝑦) ,
𝑓
{
{
πœ’ βˆ’ tanβˆ’1 (βˆ’ π‘₯ ) , 𝑓𝑦 =ΜΈ 0
{
{
𝑓
{
𝑦
{
πœ’1 = {πœ’ βˆ’ πœ‹ ,
𝑓𝑦 = 0, 𝑓π‘₯ < 0
{
{
2
{
{
πœ‹
{πœ’ + ,
𝑓𝑦 = 0, 𝑓π‘₯ > 0.
{
2
(9)
Equations (9) show that π‘₯1 tends to zero as the distance
from the curve tends to zero, and πœ’1 tends to zero as πœ’ tends
to the direction of the desired line. So we have
(6)
The magnitude of corresponding gradient is denoted as
‖𝑓‖ = βˆšπ‘“π‘₯2 + 𝑓𝑦2 . With the distance function defined above,
𝑑(π‘₯, 𝑦) is a scalar filed, and it can indicate whether or not the
sin (πœ’1 ) =
𝑓π‘₯ cos (πœ’) + 𝑓𝑦 sin (πœ’)
,
σ΅„©σ΅„© σ΅„©σ΅„©
σ΅„©σ΅„©βˆ‡π‘“σ΅„©σ΅„©
𝑓𝑦 cos (πœ’) βˆ’ 𝑓π‘₯ sin (πœ’)
.
cos (πœ’1 ) =
σ΅„©σ΅„© σ΅„©σ΅„©
σ΅„©σ΅„©βˆ‡π‘“σ΅„©σ΅„©
(10)
4
Journal of Control Science and Engineering
The derivatives of the introduced variables are shown
as follows:
π‘₯1Μ‡ =
𝑑
𝑑 (π‘₯, 𝑦) = 𝑓π‘₯ π‘₯Μ‡ + 𝑓𝑦 𝑦̇
𝑑𝑑
σ΅„© σ΅„©
= 𝑓π‘₯ 𝑉𝑔 cos (πœ’) + 𝑓𝑦 𝑉𝑔 sin (πœ’) = 𝑉𝑔 σ΅„©σ΅„©σ΅„©βˆ‡π‘“σ΅„©σ΅„©σ΅„© sin (πœ’1 ) ,
𝑑
σ΅„© σ΅„©
πœ’1Μ‡ = βˆ’π‘˜1 σ΅„©σ΅„©σ΅„©βˆ‡π‘“σ΅„©σ΅„©σ΅„© 𝑉𝑔 𝑓sat (𝑑 (π‘₯, 𝑦)) βˆ’ π‘˜2 𝑉𝑔 𝑑 (π‘₯, 𝑦)
𝑑𝑑
(11)
σ΅„© σ΅„©
σ΅„© σ΅„©
= βˆ’π‘˜1 σ΅„©σ΅„©σ΅„©βˆ‡π‘“σ΅„©σ΅„©σ΅„© 𝑉𝑔 𝑓sat (π‘₯1 ) βˆ’ π‘˜2 𝑉𝑔2 σ΅„©σ΅„©σ΅„©βˆ‡π‘“σ΅„©σ΅„©σ΅„© sin (πœ’1 ) .
Theorem 1. If π‘˜1 is a positive constant, 𝑉𝑔 β‰₯ 𝑉𝑔𝑝 , and π‘˜2 β‰₯
π‘˜2𝑝 , where 𝑉𝑔𝑝 and π‘˜2𝑝 are positive constants, then the stable
equilibrium points of system (11) equal the set {(π‘₯1 , πœ’1 ) | π‘₯1 =
0, πœ’1 = 2π‘˜πœ‹, π‘˜ ∈ 𝑍}. Furthermore, they are all uniformly
asymptotically stable.
Proof. Rewrite system (11) as xΜ‡ = 𝑔(x, 𝑑) with x = [π‘₯1 ,
πœ’1 ]𝑇 and 𝑔(x) defined as
σ΅„© σ΅„©
𝑉𝑔 σ΅„©σ΅„©σ΅„©βˆ‡π‘“σ΅„©σ΅„©σ΅„© sin (πœ’1 )
] . (12)
𝑔 (x, 𝑑) = [
σ΅„© σ΅„©
σ΅„© σ΅„©
βˆ’π‘˜1 σ΅„©σ΅„©σ΅„©βˆ‡π‘“σ΅„©σ΅„©σ΅„© 𝑉𝑔 𝑓sat (π‘₯1 ) βˆ’ π‘˜2 𝑉𝑔2 σ΅„©σ΅„©σ΅„©βˆ‡π‘“σ΅„©σ΅„©σ΅„© sin (πœ’1 )
Let xΜ‡ = 𝑔(x, 𝑑) = 0. Since 𝑉𝑔 β‰₯ 𝑉𝑔𝑝 > 0, β€–βˆ‡π‘“β€– β‰₯ πœ† >
0, the solutions to 𝑔(x) = 0 are the set {(π‘₯1 , πœ’1 ) | π‘₯1 =
0, πœ’1 = π‘˜πœ‹, π‘˜ ∈ 𝑍}. Firstly, the stable equilibrium points
are {(π‘₯1 , πœ’1 ) | π‘₯1 = 0, πœ’1 = 2π‘˜πœ‹, π‘˜ ∈ 𝑍}. Without loss of
generality, we consider the equilibrium point (0, 0). Define
the domain 𝐷 = {(π‘₯1 , πœ’1 ) | π‘₯1 ∈ 𝑅, βˆ’ πœ‹ < πœ’1 < πœ‹}.
The Lyapunov candidate function is chosen as
π‘₯1
πœ’1
0
0
𝑉 (𝑑, x) = π‘˜1 ∫ 𝑓sat (𝑦) 𝑑𝑦 + ∫ sin (𝑦) 𝑑𝑦.
π‘₯
(13)
σ΅„©σ΅„©
2 σ΅„©
σ΅„©
{
{{x ∈ R | σ΅„©σ΅„©xβˆ’x0 σ΅„©σ΅„© ≀ 𝑉𝑔𝑝 }
𝐷1 = {
{{x ∈ R2 | σ΅„©σ΅„©σ΅„©xβˆ’x σ΅„©σ΅„©σ΅„© ≀ πœ‹ }
0σ΅„©
σ΅„©
2
{
πœ‹
2
πœ‹
if 𝑉𝑔𝑝 > .
2
if 𝑉𝑔𝑝 ≀
(16)
Then, on the domain 𝐷1 , the function 𝑔(x, 𝑑) can be
expressed as
σ΅„© σ΅„©
𝑔 (x,𝑑) = σ΅„©σ΅„©σ΅„©βˆ‡π‘“σ΅„©σ΅„©σ΅„© [
𝑉𝑔 sin (πœ’1 )
].
βˆ’π‘˜1 𝑉𝑔 π‘₯1 βˆ’ π‘˜2 𝑉𝑔2 sin (πœ’1 )
(17)
It can be found that 𝑔(x, 𝑑) is continuously differentiable
on 𝐷1 . The following Jacobian matrix can be derived as [21]
𝐴=
0
βˆ’π‘‰π‘”
πœ•π‘” 󡄨󡄨󡄨󡄨
σ΅„© σ΅„©
= σ΅„©σ΅„©σ΅„©βˆ‡π‘“σ΅„©σ΅„©σ΅„© [
].
(x)󡄨󡄨
πœ•x 󡄨󡄨x=x0
βˆ’π‘˜1 𝑉𝑔 π‘˜2 𝑉𝑔2
(18)
By solving the eigenvalues of matrix 𝐴, it gets that one
of them equals β€–βˆ‡π‘“β€–π‘‰π‘” (π‘˜2 𝑉𝑔 + βˆšπ‘˜22 𝑉𝑔2 + 4π‘˜1 )/2, which is
positive. Hence, the point x0 = [0, πœ‹]𝑇 is unstable [21].
Similarly, the equilibrium points in the set {(π‘₯1 , πœ’1 ) | π‘₯1 =
0, πœ’1 = (2π‘˜ + 1)πœ‹, π‘˜ ∈ 𝑍} are all unstable.
πœ’
Since ∫0 1 𝑓sat (𝑦)𝑑𝑦 > 0 and ∫0 1 sin(𝑦)𝑑𝑦 = 1 βˆ’ cos(πœ’1 ) >
0 for all x ∈ 𝐷 βˆ’ {0}, if π‘˜1 > 0, then 𝑉(𝑑, x) > 0 on 𝐷 βˆ’ {0}.
Furthermore,
π‘₯1
𝑉 (𝑑, x) = π‘˜1 ∫ 𝑓sat (𝑦) 𝑑𝑦 + 1 βˆ’ cos (πœ’1 )
0
≀
It can be derived that π‘Š3 (π‘₯1 , πœ’1 ) is a continuous positive
definite function on the domain 𝐷. Then, the equilibrium
point (0, 0) is uniformly asymptotically stable [21]. Similarly,
the equilibrium points in the set {(π‘₯1 , πœ’1 ) | π‘₯1 = 0, πœ’1 =
2π‘˜πœ‹, π‘˜ ∈ 𝑍} are all uniformly asymptotically stable.
The following text indicates that the points in the set
{(π‘₯1 , πœ’1 ) | π‘₯1 = 0, πœ’1 = (2π‘˜ + 1)πœ‹, π‘˜ ∈ 𝑍} are unstable
when π‘˜1 , π‘˜2 , and 𝑉𝑔 are all positive constants. Without loss
of generality, we focus on the point x0 = [0, πœ‹]𝑇 and define
the domain 𝐷1 as
πœ’2
π‘˜1 2
π‘˜
π‘₯1 + 1 βˆ’ cos (πœ’1 ) ≀ 1 π‘₯12 + 1
2
2
2
(14)
= π‘Š1 (π‘₯1 , πœ’1 ) ,
𝑉 (𝑑, x) β‰₯ 1 βˆ’ cos (πœ’1 ) = 2sin2 (
πœ’1
) = π‘Š2 (π‘₯1 , πœ’1 ) ,
2
where π‘Š1 (π‘₯1 , πœ’1 ) and π‘Š2 (π‘₯1 , πœ’1 ) are continuous positive
definite functions on 𝐷. Additionally, we have the fact that
πœ•π‘‰ πœ•π‘‰
𝑉̇ (𝑑, x) =
+
𝑔 (x, 𝑑)
πœ•π‘‘
πœ•x
= 2 (π‘˜1 π‘₯1Μ‡ 𝑓sat (π‘₯1 ) + πœ’1Μ‡ sin (πœ’1 ))
=
βˆ’2π‘˜2 𝑉𝑔2
σ΅„©σ΅„© σ΅„©σ΅„© 2
σ΅„©σ΅„©βˆ‡π‘“σ΅„©σ΅„© sin (πœ’1 )
2
πœ† sin2 (πœ’1 ) = βˆ’π‘Š3 (π‘₯1 , πœ’1 ) .
≀ βˆ’2π‘˜2 𝑉𝑔𝑝
(15)
3.2. The Phase Plot of System (11) with Different Parameters π‘˜2 .
In order to analyze the dynamic performance of the nonlinear
system (11), the line style and the circle style paths are chosen
as the case study. For the line style path, the function 𝑓(π‘₯, 𝑦)
is chosen as 𝑓(π‘₯, 𝑦) = π‘Žπ‘₯ + 𝑏𝑦 + 𝑐 with β€–βˆ‡π‘“β€– = βˆšπ‘Ž2 + 𝑏2 = 1,
while for the circle style path following, the function 𝑓(π‘₯, 𝑦)
is chosen as 𝑓(π‘₯, 𝑦) = √(π‘₯ βˆ’ π‘₯0 )2 + (𝑦 βˆ’ 𝑦0 )2 βˆ’ π‘Ÿ, where
(π‘₯0 , 𝑦0 ) and π‘Ÿ represent the center and the radius of the
circle, respectively. It can be found that with the circle style
path function 𝑓(π‘₯, 𝑦) chosen as above, the magnitude of
corresponding gradient β€–βˆ‡π‘“β€– = 1. Thus Figure 2 which
presents the phase plot of system (11) with β€–βˆ‡π‘“β€– = 1 can
display the dynamics of the line- or the circle-style path
following.
Figure 2(a) shows the dynamics of the nonlinear system
(11) under different initial values. It displays the fact that if
the vehicle is far away from the desired path, the vehicle will
firstly adjust the course to an approximate invariant value
and then flies towards the path until the absolute value of the
distance 𝑑(π‘₯, 𝑦) is smaller than 𝑉𝑔 . Finally, some oscillations
appear while the vehicle is approaching the desired path.
During the proof process of Theorem 1, the positive
parameter π‘˜2 is variable if it is larger than a given positive
Journal of Control Science and Engineering
5
20
40
10
20
0
x1 (m)
x1 (m)
60
0
βˆ’10
βˆ’20
βˆ’20
βˆ’30
βˆ’40
βˆ’40
βˆ’60
βˆ’4
βˆ’2
0
πœ’1 (rad)
(0.2πœ‹, 50)
(0.2πœ‹, βˆ’50)
2
4
βˆ’50
βˆ’0.5
0
0.5
1
πœ’1 (rad)
k1 = 0.0006, k2 = 0.0006
k1 = 0.0006, k2 = 0.0008
(βˆ’0.2πœ‹, 50)
(βˆ’0.2πœ‹, βˆ’50)
(a)
k1 = 0.0006, k2 = 0.001
k1 = 0.0006, k2 = 0.0012
(b)
Figure 2: The phase plot of system (10) with β€–βˆ‡π‘“β€– = 1 and 𝑉𝑔 = 20 m/s. (a) Under four kinds of different initial values with π‘˜1 = 0.0006 and
π‘˜2 = 0.0008; (b) under four kinds of different parameter π‘˜2 .
constant, while the parameter π‘˜1 is chosen as a positive
constant to get (15) which is used to be one of the conditions
to guarantee the stability. Thus, the following study only
considers how the variable parameter π‘˜2 will influence the
performance of the proposed guidance law.
Figure 2(b) shows the different trajectories of system
(11) with the control parameters couple as (0.0006, 0.0006),
(0.0006, 0.0008), (0.0006, 0.001), and (0.0006, 0.0012) but
under the same initial conditions, respectively. The system
is convergent for the given parameters couple. However, the
convergence speeds are quite different due to the different
control parameters. With larger parameter π‘˜2 , the trajectories
in the plane of phases tend to equilibrium more directly.
Nevertheless, while implementing the control law on a real
vehicle, if the vehicle flies near the predefined path, the control law is similar to the traditional PD controller. The larger
parameter π‘˜2 will lead to high-frequency oscillations and
weaken the disturbance rejection. The following subsection
shows how to adjust the parameter π‘˜2 to improve the path
following performance.
3.3. Path Following Controller with Gain Tuned by Fuzzy
Logic (FL_PFC). As stated in the last subsection, the control
gain π‘˜2 should be adjusted to improve the path following
performance. In [8], the authors have applied the fuzzy logic
to generate the guidance law. Due to its simple structure and
comprehensible concept, we adopted the advantages of the
fuzzy logic and applied it to tune the control gain π‘˜2 . Figure 3
presents the general structure of the proposed controller. The
control parameter π‘˜2 is adjusted around a given nominal
value π‘˜20 dynamically by the fuzzy logic unit as
π‘˜2 = π‘˜20 + Ξ”π‘˜2 ,
(19)
Table 1: Rule base for the fuzzy logic unit.
Ξ”π‘˜2
𝑑 Μ‡ (π‘₯, 𝑦)
NB
NM
NS
Z
PS
PM
PB
NB
PB
Z
Z
Z
Z
Z
Z
NM
PB
PS
PS
PM
PM
PS
Z
NS
Z
NS
NS
Z
PS
PS
Z
𝑑 (π‘₯, 𝑦)
Z
Z
PS
PS
PS
PS
Z
Z
PS
Z
PS
PS
Z
NS
NS
Z
PM
NM
NS
NS
NS
NS
NS
NB
PB
Z
Z
Z
Z
Z
Z
NB
where Ξ”π‘˜2 is the output of the fuzzy logic unit for adjusting
the parameter π‘˜2 . As (13) and (15) show, while taking the
procedure of (19), the adjusting parameter π‘˜2 should be
larger than a given positive constant; then system (11) has
the asymptotically stable equilibrium points {(π‘₯1 , πœ’1 ) | π‘₯1 =
0, πœ’1 = 2π‘˜πœ‹, π‘˜ ∈ 𝑍}.
Figure 4 shows the triangular, Z-shaped, and S-shaped
membership functions used for the fuzzification of the inputs
and output, that is, the distance 𝑑(π‘₯, 𝑦), the derivative
Μ‡ 𝑦), and the adjusting parameter Ξ”π‘˜ .
of the distance 𝑑(π‘₯,
2
Here, NB, NM, NS, Z, PS, PM, and PB stand for negative
big, negative medium, negative small, zero, positive small,
positive medium, and positive big, respectively.
Table 1 presents the rule base used for adjusting the
parameter Ξ”π‘˜2 . The fuzzy rule base contains 49 rules which
will generate Ξ”π‘˜2 based distance 𝑑(π‘₯, 𝑦) and its derivative
Μ‡ 𝑦). For example, the rule (IF
with respect to the time 𝑑(π‘₯,
Μ‡
𝑑(π‘₯, 𝑦) is PS AND 𝑑(π‘₯, 𝑦) is PS THEN Ξ”π‘˜2 is NS) is based on
the consideration that if the UAV is flying near the predefined
path, the total control command 𝑒cmd should not be large.
6
Journal of Control Science and Engineering
Path following controller
Path
generator
d(x, y)
ucmd
Lyapunov-stable
based control law
k2
Μ‡
d d(x, y)
dt
Stable
inner loop
Actuator
input
UAV
Throttle
input
πœƒcmd
Ξ”k2 +
Fuzzy
logic
Altitude- and
airspeed-hold
+
States
k20
Figure 3: The structure of the proposed controller.
πœ‡(d(x, y))
1NB
NM
NS
Z
PS
PM
PB
0
βˆ’60
βˆ’40
βˆ’20
0
d(x, y) (m)
20
40
60
1NB
NM
NS
Z
PS
PM
PB
βˆ’20
βˆ’10
0
Μ‡ y) (m/s)
d(x,
10
20
30
PS
PM
PB
0.5
Μ‡ y))
πœ‡(d(x,
(a)
0.5
0
βˆ’30
(b)
πœ‡(Ξ”k2 )
1NB
NM
NS
Z
0.5
0
βˆ’8
βˆ’6
βˆ’4
βˆ’2
0
Ξ”k2
2
4
6
8
×10βˆ’4
(c)
Μ‡ 𝑦) and the output Ξ”π‘˜ , respectively.
Figure 4: The membership functions used for the fuzzification of the inputs 𝑑(π‘₯, 𝑦) and 𝑑(π‘₯,
2
With the negative small parameter Ξ”π‘˜2 , the parameter π‘˜2 and
the corresponding 𝑒cmd are reduced. The well-known minmax inference method and the center-of-area defuzzification
procedure are used to execute the fuzzy unit.
Figure 5 shows the input-output characteristics of the
fuzzy logic unit. The minimum of the output is βˆ’0.0007302.
In this paper, the control parameters π‘˜1 and π‘˜20 are chosen as
0.0006 and 0.0008, respectively. Then one has the fact that π‘˜2
is always positive with π‘˜2 > 0.00006.
4. Simulations
The simulations about the proposed controller were studied
in the Matlab/Simulink environment [19]. The Aerosonde
UAV model [22] was applied as a test vehicle while doing
the simulation, where the UAV model was designed by using
the Aerosim Aeronautical Simulation Block Set [23] which
provides a complete set of tools for development of detailed
6-degree-of-freedom nonlinear models. Figure 6 presents
Journal of Control Science and Engineering
7
×10βˆ’4
Ξ”k2
5
0
βˆ’5
βˆ’60
βˆ’40
(60, 30, βˆ’0.0007302)
βˆ’20
0
d(x
,y
) (m 20
40
)
60 βˆ’30
βˆ’20
βˆ’10
0
20
10
30
/s)
) (m
ḋ (x, y
Figure 5: Input-output surface for the fuzzy logic unit.
0
Flap
Elevator
Ailerons
Controls
Rudder
0.6
Throttle
13
Mixture
1
Ignition
[0 8 0]σ³°€
Winds
Winds
0
Reset
RST
States
Sensors
VelW
Mach
Ang Acc
Euler
AeroCoeff
PropCoeff
EngCoeff
Mass
ECEF
MSL
AGL
REarth
AConGnd
Aerosonde UAV
Pitch angle (deg)
Stop
Stop simulation
Pitch angle
Rudder
REarth
Altitude (m)
Ailerons
Altitude setpoint
Altitude
N
1000
Altitude setpoint
Eular angles
Yaw rate (rad/s)
Elevator
Yaw rate command (rad/s)
E
Smooth_Path_Following_Controller
VN
Distance
Stable inner loop controller
Smooth path following controller
Parameter k2
Aricraft states
VE
States
Figure 6: The Simulink block diagram for the simulation studies.
the Simulink model which contains the proposed smooth
path following controller, static inner loop controller, and
the UAV model used for the simulation studies. The inputs
of the smooth path following controller from the states are
the inertial position (π‘₯, 𝑦) and velocity (𝑉π‘₯ , 𝑉𝑦 ) which will be
used to generate the vehicle’s current course.
The initial position of the UAV was at the origin of the
local coordinate system, and its initial heading was set as
360 deg north, respectively. For simulation, the initial inertial
speed of the UAV was set as 25 m/s. In order to test the
performance of the proposed controller, the wind with 8 m/s
from 270 deg west was added to the model of the UAV as
8
Journal of Control Science and Engineering
50
2500
2000 Wind: 8 m/s
1500
A
0
B
D
C
E
(2100, 2100)
500
F
d(x, y) (m)
North (m)
1000
Start
(100, 4100)
(100, 100)
0
βˆ’500
βˆ’1000
βˆ’50
βˆ’100
βˆ’150
(2100, βˆ’1900)
βˆ’1500
βˆ’2000
0
1000
2000 3000
East (m)
Predefined path
FL-PFC
4000
βˆ’200
5000
0
100
200
400
500
Time (s)
600
700
800
FL-PFC
PFC
PFC
(a)
×10
1.5
300
(b)
βˆ’3
k2
1
0.5
0
0
100
200
300
400
Time (s)
500
600
700
800
500
600
700
800
FL-PFC
PFC
(c)
Vg (m/s)
40
30
20
10
0
100
200
300
400
Time (s)
FL-PFC
PFC
(d)
Figure 7: Comparison of the square path following with the methods FL_PFC and PFC. (a) The position compassion, (b) the distance 𝑑(π‘₯, 𝑦)
compassion, (c) the parameter π‘˜2 compassion, and (d) the inertial velocity 𝑉𝑔 .
shown in Figure 6. The simulation sample time was set as
50 ms. While conducting the simulations, we applied the
proposed method to follow three types of planar path, that
is, the square-, circular-, and athletics track-style paths in
Matlab environment, respectively.
Case 1 (square path following). The desired horizontal
square path was defined with four waypoints, (100, 100),
(2100, 2100), (100, 4100), and (2100, βˆ’1900), as shown in
Figure 7(a). The comparison of the path following with the
methods PFC and FL_PFC can also be seen in Figure 7(a).
Figure 7(b) is the comparison of the distance while following
the predefined square path with the controllers PFC and
FL_PFC. While executing the square line path following,
the predefined behavior that the UAV will switch course
in advance with 200 meters was adopted, which results in
the sharp changes of about 200 meters. The elliptic marks
A, B, C, D, E, and F in Figure 7(b) indicate that the
overshoot of the path following and the oscillation amplitude
by using the method FL_PFC are both smaller than those
by using the method PFC. And the maximum overshoots
and time difference of arrival (TDOA, defined as the time
difference (𝑑PFC βˆ’ 𝑑FL_PFC ) between the two methods for the
vehicle to do the switching maneuver near the waypoint)
displayed in the areas A, B, C, D, E, and F are listed in
Table 2.
Journal of Control Science and Engineering
9
600
90
Wind: 8 m/s
80
500
70 10
60
300
5
50
D(x, y) (m)
North (m)
400
0
40
30 βˆ’5
200
5 10 15
20
10
100
0
0
βˆ’300
βˆ’200
0
βˆ’100
100
200
300
βˆ’10
0
50
100
Time (s)
East (m)
Predefined circular path
FL-PFC
PFC
k2
200
FL-PFC
PFC
(a)
×10
10
150
(b)
βˆ’4
8
6
0
20
40
60
80
100
Time (s)
120
140
160
180
200
120
140
160
180
200
FL-PFC
PFC
(c)
Vg (m/s)
40
20
0
0
20
40
60
80
100
Time (s)
FL-PFC
PFC
(d)
Figure 8: Comparison of the circular path following with the methods FL_PFC and PFC. (a) The position compassion, (b) the distance
𝑑(π‘₯, 𝑦) compassion, (c) the parameter π‘˜2 compassion, and (d) the inertial velocity 𝑉𝑔 .
Figure 7(c) shows the parameterused for the methods
FL_PFC and PFC, respectively. With π‘˜2 adjusted, the course
rate command was derived to drive the UAV to follow the
path with better performance as shown in Figure 7(b).
Figure 7(d) presents the comparison of the inertial velocity
𝑉𝑔 while following the predefined path with the two proposed
methods. The inertial velocity 𝑉𝑔 is changing due to the
influence of the constant wind speed. However, the inertial
velocity 𝑉𝑔 with the method FL_PFC is smoother than
that with the method PFC. It indicates that the following
performance is modified by the fuzzy logic rule. By adjusting
the parameter π‘˜2 , the overshoots and oscillation will be
reduced, and the quickness of the path following will be
maintained. It can also be seen from the TDOA listed in
Table 2 that, by using the method FL_PFC, the vehicle can
reach the set waypoints more quickly. So if a predefined
flight path is long, the vehicle can finish it with shorter time
by using the method FL_PFC.
10
Journal of Control Science and Engineering
1000
120
100
80
North (m)
d(x, y) (m)
Wind: 8 m/s
500
Start
0
60
40
20
0
βˆ’500
0
200 400 600 800 1000 1200 1400 1600
East (m)
Predefined path
FL-PFC
βˆ’20
PFC
0
50
100
150
Time (s)
200
250
300
FL-PFC
PFC
(a)
(b)
Figure 9: Comparison of the Athletics track-style path following with the methods FL_PFC and PFC. (a) The position compassion and (b)
the distance 𝑑(π‘₯, 𝑦) compassion.
Table 2: Comparison of the maximum overshoot with the two
proposed methods.
Area A
Area B
Area C
Area D
Area E
Area F
Maximum overshoot (m)
FL_PFC
PFC
0.3404
13.54
1.981
13.54
4.576
29.26
0.606
14.76
0.3444
5.575
0.5554
10.38
TDOA (s)
𝑑PFC βˆ’ 𝑑FL_PFC
1.5
3.7
10.2
15.1
19.2
22
As Figure 7(d) and Table 2 show, while turning near
the point (2100, βˆ’1900), the maximum overshoot in the
area C presents the fact that, with the two orthogonal
components of the added wind acting on the fuselage axis
and the lateral axis of the UAV, the wind from 270 deg west
influences the turning of the UAV dramatically. With the
method PLC, the maximum overshoot in area C is 29.26 m; in
contrast, that with the method FL_PFC is only 4.576 m. This
result demonstrates the effectiveness of the path following
controller FL_PFC.
Case 2 (circular path following). The desired horizontal
circular path was defined as √π‘₯2 + (𝑦 βˆ’ 330)2 βˆ’ 250 = 0.
Figure 8(a) shows that the vehicle will follow the predefined
circular path after the approximation of transition with an
approximate invariant course value. The zoom in detail in
Figure 8(b) indicates that the vehicle can reach the predefined
path with smaller overshoot by using the method FL_PFC.
The oscillations that appear from time of 80 s to 110 s were
caused by the wind disturbance.
Case 3 (athletics track-style path following). In this case,
the athletics track-style path was adopted by combining the
line-style and circular-style paths together. The simulation
results are shown in Figure 9. Figure 9(b) indicates that,
compared with the method PFC, the method FL_PFC will
guide the UAV towards the predefined path faster together
with smaller oscillation amplitude.
5. Conclusions
A horizontal smooth path following guidance law for a small
unmanned air vehicle is presented in this paper. With the
combination of the Lyapunov stability and the inference
ability of fuzzy logic, the proposed path following controller
FL_PFC still has its stability guaranteed by the mechanism
of the Lyapunov stability arguments and adopts the simple
structure of the method PFC. With a key parameter of
the guidance law adjusted based on fuzzy logic technique,
the path following performance was improved with smaller
overshoot and oscillation amplitude and shorter reaching
time even in the presence of wind with 8 m/s. The proposed
methods were successfully implemented in following three
types of planar path, that is, the square-, circular-, and athletics track-style paths via simulations in the Matlab/Simulink
environment.
Our future work will focus on developing a 3D smooth
path following controller; the control parameters will also
be adjusted with the help of the fuzzy logic or optimization
techniques to improve the path following performance.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science
Foundation of China (no. 61503172), the National High
Journal of Control Science and Engineering
Technology Research and Development Program of China
(no. 2011AA040202), and the 2016 Provincial Program to Fostering Distinguished Young Scholars of Scientific Research in
Universities of Fujian Province.
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http://www.u-dynamics.com/.
International Journal of
Rotating
Machinery
Engineering
Journal of
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Volume 2014
The Scientific
World Journal
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International Journal of
Distributed
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Sensors
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Volume 2014
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Volume 2014
Hindawi Publishing Corporation
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Journal of
Control Science
and Engineering
Advances in
Civil Engineering
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Hindawi Publishing Corporation
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Volume 2014
Volume 2014
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Journal of
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Electrical and Computer
Engineering
Robotics
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Hindawi Publishing Corporation
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Volume 2014
Volume 2014
VLSI Design
Advances in
OptoElectronics
International Journal of
Navigation and
Observation
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Volume 2014
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Chemical Engineering
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Volume 2014
Volume 2014
Active and Passive
Electronic Components
Antennas and
Propagation
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Aerospace
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Volume 2014
International Journal of
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Modelling &
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Volume 2014
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Shock and Vibration
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Advances in
Acoustics and Vibration
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