SUGISAKAMASANORI

Journal of Advances in Artificial Life Robotics
Vol. 1(3); December (2020), pp. 106–111
ON LINE ISSN 2435-8061; ISSN-L 2435-8061
https://alife-robotics.org/jallr.html

Research Article

Wheeled Mobile Robot Robust Control Based on Hybrid
Controller Approach
Ho-Nien Shou

Department of Aviation & Communication Electronics, Air Force Institute of Technology, No.198, Jieshou W. Rd., Gangshan District, Kaohsiung,TaiwanARTICLE INFO
Article History
Received 23 Octoberr 2019
Accepted 01August 2020
Keywords
Nonholonomic mobile robot
Cerebellar model articulation controller
Lyapunov stability criterionDisturbance observer

ABSTRACTIn this paper, a novel trajectory following method based on hybrid control strategy is proposed
for the trajectory following problem of mobile robots with nonholonomic constraints. The mobilerobot controller is composed of three parts: kinematics, dynamics equations and Cerebellar Model
Articulation Controller. The control speed needed to realize the reference trajectory tracking is
obtained from the motion equation under the position tracking error. The virtual control value ofthe control speed is obtained by using Lyapunov, and the controller is designed in the dynamicequation. The cerebellar model articulation controller (CMAC) is used to approximate thenonlinearity and uncertainty of the dynamic model of the mobile robot. At the same time, thetorque controller is combined with the velocity error to form the torque controller. The influenceof the uncertain disturbance on the system is compensated on-line. In the discussion in this paper,
the nonlinear system is consider, and the Lyapunov stability criterion guarantees the global
stability of the system and the asymptotical convergence of tracking error. The simulation resultsin Matlab / Simulink environment further verify the effectiveness and superiority of the proposedcontrol algorithm.

© 2022 The Author. Published by Sugisaka Masanori at ALife Robotics Corporation Ltd.
This is an open access article distributed under the CC BY-NC 4.0 license(http://creativecommons.org/licenses/by-nc/4.0/).

Path following control only follows a designed pathregardless of the arriving time of the specific position.
According to some control theory, a control law isdesigned for the nonholonomic mobile robot system, sothat the nonholonomic mobile robot can reach andeventually follow a given path on the motion plane at agiven speed. Here, a given path is also called the expected
path, which refers to a geometric curve, 𝑓𝑓(𝑥𝑥𝑟𝑟 , 𝑦𝑦𝑟𝑟 , 𝜃𝜃𝑟𝑟 ), thecurve equation does not contain time parameters, and eachargument is not a function of time, Where 𝑥𝑥𝑟𝑟 is thereference coordinate x, 𝑦𝑦𝑟𝑟 is the reference coordinate y, and𝜃𝜃𝑟𝑟 is the WRM body direction, which is represented by thereference coordinates (𝑥𝑥, 𝑦𝑦).
The given velocity refers to the given linear velocity
𝑣𝑣𝑟𝑟 and angular velocity 𝜔𝜔𝑟𝑟 , also known as the expected
speed. The reference control input expressed in 𝑢𝑢𝑟𝑟 =[𝑣𝑣𝑟𝑟 , 𝜔𝜔𝑟𝑟 ]𝑇𝑇 . From the physical point of view, the valuechange of 𝑣𝑣𝑟𝑟 and 𝜔𝜔𝑟𝑟 is limited by the specific form of

1. Introduction
Many literatures which study nonholonomic mobilerobots are usually focus on wheeled mobile robots (WMR).
In the theoretical research of WMR’s motion control, ingeneral, only pure rolling condition is considered. In otherwords, it is assumed that no slip condition (including
lateral and longitudinal sliding) occurs. This ideal
constraint is essentially a kind of nonholonomic constraint,
therefore, WMR is a typical case of the nonholonomicsystem. In this paper, the control of WMR is studied.
According to different control objectives, the control
problems of nonholonomic systems can be divided into
three categories [1],[2],[3]: position stabilization,
trajectory tracking and path following.
Corresponding author’s E-mail: honien.shou@gmail.com

106

𝑓𝑓(𝑥𝑥𝑟𝑟 , 𝑦𝑦𝑟𝑟 , 𝜃𝜃𝑟𝑟 ), and also can not include the time parameter,and 𝑣𝑣𝑟𝑟 > 0 is required. 𝑢𝑢𝑟𝑟 and 𝑓𝑓(𝑥𝑥𝑟𝑟 , 𝑦𝑦𝑟𝑟 , 𝜃𝜃𝑟𝑟 ) can be given
in advance or generated by path generator. Hence, we canregard the path following problem as a special case of thetrajectory tracking problem since the former is much easier
to be deal with compared to the later.
The organization of this paper is in the following.
Section 1, we introduction and briefly discuss the basic
concepts of path following control. The path following
control of nonholonomic WMR are summarized theresearch motivation and main structure of this paper areillustrated in the Section 2 paragraph. In Section 2, thecontrol path following of nonholonomic WMR issummarized, and the research motivation and main
structure of this paper are described. It also describes thesecontrollers constructed from kinematics and dynamics
equations. The cerebellar model articulation controller
algorithm similar to neural network is used to design
online compensation system and path error to overcome
the uncertainty of WRM, which will be described in
Section 3. compensation system for uncertainties and thecontroller. The cerebellar neural network weight updating
algorithm and kernel space algorithm are introduced in
Section 3. The architecture of the WRM robust hybrid
controller in Section 4. Section 5 given the numericalsimulation results of WRM and conclusions.

Figure 1 WMR nonholonomic mobile robot.
Figure 1 illustrates a three wheels mobile robot, inwhich d is the distance between robot’s mass center 𝑃𝑃𝐶𝐶 , 𝑃𝑃𝑂𝑂
denotes the geometric center, 2𝑏𝑏 is the distance between
two driving wheels, and 𝑟𝑟 denotes the wheel radius. 𝑞𝑞 =[𝑥𝑥 𝑦𝑦 𝜃𝜃]𝑇𝑇 represents robot’s position and orientation,𝑢𝑢 = [𝑣𝑣 𝜔𝜔]𝑇𝑇 represents velocity and angular velocity,𝝉𝝉 = [𝜏𝜏1 𝜏𝜏2 ]𝑇𝑇 is control torque, 𝐽𝐽 denotes the inertia
moment, and 𝑚𝑚𝑎𝑎 denotes the mass of the WMR. The
parameters in each matrix appeared in Eqs. (1) and (2) are𝑀𝑀(𝑞𝑞) =𝑚𝑚𝑎𝑎
0𝑚𝑚𝑎𝑎 𝑑𝑑 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃−𝑚𝑚𝑎𝑎 𝑑𝑑 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃� ,𝐶𝐶(𝑞𝑞, 𝑞𝑞̇ ) =0𝑚𝑚𝑎𝑎
�𝐽𝐽𝑚𝑚𝑎𝑎 𝑑𝑑 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃 −𝑚𝑚𝑎𝑎 𝑑𝑑 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃
0 0 𝑚𝑚𝑎𝑎 𝑑𝑑𝜃𝜃̇ 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃�0 0 𝑚𝑚𝑎𝑎 𝑑𝑑𝜃𝜃̇ 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃 �, 𝐵𝐵(𝑞𝑞) =0 00− 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃1� 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃 �,𝐴𝐴𝑇𝑇 (𝑞𝑞) = � 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃 �,𝑆𝑆(𝑞𝑞) =𝑟𝑟−𝑑𝑑𝑏𝑏−𝑏𝑏𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃 −𝑑𝑑 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃� 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃 𝑑𝑑 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃 �01
We can express nonholonomic constraints as𝐴𝐴(𝑞𝑞)𝑞𝑞̇ = 0.
(3)
Set a n-m dimensional full rank matrix 𝑆𝑆(𝑞𝑞) as a baseset in null space 𝐴𝐴(𝑞𝑞) such that𝐴𝐴(𝑞𝑞)𝑆𝑆 𝑇𝑇 (𝑞𝑞) = 0.
(4)
We can obtain an auxiliary velocity control input 𝑢𝑢 ∈ℝ𝑛𝑛−𝑚𝑚 from Eqs. (2) and (4),𝑞𝑞̇ = 𝑆𝑆(𝑞𝑞)𝑢𝑢.
(5)
From Eq. (5) we can have𝑞𝑞̈ = 𝑆𝑆(𝑞𝑞)𝑢𝑢̇ + 𝑆𝑆(𝑞𝑞̇ )𝑢𝑢.
(6)

2. Kinematic Model and Controller Design
2.1 Model of WMR
The dynamics equations of a nonholonomic mobile
robot with n-dimensional state space and subject to m
constraints read [4],[5],[6],[7]𝑀𝑀(𝑞𝑞)𝑞𝑞̈ (𝑡𝑡) + 𝐶𝐶(𝑞𝑞, 𝑞𝑞̇ )𝑞𝑞̇ + 𝐺𝐺(𝑞𝑞) + 𝐹𝐹(𝑞𝑞̇ ) + 𝝉𝝉𝑑𝑑 = 𝐵𝐵(𝑞𝑞)𝝉𝝉 −𝐴𝐴𝑇𝑇 (𝑞𝑞)𝜆𝜆 (1)𝑞𝑞̇ = 𝑆𝑆(𝑞𝑞)𝑢𝑢(2)
where 𝑀𝑀(𝑞𝑞) ∈ ℝ𝑛𝑛×𝑛𝑛 is a symmetric, positive definite
inertia matrix, 𝐶𝐶(𝑞𝑞, 𝑞𝑞̇ ) ∈ ℝ𝑛𝑛×𝑛𝑛 is the centripetal andCoriolis matrix, 𝐺𝐺(𝑞𝑞) ∈ ℝ𝑛𝑛 is the gravitational vector, and𝐹𝐹(𝑞𝑞̇ ) ∈ ℝ𝑛𝑛 is the surface friction. To a mobile robot
moving on a smooth plane, vectors 𝐺𝐺(𝑞𝑞) and 𝐹𝐹(𝑞𝑞̇ ) areequal to zero. 𝜏𝜏𝑑𝑑 ∈ ℝ𝑛𝑛 denotes bounded unknown
disturbances, 𝝉𝝉 ∈ ℝ𝑝𝑝 denotes the control input, 𝐵𝐵(𝑞𝑞) ∈ℝ𝑛𝑛×𝑝𝑝 denotes the input transformation matrix, 𝜆𝜆 ∈ ℝ𝑚𝑚 is
the constraint force vector, and 𝐴𝐴(𝑞𝑞) ∈ ℝ𝑚𝑚×𝑛𝑛 is theconstraint matrix.

107Lemma 1: To any 𝜑𝜑 ∈ ℝ and ‖𝜑𝜑‖ ∈ ℝ , the condition
𝑓𝑓(𝜑𝜑) = 𝜑𝜑 𝑠𝑠𝑠𝑠𝑠𝑠(𝑎𝑎𝑎𝑎𝑎𝑎 𝑡𝑡𝑡𝑡𝑡𝑡(𝜑𝜑)) ≥ 0 is true, if and only if 𝜑𝜑 =0.

By inserting Eq. (6) into Eq. (1) and multiplying by
𝑆𝑆 𝑇𝑇 (𝑞𝑞) to cancel the constraint matrix 𝐴𝐴𝑇𝑇 𝜆𝜆, we obtain thedynamic equation of the nonholonomic mobile robot:
𝑆𝑆 𝑇𝑇 𝑀𝑀𝑀𝑀𝑢𝑢̇ + 𝑆𝑆 𝑇𝑇 �𝑀𝑀𝑆𝑆̇ + 𝐶𝐶𝐶𝐶�𝑢𝑢 + 𝑆𝑆 𝑇𝑇 𝝉𝝉𝑑𝑑 = 𝑆𝑆 𝑇𝑇 𝐵𝐵𝝉𝝉
(7)
After variable replacements, Eq. (7) becomes𝑀𝑀̄(𝑞𝑞)𝑢𝑢̇ + 𝐶𝐶̄ (𝑞𝑞, 𝑞𝑞̇ )𝑢𝑢 + 𝝉𝝉̄ 𝑑𝑑 = 𝝉𝝉̄(8)where 𝑀𝑀̄(𝑞𝑞) ∈ ℝ𝑝𝑝×𝑝𝑝 is the symmetric positive definite
inertia matrix, 𝐶𝐶̄ (𝑞𝑞, 𝑞𝑞̇ ) ∈ ℝ𝑝𝑝×𝑝𝑝 is matrix for centripetal andCoriolis forces, 𝑆𝑆 𝑇𝑇 𝜏𝜏𝑑𝑑 = 𝝉𝝉̄ 𝑑𝑑 ∈ ℝ𝑝𝑝 contains unstructured
unmodeled dynamic bounded unknown perturbations, and𝑆𝑆 𝑇𝑇 𝐵𝐵𝐵𝐵 = 𝝉𝝉̄ ∈ ℝ𝑝𝑝 is control input matrix；in which 𝑀𝑀̄ =0𝑚𝑚𝑎𝑎
� , 𝐶𝐶̄ = 02×2 , 𝝉𝝉̄ 𝑑𝑑 = 𝑆𝑆 𝑇𝑇 (𝑞𝑞)𝝉𝝉𝑑𝑑 , and 𝜏𝜏̄ =�
0 𝐽𝐽 − 𝑚𝑚𝑎𝑎 𝑑𝑑 2
𝑆𝑆 𝑇𝑇 (𝑞𝑞)𝐵𝐵(𝑞𝑞).

B.𝜏𝜏̄𝑑𝑑 Disturbance Observer Design

There are two main factors affecting the robustnessof the WMR system. One is the interference from theoutside world; while the other is the uncertainty of theinternal parameters of the whole system. In fact, the WMR
will be affected by friction and various noise such that
there are some differences compared to the ideal system.
The disturbance observer is designed to estimate theexternal disturbance, 𝜏𝜏̄𝑑𝑑 . The external disturbance
estimator designed for the WMR system is usually in theform of [3],[6],[7]:𝜏𝜏̄̂𝑑𝑑 = 𝑧𝑧 + 𝐿𝐿𝐿𝐿
�−1𝑧𝑧̇ = 𝐿𝐿𝑀𝑀̄ 𝑧𝑧 − 𝐿𝐿(−𝑀𝑀̄−1 𝐿𝐿𝐿𝐿 − 𝑀𝑀̄−1 𝐶𝐶̄ 𝑢𝑢 + 𝑀𝑀̄−1 𝜏𝜏̄ )(13)
where 𝜏𝜏̄̂𝑑𝑑 denotes the estimation of the unknown external
disturbance 𝜏𝜏̄𝑑𝑑 , 𝑧𝑧 is the internal state of the nonlinear
estimator, 𝐿𝐿 is the parameter of the nonlinear estimatorneeded to be solved, which is usually expressed in termsof the constant matrix.
Some assumptions are needed to control the WMR to
track the target trajectory. Let us state these assumptionsin the following.
Assumption 1: The disturbances ‖𝝉𝝉̄ 𝑑𝑑 ‖ and �𝝉𝝉̄̇ 𝑑𝑑 � arebounded, and 𝝉𝝉̄ 𝑑𝑑 is constant in the steady state, i.e.,𝑙𝑙𝑙𝑙𝑙𝑙 �𝝉𝝉̄̇ 𝑑𝑑 � = 0.

2.2 Kinematic Controller Design𝑞𝑞𝑟𝑟 = [𝑥𝑥𝑟𝑟 𝑦𝑦𝑟𝑟 𝜃𝜃𝑟𝑟 ]𝑇𝑇 is the reference orientation ofthe WMR
𝑥𝑥̇ 𝑟𝑟
𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃𝑟𝑟 −𝑑𝑑 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃𝑟𝑟 𝑣𝑣𝑟𝑟
�𝑦𝑦̇𝑟𝑟 � = � 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃𝑟𝑟 𝑑𝑑 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃𝑟𝑟 � �𝜔𝜔 �𝑟𝑟
𝜃𝜃̇𝑟𝑟
01(9)
where 𝑥𝑥𝑟𝑟 ,𝑦𝑦𝑟𝑟 ,𝜃𝜃𝑟𝑟 ,𝑣𝑣𝑟𝑟 and 𝜔𝜔𝑟𝑟 are the expectations of𝑥𝑥,𝑦𝑦,𝜃𝜃, 𝑣𝑣 and 𝜔𝜔.

A. Kinematics Controller Design

Let us define tracking error as𝑥𝑥𝑟𝑟 − 𝑥𝑥𝑥𝑥𝑒𝑒
𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃 0 𝑥𝑥𝑟𝑟 − 𝑥𝑥�𝑦𝑦𝑒𝑒 � = �− 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃 0� �𝑦𝑦𝑟𝑟 − 𝑦𝑦 � = 𝑇𝑇𝑒𝑒 �𝑦𝑦𝑟𝑟 − 𝑦𝑦 �𝜃𝜃𝑒𝑒𝜃𝜃𝑟𝑟 − 𝜃𝜃
001 𝜃𝜃𝑟𝑟 − 𝜃𝜃(10)𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃 0
where 𝑇𝑇𝑒𝑒 = �− 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃 0�
001(11)
With the differentiation of Eq. (10), we can have theattitude error with respect to time expressed as:
𝑥𝑥̇ 𝑒𝑒𝑣𝑣𝑟𝑟 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃𝑒𝑒−1 𝑦𝑦𝑒𝑒𝑣𝑣
�𝑦𝑦̇𝑒𝑒 � = � 𝑣𝑣𝑟𝑟 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃𝑒𝑒 � + � 0 −𝑥𝑥𝑒𝑒 � � �(12)𝜔𝜔𝜔𝜔𝑟𝑟0−1𝜃𝜃̇𝑒𝑒
From the above analysis, we can define the trajectory
tracking of the mobile robot kinematics model as finding
the bounded input 𝑣𝑣and 𝜔𝜔 such that for any initial error,
system (12) is bounded by the control input(𝑥𝑥𝑒𝑒 , 𝑦𝑦𝑒𝑒 , 𝜃𝜃𝑒𝑒 )𝑇𝑇andsatisfies
thecondition𝑙𝑙𝑙𝑙𝑙𝑙‖(𝑥𝑥𝑒𝑒 , 𝑦𝑦𝑒𝑒 , 𝜃𝜃𝑒𝑒 )𝑇𝑇 ‖ = 0.

𝑡𝑡→∞

Assumption 2: The first derivative of the reference linear
velocity 𝑣𝑣𝑟𝑟 and angular velocity 𝜔𝜔𝑟𝑟 are bounded and 𝑣𝑣𝑟𝑟 >0.
Lemma 2: 𝐿𝐿𝑀𝑀̄−1 is the Hurwitz matrix, and 𝜏𝜏̄̃𝑑𝑑 isasymptote convergence, thenwe express 𝜏𝜏̄𝑑𝑑 = 𝜏𝜏̄̂𝑑𝑑 .
C. The Output Layer Setting of the Cerebellar Model
Controller

Consider Eqs. (12) and (14), and introduce a new control
variable, 𝑢𝑢̄ = 𝑢𝑢𝑐𝑐 − 𝑢𝑢𝑀𝑀̄𝑢𝑢̄̇ = −𝐶𝐶̄ (𝑞𝑞, 𝑞𝑞̇ )𝑢𝑢̄ + (𝑀𝑀̄𝑢𝑢̇ 𝑐𝑐 + 𝐶𝐶̄ (𝑞𝑞, 𝑞𝑞̇ )𝑢𝑢𝑐𝑐 ) + 𝝉𝝉̄ 𝑑𝑑 − 𝝉𝝉̄= −𝐶𝐶̄ (𝑞𝑞, 𝑞𝑞̇ )𝑢𝑢̄ + 𝛤𝛤(𝑢𝑢𝑐𝑐 , 𝑢𝑢̇ 𝑐𝑐 ) + 𝝉𝝉̄ 𝑑𝑑 − 𝝉𝝉̄(14)̄̄
where𝛤𝛤(𝑢𝑢𝑐𝑐 , 𝑢𝑢̇ 𝑐𝑐 ) = 𝑀𝑀𝑢𝑢̇ 𝑐𝑐 + 𝐶𝐶 (𝑞𝑞, 𝑞𝑞̇ )𝑢𝑢𝑐𝑐 is the nonlinear
function of the mobile robot, function 𝛤𝛤(𝑢𝑢𝑐𝑐 , 𝑢𝑢̇ 𝑐𝑐 ) includes
many parameters of the mobile robot such as mass,rotation inertia and so on. It is very difficult to determine

𝑡𝑡→∞

108

disturbances and internal uncertainties act on the torqueoutput are compensated by the CMAC neural network
shown in Figure 3.

these parameters, hence we use cerebellar model controller
to approach, which can be expressed as𝛤𝛤(𝑢𝑢𝑐𝑐 , 𝑢𝑢̇ 𝑐𝑐 ) = 𝑌𝑌(𝑢𝑢𝑐𝑐 , 𝑢𝑢̇ 𝑐𝑐 )𝑊𝑊(15)where 𝑌𝑌(𝑢𝑢𝑐𝑐 , 𝑢𝑢̇ 𝑐𝑐 ) denotes the cerebellar network
output, and 𝑊𝑊is the connection weighting of the cerebellar
network output. Then we can rewrite Eq. (14) as𝑀𝑀̄𝑢𝑢̄̇ = −𝐶𝐶̄ 𝑢𝑢̄ + 𝑌𝑌𝑌𝑌 + 𝝉𝝉̄ 𝑑𝑑 − 𝝉𝝉̄(16)
3. Cerebellar Model Articulation Controller

Figure 3 Scheme of CMAC neural network control
WMR output torque to compensate the external
disturbance

Cerebellar Model Articulation Controller (CMAC) isdefined by a series of mapping [8][9]：CMAC mapping： P → Iv → Hb →Y
where P is the input vector, Iv is a set of internal variables,Hb is a set of hypercube, and Y is the output vector.
We choose 𝛿𝛿 learning rule to adjust the weight,𝐸𝐸 =1 2(17)𝑒𝑒 (𝑡𝑡)5. Simulation Results and Conclusions
In order to verify the actual control performance provided
by the proposed algorithm, we use MATLAB /SIMULINK to execute the simulation. Parameters of theWMR are list as follows: the wheel radius, 𝑟𝑟 = 0.12 𝑚𝑚,
the distance between two driving wheels, 2𝑏𝑏 = 0.6 𝑚𝑚, themass of the WMR, 𝑚𝑚𝑎𝑎 = 4 𝑘𝑘𝑘𝑘 , the distance between
robot’s mass center and geometric center, 𝑑𝑑 = 0.25 𝑚𝑚andthe inertial moment, 𝐽𝐽 = 0.25 𝑘𝑘𝑘𝑘𝑚𝑚2 . In the simulation,
the parameters of the observer and controller are designed0𝐿𝐿�, where 𝐿𝐿1 = 𝐿𝐿2 = 12𝑚𝑚, and 𝑘𝑘𝑥𝑥 = 𝑘𝑘𝑦𝑦 =as 𝐿𝐿 = � 10 𝐿𝐿2𝑘𝑘𝜃𝜃 = 1, 𝑘𝑘𝑢𝑢 = 0.3 and 𝑘𝑘𝑠𝑠 = 0.8. The initial position of thereference input is [𝑥𝑥𝑟𝑟 (0) 𝑦𝑦𝑟𝑟 (0) 𝜃𝜃𝑟𝑟 (0)] = [0 0 0].
The reference velocities are set as 𝑣𝑣𝑟𝑟 = 0.1 𝑚𝑚/𝑠𝑠 , 𝜔𝜔𝑟𝑟 =0.01 𝑟𝑟𝑟𝑟𝑟𝑟/𝑠𝑠 . The initial state of the WMR is given[𝑥𝑥(0) 𝑦𝑦(0) 𝜃𝜃(0)] =as[39.5𝑚𝑚 0.1𝑚𝑚 89.427𝑑𝑑𝑑𝑑𝑑𝑑.[]]external.The
disturbance𝜏𝜏̄ =2𝑐𝑐

where 𝑒𝑒(𝑡𝑡) = 𝑟𝑟(𝑡𝑡) − 𝑦𝑦(𝑡𝑡) is the error, c is generalization
parameter shown in Figure 2.

Figure 2 Scheme of a CMAC model
And adopt gradient decent method,𝜕𝜕𝜕𝜕𝑒𝑒(𝑡𝑡)𝛥𝛥𝑤𝑤𝑗𝑗 (𝑡𝑡) = −𝜂𝜂
= 𝜂𝜂
𝜕𝜕𝜕𝜕

𝑐𝑐

𝜋𝜋,(18)
𝑤𝑤𝑗𝑗 (𝑡𝑡) = 𝑤𝑤𝑗𝑗 (𝑡𝑡 − 1) + 𝛥𝛥𝑤𝑤𝑗𝑗 (𝑡𝑡) + 𝛼𝛼 �𝑤𝑤𝑗𝑗 (𝑡𝑡 − 1) − 𝑤𝑤𝑗𝑗 (𝑡𝑡 −2)�𝜋𝜋

�1.2 𝑠𝑠𝑠𝑠𝑠𝑠( 0.3𝑡𝑡 − 2 ) 0.51 𝑠𝑠𝑠𝑠𝑠𝑠( 0.3𝑡𝑡 + 2 ) −1.31 𝑠𝑠𝑠𝑠𝑠𝑠( 0.5𝑡𝑡)�
occurs after the WMR is moving. Learning rate of CMACcontroller is 𝜂𝜂 = 0.5, and the inertia coefficient is 𝛼𝛼 = 0.5.

(19)

where 𝜂𝜂is the learning speed, and 𝛼𝛼 is inertial coefficient.
4. The Architecture of The WRM Robust HybridController

The scheme of WMR control used in this paper isillustrated in Figure 4. By giving virtual control input toprovide virtual linear velocity and angular velocity
according to WMR’s kinematic model. The actual linear
velocity and angular velocity are given by torquecontroller , which enables the WMR to track the virtual
linear velocity and angular velocity. Then, the external

Figure 4 WRM trajectory tracking

109𝑇𝑇25xy20

Referencesrr

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rx15y1050-5-100100

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Figure 5 WRM trajectory following expected andreal path
1xy0.5eee0-0.5-1-1.5
0100

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Figure 6 WRM trajectory following path error
0.5

0.4

0.3

0.2

0.1

0-0.1-0.2-0.3
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Figure 7 WRM disturbance𝜏𝜏̄𝑑𝑑 and estimated disturbance𝜏𝜏̄̂𝑑𝑑Figure 4. 5, 6, 7 show WRM trajectory tacking, expectedand real path, following path error, disturbance andestimated disturbance.

110Authors introduction
Dr. Ho-Nien Shou
He is an Associate Professor ofDepartment of Aviation & Communication
Electronics at Air Force Institute ofTechnology in Taiwan. He received his D.
Eng. degree in Electrical Engineering,both from National Cheng Kung
University, Taiwan, in 2002. His research interest is microsatellite attitude control.

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