Hassan M. Alwan, Volkov A. Nikolaevic, Sameh F. Hasan, Kochneva O. Vladmerovna

Corresponding email: hassan.m.alwan@uotechnology.edu.iq

Corresponding email: hassan.m.alwan@uotechnology.edu.iq

**Published at : ** 18 Sep 2024

**Volume :** **IJtech**
Vol 15, No 5 (2024)

**DOI :** https://doi.org/10.14716/ijtech.v15i5.6908

Alwan, H.M., Nikolaevic, V.A., Hasan, S.F., Vladmerovna, K.O., 2024. Kinematic and Dynamic Modeling Based on Trajectory Tracking Control of Mobile Robot with Mecanum Wheels.

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Hassan M. Alwan | Mechanical Engineering Department, University of Technology, Alsinaa street, 10066, Baghdad, Iraq |

Volkov A. Nikolaevic | High School of Automation and Robotics, Peter the Great St. Petersburg Polytechnic University, 29, Polytekhnicheskaya street, St. Petersburg, 195251, Russia |

Sameh F. Hasan | Mechanical Engineering Department, University of Technology, Alsinaa street, 10066, Baghdad, Iraq |

Kochneva O. Vladmerovna | High School of Automation and Robotics, Peter the Great St. Petersburg Polytechnic University, 29, Polytekhnicheskaya street, St. Petersburg, 195251, Russia |

Abstract

The
trajectory tracking is important to make the WMR move autonomously from the
starting point to the destination along a predefined time. Implementing of trajectory tracking control
is a fundamental part to accomplish its application tasks. In this article a
new method by using a hybrid controller has been presented to solve the problem
of the trajectory tracking of four mecanum wheeled mobile robot. Proposed
controller is depending on modeling of robot kinematic and dynamic equations.
The novelty in this work is that, an optimal control system self-tuning
parameters based on an optimization algorithm for these models of the mobile
robot is utilized. The optimal control type that is used in this work is the
Linear Quadratic Regulator (LQR) controller. LQR is used to control the
actuator torque that is required in each wheel to achieve the robot task. The parameters
of the LQR controller are tuned by using Ant Colony Optimization (ACO). For
results simulation, MATLAB/ Simulink is used for circular and infinity shape
trajectories. Results show that when the robot follows a circular trajectory, the
values of position trajectory error values are reduced to small value (e_{x}=3.218
*10^{-5}m) and (e_{y}= 2.224*10^{-5}m) in x_{o}
and y_{o} directions, respectively and remained almost at these values
until the end of the simulation time. The maximum orientation error is (=
0.103rad), and convergent to zero after two seconds of the mobile robot
movement. Also when the robot follows an infinity shape trajectory, the
position trajectory error values are (e_{y}) and () are
reduced to small value -4.078 *10^{-4}m and 3.174*10^{-4}rad
respectively, while, (e_{x}) is reduced to 5.263*10^{-4} m
after about 15 seconds.

Ant colony optimization; Linear quadratic regulator controller; Robot kinematics: Trajectory tracking; Wheeled mobile robot

Introduction

The researches about the control of trajectory
tracking of the mobile robot (WMR) are increased dramatically. In their review paper (Abed,
Lutfy, and Al-Doori,* *2021) presents various trajectory planning strategies for mobile
robots using different optimization methods proposed in last year. In
their study (Song, Tang, and Li, 2024) addressed the issue of guiding a mobile robot
through a dynamic environment by combining the enhanced ant colony optimization
ACO with the dynamic window approach DWA algorithm. (Amouri, Cherfia, and Merabti,
2021) proposed
a new method to develop kinematic and dynamic models to improve the trajectory tracking accuracy. A new dynamic window algorithm DWA is proposed
by Wang, Ma, and Zhu (2022) to set the initial direction angle and
then improve its evaluation function during implementation of the trajectory
tracking to increase the efficiency and flexibility of the algorithm. A non-singular
terminal sliding mode
control (NTSMC) was used by Sun *et al.* (2020) for the mobile robot trajectory
tracking. Mathematical models of the mobile robot firstly, have been obtained with taking into account the uncertainty effect. An optimal robust control strategy is designed
by (Chai *et al.*, 2024) based on the modified backstepping method to
achieve stable, accurate and real-time trajectory tracking for the wheeled
mobile robot in the presence of unavoidable disturbances. PID and LQR controllers have been
proposed by Amudhan
*et al.* (2019) to use in path controlling of a four wheeled
omnidirectional robot. Circular and rectangular shapes trajectories
have been followed in the performance examination of proposed controllers.
Yuan *et al.,* (2019) have bee applied theoretically
and experimentally the extended state observer (ESO) relaying on the sliding
mode control (SMC) for the trajectory tracking of FMWMR. ESO-SMC based on both
kinematic and dynamic models of the WMR was applied with taking into account
the effect of the uncertainty and the disturbance. Morales *et al.* (2018) presented
the performance analysis of FMWMR experimentally by using linear quadratic
regulator (LQR) controller. Control of trajectory tracking of the omnidirectional WMR was solved by
using the model predictive control (MPC) based on only the mobile robot
kinematic model (Wang *et
al.,* 2018). A modified active disturbance
rejection control (ADRC) built by (Azar *et al.*, 2023) which includes a nonlinear state error feedback
controller and nonlinear tracking differentiator (TD) for the estimation error
correction during the trajectory tracking. In work done by Malayjerdi,
Kalani, and Malayjerdi, (2018), hybrid controller consisting of self – tuning fuzzy
PID controller based on the kinematic model of FMWMR was
utilized. In their
work, (Keek, Loh, and Chong, 2018) fractional order
proportional-integral (FOPI) controller utilized experimentally for the
trajectory tracking of the FMWMR was implemented. (Khadanga *et al. *2018) and Gao *et al.* (2017) studied the wheeled mobile robot trajectory tracking controller design. For the problem of mobile robot’s path planning
under the known environment, a path planning method of mixed artificial
potential field (APF) and ant colony optimization (ACO) based on grid map is
proposed by (Chen and Liu, 2019) to explore a new and better path to receive
stronger stability and environmental adaptability. The
kinematics and dynamics models and proposed Backstepping controller algorithm
have been derived for the trajectory tracking of the robot. Both
mathematical models for WMR were derived based on the wheels velocities and
robot mass centre point velocities. Hussein (2023) presented a new
hybrid controller based on both kinematic and dynamic models of a non-holonomic
wheeled mobile robot. The gains parameters of the hybrid controller were tuned
by applying a modified version of a grey wolf optimization. Cui *et al.* (2012) proposed and simulated results to show the
effectiveness of the backstepping-based controller with proven global stability
by selecting a Lyapunov
function and introducing a virtual control input for the built dynamical model.
In their paper (El-Shorbagy
and Hassanien, 2018) a comprehensive
review of PSO as a well-known population-based optimization technique used in
the field of WMR. They start by a brief
introduction to the behaviour of the PSO, and then introduced the other
representation, convergence properties and the applications of PSO. The aim of
their study (Heryanto and Kusumoputro, 2021) was to propose an inner loop control algorithm
for UAV using a neural network–based DIC system. The experimental results
showed that the neural network–based DIC could follow the manoeuvres of the
testing trajectory dataset with excellent performance, as indicated by an
overall mean squared error (MSE). (Mehenge *et al.,* 2023) work introduces a three-wheel holonomic
omnidirectional robot designed for mobile applications. This platform with
three omnidirectional wheels and DC motors, controlled by EMS 30A H-Bridge and
Arduino Mega 2560, it aligns kinematic calculations with mathematical models,
promising efficiency for diverse mobile manipulation robot applications. The main contribution of (Mohammed,
Al-Khafaji, and Abbas, 2023) paper is to develop an innovative algorithm
to accurately detect and identify the shape and color of objects and find their
location. The proposed algorithm utilizes the HSV color model to distinguish
between different object colors and shapes. After that, a series of filters are
applied to reduce the noise of these images to make the process of discovering
the shape and coordinates of the objects successful. (Wang *et al.* 2021) review a collision avoidance researches under unknown
environments for WMR including various types of controllers. In their approach (Pramujati
*et al.,* 2023) introduces a model-based
control scheme consists of two systems for controlling the position of a
suspended cable-driven parallel robot. Three trajectories are generated to test
the compliance of the controller with its position. The error compensation scheme
is introduced to increase the positional accuracy of the previous controller,
especially on the z-axis. Williams and Wu (2010) have established a novel method of
obstacle-avoidance motion planning for mobile robots in dynamic environments,
wherein the obstacles are moving with general velocities and accelerations, and
their motion profiles are not preknown. A hybrid system is presented in which a
global deliberate approach is applied to determine the motion in the desired
path line (DPL).
Rusdinar *et al. *(2021) developed a mobile robot with a UVC light
system installed at the top and bottom to emit UVC light. This robot can be operated
automatically as it has a magnetic line sensor and employs a fuzzy inference
system algorithm for its movement. The experiment showed that UVC light has
good sterilization and disinfection performance.

Finally, it can be concluded that despite many methods are applied in the literature for designing the trajectory
tracking control strategy of the WMR for example: Fuzzy logic controller (Rahman, Hassan, and Ihsan, 2022), Adaptive Neuro-Fuzzy Inference System ;(ANFIS) (Tilahun
*et al.* 2023), Proportion integral differential (PID)(Saleh, Hussain, and Klim, 2018), linear quadratic regulator (LQR) (Uddin, 2018), model predictive control (MPC), etc, but they have drawbacks such as
large computational burden and long convergence time when dealing with under
actuated system. In addition, sliding model control (SMC) is
another effective method, but the chattering phenomenon limits its application.

In this
work, hybrid controllers based on kinematics and dynamics are designed to solve
the trajectory tracking problem of a four mecanum wheeled mobile robot. The
type of the optimal control is the Linear Quadratic Regulator controller (LQR).
LQR is applied to design the controlled torques required by actuators and parameters of LQR controller are tuned by
using Ant Colony Optimization (ACO).

Experimental Methods

To derive mobile robot kinematic and dynamic
mathematical models, velocities of its centre in local coordinates system will be
adopted with respect to angular velocities and torques of the wheels. The
rotation of each wheel is achieved by a separate motor, while the wheels
passive rollers have been driven by the frictional force with the ground. The top view** **configuration of proposed
mobile robot (FMWMR) is shown in Figure 1a. It is noted that, the origin point of mobile
robot local coordinate system does not coincide with the whole mobile robot
mechanism mass center (point D).

Referring to i wheel of the mobile robot Figure 1b,
wheel radius is its angular velocity is and its linear velocity resulted
from the wheel rotation is Also the linear velocity of each passive roller is
referred by which is equal to and
perpendicular on the roller axis (Hasan and* *Alwan, 2022).

In
this scenario, the mobile robot is viewed as a complete mechanism with the angular velocities of its wheels serving as inputs
and the
robot’s linear and angular velocities as
outputs. In this context
the Jacobian matrix of inverse
kinematics can be expressed as (Equation 1):

FMWMR inverse kinematic equation
in robot local coordinate system is expressed as below (Equation
2):

Where: is the Matrix of angular
velocities of all robot mecanum wheels. Taking
in account that is the robot velocities in
global coordinate system,
equation of inverse kinematic can be written as follow in global coordinate
system (Equation
3):

Where matrix of robot local coordinates system rotation with
reference to global coordinates system, and will be presented (Equation 4):

The forward kinematic jacobian matrix is pseudo-inverse of (J)(Zeidis *et al.* 2019), its
formula was written as: The forward kinematics equation
in robot local coordinates system is written (Equation 5):

The forward kinematics equation for the FMWMR in
global coordinate system can be written as follow (Equation 6):

In the Lagrange formulation, the dynamic model of the robot includes the motion of the system resulting from the inertia of the entire robot mechanism and external forces. As illustrated in Figure (1a), point C represents the centre of the robot’s local coordinate frame, and (D) is the mass centre of the entire robot mechanism.

Velocity of
local coordinate system origin point of the FMWMR (point C) in robot local
coordinate system is evaluated according to the theory in as (Equation 7)**:**

where, are mass centre point velocities point (C) in
global coordinate system, is rotation angle of robot local coordinate frame
relative to global coordinate frame. Also, the velocity component of point (D)
in the local system has been expressed as (Equation 8):

Since the
robot cannot change its vertical position (z-axis), so its potential energy is
equal to zero. The kinetic energy of the whole robot mechanism includes the
kinetic energy of the four mecanum wheels and robot platform. In this
case L is only kinetic
energy of the robot and has been written as (Equation 9):

Where: m_{r}: mass of robot
platform , m_{wi}: mass of i wheel, I_{r}: Robot mass moment of
inertia about its centre of mass
point, I_{wi}: moment of inertia
of i wheel. The Lagrange equation derived from the total energy of the whole
robot will present the dynamic model and can be written as
(Equation 10):

The
general form of the robot dynamic equation can be written (Equation
11):

Mass and
inertia matrix

Matrix of carioles and
centripetal effect**Controller design for trajectory tracking**

In this section, a control system with self-tuning parameters based on
an optimization algorithm is utilized for both robot models. The type of the
optimal control that is used in this work is the Linear Quadratic Regulator
controller (LQR). It is used to design the controlled torques generated by the
wheels motors. The parameters of the LQR controller are tuned by using Ant
Colony Optimization (ACO) and this matter is presented in the next subsections.

*3.1. Linear quadratic regulator controller
(LQR)*

This
controller is considered to be a type of an optimal control system. It is used
in a lot of applications such as robot actuators control and in airplane flight
control. The main objective of LQR controller usage is getting optimal control
signal of the generated torques. The state space representation of the robot
system (Equations 20 and 21):

where (*X*)
represents the state-space vector and (U) is the torque control signal. The
feedback control law when taking (*K*) the gain vector is (Equation 22):

In this
work, the equation of the cost function is proposed Equation 23):

Where:
(R) is the symmetric positive definite matrix and (*Q*) the symmetric
positive semi-definite matrix. Elements of (R) and (*Q*) matrices
(Equations 24 and 25):

The equation of the optimal feedback is evaluated as (Equation 26):

where (*P)* is defined as the Riccati equation which is calculated (Equation
27):

The equation of the control law is introduced (Equation 28):

where to be sure that the
steady state error becomes zero.

LQR controller used in this work is adopted
to make an optimal control signal for the motors torques.

*3.2. Ant colony optimization (ACO) *

Pei,
Wang, and Zhang (2012) states
that, ACO type
of optimization is an intelligent heuristic algorithm which was developed in
1992 by Marco Dorigo which simulates from the ant foraging behaviour. The
communication between ants is relying on the pheromones and the pheromones are
proportional inversely with length of the path. During the searching of the
ants for the food in an unknown environment, ants have been moved threw a path
containing a high pheromone concentration. Concentration of pheromone is
directly proportional with the numbers of the ants that walking on this path.
Ants behaviours display the principle in which adopted to select an optimal path.
To enhance path planning efficiency, heuristic function and tabu list (tabu_{k})
are adopted into an artificial ant colony model. Within a time (t), ant k moves
from a current node i to an unvisited node j regarding the distance information
received from the target point and pheromone intensity during this path. In the
case of more than one unvisited node, the transition probability () between the nodes will be evaluated by the
ant K in accordance with the formula that shown (Equation 29) :

where U
represents the next node of ant k, is an expectation heuristic factor, is a
pheromone heuristic factor, is a pheromone concentration on the path ij at a period of time (t), (t) is an expected heuristic functio a (Equation 30):

where is a Euclidean distance between
any two adjacent nodes. After finishing the path search by all ants, pheromone
was evaporated. The formula that describes the increasing of the pheromone on
the path (Equations 31 and 32):

where () represents the rate of the pheromone
evaporation, and (t) is the time of the iterations. Value of

Where (N)
represents the intensity of the pheromone that added by the ant (K) and (L_{K})
represents the length of the path that was tracking by the ant (K) during one
iteration. There is no fixed rule to choose the ant's
number as well as the Max. number of iteration. The selection number is
considered to be acceptable because the higher number of iterations and ants
required more program execution time. Also, it is needed a computer with high
specifications.The basic
parameters for the ACO have been chosen as
follow:

Ants
Number: 30

Max. Iteration
Number: 100

The tracking error
between the desired trajectory and actual trajectory at any point can be presented as (Equation 34):

In this work, ACO is adopted in selection of optimal values of (Q) and (R) matrices elements. In simulation work, a closed loop controller (Figure 2) has been implemented. This loop is a trajectory tracking controller based on minimizing the pose error to zero. We used LQR controller where the gains were obtained by using Anti Colony Optimization ACO to control the motors torques.

**Figure ****2** Controller architecture

** **

Results and Discussion

The proposed controller has been
examined in two options of WMR movement. In first
option WMR follows a circular shape trajectory and in second option it follows
an infinity shape trajectory. The results have been simulated by using MATLAB
2019 software. The parameters of robot are taken as follow:

M=8 kg, I_{b}=0.2
kg m^{2}, I_{w}=0.1 kg m^{2}, r_{wi}=0.05 m, h_{1}=0.35
m, h_{2}=0.25 m, µ=0.01. The ranges of the
elements in the (Q) and (R) matrices fall between 0 and 30. Various values of
these elements were tested to minimize the overall cost function while ensuring
reliable controller performance. The selected values of these elements were
evaluated using ACO and are presented as (Equations 35 and 36):

In the
first case, the equations of the circular shape trajectory followed by the WMR
are described as (Equations 37 and 38):

The robot movement has been begun from initial
point with coordinates The performance of the generated trajectory is displayed in Figure 3a.
The trajectory tracking errors at any time can be presented as in Figure
3b.

**Figure 3 **The first proposed circular Shape Trajectory followed
by the WMR

After two seconds of the WMR task starting, the
values of position trajectory error values are reduced to small value (e_{x}=3.218 *10^{-5}m) and (e_{y}=
2.224*10^{-5}m) in x_{o} and y_{o} directions,
respectively and remained almost at these values until the end of the
simulation time. The maximum orientation error is (= 0.103rad),
and convergent to zero after two seconds of the mobile robot movement. Figure
4a shows the robot linear and angular velocities in global coordinate system
during the circular trajectory. As shown
in Figure 4a the robot velocities behaviour is smooth during the simulation
time. The value of the linear velocities is fluctuated about ±0.4 m/s and robot
rotation velocity is 0.4 rad/s. Generated torques from robot mecanum wheels
motors can be evaluated at any time. The torques values have been shown in Figure
4b.

In the second option, the equations
of the infinity shape trajectory followed by the WMR are described as
(Equations 39 and 40):

The matching between the desired and theoretical
trajectories of the WMR is shown in Figure 5a. The position trajectory error values after three seconds of the task
starting, are (e_{y}) and () are reduced to small value
-4.078 *10^{-4}m and 3.174*10^{-4}rad respectively, while, (e_{x})
is reduced to 5.263*10^{-4} m after about 15 seconds. The trajectory
tracking errors for the infinity shape trajectory followed by WMR at any time
can be presented as in Figure 5b. Linear and angular velocities of the center point of
mass of the WMR in the global coordinate system shown in Figure 6a. The generated torques from the wheels motors can be
evaluated at any time. The values of the torques present in Figure 6b.