Path Planning and Trajectory Tracking Control for Two-Wheel Mobile Robot

— The mobile robot is a system that can work in various environments. This means that the robot must be able to navigate without delay and avoid any obstacles placed within the boundaries of its movement. Designing mobile robots that can be intelligently managed and operate autonomously when traveling from one place to another requires at least two steps. To start with, path planning is required to prevent motion collisions. Tracking the robot's trajectory is a crucial second task. The primary goal of this study is to find the quickest and safest path between the two positions. In this work, we investigated the path planning of a mobile robot with dynamic, and dynamic obstacles with moving goal environments using RRT, BiRRT, and HA* algorithms. These algorithms are easy, computationally inexpensive, and simple to use. They have been chosen for numerous real-time path-planning applications. The DDMR's kinematic model has been utilized in this paper to control path tracking, and a PID controller has been proposed to reduce tracking deviations between the robot's actual route and the reference trajectory. This work introduced the PSO, FPA, CSA, SSA, BWOA, and proposed HBPO optimization techniques for obtaining PID parameters ( 𝒌 𝒑 , 𝒌 𝒊 , 𝒌 𝒅 ) for improved mobile robot trajectory tracking. The simulation results have been examined using three trajectory shapes: step, circular, and infinite. The simulation findings reveal that HA* outperforms the other algorithms by generating collision-free pathways that are smoother and shorter than their RRT and BiRRT equivalents. On the other hand, the proposed HBPO outperforms the other methods. The HBPO method converges quicker than the other proposed algorithms.


INTRODUCTION
Robotics is the study of the conception, design, development, control, and application of autonomous systems.It is a rapidly growing field with a wide range of applications.With the development of technology, robotics has expanded from the lab into almost every field of science, engineering, industry, manufacturing, transportation, and agriculture [1], [2], [3], [4], [5].Self-contained robots like drones, weaponry, and Unmanned Aerial Vehicles (UAVs) are utilized in military applications and everyday life.Even at home, robots are capable of doing almost any activity, including food service, room cleaning, and door control [6], [7].Autonomous Mobile Robots (AMR) are one of the technologies that are now growing, because of their relevance and uses in today's society.The mobile robot is a system that can work in various environments.This means that the robot must be able to navigate without delay and avoid any obstacles placed within the boundaries of its movement [8], [9], [10], [11].Therefore, to design mobile robots that can be intelligently managed and operate autonomously when traveling from one site to another, at least two steps are required.In the beginning, path planning is critical in order to avoid motion collisions.A critical second task is robot trajectory tracking [12].The AMR's task of path planning is crucial.In order to go safely from the start configuration to the goal configuration, it is desired to locate a collision-free motion in an environment that is full of obstacles.Mobile robots are being used in more environments, both static and dynamic.Normally, there are many feasible paths for a robot to take to reach the target from the start location.The best feasible path is chosen based on criteria like the shortest distance, smoothness of the path, minimum energy consumption, or the most frequently used criteria, the shortest distance with the minimal possible time [13], [14], [15], [16].Trajectory tracking is an essential feature of mobile robots.Mobile robots' primary purpose is to move along a specified path and decide where to go, and that information is taken by a reference path or leader robot.In order to make a robot follow a trajectory, the process of trajectory tracking involves calculating the robot's speed and steering settings at each instant of time.The positional coordinates of a particular route are represented by a group of points called a trajectory [12], In a trajectory-tracking issue, a mobile robot should follow a time-relationship graph.Geometric parameters define the desired path to trace in the path-tracking issue [17].

A. Aims and Contributions
Based on the foregoing, the main aims of this paper can be described as follows: 1. Designing various environments for a mobile robot with dynamic, and dynamic obstacles with a moving target, Then, use the RRT, BiRRT, and HA* algorithms to find the shortest route from the robot's starting point to its intended destination.
2. Investigate trajectory tracking for an autonomous nonholonomic two wheeled mobile robot with a reference trajectory and use the proportional integral derivative (PID) to regulate the robot's velocities such that tracking errors are minimized between the actual and reference trajectories.
2. Comparison of the HA* and BiRRT versus RRT algorithms in dynamic, and dynamic obstacles with moving target environments for path planning mobile robots.
3. Many proposed optimization algorithms, such as CSA, and BWOA, have been employed for the first time in this study to tune the PID controller for improving mobile robot trajectory tracking.
To propose a new technique for tuning PID controllers that combines the BWOA and PSO algorithms (HBPO) in order to improve mobile robot trajectory tracking.

B. Paper Organization and Notations
This paper is organized as follows: Section II presents related work on path planning and trajectory tracking.Section III introduces details of the modeling of two-wheel DDMR.Section IV presents robot path planning.This section has four subsections: subsections A, B, and C explain the RRT, BiRRT, and HA* algorithms, respectively, while subsection D introduces the simulation and results of path planning using these algorithms.Trajectory tracking control and optimization are discussed in Section V.This section has four subsections: Subsection A proposes wheeled mobile robot trajectory tracking based on a PID controller.Subsection B presents intelligent PID and optimization algorithms.Subsection C introduces the Hybrid Black Widow Optimization Algorithm and Particle Swarm Optimization Algorithm (HBPO), and Subsection D focuses on simulation and the results of control trajectory tracking.Subsection E discusses the results.Finally, the conclusion and future work are presented in Section VI.

II. RELATED WORKS
Many papers have focused on path planning.Finding a path from the beginning (current) state to the goal is widely described in other literature.Many path-planning strategies have been presented and tested in a variety of settings with both static and moving obstacles.Trajectory tracking is a fundamental task for mobile robots that has recently received much attention in many publications.The control of mobile robots is a vital part of engineering research due to the development of several intelligent and classical control systems.The purpose of trajectory tracking control is to develop control rules for mobile robot velocity that minimize tracking errors between the robot's real route and the reference trajectory.For many years, PID controller research has received considerable attention.This is the most wellknown method of controlling the navigation of mobile robots.
Fuzzy logic and filter smoothing based on the data from the laser scan sensor, path planning, and autonomous obstacle avoidance based on this algorithm are emphasized because they can automatically find the best path according to the size and position of the gaps between the obstacles in the dynamic environment proposed in [18].Two nature-inspired metaheuristic algorithms, the cuckoo-search and bat algorithms, address mobile robot path planning in an unknown or partially known environment [19].The use of a genetic algorithm (GA) for robot path planning in a static environment has been discussed, along with a review of the subject of path planning optimization, in [20].By enhancing the artificial potential field (APF) approach, the robot can travel the shortest distance possible while avoiding collisions with stationary obstacles [21].In [22], researchers created a new path-planning method that uses reinforcement learning and integrated environment representation to run a mobile robot with nonholonomic constraints in unpredictable dynamic settings.In [23], the particle swarm optimization (PSO) algorithm is combined with the potential field method (PFM) for static obstacles and the PFM for dynamic obstacles to provide an effective path-planning solution for the robot.A new hybrid path-planning approach that combines the A* algorithm with an adaptive window approach to conduct global path planning, real-time tracking, and obstacle avoidance has been suggested.In [24], for mobile robots in large-scale dynamic environments.In [25], described a new method based on the Bug algorithm to overcome run-time barriers.A new replanning method in [26], based on an extended rapidly exploring random tree.A new dynamic replanning approach allows the robot to correctly plan a route in complex environments.To improve the fundamental fastexploring random tree strategy, [27] used a target bias search approach and a novel metric function that took both distance and angle into account.Curve-smoothing has been used in [28] to develop a goal-biased bidirectional Rapidly Exploring Random Trees (RRTs) technique.The Hybrid A-Star search engine and visibility diagram planning have been used in a novel and effective way to discover the shortest nonholonomic path in hybrid environments for valet parking in [29].The path planning strategy proposed in [30] that combines Bi-RRT with an enhanced potential field to increase search probability and efficiency.In [31], Fast-RRT, a new RRT-based path-finding method that quickly locates a path that is close to ideal, has been used.The Fast-RRT algorithm comprises two modules: Improved RRT and Fast-Optimal.In [32], developed a trajectory planning method for mobile robots with maneuver limitations.The robot's velocity affects the maneuver limits.Virtual barriers that close off the robot-inaccessible sectors while it goes at a certain speed operate as restrictions.Any path planning algorithm is compatible with the suggested approach.In [33], proposed a modified version of the HA* technique used to navigate spherical mobile robots.Using a pendulum, the new method allows for limited and partial lateral motion.In [34], suggested an autonomous parking trajectory planning method in an unstructured environment with narrow passages.In [35], conducted research on the hybrid A-star algorithm for nonholonomic robots, focusing on two phases: the forward search phase and the analytic expansion phase.The forward search phase takes into account the robot's kinematics to plan its continuous motion on discrete grid maps.Meanwhile, the analytic expansion phase employs the Reeds-Shepp (RS) curve to enhance the algorithm's accuracy and speed.
The control of mobile robots using their trajectories has attracted the attention of numerous researchers.Trajectory tracking is an essential feature of mobile robots.The primary job of mobile robots is to move along a predetermined course and decide where to go next, with that information being taken from a reference path.Tracking control is crucial because the robot must move along the desired trajectory.Several controllers have been developed recently to operate mobile robots.The PID controller is one of many control strategies that are frequently employed in the robot field.The trajectory of an autonomous nonholonomic wheeled mobile robot using a virtual reference trajectory investigated in [36], the follower robot's velocity is controlled using a PID controller enhanced by a genetic algorithm (GA) to improve control accuracy and convergence speed.In [37], an intelligent fuzzy controller and genetic algorithm have been used to perform target tracking, obstacle avoidance, and time and path optimization.Using the Particle Swarm Optimization (PSO) technique, [38] created an optimal fuzzy PD + I controller for trajectory control of a mobile robot.PSO has been utilized to modify the controller's gains and membership functions to reduce the Integral Square Error (ISE) index, which captures linear and angular velocity errors.The goal had been to develop the best controller and improve results.In [39], A fuzzy logic-based navigation control system has been proposed for a 2-wheel mobile robot.It uses a kinematic model and a fuzzy logic controller designed in Simulink to safely navigate the environment and reach the destination.A novel trajectory tracking control system for mobile robots has been developed, combining reinforcement learning and PID control.In [40], looked into the wheeled mobile robot's navigation approach in a known area with static obstacles.The technique is based on an assessment of the visibility situation, which determines whether there is a point where the intended path and obstacles cross.In [41], presented a design for the kinematic control structure of the wheeled Mobile Robot (WMR) route planning and path following.The system utilizes Q-learning and PID techniques to track the desired trajectory of the robot [42].A variable-parameter PID controller has been proposed by [43] for a differential-drive mobile robot to follow a NURBS trajectory with time-varying velocity.The controller minimizes kinematic error by linearizing the robot's kinematic model and selecting appropriate coefficients through modeling and experimentation.In [44], A novel hybrid technique has been presented, integrating a neural network-based kinematic controller and a model reference adaptive control.The proposed controller achieves superior tracking accuracy and fast convergence compared to PID, kinematic, and adaptive dynamic controllers.Performance analysis indices confirm its effectiveness, even in the presence of parameter uncertainties and slip disturbances.

III. MODELING OF TWO-WHEEL DDMR
The robot features two parallel, conventional wheels (one on each side) that are operated by two independent actuators.It is also assumed that each wheel is perpendicular to the ground and that the wheels' contact with the ground is nonslipping and pure rolling [44].This paper considers a 2wheeled, non-holonomic mobile robot.Fig. 1 depicts the robot's design.
The quantities in Fig. 1 are described [39], [45], as:  : Linear velocity (ms -1 ),   : is the linear velocity of the right wheel (ms -1 ),   : is the linear velocity of the left wheel (ms - 1 ),  : Angular velocity (rads -1 ), : is the radius of the right and the left wheels (m), : is the distance between the right and the left wheels (m).

A. Kinematics Model of the Two-Wheel DDMR
Kinematic modeling investigates the mathematics of motion without considering the influencing factors and instead focuses on the geometric relationships that control the system [46].Fig. 2 shows the position of the differential drive mobile robot along the global coordinate axes of {O, X, Y} [47].The differential drive robot's kinematic equations in the world frame are as follows given the specified constraints [37], [39], [46], [47] .
Equation ( 1) represents the motion of a two-wheel DDMR.
The robot's nonholonomic constraint equation is as follows: The robot's dynamic function is defined as (3).
The previous equations can be written in matrix form, and the differential drive mobile robot's simplified kinematic model, which is used to design the robot, is represented by equation ( 4): By transforming these velocities components into rotational velocities (  ,   ), the above model can be improved to: Equation ( 5) is substituted into Equation ( 4) to obtain a more detailed kinematic model of WMR as (6). [ Equation ( 6) can be written in terms of   and   as (7). [ The robot's velocity is the sum of its two speeds,   and   .Also, the relationship between the robot's angular speed,  , and   , and the separation between the two wheels is as (8).
Substituting ( 1) into (8), we get an equation that explains the relationship between each wheel's angular velocity and the robot's linear and angular velocity in (9).
These two equations can be solved simultaneously to yield the required differential wheel velocities as (10). IV.

ROBOT PATH PLANNING
Path planning in autonomous mobile robot applications is challenging and depends on the environmental model, robot type, and specific application.In a 2D binary occupancy map, the robot's workspace is defined by Cartesian coordinates (x, y).The start point represents the robot's initial position, and the goal point is the desired destination.To find a collision-free path, we used MATLAB 2021a and implemented RRT, BiRRT, and Hybrid A* algorithms.These algorithms efficiently navigate the robot from start to goal while considering free space, static obstacles, dynamic obstacles, and a moving target.

A. RRT Algorithm
The fundamental RRT is a practical random samplingbased path planning technique that provides numerous advantages, such as fast speed and solid real-time performance.The RRT scheme involves iteratively growing a randomized data structure tree by adding new nodes while providing control input until it reaches the desired configuration.Rapid search space expansion combined with path modifications results in collision-free paths.The implemented algorithm can be used to resolve the uncertainty problem in state space, where the tree has grown alongside the sampled state [48], [49], [50].

B. Bidirectional RRT Algorithm
The BiRRT uses a bidirectional search tree to enhance the efficiency of node expansion.The basic RRT's tree expands in space by connecting random locations in emptiness, but it cannot select a new node when it meets a barrier [51].In comparison to the original RRT, the BiRRT generates another parallel tree in the intended growth area.Each iteration follows the same initial procedures as the RRT, which involve sampling random spots first and then expanding [52].

C. Hybrid A* Algorithm
HA* has been the subject of more research over the past decade as a result of its exceptional performance in planning for motion applications, particularly in uncharted regions [53].For non-holonomic self-driving vehicles, the HA* is among the most effective path planners.The algorithm's hybrid term employs an occupancy grid map with which the surrounding is depicted in a discrete manner to plan the vehicle's travel in a continuous domain.The HA* search algorithm enhances the standard A* search method for nonholonomic robot implementation.The key to accomplishing this is to use the vehicle's kinematics to forecast the vehicle's motion based on velocity, gear, and driving angle.This strategy assists the planner in selecting the appropriate successor node that a nonholonomic robot can follow [29].

D. Simulation and Results of Path Planning
This section compares the performance of the BiRRT and HA* methods to the reference RRT method for mobile robot path planning using MATLAB simulation.The experiments have been carried out in two environment modules: dynamic obstacles, and a moving target in a dynamic environment.Specifications of the utilized computer are: The operating system is Windows 10, which is a 64-bit operating system, and the processor is an Intel(R) Core(TM) i7-1165G7U CPU @ 2.80 GHz.The results have been presented as follow:

1) Dynamic Environment
This scenario is considered an environment whose dimensions are (100, 100) with three dynamic obstacle shapes; the initial positions of these obstacles are (40,30) moving toward (10,10), (80, 70) moving toward (10,15), and (50, 90) moving toward (10,20).(20,10) is the starting point, and various goal points are used.The maximum connection distance is 1, and the validation distance is 0.01.Fig. 3 depicts the planned paths for example goal points in this environment using the RRT algorithm (a), the BiRRT technique (b), and the HA* technique (c).In this example, the goal point is (95, 90).Fig. 4 shows (a) the length of the path in meters and (b) the time required reaching the destination in seconds.Table I provides a summary of the results obtained.The example of a goal point is explained as Fig. 3 and Fig. 4.

2) Dynamic Environment with Moving Target
This scenario is considered an environment whose dimensions are (100, 100) with three dynamic obstacles and a moving target.The initial positions of these obstacles are (40,30) moving toward (80, 40), (60,60) moving toward (20,50), and (50, 90) moving toward (80, 80).The obstacles are moving in different directions.(20,20) is the starting point, and various goal points are used.The maximum connection distance is 1, and the validation distance is 1.Fig. 5 depict the planned paths for example of goal points in this environment using the RRT algorithm (a), the BiRRT technique (b), and the HA* technique (c).In this example, the initial position of the goal point is (75, 80) and moving toward (10,10).Fig. 6 shows (a) the length of the path in meters and (b) the time required to reach the destination in seconds.Table II provides a summary of the results obtained.The example of a goal point is explained as Fig. 5 and Fig.

V. TRAJECTORY TRACKING CONTROL AND OPTIMIZATION
The problem of trajectory tracking is described as follows: It is supposed that the robot does have the position:   = [      ]  , and the target does have the position:   = [      ]  .The goal is to identify control laws for robots' linear and angular velocities (, ), such as [48] : The following equation can be used to express the distance between the robot's current location and its target position [37]: The desired angle of the trajectory  is computed as follows: Where:   and   are the desired target or the new position,   and   are the current position of the robot or the old position in the x-y plane.

A. Some Common Proposed Wheeled Mobile Robot Trajectory Tracking Based on PID Controller
The PID controller is used to track the trajectory of the autonomous wheeled mobile robot.The kinematics model in Equation (4) describes the robot's position (, , ) and its linear and angular velocities (, ).In Fig. 7, the robot travels in the  −  plane.The reference trajectory is represented by   (  ,   ), and the robot's coordinates are   (  ,   ) on the  −  plane.The tracking error vector, Ibrahim A. Hassan, Path Planning and Trajectory Tracking Control for Two-Wheel Mobile Robot denoted by  ̂ , represents the difference between the follower robot and the reference trajectory [36].11) has been used to express the distance between the robot's current location and reference position.The following formula has been used to determine the tracking error [36]: Where   is a small positive constant that is utilized to ensure that there is minimal separation between the follower robot and the reference trajectory.The error   is reduced to zero using the standard PID controller to modify the linear velocity of the follower robot   as follows [36]: Equation ( 12) has been used to express angle of the reference trajectory.The angle error between the follower robot and the reference can be expressed as: The error   is reduced to zero using the standard PID controller to modify the angular velocity of the follower robot   as follows: Fig. 8 depicts the overall block structure of a mobile robot that includes a trajectory-tracking control system.

B. Intelligent PID and Optimization Algorithms
Due to characteristics like their durability, simplicity, transparency, reliability, and high efficiency, PID controllers are especially well-liked and accepted for control in process industries.The optimization of PID control settings has been and continues to be an active field of research.The basic objectives of PID control settings are to minimize steadystate overshoot and shorten settling time.Several strategies have been employed to tune the PID.Approaches for adjusting PID parameters are classified as either classical, such as Ziegler-Nichols, Cohen-Coon, and the Gain and Phase Method, or meta-heuristics, such as the PSO, ACO, ABC, BA, and Cuckoo optimization algorithm, etc. Approaches based on meta-heuristics solve problems by offering almost optimal answers in a reasonable amount of time.Meta-heuristics have increased in popularity in recent years as a result of their effectiveness and efficiency in resolving enormous and complex problems.Each metaheuristic algorithm uses a unique combination of local and global search randomization and frequently produces varied result [54].
In this paper, hybrid BWOA-PSO (HBPO) new optimization algorithms have been presented to obtain PID parameters (  ,   ,   ) for enhancing mobile robot trajectory tracking and compared results with Black widow Optimization algorithm (BWOA), Salp Swarm algorithm (SSA), Crow Search Algorithm (CSA), Flower Pollination algorithm (FPA), and Particle Swarm Optimization (PSO).

C. Hybrid Black Widow Optimization Algorithm and Particle Swarm Optimization Algorithm (HBPO)
The combination of two optimization algorithms yields a better solution to a variety of optimization problems and results in faster convergence.In this thesis, the BWOA and PSO are combined to enhance performance.The BWOA-PSO approach has been developed to combine the benefits of the black widow optimization algorithm and particle swarm optimization.
The following details make the PSO algorithm a popular optimization algorithm: Only three parameters (inertia weight, cognitive ratio, and social ratio) control PSO, making it easy to implement and code.PSO is also adaptable enough to combine with other optimization algorithms [55].In PSO, a swarm of particles searches for the best solution in a Ddimensional search space.  = [ 1 ,  2 , … . .,   ] is the current velocity vector of each particle .
By updating the low pheromone rate search agent rather than the entire search agent before the next iteration, the BWOA enhances fitness quality and encourages a better balance between the exploitation and exploration processes.The BWOA Features are easy to use, increase convergence speed, significantly reduce trapping of local optima, have acceptable accuracy, and reduce complexity [56].The BWOA has some drawbacks in that it cannot ensure that it will find the best solution because it is a meta-heuristic algorithm.The BWOA mathematical modeling is described as follows [57]: Pheromones: play a key role in spider mating behavior.The following equation describes the pheromone rate value of black widow spiders [57]: Where:   &   : The current generation's worst and best fitness values, respectively.(): represents the  ℎ search agent's current fitness value.
In equation ( 21), the pheromone vector contains the normalized fitness in the interval [0, 1].For low pheromone rate values of 0.3 or below, the search agent update in the following equation [57]:

D. Simulation and Results of Control Trajectory Tracking
The results of tests performed to evaluate the effectiveness of the controllers and optimization algorithms mentioned in this paper.Fig. 10 depicts the block diagram of the proposed PID controllers for DDMR path tracking with an optimization algorithm to increase the performance of these controllers.The simulation experiment had been carried out on the following computer specifications: The operating system is Windows 10, which is a 64-bit operating system, and the processor is an Intel(R) Core(TM) i7-1165G7U CPU @ 2.80 GHz.Physical parameters taken from [58] include the wheel radius of 0.033m and the distance between two wheels of 0.105m.The performance of controllers is determined using the ISE criterion, which is given [59]: The integral square of the error method, as shown in Equation (5.2), has been used to simulate the objective function used in the proposed optimization algorithms [59].
The proposed algorithms are used to determine the best values for the two PID controller parameters.The swarm population size of 30 and the maximum number of iterations of 100 have been taken from [40].The parameters of the proposed algorithms are as follows: In PSO and HBPO,  1 ,  2 are 2, and w is 0.4.In SSA,  3 is rand().In CSA,  is 0.8 and  is 4. In FPA,  is 0.5 and probability (P) is 0.8.In BWOA and HBPO,  is rand().
To reduce the tracking error between the desired and actual path Several experiments have been carried out to solve this problem using the proposed algorithms-based PID controllers.The simulation results tested with three trajectory shapes, Step, Circular, Infinity, are as follows:  The BWOA, SSA, CSA, FPA, PSO, and HBPO algorithms optimize the PID controllers' parameters shows in Table III.Fig. 12 illustrates the convergence curve of the objective functions for these algorithms, with HBPO showing faster convergence.Fig. 13 and Fig. 14 display the step response and theta response for the trajectory using the HBPO algorithm, respectively.Table IV shows the specifications of the optimized two PID controller's response in terms of steady-state error (SSE), overshoot (OS), and rise time, in addition to the cost function and computation time.Because the HBPO algorithm converges faster, the two PID controller parameters computed by it have been used to examine the linear and angular velocities, right and left wheel velocities, position error, and theta error.In Fig. 15 (a), the mean linear velocity of the mobile robot is 1m/s, while the mean angular velocity is 1.3453 rad/sec in Fig. 15     The following equations can be used to describe the 2-WMR's required circle path [60] : Where:   and   represent trajectory reference.
The results of the testing of the mobile robot for following the circular path are depicted as follows: Table V shows the mean and RMS error values for each algorithm, as well as the cost function and computation time.Table VI shows the PID controllers' Graphically, the convergence curve is shown in Fig. 18.The circular and theta responses for the trajectory using the HBPO algorithm are shown in Fig. 19 and Fig. 20, respectively.Because the HBPO algorithm converges faster, the two PID controller parameters computed by it have been used to examine the linear and angular velocities, right and left wheel velocities, position error, and theta error.In Fig. 21 (a), the mean linear velocity of the mobile robot is 0.1000 m/s, while the mean angular velocity is 0.1141 rad/sec in Fig. 21 (b).Fig.
Where:   and   represent trajectory reference.
The results of the testing of the mobile robot for following the infinity path are depicted as follows: Table VII shows the mean and RMS error values for each algorithm, as well as the cost function and computation time.Table VIII shows the PID controllers' parameters.Graphically, the convergence curve is shown in Fig. 24.The infinity and theta responses for the trajectory using the HBPO algorithm are shown in Fig. 25 and Fig. 26, respectively.Because the HBPO algorithm converges faster, the two PID controller parameters computed by it have been used to examine the linear and angular velocities, right and left wheel velocities, position error, and theta error.In Fig. 27 (a), the mean linear velocity of the mobile robot is 0.2100 m/s, while the mean angular velocity is -0.0130 rad/sec in Fig. 27 (b).The following are a number of major findings that have been drawn from the results of the tests:

A. Robot Path Planning
Taking into account the physical constraints of the moving item by HA*, such as turning and acceleration rates.To recreate the steering angles for a specific car, a tree is grown from the nodes; therefore, the simulation results demonstrate that HA* outperforms the other algorithms by creating a collision-free path that is smoother and shorter than the paths of RRT and BiRRT.At percentages, the results showed that HA* outperformed RRT and BiRRT by reducing the path length by 16.25% and 10.63%, respectively, and the running time by 6.82% and 1.94%, respectively.Moreover, in a dynamic environment with a moving target, HA* demonstrated even better performance by reducing the path length by 17.91% and 10.87% compared to RRT and BiRRT, respectively, and the running time by 18.82% and 13.57%, respectively.

B. Robot Trajectory Tracking
In trajectory tracking, the results of the simulation show that HBPO performs better than the other proposed algorithms.The results demonstrated that the HBPO had been more effective and able to reduce steady-state error at percentages in step trajectory to 97.09%, 8.3176%, 48.3140%, 70.5768%, and 88.3464% compared to the BWOA, SSA, CSA, FPA, and PSO, respectively.The computation times to 9.0685%, 8.3176%, and 1.351% compared to the BWOA, SSA, and CSA, respectively.In the circular trajectory, the results in percentages of RMS theta error are 53.66%,76.7995%, 0.6236%, 91.0532%, and 82.7876% compared to the BWOA, SSA, CSA, FPA, and PSO, respectively; the percentages of RMS position error are 4.5913% compared to the FPA; and computation time is 10.0074% and 5.0004% compared to the SSA and PSO, respectively.In the infinity trajectory, the results in percentages of RMS theta error are 44.5283%,42.5477%, 51.7662%, and 7.6221% compared to the SSA, CSA, FPA, and PSO, respectively.The percentages of RMS position error are 0.9488%, 3.0779%, 2.3158%, 1.0084%, and 0.9381% compared to the BWOA, SSA, CSA, FPA, and PSO, respectively.The computation times are 1.3737% and 2.335% compared to the BWOA and CSA, respectively.

VII. CONCLUSION AND FUTURE WORKS
One of the technologies that is currently being developed is autonomous mobile robots (AMR).Due to their importance and applications in society today.A system that can operate in different environments is the mobile robot.As a result, the robot must be able to travel quickly through its environment and avoid any obstacles that are placed in its path.Therefore, at least two steps must be taken in the design of mobile robots in order for them to be intelligently controlled and work independently when they move from one location to another.To prevent motion collisions at first, path planning is essential.Tracking the robot's trajectory is a crucial second task.In this paper, we investigate the path planning of a mobile robot with dynamic, and dynamic obstacles with a moving goal Using the RRT, BiRRT, and HA* algorithms.
By producing collision-free paths that are smoother and shorter than their RRT and BiRRT comparable paths, HA* outperforms the other algorithms.To reduce tracking deviations between the robot's actual route and the reference trajectory, the DDMR's kinematic model has been utilized to control the path-tracking and PID controller.In this work, HBPO outperforms the other approaches in almost every aspect of simulation results.
In future work, the proposed work will be evaluated for its effectiveness in various scenarios and along various paths using simulated investigations on the prototype, a real mobile robot.Designing and implementing a dynamic model of DDMR with the same proposed PID and optimization algorithms that were used in this study and comparing it with a kinematic model.

6 .
Ibrahim A. Hassan, Path Planning and Trajectory Tracking Control for Two-Wheel Mobile Robot

Fig. 10 .
Fig. 10.A block diagram of the PID controller auto-tuning using optimization algorithms Fig. 11 depicts the results of the testing of the mobile robot for following the step path utilizing the suggested algorithm-based PID controllers.

Fig. 12 .Fig. 13 .Fig. 14 .
Fig. 12. Convergence curve of step trajectory (b).Fig.16(a) and Fig.16(b) demonstrate smooth velocity values for the right and left wheels without spikes.The position and theta trajectory errors for the mobile robot motion are depicted in Fig.17 (a)  and Fig.17 (b), respectively.The mean tracking error is 0.0163m for position and 0.0028 rad for theta.

Fig. 19 .
Fig. 19.Circular Trajectory 22 (a)  and Fig.22 (b) demonstrate velocity values for the right and left wheels with spikes.The position and theta trajectory errors for the mobile robot motion are depicted in Fig.23 (a)and Fig.23 (b), respectively.The mean tracking error is 0.0698m for position and mean tracking absolute error is 0.0061rad for theta.

Fig. 28 (
a) and Fig. 28 (b) demonstrate velocity values for the right and left wheels with spikes.The position and theta trajectory errors for the mobile robot motion are depicted in Fig. 29 (a) and Fig. 29 (b), respectively.The mean tracking error is 0.1054 m for position and mean tracking absolute error is 0.0032 rad for theta.

TABLE I .
SUMMARY RESULTS OF DYNAMIC ENVIRONMENT SCENARIO

TABLE III .
THE TUNED PARAMETERS OF PID CONTROLLERS USING THE PROPOSED ALGORITHMS

TABLE IV .
COST FUNCTIONS, COMPUTATION TIME, AND THE SPECIFICATIONS OF THE OPTIMIZED TWO PID CONTROLLER'S RESPONSE

TABLE V .
THE MEAN, RMS, THE COST FUNCTION, AND COMPUTATION TIME

TABLE VII .
THE MEAN, RMS, THE COST FUNCTION, AND COMPUTATION TIME