## Abstract

The demand for automation using mobile robots has been increased dramatically in the last decade. Nowadays, mobile robots are used for various applications that are not attainable to humans. Omnidirectional mobile robots are one particular type of these mobile robots, which has been the center of attention for their maneuverability and ability to track complex trajectories with ease, unlike their differential type counterparts. However, one of the disadvantages of these robots is their complex dynamical model, which poses several challenges to their control approach. In this work, the modeling of a four-wheeled omnidirectional mobile robot is developed. Moreover, an intelligent Proportional Integral Derivative (PID) neural network control methodology is developed for trajectory tracking tasks, and Particle Swarm Optimization (PSO) algorithm is utilized to find optimized controller's weights. The simulation study is conducted using Simulink and Matlab package, and the results confirmed the accuracy of the proposed intelligent control method to perform trajectory tracking tasks.

## 1 Introduction

Omnidirectional mobile robots have been utilized in numerous industries for their enhanced features such as a high degree of maneuverability, improved dexterity, and driving capacity. The high maneuverability of these robots has proven to outperform differential wheels mobile robots as they can track complex trajectory that may be challenging or even not feasible for differential wheels mobile robots. Omnidirectional mobile robots have shown the capacity to perform motions in any direction without the need to have extensive space to finish the maneuver. For all these advantages and enhanced features, the omnidirectional mobile robots have been widely adopted in service and industrial applications, such as drug delivery in pharmacies, materials delivery in a factory floor, goods arrangement and packaging in the retail industry, and many more [1].

Omnidirectional mobile robots have the capacity to perform rotational and translational motions at the same time and in an uncoupled type of motion. Various kinds of omnidirectional mobile robots were developed in the literature, such as three and four wheels omnidirectional mobile robots. The four-wheeled type can offer much higher maneuverability when compared to the three-wheeled type as it has an extra degree of freedom represented by the fourth wheel. The dynamics model and control methodologies have been proven quite challenging, as many factors affect the mobile robot system. Friction, backlash in the wheels, nonlinear behavior of the DC motors, sensor noise, positioning drift, uncertainty in model parameters have made the control method for these robots require rigorous stability and performance analysis. Several control methodologies have been proposed to take some of these factors into consideration.

In [2], dynamics of a Three-Wheeled Omnidirectional Mobile Robot (TW-OMR) was obtained and then linearized. Moreover, some insights for stability and control was provided. In [3], a proportional-integral control method was proposed to perform point following and tracking tasks. The controller's parameters were tuned using the ant colony optimization algorithm. The methodology was verified in simulation in point following motion and circular trajectory tracking tests. However, complex trajectories were not considered and verified in this study. The use of the linear control method with this highly nonlinear mobile robot system may make this method fail to meet the high maneuverability requirements of complex trajectories. High-speed trajectory tracking controller based on Takagi–Sugeno Fuzzy system was proposed in [4]. Kinematics inversion was utilized in the feedback loop to simplify the control system design. The tracking performance was evaluated with results from a classical PID control, and the proposed method showed a better outcome to follow a simple square trajectory. In [5], a review of the omnidirectional and holonomic mobile robot was provided. Insights into their dynamical and kinematics modeling were discussed. Several control methods such as PID, and Fuzzy control, were developed. However, no simulation was provided to verify these control methods. In [6], a TW-OMR dynamical model was developed. Based on this model, a computed torque control method was proposed to stabilize the mobile robot and provide accurate trajectory tracking performance. A simulation study was conducted to test the control approach, where a simple circular trajectory was utilized. In [7], a full dynamic model was derived for an omnidirectional mobile robot. Output feedback linearization control method was proposed to solve the tracking task.

In [8], the output feedback control structure with a linear controller and observer was used for tracking tasks of a TW-OMR. Disturbances on the system considered as additive uncertainty terms with the control action provide to the wheels. A laboratory-based TW-OMR was used to verify the developed control methodology. The robot was able to follow a simple circular trajectory. In [9], an adaptive backstepping control methodology was proposed to control a Four-Wheeled Omnidirectional Mobile Robot (FW-OMR). Various motions were able to attain using this method, and the performance was verified in the simulation study. In [10], a PI control method was used to control a TW-OMR. A fuzzy logic system was used to tune the PI gains instead of using try and error approach. The effectiveness of the control method was confirmed in simulation by a simple point to point and circular trajectory tracking tests. In [11], a PID control approach was used for trajectory tracking of a TW-OMR. Practical emphases were on estimating the mobile robot velocities from the robot's internal sensors rather than using an external localization system. In [12], a fuzzy control system was proposed to control a FW-OMR, and was directed towards security and surveillance applications. The encoders and sensors of the wheels' motors were used to provide the feedback signal and to estimate the robot location. The control method was verified in an experimental study and was proven effective to follow a circular trajectory. The results were compared with a simple proportional controller. A sliding mode control method was proposed in [13], to achieve trajectory tracking tasks in TW-OMR. Backstepping approach was utilized to facilitate the reachability to the control signal. Simple turning trajectory was used to verify the control method, and small tracking errors were observed. In [14], Wahhab and Al-Araji have presented design based on Convolutional Neural Network Trajectory Tracking (CNNTT) controller to control mobile robot to find optimal path in the presence of obstacles using hybrid swarm optimization. In [15], Al-Araji et al. proposed an adaptive nonlinear controller for trajectory tracking of non-holonomic mobile robot. The controller consists of feedforward multi-layer perceptron and modified Elman neural network. The Elman neural model is trained to work as orientation and position identifier, while the feed-forward multi-layer perceptron is trained off-line and the weights are on-line adapted to generate the actuating torques of mobile motors. In [16], Al-Araji et al., proposed nonlinear neural controller based on optimization algorithm to follow optimal path-tracking of mobile robot. Artificial Bee Colony and Particle Swarm Optimization algorithms are applied for finding the optimal navigation of mobile robot.

It has been noted from the literature that the control methods provided for trajectory tracking tasks in omnidirectional mobile robots are either linear or non-optimized. The performance of those methods is heavily dependent on the designer trial and error approach, which can be infeasible with a large number of parameters. Moreover, most of the control approaches reported in previous studies were only verified using a simple point to point following or circular trajectory. In practice, these robots are expected to follow complex trajectories to perform service or industrial tasks, and it is quite rare that they only follow a simple circular trajectory. Furthermore, the uncertain environment of these robots poses further challenges to their control methodologies. Therefore, in this work, an intelligent Proportional Integral Derivative (PID) based on the neural network control method is proposed for trajectory tracking tasks of a FW-OMR. The learning ability of the neural network is expected to improve the motion accuracy of these robots and make them able to overcome the uncertainties in their environment. The controller's parameters were obtained using the particle swarm optimization algorithm. A simulation study was conducted to verify the proposed control method, where a complex trajectory was tested and verified. The results have confirmed the accuracy and feasibility of the designed control approach.

## 2 Four-wheeled omnidirectional mobile robot modeling

The configuration of the FW-OMR is shown in Fig. 1. There is a 90^{°} between the wheels. It is essential to define the following notations to develop the model of the mobile robot. The robot coordinates are defined by *x*, *y*, and *θ*. The wheels velocities are

### 2.1 Robot kinematics

### 2.2 Robot dynamics

## 3 Neural network PID control method design

Neural networks have been used in modeling for their effective function approximation and learning ability. More recently, they have been applied to control various nonlinear systems. Control systems developed based on neural networks are categorized into two schemes. In the first scheme, the control action is computed directly by the neural network. In the second scheme, the neural network is used to tune the control parameters online. Both schemes require training of the network to perform the required tasks. Back propagation is one of the most famous training methods that is used to train a feedforward neural network in what is called supervised learning. It requires the differentiation of the error signal to update the network's weights. Input and output data set are also necessary for such a learning method, which could be challenging to obtain for some systems. On the other hand, metaheuristic optimization algorithms have shown promising results in solving various engineering problems and recently have been applied to train neural networks. These algorithms are based on minimizing a cost function via tuning some design parameters, and they do not require error differentiation as in the back propagation method. Therefore, they function as global optimizers and do not easily fall in local minimums as in local search methods. Thus, for these advantages, the particle swarm optimization algorithm is adopted in the current study to train the neural network [14, 15].

Every mobile robot is required to follow a trajectory to perform a certain task. The accuracy of following this trajectory is significantly important as that it has a direct effect on the safety of the people around the robot, the battery life, power consumption and heat dissipation, and the wear and tear of the robot parts. The learning ability of the neural network makes it feasible to attain the required motion accuracy with a high level of robustness. Therefore, a neural network proportional-integral-derivative (NN-PID) control method is developed in this work to enhance the trajectory tracking tasks for a FW-OMR. The closed loop control system that involves the proposed control methodology is shown in Fig. 2. The error between the reference and actual position is multiplied by the orthogonal rotational matrix to transfer the error from the world coordinates to the robot coordinates. The PSO algorithm is function as a trainer for the neural network. The cost function using in the PSO is based on the tracking error vector *e*_{σ}. The control action is produced from the NN-PID control method, and is represented by four voltages to drive the wheels' motors [16].

The structure of the closed loop control system

Citation: International Review of Applied Sciences and Engineering 14, 1; 10.1556/1848.2022.00420

The structure of the closed loop control system

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The structure of the closed loop control system

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The NN-PID control method is firstly proposed by Shu in [14–16, 20]. The NN-PID controller proposed in [14–16, 20] is represented as a neural network with three neurons in the middle layer to mimic the behavior of the continuous PID controllers. The first node in this layer corresponds to the proportional component of the PID. Similarly, the second and third nodes correspond to the integral, and derivative components, respectively. In this work, three NN-PID controllers are used, one each coordinate, i.e. NN-PID_{x}, NN-PID_{y}, and NN-PID_{θ}. A fully connected neural network layer is adopted in this work to map the output of the three NN-PID controllers to the wheel's voltages. The structure of the NN-PID control method is shown in Fig. 3. The input layer consists of three neurons with the error as input. The hidden layer has nine neurons, each three of those represent a NN-PID controller. The output layer has four neurons that represent the four wheel's voltages. A fully connected neural network is utilized to connect the hidden layer with the wheel's voltages in the final layer.

The structure of the NN-PID controller

Citation: International Review of Applied Sciences and Engineering 14, 1; 10.1556/1848.2022.00420

The structure of the NN-PID controller

Citation: International Review of Applied Sciences and Engineering 14, 1; 10.1556/1848.2022.00420

The structure of the NN-PID controller

Citation: International Review of Applied Sciences and Engineering 14, 1; 10.1556/1848.2022.00420

## 4 NN-PID training by particle swarm optimization

Particle Swarm Optimization (PSO) was firstly developed in [21, 22], and it has become extremely famous ever since. It is metaheuristic algorithms based on a simplified and efficient set of computational steps. It mimics the social behavior of swarm of particles. It has shown promising results in solving nonlinear continuous problems [23–27]. Moreover, it has shown faster convergence rate, more rapid computations, and extra accurate solutions when compared with other optimization algorithms. The particles positions represent solutions in the search space. Thus, each particle has n-dimensional vector to describe its position, where *n* is the number of the optimization parameters. The particles communicate within the swarm and follow a learning path based on their local best seen solution and also based on the best global solution by the swarm.

*m*. The PSO parameters are presented in Table 1.

PSO flowchart

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PSO flowchart

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PSO flowchart

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PSO parameters

Parameter | Value |

maximum number of iterations | 1,000 |

2 | |

2 | |

1.5 | |

swarm size | 30 |

45 | |

parameter range | [–100, 100] |

This study can be better improved or extended by incorporating other optimization techniques like the Grey-Wolf Optimization, Social Spider Optimization, Whale-Optimization Algorithm [28–31]. A comparison study can be conducted by comparing one of these recent optimization techniques with PSO algorithm.

## 5 Results and discussion

Robot parameters [17]

Parameter | Unit | Value |

m | 0.0325 | |

kg | 2.34 | |

kg·m^{2} | 0.0228 | |

Ω | 4.3111 | |

V (rad^{−1}/s) | 0.0259 | |

V.s rad^{−1} | 0.0259 | |

m | 0.089 | |

/ | 5 | |

N.s m^{−1} | 0.4978 | |

N.s m^{−1} | 0.6763 | |

N.m.s rad^{−1} | 0.0141 | |

N | 0 | |

N | 0 | |

N.m | 0 |

Cost progression with PSO iterations

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Cost progression with PSO iterations

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Cost progression with PSO iterations

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Input to hidden layer weights

Parameter | Value |

3.183994 | |

2.164797 | |

–6.76984 | |

–0.51265 | |

2.769091 | |

–8.1403 | |

–1.07712 | |

7.005729 | |

2.03942 |

Hidden to output layer weights

Weight | Value | Weight | Value | Weight | Value | Weight | Value |

2.670686 | 1.610139 | –0.26488 | –0.41302 | ||||

–4.36445 | 0.903675 | –3.51607 | 9.555913 | ||||

–11.0002 | 27.50922 | 3.772408 | –54.2146 | ||||

–13.7829 | –26.9897 | 5.721002 | 6.367803 | ||||

2.301405 | –3.54052 | 1.148764 | 0.79572 | ||||

–1.73654 | –8.72825 | 19.46894 | 30.33683 | ||||

6.857219 | –0.05555 | –2.99248 | –62.7306 | ||||

0.119107 | 3.728767 | 5.861321 | –0.00976 | ||||

48.98421 | 5.315695 | 45.37744 | 62.72158 |

First, the simulation study was carried out for the point *x,y,θ*) = (0.2,0.6,π/6). The same neural network was used in this simulation without a retraining. The results of this simulation are given in Figures 10, 11, 12 and 13. The mean square error for this simulation is calculated and presented in Table 5. Again, the mobile robot was able to follow the trajectory with high accuracy. The robot started by making a maneuver to orient towards the trajectory and then it moved until it hits the reference trajectory and it continues to follow it. In terms of the robot velocities it was similar as in the previous simulation. However, this time higher oscillation was noticed at the start of the simulation. This was expected as the neural network was not trained to start from this initial pose, therefore higher control action is provided to the motors the drive the wheels to perform the maneuvers rapidly and recover from the unexpected initial pose. The calculated mean square errors in Table 5, show higher values for the second simulation. Again, this behavior is due to the start form an untrained pose. The error increase in the *x* and *y* axis was about three times higher than the first case, while the error for the *θ* was slightly lower. Although the MSE three times increased, by analyzing the error shapes, it can be concluded that most of this error is concentrated at the initial point of the trajectory, and the remaining part of the trajectory was followed by the robot with high accuracy.

Trajectory plot

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Trajectory plot

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Trajectory plot

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Tracking errors

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Tracking errors

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Tracking errors

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Robot velocities

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Robot velocities

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Robot velocities

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Wheels velocities

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Wheels velocities

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Wheels velocities

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Mean square error of trajectory tracking results

Initial point | MSE ex | MSE ey | MSE eth |

(0, 0, π/6) | 2.3406e−05 | 2.7623e−05 | 2.5318e−05 |

(0.2, 0.6, π/6) | 7.1066e−05 | 7.2527e−04 | 2.2730e−05 |

Trajectory plot

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Trajectory plot

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Trajectory plot

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Tracking errors

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Tracking errors

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Tracking errors

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Robot velocities

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Robot velocities

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Robot velocities

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Wheels velocities

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Wheels velocities

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Wheels velocities

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## 6 Conclusion

In this work, an optimized intelligent control methodology was proposed for trajectory tracking tasks in four-wheeled omnidirectional mobile robots. Kinematics and dynamics modeling was developed to facilitate the control methodology design and development. A neural network Proportional Integral Derivative (PID) control methodology was designed for the trajectory-tracking tasks. Moreover, controller's neural network was manipulated via particle swarm optimization algorithm. The control approach robustness and effectiveness were verified in a number of simulation studies. The simulation results confirmed the high accuracy of the tracking tasks and the ability of the control method to start from an arbitrary initial point without retraining for the neural network. Future studies could be directed towards including the back propagation methodology in the feedback loop to improve motion accuracy. Moreover, a collision avoidance algorithm is worth investigation to have safe tracking. This study can be extended for future work by introducing other control strategies such as Augmented nonlinear PD controller, sliding mode control, Model reference adaptive control, robust adaptive control [32–39].

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A. J. Humaidi, S. Hasan, and A. A. Al-Jodah, “Design of second order sliding mode for glucose regulation systems with disturbance,”

, vol. 7, no. 2, pp. 243–247, 2018.*Int. J. Eng. Technol. (Uae)*