Design of a Nonlinear Fractional Order PID Neural Controller for Mobile Robot based on Particle Swarm Optimization

: The goal of this paper is to design a proposed non-linear fractional order proportional-integral-derivative neural (NFOPIDN) controller by modifying and improving the performance of fractional order PID (FOPID) controller through employing the theory of neural network with optimization techniques for the differential wheeled mobile robot multi-input multi-output (MIMO) system in order to follow a desired trajectory. The simplicity and the ability of fast tuning are important features of the particle swarm optimization algorithm (PSO) attracted us to use it to find and tune the proposed non-linear fractional order proportional-integral-derivative neural controller’s parameters and then find the best velocity control signals for the wheeled mobile robot. The simulation results show that the proposed controller can give excellent performance in terms of compared with other works (minimized mean square error equal to 0.131 for Eight-shaped trajectory and equal to 0.619 for Lissajous-curve trajectory as well as minimum number of memory units needed for the structure of the proposed NFOPIDN controller (M=2 for Eight-shaped trajectory and M=4 for Lissajous-curve trajectory) with smoothness of linear velocity signals obtained between (0 to 0.5) m/sec.


INTRODUCTION
heeled mobile robots (WMR) systems have drawn a lot of interest for the researchers recently because they are found in many applications in industry, transportations, military, security settings and other fields due to their ability of handling complex visual and information processing for artificial intelligence, loading capability and they can work in dangerous and hazardous environment .Wheeled mobile robot suffer from nonholonomic constraints (pure rolling without side slipping motion) meaning that mobile robots can move only in the direction normal to the axis of the driving wheels [1].Trajectory tracking control is important topic of the (WMR) it means to apply control signals to the (WMR) in such a way that the (WMR) follow a curve that connects its actual position and orientation with the goal position and orientation of the predefined trajectory (desired or reference trajectory).In general, Trajectory tracking control is still active region of research because as we mentioned above that (WMR) have found in various industrial applications [2].
The Motivation for this work is the problems in the mapping and localization; cognition trajectory planning; path-tracking and motion control; therefore, different trajectory tracking control approaches for (WMR) have been proposed so as to achieve the best performance for the wheeled W mobile robot including high speed, high tracking accuracy (minimized tracking error), low energy consumption and smoothness of velocity control signal obtained, such as fuzzy logic trajectory tracking controller [3 and 4]; sliding mode controller [5 and 6]; back-stepping technique [7and 8]; nonlinear PID neural trajectory tracking controller [9, 10 and 11] and cognitive trajectory tracking neural controller [12 and 13].
The fundamental essence of the contribution novelty for this paper can be described by the points listed below:  Modifying and improving the performance of fractional order PID (FOPID) controller by employing the theory of neural network techniques. The analytically derived control law based non-linear fractional order PID neural (NFOPIDN) controller has considerably numerical accuracy in terms of obtaining best actual signal and leading to minimizing tracking pose error of the wheeled mobile robot with the minimum number of memory units (M) needed based PSO algorithm. Validation of the controller adaptation performance through change the initial pose state. Verification of the controller capability of tracking different types of continuous gradient trajectories.
The remainder of the paper is organized as follows: Section two is a description of the kinematics model of the non-holonomic wheeled mobile robot.In section three, the proposed non-linear fractional order (PID) neural (NFOPIDN) controller is derived.The particle swarm optimization algorithm is explained in section four.In section five, the simulation results and discussion of the proposed controller are described finally the conclusions are presented in section six.

Kinematic Mobile Robot Model
Figure (1) shows the non-holonomic wheeled mobile robot platform which it is a vehicle with two wheels mounted on the same axis and two omni-directional castor wheels mounted in the front and rear of the vehicle.The two castor wheels carry the mechanical structure and keep the platform more stable.The wheeled mobile robot is driven by two independent DC analogue motors that are used as actuators of the left and right wheels for motion and orientation .These two wheels have the same radius denoted by r which is separated by the distance L. The center of gravity of the (WMR) is located at point c, center of axis connected the two driven wheels [14].The location of the mobile robot in the global coordinate frame [O, X, Y] can be represented by the vector as follow: q= (x c ,y c , ) T …(1) x c and y c are the coordinates of point c in the global coordinate frame , and is the robotic orientation angle measured from X-axis then these three generalized coordinates are described the mobile robot's configuration.The wheeled mobile robot has the non-holonomic constraint that the driving wheels are purely roll without slipping [15].This non-holonomic constraint can be written as in equation (2): The equations of the (WMR) in the global frame can be described in kinematics equations (3, 4 and 5) [16 and 17]: 5) Where: is the linear velocity.
is the angular velocity.In the computer simulation, the form of the pose equations can be represented in equations (6, 7 and 8): Where: x (k), y(k) and are the pose at the k th step .

Non-Linear Fractional Order Pid Neural (Nfopidn) Controller Design
A proportional-integral-derivative controller (PID controller) is widely used in industrial control systems because it is simple, reliable, strong robustness in broad operating condition and its parameters can be adjusted easily and separately.An enhanced (PID controller) has been designed in this section (non-linear fractional order PID neural controller) and the method to control the (MIMO) differential wheeled mobile robot depends on the available information of the unknown nonlinear system and the control objectives.The particle swarm optimization (PSO) algorithm has been utilized to generate the optimal parameters for the (NFOPIDN) controller so as to get the best velocity control actions which try to The feedback (NFOPIDN) controller is essential because it is important to stabilize and control the tracking pose error of the mobile robot system when the pose of the wheeled mobile robot is drifted from the desired pose.The error and control action signals are denoted in (FOPID) controller as e (t) and u (t) respectively and they are associated as shown in the equation ( 9) [18]: Where: is the proportional constant.is the integration constant.is the differentiation constant. is the fractional integral operator.
is the fractional differential operator.By taking the Laplace transform of fractional derivative and fractional integral of e (t) are given as equation (10 and 11) respectively [18]: ) After taking Laplace transform of equation ( 9) and submitting equations (10 and 11) in equation ( 9), the transfer function of (FOPID) controller will be as equation ( 12):  … ( 12) The first step in the design of the proposed (NFOPIDN) controller is converted the continuous-time (FOPID) control equation to discrete-time (FOPID) control equation by using the generating function technique with three steps:  Pre-warped Tustin transform is used as equation ( 13) that converted equation ( 12) from sdomain to z-domain. Where: T: sampling period of the system (0.2 sec).
: is gain crossover frequency of the openloop transfer function and it is taken from [19] and equal to (15 rad/sec).: : is order of derivative  Power series expansion (PSE) is used as the resulted expression which is acquired in the term of (z) with a limited (minimum) memory must necessarily use for any practical discrete-time controller. The (PSE) of the expression in the right hand side of equation ( 13) must be calculated in order to check the stability of the controller as shown in figure (3) as follows: 1-If the poles and zeros are inside the unit circle of the (z) plane the system is stable.2-If the poles and zeros are outside the unit circle of the (z) plane the system is unstable.3-If the poles and zeros are on the border of the unit circle of the (z) plane the system is critically stable.

Figure (3): Z-plane pole and zero locations.
After applying the generating function technique with the three steps, the proposed control law can be driven as follows:  The derivative term of equation ( 12) can be expanded as follows: Assuming w = in equation ( 13) yield: ) where: By substitution of w= in equation ( 15) yield: ∑ … (17) Where the coefficients are calculated from equation ( 16).It can be shown (using Maple) that the first few coefficients in (16) are as follow:  The integral term  of equation ( 12) can be expanded as follows: 

… (19)
Where:  : is the order of integral.By applying a Tustin method in equation ( 19), then: Where the term ( )  can be calculated in the same manner as ( ) in equations ( 13) and (15).The term (1 ) is the again which can be calculated as in equation ( 16).
Then, substitution of equations ( 17) and ( 21) in equation ( 12) yield the general control law becomes: The infinite numbers of memory units are needed for its realization and consequently the computational cost is increased by increasing the time.In other words, in practice the upper bound of ∞ as in ( 22) cannot be considered equal to infinity.By restricting the number of memory units to (M) the equation ( 22) will be: 24) By multiplying each side of equation ( 24) by 1 , then the equation will be: Finally, the discrete-time equation of fractional order (PID) controller will be: Based on the discrete-time equation of fractional order (PID), the proposed control law for the (NFOPIDN) controller is driven as follows: , and  can be adjusted using particle swarm optimization technique with minimum number of memory (M) needed and figure (4) is the (NFOPIDN) controller for WMR.

Particle Swarm Optimization Algorithm
The update velocity and position equations for each particle can be written as follow [20]: 33) … (34) Where: pop: the population number.
: the inertia factor.the cognitive parameter.the social parameter., : an independent random numbers with uniform probability between 0 and 1. :the velocity of the particle.:the position of the particle.
: is the best previous weight of particle.: is the best particle among all the particles in the population.The steps of PSO for (NFOPIDN) controller can be described by the following flowchart in figure (5).The non-linear fractional order proportional-integral-derivative neural (NFOPIDN) controller with eleven weights parameters and the matrix is written as an array to form a particle.First, particles are initialized randomly and updated in accordance with the following equations in order to tune the NFOPIDN parameters [10]:

… (48)
This trajectory starts from the origin with the initial desired orientation rad) and a full cycle is completed in 2 .For the simulation objective, the desired trajectory is described in equations (46 and 27) and the desired orientation angle is expressed in equation ( 28).The differential wheeled mobile robot starts from initial posture . The figures (6 and 7) are exhibited the excellent position and orientation tracking performance as compared with other works such as [10 and 11].The simulation results showed the effectiveness of the proposed controller by showing its ability to generate smoothness control velocities without spikes.The actions described in figure 8 shows that the low power is needed to mobile robot's motors.Figure 9 shows that the mean linear velocity and the angular velocity for WMR.The optimized-off-line-tuning based on particle swarm optimization algorithm is used for finding and tuning the control gains parameters of the controller which has demonstrated, as shown in table  The performance index for different values of memory M=1, 2, 3, 4, 5 and 6 can be shown in the figure (10).This figure shows that the best number of memory units for the (Eight-shaped) trajectory is equal to 2.   Another reference path (Lissajous-curve) that can be described by the equations (49, 50 and 51): For the simulation objective, the desired trajectory is described in equations (49 and 50) and the desired orientation angle is expressed in equation ( 51) .thedifferential wheeled mobile robot starts from initial posture [ 1 ,1, 1] The figures (12 and 13) showed the excellent pose tracking performance as compared with other works such as [10 and 11].The simulation results showed the effectiveness of the proposed controller by showing its ability to generate smoothness control input velocities without spikes as well as without saturation state action.The signals described in figure (14) shows that the low power is required to drive the DC motors of the mobile robot model.Figure (15) shows that the mean linear velocity of the mobile robot and the angular velocity.The optimized-off-line-tuning based on particle swarm optimization algorithm is used for finding and tuning the parameters of the (NFOPIDN) controller which has demonstrated as shown in table  The performance index for different values of m=1, 2, 3, 4, 5 and 6 can be shown in the figure (16).This figure shows that the best number of memory units for the (Lissajous-curve) trajectory is equal to 4.
pose error of differential WMR.The block diagram of the proposed controller is shown in the figure (2).∆ is defined as delay mapping Figure (2): Non-linear fractional order proportional-integral-derivative neural controller structure for mobile robot model.