The design of a swarm optimization-based fractional control for engineering application is an active research topic in the optimization analysis. This work offers the analysis, design, and simulation of a new neural network- (NN) based nonlinear fractional control structure. With suitable arrangements of the hidden layer neurons using nonlinear and linear activation functions in the hidden and output layers, respectively, and with appropriate connection weights between different hidden layer neurons, a new class of nonlinear neural fractional-order proportional integral derivative (NNFOPID) controller is proposed and designed. It is obtained by approximating the fractional derivative and integral actions of the FOPID controller and applied to the motion control of nonholonomic differential drive mobile robot (DDMR). The proposed NNFOPID controller’s parameters consist of derivative, integral, and proportional gains in addition to fractional integral and fractional derivative orders. The tuning of these parameters makes the design of such a controller much more difficult than the classical PID one. To tackle this problem, a new swarm optimization algorithm, namely, MAPSO-EFFO algorithm, has been proposed by hybridization of the modified adaptive particle swarm optimization (MAPSO) and the enhanced fruit fly optimization (EFFO) to tune the parameters of the NNFOPID controller. Firstly, we developed a modified adaptive particle swarm optimization (MAPSO) algorithm by adding an initial run phase with a massive number of particles. Secondly, the conventional fruit fly optimization (FFO) algorithm has been modified by increasing the randomness in the initialization values of the algorithm to cover wider searching space and then implementing a variable searching radius during the update phase by starting with a large radius which decreases gradually during the searching phase. The tuning of the parameters of the proposed NNFOPID controller is carried out by reducing the MS error of 0.000059, whereas the MSE of the nonlinear neural system (NNPID) is equivalent to 0.00079. The NNFOPID controller also decreased control signals that drive DDMR motors by approximately 45 percent compared to NNPID and thus reduced energy consumption in circular trajectories. The numerical simulations revealed the excellent performance of the designed NNFOPID controller by comparing its performance with that of nonlinear neural (NNPID) controllers on the trajectory tracking of the DDMR with different trajectories as study cases.
1. Introduction
Mobile robots serve platforms with huge versatility within their environment; they are not limited to one location because they can be pushed autonomously in their own circumference. In other words, it has the capability of implementing tasks without assistance from external operators [1]. Mobile robots are unique to move freely in a predefined workplace to accomplish the preferred objectives. This skill of mobility makes the mobile robots appropriate for vast application fields in unstructured and structured surroundings. The ground mobile robot can be categorized into a wheeled mobile robot (WMR) and legged mobile robot (LMR). The WMRs are prevalent because they are tailored to particular applications with reasonably small mechanical complexities and power drain [2].
Within the last decades, many different control structures were introduced into industrial societies to handle the restrictions of the classical controllers. The PID controller which has been dominated by the industrial organizations has been changed using the concept of differentiators and integrators of fractional power. It was shown that a combination of further degrees of freedom with differentiators and integrators of fractional power provided a greater degree of flexibility and performance that would otherwise be hard (even impossible) to come by the conventional PID controllers [3, 4]. The fractional-order PID (FOPID) controllers are the generalization of widely applicable PID controllers; in recent years, they have drawn much attention from both academics and industry [3–5]. Fractional controls are less sensitive to parameter changes in a controlled system. A fractional control unit can easily achieve the isodamping property. On the other hand, incorporating an integral action in the feedback loop has the advantage of eliminating steady-state errors on account of reducing relative stability of the system. It can be concluded that by designing more general controller laws in the form (1/sⁿ), (sⁿ), n ∈ R⁺, the feedback system with more favorable solutions between undesirable and constructive consequences of the above scenario could be attained and by combining these control actions.
Tuning of a fractional PID controller is difficult as five parameters have to be tuned, which means two more parameters compared to a traditional PID controller [3–5]. Some methods were proposed for the proper choice of the parameters’ values of the PID controller. The method of Ziegler–Nichols tuning strategy was acquainted in 1942 for the parameters regulation of the PID controller coefficients; this tuning technique is utilized if the model of the system is a first order plus dead time.
In recent years, methods of optimization which are theoretically different from classical optimization have been invented. They depend on specific properties and behavior of organic herd of birds and nature-inspired and neurobiological systems. These metaheuristic procedures have been developed in the last decade and are evolving as common methods for solving numerical optimization and intricate industrial case studies. Particle swarm optimization (PSO) is a metaheuristic optimization method which depends on the motion and intellect of bird’s colony behavior or fish bevy schooling. Kennedy and Eberhart initially suggested particle swarm optimization (PSO) technique in 1995 [6]. Advantages of PSO are as follows: (1) it does not need the derivative of the cost function, (2) it can be parallelized, and (3) it has fast convergence behavior.
On the other hand, the fruit fly optimization (FFO) swarm technique is one of the state-of-the-art evolutionary computation techniques based on the foraging behavior of fruit flies which was pointed out by Pan [7]. The olfactory organ of a fruit fly can collect different smells from the air and even locate the source of the food from a distance of 40 km. Subsequently, the fruit flies travel to the source of the food and use their acute visionary system to locate the food destination (minimum or maximum of the function) where their companions form a swarm and then travel in that direction. The FFO algorithm seems to be an excellent optimization algorithm; it has numerous benefits such as speed to acquire solutions, the simplicity of its structure, and ease of implementation. So, FFO was effectively used and applied in a diverse class of applications [7–9].
However, FFO algorithm suffers from some shortcomings. Firstly, there is inadequacy in the FFO algorithm concerning the searching policy, a necessary step to yield new solutions of the FFO algorithm using random information of the previous solutions. Moreover, the FFO algorithm has weak exploration ability, low convergence precision, and jumps out of the local minimum. Finally, the candidate solutions cannot be generated in a uniform manner in the domain. On the other hand, PSO experiences the premature convergence, a common phenomenon in the evolutionary methods in very sophisticated applications such as path planning and motion control of mobile robots. Also, it relies on user experience to find the optimum values of some parameters like the inertia weights and social and cognitive coefficients. Moreover, standard swarm optimization algorithms do not find the optimum solutions in a rational time [9]. Therefore, the structure of the FFO and PSO algorithms requires further improvements for attaining the optimum solutions to the real-world applications.
The motivation for the hybridization between the MAPSO and EFFO algorithms is an attempt to combine the beneficial features of MAPSO and EFFO algorithms and conduct a sequential operation for these two optimization algorithms over the progression of the process. Moreover, the hybridization between MAPSO and EFFO algorithms will overcome the limitations of the individual MAPSO and EFFO algorithms mentioned above. This hybridization will be accomplished as described later in this paper.
Many researchers have conducted research studies on motion control problem of DDMR under nonholonomic constraints, and so various kinds of controllers were demonstrated in the literature for the mobile robots to track specific trajectories. Trajectory tracking of wheeled mobile robots using hybrid visual servo equipped with onboard vision systems is described in [10]. In [11], the authors addressed the output feedback trajectory tracking problem for a nonholonomic wheeled mobile robot in the presence of parameter uncertainty, exogenous disturbances, and without velocity measurements using fuzzy logic techniques. The work in [12] focused on the localization, kinematics, and closed-loop motion control for a DDMR. The authors of [13] developed an online nonlinear optimal tracking control method for unmanned ground systems by firstly establishing the nonlinear tracking error model for unmanned ground systems (UGSs), and then the tracking control problem for UGS was converted to a continuous nonlinear optimal control problem with the help of a symplectic pseudospectral method based on the third kind of generation function. In [14], the authors proposed a kinematic-based neural network controller for nonlinear control of the DDMR with nonholonomic constraints. In [15], an iterative learning control over a wireless network for a class of unicycle type mobile robot systems is proposed, and the study included the channel noise effect and the robustness analysis of the proposed system. In [16], a sliding mode-based asymptotically stabilizing controller law has been proposed for a mobile robot. A dynamic prediction-based model predictive control method is offered in [17] for wheeled mobile robots taking into account the tire sideslip. Fuzzy based controllers for autonomous mobile robots have been argued in [18, 19], where the work in [19] dealt with unstructured environments. The work in [20] proposed a disturbance observer based on biologically inspired integral sliding mode control for trajectory tracking of mobile robots. A time-optimal velocity tracking controller for DDMR is presented in [21]. The authors in [22, 23] investigated a model predictive control (MPC) for differential drive mobile robots. Backstepping nonlinear control has been investigated on DDMR in [24]. Recently, researchers are applying a new control paradigm named active disturbance rejection control (ADRC) [25–30] on a wide range of applications [31, 32] and particularly on DDMR [33]. The authors in the literature proposed many algorithms for tuning parameters of the FOPID controller with different applications, where [34, 35] used the genetic algorithms and [36–39] utilized PSO algorithm. Others like Rajasekhar et al. [40] applied the gravitational search optimization technique based on the Cauchy and Gaussian mutation, and El-Khazali [41] exploited the artificial bee colony algorithm. Frequency-domain methods for the design of the FOPID controllers can be found in [42]. Finally, other algorithms like GA can be used to tune the FOPID controller and more complex controllers like [43].
The contributions in this research work lie in twofold:(1)Development of a MAPSO-EFFO algorithm: developing a modified adaptive particle swarm optimization (MAPSO) algorithm by adding an initial run phase with a massive number of particles. At the end of this initial running point, the smaller group of these fitness particles will be selected to continue with an adaptive PSO (APSO) algorithm. Moreover, the conventional fruit fly optimization (FFO) algorithm has been modified by increasing the randomness in the initialization values of the algorithm to cover wider searching space and then implementing a variable searching radius during the update phase by starting with a large radius which decreases gradually during the searching phase. Finally, adopting a hybridized MAPSO-EFFO algorithm by the serial blending of the MAPSO algorithm with EFFO one, i.e., the input to the EFFO algorithm is the output of the MAPSO. The hybridized MAPSO-EFFO technique is used for the evaluation of the parameters of the NNFOPID.(2)New nonlinear fractional control structure: a new NNFOPID controller is proposed in this paper which employs the structure of the neural networks (NNs). With suitable arrangements of the hidden layer neurons using sigmoid nonlinear activation and linear functions in the hidden and output layers, respectively, and with appropriate connection weights between different neurons in different layers, a new class of nonlinear neural FOPID controller is obtained by approximating the fractional derivative and integral actions of the FOPID controller. The outputs of the neural networks are the control actions used to drive the motors of the DDMR.
The paper is organized in the following structure. Section 2 gives the motivation and problem statement. Section 3 presents the fractional calculus and the theoretical background of the fractional PID controllers. The kinematic model of the DDMR is introduced in Section 4. The proposed nonlinear neural conventional and fractional PID controllers for the trajectory tracking of the DDMR are explained in Section 5. Section 6 discusses the results and simulations of the designed motion controllers for the DDMR based on different trajectories. Finally, the conclusions are given in Section 7.
2. Problem Statement
Given a nonholonomic differential drive mobile robot (DDMR) following a particular path, the tracking error occurs because of many factors like noise, disturbances, slippage, and the errors measured from sensors due to both interior and exterior causes. These issues also make the mobile robot has the difficulty to turn left or right by direction set or by using various sensors. Therefore, the DDMR kinematic model has been employed in this paper to synthesize neural network fractional-order PID (NNFOPID) controllers to regulate its speed so that it would track the required path in the plane as fast as possible with minimum mean square tracking error; this is called trajectory tracking problem. Three FOPID controllers will be designed to control the position (x and y) of the DDMR in the 2D plane and its orientation . Moreover, the aim of the proposed tracking FOPID controllers is reducing the energy consumed by the left and right motors of the DDMR. Given a reference path that needs to be followed by the DDMR, which consists of a set of positions in the 2D plane together with the orientation, i.e., , , and , the actual path consists of a set of positions , , and . Then, it is required to design a NNFOPID controller to generate the control velocities of the kinematic model of the DDMR such that the mean square error between and , and , and and is minimum with minimum peak of the left and right velocities of the DDMR. In contrast to the FOPID controller, the NNFOPID controller has more capability to capture the nonlinearity of the DDMR model due to the nature of the neural network structure employed in the design where nonlinear activation functions are used with hidden layer that has adaptable parameters (NNFOPID controller parameters).
3. Fractional Control Analysis
Fractional calculus is a part of the mathematical analysis, which demonstrates the likelihood of the differential operator orders to be the complex or real number for the differentiation and integration. Generally, the form of the fractional-order operation represented by is called as differintegral operator. The sign of controls the action of differintegral whether to be an identity operator, an integrator, or a differentiator.
The fractional integral and derivative using Grünwald–Letnikov (GL) definition follow the same procedures based on the multiderivative integer calculus. The general GL definition is stated as [42, 44]where refers to the integer part, a and b are the start and final limit values, and h is the sampling time. The utilization of GL of (1) in the computation of the output response of any fractional-order system can be illustrated as follows. Given any fractional system expressed by the fractional-order linear constant coefficients differential equation as [44],where are constant and are real numbers. Without loss of generality, the parameters and might be , and . Consider (2) with its right-hand side equal to u (t) such that
Recall GL definition in (1); then, by substituting (1) into (3), the numerical solution of (3) can be evaluated as [44]and can be evaluated in recursive manner as follows:
Now reconsider (2), where its right-hand side is equal to ǔ (t):
Thus, ǔ (t) may be calculated firstly using (1), and then the output response due to ǔ (t) is computed from the solution of (6) as
The FOPID controller increases the efficiency and the possibility of better system performance because of its five parameters. The differential equation of the FOPID controller with fractional power denoted as PIλDα is described by
Taking Laplace transform to (8), we havewhere , , and are derivative, proportional, and integral control parameters, respectively, is the order of the fractional integral, and is the fractional derivative order. It is obvious that the FOPID controller has the three standard coefficients , , and in addition to parameters λ and α, which are fractional powers for derivative and integral actions, respectively. The values of α and λ are nonintegers with the restriction of being positive real numbers.
Discretization methods of continuous-time fractional operators have been studied widely by many researchers [45, 46]. The fundamental principle to discretize a continuous fractional-order operator ( R) is to define it by what is called as the generating function s = ω (z⁻¹). Examples of such transformations are Euler, Tustin, and Simpson transformations. A more recent transformation formula is found as a weighted interpolation between the Euler and Tustin [45]. Usually, the aforementioned transformation schemes lead to a nonrational polynomial in z. To get a rational polynomial, one may find the power series expansion (PSE) of s = ω (z⁻¹) and then truncate the z-polynomial function (in the form of finite impulse response (FIR) filters) to compute the final approximation. The Tustin method is applied since it is more accurate compared to other transforms such as backward and forward difference as given below:
Based on the above analysis, the nonlinear control law that drives wheels of the DDMR will be derived in detail in the next sections. Before that, a concise review of DDMR modeling will be developed.
4. Kinematic Modeling of Nonholonomic DDMR
The position of the DDMR in the world coordinates axis {o, x, y} is illustrated in Figure 1. The kinematics model of the DDMR as shown in the figure consists of a castor wheel in the head of the cart and two driving wheels attached on one axis located at the back. The motion and the orientation of DDMR are achieved via two DC motors which form the actuators of the right and left wheels. Table 1 lists the parameters that have been used in the derivation of the DDMR kinematics.