PSO Tuned Super-Twisting Sliding Mode Controller for Trajectory Tracking Control of an Articulated Robot

Abstract

These days, robots excel at speed, precision, and reliability, surpassing human capabilities. Articulated manipulators, though, pose challenges due to their complex, nonlinear nature and susceptibility to uncertainties such as parameter changes, joint friction, and external disturbances. Designing robust trajectory tracking control for these dynamics is a key focus. This paper introduces a novel method that integrates SolidWorks modeling to create precise digital representations of the robot’s mechanical structure, facilitating easier development and simulation of control algorithms. To drive the robot joints, a permanent magnet direct current motor is used. Initially, sliding mode control (SMC) was employed, but it resulted in chattering in the control’s input response. To mitigate this issue and enhance trajectory tracking, this paper designs a super-twisting SMC (STSMC). Intelligent particle swarm optimization (PSO) is employed to obtain optimal parameter values for STSMC, ensuring consistency, stability, and robustness. A comparative analysis was conducted among PSO–STSMC, STSMC, PSO–SMC, and classical SMC. Numerical simulations revealed that the tracking error and root mean square error (RMSE) improvements were approximately $18.33\%$, $16.66\%$, and $14.29\%$ for PSO–STSMC compared to STSMC, and $79.50\%$, $78.04\%$, and $25.0\%$ compared to PSO–SMC for each of the three joints under ideal conditions, respectively. Numerical simulations demonstrate the proposed controller’s robustness to external disturbances, parameter variations, and joint friction, effectively mitigating chattering effects in the control signal. Notably, this article highlights the potential of the PSO–STSMC for practical implementation in articulated robotic systems and underscores its significance in advancing robust trajectory tracking control, providing insights into articulated robot control strategies.

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The mechanical movement and arrangement of an articulated robot closely mimic that of a human arm. A twisting joint is employed to connect the arm to the base. The arm itself comprises a variable number of rotational joints, typically ranging from 2 to 10. Each of these joints functions as an axis, enhancing the overall range of motion. In the majority of articulated robots, three or six axes are commonly utilized. Several factors, including degrees of freedom, kinematic structure, motor technology, workspace geometry, motion characteristics, and control, can be used to categorize robots [1]. Articulated manipulators are complex systems, affected by uncertainties such as parameter changes, joint friction, and external disturbances. Researchers aim to design controllers for the three-degree-of-freedom articulated robots to accurately follow trajectories despite these challenges. In [2], A backstepping global fast terminal sliding mode control (SMC) for trajectory tracking control of industrial robotic manipulators is developed. An integral of the global fast terminal sliding mode surface is first suggested to improve the dynamic performance and fast convergence of SMC and TSMC, which also obtains a finite-time convergence. A controller is later developed from the proposed sliding surface using the backstepping control method and high-order SMC to ensure the global stability of the control system; the controller provides small position and velocity control errors with less oscillations, smooth control torque, and convergence of the control errors in the short time. In spite of these advantages, the mentioned backstepping global fast terminal SMC did not consider dynamic uncertainties and disturbances and joint frictions. In [3], model-based smooth super-twisting control (MBSSTC) for cancer chemotherapy treatment is introduced. First, this paper considers three nonlinear cell-kill mathematical models: the log-kill, Norton–Simon, and $E_{\max }$ hypotheses, which are sensitive to exogenous perturbations and parametric uncertainties. To achieve precise trajectory tracking even in the face of uncertainties and disturbances, the MBSSTC is applied to the three cell-kill models. The proposed control method (MBSSTC) achieves better results compared to conventional super-twisting control (STC), and proportional-integral (PI) control methods. In spite of these contributions, the stability of the system is not proven mathematically. In addition, the trial-and-error method was used to tune the design parameters of MBSSTC; this method’s drawbacks include its tediousness and inability to identify the optimal design parameters necessary for achieving the best dynamic performance. In [4–6], SMCs are introduced for trajectory tracking control of underactuated flexible joint robots. Each control scheme ensures good performance. However, in these approaches, the authors manually computed the complex dynamic system model of the manipulator and did not consider the robustness test, so the effect of joint friction and controller parameters is acquired through a trial-and-error method, which may fail to yield optimal values. Consequently, it may fail to provide enhanced tracking accuracy and disturbance rejection. In [7], the optimal super-twisting SMC (STSMC) design is introduced for robot manipulators. Performance comparative analysis was made between social spider optimization (SSO) and particle swarm optimization (PSO) in tuning the parameters of STSMC. It is confirmed that the performance provided by SSO–based STSMC is superior to that of PSO–based STSMC. Despite these contributions, the mentioned optimal STSMC do not consider parameter uncertainties, external disturbances, and the effect of joint frictions. In [8], an improved artificial bee colony algorithm with multielite guidance (MGABC)–based SMC with a PI derivative (SMC–PID) surface is developed for the tracking system of three-link manipulators with input and output noise. The work focuses on robustness against disturbance and uncertainty in payload and noise rejection. The suggested controller shows the ability to reduce the effects of disturbances and noises, keeping the objective function (OBJF) at a minimum even in extremely noisy and disturbed systems. Under moderate disturbance and noise conditions, the percentage increase in OBJF values is $0.954\%$, and under severe conditions, it is $14.55\%$. Moreover, under moderate uncertainty conditions, the percentage increase in OBJF values is $5.726\%$, and under severe uncertainty conditions, it is $18.887\%$. This indicates that the optimized control is resilient to alterations in payload mass analysis. However, the mentioned optimized SMC–PID do not consider the actuator dynamics which reduces the model accuracy and ignores the effects of joint friction. The work in [9] employs fuzzy-based SMC for vector-controlled multiphase induction motor drive under load fluctuation. SMCs are utilized in implementing vector control systems for five-phase induction motors. They enable automatic management of motor parameters, adaptation to load variations, and mitigation of disturbances. The findings demonstrate that the suggested approach worked well for selecting motor parameters and coming to rational conclusions. However, the system’s stability was not mathematically demonstrated and the SMC design parameters were adjusted through trial and error, which made it impossible to determine the ideal design parameters required to produce the optimum dynamic performance. In [10], a fuzzy adaptive fixed-time SMC (FAFSMC) technique for trajectory tracking of a class of high-order nonlinear systems with mismatched uncertainties and external disturbances is employed. It is suggested that the fixed-time state observer be integrated with the controller to provide online data and estimate the unmeasured even states. The effectiveness of the proposed FAFSMC scheme is demonstrated by the simulation results of the three simulation examples, namely, the ship course system, two-link robotic manipulator, and three-link robotic manipulator, to the other three conventional methods for solving the trajectory tracking problem. However, there are issues in the design of the global predefined stability and the inability to identify optimal control parameters. In [11, 12], a cascaded terminal SMC and adaptive fuzzy SMC (AFSMC) with the neurofuzzy system for the trajectory tracking control of a mobile robot are introduced, respectively. Both control structures ensure good convergence and robustness of the system. However, in [11], the proposed method assumes that wheel slips and disturbances only occur in the dynamic loop, and parameter uncertainties or joint friction are not analyzed in this work, and in [12], the proposed controller tries to mitigate the chattering effect using the boundary layer method in the nonlinear component of SMC; designing the boundary layer thickness and switching gain requires human expertise, understanding of the magnitude of disturbances, and knowledge of the boundaries of system uncertainties and this approach exposes the system’s performance to a steady-state accuracy issue and the incorporation of the neurofuzzy system may increase complexity, particularly for systems such as mobile robot which have complex dynamic systems. In [13], an advanced control algorithm that combines fast integral terminal SMC (FIT-SMC), a robust exact differentiator (RED) observer, and an estimator based on feedforward neural networks (FFNNs) is introduced. By employing the proposed control method, the simulation results confirm its effectiveness for a multidegree-of-freedom (MDoF) robotic system equipped with model-based friction compensation. These findings reveal an overshoot of less than 1.5$\%$ and a settling time of 0.2950 s or less for all joints of the robotic manipulator. However, it fails to take into account the actuator dynamics for all joints. In [14], a robust controller, utilizing a nonlinear estimator, is developed to control the position and yaw of unmanned aerial vehicle (UAV) while also accounting for uncertainty estimation. The effectiveness of this controller has been evaluated within a cosimulation environment that integrates MATLAB–Simulink and Simcenter Amesim, including FPGA implementation. The results demonstrate that the robust controller outperforms a basic control version, exhibiting superior performance in both trajectory tracking and actuation efficiency. In [15], a nonlinear observer-based controller for attitude control of a ground vehicle, incorporating uncertainty estimation, is introduced. The proposed controller ensures enhanced performance and better transient behavior compared to existing controllers, particularly in the presence of parametric uncertainties and environmental disturbances. However, it fails to take into account the dynamic friction model.

Motivated by the need for enhanced trajectory tracking performance, this paper designed a new robust PSO–based STSMC by integrating SolidWorks modeling with control design to bring the accurate digital representation of the robot’s mechanical structure obtained from SolidWorks, thereby facilitating the development and simulation of control algorithms. By incorporating SolidWorks into the control framework, we ensure that the controller design and tuning process are based on a precise understanding of the robot’s dynamics and kinematics. Unlike most control approaches using the trial-and-error approach to determine the super-twisting sliding mode parameters (which are not the optimal parameters), the optimization method (PSO) is proposed to find the optimal design parameters to guarantee the rapid convergence and the overall stability of the system.

Referring to this state-of-the-art, this paper extends these results in different directions, which represent the main contributions of this work as follows:

1. Proposing a novel control strategy that integrates PSO optimization, STSMC, and accurate SolidWorks–based modeling, leading to enhanced accuracy and reliability in real-world applications. Traditional mathematical modeling methods which are used by the existing modeling method often hinge on equations and theoretical frameworks, presenting a challenge for nonexperts to grasp without visual representation.

2. To the best of our knowledge, while the control method may seem to already exist, the novel aspect lies in the integration process, starting from modeling to control design, and the combination of PSO optimization and STSMC facilitates superior trajectory tracking performance by effectively handling the combined effects, i.e., parameter uncertainties, joint frictions, and external disturbances.

The paper is organized into five main sections. Section 1 provides a detailed introduction and review of previous works regarding articulated robot control. Section 2 presents the kinematic, dynamic, and CAD software modeling of a 3 DOF articulated robot. The proposed controller, PSO–STSMC, as well as the standard controllers’ design including the stability analysis has been derived and discussed in Section 3. Section 4 addresses the results gained from the simulation with their respective discussion in two scenarios. The limitations of the study are mentioned in Section 5. Finally, the conclusion has been given in Section 6.

2. 3-DOF Articulated Manipulator System Modeling

Obtaining a thorough and accurate model of the system is the first crucial step in controlling any of the physical systems. Before applying the controller in a real-time application, this model is useful for system simulation. Kinematic analysis, essential for constructing control algorithms, has been divided into two types: forward kinematics and inverse kinematics. In the context of developing control algorithms, the dynamic behavior of the manipulator focuses on the relationship between link motion and joint actuator voltages.

2.1. Forward Kinematics

A manipulator consists of a series of links connected to each other via revolute or prismatic joints attached to the base frame, extending through the end effectors. The functional relationship between the joint variables and the location and orientation of the end effectors is described by the forward kinematic equations. Forward kinematics is the process of determining the pose of the end effectors in terms of the joint variables [16]. To model the kinematic structure of the robot manipulator, the Denavit–Hartenberg (DH) technique, which only requires four parameters, is most frequently used [17]. They are the joint angle, link offset, link length, and link twist angles abbreviated as $q_i,d_i,a_{i-1}$, and ${\alpha }_{i-1}$, respectively.

The end effector’s position and orientation matrix are obtained from the transformation matrix given in the following equation:$$\begin{array}{}\text{(1)}& T_3^0=T_1^0{T}_2^1{T}_3^2=\begin{array}{cccc}{c}_1{c}_{23}& -s_3& -s_1& l_2{c}_1{c}_2+l_3{c}_1{c}_{23}\\ s_1{s}_{23}& c_3& c_{31}& l_2{s}_1{c}_2+l_3{s}_1{c}_{23}\\ s_{23}& 0& 0& l_1+l_2{s}_2+l_3{s}_{23}\\ 0& 0& 0& 1\end{array}.\end{array}$$

The end effector’s position and orientation matrices can be evaluated from equation (1) and can be derived as$$\begin{array}{}\text{(2)}& \begin{array}{c}{p}_x\\ p_y\\ p_z\end{array}=\begin{array}{c}{l}_2{c}_1{c}_2+l_3{c}_1{c}_{23}\\ l_2{s}_1{c}_2+l_3{s}_1{c}_{23}\\ l_1+l_2{s}_2+l_3{s}_{23}\end{array}.\end{array}$$

The first three rows and columns of the transformation matrix are the end effector’s orientation matrices [18].$$\begin{array}{}\text{(3)}& R_3^0=\begin{array}{ccc}{c}_1{c}_{23}& -s_3& -s_1\\ s_1{s}_{23}& c_3& c_{31}\\ s_{23}& 0& 0\end{array}.\end{array}$$

By breaking down the elements, we have the following:

• $c_1$, $s_1$, $c_2$, $s_2$, $c_3$ , $s_3$, $c_{23}$, $s_{23}$, are $c_{31}$ represent cosine and sine values of various angles. The subscripts indicate which angle they correspond to.

• $l_1$, $l_2$, and $l_3$ represent lengths or distances.

2.2. Inverse Kinematics

Finding the joint variables using the end effector’s position and orientation constitutes the solved inverse problem. Generally, the inverse kinematics problem is more challenging than the forward kinematics problem [19]. The inverse kinematics of the 3-DOF robotic manipulator can be formulated using a geometrical approach as$$\begin{array}{}\text{(4)}& {\theta }_1=\mathrm{atan}2p_y,p_x,\text{(5)}& {\theta }_2=\mathrm{atan}\frac{p_z-l_1}{\sqrt{p_x^2+p_y^2}}-\mathrm{acos}\frac{r^2+l_2^2-l_3^2}{2l_{2}r},\text{(6)}& {\theta }_3=\mathrm{acos}\frac{p_x^2+p_y^2+{p_z-l_1}^2-l_2^2-l_3^2}{2l_2{l}_3},\end{array}$$where $\theta :\text{\hspace{0.17em}angle}$, $p_x:\mathrm{X}-\text{coordinate\hspace{0.17em}of\hspace{0.17em}}\mathrm{a}\text{\hspace{0.17em}point}$, $p_y:\mathrm{Y}-\text{coordinate\hspace{0.17em}of\hspace{0.17em}}\mathrm{a}\text{\hspace{0.17em}point}$, $p_z:\mathrm{Z}-\text{coordinate\hspace{0.17em}of\hspace{0.17em}}\mathrm{a}\text{\hspace{0.17em}point}$, and $\mathrm{atan}2:\mathrm{t}\text{he\hspace{0.17em}arctangent\hspace{0.17em}function}.$

2.3. Configuration Description of the Articulated Manipulator

The 3-DOF articulated manipulator is studied through a combination of MATLAB/Simulink and SolidWorks for all simulation analyses. This robot is entirely designed in SolidWorks software, encompassing all mechanical components, including links and joints. Figure 1 shows a complete 3D representation of a 3-DOF articulated manipulator designed using SolidWorks. The mechanical components of the robot are comprehensively outlined in the XML file. This includes details such as mass, inertia moment, center of mass, and parameters of the coordinate system in relation to the assembly environment. In STEP files, the mechanical components are presented as 3D CAD models. Finally, the Simscape Multibody link allows the complete files to be translated into MATLAB/Simulink to construct the control program for this robot system. Such a modeling approach results in a robot model that is similar to the actual dynamic model. Particular attention ought to be paid to any mates or constraints used between each subsequent element throughout the assembly operations. In turn, they later join the Simscape Multibody model and establish the level of flexibility for each component. The 3-D, 3-DOF articulated robot assembly is shown in Figure 1, which is modeled using SolidWorks 2018’s 3D CAD software.

[figure(s) omitted; refer to PDF]

2.4. Dynamic Modeling

Dynamic modeling is very important for the analysis, simulation and design of the controller for robots. The dynamics of the robot deal with the formulation of the mathematical equation of the robot arm motion and with the force acting on the robot mechanism and the acceleration it produces. For the design of the controllers (STSMC and SMC), the dynamic equation of any n-link robot is formulated as [20, 21]$$\begin{array}{}\text{(7)}& \tau =Mq\ddot{q}+Cq,\dot{q}\dot{q}+Gq+{\tau }_{fm}\dot{q}.\end{array}$$

Equation (7) provides the second-order dynamic model for robot control, i.e., the joint acceleration can be expressed as shown in equation (8), provided that $Mq$ is invertible.$$\begin{array}{}\text{(8)}& \ddot{q}=M{q}^{-1}\tau -Cq,\dot{q}\dot{q}-Gq-{\tau }_{fm}\dot{q},\end{array}$$with

• $q={q_1{q}_2{q}_3}^T\in R^{3\times 1}$: joint position vector

• $\tau \in R^{3\times 1}$: applied torque to the robot joints, $\tau$ = ${{\tau }_1{\tau }_2{\tau }_3}^T$

• $Mq\in R^{3\times 3}$: mass or inertia matrix

• $Cq,\dot{q}\in R^{3\times 3}$: Coriolis and centripetal term

• $Gq\in R^{3\times 1}$: gravity torque vector

• ${\tau }_{fm}q,\dot{q}$: joint friction terms

2.5. Manipulator’s Driver Unit Modeling

Geared motors are mostly used in robotics, particularly in applications requiring high torque. They play a crucial role in decreasing the motor’s shaft speed, thereby enhancing the motor’s output torque [22]. Since electric motors with gear transmission are typically used to operate manipulators, and hence the actuator dynamics must be taken into account [22]. The torque generated at the motor shaft is given by$$\begin{array}{}\text{(9)}& {\tau }_m={\tau }_l+J_m\frac{d^2{\theta }_m}{dt}+{\beta }_m\frac{d{\theta }_m}{dt}.\end{array}$$

Electromagnetic torque is mathematically formulated as$$\begin{array}{}\text{(10)}& {\tau }_m=K_m{I}_a.\end{array}$$

By ignoring the loss of mass and power, the reduction gear relates the position of the motor to the robot’s angular joint position as given in the following equation [23]:$$\begin{array}{}\text{(11)}& {\tau }_l=g_r\tau \text{\hspace{0.17em}and\hspace{0.17em}}q=g_r{\theta }_m,\end{array}$$where, $\tau$ is the torque of the manipulator, and $g_r$ is the gear reduction ratio. After the substitution of equations (10) and (11) into equation (9) the following expression is obtained.$$\begin{array}{}\text{(12)}& K_m{I}_a=g_r\tau +J_m{g}_r^{-1}\frac{d^{2}q}{dt}+{\beta }_m{g}_r^{-1}\frac{dq}{dt}.\end{array}$$

The electrical equations that explain the dynamics of actuators are given as$$\begin{array}{}\text{(13)}& V_a=I_a{R}_a+L_a\frac{d{I}_a}{dt}+K_b\frac{d{\theta }_m}{dt}.\end{array}$$

Since the mechanical time constant is typically significantly larger than the electric time constant, the approximation $L_{a}d{I}_a/dt\approx$ 0 can be employed to simplify the system of differential equations describing the system [23].

From equation (13), we have $$\begin{array}{}\text{(14)}& I_a=\frac{V_a}{R_a}-\frac{K_b}{R_a}{g}_r^{-1}\frac{dq}{dt}.\end{array}$$

Putting equation (14) into equation (12) yields the following:$$\begin{array}{}\text{(15)}& K_m\frac{V_a}{R_a}-\frac{K_b}{R_a}{g}_r^{-1}\frac{dq}{dt}=g_r\tau +J_m{g}_r^{-1}\frac{d^{2}q}{dt}+{\beta }_m{g}_r^{-1}\frac{dq}{dt}.\end{array}$$

Now, solving for $\tau$ as$$\begin{array}{}\text{(16)}& \tau =K_m{g}_r^{-1}{R}_a^{-1}{V}_a-J_m{g}_r^{-2}\ddot{q}+K_m{K}_b{R}_a^{-1}{g}_r^{-2}+{\beta }_m{g}_r^{-2}\dot{q}.\end{array}$$

By equating equations (7) and (16), the resulting model represents a manipulator driven by a geared PMDC motor. This entails integrating the dynamics of geared PMDC motors into the overall models of the robot manipulator as$$\begin{array}{c}\text{(17)}& Mq\ddot{q}+Cq,\dot{q}\dot{q}+Gq+{\tau }_{fm}\dot{q}=K_m{g}_r^{-1}{R}_a^{-1}{V}_a-J_m{g}_r^{-2}\ddot{q}+K_m{K}_b{R}_a^{-1}{g}_r^{-2}+{\beta }_m{g}_r^{-2}\dot{q},\\ \text{(18)}& V_a=K_m^{-1}{g}_r{R}_{a}Mq+J_m{g}_r^{-2}\ddot{q}+K_m^{-1}{g}_r{R}_{a}Cq,\dot{q}\dot{q}+K_m^{-1}{g}_r{R}_a{K}_m{K}_b{R}_a^{-1}{g}_r^{-2}+{\beta }_m{g}_r^{-2}\dot{q}\\ +K_m^{-1}{g}_r{R}_{a}Gq+K_m^{-1}{g}_r{R}_a{\tau }_{fm}\dot{q},\end{array}$$where $J_m{g}_r^{-2}$ and $K_m{K}_b{R}_a^{-1}{g}_r^{-2}+{\beta }_m{g}_r^{-2}$ are the terms of the diagonal matrices diag $J_1,J_2,J_3$ and diag ${\beta }_1,{\beta }_2,{\beta }_3$, respectively.

Where,

• $V_a$, $R_a$, $I_a$, $g_r$, $K_b$, $K_m$, $J_m$, and ${\beta }_m$, are likely representing actuator voltage, armature resistance, armature current, gear ratio, back EMF constant, motor constant, motor rotor inertia, and motor viscous friction coefficient, respectively.

Remark 1.

Equation (18) represents the dynamical model of an articulated manipulator actuated by geared electric motors. This model is utilized in the design of the proposed controllers, namely, STSMC and SMC.

3. Controller Design for the 3-DOF Articulated Manipulator

The design steps of STSMC and SMC controllers are presented for the trajectory tracking control of an articulated robot. To achieve the design objectives, the system dynamics are expressed using the error dynamic equation. Subsequently, control laws are designed to ensure the finite-time convergence of the robot’s positions and velocities.

3.1. Design of the Conventional SMC

In the control of nonlinear systems, especially for robotic manipulators, SMC has drawn a lot of interest. The reason behind this is its strong robustness to both structured and unstructured uncertainty, capability to reduce external disturbances, and simplicity of use [24]. The system trajectory is derived on the sliding surface by SMC and it reaches and remains on this surface to the equilibrium point [25]. It is a two-step approach to design the controller [4, 26]. In this paper, three independent SMC controllers are designed to control all the states of the dynamics of the robotic manipulator. The sliding surface equation is defined as [27]$$\begin{array}{}\text{(19)}& s_i={\frac{det}{dt}+c_{i}et}^{n-1},\end{array}$$where $s_i$ is the sliding surface, $c_i$ is the positive constant, n is the system’s degree, and $e_i$ is the position error signal. Since the robot manipulator is expressed by a second-order differential equation with $n=2$, therefore,the trajectory tracking error is expressed as:$$\begin{array}{}\text{(20)}& e_i=q_{di}-q_i,\end{array}$$where $e_i$, $q_{di}$, and $q_i$ are the position error and the desired and actual angular positions for joint i, respectively.

The sliding surface vector, i.e., s = ${s_1{s}_2{s}_3}^T$, is given by$$\begin{array}{}\text{(21)}& s=\dot{e}+ce,\end{array}$$where c = ${c_1{c}_2{c}_3}^T$ are the positive sliding surface parameters and e = ${e_1{e}_2{e}_3}^T$ are the error signals, which is the difference between the desired and actual angular positions.$$\begin{array}{}\text{(22)}& e=q_d-q,\end{array}$$where $q_d={q_{d1}{q}_{d2}{q}_{d3}}^T$ represents the desired joint positions.

The sliding surface time derivative is given as$$\begin{array}{}\text{(23)}& \dot{s}=\ddot{e}+c\dot{e}={\dot{q}}_d-\ddot{q}+c\dot{e}.\end{array}$$

3.1.1. Angular Position Joint Control

The first step in controlling all the states of the robot manipulator is the design of the sliding surface, and as it involves trajectory tracking control, it depends on the error.$$\begin{array}{}\text{(24)}& s_1={\dot{e}}_1+c_1{e}_1,\end{array}$$where $e_1=q_{d_1}-q_1$ and ${\dot{e}}_1={\dot{q}}_{d_1}-{\dot{q}}_1.$

After inserting this error and the error derivative equation into the sliding surface, the result becomes$$\begin{array}{}\text{(25)}& s_1={\dot{q}}_{d_1}-{\dot{q}}_1+c_1{q}_{d_1}-q_1.\end{array}$$

The first derivative of equation (25) yields$$\begin{array}{}\text{(26)}& {\dot{s}}_1={\ddot{q}}_{d_1}-{\ddot{q}}_1+c_1{\dot{q}}_{d_1}-{\dot{q}}_1.\end{array}$$

After obtaining the sliding surface and its first derivative, the subsequent step involves formulating the sliding control law for the first joint’s angular position. This law directs the trajectories to converge towards the surface and ensures continuous sliding at all times.$$\begin{array}{}\text{(27)}& Ut=u_{eq}t+u_{\text{dis}}t.\end{array}$$

The control input is computed by using the sliding surface and the system dynamics. As $\dot{s}=0$, the system becomes in the sliding condition and gives the equivalent equation, which is derived from equation (26), i.e., $u_{eq}\dot{s}=0$, as$$\begin{array}{}\text{(28)}& {\dot{s}}_1={\ddot{q}}_{d_1}-{\ddot{q}}_1+c_1{\dot{q}}_{d_1}-{\dot{q}}_1=0.\end{array}$$

Solving for ${\ddot{q}}_{d_1}$ as$$\begin{array}{}\text{(29)}& {\ddot{q}}_1={\ddot{q}}_{d_1}+c_1{\dot{q}}_{d_1}-{\dot{q}}_1.\end{array}$$

Based on equation (26), the joint actuator voltage for each joint is expressed in terms of centripetal and gravity torque as follows:$$\begin{array}{c}\text{(30)}& V_a=K_m^{-1}{g}_r{R}_{a}Mq+J\ddot{q}+K_m^{-1}{g}_r{R}_{a}Cq,\dot{q}+\beta \dot{q}+K_m^{-1}{g}_r{R}_{a}Gq+K_m^{-1}{g}_r{R}_a{\tau }_{fm}\dot{q}.\end{array}$$

To streamline equation (30) for analysis and design, the following equation is introduced, taking note that $V_a$ is equated to $U$ as a control input:$$\begin{array}{}\text{(31)}& U=H\ddot{q}+C\dot{q}+G+{\tau }_{fm},\end{array}$$in which, $\beta$, D, $C$, G, $F$, and J are $\beta =K_m{K}_b{R}_a^{-1}{g}_r^{-2}+{\beta }_m{g}_r^{-2}$, $H=K_m^{-1}{g}_r{R}_{a}Mq+J$, $C=K_m^{-1}{g}_r{R}_{a}Cq,\dot{q}+\beta$, $G=K_m^{-1}{g}_r{R}_{a}Gq$, ${\tau }_{fm}=K_m^{-1}{g}_r{R}_a{\tau }_{fm}\dot{q}$, and $J=J_m{g}_r^{-2}.$

It ought to be noted that these parameters $\beta ,J$ are positive definite diagonal matrices.

Solving for $H\ddot{q}$ from equation (31) as$$\begin{array}{}\text{(32)}& H\ddot{q}=U_{eq}-C\dot{q}-G_q-{\tau }_{fm}.\end{array}$$

By putting equation (29) into equation (32), the matrix form control signal can be formulated for the control of joint position-1 as$$\begin{array}{}\text{(33)}& \begin{array}{ccc}{H}_{11}& H_{12}& H_{13}\end{array}\begin{array}{c}{c}_1{\dot{e}}_1+{\ddot{q}}_{d1}\\ c_2{\dot{e}}_2+{\ddot{q}}_{d2}\\ c_3{\dot{e}}_3+{\ddot{q}}_{d3}\end{array}=U_{eq1}-\begin{array}{ccc}{c}_{11}& c_{12}& c_{13}\end{array}\begin{array}{c}{\dot{q}}_1\\ {\dot{q}}_2\\ {\dot{q}}_3\end{array}-G_{1}q-{\tau }_{fm1}.\end{array}$$

After rearranging the terms of equation (33) and solving for $U_{eq}$, we get$$\begin{array}{c}\text{(34)}& U_{eq1}=\begin{array}{ccc}{H}_{11}& H_{12}& H_{13}\end{array}\begin{array}{c}{c}_1{\dot{e}}_1+{\ddot{q}}_{d1}\\ c_2{\dot{e}}_2+{\ddot{q}}_{d2}\\ c_3{\dot{e}}_3+{\ddot{q}}_{d3}\end{array}+\begin{array}{ccc}{c}_{11}& c_{12}& c_{13}\end{array}\begin{array}{c}{\dot{q}}_1\\ {\dot{q}}_2\\ {\dot{q}}_3\end{array}+G_{1}q+{\tau }_{fm1}.\end{array}$$

Nevertheless, relying solely on equivalent control effort cannot ensure effective control in the presence of unforeseen disturbances or uncertainty. Consequently, additional control efforts should be strategized to mitigate the impact of unpredictable disturbances. To achieve this goal, the Lyapunov function is chosen as follows:$$\begin{array}{}\text{(35)}& Vs=\frac{1}{2}{s}^{T}s.\end{array}$$

A sufficient condition ensuring the position error tracking transitions from the reaching phase to the sliding phase is referred to as the reaching condition and is defined as$$\begin{array}{c}\text{(36)}& \dot{V}s=s^T\dot{s}<0,\text{(37)}& \dot{V}s=s^T{\ddot{q}}_{di}-{\ddot{q}}_i+c_i{\dot{e}}_i,\text{(38)}& \dot{V}s=s^T{\ddot{q}}_{di}-H{q}^{-1}{U}_{\text{dis}}+H{q}^{-1}Hq{c}_i\stackrel{.}{e_i}+{\ddot{q}}_{di}+C_i{q}_i,{\dot{q}}_i{\dot{q}}_i+{\tau }_{fmi}q,\dot{q}+G_i{q}_i\\ -C_i{q}_i,{\dot{q}}_i{\dot{q}}_i-G_{i}q-{\tau }_{fmi}\dot{q}+c_i{\dot{e}}_i,\\ \text{(39)}& \dot{V}s=s^T{\ddot{q}}_{di}-H{q}^{-1}{U}_{\text{dis}}+H{q}^{-1}Hq{c}_i\stackrel{.}{e_i}+{\ddot{q}}_{di}+c_i{\dot{e}}_i,\text{(40)}& \dot{V}s=s^T{\ddot{q}}_{di}-H{q}^{-1}{U}_{\text{dis}}-c_i\stackrel{.}{e_i}-{\ddot{q}}_{di}+c_i{\dot{e}}_i,\text{(41)}& \dot{V}s=-s^{T}H{q}^{-1}{U}_{\text{dis}}.\end{array}$$

To assure $\dot{V}s<0$, the reaching control law can be chosen as$$\begin{array}{}\text{(42)}& U_{\text{di}{\mathrm{s}}_{\mathrm{i}}}=Hq{k}_i{s_i}^{\alpha }\text{sgn}{s}_i,\text{(43)}& U_{\text{di}{\mathrm{s}}_1}=H_{1i}{k}_1{s_1}^{\alpha }\text{sgn}{s}_1,\end{array}$$where $k_1$ and $\alpha$ are positive constant values, 0 $<\alpha <$ 1 and $k_1>0$ [28].

Substituting equations (34) and (43) into equation (27), the joint-1 control input can be written as$$\begin{array}{c}\text{(44)}& U_1=\begin{array}{ccc}{H}_{11}& H_{12}& H_{13}\end{array}\begin{array}{c}{k}_1{s_1}^{\alpha }\text{sgn}{s}_1+c_1{\dot{e}}_1+{\ddot{q}}_{d1}\\ k_2{s_1}^{\alpha }\text{sgn}{s}_1+c_2{\dot{e}}_2+{\ddot{q}}_{d2}\\ k_3{s_1}^{\alpha }\text{sgn}{s}_1+c_3{\dot{e}}_3+{\ddot{q}}_{d3}\end{array}+\begin{array}{ccc}{c}_{11}& c_{12}& c_{13}\end{array}\begin{array}{c}{\dot{q}}_1\\ {\dot{q}}_2\\ {\dot{q}}_3\end{array}+G_{1}q+{\tau }_{fm1}.\end{array}$$

In a similar fashion to the design procedure of the control law of joint-1, the control law for the second and third joint angular positions is formulated as$$\begin{array}{c}\text{(45)}& U_2=\begin{array}{ccc}{H}_{21}& H_{22}& H_{23}\end{array}\begin{array}{c}{k}_1{s_1}^{\alpha }\text{sgn}{s}_1+c_1{\dot{e}}_1+{\ddot{q}}_{d1}\\ k_2{s_1}^{\alpha }\text{sgn}{s}_1+c_2{\dot{e}}_2+{\ddot{q}}_{d2}\\ k_3{s_1}^{\alpha }\text{sgn}{s}_1+c_3{\dot{e}}_3+{\ddot{q}}_{d3}\end{array}+\begin{array}{ccc}{c}_{21}& c_{22}& c_{23}\end{array}\begin{array}{c}{\dot{q}}_1\\ {\dot{q}}_2\\ {\dot{q}}_3\end{array}+G_{2}q+{\tau }_{fm2},\\ \text{(46)}& U_3=\begin{array}{ccc}{H}_{31}& H_{32}& H_{33}\end{array}\begin{array}{c}{k}_1{s_1}^{\alpha }\text{sgn}{s}_1+c_1{\dot{e}}_1+{\ddot{q}}_{d1}\\ k_2{s_1}^{\alpha }\text{sgn}{s}_1+c_2{\dot{e}}_2+{\ddot{q}}_{d2}\\ k_3{s_1}^{\alpha }\text{sgn}{s}_1+c_3{\dot{e}}_3+{\ddot{q}}_{d3}\end{array}+\begin{array}{ccc}{c}_{31}& c_{32}& c_{33}\end{array}\begin{array}{c}{\dot{q}}_1\\ {\dot{q}}_2\\ {\dot{q}}_3\end{array}+G_{3}q+{\tau }_{fm3},\end{array}$$with$$\begin{array}{}\text{(47)}& \text{sgn}s=\begin{array}{cc}1,& s>0,\\ 0,& s=0,\\ -1,& s<0.\end{array}\end{array}$$

3.2. Design of the Conventional STSMC

In this section, voltage controllers are designed based on STSMC. Super-twisting algorithm (STA) is designed to carry out continuous control with a second-order sliding mode. It is an alternative solution to mitigate the chattering effect while maintaining the same robustness and improved performance of SMC. STSMC is a nonlinear control technique known for its remarkable features in terms of robustness, accuracy, and ease of tuning and implementation. Industrial robots often employ an independent joint control (single-input-single-output [SISO]) method through a decoupled approach. In this paper, three STSMCs are designed to control all three states of the robotic manipulator dynamics. To enable the manipulator to follow a desired trajectory, a robot manipulator reference trajectory tracking control strategy is required. The initial step in designing STSMC involves defining the sliding surface, as formulated in equation (21), which follows a similar approach to SMC controller design. Considering the dynamic model of the articulated robot presented in equation (30), the proposed PSO–STSMC controller is designed using the same approach as employed in the SMC controller design. The control signal comprises two control laws: the equivalent $u_{eq}$ and the switching $u_{sw}$ control laws. In other words, the controller we aim to design can be formulated as$$\begin{array}{}\text{(48)}& V_a=u_{eq}+u_{sw}.\end{array}$$

Remark 2.

The second component of $V_a$, $u_{sw}$, is referred to as STA, a term utilized to enhance the robustness of $u_{eq}$ in the presence of disturbances and uncertainties, while simultaneously minimizing the chattering effect associated with SMC [4].

3.2.1. Angular Position Joint Control

Following a similar method employed in the design of the SMC controller, the equivalent controller part of SMC is akin to STSMC, as formulated in equation (49), and is expressed as$$\begin{array}{c}\text{(49)}& U_{eq1}=\begin{array}{ccc}{H}_{11}& H_{12}& H_{13}\end{array}\begin{array}{c}{c}_1{\dot{e}}_1+{\ddot{q}}_{d1}\\ c_2{\dot{e}}_2+{\ddot{q}}_{d2}\\ c_3{\dot{e}}_3+{\ddot{q}}_{d3}\end{array}+\begin{array}{ccc}{c}_{11}& c_{12}& c_{13}\end{array}\begin{array}{c}{\dot{q}}_1\\ {\dot{q}}_2\\ {\dot{q}}_3\end{array}+G_{1}q+{\tau }_{fm1}.\end{array}$$

However, the switching control law is derived from the STA [29]. According to this algorithm, the switching control law, suitable for situations with no prior information about the upper bounds of uncertainties, can be formulated as$$\begin{array}{}\text{(50)}& U_{sw}=k\sqrt{\text{abs}s}\text{sgn}s+w\int \text{sgn}sdt,\end{array}$$where $k=1.5\sqrt{F^{\mathrm{\ast }}},w=1.1F^{\mathrm{\ast }}$, and $F^{\mathrm{\ast }}$ represents the Lipschitz constant. This system of equations seems to define relationships between certain parameters $k_i$, $w_i$, and the Lipschitz constant $F^{\mathrm{\ast }}$ and allows for some flexibility in choosing $c_i$.

By substituting equations (49) and (50) into equation (48), the following is obtained:$$\begin{array}{c}\text{(51)}& U_1=\begin{array}{ccc}{H}_{11}& H_{12}& H_{13}\end{array}\begin{array}{c}{k}_1\sqrt{s_1}\text{sgn}{s}_1+w_1\int \text{sgn}{s}_{1}dt\\ +c_1{\dot{e}}_1+{\ddot{q}}_{d1}\\ k_2\sqrt{s_2}\text{sgn}{s}_2+w_2\int \text{sgn}{s}_{2}dt\\ +c_2{\dot{e}}_2+{\ddot{q}}_{d2}\\ k_3\sqrt{s_3}\text{sgn}{s}_3+w_3\int \text{sgn}{s}_{3}dt\\ +c_3{\dot{e}}_3+{\ddot{q}}_{d3}\end{array}+\begin{array}{ccc}{c}_{11}& c_{12}& c_{13}\end{array}\begin{array}{c}{\dot{q}}_1\\ {\dot{q}}_2\\ {\dot{q}}_3\end{array}+G_{1}q+{\tau }_{fm1}.\end{array}$$

Following the same procedure, the second and third joint angular position controls are formulated as follows:$$\begin{array}{c}\text{(52)}& U_{eq2}=\begin{array}{ccc}{H}_{21}& H_{22}& H_{23}\end{array}\begin{array}{c}{c}_1{\dot{e}}_1+{\ddot{q}}_{d1}\\ c_2{\dot{e}}_2+{\ddot{q}}_{d2}\\ c_3{\dot{e}}_3+{\ddot{q}}_{d3}\end{array}+\begin{array}{ccc}{c}_{21}& c_{22}& c_{23}\end{array}\begin{array}{c}{\dot{q}}_1\\ {\dot{q}}_2\\ {\dot{q}}_3\end{array}+G_{2}q+{\tau }_{fm2}.\end{array}$$

However, the second joint switching control law is designed from the ST algorithm, and the control law can be taken from equation (50) and represented in a column vector.

Thus, the joint-2 control input can be written as$$\begin{array}{c}\text{(53)}& U_2=\begin{array}{ccc}{H}_{21}& H_{22}& H_{23}\end{array}\begin{array}{c}{k}_1\sqrt{s_1}\text{sgn}{s}_1+w_1\int \text{sgn}{s}_{1}dt\\ +c_1{\dot{e}}_1+{\ddot{q}}_{d1}\\ k_2\sqrt{s_2}\text{sgn}{s}_2+w_2\int \text{sgn}{s}_{2}dt\\ +c_2{\dot{e}}_2+{\ddot{q}}_{d2}\\ k_3\sqrt{s_3}\text{sgn}{s}_3+w_3\int \text{sgn}{s}_{3}dt\\ +c_3{\dot{e}}_3+{\ddot{q}}_{d3}\end{array}+\begin{array}{ccc}{c}_{21}& c_{22}& c_{23}\end{array}\begin{array}{c}{\dot{q}}_1\\ {\dot{q}}_2\\ {\dot{q}}_3\end{array}+G_{2}q+{\tau }_{fm2}.\end{array}$$

Joint-3 control input can be formulated as$$\begin{array}{c}\text{(54)}& U_{eq3}=\begin{array}{ccc}{H}_{31}& H_{32}& H_{33}\end{array}\begin{array}{c}{c}_1{\dot{e}}_1+{\ddot{q}}_{d1}\\ c_2{\dot{e}}_2+{\ddot{q}}_{d2}\\ c_3{\dot{e}}_3+{\ddot{q}}_{d3}\end{array}+\begin{array}{ccc}{c}_{31}& c_{32}& c_{33}\end{array}\begin{array}{c}{\dot{q}}_1\\ {\dot{q}}_2\\ {\dot{q}}_3\end{array}+G_{3}q+{\tau }_{fm3}.\end{array}$$

However, the third joint switching control law is designed from the ST algorithm, and the control law can be taken from equation (50) and represented in a column vector.

Thus, the joint-3 control input can be given as$$\begin{array}{c}\text{(55)}& U_3=\begin{array}{ccc}{H}_{31}& H_{32}& H_{33}\end{array}\begin{array}{c}{k}_1\sqrt{s_1}\text{sgn}{s}_1+w_1\int \text{sgn}{s}_{1}dt\\ +c_1{\dot{e}}_1+{\ddot{q}}_{d1}\\ k_2\sqrt{s_2}\text{sgn}{s}_2+w_2\int \text{sgn}{s}_{2}dt\\ +c_2{\dot{e}}_2+{\ddot{q}}_{d2}\\ k_3\sqrt{s_3}\text{sgn}{s}_3+w_3\int \text{sgn}{s}_{3}dt\\ +c_3{\dot{e}}_3+{\ddot{q}}_{d3}\end{array}+\begin{array}{ccc}{c}_{31}& c_{32}& c_{33}\end{array}\begin{array}{c}{\dot{q}}_1\\ {\dot{q}}_2\\ {\dot{q}}_3\end{array}+G_{3}q+{\tau }_{fm3},\\ \text{(56)}& \begin{array}{c}{U}_i=Hq{k}_i\sqrt{abs{s}_i}\text{sgn}{s}_i+w_i\int \text{sgn}{s}_{i}dt+\\ c_i\stackrel{.}{e_i}+{\ddot{q}}_{di}+C_i{q}_i,{\dot{q}}_i{\dot{q}}_i+{\tau }_{fmi}q,\dot{q}+G_i{q}_i.\end{array}.\end{array}$$

Now, the complete STSMC law can be written in matrix form as$$\begin{array}{c}\text{(57)}& \begin{array}{c}{U}_1\\ U_2\\ U_3\end{array}=\begin{array}{ccc}{H}_{11}& H_{12}& H_{13}\\ H_{21}& H_{22}& H_{23}\\ H_{31}& H_{32}& H_{33}\end{array}\begin{array}{c}{k}_1\sqrt{s_1}\text{sgn}{s}_1+w_1\int \text{sgn}{s}_{1}dt\\ +c_1{\dot{e}}_1+{\ddot{q}}_{d1}\\ k_2\sqrt{s_2}\text{sgn}{s}_2+w_2\int \text{sgn}{s}_{2}dt\\ +c_2{\dot{e}}_2+{\ddot{q}}_{d2}\\ k_3\sqrt{s_3}\text{sgn}{s}_3+w_3\int \text{sgn}{s}_{3}dt\\ +c_3{\dot{e}}_3+{\ddot{q}}_{d3}\end{array}+\begin{array}{ccc}{c}_{11}& c_{12}& c_{13}\\ c_{21}& c_{22}& c_{23}\\ c_{31}& c_{32}& c_{33}\end{array}\begin{array}{c}{\dot{q}}_1\\ {\dot{q}}_2\\ {\dot{q}}_3\end{array}+\begin{array}{c}{G}_{1}q\\ G_{2}q\\ G_{3}q\end{array}+\begin{array}{c}{\tau }_{fm1}\\ {\tau }_{fm2}\\ {\tau }_{fm3}\end{array}.\end{array}$$

3.3. Proof of Stability Analysis

To assess the stability of the designed STC, the Lyapunov stability theorem is employed. The primary objective of the proposed control law is to track the system trajectory and ensure the global stability of the closed-loop system. The quadratic Lyapunov candidate function has been selected and is defined as$$\begin{array}{}\text{(58)}& Vs=\frac{1}{2}{s}^{T}s.\end{array}$$

Remark 3.

The stability of the robot manipulator is guaranteed if $c_i$, $k_i$, and $w_i$ have values satisfying the condition $c_i$ 0, $k_i$ 0 $\&w_i$ 0. Under this circumstance, $V$ and $\dot{V}$ will be positive and negative definite, respectively, and the sliding variables $s_1$, $s_2$, and $s_3$ will also converge to zero for all t$⟶\infty$.

Taking the time derivative of equation (58) with respect to time, we have$$\begin{array}{}\text{(59)}& \dot{V}s=s^T\dot{s}.\end{array}$$

From equation (31), ${\ddot{q}}_i$ is given by$$\begin{array}{}\text{(60)}& {\ddot{q}}_i=H{q}^{-1}{U}_i-C_i{q}_i,{\dot{q}}_i{\dot{q}}_i-G_i{q}_i-{\tau }_{fmi}\dot{q}.\end{array}$$

Provided that, $Hq$ is an invertible matrix, then from equation (26), the derivative of the sliding surface is given by$$\begin{array}{}\text{(61)}& {\dot{s}}_i={\ddot{q}}_{di}-{\ddot{q}}_i+c_i{\dot{e}}_i.\end{array}$$

By substituting equation (56) into equation (60) and equation (60) into equation (61), we have $$\begin{array}{c}\text{(62)}& {\dot{s}}_i={\ddot{q}}_{di}-H{q}^{-1}Hq{k}_i\sqrt{s_i}\text{sgn}{s}_i+w_i\int \text{sgn}{s}_{i}dt+c_i\stackrel{.}{e_i}+{\ddot{q}}_{di}+C_i{q}_i,{\dot{q}}_i{\dot{q}}_i+{\tau }_{fmi}q,\dot{q}+G_i{q}_i\\ -C_i{q}_i,{\dot{q}}_i{\dot{q}}_i-G_{i}q-{\tau }_{fmi}\dot{q}+c_i{\dot{e}}_i,\\ \text{(63)}& `{\dot{s}}_i={\ddot{q}}_{di}-k_i\sqrt{s_i}\text{sgn}{s}_i-w_i\int \text{sgn}{s}_{i}dt-c_i\stackrel{.}{e_i}-{\ddot{q}}_{di}+c_i{\dot{e}}_i,\text{(64)}& {\dot{s}}_i=-k_i\sqrt{s_i}\text{sgn}{s}_i-w_i\int \text{sgn}{s}_{i}dt,\text{(65)}& \dot{V}s=s_1{\dot{s}}_1+s_2{\dot{s}}_2+s_3{\dot{s}}_3,\text{(66)}& \dot{V}s=s_1-k_1\sqrt{s_i}\text{sgn}{s}_1-w_1\int \text{sgn}{s}_{1}dt+s_2-k_2\sqrt{s_2}\text{sgn}{s}_2-w_2\int \text{sgn}{s}_{2}dt\\ +s_3-k_3\sqrt{s_3}\text{sgn}{s}_3-w_3\int \text{sgn}{s}_{3}dt.\end{array}$$

We note that$$\begin{array}{c}\text{(67)}& s_i\text{sgn}{s}_i=s_i,\text{(68)}& \dot{V}s=-k_1\sqrt{s_1}{s}_1-w_1\int s_{1}dt-k_2\sqrt{s_2}{s}_2-w_2\int s_{2}dt-k_3\sqrt{s_3}{s}_3-w_3\int s_{3}dt,\\ \text{(69)}& \dot{V}s\le -k_1\sqrt{s_1}{s}_1+w_1\int s_{1}dt+k_2\sqrt{s_2}{s}_2+w_2\int s_{2}dt+k_3\sqrt{s_3}{s}_3+w_3\int s_{3}dt\le 0.\end{array}$$

According to equation (69), $\dot{V}s<0$, so it can be said that, as per the Lyapunov theory, the command by the super-twisting could stablize the system as long as the parameter values $k$ and $w>0$.

The following relation serves as a tuning guideline to choose the parameters of the STSMC controller using the “trial-and-error” method [30, 31]:$$\begin{array}{}\text{(70)}& \begin{array}{c}{k}_i=1.5\sqrt{F^{\mathrm{\ast }}},\\ w_i=1.1F^{\mathrm{\ast }},\\ c_i>0:\text{\hspace{0.17em}is\hspace{0.17em}free\hspace{0.17em}to\hspace{0.17em}choose}.\end{array}\end{array}$$

3.4. PSO

It is an intelligent optimization algorithm belonging to the class of optimization algorithms known as metaheuristic. It is based on the paradigm of swarm intelligence and is inspired by the social behavior of animals [32]. The inspiration for this algorithm stems from the collective behaviors observed in fish schooling and bird flocking. Due to its ease of construction, rapid convergence, and computational efficiency, this technique has gained widespread recognition and is commonly employed as an optimized tuner. In comparison to other optimization algorithms such as the genetic algorithm (GA), it is capable of identifying both local and global solutions. PSO employs initial random solutions, referred to as particles, similar to other population-based algorithms. This enables successive generations of updates to explore the optimal search space solution. The fundamental PSO method involves three phases: generating particle positions and velocities, updating particle velocities, and updating particle positions. The reduction of error variance to zero occurs when the fitness function attains its minimum value, leading to the discovery of optimal design parameters. This, in turn, enhances the dynamic performance of controlled systems [33]. While attempting to select the best value at the lowest cost, the fitness function is employed to assess the cost of each particle.$$\begin{array}{}\text{(71)}& f=\text{ITA}{\mathrm{E}}_i={\int }_0^{T}t.e_{i}tdt.\end{array}$$

The controller’s performance improves as the integral time absolute error (ITAE) decreases. In order to enhance the transient and steady-state performance of the controlled system, prioritizing minimal steady-state error over other performance indices, the ITAE performance index is employed [34].

3.5. Optimization of SMC and STSMC

Both SMC and STSMC controllers were optimized using the PSO algorithm. This algorithm takes into account the position regulation of a 3-DOF articulated robot to conserve energy by minimizing the forces (voltages) needed to maneuver the robot. Using the STSMC in equation (56), the PSO algorithm adjusts the gains represented by $c$ = diag($c_1$, $c_2$, $c_3$) and the gains $k\text{\hspace{0.17em}and\hspace{0.17em}}w$ = diag($k_1$, $k_2$, $k_3$, $w_1$, $w_2$, $w_3$).

The optimal parameters of both SMC and STSMC are then determined and summarized in Table 1. The overall block diagram of the proposed system is depicted in Figure 2. According to Figure 2, the error and change in the error of the position of the joint of an articulated robot are provided for the nonlinear controller called STSMC or SMC. Based on these inputs, the controller drives the actuator, PMDC motor, of the designed articulated robot. To find the best parameters for the designed controller, a PSO algorithm has been utilized, and the flowchart of the said optimization tool is given on the left side of Figure 2.

Table 1

Conventional SMC and STSMC’s optimal design parameters based on the PSO algorithm.

Design constants for STSMC	Value	Design constants for SMC	Value
$c_1$	29.1088	$c_1$	29.247
$k_1$	11.4222	$k_1$	9.7804
$c_2$	159.748	$c_2$	27.7894
$k_2$	9.4807	$k_2$	8.6865
$c_3$	155.256	$c_3$	31.479
$k_3$	8.2062	$k_3$	10.9286
$w_1$	2.5091	…	…
$w_2$	4.3920	…	…
$w_3$	3.1071	…	…

[figure(s) omitted; refer to PDF]

4. Simulation Results and Discusion

4.1. Model Evaluation of the Manipulator

The research is carried out using specialized hardware designed for supercomputing, including an Intel(R) Core(TM) i9-12900U CPU $@$ 3.60GHz with a clock speed of 3.60 GHz, 32.0GB of RAM, and a storage capacity of 450.0GB. This paper relies on secondary data as its primary source, sourced from [35, 36]. Prior to testing the controller’s performance, it is crucial to assess the open-loop model of the manipulator which was built on SolidWorks and after all the model is exported into MATLAB/Simulink. The evaluation is conducted by importing the manipulator’s model into the Simulink environment, where the nonlinear open-loop 3-DOF articulated manipulator is then simulated in the MATLAB/Simulink environment. The parameters used for both robot manipulators and actuators are summarized in Tables 2 and 3, respectively.

Table 2

Parameters used for the proposed robot manipulator [35, 36].

Parameter	Notation	Value
Link length 1	$l_1$	0.15 m
Link length 2	$l_2$	0.5 m
Link length 3	$l_3$	0.5 m
Mass of link-1	$m_1$	0.5 $\text{kg}$
Mass of link-2	$m_2$	0.5  $\text{kg}$
Mass of link-3	$m_3$	0.5 $\text{kg}$
Gravity	$g$	9.81 $\mathrm{m}/\mathrm{s}$

Table 3

Parameters of the actuators [35].

Motor moment of inertia	$J_{m1}=3.7\times {10}^{-4}\text{kg}\mathrm{·}{\mathrm{m}}^2$	$J_{m2}=3.7\times {10}^{-4}\text{kg}\mathrm{·}{\mathrm{m}}^2$	$J_{m3}=1.47\times {10}^{-4}\text{kg}\mathrm{·}{\mathrm{m}}^2$
Viscous friction coefficient	${\beta }_{m1}=1.3\times {10}^{-5}\mathrm{N}\mathrm{·}\mathrm{m}\mathrm{·}{\mathrm{s}}^{-1}$	${\beta }_{m2}=1.3\times {10}^{-5}\mathrm{N}\mathrm{·}\mathrm{m}\mathrm{·}{\mathrm{s}}^{-1}$	${\beta }_{m3}=2\times {10}^{-5}\mathrm{N}\mathrm{·}\mathrm{m}\mathrm{·}{\mathrm{s}}^{-1}$
Back EMF constant	$K_{b1}=2.2\times {10}^{-4}\mathrm{s}\mathrm{·}{\text{rad}}^{-1}$	$K_{b2}=2.2\times {10}^{-4}\mathrm{s}\mathrm{·}{\text{rad}}^{-1}$	$K_{b3}=2.1\times {10}^{-4}\mathrm{s}\mathrm{·}{\text{rad}}^{-1}$
Motor back EMF constant	$K_{m1}=0.21\mathrm{N}\mathrm{·}\mathrm{m}\mathrm{·}{\mathrm{A}}^{-1}$	$K_{m2}=0.21\mathrm{N}\mathrm{·}\mathrm{m}\mathrm{·}{\mathrm{A}}^{-1}$	$K_{m3}=0.23\mathrm{N}\mathrm{·}\mathrm{m}\mathrm{·}{\mathrm{A}}^{-1}$
Armature resistance	$R_1=2.8\mathrm{\Omega }$	$R_2=2.8\mathrm{\Omega }$	$R_3=4.8\mathrm{\Omega }$
Armature inductance	$L_1=3\text{\hspace{0.17em}mH}$	$L_2=3\text{\hspace{0.17em}mH}$	$L_3=2.4\text{\hspace{0.17em}mH}$
Gear reduction ratio	$g_{r1}=1/60$	$g_{r2}=1/60$	$g_{r3}=1/30$

To validate the system, constant input voltages are applied to the manipulator’s joints. Following the importation of the CAD model of the robot into MATLAB/Simulink, the simulation output for the nonlinear open-loop scenario of the 3-DOF manipulator model is generated. Input voltages ($V_1$, $V_2$, and $V_3$) are applied to joints one, two, and three, respectively. The open-loop uncontrolled dynamic model illustrates the impact of these input voltages on the output state. The simulation results for various input values are subsequently obtained, as depicted in Figures 3 and 4. The findings in Figure 3 reveal that joints, joints two and three, oscillate due to the Earth’s gravitational pull when all the input voltages of the manipulator are set to 0 V. This instability in the behavior of the two joints is attributed to gravitational forces. The robot manipulator system, evident from the simulation results in Figures 3 and 4, is inherently unstable, nonlinear, and strongly coupled. Consequently, the uncontrolled properties of the manipulator necessitate the design of a controller to stabilize it and achieve optimal performance.

[figure(s) omitted; refer to PDF]

4.2. Scenario I: Control Without Uncertainties

Under this scenario, the performance of PSO–STSMC, STSMC, PSO–SMC, and SMC has been assessed in the absence of external uncertainties such as external time-varying disturbance, parameter uncertainty, and joint friction. Figures 5, 6, 7, 8, 9, and 10 depict the dynamic response of the system controlled by PSO–STSMC, STSMC, PSO–SMC, and SMC. To evaluate the proposed PSO–based STSMC controller, a time-varying desired trajectory for each joint is selected as$$\begin{array}{}\text{(72)}& q_i^d=a_0\cos t{e}^{\cos t}-2\cos 4t+a_1,\end{array}$$where the parameter values for each joint are given in Table 4.

[figure(s) omitted; refer to PDF]

Table 4

Parameter values of the desired trajectory.

Parameter	Joint-1	Joint-2	Joint-3
$a_0$	$\pi /32$	$\pi /32$	$\pi /32$
$a_1$	0	0	0.05

In Figures 5 and 6, the zoomed window reveals that the PSO–STSMC controller exhibits a lower variance in error compared to other controllers. It achieves a steady-state response with a settling time of $0.1088\mathrm{s}$ and a rising time of $0.0423\mathrm{s}$. Furthermore, the PSO–STSMC controller results in small steady-state errors on the order of ${10}^{-5}\text{rad}$ with no overshoot. In contrast, the tuning with the trial-and-error method yields a steady-state response with a settling time of $0.6117\mathrm{s}$ and a rising time of $0.0580\mathrm{s}$ and produces steady-state errors on the order of ${10}^{-5}\text{rad}$ with no overshoot.

Similarly to joint-1 responses, from Figures 7 and 8, it is evident that the PSO–STSMC controller exhibits low variance in error. It attains a steady-state response with a settling time of $0.1073\mathrm{s}$ and a rising time of $0.0546\mathrm{s}$. In addition, the PSO–STSMC controller generates small steady-state errors on the order of ${10}^{-5}\text{rad}$ with no overshoot. However, the trial-and-error tuning method yields a steady-state response with a settling time of $0.4382\mathrm{s}$ and a rising time of $0.0543\mathrm{s}$ and produces steady-state errors on the order of ${10}^{-6}\text{rad}$ while the condition of the overshot is kept as it is. This could imply that the PSO–STSMC controller system response is fast.

The findings in Figures 11, 12, and 13, illustrate the control input voltages of the three controllers. During the initial phase, a robot manipulator experiences significant strain due to the need for a high voltage to counteract the effects of gravity. Consequently, the motor voltages respond rapidly, initially reaching a high voltage value. At the start, the STSMC controller requires a relatively higher amount of voltage to guide the actual trajectories towards the desired ones. Examining Figures 9 and 10, the zoomed window illustrates that the PSO–STSMC controller exhibits low error variance. It achieves a steady-state response with a settling time of $0.1110\mathrm{s}$ and a rising time of $0.0643\mathrm{s}$. In addition, it generates small steady-state errors on the order of ${10}^{-5}\text{rad}$ with no overshoot. In contrast, the trial-and-error tuning method results in a steady-state response with a settling time of $0.1121\mathrm{s}$ and a rising time of $0.0858\mathrm{s}$ and produces steady-state errors on the order of ${10}^{-5}\text{rad}$.

[figure(s) omitted; refer to PDF]

In a comparison of chattering, as depicted in the results from Figures 5, 6, 7, 8, 9, 10, 11, 12, and 13, it becomes evident that the proposed controller provides continuous control voltage without compromising robustness. This is achieved through the application of a reaching control law based on PSO–STSMC. In contrast, the PSO–SMC controller produces high-frequency control voltages, leading to an undesired chattering effect. The ability of PSO–STSMC to reduce chattering is more pronounced, even when a power rate reaching law is employed, as observed in PSO–SMC.

Table 5 presents the time domain performance specifications of the designed controllers. Simulation results in Figures 14, 15, and 16 demonstrated that the sliding surface is stable as proven by using the Lyapunov stability criterion and the error dies out asymptotically, and the PSO–STSMC sliding surface demonstrates notably improved performance, indicating the efficacy of the proposed control strategy employed. This achievement can be elucidated through several key observations: stability and robustness, transient response, tracking accuracy, chattering reduction, and energy efficiency.

Table 5

Tracking performance of the controllers without uncertainty.

Controller type	State variables	Rise time (sec)	Settling time (sec)	Steady-state error (rad)	RMSE (rad)	ITAE (rad)
PSO–STSMC	$q_1$	0.0423	0.1088	$-1.4348\times {10}^{-5}$	$4.8787\times {10}^{-4}$	$6.9720\times {10}^{-4}$
	$q_2$	0.0546	0.1073	$-5.6650\times {10}^{-6}$	$5.4649\times {10}^{-4}$	$5.4630\times {10}^{-4}$
	$q_3$	0.0643	0.1110	$3.4046\times {10}^{-5}$	$1.2000\times {10}^{-3}$	$1.6828\times {10}^{-3}$
Classical STSMC	$q_1$	0.0580	0.6117	$-5.2921\times {10}^{-5}$	$5.9737\times {10}^{-4}$	$7.1296\times {10}^{-4}$
	$q_2$	0.0543	0.4382	$-1.0751\times {10}^{-4}$	$6.5577\times {10}^{-4}$	$6.5067\times {10}^{-4}$
	$q_3$	0.0858	0.1121	$-1.0903\times {10}^{-4}$	$1.4000\times {10}^{-3}$	$1.7342\times {10}^{-3}$
PSO–SMC	$q_1$	0.0702	0.1424	$2.5330\times {10}^{-4}$	$1.0000\times {10}^{-3}$	$3.9554\times {10}^{-3}$
	$q_2$	0.0901	0.1718	$1.2313\times {10}^{-4}$	$1.2000\times {10}^{-3}$	$3.4580\times {10}^{-3}$
	$q_3$	0.0751	0.1415	$2.4385\times {10}^{-4}$	$1.6000\times {10}^{-3}$	$2.3207\times {10}^{-3}$
Classical SMC	$q_1$	0.2649	0.4726	$-1.7587\times {10}^{-4}$	$2.1000\times {10}^{-3}$	$3.5864\times {10}^{-3}$
	$q_2$	0.2509	0.4515	$-3.5944\times {10}^{-4}$	$2.2000\times {10}^{-3}$	$2.9180\times {10}^{-3}$
	$q_3$	0.208	0.3850	$-4.1132\times {10}^{-4}$	$3.1000\times {10}^{-3}$	$6.3888\times {10}^{-3}$

[figure(s) omitted; refer to PDF]

4.3. Scenario II: Control With Uncertainties

In this scenario, the proposed controller’s ability to handle disturbance rejection, parameter variation, and joint friction tests has been assessed. The dynamics of a manipulator can change when an industrial robot handles objects of different masses, leading to uncertainties in the manipulator model. It is important to note that these factors may occur individually or in combination, and their overall impact tends to hinder performance from the nominal design. In this section, the robustness of the controllers is assessed through three simulation cases. First, the evaluation involves the system model with external disturbances. Second, the system model with parametric uncertainty is considered. Third, the analysis includes the consideration of static friction.

4.3.1. Disturbance Consideration

The external disturbances are modeled as sinusoidal signals. They can be conceptualized as external voltages injected at the controller output, potentially arising from power supply fluctuations and disturbance voltage signals. At t = 1.5 s, a disturbance signal, $4\sin t$, is introduced to the control signal of each joint of the manipulator. Retuning or adjusting the control parameters may be necessary to ensure the effectiveness and stability of the control system in the presence of disturbances. However, in this scenario, for the sake of robustness comparison, the control parameters are kept the same between scenarios with and without disturbances, as they were initially tuned for optimal performance under nominal conditions. Upon closer examination of the magnified window, it is evident that the peak error of the PSO–STSMC control is lower compared to that of STSMC and PSO–SMC. Furthermore, from the results in Figures 17, 18, and 19, PSO–STSMC exhibits the fastest recovery time after the disturbance. However, Figure 19 shows that PSO–SMC recovers from the error with an increase in the chattering effect of the control signal. In the disturbance rejection analysis, improvements of $74.46\%$, $85.36\%$, and $37.5\%$ are achieved by PSO–STSMC over the classical STSMC controller in ITAE values for link-1, link-2, and link-3, respectively.

[figure(s) omitted; refer to PDF]

4.3.2. Parameter Uncertainty Consideration

In various scenarios, robotic arms may encounter different loads with varying masses. To assess the robustness of the controller, the masses of the links are altered by a certain percentage. This change in the mass of the robot link, resulting from an unknown payload carried by the manipulator’s end effector, is treated as parametric uncertainty. In the simulation, parameter uncertainty is represented by taking $20\%$ of the mass of the manipulator’s nominal parameters.

4.3.3. Joint Friction Consideration

Certainly, frictional forces exert an influence on every robotic mechanism, and modeling these forces, at least approximately, is crucial to ensure that dynamic equations accurately reflect the reality of the actual plant [37]. The simplified friction model includes only the coulomb and viscous elements. The significance of this model lies in its simplicity, involving just two parameters, making it suitable for adoption as described in [38, 39].$$\begin{array}{}\text{(73)}& {\tau }_{fm}{\dot{q}}_i={\tau }_{ci}\text{sgn}{\dot{q}}_i+{\tau }_{vi}{\dot{q}}_i,\end{array}$$where ${\tau }_{fm}$ is the joint frictional force and ${\tau }_c$ and ${\tau }_v$ are the coulomb friction and viscous damping constants, respectively. Figures 20, 21, and 22 present the position tracking of the designed controllers for all considered joints in consideration of frictions. In the presence of joint friction, all controllers attempt corrective actions to mitigate the impact of joint friction, with minimal increase in chattering observed in the control signal, even in the case of the SMC controller. Considering joint friction, PSO–STSMC achieves improvements of $16.50\%$, $27.85\%$, and $1.16\%$ over the classical STSMC controller in ITAE values for link-1, link-2, and link-3, respectively. Table 6 provides a summary of the performance of the designed controllers under the influence of friction.

[figure(s) omitted; refer to PDF]

Table 6

ITAE of the designed controllers under joint friction.

Controller types	ITAE1	ITAE2	ITAE3
PSO–STSMC	0.00086	0.00057	0.00171
Classical STSMC	0.00103	0.00079	0.00173
PSO–SMC	0.00396	0.00347	0.00231
Classical SMC	0.00420	0.00358	0.00636

The simulation results, Figures 23, 24, and 25, indicate that, even when the system experiences joint friction, the actual angular positions follow their reference trajectories for all control techniques. However, the proposed controller, PSO–STSMC, exhibits superior robustness characteristics when the system encounters parameter uncertainty in system parameters. Specifically, during the consideration of parameter uncertainty, enhancements of $36.36\%$, $3.17\%$, and $5.55\%$ are achieved by PSO–STSMC over the classical STSMC controller in ITAE values for link-1, link-2, and link-3, respectively. Table 7 summarizes the performance of the designed controllers under uncertainty.

[figure(s) omitted; refer to PDF]

Table 7

ITAE of the designed controllers under parameter uncertainty.

Controller types	ITAE1	ITAE2	ITAE3
PSO–STSMC	0.0007	0.00061	0.0017
Classical STSMC	0.0011	0.00063	0.0018
PSO–SMC	0.0039	0.0035	0.0025
Classical SMC	0.0035	0.0027	0.0063

4.3.4. Combined Effect (Disturbance, Parameter Uncertainty, and Joint Friction) Consideration

Figures 26, 27, and 28 display the trajectory tracking profile of the system’s response when exposed to external disturbances, parameter uncertainties, and joint frictions simultaneously. Following the introduction of disturbances, parameter variations, and joint frictions after $1.5\sec$ seconds, it can be observed that the reaching time is nearly consistent with the previous simulation results. However, a slight change is noticed in the tracking error (ITAE). Examining the zoomed sections in Figures 26, 27, and 28, the impact of external disturbances, parameter uncertainty, and joint friction is observed at the beginning of $1.5\sec$ seconds. Nevertheless, all the controllers tolerate these influences, and the errors are corrected within a short time, demonstrating the controllers’ robustness. Notably, PSO–SMC addresses minimal error by increasing the chattering in the control signal, while PSO–STSMC remedies the error with a minor magnitude increment relative to PSO–SMC.

[figure(s) omitted; refer to PDF]

5. Limitation of the Study

Tuning the parameters of the PSO algorithm and the STSMC can be challenging and may require a significant number of function evaluations to converge to an optimal solution, which can be computationally expensive, especially for high-dimensional or real-time control problems. Like many optimization algorithms, PSO is sensitive to its initialization parameters. Suboptimal initialization can lead to slower convergence or even premature convergence to suboptimal solutions.

6. Conclusions and Future Works

This paper introduces the development of a trajectory tracking control scheme for a 3-DOF articulated manipulator using PSO–based STSMC. The suggested PSO–STSMC controller is evaluated through simulation results, considering model parameter uncertainties, time-varying disturbances, and joint frictions. The simulations were conducted in the MATLAB/Simulink environment, utilizing the CAD-imported model. The effectiveness of the proposed control schemes is assessed based on ITAE, root mean square error (RMSE), and time domain specifications.

The PSO–STSMC controller demonstrates notable improvements, with a percentage RMSE improvement of approximately $18.33\%$, $16.66\%$, and $14.29\%$ compared to classical STSMC, and $79.50\%$, $78.04\%$, and $25.0\%$ compared to PSO–SMC for each joint, respectively, under nominal conditions. It exhibits small steady-state error, rapid response, precise tracking accuracy, and reduced chattering behavior compared to conventional controllers. In addition, it effectively decreases control signal chattering and enhances control input smoothness.

In disturbance rejection tests at the system input, the PSO–STSMC controller shows a significant improvement in ITAE values, with enhancements of $77.78\%$, $87.23\%$, and $53.13\%$ over the PSO–SMC controller for joint-1, joint-2, and joint-3, respectively. Similarly, for parameter uncertainty consideration, the PSO–STSMC controller outperforms the PSO–SMC controller by $82.05\%$, $82.57\%$, and $32.00\%$ for joint-1, joint-2, and joint-3, respectively. The analysis demonstrates that the PSO–STSMC controller exhibits promising dynamic response and robustness compared to classical STSMC, PSO–SMC, and SMC controllers for controlling this manipulator system.

The authors of this paper recommend that future researchers in this area focus on (i) extending the system’s degree of freedom using the control approaches discussed in this paper for specific applications and (ii) implementing real-time control of the manipulator based on the proposed controller.

Funding

The authors did not receive support from any organization for this article.

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