SIMPLIFIED EXPONENTIAL REACHING LAW BASED SLIDING MODE CONTROL OF TWO-LINK PLANAR ROBOT

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Introduction
Robotic technology is rapidly developing in automation industries worldwide, enhancing efficiency and quality. It is used in handling hazardous materials, building satellites, and performing repetitive tasks in industries like drilling, mining, and medical equipment placement. The list of robotic applications is tedious to summarize.
Robotic manipulator arm is an industrial-scale robotic arm with rigid links connected by moveable joints for motion-control that can be opened or closed in motion [1,2,3]. Robots imitate human movement using brain, feet, and arms, acquiring aptitude and imagination through manual or autonomous operation of robotic arms [4]. Computer systems control robotic manipulators [5], joysticks, graphical user interface [6], accelerometers [2], sensors [7], and through the internet for remote applications [8].
Planar robots are used in parallel planes, differing from spherical robots with circular links or spatial robots with spatial links [9].
More sophisticated methods of controlling/directing a robotic arm include, haptic technology [10], electromyography, and the human mind [11] have recently been developed. Haptic technology provides tactile feedback through forces, vibrations, or motion, enhancing user experience through force, sensation, and acceleration detection using an accelerometer [10]. Electromyography controls robotic arm using muscular tension signal via Bluetooth [12]. Human mind-controlled robotic arm allows daily tasks for paralyzed individuals by visualizing a moving arm, enabling them to function without invasive techniques [11]. Robots are controlled by microcontrollers that repeat tasks until power is turned off or until completed [11].
Technological research communities have historically examined control systems [13]. Beyond linear control algorithms (PID), many resilient and adaptive control solutions have been documented to deal with the non-linearity and ambiguous behavior of robotic manipulators. A few examples of these control strategies are feedback linearization, variable structure control (VSC), Passivity-based control (PBC), and Disturbance-Observer-Based Control (DOBC) [14]. By cancelling out the nonlinear dynamics of a nonlinear system, feedback linearization converts it into an analogous linear system. For this transformation, precise parameter values and state values are needed. Output linearization and Input/State linearization are the two primary types of feedback linearization [15].
VSC stabilizes system behavior through nonlinear feedback, initializing uncertainties on the switching surface, and initiating sliding mode, maintaining trajectories independent of model errors on the sliding manifold [14]. DOBC is a method for guiding robotic manipulators, calculating disturbance torques and perturbing the nominal model. It enhances system performance by treating coupling torques as unknown external torques, enabling joint control independently in multi-link robotic systems [14].
Due to the nonlinearities and input couplings in the dynamics of robotic systems, high-accuracy trajectory tracking is very challenging even with robust nonlinear controllers. Also, the complexity of these controllers lead to high computational burden. On the other hand, the simple linear controllers are non-robust and require sensors of high accuracy to perform the required task.
Therefore, a simple controller which combines the desired properties of linear (Proportional and Derivative (PD)) and nonlinear control scheme (sliding mode control (SMC)) is required to precisely track the desired trajectory with minimal error.

Literature Review
Due to the two-link robot's nonlinear nature, specific control systems are frequently needed to hold a position or carry out a task [16]. SMC has proven to be more effective for researchers than PID [17]. Others have employed PID feedback loops with holographic neural networks and artificial neural networks to perform the task of paper folding [8]. Improved SMC performance using Lagrange technique and Self-Tuning mechanism outperforms conventional SMC in Signum function and Saturation function [2]. A double pendulum system that used the Lagrange technique as its dynamic model was controlled by PID controllers. The recommended control strategies performed better than expected [18]. Adams software was used to create a virtual manipulator prototype, with satisfactory tracking ability for joint angles using SMC and MATLAB simulations [15]. The dynamic model in [16] was reformulated using the Lagrange technique. SMC and PID were employed to regulate the robot's movements. The PID controller needs more time than the SMC controller to move the arm into the appropriate position. The FOPID-SMC smoothing method was tested by [19] in comparison to the traditional SMC control method. Control scheme reduced conventional control overshoot to 1.139% using FOPID-SMC for joints and links. [20] looked at the manipula-tor for a planar two-link stiff flexible coupling. [21] propose exponential reaching law sliding mode control for dual-arm robots, addressing modeling, parametric uncertainty, and system resilience using efficient, superior transient performance control. Improved Sliding Mode Controller Performance through Power Rate Exponential Reaching Law was the focus of [22] research. Improved PRRL (PRERL) using Power Rate Exponential Reaching Law confirmed effective in simulation tests on robotic arm set point control problem. SMC of Manipulator Based on Improved Reaching Law and Sliding Surface was proposed by [23]. The control strategy for the 6-DOF UR5 manipulator is developed using dynamic models, reaching law, and sliding surfaces. Experiments show it increases convergence speed, reduces chattering, and improves system convergence.

Methodology
This section covers the dynamic modelling which involves the use of lagrange method. It also covers the controller design of the robotic system.

Dynamic model of the Two-Link Planar Robot
FIg. 1 below shows the representation of a typical twolink planar robot with the forces involved. Figure 1: A Two-link planar robot manipulator simplified model [15] where; q i =joint angles (rad),m i =mass of link i (kg), l i =length of link i (kg) I i =moment of inertia of link i (k gm 2 ), l c i =distance from the previous joint to the center of mass of link i (kg) The dynamics equation of a robot can be written in the general form as, where; H(q) =mass inertia, C(q,q) = Centripetal and Coriolis force on the manipulator, g(q) = gravitational force acting on the manipulator andq = acceleration, q= velocity, q = displacement of the joints of the manipulator, and τ =joint driving torque [25].
The following equation is the Lagrange equation used to determine the dynamics of the planar robot.
Thus, the kinetic energy of the two-link robot is of the following form as reported by [25].
where P = P(q) is the potential energy and is independent ofq By assuming that the mass of the entire object is concentrated in its center of mass, it is possible to calculate the potential energy of the two-link robot as where the vector r ci indicates the coordinates of the center of mass of link i and the vector g indicates the direction of gravity in the inertial frame hence, the twolink robot's total potential energy is The following can be used to derive the Euler-Lagrange equations for such a system. Since For the two-link planar robot, the velocity of link 1 and link 2 is v c1 = J vc1q (7) where; Therefore, the kinetic energy's translational component is The potential energy of the manipulator is simply the total of that of the two linkages, which comes next. The mass, gravitational acceleration, and height of the center of mass are multiplied by each link's mass to determine its potential energy. Thus, P = (m 1 l c1 + m 2 l 1 )g sin(q 1 ) + m 2 l c2 sin(q 1 + q 2 ) (12) Hence, the functions φ k becomes (13) where φ 1 and φ 2 are generalized forces on link 1 and 2, respectively.
The matrix C(q,q) is given, in this situation as In general, the system's motion is described by the nonlinear equation in the following form: The norminal values used for the dynamic model are m 1 = m 2 = 1k g, I i = 1 12 k gm 2 , l 1 = l 2 = 1m, and l c1 = 0.5(k g)

Controller Design
Here, the PD controller will be first discussed followed by the simplified exponential reaching law SMC. Finally, the disturbance models that will be used to evaluate the robustness of the proposed controller will be presented.

The PD Controller
In this work, the conventional PD is utilized, with control input u(t) as displayed in Fig. 2 U PD = K p e i + K dė i From 17, is the tracking error, θ di and θ i are the desired and measured joint angles in radian for i = 1, 2, 3, ....,k P is the proportional gain and k d is the derivative gain of the controller Figure 2: The PD Controller Block Diagram [26] The proportional derivative controller was employed to increase the stability of the system without altering the steady-state error. The following gain parameters were used for the controller to achieve this objective; k P = 2000I, and k D =100I.

The Simplified Exponential Reaching Law (SSMC) Design
Consider the control law from equation 1 as, whereθ is the angle signal, u(t) is the control input, There are two steps to designing the sliding mode controller. First, the sliding mode function is designed as and the sliding surface is C must satisfy the Hurwitz criterion, C > 0 The tracking error is Hence, the second derivative of equation 22 is ; Differentiating equation 21 and substituting in equation 23ṡ The exponential reaching law is given bẏ whereṡ(t) = −ks is the exponential term, and its solution is s = s(0)e −kt The proportional rate term −ks forces the state to approach the switching manifolds faster when s is large, as is evident.
Equating (24) with (25) gives Substituting (26) into eqn. (1) gives + H ηs gn s(t) + ks(t) (27) Equation 27 is the control law for the exponential reaching law based SMC (RSMC). Using the exponential rate reaching law, the sliding mode has two phases: the reaching phase and the sliding phase. The sliding phase ensures systems equilibrium while the reaching phase is responsible for speed of convergence thereby maintaining a steady manifold. Therefore in order to achieve the simplified exponential reaching law based SMC (SSMC), the first two terms of ((27)) are ignored. This will not affect the performance of the controller, rather it will enhance its performance while reducing the complexity. Thus, the SSMC control law is given as follows: where k is positive constant which is the parameter of the exponential reaching term and η is the parameter of the constant reaching term. Equation 28 is used as the input to the controller in order to mitigate the effect of chattering in the SMC

The Frictional Disturbance Model
In this work, friction is considered a form of disturbance. Thus, model of friction consist of both Coulomb and viscous frictions and is expressed as where k v = viscosity coefficient,θ = joint velocity, and f c = Coulomb friction force.
Since Torque τ, is given as where r = distance, f (θ) = frictional force and θ=angular displacement Hence, the dynamic model of the robot coupled with disturbance in the form of friction is given as; where τ f = Disturbance in the form of Friction.
Parameter selected for the coefficient of friction are K v = 2 and f c = 1.

Result and Discussions
Synthesis of the controllers can also serve as a parametric measure of analysing the performance of these controllers. The SSMC was synthesized to have minimal steps of algorithm as against the RSMC. This was achieved as the algorithm steps in the SSMC are quite short as compared to the ones in RSMC. The RSMC has longer and complex steps of algorithm. The direct implication of this long or complex algorithm is the issue of the computational burden. The SSMC proves to be of superior response compared to the RSMC and this was attributed to the complexity of the control algorithm.

System Performance without Disturbance
Figs. 3−8 presents the result of the controllers without the system being subjected to disturbance. Fig. 3 presents the result of the tracking position performance of the controllers while tracking a trajectory by link 1.
The proposed scheme's tracking performance is quite better compared to the PD and the RSMC, as it was able to track the trajectory with minimal error.       Fig. 7 presents the tracking error on link 1. PD control scheme presented higher magnitude of error and the peak-peak is wider compared to the SMC and the ERL control schemes. This means that the PD control scheme deviated much more, followed by the RSMC and then the SSMC.  It is obvious that the PD control scheme presented higher magnitude of error and the peak-peak is wider compared to the RSMC and the SSMC control schemes. This means that the PD control scheme deviated much more, followed by the RSMC and then the SSMC. It is evident that the SSMC has presented a better tracking position and speed performance compared to the RSMC and PD control schemes. With regards to the error, the PD presented much more error, followed by the RSMC and then the SSMC. This shows that the SSMC surpasses both the PD and the RSMC in tracking, speed and Error.

System Performance with Disturbance
The speed and position tracking performance with disturbance is shown in Fig. 9 − 14 below. Fig. 9 presents the position tracking of link 1. It is observed that the PD was slightly able to track the target position, the RSMC did the same at the beginning and finally deviated while the SSMC was able to follow the track thoughout. This shows that the SSMC was able to perform better with minimal error compared to both the RSMC and PD. Figure 9: Position Tracking of Link 1 with Disturbance. Fig. 10 presents the result of the speed tracking performance of the controllers while tracking a trajectory on link 1. All the controllers initiated with a constant speed with much deviation from PD and RSMC, and SSMC has minimum deviation. At mid-point, only the PD was able to slightly maintain the speed but the rest could not maintain the speed of the tracking. This shows that the PD has minimum deviation, followed by the SSMC and then the RSMC.     Fig. 13 presents the tracking error on link 1. PD control scheme presented higher magnitude of error and the peak-peak is wider compared to the RSMC and the SSMC control schemes. This means that the PD control scheme deviated much more, followed by the RSMC and then the SSMC. The SSMC had the minimum in this regard.  It is obvious that the PD control scheme presented higher magnitude of error and the peak-peak is wider compared to the RSMC and the SSMC control schemes. This means that the PD control scheme deviated much more, followed by the RSMC and then the SSMC. The SSMC has the minimum amount of error on this link. A quantitative analysis of the tracking performance of the robot carried out on all the three controllers in terms of the root mean square of error (RMSE) in the trajectory tracking was performed and is presented in Table 1 and 2. The graphical presentation is in Fig. 15 and 16 above. From Fig. 15, PD has more error on both when it is subjected to disturbance and when it is not. It can be seen that PD gives more error in the trajectory tracking than the RSMC controller, while the SSMC has the minimum error, and this means that the SSMC has better tracking performance in terms of the error on link 1. Table 1 depicts the error on link 1. Being SSMC having the minimum error, it is obvious that RSMC and PD are not as accurate as SSMC due to their higher magnitude of error. Since RSMC and PD have higher magnitude of change in error to SSMC, SSMC can be said to be more robust than the other controllers on link 1.

Summary
Effort was made to reduce the complexity of the RSMC as given by (28) which reduces the computational burden. The position and speed tracking of the SSMC was found to be better being that the controller was able to track the desired position and speed almost closer to the ideal position with minimum deviation (error) as compared to the PD and RSMC. This can be seen in Fig. 3 to Fig. 6 and also Figs. 7 and 8. Furthermore, the proposed controller maintained its lead interns of tracking the position and speed even with the presence of disturbance compared to both PD and RSMC, as can be seen in Fig. 9 to 12. For the tracking error with and without disturbance on link 1 and link 2 respectively, it can be seen from both tables and Fig. 15 and Fig. 16 that the proposed controller recorded the least errors in all cases.

Conclusion
Modelling and synthesis of the control scheme using the simplified exponential reaching law based sliding mode controller (SSMC) for controlling the system was achieved. The results of the proposed control system (SSMC) with that of the conventional PD controller and the RSMC controller were compared. RMSE with disturbance and without disturbance on both links for SSMC was found to be the minimum and hence better as compared to RSMC and PD (See Fig. 15, Fig. 16, Table 1 and 2). Based on the result obtained, SSMC has the minimum error among the three control schemes and this shows that SSMC can be found to be more accurate. The change in error experienced with and without disturbance by SSMC on both links was also found to be the least, followed by RSMC and then PD (See Fig.  15, Fig. 16, Table 1 and 2). This shows that SSMC is more robust as compared to the other controllers since it is able to maintain or have minimum change in error. It can be concluded that SSMC control scheme has proven to have an improved performance with simplicity and robustness as it was able to maintain minimum tracking error and change in error. The work can be extended to other advanced controllers with fine-tuned controller parameters for improved performance.