# Neural Dynamics Variations Observer Designed for Robot Manipulator Control Using a Novel Saturated Control Technique

Authors: Francisco G. Rossomando; Emanuel Serrano; Carlos M. Soria; Gustavo Scaglia
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This work presents a novel controller for the dynamics of robots using a dynamic variations observer. The proposed controller uses a saturated control law based onsintg−1. function instead of tanh.. Besides, this function is an alternative to the use of tanh. in saturation control, since it reaches its maximum value more gradually than the hyperbolic tangent function. Using this characteristic, the transition between states is smoother, with similar accuracy to tanh.. The controller is designed using a saturated SMC (sliding mode controller) and a dynamic variations observer based on GRNN (general regression neural network). The originality of this work is the use of a combination of adaptive GRNN with a sliding mode controller (SMC) including a new saturation function. Finally, experiments based on trajectory tracking demonstrate the robustness and simplicity of this method.


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## Body

## 1. Introduction

Several years ago, the work of many researchers resulted in the development of various controllers for robots and mechatronic systems [1–6]. These algorithms were the first in the literature whose stability was proved using different theories [7–9] or using finite-time stability [10–13].Moreover, some of these algorithms were implemented on different robot arms [14–16] and subsequent authors proposed different variants of adaptive control for robot manipulators [17–20]. Only a few of those algorithms solve the particular problems present in industrial robots; in this research work, our main objective is to reduce the lack of fidelity present in robot’s arms that could vitiate any real benefits from a model-based controller. Because of this specific situation, some authors use some type of estimators to identify such infidelities between models (plant controller) or disturbances to improve controller efficiency. In this case, Mohammadi et al. [21] designed a general systematic disturbance observer for robot manipulators, where the proposed estimator doesn’t have restrictions on the number of DOF, configuration, and joints. The disturbance observer has been designed based on the Linear Matrix Inequality (LMI) theory. This method presents different types of convergence depending on the disturbance characteristics, and this technique was designed to minimize large and fast disturbances while the controller drives the output to the desired reference. For efficient operation, the disturbances must be bounded. Also, as stated by Feemster et al. [22], it is necessary that the external disturbance has to be bounded and its period known.Na et al. [23] proposed an unknown system dynamic estimator (USDE) based on simple algebraic calculations. This estimator can be modified by an unknown disturbance estimator (UDE) for external disturbances. Both estimators can adjust their parameters to approximate the unknown dynamics and disturbances, respectively. This work proposes these estimators in the design of composite controllers for trajectory tracking, but the estimations do not consider the accelerations of the joints.Xie et al. [24] proposed a kinematic controller based on global asymptotic stability, which is combined with the speed-based adaptive dynamic compensation controller design. This work uses the sintg−1. function only in its kinematic controller.Bussola et al. [25] used a redundant SCARA architecture to execute different tasks and compared the performance with a classical SCARA in different situational tasks. This control proposes to use nonlinear control in both cases.In another work, Freire et al. [26] proposed an adaptive PID control to control a manipulator robot in trajectory tracking. To adjust the PID gains, an ANN to retropropagate the trajectory error is applied. Similarly, Sharma et al. [27] used a recurrent neural network to implement an adaptive PID controller.Al-Khedher and Alshamasin [28] used artificial neural networks (ANN) for the control of a SCARA robot, and its performance is compared with a classical PD controller. An alternative type of solution has been proposed for robot manipulators control problems in Rossomando and Soria [29]; their work is based on the design of an SMC adaptive controller in discrete time on a discretized model.Jha and Biswal [30] utilized an ANN model to solve the inverse kinematics problem of a SCARA manipulator; the ANN is based on a multilayer perceptron using a gradient descendent rule to find the weights tuning law.Yi and Zhai [31] showed a novel control technique for chattering-free in trajectory tracking for the robot’s manipulators when external disturbances and different uncertainties are acting on its dynamics. To reduce these undesirable perturbations, a second-order fast nonsingular terminal sliding mode (SOFNTSM) control is designed. This proposal guarantees robustness and ensures convergence. Also, using a simulation technique demonstrates the effectiveness of the control method. More recently, Rosas et al. [32] developed an active compensation of disturbances (ADRC) where the effectiveness of this proposal is applied on a SCARA robot arm.This paper addresses the role of the disturbance observers in nonlinear electromechanical systems, in a particular case of a robot manipulator. Prior information in dynamic information is manifested in terms of nonlinear matrices (inertia, friction, and Coriolis), and this information is applied to design the control action. However, there are differences between the real and known dynamics that can affect the robot performance in trajectory tracking. To overcome this problem, a neuralGRΝΝ estimator is implemented. This observation technique can obtain good estimations of the dynamical differences to reduce the control error. Also, the main contribution of this proposal is the implementation of the sintg−1. function as saturation function in the control law, which is smoother than tanh. or sat. functions. Due to this characteristic, the transition between states can improve the effectiveness of the controller, and similar error levels can be obtained. Also, this bounded function can prevent neural parameters from drifting due to control errors from values outside the saturation’s maximum value. These problems sometimes are occasioned by communication errors between the robot’s control unit and the PC. Being this last consideration the main novelty of this work. To probe the theoretical results, some experiments on the SCARA Bosch SR-800 arm system were performed. The convergence of the presented control technique and the demonstration of the closed loop system stability is another contribution of this work.This research work is organized as follows: the section “Model Description” presents a brief description of a robot manipulator model SCARA type. The section “Controller Design” shows the design of the controller based on inverse nonlinear sliding mode control. An overview ofGRNN and a function approximation of the different dynamics is presented in the section “GRNN Function Approximation and Adaptive Tuning Laws.” Also, in this section, the adaptive tuning laws for the GRNN are obtained. Finally, realistic simulation results and conclusions are given in the sections “Results and Discussion” and “Conclusions,” respectively.


## 2. Model Description

Consider the robot manipulator SCARA type model (Figure1) described in studies by Rossomando et al. [29] and Freire et al. [26],(1)Mqq¨+Cq,q˙q˙+fq˙=u,where q=q1,q2T are the generalized coordinates of position of the robot arm and u are the normalized command signals arranged as components of the control signal torque vector u=u1,u2T. Moreover, Mq is the inertia matrix, Cq,q˙ is the Coriolis matrix and fq˙ is defined as the friction term.Figure 1
Representation of the 2-DOF robot arm.Recalling the termsCq,q˙q˙+fq˙=Nq,q˙ and replacing in the following equation:(2)Mqq¨+Nq,q˙=u.Different robotic arms own intrinsic dynamic properties; these properties will be taken into account for the implementation of the control law. The structural properties of the robot arm are as follows [33]:(1)
Mq is a symmetric and positive definite matrix(2)
M˙q−Cq,q˙∈ℝn×n is a skew-symmetric matrixThe known manipulator dynamics is represented by(3)M^qq¨+N^q,q˙=u,where equation (3) will be used in the controller design, and the estimated matrices are defined as follows:(4)M^q=1.7277+0.1908cosq20.0918+0.0954cosq20.0918+0.0954cosq20.0918,C^q,q˙=31.8192−0.0954sinq2q˙2−0.0954sinq2q˙2+q˙10.3418sinq2q˙112.5783,f^q˙=1.0256signq˙11.7842signq˙2.


## 3. Controller Design

### 3.1. Neuro Adaptive SMC Controller

In this section, the trajectory tracking control scheme is described. The control scheme is based on the generalized coordinates. First, the equation of motion is represented in terms of joint coordinates. Then, the trajectory tracking control scheme with adaptive neural SMC control is proposed. To start this analysis, the control error signal is defined as(5)e=q−qref.Now, in order to apply the control law, a sliding surface is used for the control error signal:(6)r=diagddt+α1,ddt+α2e=e˙+αe,where α is a diagonal matrix with positive constants αi>0 and each αi is defined as a design parameter. Deriving equation (6), it leads to(7)r˙=e¨+αe˙.Using equation (7) and replacing equation (5),(8)q¨−q¨ref=r˙−αe˙.From equation (6) and rearranging equation (8),(9)q¨=r˙−αr−αe+q¨ref=r˙−αr+α2e+q¨ref.Replacing equation (9) in equation (2),(10)r˙−αr+α2e+q¨ref=M−1u−M−1Nq,q˙.Obtainingr˙ from equation (10),(11)r˙=−M−1Nq,q˙+M−1u+αr−α2e−q¨ref.


### 3.2. Saturation Definition

In this subsection, an approximation ofsintg−1. function is proposed. The main objective of this work is to propose an alternative of the tanh. function as a smooth saturation function. Figure 2 shows sintg−1., sat., and tanh. functions, as the sintg−1. function reaches its max value more gradually than the hyperbolic tangent function tanh., and since it presents a smoother transition, this characteristic can be used to take advantage of control applications. In [24], a sintg−1. function was used in a kinematics controller for mobile robot applications, being the tracking accuracy of the robot improved. Now, expressing the sintg−1. function as(12)sintg−1b2ri=b2ri1+b2ri2.Figure 2
sat., tanh., and sintg−1. functions.Equation (12) changes from trigonometric to an algebraic function. An algebraic function is simpler to implement in microcontrollers and industrial computers.Assumption 1.
It is assumed that the quantity1+b2ri2 is nonnegative along the evolution of the system.
The control law is expressed as(13)u=M^M^−1N^q,q˙+q¨ref−b1sintg−1b2r−ξεsignr,where ξε∈ℜ2×2, and it is expressed as ξε=diagξε1,ξε2. The proposed control structure is shown in Figure 3. Now, closing the control loop and replacing equation (13) in equation (11), we lead to(14)r˙=−b1sintg−1b2r+αr−α2e−ξεsignr+δt.
From equation (14), δt denotes the error due to the dynamic difference, and it is expressed as(15)δt=−M−1Nq,q˙+M−1qu−−M^−1N^q,q˙+M^−1qu=−M−1Nq,q˙+M^−1N^q,q˙+M−1q−M^−1qu,where δ is considered bounded by(16)δt≤δ0,where δ0 is a positive constant value.Figure 3
Control structure of the robot arm using a neural observer.


### 3.3. GRNN Function Approximation and Adaptive Tuning Laws

In this subsection, a general regression neural network (GRNN) identification technique is presented, where the GRNN is a single-pass neural network which uses a Gaussian activation function in the hidden layer, and it can be used to approximate any continuous function [34].The variableδt represents the unknown parameters due to mass variation of the payload and can be represented as(17)δt−φMN∗TΨq¯+φM∗Tχqu≤ε<∞,or(18)δt=φMN∗TΨq¯+φM∗Tχqu+ε,where q¯ is the regressor vector and is defined as q¯=q,q˙. The neuron number is defined as m, and the vector Ψ.∈ℜm×1 is a functions vector where each element is defined as Ψi=expq¯−z/2ρ/∑j=1mexpq¯−zj/2ρj and z∈ℜdimq¯×1 and ρ∈ℜm×1 define the centers and the spread of each exponential function, respectively. In the same way, the matrix χ.∈ℜm×n is defined as χi=expq−z/2ρ/∑j=1mexpq−zj/2ρj and z∈ℜdimq×1 and ρ∈ℜm×1 define the centers and the spread of each exponential function, respectively. Besides, φMN∗∈Rn×m and φM∗∈ℜn×mI are the optimal parameter matrix; each element of φMN∗ and φM∗ is constant and unknown. The vector ε is the approximation error and εi≤ξεi for i=1,2. Being n = 2 the model output number, the estimation function is chosen as follows:(19)δ^t=φ^MNTΨq¯+φ^MTχqu.The GRNN structure has the capacity of approximating any real continuous vector function; in [34, 35], there are different references about this identification technique. Using the estimated output of GRNN equation (19) and taking the difference between equation (18) and equation (19), we obtain(20)δt−δ^t=δ˜t.The approximation errorδ˜t can be approximated by(21)δ˜t=φMN∗TΨq¯+φM∗Tχqu+ε−φ^MNTΨq¯+φ^MTχqu.Taking into account thatφMN∗T=φ˜MNT+φ^MNT and φM∗T=φ˜MT+φ^MT and replacing them in equation (21),(22)δ˜t=φ˜MNTΨq¯+φ˜MTχqu+ε.Replacing the proposed control law (13) in the robot dynamics (11) and also the neural approximation (22), it leads to(23)r˙=−b1sintg−1b2r+δ˜t+αr−α2e−ξεsignr=⋯=−b1sintg−1b2r+φ˜MNTΨq¯+φ˜MTχqu+ε+αr−α2e−ξεsignr.Theorem: the closed-loop system (23) using the proposed control law (13) using the neural approx. (22) has asymptotic stability.


#### 3.3.1. Demonstration

The first step is defining a definite positive Lyapunov’s function candidate (LFC) to demonstrate the convergence of the proposed control method.Defining the LFC as(24)ℒ=∑i21b21+b2ri2+12ei2+12b3φ˜MNiTφ˜MNi+12b4φ˜MiTφ˜Mi.Deriving equation (24),(25)ℒ˙=∑i2b2ri1+b2ri2r˙i+eie˙i+1b3φ˜MNiTφ˜˙MNi+1b3φ˜MiTφ˜˙Mi.Replacing equation (30) in equation (32),(26)ℒ˙=∑i2−b1b2ri1+b2ri22+b2ri1+b2ri2φ˜MNiTΨq¯+φ˜MiTχqu+εi+⋯+b2ri1+b2ri2αiri−αi2ei−b2ri1+b2ri2ξεsignri+eiri−αiei+⋯+1b3φ˜MNiTφ˜˙MNi+1b3φ˜MiTφ˜˙Mi.Rearranging equation (33),(27)ℒ˙≤∑i2−b1b2ri1+b2ri22+b2ri1+b2ri2φ˜MNiTΨq¯+⋯+b2ri1+b2ri2φ˜MiTχqu+b2ri1+b2ri2αiri−αi2ei−⋯−b2ri1+b2ri2ξεisignri+b2ri1+b2ri2εi+⋯+eiri−αiei+1b3φ˜MNiTφ˜˙MNi+1b3φ˜MiTφ˜˙Mi.Rearranging equation (34),(28)ℒ˙≤∑i2−b1b2ri1+b2ri22+b2ri1+b2ri2εi+⋯+b2ri1+b2ri2αiri−αi2ei−ξεib2ri1+b2ri2+eiri−αiei+∑i2φ˜MNiTtb2ri1+b2ri2Ψq¯+1b3φ˜˙MNi+⋯+∑i2φ˜MiTtb2ri1+b2ri2χqu+1b4φ˜˙Mi.Taking into account the last terms between brackets in equation (28) and defining φ˜˙MNi as(29)φ˜˙MNi=−b3b2ri1+b2ri2Ψq¯.In the same way, forφ˜˙Mi is obtained in the next equation:(30)φ˜˙Mi=−b4b2ri1+b2ri2χqu.The subindexi in equation (29) and equation (30) denotes the vector weights adjusted by ri; replacing the same equation in (28) leads to(31)ℒ˙i=−b1b2ri1+b2ri22+b2ri1+b2ri2αiri−αi2ei+eiri−αiei+Δi,where Δi=εib2ri−ξεib2ri1/1+b2ri2≤0.Now, it is necessary to analyze two cases, whenri=0 and ri≠0.Analyzing the first caseri=0, all the terms with b2ri/1+b2ri2 are equal to zero. Thus, ℒ˙i can be reduced to(32)ℒ˙i=−αiei2≤0.This expression denotes that the control error is bounded whenri=0. Analyzing the second case ri≠0, in this instance, b2ri/1+b2ri2≠0; rewriting equation (31),(33)ℒ˙i=−b1−αi1+b2ri2b2b2ri1+b2ri22+−αi2eib2ri1+b2ri2+eiri−αiei2+Δi.Doing the same operation for the second term,(34)ℒ˙i=−b1−αi1+b2ri2b2b2ri1+b2ri22−⋯−αi2−1+b2ri2b2eib2ri1+b2ri2−αiei2+Δi.Defining the auxiliary vectorEi=eib2ri/1+b2ri2T and rewriting (34) as(35)ℒ˙i=−EiTΘiEi−αiei2+Δi,where Θi is expressed as(36)Θi=αi2αi22−1+b2ri22b2αi22−1+b2ri22b2b1−αi1+b2ri2b2.It is necessary to demonstrate that the matrixΘi is a positive definite to demonstrate the convergence of the presented control technique. Taking into account the Sylvester’s criterion, which is a simple way to determine if Θi is a definite positive matrix.Now computing if the upper left 1-by-1 corner ofΘi has a positive determinant. In this case,(37)αi2>0.Consequently, if the upper left 2-by-2 corner ofΘi has a positive determinant, where(38)αi2b1−αi1+b2ri2b2−αi22−1+b2ri22b22>0.Expanding the abovementioned equation (38) and rearranging,(39)αi2b1−αi21+b2ri22b2−αi44−αi221+b2ri2b2+1+b2ri24b22=⋯=αi2b1−αi44−14b22−ri24>0.Then, equation (39) can be expressed as(40)αi2b1−αi3>1b22+ri2.From definition ofℒi (equation (24)),(41)ℒi>1b21+b2ri2.Obtainingri from the abovementioned equation (41),(42)b2ℒi2−1b2>ri.Replacing in equation (40),(43)αi2b1−αi3>b2ℒi2−1b22+1b22,with condition (i) and condition (ii) and the matrix Θi which is a definite positive; thus, ℒ˙i can be the upper bound by the following expression:(44)ℒ˙i≤−αiei2,forαi2b1−αi3>ℒi2.From equation (44) ℒ˙i≤0, for all t∈0,∞, the sufficient condition for equation (45) can be obtained:(45)ℒ˙i≤−αiei2,forαi2b1−αi3>ℒi02.From the definition of LFC, equation (24), Vi0 is expressed as(46)ℒi0=1b21+b2ri02+12ei20≤1b2+ri0+12ei20.Using the abovementioned expression, equation (46) can be rewritten as(47)ℒ˙i≤−αiei2,forαi2b1−αi3>1b2+e˙i0+αiei0+12ei202.Based on previous results and considering equations (24), (37), (39), and (47) and Barbalat’s lemma [36], these results guarantee the boundedness of the error control signal ei during the closed loop operation.


## 3.1. Neuro Adaptive SMC Controller

In this section, the trajectory tracking control scheme is described. The control scheme is based on the generalized coordinates. First, the equation of motion is represented in terms of joint coordinates. Then, the trajectory tracking control scheme with adaptive neural SMC control is proposed. To start this analysis, the control error signal is defined as(5)e=q−qref.Now, in order to apply the control law, a sliding surface is used for the control error signal:(6)r=diagddt+α1,ddt+α2e=e˙+αe,where α is a diagonal matrix with positive constants αi>0 and each αi is defined as a design parameter. Deriving equation (6), it leads to(7)r˙=e¨+αe˙.Using equation (7) and replacing equation (5),(8)q¨−q¨ref=r˙−αe˙.From equation (6) and rearranging equation (8),(9)q¨=r˙−αr−αe+q¨ref=r˙−αr+α2e+q¨ref.Replacing equation (9) in equation (2),(10)r˙−αr+α2e+q¨ref=M−1u−M−1Nq,q˙.Obtainingr˙ from equation (10),(11)r˙=−M−1Nq,q˙+M−1u+αr−α2e−q¨ref.


## 3.2. Saturation Definition

In this subsection, an approximation ofsintg−1. function is proposed. The main objective of this work is to propose an alternative of the tanh. function as a smooth saturation function. Figure 2 shows sintg−1., sat., and tanh. functions, as the sintg−1. function reaches its max value more gradually than the hyperbolic tangent function tanh., and since it presents a smoother transition, this characteristic can be used to take advantage of control applications. In [24], a sintg−1. function was used in a kinematics controller for mobile robot applications, being the tracking accuracy of the robot improved. Now, expressing the sintg−1. function as(12)sintg−1b2ri=b2ri1+b2ri2.Figure 2
sat., tanh., and sintg−1. functions.Equation (12) changes from trigonometric to an algebraic function. An algebraic function is simpler to implement in microcontrollers and industrial computers.Assumption 1.
It is assumed that the quantity1+b2ri2 is nonnegative along the evolution of the system.
The control law is expressed as(13)u=M^M^−1N^q,q˙+q¨ref−b1sintg−1b2r−ξεsignr,where ξε∈ℜ2×2, and it is expressed as ξε=diagξε1,ξε2. The proposed control structure is shown in Figure 3. Now, closing the control loop and replacing equation (13) in equation (11), we lead to(14)r˙=−b1sintg−1b2r+αr−α2e−ξεsignr+δt.
From equation (14), δt denotes the error due to the dynamic difference, and it is expressed as(15)δt=−M−1Nq,q˙+M−1qu−−M^−1N^q,q˙+M^−1qu=−M−1Nq,q˙+M^−1N^q,q˙+M−1q−M^−1qu,where δ is considered bounded by(16)δt≤δ0,where δ0 is a positive constant value.Figure 3
Control structure of the robot arm using a neural observer.


## 3.3. GRNN Function Approximation and Adaptive Tuning Laws

In this subsection, a general regression neural network (GRNN) identification technique is presented, where the GRNN is a single-pass neural network which uses a Gaussian activation function in the hidden layer, and it can be used to approximate any continuous function [34].The variableδt represents the unknown parameters due to mass variation of the payload and can be represented as(17)δt−φMN∗TΨq¯+φM∗Tχqu≤ε<∞,or(18)δt=φMN∗TΨq¯+φM∗Tχqu+ε,where q¯ is the regressor vector and is defined as q¯=q,q˙. The neuron number is defined as m, and the vector Ψ.∈ℜm×1 is a functions vector where each element is defined as Ψi=expq¯−z/2ρ/∑j=1mexpq¯−zj/2ρj and z∈ℜdimq¯×1 and ρ∈ℜm×1 define the centers and the spread of each exponential function, respectively. In the same way, the matrix χ.∈ℜm×n is defined as χi=expq−z/2ρ/∑j=1mexpq−zj/2ρj and z∈ℜdimq×1 and ρ∈ℜm×1 define the centers and the spread of each exponential function, respectively. Besides, φMN∗∈Rn×m and φM∗∈ℜn×mI are the optimal parameter matrix; each element of φMN∗ and φM∗ is constant and unknown. The vector ε is the approximation error and εi≤ξεi for i=1,2. Being n = 2 the model output number, the estimation function is chosen as follows:(19)δ^t=φ^MNTΨq¯+φ^MTχqu.The GRNN structure has the capacity of approximating any real continuous vector function; in [34, 35], there are different references about this identification technique. Using the estimated output of GRNN equation (19) and taking the difference between equation (18) and equation (19), we obtain(20)δt−δ^t=δ˜t.The approximation errorδ˜t can be approximated by(21)δ˜t=φMN∗TΨq¯+φM∗Tχqu+ε−φ^MNTΨq¯+φ^MTχqu.Taking into account thatφMN∗T=φ˜MNT+φ^MNT and φM∗T=φ˜MT+φ^MT and replacing them in equation (21),(22)δ˜t=φ˜MNTΨq¯+φ˜MTχqu+ε.Replacing the proposed control law (13) in the robot dynamics (11) and also the neural approximation (22), it leads to(23)r˙=−b1sintg−1b2r+δ˜t+αr−α2e−ξεsignr=⋯=−b1sintg−1b2r+φ˜MNTΨq¯+φ˜MTχqu+ε+αr−α2e−ξεsignr.Theorem: the closed-loop system (23) using the proposed control law (13) using the neural approx. (22) has asymptotic stability.


### 3.3.1. Demonstration

The first step is defining a definite positive Lyapunov’s function candidate (LFC) to demonstrate the convergence of the proposed control method.Defining the LFC as(24)ℒ=∑i21b21+b2ri2+12ei2+12b3φ˜MNiTφ˜MNi+12b4φ˜MiTφ˜Mi.Deriving equation (24),(25)ℒ˙=∑i2b2ri1+b2ri2r˙i+eie˙i+1b3φ˜MNiTφ˜˙MNi+1b3φ˜MiTφ˜˙Mi.Replacing equation (30) in equation (32),(26)ℒ˙=∑i2−b1b2ri1+b2ri22+b2ri1+b2ri2φ˜MNiTΨq¯+φ˜MiTχqu+εi+⋯+b2ri1+b2ri2αiri−αi2ei−b2ri1+b2ri2ξεsignri+eiri−αiei+⋯+1b3φ˜MNiTφ˜˙MNi+1b3φ˜MiTφ˜˙Mi.Rearranging equation (33),(27)ℒ˙≤∑i2−b1b2ri1+b2ri22+b2ri1+b2ri2φ˜MNiTΨq¯+⋯+b2ri1+b2ri2φ˜MiTχqu+b2ri1+b2ri2αiri−αi2ei−⋯−b2ri1+b2ri2ξεisignri+b2ri1+b2ri2εi+⋯+eiri−αiei+1b3φ˜MNiTφ˜˙MNi+1b3φ˜MiTφ˜˙Mi.Rearranging equation (34),(28)ℒ˙≤∑i2−b1b2ri1+b2ri22+b2ri1+b2ri2εi+⋯+b2ri1+b2ri2αiri−αi2ei−ξεib2ri1+b2ri2+eiri−αiei+∑i2φ˜MNiTtb2ri1+b2ri2Ψq¯+1b3φ˜˙MNi+⋯+∑i2φ˜MiTtb2ri1+b2ri2χqu+1b4φ˜˙Mi.Taking into account the last terms between brackets in equation (28) and defining φ˜˙MNi as(29)φ˜˙MNi=−b3b2ri1+b2ri2Ψq¯.In the same way, forφ˜˙Mi is obtained in the next equation:(30)φ˜˙Mi=−b4b2ri1+b2ri2χqu.The subindexi in equation (29) and equation (30) denotes the vector weights adjusted by ri; replacing the same equation in (28) leads to(31)ℒ˙i=−b1b2ri1+b2ri22+b2ri1+b2ri2αiri−αi2ei+eiri−αiei+Δi,where Δi=εib2ri−ξεib2ri1/1+b2ri2≤0.Now, it is necessary to analyze two cases, whenri=0 and ri≠0.Analyzing the first caseri=0, all the terms with b2ri/1+b2ri2 are equal to zero. Thus, ℒ˙i can be reduced to(32)ℒ˙i=−αiei2≤0.This expression denotes that the control error is bounded whenri=0. Analyzing the second case ri≠0, in this instance, b2ri/1+b2ri2≠0; rewriting equation (31),(33)ℒ˙i=−b1−αi1+b2ri2b2b2ri1+b2ri22+−αi2eib2ri1+b2ri2+eiri−αiei2+Δi.Doing the same operation for the second term,(34)ℒ˙i=−b1−αi1+b2ri2b2b2ri1+b2ri22−⋯−αi2−1+b2ri2b2eib2ri1+b2ri2−αiei2+Δi.Defining the auxiliary vectorEi=eib2ri/1+b2ri2T and rewriting (34) as(35)ℒ˙i=−EiTΘiEi−αiei2+Δi,where Θi is expressed as(36)Θi=αi2αi22−1+b2ri22b2αi22−1+b2ri22b2b1−αi1+b2ri2b2.It is necessary to demonstrate that the matrixΘi is a positive definite to demonstrate the convergence of the presented control technique. Taking into account the Sylvester’s criterion, which is a simple way to determine if Θi is a definite positive matrix.Now computing if the upper left 1-by-1 corner ofΘi has a positive determinant. In this case,(37)αi2>0.Consequently, if the upper left 2-by-2 corner ofΘi has a positive determinant, where(38)αi2b1−αi1+b2ri2b2−αi22−1+b2ri22b22>0.Expanding the abovementioned equation (38) and rearranging,(39)αi2b1−αi21+b2ri22b2−αi44−αi221+b2ri2b2+1+b2ri24b22=⋯=αi2b1−αi44−14b22−ri24>0.Then, equation (39) can be expressed as(40)αi2b1−αi3>1b22+ri2.From definition ofℒi (equation (24)),(41)ℒi>1b21+b2ri2.Obtainingri from the abovementioned equation (41),(42)b2ℒi2−1b2>ri.Replacing in equation (40),(43)αi2b1−αi3>b2ℒi2−1b22+1b22,with condition (i) and condition (ii) and the matrix Θi which is a definite positive; thus, ℒ˙i can be the upper bound by the following expression:(44)ℒ˙i≤−αiei2,forαi2b1−αi3>ℒi2.From equation (44) ℒ˙i≤0, for all t∈0,∞, the sufficient condition for equation (45) can be obtained:(45)ℒ˙i≤−αiei2,forαi2b1−αi3>ℒi02.From the definition of LFC, equation (24), Vi0 is expressed as(46)ℒi0=1b21+b2ri02+12ei20≤1b2+ri0+12ei20.Using the abovementioned expression, equation (46) can be rewritten as(47)ℒ˙i≤−αiei2,forαi2b1−αi3>1b2+e˙i0+αiei0+12ei202.Based on previous results and considering equations (24), (37), (39), and (47) and Barbalat’s lemma [36], these results guarantee the boundedness of the error control signal ei during the closed loop operation.


## 3.3.1. Demonstration

The first step is defining a definite positive Lyapunov’s function candidate (LFC) to demonstrate the convergence of the proposed control method.Defining the LFC as(24)ℒ=∑i21b21+b2ri2+12ei2+12b3φ˜MNiTφ˜MNi+12b4φ˜MiTφ˜Mi.Deriving equation (24),(25)ℒ˙=∑i2b2ri1+b2ri2r˙i+eie˙i+1b3φ˜MNiTφ˜˙MNi+1b3φ˜MiTφ˜˙Mi.Replacing equation (30) in equation (32),(26)ℒ˙=∑i2−b1b2ri1+b2ri22+b2ri1+b2ri2φ˜MNiTΨq¯+φ˜MiTχqu+εi+⋯+b2ri1+b2ri2αiri−αi2ei−b2ri1+b2ri2ξεsignri+eiri−αiei+⋯+1b3φ˜MNiTφ˜˙MNi+1b3φ˜MiTφ˜˙Mi.Rearranging equation (33),(27)ℒ˙≤∑i2−b1b2ri1+b2ri22+b2ri1+b2ri2φ˜MNiTΨq¯+⋯+b2ri1+b2ri2φ˜MiTχqu+b2ri1+b2ri2αiri−αi2ei−⋯−b2ri1+b2ri2ξεisignri+b2ri1+b2ri2εi+⋯+eiri−αiei+1b3φ˜MNiTφ˜˙MNi+1b3φ˜MiTφ˜˙Mi.Rearranging equation (34),(28)ℒ˙≤∑i2−b1b2ri1+b2ri22+b2ri1+b2ri2εi+⋯+b2ri1+b2ri2αiri−αi2ei−ξεib2ri1+b2ri2+eiri−αiei+∑i2φ˜MNiTtb2ri1+b2ri2Ψq¯+1b3φ˜˙MNi+⋯+∑i2φ˜MiTtb2ri1+b2ri2χqu+1b4φ˜˙Mi.Taking into account the last terms between brackets in equation (28) and defining φ˜˙MNi as(29)φ˜˙MNi=−b3b2ri1+b2ri2Ψq¯.In the same way, forφ˜˙Mi is obtained in the next equation:(30)φ˜˙Mi=−b4b2ri1+b2ri2χqu.The subindexi in equation (29) and equation (30) denotes the vector weights adjusted by ri; replacing the same equation in (28) leads to(31)ℒ˙i=−b1b2ri1+b2ri22+b2ri1+b2ri2αiri−αi2ei+eiri−αiei+Δi,where Δi=εib2ri−ξεib2ri1/1+b2ri2≤0.Now, it is necessary to analyze two cases, whenri=0 and ri≠0.Analyzing the first caseri=0, all the terms with b2ri/1+b2ri2 are equal to zero. Thus, ℒ˙i can be reduced to(32)ℒ˙i=−αiei2≤0.This expression denotes that the control error is bounded whenri=0. Analyzing the second case ri≠0, in this instance, b2ri/1+b2ri2≠0; rewriting equation (31),(33)ℒ˙i=−b1−αi1+b2ri2b2b2ri1+b2ri22+−αi2eib2ri1+b2ri2+eiri−αiei2+Δi.Doing the same operation for the second term,(34)ℒ˙i=−b1−αi1+b2ri2b2b2ri1+b2ri22−⋯−αi2−1+b2ri2b2eib2ri1+b2ri2−αiei2+Δi.Defining the auxiliary vectorEi=eib2ri/1+b2ri2T and rewriting (34) as(35)ℒ˙i=−EiTΘiEi−αiei2+Δi,where Θi is expressed as(36)Θi=αi2αi22−1+b2ri22b2αi22−1+b2ri22b2b1−αi1+b2ri2b2.It is necessary to demonstrate that the matrixΘi is a positive definite to demonstrate the convergence of the presented control technique. Taking into account the Sylvester’s criterion, which is a simple way to determine if Θi is a definite positive matrix.Now computing if the upper left 1-by-1 corner ofΘi has a positive determinant. In this case,(37)αi2>0.Consequently, if the upper left 2-by-2 corner ofΘi has a positive determinant, where(38)αi2b1−αi1+b2ri2b2−αi22−1+b2ri22b22>0.Expanding the abovementioned equation (38) and rearranging,(39)αi2b1−αi21+b2ri22b2−αi44−αi221+b2ri2b2+1+b2ri24b22=⋯=αi2b1−αi44−14b22−ri24>0.Then, equation (39) can be expressed as(40)αi2b1−αi3>1b22+ri2.From definition ofℒi (equation (24)),(41)ℒi>1b21+b2ri2.Obtainingri from the abovementioned equation (41),(42)b2ℒi2−1b2>ri.Replacing in equation (40),(43)αi2b1−αi3>b2ℒi2−1b22+1b22,with condition (i) and condition (ii) and the matrix Θi which is a definite positive; thus, ℒ˙i can be the upper bound by the following expression:(44)ℒ˙i≤−αiei2,forαi2b1−αi3>ℒi2.From equation (44) ℒ˙i≤0, for all t∈0,∞, the sufficient condition for equation (45) can be obtained:(45)ℒ˙i≤−αiei2,forαi2b1−αi3>ℒi02.From the definition of LFC, equation (24), Vi0 is expressed as(46)ℒi0=1b21+b2ri02+12ei20≤1b2+ri0+12ei20.Using the abovementioned expression, equation (46) can be rewritten as(47)ℒ˙i≤−αiei2,forαi2b1−αi3>1b2+e˙i0+αiei0+12ei202.Based on previous results and considering equations (24), (37), (39), and (47) and Barbalat’s lemma [36], these results guarantee the boundedness of the error control signal ei during the closed loop operation.


## 4. Results and Discussion

### 4.1. Experiments

The experiments were performed on the SCARA Bosch SR-800 (Figure1) arm system. A CPU Intel Dual Core, with 2.6 GHz and 4 GB RAM memory to control the robot arm, is used. Debian Linux has been installed in the CPU and it uses RTAI (Real Time Application Interface). The approach proposed in this paper is applied to the robotic manipulator Bosch SR-800 by controlling the power unit that applies the calculated torques (control actions) to the joints of the robot arm. The first experiment shows different experiments with different neural networks. For performance comparison purposes, in addition to the use of the different saturated controllers to control the robotic manipulator, a nonadaptive SMC controller was also applied to the same robot for reference. In the second set of experiments, different saturated functions were used.


### 4.2. Adaptive Neural Control Comparison

The first set of experiments was made with different neural networks to compensate for the dynamic variations.The objective of this approach is to contribute to the use of a new saturation function and prove its efficiency in an adaptive control technique. In the first set of experiments, the basic form of Lissajous “8” is considered as a reference trajectory. The experiment was carried out using three different adaptive techniques using the single saturation functionsintg−1., where these controllers were applied on the robot arm to demonstrate their effectiveness. The controllers are based on saturated SMC, the first one uses a nonadaptive SMC, which does not use any on-line adjustment, and the second one, an MLP adaptive controller and the last one, a GRNN adaptive controller based on SMC.Figure4 shows the trajectories followed by the robot arm using each one of the control strategies. Figure 5 shows the reference trajectory for each joint (q1 and q2) and the followed trajectory. Figure 6 shows the square norm of the control errors generated by each control strategy (the error norm is defined by e=sqrteq12+eq22). The experimental results prove that the highest error was obtained by a nonadaptive SMC controller, which has no on-line calibration. In this case, the effects of the model uncertainties on the error can be observed clearly. The methods that compensate the model uncertainties have a lower error than the previous case. Finally, the lowest error was obtained by the GRNN adaptive SMC strategy, which reduces the trajectory error caused by the model uncertainties. In Figure 7, the control actions of the GRNN adaptive SMC are shown, and the chattering phenomenon can be seen. This phenomenon can be reduced using a low pass filter previous to the electromechanical actuator.Figure 4
Trajectory followed for the different strategies.Figure 5
Reference (dashed line) and trajectory followed (solid line).
(a)(b)Figure 6
Square norm of the trajectory errors generated by each control strategy.Figure 7
Control actions (adaptive + neural) and neural compensation using adaptive GRNN.Figure8 presents the distribution of the errors obtained along with the experiment. In the 1st row, it shows the error distribution using nonadaptive SMC control. The 2nd row presents the error distribution using adaptive GRΝΝ-SMC control. Finally, in the 3rd row, the error distribution using adaptive MLP-SMC control is shown. In Table 1, the performance indicators ΙΑΕ and ΙΤΑΕ are presented (see equation (48) where T is the total time of the experiments). These indicators are applied on the proposed controllers based on using only sintg−1. function, where the indicators show that the GRNN-SMC controller gets the smallest error as a result:(48)IAE=∫0Teq1+eq2dt,ITAE=∫0Tteq1+eq2dt.Figure 8
Distribution of the errors obtained along the experiment. (a, b) The error distribution using a nonadaptive SMC controller. (c, d) The error distribution using an adaptiveGRNN-SMC controller. (e, f) The error distribution using an adaptive MLP-SMC controller.
(a)(b)(c)(d)(e)(f)Table 1
The performance indexes IAE and ITAE of different controllers usingsintg−1..
ControllerIAEITAENonadaptive SMC569.33241.4233e + 04Adapt. GRNN-SMC424.76101.0619e + 04Adapt. MLP-SMC473.81361.1845e + 04


### 4.3. Dynamical Comparison between Functions

In the second set of the experiment, a dynamical comparison between functionssat., tanh., and sintg−1. was made. Figure 9 shows the trajectories performed by the robot arm using each one of the control strategies. Figure 10 shows the reference (dashed line) and trajectory followed (solid line), being the maximum error during the initial time. Figure 11 shows the square norm of the control errors generated by each saturation type applied in the control law. In this figure, it is appreciated that two last strategies have got similar errors. In Figure 12, the control actions based on the adaptive GRNN-SMC controller are shown. Both signals are polluted by the chattering effect, but in a percentage of about 10 with respect to the maximum value of the control signal, higher values are dangerous for the electromechanical systems. Figure 13 presents the distribution of the errors obtained along with the experiment. Figures 13(a) and 13(b) show the error distribution using the sat. function. Figures 13(c) and 13(d) present the error distribution using the tanh. function. Figures 13(e) and 13(f) show the error distribution using the sintg−1. function. Finally, in Table 2, the performance indicators ΙΑΕ and ΙΤΑΕ are presented. These indicators are applied on alternative controllers based on adaptive GRNN-SMC using different saturation types sat., tanh., and sintg−1.. The denoted indicators show similar results as saturation functions in control. Thus, such functions could be used in different applications when a variable must not exceed a certain maximum value.Figure 9
Trajectory followed for the different strategies.Figure 10
(a) Joint q1. (b) Joint q2. Reference (dashed line) and trajectory followed (solid line).
(a)(b)Figure 11
Square norm of the trajectory errors generated by each control strategy.Figure 12
(a) Torque 1. (b) Torque 2. Control actions (adaptive + neural) and neural compensation using adaptive GRNN.
(a)(b)Figure 13
Distribution of the errors obtained along the experiment. (a-b) The error distribution using thesat. function. (c-d) The error distribution using the tanh. function. (e-f) The error distribution using the sintg−1. function.
(a)(b)(c)(d)(e)(f)Table 2
Performance indexes IAE and the ITAE of different controllers based on adaptive GRNN-SMC usingsat., tanh., and sintg−1..
ControllerIAEITAEsat.482.69921.2067e + 04tanh.424.76101.0619e + 04sintg−1.442.60081.1065e + 04Based on these results, the viability of the proposed controller is demonstrated. The good results obtained, despite the nonlinearities and uncertainties of the model, together with the variations in the dynamics of the robot, show the level of robustness of the proposed GRNN-SMC controller. In addition, the stability of the closed loop system was demonstrated analytically and the adjustment laws were obtained using Lyapunov’s theory. Another interesting observation is that the selection of the neural net structure of the estimator can improve the accuracy of the proposed control.In this proposal for intelligent control, it can be considered as a general solution for the control of nonlinear systems. Its uses can be highlighting in robotic systems, in situations where the dynamics are variable or when there are uncertainties associated with the dynamic model.


## 4.1. Experiments

The experiments were performed on the SCARA Bosch SR-800 (Figure1) arm system. A CPU Intel Dual Core, with 2.6 GHz and 4 GB RAM memory to control the robot arm, is used. Debian Linux has been installed in the CPU and it uses RTAI (Real Time Application Interface). The approach proposed in this paper is applied to the robotic manipulator Bosch SR-800 by controlling the power unit that applies the calculated torques (control actions) to the joints of the robot arm. The first experiment shows different experiments with different neural networks. For performance comparison purposes, in addition to the use of the different saturated controllers to control the robotic manipulator, a nonadaptive SMC controller was also applied to the same robot for reference. In the second set of experiments, different saturated functions were used.


## 4.2. Adaptive Neural Control Comparison

The first set of experiments was made with different neural networks to compensate for the dynamic variations.The objective of this approach is to contribute to the use of a new saturation function and prove its efficiency in an adaptive control technique. In the first set of experiments, the basic form of Lissajous “8” is considered as a reference trajectory. The experiment was carried out using three different adaptive techniques using the single saturation functionsintg−1., where these controllers were applied on the robot arm to demonstrate their effectiveness. The controllers are based on saturated SMC, the first one uses a nonadaptive SMC, which does not use any on-line adjustment, and the second one, an MLP adaptive controller and the last one, a GRNN adaptive controller based on SMC.Figure4 shows the trajectories followed by the robot arm using each one of the control strategies. Figure 5 shows the reference trajectory for each joint (q1 and q2) and the followed trajectory. Figure 6 shows the square norm of the control errors generated by each control strategy (the error norm is defined by e=sqrteq12+eq22). The experimental results prove that the highest error was obtained by a nonadaptive SMC controller, which has no on-line calibration. In this case, the effects of the model uncertainties on the error can be observed clearly. The methods that compensate the model uncertainties have a lower error than the previous case. Finally, the lowest error was obtained by the GRNN adaptive SMC strategy, which reduces the trajectory error caused by the model uncertainties. In Figure 7, the control actions of the GRNN adaptive SMC are shown, and the chattering phenomenon can be seen. This phenomenon can be reduced using a low pass filter previous to the electromechanical actuator.Figure 4
Trajectory followed for the different strategies.Figure 5
Reference (dashed line) and trajectory followed (solid line).
(a)(b)Figure 6
Square norm of the trajectory errors generated by each control strategy.Figure 7
Control actions (adaptive + neural) and neural compensation using adaptive GRNN.Figure8 presents the distribution of the errors obtained along with the experiment. In the 1st row, it shows the error distribution using nonadaptive SMC control. The 2nd row presents the error distribution using adaptive GRΝΝ-SMC control. Finally, in the 3rd row, the error distribution using adaptive MLP-SMC control is shown. In Table 1, the performance indicators ΙΑΕ and ΙΤΑΕ are presented (see equation (48) where T is the total time of the experiments). These indicators are applied on the proposed controllers based on using only sintg−1. function, where the indicators show that the GRNN-SMC controller gets the smallest error as a result:(48)IAE=∫0Teq1+eq2dt,ITAE=∫0Tteq1+eq2dt.Figure 8
Distribution of the errors obtained along the experiment. (a, b) The error distribution using a nonadaptive SMC controller. (c, d) The error distribution using an adaptiveGRNN-SMC controller. (e, f) The error distribution using an adaptive MLP-SMC controller.
(a)(b)(c)(d)(e)(f)Table 1
The performance indexes IAE and ITAE of different controllers usingsintg−1..
ControllerIAEITAENonadaptive SMC569.33241.4233e + 04Adapt. GRNN-SMC424.76101.0619e + 04Adapt. MLP-SMC473.81361.1845e + 04


## 4.3. Dynamical Comparison between Functions

In the second set of the experiment, a dynamical comparison between functionssat., tanh., and sintg−1. was made. Figure 9 shows the trajectories performed by the robot arm using each one of the control strategies. Figure 10 shows the reference (dashed line) and trajectory followed (solid line), being the maximum error during the initial time. Figure 11 shows the square norm of the control errors generated by each saturation type applied in the control law. In this figure, it is appreciated that two last strategies have got similar errors. In Figure 12, the control actions based on the adaptive GRNN-SMC controller are shown. Both signals are polluted by the chattering effect, but in a percentage of about 10 with respect to the maximum value of the control signal, higher values are dangerous for the electromechanical systems. Figure 13 presents the distribution of the errors obtained along with the experiment. Figures 13(a) and 13(b) show the error distribution using the sat. function. Figures 13(c) and 13(d) present the error distribution using the tanh. function. Figures 13(e) and 13(f) show the error distribution using the sintg−1. function. Finally, in Table 2, the performance indicators ΙΑΕ and ΙΤΑΕ are presented. These indicators are applied on alternative controllers based on adaptive GRNN-SMC using different saturation types sat., tanh., and sintg−1.. The denoted indicators show similar results as saturation functions in control. Thus, such functions could be used in different applications when a variable must not exceed a certain maximum value.Figure 9
Trajectory followed for the different strategies.Figure 10
(a) Joint q1. (b) Joint q2. Reference (dashed line) and trajectory followed (solid line).
(a)(b)Figure 11
Square norm of the trajectory errors generated by each control strategy.Figure 12
(a) Torque 1. (b) Torque 2. Control actions (adaptive + neural) and neural compensation using adaptive GRNN.
(a)(b)Figure 13
Distribution of the errors obtained along the experiment. (a-b) The error distribution using thesat. function. (c-d) The error distribution using the tanh. function. (e-f) The error distribution using the sintg−1. function.
(a)(b)(c)(d)(e)(f)Table 2
Performance indexes IAE and the ITAE of different controllers based on adaptive GRNN-SMC usingsat., tanh., and sintg−1..
ControllerIAEITAEsat.482.69921.2067e + 04tanh.424.76101.0619e + 04sintg−1.442.60081.1065e + 04Based on these results, the viability of the proposed controller is demonstrated. The good results obtained, despite the nonlinearities and uncertainties of the model, together with the variations in the dynamics of the robot, show the level of robustness of the proposed GRNN-SMC controller. In addition, the stability of the closed loop system was demonstrated analytically and the adjustment laws were obtained using Lyapunov’s theory. Another interesting observation is that the selection of the neural net structure of the estimator can improve the accuracy of the proposed control.In this proposal for intelligent control, it can be considered as a general solution for the control of nonlinear systems. Its uses can be highlighting in robotic systems, in situations where the dynamics are variable or when there are uncertainties associated with the dynamic model.


## 5. Conclusions

This work presents a saturated control of two techniques using neural networks based on the classicalsat.,tanh., and sintg−1. functions and also provides the stability proof for a saturated control applied to a robot manipulator.Comparative experiments of all these control variants have been performed on a standard industrial SCARA manipulator; in all cases, the proposed control gets satisfactory results; in the third case, it has an average error and can be used as an alternative oftanh. or sat. function as saturated control.The feasibility of applying an uncertainty estimator with a control based on sliding mode control, including its asymptotic stability, was also demonstrated.The experiments showed that the trajectory followed by the manipulator can be controlled precisely by the use of the proposed method and, at the same time, guarantee the asymptotical convergence.



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*Source: 3240210-2020-05-21.xml*