A Dual Neural Network as an Identifier for a Robot Arm

Abstract

A novel dual recurrent neural network is presented and is used to identify the dynamics for a robot arm with three-Degrees of freedom (DoF) and trained with a filtered error algorithm. The dual neural network has a structure of two recurrent neural networks working simultaneously fighting each other to obtain the best identification values, being the criteria for the selection of the vest values: the standard deviation for the identification error. The neural identifier provides important information to a nonlinear block control transformation form acting as a control law to solve the trajectory tracking problem for the robotic plant's behavior.

1. Introduction

Many modern control systems require for their proper operation an accurate mathematical model; additionally, to control an actual robot, some technique to construct its model online is required.

A control system of high robustness for a robotic plant must be able to work under strong conditions of uncertainty in plant modelling, external perturbations and stringent nonlinearities.

It is known that a robot arm has stringent nonlinear dynamics, experiences parametric variations during operation (due to heating, hysteresis and mechanical wear) and is susceptible to external disturbances (e.g., shocks, air blasts, changes in the characteristics of objects handled, etc.). Even more, in this kind of plant, normally it is possible the complete access to the system states; nevertheless, this information might be noisy and contaminated with measurement errors and delays.

For most nonlinear systems, getting an accurate and reliable mathematical model represents a challenge due to their complex physical structure, hidden parameters and highly coupled dynamics [1]; artificial neural networks (NNs) have become an attractive tool in constructing complex models of nonlinear systems [2], which is called ‘system identification’ [3]. To obtain an effective control law for a robot arm a system identification is required. The use of recurrent neural networks (RNNs) for identification and control has increased recently [4] because of their excellent adaptability and learning capabilities in the presence of external disturbances and uncertainty in modelling, such as those occurring in a robotic plant.

In the recent robust control literature, numerous approaches have been proposed in order to achieve satisfactory control performance in nonlinear systems. Among these, nonlinear block control (NBC) transformation (combined with sliding modes) is a very good methodology [5], although NBC requires for its operation the information provided by a strong mathematical model (i.e., requires a RNN as an identifier).

Some works suggest the use of multiple NNs as an identification system with a switching algorithm applied to select the best model, as in [6, 7]. Nevertheless, very few of them use RNNs to be used in NBC or backstepping control systems.

In particular, in [6] an identifier-based adaptive control is proposed which acts as an indirect adaptive control with multiple models and NNs, using a hysteresis switching algorithm to select the best model. From [6], the following question arises: how would the NN perform when selecting the best data instead of selecting the best model? This can be investigated taking into account another switching law.

In [8], two parallel NNs are used to feed a conventional sliding mode (SM) system to control a two-DoF robot arm. The NNs have a feed-forward structure and are trained with the gradient descendent algorithm. Nowadays, it is well known that RNNs give the best results when combined with SM (particularly with high-order SM).

In [9], a RHONN structure is used to identify the robot arm dynamics to obtain a solution to the trajectory tracking problem for robotic manipulators. The RHONN learning is performed online by a Kalman filtering algorithm. [9] has a dynamical single NN, as occurs in [10].

1.1 Main Contribution

The main contribution of this work is to provide an improved dual recurrent neural network (dual-RNN) which has a novel and effective switching system for selecting the best identification values for the behaviour of a robotic plant while also improving the controller performance as a closed-loop system.

In this work, the dual-RNN is proposed with the following design:

• It is structured by two NNs - one of them is a first-order RNN and the other is a second-order RNN - both working simultaneously.

• It is used with a switching law to select online the best data (important: not to select the best model).

• The criteria to select the best data are based on the standard deviation of the identification error.

• Both RNNs are trained with the filtered error algorithm in continuous time.

• It is fed back through a NBC; then, the dual-RNN is inside a closed-loop control system.

By way of overview, Figure 1 shows the closed-loop control scheme in a block diagram, where can be seen the role of the dual-RNN working within the control system.

Figure 1.

Identification and control scheme

1.2 General Outline

Section 2 presents the characterization and mathematical model for an actual robot arm which is the plant to be identified. In Section 3, the dual-RNN as an identifier for the robot arm of section 2 is treated in detail. Section 4 is devoted to obtaining a system control signal which is needed for the dual-RNN's operation. In Section 5, simulation results are presented. Finally, conclusions are set forth in Section 6.

2. Robot Arm

The actual robot considered here was designed and constructed by the main author and is presented in Figure 2, where it can be seen that it has a closed housing structure for working in dusty industrial environments, being constructed of nylamid, aluminium and steel.

Figure 2.

Actual robot arm constructed in nylamid, aluminium and steel

The mechanical configuration of the robot is a three-DoF selective compliant articulated robot for assembly (SCARA) [11] which has first and second joints of revolute type (it allows for the relative rotation between the two links), and linear relative motion can be performed by a third joint of prismatic type. Its parameters are shown in Table 1.

Table 1.

Parameters of the SCARA robot arm

Link number	Total length	Centroid length	Total mass	Inertia average
i(DoF)	L[m]	Lc[m]	m[kg]	I[kg⋅m2]
1	0.215	0.062	2.75	0.5
2	0.25	0.095	1.18	0.6
3	0.13	―	0.5	―

2.1 Direct Kinematics

The Denavit-Hartenberg (DH) convention [12] is a commonly used method for selecting frames of reference in robotic plants to obtain a kinematics mathematical model. The DH convention involves the following homogeneous transformation matrix:

$\begin{array}{cc}{}^{i-1}{H}_i=R_{{\stackrel{⌣}{z}}_{i-1}\mathrm{,}{\theta }_i}{T}_{{\stackrel{⌣}{z}}_{i-1}\mathrm{,}{d}_i}{T}_{{\stackrel{⌣}{x}}_i\mathrm{,}{l}_i}{R}_{{\stackrel{⌣}{x}}_i\mathrm{,}{\alpha }_i}=& \\ & \\ \left[\begin{array}{cccc}cos({\theta }_i)& -sin({\theta }_i)cos({\alpha }_i)& sin({\theta }_i)sin({\alpha }_i)& l_{i}cos({\theta }_i)\\ sin({\theta }_i)& cos({\theta }_i)cos({\alpha }_i)& -cos({\theta }_i)sin({\alpha }_i)& l_{i}sin({\theta }_i)\\ 0& sin({\alpha }_i)& cos({\alpha }_i)& d_i\\ 0& 0& 0& 1\end{array}\right]& \end{array}$

(1)

where θ, d, l and α are parameters associated with a link and joint i, their values coming from specific aspects of the geometric relationship between two coordinate frames for the specific mechanical configuration of the robot arm; ${\stackrel{⌣}{x}}_i$ , ${\stackrel{⌣}{y}}_i$ and ${\stackrel{⌣}{z}}_i$ are the axes of the i-th coordinate system; $R_{{\stackrel{⌣}{z}}_{i-1}\mathrm{,}{\theta }_i}$ represents the rotation matrix with respect to ${\stackrel{⌣}{z}}_{i-1}$ is a translation matrix with respect to ${\stackrel{⌣}{z}}_{i-1};T_{{\stackrel{⌣}{x}}_i\mathrm{,}{l}_i}$ is a translation matrix with respect to ${\stackrel{⌣}{x}}_i$ ; and $R_{{\stackrel{⌣}{x}}_i\mathrm{,}{\alpha }_i}$ is a rotation matrix with respect to ${\stackrel{⌣}{x}}_i$ (see [13]).

For a three-DoF robot arm i has three values (1, 2, 3), which must be included in Equation (1), and from which we must obtain three homogeneous transformation matrices (⁰H₁, ¹H₂ and ²H₃). To feed (1) with the values of θ, d, l and α, Table 2 was constructed with the DH parameters of the actual robot.

Table 2.

Denavit-Hartenberg parameters

i	θi	di	li	αi
1	q 1	0	l 1	0
2	q 2	0	l 2	π/2
3	0	q 3	0	0

Now, to go from system 0 to system 3 we compute ⁰H₃ =⁰H₁x¹H₂x²H₃ to obtain the position and orientation of the end-effector using the information of Tables 1 and 2, such that we get:

$\begin{array}{c}{}^0{H}_3=\left[\begin{array}{cccc}cos(q_1+q_2)& sin(q_1+q_2)& 0& {\varsigma }_1\\ sin(q_1+q_2)& -cos(q_1+q_2)& 0& {\varsigma }_2\\ 0& 0& -1& -q_3\\ 0& 0& 0& 1\end{array}\right]\end{array}$

(2)

with ${\varsigma }_1=0.25cos\left(q_1+q_2\right)+0.215cos\left(q_1\right)$

and ${\varsigma }_2=0.25sin\left(q_1+q_2\right)+0.215sin\left(q_1\right)$

where q₁ [rad], q₂ [rad] and q₃ [m] are the articular plant positions.

The direct kinematics for the robot are obtained from the last column of (2) and are given in numerical form as:

$X\left(q\left(t\right)\right)=\left[\begin{array}{c}\stackrel{⌣}{x}\\ \stackrel{⌣}{y}\\ \stackrel{⌣}{z}\end{array}\right]=\left[\begin{array}{c}0.215cos\left(q_1\right)+0.25cos\left(q_1+q_2\right)\\ 0.215sin\left(q_1\right)+0.25sin\left(q_1+q_2\right)\\ -q_3\end{array}\right]$

(3)

where $\stackrel{⌣}{x},\stackrel{⌣}{y}$ and $\stackrel{⌣}{z}$ are the spacial Cartesian coordinates of the end-effector and q= [q₁ q₂ q₃] ^T .

2.2 Inverse Kinematics

We can obtain the inverse kinematics by solving the Equation (3) for q₁, q₂ and q₃. This can be done including the Moore-Penrose pseudo-inverse Jacobian matrix J^T (JJ^T)⁻¹ in the first time-derivative of q as follows:

$\stackrel{.}{q}=J^T{\left(J{J}^T\right)}^{-1}\stackrel{.}{X}$

(4)

where J is the Jacobian matrix obtained from $J=\frac{\partial }{\partial q}X$ . To obtain (4), recall the chain rule: $\stackrel{.}{X}=J\stackrel{.}{q}$ .

Computing and integrating (4), the inverse kinematics are obtained and given by:

$\begin{array}{cc}{\left[\begin{array}{ccc}{q}_1& q_2& q_3\end{array}\right]}^T=& \\ & \\ \left[\begin{array}{c}{{\displaystyle tan}}^{-1}\left(\stackrel{⌣}{y}/\stackrel{⌣}{x}\right)-{{\displaystyle tan}}^{-1}\left[0.25sin\left(q_2\right)/\left(0.215+0.25cos\left(q_2\right)\right)\right]\\ {{\displaystyle cos}}^{-1}\left[\left({\stackrel{⌣}{x}}^2+{\stackrel{⌣}{y}}^2-0.108725\right)/0.1075\right]\\ -\stackrel{⌣}{z}\end{array}\right]& \end{array}$

(5)

It can be observed that $q_2={\cos }^{-1}\left[\left({\stackrel{⌣}{x}}^2+{\stackrel{⌣}{y}}^2-0.108725\right)/0.1075\right]$ so (5) is an exclusive function of X.

2.3 Attitude of the End-effector

The end-effector orientation is given by:

$\left[\begin{array}{c}\gamma \\ \varphi \\ \psi \end{array}\right]=\left[\begin{array}{c}{{\displaystyle \sin }}^{-1}\left(-r_{31}\right)\\ {{\displaystyle \cos }}^{-1}\left(r_{11}/\cos \left(\gamma \right)\right)\\ {{\displaystyle \cos }}^{-1}\left(r_{33}/\cos \left(\gamma \right)\right)\end{array}\right]$

(6)

This expression is obtained from Equation (2), which generates a sub-matrix $\in {ℝ}^{3\times 3}$ with r_ij elements (i = j = 1,2,3), where γ, φ and ψ are for the pitch, roll and yaw movement, respectively [14].

Considering the parameters of the SCARA arm (Table 1), the solution to (6) gives $\gamma =0^{\circ },\varphi =q_1+q_2$ and ψ = 180. As such, the roll movement is the only one present in this mechanical robot arm configuration.

2.4 Dynamics

By using the Euler-Lagrange approach [15], a total torque vector $\tau \in {ℝ}^3$ is obtained and is given by:

$\tau =M\left(q\right)\stackrel{..}{q}+\stackrel{.}{M}\left(q\right)\stackrel{.}{q}-\frac{\partial }{\partial q}\frac{1}{2}{\stackrel{.}{q}}^{T}M\left(q\right)\stackrel{.}{q}+\frac{\partial U\left(q\right)}{\partial q}+F_f^{\tau }\left(q\mathrm{,}\stackrel{.}{q}\right)$

(7)

where $M\left(q\right)\in {ℝ}^{3\times 3}$ is the inertia matrix, $U\in {ℝ}^3$ is the potential energy vector and $F_f^{\tau }\in {ℝ}^3$ is the torque vector due to friction.

Introducing the matrix $C\left(q\mathrm{,}\stackrel{.}{q}\right)\in {ℝ}^{3\times 3}$ which represents the Coriolis and centripetal enforces as:

$C\left(q\mathrm{,}\stackrel{.}{q}\right)=\stackrel{.}{M}\left(q\right)\stackrel{.}{q}-\frac{\partial }{\partial q}\frac{1}{2}{\stackrel{.}{q}}^{T}M\left(q\right)\stackrel{.}{q}$

and a gravitational vector $F_g^{\tau }\left(q\right)\in {ℝ}^3$ as

$F_g^{\tau }\left(q\right)=\frac{\partial U(\left(q\right)}{\partial q}$

the dynamics for the robot are given by:

$\tau =M\left(q\right)\stackrel{..}{q}+C\left(q\mathrm{,}\stackrel{.}{q}\right)\stackrel{.}{q}+F_g^{\tau }\left(q\right)+F_f^{\tau }\left(q\mathrm{,}\stackrel{.}{q}\right)$

(8)

In order to have a numerical dynamics model of the actual robot, we will convert (7) into the following seven-step algorithm taking data from Table 1:

1. To obtain the forward kinematics for each centroid joint using Equation (3).

2. To obtain the linear speed v from:

   $v_i=\frac{d}{dt}{\left[\begin{array}{ccc}{\stackrel{⌣}{x}}_i& {\stackrel{⌣}{y}}_i& {\stackrel{⌣}{z}}_i\end{array}\right]}^T$
   and squaring v by $v_i^T{v}_i=\begin{array}{c}{\stackrel{.}{\stackrel{⌣}{x}}}_i^2\cdot {\stackrel{.}{\stackrel{⌣}{y}}}_i^2\cdot {\stackrel{.}{\stackrel{⌣}{z}}}_i^2\end{array}$

3. To obtain the kinetic energy (K) present in each i-th link via computing

   $K_i\left(q_i\mathrm{,}{\stackrel{.}{q}}_i\right)=\frac{1}{2}{m}_i{v}_i^T{v}_i+\frac{1}{2}{I}_i{\left[{\displaystyle \sum _i^n}{\stackrel{.}{q}}_i\right]}^2$

4. To obtain the potential energy (U) by U_i (q)=m_igL _cih_i where g = 9.81m/s² and h_i are the height of the i-th link.

5. To obtain the Lagrangian (L) with $L\left(q\mathrm{,}\stackrel{.}{q}\right)=K\left(q\mathrm{,}\stackrel{.}{q}\right)-U\left(q\mathrm{,}\stackrel{.}{q}\right)$ .

6. To apply the Euler-Lagrange movement equations to obtain the i-th link torque as:

   ${\tau }_i=\frac{d}{dt}\frac{\partial L\left(q\mathrm{,}\stackrel{.}{q}\right)}{\partial {\stackrel{.}{q}}_i}-\frac{\partial L\left(q\mathrm{,}\stackrel{.}{q}\right)}{\partial q_i}+F_f^{\tau }\left(\stackrel{.}{q}\mathrm{,}\tau \right)$

7. To rearrange the entire resulting expression according to the form that possesses Equation (8). After this, we have:

   $\begin{array}{c}M\left(q\right)=\left[\begin{array}{ccc}{M}_{11}& M_{12}& 0\\ M_{21}& M_{22}& 0\\ 0& 0& M_{33}\end{array}\right]\end{array}$
   where:
   $\begin{array}{c}\hfill M_{11}=m_1{L}_{c1}^2+m_2{L}_{c2}^2+m_2{L}_1^2+m_3{L}_2^2+\hfill \\ \hfill +m_3{L}_1^2+I_1+I_2+\left(2m_2{L}_1{L}_{c2}+2m_3{L}_1{L}_2\right)cos\left(q_2\right);\hfill \\ \hfill M_{12}=M_{21}=m_2{L}_{c2}^2+m_3{L}_2^2+I_2+\hfill \\ \hfill +\left(m_2{L}_1{L}_{c2}+m_3{L}_1{L}_2\right)cos\left(q_2\right);\hfill \\ \hfill M_{22}=m_2{L}_{c2}^2+m_3{L}_2^2+I_2;M_{33}=m_3.\hfill \\ \hfill C\left(q\mathrm{,}\stackrel{.}{q}\right)=\left[\begin{array}{ccc}{C}_{11}& C_{12}& 0\\ C_{21}& 0& 0\\ 0& 0& 0\end{array}\right]\hfill \end{array}$
   where:
   $\begin{array}{c}\hfill C_{11}=-\left(2m_2{L}_1{L}_{c2}+2m_3{L}_1{L}_2\right)sin\left(q_2\right){\stackrel{.}{q}}_2;\hfill \\ \hfill C_{12}=-\left(m_2{L}_1{L}_{c2}+m_3{L}_1{L}_2\right)sin\left(q_2\right){\stackrel{.}{q}}_2;\hfill \\ \hfill C_{21}=\left(m_2{L}_1{L}_{c2}+m_3{L}_1{L}_2\right)sin\left(q_2\right){\stackrel{.}{q}}_1;\hfill \\ \hfill F_g^{\tau }\left(q\right)={\left[\begin{array}{ccc}0& 0& -m_{3}g\end{array}\right]}^T\hfill \\ \hfill F_f^{\tau }\left(q\mathrm{,}\stackrel{.}{q}\right)={\left[\begin{array}{ccc}{f}_1& f_2& f_3\end{array}\right]}^T\hfill \end{array}$
   where f₁, f₂ and f₃ must be obtained heuristically according to the friction model presented in [12]. Thus:
   $\begin{array}{c}\hfill f_1=1.86\left[kg\cdot m^2/s\right]{\stackrel{.}{q}}_1+1.93\left[kg\cdot m^2/s^2\right]\text{sign\hspace{0.17em}}\left({\stackrel{.}{q}}_1\right)\hfill \\ \hfill f_2=0.16\left[kg\cdot m^2/s\right]{\stackrel{.}{q}}_2+0.3\left[kg\cdot m^2/s^2\right]\text{sign\hspace{0.17em}}\left({\stackrel{.}{q}}_2\right)\hfill \\ \hfill f_3=0.1\left[kg/s\right]{\stackrel{.}{q}}_3+0.2\left[kg\cdot m/s^2\right]\hspace{0.17em}\text{sign}\hspace{0.17em}\left({\stackrel{.}{q}}_3\right)\hfill \end{array}$

Finally, substituting the parameters of Table 1 we know the numerical values for M, C, $F_g^{\tau }$ and $F_f^{\tau }$ , which are given by:

$\begin{array}{c}\hfill \begin{array}{c}M(q)=\\ \left[\begin{array}{ccc}1.23013+0.102cos\left(q_2\right)& 0.6419+0.05098cos\left(q_2\right)& 0\\ 0.6419+0.05098cos\left(q_2\right)& 0.642& 0\\ 0& 0& 0.5\end{array}\right]\end{array}\hfill \\ \hfill C\left(q\mathrm{,}\stackrel{.}{q}\right)=\left[\begin{array}{ccc}-0.101953sin\left(q_2\right){\stackrel{.}{q}}_2& -0.050975sin\left(q_2\right){\stackrel{.}{q}}_2& 0\\ 0.050975sin\left(q_2\right){\stackrel{.}{q}}_1& 0& 0\\ 0& 0& 0\end{array}\right]\hfill \\ \hfill F_g^{\tau }\left(q\right)={\left[\begin{array}{ccc}0& 0& -4.905\end{array}\right]}^T\hfill \\ \hfill F_f^{\tau }\left(q\mathrm{,}\stackrel{.}{q}\right)=\left[\begin{array}{c}1.86{\stackrel{.}{q}}_1+1.93\hspace{0.17em}\text{sign}\hspace{0.17em}\left({\stackrel{.}{q}}_1\right)\\ 0.16{\stackrel{.}{q}}_2+0.3\hspace{0.17em}\text{sign}\hspace{0.17em}\left({\stackrel{.}{q}}_2\right)\\ 0.1{\stackrel{.}{q}}_3+0.2\hspace{0.17em}\text{sign}\hspace{0.17em}\left({\stackrel{.}{q}}_3\right)\end{array}\right]\hfill \end{array}$

2.5 State Space Representation

The dynamics equation in the state space is given by:

$\begin{array}{c}{\stackrel{.}{\chi }}_1={\chi }_2\\ {\stackrel{.}{\chi }}_2=-M^{-1}(q)\left[C\left(q\mathrm{,}\stackrel{.}{q}\right)\stackrel{.}{q}+F_g\left(q\right)+F_f\left(q\mathrm{,}\stackrel{.}{q}\right)\right]+M^{-1}\left(q\right)\tau \\ y={\left[\begin{array}{cc}{\chi }_1& {\chi }_2\end{array}\right]}^T+\delta \left(t\right)\end{array}$

9 where f₁=[q₁ q₂ q₃] ^T and ${\chi }_2={\left[\begin{array}{ccc}{\stackrel{.}{q}}_1& {\stackrel{.}{q}}_2& {\stackrel{.}{q}}_3\end{array}\right]}^T$ are the articular position and articular velocity respectively; y is the actual system output; and $\delta \left(t\right)\in {ℝ}^6$ is a function which represents system noise and external perturbations. In a closed-loop control system, χ₁ and χ₂ become contaminated with δ(t).

The function of the robot's mathematical model in a control system implemented in the simulation stage is to have an identifiable pattern for the dual-RNN and to obtain closed-loop results.

3. Dual-RNN

The identification process is required when a control law such as the NBC transformation form needs the appropriate plant model.

The function of the NN within the system is to identify the dynamic plant behaviour. The problem with neural identification is to select an appropriate model for the task as well as the adjustment of its parameters according to some adaptive law so that the response of the neuronal identifier to an input signal approximates the system response for the same input [16].

3.1 Dual-RNN Structure

The dual-RNN is composed of two RNNs - one of them is a first-order RNN and the other is a second-order RNN-both working simultaneously and competing with each other to obtain the best identification values, which are selected by a switching system and according to certain criteria.

Each of the RNNs has a similar structure to that of the state space representation for the robot's dynamic model (see Equation (9)) and they are given in decoupled form by:

$\begin{array}{c}{\left[{\stackrel{.}{x}}_1^i\right]}_1=-a_1^i{\left[x_1^i\right]}_1+{\left[{\Theta }_1^i{Z}_1^i\right]}_1+g_i{\left[x_2^i\right]}_1\\ {\left[{\stackrel{.}{x}}_2^i\right]}_1=-a_2^i{\left[x_2^i\right]}_1+{\left[{\Theta }_2^i{Z}_2^i\right]}_1+h_i{u}_i\end{array}$

(10)

$\begin{array}{c}{\left[{\stackrel{.}{x}}_1^i\right]}_2=-a_1^i{\left[x_1^i\right]}_2+{\left[{\Theta }_1^i{Z}_1^i\right]}_2+g_i{\left[x_2^i\right]}_2\\ {\left[{\stackrel{.}{x}}_2^i\right]}_2=-a_2^i{\left[x_2^i\right]}_2+{\left[{\Theta }_2^i{Z}_2^i\right]}_2+h_i{u}_i\end{array}$

(11)

where [·]₁ represents a variable or a function coming from the first-order RNN; [·]₂ represents a variable or a function coming from the second-order RNN; x₁ ⁱ are the NN first-block states which estimate the angular position for the plant; x₂ ⁱ are the NN second-block states which estimate the angular velocity for the plant; ${\Theta }_1^i=\left[\begin{array}{ccc}{\left({\theta }_1^i\right)}_1& \cdots & {\left({\theta }_1^i\right)}_n\end{array}\right]$ are the first-block synaptic weight vectors; ${\Theta }_2^i=\left[\begin{array}{ccc}{\left({\theta }_2^i\right)}_1& \cdots & {\left({\theta }_2^i\right)}_n\end{array}\right]$ are the second-block synaptic weight vectors; $Z_1^i={\left[\begin{array}{ccc}{\left(z_1^i\right)}_1& \cdots & {\left(z_1^i\right)}_n\end{array}\right]}^T$ are the first-block vectorial activation functions; $Z_2^i={\left[\begin{array}{ccc}{\left(z_2^i\right)}_1& \cdots & {\left(z_2^i\right)}_n\end{array}\right]}^T$ are the second-block vectorial activation functions; u_i represents the direct input signal for the dual-RNN (i.e., the control signal); a₁ ⁱ , a₂ ⁱ , g_i and h_i are constant gains; and i =1,2,3 is the number of the robot link represented by the dual-RNN.

It is important to mention that a dual-RNN given by equations (10) and (11) absorbs non-modelled dynamics, parametric variations, system noise and external perturbations through the ΘⁱZ ⁱ product, as well as that the control signal u indirectly affects the first-block states.

3.2 Neural Activation Functions

If equations (10) and (11) represent a dynamic NN and have vectorial activation functions given by $Z_1^i,Z_2^i\in R^n$ , then Θ₁ ⁱ , ${\Theta }_2^i\in R^n$ are transposed synaptic weight vectors. Vectorial activation functions are proposed with seven elements each (n=7), and their structures were obtained in a heuristic way from numerous experiments performed leading up to this work.

The best activation functions obtained for the first-order RNN are as follows:

$\begin{array}{ccccccc}{Z}_1^1=[S(q_1)& S(q_2)& S(q_3)& S({\stackrel{.}{q}}_1)& 0& 0& {0]}^T\\ Z_1^2=[S(q_1)& S(q_2)& S(q_3)& S({\stackrel{.}{q}}_2)& 0& 0& {0]}^T\\ Z_1^3=[S(q_1)& S(q_2)& S(q_3)& S({\stackrel{.}{q}}_3)& 0& 0& {0]}^T\\ Z_2^1=[S({\stackrel{.}{q}}_1)& S({\stackrel{.}{q}}_2)& S({\stackrel{.}{q}}_3)& S(q_1)& 0& 0& {0]}^T\\ Z_2^2=[S({\stackrel{.}{q}}_1)& S({\stackrel{.}{q}}_2)& S({\stackrel{.}{q}}_3)& S(q_2)& 0& 0& {0]}^T\\ Z_2^3=[S({\stackrel{.}{q}}_1)& S({\stackrel{.}{q}}_2)& S({\stackrel{.}{q}}_3)& S(q_3)& 0& 0& {0]}^T\end{array}$

(12)

and the best activation functions obtained for the second-order RNN are as follows:

$\begin{array}{cccc}{Z}_1^1=[S(q_1)& S(q_2)& S(q_3)& S(q_1)S(q_1)\\ & S(q_1)S(q_2)& S(q_1)S(q_3)& {0]}^T\\ Z_1^2=[S(q_1)& S(q_2)& S(q_3)& S(q_2)S(q_1)\\ & S(q_2)S(q_2)& S(q_2)S(q_3)& {0]}^T\\ Z_1^3=[S(q_1)& S(q_2)& S(q_3)& S(q_3)S(q_1)\\ & S(q_3)S(q_2)& S(q_3)S(q_3)& {0]}^T\\ Z_2^1=[S({\stackrel{.}{q}}_1)& S({\stackrel{.}{q}}_1)S({\stackrel{.}{q}}_1)& S({\stackrel{.}{q}}_1)S({\stackrel{.}{q}}_2)& S({\stackrel{.}{q}}_1)S({\stackrel{.}{q}}_3)\\ & S({\stackrel{.}{q}}_1)S(q_1)& S({\stackrel{.}{q}}_1)S(q_2)& S({\stackrel{.}{q}}_1)S(q_3{)]}^T\\ Z_2^2=[S({\stackrel{.}{q}}_2)& S({\stackrel{.}{q}}_2)S({\stackrel{.}{q}}_1)& S({\stackrel{.}{q}}_2)S({\stackrel{.}{q}}_2)& S({\stackrel{.}{q}}_2)S({\stackrel{.}{q}}_3)\\ & S({\stackrel{.}{q}}_2)S(q_1)& S({\stackrel{.}{q}}_2)S(q_2)& S({\stackrel{.}{q}}_2)S(q_3{)]}^T\\ Z_2^3=[S({\stackrel{.}{q}}_3)& S({\stackrel{.}{q}}_3)S({\stackrel{.}{q}}_1)& S({\stackrel{.}{q}}_3)S({\stackrel{.}{q}}_2)& S({\stackrel{.}{q}}_3)S({\stackrel{.}{q}}_3)\\ & S({\stackrel{.}{q}}_3)S(q_1)& S({\stackrel{.}{q}}_3)S(q_2)& S({\stackrel{.}{q}}_3)S(q_3{)]}^T\end{array}$

(13)

where S (·) represents the tanh (·) function whose products give the second RNN its character as second-order.

3.3 RHONN Training Algorithm

Consider the system described as follows:

$\begin{array}{c}{\stackrel{.}{\chi }}_l=-a{\chi }_l+{\Theta }_l^{*}{Z}_l\\ {\stackrel{.}{x}}_l=-a{x}_l+{\Theta }_l^{*}{Z}_l\end{array}$

(14)

where Θ _l is the estimator of the NN synaptic weights Θ _l ,x_l represents the plant states and x_l the NN identifier. The difference between both is the identification error, represented as:

${є}_l=x_l-{\chi }_l$

(15)

being due to meet:

${\dot{є}}_l=-a{є}_l+\left[{\mathrm{\Theta }}_l-{\mathrm{\Theta }}_l^{*}\right]Z_l$

(16)

Next, the NN synaptic weights are updated by the training law:

${\stackrel{.}{\Theta }}_l=-{\left[{\gamma }_l{Z}_l{є}_l\right]}^T$

(17)

which is called the ‘filtered error algorithm’ [17], where γ represents a learning rate (i.e., an adaptive gain). On the one hand, a very high rate of learning will make the system unstable. On the other hand, a very low rate of learning will make the system slow to learn. This last parameter is obtained in a heuristic way.

To train (10)–(11) l = 1- 6, (15) can be expressed as:

$\begin{array}{cc}{є}_1^i=x_1^i-{\chi }_1^i& \left[rad\right]\\ {є}_2^i=x_2^i-{\chi }_2^i& \left[rad/s\right]\end{array}$

(18)

and (17) can be expressed as:

${\stackrel{.}{\Theta }}_j^i=-{[{\gamma }_l{Z}_j^i{є}_j^i]}^T$

(19)

with i = 1,2,3, j = 1,2 and taking into account that $\chi ={\left[\begin{array}{cc}{\chi }_1& {\chi }_2\end{array}\right]}^T={\left[\begin{array}{cccccc}{q}_1& q_2& q_3& {\stackrel{.}{q}}_1& {\stackrel{.}{q}}_2& {\stackrel{.}{q}}_3\end{array}\right]}^T$ according to (9).

Through Θ, χ becomes an indirect input for the dual-RNN.

3.4 Switching System

The information delivered to the control law (which is discussed in the next section) from the identifier is: ${\Theta }_j^i{Z}_j^i$ , ${\stackrel{.}{\Theta }}_1$ and ${\stackrel{.}{x}}_1$ . This information is produced in a dual form for each item, but just one element of this double data is selected online using certain criteria.

In this work, a novel criterion to select the best values is proposed: take the data coming from the minimum standard deviation of the identification error, expressed by the following laws for switching:

${\Theta }_j^i{Z}_j^i=\{\begin{array}{c}\begin{array}{cc}{\left[{\Theta }_j^i{Z}_j^i\right]}_1& if{\left[{\sigma }_{{є}_j^i}\right]}_1=min\left({\left[{\sigma }_{{є}_j^i}\right]}_1\mathrm{,}{\left[{\sigma }_{{є}_j^i}\right]}_2\right)\\ {\left[{\Theta }_j^i{Z}_j^i\right]}_2& if{\left[{\sigma }_{{є}_j^i}\right]}_2=min\left({\left[{\sigma }_{{є}_j^i}\right]}_1\mathrm{,}{\left[{\sigma }_{{є}_j^i}\right]}_2\right)\end{array}\end{array}$

(20)

${\Theta }_1^i=\{\begin{array}{c}\begin{array}{cc}{\left[{\Theta }_1^i\right]}_1& if{\left[{\sigma }_{{є}_j^i}\right]}_1=min\left({\left[{\sigma }_{{є}_j^i}\right]}_1\mathrm{,}{\left[{\sigma }_{{є}_j^i}\right]}_2\right)\\ {\left[{\Theta }_1^i\right]}_2& if{\left[{\sigma }_{{є}_j^i}\right]}_2=min\left({\left[{\sigma }_{{є}_j^i}\right]}_1\mathrm{,}{\left[{\sigma }_{{є}_j^i}\right]}_2\right)\end{array}\end{array}$

(21)

${\stackrel{.}{x}}_1^i=\{\begin{array}{c}\begin{array}{cc}{\left[{\stackrel{.}{x}}_1^i\right]}_1& if{\left[{\sigma }_{{є}_j^i}\right]}_1=min\left({\left[{\sigma }_{{є}_j^i}\right]}_1\mathrm{,}{\left[{\sigma }_{{є}_j^i}\right]}_2\right)\\ {\left[{\stackrel{.}{x}}_1^i\right]}_2& if{\left[{\sigma }_{{є}_j^i}\right]}_2=min\left({\left[{\sigma }_{{є}_j^i}\right]}_1\mathrm{,}{\left[{\sigma }_{{є}_j^i}\right]}_2\right)\end{array}\end{array}$

(22)

where ${\sigma }_{{є}_j^i}$ is the standard deviation of the identification error for the i-th state of the j-th block.

Each i,j-th switch acts independently. As such, the dual-RNN output contains information from both the first-order and the second-order RNNs simultaneously. This ensures that the identification process significantly improves, i.e., the dual-RNN identification error (18) is lower than that of any single RNN, the obtained synaptic weights are more accurate and the ΘZ term is closer to the sum of the non-modelled dynamics, parametric variations, external perturbations and noise.

4. Control

The dual-RNN (10)–(11) needs a control signal u for its operation. A NBC transformation form generates this u and is an ideal controller for this work due to the fact that the NBC is designed from a block structure, such as that given by Equation (9).

In nonlinear control theory, there already exists an algorithm to construct a NBC with a similar structure to that of (10) or (11), and from which can be built a driver for the system:

$\stackrel{.}{x}=f\left(x\mathrm{,}t\right)+B\left(x\mathrm{,}t\right)u+\varphi \left(x\mathrm{,}t\right)$

(23)

where x represents the states of the system; f(x,t) is a function containing a feedback signal; u is the control signal; φ(x,t) is a function that necessarily does not affect the feedback or the system control term. As such, it represents parametric variations, unmodelled dynamics and disturbances affecting the system, which are calculated through (10)–(11) by the product of the synaptic weights 0 and activation functions Z.

Next, since we have an identification system similar to (23) according to the NBC transformation form technique [18], and taking into account that we have a relative degree system r=2, the following expression is obtained:

$\begin{array}{c}e=q-r\left(t\right)\\ {\stackrel{.}{x}}_d=-B_0^{-1}\left[{\Theta }_1{Z}_1+K_{0}e+\stackrel{.}{r}\left(t\right)\right]\\ \stackrel{.}{e}=\stackrel{.}{q}-{\stackrel{.}{x}}_d\\ {\stackrel{..}{x}}_d=-B_1^{-1}\left[{\stackrel{.}{\Theta }}_{1}tanh\left({\stackrel{.}{x}}_1\right)-K_0^{2}e+K_0{B}_0\stackrel{.}{e}-\stackrel{..}{r}\left(t\right)\right]\\ u_{eq}=-B_1^{-1}\left[{\Theta }_2{Z}_2+{\stackrel{..}{x}}_d+K_1\stackrel{.}{e}\right]\end{array}$

(24)

where $e\in {ℝ}^3$ is the tracking error; $r\left(t\right)\in {ℝ}^3$ is the reference signal; $q\in {ℝ}^3$ represents the position of the robot; $x_d\in {ℝ}^3$ is the calculated desired position of the robot; B₀, B₁, K₀ and K₁ are 3 × 3 constant matrices which ensure system stability; ${\Theta }_j{Z}_j={\left[\begin{array}{ccc}{\Theta }_j^1{Z}_j^1& {\Theta }_j^2{Z}_j^2& {\Theta }_j^3{Z}_j^3\end{array}\right]}^T$ with j= 1,2, (come from (20)); ${\stackrel{.}{\Theta }}_1\in {ℝ}^3$ come from (21); ${\stackrel{.}{x}}_1\in {ℝ}^3$ come from (22); and $u_{eq}\in {ℝ}^3$ is the control signal delivered by the NBC. For a detailed procedure in obtaining (24), see [19, 20].

The dual-RNN is fed with:

$u=sat\hspace{0.17em}\left[u_{eq}\right]$

(25)

with upper limits of +24 Nm and lower limits of −24 Nm, which are the operational ranges of the plant. The saturation function in (25) acts as a conventional SM.

5. Simulation

The implementation of the system is made in MATLAB-Simulink (Copyrights and registered trademarks of The MathWorks, Inc).

A block diagram was constructed, as shown in Figure 1, to obtain simulation results for the experiments designed in this work.

5.1 Simulation Conditions

For the block model made in Simulink, we have work conditions, listed below.

Equations (10)–(11) and (19) are involved in the subsystem dual-RNN of Figure 1, with the following conditions and parameters:

• Neural adaptive gains in the order of 5 × 10⁶ (γ _l in (19)).

• State gains in the order of 750 (a_jⁱ in (10)–(11)).

• Input gains in the order of unity (g_i and h_i in (10)–(11)).

• Feedback gains in the order of unity.

• Initial conditions for the NN at the origin.

• Computer sampling time is 0.0001 s.

• Computer solver: Bogacki-Shampine.

In (9), δ(t) is defined as the system noise and external perturbations. For simulation purposes, let us define δ ^Level with Level = Light, Low Medium, High, Limit as a band-limited white noise function, with a computer sampling time of 0.0005 s, and whose values are given in Table 3 (any seed for the white noise function works out). Moreover, define η (t ₀, t ₁) as the external perturbation induced by a torque value of t in Nm from the initial time t₀[s] to the final time t₁[s]. Accordingly, $\delta \left(t\right)={\delta }^{Level}+{\eta }^{\tau }\left(t_0\mathrm{,}{t}_1\right)$ .

Table 3.

Noise levels

${\delta }^{Level}$	Noise power
${\delta }^{Light}$	$1.2\times {10}^{-8}Nm$
${\delta }^{Low}$	$2\times {10}^{-7}Nm$
${\delta }^{Medium}$	$5\times {10}^{-6}Nm$
${\delta }^{High}$	$2\times {10}^{-5}Nm$
${\delta }^{Limit}$	$1.2\times {10}^{-4}Nm$

We should not confuse units for the noise power or the perturbation torque with units of $\delta \left(t\right)\in {ℝ}^6$ [rad] and ${\delta }_2^{Level}+{\eta }_2^{\tau }\left[rad/s\right].{\delta }^{Level}$ is a vibration while η^τ is a step value.

In what follows, we considered the notation for the initial conditions of the robot as ${\chi }_1^i\left(t\right)=q_i$ and ${\chi }_2^i\left(t\right)={\stackrel{.}{q}}_i$ .

For the motor moving the third link of the robot, 1 rad is equivalent to 1.9times10-²m of prismatic displacement. For convenience, in this section q₃ and $q_3$ and ${\stackrel{.}{q}}_3$ are expressed in [rad] and [rad/s] respectively.

5.2 Simulation Results I

It is important to observe the behaviour of the standard deviation for the identification error (σ_∊), since this is the criteria to select the best identification data.

The reference signal r(t) that must track the robot arm in the experiments of Simulation Results I and II is disclosed in Table 4.

Table 4.

Reference signals to track for subsections 5.2 and 5.3

Linknumber	Referencesignal	Refamplitude	Reffrequency
(DoF)	(Ref)	[rad]	[ $rad/s$ ]
First	Sine wave	0.9	0.628
Second	Sine wave	0.9	0.94
Third	Sine wave	0.9	1.13

Figures 3–8 show ${\left[{\sigma }_{{є}_j^i}\right]}_1$ (blue line) vs ${\left[{\sigma }_{{є}_j^i}\right]}_2$ (brown line) working simultaneously for selecting online 2 the output for the dual-RNN.

Figure 3.

σ_∊ for the first state of the first block

Several operating conditions for the simulation were taken into account, as specified below.

In Figure 3 can be seen ${\sigma }_{{є}_1^1}$ with $\delta \left(t\right)={\delta }^{High}+{\eta }^{1.1}\left(\mathrm{1,1.5}\right)$ , and ${\chi }_1^1\left(0\right)=pi/6$ . We can observe that RNN2 has the minimum values except for a period from 2.3 to 4.5 s.

In Figure 4 can be seen ${\sigma }_{{є}_1^2}$ with $\delta \left(t\right)={\delta }^{High}+{\eta }^{0.9}\left(\mathrm{2,2.5}\right)$ , and ${\chi }_1^2\left(0\right)=pi/6\left[rad\right]$ . We can see that RNN2 has the minimum values except for a period from 2.1 to 2.7 s.

Figure 4.

σ_∊ for the second state of the first block

In Figure 5 can be seen ${\sigma }_{{є}_1^3}$ with $\delta \left(t\right)={\delta }^{Low}+{\eta }^{4.705}\left(\mathrm{6,6.5}\right)$ , and ${\chi }_1^3\left(0\right)=0\left[rad\right]$ . It can be seen that RNN1 fails to identify when the disturbance reaches it; nevertheless, RNN2 becomes the system backup.

Figure 5.

σ_∊ for the third state of the first block

In Figure 6 can be seen ${\sigma }_{{є}_2^1}$ with $\delta \left(t\right)={\delta }^{Low}+{\eta }^5\left(\mathrm{2,2.5}\right)$ , and ${\chi }_2^1\left(0\right)=0\left[rad/s\right]$ . In contrast to the rest of the Figures, this one shows that RNN1 has the minimum values except for a period from 4.7 to 6.8 s.

Figure 6.

σ_∊ for the first state of the second block

In Figure 7 can be seen ${\sigma }_{{є}_2^2}$ with $\delta \left(t\right)={\delta }^{High}+{\eta }^1\left(\mathrm{2,2.5}\right)$ , and ${\chi }_2^2\left(0\right)=\pi /4\left[rad/s\right]$ . After 2 s, RNN2 generally has the minimum values.

Figure 7.

σ_∊ for the second state of the second block

In Figure 8 can be seen ${\sigma }_{{є}_2^3}$ with $\delta \left(t\right)={\delta }^{High}+{\eta }^0\left(\mathrm{0,0}\right)$ , and ${\chi }_2^3\left(0\right)=0\left[rad\right]$ . Before four seconds, there is a competitive struggle between RNN1 and RNN2.

Figure 8.

σ_∊ for the third state of the third block

For all the Figures, every time there is an intersection of lines there is a switching in the output data of the dual-RNN.

In the long run, there is a clear gap between the two lines due to the feedback effect on the dual-RNN. However, the movements of an industrial robot, take less than a second for pick and place processes, welding processes, among others.

5.3 Simulation Results II

To verify the best performance of the dual-RNN with respect to a traditional RNN, is necessary to replace the dual-RNN block in Figure 1 with RNN1 or RNN2, and so obtain results for each in a separate manner. This ensures that the feedback of either one of them does not affect the performance of the other. Table 5 comprises three sections: the top is for the dual-RNN data, the middle is for the RNN1 data (without feedback from RNN2), and the one below is for the RNN2 data (without feedback from RNN1). The data are obtained for different simulation times and there is an averages row for each, where we can observe that the minimum values correspond with those for the dual-RNN. Due to the foregoing, it follows that the dual-RNN as an identification system gives better performance than a single first- or second-order RNN.

Table 5.

${\sigma }_{є}$ vs ${\left[{\sigma }_{є}\right]}_1$ and ${\left[{\sigma }_{є}\right]}_2$

σε	0.5s	1s	2s	3s
σε11	0.04317	0.04071	0.03979	0.03918
σε21	0.04083	0.03984	0.03936	0.03868
σε31	0.04103	0.03863	0.03872	0.03841
σε12	0.04150	0.04071	0.04214	0.04201
σε22	0.04668	0.04269	0.04261	0.04276
σε32	0.04675	0.04588	0.04874	0.04857
Average	0.04333	0.04141	0.04189	0.04160

[σε]1	0.5 s	1s	2s	3s
[σε11]1	0.04453	0.04107	0.04132	0.04181
[σε21]1	0.04813	0.04508	0.04398	0.04413
[σε31]1	0.04468	0.04086	0.03875	0.03899
[σε12]1	0.05324	0.05022	0.04975	0.05131
[σε22]1	0.04667	0.04279	0.04285	0.04369
[σε32]1	0.04674	0.04566	0.05098	0.06332
Average	0.04733	0.04428	0.04461	0.04721

[σε]1	0.5s	1s	2s	3s
[σε11]2	0.04239	0.04025	0.03956	0.03902
[σε21]2	0.04007	0.03945	0.03916	0.03855
[σε31]2	0.04028	0.03823	0.03840	0.03819
[σε12]2	0.04144	0.04069	0.04213	0.04200
[σε22]2	0.04811	0.04722	0.04621	0.04521
[σε32]2	0.04908	0.04762	0.04890	0.05289
Average	0.04356	0.04224	0.04239	0.04264

5.4 Simulation Results III

The dual-RNN as identifier for the robot arm must adjust the synaptic weights online to obtain the correct neural states: x₁^1,2,3 are the states for the position and x₂^1,2,3 are the states for the velocity. At any given moment, the robot has its own position and joint velocity represented by q₁₂₃ and ${\stackrel{.}{q}}_{\mathrm{1,2,3}}$ , respectively. The function of the identifier is to obtain online x₁^1,2,3 as close as possible to q₁₂₃ and x₂^1,2,3 to q_{1 2 3}. This subsection is devoted to showing how well the identification process performs for the robot's position.

The reference signal r(t) that must track the robot in the experiments of this subsection is shown in Table 6.

Table 6.

Reference signals for Simulation Results III

Linknumber(DoF)	Referencesignal(Ref)	Refamplitude[rad]	Reffrequency[rad/s]
First	Sawtooth	0.75	2
Second	Square wave	0.6	3
Third	Sawtooth	0.45	4

The first link is working with δ(t)=δ ^Light + η⁰(0,0) and χ₁¹(0)=-π/2 [rad, and the results for this identification process are given by Figures 9 and 10. In Figure 9, it is not possible to distinguish between the two lines because, with a light noise level, x₁¹ tracks q ₁ in a nearly perfect way. Figure 10 is a zoomed view of the transient of Figure 9, where the ending of the transient before 0.015 s can be seen, which indicates a nearly immediate identification for the first state.

Figure 9.

The dual-RNN tracking the robot movements for the first link working with light noise levels

Figure 10.

Zoomed view of the transient of Figure 9

The second link is working with δ(t)=δ ^Medium + η⁰(0,0) and χ₁²(0)=-π/2 [rad], and the results for this identification process are given by Figures 11 and 12. It is now possible to distinguish the two lines (the state of the robot (q₂) in blue and the state of the identifier (x1²) in green). This is due to the medium noise level, which has a noise power near to δ ^light x400. Nevertheless, as can be seen in Figure 12, the identification time is similar to that with a light noise level. Even more, after the transient the two lines remain attached each other.

Figure 11.

The dual-RNN tracking the robot movements for the second link working with medium noise levels

Figure 12.

Zoom into the transient of Figure 11

The third link is working with δ(t)=δ ^Limit + η⁰(0,0) and χ₁³(0)=-π/2 [rad], and the results for this identification process are given by Figure 13, where one can see the effect of a strong noise level when positioning the robot. Even so, the dual-NN tries to adjust to the movements of the robot and the identification process is not lost. It is worth mentioning that δ ^Limit =δ ^Light × 10000.

Figure 13.

The dual-RNN tracking the robot movements for the third link working with extra-high noise levels

5.5 Additional Results I

With the aim of demonstrating the performance of the control system, tracking results are presented. It is not intended to show the benefits of the driver.

In Figure 14, the second link is tracking the reference signal given in Table 4 with δ(t)=δ ^High + η^1.5(3,3.5), and χ₁²(0) = π/9[rad]

Figure 14.

Tracking the reference signal under high levels of noise

In Figure 15, the second link is tracking the reference signal given in Table 4 with δ(t)=δ ^Low +η⁰(0,0), and χ₁²(0)=π/9 [rad].

Figure 15.

Tracking the reference signal under low levels of noise

It can be seen that the identification process is more difficult for the operating conditions shown in Figure 14 compared to those shown in Figure 15, since with high noise levels the robot's movements are rougher.

5.6 Additional Results II

Even though the aim of this work is not to present the benefits of any control system, in order to give the reader a point of comparison regarding the performance of the dual-RNN operating within the control system, we provide below a performance comparison of the system given in Figure 1 (hereinafter called ‘dual-CS’) against a classic, but enhanced, PID controller, the parameters of which are given in Table 7 and which is tuned heuristically. This enhanced PID controller generates a control signal from a high speed computed control system, which makes it a kind of computed torque control system. For a classic non-enhanced PID, it is virtually impossible to track hard reference signals combined with the highly nonlinear dynamics of the plant.

Table 7.

Proportional, integral and derivative gains for the PID controller

Link number(DoF)	Kp	Ki	Kd
First				10	2.05	1.3
Second	10	9.25	5.2
Third	10	5.5	2.15

The noise conditions under which the experiments are performed in this subsection are δ ^Medium and the reference signal r(t) that must track the robot arm is disclosed in Table 8.

Table 8.

Reference signals for Additional Results II

Linknumber(DoF)	Referencesignal(Ref)	Refamplitude[rad]	Reff(1,2)T(3)[rad/s](1,2); [s](3)
First	Square wave	0.5	1.5(1)
Second	Sawtooth	0.4	π(2)
Third	Vectorial sequence	[3, 1, 4, 1.5, 1]	[0, 1, 3.5, 6, 7.5](3)

Figure 16 shows the performance of the dual-CS in tracking a square wave as the reference signal. Here, one can observe excellent controller behaviour, even with a medium noise level. Nevertheless, due to the noise - suddenly, for short periods of time - strange behaviour is exhibited in the signal tracking (see in the vicinity of 0.5 s and 8.5 s).

Figure 16.

The dual-CS tracking a square wave

Figure 17 shows the performance of the PID controller defined by the first row of Table 7 tracking the same reference signal of Figure 16. It can be seen that the noise does not allow the controller to produce the characteristic PID tracking form for a square wave, which is the same as that produced for a step as the reference signal (a soft rounded overshoot no longer than 25% of the amplitude).

Figure 17.

The PID controller tracking a square wave

As occurs in Figure 16, strange behaviour is exhibited in the signal tracking (see in the vicinity of 0.5 s, 2.2 s and 6.2 s).

Figure 18 shows the performance of the dual-CS tracking a sawtooth as the reference signal. As in Figure 16, the dual-CS's performance is remarkable. This kind of reference signal does not produce strange movement for the robot positioning. The return from the overshoot is excellent and the transient is short enough.

Figure 18.

The dual-CS tracking a sawtooth

Figure 19 shows the performance of the PID controller defined by the second row of Table 7 tracking the same reference signal of Figure 18. It can be seen that the controller cannot deal with the noise level, with this kind of reference signal nor with the first and third link movements acting as external perturbations for the second link.

Figure 19.

The PID controller tracking a sawtooth

Figure 20 shows the performance of the dual-CS tracking a vectorial sequence as the reference signal. Conclusions for this result are similar to those of Figures 16 and 18, but we can add to the above that do not exist an overshoot when changing the ramp slope (see for 1 s, 3.5 s and 6 s).

Figure 20.

The dual-CS tracking a vectorial sequence

Figure 21 shows the performance of the PID controller defined by the third row of Table 7 tracking the same reference signal of Figure 20. Conclusions are similar to those of Figures 17 and 19 and it can be seen that after 7.5 s the control is lost.

Figure 21.

The PID controller tracking a vectorial sequence

6. Conclusions

The dual-RNN is capable of working under different operating conditions, such as strong, high, medium, low and light noise levels, bounded external perturbations and initial conditions for the robot out from the origin.

When a RNN inside the dual-RNN fails to identify, the other RNN becomes the backup for the system (see Figure 5).

The dual-RNN's performance was better than that of a traditional single RNN (see Table 5); as such, this novel dual-RNN becomes an improved identifier for a robot arm.

Furthermore, the dual-RNN gives more robustness to the control system than would be the case for a single RNN.

The authors would recommend the implementation of the dual-RNN if you have a robotic system working inside a factory or a noisy environment, as well as when the tasks of the robot include high-speed or short movements, such as pick and place or welding processes.

The dual-CS, which uses the dual-RNN as an identifier, is stronger and more robust compared with a classical PID controller, both working within similar conditions: the same reference signal and the same noise levels. The use of the dual-RNN is not limited by working within a dual-CS but can be the identifier for any control system that requires it.