#### Abstract

In this paper, a model reference adaptive control (MRAC) principle for a one-degree-of-freedom rigid-link electrically driven robot is presented. The proposed control methodology addresses the problem of trajectory tracking with parameter uncertainties in the dynamic model of the system and proposes adaptation laws for the electrical and mechanical parameters. Closed-loop stability is rigorously discussed, proving that the tracking error trajectories converge to the origin exponentially. With the aim of performing experimental comparisons, two control schemes are also revisited theoretically and experimentally: one is an algorithm previously reported in the literature and the other is an adaptive controller derived under the assumption that the electrical dynamics of the actuator are negligible. All the discussed controllers have been implemented in an experimental setup consisting in a rigid-link robot actuated with brushed DC motor. The comparison indicates that better results are obtained with the new MRAC scheme.

#### 1. Introduction

Adaptive control is a design technique focused on high-performance applications in control of dynamical systems, in order to deal with parametric uncertainties characterized by a set of unknown constant parameters. On the other hand, the design of adaptive controllers requires knowing the structure of the dynamic model of the system [1].

An advantage of adaptive control, considering that the parameters of the plant changes during operation, is that the controller compensates the changes in the model parameters achieving better performance. In a nonadaptive system, where the parameters of the plant are considered constants, these changes can compromise the good performance of the overall system. Another advantage of the adaptive control systems is that the knowledge of the plant parameters to design the controller is not necessary [2].

Direct current (DC) motors are widely used as actuators in different kind of robots. Due to its dynamics, DC motors can be described appropriately by a linear dynamic model, which is sensitive to parametric variations, noise interference, and parametric uncertainties. Therefore, conventional control schemes can not yield a convenient performance in the presence of the aforementioned disturbances [3]. Taking into account these facts, this document presents a new adaptive control scheme in addition to a theoretical and experimental comparative study of adaptive algorithms for the trajectory-tracking control of a one-degree-of-freedom (DOF) rigid-link electrically driven (RLED) robot.

In [4], an adaptive tracking controller for a DC motor was presented. Simulation results are given, where the DC motor shaft follows a desired sinusoidal trajectory with an acceptable tracking performance. In [5], a summary of previous works on trajectory-tracking control for manipulator robots was presented, in which the electrical dynamics of the actuators are neglected in the design procedure. In [6], the idea of considering the electrical dynamics of the actuators as a part of the system dynamics was introduced, and a robust controller for the trajectory tracking was presented. This controller guarantees that the tracking error is uniformly ultimately bounded. In addition, an adaptive controller for a rigid robot was presented in [5], where the electrical dynamics of the actuators are considered in the controller design. In [7], the electrical dynamics were added to the controller design for an adaptive algorithm previously reported in literature. This is for the position regulation problem on rigid robots actuated with a brushed DC motor. However, only simulation results were provided. In [8], a formal stability analysis was introduced for a voltage based controller designed for manipulator robots, although the problem of parametric uncertainties was not addressed.

Let us notice that much of the recent literature in the control of DC motors does not take into account the load dynamics but pays attention to other important problems with practical relevance. In [9, 10], the velocity control of a DC motor was addressed by taking into account the dynamics of a DC-DC power converter. Those approaches were supported by real-time experiments. The integration of barrier Lyapunov function to avoid the violation of state constraints was used in [11] for the development of DC motor controller. In [12], an output-feedback control was proposed for the positioning problem of a lineal actuator driven by a DC motor, where the saturation of the control input and the presence of nonlinear friction are considered in the development of the new methodology.

The application of model reference adaptive control (MRAC) for a RLED robot is introduced in this document. In general, the MRAC-based adaptive controllers are designed using a reference model which describes the desired characteristics of the plant to be controlled. The use of a reference model facilitates the analysis of the adaptive system and provides a stability framework, since the controller design and stability analysis was performed in two steps. Regarding this, the MRAC scheme was analyzed in [13], in the presence of unmodelled dynamics, where the analysis showed the existence of robustness and this fact was confirmed by means of simulations. In [14], the application of a MRAC scheme for an induction motor was presented, shown to be robust to parametric variations. In [15], a MRAC-based controller was presented in order to solve the speed control problem of a DC motor. In this case, the classic adaptation scheme is replaced with fuzzy linear adaptation to improve the system performance at low speed and variable load conditions. Nevertheless, results are only validated with simulations. In [16], an adaptive output-feedback controller that belongs to the class of MRAC schemes was presented. This algorithm is applied to manipulators neglecting the actuator dynamics and uses a linear observer to estimate the states for a manipulator. In [17], a MRAC scheme was presented in order to control robot manipulators. However, the electrical dynamics of the actuators are not considered in the controller design. More recently, the velocity control of DC motor was explored in [18] by using a MRAC scheme together with a variable structure extension. However, electrical dynamics were also neglected. Other interesting applications of the MRAC philosophy for nonlinear systems can be found in, for example, [19–22].

More recently, intelligent control techniques have been applied to control this kind of systems. For instance, fuzzy and neural networks based controllers were introduced in [23–32]. However, in most cases, only simulation results are provided. In [33–35], some sliding-mode based controllers were presented for robot manipulators, where the electrical dynamics of the actuator are neglected. However, in [36], a neural-network-based terminal sliding-mode control scheme was proposed for robotic manipulators including actuator dynamics. In [37], a robust optimal voltage control of electrically driven robot manipulators was presented. Particle swarm optimization is used to optimize the control design parameters. Also, a comparative study between voltage and torque based controllers is presented, which confirms the superiority of the voltage control strategy. In [38], a robust discrete repetitive control of electrically driven robot manipulators was introduced, where the control problem is the tracking of a periodic trajectory. However, in all of the aforementioned work, only simulation results were given, making comparing with the controllers presented in this document difficult.

The main contributions of this document are the inclusion of the electrical dynamics of the actuator in the control law and the problem of parametric uncertainties, which are considered in a united form, in addition to the design and analysis of a new adaptive control algorithm for the trajectory-tracking control of a 1-DOF RLED robot. Specifically, the novel adaptive algorithm presented is based on a MRAC principle. It is noteworthy that the application of the MRAC scheme for the control of RLED robots deserves a deeper study since only a few studies have been reported.

Additionally, an experimental performance comparison of our solution with respect to some other adaptive algorithms is presented. For the comparative study, this paper revisits two adaptive controllers: an algorithm previously reported in literature [39] and an adaptive controller derived under the assumption that the electrical dynamics of the actuator is negligible. The new MRAC scheme and the others are tested in real time in an experimental platform consisting in a 1-DOF RLED robot which is affected by gravitational force.

This paper is organized as follows: Section 2 is devoted to summarizing the RLED robot dynamic model. In addition, a brief discussion about the control goal is presented. The proposed MRAC scheme is described in Section 3. An adaptive controller previously reported in literature is given in Section 4. In Section 5, an adaptive controller obtained from the assumption that the electrical dynamics of the actuator are negligible is presented. A brief description of the experimental platform, results of the real-time implementation of the new controller, and the performance comparison are presented in Section 6. Finally, in Section 7, some concluding remarks are given.

#### 2. RLED Robot Dynamic Model and Control Goal

The dynamic model of a 1-DOF RLED robot actuated by a brushed DC motor is presented as in [39] and is written aswhere is the grouped rotor inertia, is the grouped viscous friction, is the constant of grouped gravitational load, is the inductance, is the electric resistance, and is the back electromotive force constant, all these being positive constants. Furthermore, , , and represent position, velocity, and acceleration, respectively. Finally, and represent current and the input voltage, respectively.

It is possible to represent (1)-(2) as a linear regression model, which separates the constant parameters from the states. This representation is especially useful in parametric identification and in the design of adaptive controllers. Then, the mechanical part of the system model described in (1) is rewritten in a linear regression form aswherewith in (4) being the regression matrix of the mechanical part and in (5) being the vector of constant parameters of the mechanical part.

Similarly, the electrical part of the system model described in (2) can be represented aswherewith in (7) being the regression matrix of the electrical part and in (8) being the vector of constant parameters of the electrical part.

Let us consider a time-varying twice differentiable signal , which denotes the desired trajectory of the joint position. Then, the control goal consists in ensuring that the trajectory-tracking error satisfiesThis paper introduces three controllers that ensure limit (9).

#### 3. Proposed MRAC Principle

A new MRAC-based algorithm is presented in this section, where, in the first stage of design, a control action is proposed for the tracking between the real model and the reference model. After this, a complementary control action is defined for the tracking between the reference model and a desired trajectory. Hence, the control objective defined in (9) is satisfied.

##### 3.1. Reference Model

The proposed reference model for systems (1)-(2) is defined aswhere is the estimated grouped rotor inertia, is the estimated grouped viscous friction, is the estimated constant of grouped gravitational load, is the estimated inductance, is the estimated electric resistance, and is the estimated back electromotive force constant, all these being positive constants. Furthermore, and represent velocity and acceleration of the reference model, respectively. Also, and represent current and the input voltage of the reference model, respectively. Finally, is an auxiliary control signal to be defined. Note that represents also the input of systems (1)-(2).

It is possible to express the reference dynamic model described in (10)-(11) as a linear regression model as well. Then, the mechanical part of the reference model in (10) is rewritten in a linear regression form aswherewith being the regression matrix of the mechanical part of the reference model and being the vector of estimated parameters of the mechanical part of the reference model.

Similarly, the electrical part of the reference model described in (11) can be expressed aswherewith being the regression matrix of the electrical part of the reference model and being the vector of estimated parameters of the electrical part of the reference model.

##### 3.2. Error Model

Now we proceed to compute the open-loop error dynamics of the electrical and mechanical part. The error model of the mechanical part is given by the difference between the mechanical part of the reference model in (12) and the mechanical part of the real model in (3). Analogously, the error model of the electrical part is given by the difference between the electrical part of the reference model in (15) and the electrical part of the real model in (6).

The error model is expressed aswhere .

The adaptation errors are defined as follows:

In order to define the error model in terms of the parameter adaptation error, it is possible to substitute the expression of from (19) into the error model in (17), such thatWriting explicitly the first two terms in (21), doing suitable cancellations, and defining the position error between the reference model and the real model asthe error model of the mechanical part can be written asIt is worth remarking that if the parameter estimation error of the mechanical part is equal to zero in (23), then the remaining elements describe the error model for a reference model with known parameters.

Similarly proceeding, the error model of the electrical part in (18) can be rewritten asAs well as for the mechanical part of the error model, notice that if the parameter estimation error of the electrical part is equal to zero in (24), then the remaining elements in (24) describe the error model for a reference model with known parameters. Systems (23)-(24) describe the tracking error dynamics between the reference model (10)-(11) and the actual model (1)-(2).

##### 3.3. Controller for the Tracking between Reference Model and Real Model

As can be seen from the error dynamics (23)-(24), the input must be designed such that the convergence of the electric current error and the position error is ensured. In other words, the control signal is introduced in order to ensure the tracking between the reference model (10)-(11) and the actual model (1)-(2). With this aim, is proposed as feed-forward compensation in terms of the electrical dynamics plus a proportional-integral part in terms of a change of variable ; that is,where and are positive constants and the change of variable,is defined in order to ensure that the current error introduces the compensation to the mechanical error dynamics (23). In other words, the proportional part and the integral part , withare introduced to ensure that . On the other hand, the compensation given asintroduces a proportional term in and a filtered derivative compensation defined aswith , , and being positive constants.

Taking into account the definition of the electrical adaptation error in (20), in particular, the definitions of and , the control signal in (25) can be expressed asSubstituting the definitions of in (26) and in (28) into (23), the expression of the mechanical part of the error model is rewritten asAnalogously, by substituting the control signal in (30) into the the electrical part of the error model in (24), this yieldswhereThe closed-loop systems (31)-(32) result from applying the controller in (25) to the open-loop system in (23)-(24).

##### 3.4. Adaptation Law

We propose the following adaptation lawswhere is a positive constant sufficiently small, is defined in (13), is defined in (33), andare defined as the adaptation gains of the mechanical and electrical part, respectively.

##### 3.5. Analysis for Tracking between the Reference Model and Real Model

We propose the following energy function: with defined in (22), in (26), in (27), in (29), in (19), and in (20).

It is possible to prove that, for small enough, in (37) is a positive definite and radially unbounded function in terms of the states of the reference model error described by (27), (29), (31), and (32), as well as the adaptation laws in (34)-(35).

To ensure that the tracking error between the reference model and the actual model, which is expressed by , is a bounded signal, it is required that in (37) be a decreasing function, that is, the time derivative to be negative semidefinite.

By performing all the suitable substitutions and conducting the algebra, the time derivative of the function (37) is given bywhich can be expressed in a matrix form aswhereSince (39) is a quadratic form, by proving that in (41) is a positive definite matrix, we ensure that is a negative semidefinite function. We can prove this fact by using Sylvester’s criterion to determine conditions to ensure that is positive definite. Then, it can be shown that, by selecting sufficiently large values for and , as well as sufficiently small values for and , there exist conditions for to be a positive definite matrix. Finally, it is possible to invoke Barbalat’s lemma [40] to prove thatwhere is defined in (40). Hence, the tracking error between the reference model and the real model tends to zero as time increases.

##### 3.6. Controller for Desired Trajectory Tracking

Once the tracking error between the reference model and the actual model is proven to be convergent, we propose the control voltage in order to ensure that the reference model position tracks a desired time-varying trajectory . With this aim, the tracking error signals are defined asand the control voltage is proposed aswhere is defined in (25) and

Proceeding to obtain the closed-loop equations in terms of in (43) and in (44), we substitute (47) into (43), where, solving for , we obtainwhere and are positive constants. Solving (10) for and substituting into the second time derivative of the tracking error in (44), one obtainsSubstituting (48) into (49) and after some simplifications, we obtain the tracking error dynamics of the mechanical part as follows:

Similarly, to obtain the closed-loop equation for the electrical part in terms of the tracking error , we calculate the time derivative of (43), and multiplying by , this yieldsSolving (11) for , substituting into (51), and performing the algebra, we obtain the error dynamics of the electrical part as follows:

The system formed by (50) and (52) is linear and time-varying. It can be demonstrated that if , , , , and are strictly positives for all , we have thatwith exponential rate of convergence. It is noteworthy that the limits (42) and (53) imply that the control goal in (9) is satisfied.

Finally, it is worth remarking that the total control voltage to be applied to the open-loop system dynamics in (1)-(2) is given in (45), with the auxiliary control signals in (25) which ensures the tracking between reference model and real model, in (46) which introduces the desired dynamics to the system, the parameter adaptation laws in (34), in (35), and definitions therein.

#### 4. Adaptive Controller Proposed by Dawson, Hu, and Burg (1998)

In this section, an adaptive controller proposed in [39] is presented and the stability analysis is briefly reviewed. The purpose of presenting this algorithm is to make a comparison with respect to the MRAC scheme introduced in this paper. Thus, the trajectory-tracking adaptive controller is written aswhere is a positive constant, is the filtered tracking error given bywith being a positive constant, stands for the trajectory-tracking error defined aswhere is the desired trajectory, is the current error defined bywhere is the desired current, which is expressed aswith the positive constant ,defined as the regressor of the mechanical part, andis an estimation of the parameters of the mechanical part . Similarly,represents the estimation of , which is a vector of grouped parameters, containing parameters from both mechanical and electrical part. Likewise,with being the regressor of the vector of grouped parameters.

##### 4.1. Adaptation Law

By defining the adaptation errors asthe adaptation laws for the update of the mechanical parameters and grouped electrical parameters are given bywhereare the adaptation gains of the mechanical and electrical parts, respectively.

The closed-loop system is obtained by substituting the control action defined in (54) into the dynamic model (1)-(2). By expressing in terms of the filtered tracking error , the current error , and the adaptation errors and , the closed-loop system is given by

##### 4.2. Stability Analysis

To investigate stability of the equilibrium point at the origin of system (67), the following Lyapunov candidate function is proposed:where the time derivative of along the trajectories of the closed-loop system (67) is given byHence, in (69) is negative or zero. Then, the origin of the state space is Lyapunov stable. Invoking Barbalat’s lemma is possible to prove thatis satisfied. Also it can be shown that the tracking error approaches zero as time increases; that isSee [39] for further details about the stability analysis.

#### 5. Adaptive PD Controller Neglecting Electrical Dynamics

In this section, an adaptive PD controller is presented. This algorithm is derived under the assumption that the electrical dynamics of the actuator are neglected. In literature, this is based on the assumption that the inductance is sufficiently small, such that the current can be approached bySubstituting (72) into the mechanical part of model (1), the expressionis obtained. Note that system (73) can be controlled by using a PD-type scheme plus adaptive compensation to ensure convergence of the error .

By assuming that the current error converges quickly to zero, the resistance is compensated by the controller, and redefining the parameters of model (73) as our adaptive controller for system (73) is written as follows:withand is obtained from (56) and and are defined as positive constants. Note that the controller (75) has a PD-type structure in the position error .

##### 5.1. Adaptation Law

The adaptation law for the update of the mechanical parameters is given bywhere is the adaptation gains of the mechanical part and is a strictly positive constant.

##### 5.2. Stability Analysis

By defining the adaptation error of the mechanical part asthe closed-loop system formed by the substitution of the controller in (75) into the open-loop system in (73) can be written as It is possible to show that the origin of state space is an equilibrium point.

Let us consider that the constant involved in the adaptation law (77) satisfies

Furthermore, under (81),is a candidate Lyapunov function for the state space origin of system (80). The time derivative of along the closed-loop system trajectories is given in a matrix form byIt is possible to show that (81) is a sufficient condition that guarantees the positive definiteness of the matrix in (83) and, invoking Barbalat’s lemma, it is possible to prove thatis satisfied.

#### 6. Experimental Evaluation

In this section, a brief description of the experimental setup and the results obtained from the implementation of the described adaptive controllers are presented.

##### 6.1. Experimental Platform

For the experimental validation of the adaptive controllers presented in this paper, the testbed was constructed using a brushed DC motor from* Advanced Motion Controls* model MBR2303NI (*Nema 34 brushed direct current motor*), mounted on an aluminum plate and a pendular load attached to the shaft. See Figure 1 for a picture of the experimental platform. This experimental setup was instrumented with the following components:(i)A PC with (ii) A servoamplifier* Advanced Motion Controls* model* 30A20ACV*,(iii) An optical encoder* US Digital* model* HB6M-2500-250-IE-D-H* with a resolution of 2500 pulses per revolution(iv) An inductive filter* Advanced Motion Controls* model* BFC10010*(v) A current sensor* F.W. Bell* model* NT-5*(vi) A DAQ* Sensoray 626*. The control algorithms were implemented in and the PC interacts with the DAQ through the* Real-Time Windows Target* libraries at 1 millisecond of sample time. Finally, Figure 1 shows a diagram of the overall experimental platform.

The desired trajectory was selected asThe physical meaning of this trajectory is a displacement from the downward position to the upward position of the joint, passing through the point where the load is more affected by the gravitational force.

##### 6.2. Control Gains

The selected control gains and adaptation gains are shown in Tables 1–3. Specifically, Table 1 shows the gains for the proposed MRAC algorithm in (45), Table 2 shows the gains for the adaptive controller in (54) previously introduced by [39], and Table 3 contains the gains for the adaptive PD controller in (75).

The tuning process was carried out in an experimental form, starting with sufficiently small values. Then, each gain was increased looking for an improvement of the trajectory-tracking performance. Related to the control gains, small values were preferred; this is in order to exhibit the adaptation performance of each algorithm. In general, it is possible to say that all the controllers were tuned in a similar manner.

##### 6.3. Experimental Results

The following shows the obtained experimental results. Figures 2-3 depict the comparison between the desired, the reference, and the actual signals for the position and electric current, respectively, obtained from the application of the proposed model reference adaptive controller in (45). As can be seen, the obtained experimental results validate the theory, since the tracking between the reference and the desired signals is achieved while the tracking between the actual and the reference signals is also accomplished.

Additionally, Figures 4–6 show the performance comparison of the proposed model reference adaptive controller (MRAC) in (45), the adaptive controller proposed by Dawson, Hu, and Burg (DHB) in (54), and the adaptive PD (APD) controller neglecting the electrical dynamics in (75). Specifically, Figure 4 depicts the comparison between the actual position and the desired position in (85) for the aforementioned adaptive controllers. In Figure 5, the comparison of the electric current signals can be seen. Finally, Figure 6 depicts the applied control voltage generated for each control algorithm.

It can be observed from Figure 4 that the MRAC algorithm converges much faster than the others. This can be possibly attributed to the overparameterization of the DHB algorithm and to the simplicity of the APD controller, since the DHB adapts nine parameters against only six parameters adapted by the MRAC, making the MRAC adaptation laws easier to tune. On the other hand, the fact that the APD neglects the dynamics of the electric actuator produces higher error values. It was also observed in Figure 4 that the APD algorithm presents better tracking performance than the DHB controller during the first seconds of experiment, which may be due to the tuning of the DHB controller and the fact that the APD only responds to the magnitude of the position error, without the delay introduced by the electric dynamics of the actuator. However, after the first seconds of experiment, the tracking performance of the DHB becomes much better than the APD.

Furthermore, in Figure 7, the time evolution of the estimated parameters , , and of the mechanical part is shown for the MRAC and the DHB algorithms. Similarly, in Figure 8, the time evolution of the estimated parameters , , and of the electrical part is shown. The mechanical parameters of the APD controller in (75) are not shown in Figure 7 because their values are out of scale and do not provide any meaningful information to the comparison. Likewise, the estimated values of the lumped parameters in (61) from the DHB controller are also omitted.

It is worth remembering that matrix in (41) should be positive definite in order to ensure that the trajectories in (40) converge to zero as time approaches infinity. With the aim of demonstrating that this condition is satisfied, it is possible to use the final values of the estimated parameters in Figures 7 and 8 for [s], that is , , , and , in addition to the control gains shown in Table 1 for the MRAC algorithm. Hence, by selecting an appropriate value for in (41), for example , the matrix is positive definite with the eigenvalues , , , and . Therefore, if rough estimates of the system parameters are available, then the provided stability analysis, specifically, the conditions to ensure the positive definiteness of matrix in (41), can be used as tuning rules for the controller gains.

With the aim of complementing the results of the experimental comparison shown in Figures 4–6, Figure 9 depicts the comparison between the error signals for the three controllers, where the exponential convergence rate of the trajectory-tracking error can be seen. Furthermore, Table 4 presents a comparative of the maximum error obtained during the experiments for [s]. Specifically, we compute for each controller during the mentioned time interval. Table 4 also shows the maximum percentage of variation for the tracking error with respect to the total displacement of the desired trajectory ; that is,Table 4 also shows the RMS value of the applied control voltages for each algorithm, for a time interval of [s].

The results shown in Table 4 suggest that the controller MRAC performs better than the rest of the controllers of the comparison, yielding the smallest maximum deviation with respect to the desired trajectory. The controller DHB shows the second best tracking performance on this study. The APD controller presents the worst tracking performance.

Note that the RMS value of the applied voltage given in Table 4 is similar for the three controllers. Therefore, it is possible to say that the proposed MRAC algorithm yields better performance with practically the same applied energy.

It was also observed that the MRAC controller and the DHB scheme (controllers with armature current feedback) performed much better than the APD algorithm giving an improvement of 80% and 75%, respectively. However, from the engineering point of view, the APD adaptive controller may be preferred in some cases, since is a simpler controller with respect to those that use armature current feedback. A possible advantage of the APD controller is that the applied control voltage is a noise free signal (see Figure 6) as a result of not including the dirty current measurement .

#### 7. Concluding Remarks

This paper has dealt with the trajectory-tracking control of a RLED robot with 1-DOF, for which the parameters of the dynamic model are unknown. A new controller has been proposed based on a MRAC principle. Asymptotic tracking of the desired trajectory has been proved theoretically and practically. The new controller has been compared with respect to an adaptive controller previously reported in literature and a controller designed by neglecting the electrical part of the system.

The new scheme showed better performance even with the adaptation of less parameters with respect to the DHB scheme, avoiding with this the overparameterization of the model. In other words, the new MRAC only requires adaptation of the six parameters present in the system, while the DHB scheme is based on adaption of nine parameters.

The results for the three controllers given in Figure 9 show that the position error signal is kept in a certain bound after some finite time (practical stability). The reasons are the quantization disturbance of position measurement, noisy current measurement, and discrete implementation, among others.

The experimental comparison indicated that, in general, better results in the tracking performance are achieved if current feedback is added to the controller. This is clearly observed from Figure 9. The reason is that the assumption of neglecting the electrical dynamics is not valid for some operating conditions. This study suggests that good performance can be obtained with feedback of position, velocity, and a “small amount” of current feedback.

Finally, as future work, the noise reduction in the control signal can be investigated, which would improve the tracking performance. This can be done by using an estimation of the current signal instead of using a noisy measurement of such signal, avoiding the requirement of a current sensor. In this sense, some preliminary work is presented in [41, 42], where the adaptive control of robot manipulators is considered including the actuator dynamics.

#### Competing Interests

The authors declare that they have no competing interests.

#### Acknowledgments

This work was partially supported by CONACyT, Project no. 176587, and SIP-IPN, Mexico.