Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2017 / Article
Special Issue

Applied Mathematics to Mobile Robotics and their Applications

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Research Article | Open Access

Volume 2017 |Article ID 8501098 | 13 pages | https://doi.org/10.1155/2017/8501098

Cascade Sliding Control for Trajectory Tracking of a Nonholonomic Mobile Robot with Adaptive Neural Compensator

Academic Editor: Francisco Valero
Received31 Mar 2017
Revised12 Jul 2017
Accepted24 Jul 2017
Published19 Sep 2017

Abstract

A design of sliding mode controllers (SMC) with adaptive capacity is presented. This control technique is formed by two cascaded SMC controllers, one of them having an adaptive neural compensator (ANC); both are put on a WMR (wheeled mobile robot). The mobile robot is divided into a kinematics and a dynamics structure; the first SMC controller acts only on the kinematic structure and the SMC with neural adaptive compensator on the other one. The dynamic SMC was designed applying an inverse dynamic controller and using the model dynamics of the WMR. The adaptive neural compensation (ANC) was used in order to reduce the control error caused by the dynamics variations but it conveys a residual approximation error, so a sliding part was designed to cancel such error. This technique allows achieving the control objective despite parameter variations and external disturbances that take place in the dynamics; on the other hand, the ANC can adjust its neural parameters to reduce the dynamics variations of the WMR and thus improve the trajectory tracking control. Problems of convergence and stability are treated and design rules based on Lyapunov’s theorem are given.

1. Introduction

1.1. Previous Works

The problem of trajectory tracking is a classic control one; its importance lies in the actual applications that these algorithms have. Although some of these problems have been solved before, there is always the possibility of somehow improving the algorithms previously designed.

The biggest advantage in using SMC technique is the low sensitivity in front of changes in the process dynamics and external disturbances, which eliminates the necessity of an accurate model. Recent works about disturbance rejections and SMC applied to robots control are dealt with in papers such as [1], where a spherical robot using Omni wheels uses a cascade sliding mode control (CSMC) to control its position.

The structure of the CSMC is formed by a cascade combination of states of the dynamic model and a set of sliding surfaces. This work shows that the problem of constant speed can be avoided in order to achieve the position control without adaptive control techniques. An application of SMC in WMR is presented in [2]; the trajectory tracking problem is investigated for a nonholonomic wheeled mobile robot (NWMR) with parameter uncertainties and external disturbances. In this strategy, combining the kinematic model with the dynamic model, the actuator voltage is employed as the control input, and the uncertainties are approximated by a fuzzy logic system. The control system is stable and the tracking position errors converge asymptotically to zeros. Simulation results demonstrate the effectiveness of the proposed strategy.

In similar form, Ashrafiuon et al. [3] presented a nonlinear sliding mode control law that is employed to stabilize the error dynamics. It is shown that the control law is uniformly asymptotically stable if unknown disturbances and modeling uncertainties are bounded. The framework is applied to differential drive mobile robots, air vehicles operating in the vertical plane, and marine vehicles. Simulations are presented for models of in-house mobile robots and surface vessels subject to unknown disturbances.

Another design for SMC controllers is proposed by [4], who uses a controller based on linear algebra for mobile robots. The results presented show a good performance, while the oscillations in trajectory tracking are reduced.

In the work presented by [5], the tracking control of a two-wheeled mobile robot is considered to underpin the developed theoretical results based on SMC. Model-based tracking control of a wheeled mobile robot (WMR) is first transferred to a stabilization problem for the corresponding tracking error system, and then the developed theoretical results are applied to show that the tracking error system is globally asymptotically stable even in the presence of matched and mismatched uncertainties. These works [35] do not present adaptive capacity combined with SMC in trajectory tracking control.

In another work [6], a practical control law for wheeled mobile robots is proposed in order to improve the transient performance and decrease the tracking errors. The nominal system is governed using a controller derived under the back-stepping framework. Such a design can effectively reach the system’s tracking objective and enhance robustness via properly configured parameters.

Another case of cascade-control algorithm based on a sliding mode is proposed by [7]. It is implemented for trajectory tracking control of a hydraulically driven 6-DOF (six degrees of freedom) robot manipulator. The proposed controller, which consists of two control loops, is applied to separate the hydraulic dynamics from the mechanical part so that the designed controller takes into account not only the mechanical dynamics but also the hydraulic dynamics of the manipulator. Experimental results demonstrate the satisfactory position tracking behavior of the parallel manipulator using a cascade controller based on a sliding mode.

In the meantime, a two-time-scale filter is applied to the system function to estimate the disturbances, essentially improving the system’s precision. Experimental results demonstrated that the proposed control is practical for WMRs in tracking control. Instead in [8] the authors present a SMC for the whole structure of the mobile robot and the results show a good performance as well as asymptotic convergence demonstrated using Lyapunov’s theory. But the control law is not trivial to implement for another kind of mobile robot. Similar research is shown in [9], where an adaptive neural network DSC (dynamic surface control) formulated on disturbance observer for the WMR (wheeled mobile robot) with parameters variations and external disturbances has been implemented. The observer is employed to estimate the unknown disturbances, and the neural approximation is used to identify the nonlinear model and all possible uncertainties of the system. Then, the Lyapunov theory is used to demonstrate the convergence of the proposed control system. Simulation and experiments results are included to demonstrate the adaptive capability and robustness of the proposed control approach.

In [10], an adaptive exponential SMC is proposed as a possible solution to minimize the chattering effect, external disturbances, and parameters variations in trajectory tracking of a wheeled mobile robot. The results obtained via experimentation show the effectiveness of the adaptive exponential SMC against disturbances and parameters variations.

Another important consideration in SMC is how to reduce the chattering effect; in [11] passivity-based adaptive and nonadaptive chattering-free sliding mode controllers are proposed assuming that the norm upper bound of the derivative of the sliding surface is available.

In [12], a method based on an integral SMC applied in the trajectory tracking of WMR is presented. The control technique is designed to solve the reaching phase problem with the elimination of matched disturbances and minimize the unmatched one. The proposed controller has two parts: first one is a high-level stabilization controller to compensate the known system and second one is a supervisory controller to improve the trajectory tracking in the presence of disturbances. The simulation results are based on the kinematics model of the nonholonomic mobile robot.

1.2. Contribution

In this work, an adaptive neural compensation (ANC) is considered. The ANC can be used by static controllers, obtaining as a result an adaptive control over the whole robotic system to be controlled. In this case, the ANC is applied over an inverse dynamic controller, and the complete control system shows a similar behavior of an adaptive controller system.

The paper [13] describes a method for the design of the discrete time dynamic compensation for a WMR with parameters variations, and the application of this method to trajectory tracking control. This method uses a single NN-RBF for dynamic compensation and is designed using Lyapunov’s criterion. The stability analysis demonstrates a Uniformly Ultimately Bounded (UUB) control method.

Another work [14] shows a control technique based on a SMC and adaptive neural control applied over a WMR, but the sliding control is acting over the whole dynamic part of WMR.

This paper deals with the adaptive trajectory tracking control using SMC, and its design is based on a nominal robot dynamics. The control system has two parts. The first one uses the robot kinematics and SMC. The second one uses the WMR dynamics to design a nominal inverse control with ANC.

The main assumption of this work is that the most important parametric variations and uncertainties of the WMR appear in the dynamic part. The control technique uses a neural net based on RBF (radial basis functions) in combination with SMC and inverse dynamics. The control part based on inverse dynamics is designed on the nominal model of the WMR.

The adaptation laws of the ANC are obtained using Lyapunov’s stability criterion, obtaining an asymptotic stability of the proposed control system.

The combination of cascade SMC and ANC technique presents the following advantages:(1)The focus of this technique is the application of two sliding controllers in cascade to reduce problems on the perfect velocity matching of the WMR caused by parameters or dynamics variations. And it can be applied in WMR systems or nonlinear systems when their parameters vary or are unknown.(2)The ANC does not need to learn the whole robot structure; it adjusts the dynamic variations to approximate the nominal to the real dynamics of the WMR.(3)The ANC is easy to implement and it can be applied in different controllers without adaptive capacity, obtaining an adaptive control technique for the proposed control problem.

1.3. Organization

The paper is presented as follows. In Section 2, a description of the WMR system is given and the robot model (kinematic and dynamic parts) is proposed. Explicit expressions of the cascade SMC and the ANC compensation are derived in Sections 3 and 4, respectively. These expressions are used to obtain the neural adjustment law and prove the convergence of the proposed control system (Section 5). Several experimental results are presented in Section 6 for the evaluation of the proposed control technique and demonstrate the effectiveness of the adaptive capability. Finally, Section 7 concludes the paper.

2. Robot System Description

Figure 1 shows a typical structure of a nonholonomic mobile robot. It consists of a robot body (CPU, sensors, and motors) with two driving wheels mounted on the same axis and two motors for each one and a free rear wheel (or castor wheel). Both motors are the actuators of motion and orientation and provide the torque to the wheels.

For WMR control, the variables of interest in inertial Cartesian frame ( plane) are , , , , and , where , are the coordinates of the point of interest , the angle indicates the WMR direction, and the pair , is the linear and angular velocities, respectively, of the WMR.

The WMR parameters are described as follows: denotes the mass center of the WMR. is the distance between the free wheel (castor wheel) referred from the axis of the traction wheels. is the distance between the point and the axis of the traction wheels.

The robot’s model is divided taking into account the kinematics and the dynamics models, as shown in Figure 2. The kinematic and dynamic models were formulated for [15] and can be expressed as follows.

Kinematics Model

Dynamic ModelThe identified parameters vector is described in the Appendix and the model uncertainties of the WMR are the pair and the pair , where the pair is dependent on WMR friction and slip velocities and some of these uncertainties influences on mobile robots are described in [16]. On the other hand, the pair is dependent on robot’s parameters as load weight, mass, inertia, wheel diameter, tires properties (static and dynamic friction), electric motor, its servos, and others.

3. Sliding Mode Kinematics Controller

To obtain the kinematics controller, the robot’s kinematics model is necessary to know. The robot’s kinematics model (2) can be rewritten as follows:For (3) to be controllable, it requires that must be nonsingular.

Assumption 1. is a nonmodeled kinematics vector which is bounded, where is the upper bound.where They are the position output vector, the velocities input vector, and the vector of nonmodeled kinematics. The trajectory tracking error can be expressed asThe main objective of the control law, which can be obtained by the SMC, is the convergence of any state trajectory of nonlinear system to specified surface (defined by designer) into space state and maintain it in the subsequent time [1719].
A sliding surface for a MIMO system can be obtained by position error (6) and defining asThe derivative of (7) is is a strictly positive constant and denotes a row of vector. In the design of the kinematics SMC, first, an ideal equivalent control law must be determined. The ideal control law is obtained from (8) equaling zero asReplacing (3) with (8), equaling zero, the following is obtained:Similar case was studied in [17], defining the control law that guarantees the sliding condition (9), which is composed of an ideal control , but the term is unknown; taking into account Assumption 1, is bounded. For control purposes, it can be approximated by . Thus, the control law is defined byThen the values of , , and areThe term was included to guarantee the convergence of (11); now we define Lyapunov’s function candidate likewhere the superscript denotes the analysis performed with respect to the kinematic controller; now differentiating (13) with respect to time and replacing (3) lead toReplacing (11) with (14), Then, the following is obtained:

From Assumption 1   and is negative definite; now integrating both sides of (16) results inBecause is bounded and is also bounded and nonincreasing, it can be obtained thatBy Barbalat’s lemma [17], it can be shown that limit . That is, tends to zero as t tends to infinity, being a stable control system. Therefore, the tracking error of the SMC based on WMR kinematics tends to zero as tends to zero.

4. Sliding Mode Controller with Adaptive Neural Compensation

4.1. Problem Formulation

At this point, new solutions based on SMC and ANC for WMR considering parameter variations in its dynamics are proposed. The aim of this work is to reduce the trajectory tracking error produced by the uncertainties in the WMR model and parameter variations in the dynamics, but maintaining an adequate mechanical performance and improving the performance of the WMR during trajectory tracking.

The proposed strategy in this matter combines inverse dynamics control and adaptive neural compensation (ANC) with SMC, where the inverse dynamics controller is built with the known part of the robot’s dynamics, the adaptive neural part compensates dynamic variations of the robot, and SMC eliminates the error introduced by the neural network.

This point aims to design a combination of SMC and ANC which ensure the global convergence of controlled variables for the control system for a given reference trajectory .

Now, the system model (2) is expressed in compact form aswherewhere , denote row vectors of and , respectively, is the state vector, and with is the control action input vector.

Assumption 2. Dynamic variations in (19) produce another pair of additive functions represented by and , which represent nonlinear unknown functions and are considered as uncertainties.
The WMR dynamics from (1) without its variation is given by

Assumption 3. The matrix is nonsingular; that is, exists and has bounded norm; this is equivalent to assumingwhere and are the lower and upper eigenvalues of g, respectively.

4.2. Implementation of SMC with Adaptive Neural Compensation (ANC)

The objective of this issue is to design a control law that combines SMC with adaptive neural compensation. And this control law must ensure the convergence of all controlled variables. The SMC with ANC which is designed based on the NN-RBF structure can deal with this kind of control problem [20].

For a good trajectory tracking, the output must follow the desired trajectory ; therefore, the tracking error should converge to zero.

Now, we define the tracking error asA sliding surface for the WMR system can be written in terms of the control error,The derivate of (24) isTo ensure that the state variables remain on the sliding surface, it is necessary for to equal zero,Based on (26), the equivalent control law can be obtained, and in the same way of the previous section, we included the term to guarantee the convergence, and the obtained control law is expressed asNow, it is necessary to define a control law for the nonlinear system (21), define a new control law that guarantees robust performance and the convergence, and also ensure the sliding condition [20]. This new control law is implemented by (27) and a discontinuous term . It is defined byFrom (20), because is a diagonal matrix and the term is constituted by a strictly positive constant and a discontinuity . The function is defined byTo prove the convergence of (28), a Lyapunov’s function candidate is defined asDifferentiating (30) with respect to time is expressed by and replacing (28) with (31),Equation (32) proves the convergence to zero of the designed control law (28). The next step is considering the model indicated in (19) with dynamic variations; the nonmodeled structure can be expressed as

Substituting the dynamic part of the WMR (33) into (26) and applying the proposed control action (28), the following is obtained:

In (34), the terms corresponding to and disturbances are undesired; to deal with this problem, the term with adaptive capacity is added, which would compensate and variations in the dynamics of the system. And the term would eliminate the undesired errors (approximation error of NN-RBF and the uncertainties) that the adaptive term cannot compensate.

Taking into account these considerations, from (28), the control law that combines SMC and ANC is expressed aswhere is a compensation variable which can be approximated by the neural function and it is the main structure of the ANC; the compensation variable can be represented as function of the optimal neural parameters, indicated bywhere , , and are optimal parameter vectors of the weights , the input weights vector , and the radial basis functions , respectively; and are the optimal values of the respective vectors (centers and widths ); and is the neural approximation error generated by the ANC, and it is bounded by .

Other neural parameters of the ANC are defined by and : the first one is the regressor vector and can be defined as and the second one is the number of RBF neurons defined by empirical criterion [21]. The vector of RBF functions is ; each vector component is defined by with in the center of the receptive field and is the width of the Gaussian function.

The optimal parameters of (36) are constant and unknown. These parameters are used as analytical tool to estimate their real values. The estimate neural parameters are , , and , and they are estimated parameter vectors of , , and , respectively; and and are the corresponding estimation of and , respectively.

Defining the neural parameter approximation error (weights) like , and function error like , the neural compensation expressed in (36) can be written aswhere and represent the learning error of each neural term, and the terms (product of two errors) like and can be considered into .

Taking into account [22, 23], the neural approximations can uniformly approximate any continuous function. Now, it is possible to consider that and .

To obtain the closed loop equation, it is necessary to consider the proposed sliding surface (25), and replacing the robotic system (19) and also control law (35) with neural compensation (38), the closed loop equation becomes

Using a Taylor’s series to approximate the function difference and taking the point and , the expansion can be described bywhere represents the high-order term in a Taylor’s series approximation and and are derivatives of with respect to and at . They are expressed asFrom (40), the following is obtained: From (42), the high-order term is bounded bywhere , , and are constants used to determine the boundedness of , due to the fact that the radial basis functions (gauss bell functions) and its derivatives are bounded by , , and . Replacing (42) with (39), the following can be obtained:

Now, considering all terms that contain high-order terms in (44) and including the neural approximation error, they can be expressed as “,” and it is bounded by a constant, as shown in (45):

5. Neural Parameters Tuning

To achieve a stable adjustment law, a Lyapunov’s candidate function must be proposed and it must consider all neural parameters (error, neural weights, spreads, and centers); the proposed function is where , , , and are nonnegative constants, respectively. is outputs number and is neurons number . In this analysis, to reduce any possible cause of chattering effect, a boundary layer thickness is defined, and can be computed as follows:In this analysis, the sign function was changed by a saturation function, in similar way as done by [14, 24]. This boundary layer (BL) smoothes out the control behavior and ensures that the system states remain within this layerNow, the derivative of (46) with respect to time is written in (49), noting that inside the BL and outside. Then (49) can be expressed asSubstituting (44) in (49), the following is obtained:Rearranging (50),Now grouping terms in (51) leads to Now , , , and are selected asConsidering (53) into (52), then (52) can be rewritten asFrom (54), it follows that We integrate both sides of (55) which is defined asBecause is bounded, and is also bounded and nonincreasing, the following can be obtained:Therefore, by Barbalat’s lemma [17], it can be shown that . That is, tends to zero as tends to infinity.

As a result, the proposed control is stable. Moreover, the control error of the dynamic part will converge to zero asymptotically according to .

From (53), considering , , , and , the neural adjusting rules areEquations (58)–(61) are addressed to the tuning online of the dynamic compensation. Equations (58) and (59) adjust the neural weights, and (60) and (61) are responsible for setting the widths and the centers of each RBF vector of the ANC.

6. Experimental Results and Discussion

This section analyzes the performance of the proposed control technique, a single experiment is executed, and the corresponding results are presented below. To implement the experiment, a pioneer 2DX is used. The robot’s parameters are indicated in Table 1. The control action commands of the WMR are linear and angular velocities. Figure 3 shows a pioneer 2DX, and it has an 800 MHz Intel-Pentium with 512 Mb of memory RAM onboard CPU in which the proposed control technique is implemented.


ParametersPioneer 3DXPioneer 2DXUnits

0.240890.3037s
0.24240.2768s
−9.3603e−4−4.018e−4s⋅m/rad2
0.996290.9835
−3.725e−3−3.818e−3s/m
1.09151.0725

In the experiment, the odometric sensors are used to sense the robot position; this experiment tries to demonstrate the advantages of CSMC-AMC method with respect to other control techniques from the literature.

The experiment is performed in the following manner: the robot must follow a predetermined trajectory (in this case a pentagon), where the dynamic controller is calibrated with different parameters not corresponding to the pioneer 2DX robot on the nominal model part ( and in (33)) (being the parameters of the 3DX pioneer robot).

The ANC (adaptive neural compensator) is activated at 50 sec. after initiating the experiment. Such compensator reduces the control error produced by the difference between the real and nominal dynamical structure, while the sliding part is used to eliminate the residual error added by the neural approximation, reducing the control error , and therefore the error position too. The registered results in the experiment are presented in Figures 49.

The response driven by the proposed control technique shows a larger value of the trajectory error because the robot is far away from the reference trajectory in the initial moment. Since the ANC can compensate these phenomena through learning, it can be seen that after 50 secs. of the initial time, there is a transitory error due to the initial commutation of the ANC, but then the tracking errors are continuously reduced and a better tracking performance of the proposed control technique is obtained. Figure 6 shows the control actions and the WMR output speeds of the CSMC-ANC technique, while Figure 5 depicts the desired and followed trajectory by the WMR with CSMC-ANC.

In Figure 7, the trajectory errors for the experiments using the proposed CSMC-ANC technique to follow the desired reference trajectory are shown. The trajectory error is defined as the error norm of the difference between the instantaneous distance of the WMR and the desired reference. It does easily verify that the error at final time is less than the initial position error. Figures 8 and 9 show the neural weights evolution during the experiment. In both figures, it can be observed that the network starts to set the weights from 50 sec., that is, the instant when the ANC compensator is activated.

In this work, a control technique is presented that uses two SMC based controllers to control the trajectory of a mobile robot. The design of the proposed control technique was done to avoid the chattering effect. Therefore, the controller uses an adaptive part (ANC) to reduce the effect produced by the discontinuous function in the control action; this discontinuity is the main cause of this effect. And to ensure the reduction, the discontinuous function was replaced by the saturation function equation (47) smoothing the output response. Further, the cascade SMC with ANC is robust and efficient with respect to the dynamical variations of the WMR. In addition, it is effective in reducing the control error produced by the parameter variations or model uncertainties. The proposed controller designed in this research work for a WMR does not use the real dynamics for its implementation.

In the literature, there are many mathematical formulations for WMR dynamics not concerned with perfect accuracy, but the response of the cascade SMC-ANC as a control action to these dynamic variations as well as possible parameter variations of the WMR improves the system robustness.

The proposed control technique design methodology presented in this research work admits two significant characteristics, with important implications for WMR control design applications. The first one, the pair CSMC-ANC, is stable and its control error tends to zero (Section 6). And second one can be considered as a possible solution for the control of WMR and of any nonlinear systems, when the uncertainties are present in the robot dynamics or there are parameter variations in the robot.

7. Conclusions

In this research work, the problem of using two SMC controllers connected in cascaded has been considered. One of them uses an adaptive neural compensation (ANC). This technique was applied to nonlinear systems with nonmodeled uncertainties.

In the proposed technique, it is necessary to highlight that the controller design of the dynamic part was built using the combination of a feedback linearization controller and a SMC with an adaptive neural compensation (ANC). This control technique guarantees asymptotic convergence through a Lyapunov-based adaptive neural compensation.

The ANC in combination with SMC can reduce the control error caused by parameters variations of the WMR that affects the feedback linearization built with the known dynamics. This compensation modifies the control action to reduce the effects of dynamic variations of the WMR or model uncertainties. The main advantage of this technique is that the ANC does not need to learn all the dynamics of the system structure, and it was designed to compensate only the dynamic variations and all possible uncertainties in the model. All other errors tend to zero by the action of the CSMC.

By experimental results, the proposed method has been shown to reduce trajectory errors more than the conventional SMC.

Appendix

Parameters Description

The identified parameters [15] can be described by In these relations, is the robot mass; is the radius of the left and right wheels; kb is equal to the electromotoric force constant multiplied by the reduction constant; Ra is the electric resistance; ka is the constant of torque multiplied by the reduction constant; , , y are positive constants; Ie and Be are the moment of inertia and the viscous friction coefficient both belonging to the combination of motor, gearbox, and wheel; and Rt is the nominal radius of the wheel.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was funded by the National Council of Scientific and Technological Research (CONICET) PIO-CONICET Project no. 15020150100084CO.

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Copyright © 2017 Flavio Capraro et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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