Position tracking control of nonholonomic mobile robots via $H_\infty$-based adaptive fractional-order sliding mode controller

1.   Introduction

Recently, there has been a lot of interest in the position tracking control problem of nonholonomic wheeled mobile robots because of their wide range of applications in the medical areas, organizations, military operations, and many other sectors ^[1,2,3,4]. The dynamical structure of nonholonomic mobile robots are extremely nonlinear, coupled, and time-varying. To manage these nonlinear properties, uncertainties, and outside disturbances, a variety of classical controllers have been developed in the literary texts to govern these dynamical systems, including PID controllers, back-stepping based controllers, adaptive controllers, sliding mode-based controllers, and many others (see ^{[5,6,7,8,9,10]}).

Among these, sliding mode controllers (SMC) (see the works^[11,12,13]) are the most often used control technique due to their inherent robustness against outside disturbances and uncertainty. SMC employs a sliding surface to guarantee that tracking errors converge toward zero. The sliding mode control scheme's inherent adaptability feature is that it operates irrespective of the system's dynamics whenever the system is working on the sliding manifold. In SMCs, linear and nonlinear sliding surfaces are increasingly used for improved controller performance. In the literature, linear sliding surfaces have been used to introduce linear sliding mode controllers (LSMC) ^[14]. In LSMC, only asymptotic convergence of dynamical states is confirmed but there is no evidance that analyse the fixed time convergence property of dynamical system so to resolve this issue, a terminal sliding mode controllers (TSMC) ^[15] have been presented using non-linear sliding manifolds. In ^[16], a fixed-time non-singular fast TSM control for wheeled mobile robots was presented using a disturbance observer.

For efficient performance of the dynamical system, various combinations of sliding mode controllers with fractional calculus have been reported in the literature for improved and accurate controller performance. Fractional-order controllers perform better than integer-order controllers due to their faster order convergence. Applying the fractional-order derivative to LSMC, referred to as the fractional-order sliding mode controller (FoSMC) ^[17,18], is the first step in the work on integrating the fractional-order derivative with SMC. The tracking capability of these controllers is better than that of conventional sliding mode controllers due to their higher-order convergence. Eray and Tokat in ^[19] discusses the design of FoSMC for the robot manipulator trajectory tracking problem with a time-varying sliding surface. The authors of this study demonstrate the tracking errors' asymptotic convergence toward their system states. In ^[20], a fractional adaptability rule for a sliding mode control approach for a MIMO nonlinear dynamical system was presented while in ^[17], a linked FoSMC strategy using obstacle avoidance for the control of a four-wheeled steerable mobile robot was designed. A new fractional-order global sliding mode control method for these systems under outside disruptions is presented in ^[21].

Although fractional-order sliding mode controllers give efficient responses to the dynamical system's states, the negative influence of uncertainties and external disturbances causes a degrading performance of the overall system. So, to overcome these aspects, $H_\infty$-based robust controllers ^[22] have been designed in the literature. An $H_\infty$-based position tracking control method for robot manipulators employing a wavelet neural network-based TSMC was presented by Panwar ^[23]. The integration of the $H_\infty$-based controller with sliding mode controller gave an outstanding performance of the dynamical system with controlled gain values and significantly faster response of system states towards their trajectories.

Inspired by the prior work, this study offers a novel combination of $H_\infty$-based controllers, neural networks, and fractional-order SMC controllers for enhanced performance of nonholonomic wheeled mobile robots. A fractional-order sliding manifold is presented first in the control methodology, and then an $H_\infty$-based adaptive fractional-order sliding mode novel controller is designed for the control of nonholonomic wheeled mobile robots with system uncertainties and external disturbances. The main contribution of the presented work is as follows:

(1) The position tracking issue of a nonholonomic wheeled mobile robotic system in the presence of outside disruptions and system uncertainty is examined.

(2) An efficient combination of a fractional-order sliding mode control technique with neural network-based $H_\infty$ trajectory tracking control approach is proposed based on a fractional-order sliding manifold.

(3) The stability of the dynamical system and the resulting convergence of the position's tracking errors toward their equilibrium are examined by using the Lyapunov stability criterion.

(4) A comparative simulation study is conducted on a 3-dof nonholonomic wheeled mobile robot to analyse the performance of the the designed technique with the existing control techniques.

This paper's remaining sections are organized as follows: A dynamical model for a nonholonomic wheeled mobile robot is provided in Section 2. The control technique is given in Section 3, and the stability analysis is presented in Section 4. A simulation study is presented in Section 5, and the article is concluded in Section 6.

2.   Dynamic model of nonholonomic wheeled mobile robot

The Euler-Lagrange equation for nonholonomic wheeled mobile robot dynamics is stated as

where

denotes the cordinates for the mobile base,

denotes the inertia matrix,

denotes the centripetal or coriolis matrix,

denotes the friction to the dynamical system,

denotes the bounded disturbance,

transforms the states of the wheeled mobile robot in the earth's reference frame,

denotes the control torque input,

denotes the constraint matrix associated with

Langrange multiplier.

The wheeled mobile robots are subjected to the following nonholonomic kinematic constraint:

This constraint is the limitations on the dynamical equation of the wheeled mobile robot system to the manifold

From Eq (2.2), we can obtain a full-rank matrix

as

From the constraints given in Eqs (2.2) and (2.3), we have a new vector

that satisfies the following condition:

So, by controlling this new vector

in the task space, the whole position and orientation of the wheeled mobile robot can be controlled.

Differentating Eq (2.4) w.r.t $'t'$, we get

Utilizing Eqs (2.4) and (2.5) in Eq (2.1) and multiplying the resultant equation by $T^T$, we obtain

with

Let the above-constrained dynamics Eq (2.6) of a nonholonomic wheeled mobile robot satisfy the following properties.

Property 1. $\bar M_g$ is a bounded, symmetric, and positive-definite matrix.

Property 2. The dynamical terms

and

are bounded for some constants $g_i > 0$ for $i = 2, 3$ with

for $t > 0$.

Property 3. The term

assures the skew-symmetric property, i.e.,

Assumption 1. All the Jacobian matrices are uniformly bounded and continuous if

is consistent and uniformly bounded.

3.   Controller structure

3.1.   Sliding surface

Consider the following sliding surface:

where

gives the position tracking errors,

gives the desired trajectory,

where $A _1, A _2$ are positive values, and

is a sliding variable. Differentiating the above equation, we get

Putting

into above equation, we will get

Putting the value of $\ddot{\chi}(t)$ from Eq (2.6), Eq (3.2) is reduced to

The reduced dynamical equation in terms of $r(t)$ is given as

Taking

and using Eq (3.1), Eq (3.3) is reduced to the following equation:

Rewriting the above equation, we get

where

is the nonlinear function of the dynamical structure that will be approximated by a radial basis function neural network (RBFNN). The architecture of the RBFNN is given in the next subsection.

3.2.   RBFNN

Due to the adaptive nature of the RBFNN, it is utilized to reproduce the non-linear part of the manipulator's dynamics. Let the function approximation on a simply connected compact set of the continuous function $h(t)$ be

where,

demonstrates the weight matrix, and it will be updated online in an adaptive manner

denotes predefine basis array,

denotes reconstruction error, and N denotes the number of nodes used in the structure of neural networks. So, we have

for some $\psi_{N} > 0$.

For larger values of N, $\psi(t)$ may be reduced to be very small. The RBFNN uses the Gaussian function $\xi(t)$, which is given by

where $i = 1, 2, ..., N$ and $c_i$, $\sigma_{i}$ represent the center vectors and width vectors, respectively.

Putting the value of $h(t)$ from (3.6) into (3.5), the reduced-error dynamical equation is then given by

In addition to this, it is not sufficient to use only the neural network architecture to handle the dynamical system. The effect of environmental constraints that are unknown can be very serious for the nonholonomic wheeled mobile robot system. Therefore, a switching control law during the sliding phase should be added to control the dynamical system that ensures robustness against external disturbances. To construct the robust dynamical system's control technique, an $H_{\infty}$-based tracking controller is presented in this work. The structure of the $H_{\infty}$-based tracking control technique is given in the next subsection.

3.3.   $H_{\infty}$-based tracking control technique

To attenuate the effects caused by unmodeled dynamics, disturbances, and approximate errors, the advantages of the $H_{\infty}$-based tracking control technique are utilized in this work. The $H_{\infty}$-based tracking control technique is a robust technique and the efficient combination of this technique with neural networks and fractional-order SMC controllers give enhanced performance of the wheeled mobile robot with less chattering in the control phase.

To impose the $H_{\infty}$-based tracking control technique on the dynamical system, let the overall system satisfy the given condition:

where $P$, $Q$ are positive definite matrices with predescribed attenuation level $\pi > 0$ and $\mathcal{T} \in (0, \infty)$.

3.4.   $H_{\infty}$-based adaptive fractional order sliding mode controller

To obtain the reference trajectories, the proposed $H_{\infty}$-based adaptive fractional-order sliding mode controller is given by

where

and $P_1$ and $P_2$ are positive-definite gain matrices.

4.   Stability analysis

Theorem 4.1. The system states of the reduced-error dynamics of the nonholonomic wheeled mobile robot system as given in Eq (3.8) converges toward zero along with the boundedness of signals if the controller is chosen as described in Eq (3.10) with the update law as follows:

where

Proof. Consider the candidate Lyapunov function as

where

Differentating Eq (4.2) w.r.t. $'t'$, we get

Taking the value of $\bar{M}_g \dot{r}(t)$ from Eq (3.8) with

we have

Using property 3 and taking the value of $T^{T}U$, Eq (4.4) becomes

Using the update law as given in Eq (4.1) and simplifying Eq (4.5), we have

Simplifying the above equation, we have

and

Rewriting the above equation by using some basic concepts, we have the following equation:

where

for $t > 0$.

This completes the proof. □

Now, adding and subtracting $\frac{\pi^2 \|B\|^2}{2}$ in the above equation, we get

After simplification of the above equation,

and

Taking the value of $\dot{\bar{\chi}}$ from Eq (3.1), we get

and

Putting

in the above equation, we get

Now, integrating above equation between the limit $0$ to $\mathcal{T}$, we have

Taking the value of $\int_{0}^{\mathcal{T}} \bar{\chi}^T P \bar{\chi} (t)$ from Eq (3.5), we have

Since,

by choosing

and

we can conclude that

which implies the following equation:

As $g_3$ and $B$ belong to $L^2[0, t]$ with $t > 0$, $|V(t)| < \infty$ implies that sliding variables and trajectory tracking errors are both bounded. Utilizing the Lyapunov criteria, it can be stated that the system states are stable in the Lyapunov sense with convergence of trajectory tracking errors toward their equilibrium.

5.   Simulation

A simulation investigation is conducted on a 3-dof nonholonomic wheeled mobile robot to demonstrate the effective performance of the developed controller. The dynamic framework and parameters employed in this investigation for the nonholonomic wheeled mobile robot position tracking issue are taken from ^[24]. The non-holonomic constraint applied on mobile robot system is considered as:

The simulation study on the nonholonomic wheeled mobile robot is carried out using MATLAB. The ODE45 MATLAB solver is utilized to solve ordinary differential equations. For calculating the fractional-order derivative, the definition of the Grunwald-Letnikov (GL) derivative ^[25] has been used. For comparison with the existing control scheme, a non-singular terminal sliding mode type controller is chosen. The architecture of this controller is taken from ^[22], and during the simulation study, the name for this controller is written as existing controller. The case "proposed controller with $\Delta = 0$" represents the case in which the estimation of disturbances and parametric uncertainties is negligible, i.e., in this case simulation results are carried out with the adaptive fractional-order sliding mode controller without $H_\infty$ tracking control to show the robustness of the designed control scheme.

To show the efficiency of the proposed control scheme, the performance of the designed controller alongside the existing controller and proposed controller with $\Delta = 0$ for a nonholonomic wheeled mobile robot system is given in Figures 1–5. The position and velocity tracking errors are shown in Figures 1 and 2. From this figure, it can be shown that the trajectory tracking errors converge quickly with fewer fluctuations as compared to the existing controller, while for the case "proposed controller with $\Delta = 0$" a lot of fluctuations are seen that show the robustness of the designed controller as in the designed controller, for which the $H_\infty$ tracking control technique is utilized to handle these negative after effects. In Figure 3, the position tracking performance is given. This figure shows that for the proposed control approach, the dynamical system tracks the desired trajectory very efficiently by taking less time to reach the trajectory.

In Figure 4, velocity tracking performance is given, which shows that the reference velocity is achieved very smoothly. Control input torque for the proposed scheme is shown in Figure 5.

From these figures, we come to the conclusion that the proposed controller efficiently tracks the reference trajectory in a robust manner as compared to the existing controller.

Further, to compare the performance of the controller statistically, the $L^{2}$-norm is presented in tabular form by comparing these parameters with the existing controller. The formula used for the $L^{2}$-norm is

Table 1 presents the $L^{2}$-norm for position tracking errors for existing controller, proposed controller with $\Delta = 0$, and proposed controller comparatively that shows $L^{2}$-norm for proposed controller is very less. The lesser value of $L^{2}$-norm shows more efficient performance of the controller. Hence it can be concluded from $L^{2}$-norm that the performance of the proposed controller is enhanced as compared to the existing controller given in ^[22].

6.   Conclusions

In this article, an $H_\infty$-based adaptive fractional-order sliding mode controller was designed for the position tracking problem of nonholonomic wheeled mobile robots. In the designed controller, $H_\infty$-based trajectory tracking control with fractional-order sliding mode control and neural networks are utilized to enhance the performance of the system. The $H_\infty$ trajectory tracking performance enhances the performance of the dynamical system by compensating the negative aspects of external disturbances and parametric uncertainties in a robust way. To reproduce the non-linear dynamical part of the dynamical structure, the radial basis neural network was utilized. In order to analyze the convergence of tracking errors, the Lyapunov stability criterion was used to show the stability of the overall dynamical system. To show the efficiency and robustness of the presented controller, a simulation analysis was carried out by drawing comparisons to the obtained results with the existing techniques. The overall analysis including simulation results and statistical analysis, shows that the designed control method efficiently tracks the intended trajectory with a shorter reaching time.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article

Acknowledgments

The authors would express a special thank to the NIT Kurukshetra, Meerut College Meerut and MJPRU Bareilly for providing the necessary support for this research. The research work is financially supported by the Higher Education Department, U. P. Government under the Research and Development Scheme received vide letter No. 39/2024/242/sattar-04-2024-001-04(33)/2023.

Conflict of interest

There are no conflicts of interest.