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This paper presents energy reduction with anticontrol of chaos for nonholonomic mobile robot system. Anticontrol of chaos is also called chaotification, meaning to chaotify an originally non-chaotic system, and in this paper error of mobile robot system has been synchronized with chaotic gyroscope for reducing energy and increasing performance. The benefits of chaos synchronization with mechanical systems have led us to an innovation in this paper. The main purpose is that the control system in the presence of chaos work with lower control cost and control effort has been reduced. For comparison of proposed method, the feedback linearization controller has also been designed for mobile robot with noise. Finally, the efficacies of the proposed method have been illustrated by simulations, energy of control signals has been calculated, and effect of Alpha (

Wheeled mobile robots (WMRs) have been an active area of research and development over the past three decades [

The tracking control approaches for the mobile robots are mainly divided into six types: (1) feedback linearization [

Tracking control by using synchronization of a mechanical system with a chaotic system is presented in this paper and it is compared with error and with noise.

Many natural and man-made systems can be represented by means of a graph in mathematical terms, and these graphs are called complex networks. Most real-world industrial systems such as mobile robot are typical complex networks. Collective motions of complex networks have recently been the subject of considerable interest within the science and technology communities. Especially, one of the interesting and significant phenomena in complex dynamical networks is the synchronization of all dynamical nodes in a network [

Dynamic chaos is a very interesting nonlinear effect which has been intensively studied during the last three decades. Chaos control can be mainly divided into two categories [

For common tracking control, error converges to zero, but, generally, there are three choices for error: error converges to zero, error is synchronized with periodic signals, and error is synchronized with chaotic system.

In this paper, error of mobile robot system is synchronized with chaotic gyroscope for reducing energy and increasing performance. For comparison of proposed method, the feedback linearization controller is also designed for mobile robot with Gaussian and Rayleigh noises.

The paper is organized in the following manner. Nonholonomic mobile robot dynamics and its state-space representation and the constraint equations of the mobile robot are described in Section

The kinematic model of the mobile robot is given as follows:

Illustration of the error transformation where the following vehicle follows the path of the leading vehicle at distance

Assume that a feasible reference

In Figure

Differentiating (

Here, the tracking control problem is to find appropriate control laws for

Consider a “square” system (where the number of inputs is equal to the number of outputs =

Let

Let

Now, we obtain a decoupled set of equations:

Considering the kinematic model of mobile robot given by (

Block diagram of control system.

Let

The state has been extended:

Take

To obtain the control,

In previous section, error signals are converged to zero but in this section the error signals track the small ratio of amplitude of chaotic gyroscope system, as shown in Figure

Block diagram of control system.

Feedback linearization technique in previous section will be used for error system is given by (

Let

The state has been extended:

Take

To obtain the control,

The symmetric gyroscope mounted on a vibrating base is shown in Figure

A schematic diagram of a symmetric gyroscope.

This gyro system exhibits complex dynamics and has been studied by [

(a) Time series of

As in previous section, instead of chaos system, noise is entered to control block diagram. It means that error system instead of converging to zero, is faced with noise, as shown in Figure

Block diagram of control system.

System has two inputs and two outputs; in anti-control of chaos for desired references, the two states gyroscope system is used, and for noise, Gaussian noise and Rayleigh noise are the desired references for error system.

Equations are similar to Section

Control of the mobile robot dynamic model is applied to system with circle feasible reference (desired), and results are shown in Figure

(a) Control signals for circle reference and (b) path following.

Anti-control of chaos for the mobile robot dynamic model is applied to system with circle feasible reference (desired), and results are shown in Figure

(a) Control signals for circle reference for anti-control of chaos with

Control of the mobile robot with noise is applied to system with circle feasible reference (desired), and results are shown in Figure

(a) Control signals for circle wave reference for noise with

The major purpose of this paper is energy comparison between control signals. These control signals are for control of mobile robot and error system synchronization with chaos and control of error system with noise. In this paper, energy means square of signals integral (area under the curve); energy is calculated as follows:

The comparison of energy is shown in Table

Energy comparison.

Control | Feedback linearization control | Anti-control of chaos | Noise |
---|---|---|---|

Circle | 15683 | 592.0880 | 674.5611 |

In this section, we show the effect of Alpha (

Effect of Alpha (

In this paper, energy reduction with anti-control of chaos for nonholonomic mobile robot system was checked, and the mobile robot system was, controlled with three different ways and feedback linearization controller was used for controlling with circle feasible references. Error of mobile robot system was synchronized with chaotic system instead of zero tracking and was compared with noise instead of chaotic system. Energy improving for anti-control of chaos is visible. Finally, this mobile robot control with chaos synchronization also provides some new insights for controlling and the possible applications in the real-world engineering systems.