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Several research studies are conducted based on the control of wheeled mobile robots. Nonholonomy constraints associated with wheeled mobile robots have encouraged the development of highly nonlinear control techniques. Nonholonomic wheeled mobile robot systems might be exposed to numerous payloads as per the application requirements. This can affect statically or dynamically the complete system mass, inertia, the location of the center of mass, and additional hardware constraints. Due to the nonholonomic and motion limited properties of wheeled mobile robots, the precision of trajectory tracking control is poor. The nonholonomic wheeled mobile robot tracking system is therefore being explored. The kinematic model and sliding mode control model are analyzed, and the trajectory tracking control of the robot is carried out using an enhanced variable structure based on sliding mode. The shear and sliding mode controls are designed, and the control stability is reviewed to control the trajectory of a nonholonomic wheeled mobile robot. The simulation outcomes show that the projected trajectory track control technique is able to improve the mobile robot’s control, the error of a pose is small, and the linear velocity and angular speed can be controlled. Take the linear and angular velocity as the predicted trajectory.

A mobile robot is a dynamic nonlinear device, with the benefits of unmanned driving and different controlling practices [

Pappalardo and Guida [

Ernesto et al. [

The parametric uncertainties in the actuator were approximated by the Arab and Fateh [

A large number of application domains extending from surveillance, planetary exploration, patrolling, emergency rescue operations, reconnaissance, petrochemical applications, industrial automation, construction, entertainment, museum guides, personal services, intervention in extreme environments, transportation, medical care, and other industrial and nonindustrial applications are making use of wheeled mobile robots since the past two decades. Each application faces unique limitations on the robot design procedure which is reduced by selecting the most convenient drive configuration interpreting all the compromises.

The shear feature and sliding mode controller were established in accordance with the sliding mode control theory in accordance with the study of the mobile robot kinematic model. The control stability is studied to diminish the posing fault, and the trajectory of nonholonomic mobile robot control is carried out.

The trajectory tracking problem of nonholonomic WMR is to propose a virtual mobile robot model which can perfectly track the ideal space trajectory and then let the actual robot track. Depending on the spatial coordinates, the desired movement of the nonholonomic robot can be expressed [

Representation of nonholonomic wheeled mobile robot.

Figure

As a characteristic nonholonomic control scheme, the motion of WMR is not only limited by the basic requirements of starting point and ending point but also meets the special requirements of velocity direction or trajectory shape. In addition, WMR is an essentially nonlinear under actuated drift-free dynamic system, which brings greater challenges to its control [

The motion behavior of the mobile robot is constrained by kinematics, which is determined by the structural characteristics of the robot and its motion model. In practical applications, the system is always subject to various constraints, such as geometric constraints and motion constraints, steady and unsteady constraints, double-sided constraints and single-sided constraints, integrable constraints, and nonintegrable constraints [

where

If this form of constraint can be transformed into a form:

then the constraint form is integrable. Even if

A nonholonomic constraint is defined as a time derivative including variables (coordinates) in the generating system and cannot be integrated. Nonholonomic constraint is a characteristic of the mechanical system. When the dimension of the control variable is lower than the state variable, the control variable can control the state variable [

If the constraint of Equation (

Mobile robots through nonholonomic components are called nonholonomic mobile robots. It belongs to the category of nonholonomic constraints, and its nonholonomic constraints mainly affect the motion characteristics of nonholonomic systems. The nonholonomic constraints received by wheeled mobile robots are generally expressed as first-order differential equations.

In the plane, the position and attitude of the robot can be represented by the following state vectors:

To keep the balance in the plane, this type of robot usually has one or more castors as additional support only. The driving control is completed by motors which are separately distributed in two wheels. In the universal organize scheme, the motion equation of the robot can be formulated as:

Its nonholonomic motion constraints are as follows:

Equation (

where

There are many ways to make the robot move to the ideal position by using the plane changing trajectory. The inverse kinematic challenge is easier for the robot to plane the target trajectory

The positive and negative signs represent the ideal driving direction (+ is the forward direction, and - is the reverse direction). Every point on the track has a tangent angle defined as follows:

where

By using the Equations (

When the system does not satisfy the linear parameterization condition or when the system model has both structural and parametric uncertainties, a sliding mode control algorithm can be used. If the upper bound function can be found for the unknown dynamic characteristics, the sliding mode control algorithm can achieve the desired control effect by strengthening the control input, that is, using very strong control variables to deal with the unknown dynamic characteristics. Sliding mode control has strong control performance for nonlinear systems affected by uncertain dynamics, which makes the system move according to the designed sliding mode trajectory. The control method has the advantages of a simple calculation method and strong anti-interference ability.

Sliding mode control for nonlinear systems influenced by unknown factors is an easy and efficient way of control. The problem can be reduced to a complex one by simple sliding mode control. The nonlinear method for the single input output can be expressed as follows:

In order to make the trajectory tracking error

Among them,

where

The strategy of sliding mode controller mainly comprises 2 parts: the equivalent control variable

Design

The sliding mode control law

For uncertain systems, using the sliding mode variable structure control method can often get the desired results. Compared with the traditional control algorithm, the sliding mode production variable structure control method is discrete; that means, the high-frequency switch control is used to switch between several control laws, which can make the system oscillate with small amplitude and high frequency near the set space surface. Such motion is defined as sliding mode motion. Since various control laws in the variable control method for sliding mode are set in advance and are not disturbed and do not rely on device parameters, this method is highly robust. Then, the concept of sliding mode control combines a trajectory tracking controller with global asymptotic stability.

Any

Based on Equation (

It is concluded that as long as

Through the design of sliding mode controller,

To weaken the jitter, the exponential reaching law

Suppose

From Equation (

When

Because the sliding mode variable structure control requires that the state quantity of the motion on the sliding surface must be the end point, when the moving point reaches the sliding surface, it must meet the following requirements:

By deriving Equation (

Because

The Lyapunov function candidate be

When

The time derivative of

Additional universalization of

So,

The time to range

When

The term

By Equations (

Equation (

Yet over,

Correspondingly, the time to influence

Equations (

Considering that line and circle are common expected trajectory forms in practical application, this paper selects linear and circular trajectories to simulate and verify the above tracking control law and carries out the corresponding trajectory tracking controller simulation for nonholonomic WMR. The excessive acceleration caused by excessive speed will destroy the nonholonomic constraints of the robot, which is easy to cause the lateral movement or longitudinal sliding of the mobile robot. To avoid the above problems, the speed of the mobile robot must be limited. The maximum speed is 1.5 m/s, and the extreme angular velocity is 1.5 rad/s. The experiments of WMR are considered as follows: the hardware components for the robotic stages are nominated as ^{2}, ^{2}, ^{2}, ^{2}, ^{2}, and

When the trajectory is a straight line, the control input of the mobile robot is selected as follows: the primary linear velocity is 0.4 m/s, the angular velocity is 0.3 rad/s, and the controller parameters are set to 1.5. The simulation results are shown in Figures

Track tracking effect.

Change curve of pose error.

Linear velocity variation curve.

Variation curve of angular velocity.

It can be seen from Figure

When the trajectory is circular, the control input of the mobile robot is selected as follows: the primary linear velocity is 1 m/s, the angular velocity is 1 rad/s, and the controller parameters are set to 1. Figures

Track tracking effect.

Change curve of pose error.

Linear velocity variation curve.

Variation curve of angular velocity.

From Figures

In summary, it can be seen from the results of the above simulation of two tracking types on a nonholonomic mobile robot that the nonholonomic mobile robot system can progressively pursue the reference trajectory under the effect of a designed trajectory tracking law, the position error can converge to zero quickly, and the input of the speed control can converge also to the referee quickly. The efficacy of the proposed law on control is verified.

In this paper, the nonholonomic WMR trajectory control system based on improved variable structure of sliding mode is projected. Based on analyzing the principle of sliding mode control, the improved method is designed by adding stability analysis to reduce the pose error of trajectory tracking control. The experimental results verify the high control precision effect of the designed method and provide a basic theoretical reference for practical application.

Data are available on request

The authors declare no conflicts of interest.