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This paper presents a predictive control of omnidirectional mobile robot with three independent driving wheels based on kinematic and dynamic models. Two predictive controllers are developed. The first is based on the kinematic model and the second is founded on the dynamic model. The optimal control sequence is obtained by minimizing a quadratic performance criterion. A comparison has been done between the two controllers and simulations have been done to show the effectiveness of the predictive control with the kinematic and the dynamic models.

In recent years, researches on mobile robots have increased significantly due to their flexible motions. With the complex environment for robots applications, higher requirements are put forward on the robots movements’ forms and movements’ abilities. Omnidirectional mobile robots compared with two-wheeled and four-wheeled mobile robots [

In the other side, there are many methods to address the optimal control law of the mobile robot. In [

Trajectory tracking of mobile robots and path following are two of the major problems of motion control of autonomous vehicles as it can be seen in [

MPC (model based predictive control) techniques are nowadays pretty popular in academic as well as in industrial control engineering [

Nevertheless, for computational reasons, MPC applications widely have been narrowed to linear models; consequently, in context of mobile robot, most model based predictive controllers utilize a linear model of robot’s kinematic to predict future system outputs [

This work includes an omnidirectional mobile robot modeling using the kinematic and the dynamic of the robot. We develop two predictive controllers based on, respectively, the kinematic and dynamic models.

The rest of this paper is organized as follows. In Section

Figure

The model of the three-wheeled mobile robot.

Due to structural symmetry, the vehicle has the property that the center of geometry coincides with the center of mass.

The parameter_{1},_{2},_{3} denote the translational velocities of each wheel.

Wheels are numbered 1, 2, and 3 and are assigned clockwise with the angle between any two neighboring wheels being

We then linearize the system model by calculating an error model with respect to a reference trajectory and we develop the kinematic model in Taylor series around the point

The derivation of the dynamic model is begun by the application of Newton’s second law:

We have the following relation between

In addition, the dynamic characteristics of the driven system of each wheel are given by the following [

The inverse kinematic equations for the mobile robot with respect to the mobile robot frame that relates

We have to formulate (

To transform the previous equation system to a state space model, we take the following state variables:

Defining

The discrete model has the following equation:

The basic idea of predictive control consists in calculating, at each sampling instant, a control sequence on a prediction horizon aimed at minimizing a quadratic cost function. The control algorithm is based on the following:

The use of a model to predict, on a future horizon, the output of the process

Computing the control sequence which minimizes a performance criterion which involves a sequence of the predicted output

We define the error vectors;

We first introduce these vectors:

We write from (

Using the vectors (

Introducing the relation of (

The resolution of (

The aims of the simulations are to examine the effectiveness and performance of the predictive controllers based on the proposed kinematic and dynamic models. We have applied the above control strategy to the omnidirectional mobile robot. It was assumed that the control sampling period is 100 ms. The physical parameters of the mobile robot are as follows:

We use the state variables at each sample time to calculate the state matrices of the robot because they change at each sample time depending on the state variables. We discretize it at each sample time; we have then an online variation of the model. We use the model predictive control to track the desired trajectory. We used here the prediction horizon

The translational velocity of the robot in the absolute coordinate system is

So that we note that the omnidirectional mobile robot can separately achieve the translational motion and the rotational motion around the gravity center in the two-dimensional plane. To confirm this holonomic property we have fixed the desired rotation angle

The path following is about following a predefined path which does not involve time as a constraint. Thus, the important thing is to follow the path and to reach the goal. The speed of the robot is a minor thing. On the contrary, trajectory tracking involves time as a constraint.

The path following, the rotation angular, and the linear velocity are presented in Figures

Lemniscate path following with the predictive control joined with the kinematic model.

Linear velocity of the kinematic predictive control.

Rotation angle with the kinematic predictive control.

Lemniscate path following with the predictive control joined with the dynamic model.

Linear velocity with the dynamic predictive control.

Rotation angle with the predictive dynamic control.

Figure

We used the cubic splines with a bang-bang speed profile and the trajectory reference would have these formulae:

The

The

Linear velocity in the kinematic predictive control.

Rotation angle in the case of kinematic predictive control.

The

The

Rotation angle in the case of dynamic predictive control.

Linear velocity in the case of dynamic predictive control.

The aim of the simulation results is to examine the effectiveness and performance of the proposed predictive control with the kinematic model and the dynamic model of the omnidirectional mobile robot. So we proposed two cases: the path following with no constraint of time and the trajectory tracking with a fixed time. The simulation results show that the predictive control is capable of steering the omnidirectional mobile platform to exactly track the desired cubic splines trajectory and follow the lemniscates path. Although the kinematic model is easier in formulation, the dynamic model uses more details that makes it better than the kinematic model since it is more reliable, fast, and exact.

This paper has presented an application of the model predictive control to solve problem of trajectory tracking and path following of omnidirectional mobile robot. Two different models are considered: the kinematic model and the dynamic model. Control algorithm has been applied for the path following using lemniscates path and trajectory tracking using cubic splines with a bang-bang velocity profile. These two cases are performed by using simulation program. The results show the effectiveness of the dynamic model in comparison with the kinematic one to track the trajectory and path without posture errors and to converge to the desired control value (linear velocity) and the desired rotation angle rapidly and without oscillations.

There are no conflicts of interest in this work.