The Kinematic Model and the Simulation of an Omidirectional Mobile Robot

1. Introduction

Mobile robots represent one of the most important fields of research today, because of their great utility in various fields of work. A very efficient type of mobile robot is the omnidirectional one. It has a configuration as presented in Fig.1, consisting of three arms positioned at 120 degrees relatively to each other. At the end of each arm there is a wheel perpendicularly on it. Each wheel consists of two circles of roll modules that allow the wheel to slide in the direction of the arm (Fig. 2). The main advantage of this kind of mobile robot with respect to the classical configuration, that includes two fixed parallel active wheels and a mobile one, is that the omnidirectional one can move in any direction in the plane at any given moment, without being influenced by the current position of the robot.

Figure 1 – The robot configuration Figure 2 – The structure of the wheel

The classical configuration doesn’t allow the robot to move in any direction, in most cases a rotation being necessary before the desired movement could be performed. The only disadvantage of the omnidirectional model would be the structure that is a bit more complex, and that can result in a higher production cost.

2. The kinematic model

The kinematic model of the omnidirectional mobile robot will compute the rotation speed of each wheel in order for the center of the robot to move straight from one point to another, at a given velocity and with an added rotation of the entire robot around its axis with a given angle. In order to achieve the kinematic model, four orthogonal coordinates: three of them attached to the motor arm (XaYs for the QA arm; XbYb for the QB arm; XcYc for the QC arm) and a general one, that is XY, are used. All these systems originate in the intersection point of the three arms of the robot, the Q point in the figure.

In order to calculate the kinematic model of the robot, the following presumpsions were made:

a) in the X1Q1Y1 system the coordinates of the (x1,y1) points are known; (the parametrical

Figura 3 – The graphical layout for determing Figura 4 – Graphical help for calculating

the kinematic model of the robot the kinematic model

equations of point Q trajectory) the L length of the arm: QA=QB=QC; the r radius of wheels and the rotation angle θ, as can be seen in figure 4.

b) the following variants are unknown a1) the angular velocity ωa, ωb, ωc of the three wheels;

b1) the v1a, v1b, v1c velocities; c1) the v2a, v2b, v2c velocities.

In order to obtain the kinematic model, the following steps have to be made.

Step 1: The XQY system is rotated with 30o counterclockwise, until it becomes the new XaQYa system, obtaining equation (1). This equation can be projected on the X axis and respectively on Y axis, thus obtaining (2).

(1) (2)

After having completed the necessary operations, system (2) becomes

(3)

Step 2: The XQY system is rotated with 150o counterclockwise, until it transforms into the new XbQYb system, obtaining equation (4). This equation can be projected on the X and respectively on Y axis, thus obtaining (5).

(4) (5)

By doing the necessary calculations, system (5) becomes: (