Design, workspace analysis and inverse kinematics problem of Delta parallel robot

Parallel robots have a number of advantages over the traditional serial robots due to their particular architecture [1-10]. In this paper a structure of Delta parallel robot will be presented. An analysis about the kinematics problems of this structure will be made. Also, it will be presented the work space of Delta robot and it will be realized a GUI (Graphical User Interface) simulation of this structure in Matlab.


Introduction
Parallel robots have a number of advantages over the traditional serial robots due to their particular architecture [1][2][3][4][5][6][7][8][9][10].In this paper a structure of Delta parallel robot will be presented.An analysis about the kinematics problems of this structure will be made.Also, it will be presented the work space of Delta robot and it will be realized a GUI (Graphical User Interface) simulation of this structure in Matlab.

The description, parameterization and CAD model for Delta parallel robot
Geometric parameters for DELTA parallel robot arte presented in Table 1 and Fig. 1.This type of robot has in its structure three rotation motors positioned at 120 degrees one towards other.
The CAD model for Delta parallel robot was real-   L i , R i , l i , r i , q i , i = 1…3, mm Parameterization:

Inverse kinematics problem (IKP) for Delta parallel robot
Inverse kinematics problem results from the determination of angle values when the position of the characteristic point or the final effector (TCP -Tool Centre Point), respective the general coordinates: For the solution of angle the next equations will be used Variables , and will be used in final equation that will solve angle, the angle from the first link motor.For angle appears to be two solutions http://dx.doi.org/10.5755/j01.mech.17.3.506 Fig. 3 Representation of the two solutions for the first motor link of Delta parallel robot ( 4) and ( 5) As it can be observed two solutions are admitted for two possible position solutions for the first motor link (Fig. 3).From these two solutions it will be chosen the first solution because with this one the result of the equation is the correct value for our angle.The same procedure will be applied to compute next two angles of the other two motor links.
In conclusion, equations represent the , , analytic solutions for inverse kinematic models of Delta parallel robot.Table 2 presents the computing algorithm for inverse kinematic problem of Delta parallel robot with rotation actuators., , q q q (4), ( 5) Output variables 1 2 3 , , q q q (4), ( 5)

Workspace analysis for Delta parallel robot
One of the most important issues in the process of design of the parallel robots is to determine their workspace.For parallel robots, this issue may be even more critical since parallel robots will sometimes have a rather limited workspace.Various numerical methods for workspace determination of the parallel robots have been developed in recent years.
The following figures visualize the 3D robot workspace (Fig. 5, a -d).
The majority of numerical methods used for parallel manipulator workspace boundary determination typically rely on manipulator's pose parameter discretization.With the discretization approach, the workspace is envisioned as the uniform grid of nodes in a Cartesian or polar coordinate system.
Each node is then examined in order to determine whether it belongs to the workspace or not.The accuracy of the workspace boundary in this case depends on the sampling step, used to create the grid.

Conclusions
A description of the workspace of the parallel robot is provided based on the analysis of the robot.The kinematics and workspace analyses presented in this paper can greatly benefit the design, trajectory planning and control of such a parallel robot.For the simulation, we used an evaluation model from the Matlab/SimMechanics.Especially nonexperts will benefit from the proposed visualization tools, as they facilitate the modeling and the interpretation of results.ment of mobile minirobots for in pipe inspection tasks, Mechanika 6(68): 60-64.6.

S u m m a r y
The constant-translation workspace, which is the set of all feasible orientations of the mobile platform for a given position, was studied.The design of the robot concerns geometric parameters calculation of the robot so that the moving platform would be found within the given workspace.

Fig. 1 Fig. 2
Fig. 1 Geometric parameters for Delta parallel robot structure: a, b and c

Fig. 4
Fig. 4 Graphical user interface for solving the inverse kinematic problem of Delta parallel robot In Fig. 4 graphical user interface realized in Matlab for solving the inverse kinematic problem of Delta parallel robot is presented.In Table3is presented a set of numerical and experimental results obtained with help of the graphical user interface presented in Fig.4.The developed GUI im-

Table 2
Algorithm variables for solving inverse kinematic problem of delta parallel robot with rotation actuators

Table 3
Numerical solutions for inverse kinematic problem of delta parallel robot with rotation actuators Input values: Operational coordinates Output values: Angle values from motor links X, mm Y, mm Z, mm 1 q , º