Dynamic Analysis of Gough-Stewart Platform Manipulator

A novel derivation to evaluate all the controlled forces which cau se by the motors and effected along the prismatic joints on the legs of the Gough-Stewart platform manipulator based on the virtual wor k method is proposed in this paper. In this paper the manipulator can be c onsidered as a multibody mechanism with rigid elements. It can be assumed that the manipulator motion was known. The aim of the dyna mic analysis in this paper is to evaluate all the controlled forces which necessary to implement the manipulator programming motion


Introduction.
In general, Gough-Stewart platform manipulator is a six degree of freedom with two main bodies [3].The fixed body is called the base, while another body is regarded as movable and is called the moving plate.These two bodies are connected together by six extensible legs.In this paper we assumed that every leg of the legs of the manipulator is consists of two parts connected together with a prismatic kinematic joint (p).The prismatic joints are affected under the controlled forces which cause by the motors.All the legs are connected with the base by spherical kinematic joints (s) in the points i A , while they are connected with the moving plate by spherical joints with fingers in the points i B , as shown in Figure1.
Force and moment analysis for any robotic system are useful to choose the suitable motors for implementation the programming motion [5].Several methods are proposed to derive dynamic equations of the Gough-Stewart manipulator.
The motivation of this paper is to derive a mathematical formulation for evaluation all the controlled forces in the prismatic kinematic joints of the Gough-Stewart platform manipulator.

Position equation
In order to describe the motion of the moving plate of the manipulator relative to the base, we assumed that the moving plate attached to the base as shown in Figure 1, so the position vector of the point A i in the Global coordinate system can be written as: ( ) and the position vector of the point B i in the local coordinates system can be written as: ( ) The rotation of transformation between the local (moving plate) and the global coordinate systems can be represented by Euler's rotation matrix with ψ as pitch angle, θ as yaw angle and ϕ as a roll angle [1].Thus R 01 is a matrix of coordinate transformation from the local to the global coordinates can be written as: Generally, kinematic relations of manipulator are expressed by the loop equations.In Figure 1 It is assumed that, the geometrical parameters of the moving plate centre can be represented as: , and the legs vectors as: Thus the position equation will be written as:

Forces and moments effected on
the moving plate of the manipulator.

Gravity force.
It can be considered that 1 G is the gravity force vector of the moving plate in the Global coordinate system, so its projector on the Global coordinate is There is a moment that caused by the gravity force, its vector relative to the beginning of the local coordinate system can be given as:

Working force.
It can be assumed that the moving plate affected under a working force with a vector H P .This force will be caused a moment with a vector relative to the beginning of the local coordinate system { } H P M 01 .

Controlled forces(motors forces).
Every leg of the legs of the manipulator affected under a controlled force which effected along the prismatic joint (along the leg).This force causes by the motor i Q 4. Inertia forces and moments evaluation.
The formulation of the inertia force of the moving plate of the manipulator can be written as: PDF created with pdfFactory Pro trial version www.pdffactory.comW is the vector projector of the acceleration of the moving plate centre on the local coordinate system [7].Inertia moment of the moving plate relative to the beginning of the local coordinate system can be obtained as: In the above equation 1 C J -inertia tenser of the moving plate relative to its mass centre; 1 ω -moving plate angular velocity; 1 ε -moving plate angular acceleration and 1 C r -radius vector of the moving plate centre.Thus the projector of the inertia moment on the local coordinate system can be written as: Similarly the inertia forces and moments for the legs of the manipulator can be written as: But we assumed that the mass, inertia forces and moments of the legs is very little, so we will neglect them in the evaluations of the present work.

Virtual work evaluation.
The moving plate of the manipulator affected under the total force which consists of its weight, working force and inertia force.The total force vector is The total force causes a total moment relative to beginning of the local coordinate system as follow: Thus the summation of the virtual work of the moving plate by any little transformation and orientation of the moving plate can be obtained as: In the above equation PDF created with pdfFactory Pro trial version www.pdffactory.com Equation ( 9) can be written in a following form: (10) In the above equation ) If we assume that all the forces and moments effected on the moving plate can be written as: and the geometrical parameters of the moving plate centre can be represented as: the virtual work will be written as:

The controlled forces evaluation.
In the static equilibrium condition the virtual work represented in equation ( 12) equal to virtual work which causes by the controlled forces which can given in the following form [9] : From equation (1) we can derive the following relation: PDF created with pdfFactory Pro trial version www.pdffactory.comand, Thus the controlled forces (14) can be formulated as follows: From the above equation can evaluate all the controlled forces which effected on the prismatic joints of the manipulator.

Example of analysis:
In this example a manipulator with the following parameters is considered: a. Coordinates of points i A of the base (in the Global coordinates system) are as in the matrix: The results for all points are as explained in Table 2.
e.The platform mass is m = 15Kg, tenser of platform Inertia is and the forces and moments vector which applied to the platform is: f.The movement of the platform throw the track points (1,2,3,4,5,6) with acceleration and angular velocity of point P as are follow:  3.In this table it can be found that there is a singularity position in the point 4, it can be seen that, the forces are very high in this position.

Conclusions
In this paper, an approach for the dynamic analysis of the Gough-Stewart platform manipulator is proposed based on the virtual work.The mathematical simulation for the robot arms in this paper is a novel method and it has been derive by the authors.The stiffness of the arms caused by the motion of the platform was neglected.The joints of the manipulator's arms have been assumed as ideal joints (without friction forces).It has been proved that all the controlled forces can be evaluated without using the reactions in the kinematic joints .

.
The results of the motorized forces are as shown in the Table Dynamics analysis of a 3-DOF parallel manipulator with R-P-S joint structure".Journal of mechanisms and machine theory, Vol.42, Issue 5, 2007.p 541-557.

Figure ( 1 O
Figure (1) Position of the robot joints

Figure ( 2 )
Figure (2) The rotation of the axis

Table ( 1) Platform center coordinates and its angles of orientation
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