On the Dynamics of a Manipulator Robot System

ON THE DYNAMICS OF A MANIPULATOR ROBOT SYSTEM

D. BADOIU1

1Petroleum-Gas University of Ploiesti, ROMANIA,

Abstract: The paper presents some results concerning the dynamic study of a manipulator robot with a plane movement, which has two rotation component modules, when the elasticity of the links is taken into account. The movement equations are obtained by considering that the components of the links’ elastic deformations overlap the movement of the robot when the links are considered to be rigid. The simulation results are compared with some results obtained on an experimental robot system.

Keywords: robot, dynamics, elasticity, coordinates variation

1. THEORETICAL CONSIDERATIONS

At present, the robots with an open articulated structure, widely used in industrial applications, are characterized by low operating speeds and by the fact that the maximum weight that can be handled in good conditions is very small in comparison with the total weight of the robot. For these robots, their control system do not take into account the influence of the component links’ elasticity. In fact, for increasing the operating speeds and for an optimum design of the component links, it is necessary that the dynamic model corresponding to the mechanisms of these robots takes into account the links’ elasticity.

In this paper, some results concerning the dynamic study of a manipulator robot with a plane movement, which has two rotation component modules (fig. 1), when the elasticity of the links is taken into account, are presented.

Figure 1: Robot with two rotation modules

The movement equations are obtained by considering that the components of the links’ elastic deformations overlap the movement of the robot when the links are considered to be rigid. Finally, the simulation results are compared with some results obtained on an experimental robot system.

The system of equations corresponding to the movement of the robot when the links are considered to be rigid has the following form [1]:

(1)

where: , is the inertia matrix, is a vector, whose elements depend on the coordinates and and their derivatives with time, and is a vector, whose elements depend on the forces and moments that act on the robot’s structure. A detailed presentation of the expressions of the elements that appear into (1) is given in [1].

The elastic deformation of the component links is expressed depending on the displacements and the rotations corresponding to the end points of each link :

(2)

where: , and , represent the displacements along the axis and axis, respectively, and , are the rotations along the direction of the axis.

It is obvious that: ; ; .

The vector that contains all these displacements and rotations has the following expression:

(3)

The system of equations corresponding to the variation of the components of the vector , which overlap the movement of the robot on the whole, can be determined using the Lagrange formalism:

(4)

In the relation (4), is the total kinetic energy:

(5)

where: is the density of i link’s material and is the speed of a current point P on the i element.

The speed is given by: , where P’ is the position of the point P, when the elastic deformation of the link i is considered, and the vector can be calculated with the relation:

(6)

where: , are the rotation matrices ( and ) and the vector can be expressed with the following relation:

(7)

where: , are the matrices which contain the shape functions [4].

The relation (6) can be rewritten in the following form:

(8)

where: the vector depends only on and and:

(9)

where: the matrix verify the relation: , and the matrix is given by:

(10)

In the relation (4), U is the total potential energy corresponding to the elastic deformation of the component links (we assume that the movement of the robot is on a horizontal plane):

where: ;(11)

where: is the rigidity matrix [4].

The vector Q, in the relation (4), contains the generalized forces corresponding to the forces and moments which act on the component elements of the robot. If we consider only the motor moments and , then:

(12)

After the calculus is made, the system of equations (4) takes the following compact form:

(13)

where:

(14)

(15)

(16)

(17)

where:

(18)

The system of equations (13) was solved using the Newmark’s method [6].

2. NUMERICAL EXAMPLE

Starting from the method explained above, a computer program has been created. The simulation results have been compared with the measurements results obtained on an experimental system robot [2].

The elements of the experimental robot are of steel and have the following characteristics:

- the links’ length: ; ;

- the area of the transversal sections: ; ;

- the sectional inertia moments: ; ;

- the mass of the first acting motor, concentrated in : ;

- the mass of the second acting motor, concentrated in : ;

The mass of the manipulated object was 0,5 kg and its inertia moment on the direction of the axis (fig. 1) was: .

Next, some results are presented, when the motor moments are defined by the following expressions:

(19)

In the figures 2 and 3 the variation of the generalized coordinates and , when the links’ elasticity is considered, is represented. and correspond to the case when the links are considered to be rigid. With dot points are represented the experimental results.

Figure 2: The variation of the coordinate

Figure 3: The variation of the coordinate

3. CONCLUSIONS

The figures 2 and 3 emphasizes that there is a good concordance between the experimental results and the results obtained after simulations. The variation of the coordinates and (fig. 2) emphasizes the influence of the links’ elasticity on the movement of the robot system. The figure 3 shows that although the motor moment is equal to zero, the coordinate varies during the movement of the robot, due to the dynamic coupling between the component links.

REFERENCES

[1]Dombre E., Khalil W.: Modelisation et commande des robots, Ed. Hermes, Paris, 1988.

[2]Chedmail, P., Bardiaux, J.C.: Experimental validation of a plane flexible robot modelling, IFAC Theory of Robots, Viena, 1986.

[3]Badoiu D.: Analiza structurala si cinematica a mecanismelor, Ed. Tehnica, Bucuresti, 2001.

[4]Zienkiewicz, O.C.: The Finite Element Method in Engineering Science, McGraw-Hill, London, 1971.

[5]Craig J.J.: Introduction to robotics: mechanics and control, Addison-Wesley, 1986.

[6]Posea, N.: Calculul dinamic al structurilor, Editura Tehnica, Bucuresti, 1991.

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