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cand. tech. sci. Semakhin A.M.

Kurgan State University, Russia

THE MATHEMATICAL DESCRIPTION OF THE TRAJECTORY OF MOVING GRIPPING DEVICE OF INDUSTRIAL ROBOT

 

The analysis of production of the enterprises of mechanical engineering and instrument making with small-scale type of manufacture has shown, that time of processing of a detail makes no more than 10 % from time of manufacturing. The rest of the time is necessary on processes of loading-unloading, moving and storage /1/.

Reduction of auxiliary time is a topical problem.

The industrial system in structure of the numerically controlled machine tool, the industrial robot for loading-unloading of the basic process equipment, work-storage unit and workpiece storage is investigated. It is necessary to develop the mathematical description and to execute calculation of a trajectory of moving gripping device of the industrial robot in view of obstacles of a working zone of the NC machine tool.

For the description of movement gripping device of the industrial robot in base system of coordinates representation Denavit-Hartenberg, consisting in formation of a homogeneous matrix of transformation is used. It allows to transform consistently coordinates of gripper from the system of readout connected with last link, to base system of coordinates /2/.

The homogeneous matrixes of transformation connecting i and i-1 look like system of coordinates.

For a rotary joint

                                               (1)

For a forward joint

          (2)

where  - a homogeneous matrix of elementary turn of axis  on corner .

 - homogeneous matrix of shift along axis  on distance .

 - homogeneous matrix of elementary shift along axis  on distance .

 - homogeneous matrix of elementary turn around of axis  on corner .

For transition from  systems of coordinates to  expression is fair

                                                                  (3)

where  - radius-vector of the point set in system of coordinates  of a part, .

 - matrix of transformation  of a link.

As any point  it is accepted the material point , expressing gripping device of the industrial robot.

Let's consider a trajectory of movement of material point  during the initial moment of time . Process of transition of system of coordinates  in system of coordinates  is expressed in the form of the equation

                                                                           (4)

where  - product of matrixes of transformation from the previous link to the subsequent link, since 0 link and finishing n link.

 - radius-vector of a point n à link.

Differentiating the equation 4 on , we receive the general differential equation describing possible trajectories of moving the gripper of the industrial robot with a detail concerning coordinate system .

         (5)

where  - the radius-vector defining position of a point .

For rotary and forward pairs

;                                                                                   (6)

The equation 5 is the generalized differential equation describing a trajectory of movement gripping device of industrial robot.

For maintenance of a unobstructed conclusion gripping device of industrial robot from a zone of loading of the basic process equipment the determinant pays off /3/.

                                                     (7)

where  - determinant of private derivatives  and  on X, Y.

 - determinant of private derivatives  and  on Y, Z.

 - determinant of private derivatives  and  on Z, X.

The determinant  is calculated under the formula

                                           (8)

where , - private derivatives  on X, Y. :  - surface of safety (elliptic paraboloid) zone of loading of the NC machine tool.

,  - private derivatives  on X, Y. :  - the guiding plane.

If determinant  industrial robot cannot deduce gripping device with a detail from a zone of loading of the NC machine tool on a curve received by crossing  and . At  it is impossible to work out the differential equation which decision coincides with crossing  and . If determinant  gripping device can move on crossing surfaces  and .

The differential equation of a curve on which moves the gripping device of industrial robot looks like

                                                (9)

 

The Cauchy problem is solved for definition of the trajectory of moving gripping device of industrial robot. The Cauchy problem is solved by Runge-Kutta method of the fourth order.

 

References:

 

1. V.A.Egorov, V.D.Luzanov, S.M.Shcherbakov Transport store systems for flexible manufacturing systems. - L.: Mechanical engineering, 1989. - 293 p.

2. K. S. Fu, R.C. Gonzalez, C. S. G. Lee Robotics: Control, Sensing, Vision and Intelligence. –M. Mir, 1989. – 624 c.

3. Pulatov S.I., Madrahimov Z.T. Models and methods of functional-parametrical interaction of the industrial robot with the processing equipment in structure of flexible manufacturing systems. - Tashkent, 1990. - 50 p.