Analytical determination of position of the delta cable system.

Stollmann, Vladimir ; Vacek, Vladimir

1. INTRODUCTION

Conception solution of Delta cable systems was created at the Faculty of Forestry of the Technical University in Zvolen. It represents the locomotive mechanism of the deltastats, which are a new technical means intended for transport and handling of bulky and heavy objects on large distances, mainly for automization and robotization of forest operations (Stollmann & Belanova 2006). A typical solution of the Delta cable system to application in forest area is depicted in the Fig 1, where:--poles with firmly fasten cables, 2--cable carriage, 3--hauling road.

[FIGURE 1 OMITTED]

2. KINEMATIC SCHEME

The Delta cable systems are characterized by parallel kinematical structure, which is in nowadays robotization technique considered a modern structural element (Smrcek & Hajtinger 1996, Barborak et al. 2006). The Delta systems enable navigation and at the same time an orientation of working floor in space and require the use of six cables, which are firmly fasten on three posts--see Fig 2, where 1--working floor.

For applying of the Delta cable systems in the sets of robots (deltastats) it is typical the gravitational establishment of the robotic working unit to vertical position. This enables to reduce the number of cables of the Delta cable system to there pieces see Fig 4.

[FIGURE 2 OMITTED]

3. STRUCTURE OF DRIVE

The Delta cable system represents a locomotive mechanism of the deltastats, which in its substance are airborne robotic systems. The structure of the De lta cable system drive is given in Fig 3, where: 1--control unit, 2--drive unit, 3--brake, 4 clutch, 5--drum, 6--length of cable sensing unit, 7--cable carriage.

Its characteristic sign is location of driving units and cable drums directly in the carriage.

The movement of cable carriage 2 (Fig 1) is performed by means of steel cables. The steel cables are firmly fastened to posts and they are wound on the drums, which are located in the cable carriage 7 (Fig 3).

[FIGURE 3 OMITTED]

4. DEFINING OF NAVIGATION TASK

Determining of the carriage position 2 (Fig 1) in the space is a key task for the need of navigation of a carriage to the target point. We can consider various methods of the task solution, out of which we are choosing the odometric method for its simple technical realization. In the case of the Delta cable systems, this method requires the measuring the length of cables, where we can use the measuring system consisting of measuring pulley and incremental readers (Sakal et al. 1999).

5. ANALYTICAL SOLUTION OF NAVIGATION TASK

[FIGURE 4 OMITTED]

Let presuppose that we known the lengths of cables [r.sub.1] [r.sub.2] [r.sub.3] at the point X and the coordinates of the points A, B, C--see fig 4. Our task is to determine the position of the carriage i.e. the coordinates of the points X. We are obtaining the searched solution as an intersection of three spherical surfaces. The equations of spherical surfaces are:

[(x - [x.sub.1]).sup.2] + [(y - y.sub.1]).sup.2] + [(z - [z.sub.1]).sup.2] = [r.sub.1.sup.2] (1)

[(x - [x.sub.2]).sup.2] + [(y - y.sub.2]).sup.2] + [(z - [z.sub.2]).sup.2] = [r.sub.2.sup.2] (2)

[(x - [x.sub.3]).sup.2] + [(y - y.sub.3]).sup.2] + [(z - [z.sub.3]).sup.2] = [r.sub.3.sup.2] (1)

By powering the equations (1), (2) and (3) we get:

[x.sup.2] - [2xxx.sub.1] + [x.sup.2.sub.1] + [y.sup.2] - [2yy.sub.1] + [y.sup.2.sub.1] + [z.sup.2] - [2zz.sub.1] + [z.sup.2.sub.1] = [r.sup.2.sub.1] (4)

[x.sup.2] - [2xxx.sub.2] + [x.sup.2.sub.2] + [y.sup.2] - [2yy.sub.2] + [y.sup.2.sub.2] + [z.sup.2] - [2zz.sub.2] + [z.sup.2.sub.2] = [r.sup.2.sub.2] (5)

[x.sup.2] - [2xxx.sub.3] + [x.sup.2.sub.1] + [y.sup.2] - [2yy.sub.3] + [y.sup.2.sub.1] + [z.sup.2] - [2zz.sub.3] + [z.sup.2.sub.1] = [r.sup.2.sub.3] (6)

By subtracting of the equations (4), (5) and (4), (6) we get:

2x ([x.sub.2] - [x.sub.1]) + [x.sup.2.sub.1] - [x.sup.2.sub.2] + 2y([y.sub.2] - [y.sup.1]) + [y.sup.2.sub.1] - [y.sup.2.sub.2] + + 2z ([z.sub.2] - [z.sub.1]) + [z.sup.2.sub.1] - [z.sup.2.sub.2] = [r.sup.2.sub.1] - [r.sup.2.sub.2] (7)

2x ([x.sub.3] - [x.sub.1]) + [x.sup.2.sub.1] - [x.sup.2.sub.3] + 2y ([y.sub.3] - [y.sub.1]) + [y.sup.2.sub.1] - [y.sup.2.sub.3] + +2z([z.sub.3] - [z.sub.1]) + [z.sup.2.sub.1] - [z.sup.2.sub.3] = [r.sup.2.sub.1] - [r.sup.2.sub.3] (8)

Let substitute:

[a.sub.1] = 2([x.sub.2] - [x.sub.1]) [a.sub.2] = 2([x.sub.3] - [x.sub.1]) (9)

[b.sub.1] = 2([y.sub.2] - [y.sub.1]) [b.sub.2] = 2([y.sub.3] - [y.sub.1]) (10)

[c.sub.1] = 2([z.sub.2] - [z.sub.1]) [c.sub.2] = 2([z.sub.3] - [z.sub.1]) (11)

[d.sub.1] = [r.sup.2.sub.1] - [r.sup.2.sub.2] - [x.sup.2.sub.1] + [x.sup.2.sub.2] - [y.sup.2.sub.1] [y.sup.2.sub.2] - [z.sup.2.sub.1] - [z.sup.2.sub.2] (12)

[d.sub.2] = [r.sup.2.sub.1] - [r.sup.2.sub.3] - [x.sup.2.sub.1] + [x.sup.2.sub.3] - [y.sup.2.sub.1] [y.sup.2.sub.3] - [z.sup.2.sub.1] - [z.sup.2.sub.3] (13)

Equations (7), (8) can be written as:

[a.sub.1] x + [b.sub.1] y + [c.sub.1] z = [d.sub.1] (14)

[a.sub.2] x + [b.sub.2] y + [c.sub.2] z = [d.sub.2] (15)

If we expressed the variables y, z in the equations (14), (15) by the variable x then we have:

[b.sub.1] y + [c.sub.1] z = [d.sub.1] - [a.sub.1]x (16)

[b.sub.2]y + [c.sub.2]z = [d.sub.2] - [a.sub.2]x (17)

This system of linear equations (16), (17) can be solved by the Cramer's rule where D, [D.sub.y] [D.sub.z] are determinants (Matejdes.2005):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

Let denote:

[q.sub.2] = [c.sub.2][d.sub.1] - [c.sub.1][d.sub.2]/[b.sub.1] [c.sub.2] - [b.sub.2][c.sub.1] [k.sub.2] = [a.sub.1][c.sub.2] - [a.sub.2][c.sub.1]/[b.sub.1][c.sub.2] - [b.sub.2][c.sub.1] (19)

Then equation (18)can be rewritten as:

y = [q.sub.2] - [k.sub.2]x (20)

Similarly:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

z = [q.sub.3] - [k.sub.3]x (23)

By substituting relations (20) and (23) into the equation (1) we obtain following quadratic equation with one variable x:

[(x - [x.sub.1]).sup.2] + ([q.sub.2] - [[k.sub.2] x - [y.sub.1]).sup.2] + ([q.sub.3] - [[k.sub.3]x - [z.sub.1]).sup.2] = [r.sup.2.sub.1] (24)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

after modification:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

Let denote:

a = 1 + [k.sup.2.sub.2] + [k.sup.2.sub.3] (27)

b = -2[x.sub.1] - 2[k.sub.2][q.sub.2] + 2[k.sub.2][y.sub.1] - 2[k.sub.3][q.sub.3] + 2[k.sub.3][z.sub.1] (28)

c = [x.sup.2.sub.1] + [q.sup.2.sub.2] -2[q.sub.2][y.sub.1] + [y.sup.2.sub.1] + [q.sup.2.sub.1] +2[q.sub.3][z.sub.1] + [z.sup.2.sub.1] - [r.sup.2.sub.1] (29)

Then the equation (26) can be expressed in the form:

a[x.sup.2] + bx + c = 0 (30)

Two solutions of this equation are given by the formula:

[x.sub.1,2] = -b [+ or -] [square root of [b.sup.2] - 4ac]/2a (31)

The answer to the navigation task has two solutions. The x coordinate of these points is given by the relation (31) and their relating coordinates y, z are determined from the equations (20) and (23).

6. CONCLUSION

In the article is described an analytical solution of navigation task of the Delta cable systems by odometric method. The merit of navigation by this method is its simplicity, reliability, including its independence from satellite systems. The results are two solutions. The both are real in the practice and have a practical meaning at the deltastats operations. We suppose that certain disadvantage of their use can be their lower accuracy. Therefore, in the future we would like to focus on its evaluation.

The contribution was elaborated within the research project VEGA No. 1/3523/06 The research of new technical and technological principles for wood concentration and research project KEGA No. 3/6448/08.

Translation: PhDr. Olga Lejsalova, CSc.

7. REFERENCES

Barborak O. ; Liptak P., & Jozefek M. (2006). Robotization one of the key trends of expert preparation in specialized machime amdproduction technology. TU Kosice, ISBN 808073-560-3, pp. 35-98

Matejdes, M. (2005): Applied Mathematics. Zvolen: Matcentrum, ISBN 80-89077-01-3, pp. 558

Sakal, P.; Bozek, P.; Nemlaha, E. (1999): Numeric control systems (in Slovak). Tripsoft Trnava, ISBN 80-968294-1-6,. pp.154

Smrcek, J.; Hajtinger, K. (1996). The analysis of the robots implementation possibilities in the non-standard environment. In: Int. Cong. "MATAR 96--Machine Tools, Automation and Robotics in Mechanical Engineering", Praha, ISBN 80-364-2568-6, pp. 185-189

Stollmann,V.; Belanova,K. (2006). Deltastats (in Slovak). Patent application PP 0079/2006, Urad priemyselneho vlastnictva Banska Bystrica