# Aspects concerning virtual models for a double hexapodal platform.

1. INTRODUCTION

The Stewart-Gough platforms and their kinematics are well-known --they were used in 1965 in the field of flight simulators and then in different fields, such as aerospace--for the orientation of the telescopes, satellite antennas; surgery (interventions on brain, eye, etc.), micro machining, micro and nano technologies (the platforms with piezo actuators), micro manipulations (cellular biology applications, genetics). The direct and inverse kinematics of the platform together with new calibration methods were developed (Dietmaier, 1998); (Jakobovic & Jelenkovic, 2002); (Xiao-Shan et al., 2005), with the aim of motion controlling and improving precision. A certain drawback of this kind of structure is the limited operating space. To extend it, the superposition of two or more hexapodal platforms, obtaining so staged structures with combined kinematics, parallel--serial kinematics is proposed. Structures composed by several modules were carried out, but especially with three legs per module, being easier to be controlled. The double hexapodal robot ROBEX is based on two Stewart-Gough platforms, having specific constructive parameters, an assembling procedure making it modular and having a particular condition imposed in order to reduce the redundancy of the system.

The main goal of this paper is to develop kinematics for ROBEX, a system with extended operating space, but less robust and accurate. Virtual models and control software development are also presented. Medical and especially surgical applications are foreseen.

Further research will concern with the operating space analysis and with developments for "n" modules, based on miniaturized hexapods, using miniature joints and special actuators.

2. THEORETICAL CONSIDERATION

In order to extend the mobility of a single hexapod, a structure with two staged modules was proposed, as in Fig. 1. To control this structure, 12 parameters are necessary: the lengths of the 12 legs. For a structure with n modules, we have 6n degrees of freedom (DOF)--the robot is of "snake" type and it is hyper redundant.

This creates difficulties regarding the control: to establish the optimal configuration of the whole structure from an infinite number of possible configurations and practical difficulties too: to simultaneously control a large number of axes.

Taking into account these reasons, a simplified control method was proposed: to use the same set of control parameters for all hexapods, they having in this way identical configurations. This leads to six DOF, regardless the number of the hexapods. They are disposed on an arc of circle in plane or on a helix in space. This supplementary condition introduces an important limitation, because the obstacle avoidance isn't possible, but in certain applications this requirement isn't necessary. The next computation is carried out for two identical sizes hexapods.

[FIGURE 1 OMITTED]

To define the platform position three parameters are used: coordinates x, y, z of the platform centre; to define the orientation of the platform, three independent angles (Euler's angles) were introduced:

--[psi]--the rotation angle around the fixed vertical axis

--[theta]--the tilt angle of the platform around a horizontal axis that rotates itself with angle [psi]

--[phi]--the proper rotation angle of the platform around the axis that passes through its centre and is perpendicular on it.

In order to establish the vertexes coordinates of the hexagons [B.sub.1i] and [B.sub.2i], i = 1, .., 6, that are not necessary regular hexagons, a matrices method is used, [V] being the positioning vector, and [[psi]], [[theta]], [phi] the elementary rotation matrices (Dudita et al., 1989); (Vukobratovic, 1989).

The next coordinates systems are defined (Fig. 1): the fixed system [O.sub.0][x.sub.0][y.sub.0][z.sub.0], the system [O.sub.1][x.sub.1][y.sub.1][z.sub.1], linked to the intermediate mobile platform 1 and the system [O.sub.2][x.sub.2][y.sub.2][z.sub.2], linked to the final mobile platform 2.

Starting from the position and the orientation of the platform 2, described by its centre coordinates [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in the system [O.sub.0][x.sub.0][y.sub.0][z.sub.0] and the orientation angles [[psi].sup.0.sub.2] = [PSI], [[theta].sup.0.sub.2] = [THETA], [[phi].sup.0.sub.2] = [PHI] in the same system, the analogue parameters have to be determined for:

--the intermediate platform 1 in the system [O.sub.0][x.sub.0][y.sub.0][z.sub.0]: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], respectively [[psi].sup.0.sub.1], [[theta].sup.0.sub.1], [[phi].sup.0.sub.1]

--the final platform 2 in the system [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], respectively [[psi].sup.1.sub.2], [[theta].sup.1.sub.2], [[phi].sup.1.sub.2].

For a given point P we have the next relation between its coordinates in vector representation [[V.sup.0.sub.P]], [[V.sup.1.sub.P]], [[V.sup.2.sub.P]] in the systems [O.sub.0][x.sub.0][y.sub.0][z.sub.0], [O.sub.1][x.sub.1][y.sub.1][z.sub.1], respectively [O.sub.2][x.sub.2][y.sub.2][z.sub.2]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

For identical configurations, imposed by the supplementary condition mentioned above, we have identical distances and identical angles.

The final relation results:

[[psi]] x [[theta]] x [[phi]] = [square root of [[PSI]] x [[THETA]] x [[PHI]]] (4)

Using the same method, the relation (4) can be generalized for a system with n modules:

[[psi]] x [[theta]] x [[phi]] = n[square root of [[PSI]] x [[THETA]] x [[PHI]]] (5)

The coordinates of the points [B.sub.1i] and [B.sub.2i], i = 1, .., 6 are then determined:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

From the relations (6) and (7) the lengths of the legs are deduced, namely the control parameters.

3. MODELLING AND CONTROL

The control and simulation software of the ROBEX was performed in LabVIEW. In Fig. 2, the default configuration and a configurations defined by the angles [[PSI]], [[THETA]], [[PHI]]] of the double hexapodal structure are shown (in x-y projection in the left graph and in x-z projection in the right graph). Complex computations are required to obtain the above representations. The inverse kinematics relations involving matrices and vectors were transposed in application software performed in the graphical development environment LabVIEW.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

4. THE EMBODIMENT OF THE MODEL

The construction of ROBEX is based on the LM 1247 linear electrical actuator produced by FAULHABER having 6 urn theoretical resolution. Special designed connecting elements are used for modular assembling; another function of these elements is to assure the co planarity between the centres of the universal joints from the base platform of the upper module and of the universal joints from the mobile platform of the lower module (fig. 3).

The functional model, in a simplified representation, contains: the mechanical structure, the actuating section, including the power supply and the external controller (a PC or equivalent) and the control software--a LabVIEW application. The controller generates the positioning algorithms and the motion paths, which are transmitted to the actuating section, based on 12 LM1247-020-01 motors and MCLM 3003/06S motion controllers that carry out the displacement of the mechanical part. The PC is linked to the actuating section by a RS232 serial interface, using a set of controls in ASCII format.

5. CONCLUSION

In this article the inverse kinematics for a double hexapodal structure is presented, the control parameters being obtained. The double structure assumes that both modules have identical configurations, hence the same control parameters. Using the LabVIEW features, graphical representations of the systems motion are also presented. The virtual models will be used for the physical embodiment, the project ROBEX, grant 71-024/2007 funded by ANCS / CNMP being in course.

6. REFERENCES

Dietmaier P. (1998). The Stewart-Gough Platform of General Geometry Can Have 4[degrees] Real Postures Available from: http://www.mech.tugraz.at Accessed: 2007-10-25

Dudita, F.; Diaconescu, D. & Gogu, G. (1989). Mecanisme articulate. Inventica. Cinematica--Articulated mechanisms. Inventics. Kinematics, Editura Tehnica--Technical Publishing House, ISBN 973-31-0119-2, Bucuresti

Jakobovic D. & Jelenkovic L. (2002). The Forward and Inverse Kinematics Problems for Stewart Parallel mechanisms, Available from: http://www.zemris.fer.hr Accessed: 2006-05-26

Vukobratovic M. (1989). Applied Dynamics of Manipulation Robots, Springer Verlag, ISBN 3-540-51468-6, Berlin Heidelberg New York

Xiao-Shan, G.; Lei, D.; Qizheng, L. & Gui-Fang, Z. (2005). Generalized Stewart-Gough platforms and their direct kinematics, IEEE Transaction on Robotics, Vol. 21, No. 2, April 2005, 141-151, 10.1109/TRO.2004.835456

The Stewart-Gough platforms and their kinematics are well-known --they were used in 1965 in the field of flight simulators and then in different fields, such as aerospace--for the orientation of the telescopes, satellite antennas; surgery (interventions on brain, eye, etc.), micro machining, micro and nano technologies (the platforms with piezo actuators), micro manipulations (cellular biology applications, genetics). The direct and inverse kinematics of the platform together with new calibration methods were developed (Dietmaier, 1998); (Jakobovic & Jelenkovic, 2002); (Xiao-Shan et al., 2005), with the aim of motion controlling and improving precision. A certain drawback of this kind of structure is the limited operating space. To extend it, the superposition of two or more hexapodal platforms, obtaining so staged structures with combined kinematics, parallel--serial kinematics is proposed. Structures composed by several modules were carried out, but especially with three legs per module, being easier to be controlled. The double hexapodal robot ROBEX is based on two Stewart-Gough platforms, having specific constructive parameters, an assembling procedure making it modular and having a particular condition imposed in order to reduce the redundancy of the system.

The main goal of this paper is to develop kinematics for ROBEX, a system with extended operating space, but less robust and accurate. Virtual models and control software development are also presented. Medical and especially surgical applications are foreseen.

Further research will concern with the operating space analysis and with developments for "n" modules, based on miniaturized hexapods, using miniature joints and special actuators.

2. THEORETICAL CONSIDERATION

In order to extend the mobility of a single hexapod, a structure with two staged modules was proposed, as in Fig. 1. To control this structure, 12 parameters are necessary: the lengths of the 12 legs. For a structure with n modules, we have 6n degrees of freedom (DOF)--the robot is of "snake" type and it is hyper redundant.

This creates difficulties regarding the control: to establish the optimal configuration of the whole structure from an infinite number of possible configurations and practical difficulties too: to simultaneously control a large number of axes.

Taking into account these reasons, a simplified control method was proposed: to use the same set of control parameters for all hexapods, they having in this way identical configurations. This leads to six DOF, regardless the number of the hexapods. They are disposed on an arc of circle in plane or on a helix in space. This supplementary condition introduces an important limitation, because the obstacle avoidance isn't possible, but in certain applications this requirement isn't necessary. The next computation is carried out for two identical sizes hexapods.

[FIGURE 1 OMITTED]

To define the platform position three parameters are used: coordinates x, y, z of the platform centre; to define the orientation of the platform, three independent angles (Euler's angles) were introduced:

--[psi]--the rotation angle around the fixed vertical axis

--[theta]--the tilt angle of the platform around a horizontal axis that rotates itself with angle [psi]

--[phi]--the proper rotation angle of the platform around the axis that passes through its centre and is perpendicular on it.

In order to establish the vertexes coordinates of the hexagons [B.sub.1i] and [B.sub.2i], i = 1, .., 6, that are not necessary regular hexagons, a matrices method is used, [V] being the positioning vector, and [[psi]], [[theta]], [phi] the elementary rotation matrices (Dudita et al., 1989); (Vukobratovic, 1989).

The next coordinates systems are defined (Fig. 1): the fixed system [O.sub.0][x.sub.0][y.sub.0][z.sub.0], the system [O.sub.1][x.sub.1][y.sub.1][z.sub.1], linked to the intermediate mobile platform 1 and the system [O.sub.2][x.sub.2][y.sub.2][z.sub.2], linked to the final mobile platform 2.

Starting from the position and the orientation of the platform 2, described by its centre coordinates [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in the system [O.sub.0][x.sub.0][y.sub.0][z.sub.0] and the orientation angles [[psi].sup.0.sub.2] = [PSI], [[theta].sup.0.sub.2] = [THETA], [[phi].sup.0.sub.2] = [PHI] in the same system, the analogue parameters have to be determined for:

--the intermediate platform 1 in the system [O.sub.0][x.sub.0][y.sub.0][z.sub.0]: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], respectively [[psi].sup.0.sub.1], [[theta].sup.0.sub.1], [[phi].sup.0.sub.1]

--the final platform 2 in the system [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], respectively [[psi].sup.1.sub.2], [[theta].sup.1.sub.2], [[phi].sup.1.sub.2].

For a given point P we have the next relation between its coordinates in vector representation [[V.sup.0.sub.P]], [[V.sup.1.sub.P]], [[V.sup.2.sub.P]] in the systems [O.sub.0][x.sub.0][y.sub.0][z.sub.0], [O.sub.1][x.sub.1][y.sub.1][z.sub.1], respectively [O.sub.2][x.sub.2][y.sub.2][z.sub.2]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

For identical configurations, imposed by the supplementary condition mentioned above, we have identical distances and identical angles.

The final relation results:

[[psi]] x [[theta]] x [[phi]] = [square root of [[PSI]] x [[THETA]] x [[PHI]]] (4)

Using the same method, the relation (4) can be generalized for a system with n modules:

[[psi]] x [[theta]] x [[phi]] = n[square root of [[PSI]] x [[THETA]] x [[PHI]]] (5)

The coordinates of the points [B.sub.1i] and [B.sub.2i], i = 1, .., 6 are then determined:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

From the relations (6) and (7) the lengths of the legs are deduced, namely the control parameters.

3. MODELLING AND CONTROL

The control and simulation software of the ROBEX was performed in LabVIEW. In Fig. 2, the default configuration and a configurations defined by the angles [[PSI]], [[THETA]], [[PHI]]] of the double hexapodal structure are shown (in x-y projection in the left graph and in x-z projection in the right graph). Complex computations are required to obtain the above representations. The inverse kinematics relations involving matrices and vectors were transposed in application software performed in the graphical development environment LabVIEW.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

4. THE EMBODIMENT OF THE MODEL

The construction of ROBEX is based on the LM 1247 linear electrical actuator produced by FAULHABER having 6 urn theoretical resolution. Special designed connecting elements are used for modular assembling; another function of these elements is to assure the co planarity between the centres of the universal joints from the base platform of the upper module and of the universal joints from the mobile platform of the lower module (fig. 3).

The functional model, in a simplified representation, contains: the mechanical structure, the actuating section, including the power supply and the external controller (a PC or equivalent) and the control software--a LabVIEW application. The controller generates the positioning algorithms and the motion paths, which are transmitted to the actuating section, based on 12 LM1247-020-01 motors and MCLM 3003/06S motion controllers that carry out the displacement of the mechanical part. The PC is linked to the actuating section by a RS232 serial interface, using a set of controls in ASCII format.

5. CONCLUSION

In this article the inverse kinematics for a double hexapodal structure is presented, the control parameters being obtained. The double structure assumes that both modules have identical configurations, hence the same control parameters. Using the LabVIEW features, graphical representations of the systems motion are also presented. The virtual models will be used for the physical embodiment, the project ROBEX, grant 71-024/2007 funded by ANCS / CNMP being in course.

6. REFERENCES

Dietmaier P. (1998). The Stewart-Gough Platform of General Geometry Can Have 4[degrees] Real Postures Available from: http://www.mech.tugraz.at Accessed: 2007-10-25

Dudita, F.; Diaconescu, D. & Gogu, G. (1989). Mecanisme articulate. Inventica. Cinematica--Articulated mechanisms. Inventics. Kinematics, Editura Tehnica--Technical Publishing House, ISBN 973-31-0119-2, Bucuresti

Jakobovic D. & Jelenkovic L. (2002). The Forward and Inverse Kinematics Problems for Stewart Parallel mechanisms, Available from: http://www.zemris.fer.hr Accessed: 2006-05-26

Vukobratovic M. (1989). Applied Dynamics of Manipulation Robots, Springer Verlag, ISBN 3-540-51468-6, Berlin Heidelberg New York

Xiao-Shan, G.; Lei, D.; Qizheng, L. & Gui-Fang, Z. (2005). Generalized Stewart-Gough platforms and their direct kinematics, IEEE Transaction on Robotics, Vol. 21, No. 2, April 2005, 141-151, 10.1109/TRO.2004.835456

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Author: | Margaritescu, Mihai; Moldovanu, Alexandru; Roat, Constantin; Brisan, Cornel |
---|---|

Publication: | Annals of DAAAM & Proceedings |

Article Type: | Report |

Geographic Code: | 4EUAU |

Date: | Jan 1, 2009 |

Words: | 1423 |

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