Developed Motion of Robot End-Effector of Timelike Ruled Surfaces With Spacelike Rulings (The First Case)

Yıl 2020, Cilt: 2 Sayı: 1, 9 - 21, 04.03.2020

Gülnur Şaffak Atalay , Emin Kasap

Öz

The trajectory of a robot end effector is described by a
ruled surface and a spin angle about the ruling of the ruled surface. In this
paper, we analyzed the problem of describing trajectory of a robot end-effector
by a timelike ruled surface with spacelike ruling. We obtained the developed frame
 by rotating the
generator frame  at an Darboux
angle  in the plane , which is on the striction curve  of the timelike
ruled surface . Afterword, natural frame, tool frame and surface
frame which is necessary for the movements of robot are defined derivative
formulas of the frames are founded by calculating the Darboux vectors. Tool frame
 are constituted
by means of this developed frame. Thus, robot end effector motion is defined
for the timelike ruled surface  generated by
the orientation vector . Also, by using Lancret curvature of the surface and
rotation angle (Darboux angle) in the developed frame the robot end-effector
motion is developed. Therefore, differential properties and movements an
different surfaces in Minkowski space is analyzed by getting the relations for
curvature functions which are characterized a timelike ruled surface with spacelike
directix. Finally, to be able to get a member of trajectory surface family
which has the same trajectory curve is shown with the examples in every
different choice of the Darboux angle which is used to described the developed
frame. 

     

Anahtar Kelimeler

Robot end-effector, Timelike ruled surface

Kaynakça

• [1] A. Turgut and H. H. Hacısalihoğlu, “Timelike ruled surfaces in the Minkowski 3-space,” Far East Journal of Mathematical Sciences, vol. 5, no. 1, pp. 83–90, 1997.
• [2] A. Turgut and H. H. Hacısalihoǧlu, “Timelike ruled surfaces in the Minkowski 3-space. II,” Turkish Journal of Mathematics, vol. 22, no. 1, pp. 33–46, 1998.
• [3] A. O. Öğrenmis, H. Balgetir, and M. Ergüt, “On the ruled surfaces in Minkowski 3-space ,” Journal of Zhejiang University: Science A, vol. 7, no. 3, pp. 326–329, 2006.
• [4] B. S. Ryuh, Robot trajectory planing using the curvature theory of ruled surfaces, Doctoral dissertion, Purdue Universty, West Lafayette, Ind, USA, 1989.
• [5] McCarthy, J. M. and Roth, B., The curvature theory of line trajectories in spatial kinematics, ASME Journal of Mechanical Design, 103 (1981), No.4, 718-724.
• [6] B. S. Ryuh and G. R. Pennock, ‘Accurate motion of a robot end-effector using the curvature theory of ruled surfaces, Journal of mechanisms, Transmissions, and Automation in Design, vol. 110, no. 4, pp. 383-388, 1988.
• [7] B.S.Ryuh, K.M.Lee and M.J.Moon , A Study on the Dual Curvature Theory of a Ruled Surface for the Precision Control of a Robot Trajectory , A Scientific and Technical Publishing Company, 2006. [8] C.H.Chu* , W.N.Huang* , Y.Y.Hu, Machining accuracy improvement in five-axis flank milling of ruled surfaces, Volume 48, pp. 914-921,2008.
• [9] J. H. Kim, B.S.Ryuh and G.R. Pennock, Development of a trajectory generation method for a five-axis NC machine, Mechanism and Machine Theory 36 (2001) 983-996.
• [10] A.Gasparetto and V.Zanotto, A New Method for Smooth Trajectory Planning of Robot Manipulators, Mechanism and Machine Theory 42(2007)455-471.
• [11] F.L.Litvin and X.C.Gao, “Analytical representation of trajectory of manipulators, trends and developments in mechanisms, machines, and robotics,” in the ASME Design Technology Conferences, the 20th Biennial Mechanisms Conference, vol.15-3, pp.481-485, Kissimmee, Fla, USA, September 1988. [12] R. Paul, Manipulator Cartesian path control, IEEE Transactions on Systems, Man and Cybernetics, vol.9, no. 11, pp.702-711,1979.
• [13] C. Ekici, Y. Ünlütürk, M.Dede and B.S.Ryuh, On Motion of Robot End-Effector Using the Curvature Theory of Timelike Ruled Surfaces with Timelike Rulings, Hindawi Publishing Corporation, Mathematical Problems in Engineering, Volume 2008, Article ID 362783, 19 pages doi:10.1115/2008/362783. [14] T. Turhan and N.Ayyıldız, On Curvature Theory of Ruled Surfaces with Lightlike Ruling in Minkowski 3-Space, Int. Journal of Mathematical Sciences and Applications, Vol.1,No.3, September 2011.
• [15] Sahiner B., Kazaz M., and Ugurlu H.H., A Study on Motion of a Robot End-Effector Using the Curvature Theory of Dual Unit Hyperbolic Spherical Curves , Filomat 30:3 (2016), 791–802. DOI 10.2298/FIL1603791S.
• [16] B. O'Neill, Semi-Riemannian Geometry, Academic Press, New York-London, 1983.
• [17] Ratcliffe, J. G. , Foundations of Hyperbolik Manifolds, Springer- Vergal New York, Inc., 736 p, 1994.