**Commenced**in January 2007

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##### Study on Robot Trajectory Planning by Robot End-Effector Using Dual Curvature Theory of the Ruled Surface

**Authors:**
Y. S. Oh,
P. Abhishesh,
B. S. Ryuh

**Abstract:**

This paper presents the method of trajectory planning by the robot end-effector which accounts for more accurate and smooth differential geometry of the ruled surface generated by tool line fixed with end-effector based on the methods of curvature theory of ruled surface and the dual curvature theory, and focuses on the underlying relation to unite them for enhancing the efficiency for trajectory planning. Robot motion can be represented as motion properties of the ruled surface generated by trajectory of the Tool Center Point (TCP). The linear and angular properties of the six degree-of-freedom motion of end-effector are computed using the explicit formulas and functions from curvature theory and dual curvature theory. This paper explains the complete dualization of ruled surface and shows that the linear and angular motion applied using the method of dual curvature theory is more accurate and less complex.

**Keywords:**
Dual curvature theory,
robot end effector,
ruled surface,
TCP,
tool center point.

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1129239

**References:**

[1] Litvin, F. L. and Gao, X. C., “Analytical Representation of Trajectory of Manipulators,” in Trends and Developments in Mechanisms, Machines, and Robotics, The 1988 ASME Design Technology Conferences-20th Biennial Mechanisms Conference, Kissimmee, Florida., 1988, Sept. 25-28, DE-Vol. 15-3, pp. 481-485.

[2] McCathy, J. M. and Roth, B., “The Curvature Theory of Line Trajectories in Spatial Kinematics,” ASME Journal of Mechanism Design, vol. 103, No.4, 1981.

[3] Ryuh, B. S. and Pennock, G. R., “Accurate Motion of a Robot End-Effector using the Curvature Theory of a ruled surfaces,” Journal of Mechanisms, Transmissions, and Automation in Design, Trans. ASME, Dec. 1988, vol. 110, No.4, pp. 383-388.

[4] Ryuh, B. S., “Robot Trajectory planning using the curvature theory of ruled surfaces,” Doctoral dissertation, 1989, Purdue University, West Lafayette, Indiana, pp. 143.

[5] Ryuh, B. S. and Pennock, G. R., “Trajectory planning using the Furguson curve model and curvature theory of a ruled surface,” Journal of Mechanical Design, Transactions of ASME, 1990, Sep. Vol. 112, No. 4, pp. 377-383.

[6] DoCamo, M. P., “Differential Geometry of Curves and Surfaces,” Prentice-Hall, New Jersey, 1976, pp. 503.

[7] Veldkamp, G. R., “On the Use of Dual Numbers, Vectors and Matrices in Instantaneous Spatial Kinematics,” Mechanism and Machine Theory,1976, Vol. 11, No. 2, pp. 141-158.

[8] Kirson, Y., “High Order Curvature Theory in Space Kinematics,” Doctoral dissertation, University of California, Berkeley, 1975, pp.140.

[9] Guggerhemier, H., “Differential Geometry,” Dover Publications, 1977, pp.-378.

[10] Cumali Ekici, Yasin Unluturk, Mustafa Dede, Ryuh, B. S., “On Motion of Robot End-Effector Using the Curvature Theory of Timelike Ruled Surfaces with Timelike Rulings,” Mathematical Problems in Engineering, Vol. 2008, pp.12.

[11] Kotelnikov, A. P., “Screw Calculus and Some Applications to Geometry and Mechanics,” Annals of Imperial University of Kazan, 1895.

[12] Yücesan, A., Ayyıldız, N., Çöken, A. C., “On rectifying dual space curves,” Revista Mathemática Computense, 2007, Vol. 20(2), pp. 497-506.