An Analytic Closed-form Solution for Trajectory Generation on a Path along an Arc of a Circle

Barghi Jond, Hossein; V. Nabiyev, Vasif; Akbarimajd, Adel

Document Type : Review Article

Authors

¹ Young Researchers and Elite Club, Ahar Branch, Islamic Azad University, Ahar, Iran

² Department of Computer Engineering, Karadeniz Technical University, Trabzon, Turkey

³ Department of Electrical and Computer Engineering, University of Mohaghegh Ardabili, Ardabil, Iran

Abstract

A polynomial trajectory is a time-traveled distance function used to describe trajectory of the robot. Optimal high-degree polynomial trajectories considering initial and the final velocity conditions besides the acceleration constraints are desired. In this paper, a trajectory optimization problem aiming travel maximum distance for a robot that follows an arc based path is formulated. Along the path, the robot requires observing initial and final zero velocity conditions as well as certain acceleration limits. A high-degree polynomial equation along the trajectory is proposed inside of the optimization problem. The closed-form solution of the problem had been obtained analytically. The solution includes the coefficients of the any high-degree trajectory polynomial equation where the coefficients are obtained in closed-form. Simulations several experiments show that the resulting high-degree trajectories satisfy the initial and final zero velocity conditions as well as acceleration constraint.

Keywords

References

[1] R.G. Lerner, G.L. Trigg, “Encyclopaedia of Physics,” (second Edition), VHC Publishers, 1991.

[2] T.W.B. Kibble, “Classical Mechanics,” European Physics Series, 1973.

[3] J. T. Betts, “Survey of Numerical Methods for Trajectory Optimization,” Journal of Guidance, Control, and Dynamics, Vol. 21, No. 2, 1998.

[4] A. Gasparetto and V. Zanotto, “A new method for smooth trajectory planning of robot manipulators,” Mechanism and Machine Theory 42(4), 455-471, 2007.

[5] M.H. Korayem, V. Azimirad, B. Tabibian and M. Abolhasani, “Analysis and experimental study of non-holonomic mobile manipulator in presence of obstacles for moving boundary condition,” Acta Astronautica 67(7-8), 659-672, 2010.

[6] M.R. Azizi and D. Naderi, “Dynamic modeling and trajectory planning for a mobile spherical robot with a 3Dof inner mechanism,” Mechanism and Machine Theory 64, 251-261, 2013.

[7] E. Red, “A dynamic optimal trajectory generator for Cartesian Path following,” Robotica 18(5), 451-458, 2000.

[8] P. Huang, Y. Xu, “PSO-Based Time-Optimal Trajectory Planning for Space Robot with Dynamic Constraints,” Proc. IEEE Int. Conf. Robotics and Biomimetics, 1402-1407, 2006.

[9] F.J. Abu-Dakka, F.J. Valero, J.L. Suner and V. Mata, “Direct approach to solving trajectory planning problems using genetic algorithms with dynamics considerations in complex environments,” Robotica, 2014.

[10] G. Fang and M.W.M.G. Dissanayake, “A neural network-based method for time-optimal trajectory planning,” Robotica 16(2), 143-158, 1998.

[11] J. H. Reif, “Complexity of the Mover’s Problem and Generalizations,” 20th Annual IEEE Symposium on Foundations of Computer Science, San Juan, Puerto Rico, pp. 421–427, October 1979.

[12] J. C. Latombe, “Robot Motion Planning,” Kluwer Academic Publishers, Boston, MA, 1991.

[13] M. Boryga, A. Grabos, “Planning of manipulator motion trajectory with higher-degree polynomials use,” Mechanism and Machine Theory, vol. 44, pp. 1400–1419, 2009.

[14] A. Elnagar and A. Hussein, “On optimal constrained trajectory planning in 3D environments,” Robotics and Autonomous Systems, Vol. 33, pp. 195–206, 2000.

[15] E. Velenis and P. Tsiotras, “Optimal Velocity Proﬁle Generation for Given Acceleration Limits: Theoretical Analysis,” American Control Conference, Portland, OR, USA, June 2005.

[16] D. A. Simakis, “Vehicle Guidance and Control along Circular Trajectories,” Naval Postgraduate School, Monterey, California, USA, 1992.

[17] S. S. Ponda, R. M. Kolacinski and E. Frazzoli, “Trajectory Optimization for Target Localization Using Small Unmanned Aerial Vehicles,” AIAA Guidance, Navigation, and Control Conference, Chicago, Illinois, August 2009.

[18] I. M. Garc´ıa, “Nonlinear Trajectory Optimization with Path Constraints Applied to Spacecraft Reconﬁguration Maneuvers,” MSc thesis, Massachusetts Institute of Technology, 2005.

[19] Y. Guan, K. Yokoi, O. Stasse and A. Kheddar, “On Robotic Trajectory Planning Using Polynomial Interpolations,” IEEE International Conference on Robotics and Biomimetics, Shatin, 111-116, 2005.

[20] H. B. Jond, A. Akbarimajd, N. G. Ozmen, “Time-Distance Optimal Trajectory Planning for Mobile Robots on Straight and Circular Paths,” Journal of Advances in Computer Research, Vol. 5, No. 2, pp. 23-36, May 2014.

[21] H. B. Jond, V. V. Nabiyev, A. Akbarimajd, “Planning of Mobile Robots under Limited Velocity and Acceleration,” 22nd Signal Processing and Communications Applications Conference, Trabzon, Turkey, pp.1579-1582, 2014.

[22] R. L. Williams II, “Simplified Robotics Joint-Space Trajectory Generation with a via Point Using a Single Polynomial,” Journal of Robotics, Vol. 2013.

Majlesi Journal of Electrical Engineering

An Analytic Closed-form Solution for Trajectory Generation on a Path along an Arc of a Circle

References

References

Volume 10, Issue 1 - Serial Number 1
March 2016

An Analytic Closed-form Solution for Trajectory Generation on a Path along an Arc of a Circle

References

References

Volume 10, Issue 1 - Serial Number 1March 2016

Volume 10, Issue 1 - Serial Number 1
March 2016