Regularization of the Movement of a Material Point Along a Flat ‎Trajectory: Application to Robotics Problems

Document Type : Research Paper

Authors

1 L.N. Gumilyov Eurasian National University, 2, Satpaev Str., Nur-Sultan, 010008, Kazakhstan

2 Institute of Information and Computational Technologies,125, Pushkin str., Almaty, 050000, Kazakhstan

3 Al Farabi Kazakh National University, 71/23, al-Farabi Ave., Almaty, 050040, Kazakhstan

Abstract

A control problem of the robot’s end-effector movement along a predefined trajectory is considered. The aim is to reduce the work against resistance forces and improve the comfortability of the motion. The integral of kinetic energy and weighted inertia forces for the whole period of motion is introduced as a cost functional. By applying variational methods, the problem is reduced to a system of quasilinear ordinary differential equations of the fourth order. Numerical examples of solving the problem for movement along straight, circular and elliptical trajectories are presented. For the sake of clarity, the problem is studied for a specific kind of a 3D printer in the 2DoF approximation. However, in the case of negligible masses of moving elements compared the mass of an end-effector, the solution is universal, i.e., it remains the same for given trajectories.

Keywords

Main Subjects

[1] Patle, B.K., Babu L.G., Pandey, A., et al., A review: On path planning strategies for navigation of mobile robot, Defence Technology, 15(4), 2019, 582-606.
[2] Rosyid, A., El-Khasawneh, B., Alazzam, A., Review article: Performance measures of parallel kinematics manipulators, Mechanical Sciences, 11, 2020, 49-73.
[3] Moradi, M., Naraghi, M., Nikoobin, A., Indirect Optimal Trajectory Planning of Robotic Manipulators with the Homotopy Continuation Technique, Proceeding of the 2nd RSI/ISM International Conference on Robotics and Mechatronics (ICRoM): Tehran, 2014, 286-291.
[4] Gosselin, C.M., Angeles, J., A globe performance index for the kinematic optimization of robotic manipulators, ASME Journal of Mechanical Design, 113(3), 1991, 220–226.
[5] Huang, T., Li, Z.X., Li, M., Chetwynd, D.G., Gosselin, C.M., Conceptual design and dimensional synthesis of a novel 2-DOF translational parallel robot for pick-and-place operations, ASME Journal of Mechanical Design, 126(3), 2004, 449–455.
[6] Huang, T., Li, M., Li, Z., Chetwynd, D.G., Whitehouse, D.J., Optimal kinematic design of 2-DOF parallel manipulators with well-shaped workspace bounded by a specified conditioning index, IEEE Transactions on Robotics and Automation, 20(3), 2004, 538–543.
[7] Miller, K., Optimal design and modeling of spatial parallel manipulators, The International Journal of Robotics Research, 23(2), 2004, 127–140.
[8] Stock, M., Miller, K., Optimal kinematic design of spatial parallel manipulators: application to linear delta robot, ASME Journal of Mechanical Design, 125(2), 2003, 292–301.
[9] Nabat, V., Rodriguez, M.O., Company, O., Krut, S., Pierrot, F., Par4: very high speed parallel robot for pick-and-place, Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, 2005, 1202–1207.
[10] Liu, X.J., Wang, J., A new methodology for optimal kinematic design of parallel mechanisms, Mechanism and Machine Theory, 42(9), 2007, 1210–1224.
[11] Liu, X.J., Wang, Q.M., Wang, J., Kinematics, dynamics and dimensional synthesis of a novel 2-DoF translational manipulator, Journal of Intelligent and Robotic Systems, 41, 2004, 205–224.
[12]Gharaaty, S., Tingting, Shu, Joubair, A., et al. Online pose correction of an industrial robot using an optical coordinate measure machine system, International Journal of Advanced Robotic Systems, 15, 2018, 1-16.
[13] Qi, H., Liwen, G., Jianxin, W., Dynamic Performance Evaluation of a 2-DoF Planar Parallel Mechanism, International Journal of Advanced Robotic Systems, 9, 2012, 1-11.
[14] Huang, T., Liu, S., Mei, J., et al., Optimal design of a 2-DoF pick-and-place parallel robot using dynamic performance indices and angular constraints, Mechanism and Machine Theory, 70, 2013, 246-253.
[15] Kim, H.S., Ngoc, Ph.V.B, Dynamics Analysis of a 2-DOF Planar Translational Parallel Manipulator, Journal of the Korean Society of Manufacturing Technology Engineers, 22(2), 2013, 185-191.
[16] Qi, H., Liwen, G., Jianxin, W., Dynamic Performance Evaluation of a 2-DoF Planar Parallel Mechanism, International Journal of Advanced Robotic Systems, 9, 2012, 1-10.
[17] Malyshev, D.I., Posypkin, M.A., Rybak, L.A., Usov, A.L., Analysis of the robot 's work area DexTAR dexterous twin-arm robot, International Journal of Open Information Technologies, 6(7), 2018, 15-20.
[18] Mukanova, B.G., Akhmetzhanov, M.A. , Azimova, D. N., Regularization of the movement of a material point along a flat trajectory: application to robotics problems, Preprint, 2020, https://arxiv.org/abs/2007.01821.
[19] Rosyid, A., El-Khasawneh, B., Alazzam, A., Review article: Performance measures of parallel kinematics manipulators, Mechanical Sciences,11, 2020, 49-73.
[20] Zhan, Zh., Zhang , X., Jian, Zh., Zhang, H., Error modelling and motion reliability analysis of a planar parallel manipulator with multiple uncertainties, Mechanism and Machine Theory, 124, 2018, 55-72.
[21] Stan, S., Mătieş, V., Bălan, R., Optimization of a 2 DOF Micro Parallel Robot Using Genetic Algorithms, Frontiers in Evolutionary Robotics, Vienna, 2008, 596.
[22] Merlet, J.P., Parallel robots, 2nd ed., Springer, 2006.
[23] Kolmogorov, A.N., Fomin, S.V., Elements of the Theory of Functions and Functional Analysis, 7th ed., Moscow, 2004.
[24] Sobolev, S.L., Some applications of functional analysis in mathematical physics, Nauka, Moscow, 1988.
[25] Vasilyev, F.P., Methods of solving extreme problems, Nauka, Moscow, 1981.