# Automatic formulation of falling multiple flexible-link robotic manipulators using 3×3 rotational matrices

Document Type: Full Length Article

Author

Assistant Professor, Department of Mechanical Engineering, Shahid Bahonar University of Kerman, Kerman, Iran

10.22064/tava.2017.35085.1032

Abstract

In this paper, the effect of normal impact on the mathematical modeling of flexible multiple links is investigated. The response of such a system can be fully determined by two distinct solution procedures. Highly nonlinear differential equations are exploited to model the falling phase of the system prior to normal impact; and algebraic equations are used to model the normal collision of this open-chain robotic system. To avoid employing the Lagrangian method which suffers from too many differentiations, the governing equations of such complicated system are acquired via the Gibbs-Appell (G-A) methodology. The main contribution of the present work is the use of an automatic algorithm according to 3×3 rotational matrices to obtain the system’s motion equations more efficiently. Accordingly, all mathematical formulations are completed by the use of 3×3 matrices and 3×1 vectors only. The dynamic responses of this system are greatly reliant on the step sizes. Therefore, as well as solving the obtained differential equations by using several ODE solvers, a computer program according to the Runge-Kutta method was also developed. Finally, the computational counts of both algorithms i.e., 3×3 rotational matrices and 4×4 transformation matrices are compared to prove the efficiency of the former in deriving the motion equations.

Highlights

• Modelling of finite and impulsive motions for a multi-flexible-link system is presented.
• The recursive Gibbs-Appell formulation is used to derive the motion equations.
• An algorithm based on 3×3 rotational matrices is applied to derive dynamic equations.
• Joints’ pre- and post-collision velocities are found related using Newton’s kinematic impact law.
• Simulation results for single and double elastic links are presented in time domain.

Keywords

Main Subjects

### References

[1] J. Wittenburg, Dynamics of systems of rigid bodies, Vieweg+Teubner Verlag, 1977.

[2] C.C. Chang, S.T. Peng, Impulsive motion of multibody systems, Multibody System Dynamics, 17 (2007) 47-70.

[3] Y. Hurmuzlu, D.B. Marghitu, Rigid body collisions of planar kinematic chains with multiple contact points, The International Journal of Robotics Research, 13 (1994) 82-92.

[4] A. Rodriguez, A. Bowling, Solution to indeterminate multipoint impact with frictional contact using constraints, Multibody System Dynamics, 28 (2012) 313-330.

[5] M. Mahmoodi, M. Kojouri Manesh, M. Eghtesad, M. Farid, S. Movahed, Adaptive passivity-based control of a flexible-joint robot manipulator subject to collision, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 228 (2014) 840-849.

[6] A.M. Shafei, H.R.Shafei, Dynamic behavior of flexible multiple links captured inside a closed space, ASME Journal of Computational and Nonlinear Dynamics, 11 (2016) 1-13.

[7] A. Tornambè, Modeling and control of impact in mechanical systems: Theory and experimental results, IEEE Transactions on Automatic Control, 44 (1999) 294-309.

[8] S. Liu, L. Wu, Z. Lu, Impact dynamics and control of a flexible dual-arm space robot capturing an object, Applied Mathematics and Computation, 185 (2007) 1149-1159.

[9] S.A. Modarres Najafabadi, J. Kövecses, J. Angeles, Energy analysis and decoupling in three-dimensional impacts of multibody systems, ASME Journal of Applied Mechanics, 74 (2007) 845-851.

[10] M. Hajiaghamemar, M. Seidi, J.R. Ferguson, V. Caccese, Measurement of head impact due to standing fall in adults using anthropomorphic test dummies, Annals of Biomedical Engineering, 43 (2015) 2143-2152.

[11] Z. Kariž, G.R. Heppler, A controller for an impacted single flexible link, Journal of Vibration and Control, 6 (2000) 407-428.

[12] D. Boghiu, D.B. Marghitu, The control of an impacting flexible link using fuzzy logic strategy, Journal of Vibration and Control, 4 (1998) 325-341.

[13] Y.A. Khulief, A.A. Shabana, A continuous force model for the impact analysis of flexible multibody systems, Mechanism and Machine Theory, 22 (1987) 213-224.

[14] A.S. Yigit, A.G. Ulsoy, R.A. Scott, Dynamics of a radially rotating beam with impact, Part 1: Theoretical and computational model, Journal of Vibration and Acoustics, 112 (1990) 65-70.

[15] A.S. Yigit, A.G. Ulsoy, R.A. Scott, Dynamics of a radially rotating beam with impact, Part 2: Experimental and simulation results, Journal of Vibration and Acoustics, 112 (1990) 71-77.

[16] A.S. Yigit, A.G. Ulsoy, R.A. Scott, Spring-dashpot models for the dynamics of a radially rotating beam with impact, Journal of Sound and Vibration, 142 (1990) 515-525.

[17] A.S. Yigit, The effect of flexibility on the impact response of a two-link rigid-flexible manipulator, Journal of Sound and Vibration, 177 (1994) 349-361.

[18] B.V. Chapnik, G.R. Heppler, J.D. Aplevich, Modeling impact on a one-link flexible robotic arm, IEEE Transactions on Robotics and Automation, 7 (1991) 479-488.

[19] Y.A. Khulief, Modeling of impact in multibody systems: An overview, Journal of Computational and Nonlinear Dynamics, 8 (2013) 1-15.

[20] Q. Yu, I.M. Chen, A general approach to the dynamics of nonholonomic mobile manipulator systems, ASME Journal of Dynamic Systems, Measurement and Control, 124 (2002) 512-521.

[21] Q. Sun, M. Nahon, I. Sharf, An inverse dynamics algorithm for multiple flexible-link manipulators, Journal of Vibration and Control, 6 (2000) 557-569.

[22] A. Mohan, S.K. Saha, A recursive, numerically stable, and efficient simulation algorithm for serial robots with flexible links, Multibody System Dynamics, 21 (2009) 1-35.

[23] U. Lugris, M.A. Naya, A. Luaces, J. Cuadrado, Efficient calculation of the inertia terms in floating frame of reference formulations for flexible multibody dynamics, Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, 223 (2009) 147-157.

[24] U. Lugris, M.A. Naya, F. Gonzalez, J. Cuadrado, Performance and application criteria of two fast formulations for flexible multibody dynamics, Mechanics Based Design of Structures and Machines, 35 (2007) 381-404.

[25] Y.L. Hwang, Recursive Newton-Euler formulation for flexible dynamic manufacturing analysis of open-loop robotic systems, Int J Adv Manuf Technol, 29 (2006) 598-604.

[26] D.S. Bae, E.J. Haug, A recursive formulation for constrained mechanical system dynamics: Part I. Open loop systems, Journal of Structural Mechanics, 15 (1987) 359-382.

[27] T.M. Wasfy, A.K. Noor, Computational strategies for flexible multibody systems, Applied Mechanics Reviews, 56 (2003) 553-613.

[28] M.H. Korayem, A.M. Shafei, Application of recursive Gibbs–Appell formulation in deriving the equations of motion of N-viscoelastic robotic manipulators in 3D space using Timoshenko beam theory, Acta Astronautica, 83 (2013) 273-294.

[29] M.H. Korayem, A.M. Shafei, F. Absalan, B. Kadkhodaei, A. Azimi, Kinematic and dynamic modeling of viscoelastic robotic manipulators using Timoshenko beam theory: theory and experiment, Int J Adv Manuf Technol, 71 (2014) 1005-1018.

[30] M.H. Korayem, A.M. Shafei, M. Doosthoseini, F. Absalan, B. Kadkhodaei, Theoretical and experimental investigation of viscoelastic serial robotic manipulators with motors at the joints using Timoshenko beam theory and Gibbs–Appell formulation, Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, (2015) 1464419315574406.

[31] M.H. Korayem, A.M. Shafei, H.R. Shafei, Dynamic modeling of nonholonomic wheeled mobile manipulators with elastic joints using recursive Gibbs–Appell formulation, Scientia Iranica: Transactions B: Mechanical Engineering, 19 (2012) 1092-1104.

[32] M.H. Korayem, A.M. Shafei, E. Seidi, Symbolic derivation of governing equations for dual-arm mobile manipulators used in fruit-picking and the pruning of tall trees, Computers and Electronics in Agriculture, 105 (2014) 95-102.

[33] M.H. Korayem, A.M. Shafei, A new approach for dynamic modeling of n-viscoelastic-link robotic manipulators mounted on a mobile base, Nonlinear Dynamics, 79 (2015) 2767-2786.

[34] M.H. Korayem, A.M. Shafei, Motion equation of nonholonomic wheeled mobile robotic manipulator with revolute–prismatic joints using recursive Gibbs–Appell formulation, Applied Mathematical Modelling, 39 (2015) 1701-1716.

[35] M.H. Korayem, A.M. Shafei, S.F. Dehkordi, Systematic modeling of a chain of N-flexible link manipulators connected by revolute–prismatic joints using recursive Gibbs-Appell formulation, Archive of Applied Mechanics, 84 (2014) 187-206.

[36] M. Förg, F. Pfeiffer, H. Ulbrich, Simulation of unilateral constrained systems with many bodies, Multibody System Dynamics, 14 (2005) 137-154.

[37] H. Gattringer, H. Bremer, M. Kastner, Efficient dynamic modeling for rigid multi-body systems with contact and impact: An O(n) formulation, Acta Mechanica, 219 (2011) 111-128.

[38] D. Tlalolini, Y. Aoustin, C. Chevallereau, Design of a walking cyclic gait with single support phases and impacts for the locomotor system of a thirteen-link 3D biped using the parametric optimization, Multibody System Dynamics, 23 (2009) 33-56.

[39] A.M. Shafei, H.R. Shafei, Oblique impact of multi-flexible-link systems, Journal of Vibration and Control, (2016) 1077546316654854.

[40] H. Zohoor, S.M. Khorsandijou, Dynamic model of a flying manipulator with two highly flexible links, Applied Mathematical Modelling, 32 (2008) 2117-2132.