Abstract
This work presents a systematic method for the dynamic modeling of multi-rigid links confined within a closed environment. The behavior of the system can be completely characterized by two different mathematical models: a set of highly coupled differential equations for modeling the confined multi-link system when it has no impact with surrounding walls; and a set of algebraic equations for expressing the collision of this open kinematic chain system with the confining surfaces. In order to avoid the Lagrangian formulation (which uses an excessive number of total and partial derivatives in deriving the governing equations of multi-rigid links), the motion equations of such a complex system are obtained according to the recursive Gibbs–Appell formulation. The main feature of this paper is the recursive approach, which is used to automatically derive the governing equations of motion. Moreover, in deriving the motion equations, the manipulators are not limited to planar motions only. In fact, for systematic modeling of the motion of a multi-rigid-link system in 3D space, two imaginary links are added to the \(n\)-real links of a manipulator in order to model the spatial rotations of the system. Finally, a 2D and a 3D case studies are simulated to demonstrate the effectiveness of the proposed approach.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Wittenburg, J.: Dynamics of Systems of Rigid Bodies. Teubner, Stuttgart (1977)
Chang, C.C., Peng, S.T.: Impulsive motion of multibody systems. Multibody Syst. Dyn. 17, 47–70 (2007)
Hurmuzlu, Y., Marghitu, D.B.: Rigid body collision of planar kinematic chain with multiple contact points. Int. J. Robot. Res. 13, 82–92 (1994)
Rodriguez, A., Bowling, A.: Solution to indeterminate multipoint impact with frictional contact using constraints. Multibody Syst. Dyn. 28, 313–330 (2012)
Zhang, H., Brogliato, B., Liu, C.: Dynamics of planar rocking-blocks with Coulomb friction and unilateral constraints: comparisons between experimental and numerical data. Multibody Syst. Dyn. 32, 1–25 (2014)
Glocker, C.: Energetic consistency conditions for standard impacts Part I: Newton-type inequality impact laws and Kane’s example. Multibody Syst. Dyn. 29, 77–117 (2013)
Agarwal, A., Shah, S.V., Bandyopadhyay, S., Saha, S.K.: Dynamics of serial kinematic chains with large number of degrees-of-freedom. Multibody Syst. Dyn. 32, 273–298 (2014)
Chenut, X., Fisette, P., Samin, J.-C.L.: Recursive formalism with a minimal dynamic parameterization for the identification and simulation of multibody systems. Application to the human body. Multibody Syst. Dyn. 8, 117–140 (2002)
Mata, V., Provenzano, S., Valero, F., Cuadrado, J.I.: Serial-robot dynamics algorithms for moderately large number of joints. Mech. Mach. Theory 37, 739–755 (2002)
Seidi, M., Hajiaghamemar, M., Caccese, V.: Evaluation of effective mass during head impact due to standing falls. Int. J. Crashworthiness 20, 134–141 (2015)
Anderson, K.S., Critchley, J.H.: Improved ‘order-n’ performance algorithm for the simulation of constrained multi-rigid-body dynamic systems. Multibody Syst. Dyn. 9, 185–212 (2003)
Mohan, A., Saha, S.K.: A recursive, numerically stable, and efficient simulation algorithm for serial robots. Multibody Syst. Dyn. 17, 291–319 (2007)
Naudet, J., Lefeber, D., Daerden, F., Terze, Z.: Forward dynamics of open-loop multibody mechanisms using an efficient recursive algorithm based on canonical momenta. Multibody Syst. Dyn. 10, 45–59 (2003)
Korayem, M.H., Shafei, A.M.: 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 Astronaut. 83, 273–294 (2013)
Korayem, M.H., Shafei, A.M., Absalan, F., Kadkhodaei, B., Azimi, A.: Kinematic and dynamic modeling of viscoelastic robotic manipulators using Timoshenko beam theory: theory and experiment. Int. J. Adv. Manuf. Technol. 71, 1005–1018 (2014)
Korayem, M.H., Shafei, A.M., Doosthoseini, M., Absalan, F., Kadkhodaei, B.: Theoretical and experimental investigation of viscoelastic serial robotic manipulators with motors at the joints using Timoshenko beam theory and Gibbs–Appell formulation. Proc. Inst. Mech. Eng., Part K: J Multi-Body Dyn. (2015). doi:10.1177/1464419315574406
Korayem, M.H., Shafei, A.M., Shafei, H.R.: Dynamic modeling of nonholonomic wheeled mobile manipulators with elastic joints using recursive Gibbs–Appell formulation. Sci. Iran. Trans. B: Mech. Eng. 19, 1092–1104 (2012)
Korayem, M.H., Shafei, A.M., Seidi, E.: Symbolic derivation of governing equations for dual-arm mobile manipulators used in fruit-picking and the pruning of tall trees. Comput. Electron. Agric. 105, 95–102 (2014)
Korayem, M.H., Shafei, A.M.: A new approach for dynamic modeling of n-viscoelastic-link robotic manipulators mounted on a mobile base. Nonlinear Dyn. 79, 2767–2786 (2015)
Korayem, M.H., Shafei, A.M.: Motion equation of nonholonomic wheeled mobile robotic manipulator with revolute–prismatic joints using recursive Gibbs–Appell formulation. Appl. Math. Model. 84, 187–206 (2014)
Korayem, M.H., Shafei, A.M., Dehkordi, S.F.: Systematic modeling of a chain of \(N\)-flexible link manipulators connected by revolute–prismatic joints using recursive Gibbs–Appell formulation. Arch. Appl. Mech. 39, 1701–1716 (2015)
Naudet, J., Lefeber, D., Daerden, F., Terze, Z.: Forward dynamics of open-loop multibody mechanisms using an efficient recursive algorithm based on canonical momenta. Multibody Syst. Dyn. 10, 45–59 (2003)
Förg, M., Pfeiffer, F., Ulbrich, H.: Simulation of unilateral constrained systems with many bodies. Multibody Syst. Dyn. 14, 137–154 (2005)
Gattringer, H., Bremer, H., Kastner, M.: Efficient dynamic modeling for rigid multi-body systems with contact and impact: an \(\mathrm{o}(n)\) formulation. Acta Mech. 219, 111–128 (2011)
Lot, R., Dalio, M.: A symbolic approach for automatic generation of the equations of motion of multibody systems. Multibody Syst. Dyn. 12, 147–172 (2004)
Westervelt, E., Grizzle, J., Chevallereau, C., Choi, J., Morris, B.: Feedback Control of Dynamic Bipedal Robot Locomotion (Control and Automation). CRC Press, Boca Raton (2007)
Flores, P., Ambrósio, J.: On the contact detection for contact-impact analysis in multibody systems. Multibody Syst. Dyn. 24, 103–122 (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Shafei, A.M., Shafei, H.R. A systematic method for the hybrid dynamic modeling of open kinematic chains confined in a closed environment. Multibody Syst Dyn 38, 21–42 (2016). https://doi.org/10.1007/s11044-015-9496-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11044-015-9496-1