Improved Approximation Approach for Folding Analyses of Structures with Kinematic Indeterminacy
Publication: Journal of Engineering Mechanics
Volume 148, Issue 5
Abstract
The kinematic analysis method based on generalized inverse theory has been used in the engineering field. However, the traditional numerical procedure is found to be extraordinarily time consuming when the number of unknowns increases. This paper presents an improved approximation approach (IAA) of the kinematic analysis method to trace the rigid motion of mechanisms with both high accuracy and efficiency. It simplifies the procedure of obtaining the generalized inverse into the process of solving linear equations. Accordingly, the whole calculation can be executed with sparse matrix format. To demonstrate the progress of the IAA, the numerical examples are examined through a comparison with the conventional methods. The results indicate that IAA shows distinct improvements in precision and efficiency. The improvements are found to be clear and effective, especially for systems with multiple DOFs.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
All data, models, and code generated or used during the study appear in the published article.
Acknowledgments
This work was supported by JSPS KAKENHI Grant No. JP19K23547.
References
Bunch, J. R., and D. J. Rose. 2014. Sparse matrix computations. Cambridge, MA: Academic Press.
Chai, T. J., and C. S. Tan. 2019. “Review on deployable structure.” IOP Conf. Ser.: Earth Environ. Sci. 220 (1): 012034. https://doi.org/10.1088/1755-1315/220/1/012034.
Courrieu, P. 2005. “Fast computation of Moore-Penrose inverse matrices.” Neural Inf. Process. Lett. Rev. 8 (2): 25–29.
Doty, K. L., C. Melchiorri, and C. Bonivento. 1993. “A theory of generalized inverses applied to robotics.” Int. J. Rob. Res. 12 (1): 1–19. https://doi.org/10.1177/027836499301200101.
Ebrahimi, H., D. Mousanezhad, B. Haghpanah, R. Ghosh, and A. Vaziri. 2017. “Lattice materials with reversible foldability.” Adv. Eng. Mater. 19 (2): 1600646. https://doi.org/10.1002/adem.201600646.
Frizzell, K., R. Flores, A. Riedl, and D. C. Conner. 2018. “Modifiable intuitive robot controller: Computer vision-based controller for various robotic designs.” In Proc., SoutheastCon 2018, 1–7. New York: IEEE.
Golub, G. H., and C. Reinsch. 1970. “Singular value decomposition and least squares solutions.” Numer. Math. 14 (5): 403–420. https://doi.org/10.1007/BF02163027.
Hangai, Y., and K. Kawaguchi. 1991. Shape analysis: Generalized inverses and applications. [In Japanese.] Tokyo: Baifukan Press.
Hangai, Y., and M. Wu. 1999. “Analytical method of structural behaviours of a hybrid structure consisting of cables and rigid structures.” Eng. Struct. 21 (8): 726–736. https://doi.org/10.1016/S0141-0296(98)00027-3.
Hussain, I., G. Salvietti, G. Spagnoletti, M. Malvezzi, D. Cioncoloni, S. Rossi, and D. Prattichizzo. 2017. “A soft supernumerary robotic finger and mobile arm support for grasping compensation and hemiparetic upper limb rehabilitation.” Rob. Auton. Syst. 93 (Jul): 1–12. https://doi.org/10.1016/j.robot.2017.03.015.
Jovichikj, R., A. Yaşir, and G. Kiper. 2018. “Reconfigurable deployable umbrella canopies.” In Proc., 2018 Int. Conf. on Reconfigurable Mechanisms and Robots, 1–6. New York: IEEE.
Kantún-Montiel, G. 2014. “Outer generalized inverses with prescribed ideals.” Linear Multilinear Algebra 62 (9): 1187–1196. https://doi.org/10.1080/03081087.2013.816302.
Katsikis, V. N., and D. Pappas. 2008. “Fast computing of the Moore-Penrose inverse matrix.” Electron. J. Linear Algebra 117 (1): 637–650. https://doi.org/10.13001/1081-3810.1287.
Katsikis, V. N., D. Pappas, and A. Petralias. 2011. “An improved method for the computation of the Moore–Penrose inverse matrix.” Appl. Math. Comput. 217 (23): 9828–9834. https://doi.org/10.1016/j.amc.2011.04.080.
Kawaguchi, K. 2011. Generalized inverse and its applications to structural engineering. [In Japanese.] Tokyo: Corona Publishing.
Kawaguchi, K., Y. Hangai, and K. Miyazaki. 1993. “The dynamic analysis of kinematically indeterminate frameworks.” In Proc., IASS-MSU Symp., 569–576. Istanbul, Turkey: Mimar Sinan Üniversitesi.
Kawaguchi, K., K. Nabana, and Y. Hangai. 1996. “Considerations for folding analysis of frame structures.” In Proc., 2nd Int. Conf. on Mobile and Rapidly Assembled Structures, 383–394. Southampton, UK: Wessex Institute of Technology Press.
McCullagh, P. 2019. Generalized linear models. New York: Routledge.
Merrill, D., and M. Garland. 2016. “Merge-based parallel sparse matrix-vector multiplication.” In Proc., Int. Conf. for High Performance Computing, Networking, Storage and Analysis. New York: IEEE.
Mirrazavi Salehian, S. S., N. Figueroa, and A. Billard. 2018. “A unified framework for coordinated multi-arm motion planning.” Int. J. Rob. Res. 37 (10): 1205–1232. https://doi.org/10.1177/0278364918765952.
Nashed, M. Z. 2014. “Generalized inverses and applications.” In Proc., Advanced Seminar Sponsored by the Mathematics Research Center, the University of Wisconsin—Madison, October 8-10, 1973. Amsterdam, Netherlands: Elsevier.
Pellegrino, S. 1993. “Structural computations with the singular value decomposition of the equilibrium matrix.” Int. J. Solids Struct. 30 (21): 3025–3035. https://doi.org/10.1016/0020-7683(93)90210-X.
Penrose, R. 1955. “A generalized inverse for matrices.” In Proc., Mathematical Cambridge Philosophical Society, 406–413. Cambridge, UK: Cambridge University Press.
Penrose, R. 1956. “On best approximate solutions of linear matrix equations.” In Proc., Mathematical Cambridge Philosophical Society, 17–19. Cambridge, UK: Cambridge University Press.
Petković, M. D., and P. S. Stanimirović. 2011. “Iterative method for computing the Moore–Penrose inverse based on Penrose equations.” J. Comput. Appl. Math. 235 (6): 1604–1613. https://doi.org/10.1016/j.cam.2010.08.042.
Ramos, A. S., and G. H. Paulino. 2016. “Filtering structures out of ground structures—A discrete filtering tool for structural design optimization.” Struct. Multidiscip. Optim. 54 (1): 95–116. https://doi.org/10.1007/s00158-015-1390-1.
Rodrigues, G. V., L. M. Fonseca, M. A. Savi, and A. Paiva. 2017. “Nonlinear dynamics of an adaptive origami-stent system.” Int. J. Mech. Sci. 133 (Nov): 303–318. https://doi.org/10.1016/j.ijmecsci.2017.08.050.
Soleymani, F. 2015. “An efficient and stable Newton-type iterative method for computing generalized inverse AT, S.” Numerical Algorithms 69 (3): 569–578. https://doi.org/10.1007/s11075-014-9913-1.
Stanimirović, P. S., D. Pappas, V. N. Katsikis, and I. P. Stanimirović. 2012. “Full-rank representations of outer inverses based on the QR decomposition.” Appl. Math. Comput. 218 (20): 10321–10333. https://doi.org/10.1016/j.amc.2012.04.011.
Stanimirović, P. S., and M. B. Tasić. 2008. “Computing generalized inverses using LU factorization of matrix product.” Int. J. Comput. Math. 85 (12): 1865–1878. https://doi.org/10.1080/00207160701582077.
Sun, B., and Z. Li. 2019. “An additional ultra-soft grid method for deformation and kinematic computation of scissor-type structures.” Int. J. Mech. Sci. 153 (Jan): 230–239. https://doi.org/10.1016/j.ijmecsci.2019.02.005.
Tachi, T. 2010. “Geometric considerations for the design of rigid origami structures.” In Vol. 12 of Proc., Int. Association for Shell and Spatial Structures (IASS) Symp., 458–460. Beijing: China Architecture & Building Press.
Tachi, T., and T. C. Hull. 2017. “Self-foldability of rigid origami.” J. Mech. Rob. 9 (2): 021008. https://doi.org/10.1115/1.4035558.
Tang, Z., and J. S. Dai. 2018. “Bifurcated configurations and their variations of an 8-bar linkage derived from an 8-kaleidocycle.” Mech. Mach. Theory 121 (Mar): 745–754. https://doi.org/10.1016/j.mechmachtheory.2017.10.012.
Thomas King, J., and D. Chillingworth. 1979. “Approximation of generalized inverses by iterated regularization.” Numer. Funct. Anal. Optim. 1 (5): 499–513. https://doi.org/10.1080/01630567908816031.
Toutounian, F., and A. Ataei. 2009. “A new method for computing Moore–Penrose inverse matrices.” J. Comput. Appl. Math. 228 (1): 412–417. https://doi.org/10.1016/j.cam.2008.10.008.
wo80. 2020. “Math.NET numerics and CSparse.” Accessed September 17, 2021. https://github.com/wo80/CSparse.NET/wiki/Math.NET-Numerics-and-CSparse.
Wu, M., T. Zhang, P. Xiang, and F. Guan. 2018. “Single-layer deployable truss structure driven by elastic components.” J. Aerosp. Eng. 32 (2): 04018144. https://doi.org/10.1061/(ASCE)AS.1943-5525.0000977.
Xiang, P., M. Wu, and R. Q. Zhou. 2015. “Dynamic analysis of deployable structures using independent displacement modes based on Moore-Penrose generalized inverse matrix.” Struct. Eng. Mech. 54 (6): 1153–1174. https://doi.org/10.12989/sem.2015.54.6.1153.
Zhang, T., and K. Kawaguchi. 2021. “Folding analysis for thick origami with kinematic frame models concerning gravity.” Autom. Constr. 127 (Jul): 103691. https://doi.org/10.1016/j.autcon.2021.103691.
Zhang, T., K. Kawaguchi, and M. Wu. 2018. “A folding analysis method for origami based on the frame with kinematic indeterminacy.” Int. J. Mech. Sci. 146 (Oct): 234–248. https://doi.org/10.1016/j.ijmecsci.2018.07.036.
Zhang, T., K. Kawaguchi, and M. Wu. 2019a. “Concept and preliminary analysis of novel movable structural system for cable-stayed footbridge.” J. Bridge Eng. 24 (4): 04019021. https://doi.org/10.1061/(ASCE)BE.1943-5592.0001378.
Zhang, T., K. Kawaguchi, and M. Wu. 2019b. “Optimization of frame structures with kinematical indeterminacy for optimum folding.” J. Eng. Mech. 145 (9): 04019072. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001646.
Zhang, T., M. Wu, and F. Guan. 2016. “Deployment study on a single-layer deployable truss structure driven by elastic components.” J. Int. Assoc. Shell Spatial Struct. 57 (4): 285–294. https://doi.org/10.20898/j.iass.2016.190.855.
Information & Authors
Information
Published In
Copyright
© 2022 American Society of Civil Engineers.
History
Received: May 28, 2020
Accepted: Jan 17, 2022
Published online: Mar 10, 2022
Published in print: May 1, 2022
Discussion open until: Aug 10, 2022
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited by
- Tianhao Zhang, Ken’Ichi Kawaguchi, Lead the folding motion of the thick origami model under gravity, Journal of Asian Architecture and Building Engineering, 10.1080/13467581.2022.2145210, (1-13), (2022).