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Inverse kinematic solutions of 6-D.O.F. biopolymer segments

Published online by Cambridge University Press:  13 April 2016

Jin Seob Kim  and
Gregory S. Chirikjian
Jin Seob Kim
Affiliation:
Department of Mechanical Engineering, Johns Hopkins University Baltimore, MD 21218, USA. E-mail: [email protected]
Gregory S. Chirikjian*
Affiliation:
Department of Mechanical Engineering, Johns Hopkins University Baltimore, MD 21218, USA. E-mail: [email protected]
*
*Corresponding author. E-mail: [email protected]
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Summary

We present two methods to find all the possible conformations of short six degree-of-freedom segments of biopolymers which satisfy end constraints in position and orientation. One of our methods is motivated by inverse kinematic solution techniques which have been developed for “general” 6R serial robotic manipulators. However, conventional robot kinematics methods are not directly applicable to the geometry of polymers, which can be treated as a degenerate case where all the “link lengths” are zero. Here, we propose a method which extends the elimination method of Kohli and Osvatic. This method can be applied directly to the geometry of biopolymers. We also propose a heuristic method based on a Lie-group-theoretic description. In this method, we utilize inverse iterations of the Jacobian matrix to obtain all conformations which satisfy end constraints. This can be easily implemented for both the general 6R manipulator and polymers. Although the extended elimination method is computationally faster than the Jacobian method, in cases where some of the joint angles are 180° (i.e., where the elimination method fails), we combine these two methods effectively to obtain the full set of inverse kinematic solutions. We demonstrate our approach with several numerical examples.

Keywords

Biopolymer structure6R manipulatorInverse kinematicsEigenvalue problemLie-group-theoretic methodJacobian matrix
Type
Articles
Information
Robotica , Volume 34 , Issue 8 , August 2016 , pp. 1734 - 1753
Copyright
Copyright © Cambridge University Press 2016 

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References

1. Favrin, G., Irbäck, A. and Sjunnesson, F., “Monte Carlo update for chain molecules: Biased Gaussian steps in torsional space,” J. Chem. Phys. 114, 8154–8158 (2001).Google Scholar
2. Ulmschneider, J. P. and Jorgensen, W. L., “Monte Carlo backbone sampling for polypeptides with variable bond angles and dihedral angles using concerted rotations and a Gaussian bias,” J. Chem. Phys. 118, 4261–4271 (2003).Google Scholar
3. Ulmschneider, J. P. and Jorgensen, W. L., “Monte Carlo backbone sampling for nucleic acids using concerted rotations including variable bond angles,” J. Phys. Chem. 108 (43), 16883–16892 (2004).Google Scholar
4. Frenkel, D. and Smit, B., Understanding Molecular Simulation (Academic Press, San Diego, 2002).Google Scholar
5. Gō, N. and Scheraga, H. A., “Ring closure and local conformational deformations of chain molecules,” Macromolecules 3, 178–187 (1970).Google Scholar
6. Gō, N. and Scheraga, H. A., “Ring closure in chain molecules with Cn , I, or S 2n symmetry,” Macromolecules 6, 273–281 (1973).Google Scholar
7. Gō, N. and Scheraga, H. A., “Calculation of the conformation of cyclo-hexaglycyl,” Macromolecules 6, 525–535 (1973).CrossRefGoogle Scholar
8. Dodd, L., Boone, T. and Theodorou, D., “A concerted rotation algorithm for atomistic Monte Carlo simulations of polymer melts and glasses,” Mol. Phys. 78, 961–996 (1993).CrossRefGoogle Scholar
9. Knapp, E., “Long time dynamics of a polymer with rigid body monomer units relating to a protein model: Comparison with the Rouse model,” J. Comput. Chem. 13, 793–798 (1992).CrossRefGoogle Scholar
10. Knapp, E. and Irgens-Defregger, A., “Off-lattice Monte Carlo method with constraints: Long-time dynamics of a protein model without nonbonded interactions,” J. Comput. Chem. 14, 19–29 (1993).Google Scholar
11. Coutsias, E. A., Seok, C., Jacobson, M. P. and Dill, K. A., “A kinematic view of loop closure,” J. Comput. Chem. 25, 510–528 (2004).CrossRefGoogle ScholarPubMed
12. Mak, C. H., “RNA conformational sampling: 1. Single-nucleotide loop closure,” J. Comput. Chem. 29, 926–933 (2008).CrossRefGoogle Scholar
13. Mak, C. H., Chung, W-Y. and Markovskiy, N. D., “RNA conformational sampling II: Arbitrary length multinucleotide loop closure,” J. Chem. Theory Comput. 7, 1198–1207 (2011).Google Scholar
14. Wedemeyer, W. and Scheraga, H. A., “Exact analytical loop closure in prsotein using polynomial equations,” J. Comput. Chem. 20, 819–844 (1999).Google Scholar
15. Lee, H. and Liang, C., “A new vector theory for the analysis of spatial mechanism,” Mech. Mach. Theory 23, 209–217 (1988).Google Scholar
16. Lee, H. and Liang, C., “Displacement analysis of the general spatial 7-link 7R mechanism,” Mech. Mach. Theory 23, 219–226 (1988).Google Scholar
17. Manocha, D. and Canny, J., “Efficient inverse kinematics for general 6R manipulators,” IEEE Trans. Robot. Autom. 10, 648–657 (1994).Google Scholar
18. Wu, M. G. and Deem, M. W., “Analytical rebridging Monte Carlo: Application to cis/trans isomerization in proline-containing, cyclic peptides,” J. Chem. Phys. 111, 6625–6632 (1999).Google Scholar
19. Duffy, J. and Crane, C., “A displacement analysis of the general spatial 7R mechanisms,” Mech. Mach. Theory 15, 153–169 (1980).Google Scholar
20. Tsai, L-W. and Morgan, A., “Solving the kinematics of the most general six- and five-degree-of-freedom manipulators by continuation methods,” ASME J. Mech. Transm. Autom. Des. 107, 189–200 (1985).Google Scholar
21. Manseur, R. and Doty, K. L., “A robot manipulator with 16 real inverse kinematic solution sets,” Int. J. Robot. Res. 8 (5), 75–79 (1989).Google Scholar
22. Wampler, C. and Morgan, A., “Solving the 6R inverse position problem using a generic-case solution methodology,” Mech. Mach. Theory 26 (1), 91–106 (1991).Google Scholar
23. Raghavan, M. and Roth, B., “Kinematic analysis of the 6R manipulator of general geometry,” In: Proceedings of the 5th International Symposium on Robotics Research (Miura, H. and Airmoto, S., eds.) (MIT Press, Cambridge, MA, 1990) pp. 263–270.Google Scholar
24. Raghavan, M. and Roth, B., “Kinematic analysis of the 6R manipulator and related linkages,” ASME J. Mech. Des. 115, 502–508 (1993).Google Scholar
25. Manocha, D., Zhu, Y. and Wright, W., “Conformational analysis of molecular chains using nano-kinematics,” Comput. Appl. Biosci. 11, 71–86 (1995).Google Scholar
26. Kohli, D. and Osvatic, M., “Inverse kinematics of the general 6R and 5R, P serial manipulators,” ASME J. Mech. Des. 115, 922–931 (1993).Google Scholar
27. Ghazvini, M., “Reducing the Inverse Kinematics of Manipulators to the Solution of a Generalized Eigenproblem,” In: Computational Kimematics (Angeles, J., et al., ed.) (Kluwer Academic Publishers, Springer, Netherlands, 1993) pp. 15–26.Google Scholar
28. Nielson, J. and Roth, B., “On the kinematic analysis of robotic mechanisms,” Int. J. Robot. Res. 12 (12), 1147–1160 (1999).Google Scholar
29. Husty, M. L., Pfurner, M. and Schröcker, H-P., “A new and efficient algorithm for the inverse kinematics of a general serial 6R manipulator,” Mech. Mach. Theory 42 (1), 66–81 (2007).CrossRefGoogle Scholar
30. Rudny, T., “Solving inverse kinematics by fully automated planar curves intersecting,” Mech. Mach. Theory 74, 310–318 (2014).Google Scholar
31. Spong, M. and Vidyasagar, M., Robot Dynamics and Control (John Wiley and Sons, New York, 1989).Google Scholar
32. Golub, G. and Van Loan, C., Matrix Computations (The Johns Hopkins University Press, Baltimore, 1996).Google Scholar
33. Anderson, E. et al., LAPACK User's Guide (SIAM, Philadelphia, PA, 1999).Google Scholar
34. McCarthy, M., An Introduction to Theoretical Kinematics (MIT Press, Cambridge, MA, 1990).Google Scholar
35. Murray, M., Li, Z. and Sastry, S., A Mathematical Introduction to Robotic Manipulation (CRC Press, Boca Raton, 1994).Google Scholar
36. Chirikjian, G. S. and Kyatkin, A. B., Engineering Applications of Noncommutative Harmonic Analysis (CRC Press, Boca Raton, FL, 2001).Google Scholar
37. Sommese, A. J. and Wampler, C. W., The Numerical Solution to Systems of Polynomials Arising in Engineering and Science (World Scientific, Singapore, 1985).Google Scholar
38. Morgan, A., Solving Polynomial Systems Using Contiuation For Engineering and Scientific Problems (Prentice-Hall, New Jersey, 1987).Google Scholar
39. Ebert-Uphoff, I. and Chirikjian, G. S., “Inverse kinematics of discretely actuated hyper-redundant manipulators using workspace densities,” Proceedings of the IEEE International Conference on Robotics and Automation (Minneapolis, MN, 1996) pp. 139–145.Google Scholar
40. Chirikjian, G. S., “Inverse kinematics of binary manipulators using a continuum model,” J. Intell. Robot. Syst. 19, 5–22 (1997).Google Scholar
41. Suthakorn, J. and Chirikjian, G. S., “A new inverse kinematics algorithm for binary manipulators with many actuators,” Adv. Robot. 15 (2), 225–244 (2001).Google Scholar
42. Wang, Y. and Chirikjian, G. S., “Workspace generation of hyper-redundant manipulators as a diffusion process on SE(N),” IEEE Trans. Robot. Autom. 20 (3), 399–408 (2004).Google Scholar
43. Kyatkin, A. B. and Chirikjian, G. S., “Computation of robot configuration and workspaces via the fourier transform on the discrete motion group,” Int. J. Robot. Res. 18 (6), 601–615 (1999).Google Scholar
44. Wang, Y., “A fast workspace-density-driven inverse kinematics method for hyper-redundant manipulators,” Robotica 24, 649–655 (2006).Google Scholar
45. Chirikjian, G. S., “Conformational statistics of macromolecules using generalized convolution,” Comput. Theor. Polym. Sci. 11, 143–153 (2001).Google Scholar
46. Kim, J. S. and Chirikjian, G. S., “A unified approach to conformational statistics of classical polymer and polypeptide models,” Polymer 46, 11904–11917 (2005).Google Scholar
47. Whitney, D., “Resolved motion rate control of manipulators and human prostheses,” IEEE Trans. Man-Mach. Syst. MMS–10 (2), 47–53 (1969).Google Scholar
48. Uicker, J., Denavit, J. and Hartenberg, R., “An iterative method for the displacement analysis of spatial mechanisms,” ASME J. Appl. Mech. 107, 189–200 (1954).Google Scholar
49. Isobe, T., Nagasaka, K. and Yamamoto, S., “A new approach to kinematic control of simple manipulators,” IEEE Trans. Syst. Man Cybern. 22 (5), 1116–1124 (1992).Google Scholar
50. Wang, L. and Chen, C., “A combined optimization method for solving inverse kinematics problem of mechanical manipulators,” IEEE Trans. Robot. Autom. 7 (4), 489–499 (1991).CrossRefGoogle Scholar
51. Olsen, A. L. and Petersen, H. G., “Inverse kinematics by numerical and analytical cyclic coordinate descent,” Robotica 29 (4), 619–626 (2011).Google Scholar
52. Canutescu, A. and Dunbrack, R., “Cyclic coordinate descent: A robotic algorithm for protein loop closure,” Protein Sci. 12, 963–972 (2003).Google Scholar
53. Boomsma, W. and Hamelryck, T., “Full cyclic coordinate descent: Solving the protein loop closure problem in C α space,” BMS Bioinformatics 6, 159 (2005).Google Scholar
54. Al-Nasr, K. and He, J., “An effective convergence independent loop closure method using forward-backward cyclic coordinate descent,” Int. J. Data Min. Bioinformatics 3 (3), 346–361 (2009).Google Scholar
55. Kim, J. S. and Chirikjian, G. S., “Conformational analysis of stiff chiral polymers with end-constraints,” Mol. Simul. 32 (14), 1139–1154 (2006).Google Scholar
56. Park, F. C., “Distance metrics on the rigid-body motions with applications to mechanism design,” J. Mech. Des. 117, 48–54 (1995).Google Scholar
57. Chirikjian, G. S. and Zhou, S., “Metrics on motion and deformation of solid models,” J. Mech. Des. 120 (2), 252–261 (1998).Google Scholar
58. Chirikjian, G. S. and Yan, Y., “Mathematical aspects of molecular replacement. II. Geometry of motion spaces,” Acta Cryst. A68, 208–221 (2012).Google Scholar
59. Alberts, B., Johnson, A., Lewis, J., Raff, M., Roberts, K. and Walter, P., Molecular Biology of the Cell (Garland Science, New York, 2000).Google Scholar
60. Ulmschneider, J. P. and Jorgensen, W. L., “Polypeptide folding using Monte Carlo sampling, concerted rotation, and continuum solvation,” J. Am. Chem. Soc. 126, 1849–1857 (2004).Google Scholar
61. Ulmschneider, J. P. and Jorgensen, W. L., “Monte Carlo vs molecular dynamics for all-atom polypeptide folding simulations,” J. Phys. Chem. B 110, 16733–16742 (2006).Google Scholar