Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T15:46:13.407Z Has data issue: false hasContentIssue false
  • English
  • Français

Kinematic topology and constraints of multi-loop linkages

Published online by Cambridge University Press:  02 August 2018

Andreas Müller
Andreas Müller*
Affiliation:
Institute of Robotics, Johannes Kepler University, Linz, Austria
*
*Corresponding author. E-mail: a.mueller@jku.at
Get access Rights & Permissions [Opens in a new window]

Summary

Modeling the instantaneous kinematics of lower pair linkages using joint screws and the finite kinematics with Lie group concepts is well established on a solid theoretical foundation. This allows for modeling the forward kinematics of mechanisms as well the loop closure constraints of kinematic loops. Yet there is no established approach to the modeling of complex mechanisms possessing multiple kinematic loops. For such mechanisms, it is crucial to incorporate the kinematic topology within the modeling in a consistent and systematic way. To this end, in this paper a kinematic model graph is introduced that gives rise to an ordering of the joints within a mechanism and thus allows to systematically apply established kinematics formulations. It naturally gives rise to topologically independent loops and thus to loop closure constraints. Geometric constraints as well as velocity and acceleration constraints are formulated in terms of joint screws. An extension to higher order loop constraints is presented. It is briefly discussed how the topology representation can be used to amend structural mobility criteria.

Keywords

TopologyKinematicsConstraintsMobilityJacobianScrewsLie Groups
Type
Articles
Information
Robotica , Volume 36 , Issue 11 , November 2018 , pp. 1641 - 1663
Copyright
Copyright © Cambridge University Press 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Alizade, R., Bayram, C. and Gezgin, E., “Structural synthesis of serial platform manipulators,” Mech. Mach. Theory 42, 580–599 (2007).Google Scholar
2. Basu, D. and Ghosal, A., “Singularity analysis of platform-type multi-loop spatial mechanisms,” Mech. Mach. Theory 32 (3), 375–389 (1997).Google Scholar
3. Brockett, R. W., “Robotic Manipulators and the Product of Exponentials Formula,” Mathematical Theory of Networks and Systems, Lecture Notes in Control and Information Sciences, Vol. 58, pp 120–129 (1984).Google Scholar
4. de Bustos, I. F., et al., “Second order mobility analysis of mechanisms using closure equations,” In: Meccanica 47 (7), 1695–1704 (Springer, The Netherlands, 2012).Google Scholar
5. Chen, C., “The order of local mobility of mechanisms,” Mech. Mach. Theory 46, 1251–1264 (2011).Google Scholar
6. Davidson, J. K. and Hunt, K. H., Robots and Screw Theory: Applications of Kinematics and Statics to Robotics (Oxford University Press, Oxford, 2004).Google Scholar
7. Davis, T. H., “Mechanical networks – I,II,III,” Mech. Mach. Theory 18 (2), 95–112 (1983).Google Scholar
8. Davies, T. H., “Kirchhoff's circulation law applied to multi-loop kinematic chains,” Mech. Mach. Theory 16 (3), 171–183 (1981).Google Scholar
9. Davis, T. H., “A network approach to mechanisms and machines: Some lessons learned,” Mech. Mach. Theory 89, 14–27 (2015).Google Scholar
10. Ding, H., Zhao, J. and Huang, Z., “Unified structural synthesis of planar simple and multiple joint kinematic chains,” Mech. Mach. Theory 45 (4), 555–568 (2012).Google Scholar
11. Featherstone, R. Rigid Body Dynamics Algorithms (Springer Science and Business Media, New York, 2008).Google Scholar
12. Gogu, G., “Mobility of mechanism: A critical review,” Mech. Mach. Theory 40, 1068–1097 (2005).Google Scholar
13. Gogu, G., Structural Synthesis of Parallel Robots – Part 1: Methodology (Springer, The Netherlands, 2008).Google Scholar
14. Jain, A., “Graph theoretic foundations of multibody dynamics, Part I: Structural properties,” Multibody Syst. Dyn. 26, 307–333 (2011).Google Scholar
15. Hervé, J. M., “Analyse structurelle des mécanismes par groupe des déplacements,” Mech. Mach. Theory 13, 437–450 (1978).Google Scholar
16. Hervé, J. M., “Intrinsic formulation of problems of geometry and kinematics of mechanisms,” Mech. Mach. Theory 17 (3), 179–184 (1982).Google Scholar
17. Kieffer, J., “Differential analysis of bifurcations and isolated singularities of robots and mechanisms,” IEEE T. Rob. Automat. 10 (1), 1–10 (1994).Google Scholar
18. Laman, G., “On graphs and the rigidity of plane skeletal structures,” J. Eng. Math. 4, 331–340 (1970).Google Scholar
19. Lee, A. and Streinu, I., “Pebble Game Algorithms and (k, l)-Sparse Graphs,” Proceedings of the European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05) (2005).Google Scholar
20. Lee, A. and Streinu, I., “Pebble game algorithms and sparse graphs,” Discrete Math. 308 (8), 1425–1437 (1425).Google Scholar
21. Lerbet, J., “Analytic geometry and singularities of mechanisms,” ZAMM, Z. Angew. Math. Mech. 78 (10b), 687–694 (1999).Google Scholar
22. Merlet, J.-P., Optimal design of robots, Robotics: Science and Systems (Cambridge, MA, USA, 2005).Google Scholar
23. Müller, A., “On the terminology and geometric aspects of redundantly actuated parallel manipulators,” Robotica 31 (1), 137–147 (2013).Google Scholar
24. Müller, A., “Higher derivatives of the kinematic mapping and some applications,” Mech. Mach. Theory 76, 70–85 (2014).Google Scholar
25. Müller, A., “Representation of the kinematic topology of mechanisms for kinematic analysis,” Mech. Sci. 6, 1–10, (2015).Google Scholar
26. Müller, A., “Local kinematic analysis of closed-loop linkages -mobility, Singularities, and Shakiness,” ASME J. Mech. Rob. 8, 041013–1 (2016).Google Scholar
27. Müller, A., “Recursive higher-order constraints for linkages with lower kinematic pairs,” Mech. Mach. Theory 100, 33–43 (2016).Google Scholar
28. Müller, A., “Coordinate mappings for rigid body motions,” ASME J. Comput. Nonlinear Dyn. 12 (2), (2016).Google Scholar
29. Müller, A. and Shai, O., “Constraint graphs for combinatorial mobility determination,” Mech. Mach. Theory 108, 260–275 (2017).Google Scholar
30. Murray, R. M., Li, Z. and Sastry, S. S., A Mathematical Introduction to Robotic Manipulation (CRC Press, Boca Raton, 1994).Google Scholar
31. Park, F. C. and Lynch, K. M., Modern Robotics: Mechanics, Planning and Control (Cambridge University Press, Cambridge, 2017).Google Scholar
32. Ploen, S. R. and Park, F. C., “Coordinate-invariant algorithms for robot dynamics,” IEEE Tran. Rob. Automat. Soc. 15 (6), 1130–1135 (1999).Google Scholar
33. Rico, J. M., Gallardo, J. and Duffy, J., “Screw theory and higher order kinematic analysis of open serial and closed chains,” Mech. Mach. Theory 34 (4), 559–586 (1999).Google Scholar
34. Rico, J. M., Gallargo, J. and Ravani, B., “Lie algebra and the mobility of kinematic chains,” J. Robot. Syst. 20 (8), 477–499 (2003).Google Scholar
35. Rico, J. M. and Ravani, B., “On mobility analysis of linkages using group theory,” ASME J. Mech. Des. 125, 70–80 (2003).Google Scholar
36. Rico, J. M., Aguilera, L. D., Gallardo, J., Rodriguez, R., Orozco, H. and Barrera, J. M., “A more general mobility criterion for parallel platforms,” ASME J. Mech. Des. 128, 207–219 (2004).Google Scholar
37. Rojas, N. and Thomas, F., Forward Kinematics of the General Triple-Arm Robot Using a Distance-Based Formulation, In: Proc. Computational Kinematics (Zeghloul, Said, Romdhane, Lotfi and Larib, Med Amine, eds.) (Springer International Publishing, Cham, Switzerland, 2017) pp. 257–26.Google Scholar
38. Pozhbelko, V., “A unified structure theory of multibody open-, closed-, and mixed-loop mechanical systems with simple and multiple joint kinematic chains,” Mech. Mach. Theory 100, 1–16 (2016).Google Scholar
39. Schulze, B., Sljoka, A. and Whiteley, W., “How does symmetry impact the flexibility of proteins,” Philos. Trans. A Math. Phys. Eng. Sci. 372, (2014).Google Scholar
40. Selig, J., Geometric Fundamentals of Robotics (Monographs in Computer Science Series) (Springer-Verlag, New York, 2005).Google Scholar
41. Sljoka, A., Algorithms in Rigidity Theory with Applications to Protein Flexibility and Mechanical Linkages (York University, Toronto, 2012).Google Scholar
42. Sljoka, A., Shai, O. and Whiteley, W., “Checking Mobility and Decomposition of Linkages Via Pebble Game Algorithm,” ASME Design Engineering Technical Conferences (IDETC), August 28–31 (2011) Washington, USA.Google Scholar
43. Shai, O., Sljoka, A. and Whiteley, W., “Directed graphs,” Decompositions, Spatial Rigidity, Discrete Appl. Math. 161, 3028–3047 (2013).Google Scholar
44. Sugimoto, K., “Kinematic analysis and derivation of equations of motion for mechanisms with loops of different motion spaces (theoretical analysis),” JSME Int. J. Series C Mech. Syst., Mach. Elements Manuf. 44 (3), 610–617 (2001).Google Scholar
45. Talabă, D., “Mechanical models and the mobility of robots and mechanisms,” Robotica 33 (1), 181–193 (2015).Google Scholar
46. Uicker, J. J., Ravani, B. and Sheth, P. N. Matrix Methods in the Design Analysis of Mechanisms and Multibody Systems (Cambridge University Press, Cambridge, 2013).Google Scholar
47. Wei, G., Dai, S., Wang, S. and Luo, H., “Kinematic analysis and prototype of a metamorphic anthropomorphic hand with a reconfigurable palm,” Int. J. Human. Robot. 08 (3), 459–479 (2011).Google Scholar
48. White, N. and Whiteley, W., “The algebraic geometry of motions of bar-and-body frameworks,” SIAM J. Algebr. Discrete Meth. 8 (1), 1–32 (1987).Google Scholar
49. Whiteley, W., “Counting out to the flexibility of molecules,” Phys. Biol. 2, S116–S126 (2005).Google Scholar
50. Whiteley, W., “Rigidity and Scene Analysis,” In: Handbook of Discrete and Computational Geometry, 2nd ed. (Goodman, J. E. and O'Rourke, J., eds.) (Chapman and Hall/CRC, 2004).Google Scholar
51. Wittenburg, J., Dynamics of Systems of Rigid Bodies B. G. Teubner, Stuttgart (1977).Google Scholar
52. Wittenburg, J., Dynamics of Multibody Systems 2nd ed. (Springer-Verlag, Berlin Heidelberg, 2008).Google Scholar
53. Wohlhart, K., “Screw Spaces and Connectivities in Multiloop Linkages,” In: On Advances in Robot Kinematics (Lenarčič, J. and Galletti, C., eds.) (Springer, Dordrecht, 2004) pp. 97–104.Google Scholar
54. Yan, H.-S., Creative Design of Mechanical Devices (Springer, 1998).Google Scholar
55. Zlatanov, D., Bonev, I. A. and Gosselin, C. M., “Constraint Singularities as C-Space Singularities,” In: Advances in Robot Kinematics, Theory And Application (Lenarčič, J. and Thomas, F., eds.) (Springer, Dordrecht, 2002) 183–192.Google Scholar