When the leg rods of a fully in-parallel manipulator are fixed in their lengths, it is usual that the device can be assembled in several distinct ways. Sometimes it happens that motion between such assemblies can take place such that the linkage is never at a special configuration; that is, a configuration where the moving-platform body acquires uncontrollable freedom relative to the base. The possibility of such motion has implications for control. Focusing on 3–3 devices, we present a geometric explanation of how these motions arise, and give a sufficient condition for their existence. For the 3–3 planar-motion device, we show that never-special assembly changing motions can be excluded by making platform and base triangles similar, and we conjecture that appropriate, perhaps identical, specialization for the octahedral manipulator has the same effect.
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References
1.
Bruce, J. W., and Giblin, P. J. 1992. Curves and Singularities: A Geometrical Introduction to Singularity Theory, 2nd ed. Cambridge University Press .
2.
Burdick, J. W. 1988. Kinematic analysis and design of redundant manipulators. PhD thesis, Stanford University.
3.
Charentus, S., and Renaud, M. 1989. Modeling and control of a modular, redundant robot manipulator. Technical Report 89255 (June), Laboratoire d’Automatique et d’Analyse des Systèmes, Centre National de la Recherche Scientifique.
4.
Daniel, R. W., and McAree, P. R. 1998. Fundamental limits of performance for force-reflecting teleoperation . Intl. J. Robot. Res.17(8).
5.
Gosselin, C. M., and Merlet, J.-P. 1994. The direct kinematics of planar manipulators: Special architectures and number of solutions . Mech. Machine Theory29(8): 1083–1097 .
6.
Gosselin, C. M.Sefrioui, J., and Richard, M. J. 1992. Solutions polynomiales au probleme de la cinematique directe des manipulateurs paralleles plans a trios degres de liberte . Mech. Machine Theory27(2): 107–119 .
7.
Hartenberg, R. S., and Denavit, J. 1964. Kinematic Synthesis of Linkages (series in mechanical engineering.) New York: McGraw-Hill .
8.
Hunt, K. H. 1983. Structural kinematics of in-parallelactuated robot arms . Trans. ASME J. Mech. Trans. Automat. Design105: 705–712 .
9.
Hunt, K. H. 1990. Kinematic Geometry of Mechanisms. Oxford: Clarendon Press .
10.
Hunt, K. H., and McAree, P. R. 1998. The octahedral manipulator: Geometry and mobility . Intl. J. Robot. Res.17(8): 868–885 .
11.
Hunt, K. H., and Primrose, E. J. F. 1993. Assembly configurations of some in-parallel-actuated manipulators . Mech. Machine Theory28(1): 31–42 .
12.
Innocenti, C., and Parenti-Castelli, V. 1998. Singularity-free evolution from one configuration to another in serial and fully-parallel manipulators . Trans. ASME J. of Mech. Design120: 73–99 .
13.
Kailath, T. 1980. Linear Systems. Englewood Cliffs, NJ: Prentice-Hall .
14.
Lee, J., Duffy, J., and Hunt, K. H. 1998. A practical quality index based on the ocahedral manipulator . Intl. J. Robot. Res.17(10): 1081–1090 .
15.
McAree, P. R., and Daniel, R. W. 1996. A fast, robust solution to the Stewart platform forward kinematics . J. Robot. Sys.13(7): 407–427 .
16.
McAree, P. R., Gibson, C. G., Samuel, A. E., and Hunt, K. H. 1991. Dexterity measure for the kinematic control of a multifinger, multifreedom robot hand . Intl. J. Robot. Res.10(5): 439–453 .
17.
Merlet, J.-P. 1990. Les robots parallèles. Traité des Nouvelles Technologies, Série Robotique. France: Hermes .
18.
Merlet, J.-P. 1992. Direct kinematics and assembly modes of parallel manipulators . Intl. J. of Robot. Res.11(2): 150–162 .
19.
Müller, R. 1903. Zur Lehrevonder Moment an begegungeines starren eben Systems: Eine Eigenschaft der Burmesterschen Punkte (The theory of the momentary motion of the rigid plane system: A property of the Burmester points.) Z. Math. Phys. (Special Report): 210–216 . (Trans. 1962, Kansas State University Bulletin 46: 6).
20.
Nanua, P., Waldron, K. J., and Murthy, V. 1990. Direct kinematic solution of a Stewart platform . IEEE Trans. Robot. Automat.6(4): 438–444 .
21.
Pai, D. K., and Leu, M. C. 1989 (Scottsdale, AZ). Generic singularities of robot manipulators . Proc. of the IEEE Intl. Conf. on Robot. and Automat. Los Alamitos, CA: IEEE.
22.
Pieper, D. 1968. The kinematics of manipulators under computer control. PhD thesis, Stanford University.
23.
Poston, T., and Stewart, I. 1978. Catastrophe Theory and Its Applications. Pitman .
24.
Roberts, S. 1879. On three-bar motion in place space . Proc. London Math. Soc. 7-7 .
25.
Schoenflies, A. 1886. Geometrie der Bewegung in synthetischer Darstellung. Leipzig: Teubner . (See also the French trans., La Géomérie du Mouvement, Paris 1893.)
26.
Strang, G. 1988. Linear Algebra and Its Applications, 3rd ed. Orlando, FL: Harcourt Brace Jovanovich .
27.
Wampler, C. W. 1996. Forward displacement analysis of general six-in-parallel SPS (Stewart) platform manipulators using soma coordinates . Mech. Machine Theory31(3): 331–337 .
28.
Wenger, P. 1998. Classification of 3R positioning manipulators . Trans. ASME J. Mech. Design120(2): 327–332 .
29.
Wenger, P., and Chablat, D. 1998. Workspace and assembly modes in fully parallel manipulators: A descriptive study . Proc. of the 6th. Intl. Symp. on Advances in Robot Kinematics. Salzburg, Austria.
30.
Wenger, P., and Omri, J. E. 1995 (San Feliu de Glixol, Spain). How to recognize simply a nonsingular posture changing 3-DOF manipulator . Proc. of the 7th. Intl. Conf. on Adv. Robot.
31.
Whitney, H. 1955. On singularities of mappings of Euclidean space I: Mappings of the plane to the plane . Ann. Math62: 374–410 .
