Abstract
In this chapter, the classical line geometry is modeled in ℝ3,3, where lines are represented by null vectors, and points and planes by null 3-blades. The group of 3D special projective transformations SL(4) when acting upon points in space induces a Lie group isomorphism, with SO(3,3) acting upon lines.
As an application of the use of the ℝ3,3 model of line geometry, this chapter analyzes the inverse singularity analysis of generalized Stewart platforms, using vectors of ℝ3,3 to encode the force and torque wrenches to classify their singular configurations.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Editorial note: See also Chap. 14.
References
Ball, R.: The Theory of Screws: A Study in the Dynamics of a Rigid Body. Hodges, Foster (1876)
Basu, D., Ghosal, A.: Singularity analysis of platform-type multi-loop spatial mechanism. Mech. Mach. Theory 32, 375–389 (1997)
Ben-Horin, P., Shoham, M.: Singularity analysis of a class of parallel robots based on Grassmann–Cayley algebra. Mech. Mach. Theory 41, 958–970 (2006)
Busemann, H.: Projective Geometry and Projective Metrics. Academic Press, New York (1953)
Cayley, A.: On the six coordinates of a line. Trans. Camb. Philos. Soc. 5, 290–323 (1869)
Collins, C., Long, G.: Singularity analysis of an in-parallel hand controller for force-reflected teleoperation. IEEE J. Robot. Autom. 11, 661–669 (1995)
Dandurand, A.: The rigidity of compound spatial grids. Struct. Topol. 10, 41–56 (1984)
Fang, Y., Tsai, L.: Structure synthesis of a class of 4-DoF and 5-DoF parallel manipulators with identical limb structures. Int. J. Robot. Res. 21, 799–810 (2002)
Featherstone, R.: Robot Dynamics Algorithms. Springer, Berlin (1987)
Gao, X., Lei, D., Liao, Q., Zhang, G.: Generalized Stewart–Gough platforms and their direct kinematics. IEEE Trans. Robot. Autom. 21, 141–151 (2005)
Gosselin, C., Angeles, J.: Singularity analysis of closed-loop kinematic chains. IEEE Trans. Robot. Autom. 6, 281–290 (1990)
Hodge, W., Pedoe, D.: Methods of Algebraic Geometry, vol. 1. Cambridge University Press, Cambridge (1952)
Huang, Z., Chen, L., Li, W.: The singularity principle and property of Stewart parallel manipulator. J. Robot. Syst. 20, 163–176 (2003)
Hunt, K.: Kinematic Geometry of Mechanisms. Oxford University Press, Oxford (1978)
Jenner, W.: Rudiments of Algebraic Geometry. Oxford University Press, Oxford (1963)
Li, H.: Invariant Algebras and Geometric Reasoning. World Scientific, Singapore (2008)
Long, G.: Use of the cylindroid for the singularity analysis of rank 3 robot manipulator. Mech. Mach. Theory 32, 391–404 (1997)
Maxwell, E.: General Homogeneous Coordinates in Spaces of Three Dimensions. Cambridge University Press, Cambridge (1951)
Merlet, J.: Singular configurations of parallel manipulators and Grassmann geometry. Int. J. Robot. Res. 8, 45–56 (1989)
Merlet, J.: Parallel Robots. 2nd edn. Springer, Heidelberg (2006)
Park, F., Kim, J.: Singularity analysis of closed kinematics chains. J. Mech. Des. 121, 32–38 (1999)
Pottmann, H.: Computational Line Geometry. Springer, Heidelberg (2001)
Semple, J., Roth, L.: Introduction to Algebraic Geometry. Oxford University Press, Oxford (1949)
Sommerville, D.: Analytic Geometry of Three Dimensions. Cambridge University Press, Cambridge (1934)
Stewart, D.: A platform with six degrees of freedom. Proc. Inst. Mech. Eng. 180, 371–378 (1965)
Study, E.: Geometrie der Dynamen. Leipzig (1903)
Woo, L., Freudenstein, F.: Application of line geometry to theoretical kinematics and the kinematic analysis of mechanical systems. J. Mech. 5, 417–460 (1970)
Yang, A.: Calculus of screws. In: Spillers, W. (ed.) Basic Questions of Design Theory. Elsevier, Amsterdam (1974)
Zlatanov, D., Fenton, R., Benhabib, B.: Identification and classification of the singular configurations of mechanisms. Mech. Mach. Theory 33, 743–760 (1998)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag London Limited
About this chapter
Cite this chapter
Li, H., Zhang, L. (2011). Line Geometry in Terms of the Null Geometric Algebra over ℝ3,3, and Application to the Inverse Singularity Analysis of Generalized Stewart Platforms. In: Dorst, L., Lasenby, J. (eds) Guide to Geometric Algebra in Practice. Springer, London. https://doi.org/10.1007/978-0-85729-811-9_13
Download citation
DOI: https://doi.org/10.1007/978-0-85729-811-9_13
Publisher Name: Springer, London
Print ISBN: 978-0-85729-810-2
Online ISBN: 978-0-85729-811-9
eBook Packages: Computer ScienceComputer Science (R0)