Research Papers

A Distance Geometry Approach to the Singularity Analysis of 3R Robots

[+] Author and Article Information
Federico Thomas

Institut de Robòtica i Informàtica Industrial,
Llorens Artigas 4-6,
Barcelona 08028, Spain
e-mail: fthomas@iri.upc.edu

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received June 23, 2014; final manuscript received December 23, 2014; published online August 18, 2015. Assoc. Editor: Yuefa Fang.

J. Mechanisms Robotics 8(1), 011001 (Aug 18, 2015) (11 pages) Paper No: JMR-14-1146; doi: 10.1115/1.4029500 History: Received June 23, 2014

This paper shows how the computation of the singularity locus of a 3R robot can be reduced to the analysis of the relative position of two coplanar ellipses. Since the relative position of two conics is a projective invariant and the basic projective geometric invariants are determinants, it is not surprising that, using distance geometry, the computation of the singularity locus of a 3R robot can be fully expressed in terms of determinants. Geometric invariants have the benefit of simplifying symbolic manipulations. This paper shows how their use leads to a simpler characterization, compared to previous approaches, of the cusps and nodes in the singularity loci of 3R robots.

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Wenger, P. , 2007, “Cuspidal and Non-Cuspidal Robot Manipulators,” Robotica, 25(6), pp. 677–689. [CrossRef]
Wenger, P. , 2000, “Some Guidelines for the Kinematic Design of New Manipulators,” Mech. Mach. Theory, 35(3), pp. 437–449. [CrossRef]
Paul, R. , and Zhang, H. , 1986, “Computationally Efficient Kinematics for Manipulators With Spherical Wrists Based on the Homogeneous Transformation Representation,” Int. J. Rob. Res., 5(2), pp. 32–44. [CrossRef]
Husty, M. , Ottaviano, E. , and Ceccarelli, M. , 2008, “A Geometrical Characterization of Workspace Singularities in 3R Manipulators,” Advances in Robot Kinematics: Analysis and Design, J. Lenarčič , and Ph. Wenger , eds., Springer, Dordrecht, The Netherlands, pp. 411–418.
Pieper, D. L. , 1968, “The Kinematics of Manipulators Under Computer Control,” Ph.D. thesis, Department of Mechanical Engineering, Stanford University, Stanford, CA.
Kovács, P. , and Hommel, G. , 1993, “On the Tangent-Half-Angle Substitution,” Computational Kinematics, J. Angeles , G. Hommel , and P. Kovács , eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, pp. 27–39.
Smith, D. R. , and Lipkin, H. , 1993, “Higher Order Singularities of Regional Manipulators,” IEEE International Conference on Robotics and Automation, Atlanta, GA, May 2–6, Vol. 1, pp. 194–199.
Smith, D. R. , and Lipkin, H. , 1990, “Kinematic Analysis of Solvable Manipulators Using Conic Sections,” 21st ASME Mechanisms Conference, Chicago, IL, Sept. 16–19, pp. 16–19.
Ceccarelli, M. , 1989, “On the Workspace of 3R Robot Arms,” 5th IFToMM International Symposium on Theory and Practice of Mechanism, Bucharest, Romania, July 6–11, Vol. II–1, pp. 37–46.
Zein, M. , 2007, “Analyse cinématique des manipulateurs sériels 3R orthogonaux et des manipulateurs parallèles plans,” Ph.D. dissertation, École Central de Nantes, Université de Nantes, Nantes, France.
Bamberger, H. , Wolf, A. , and Shoham, M. , 2008, “Assembly Mode Changing in Parallel Mechanisms,” IEEE Trans. Rob., 24(4), pp. 765–772. [CrossRef]
Saramago, S. F. P. , Ottaviano, E. , and Ceccarelli, M. , 2002, “A Characterization of the Workspace Boundary of Three-Revolute Manipulators,” ASME Paper No. DETC2002/MECH-34342.
Wenger, P. , 1997, “Design of Cuspidal and Non-Cuspidal Robot Manipulators,” IEEE International Conference on Robotics and Automation, Albuquerque, NM, Apr. 20–25, pp. 2172–2177.
Baili, M. , 2004, “Analyse et classification de manipulateurs 3R à axes orthogonaux,” Ph.D. dissertation, École Central de Nantes, Université de Nantes, Nantes, France.
Donelan, P. , and Müller, A. , 2011, “General Formulation of the Singularity Locus for a 3-DOF Regional Manipulator,” IEEE International Conference on Robotics and Automation (ICRA), Shanghai, China, May 9-13, pp. 3958–3963.
Thomas, F. , 2014, “Computing Cusps of 3R Robots Using Distance Geometry,” 14th International Symposium on Advances in Robot Kinematics (ARK2014), Ljubljana, Slovenia, June 29–July 3.
Thomas, F. , and Ros, L. , 2005, “Revisiting Trilateration for Robot Localization,” IEEE Trans. Rob., 21(1), pp. 93–101. [CrossRef]
Faucette, W. M. , 1996, “A Geometric Interpretation of the Solution of the General Quartic Polynomial,” Am. Math. Mon., 103(1), pp. 51–57. [CrossRef]
Bôcher, M. , 1915, Plane Analytic Geometry, Henry Holt and Co., New York, pp. 176–188.
Choi, Y.-K. , Wang, W. , Liu, Y. , and Kim, M.-S. , 2006, “Continuous Collision Detection for Two Moving Elliptic Disks,” IEEE Trans. Rob., 22(2), pp. 213–224. [CrossRef]
Sommerville, D. M. Y. , 1961, Analytical Conics, G. Bell & Sons, London, UK, p. 274.
Richter-Gebert, J. , 2011, Perspectives on Projective Geometry: A Guided Tour Through Real and Complex Geometry, Springer, Dordrecht, The Netherlands, p. 191.
Salmon, G. , 1869, A Treatise on Conic Sections, Chelsea Publishing Co., New York.
Elizalde, B. , Alberich-Carramiñana, M. , and Thomas, F. , “On the Relative Position of Two Coplanar Ellipses” (unpublished).
Dickson, L. E. , 1914, Elementary Theory of Equations, Wiley, New York.
Blinn, J. F. , 2002, “Polynomial Discriminants. I. Matrix Magic,” IEEE Comput. Graphics Appl., 20(6), pp. 94–98. [CrossRef]
Dolgachev, I. V. , 2012, Classical Algebraic Geometry: A Modern View, Cambridge University Press, Cambridge, UK, p. 107.
Srinivasiengar, C. N. , 1927, “On the Conditions for the Double Contact of Two Conics,” J. Mysore Univ., 1(2), pp. 110–111.
Sylvester, J. J. , 1904, “Additions to the Articles'On a New Class of Theorems' and ‘On Pascal's Theorem,’” Philos. Mag., 37(251), pp. 363–370, 1850 (reprinted in 1904, J. J. Sylvester's Mathematical Papers, Vol. 1, Cambridge University Press, Cambridge, UK, pp. 145–151).
Morris, R. , 1997, “The Use of Computer Graphics for Solving Problems in Singularity Theory,” Visualization and Mathematics, Experiments, Simulations and Environments, H. C. Hege , and K. Polthier , eds., Springer, Dordrecht, The Netherlands, pp. 53–66.
Thomas, F. , and Wenger, P. , 2011, “On the Topological Characterization of Robot Singularity Loci. A Catastrophe-Theoretic Approach,” IEEE International Conference on Robotics and Automation (ICRA), Shanghai, China, May 9–13, pp. 3940–3945.


Grahic Jump Location
Fig. 1

A 3R robot and associated notation

Grahic Jump Location
Fig. 3

Left: the equivalent bar-and-joint framework associated with the 3R robot in Fig. 1. Right: this framework can be split by the plane defined by P3, P4, and P7 into two subassemblies, each containing a tetrahedron (shown as a shaded volume) and a triangle.

Grahic Jump Location
Fig. 5

Center: plot of the curves defined by det(A)=0 (in gray) and Δ = 0 (in green). These curves segment the plane into regions where the spatial relationship between A and B is the same. We are only interested in the region where A is a real ellipse (that is, the region where det(A)≤0). Left column: spatial relationships between A (in red) and B (in blue) associated with different regions of this plane. Right column: spatial relationships between A and B in different points of the singularity locus.

Grahic Jump Location
Fig. 4

The Denavit–Hartenberg parameters of the robot used as an example and its schematic representation including orthogonal sections of its singularity locus in the robot's workspace

Grahic Jump Location
Fig. 6

Shaded depth map of  log (abs(Δ(s1,7,s2,7))). The robot's singularity locus appears as valleys of this map. The two points marked with white dots correspond to configurations unreachable by the robot where A and B have a double contact in the complex domain.

Grahic Jump Location
Fig. 7

What it seemed to be a higher order singularity in Fig. 5, it is revealed to be a node close to meet two cusps

Grahic Jump Location
Fig. 8

The singularity locus shown in Fig. 5 (center) mapped onto the robot's workspace (ρ,z). The two singularities at ρ = 0 correspond to the two tangencies between the curves defined by Δ = 0 and det(A)=0 in the distance space (s1,7, s2,7).

Grahic Jump Location
Fig. 9

Plot of δ2 = 0 (in red) and δ3 = 0 (in green). Observe how both curves intersect at the cusps of the singularity locus (light gray).

Grahic Jump Location
Fig. 10

Left: plot of the line (in red) and ellipse (in green) resulting from substituting λ1 = -2.3439 in Eqs. (33) and (34), respectively. Right: the same for λ2 = -0.5461. Observe in the first case the intersection points do not lie in the singularity locus represented in light gray.




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