American Institute of Mathematical Sciences

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June  2014, 6(2): 141-166. doi: 10.3934/jgm.2014.6.141

Geometric characterization of the workspace of non-orthogonal rotation axes

Bertold Bongardt 1,

1. 

DFKI GmbH, Robotics Innovation Center, Robert-Hooke-Straße 1, 28359 Bremen, Germany

Received  August 2013 Revised  May 2014 Published  June 2014

In this article, a novel characterization of the workspace of 3R chains with non-orthogonal, intersecting axes is derived by describing the set of singular orientations as two tori that separate two-solvable from non-solvable orientations within $SO(3)$. Therefore, the tori provide the boundary of the workspace of the axes' constellation. The derived characterization generalizes a recent result obtained by Piovan and Bullo. It is based on a specific, novel representation of rotations, called unit ball representation, which allows to interpret the workspace characterization with ease. In an appendix, tools for dealing with angles and rotations are introduced and the equivalence of unit quaternion representation and unit ball representation is described.
Keywords: rotation representation, rotation decomposition, kinematic analysis, Non-orthogonal Euler angles, singularity analysis..
Mathematics Subject Classification: Primary: 70B10; Secondary: 22E70, 15A1.
Citation: Bertold Bongardt. Geometric characterization of the workspace of non-orthogonal rotation axes. Journal of Geometric Mechanics, 2014, 6 (2) : 141-166. doi: 10.3934/jgm.2014.6.141
References:
[1]

B. Alpern, L. Carter, M. Grayson and C. Pelkie, Orientation maps: Techniques for visualizing rotations (a consumer's guide),, in VIS '93: Proceedings of the 4th conference on Visualization, (1993), 183. Google Scholar

[2]

S. Bai and J. Angeles, A unified input-output analysis of four-bar linkages,, Mechanism and Machine Theory, 43 (2008), 240. doi: 10.1016/j.mechmachtheory.2007.01.002. Google Scholar

[3]

O. A. Bauchau and L. Trainelli, The Vectorial Parameterization of Rotation,, Nonlinear Dynamics, 32 (2003), 71. doi: 10.1023/A:1024265401576. Google Scholar

[4]

B. Bongardt, Sheth-Uicker Convention Revisited - A Normal Form for Specifying Mechanisms,, Technical report, (2012). Google Scholar

[5]

P. B. Davenport, Rotations about nonorthogonal axes,, AIAA Journal, 11 (1973), 853. Google Scholar

[6]

L. Dorst, D. Fontijne and S. Mann, Geometric Algebra - An Object-Oriented Approach to Geometry,, Morgan Kaufmann Series in Computer Graphics, (2007). Google Scholar

[7]

F. Freudenstein, Approximate synthesis of four-bar linkages,, Transaction ASME, 77 (1955), 853. Google Scholar

[8]

K. C. Gupta, Kinematic analysis of manipulators using the zero reference position description,, International Journal of Robotics Research, 5 (1986), 5. doi: 10.1177/027836498600500202. Google Scholar

[9]

A. J. Hanson, Visualizing Quaternions,, Morgan Kaufmann, (2007). doi: 10.1145/1281500.1281634. Google Scholar

[10]

M. J. D. Hayes, K. Parsa and J. Angeles, The effect of data-set cardinality on the design and structural errors of four-bar function-generators,, in 10th World Congress on the Theory of Machines and Mechanisms, (1999), 437. Google Scholar

[11]

M. Husty, E. Ottaviano and M. Ceccarelli, A Geometrical characterization of workspace singularities in 3r manipulators,, in Advances in Robot Kinematics: Analysis and Design, (2008), 411. doi: 10.1007/978-1-4020-8600-7_43. Google Scholar

[12]

M. L. Husty, M. Pfurner and H.-P. Schröcker, A new and efficient algorithm for the inverse kinematics of a general serial 6R manipulator,, Mechanism and Machine Theory, 42 (2007), 66. doi: 10.1016/j.mechmachtheory.2006.02.001. Google Scholar

[13]

F. Klein, Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis,, 1932., (). Google Scholar

[14]

J. B. Kuipers, Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace and Virtual Reality,, Princeton University Press, (2002). Google Scholar

[15]

Z. Liu and J. Angeles, Least-square optimization of planar and spherical four-bar function generator under mobility constraints,, Journal of Mechanical Design, 114 (1992), 569. doi: 10.1115/1.2917045. Google Scholar

[16]

C. D. Mladenova and I. M. Mladenov, Vector decomposition of finite rotations,, Reports on Mathematical Physics, 68 (2011), 107. doi: 10.1016/S0034-4877(11)60030-X. Google Scholar

[17]

R. M. Murray, S. S. Sastry and L. Zexiang, A Mathematical Introduction to Robotic Manipulation,, CRC Press, (1994). Google Scholar

[18]

R. P. Paul and C. N. Stevenson, Kinematics of robot wrists,, International Journal of Robotics Research, 2 (1983), 31. doi: 10.1177/027836498300200103. Google Scholar

[19]

M. Pfurner, Analysis of Spatial Serial Manipulators Using Kinematic Mapping,, PhD thesis, (2006). Google Scholar

[20]

G. Piovan and F. Bullo, On coordinate-free rotation decomposition: Euler angles about arbitrary axes,, IEEE Transactions on Robotics, 28 (2012), 728. doi: 10.1109/TRO.2012.2184951. Google Scholar

[21]

J. M. Selig, Geometric Fundamentals of Robotics, 2nd ed.,, Springer, (2005). Google Scholar

[22]

J. R. Shewchuk, Lecture Notes on Geometric Robustness,, 2009., (). Google Scholar

[23]

M. D. Shuster and F. L. Markley, Generalization of the euler angles,, The Journal of the Astronautical Sciences, 51 (2003), 123. Google Scholar

[24]

J. Stillwell, Naive Lie Theory,, Undergraduate Texts in Mathematics, (2008). doi: 10.1007/978-0-387-78214-0. Google Scholar

[25]

A. T. Yang and F. Freudenstein, Application of dual-number quaternion algebra to the analysis of spatial mechanisms,, Journal of Applied Mechanics, 31 (1964), 300. doi: 10.1115/1.3629601. Google Scholar

show all references

References:
[1]

B. Alpern, L. Carter, M. Grayson and C. Pelkie, Orientation maps: Techniques for visualizing rotations (a consumer's guide),, in VIS '93: Proceedings of the 4th conference on Visualization, (1993), 183. Google Scholar

[2]

S. Bai and J. Angeles, A unified input-output analysis of four-bar linkages,, Mechanism and Machine Theory, 43 (2008), 240. doi: 10.1016/j.mechmachtheory.2007.01.002. Google Scholar

[3]

O. A. Bauchau and L. Trainelli, The Vectorial Parameterization of Rotation,, Nonlinear Dynamics, 32 (2003), 71. doi: 10.1023/A:1024265401576. Google Scholar

[4]

B. Bongardt, Sheth-Uicker Convention Revisited - A Normal Form for Specifying Mechanisms,, Technical report, (2012). Google Scholar

[5]

P. B. Davenport, Rotations about nonorthogonal axes,, AIAA Journal, 11 (1973), 853. Google Scholar

[6]

L. Dorst, D. Fontijne and S. Mann, Geometric Algebra - An Object-Oriented Approach to Geometry,, Morgan Kaufmann Series in Computer Graphics, (2007). Google Scholar

[7]

F. Freudenstein, Approximate synthesis of four-bar linkages,, Transaction ASME, 77 (1955), 853. Google Scholar

[8]

K. C. Gupta, Kinematic analysis of manipulators using the zero reference position description,, International Journal of Robotics Research, 5 (1986), 5. doi: 10.1177/027836498600500202. Google Scholar

[9]

A. J. Hanson, Visualizing Quaternions,, Morgan Kaufmann, (2007). doi: 10.1145/1281500.1281634. Google Scholar

[10]

M. J. D. Hayes, K. Parsa and J. Angeles, The effect of data-set cardinality on the design and structural errors of four-bar function-generators,, in 10th World Congress on the Theory of Machines and Mechanisms, (1999), 437. Google Scholar

[11]

M. Husty, E. Ottaviano and M. Ceccarelli, A Geometrical characterization of workspace singularities in 3r manipulators,, in Advances in Robot Kinematics: Analysis and Design, (2008), 411. doi: 10.1007/978-1-4020-8600-7_43. Google Scholar

[12]

M. L. Husty, M. Pfurner and H.-P. Schröcker, A new and efficient algorithm for the inverse kinematics of a general serial 6R manipulator,, Mechanism and Machine Theory, 42 (2007), 66. doi: 10.1016/j.mechmachtheory.2006.02.001. Google Scholar

[13]

F. Klein, Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis,, 1932., (). Google Scholar

[14]

J. B. Kuipers, Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace and Virtual Reality,, Princeton University Press, (2002). Google Scholar

[15]

Z. Liu and J. Angeles, Least-square optimization of planar and spherical four-bar function generator under mobility constraints,, Journal of Mechanical Design, 114 (1992), 569. doi: 10.1115/1.2917045. Google Scholar

[16]

C. D. Mladenova and I. M. Mladenov, Vector decomposition of finite rotations,, Reports on Mathematical Physics, 68 (2011), 107. doi: 10.1016/S0034-4877(11)60030-X. Google Scholar

[17]

R. M. Murray, S. S. Sastry and L. Zexiang, A Mathematical Introduction to Robotic Manipulation,, CRC Press, (1994). Google Scholar

[18]

R. P. Paul and C. N. Stevenson, Kinematics of robot wrists,, International Journal of Robotics Research, 2 (1983), 31. doi: 10.1177/027836498300200103. Google Scholar

[19]

M. Pfurner, Analysis of Spatial Serial Manipulators Using Kinematic Mapping,, PhD thesis, (2006). Google Scholar

[20]

G. Piovan and F. Bullo, On coordinate-free rotation decomposition: Euler angles about arbitrary axes,, IEEE Transactions on Robotics, 28 (2012), 728. doi: 10.1109/TRO.2012.2184951. Google Scholar

[21]

J. M. Selig, Geometric Fundamentals of Robotics, 2nd ed.,, Springer, (2005). Google Scholar

[22]

J. R. Shewchuk, Lecture Notes on Geometric Robustness,, 2009., (). Google Scholar

[23]

M. D. Shuster and F. L. Markley, Generalization of the euler angles,, The Journal of the Astronautical Sciences, 51 (2003), 123. Google Scholar

[24]

J. Stillwell, Naive Lie Theory,, Undergraduate Texts in Mathematics, (2008). doi: 10.1007/978-0-387-78214-0. Google Scholar

[25]

A. T. Yang and F. Freudenstein, Application of dual-number quaternion algebra to the analysis of spatial mechanisms,, Journal of Applied Mechanics, 31 (1964), 300. doi: 10.1115/1.3629601. Google Scholar

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