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

Geometric characterization of the workspace of non-orthogonal rotation axes

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.
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.

[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.

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[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.

[9]

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

[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.

[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.

[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.

[13]

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

[14]

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

[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.

[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.

[17]

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

[18]

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

[19]

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

[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.

[21]

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

[22]

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

[23]

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

[24]

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

[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.

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.

[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.

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[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.

[9]

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

[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.

[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.

[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.

[13]

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

[14]

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

[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.

[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.

[17]

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

[18]

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

[19]

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

[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.

[21]

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

[22]

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

[23]

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

[24]

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

[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.

[1]

Zhen Lei. Rotation-strain decomposition for the incompressible viscoelasticity in two dimensions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2861-2871. doi: 10.3934/dcds.2014.34.2861

[2]

Palle E. T. Jorgensen and Steen Pedersen. Orthogonal harmonic analysis of fractal measures. Electronic Research Announcements, 1998, 4: 35-42.

[3]

Danny Calegari, Alden Walker. Ziggurats and rotation numbers. Journal of Modern Dynamics, 2011, 5 (4) : 711-746. doi: 10.3934/jmd.2011.5.711

[4]

Arek Goetz. Dynamics of a piecewise rotation. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 593-608. doi: 10.3934/dcds.1998.4.593

[5]

Xavier Buff, Nataliya Goncharuk. Complex rotation numbers. Journal of Modern Dynamics, 2015, 9: 169-190. doi: 10.3934/jmd.2015.9.169

[6]

Michel Laurent, Arnaldo Nogueira. Rotation number of contracted rotations. Journal of Modern Dynamics, 2018, 12: 175-191. doi: 10.3934/jmd.2018007

[7]

Congming Li, Jisun Lim. The singularity analysis of solutions to some integral equations. Communications on Pure & Applied Analysis, 2007, 6 (2) : 453-464. doi: 10.3934/cpaa.2007.6.453

[8]

Tyrus Berry, Timothy Sauer. Consistent manifold representation for topological data analysis. Foundations of Data Science, 2019, 1 (1) : 1-38. doi: 10.3934/fods.2019001

[9]

Christopher Cleveland. Rotation sets for unimodal maps of the interval. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 617-632. doi: 10.3934/dcds.2003.9.617

[10]

David Cowan. A billiard model for a gas of particles with rotation. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 101-109. doi: 10.3934/dcds.2008.22.101

[11]

Mads Kyed. On a mapping property of the Oseen operator with rotation. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1315-1322. doi: 10.3934/dcdss.2013.6.1315

[12]

Jifeng Chu, Meirong Zhang. Rotation numbers and Lyapunov stability of elliptic periodic solutions. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1071-1094. doi: 10.3934/dcds.2008.21.1071

[13]

Katja Polotzek, Kathrin Padberg-Gehle, Tobias Jäger. Set-oriented numerical computation of rotation sets. Journal of Computational Dynamics, 2017, 4 (1&2) : 119-141. doi: 10.3934/jcd.2017004

[14]

Ingrid Beltiţă, Anders Melin. The quadratic contribution to the backscattering transform in the rotation invariant case. Inverse Problems & Imaging, 2010, 4 (4) : 599-618. doi: 10.3934/ipi.2010.4.599

[15]

Deissy M. S. Castelblanco. Restrictions on rotation sets for commuting torus homeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5257-5266. doi: 10.3934/dcds.2016030

[16]

Héctor E. Lomelí. Heteroclinic orbits and rotation sets for twist maps. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 343-354. doi: 10.3934/dcds.2006.14.343

[17]

Salvador Addas-Zanata. Stability for the vertical rotation interval of twist mappings. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 631-642. doi: 10.3934/dcds.2006.14.631

[18]

Anna Belova. Rigorous enclosures of rotation numbers by interval methods. Journal of Computational Dynamics, 2016, 3 (1) : 81-91. doi: 10.3934/jcd.2016004

[19]

Paolo Antonelli, Daniel Marahrens, Christof Sparber. On the Cauchy problem for nonlinear Schrödinger equations with rotation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 703-715. doi: 10.3934/dcds.2012.32.703

[20]

Suddhasattwa Das, Yoshitaka Saiki, Evelyn Sander, James A. Yorke. Solving the Babylonian problem of quasiperiodic rotation rates. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 2279-2305. doi: 10.3934/dcdss.2019145

2018 Impact Factor: 0.525

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]