# Another Map from P^7 to the Study Quadric

Journal article

Selig, Jon (2020). Another Map from P^7 to the Study Quadric.

*Computer Aided Geometric Design.*77, p. 101816.

Authors | Selig, Jon |
---|---|

Abstract | In [5] Pfurner, Schröcker and Husty introduced a mapping from P^7 to the Study quadric. In [9], it was shown that this map could be thought of as the composition of an extended version of the inverse Cayley map based on the 6x6 adjoint representation of the group, and the Cayley map itself. Here, the analogous map using the Cayley map based on the standard 4x4 representation of SE(3) is studied. It is shown that mapping a general line in P^7 results in a motion with cubic trajectories. A different view of the map is then studied. A birational map between the Study quadric and the variety defined by the adjoint representation of the group is given. The new map is then the composition of the map from the Study quadric, extended to all P^7, with the map from the P^17 back to the Study quadric. The effect of the new map on symmetric subspaces of SE(3) is also considered. Lastly, an example is given showing how the interpolation techniques can be extended to point constraints. That is, where a point on the body is required to pass through a sequence of successive points in space. |

Keywords | Interpolation of rigid-body motions; Dual quaternions |

Year | 2020 |

Journal | Computer Aided Geometric Design |

Journal citation | 77, p. 101816 |

Publisher | Elsevier |

ISSN | 0167-8396 |

Digital Object Identifier (DOI) | doi:10.1016/j.cagd.2020.101816 |

Web address (URL) | https://www.sciencedirect.com/science/article/pii/S0167839620300030 |

Publication dates | |

Online | 23 Jan 2020 |

Print | 01 Feb 2020 |

Publication process dates | |

Accepted | 08 Jan 2020 |

Deposited | 09 Jan 2020 |

Accepted author manuscript | License CC BY-NC-ND 4.0 File Access Level Open |

https://openresearch.lsbu.ac.uk/item/88w5q

## Restricted files

### Accepted author manuscript

##### 26

total views##### 2

total downloads##### 2

views this month##### 0

downloads this month

## Export as

## Related outputs

##### On Plane-Symmetric Rigid-Body Motions

Bayril, D. and Selig, J. (2020). On Plane-Symmetric Rigid-Body Motions.*Journal of Geometry.*111 (29).

##### On the Plane Symmetric Bricard Mechanism

Selig, J. (2020). On the Plane Symmetric Bricard Mechanism. Lenarčič, J. and Siciliano, B. (ed.)*17 International Symposium on Advances in Robot Kinematics.*Ljubljana, Slovenia 28 Jun - 02 Jul 2020 Springer Verlag.

##### Rigid Body Dynamics using Equimomental Systems of Point-Masses

Laus, L. and Selig, J. (2019). Rigid Body Dynamics using Equimomental Systems of Point-Masses.*Acta Mechanica.*pp. 1-16.

##### Parallel Robots with Homokinetic Joints:The Zero-Torsion Case

Wu, Y, Selig, J and Carricato, M (2019). Parallel Robots with Homokinetic Joints:The Zero-Torsion Case.*15th IFToMM World Congress.*Krakov, Poland 30 Jun - 04 Jul 2019 doi:10.1007/978-3-030-20131-9_27

##### On the Dynamics of a Ball Rolling on a Tipping Plane

Laus, LP and Selig, JM (2019). On the Dynamics of a Ball Rolling on a Tipping Plane.*15th IFToMM World Congress,.*Krakov, Poland, 30 Jun - 04 Jul 2019 doi:10.1007/978-3-030-20131-9_194

##### Double Bennett Mechanisms with Assembly Modes of Different Dimensions

Selig, JM and Li, Z (2018). Double Bennett Mechanisms with Assembly Modes of Different Dimensions.*4th International Conference on Reconfigurable Mechanisms and Robots.*Delft 20 - 22 Jun 2018 Institute of Electrical and Electronics Engineers (IEEE). doi:10.1109/REMAR.2018.8449894

##### Displacement Varieties for Some PUP Linkages

Selig, JM (2018). Displacement Varieties for Some PUP Linkages.*16th International Symposium on Advances in Robot Kinematics.*Bologna, Italy 01 - 05 Jul 2018 Springer. doi:10.1007/978-3-319-93188-3_4

##### Some rational vehicle motions

Selig, JM (2013). Some rational vehicle motions.*6th International Workshop on Computational Kinematics (CK2013).*Barcelona, Spain 12 - 15 May 2013 London South Bank University. doi:10.1007/978-94-007-7214-4-3

##### Equimomental systems and robot dynamics

Selig, JM (2015). Equimomental systems and robot dynamics.*IMA Conference on Mathematics of Robotics.*St Anne’s College, University of Oxford 09 - 11 Sep 2015 London South Bank University. pp. ? - ? (8)

##### Some Remarks on the RRR Linkage

Selig, JM (2014). Some Remarks on the RRR Linkage. in: Lenarcic, J and Khatib, O (ed.) Advances in robot kinematics: analysis and design Switzerland Springer International Publishing. pp. 77 - 85 (9)##### The complex of lines from successive points and the horopter

Selig, JM (2008). The complex of lines from successive points and the horopter.*IEEE International Conference on Robotics and Automation.*California 19 - 23 May 2008 London South Bank University. pp. 2380 - 2385 doi:10.1109/ROBOT.2008.4543569

##### A screw theory of static beams

Selig, JM and Ding, X (2001). A screw theory of static beams.*IEEE International Conference on Intelligent Robots and Systems.*San Diego, CA, USA 29 Oct - 03 Nov 2001 London South Bank University. doi:10.1109/IROS.2001.973376

##### Mobile Icosapods

Gallet, M, Nawratil, G, Schicho, J and Selig, JM (2017). Mobile Icosapods.*Advances in Applied Mathematics.*88 (July), pp. 1-25.

##### Motion Interpolation in Lie Subgroups and Symmetric Subspaces

Selig, JM, Wu, Y and Carricato, M (2017). Motion Interpolation in Lie Subgroups and Symmetric Subspaces.*7th IFToMM International Workshop of Computational Kinematics.*Futurescope, Poitiers, France 22 - 24 May 2017 Springer. doi:10.1007/978-3-319-60867-9_53

##### Hyperbolic pseudoinverses for kinematics in the Euclidean group

Selig, JM (2017). Hyperbolic pseudoinverses for kinematics in the Euclidean group.*SIAM Journal on Matrix Analysis and Applications.*38 (4), pp. 1541-1559.

##### Note on the Principle of Transference

Selig, JM (1986). Note on the Principle of Transference.*Design Engineering Technical Conference.*Columbus, Ohio London South Bank University.

##### Scaling Direct Drive Robots

Wallace, R and Selig, JM (1995). Scaling Direct Drive Robots.*Robotics and Automation, 1995 IEEE International Conference on.*Nagoya, Japan 21 - 27 May 1995 Wallace.

##### Clifford algebra of points, lines and planes

Selig, JM (1999).*Clifford algebra of points, lines and planes.*London South Bank University School of Computing, Infromations Systems and Mathematics, South Bank University, London.

##### Some remarks on the statistics of pose estimation

Selig, JM (2000).*Some remarks on the statistics of pose estimation.*School of Computing, Information Systems and Mathematics, South Bank University, London.

##### Some mathematical problems in robotics

Selig, JM (2000).*Some mathematical problems in robotics.*School of Computing, Information Systems and Mathematics, South Bank University, London.

##### Introduction to Polynomial Invariants of Screw Systems

Donelan, P and Selig, JM (2007).*Introduction to Polynomial Invariants of Screw Systems.*arXiv.

##### Centrodes and Lie algebra

Selig, JM (2007). Centrodes and Lie algebra.*2th IFT oMM World Congress.*Besancon France 18 - 21 Jun 2007 London South Bank University. pp. ? - ? (6)

##### Cayley maps for SE(3)

Selig, JM (2007). Cayley maps for SE(3).*12th International Federation for the Promotion of Mechanism and Machine Science World Congress.*Besancon France London South Bank University. pp. ? - ? (6)

##### A geometric Newton-Raphson method for Gough-Stewart platforms

Selig, JM and Li, H (2009). A geometric Newton-Raphson method for Gough-Stewart platforms.*Fifth International Workshop on Computational Kinematics.*University of Duisburg-Essen, Germany 06 - 08 May 2009 London South Bank University. pp. 183 - 190 doi:10.1007/978-3-642-01947-0_23

##### Some rigid-body constraint varieties generated by linkages

Selig, JM (2012). Some rigid-body constraint varieties generated by linkages. in: Lenarcic, J and Husty, M (ed.) Latest Advances in Robot Kinematics Dordrecht Springer.##### A screw syzygy with applications to robot singularity computation

Selig, JM and Donelan, P (2008). A screw syzygy with applications to robot singularity computation. in: Advances in Robot Kinematics: Analysis and Design Netherlands Springer Science+Business Media B.V. pp. 147 - 154##### Three problems in robotics

Selig, JM (2002). Three problems in robotics.*Symposium Commemorating the Legacy, Works, and Life of Sir Robert Stawell Ball.*University of Cambridge 09 - 11 Jul 2000 London South Bank University. doi:10.1243/0954406021524927

##### On the geometry of the homogeneous representation for the group of proper rigid-body displacements

Selig, JM (2013). On the geometry of the homogeneous representation for the group of proper rigid-body displacements.*Romanian Journal of Technical Sciences - Applied Mechanics.*58 (1-2), pp. 5 - 28.

##### A Class of Explicitly Solvable Vehicle Motion Problems

Selig, JM (2015). A Class of Explicitly Solvable Vehicle Motion Problems.*IEEE Transactions on Robotics.*31 (3), pp. 766 - 777 (11).

##### Persistent rigid-body motions and Study's "Ribaucour" problem

Selig, JM and Carricato, M (2016). Persistent rigid-body motions and Study's "Ribaucour" problem.*Journal of Geometry.*108 (1), pp. 149-169.

##### On the use of the Klein quadric for geometric incidence problems in two dimensions

Rudnev, M and Selig, JM (2016). On the use of the Klein quadric for geometric incidence problems in two dimensions.*Siam Journal on Discrete Mathematics.*30 (2), pp. 934 - 954 (21).

##### Some Mobile Overconstrained Parallel Mechanisms

Selig, JM (2016). Some Mobile Overconstrained Parallel Mechanisms.*Advances in Robot Kinematics.*Grasse, France 27 - 30 Jun 2016

##### Rational interpolation of car motions

Selig, JM (2015). Rational interpolation of car motions.*Journal of Mechanisms and Robotics.*7 (3).

##### On the line geometry of rigid-body inertia

Selig, JM and Martins, D (2014). On the line geometry of rigid-body inertia.*Acta Mechanica.*225 (11), pp. 3073 - 3101.

##### Characterisation of Frenet-Serret and Bishop motions with applications to needle steering

Selig, JM (2013). Characterisation of Frenet-Serret and Bishop motions with applications to needle steering.*Robotica.*31 (6), pp. 981 - 992.

##### Half-turns and line symmetric motions

Selig, JM and Husty, M (2011). Half-turns and line symmetric motions.*Mechanism and Machine Theory.*46 (2), pp. 156 - 167.

##### On the instantaneous acceleration of points in a rigid body

Selig, JM (2011). On the instantaneous acceleration of points in a rigid body.*Mechanism and Machine Theory.*46 (10), pp. 1522 - 1535.

##### On the geometry of point-plane constraints on rigid-body displacements

Selig, JM (2011). On the geometry of point-plane constraints on rigid-body displacements.*Acta Applicandae Mathematicae.*116 (2), pp. 133 - 155.

##### Rigid body dynamics using Clifford algebra

Selig, JM and Bayro-Corrochano, E (2010). Rigid body dynamics using Clifford algebra.*Advances in Applied Clifford Algebras.*20 (1), pp. 141 - 154.

##### Quadratic constraints on rigid-body displacements

Selig, JM (2010). Quadratic constraints on rigid-body displacements.*Journal of Mechanisms and Robotics.*2 (4).

##### Exponential and cayley maps for dual quaternions

Selig, JM (2010). Exponential and cayley maps for dual quaternions.*Advances in Applied Clifford Algebras.*20 (3-4), pp. 923 - 936.

##### Manipulating robots along helical trajectories

Selig, JM and Ovseevitch, AI (2009). Manipulating robots along helical trajectories.*Robotica.*14 (3), pp. 261-267.

##### A screw theory of Timoshenko beams

Selig, JM and Ding, X (2009). A screw theory of Timoshenko beams.*Journal of Applied Mechanics, Transactions ASME.*76 (3), pp. 1 - 7.

##### Curves of stationary acceleration in SE(3)

Selig, JM (2007). Curves of stationary acceleration in SE(3).*IMA Journal of Mathematical Control and Information.*24 (1), pp. 95 - 113.

##### Active versus passive transformations in robotics

Selig, JM (2006). Active versus passive transformations in robotics.*IEEE Robotics and Automation Magazine.*13 (1), pp. 79 - 84.

##### Interpolated rigid-body motions and robotics

Selig, JM and Yuanqing, W (2006). Interpolated rigid-body motions and robotics.*Proceedings of the 2006 IEEE/RSJ International Conference on Intelligent Robots and Systems.*Beijing, China 09 - 15 Oct 2006 Institute of Electrical and Electronics Engineers (IEEE). pp. 1086 - 1091 doi:10.1109/IROS.2006.281815

##### Dynamics of vibratory bowl feeders

Selig, JM and Dai, JS (2005). Dynamics of vibratory bowl feeders.*Proceedings of the 2005 IEEE International Conference on Robotics and Automation April 2005.*Barcelona, Spain Apr 2005 Institute of Electrical and Electronics Engineers (IEEE). doi:10.1109/ROBOT.2005.1570617

##### On the compliance of coiled springs

Ding, X and Selig, JM (2004). On the compliance of coiled springs.*International Journal of Mechanical Sciences.*46 (5), pp. 703 - 727.

##### Structure of the spatial stiffness matrix

Selig, JM and Ding, X (2002). Structure of the spatial stiffness matrix.*International Journal of Robotics and Automation.*17 (1), pp. 1 - 16.

##### Diagonal spatial stiffness matrices

Selig, JM and Ding, X (2002). Diagonal spatial stiffness matrices.*International Journal of Robotics and Automation.*17 (2), pp. 100 - 106.

##### Theory of vibrations in Stewart platforms

Selig, JM and Ding, X (2001). Theory of vibrations in Stewart platforms.*IEEE International Conference on Intelligent Robots and Systems.*San Diego, CA USA Institute of Electrical and Electronics Engineers (IEEE). doi:10.1109/IROS.2001.976395

##### Spatial stiffness matrix from simple stretched springs

Selig, JM (2000). Spatial stiffness matrix from simple stretched springs.*IEEE International Conference on Robotics & Automation.*San Francisco Ca. USA 24 - 28 Apr 2000 Institute of Electrical and Electronics Engineers (IEEE). doi:10.1109/ROBOT.2000.845218

##### Clifford algebra of points, lines and planes

Selig, JM (2000). Clifford algebra of points, lines and planes.*Robotica.*18 (5), pp. 545 - 556.