Access the full text.
Sign up today, get DeepDyve free for 14 days.
B. Lavenda (2017)
The Optical Properties of GravityJournal of Modern Physics, 08
E. Hairer, S. Ncrsett, G. Wanner (1993)
Solving ordinary differential equations h nonstiff problems
L. Collaboration, Virgo Abadie, B. Abbott, R. Abbott, M. Abernathy, T. Accadia, F. Acernese, C. Adams, R. Adhikari, P. Ajith, B. Allen, G. Allen, E. Ceron, R. Amin, S. Anderson, W. Anderson, F. Antonucci, S. Aoudia, M. Arain, M. Araya, M. Aronsson, K. Arun, Y. Aso, S. Aston, P. Astone, D. Atkinson, P. Aufmuth, C. Aulbert, S. Babak, P. Baker, G. Ballardin, S. Ballmer, D. Barker, S. Barnum, F. Barone, B. Barr, P. Barriga, L. Barsotti, M. Barsuglia, M. Barton, I. Bartos, R. Bassiri, M. Bastarrika, J. Bauchrowitz, T. Bauer, B. Behnke, M. Beker, M. Benacquista, A. Bertolini, J. Betzwieser, N. Beveridge, P. Beyersdorf, S. Bigotta, I. Bilenko, G. Billingsley, J. Birch, S. Birindelli, R. Biswas, M. Bitossi, M. Bizouard, E. Black, J. Blackburn, L. Blackburn, D. Blair, B. Bland, M. Blom, A. Blomberg, C. Boccara, O. Bock, T. Bodiya, R. Bondarescu, F. Bondu, L. Bonelli, R. Bork, M. Born, S. Bose, L. Bosi, M. Boyle, S. Braccini, C. Bradaschia, P. Brady, V. Braginsky, J. Brau, J. Breyer, D. Bridges, A. Brillet, M. Brinkmann, V. Brisson, M. Britzger, A. Brooks, D. Brown, R. Budzy'nski, T. Bulik, H. Bulten, A. Buonanno, J. Burguet-Castell, O. Burmeister, D. Buskulic, R. Byer, L. Cadonati, G. Cagnoli, E. Calloni, J. Camp, E. Campagna, P. Campsie, J. Cannizzo, K. Cannon, B. Canuel, J. Cao, C. Capano, F. Carbognani, S. Caride, S. Caudill, M. Cavaglià, F. Cavalier, R. Cavalieri, G. Cella, C. Cepeda, E. Cesarini, T. Chalermsongsak, E. Chalkley, P. Charlton, É. Chassande-Mottin, S. Chelkowski, Y. Chen, A. Chincarini, N. Christensen, S. Chua, C. Chung, D. Clark, J. Clark, J. Clayton, F. Cleva, E. Coccia, C. Colacino, J. Colas, A. Colla, M. Colombini, R. Conte, D. Cook, T. Corbitt, C. Corda, N. Cornish, A. Corsi, C. Costa, J. Coulon, D. Coward, D. Coyne, J. Creighton, T. Creighton, A. Cruise, R. Culter, A. Cumming, L. Cunningham, E. Cuoco, K. Dahl, S. Danilishin, R. Dannenberg, S. D’Antonio, K. Danzmann, A. Dari, K. Das, V. Dattilo, B. Daudert, M. Davier, G. Davies, A. Davis, E. Daw, R. Day, T. Dayanga, R. Rosa, D. DeBra, J. Degallaix, M. Prete, V. Dergachev, R. Derosa, R. DeSalvo, P. Devanka, S. Dhurandhar, L. Fiore, A. Lieto, I. Palma, M. Emilio, A. Virgilio, M. D'iaz, A. Dietz, F. Donovan, K. Dooley, E. Doomes, S. Dorsher, Ewan Douglas, M. Drago, R. Drever, J. Driggers, J. Dueck, J. Dumas, T. Eberle, M. Edgar, M. Edwards, A. Effler, P. Ehrens, R. Engel, T. Etzel, M. Evans, T. Evans, V. Fafone, S. Fairhurst, Y. Fan, B. Farr, D. Fazi, H. Fehrmann, D. Feldbaum, I. Ferrante, F. Fidecaro, L. Finn, I. Fiori, R. Flaminio, M. Flanigan, K. Flasch, S. Foley, C. Forrest, E. Forsi, N. Fotopoulos, J. Fournier, J. Franc, S. Frasca, F. Frasconi, M. Frede, M. Frei, Z. Frei, A. Freise, R. Frey, T. Fricke, D. Friedrich, P. Fritschel, V. Frolov, P. Fulda, M. Fyffe, L. Gammaitoni, J. Garofoli, F. Garufi, G. Gemme, E. Génin, A. Gennai, I. Gholami, S. Ghosh, J. Giaime, S. Giampanis, K. Giardina, A. Giazotto, C. Gill, E. Goetz, L. Goggin, G. Gonz'alez, M. Gorodetsky, S. Gossler, R. Gouaty, C. Graef, M. Granata, A. Grant, S. Gras, C. Gray, R. Greenhalgh, A. Gretarsson, C. Greverie, R. Grosso, H. Grote, S. Grunewald, G. Guidi, E. Gustafson, R. Gustafson, B. Hage, P. Hall, J. Hallam, D. Hammer, G. Hammond, J. Hanks, C. Hanna, J. Hanson, J. Harms, G. Harry, I. Harry, E. Harstad, K. Haughian, K. Hayama, J. Heefner, H. Heitmann, P. Hello, I. Heng, A. Heptonstall, M. Hewitson, S. Hild, E. Hirose, D. Hoak, K. Hodge, K. Holt, D. Hosken, J. Hough, E. Howell, D. Hoyland, D. Huet, B. Hughey, S. Husa, S. Huttner, T. Huynh--Dinh, D. Ingram, R. Inta, T. Isogai, A. Ivanov, P. Jaranowski, W. Johnson, D. Jones, G. Jones, R. Jones, L. Ju, P. Kalmus, V. Kalogera, S. Kandhasamy, J. Kanner, E. Katsavounidis, K. Kawabe, S. Kawamura, F. Kawazoe, W. Kells, D. Keppel, A. Khalaidovski, F. Khalili, E. Khazanov, C. Kim, H. Kim, P. King, D. Kinzel, J. Kissel, S. Klimenko, V. Kondrashov, R. Kopparapu, S. Koranda, I. Kowalska, D. Kozak, T. Krause, V. Kringel, S. Krishnamurthy, B. Krishnan, A. Kr'olak, G. Kuehn, J. Kullman, R. Kumar, P. Kwee, M. Landry, M. Lang, B. Lantz, N. Lastzka, A. Lazzarini, P. Leaci, J. Leong, I. Leonor, N. Leroy, N. Letendre, J. Li, T. Li, H. Lin, P. Lindquist, N. Lockerbie, D. Lodhia, M. Lorenzini, V. Loriette, M. Lormand, G. Losurdo, P. Lu, J. Luan, M. Lubinski, A. Lucianetti, H. Luck, A. Lundgren, B. Machenschalk, M. Macinnis, J. Mackowski, M. Mageswaran, K. Mailand, E. Majorana, C. Mak, N. Man, I. Mandel, V. Mandic, M. Mantovani, F. Marchesoni, F. Marion, S. M'arka, Z. M'arka, E. Maros, J. Marque, F. Martelli, I. Martin, R. Martin, J. Marx, K. Mason, A. Masserot, F. Matichard, L. Matone, R. Matzner, N. Mavalvala, R. McCarthy, D. McClelland, S. Mcguire, G. Mcintyre, G. McIvor, D. McKechan, G. Meadors, M. Mehmet, T. Meier, A. Melatos, A. Melissinos, G. Mendell, D. Men'endez, R. Mercer, L. Merill, S. Meshkov, C. Messenger, M. Meyer, H. Miao, C. Michel, L. Milano, J. Miller, Y. Minenkov, Y. Mino, S. Mitra, V. Mitrofanov, G. Mitselmakher, R. Mittleman, B. Moe, M. Mohan, S. Mohanty, S. Mohapatra, D. Moraru, J. Moreau, G. Moreno, N. Morgado, A. Morgia, T. Morioka, K. Mors, S. Mosca, V. Moscatelli, K. Mossavi, B. Mours, C. Mowlowry, G. Mueller, S. Mukherjee, A. Mullavey, H. Muller-Ebhardt, J. Munch, P. Murray, T. Nash, R. Nawrodt, J. Nelson, I. Neri, G. Newton, A. Nishizawa, F. Nocera, D. Nolting, E. Ochsner, J. O'Dell, G. Ogin, R. Oldenburg, B. O'reilly, R. O’Shaughnessy, C. Osthelder, D. Ottaway, R. Ottens, H. Overmier, B. Owen, A. Page, G. Pagliaroli, L. Palladino, C. Palomba, Y. Pan, C. Pankow, F. Paoletti, M. Papa, S. Pardi, M. Pareja, M. Parisi, A. Pasqualetti, R. Passaquieti, D. Passuello, P. Patel, M. Pedraza, L. Pekowsky, S. Penn, C. Peralta, A. Perreca, G. Persichetti, M. Pichot, M. Pickenpack, F. Piergiovanni, M. Pietka, L. Pinard, I. Pinto, M. Pitkin, H. Pletsch, M. Plissi (2010)
Predictions for the rates of compact binary coalescences observable by ground-based gravitational-wave detectorsClassical and Quantum Gravity, 27
M. Boyle (2013)
Angular velocity of gravitational radiation from precessing binaries and the corotating framePhysical Review D, 87
C. Bottasso, M. Borri (1998)
Integrating Finite RotationsComputer Methods in Applied Mechanics and Engineering, 164
Xiaoshan Lin, T. Ng (1995)
Contact detection algorithms for three-dimensional ellipsoids in discrete element modellingInternational Journal for Numerical and Analytical Methods in Geomechanics, 19
L. Dorst, Joan Lasenby (2011)
Guide to Geometric Algebra in Practice
Scott Johnson, John Williams, B. Cook (2008)
Quaternion‐based rigid body rotation integration algorithms for use in particle methodsInternational Journal for Numerical Methods in Engineering, 74
Dimin Wu, Zhengzhi Wang (2012)
Strapdown Inertial Navigation System Algorithms Based on Geometric AlgebraAdvances in Applied Clifford Algebras, 22
F. McRobie, Joan Lasenby (1999)
Simo-Vu Quoc rods using Clifford algebraInternational Journal for Numerical Methods in Engineering, 45
W. Press, S. Teukolsky, W. Vetterling, B. Flannery (2007)
Numerical Recipes 3rd Edition: The Art of Scientific Computing
L. Candy, Joan Lasenby (2011)
Attitude and Position Tracking
B. Gossick (1970)
A history of vector analysis: The evolution of the idea of a vectorial system: by Michael J. Crowe. 270 pages, diagrams, 6 x 9 in. Notre Dame, Indiana, University of Notre Dame Press, 1967.
B. Hall (2003)
Lie Groups, Lie Algebras, and Representations
(1984)
General Relativity, 1st edn
M. Boyle, Lawrence Kidder, S. Ossokine, H. Pfeiffer (2014)
Gravitational-wave modes from precessing black-hole binariesarXiv: General Relativity and Quantum Cosmology
C. Kilmister, D. Hestenes, G. Sobczyk (1984)
Clifford Algebra to Geometric CalculusThe Mathematical Gazette, 69
R. O’Shaughnessy, J. Kaplan, V. Kalogera, K. Belczynski (2005)
Bounds on Expected Black Hole Spins in Inspiraling BinariesThe Astrophysical Journal, 632
E. Zupan, M. Saje (2011)
Integrating rotation from angular velocityAdv. Eng. Softw., 42
Robin Miller (1983)
A new strapdown attitude algorithmJournal of Guidance Control and Dynamics, 6
M. Ignagni (1990)
Errata: Optimal Strapdown Attitude Integration AlgorithmsJournal of Guidance Control and Dynamics, 13
T. Vold (1993)
An introduction to geometric algebra with an application in rigid body mechanicsAmerican Journal of Physics, 61
J. Simo, K. Wong (1991)
Unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentumInternational Journal for Numerical Methods in Engineering, 31
Clifford Algebras Michael Boyle Cornell Center for Astrophysics and Planetary Science Cornell University Ithaca NY 14853 USA e-mail: boyle@astro
E. Hairer, C. Lubich, G. Wanner (2004)
Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations
Kyung Lee, Frank Raymond
Lie Groups
A. Lew, J. Marsden, M. Ortiz, Matthew West (2004)
Variational time integratorsInternational Journal for Numerical Methods in Engineering, 60
O. Walton, R. Braun (1993)
Simulation of rotary-drum and repose tests for frictional spheres and rigid sphere clusters
(2001)
Algebraic Topology, 1st edn
W. Magnus (1954)
On the exponential solution of differential equations for a linear operatorCommunications on Pure and Applied Mathematics, 7
E. Zupan, D. Zupan (2014)
On higher order integration of angular velocities using quaternionsMechanics Research Communications, 55
E. Jones, T. Oliphant, Pearu Peterson (2001)
SciPy: Open Source Scientific Tools for Python
P. Clifford
Applications of Grassmann's Extensive AlgebraAmerican Journal of Mathematics, 1
BP Abbott (2016)
The optical properties of gravityPhys. Rev. Lett., 116
A. Neumaier (2001)
Introduction to Numerical Analysis
M. Ignagni (1990)
Optimal strapdown attitude integration algorithmsJournal of Guidance Control and Dynamics, 13
C. Doran, D. Hestenes, F. Sommen, N. Acker (1993)
Lie-groups as Spin groups.Journal of Mathematical Physics, 34
F. Grassia (1998)
Practical Parameterization of Rotations Using the Exponential MapJ. Graphics, GPU, & Game Tools, 3
D. Hestenes (1983)
Celestial mechanics with geometric algebraCelestial mechanics, 30
Geir Bogfjellmo, Håkon Marthinsen (2013)
High-Order Symplectic Partitioned Lie Group MethodsFoundations of Computational Mathematics, 16
(1999)
Proceedings of the NSF/CBMS regional conference on numerical analysis of Hamiltonian differential equations
L. Blanchet (2002)
Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact BinariesLiving Reviews in Relativity, 5
A. Treven, M. Saje (2015)
Integrating rotation and angular velocity from curvatureAdv. Eng. Softw., 85
(1999)
Submitted to The Astrophysical Journal. Spin–Orbit Misalignment in Close Binaries with Two Compact Objects
Ravishankar Shivarama, E. Fahrenthold (2004)
Hamilton's Equations with Euler Parameters for Rigid Body Dynamics Modeling. Chapter 3
M. Ignagni (1996)
Efficient Class of Optimized Coning Compensation AlgorithmsJournal of Guidance Control and Dynamics, 19
C. Kane, J. Marsden, M. Ortiz, Matthew West (2000)
Variational Integrators and the Newmark Algorithm for Conservative and Dissipative Mechanical SystemsInternational Journal for Numerical Methods in Engineering, 49
C. Doran, A. Lasenby (2003)
Geometric Algebra for Physicists
A. Munjiza, J. Latham, N. John (2003)
3D dynamics of discrete element systems comprising irregular discrete elements—integration solution for finite rotations in 3DInternational Journal for Numerical Methods in Engineering, 56
(2012)
Kinematics in conformal geometric algebra with applications in strapdown inertial navigation
J. Simo (1985)
A finite strain beam formulation. The three-dimensional dynamic problem. Part IComputer Methods in Applied Mechanics and Engineering, 49
O. Woodman (2007)
An introduction to inertial navigation
(2017)
Symmetry Groups and Their Applications, Pure and Applied Mathematics
P. Grandclément, M. Ihm, V. Kalogera, K. Belczynski (2003)
Searching for Gravitational Waves from the Inspiral of Precessing Binary Systems: Astrophysical Expectations and Detection Efficiency of "Spiky'' Templates.Physical Review D, 69
A common problem in physics and engineering is determination of the orientation of an object given its angular velocity. When the direction of the angular velocity changes in time, this is a nontrivial problem involving coupled differential equations. Several possible approaches are examined, along with various improvements over previous efforts. These are then evaluated numerically by comparison to a complicated but analytically known rotation that is motivated by the important astrophysical problem of precessing black-hole binaries. It is shown that a straightforward solution directly using quaternions is most efficient and accurate, and that the norm of the quaternion is irrelevant. Integration of the generator of the rotation can also be made roughly as efficient as integration of the rotation. Both methods will typically be twice as efficient as naive vector- or matrix-based methods. Implementation by means of standard general-purpose numerical integrators is stable and efficient, so that such problems can be readily solved as part of a larger system of differential equations. Possible generalization to integration in other Lie groups is also discussed.
Advances in Applied Clifford Algebras – Springer Journals
Published: May 27, 2017
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.