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The integration of angular velocity. (English) Zbl 1410.70005

Summary: 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.

MSC:

70B10 Kinematics of a rigid body
15A66 Clifford algebras, spinors
65L05 Numerical methods for initial value problems involving ordinary differential equations

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[1] Abbott, B.P., et al.: The optical properties of gravity. Phys. Rev. Lett. 116, 061102 (2016). doi:10.1103/Phys-RevLett.116.061102 · doi:10.1103/Phys-RevLett.116.061102
[2] Blanchet, L.: Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries Living Rev. Relativ. 17(2), 2014. doi:10.12942/lrr-2014-2 · Zbl 1316.83003
[3] Bogfjellmo, G., Marthinsen, H.: High order symplectic partitioned Lie group methods (2013). arXiv:1303.5654 [grqc] · Zbl 1346.65069
[4] Bottasso, C.L., Borri, M.: Integrating finite rotations. Comput. Methods Appl. Mech. Eng. 164, 307 (1998) · Zbl 0961.74029 · doi:10.1016/S0045-7825(98)00031-0
[5] Boyle, M.: Angular velocity of gravitational radiation from precessing binaries and the corotating frame. Phys. Rev. D 87, 104006 (2013)
[6] Boyle, M.; Lindblom, L.; Pfeiffer, HP; Scheel, MA; Kidder, LE, No article title, Phys. Rev. D, 75, 024006 (2007) · doi:10.1103/PhysRevD.75.024006
[7] Boyle, M., Kidder, L.E., Ossokine, S., Pfeiffer, H.P.: Gravitational-wave modes from precessing black-hole binaries’ (2014). arXiv:1409.4431 [gr-qc]
[8] Buonanno, A.; Chen, Y.; Vallisneri, M., No article title, Phys. Rev. D, 67, 104025 (2003) · doi:10.1103/PhysRevD.67.104025
[9] Candy, L.P.: Kinematics in conformal geometric algebra with applications in strapdown inertial navigation., Ph.D. thesis. University of Cambridge, Great Britain (2012)
[10] Candy, L., Lasenby, J.: Attitude and position tracking. In: Dorst, L., Lasenby, J. (eds.) Guide to Geometric Algebra in Practice, pp. 105-125. Springer London (2011) · Zbl 1291.70003
[11] Clifford, W.K.: Applications of grassmann’s extensive algebra. Am. J. Math. 1, 350 (1878) · JFM 10.0297.02
[12] Crouch, PE; Grossman, R., No article title, J. Nonlinear Sci., 3, 1 (1993) · Zbl 0798.34012 · doi:10.1007/BF02429858
[13] Crowe, M.J.: A History of Vector Analysis: The Evolution of the Idea of a Vectorial System. Dover, New York (1985) · Zbl 0633.01001
[14] Doran, C., Lasenby, A.: Geometric algebra for physicists, 4th edn. Cambridge University Press, Cambridge (2010) · Zbl 1135.53001
[15] Doran, C., Hestenes, D., Sommen, F., Acker, N.V.: Lie groups as spin groups. J. Math. Phys. 34, 3642 (1993) · Zbl 0810.15014 · doi:10.1063/1.530050
[16] Duistermaat, J.J., Kolk, J.A.C.: Lie Groups. Springer, Berlin (2000) (see Sec. 1.5) · Zbl 0955.22001
[17] Grandclément, P., Ihm, M., Kalogera, V., Belczynski, K.: Searching for gravitational waves from the inspiral of precessing binary systems: astrophysical expectations and detection efficiency of “spiky” templates. Phys. Rev. D 69, 102002 (2004) · doi:10.1103/PhysRevD.69.102002
[18] Grassia, F.S.: Practical parameterization of rotations using the exponential map. J. Graph Tools 3, 29 (1998) · doi:10.1080/10867651.1998.10487493
[19] Hairer, E., Wanner, G., Nørsett, S.P.: Solving Ordinary Differential Equations I: Nonstiff Problems, Springer Series in Computational Mathematics, vol. 8. Springer, Berlin (1993) · Zbl 0789.65048
[20] Hairer, E., Wanner, G., Lubich, C.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn. Springer, New York (2006) · Zbl 1094.65125
[21] Hall, B.: Lie Groups, Lie Algebras, and Representations, Graduate Texts in Mathematics, vol. 222. Springer International Publishing (2015) · Zbl 1316.22001
[22] Hatcher, A.: Algebraic Topology, 1st edn. Cambridge University Press, New York (2001) · Zbl 1044.55001
[23] Hestenes, D.: Celestial mechanics with geometric algebra. Celestial Mech. 30, 151 (1983) · Zbl 0529.70011 · doi:10.1007/BF01234303
[24] Hestenes, D., Sobczyk, G.: Clifford Algebra to Geometric Calculus. Kluwer Academic Publishers, Norwell (1987) · Zbl 0541.53059
[25] Ignagni, M.B.: Optimal strapdown attitude integration algorithms. J. Guid. Control Dyn. 13, 363 (1990) · Zbl 0701.70021 · doi:10.2514/3.20558
[26] Ignagni, M.B.: Errata: Optimal strapdown attitude integration algorithms. J. Guid. Control Dyn. 13, 0576b (1990) · Zbl 0701.70021
[27] Ignagni, M.B.: Efficient class of optimized coning compensation algorithms. J. Guid. Control Dyn. 19, 424 (1996) · Zbl 0858.70018 · doi:10.2514/3.21635
[28] Johnson, S.M., Williams, J.R., Cook, B.K.: Quaternion-based rigid body rotation integration algorithms for use in particle methods. Int. J. Numer. Methods Eng. 74, 1303 (2008) · Zbl 1159.70300 · doi:10.1002/nme.2210
[29] Jones, E., Oliphant, T., Peterson, P. et al.: SciPy: open source scientific tools for Python (online) (2001). Accessed 11 March 2016
[30] Kalogera, V.: Spin-orbit misalignment in close binaries with two compact objects. Astrophys J 541, 319 (2000) · doi:10.1086/309400
[31] Kane, C., Marsden, J.E., Ortiz, M., West, M.: Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems. Int. J. Numer. Methods Eng. 49, 1295 (2000) · Zbl 0969.70004 · doi:10.1002/1097-0207(20001210)49:10<1295::AID-NME993>3.0.CO;2-W
[32] L. S. Collaboration; V. Collaboration.: Predictions for the rates of compact binary coalescences observable by ground-based gravitational-wave detectors. Class. Quantum Gravity 27, 173001 (2010) · Zbl 1343.65095
[33] Lew, A., Marsden, J.E., Ortiz, M., West, M.: Variational time integrators. Int. J. Numer. Methods Eng. 60, 153 (2004) · Zbl 1060.70500 · doi:10.1002/nme.958
[34] Lin, X., Ng, T.: Contact detection algorithms for three-dimensional ellipsoids in discrete element modelling. Int. J. Numer. Anal. Methods Geomech. 19, 653 (1995) · Zbl 0834.73076 · doi:10.1002/nag.1610190905
[35] Magnus, W.: On the exponential solution of differential equations for a linear operator. Commun. Pure Appl. Math. 7, 649 (1954) · Zbl 0056.34102 · doi:10.1002/cpa.3160070404
[36] McRobie, F.A., Lasenby, J.: Simo-Vu Quoc rods using Clifford algebra. Int. J. Numer. Methods Eng. 45, 377 (1999) · Zbl 0940.74081 · doi:10.1002/(SICI)1097-0207(19990610)45:4<377::AID-NME586>3.0.CO;2-P
[37] Miller, W.: Symmetry Groups and Their Applications, Pure and Applied Mathematics. Academic Press, New York (1972) (see Lemma 5.3) · Zbl 0306.22001
[38] Miller, R.B.: A new strapdown attitude algorithm. J. Guid. Control Dyn. 6, 287 (1983) · doi:10.2514/3.19831
[39] Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation, 1st edn. W.H. Freeman, San Francisco (1973)
[40] Munjiza, A., Latham, J.P., John, N.W.M.: Dynamics of discrete element systems comprising irregular discrete elements—integration solution for finite rotations in 3D. Int. J. Numer. Methods Eng. 56, 35 (2003) · Zbl 1027.70002 · doi:10.1002/nme.552
[41] Munthe-Kaas, H.: Proceedings of the NSF/CBMS regional conference on numerical analysis of Hamiltonian differential equations. Appl. Numer. Math. 29, 115 (1999) · Zbl 0934.65077 · doi:10.1016/S0168-9274(98)00030-0
[42] O’Shaughnessy, R., Kaplan, J., Kalogera, V., Belczynski, K.: Bounds on expected black hole spins in inspiraling binaries. Astrophys. J. 632, 1035 (2005) · doi:10.1086/444346
[43] Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes 3rd Edition: The Art of Scientific Computing, 3rd edn. Cambridge University Press, New York (2007) · Zbl 1132.65001
[44] Shivarama, R., Fahrenthold, E.P.: Hamilton’s equations with Euler parameters for rigid body dynamics modeling. J. Dyn. Syst. Meas. Control 126, 124 (2004) · Zbl 1065.74072 · doi:10.1115/1.1649977
[45] Simo, J.C.: A finite strain beam formulation. The three-dimensional dynamic problem. Comput. Methods Appl. Mech. Eng. 49, 55 (1985) · Zbl 0583.73037 · doi:10.1016/0045-7825(85)90050-7
[46] Simo, J.C., Wong, K.K.: Unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum. Int. J. Numer. Methods Eng. 31, 19 (1991) · Zbl 0825.73960 · doi:10.1002/nme.1620310103
[47] Stoer , J., Bulirsch, R.: Introduction to Numerical Analysis, Texts in Applied Mathematics, vol. 12. Springer, New York (2002) · Zbl 1004.65001
[48] Treven, A., Saje, M.: Integrating rotation and angular velocity from curvature. Adv. Eng. Softw. 85, 26 (2015) · doi:10.1016/j.advengsoft.2015.02.010
[49] Vold, T.G.: Introduction to geometric algebra with an application to rigid body mechanics. Am. J. Phys. 61, 491 (1993) · Zbl 1219.70007 · doi:10.1119/1.17201
[50] Wald, R.M.: General Relativity, 1st edn. University of Chicago Press, Chicago (1984) · Zbl 0549.53001 · doi:10.7208/chicago/9780226870373.001.0001
[51] Walton, O.R., Braun, R.L.: Simulation of rotary-drum and repose tests for frictional spheres and rigid sphere clusters. In: Plasynski, S.I., Peters, W.C., Roco, M.C. (eds.) Flow of Particulates and Fluids: Proceedings, Joint DOE/NSF Workshop on Flow of Particulates and Fluids, Ithaca. National Technical Information Service (1993)
[52] Woodman, O.J.: An Introduction to Inertial Navigation, Technical Report UCAM-CL-TR-696. University of Cambridge, Computer Laboratory, Cambridge (2007)
[53] Wu, D., Wang, Z.: Strapdown inertial navigation system algorithms based on geometric algebra. Adv. Appl. Clifford Algebras 22, 1151 (2012) · Zbl 1255.83113 · doi:10.1007/s00006-012-0326-8
[54] Zupan, E., Saje, M.: Integrating rotation from angular velocity. Adv. Eng. Softw. 42, 723 (2011) · Zbl 1343.65095 · doi:10.1016/j.advengsoft.2011.05.010
[55] Zupan, E., Zupan, D.: On higher order integration of angular velocities using quaternions. Mech. Res. Commun. 55, 77 (2014) · Zbl 1329.57016 · doi:10.1016/j.mechrescom.2013.10.022
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