Motor parameterization. (English) Zbl 1397.51014
Summary: In this paper, we consider several parameterizations of rigid transformations using motors in 3-D conformal geometric algebra. In particular, we present parameterizations based on the exponential, outer exponential, and Cayley maps of bivectors, as well as a map based on a first-order approximation of the exponential followed by orthogonal projection onto the group manifold. We relate these parameterizations to the matrix representations of rigid transformations in the 3-D special Euclidean group. Moreover, we present how these maps can be used to form retraction maps for use in manifold optimization, retractions being approximations of the exponential map that preserve the convergence properties of the optimization method while being less computationally expensive, and, for the presented maps, also easier to implement.
MSC:
| 51N99 | Analytic and descriptive geometry |
| 15A66 | Clifford algebras, spinors |
| 70E15 | Free motion of a rigid body |
| 51N20 | Euclidean analytic geometry |
References:
| [1] | Absil, P-A; Baker, CG; Gallivan, KA, Trust-region methods on Riemannian manifolds, Found. Comput. Math., 7, 303-330, (2007) · Zbl 1129.65045 · doi:10.1007/s10208-005-0179-9 |
| [2] | Absil, P.-A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton (2008) · Zbl 1147.65043 · doi:10.1515/9781400830244 |
| [3] | Adler, RL; Dedieu, JP; Margulies, JY; Martens, M; Shub, M, Newton’s method on Riemannian manifolds and a geometric model for the human spine, IMA J. Numer. Anal., 22, 359-390, (2002) · Zbl 1056.92002 · doi:10.1093/imanum/22.3.359 |
| [4] | Baker, C.G.: Riemannian Manifold Trust-Region Methods with Applications to Eigenproblems. PhD thesis, College of Arts and Sciences, Florida State University (2008) |
| [5] | Bayro-Corrochano, E, Motor algebra approach for visually guided robotics, Pattern Recognit., 35, 279-294, (2002) · Zbl 0988.68194 · doi:10.1016/S0031-3203(00)00183-7 |
| [6] | Bayro-Corrochano, E; Zhang, Y, The motor extended Kalman filter: a geometric approach for rigid motion estimation, J. Math. Imaging Vis., 13, 205-228, (2000) · Zbl 0969.68172 · doi:10.1023/A:1011293515286 |
| [7] | Boumal, N; Mishra, B; Absil, P-A; Sepulchre, R, Manopt, a Matlab toolbox for optimization on manifolds, J. Mach. Learn. Res., 15, 1455-1459, (2014) · Zbl 1319.90003 |
| [8] | Celledoni, E; Owren, B, Lie group methods for rigid body dynamics and time integration on manifolds, Comput. Methods Appl. Mech. Eng., 192, 421-438, (2003) · Zbl 1018.70007 · doi:10.1016/S0045-7825(02)00520-0 |
| [9] | Chasles, M.: Notes sur les Propriétés Générales du Système de Deux Corps Semblades entr’eux. Bulletin de Sciences Mathematiques, Astronomiques Physiques et Chimiques, pp. 321-326 (1830) |
| [10] | Clifford, WK, Preliminary sketch of biquaternions, Proc. Lond. Math. Soc., 4, 361-395, (1873) · JFM 05.0280.01 |
| [11] | Davidson, J.K., Hunt, K.H.: Robots and Screw Theory: Applications of Kinematics and Statics to Robotics. Oxford University Press Inc., New York (2004) · Zbl 1070.68138 |
| [12] | Doran, C., Lasenby, A.: Geometric Algebra for Physicists. Cambridge University Press, Cambridge (2003) · Zbl 1078.53001 · doi:10.1017/CBO9780511807497 |
| [13] | Dorst, L., Fontijne, D., Mann, S.: Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry. Morgan Kaufmann Publishers Inc., San Francisco (2007) |
| [14] | Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer series in computational mathematics. Springer, Berlin (2006) · Zbl 1094.65125 |
| [15] | Helmstetter, J, Exponentials of bivectors and their symplectic counterparts, Adv. Appl. Clifford Algebras, 18, 689-698, (2008) · Zbl 1177.15028 · doi:10.1007/s00006-008-0094-7 |
| [16] | Iserles, A; Munthe-Kaas, HZ; Nørsett, SP; Zanna, A, Lie-group methods, Acta Numer., 9, 215-365, (2000) · Zbl 1064.65147 · doi:10.1017/S0962492900002154 |
| [17] | Li, H.: Parameterization of 3D Conformal Transformations in Conformal Geometric Algebra. Springer, London, pp. 71-90 (2010) · Zbl 1202.68450 |
| [18] | Lounesto, P.: Cayley transform, outer exponential and spinor norm. In: Proceedings of the Winter School “Geometry and Physics”. Circolo Matematico di Palermo, pp. 191-198 (1987) · Zbl 0652.15022 |
| [19] | Lounesto, P.: Clifford Algebras and Spinors. London Mathematical Society lecture note series, 2nd edn. Cambridge University Press, Cambridge (2001) · Zbl 0973.15022 · doi:10.1017/CBO9780511526022 |
| [20] | Ma, Y; Košecká, J; Sastry, S, Optimization criteria and geometric algorithms for motion and structure estimation, Int. J. Comput. Vis., 44, 219-249, (2001) · Zbl 0998.68197 · doi:10.1023/A:1012276232049 |
| [21] | McCarthy, J.M., Soh, G.S.: Geometric design of linkages. Springer, New York, NY (2011) · Zbl 1227.70003 |
| [22] | Riesz, M.: Clifford Numbers and Spinors (Chapters I-IV). Springer, Dordrecht, pp. 1-196 (1993) · Zbl 1018.70007 |
| [23] | Sarkis, M; Diepold, K, Camera-pose estimation via projective Newton optimization on the manifold, IEEE Trans. Image Process., 21, 1729-1741, (2012) · Zbl 1373.94362 · doi:10.1109/TIP.2011.2177845 |
| [24] | Selig, J.M.: Cayley maps for se(3). In: 12th International Federation for the Promotion of Mechanism and Machine Science World Congress, p. 6 (2007) |
| [25] | Selig, JM, Exponential and Cayley maps for dual quaternions, Adv. Appl. Clifford Algebras, 20, 923-936, (2010) · Zbl 1215.70007 · doi:10.1007/s00006-010-0229-5 |
| [26] | Smith, S.T.: Geometric Optimization Methods for Adaptive Filtering. PhD thesis, Harvard University (1993) · Zbl 0988.68194 |
| [27] | Tingelstad, L; Egeland, O, Automatic multivector differentiation and optimization, Adv. Appl. Clifford Algebras, 27, 707-731, (2017) · Zbl 1367.65034 · doi:10.1007/s00006-016-0722-6 |
| [28] | Tingelstad, L; Egeland, O, Motor estimation using heterogeneous sets of objects in conformal geometric algebra, Adv. Appl. Clifford Algebras, 27, 2035-2049, (2017) · Zbl 1410.70009 · doi:10.1007/s00006-016-0692-8 |
| [29] | Townsend, J; Koep, N; Weichwald, S, Pymanopt: a python toolbox for optimization on manifolds using automatic differentiation, J. Mach. Learn. Res., 17, 1-5, (2016) · Zbl 1416.65580 |
| [30] | Valkenburg, R; Dorst, L; Dorst, L (ed.); Lasenby, J (ed.), Estimating motors from a variety of geometric data in 3D conformal geometric algebra, 25-45, (2011), London · Zbl 1290.68126 · doi:10.1007/978-0-85729-811-9_2 |
| [31] | Vandereycken, B.: Riemannian and multilevel optimization for rank-constrained matrix problems. PhD thesis, Department of Computer Science, KU Leuven (2010) · Zbl 1290.68126 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
