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Research Papers

A Reparametrization of the Rotation Matrix in Rigid-Body Dynamics

[+] Author and Article Information
Xiaoqing Zhu

School of Electronic Information
and Control Engineering,
Beijing University of Technology,
Beijing 100124, China
Centre for Intelligent Machines,
Department of Mechanical Engineering,
McGill University,
Montréal, QC H3A 2K6, Canada
e-mails: alex.zhuxq@gmail.com; alexzhu@cim.mcgill.ca

Jorge Angeles

Professor
Fellow ASME
Centre for Intelligent Machines,
Department of Mechanical Engineering,
McGill University,
Montréal, QC H3A 2K6, Canada
e-mail: angeles@cim.mcgill.ca

To avoid confusion, the same notation used in Ref. [24] is adopted here.

These two matrix norms are based on the singular values of the matrix in question, which are frame-invariant.

Redefinition of Q(0) as Q0 ≠ 1 amounts to multiplying Q(t) by Q0 from the left, for t ≥ T.

Given that the condition number is bounded from below by 1, but unbounded from above, we prefer to plot its reciprocal, which is bounded from above by 1, from below by 0.

No need of subscripts here, because the system under study comprises one single body, the wheel.

ϖ is read “varpi”.

The simulation video is available on http://www.cim.mcgill.ca/rmsl/Index/index.htm, under Research\Dynamics.

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received September 22, 2014; final manuscript received March 6, 2015; published online March 31, 2015. Assoc. Editor: Alexander F. Vakakis.

J. Appl. Mech 82(5), 051003 (May 01, 2015) (9 pages) Paper No: JAM-14-1435; doi: 10.1115/1.4030006 History: Received September 22, 2014; Revised March 06, 2015; Online March 31, 2015

The parametrization of a rigid-body rotation is a classical subject in rigid-body dynamics. Euler angles, the rotation matrix and quaternions are the most common representations. However, Euler angles are known to be prone to singularities, besides not being frame-invariant. The full 3 × 3 rotation matrix conveys all the motion information, but poses the problem of an excessive number of parameters, nine, to represent a transformation that entails only three independent parameters. Quaternions are singularity-free, and thus, ideal to study rigid-body kinematics. However, quaternions, comprising four components, are subject to one scalar constraint, which has to be included in the mathematical model of rigid-body dynamics. The outcome is that the use of quaternions imposes one algebraic constraint, even in the case of mechanically unconstrained systems. An alternative parametrization is proposed here, that (a) comprises only three independent parameters; (b) is fairly robust to representation singularities; and (c) satisfies the quaternion scalar constraint intrinsically. To illustrate the concept, a simple, yet nontrivial case study is included. This is a mechanical system composed of a rigid, toroidal wheel rolling without slipping or skidding on a horizontal surface. The simulation algorithm based on the proposed parametrization and fundamentally on quaternions, together with the invariant relations between the quaternion rate of change and the angular velocity, is capable of reproducing the falling of the wheel under deterministic initial conditions and a random disturbance acting on the tilting axis. Finally, a comparative study is included, on the numerical conditioning of the parametrization proposed here and that based on Euler angles. Ours shows as broader well-conditional region than Euler angles offer. Moreover, the two parametrizations exhibit an outstanding complementarity: where the conditioning of one degenerates, the other comes to rescue.

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References

Stuelpnagel, J., 1964, “On the Parameterization of the Three-Dimensional Rotation Group,” SIAM Rev., 6(4), pp. 422–430. [CrossRef]
Pina, E., 1983, “A New Parameterization of the Rotation Matrix,” Am. J. Phys., 51(4), pp. 375–379. [CrossRef]
Tilma, T., and Sudarshan, E., 2002, “Generalized Euler Angle Parameterization for SU(N),” J. Phys. A: Math. Gen., 35(48), p. 10467. [CrossRef]
Tilma, T., and Sudarshan, E., 2004, “Generalized Euler Angle Parameterization for U(N) With Applications to SU(N) Coset Volume Measures,” J. Geom. Phys., 52(3), pp. 263–283. [CrossRef]
Dewey, J., Helman, M., Knott, S., Turco, E., and Hutton, D., 1989, “Kinematics of the Western Mediterranean,” Geol. Soc., London, Special Pub., 45(1), pp. 265–283. [CrossRef]
Colombet, P., Robinson, J., Christel, P., Franceschi, J., and Djian, P., 2007, “Using Navigation to Measure Rotation Kinematics During ACL Reconstruction,” Clin. Orthop. Related Res., 454(1), pp. 59–65. [CrossRef]
Danielson, D. A., and Hodges, D. H., 1987, “Nonlinear Beam Kinematics by Decomposition of the Rotation Tensor,” ASME J. Appl. Mech., 54(2), pp. 258–262. [CrossRef]
Ishii, T., Mukai, Y., Hosono, N., Sakaura, H., Nakajima, Y., Sato, Y., Sugamoto, K., and Yoshikawa, H., 2004, “Kinematics of the Upper Cervical Spine in Rotation: In Vivo Three-Dimensional Analysis,” Spine, 29(7), pp. E139–E144. [CrossRef] [PubMed]
Baruh, H., 1999, Analytical Dynamics, WCB/McGraw-Hill, Boston.
Pio, R. L., 1966, “Euler Angle Transformations,” IEEE Trans. Autom. Control, 11(4), pp. 707–715. [CrossRef]
Evans, D. J., 1977, “On the Representation of Orientation Space,” Mol. Phys., 34(2), pp. 317–325. [CrossRef]
Burger, H., 1995, “Use of Euler-Rotation Angles for Generating Antenna Patterns,” IEEE Trans. Antennas Propag., 37(2), pp. 56–63. [CrossRef]
Horn, A., 1954, “Doubly Stochastic Matrices and the Diagonal of a Rotation Matrix,” Am. J. Math., 76(3), pp. 620–630. [CrossRef]
Raptis, I. A., Valavanis, K. P., and Moreno, W. A., 2011, “A Novel Nonlinear Backstepping Controller Design for Helicopters Using the Rotation Matrix,” IEEE Trans. Control Syst. Technol., 19(2), pp. 465–473. [CrossRef]
Rusydi, M. I., Okamoto, T., Ito, S., and Sasaki, M., 2014, “Rotation Matrix to Operate a Robot Manipulator for 2D Analog Tracking Objects Using Electrooculography,” Robotics, 3(3), pp. 289–309. [CrossRef]
Zhang, Y., and Xiao, D., 2014, “An Image Encryption Scheme Based on Rotation Matrix Bit-Level Permutation and Block Diffusion,” Commun. Nonlinear Sci. Num. Simul., 19(1), pp. 74–82. [CrossRef]
Shoemake, K., 1985, “Animating Rotation With Quaternion Curves,” 12th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH'85), San Francisco, CA, July 22–26, pp. 245–254. [CrossRef]
Bar-Itzhack, I. Y., 2000, “New Method for Extracting the Quaternion From a Rotation Matrix,” J. Guid., Control, Dyn., 23(6), pp. 1085–1087. [CrossRef]
Lekkas, A., and Fossen, T. I., 2013, “A Quaternion-Based LOS Guidance Scheme for Path Following of AUVs,” 9th IFAC Conference on Control Applications in Marine Systems, Osaka, Japan, Sept. 17–20, pp. 245–250. [CrossRef]
Wu, S., Wu, G., Tan, S., and Wu, Z., 2013, “Quaternion-Based Adaptive Terminal Sliding Mode Control for Spacecraft Attitude Tracking,” 10th IEEE International Conference on Control and Automation (ICCA), Hangzhou, China, June 12–14, pp. 913–917. [CrossRef]
Argyle, M. E., Beach, J. M., Beard, R. W., McLain, T. W., and Morris, S., 2014, “Quaternion Based Attitude Error for a Tailsitter in Hover Flight,” American Control Conference (ACC), Portland, OR, June 4–6, pp. 1396–1401. [CrossRef]
Fister, I., Yang, X., Brest, J., and Fister, I., Jr., 2013, “Modified Firefly Algorithm Using Quaternion Representation,” Expert Syst. Appl., 40(18), pp. 7220–7230. [CrossRef]
Terze, Z., Muller, A., and Zlatar, D., 2014, “Redundancy-Free Integration of Rotational Quaternions in Minimal Form,” ASME Paper No. DETC2014-35118. [CrossRef]
Angeles, J., 2014, Fundamentals of Robotic Mechanical Systems: Theory, Methods, and Algorithms, 4th ed., Springer, New York.
Synge, J. L., 1960, “Classical Dynamics,” Principles of Classical Mechanics and Field Theory, Springer, Berlin, pp. 1–225.
Akcoglu, M. A., Bartha, P. F., and Ha, D. M., 2011, Analysis in Vector Spaces, Wiley, Hoboken, NJ.
Ascher, U. M., and Petzold, L. R., 1998, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, Vol. 61, SIAM, Toronto.
Golub, G. H., and Van Loan, C. F., 2012, Matrix Computations, Vol. 3, Johns Hopkins University Press, Baltimore, MD.
Angeles, J., and Lee, S. K., 1988, “The Formulation of Dynamical Equations of Holonomic Mechanical Systems Using a Natural Orthogonal Complement,” ASME J. Appl. Mech., 55(1), pp. 243–244. [CrossRef]

Figures

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Fig. 1

A rigid-body rolling without slipping

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Fig. 2

Parametrization of the unit vector e

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Fig. 3

A plot of 1/κF(N) versus β and φ

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Fig. 4

Reciprocals of κF(N) and κF(U), the former at φ=π

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Fig. 5

Geometry of a toroidal wheel: (a) front view and (b) cross section of the wheel and its convex hull

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Fig. 6

The wheel at an arbitrary pose

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Fig. 7

A description of the wheel kinematics

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Fig. 8

Contact point analysis

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Fig. 11

Fall down process

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Fig. 9

Simulation results for one given θ(0) and various values of ϖ(0):…,π/3rad/s;-·-·,2π/3rad/s; – – –, π/2 rad/s; —, 4π/3 rad/s

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Fig. 10

Simulation results for one given ω(0) and various values of θ(0): ⋯, 10 deg; –⋅–, 20 deg; – – –, 30 deg; —, 40 deg

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