A survey on the computation of quaternions from rotation matrices

The parameterization of rotations is a central topic in many theoretical and applied fields such as rigid body mechanics, multibody dynamics, robotics, spacecraft attitude dynamics, navigation, 3D image processing, computer graphics, etc. Nowadays, the main alternative to the use of rotation matrices, to represent rotations in R3, is the use of Euler parameters arranged in quaternion form. Whereas the passage from a set of Euler parameters to the corresponding rotation matrix is unique and straightforward, the passage from a rotation matrix to its corresponding Euler parameters has been revealed to be somewhat tricky if numerical aspects are considered. Since the map from quaternions to 3x3 rotation matrices is a 2-to-1 covering map, this map cannot be smoothly inverted. As a consequence, it is erroneously assumed that all inversions should necessarily contain singularities that arise in the form of quotients where the divisor can be arbitrarily small. This misconception is herein clarified. This paper reviews the most representative methods available in the literature, including a comparative analysis of their computational costs and error performances. The presented analysis leads to the conclusion that Cayley's factorization, a little-known method used to compute the double quaternion representation of rotations in four dimensions from 4x4 rotation matrices, is the most robust method when particularized to three dimensionsPreprin

Some Results of the Educational Experiment APIS (Cervantes Mission on Board ISS)

Some results of the analysis of the pictures taken along the performance of the Análisis de Propiedades Inerciales de Sólidos, Analysis of the Inertia Properties of Solid Bodies (APIS) experiment carried out in the Cervantes mission on board ISS, are presented. APIS was an educational experiment devoted to take advantage of the unique conditions of absence of relative gravity forces of a space platform such as ISS, to show some of the characteristics of the free rotational motion of a solid body, which are impossible to carry out on earth. This field of experimental research has application to aerospace engineering science (e.g. attitude control of spacecrafts), to astrophysical sciences (e.g. state of rotation and tumbling motions of asteroids) and to engineering education. To avoid the effect of the ambient atmosphere loads on the motion, the test body is placed inside a sphere, which reduces the effect of the aerodynamic forces to just friction. The drastic reduction of the effect of the surrounding air during the short duration of the experimental sequences allows us to compare the actual motion with the known solutions for the solid body rotation in vacuum. In this paper, some selected, relevant sequences of the sphere enclosing a body with a nominal cylindrical inertia tensor, put into rotation by the astronaut, are shown; the main problems to extract the information concerning the characteristic parameters of the motion are outlined, and some of the results obtained concerning the motion of the test probe are included, which show what seems to be a curious and unexpected solution of the Euler equations for the solid body rotation in vacuum, without energy dissipation, when the angular momentum is almost perpendicular to the axisymmetry axis

A Method for Fast Diagonalization of a 2x2 or 3x3 Real Symmetric Matrix

A method is presented for fast diagonalization of a 2x2 or 3x3 real symmetric matrix, that is determination of its eigenvalues and eigenvectors. The Euler angles of the eigenvectors are computed. A small computer algebra program is used to compute some of the identities, and a C++ program for testing the formulas has been uploaded to arXiv.Comment: Corrected formula 4.1

Elastic Rod Model of a Supercoiled DNA Molecule

We study the elastic behaviour of a supercoiled DNA molecule. The simplest model is that of a rod like chain, involving two elastic constants, the bending and the twist rigidities. We show that this model is singular and needs a small distance cut-off, which is a natural length scale giving the limit of validity of the model, of the order of the double helix pitch. The rod like chain in presence of the cutoff is able to reproduce quantitatively the experimentally observed effects of supercoiling on the elongation-force characteristics, in the small supercoiling regime. An exact solution of the model, using both transfer matrix techniques and its mapping to a quantum mechanics problem, allows to extract, from the experimental data,the value of the twist rigidity. We also analyse the variation of the torque and the writhe to twist ratio versus supercoiling, showing analytically the existence of a rather sharp crossover regime which can be related to the excitation of plectonemic-like structures. Finally we study the extension fluctuations of a stretched and supercoiled DNA molecule, both at fixed torque and at fixed supercoiling angle, and we compare the theoretical predictions to some preliminary experimental data.Comment: 29 pages Revtex 5 figure

Modified Gibbs's representation of rotation matrix

A modified Gibbs's rotation matrix is derived and the connection with the Euler angles, quaternions, and Cayley$-$Klein parameters is established. As particular cases, the Rodrigues and Gibbs parameterizations of the rotation are obtained. The composition law of two rotations from the quaternion representation is presented showing a convenient expression for calculating the successive rotations.Comment: 17 page