Use Of Rotation Matrices Research Articles

A geometric algebra algorithm for the closed-form forward displacement analysis of 3-PPS parallel mechanisms

This paper presents an efficient graphical representation geometric algebra approach for the forward displacement analysis (FDA) of 3-PPS parallel mechanisms (PMs). Our approach based on conformal geometric algebra (CGA) reveals interesting intuitions and insight of the modelling and solution for FDA of this type PMs and reduces considerably computational burden and time. In this work, we first deduce the position of an arbitrary joint among three spherical joints on the moving platform through motors and represent it in terms of only one variable. Then, we formulate the position of either one joint of two remained spherical joints using the intersection of a sphere and two planes under the CGA framework. To obtain the geometric formulation of the third spherical joint position, we construct the outer product of two spatial balls with centers at two formulated spherical joints centers and two known planes. The positions of three spherical joints are all explicit geometric formulations of geometric entities of the parallel mechanisms. We achieve a coordinate-invariant equation expressed in terms of geometric entities via inner product of the last joint by itself. The symbolic 8th-degree univariate polynomial input-output equation is first proposed and all 8 sets of closed-form solutions can be obtained. The algorithm is the geometric algebra computation in a clear and coordinates-free way that avoids the use of rotation matrices, and complex algebraic modelling and nonlinear and multivariable elimination computations as most current approaches do. Finally, two numerical examples have been applied to demonstrate the efficiency of the algorithm comparing with the Dixon resultant method. The results show that this algorithm can simplify complexity of the problem and reduce computation time dramatically with strong geometrical intuition. The achieved symbolic univariate polynomial input-output equation can be applied directly to solve the DPA for all types of 3-PPS PMs instead of using any other methods. This work provides a novel geometric modeling and solution approach for the forward displacement analysis of spatial parallel mechanism and be of greatly useful for roboticists or engineers to design and develop 3-PPS PMs for kinds of applications.

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, three-dimensional image processing, and computer graphics. 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 3 × 3 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 4 × 4 rotation matrices, is the most robust method when particularized to three dimensions.

Use of MEMS accelerometers/inclinometers as a geotechnical monitoring method for ground subsidence

Accelerometer and inclinometer are inertial sensors capable of measuring corresponding magnitude of Earth gravitational field along the direction of each axis. By means of rotation matrices related to inertial navigation methods, the output values of a three-dimensional accelerometer or a two-dimensional inclinometer can be transformed and processed into the azimuth and dip angle of the monitored target. With the rapid growth in development and cost reduction of Micro Electro Mechanical Systems (MEMS) and Wireless Sensor Network (WSN) in recent years, the engineers are able to carry out real-time wireless geotechnical monitoring during construction. In this paper, we set up a one-day measurement implemented by a self- developed wireless MEMS monitoring system on the surface in the construction site of South Hongmei Road super high way tunnel in Shanghai, by making use of rotation matrices in specific ways, the raw data are processed to expressions of three-dimensional normal vectors that represent the change of the ground. After unifying the vectors in the same coordinate system, we conduct a brief ground settlement analysis by means of an evaluation of the dip angles in the cross section and the azimuths of the sensor nodes. ARTICLE INFO

A universal framework for modelling measured velocity in laser vibrometry with applications

This paper presents a novel, universally applicable framework for modelling measured velocity in laser vibrometry systems. The framework is introduced generically before demonstration of its application to three scanning vibrometer systems, each configured to measure vibration of a tracked point on a rotating target. The novelty in this vectorial framework lies in the combination of its elements, which include vector descriptions of target velocity, optical device velocity at deflection points, laser beam orientations, incorporating reflection and refraction, and surface normals. Initial alignment and a full set of inevitable misalignments are incorporated by the modification of position vectors and the use of rotation matrices. Inclusion of components of measured velocity associated with moving optical devices is an important feature of the framework. The models derived and their validation against published data demonstrate how this versatile framework can be applied to any optical configuration measuring target motions with any level of complexity. The individual models are explored extensively and quantitatively through simulation. Small but inevitable misalignments are shown to generate measurable low order velocity components and their effects on the sensitivities to in-plane and out-of-plane components of target vibration are quantified.

An efficient Newton method for general motorcycle kinematics

This paper presents a detailed study of the kinematics of single-track vehicles, with a special emphasis on motorcycles. We consider a general class of tyre profiles as well as general vehicle geometry. We show that the kinematic problem may be reduced to the problem of finding the zero of a (single) nonlinear equation in the pitch angle which may then be solved using a safeguarded Newton method, providing rapid convergence. Special care, enabled by the systematic use of rotation matrices, is taken to understand the range of pitch angles for which all quantities in the equation are well defined. The paper provides a fast and numerically reliable algorithm that can be used within analysis tools such as those involving numerical integration of system dynamics.

Efficiency of Rotational Operators for Geometric Manipulation of Chain Molecules

Geometric manipulation of molecules is an essential elementary component in computational modeling programs for molecular structure, stability, dynamics, and design. The computational complexity of transformation of internal coordinates to Cartesian coordinates was discussed before.1 The use of rotation matrices was found to be slightly more efficient than that of quaternion although quaternion operators have been widely advertised for rotational operations, especially in molecular dynamics simulations of liquids where the orientation is a dynamical variable.2 The discussion on computational efficiency is extended here to a more general case in which bond angles and sidechain torsion angles are allowed to vary. The algorithm of Thompson3 is derived again in terms of quaternions as well as rotation matrices, and an algorithm with optimal efficiency is described. The algorithm based on rotation matrices is again found to be slightly more efficient than that based on quaternions.

Sequential Tilt and Tilted Power of Thin Lenses

The purposes of this article were, for any sequence of tilts applied to a thin lens, to calculate an equivalent turn and single tilt, to show how to use the equivalent turn and tilt to calculate the tilted power of the lens, and, given the desired tilted power, to calculate the power of the untilted lens necessary to compensate for the effects of tilt. The untilted lens may be stigmatic (spherical) or astigmatic (spherocylindrical). The analysis makes use of rotation matrices to represent rotation in space and previous work in third-order optics on oblique central refraction. Equations are presented for calculating the combination of turn and tilt that is equivalent to any sequence of tilts. They are specialized for the particular case of combinations of faceform and pantoscopic tilts and allow the decomposition of an arbitrary tilt into a combination of turn and pantoscopic and faceform tilts. The equations also lead to a procedure for calculating or compensating for the tilted power of a sequentially tilted thin lens. Previous work on the effect of tilt on thin lenses has been generalized to handle combinations of arbitrary tilts.

Angle calculations for a three-circle goniostat

A general procedure for angle calculations for a three-circle goniostat has been developed. This new approach is based on a vector description of the transformation of a reciprocal-lattice vector under the action of a rotation shaft. It does not invoke the use of rotation matrices and applies equally well to cases where the directions of the rotation axes do not conform with coordinate axes of the laboratory system adopted for the analysis.

More applications of Euler rotation angles

Discusses various uses of rotation matrices. The author has used rotation matrices to find the patterns with elements in a conformal array that requires one to rotate not only the direction, but the polarization directions, as well. A second useful thing to do with rotation matrices is to analytically rotate objects in space. In computer graphics, it may be neccessary to rotate the object. Many computer graphics texts have useful discussions of object rotations using rotation matrices. A third related use of rotation matrices is to understand antenna positioners. One can describe the rotations of a positioner by rotation matrices, and calculate the effects of rotation not only on the directing of the antenna, but on the polarization directions, as well.

The use of rotation matrices in the mathematical description of molecular orientations in polymers

The orientational distribution of chain segments or crystals in polymeric materials is most conveniently described as a series expansion of rotation matrix elements. Rotation matrices are representations of rotation operators, and their elements form a complete set of orthogonal functions. Their use in the mathematical description of molecular orientations therefore yields the advantage that the moments in the series expansion of the orientational distribution, also called order parameters, are the averages of the rotation matrix elements themselves. Furthermore, rotations between molecular and macroscopic frames can easily be carried out to link molecular orientation to a macroscopic observable. In the third place, various symmetries reduce the number of non-vanishing order parameters in a straightforward way. In the case of fiber symmetry and axially symmetric molecular elements, the rotation matrix elements reduce to Legendre polynomials.

Comparative steady-state performance of crossbonded cable systems

The paper gives a universal system of steady-state analysis for any crossbonded and/or transposed cable system with unequal lays. It first introduces a nomenclature which enables any conductor or sheath to be identified in geometric and/or longitudinal position in any lay and/or section both for the general case and for four specific cases of crossbonding. It is then shown that the effect of unequal lays on sheath voltages, currents and losses may always be analysed, provided that there is rigorous adherence to the system of nomenclature. The paper is restricted to cases of unbalanced conductor currents in the steady state and does not deal with the additional variables and complexities when the various crossbonded systems have multiple-earthed points and are under external or internal unbalanced fault conditions. A large selection of graphical results is presented for sheath currents, losses and longitudinal sheath voltages of the three main cases of crossbonding. In the particular case of trefoil spacing, taken as an example, the influence of variation of lay length between 0.6 and 1.4 p.u. on the sheath currents and longitudinal induced voltages is shown. The results are discussed in detail since comparative performances for these various systems have not previously been presented. The conclusions extend also the analytical methods, it being shown that all crossbonded systems of whatever complexity can be analysed by the use of rotation matrices and the notation for nonuniform lay length; it follows en passant that normal symmetrical-component methods do not yield a solution, owing to the crossbonded systems having multiple unbalances and six degrees of freedom.

Stereochemical criteria for polypeptide and protein chain conformations

A large number of configurations are possible for a polypeptide structure depending upon the relative orientations of the two peptide groups linked at anα-carbon atom. If the linkages at theα-carbon atoms are identical, then the structure assumes a regular helical form. Such a regular helical structure can be specified by two parametersφ, φ′ which are the angles of rotation of the two groups about the N—αC andαC—C′ bonds meeting at anα-carbon atom. In this paper, a method of evaluating the helical parameters of a structure, namely, the number of residues per turnn, the unit translation along the axis of the helixh and the direction cosines of the helical axis with respect to a suitably chosen co-ordinate system, is described making use of rotation matrices. The evaluation has been done for three different values of the angle $$N\widehat{aC}C'$$ at theα-carbon atom, namely, 105°, 110° and 115°. The results are shown graphically in the form of curves for constantn and constanth in theφ − φ′ plane.