Euler Parameters Research Articles

An Accurate Singularity-Free Formulation of a Three-Dimensional Curved Euler–Bernoulli Beam for Flexible Multibody Dynamic Analysis

An accurate singularity-free formulation of a three-dimensional curved Euler–Bernoulli beam with large deformations and large rotations is developed for flexible multibody dynamic analysis. Euler parameters are used to characterize orientations of cross sections of the beam, which can resolve the singularity problem caused by Euler angles. The position of the centroid line of the beam is integrated from its slope, and position vectors of nodes of beam elements are no longer used as generalized coordinates. Hence, the number of generalized coordinates for each node is minimized. Euler parameters instead of position vectors are interpolated in the current formulation, and a new C1-continuous interpolation function is developed, which can greatly reduce the number of elements. Governing equations of the beam and constraint equations are derived using Lagrange's equations for systems with constraints, which are solved by the generalized- α method for resulting differential-algebraic equations (DAEs). The current formulation can be used to calculate static and dynamic problems of straight and curved Euler–Bernoulli beams under arbitrary, concentrated and distributed forces. The stiffness matrix and generalized force in the current formulation are much simpler than those in the geometrically exact beam formulation (GEBF) and absolute node coordinate formulation (ANCF), which makes it more suitable for static equilibrium problems. Numerical simulations show that the current formulation can achieve the same accuracy as the GEBF and ANCF with much fewer elements and generalized coordinates.

An Ordinary Differential Equation Formulation for Multibody Dynamics: Holonomic Constraints

A method is presented for formulating and numerically integrating ordinary differential equations (ODEs) of motion for holonomically constrained multibody systems. Tangent space coordinates are defined as independent generalized coordinates that serve as state variables in the formulation, yielding ODEs of motion. Orthogonal dependent coordinates are used to enforce kinematic constraints at position, velocity, and acceleration levels. Criteria that assure accuracy of constraint satisfaction and well conditioning of the reduced mass matrix in the equations of motion are used as the basis for redefining local coordinates on the constraint manifold, as needed, transparent to the user and at minimal computational cost. The formulation is developed for holonomically constrained multibody models that are based on essentially any form of generalized coordinates. A spinning top with Euler parameter orientation coordinates is used as a model problem to analytically reduce Euler's equations of motion to ODEs. Numerical results using a fourth-order Nystrom integrator verify that accurate results are obtained, satisfying position, velocity, and acceleration constraints to computer precision. A computational algorithm for implementing the approach with state-of-the-art explicit numerical integrators is presented and used in solution of three examples, one planar and two spatial. Performance of the method in satisfying all three forms of kinematic constraint, based on error tolerances embedded in the formulation, is verified.

Equations and algorithms for determining the inertial attitude and apparent velocity of a moving object in quaternion and biquaternion 4D orthogonal operators

We consider equations and algorithms describing the operation of strapdown inertial navigation systems (SINS) intended for determining the inertial attitude parameters (the Rodrigues–Hamilton (Euler) parameters) and the apparent velocity of a moving object. The construction of these equations and algorithms is based on the Kotelnikov–Study transference principle, Hamiltonian quaternions and Clifford biquaternions, and differential equations in four-dimensional (quaternion and biquaternion) orthogonal operators.

Reconfiguration analysis of a 4-DOF 3-RER parallel manipulator with equilateral triangular base and moving platform

This paper deals with the reconfiguration analysis of a 4-DOF (degrees-of-freedom) 3-RER parallel manipulator (PM) with equilateral triangular base and moving platform (ETBP), which is a special case of a 4-DOF PM in the literature. The 4-DOF 3-RER PM is composed of a base and a moving platform connected by three 3-RER legs, each of which is a serial kinematic chain composed of a revolute (R) joint, a planar (E) joint and an R joint in sequence. At first, a set of constraint equations of the 3-RER PM with ETBP is derived with the orientation of the moving platform represented using a Euler parameter quaternion (also Euler-Rodrigues quaternion) and then solved in closed form. It is found that the 3-RER PM with ETBP has three 4-DOF operation modes if both the base and moving platform are identical or two 4-DOF operation modes if the base and moving platform are not identical. The motion characteristics of the moving platform are obtained using the kinematic interpretation of Euler parameter quaternions with certain number of constant zero components, which was presented in a recent paper by the author of this paper. The transition configurations among different operation modes are also identified.

A New Singularity-Free Formulation of a Three-Dimensional Euler–Bernoulli Beam Using Euler Parameters

In this investigation, a new singularity-free formulation of a three-dimensional Euler–Bernoulli beam with large deformations and large rotations is developed. The position of the centroid line of the beam is integrated from its slope, which can be easily expressed by Euler parameters. The hyperspherical interpolation function is used to guarantee that the normalization constraint equation of Euler parameters is always satisfied. Each node of a beam element has only four nodal coordinates, which are significantly fewer than those in an absolute node coordinate formulation (ANCF) and the finite element method (FEM). Governing equations of the beam and constraint equations are derived using Lagrange's equations for systems with constraints, which are solved by a differential-algebraic equation (DAE) solver. The current formulation can be used to calculate the static equilibrium and linear and nonlinear dynamics of an Euler–Bernoulli beam under arbitrary, concentrated, and distributed forces. While the mass matrix is more complex than that in the ANCF, the stiffness matrix and generalized forces are simpler, which is amenable for calculating the equilibrium of the beam. Several numerical examples are presented to demonstrate the performance of the current formulation. It is shown that the current formulation can achieve the same accuracy as the ANCF and FEM with a fewer number of coordinates.

Homothetic Cayley Formula and its Applications

In this study the homothetic Cayley Formula has been defined for an antisymmetric matrix and the relation between an antisymmetric matrix and a vector has been given. Hence Rodrigues and Euler parameters have been obtained for homothetic motions. On the other hand, some results and applications have been achieved by using homothetic rotation matrix.

Reconfiguration Analysis of a Two Degrees-of-Freedom 3-4R Parallel Manipulator With Planar Base and Platform1

This paper deals with a 2-DOF (degrees-of-freedom) 3-4R parallel manipulator (PM) with planar base and platform—a novel PM with multiple operation modes (or disassembly free reconfigurable PM) that can use the minimum number of actuated joints. At first, a set of constraint equations of the 3-4R PM are derived with the orientation of the moving platform represented using a Euler parameter quaternion (also Euler–Rodrigues quaternion) and then solved using the algebraic geometry method. It is found that this 3-4R PM has six 2-DOF operation modes, including the two expected spherical translation mode and sphere-on-sphere rolling mode when the PM was synthesized. The motion characteristics of the moving platform are obtained using the kinematic interpretation of Euler parameter quaternions with certain number of constant zero components, which was presented in a recent paper by the first author of this paper, instead of the eigenspace-based approach in the literature. The transition configurations, which are constraint singular configurations, among different operation modes are also presented. This work provides a solid foundation to the development and control of the 2-DOF 3-4R PM with both 2-DOF spherical translation mode and 2-DOF sphere-on-sphere rolling mode.

Snap-Through Phenomenon and Self-Contact of Spatial Elastica Subjected to Mid-Torque

This paper presents the snap-through phenomenon and effect of self-contact of the spatial elastica subjected to mid-length torque. One end of the elastica is clamped while the other end is placed in a sleeve joint. The total arc-length of the elastica can be varied by sliding the end through the sleeve joint. At a certain value of total arc-length, the sleeve joint is clamped and an external torque is applied at the mid-length of the elastica. The system of governing differential equations is derived from the equilibrium of an elastica segment and geometric relations of the inextensible elastica. The transformation matrix formulated in terms of Euler parameters is utilized to avoid the kinematic singularity. To display the behavior of the elastica, the system of differential equations needs to be integrated numerically from one end to the other end. The integration is performed so that the boundary conditions and some constraint conditions of the problem are satisfied, i.e., a shooting method is used. The effect of self-contact is taken into account by considering the contact force as a point load applying at contact point. From the results, the snap-through phenomenon, effect of self-contact and equilibrium configurations are highlighted herein.

Euler–Rodrigues formula variations, quaternion conjugation and intrinsic connections

This paper reviews the Euler–Rodrigues formula in the axis–angle representation of rotations, studies its variations and derivations in different mathematical forms as vectors, quaternions and Lie groups and investigates their intrinsic connections. The Euler–Rodrigues formula in the Taylor series expansion is presented and its use as an exponential map of Lie algebras is discussed particularly with a non-normalized vector. The connection between Euler–Rodrigues parameters and the Euler–Rodrigues formula is then demonstrated through quaternion conjugation and the equivalence between quaternion conjugation and an adjoint action of the Lie group is subsequently presented. The paper provides a rich reference for the Euler–Rodrigues formula, the variations and their connections and for their use in rigid body kinematics, dynamics and computer graphics.

A constraint violation suppressing formulation for spatial multibody dynamics with singular mass matrix

The constraint violation problem of spatial multibody systems is analyzed in this paper. The mass matrix is singular in the equations of motion when the Euler parameters with the normalization constraints are used to describe the orientation of the spatial rigid body. The constrained and weighted least-squares-based geometrical projection method is implemented to suppress the constraint violation during numerical integration, and the explicit correction formulation can be obtained by the block matrix inversion scheme. The mass matrix weighted correction formulation gives the physically consistent energy norm, but it needs the mass matrix to be positive definite. To extend the physically consistent correction formulation for solving spatial multibody systems’ constraint violation problems with a singular mass matrix, a Modified Mass-Orthogonal Projection Method (MMOPM) and a Generalized Physical Orthogonal Projection Method (GPOPM) are proposed. MMOPM modifies the mass matrix directly by adding a penalty factor matrix which appears in the mass-orthogonal projection method and leads to a positive definite weight matrix that satisfies the block matrix inversion scheme condition. GPOPM is a generalization of the physical orthogonal projection method where the constrained least-squares method is weighted by the positive semi-definite mass matrix and the correction formulation is given by using the generalized block matrix inversion scheme. Numerical results show the feasibility and accuracy of the presented MMOPM and GPOPM. The constraints in position and velocity can reach machine precision during numerical integration. The elimination of violation of position constrains may require few iterations, while the violation of velocity constraints is removed in one step, and GPOPM is more accurate in velocity correction than MMOPM.

Real-Time Dynamic Simulations of Large Road Vehicles Using Dense, Sparse, and Parallelization Techniques

This article presents three multibody formulations with improved efficiency in order to achieve real-time simulations for the forward dynamic of two real-life road vehicles. The bigger is a semitrailer truck with 40 degrees of freedom (DOF). Two topological and semirecursive formulations are used as well as a global formulation based on the use of Euler parameters and flexible joints. The first semirecursive formulation carries out a double velocity transformation and the integration is done by means of the explicit fourth order Runge–Kutta method. The second semirecursive formulation and the global one use a penalty scheme at position level and orthogonal projections at velocity and acceleration levels. In both cases the integrator was the implicit Hilbert–Huges–Taylor (HHT) method. The double velocity transformation method involves the coordinate partitioning of the constraint Jacobian matrix which leads to the costly solution of a redundant but consistent with the constraints linear system of equations. The choice of a unique set of independent coordinates may not be valid for a complete simulation and additional repartitioning would be required. Based on previous experience and as the examples show in this article, a careful initial choice of the independent coordinates can remain valid for complete simulations involving common maneuvers. This represents a numerical advantage for dense matrix methods and can be further exploited if sparse matrix techniques are employed. This has been the case for both of the vehicles used, reaching real-time simulations even with the semitrailer truck. The implicit semirecursive formulation involves the numerical evaluation of the stiffness and damping matrices, which hamper obtaining real-time simulations. For the semitrailer truck, this computation represents the 76% of the total simulation time. The numerical computation of these matrices is carried out by columns and its algorithm is straightforwardly parallelizable. Using a quad-core processor and with a simple and efficient OpenMP implementation, it has been possible to achieve a speedup of 3.25 reducing the simulation times under the real-time limit. The sparse matrices of Euler parameters formulation show very different sparsity degrees, difference that grows with the size of the multibody model. This poses a challenge to sparse matrix implementations in order to be able to efficiently perform matrix operations without increasing fillings or handling zero entries. This has been successfully accomplished using a new sparse matrix representation. This one is not a feature of general purpose sparse software, requiring at some stages the implementation of our own algorithms. Reductions in time of three orders of magnitude have led to real-time simulations even with the semitrailer truck.

On the rotational equations of motion in rigid body dynamics when using Euler parameters.

Many models of three-dimensional rigid body dynamics employ Euler parameters as rotational coordinates. Since the four Euler parameters are not independent, one has to consider the quaternion constraint in the equations of motion. This is usually done by the Lagrange multiplier technique. In the present paper, various forms of the rotational equations of motion will be derived, and it will be shown that they can be transformed into each other. Special attention is hereby given to the value of the Lagrange multiplier and the complexity of terms representing the inertia forces. Particular attention is also paid to the rotational generalized external force vector, which is not unique when using Euler parameters as rotational coordinates.

Comparative study of quantum discord and geometric discord for generic bipartite states

The characterization of quantum discord (QD) and geometric discord (GD) has mostly concentrated on two-qubit states since the minimization in both discords is a daunting task for high-dimensional states. Numerical calculations of both discords are carried out for a generic bipartite state. When one-dimensional orthogonal projectors for a local measurement on n-dimensional Hilbert space are realized by the generators and the Euler angles of SU (n), the optimal measurements have a figure of merit that includes n(n - 1) Euler parameters. As an representative example, such projectors and two kinds of algorithms are used to estimate both discords for two-qutrit mixed states in recent literature. The generalized negativity as a measure of quantum entanglement is calculated for reference purposes. For those states with one parameter the discords and the negativity respectively display the nonlinear and the linear function of the parameter, with different turning points. However, they are positively correlated in the suitable ranges of the parameter for those states. The hierarchy of those quantities is discussed as well. Those shed new light on the understanding of QDs and quantum entanglement of mixed states in high-dimensions.

Derivation of paleolongitude from the geometric parametrization of apparent polar wander path: Implication for absolute plate motion reconstruction

AbstractObtaining ancient longitude position of continents in the past has always been a challenge for plate tectonic reconstructions. Paleomagnetism has been commonly used to reconstruct paleolatitudes and relative rotations but not paleolongitudes. In this work, we present a synthesized method to derive paleolongitude by geometrically parametrizing apparent polar wander path (APWP). Great and small circle modeling are implemented concurrently to the identified APWP tracks to calculate the paleomagnetic Euler parameters (stage rotation pole and rotation angle). From the Euler parameters of the optimal fitting option, the absolute motion history can be restored for the reference geometries. Using our method as well as the results from relative plate motion studies, we reevaluate the dispersion history of East Gondwana since 140 Ma. To further test the validity of our method, we compare the predictions from four other absolute motion models mainly in paleolatitude movement, longitudinal variation, and great circle distance, which suggest the most similarity with the global hybrid reference frame.

Quaternion regularization and trajectory motion control in celestial mechanics and astrodynamics: II

Problems of regularization in celestial mechanics and astrodynamics are considered, and basic regular quaternion models for celestial mechanics and astrodynamics are presented. It is shown that the effectiveness of analytical studies and numerical solutions to boundary value problems of controlling the trajectory motion of spacecraft can be improved by using quaternion models of astrodynamics. In this second part of the paper, specific singularity-type features (division by zero) are considered. They result from using classical equations in angular variables (particularly in Euler variables) in celestial mechanics and astrodynamics and can be eliminated by using Euler (Rodrigues-Hamilton) parameters and Hamilton quaternions. Basic regular (in the above sense) quaternion models of celestial mechanics and astrodynamics are considered; these include equations of trajectory motion written in nonholonomic, orbital, and ideal moving trihedrals whose rotational motions are described by Euler parameters and quaternions of turn; and quaternion equations of instantaneous orbit orientation of a celestial body (spacecraft). New quaternion regular equations are derived for the perturbed three-dimensional two-body problem (spacecraft trajectory motion). These equations are constructed using ideal rectangular Hansen coordinates and quaternion variables, and they have additional advantages over those known for regular Kustaanheimo-Stiefel equations.

A supplementary note on constructing the general Earth's rotation theory

AbstractRepresenting a post-scriptum supplementary to a previous paper of the authors Brumberg & Ivanova (2011) this note aims to simplify the practical development of the Earth's rotation theory, in the framework of the general planetary theory, avoiding the non–physical secular terms and involving the separation of the fast and slow angular variables, both for planetary–lunar motion and Earth's rotation. In this combined treatment of motion and rotation, the fast angular terms are related to the mean orbital longitudes of the bodies, the diurnal and Euler rotations of the Earth. The slow angular terms are due to the motions of pericenters and nodes, as well as the precession of the Earth. The combined system of the equations of motion for the principal planets and the Moon and the equations of the Earth's rotation is reduced to the autonomous secular system with theoretically possible solution in a trigonometric form. In the above–mentioned paper, the Earth's rotation has been treated in Euler parameters. The trivial change of the Euler parameters to their small declinations from some nominal values may improve the practical efficiency of the normalization of the Earth's rotation equations. This technique may be applied to any three-axial rigid planet. The initial terms of the corresponding expansions are given in the Appendix.

Euler parameters kinetic singularity

Euler parameters do not suffer from the kinematic singularity associated with the three-parameter representation of the rotation. For this reason, the four Euler parameters are widely used in describing the motion of rigid bodies in space. These parameters are related by an algebraic equation which must be satisfied at the position, velocity, and acceleration levels. However, the constraint forces associated with Euler parameter constraints are always zero and lead to a kinetic singularity when a certain solution procedure is used to solve for the accelerations. It is shown in this paper that when the equations are formulated in terms of Euler parameters, the use of a solution procedure for the accelerations that requires the elimination of Lagrange multipliers leads to a singular coefficient matrix. Since Lagrange multipliers associated with Euler parameters constraints always define zero constraint forces, a nonsingular coefficient matrix can be identified and used to solve for Lagrange multipliers associated with other joint constraints. The proposed solution procedure requires the inverse of a 3 × 3 constant matrix for each rigid body in the system and the LU solution of a system of algebraic equations that has dimension equal to the number of constraint equations excluding Euler parameters constraints.

Characterization of microstructures using contour tree connectivity for fluid flow analysis

Quantifying the connectivity of material microstructures is important for a wide range of applications from filters to biomaterials. Currently, the most used measure of connectivity is the Euler number, which is a topological invariant. Topology alone, however, is not sufficient for most practical purposes. In this study, we use our recently introduced connectivity measure, called the contour tree connectivity (CTC), to study microstructures for flow analysis. CTC is a new structural connectivity measure that is based on contour trees and algebraic graph theory. To test CTC, we generated a dataset composed of 120 samples and six different types of artificial microstructures. We compared CTC against the Euler parameter (EP), the parameter for connected pairs, the nominal opening dimension (dnom) and the permeabilities estimated using direct pore scale modelling. The results show that dnom is highly correlated with permeability (R2=0.91), but cannot separate the structural differences. The groups are best classified with feature combinations that include CTC. CTC provides new information with a different connectivity interpretation that can be used to analyse and design materials with complex microstructures.

A Numerical Approach for Simulating a Self-Contacted Elastic Rod

A challenging problem in the numerical simulation of a self-contacted elastic rod is the change of topological structure during the contacting process. In this article, a numerical approach is introduced to simulate a self-contacted Cosserat rod. By introducing the angular velocity and Euler parameters as the state variables, an improved Lagrange equation is introduced. A finite element method is built to deal with the simplified model, and numerical examples are given.

Rotational equations usable for railway wheelsets

The Euler equation is a correct way for writing rotational moments of solids. But it is simple only if written in rotating frames. Applying it to railway wheelsets is difficult because it necessitates using the Euler angles, or Euler parameters, combined to rotation matrices or, numerically more stable, quaternions. Euler angles can be avoided in railway specific codes, by writing dynamical equations in track frames. However, academic literature [Landau LD, Lifshitz EM. Mechanics (Institute of Physical Problems, USSR Academy of Sciences, Moscow), Vol. 1, Course of theoretical physics. 21st English ed. Oxford (UK): Elsevier; 1960; Shabana AA, Zaazaa KE, Sugiyama H. Railroad vehicle dynamics. CRC Press; 2008.] does not provide simple solutions as to how properly writing equations of gyroscopic moments in no rotating frames. This paper describes how it is possible, owing to an approximation validated for railway applications, to avoid Euler angles and rotation matrices, while correctly taking into account gyroscopic effects. Using a most severe example, emphasising gyroscopic effects, it is demonstrated that a fast specific code using the approximation provides results equivalent to those of an multi body system generalised code with no approximation.