Quaternion Rotation Research Articles

Quaternion Filtering Based on Quaternion Involutions and its Application in Signal Processing

The quaternion gradient plays an important role in quaternion signal processing, and has undergone several modifications. Recently, three methods for obtaining the quaternion gradient have been proposed based on generalized HR calculus, the quaternion product, and quaternion involutions, respectively. The first method introduces the quaternion rotation, which is difficult to calculate and often relies on lookup tables; the second is cumbersome because it depends on all the real and imaginary parts of the variables and parameters; the third transforms the gradient from the quaternion domain to the real domain, and uses real derivatives and the real chain rule. In this paper, we generalize the quaternion involutions method to calculate the quaternion gradient of the quaternion matrix function. Several examples are presented in which the proposed gradient is applied to the adaptive filter, Kalman filter, and kernel filters to solve the associated optimization problems.

Eigen-Decomposition of Quaternions

This paper introduces biquaternion eigen-decomposition theory (via Peirce decomposition) with respect to a selected quaternion with a non-zero vector part. The eigen-decomposition allows evaluation of polynomials and power series with real coefficients as functions of quaternions. This extension of analytic functions to functions of quaternions requires only standard complex function evaluation. The theory also applies to quaternion rotations. The theory uses biquaternion calculations indicated by matrix methods via the algebraic isomorphism between Hamilton’s biquaternions and appropriate $$4\times 4$$ complex matrices. The isomorphism preserves algebraic structure. In particular, the left and right biquaternion multiplication by the selected quaternion maps to left and right matrix multiplication, respectively. This unifies the representation of the left and right quaternion multiplication as a linear map into a single matrix form. This matrix, as a linear operator, acts on matrices, so that the eigenvectors have matrix form that maps into the biquaternions. Use of an alternate quaternion basis results in a similarity transform of the representation matrix, preserving eigenvalues across change of basis. The similarity transform allows simple eigenvector calculation. The matrix for the selected quaternion has two identical, complex conjugate pairs of eigenvalues. Each pair corresponds to two complex conjugate pairs of eigenvector biquaternions, an idempotent pair and a nilpotent pair. Idempotent and nilpotent eigenvectors correspond to the commuting and non-commuting parts, respectively, of quaternion multiplication.

A weak form quadrature element formulation of geometrically exact shells incorporating drilling degrees of freedom

Geometrically nonlinear analysis of shell structures is conducted using weak form quadrature elements. A new geometrically exact shell formulation incorporating drilling degrees of freedom is established wherein rotation quaternions in combination with a total Lagrange updating scheme are employed for rotation description. An extended kinematic condition to serve as the drilling rotation constraint, derived from polar decomposition of modified mid-surface deformation gradient, is exactly satisfied in the formulation. Several benchmark examples are presented to illustrate the versatility and robustness of the present formulation.

Why and How to Avoid the Flipped Quaternion Multiplication

Over the last decades quaternions have become a crucial and very successful tool for attitude representation in robotics and aerospace. However, there is a major problem that is continuously causing trouble in practice when it comes to exchanging formulas or implementations: there are two quaternion multiplications commonly in use, Hamilton’s multiplication and its flipped version, which is often associated with NASA’s Jet Propulsion Laboratory. This paper explains the underlying problem for the popular passive world-to-body usage of rotation quaternions, and promotes an alternative solution compatible with Hamilton’s multiplication. Furthermore, it argues for discontinuing the flipped multiplication. Additionally, it provides recipes for efficiently detecting relevant conventions and migrating formulas or algorithms between them.

Full-Scale Reconstruction for Transmission Line Galloping Curves Based on Attitudes Sensors

Caused by strong winds or nonuniform icing, conductor galloping is one of the major hazards that should be monitored in a timely fashion. In this paper, we proposed a new full-scale reconstruction method for transmission line curves based on the attitude sensors that uses only the tangential information and arc-length constraint. Meanwhile, a comparatively low-complexity algorithm is presented: (1) quaternion rotation to generate the tangent vectors; (2) a modified B-spline global interpolation to evaluate the curvature information; and (3) a linear mapping method to correct reconstruction curves. Moreover, sensors placement method is put forward according to the galloping features. Finally, both simulation and experimental results demonstrate that the proposed method can accurately reconstruct the galloping curves in a relatively short time, which fulfills the requirement of the real-time galloping monitoring system.

Quantitative suggestions for build orientation selection

Orientation determination is an essential process planning task in additive manufacturing (AM) since it directly affects the part quality, part texture, mechanical properties, build time, fabrication cost, etc. Evaluation method provides a simple and effective way to determine the optimum orientation of a part. However, the candidates are predesigned in the evaluation method, which results in limited candidate space and makes the qualities of evaluation results highly dependent on the qualities of the predesigned candidates; bad outcomes will be obtained due to the poor candidates. To this end, a feedback multi-attribute decision-making (MADM) model is proposed in this work. The feedback MADM model is an integration of two sub-models: MADM model and proportional-integral-derivative (PID) control model. MADM model aims to calculate the score of a given build orientation. Three criteria, surface roughness, support volume, and build time, are considered in this model, and the ordered weighted averaging (OWA) operator is applied for aggregation. In the PID control model, the finite candidate space is first expanded to infinity by quaternion rotation, then PID controller is applied to match the build orientation with the user’s requirements, that is, search for the part orientation whose score is consistent with the user expected score in the infinite alternative orientations. Four parts with different geometric structures are tested in the experiments, and evaluation and control features of the feedback MADM model are discussed.

A weak form quadrature element formulation for geometrically exact thin shell analysis

The present paper addresses a weak form quadrature element formulation for the geometrically exact thin shell model in which the Kirchhoff-Love hypothesis is adopted. The displacement derivative continuity conditions are enforced by the reconstruction of rotation variables at the edges of elements. By the utilization of rotation quaternions, a total Lagrange updating scheme is implemented for edge constraint director rotations. Several numerical examples are presented to illustrate the effectiveness of the proposed formulation and the significant reduction in the number of degrees of freedom in geometrically nonlinear thin shell analysis with large displacements and rotations.

A finite element/quaternion/asymptotic numerical method for the 3D simulation of flexible cables

In this paper, a method for the quasi-static simulation of flexible cables assembly in the context of automotive industry is presented. The cables geometry and behavior encourage to employ a geometrically exact beam model. The 3D kinematics is then based on the position of the centerline and on the orientation of the cross-sections, which is here represented by rotational quaternions. Their algebraic nature leads to a polynomial form of equilibrium equations. The continuous equations obtained are then discretized by the finite element method and easily recast under quadratic form by introducing additional slave variables. The asymptotic numerical method, a powerful solver for systems of quadratic equations, is then employed for the continuation of the branches of solution. The originality of this paper stands in the combination of all these methods which leads to a fast and accurate tool for the assembly process of cables. This is proved by running several classical validation tests and an industry-like example.

Quaternion Regularization of the Equations of the Perturbed Spatial Restricted Three-Body Problem: I

We develop a quaternion method for regularizing the differential equations of the perturbed spatial restricted three-body problem by using the Kustaanheimo–Stiefel variables, which is methodologically closely related to the quaternion method for regularizing the differential equations of perturbed spatial two-body problem, which was proposed by the author of the present paper. A survey of papers related to the regularization of the differential equations of the two- and threebody problems is given. The original Newtonian equations of perturbed spatial restricted three-body problem are considered, and the problem of their regularization is posed; the energy relations and the differential equations describing the variations in the energies of the system in the perturbed spatial restricted three-body problem are given, as well as the first integrals of the differential equations of the unperturbed spatial restricted circular three-body problem (Jacobi integrals); the equations of perturbed spatial restricted three-body problem written in terms of rotating coordinate systems whose angular motion is described by the rotation quaternions (Euler (Rodrigues–Hamilton) parameters) are considered; and the differential equations for angular momenta in the restricted three-body problem are given. Local regular quaternion differential equations of perturbed spatial restricted three-body problem in the Kustaanheimo–Stiefel variables, i.e., equations regular in a neighborhood of the first and second body of finite mass, are obtained. The equations are systems of nonlinear nonstationary eleventhorder differential equations. These equations employ, as additional dependent variables, the energy characteristics of motion of the body under study (a body of a negligibly small mass) and the time whose derivative with respect to a new independent variable is equal to the distance from the body of negligibly small mass to the first or second body of finite mass. The equations obtained in the paper permit developing regular methods for determining solutions, in analytical or numerical form, of problems difficult for classicalmethods, such as the motion of a body of negligibly small mass in a neighborhood of the other two bodies of finite masses.

Modelling and solving the position tracking problem of remote-controlled gastrointestinal drug-delivery capsules

Abstract A theoretical basis is presented to develop a real-time position tracking system for a remote-controlled drug-delivery capsule that would enable the targeted delivery of drugs to specific locations in the gastrointestinal tract. Tracking is accomplished by dual magnetic vector detection over time, using two separate sets of magnetic field sensing devices. One is used to detect the alternating magnetic field excited externally with certain frequencies, while the other is used to detect the geomagnetic field. A mathematical model of the magnetic flux density in space and time is constructed using the Biot–Savart law. Additionally, the earth’s magnetic field and quaternion rotation theory are also used in the model to compensate for the constantly changing spatial orientation of the capsule as it travels through the gastrointestinal tract. Based on the model, an improved artificial bee colony algorithm is used to solve the inverse magnetic field problem. Firstly, chaotic sequencing improves the initial solution diversity, and a ranked selection strategy is applied. At later stages, the Levenberg-Marquardt algorithm is introduced in order to accelerate convergence. To verify the theoretical basis presented above, a prototype of the tracking system is developed. Calculations verification results in a convergence rate of 100% with an average of 179 iterations. The prototype testing shows that the dual magnetic vector detection method simplifies the solution of the inverse magnetic field problem, shortens the tracking time for each round of data, and increases the solution accuracy.

Visualizing Quaternion Multiplication

Quaternion rotation is a powerful tool for rotating vectors in 3-D; as a result, it has been used in various engineering fields, such as navigations, robotics, and computer graphics. However, understanding it geometrically remains challenging, because it requires visualizing 4-D spaces, which makes exploiting its physical meaning intractable. In this paper, we provide a new geometric interpretation of quaternion multiplication using a movable 3-D space model, which is useful for describing quaternion algebra in a visual way. By interpreting the axis for the scalar part of quaternion as a 1-D translation axis of 3-D vector space, we visualize quaternion multiplication and describe it as a combined effect of translation, scaling, and rotation of a 3-D vector space. We then present how quaternion rotation formulas and the derivative of quaternions can be formulated and described under the proposed approach.

Algorithms for the orientation of a moving object with separation of the integration of fast and slow motions

Abstract Equations and algorithms for determining the orientation of a moving object in inertial and normal geographic coordinate systems are considered with separation of the integration of the fast and slow motions into ultrafast, fast and slow cycles. Ultrafast cycle algorithms are constructed using a Riccati-type kinematic quaternion equation and the Picard method of successive approximations, and the increments in the integrals of the projections of the absolute angular velocity vector of the object onto the coordinate axes (quasicoordinates) associated with them are used as input information. The fast cycle algorithm realizes the calculation of the classical rotation quaternion of an object on a step of the fast cycle in an inertial system of coordinates. The slow cycle algorithm is used in calculating the orientation quaternion of an object in the normal geographic coordinate system and aircraft angles. Results of modelling different versions of the fast and ultrafast cycle algorithms for calculating the inertial orientation of an object are presented and discussed. The experience of the authors in developing algorithms for determining the orientation of moving objects using a strapdown inertial navigation system is described and results obtained by them earlier in this field are developed and extended.

Lie Group Forward Dynamics of Fixed-Wing Aircraft With Singularity-Free Attitude Reconstruction on SO(3)

This paper proposes an approach to formulation and integration of the governing equations for aircraft flight simulation that is based on a Lie group setting, and leads to a nonsingular coordinate-free numerical integration. Dynamical model of an aircraft is formulated in Lie group state space form and integrated by ordinary-differential-equation (ODE)-on-Lie groups Munthe-Kaas (MK) type of integrator. By following such an approach, it is assured that kinematic singularities, which are unavoidable if a three-angles-based rotation parameterization is applied for the whole 3D rotation domain, do not occur in the proposed noncoordinate formulation form. Moreover, in contrast to the quaternion rotation parameterization that imposes additional algebraic constraint and leads to integration of differential-algebraic equations (DAEs) (with necessary algebraic-equation-violation stabilization step), the proposed formulation leads to a nonredundant ODE integration in minimal form. To this end, this approach combines benefits of both traditional approaches to aircraft simulation (i.e., three angles parameterization and quaternions), while at the same time it avoids related drawbacks of the classical models. Besides solving kinematic singularity problem without introducing DAEs, the proposed formulation also exhibits numerical advantages in terms of better accuracy when longer integration steps are applied during simulation and when aircraft motion pattern comprises steady rotational component of its 3D motion. This is due to the fact that a Lie group setting and applied MK integrator determine vehicle orientation on the basis of integration of local (tangent, nonlinear) kinematical differential equations (KDEs) that model process of 3D rotations (i.e., vehicle attitude reconstruction on nonlinear manifold SO(3)) more accurately than “global” KDEs of the classical formulations (that are linear in differential equations part in the case of standard quaternion models).

Improved quaternion-based integration scheme for rigid body motion

Rotation quaternions are frequently used for describing the orientation of non-spherical rigid bodies. Their compact representation by four numbers and disappearance of numerical problems, such as gimbal lock, are reasons for using them. We describe an improvement of a predictor–corrector direct multiplication (PCDM) method for numerically integrating the rigid body equations of motion with rotation quaternions. The method only uses quaternions to describe the orientation, so no rotation matrices are needed in the implementation. A predictor–corrector approach is used to update the quaternions each time step, such that no renormalization is needed at the end of the time step. The PCDM method suggested by Zhao and Van Wachem is improved such that forces and torques are calculated at the correct time using position and orientation information at that same time. This is achieved by using a leapfrog approach in the improved version, in which the linear and angular velocities and rotation quaternions are defined at half time steps, while whole time step information of these quantities is calculated as part of the improved integration scheme. The improved PCDM scheme is compared with the original implementation for rotational kinetic energy conservation, accuracy of object orientation and angular velocity, and rate of convergence for different time steps. With the modifications that we propose, the improved method has a true second-order rate of convergence, without the need for explicit renormalization of the quaternions. Furthermore, the method is applicable to problems with position and velocity dependent torques, while still only a single force/torque evaluation is needed per time step. For objects experiencing torque, the improved PCDM method performs better than the original method, now showing a true second-order rate of convergence, and much smaller errors in the prediction of object orientation and angular velocity while still requiring only a single torque evaluation per time step.

Joint fingerprinting and decryption method for color images based on quaternion rotation with cipher quaternion chaining

This paper addresses the problem of unauthorized redistribution of multimedia content by malicious users (pirates). In this method three color channels of the image are considered a 3D space and each component of the image is represented as a point in this 3D space. The distribution side uses a symmetric cipher to encrypt perceptually essential components of the image with the encryption key and then sends the encrypted data via multicast transmission to all users. The encryption involves rotation, and translation of points in 3D color space using quaternion algebra. Each user has a unique decryption key which is different from the encryption key. The differences between the common encryption key and the individual user’s decryption key cause the decrypted image to contain minor changes which are user’s fingerprint. A computer-based simulation was conducted to examine the method’s robustness against noise, compression, and collusion attacks.

Singularity-free time integration of rotational quaternions using non-redundant ordinary differential equations

A novel ODE time stepping scheme for solving rotational kinematics in terms of unit quaternions is presented in the paper. This scheme inherently respects the unit-length condition without including it explicitly as a constraint equation, as it is common practice. In the standard algorithms, the unit-length condition is included as an additional equation leading to kinematical equations in the form of a system of differential-algebraic equations (DAEs). On the contrary, the proposed method is based on numerical integration of the kinematic relations in terms of the instantaneous rotation vector that form a system of ordinary differential equations (ODEs) on the Lie algebra $\mathit{so}(3)$ of the rotation group $\mathit{SO}(3)$ . This rotation vector defines an incremental rotation (and thus the associated incremental unit quaternion), and the rotation update is determined by the exponential mapping on the quaternion group. Since the kinematic ODE on $\mathit{so}(3)$ can be solved by using any standard (possibly higher-order) ODE integration scheme, the proposed method yields a non-redundant integration algorithm for the rotational kinematics in terms of unit quaternions, avoiding integration of DAE equations. Besides being ‘more elegant’—in the opinion of the authors—this integration procedure also exhibits numerical advantages in terms of better accuracy when longer integration steps are applied during simulation. As presented in the paper, the numerical integration of three non-linear ODEs in terms of the rotation vector as canonical coordinates achieves a higher accuracy compared to integrating the four (linear in ODE part) standard-quaternion DAE system. In summary, this paper solves the long-standing problem of the necessity of imposing the unit-length constraint equation during integration of quaternions, i.e. the need to deal with DAE’s in the context of such kinematical model, which has been a major drawback of using quaternions, and a numerical scheme is presented that also allows for longer integration steps during kinematic reconstruction of large three-dimensional rotations.

The Foldability of Cylindrical Foldable Structures Based on Rigid Origami

Foldable structures, a new kind of space structures developed in recent decades, can be deployed gradually to a working configuration and also can be folded for transportation, thus have potentially broad application prospects in the fields of human life, military, aerospace, building structures, and so on. Combined with the technology of origami folding, foldable structures derive more diversified models, and the foldable structures in cylindrical shape are mainly studied in this paper. Some researchers use the theory of quaternion representing spatial fixed-point rotation and construct the rotating vector model to obtain the quaternion rotation sequence method (QRS method) analyzing origami, but the method is very limited and not suitable for the cylindrical foldable structures. In order to solve the problem, a new method is developed, which combines the QRS method and the dual quaternion method. After analyzing the folding angle via the QRS method for multivertex crease system and calculating the coordinates of all vertices via the dual quaternion, the rigid foldability can be checked. Finally, two examples are carried out to confirm validity and versatility of the method.

Motion trajectory of human arms based on the dual quaternion with motion tracker

In this paper, we present an algorithm for human motion capture of the real-time motion trajectory of human arms based on wireless inertial 3D motion trackers. It aims to improve the accuracy of inertial motion captures and quickly reconstruct some human movements. To evaluate the performance of the proposed dual quaternion algorithm, we present the prototype design. The wireless inertial measurement system and Kinect device are introduced simultaneously in capturing human motion. The dual quaternion algorithm incorporates features of the quaternion rotation and translation. So the singular points of Euler angles can be avoided. Dual quaternion algorithm and DCM(direction cosine matrix) are used to reconstruct human arm movements respectively. Compared with the computing speed in Matlab, the speed of the dual quaternion is faster than it of DCM. To the end, we propose a 3D ADAMS human robotic model for simulating the motion trajectory using dual quaternion algorithm. The results show that the dual quaternion can achieve capabilities of a positive DCM solving, which completed between body segments rotating and translating the coordinate system transformation. Also it can effectively drive in real-time a human model to animate movement, and provide a good algorithm.

Real Time 3D-Handwritten Character and Gesture Recognition for Smartphone

This Paper presents two different models for smartphone based 3D-handwritten character and gesture recognition. Smartphones available today are equipped with inbuilt sensors like accelerometer, gyroscope, and gravity sensor, which are able to provide data on motion of the device in 3D space. The sensor readings are easily accessible through native operating system provided by the smartphone. The accelerometer and gyroscope sensors data have been used in the models to make the systems more expressive and to disambiguate recognition. The acceleration generated by the hand motions are generated by the 3-axes accelerometer present in smartphone. Data from gyroscope is used to obtain the quaternion rotation matrix, which is used to remove the tilting and inclination offset. An automatic segmentation algorithm is implemented to identify individual gesture or character in a sequence. First model is based on training and evaluation mode, while second one is based on training-less algorithm. The recognition models as presented in this paper can be used for both user dependent and user independent 3d-character and gesture recognition. Results shows that, Model I gives an efficient recognition rate ranging from 90% to 94% obtained with minimal training sequence, while the Model II gives a recognition rate of 88%.

Enabling quaternion derivatives: the generalized HR calculus.

Quaternion derivatives exist only for a very restricted class of analytic (regular) functions; however, in many applications, functions of interest are real-valued and hence not analytic, a typical case being the standard real mean square error objective function. The recent HR calculus is a step forward and provides a way to calculate derivatives and gradients of both analytic and non-analytic functions of quaternion variables; however, the HR calculus can become cumbersome in complex optimization problems due to the lack of rigorous product and chain rules, a consequence of the non-commutativity of quaternion algebra. To address this issue, we introduce the generalized HR (GHR) derivatives which employ quaternion rotations in a general orthogonal system and provide the left- and right-hand versions of the quaternion derivative of general functions. The GHR calculus also solves the long-standing problems of product and chain rules, mean-value theorem and Taylor's theorem in the quaternion field. At the core of the proposed GHR calculus is quaternion rotation, which makes it possible to extend the principle to other functional calculi in non-commutative settings. Examples in statistical learning theory and adaptive signal processing support the analysis.