scholarly journals An Algorithm for the Factorization of Split Quaternion Polynomials

2021 ◽  
Vol 31 (3) ◽  
Author(s):  
Daniel F. Scharler ◽  
Hans-Peter Schröcker
Keyword(s):  
Hyperbolic Plane ◽  
Dual Quaternion ◽  
Real Polynomial ◽  
Split Quaternion ◽  
Split Quaternions ◽  
Dual Quaternions ◽  
Factorization Theory ◽  
Real Polynomials ◽  
Theory Of Motion ◽  
Univariate Polynomials

AbstractWe present an algorithm to compute all factorizations into linear factors of univariate polynomials over the split quaternions, provided such a factorization exists. Failure of the algorithm is equivalent to non-factorizability for which we present also geometric interpretations in terms of rulings on the quadric of non-invertible split quaternions. However, suitable real polynomial multiples of split quaternion polynomials can still be factorized and we describe how to find these real polynomials. Split quaternion polynomials describe rational motions in the hyperbolic plane. Factorization with linear factors corresponds to the decomposition of the rational motion into hyperbolic rotations. Since multiplication with a real polynomial does not change the motion, this decomposition is always possible. Some of our ideas can be transferred to the factorization theory of motion polynomials. These are polynomials over the dual quaternions with real norm polynomial and they describe rational motions in Euclidean kinematics. We transfer techniques developed for split quaternions to compute new factorizations of certain dual quaternion polynomials.

scholarly journals Four-Pose Synthesis of Angle-Symmetric 6R Linkages

2015 ◽  
Vol 7 (4) ◽  
Author(s):  
Gábor Hegedüs ◽  
Josef Schicho ◽  
Hans-Peter Schröcker
Keyword(s):  
Rotation Angle ◽  
Dual Quaternions ◽  
Revolute Joints ◽  
Factorization Theory ◽  
Theory Of Motion ◽  
Kinematic Loops ◽  
Parametric Families ◽  
Synthesis Approach ◽  
Relative Motions

We use the recently introduced factorization theory of motion polynomials over the dual quaternions and cubic interpolation on quadrics for the synthesis of closed kinematic loops with six revolute joints that visit four prescribed poses. The resulting 6R linkages are special in the sense that the relative motions of all links are rational. They exhibit certain elegant properties such as symmetry in the rotation angles and, at least in theory, full-cycle mobility. Our synthesis approach admits either no solution or two one-parametric families of solutions. We suggest strategies for picking good solutions from these families. They ensure a fair coupler motion and optimize other linkage characteristics such as total rotation angle or linkage size. A comprehensive synthesis example is provided.

scholarly journals Factorization of Dual Quaternion Polynomials Without Study’s Condition

2021 ◽  
Vol 31 (2) ◽  
Author(s):  
Johannes Siegele ◽  
Martin Pfurner ◽  
Hans-Peter Schröcker
Keyword(s):  
Necessary Condition ◽  
Dual Numbers ◽  
Dual Quaternion ◽  
Real Polynomial ◽  
Real Motion ◽  
Dual Quaternions ◽  
Factorization Algorithms

AbstractIn this paper we investigate factorizations of polynomials over the ring of dual quaternions into linear factors. While earlier results assume that the norm polynomial is real (“motion polynomials”), we only require the absence of real polynomial factors in the primal part and factorizability of the norm polynomial over the dual numbers into monic quadratic factors. This obviously necessary condition is also sufficient for existence of factorizations. We present an algorithm to compute factorizations of these polynomials and use it for new constructions of mechanisms which cannot be obtained by existing factorization algorithms for motion polynomials. While they produce mechanisms with rotational or translational joints, our approach yields mechanisms consisting of “vertical Darboux joints”. They exhibit mechanical deficiencies so that we explore ways to replace them by cylindrical joints while keeping the overall mechanism sufficiently constrained.

Robust and efficient forward, differential, and inverse kinematics using dual quaternions

2020 ◽  
pp. 027836492093194
Author(s):  
Neil T Dantam
Keyword(s):  
Inverse Kinematics ◽  
Forward Kinematics ◽  
Robot Kinematics ◽  
Dual Numbers ◽  
Dual Quaternion ◽  
Implicit Representation ◽  
Alternative Representation ◽  
Transformation Matrices ◽  
Dual Quaternions ◽  
Efficient Representation

Modern approaches for robot kinematics employ the product of exponentials formulation, represented using homogeneous transformation matrices. Quaternions over dual numbers are an established alternative representation; however, their use presents certain challenges: the dual quaternion exponential and logarithm contain a zero-angle singularity, and many common operations are less efficient using dual quaternions than with matrices. We present a new derivation of the dual quaternion exponential and logarithm that removes the singularity, we show an implicit representation of dual quaternions offers analytical and empirical efficiency advantages compared with both matrices and explicit dual quaternions, and we derive efficient dual quaternion forms of differential and inverse position kinematics. Analytically, implicit dual quaternions are more compact and require fewer arithmetic instructions for common operations, including chaining and exponentials. Empirically, we demonstrate a 30–40% speedup on forward kinematics and a 300–500% speedup on inverse position kinematics. This work relates dual quaternions with modern exponential coordinates and demonstrates that dual quaternions are a robust and efficient representation for robot kinematics.

Kinematic calibration of serial robot using dual quaternions

2019 ◽  
Vol 46 (2) ◽  
pp. 247-258 ◽  
Author(s):  
Guozhi Li ◽  
Fuhai Zhang ◽  
Yili Fu ◽  
Shuguo Wang
Keyword(s):  
Kinematic Analysis ◽  
Quaternion Algebra ◽  
Error Model ◽  
Kinematic Calibration ◽  
Dual Quaternion ◽  
Robot Calibration ◽  
Geometrical Meaning ◽  
Content Type ◽  
Dual Quaternions ◽  
Serial Robot

Purpose The purpose of this paper is to propose an error model for serial robot kinematic calibration based on dual quaternions. Design/methodology/approach The dual quaternions are the combination of dual-number theory and quaternion algebra, which means that they can represent spatial transformation. The dual quaternions can represent the screw displacement in a compact and efficient way, so that they are used for the kinematic analysis of serial robot. The error model proposed in this paper is derived from the forward kinematic equations via using dual quaternion algebra. The full pose measurements are considered to apply the error model to the serial robot by using Leica Geosystems Absolute Tracker (AT960) and tracker machine control (T-MAC) probe. Findings Two kinematic-parameter identification algorithms are derived from the proposed error model based on dual quaternions, and they can be used for serial robot calibration. The error model uses Denavit–Hartenberg (DH) notation in the kinematic analysis, so that it gives the intuitive geometrical meaning of the kinematic parameters. The absolute tracker system can measure the position and orientation of the end-effector (EE) simultaneously via using T-MAC. Originality/value The error model formulated by dual quaternion algebra contains all the basic geometrical parameters of serial robot during the kinematic calibration process. The vector of dual quaternion error can be used as an indicator to represent the trend of error change of robot’s EE between the nominal value and the actual value. The accuracy of the EE is improved after nearly 20 measurements in the experiment conduct on robot SDA5F. The simulation and experiment verify the effectiveness of the error model and the calibration algorithms.

scholarly journals Some Fixed Points Results of Quadratic Functions in Split Quaternions

2016 ◽  
Vol 2016 ◽  
pp. 1-5 ◽  
Author(s):  
Young Chel Kwun ◽  
Mobeen Munir ◽  
Waqas Nazeer ◽  
Shin Min Kang
Keyword(s):  
Fixed Points ◽  
Quadratic Polynomial ◽  
Quadratic Functions ◽  
Split Quaternion ◽  
Quadratic Polynomials ◽  
Split Quaternions

We attempt to find fixed points of a general quadratic polynomial in the algebra of split quaternion. In some cases, we characterize fixed points in terms of the coefficients of these polynomials and also give the cardinality of these points. As a consequence, we give some simple examples to strengthen the infinitude of these points in these cases. We also find the roots of quadratic polynomials as simple consequences.

scholarly journals On the Consimilarity of Split Quaternions and Split Quaternion Matrices

2016 ◽  
Vol 24 (3) ◽  
pp. 189-207 ◽  
Author(s):  
Hidayet Hüda Kösal ◽  
Mahmut Akyiğit ◽  
Murat Tosun
Keyword(s):  
Split Quaternion ◽  
Solvability Conditions ◽  
Split Quaternions ◽  
General Solutions ◽  
Quaternion Matrices

AbstractIn this paper, we introduce the concept of consimilarity of split quaternions and split quaternion matrices. In his regard, we examine the solvability conditions and general solutions of the equationsandin split quaternions and split quaternion matrices, respectively. Moreover, coneigenvalue and coneigenvector are defined for split quaternion matrices. Some consequences are also presented.

scholarly journals STRICTLY REAL FUNDAMENTAL THEOREM OF ALGEBRA USING POLYNOMIAL INTERLACING

2021 ◽  
pp. 1-7
Author(s):  
SOHAM BASU
Keyword(s):  
Fundamental Theorem ◽  
Quadratic Polynomial ◽  
Complex Numbers ◽  
Real Analysis ◽  
Real Polynomial ◽  
Bivariate Polynomials ◽  
Real Polynomials ◽  
Fundamental Theorem Of Algebra ◽  
Polynomial Factor ◽  
Real Quadratic

Abstract Without resorting to complex numbers or any advanced topological arguments, we show that any real polynomial of degree greater than two always has a real quadratic polynomial factor, which is equivalent to the fundamental theorem of algebra. The proof uses interlacing of bivariate polynomials similar to Gauss's first proof of the fundamental theorem of algebra using complex numbers, but in a different context of division residues of strictly real polynomials. This shows the sufficiency of basic real analysis as the minimal platform to prove the fundamental theorem of algebra.

scholarly journals ON BÄCKLUND TRANSFORMATIONS WITH SPLIT QUATERNIONS

2020 ◽  
pp. 423
Author(s):  
Muhammed Talat Sariaydin
Keyword(s):  
Mathematical Model ◽  
Minkowski Space ◽  
Bäcklund Transformations ◽  
Split Quaternion ◽  
Split Quaternions ◽  
Backlund Transformations ◽  
Null Curves ◽  
Basic Concepts ◽  
The Mathematical Model ◽  
End Results

The present paper deals with the introduction of Bäcklund Transformations with split quaternions in Minkowski space. Firstly, we tersely summarized the basic concepts of split quaternion theory and Bishop Frames of non-null curves in Minkowski space. Then, for Bäcklund transformations defined with each case of non-null curves, we give relationships between Bäcklund transformations and split quaternions. It is also presented some special propositions for transformations constructed with split quaternions. At the end, results obtained with the mathematical model have been evaluated.

Sign in / Sign up

Export Citation Format

Share Document