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.

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.

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 Dual Quaternion Functions and Its Applications

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Su Jin Lim ◽  
Kwang Ho Shon
Keyword(s):  
Extension Problem ◽  
Dual Numbers ◽  
Dual Quaternion ◽  
Regular Functions ◽  
Dual Identity

A dual quaternion is associated with two quaternions that have basis elementse0,e1,e2,e3, andε. Dual numbers are often written in the formz=ζ+εζ*, whereεis the dual identity and has the propertiesε2=0  (ε≠0). We research the properties of some regular functions with values in dual quaternion and give applications of the extension problem for dual quaternion functions.

scholarly journals On Sign Pattern Matrices that Allow or Require Algebraic Positivity

2019 ◽  
Vol 35 ◽  
pp. 331-356
Author(s):  
Diane Christine Pelejo ◽  
Jean Leonardo Abagat
Keyword(s):  
The Other ◽  
Necessary Condition ◽  
Sign Pattern ◽  
Real Polynomial ◽  
Positive Matrix ◽  
Open Problems ◽  
Other Hand ◽  
The Matrix ◽  
Pattern Matrix ◽  
Square Matrix

A square matrix M with real entries is algebraically positive (AP) if there exists a real polynomial p such that all entries of the matrix p(M) are positive. A square sign pattern matrix S allows algebraic positivity if there is an algebraically positive matrix M whose sign pattern is S. On the other hand, S requires algebraic positivity if matrix M, having sign pattern S, is algebraically positive. Motivated by open problems raised in a work of Kirkland, Qiao, and Zhan (2016) on AP matrices, all nonequivalent irreducible 3 by 3 sign pattern matrices are listed and classify into three groups (i) those that require AP, (ii) those that allow but not require AP, or (iii) those that do not allow AP. A necessary condition for an irreducible n by n sign pattern to allow algebraic positivity is also provided.

Dual Quaternion Synthesis of Constrained Robotic Systems

2003 ◽  
Vol 126 (3) ◽  
pp. 425-435 ◽  
Author(s):  
Alba Perez ◽  
J. M. McCarthy
Keyword(s):  
Parallel Robots ◽  
Robotic Systems ◽  
Dual Quaternion ◽  
Kinematic Synthesis ◽  
End Effector ◽  
Reference Position ◽  
Dual Quaternions ◽  
Design Variables ◽  
Finite Set ◽  
Serial Chains

This paper presents a dual quaternion methodology for the kinematic synthesis of constrained robotic systems. These systems are constructed from one or more serial chains such that each chain imposes at least one constraint on the movement of the workpiece. Serial chains that have constrained workspaces can be synthesized by evaluating the kinematics equations of the chain on a finite set of task positions. In this case, the end-effector positions are known and the Denavit-Hartenberg parameters become design variables. Here we reformulate the kinematics equations in terms of successive screw displacements so the design variables are the coordinates defining the joint axes of the chain in a reference position. Then, dual quaternions defining these transformations are introduced to simplify the structure of the design equations. The result is a synthesis formulation that can be applied to a broad range of constrained serial chains, which can in turn be assembled into constrained parallel robots. We demonstrate the formulation and solution of the dual quaternion design equations for the spatial RPRP chain.

scholarly journals Dual-Quaternion Analytic LQR Control Design for Spacecraft Proximity Operations

Sensors ◽  
2021 ◽  
Vol 21 (11) ◽  
pp. 3597
Author(s):  
Kyl Stanfield ◽  
Ahmad Bani Younes
Keyword(s):  
Optimal Control ◽  
Monte Carlo ◽  
Monte Carlo Simulations ◽  
Linear Quadratic Regulator ◽  
Small Error ◽  
Control Law ◽  
Dual Quaternion ◽  
Linear Quadratic ◽  
Proximity Operations ◽  
Dual Quaternions

Proximity operations offer aggregate capability for a spacecraft operating in close proximity to another spacecraft, to perform on-orbit satellite servicing, or to a space object to perform debris removal. To utilize a spacecraft performing such advanced maneuvering operations and perceiving of the relative motion of a foreign spacecraft, these trajectories must be modeled accurately based on the coupled translational and rotational dynamics models. This paper presents work towards exploiting the dual-quaternion representations of spacecraft relative dynamics for proximity operations and developing a sub-optimal control law for efficient and robust maneuvers. A linearized model using dual-quaternions for the proximity operation was obtained, and its stability was verified using Monte Carlo simulations for the linear quadratic regulator solution. A sub-optimal control law using generalized higher order feedback gains in dual-quaternion form was developed based on small error approximations for the proximity operation and also verified through Monte Carlo simulations. Necessary information needed to understand the theory behind the use of the dual-quaternion is also overviewed within this paper, including the validity of using the dual-quaternions against their Cartesian or quaternion equivalents.

scholarly journals A Novel Dual Quaternion Based Cost Effcient Recursive Newton-Euler Inverse Dynamics Algorithm

2019 ◽  
Vol 1 (2) ◽  
pp. 144-168
Author(s):  
Cristiana Miranda de Farias
Keyword(s):  
Feedback Linearization ◽  
Inverse Dynamics ◽  
Joint Space ◽  
Linearization Method ◽  
Torque Control ◽  
Dual Quaternion ◽  
Serial Manipulators ◽  
Dual Quaternions ◽  
Forward Kinematic ◽  
Computed Torque

In this paper, the well known recursive Newton-Euler inverse dynamics algorithm for serial manipulators is reformulated into the context of the algebra of Dual Quaternions. Here we structure the forward kinematic description with screws and line displacements rather than the well established Denavit-Hartemberg parameters, thus accounting better efficiency, compactness and simpler dynamical models. We also present here the closed solution for the dqRNEA, and to do so we formalize some of the algebra for dual quaternion-vectors and dual quaternion-matrices. With a closed formulation of the dqRNEA we also create a dual quaternion based formulation for the computed torque control, a feedback linearization method for controlling a serial manipulator's torques in the joint space. Finally, a cost analysis of the main Dual Quaternions operations and of the Newton-Euler inverse dynamics algorithm as a whole is made and compared with other results in the literature.

scholarly journals Some Characterizations for a Quaternion-Valued and Dual Variable Curve

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 125
Author(s):  
Muge Karadag ◽  
Ali Sivridag
Keyword(s):  
Real Variable ◽  
Dual Variable ◽  
Dual Quaternion ◽  
Harmonic Curvature ◽  
Dual Quaternions ◽  
Long Time ◽  
Real Quaternions

Quaternions, which are found in many fields, have been studied for a long time. The interest in dual quaternions has also increased after real quaternions. Nagaraj and Bharathi developed the basic theories of these studies. The Serret–Frenet Formulae for dual quaternion-valued functions of one real variable are derived. In this paper, by making use of the results of some previous studies, helixes and harmonic curvature concepts in Q D 3 and Q D 4 are considered and a characterization for a dual harmonic curve to be a helix is given.

scholarly journals Immersion and Invariance Adaptive Control for Spacecraft Pose Tracking via Dual Quaternions

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Xiaoping Shi ◽  
Xuan Peng ◽  
Yupeng Gong
Keyword(s):  
Relative Motion ◽  
Direction Cosine ◽  
Dual Quaternion ◽  
Unknown Parameters ◽  
Estimation Errors ◽  
Performance Improvements ◽  
Pose Tracking ◽  
Immersion And Invariance ◽  
Dual Quaternions ◽  
Control Scheme

This paper addresses the simultaneous attitude and position tracking of a target spacecraft in the presence of general unknown bounded disturbances in the framework of dual quaternions, which provides a concise and integrated description of the coupled rotational and translational motions. By virtue of the newly introduced dual direction cosine matrix, the dimension of the dual quaternion-based relative motion dynamics written in vector/matrix form can be lowered to six. Treating the disturbances as unknown parameters, a modular adaptive pose tracking control scheme composed of two separately designed parts is then derived. One part is the adaptive disturbance estimator designed based on the immersion and invariance theory. Driven by the disturbance estimation errors, it can realize exponential convergence of the estimations and has the nice “parameter lock” property, which can hardly be expected in the conventional certainty equivalent adaptive controllers. The other part is a proportional-derivative-like pose tracking controller where the estimated disturbances are directly used. The closed-loop stability of the relative motion system under different kinds of disturbances is proven by Lyapunov stability analysis. Simulations and comparisons with two previous dual quaternion-based controllers demonstrate the novel features and performance improvements of the proposed control scheme.

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