scholarly journals A Multiplication Technique for the Factorization of Bivariate Quaternionic Polynomials

2021 ◽  
Vol 32 (1) ◽  
Author(s):  
Johanna Lercher ◽  
Hans-Peter Schröcker
Keyword(s):  
Closed Loop ◽  
Necessary Condition ◽  
Real Polynomial ◽  
Bivariate Polynomials ◽  
Skew Field ◽  
Revolute Joints ◽  
Univariate Polynomials ◽  
Central Result ◽  
Factorization Condition ◽  
Quaternionic Polynomials

AbstractWe consider bivariate polynomials over the skew field of quaternions, where the indeterminates commute with all coefficients and with each other. We analyze existence of univariate factorizations, that is, factorizations with univariate linear factors. A necessary condition for existence of univariate factorizations is factorization of the norm polynomial into a product of univariate polynomials. This condition is, however, not sufficient. Our central result states that univariate factorizations exist after multiplication with a suitable univariate real polynomial as long as the necessary factorization condition is fulfilled. We present an algorithm for computing this real polynomial and a corresponding univariate factorization. If a univariate factorization of the original polynomial exists, a suitable input of the algorithm produces a constant multiplication factor, thus giving an a posteriori condition for existence of univariate factorizations. Some factorizations obtained in this way are of interest in mechanism science. We present an example of a curious closed-loop mechanism with eight revolute joints.

scholarly journals Design and Kinematic Analysis of a Novel 2-DoF Closed-Loop Mechanism for the Actuation of Machining Robots

2021 ◽  
Author(s):  
Angelica Ginnante ◽  
François Leborne ◽  
Stéphane Caro ◽  
Enrico Simetti ◽  
Giuseppe Casalino
Keyword(s):  
Degrees Of Freedom ◽  
Closed Loop ◽  
Parallel Mechanisms ◽  
Serial Manipulators ◽  
Hard Materials ◽  
Serial Robots ◽  
Revolute Joints ◽  
Machining Robot ◽  
Kinematic Models ◽  
Kinematic Performance

Abstract The essential characteristics of machining robots are their stiffness and their accuracy. For machining tasks, serial robots have many advantages such as large workspace to footprint ratio, but they often lack the stiffness required for accurately milling hard materials. One way to increase the stiffness of serial manipulators is to make their joints using closed-loop or parallel mechanisms instead of using classical prismatic and revolute joints. This increases the accuracy of a manipulator without reducing its workspace. This paper introduces an innovative two degrees of freedom closed-loop mechanism and shows how it can be used to build serial robots featuring both high stiffness and large workspace. The design of this mechanism is described through its geometric and kinematic models. Then, the kinematic performance of the mechanism is analyzed, and a serial arrangement of several such mechanisms is proposed to obtain a potential design of a machining robot.

scholarly journals Regularization of the Spatial Inverse Positioning Problem of Revolute Joint Manipulators

2017 ◽  
Vol 61 (3) ◽  
pp. 279
Author(s):  
Dániel András Drexler
Keyword(s):  
Inverse Kinematics ◽  
Closed Loop ◽  
Singular Values ◽  
Singular Configurations ◽  
Revolute Joints ◽  
Singular Direction ◽  
Inverse Kinematics Problem ◽  
Singular Configuration ◽  
Damped Least Squares ◽  
Kinematic Singularities

Inverse kinematics is a central problem in robotics, and its solution is burdened with kinematic singularities, i.e. the task Jacobian of the problem is singular. A subproblem of the general inverse kinematics problem, the inverse positioning problem is considered for spatial manipulators consisting of revolute joints, and a regularization method is proposed that results in a regular task Jacobian in singular configurations as well, provided that the manipulator’s geometry makes movement in singular directions possible. The conditions of regularizability are investigated, and bounds on the singular values of the regularized task Jacobian are given that can be used to create stable closed-loop inverse kinematics algorithms. The proposed method is demonstrated on the inverse positioning problem of an elbow manipulator and compared to the Damped Least Squares and the Levenberg-Marquardt methods, and it is shown that only the proposed method can leave the singular configuration in the singular direction.

A Rolling 8-Bar Linkage Mechanism

2015 ◽  
Vol 7 (4) ◽  
Author(s):  
Yaobin Tian ◽  
Yan-An Yao ◽  
Jieyu Wang
Keyword(s):  
Kinematic Analysis ◽  
Degrees Of Freedom ◽  
Closed Loop ◽  
Rolling Direction ◽  
Linkage Mechanism ◽  
Zero Moment Point ◽  
Revolute Joints ◽  
Straight Line ◽  
Polygonal Region ◽  
Zero Moment

In this paper, a rolling mechanism constructed by a spatial 8-bar linkage is proposed. The eight links are connected with eight revolute joints, forming a single closed-loop with two degrees of freedom (DOF). By kinematic analysis, the mechanism can be deformed into planar parallelogram or spherical 4-bar mechanism (SFM) configuration. Furthermore, this mechanism can be folded onto a plane at its singularity positions. The rolling capability is analyzed based on the zero-moment-point (ZMP) theory. In the first configuration, the mechanism can roll along a straight line. In the second configuration, it can roll along a polygonal region and change its rolling direction. By alternatively choosing one of the two configurations, the mechanism has the capability to roll along any direction on the ground. Finally, a prototype was manufactured and some experiments were carried out to verify the functions of the mechanism.

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 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 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.

Command-Shaping Control System for Double-Pendulum Gantry Cranes

2011 ◽  
Author(s):  
Ziyad N. Masoud ◽  
Khaled A. Alhazza
Keyword(s):  
Closed Loop ◽  
Single Mode ◽  
Feedback System ◽  
Mode Frequency ◽  
Gantry Crane ◽  
Necessary Condition ◽  
Double Pendulum ◽  
Double Step ◽  
Scaled Model ◽  
Command Shaping

Traditionally, multi-mode command-shaping controllers are tuned to the system frequencies. This work suggests an opposite approach. A frequency-modulation (FM) strategy is developed to tune the system frequencies to match the frequencies eliminated by a single-mode command-shaper. The shaper developed in this work is based on a double-step command-shaping strategy. Using the FM Shaper, a simulated feedback system is used to modulate the closed-loop frequencies of a simulated double-pendulum model to the point where the closed-loop second mode frequency becomes an odd multiple of the closed-loop first mode frequency, which is the necessary condition for a satisfactory performance of a single-mode command-shaper. The double-step command-shaper is based on the closed-loop first mode frequency. The input commands to the plant of the simulated closed-loop system are then used to drive the actual double-pendulum. Performance is validated experimentally on a scaled model of a double-pendulum gantry crane.

Closed-Loop Tape Springs as Fully Compliant Mechanisms: Preliminary Investigations

2004 ◽  
Author(s):  
Christine Vehar ◽  
Sridhar Kota ◽  
Robert Dennis
Keyword(s):  
Degrees Of Freedom ◽  
Closed Loop ◽  
Compliant Mechanisms ◽  
Variable Stiffness ◽  
Small Radius ◽  
Constant Thickness ◽  
Revolute Joints ◽  
Spring Mechanism ◽  
Actuation Scheme ◽  
Straight Segments

The paper introduces tape springs as elements of fully compliant mechanisms. The localized folds of tape springs serve as compact revolute joints, with a very small radius and large range of motion, and the unfolded straight segments serve as links. By exploiting a tape spring’s ability to function as both links and joints, we present a new method of realizing fully compliant mechanisms with further simplification in their construction. Tape springs, typically found in carpenter tape rules, are thin-walled strips having constant thickness, zero longitudinal curvature, and a constant transverse curvature. The paper presents a closed-loop tape spring mechanism. By representing its folds as idealized revolute joints and its variable length links as sliding joints connecting rigid links, we present a modified Gruebler’s equation to determine its kinematic and idle degrees of freedom. To realize practical utility of tape spring mechanisms, we propose a simple actuation scheme incorporating shape memory alloy (SMA) wire actuators and successfully demonstrate its performance with a proof-of-concept prototype. The paper also presents potential applications for actuated tape spring mechanisms including a large displacement translational mechanism, planar positioning mechanisms, bi-stable, multi-stable, and variable stiffness mechanisms.

Forward Position Analysis of Two 3-R Robots Manipulating a Planar Linkage Payload

1995 ◽  
Vol 117 (2A) ◽  
pp. 292-297 ◽  
Author(s):  
G. R. Pennock ◽  
K. G. Mattson
Keyword(s):  
Closed Loop ◽  
Degree Of Freedom ◽  
Sixth Order ◽  
Position Analysis ◽  
Revolute Joints ◽  
Order Polynomial ◽  
Cooperating Robots ◽  
Position Problem ◽  
Three Degree Of Freedom ◽  
Insight Into

This paper presents the forward position analysis of two planar three degree-of-freedom robots, with all revolute joints, manipulating a single degree-of-freedom closed-loop linkage payload. Kinematic constraint relations are developed which provide geometric insight into the cooperating robot-payload system and are important in the control of the two robots. For illustrative purposes, the payload that is considered here is a planar four-bar linkage. The paper shows that the orientation of a specified link in the payload can be described by a sixth-order polynomial. This polynomial is an important contribution, not only to the kinematics of the cooperating robots, but to the multiple-input closed-loop nine-bar linkage formed by the two robots and the payload. The polynomial contains important information regarding the assembly configurations and the stationary configurations of the system. The paper shows that zero, two, four, or six assembly configurations are possible and that each configuration corresponds to a different circuit of the system. Graphical methods are utilized to provide geometric insight into the assembly and stationary configurations and to check the results obtained from the sixth-order polynomial. A numerical example is included which demonstrates the importance of the polynomial in solving the forward position problem, and in determining the number of assembly configurations.

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