19 results on '"Dual quaternion"'
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2. A dual quaternion approach to efficient determination of the maximal singularity-free joint space and workspace of six-DOF parallel robots
- Author
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Yang, XiaoLong, Wu, HongTao, Chen, Bai, Li, Yao, and Jiang, SuRong
- Published
- 2018
- Full Text
- View/download PDF
3. Coordinate-invariant rigid-body interpolation on a parametric C1 dual quaternion curve
- Author
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Allmendinger, Felix, Charaf Eddine, Sami, and Corves, Burkhard
- Published
- 2018
- Full Text
- View/download PDF
4. A dual quaternion solution to the forward kinematics of a class of six-DOF parallel robots with full or reductant actuation
- Author
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Yang, XiaoLong, Wu, HongTao, Li, Yao, and Chen, Bai
- Published
- 2017
- Full Text
- View/download PDF
5. A dual quaternion approach to efficient determination of the maximal singularity-free joint space and workspace of six-DOF parallel robots
- Author
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Bai Chen, Xiaolong Yang, Yao Li, Surong Jiang, and Hongtao Wu
- Subjects
0209 industrial biotechnology ,Computer science ,Mechanical Engineering ,Parallel manipulator ,Boundary (topology) ,Bioengineering ,02 engineering and technology ,Kinematics ,Workspace ,Topology ,Computer Science Applications ,Computer Science::Robotics ,symbols.namesake ,020303 mechanical engineering & transports ,020901 industrial engineering & automation ,0203 mechanical engineering ,Mechanics of Materials ,Position (vector) ,Orientation (geometry) ,Jacobian matrix and determinant ,symbols ,Dual quaternion - Abstract
The avoidance of singularities is critical to design and control of parallel robots. This paper aims at efficient determination of the maximal singularity-free joint space and workspace of a class of six-DOF parallel robots with six kinematic chains of same type. We represent the singularity-free joint space by a 6-cube and determine it firstly. The singularity-free workspace is generated by continuous motion of all active joints in the singularity-free joint space. As a result, the boundary of the workspace can be obtained with simultaneous consideration of position and orientation of the mobile platform. The size relation between the maximal singularity-free joint space and workspace is discussed. To efficiently determine the singularity-free joint space and workspace, we propose dual quaternion-based Jacobian matrices and construct an efficient algorithm. The algorithm detects singularities in a given joint space and simultaneously calculates its corresponding workspace. The computational costs of the proposed algorithm and the traditional one are compared using a 6-U P S parallel robot, leading to 9 seconds and 458 seconds respectively. Finally, both the maximal singularity-free joint space and workspace of a 6- P US parallel robot are determined to further demonstrate the effectiveness of the new approach.
- Published
- 2018
6. Coordinate-invariant rigid-body interpolation on a parametric C1 dual quaternion curve
- Author
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Felix Allmendinger, Burkhard Corves, and Sami Charaf Eddine
- Subjects
0209 industrial biotechnology ,Computer science ,Mechanical Engineering ,010102 general mathematics ,Coordinate system ,Bioengineering ,02 engineering and technology ,Rigid body ,Mathematical proof ,01 natural sciences ,Computer Science Applications ,020901 industrial engineering & automation ,Mechanics of Materials ,0101 mathematics ,Invariant (mathematics) ,Dual quaternion ,Algorithm ,ComputingMethodologies_COMPUTERGRAPHICS ,Parametric statistics ,Interpolation - Abstract
We present a method to generate first-order continuous rigid-body motion by interpolation. The input is a sequence of rigid-body poses at given timesteps, which the body is required to pass through (key poses). Different from frequently employed interpolation schemes, the generated rigid-body motion is unique no matter what reference coordinate systems are chosen. Our method is novel in that the user can optionally prescribe key velocity data, too. If key velocities are not prescribed, parametric velocities are computed and incorporated into the interpolating function. The parameters allow to subsequently adjust the rigid-body trajectory. Another purpose of this article is a comprehensive derivation of coordinate-invariant interpolation along with a concise collection of proofs. The derivation enables the reader to straight-forwardly implement this method. Numerical examples are given to highlight the benefits and motivate the implementation.
- Published
- 2018
7. A dual quaternion solution to the forward kinematics of a class of six-DOF parallel robots with full or reductant actuation
- Author
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Bai Chen, Xiaolong Yang, Hongtao Wu, and Yao Li
- Subjects
0209 industrial biotechnology ,Robot kinematics ,Forward kinematics ,Inverse kinematics ,Mechanical Engineering ,Parallel manipulator ,Bioengineering ,02 engineering and technology ,Kinematics ,Computer Science Applications ,Computer Science::Robotics ,020303 mechanical engineering & transports ,020901 industrial engineering & automation ,Generalized coordinates ,0203 mechanical engineering ,Mechanics of Materials ,Control theory ,Kinematics equations ,Dual quaternion ,Mathematics - Abstract
The forward kinematics is the basis of the design and control of the parallel robots. This paper aims to provide an efficient solution to the forward kinematics of a class of six-degrees-of-freedom parallel robots for real-time applications. With a unit dual quaternion used as the generalized coordinates of the robot system, the forward kinematic equations are derived to be a set of quadratic ones. An efficient algorithm is proposed to get the actual solution to them. The convergence and singularity problems of the new algorithm have been discussed. We have provided a convergence strategy and revealed the internal relation of the singularity with that of the parallel robot, proving the feasibility of the algorithm and giving the working condition in the practical applications. The new algorithm have been compared to the Newton's method for an 8-U P S parallel robot, resulting in the time consumptions of 0.2187 milliseconds and 14.25 milliseconds respectively. And then we perform a simulation of the state-feedback control for an 8- P US parallel robot. The two examples present the applications of the new algorithm and demonstrate its validity and efficiency.
- Published
- 2017
8. The Serret-Frenet formulae for dual quaternion-valued functions of a single real variable
- Author
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Rifat Güneş, A.Idot. Sidotvridotdaǧ, and Sadık Keleş
- Subjects
Algebra ,Mechanics of Materials ,Mechanical Engineering ,Frenet–Serret formulas ,Mathematical analysis ,Real variable ,Bioengineering ,Dual quaternion ,Computer Science Applications ,Dual (category theory) ,Mathematics - Abstract
Dual quaternion-valued functions of a single real variable define a curve in ID4. The Serret-Frenet formulae for a curve in ID3, well known [1], it is rederived with the help of dual spatialquaternions. Making use of these formulae, we derived the Serret-Frenet formulae for dual quaternion-valued functions in ID4.
- Published
- 1994
9. The Serret-Frenet formulae for dual quaternion-valued functions of a single real variable
- Author
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Sivridaǧ, A.İ., Güneş, R., and Keleş, S.
- Abstract
Dual quaternion-valued functions of a single real variable define a curve in ID4. The Serret-Frenet formulae for a curve in ID3, well known [1], it is rederived with the help of dual spatialquaternions. Making use of these formulae, we derived the Serret-Frenet formulae for dual quaternion-valued functions in ID4.
- Published
- 1994
- Full Text
- View/download PDF
10. A variable-DOF single-loop 7R spatial mechanism with five motion modes
- Author
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Xianwen Kong
- Subjects
0209 industrial biotechnology ,Plane (geometry) ,Mechanical Engineering ,Motion (geometry) ,Control reconfiguration ,Bioengineering ,Geometry ,02 engineering and technology ,Kinematics ,Revolute joint ,Topology ,Translation (geometry) ,Computer Science Applications ,Computer Science::Robotics ,020303 mechanical engineering & transports ,020901 industrial engineering & automation ,0203 mechanical engineering ,Mechanics of Materials ,Configuration space ,Dual quaternion ,Computer Science::Information Theory ,Mathematics - Abstract
This paper is about the construction and reconfiguration analysis of a novel variable-DOF (or kinematotropic) single-loop 7R spatial mechanism, which is composed of seven R (revolute) joints. Firstly, the novel variable-DOF single-loop 7R spatial mechanism is constructed from a general variable-DOF single-loop 7R spatial mechanism and a plane symmetric Bennett joint 6R mechanism for circular translation. The reconfiguration analysis is then carried out in the configuration space by solving a set of kinematic loop equations based on dual quaternions and the natural exponential function substitution using tools from algebraic geometry. The analysis shows that the variable-DOF single-loop 7R spatial mechanism has five motion modes, including a 2-DOF planar 5R mode, two 1-DOF spatial 6R modes, and two 1-DOF spatial 7R modes and can transit between the 2-DOF planar 5R mode and each of the other motion modes through two transition configurations. There are two transition configurations from which the mechanism can switch among its four 1-DOF motion modes.
- Published
- 2018
11. A new forward kinematic algorithm for a general Stewart platform
- Author
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Wanyong Zhou, Huadong Liu, Wu Yi Chen, and Xinyou Li
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Forward kinematics ,Inverse kinematics ,Computer science ,Mechanical Engineering ,Parallel manipulator ,Inverse ,Bioengineering ,Stewart platform ,Kinematics ,Serial manipulator ,Computer Science Applications ,Computer Science::Robotics ,Mechanics of Materials ,Control theory ,Dual quaternion ,Algorithm ,ComputingMethodologies_COMPUTERGRAPHICS - Abstract
In this paper, a new forward kinematic algorithm based on dual quaternion for a six-degree-of-freedom (DOF) general Stewart Platform is proposed. The algorithm is established after taking into account the pose parameters of joints and the D–H parameters of each branch chain, which yields a precise mathematic model with a total of 174 geometric parameters. The validity of the algorithm is tested using an inverse kinematic algorithm of a general serial manipulator. The forward kinematic algorithms for the 6-6R and 6-2RP3R mechanisms can be expressed by a unified mathematic model. The proposed algorithm can solve the difficult problem of forward kinematics for the unsolved 6-6R parallel manipulator and can also be applied to error analysis, error synthesis, kinematic calibration, stiffness analysis and the accurate kinematic simulation by analysing the effect of geometric parameters on the platform.
- Published
- 2015
12. Synthesis of multi-mode single-loop Bennett-based mechanisms using factorization of motion polynomials
- Author
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Jingjun Yu, Kai Liu, and Xianwen Kong
- Subjects
0209 industrial biotechnology ,Computer science ,Mechanical Engineering ,Motion (geometry) ,Bioengineering ,02 engineering and technology ,Kinematics ,Algebraic geometry ,Topology ,Computer Science Applications ,020303 mechanical engineering & transports ,020901 industrial engineering & automation ,0203 mechanical engineering ,Factorization ,Mechanics of Materials ,Algebra representation ,Algebraic number ,Dual quaternion ,Rotation (mathematics) - Abstract
This paper systematically deals with the synthesis of multi-mode single-loop 6R, 7R and 8R Bennett-based mechanisms from an algebraic viewpoint. Based on the factorization of motion polynomials over dual quaternions, an algebraic method is proposed to synthesize multi-mode single-loop 6R, 7R and 8R Bennett-based mechanisms. Using this method, several multi-mode single-loop Bennett-based mechanisms with different number of joints are constructed depending on explicit poses of joint axes. Then motion mode analysis of the 7R mechanism is carried out by formulating and solving a set of kinematic loop equations using tools from algebraic geometry. The analysis demonstrates that this multi-mode 7R mechanism has four motion modes, including a two degree-of-freedom (DOF) double Bennett mode, a 2-DOF hybrid mode, a 1-DOF rotation mode and a 1-DOF spatial 7R mode. Meanwhile, multimode characteristics of the single-loop 6R and 8R mechanisms also are concisely demonstrated in light of reconfiguration analysis. This work provides an algebraic representation framework for further investigation on multi-mode mechanisms that composed of two or more single-loop overconstraint mechanisms.
- Published
- 2021
13. Hyper Dual Quaternions representation of rigid bodies kinematics
- Author
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Avraham Cohen and Moshe Shoham
- Subjects
0209 industrial biotechnology ,Robot kinematics ,Computer science ,Automatic differentiation ,Mechanical Engineering ,Mathematical analysis ,Dual number ,Bioengineering ,02 engineering and technology ,Kinematics ,Computer Science Applications ,020303 mechanical engineering & transports ,020901 industrial engineering & automation ,0203 mechanical engineering ,Mechanics of Materials ,Feature (computer vision) ,Representation (mathematics) ,Dual quaternion ,Quaternion - Abstract
Hyper-Dual Quaternions, HDQ, are first introduced in this paper. Utilizing the Hyper-Dual Numbers, HDN, Dual Quaternions, DQ, and hyper-dual angle, the general expression of HDQ is obtained. Rigid single- and multi-body kinematics such as in a serial robot kinematics, are then expressed in HDQ form. Taking advantage of the dual numbers’ “automatic differentiation” feature, HDQ encompasses both body pose (translational and rotational) and body velocities in a compact form with no need for further pose differentiation.
- Published
- 2020
14. Operation mode analysis of lower-mobility parallel mechanisms based on dual quaternions
- Author
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Kai Liu, Xianwen Kong, and Jingjun Yu
- Subjects
0209 industrial biotechnology ,Computer science ,Mechanical Engineering ,Constraint (computer-aided design) ,Motion (geometry) ,Bioengineering ,Prime decomposition ,02 engineering and technology ,Kinematics ,Topology ,Computer Science Applications ,020303 mechanical engineering & transports ,020901 industrial engineering & automation ,Operator (computer programming) ,0203 mechanical engineering ,Mechanics of Materials ,Dual quaternion ,Constant (mathematics) ,Variable (mathematics) - Abstract
This paper aims to provide an efficient way for analyzing operation modes and revealing corresponding motion characteristics of lower-mobility parallel mechanisms (LMPMs) using unit dual quaternions. Unit dual quaternions are first classified into 132 cases corresponding to 132 elementary operation modes based on the number of constant zero components. Meanwhile, the kinematic interpretation of these cases is presented in detail. Then a general method for analyzing operation modes and revealing motion characteristics of LMPMs is proposed according to unit dual quaternions and geometrical constraints. By this means, operation modes of LMPMs with complicated constraint conditions can also be analyzed, where a prime decomposition of the ideal corresponding to constraint equations in this condition is infeasible. Taken a 3-RSR LMPM and a 3-RPS LMPM as examples, the operation modes and motion characteristics can be obtained by the proposed approach. It is shown that the former LMPM has seven operation modes including two elementary operation modes and five extra operation modes. Under certain operation modes, the zero-torsion motion of the 3-RSR LMPM can not even be achieved. On the other hand, the latter has two operation modes in which the parasitic motion exists. To gain a deeper insight into the physical meaning of the operation modes, axodes are analyzed and drawn by mean of the velocity operator. It is demonstrated that the 3-RSR LMPM can realize an equal-diameter spherical pure rolling movement with variable diameters and the 3-RPS LMPM can achieve a rolling movement accompanied by sliding.
- Published
- 2019
15. Half-turns and line symmetric motions
- Author
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J. M. Selig and Manfred L. Husty
- Subjects
Quadric ,Ruled surface ,Mechanical Engineering ,Mathematical analysis ,Lie group ,Motion (geometry) ,Bioengineering ,Rigid body ,Intersection (Euclidean geometry) ,Computer Science Applications ,Mechanics of Materials ,Line (geometry) ,Dual quaternion ,Mathematics - Abstract
A line symmetric motion is the motion obtained by reflecting a rigid body in the successive generator lines of a ruled surface. In this work we review the dual quaternion approach to rigid body displacements, in particular the representation of the group SE(3) by the Study quadric. Then some classical work on reflections in lines or half-turns is reviewed. Next two new characterisations of line symmetric motions are presented. These are used to study a number of examples one of which is a novel line symmetric motion given by a rational degree five curve in the Study quadric. The rest of the paper investigates the connection between sets of half-turns and linear subspaces of the Study quadric. Line symmetric motions produced by some degenerate ruled surfaces are shown to be restricted to certain 2-planes in the Study quadric. Reflections in the lines of a linear line complex lie in the intersection of the Study quadric with a 4-plane.
- Published
- 2011
16. Classification of angle-symmetric 6R linkages
- Author
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Josef Schicho and Zijia Li
- Subjects
FOS: Computer and information sciences ,Computer Science - Symbolic Computation ,0209 industrial biotechnology ,Pure mathematics ,Property (programming) ,Chebychev–Grübler–Kutzbach criterion ,Motion (geometry) ,Bioengineering ,Geometry ,02 engineering and technology ,Linkage (mechanical) ,Symbolic Computation (cs.SC) ,Type (model theory) ,law.invention ,Mathematics - Algebraic Geometry ,Computer Science - Robotics ,020901 industrial engineering & automation ,Reflection symmetry ,0203 mechanical engineering ,law ,FOS: Mathematics ,Algebraic Geometry (math.AG) ,Mathematics ,Mechanical Engineering ,Computer Science Applications ,020303 mechanical engineering & transports ,Mechanics of Materials ,Dual quaternion ,Robotics (cs.RO) ,Rotation (mathematics) - Abstract
In this paper, we consider a special kind of overconstrained 6R closed linkage which we call angle-symmetric 6R linkage. These are linkages with the property that the rotation angles are equal for each of the three pairs of opposite joints. We give a classification of these linkages. It turns out that there are three types. First, we have the linkages with line symmetry. The second type is new. The third type is related to cubic motion polynomials.
- Published
- 2013
17. Factorization of rational curves in the study quadric
- Author
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Hans-Peter Schröcker, Josef Schicho, and Gábor Hegedüs
- Subjects
0209 industrial biotechnology ,Quadric ,Group (mathematics) ,Mechanical Engineering ,010102 general mathematics ,Euler's factorization method ,Motion (geometry) ,Bioengineering ,02 engineering and technology ,01 natural sciences ,Computer Science Applications ,Algebra ,020901 industrial engineering & automation ,Factorization ,Mechanics of Materials ,Factorization of polynomials ,Rational motion ,0101 mathematics ,Dual quaternion ,Mathematics - Abstract
For every generic rational curve C in the group of Euclidean displacements we construct a linkage such that the constrained motion of one of the links is exactly C. Our construction is based on the factorization of polynomials over dual quaternions.
- Published
- 2013
18. On point-line geometry and displacement
- Author
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Yi Zhang and Kwun-Lon Ting
- Subjects
Mechanical Engineering ,Mathematical analysis ,Duality (mathematics) ,Motion (geometry) ,Bioengineering ,Topology ,Displacement (vector) ,Computer Science Applications ,Transformation (function) ,Mechanics of Materials ,Screw theory ,Point (geometry) ,Dual quaternion ,Rigid transformation ,Mathematics - Abstract
A framework and the relevant algebraic treatment concerning point-line positions and displacements are explored using dual quaternion algebra. A screw or a dual vector is used to represent a point-line and the pitch is used to measure the endpoint location along the point-line. A point-line operator is developed to express a point-line transformation. The transformation allows a point-line to be considered as an independent rigid element having a completely specified motion without invoking rigid body transformation.
- Published
- 2004
19. Forward displacement analysis of general six-in-parallel sps (Stewart) platform manipulators using soma coordinates
- Author
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Charles W. Wampler
- Subjects
Engineering ,business.industry ,Mechanical Engineering ,Mathematical analysis ,Bioengineering ,Stewart platform ,Base (topology) ,Upper and lower bounds ,Displacement (vector) ,Computer Science Applications ,law.invention ,Invertible matrix ,Mechanics of Materials ,Simple (abstract algebra) ,law ,Calculus ,Representation (mathematics) ,Dual quaternion ,business - Abstract
General six-in-parallel SPS platform manipulators are constructed of six telescoping legs, each connecting a stationary base platform to a moving platform via spherical joints. These are often termed “generalized Stewaart platforms”. given the legnths of teh six legs, teh forward displacement problem is to find the location of the end platform relative to the base platform. It was first demonstrated numerically that the problem may in general have at most 40 nonsingular solutions and this bound has been verified using several different mathematical arguments. The problem is reformulated in this report using a classical representation of rigid-body displacements: Study's soma coordinates, or equivalently, dual quaternions. This provides a much simpler analytical proof of the upper bound of 40. Moreover, the simple form of the equations may be useful in further studies of the problem.
- Published
- 1996
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