Kinematic Analysis of Conventional and Multi-Mode Spatial Mechanisms Using Dual Quaternions

Volume 5B: 40th Mechanisms and Robotics Conference ◽

10.1115/detc2016-59194 ◽

2016 ◽

Cited By ~ 3

Author(s):

Xianwen Kong

Keyword(s):

Algebraic Geometry ◽

Kinematic Analysis ◽

Fundamental Issue ◽

Input Output ◽

Plane Symmetric ◽

Spatial Mechanism ◽

Dual Quaternions ◽

Reconfigurable Mechanisms ◽

Kinematic Loop ◽

Multi Mode

Although kinematic analysis of conventional mechanisms is a well-documented fundamental issue in mechanisms and robotics, the emerging reconfigurable mechanisms and robots (or mechanisms and robots with multiple operation modes) require re-examining this fundamental issue. Recent advances in mathematics, especially algebraic geometry and numerical algebraic geometry, make it possible to develop an efficient method for the kinematic analysis of not only conventional mechanisms and robots but also reconfigurable mechanisms and robots. This paper first presents a method for setting up a set of kinematic loop equations for mechanisms using dual quaternions. Using this approach, a set of kinematic loop equations of a spatial mechanism is composed of six equations. The effectiveness of the proposed kinematic loop equations is then demonstrated by deriving the explicit input-output equations of a line symmetric 1-DOF (degree-of-freedom) 7R single-loop spatial mechanism, the re-configuration analysis of a novel multi-mode 1-DOF 7R spatial mechanism. In the former case, an explicit input-output equation of degree 8 is derived. In the latter case, it is found that the 7R multi-mode mechanism has three motion modes, including a planar 4R mode, an orthogonal Bricard 6R mode, and a plane symmetric 6R mode. Unlike the 7R multi-mode mechanisms in the literature, the 7R multi-mode mechanism presented in this paper does not have a 7R mode in which all the seven R joints can move simultaneously.

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Reconfiguration Analysis of Multimode Single-Loop Spatial Mechanisms Using Dual Quaternions

Journal of Mechanisms and Robotics ◽

10.1115/1.4037111 ◽

2017 ◽

Vol 9 (5) ◽

Cited By ~ 10

Author(s):

Xianwen Kong

Keyword(s):

Algebraic Geometry ◽

Single Degree Of Freedom ◽

Plane Symmetric ◽

Dual Quaternions ◽

Revolute Joints ◽

Spatial Mechanisms ◽

Reconfigurable Mechanisms ◽

Motion Modes ◽

Kinematic Loop ◽

And Robotics

Although kinematic analysis of conventional mechanisms is a well-documented fundamental issue in mechanisms and robotics, the emerging reconfigurable mechanisms and robots pose new challenges in kinematics. One of the challenges is the reconfiguration analysis of multimode mechanisms, which refers to finding all the motion modes and the transition configurations of the multimode mechanisms. Recent advances in mathematics, especially algebraic geometry and numerical algebraic geometry, make it possible to develop an efficient method for the reconfiguration analysis of reconfigurable mechanisms and robots. This paper first presents a method for formulating a set of kinematic loop equations for mechanisms using dual quaternions. Using this approach, a set of kinematic loop equations of spatial mechanisms is composed of six polynomial equations. Then the reconfiguration analysis of a novel multimode single-degree-of-freedom (1DOF) 7R spatial mechanism is dealt with by solving the set of loop equations using tools from algebraic geometry. It is found that the 7R multimode mechanism has three motion modes, including a planar 4R mode, an orthogonal Bricard 6R mode, and a plane symmetric 6R mode. Three (or one) R (revolute) joints of the 7R multimode mechanism lose their DOF in its 4R (or 6R) motion modes. Unlike the 7R multimode mechanisms in the literature, the 7R multimode mechanism presented in this paper does not have a 7R mode in which all the seven R joints can move simultaneously.

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A Single-Loop 7R Spatial Mechanism That Has Three Motion Modes With the Same Instantaneous DOF but Different Finite DOF

Volume 5B: 42nd Mechanisms and Robotics Conference ◽

10.1115/detc2018-85125 ◽

2018 ◽

Author(s):

Xianwen Kong ◽

Andreas Müller

Keyword(s):

Higher Order ◽

Mobility Analysis ◽

Single Loop ◽

Plane Symmetric ◽

Spatial Mechanism ◽

Motion Modes ◽

Regular Configuration ◽

Kinematic Loop ◽

Multi Mode ◽

Motion Mode

Multi-mode mechanisms, including kinematotropic mechanisms, are a class of reconfigurable mechanisms that can switch motion modes with the same or different DOF (degree-of-freedom). For most of the multi-mode mechanisms reported in the literature, the instantaneous (or differential) DOF and finite DOF in a motion mode are equal. In this paper, we will discuss the construction, reconfiguration analysis, and higher-order mobility analysis of a multi-mode single-loop 7R mechanism that has three motion modes with the same instantaneous DOF but different finite DOF. Firstly, the novel multi-mode single-loop 7R spatial mechanism is constructed by inserting one revolute (R) joint into a plane symmetric Bennett joint-based 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 multi-mode single-loop 7R spatial mechanism has three motion modes, including a 2-DOF planar 5R mode and two 1-DOF spatial 6R modes and can transit between each pair of motion modes through two transition configurations. The higher-order mobility analysis shows that the 7R mechanism has two-instantaneous DOF at a regular configuration of any motion mode and three instantaneous DOF in a transition configuration. The infinitesimal motions that are not tangential to finite motions are of second-order in transition configurations between 2-DOF motion mode 1 and 1-DOF motion modes 2 or 3 or first-order in transition configurations between 1-DOF motion modes 2 and 3.

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A Double-Faced 6R Single-Loop Overconstrained Spatial Mechanism

Journal of Mechanisms and Robotics ◽

10.1115/1.4039224 ◽

2018 ◽

Vol 10 (3) ◽

Cited By ~ 1

Author(s):

Xianwen Kong ◽

Xiuyun He ◽

Duanling Li

Keyword(s):

Kinematic Analysis ◽

Isosceles Triangle ◽

The Other ◽

Dual Quaternion ◽

Single Loop ◽

Plane Symmetric ◽

Spatial Mechanism ◽

Revolute Joints ◽

Full Turn ◽

Overconstrained Mechanisms

This paper deals with a 6R single-loop overconstrained spatial mechanism that has two pairs of revolute joints with intersecting axes and one pair of revolute joints with parallel axes. The 6R mechanism is first constructed from an isosceles triangle and a pair of identical circles. The kinematic analysis of the 6R mechanism is then dealt with using a dual quaternion approach. The analysis shows that the 6R mechanism usually has two solutions to the kinematic analysis for a given input and may have two circuits (closure modes or branches) with one or two pairs of full-turn revolute joints. In two configurations in each circuit of the 6R mechanism, the axes of four revolute joints are coplanar, and the axes of the other two revolute joints are perpendicular to the plane defined by the above four revolute joints. Considering that from one configuration of the 6R mechanism, one can obtain another configuration of the mechanism by simply renumbering the joints, the concept of two-faced mechanism is introduced. The formulas for the analysis of plane symmetric spatial triangle are also presented in this paper. These formulas will be useful for the design and analysis of multiloop overconstrained mechanisms involving plane symmetric spatial RRR triads.

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Variable degree-of-freedom spatial mechanisms composed of four circular translation joints

Journal of Mechanisms and Robotics ◽

10.1115/1.4050152 ◽

2021 ◽

pp. 1-16

Author(s):

Xianwen Kong

Keyword(s):

Algebraic Geometry ◽

Screw Theory ◽

Degree Of Freedom ◽

Variable Degree ◽

Circular Trajectory ◽

Spatial Mechanism ◽

Spatial Mechanisms ◽

Motion Modes ◽

Multi Mode ◽

Motion Mode

Abstract This paper deals with the construction and reconfiguration analysis of a spatial mechanism composed of four circular translation (G) joints. Two links connected by a G joint, which can be in different forms such as a planar parallelogram, translate along a circular trajectory with respect to each other. A spatial 4G mechanism, which is composed of four G joints, usually has 1-DOF (degree-of-freedom). Firstly, a 2-DOF spatial 4G mechanism is constructed. Then a novel variable-DOF spatial 4G mechanism is constructed starting from the 2-DOF 4G mechanism using the approach based on screw theory. Finally, the reconfiguration analysis is carried out in the configuration space using dual quaternions and tools from algebraic geometry. The analysis shows that the variable-DOF spatial 4G mechanism has one 2-DOF motion mode and one to two 1-DOF motion modes and reveals how the 4G mechanism can switch among these motion modes. By removing one link from two adjacent G joints each and two links from each of the remaining two G joints, we can obtain a queer-rectangle and a queer-parallelogram, which are the generalization of the queer-square or derivative queer-square in the literature. The approach in this paper can be extended to the analysis of other types of coupled mechanisms using cables and gears and multi-mode spatial mechanisms involving G joints.

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Kinematic design and motion analysis of spatial rapier drive mechanisms used in weaving machines

Textile Research Journal ◽

10.1177/0040517514534754 ◽

2014 ◽

Vol 84 (19) ◽

pp. 2065-2073 ◽

Cited By ~ 1

Author(s):

Recep Eren ◽

Mesrur Erturk ◽

Barıs Hascelik

Keyword(s):

Motion Analysis ◽

Kinematic Analysis ◽

Gear Ratio ◽

Main Shaft ◽

Drive Mechanism ◽

Spatial Mechanism ◽

Kinematic Design ◽

Shaft Angle

This paper presents an approach for the kinematic design of a rapier drive mechanism containing a spatial mechanism and analyses rapier motion curve. Kinematic design and analysis equations are derived and then the link lengths of the spatial mechanism are calculated in order to satisfy the critical rapier positions inside and outside the shed. In this way, the portions of one loom revolution, during which the rapiers are inside and outside the shed, are determined. The rapier motion curve is obtained by using kinematic analysis equations. It is shown that the position of the oscillating link in the spatial mechanism and the loom main shaft angle at which the rapier enters the shed have the most significant effect on the rapier motion curve. The gear ratio has also some effect on the rapier motion curve. Different rapier motion curves are obtained by changing these parameters and the suitability of these curves for rapier motion is discussed.

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Closed-Form Displacement Relations of a Five-Link R-R-C-C-R Spatial Mechanism

Journal of Engineering for Industry ◽

10.1115/1.3427879 ◽

1971 ◽

Vol 93 (1) ◽

pp. 221-226 ◽

Cited By ~ 6

Author(s):

A. H. Soni ◽

P. R. Pamidi

Keyword(s):

Closed Form ◽

Polynomial Equation ◽

Input Output ◽

Degree Polynomial ◽

Spatial Mechanism ◽

Dual Number ◽

Numerical Example

Using (3 × 3) matrices with dual-number elements, closed form displacement relationships are derived for a spatial five-link R-R-C-C-R mechanism. The input-output closed form displacement relationship is an eighth degree polynomial equation. A numerical example is presented.

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Kinematic Analysis of Spatial Mechanisms by Means of Screw Coordinates. Part 2—Analysis of Spatial Mechanisms

Journal of Engineering for Industry ◽

10.1115/1.3427919 ◽

1971 ◽

Vol 93 (1) ◽

pp. 67-73 ◽

Cited By ~ 29

Author(s):

M. S. C. Yuan ◽

F. Freudenstein ◽

L. S. Woo

Keyword(s):

Computer Program ◽

Kinematic Analysis ◽

Static Force ◽

Single Degree Of Freedom ◽

Numerical Examples ◽

Spatial Mechanism ◽

Single Degree ◽

Spatial Mechanisms ◽

Torque Analysis ◽

Basic Concepts

The basic concepts of screw coordinates described in Part I are applied to the numerical kinematic analysis of spatial mechanisms. The techniques are illustrated with reference to the displacement, velocity, and static-force-and-torque analysis of a general, single-degree-of-freedom spatial mechanism: a seven-link mechanism with screw pairs (H)7. By specialization the associated computer program is capable of analyzing many other single-loop spatial mechanisms. Numerical examples illustrate the results.

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Kinematic calibration of serial robot using dual quaternions

Industrial Robot the international journal of robotics research and application ◽

10.1108/ir-10-2018-0221 ◽

2019 ◽

Vol 46 (2) ◽

pp. 247-258 ◽

Cited By ~ 3

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.

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