scholarly journals Position-Singularity Analysis of a Class of the 3/6-Gough-Stewart Manipulators Based on Singularity-Equivalent-Mechanism

10.5772/45664 ◽  
2012 ◽  
Vol 9 (1) ◽  
pp. 9 ◽  
Author(s):  
Hui Zhou ◽  
Yi Cao ◽  
Baokun Li ◽  
Meiping Wu ◽  
Jinghu Yu ◽  
...  
Keyword(s):  
Screw Theory ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Parallel Manipulators ◽  
Singularity Analysis ◽  
Polynomial Expression ◽  
Moving Platform ◽  
Singularity Locus ◽  
Three Dimensional Space ◽  
Equivalent Mechanism

This paper addresses the problem of identifying the property of the singularity loci of a class of 3/6-Gough-Stewart manipulators for general orientations in which the moving platform is an equilateral triangle and the base is a semiregular hexagon. After constructing the Jacobian matrix of this class of 3/6-Gough-Stewart manipulators according to the screw theory, a cubic polynomial expression in the moving platform position parameters that represents the position-singularity locus of the manipulator in a three-dimensional space is derived. Graphical representations of the position-singularity locus for different orientations are given so as to demonstrate the results. Based on the singularity kinematics principle, a novel method referred to as ‘singularity-equivalent-mechanism' is proposed, by which the complicated singularity analysis of the parallel manipulator is transformed into a simpler direct position analysis of the planar singularity-equivalent-mechanism. The property of the position-singularity locus of this class of parallel manipulators for general orientations in the principal-section, where the moving platform lies, is identified. It shows that the position-singularity loci of this class of 3/6-Gough-Stewart manipulators for general orientations in parallel principal-sections are all quadratic expressions, including a parabola, four pairs of intersecting lines and infinite hyperbolas. Finally, the properties of the position-singularity loci of this class of 3/6-Gough-Stewart parallel manipulators in a three-dimensional space for all orientations are presented.

The Singularity Analysis of 3-RPS Parallel Manipulator

2006 ◽  
Author(s):  
Yanwen Li ◽  
Zhen Huang ◽  
Lumin Wang
Keyword(s):  
Parallel Manipulator ◽  
Screw Theory ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Singularity Analysis ◽  
Necessary Condition ◽  
Line Geometry ◽  
Simple Singularity ◽  
Sufficient And Necessary Condition ◽  
Three Dimensional Space

This paper firstly introduces a kinematic principle of singularity. It is a sufficient and necessary condition to identify singularity. Using the condition this paper systematically studies the singularity of 3-RPS parallel manipulator. A simple singularity equation is derived and the complete singularity loci in the three-dimensional space are illustrated. In order to analyze the singularity property and verify the correctness of the derived equation the line-geometry and the constraint screw theory are used. Some important singularity properties and the distribution characteristics are presented.

Geometrical Properties of Modelled Robot Elasticity: Part II — Center of Elasticity

1992 ◽  
Author(s):  
Harvey Lipkin ◽  
Timothy Patterson
Keyword(s):  
Screw Theory ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Compliance Matrix ◽  
Geometrical Properties ◽  
Robot Systems ◽  
Pass Through ◽  
Generalized Center ◽  
Three Dimensional Space ◽  
Elastic Characteristics

Abstract The elastic characteristics of many robot systems can be modeled by a 6 × 6 stiffness or compliance matrix. Several new and important results are presented via screw theory: i) A generalized center-of-elasticity is proposed based on Ball’s (1900) principal screws and its properties are investigated, ii) If a compliant axis exists, it is shown to pass through the center. iii) The perpendicular vectors from the center to the wrench-compliant axes are coplanar and sum to zero. A similar result holds for the twist-compliant axes, iv) Linear and rotational properties are characterized by dual ellipsoids in three-dimensional space. These elements simplify the understanding of complex elastic properties.

FORCE-UNCONSTRAINED POSES OF THE 3-PRR AND 4-PRR PLANAR PARALLEL MANIPULATORS

2005 ◽  
Vol 29 (4) ◽  
pp. 617-628 ◽  
Author(s):  
Flavio Firmani ◽  
Ron P. Podhorodeski
Keyword(s):  
Parallel Manipulator ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Parallel Manipulators ◽  
Task Space ◽  
Planar Parallel Manipulator ◽  
Order Polynomial ◽  
Position And Orientation ◽  
Planar Parallel Manipulators ◽  
Three Dimensional Space

Force-unconstrained (singular) poses of the 3-PRR planar parallel manipulator (PPM), where the underscore indicates the actuated joint, and the 4-PRR, a redundant PPM with an additional actuated branch, are presented. The solution of these problems is based upon concepts of reciprocal screw quantities and kinematic analysis. In general, non-redundant PPMs such as the 3-PRR are known to have two orders of infinity of force-unconstrained poses, i.e., a three-variable polynomial in terms of the task-space variables (position and orientation of the mobile platform). The inclusion of redundant branches eliminates one order of infinity of force-unconstrained configurations for every actuated branch beyond three. The geometric identification of force-unconstrained poses is carried out by assuming one variable for each order of infinity. In order to simplify the algebraic procedure of these problems, the assumed or “free” variables are considered to be joint displacements. For both manipulators, an effective elimination technique is adopted. For the 3-PRR, the roots of a 6th-order polynomial determine the force-unconstrained poses, i.e., surfaces in a three dimensional space defined by the task-space variables. For the 4-PRR, a 64th-order polynomial determines curves of force-unconstrained poses in the same dimensional space.

Kinematic Analysis of a Three-Degree of Freedom Compliant Platform

2013 ◽  
Vol 135 (5) ◽  
Author(s):  
Julio C. Correa ◽  
Carl Crane
Keyword(s):  
Kinematic Analysis ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Degree Of Freedom ◽  
Compliant Joints ◽  
Central Platform ◽  
Moving Platform ◽  
Three Degree Of Freedom ◽  
Three Dimensional Space

This paper addresses the kinematic analysis of a three-degree of freedom (DOF) compliant platform able to move in three dimensional space. The device is formed by the actuators, a central moving platform, and compliant joints. The actuators are three binary links. The moving platform is an equilateral plate. Springs connect the free end of each actuator with each vertex of the central platform. In this way, the motion of the actuators is transmitted to the moving platform. Compliant joints increase the complexity of the motion of the central platform and few studies have been carried out. This paper focuses on the forward and reverse analyses for the platform and the derivation of equations that relate the velocity of the moving platform with the velocity of the actuators.

Kinematic analysis of a novel 2(3-RUS) parallel manipulator

Robotica ◽  
2015 ◽  
Vol 34 (10) ◽  
pp. 2241-2256 ◽  
Author(s):  
Róger E. Sánchez-Alonso ◽  
José-Joel González-Barbosa ◽  
Eduardo Castillo-Castaneda ◽  
Jaime Gallardo-Alvarado
Keyword(s):  
Kinematic Analysis ◽  
Parallel Manipulator ◽  
Screw Theory ◽  
Kinematic Model ◽  
Three Dimensional ◽  
Parallel Manipulators ◽  
Analytical Form ◽  
Planar Geometry ◽  
Moving Platform

SUMMARYThis paper introduces a novel 6-DOF parallel manipulator, which is composed of two 3-RUS parallel manipulators that share a common three-dimensional moving platform. Semi-analytical form solutions are easily obtained to solve the forward displacement analysis of the robot using the non-planar geometry of the moving platform, whereas the velocity, acceleration, and singularity analyses are performed using screw theory. A case study is included to show the application of the kinematic model, which is verified with the aid of a commercially available software. Simple kinematic analysis and reduced singular regions are the main benefits of the proposed parallel manipulator.

Position-singularity analysis of a special class of the Stewart parallel mechanisms with two dissimilar semi-symmetrical hexagons

Robotica ◽  
2012 ◽  
Vol 31 (1) ◽  
pp. 123-136 ◽  
Author(s):  
Baokun Li ◽  
Yi Cao ◽  
Qiuju Zhang ◽  
Zhen Huang
Keyword(s):  
Special Class ◽  
Screw Theory ◽  
Singularity Analysis ◽  
Parallel Mechanisms ◽  
Geometric Characteristics ◽  
Characteristic Plane ◽  
The Matrix ◽  
Grassmann Line Geometry ◽  
Moving Platform ◽  
Singularity Locus

SUMMARYIn this paper, for a special class of the Stewart parallel mechanism, whose moving platform and base one are two dissimilar semi-symmetrical hexagons, the position-singularity of the mechanism for a constant-orientation is analyzed systematically. The force Jacobian matrix [J]T is constructed based on the principle of static equilibrium and the screw theory. After expanding the determinant of the simplified matrix [D], whose rank is the same as the rank of the matrix [J]T, a cubic symbolic expression that represents the 3D position-singularity locus of the mechanism for a constant-orientation is derived and graphically represented. Further research shows that the 3D position-singularity surface is extremely complicated, and the geometric characteristics of the position-singularity locus lying in a general oblique plane are very difficult to be identified. However, the position-singularity locus lying in the series of characteristic planes, where the moving platform coincides, are all quadratic curves compromised of infinite many sets of hyperbolas, four pairs of intersecting lines and a parabola. For some special orientations, the quadratic curve can degenerate into two lines or even one line, all of which are parallel to the ridgeline. Two theorems are presented and proved for the first time when the geometric characteristics of the position-singularity curves in the characteristic plane are analyzed. Moreover, the kinematic property of the position-singularity curves is obtained using the Grassmann line geometry and the screw theory. The theoretical results are demonstrated with several numeric examples.

Orientation workspace analysis of a special class of the Stewart–Gough parallel manipulators

Robotica ◽  
2010 ◽  
Vol 28 (7) ◽  
pp. 989-1000 ◽  
Author(s):  
Yi Cao ◽  
Zhen Huang ◽  
Hui Zhou ◽  
Weixi Ji
Keyword(s):  
Special Class ◽  
Three Dimensional ◽  
Parallel Manipulators ◽  
Discretization Method ◽  
Practical Orientation ◽  
Workspace Analysis ◽  
Half Angle ◽  
Polynomial Expression ◽  
Singularity Locus ◽  
Orientation Workspace

SUMMARYThe workspace of a robotic manipulator is a very important issue and design criteria in the context of optimum design of robots, especially for parallel manipulators. Though, considerable research has been paid to the investigations of the three-dimensional (3D) constant orientation workspace or position workspace of parallel manipulators, very few works exist on the topic of the 3D orientation workspace, especially the nonsingular orientation workspace and practical orientation workspace. This paper addresses the orientation workspace analysis of a special class of the Stewart–Gough parallel manipulators in which the moving and base platforms are two similar semisymmetrical hexagons. Based on the half-angle transformation, a polynomial expression of 13 degree that represents the orientation singularity locus of this special class of the Stewart–Gough parallel manipulators at a fixed position is derived and graphical representations of the orientation singularity locus of this special class of the Stewart–Gough manipulators are illustrated with examples to demonstrate the result. Exploiting this half-angle transformation and the inverse kinematics solution of this special class of the Stewart–Gough parallel manipulators, a discretization method is proposed for computing the orientation workspace of this special class of the Stewart–Gough parallel manipulators taking limitations of active and passive joints and the link interference all into consideration. Based on this algorithm, this paper also presents a new discretization method for computing the nonsingular orientation workspace of this class of the manipulators, which not only can satisfy all the kinematics demand of this class of the manipulators but also can guarantee the manipulator is nonsingular in the whole orientation workspace, and the practical orientation workspace of this class of the manipulators, which not only can guarantee the manipulator is nonsingular and will never encounter any kinematic interference but also can satisfy the demand of the orientation workspace with a regular shape in practical application, respectively. Examples of a 6/6-SPS Stewart–Gough parallel manipulator of this special class are given to demonstrate these theoretical results.

Structure and property of the singularity loci of the 3/6-Stewart-Gough platform for general orientations

Robotica ◽  
2005 ◽  
Vol 24 (1) ◽  
pp. 75-84 ◽  
Author(s):  
Z. Huang ◽  
Y. Cao ◽  
Y. W. Li ◽  
L. H. Chen
Keyword(s):  
Singularity Analysis ◽  
Planar Mechanism ◽  
Structure And Property ◽  
Locus Equation ◽  
Polynomial Expression ◽  
Singularity Locus ◽  
Straight Lines ◽  
Stewart Gough Platform ◽  
Similar Issue ◽  
Equivalent Mechanism

This paper focuses on the structure and property of the singularity loci of the 3/6-Stewart-Gough platform for general orientations. Based on the singularity kinematics principle, a planar singularity-equivalent-mechanism is proposed, by which the complicated singularity analysis of that parallel mechanism is transformed into a simpler position analysis of the planar mechanism. All the possible positions of the planar mechanism form the singularity loci of the 3/6-Stewart-Gough manipulator. The result shows that the singularity equation become quite simple moreover the structure and property of the singularity loci are also identified and explained. For the most general orientations of the typical 3/6-Stewart-Gough platform, the singularity locus equation is a special irresolvable polynomial expression of degree three, which in infinite parallel principal sections includes a parabola, four pairs of intersecting straight lines and infinity of hyperbolas. This result is beneficial to analysis of the similar issue of other Stewart-Gough manipulators.

Extension of the Spatial PDI Model to Three Dimensions

1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien
Keyword(s):  
Population Density ◽  
Dimensional Space ◽  
Finite Lattice ◽  
Three Dimensional ◽  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Density Index ◽  
Three Dimensional Space

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.

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