Determination of particular singular configurations of Stewart platform type of fixator by the stereographic projection method

2021 ◽  
pp. 1-19
Author(s):  
Doğan Dönmez ◽  
İbrahim Deniz Akçalı ◽  
Ercan Avşar ◽  
Ahmet Aydın ◽  
Hüseyin Mutlu
Keyword(s):  
Projection Method ◽  
Stereographic Projection ◽  
Stewart Platform ◽  
Singular Configurations

Evaluation and Representation of the Theoretical Orientation Workspace of the Gough–Stewart Platform

2009 ◽  
Vol 1 (2) ◽  
Author(s):  
Qimi Jiang ◽  
Clément M. Gosselin
Keyword(s):  
Theoretical Orientation ◽  
Robotic Manipulators ◽  
Stewart Platform ◽  
Closed Region ◽  
Leg Length ◽  
Fixed Frame ◽  
Cartesian Space ◽  
Orientation Workspace

The evaluation and representation of the orientation workspace of robotic manipulators is a challenging task. This work focuses on the determination of the theoretical orientation workspace of the Gough–Stewart platform with given leg length ranges [ρimin,ρimax]. By use of the roll-pitch-yaw angles (ϕ,θ,ψ), the theoretical orientation workspace at a prescribed position P0 can be defined by up to 12 workspace surfaces. The defined orientation workspace is a closed region in the 3D orientation Cartesian space Oϕθψ. As all rotations R(x,ϕ), R(y,θ), and R(z,ψ) take place with respect to the fixed frame, any point of the defined orientation workspace provides a clear measure for the platform to, respectively, rotate in order around the (x,y,z) axes of the fixed frame. An algorithm is presented to compute the size (volume) of the theoretical orientation workspace and intersectional curves of the workspace surfaces. The defined theoretical orientation workspace can be applied to determine a singularity-free orientation workspace.

Experimental Investigations on the Contour Generation of a Reconfigurable Stewart Platform

2013 ◽  
pp. 291-301
Author(s):  
G. Satheesh Kumar ◽  
T. Nagarajan
Keyword(s):  
Error Analysis ◽  
Natural Frequency ◽  
Vibration Isolation ◽  
Stewart Platform ◽  
Design Tool ◽  
Experimental Investigations ◽  
Variable Geometry ◽  
Optimum Geometry ◽  
Complete Set

Reconfiguration of Stewart platform for varying tasks accentuates the importance for determination of optimum geometry catering to the specified task. The authors in their earlier work (Satheesh et al., 2008) have indicated the non availability of an efficient holistic methodology for determining the optimum geometry. Further, they have proposed a solution using the variable geometry approach through the formulation of dimensionless parameters in combination with generic parameters like configuration and joint vector. The methodology proposed provides an approach to develop a complete set of design tool for any new reconfigurable Stewart platform for two identified applications viz., contour generation and vibration isolation. This paper details the experimental investigations carried out to validate the analytical results obtained on a developed Stewart platform test rig and error analysis is performed for contour generation. The experimental natural frequency of the developed Stewart platform has also been obtained.

Kinematic Analysis and Prototyping of a Partially Decoupled 4-DOF 3T1R Parallel Manipulator

2006 ◽  
Vol 129 (6) ◽  
pp. 611-616 ◽  
Author(s):  
Pierre-Luc Richard ◽  
Clément M. Gosselin ◽  
Xianwen Kong
Keyword(s):  
Kinematic Analysis ◽  
Parallel Manipulator ◽  
Quadratic Equation ◽  
Design Stage ◽  
Forward Displacement ◽  
Singular Configurations ◽  
The Family ◽  
Moving Platform ◽  
Forward Displacement Analysis

A four-degree-of-freedom (DOF) 3T1R parallel manipulator is presented in this paper. This manipulator generates the family of so-called Schönflies motions, SCARA motions or 3T1R motions, in which the moving platform can translate in all directions and rotate around an axis of a fixed direction. The kinematic analysis of this architecture is presented, including the study of the constraint singular configurations, kinematic singular configurations, and the determination of the workspace. A prototype (the Quadrupteron) is also presented and demonstrated. The characteristics of the proposed prototype are (a) there is no constraint singularity, (b) its input-output equations are partially decoupled, (c) its kinematic singular configurations can be expressed using an equation in the angle of rotation of the moving platform and are thus easy to avoid at the design stage, and (d) its forward displacement analysis requires the solution of a univariate quadratic equation and can therefore be solved efficiently.

Determination of Constant Orientation Workspace of a Stewart Platform by Geometrical Method

2015 ◽  
Vol 813-814 ◽  
pp. 997-1001 ◽  
Author(s):  
S. Gokul Narasimhan ◽  
R. Shrivatsan ◽  
K. Venkatasubramanian ◽  
Anjan Kumar Dash
Keyword(s):  
Algebraic Method ◽  
Parallel Manipulators ◽  
Stewart Platform ◽  
Closed Form Expression ◽  
Robot Design ◽  
Form Expression ◽  
Novel Method ◽  
Workspace Optimization ◽  
Orientation Workspace

Determination of workspace is one of the main considerations in the design of any robot since the workspace geometry is considered a fundamental issue for robot design. This also plays a crucial role in trajectory planning. Among parallel manipulators, 6-DOF Stewart platforms is the most researched and widely used robot. However, till date there is no closed form expression of workspace volume for Stewart platform. In this paper, a novel method is proposed to find out the workspace volume of Stewart platform. In this paper, individual workspace of each leg of the manipulator (P-U-S) is determined and then translated by a common distance towards their geometrical center thus generating constant orientation workspace. To determine the workspace volume, geometric intersection of the six spheres is computed. This results in workspace of definite shape and size, whose volume is calculated using simple formulae. It is observed that the geometric way of determination of workspace area is computationally less tedious than the algebraic method. This also helps a lot for workspace optimization of such manipulators.

Note on the determination of the optic axial angle of a crystal in thin-section by the Mallard-Becke method

1913 ◽  
Vol 16 (77) ◽  
pp. 348-351 ◽  
Author(s):  
Harvey Collingridge
Keyword(s):  
Thin Section ◽  
Stereographic Projection ◽  
Original Method ◽  
Graphic Method ◽  
Camera Lucida ◽  
Axial Angle ◽  
Linear Distance

Mallard's original method was based on the measurement of the linear distance, as determined by an eyepiece-micrometer, between the optic axes in a section of the crystal at right angles to the acute biseetrix viewed in convergent light.Professor F. Becke implored on this method by utilizing sections which were not at right angles to the acute hissetrix, but in which both optic axes were visible in the field. He projected both axes by means of an Abbe camera lucida on to a revolving drawing-table, and by means of the Mallard equation plotted the axes on a stereographic projection and thus obtained the optic axial angle, the angles of course being corrected for refraction to the true angles in the crystal section. Professor Becke subsequently, by utilizing the Biot-Fresnel law, formulated a graphic method of obtaining the optic axial angle from a section in which only one axis was visible.

The Inverse Problem of Kinematics as the Tool for Determination of Motion Trajectories Exercised by Worm Drives of a Stewart Platform

2013 ◽  
Vol 837 ◽  
pp. 357-362 ◽  
Author(s):  
Andrzej Dymarek ◽  
Tomasz Dzitkowski
Keyword(s):  
Inverse Problem ◽  
Differential Equations ◽  
Stewart Platform ◽  
Engineering Process ◽  
Disabled People ◽  
Motion Trajectories ◽  
Vehicle Simulator

The paper presents how to find out motion trajectories for individual platforms incorporated into a Stewart platform. The differential equations for motion of actuators are resolved under the assumed initial preconditions. Also are presented graphs for motions of end tips of actuators incorporated into the platform structure. The study makes up one of indispensable stages within the engineering process aimed to design a vehicle simulator for disabled people to help them learning how to drive a car.

Least-squares Analysis of Fabric Data: A Note on Conical, Cylindroidal and Near–cylindroidal Folds

1971 ◽  
Vol 8 (6) ◽  
pp. 694-697 ◽  
Author(s):  
C. S. Venkitasubramanyan
Keyword(s):  
Least Squares ◽  
Stereographic Projection ◽  
Rotation Axis ◽  
Computational Procedure ◽  
Apical Angle ◽  
Small Circle ◽  
Best Fitting ◽  
Fabric Diagrams ◽  
The Mean

A cylinder and a plane may be considered as special limiting cases for a right circular cone as the semi-apical angle approaches 0° and 90° respectively (Loudon 1964, Kelley 1966). If these forms are viewed as surfaces generated by an array of lines in space, the rotation axis for the array (the axis of the "cone") can be determined from the orientations of the surface-generating lines by a single computational procedure, using least-squares criterion. The mean angle between the rotation axis and the surface-generating lines will be the semi-apical angle of the cone. However, if this method for determination of the semi-apical angle of the cone, and therefore the best-fitting small circle, is extended to fabric diagrams, in which an array of lines may only statistically describe a great circle or small circle on a stereographic projection, ambiguities arise in certain cases and the semi-apical angle obtained may not be the true semi-apical angle. The difficulty arises because the poles to foliation surfaces are arbitrarily assigned "senses".

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