scholarly journals A Linear Relaxation Method for Computing Workspace Slices of the Stewart Platform

2012 ◽  
Vol 5 (1) ◽  
Author(s):  
Oriol Bohigas ◽  
Montserrat Manubens ◽  
Lluís Ros
Keyword(s):  
Complex Shape ◽  
Three Dimensional ◽  
Relaxation Method ◽  
Stewart Platform ◽  
Large Dimension ◽  
Linear Relaxation ◽  
Connected Components ◽  
Cartesian Space ◽  
Orientation Workspace ◽  
Unified Method

The workspace of a Stewart platform is a complex six-dimensional volume embedded in the Cartesian space defined by six pose parameters. Because of its large dimension and complex shape, this volume is difficult to compute and represent, and comprehension on its structure is being gained by studying its three-dimensional slices. While successful methods have been given to determine the constant-orientation slice, the computation and appropriate visualization of the constant-position slice (also known as the orientation workspace) has proved to be a challenging task. This paper presents a unified method for computing both of such slices, and any other ones defined by fixing three pose parameters, on general Stewart platforms possibly involving mechanical limits on the active and passive joints. Advantages over existing methods include, in addition to the previous, the ability to determine all connected components of the workspace, and any motion barriers present in its interior.

scholarly journals A Unified Method for Computing Position and Orientation Workspaces of General Stewart Platforms

2011 ◽  
Author(s):  
Oriol Bohigas ◽  
Llui´s Ros ◽  
Montserrat Manubens
Keyword(s):  
Complex Shape ◽  
Three Dimensional ◽  
Stewart Platform ◽  
Large Dimension ◽  
Connected Components ◽  
Cartesian Space ◽  
Position And Orientation ◽  
Stewart Platforms ◽  
Orientation Workspace ◽  
Unified Method

The workspace of a Stewart platform is a complex six-dimensional volume embedded in the Cartesian space defined by six pose parameters. Because of its large dimension and complex shape, such workspace is difficult to compute and represent, so that comprehension on its structure is being gained by studying its three-dimensional slices. While successful methods have been given to determine the constant-orientation slice, the computation and appropriate visualization of the constant-position slice (also known as the orientation workspace) has proved to be a challenging task. This paper presents a unified method for computing both of such slices, and any other ones defined by fixing three pose parameters, on general Stewart platforms involving mechanical limits on the active and passive joints. Additional advantages over previous methods include the ability to determine all connected components of the workspace, and any motion barriers present in its interior.

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.

Contracting ducts of finite length

1951 ◽  
Vol 2 (4) ◽  
pp. 254-271 ◽  
Author(s):  
L. G. Whitehead ◽  
L. Y. Wu ◽  
M. H. L. Waters
Keyword(s):  
Stream Function ◽  
Finite Length ◽  
Velocity Potential ◽  
Three Dimensional ◽  
Relaxation Method ◽  
Two Dimensional ◽  
Dimensional Flow ◽  
Hodograph Plane ◽  
Two Dimensional Flow ◽  
Working Section

SummmaryA method of design is given for wind tunnel contractions for two-dimensional flow and for flow with axial symmetry. The two-dimensional designs are based on a boundary chosen in the hodograph plane for which the flow is found by the method of images. The three-dimensional method uses the velocity potential and the stream function of the two-dimensional flow as independent variables and the equation for the three-dimensional stream function is solved approximately. The accuracy of the approximate method is checked by comparison with a solution obtained by Southwell's relaxation method.In both the two and the three-dimensional designs the curved wall is of finite length with parallel sections upstream and downstream. The effects of the parallel parts of the channel on the rise of pressure near the wall at the start of the contraction and on the velocity distribution across the working section can therefore be estimated.

THREE DIMENSIONAL INTERPRETATION OF GRAVITATIONAL ANOMALIES

Geophysics ◽  
1952 ◽  
Vol 17 (2) ◽  
pp. 344-364 ◽  
Author(s):  
Fraser S. Grant
Keyword(s):  
Gravitational Field ◽  
Mass Distribution ◽  
Three Dimensional ◽  
Relaxation Method ◽  
The Body ◽  
Multipole Moments ◽  
Size And Shape ◽  
Surface Field ◽  
Derivatives Of

A method is developed for determining the approximate size and shape of the three‐dimensional mass distribution that is required to produce a given gravitational field. The first few reduced multipole moments of the distribution are calculated from the derivatives of the surface field, and the approximative structure is determined from the values of these moments and a knowledge of the density contrast between the body and its surroundings. A system of classification of problems by symmetry is introduced and its practical usage discussed. A relaxation method is described which may be used to adjust the initial solution systematically to give agreement over the whole field. A descriptive discussion is appended.

scholarly journals Regions of Feasible Point-to-Point Trajectories in the Cartesian Workspace of Fully-Parallel Manipulators

1999 ◽  
Author(s):  
Damien Chablat ◽  
Philippe Wenger
Keyword(s):  
Parallel Manipulator ◽  
Cartesian Product ◽  
Parallel Manipulators ◽  
Joint Space ◽  
Inverse Kinematic ◽  
Connected Components ◽  
Feasible Point ◽  
Cartesian Space ◽  
Point Trajectories ◽  
Point To Point

Abstract The goal of this paper is to define the n-connected regions in the Cartesian workspace of fully-parallel manipulators, i.e. the maximal regions where it is possible to execute point-to-point motions. The manipulators considered in this study may have multiple direct and inverse kinematic solutions. The N-connected regions are characterized by projection, onto the Cartesian workspace, of the connected components of the reachable configuration space defined in the Cartesian product of the Cartesian space by the joint space. Generalized octree models are used for the construction of all spaces. This study is illustrated with a simple planar fully-parallel manipulator.

Determination of the Workspace of 6-DOF Parallel Manipulators

1989 ◽  
Author(s):  
C. Gosselin
Keyword(s):  
Parallel Manipulator ◽  
Graphical Representation ◽  
Three Dimensional ◽  
Parallel Manipulators ◽  
Degree Of Freedom ◽  
Geometrical Properties ◽  
Cartesian Space ◽  
Six Degree Of Freedom

Abstract This paper presents an algorithm for the determination of the workspace of parallel manipulators. The method described here, which is based on geometrical properties of the workspace, leads to a simple graphical representation of the regions of the three-dimensional Cartesian space that are attainable by the manipulator with a given orientation of the platform. Moreover, the volume of the workspace can be easily computed by performing an integration on its boundary, which is obtained from the algorithm. Examples are included to illustrate the application of the method to a six-degree-of-freedom fully-parallel manipulator.

A 6-DOF Ship-Borne Antenna Platform With Large Orientation Workspace

2020 ◽  
Author(s):  
Yuhang He ◽  
Weijia Li ◽  
Yaozhong Wu ◽  
Jinbo Wu ◽  
Zhiyuan Cheng
Keyword(s):  
Particle Swarm Optimization ◽  
Particle Swarm ◽  
Weight Ratio ◽  
Stewart Platform ◽  
Design Parameters ◽  
Parallel Mechanisms ◽  
Swarm Optimization ◽  
Pitch Range ◽  
Orientation Workspace ◽  
Load Weight

Abstract Compared with traditional antenna platform with two axes, Stewart platform can search airspace with no tracking blind district. And the advantages of high accuracy, high stiffness and high load-weight ratio also make it be a better solution for antenna platforms. This paper designed a 6-DOF ship-borne antenna platform based on the Stewart platform to overcome the difficulties that to realize a large orientation workspace (azimuth range is from 0° to 360°, pitch range is from 0° to 100°) under the compact dimensions of parallel mechanisms. A novel joint structure has been proposed which can provide a larger rotation angle than common Hooke joints to realize the large orientation workspace without the inter-mechanism interference. In addition, this paper defined the concept of working height and working radius then proposed a trajectory based on that to obtain the complete pose (translation and orientation) of antenna platform by azimuth and pitch angles. After that, the particle swarm optimization algorithm is employed to seek the optimal geometrical design parameters. A prototype of the 6-DOF ship-borne antenna platform adopted the particle swarm optimization results has been constructed. And the results show that it not noly meets the design requirements, but also provides a good performance.

scholarly journals Kinematics of Stewart Platform Explains Three-Dimensional Movement of Honeybee’s Abdominal Structure

2019 ◽  
Vol 19 (3) ◽  
Author(s):  
Youjian Liang ◽  
Jieliang Zhao ◽  
Shaoze Yan ◽  
Xin Cai ◽  
Yibo Xing ◽  
...  
Keyword(s):  
Three Dimensional ◽  
Stewart Platform

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.

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