The Duality Between the Singularity of Bricard Mechanism and the Singularity of Stewart Platform

2014 ◽  
Author(s):  
Michael Slavutin ◽  
Offer Shai ◽  
Andreas Müller
Keyword(s):  
Mechanical Systems ◽  
Dual Graph ◽  
Original System ◽  
Stewart Platform ◽  
Dual System ◽  
Graph Representation ◽  
Parallel Mechanisms ◽  
Serial Chain ◽  
One Step ◽  
Specific Configuration

The duality (known also as symmetry) between serial chain manipulators and fully parallel mechanisms is well known in the literature. This paper takes this idea one step further, by introducing a systematic method that transforms mechanical systems into other and different mechanical systems so that the wrench screws in the original system gives rise to the relative twist screws in the second system. The mathematical foundation of this work relies on using the BB graph, a variant of graph representation widely used in mechanisms, possessing both the topology and geometry of the original system. From the dual graph of the latter it is possible to construct the dual system at a specific configuration. Relying on the equivalence between the dual systems, it is proved that if the screw system of a mechanism is at the singular position, so is that of its dual. This idea is demonstrated by showing the dual system of a Bricard mechanism, which is a 6/6 Stewart Platform in the singular position. The paper also shows that the cyclohexane molecule is dual to the 6/3 Stewart platform at the singular position, providing another perspective of the known mobility of this molecule.

A systematic approach to the instantaneous duality of mechanisms and its application

2015 ◽  
Vol 230 (3) ◽  
pp. 437-444
Author(s):  
Andreas Müller ◽  
Offer Shai
Keyword(s):  
Original System ◽  
Parallel Manipulators ◽  
Stewart Platform ◽  
Dual System ◽  
Dual Systems ◽  
Topological Graph ◽  
Dual Graphs ◽  
Constraint Graph ◽  
One Step ◽  
Serial Chains

The instantaneous duality (also known as symmetry) between serial chains and fully parallel manipulators is well known in the literature. This paper takes the idea one step further, by introducing a systematic method that transforms one mechanical system into another. This duality concept rests on the concept of dual graphs to define the kinematics of the dual system. The mechanism structure can be represented in two essentially different ways: its kinematic topology or its constraint system. The first is embodied by the topological graph and the second by the constraint graph. The dual to a topological graph is a constraint graph and vice versa. Hence, there are various ways to introduce a dual topology. The dual kinematics is defined by instantaneously identifying the twist screws in the original system with the wrench screws of the dual. This identification allows carrying over statements from the original to the dual system. In particular, it is shown that the singularities can be easily established in the dual if they are known in the original system. This concept of transference is a powerful tool since a variety of dual systems can be assigned to a given system. This idea is demonstrated for a Bricard mechanism that is instantaneously dual to a 6/6 Stewart platform at a singular position, and in another configuration (resembling the cyclohexane molecule) it is dual to the 6/3 Stewart platform at the singular position. This provides another perspective of the known mobility of this molecule.

Improving Techniques in Statically Equivalent Serial Chain Modeling for Center of Mass Estimation

2014 ◽  
Author(s):  
Bingjue Li ◽  
Andrew P. Murray ◽  
David H. Myszka
Keyword(s):  
Center Of Mass ◽  
Original System ◽  
Rigid Bodies ◽  
Joint Angles ◽  
Practical Applications ◽  
Serial Chain ◽  
System Of Rigid Bodies ◽  
Final Development ◽  
Data Error ◽  
Insight Into

Any articulated system of rigid bodies defines a Statically Equivalent Serial Chain (SESC). The SESC is a virtual chain that terminates at the center of mass (CoM) of the original system of bodies. A SESC may be generated experimentally without knowing the mass, CoM, or length of each link in the system given that its joint angles and overall CoM may be measured. This paper presents three developments toward recognizing the SESC as a practical modeling technique. Two of the three developments improve utilizing the technique in practical applications where the arrangement of the joints impacts the derivation of the SESC. The final development provides insight into the number of poses needed to create a usable SESC in the presence of data collection errors. First, modifications to a matrix necessary in computing the SESC are proposed. Second, the problem of generating a SESC experimentally when the system of bodies includes a mass fixed in the ground frame are presented and a remedy is proposed for humanoid-like systems. Third, an investigation of the error of the experimental SESC versus the number of data readings collected in the presence of errors in joint readings and CoM data is conducted. By conducting the method on three different systems with various levels of data error, a general form of the function for estimating the error of the experimental SESC is proposed.

scholarly journals Calibration Method for a Parallel Mechanism Type Machine Tool by Response Surface Methodology –Consideration via Simulation on a Stewart Platform Mechanism–

2010 ◽  
Vol 4 (4) ◽  
pp. 355-363 ◽  
Author(s):  
Hiroshi Yachi ◽  
◽  
Hiroshi Tachiya
Keyword(s):  
Response Surface Methodology ◽  
Response Surface ◽  
Parallel Mechanism ◽  
Measurement Data ◽  
Calibration Method ◽  
Stewart Platform ◽  
Parallel Mechanisms ◽  
Data Set ◽  
Efficient Data ◽  
Efficient Measurement

This paper proposes a calibration method for parallel mechanisms usingResponse Surface Methodology. This method is a statistical approach to estimating an unknown input-output relationship using a small set of efficient data collected on an intended system. Although identifying locations causing positional errors in a parallel mechanism and precisely measuring the position and posture of the output point are difficult, the proposed calibration method based onResponse Surface Methodologyaims to compensate for positional and postural errors, without indentifying the locations causing these errors, by using a small yet efficient measurement data set. This study analyzes the effectiveness of the method we propose by applying it to a Stewart platform, which is a typical spatial 6-DOF parallel mechanism.

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 Structural Reanalysis Based on FRFs Using Sherman–Morrison–Woodbury Formula

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Jun Ren ◽  
Qianghao Zhang
Keyword(s):  
Degrees Of Freedom ◽  
Salient Feature ◽  
Original System ◽  
Modal Testing ◽  
Structural Dynamic ◽  
Structural Dynamic Modification ◽  
Modal Model ◽  
Dynamic Modification ◽  
One Step ◽  
Stiffness And Damping

Structural dynamic modification is a popular approach to obtain desire frequencies and dynamic characteristics. It has been observed that reanalyzing the modified structure usually involves complicated calculations when modifications are concerned with numerous degrees of freedom (DOFs), especially adding substructures to these DOFs. This paper proposed a method to reanalyze the frequency response functions (FRFs) of structures with multiple co-ordinates modifications. Two different cases are taken into consideration in the modifications, including adding (or decreasing) masses, stiffness, and damping, as well as adding spring-mass substructures, which makes the method more practical. This method is developed by employing Sherman–Morrison and Woodbury (SMW) formula based on the FRFs related to the modifications coordinates of the original system. The advantage of this method is that neither a physical model nor a modal model is required; instead, it needs only the FRFs, which can be directly measured by experimental modal testing. Another salient feature of this proposed strategy is that the FRFs of the modified structure can be calculated in only one step. Validation of this proposed method is demonstrated using various numerical examples. It is shown that the method is very effective and can be considered for real applications.

scholarly journals Collision-free workspace of parallel mechanisms based on an interval analysis approach

Robotica ◽  
2016 ◽  
Vol 35 (8) ◽  
pp. 1747-1760 ◽  
Author(s):  
MohammadHadi FarzanehKaloorazi ◽  
Mehdi Tale Masouleh ◽  
Stéphane Caro
Keyword(s):  
Interval Analysis ◽  
Parallel Mechanism ◽  
Spherical Shape ◽  
Stewart Platform ◽  
Analysis Approach ◽  
Parallel Mechanisms ◽  
Planar Case ◽  
End Effector ◽  
Mechanical Interference

SUMMARYThis paper proposes an interval-based approach in order to obtain the obstacle-free workspace of parallel mechanisms containing one prismatic actuated joint per limb, which connects the base to the end-effector. This approach is represented through two cases studies, namely a 3-RPR planar parallel mechanism and the so-called 6-DOF Gough–Stewart platform. Three main features of the obstacle-free workspace are taken into account: mechanical stroke of actuators, collision between limbs and obstacles and limb interference. In this paper, a circle(planar case)/spherical(spatial case) shaped obstacle is considered and its mechanical interference with limbs and edges of the end-effector is analyzed. It should be noted that considering a circle/spherical shape would not degrade the generality of the problem, since any kind of obstacle could be replaced by its circumscribed circle/sphere. Two illustrative examples are given to highlight the contributions of the paper.

Some Duality Results in the Theory of Queues

1969 ◽  
Vol 6 (01) ◽  
pp. 99-121 ◽  
Author(s):  
Irwin Greenberg
Keyword(s):  
Service Time ◽  
Busy Period ◽  
Original System ◽  
Dual System ◽  
Duality Results ◽  
System A ◽  
Duality Relations

When the interarrival and service time distributions of a queue are interchanged a new queue is obtained which can be considered as the dual of the original. Another dual system, a dam, can also be associated with the original queue. Events defined for the original system can be transformed into events defined for the duals and conversely, and hence, probabilities obtained for one system can be extended to the others. In this paper several duality relations are derived, with particular emphasis on results pertaining to a single busy period. Examples are given, most of which refer to the M/G/1 – G/M/l queues.

Smooth Decomposition Analysis and Order Reduction of Nonlinear Mechanical Systems Under Random Excitation

2012 ◽  
Author(s):  
Sergio Bellizzi ◽  
Rubens Sampaio
Keyword(s):  
Covariance Matrix ◽  
Decomposition Method ◽  
Equations Of Motion ◽  
Galerkin Approximation ◽  
Mechanical Systems ◽  
Decomposition Analysis ◽  
Original System ◽  
Random Excitation ◽  
Order Structure ◽  
Reduced Order

This paper presents a possible alternative procedure to the Karhunen-Loève approach to construct reduced order models which capture accurately the dynamics of nonlinear discrete mechanical systems under random excitation. This procedure combines the Smooth Decomposition method and the Petrov-Galerkin approximation. The smooth decomposition method is a multivariate-data analysis method characterizing coherent structures (the smooth modes) as the eigenvectors of the generalized eigenproblem defined from the covariance matrix of the displacement field and the covariance matrix of the velocity field. The Petrov-Galerkin approximation is used to project the dynamics in a subspace generated by a set of the smooth modes. The Petrov-Galerkin approximation preserves the second order structure of the equations of motion. The procedure is considered for a mechanical system including a strongly nonlinear end-attachment. The efficiency of the approach is analyzed comparing the power spectral density functions of the reduced-order model and of the original system.

A bicausal bond graph representation of operational amplifiers

2003 ◽  
Vol 217 (1) ◽  
pp. 49-58 ◽  
Author(s):  
P J Gawthrop ◽  
D Palmer
Keyword(s):  
Operational Amplifier ◽  
Mechanical Systems ◽  
Graph Model ◽  
Bond Graph ◽  
Electromechanical System ◽  
Graph Representation ◽  
Operational Amplifiers ◽  
Electronic Circuits ◽  
Graph Interpretation ◽  
Virtual Earth

The virtual earth concept, well known to designers of active electronic circuits with operational amplifier components, is shown to have a novel bicausal bond graph interpretation. This leads to simplified bond graph modelling of such circuits. Some simple operational amplifier circuits, together with a more complex active filter, are used to illustrate the approach. A complex electromechanical system shows that the method is useful in creating a unified bond graph model of systems involving both analogue electronic and mechanical systems.

The Maximal Singularity-Free Workspace of the Gough–Stewart Platform for a Given Orientation

2008 ◽  
Vol 130 (11) ◽  
Author(s):  
Qimi Jiang ◽  
Clément M. Gosselin
Keyword(s):  
Case Studies ◽  
Parallel Mechanism ◽  
Stewart Platform ◽  
Parallel Robots ◽  
Analytic Method ◽  
Robot Design ◽  
Parallel Mechanisms ◽  
Leg Length ◽  
Maximal Singularity ◽  
The Relationship

The maximal singularity-free workspace of parallel mechanisms is a desirable criterion in robot design. However, for a 6DOF parallel mechanism, it is very difficult to find an analytic method to determine the maximal singularity-free workspace around a prescribed point for a given orientation. Hence, a numerical algorithm is presented in this paper to compute the maximal singularity-free workspace as well as the corresponding leg length ranges of the Gough–Stewart platform. This algorithm is based on the relationship between the maximal singularity-free workspace and the singularity surface. Case studies with different orientations are performed to demonstrate the presented algorithm. The obtained results can be applied to the geometric design or parameter (leg length) setup of this type of parallel robots.

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