New Mechanical Generators of Schoenflies Motion Implementing Bennett Linkages

2012 ◽  
Author(s):  
Chung-Ching Lee ◽  
Jacques M. Hervé
Keyword(s):  
Rigid Body ◽  
Special Design ◽  
Motion Group ◽  
Open Chain ◽  
Hybrid Topology ◽  
Parallel Arrangement ◽  
Axis Parallel ◽  
Mechanical Generator ◽  
Product Closure

Based on the Bennett 4R chain, we construct a rotating loop by fixing one R axis to the frame and the fixed R becomes a coaxial double R pair. The R pair opposite to the fixed double R is replaced by a spherical S pair which can be equivalent to a (RRR) open chain with non-coplanar intersecting axes. In the (RRR) sub-chain, we choose special axes and derive R|- R|(R(RRR)R chain moving with 2 DoFs. That moving R becomes a coaxial double R with the addition of another rigid body and the obtained chain with hybrid topology generates a 3-dof motion, which is mathematically modeled by a 3D submanifold of a 4D group of X motions. Because of the product closure in an X-motion group, adding an H pair with any pitch and an axis parallel to the fixed R axis leads to a mechanical generator of a 4D X-motion group. Then, parallel arrangement of two generators of the same X motion gives a new parallel generator of X motion, which can be actuated by four fixed R pairs; the two Hs must have distinct pitches. A special design with four collinear actuated axes is revealed too.

New Schoenflies-Motion Manipulator Implementing Isosceles Triangle and Delassus Parallelogram

2014 ◽  
Author(s):  
Chung-Ching Lee ◽  
Jacques M. Hervé
Keyword(s):  
Computer Simulations ◽  
Lie Group ◽  
Parallel Manipulator ◽  
Isosceles Triangle ◽  
Practical Applications ◽  
Hybrid Topology ◽  
Pick And Place ◽  
Parallel Arrangement ◽  
Spatial Translation ◽  
Product Closure

Schoenflies (X) motion is a 4D displacement Lie group including a spatial translation and any rotation whose axis is parallel to a given direction. Delassus parallelogram has four parallel screw (H) pairs with related pitches and the isosceles triangle is a special HHHP. After merging these two chains, an HHH-//-HHH generator of 2-DoF translation along a right helicoid is derived. It produces a 2-DoF motion mathematically modeled by a 2D submanifold of a 4D group of X-motion. Because of the product closure in an X-group, the 4-DoF generator with HHH-//-HHH loop serving as a subchain is revealed by adding two H pairs with axes parallel to fixed H axes. Parallel arrangement of two generators of the same X motion results in a new Schoenflies-motion manipulator with hybrid topology for 4-DoF pick-and-place operations. Four fixed H pairs (two double Hs) can actuate this manipulator and the two coaxial Hs must have distinct pitches. In addition, the possible design choices of special architectures are introduced for practical applications. Computer simulations of the new parallel manipulator with Schoenflies motion verify the effectiveness.

Synthesis of Planar, Shape-Changing Rigid-Body Mechanisms

2006 ◽  
Author(s):  
Brian M. Korte ◽  
Andrew P. Murray ◽  
James P. Schmiedeler
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Shape Change ◽  
Piecewise Linear ◽  
Single Chain ◽  
Open Chain ◽  
Revolute Joints ◽  
Planar Shape ◽  
Rigid Body Displacement ◽  
Planar Linkages

This paper presents a procedure to synthesize planar linkages, composed of rigid links and revolute joints, capable of approximating a shape change defined by a set of curves. These “morphing curves” differ from each other by a combination of rigid-body displacement and shape change. Rigid link geometry is determined through analysis of piecewise linear curves to achieve shape-change approximation, and increasing the number of links improves the approximation. A mechanism is determined through connecting the rigid links into a single chain and adding dyads to eliminate degrees of freedom. The procedure is applied to two open-chain examples.

Geometric Insight Into the Dynamics of a Rigid Body Using the Spatial Triangle of Screws

2001 ◽  
Author(s):  
Gordon R. Pennock ◽  
Patrick J. Meehan
Keyword(s):  
Rigid Body ◽  
Closed Loop ◽  
Equations Of Motion ◽  
Analytical Techniques ◽  
Rate Of Change ◽  
Time Rate ◽  
Open Chain ◽  
Spatial Mechanisms ◽  
Kinetics Of ◽  
Insight Into

Abstract Geometric relationships between the velocity screw and momentum screw are presented, and the dual angle between these two screws is shown to provide important insight into the kinetics of a rigid body. Then the centripetal screw is defined, and the significance of this screw in a study of the dynamics of a rigid body is explained. The dual-Euler equation, which is the dual form of the Newton-Euler equations of motion, is shown to be a spatial triangle. The vertices of the triangle are the centripetal screw, the time rate of change of momentum screw, and the force screw. The sides of the triangle are three dual angles between the three vertices. The spatial triangle provides valuable geometrical insight into the dynamics of a rigid body and is believed to be a meaningful alternative to existing analytical techniques. The authors believe that the work presented in this paper will prove useful in a dynamic analysis of closed-loop spatial mechanisms and multi-rigid body open-chain systems.

Dynamic Analysis of a Multi-Rigid-Body Open-Chain System

1983 ◽  
Vol 105 (1) ◽  
pp. 28-34 ◽  
Author(s):  
G. R. Pennock ◽  
A. T. Yang
Keyword(s):  
Rigid Body ◽  
Robot Manipulators ◽  
Dynamic Equations ◽  
Newtonian Mechanics ◽  
Open Chain ◽  
Analytical Technique ◽  
Chain System ◽  
Joint Forces ◽  
Reaction Forces ◽  
Common Effort

In this paper we present an analytical technique, based on Newtonian mechanics with screw calculus and dual-number matrices, to derive the dynamic equations of a multi-rigid-body open-chain system. Next, we outline a systematic procedure to derive closed-form expressions for the joint forces and torques and the reaction forces and moments exerted on each member in the system. Finally, we illustrate the procedure with two examples of robot manipulators. It is hoped that the analytical technique presented here will provide a meaningful alternative, or serve as a complement to existing methods, in our common effort to advance the design of robot manipulators.

scholarly journals Dynamic model of multi-rigid-body systems based on particle dynamics with recursive approach

2005 ◽  
Vol 2005 (4) ◽  
pp. 365-382 ◽  
Author(s):  
Hazem Ali Attia
Keyword(s):  
Rigid Body ◽  
Dynamic Model ◽  
Equations Of Motion ◽  
Particle Dynamics ◽  
Rigid Bodies ◽  
Constrained System ◽  
Open Chain ◽  
Chain System ◽  
Closed Chain ◽  
Serial Chains

A dynamic model for multi-rigid-body systems which consists of interconnected rigid bodies based on particle dynamics and a recursive approach is presented. The method uses the concepts of linear and angular momentums to generate the rigid body equations of motion in terms of the Cartesian coordinates of a dynamically equivalent constrained system of particles, without introducing any rotational coordinates and the corresponding rotational transformation matrix. For the open-chain system, the equations of motion are generated recursively along the serial chains. A closed-chain system is transformed to open-chain by cutting suitable kinematical joints and introducing cut-joint constraints. An example is chosen to demonstrate the generality and simplicity of the developed formulation.

scholarly journals Kinematics of Serial Manipulators

2020 ◽  
Author(s):  
Ivan Virgala ◽  
Michal Kelemen ◽  
Erik Prada
Keyword(s):  
Rigid Body ◽  
Numerical Solutions ◽  
Kinematic Model ◽  
Inverse Kinematic ◽  
Body Motion ◽  
Robot Kinematics ◽  
Kinematic Modeling ◽  
Open Chain ◽  
Serial Manipulators ◽  
Serial Robot

This book chapter deals with kinematic modeling of serial robot manipulators (open-chain multibody systems) with focus on forward as well as inverse kinematic model. At first, the chapter describes basic important definitions in the area of manipulators kinematics. Subsequently, the rigid body motion is presented and basic mathematical apparatus is introduced. Based on rigid body conventions, the forward kinematic model is established including one of the most used approaches in robot kinematics, namely the Denavit-Hartenberg convention. The last section of the chapter analyzes inverse kinematic modeling including analytical, geometrical, and numerical solutions. The chapter offers several examples of serial manipulators with its mathematical solution.

scholarly journals Point-joint coordinate formulation for the dynamic analysis of generalised planar linkages

2005 ◽  
Vol 46 (4) ◽  
pp. 575-589
Author(s):  
Hazem Ali Attia
Keyword(s):  
Rigid Body ◽  
Dynamic Analysis ◽  
Equations Of Motion ◽  
Translational Motion ◽  
Constraint Forces ◽  
Constrained System ◽  
Open Chain ◽  
Closed Chain ◽  
Efficient Integration ◽  
Planar Linkages

AbstractThis paper presents a two-step formulation for the dynamic analysis of generalised planar linkages. First, a rigid body is replaced by a dynamically equivalent constrained system of particles and Newton's second law is used to study the motion of the particles without introducing any rotational coordinates. The translational motion of the constrained particles represents the general motion of the rigid body both translationally and rotationally. The simplicity and the absence of any rotational coordinates from the final form of the equations of motion are considered the main advantages of this formulation. A velocity transformation is then used to transform the equations of motion to a reduced set in terms of selected relative joint variables. For an open-chain, this process automatically eliminates all of the non-working constraint forces and leads to efficient integration of the equations of motion. For a closed-chain, suitable joints should be cut and some cut-joint constraint equations should be included. An example of a closed-chain is used to demonstrate the generality and efficiency of the proposed method.

Geometric Insight Into the Dynamics of a Rigid Body Using the Spatial Triangle of Screws

2002 ◽  
Vol 124 (4) ◽  
pp. 684-689 ◽  
Author(s):  
Gordon R. Pennock ◽  
Patrick J. Meehan
Keyword(s):  
Rigid Body ◽  
Closed Loop ◽  
Equations Of Motion ◽  
Analytical Techniques ◽  
Rate Of Change ◽  
Time Rate ◽  
Open Chain ◽  
Spatial Mechanisms ◽  
Kinetics Of ◽  
Insight Into

Geometric relationships between the velocity screw and momentum screw are presented, and the dual angle between these two screws is shown to provide important insight into the kinetics of a rigid body. Then the centripetal screw is defined, and the significance of this screw in a study of the dynamics of a rigid body is explained. The dual-Euler equation, which is the dual form of the Newton-Euler equations of motion, is shown to be a spatial triangle. The vertices of the triangle are the centripetal screw, the time rate of change of momentum screw, and the force screw. The sides of the triangles are three dual angles between the three vertices. The spatial triangle provides valuable geometrical insight into the dynamics of a rigid body and is believed to be a meaningful alternative to existing analytical techniques. The authors believe that the work presented in this paper will prove useful in a dynamic analysis of closed-loop spatial mechanisms and multi-rigid body open-chain systems.

Kinematic Synthesis of Binary Actuated Mechanisms for Rigid Body Guidance

2000 ◽  
Author(s):  
Michael S. Hanchak ◽  
Andrew P. Murray
Keyword(s):  
Rigid Body ◽  
Design Space ◽  
Kinematic Synthesis ◽  
Prismatic Joint ◽  
Revolute Joints ◽  
Binary State ◽  
Parallel Arrangement ◽  
Position Problem ◽  
Position Synthesis ◽  
Kinematic Solution

Abstract This paper presents a method for designing mechanisms composed of Revolute-Binary state prismatic-Revolute (RBR) chains for rigid body guidance. Where a prismatic joint allows for any distance between two revolute joints, a binary state prismatic joint reaches two distances precisely. A single RBR chain can be designed to reach six positions. A parallel arrangement of three RBR chains can be assembled at the six positions but, in general, is not a viable kinematic solution. By requiring the arrangement of three RBR chains to share specific fixed and moving pivots, called an N-type arrangement, four positions are reachable. Further, the design space is quickly searchable for singularity-free solutions. Examples illustrate a solution to a four position synthesis problem and a ten position problem using a serial assembly of these mechanisms.

Approximation of Finite Rigid Body Motions

2009 ◽  
Author(s):  
Andreas Mu¨ller
Keyword(s):  
Rigid Body ◽  
Euler Equations ◽  
Time Derivative ◽  
Time Integration ◽  
Body Motion ◽  
Motion Group ◽  
Time Stepping ◽  
Approximation Formulas ◽  
Rigid Body Motions ◽  
Local Parameters

It is well-known that there is no integrable relation between the twist of a rigid body and its finite motion, since the angular velocity components are non-holonomic velocity coordinates. Moreover, the reconstruction of the body’s motion requires to solve a set of differential equations on the rigid body motion group. This is usually avoided by introducing local parameters (e.g. Euler angles) so that the problem becomes an ordinary differential equation on a vector space (e.g. kinematic Euler equations). In this paper the original problem on the motion group is treated. A family of approximation formulas is presented that allow reconstructing large rigid body motions from a given velocity field up to a desired order. It is shown that a k-th order accurate reconstruction requires the first k – 1 time derivative of the velocity. As an application the reconstruction formulas are used for the rotation update in a momentum preserving time stepping scheme for time integration of the dynamic Euler equations.

Sign in / Sign up

Export Citation Format

Share Document