Planar Linkage Synthesis for Mixed Exact and Approximated Motion Realization via Kinematic Mapping

2015 ◽  
Author(s):  
Ping Zhao ◽  
Xin Ge ◽  
Bin Zi ◽  
Q. J. Ge
Keyword(s):  
Null Space ◽  
Parallel Manipulators ◽  
Surface Fitting ◽  
Realization Problem ◽  
Constraint Equations ◽  
Kinematic Mapping ◽  
Motion Approximation ◽  
Mapping Theory ◽  
Discrete Motion ◽  
Fitting Problem

It has been well established that kinematic mapping theory could be applied in mechanism synthesis area, where discrete motion approximation problem could be converted to surface fitting problem of a group of discrete points in hyperspace. In this paper, we applied kinematic mapping theory to planar discrete motion synthesis of an arbitrary number of approximated poses as well as up to four exact poses. A simultaneous type and dimensional synthesis approach for mixed exact and approximate motion realization problem for three types of planar dyad chains (RR, RP, PR) is presented. For all three types of dyads, N given approximated poses are utilized to formulate a general quadratic surface fitting problem in hyperspace, while up to four prescribed poses could be imposed as pose-constraint equations to this surface fitting system such that they could be exactly realized. The surface fitting problem is converted to a linear system with two quadratic constraint equations, which could be solved by null space analysis technique. On the other hand, the given exact poses are formulated as linear pose-constraint equations and added back to the system, where both type and dimensions of the resulting optimal dyads could be determined by the solution. These optimal dyads could then be implemented as different types of four-bar linkages or parallel manipulators. The result is a novel algorithm that is simple and efficient, which allows for N-pose motion approximation of planar dyads containing both revolute and prismatic joints, as well as handling of up to four prescribed poses to be realized precisely.

Planar Linkage Synthesis for Mixed Exact and Approximated Motion Realization Via Kinematic Mapping

2016 ◽  
Vol 8 (5) ◽  
Author(s):  
Ping Zhao ◽  
Xin Ge ◽  
Bin Zi ◽  
Q. J. Ge
Keyword(s):  
Null Space ◽  
Parallel Manipulators ◽  
Surface Fitting ◽  
Dimensional Synthesis ◽  
Constraint Equations ◽  
Kinematic Mapping ◽  
Motion Approximation ◽  
Mapping Theory ◽  
Discrete Motion ◽  
Fitting Problem

It has been well established that kinematic mapping theory could be applied to mechanism synthesis, where discrete motion approximation problem could be converted to a surface fitting problem for a group of discrete points in hyperspace. In this paper, we applied kinematic mapping theory to planar discrete motion synthesis of an arbitrary number of approximated poses as well as up to four exact poses. A simultaneous type and dimensional synthesis approach is presented, aiming at the problem of mixed exact and approximate motion realization with three types of planar dyad chains (RR, RP, and PR). A two-step unified strategy is established: first N given approximated poses are utilized to formulate a general quadratic surface fitting problem in hyperspace, then up to four exact poses could be imposed as pose-constraint equations to this surface fitting system such that they could be strictly satisfied. The former step, the surface fitting problem, is converted to a linear system with two quadratic constraint equations, which could be solved by a null-space analysis technique. On the other hand, the given exact poses in the latter step are formulated as linear pose-constraint equations and added back to the system, where both type and dimensions of the resulting optimal dyads could be determined by the solution. These optimal dyads could then be implemented as different types of four-bar linkages or parallel manipulators. The result is a novel algorithm that is simple and efficient, which allows for N-pose motion approximation of planar dyads containing both revolute and prismatic joints, as well as handling of up to four prescribed poses to be realized precisely.

Kinematic-Mapping Based Solution to Various Mixed-Exact-and-Approximated Problem in Planar Motion Synthesis

2016 ◽  
Author(s):  
Ping Zhao ◽  
Xiangyun Li ◽  
Bin Zi ◽  
Q. J. Ge
Keyword(s):  
Null Space ◽  
Constraint Equation ◽  
Parallel Manipulators ◽  
Linear Constraint ◽  
Surface Fitting ◽  
Motion Synthesis ◽  
Analysis Technique ◽  
Additional Task ◽  
Kinematic Mapping ◽  
Fitting Problem

The design of mechanisms that lead a rigid-body through a set of prescribed discrete poses is usually referred to as “motion synthesis”. In practical motion synthesis cases, aside of realizing a set of given poses, various types of geometric constraint conditions could also require to be satisfied, e.g. defining the coordinates of the center/circle points of dyad linkages, setting the ground line/coupler line for four-bar linkages, realization of additional task positions, etc. Some of these constraint conditions require to be realized exactly while others might allow approximation. To solve this mixed-exact-and-approximated problem, this paper proposed a kinematic-mapping-based approach, which builds on the previous work of the realization of an arbitrary number of approximated poses as well as up to four exact poses. We now have found that the aforementioned various types of constraint conditions could be converted to each other through a general linear constraint equation. Thus, those “approximated conditions” could be uniformly converted to several prescribed discrete poses so as to be formulated as a general approximated motion synthesis problem, which is actually a general quadratic surface fitting problem in kinematic-mapping space, while up to four “exact conditions” could be imposed as linear constraint equations to this surface fitting system such that they could be exactly realized. Through null-space analysis technique, both type and dimensions of the resulting optimal dyad linkages could be determined by the solution of this surface-fitting problem with constraints. These optimal dyads could then be implemented as different types of four-bar linkages or parallel manipulators.

A Task Driven Approach to Simultaneous Type Synthesis and Dimensional Optimization of Planar Parallel Manipulator Using Algebraic Fitting of a Family of Quadrics

2013 ◽  
Author(s):  
Xiangyun Li ◽  
Ping Zhao ◽  
Q. J. Ge ◽  
Anurag Purwar
Keyword(s):  
Least Squares ◽  
Parallel Manipulator ◽  
Parallel Manipulators ◽  
Surface Fitting ◽  
Image Space ◽  
Synthesis Process ◽  
Dimensional Synthesis ◽  
Planar Parallel Manipulator ◽  
Kinematic Mapping ◽  
The Given

This paper studies the rigid body guidance problem for 3-DOF planar parallel manipulators (PPM) with three-triad assembly. We present a novel, unified, and simultaneous type and dimensional synthesis approach to planar parallel manipulator synthesis by using kinematic mapping, surface fitting, and least squares techniques. Novelty of our approach lies in linearization of a highly non-linear problem and the fact that the nature of the given motion or displacement drives the synthesis process without assuming triad topology or their geometry. It has been well established that by using planar quaternions and kinematic mapping, workspace related constraints of planar dyads or triads can be represented as algebraic constraint manifolds in the image space of planar displacements. The constraints associated with planar RR-, PR- and RP-dyads correspond to a single quadric in the image space, while that of each of the six planar triads (RRR, RPR, PRR, PPR, RRP and RPP) map to a pair of quadrics and the space between them. Moreover, the quadrics associated with RRR- and RPR-triads are of the same type as that of RR dyads, of PRR- and PPR-triads as that of PR-, and RRP- and RPP-triads as that of RP-dyad. This simplification nicely extends a dyad synthesis problem to a triad synthesis one. The problem is formulated as the least-squares error minimization problem to find a trinity of quadrics that best fit the image points of task displacements. The fitting error corresponding to each single quadric of the trinity is regarded as variation (thickness) of that quadric, which turns that quadric into a pair of quadrics. Hence, three dyads with minimal surface fitting errors can be converted to three triads in the Cartesian space.

scholarly journals Operation Modes and Singularities of 3-PRS Parallel Manipulators With Different Arrangements of P-Joints

2015 ◽  
Author(s):  
Latifah Nurahmi ◽  
Stéphane Caro ◽  
Philippe Wenger
Keyword(s):  
Algebraic Approach ◽  
Parallel Manipulators ◽  
Joint Space ◽  
Euclidean Group ◽  
Constraint Equations ◽  
Operation Modes ◽  
Prismatic Joints ◽  
Kinematic Mapping ◽  
The Subject ◽  
Orientation Workspace

The subject of this paper is about the study of the operation modes and the singularity conditions of the 3-PRS parallel manipulator with different arrangements of prismatic joints. The three prismatic joints of the PRS legs are attached to the base with an angle α between the horizontal plane of the base and their directions. By using an algebraic approach, namely the Study kinematic mapping of the Euclidean group SE(3), the mechanisms are described by a set of eight constraint equations. A primary decomposition is computed over a set of eight constraint equations and reveals that the 3-PRS manipulators with different arrangements of prismatic joints have identical operation modes, namely x0 = 0 and x3 = 0. Both operation modes are analysed. The singularity conditions are obtained by deriving the determinant of the Jacobian matrix of the eight constraint equations. All the singular configurations are mapped onto the joint space and are geometrically interpreted. The singularity loci of the 3-PRS parallel manipulators are also traced in its orientation workspace for different values of angle α.

Coordinate Reduction in the Dynamics of Constrained Multibody Systems—A New Approach

1988 ◽  
Vol 55 (4) ◽  
pp. 899-904 ◽  
Author(s):  
S. K. Ider ◽  
F. M. L. Amirouche
Keyword(s):  
Null Space ◽  
Computationally Efficient ◽  
New Approach ◽  
Constraint Equations ◽  
Kane’S Equations ◽  
Linearly Independent ◽  
Kane's Equations ◽  
Constrained Dynamical Systems ◽  
Coordinate Reduction ◽  
Algebraic Constraint

In this paper a new theorem for the generation of a basis for the null space of a rectangular matrix, with m linearly independent rows and n (n > m) columns is presented. The method is based on Gaussian row operations to transform the constraint Jacobian matrix to an uptriangular matrix. The Gram-Schmidt process is then utilized to identify basis vectors orthogonal to the uptriangular matrix. A complement orthogonal array which forms the basis for the null space for which the algebraic constraint equations are satisfied is then formulated. An illustration of the theorem application to constrained dynamical systems for both Lagrange and Kane’s equations is given. A numerical computer algorithm based on Kane’s equations with embedded constraints is also presented. The method proposed is well conditioned and computationally efficient and inexpensive.

Analysis of Active Joint Failure in Parallel Robot Manipulators

2004 ◽  
Vol 126 (6) ◽  
pp. 959-968 ◽  
Author(s):  
Mahir Hassan ◽  
Leila Notash
Keyword(s):  
Jacobian Matrix ◽  
Null Space ◽  
Robot Manipulators ◽  
Parallel Manipulators ◽  
Parallel Robot ◽  
Task Space ◽  
Active Joint ◽  
Joint Failure ◽  
Space Vectors ◽  
Six Degree Of Freedom

In this study, the effect of active joint failure on the mobility, velocity, and static force of parallel robot manipulators is investigated. Two catastrophic active joint failure types are considered: joint jam and actuator force loss. To investigate the effect of failure on mobility, the Gru¨bler’s mobility equation is modified to take into account the kinematic constraints imposed by various branches in the manipulator. In the case of joint jam, the manipulator loses the ability to move and apply force in a specific portion of its task space; while in the case of actuator force loss, the manipulator gains an unconstrained motion in a specific portion of the task space in which an externally applied force cannot be resisted by the actuator forces. The effect of joint jam and actuator force loss on the velocity and on the force capabilities of parallel manipulators is investigated by examining the change in the Jacobian matrix, its inverse, and transposes. It is shown that the reduced velocity and force capabilities after joint jam and loss of actuator force could be determined using the null space vectors of the transpose of the Jacobian matrix and its inverse. Computer simulation is conducted to demonstrate the application of the developed methodology in determining the post-failure trajectory of a 3-3 six-degree-of-freedom Stewart-Gough manipulator, when encountering active joint jam and actuator force loss.

A Task Driven Approach to the Synthesis of Spherical 4R Linkages Using Algebraic Fitting Method

2013 ◽  
Author(s):  
Ping Zhao ◽  
Xiangyun Li ◽  
Anurag Purwar ◽  
Kartik Thakkar ◽  
Q. J. Ge
Keyword(s):  
Linear Method ◽  
Geometric Constraints ◽  
Motion Generation ◽  
Constraint Manifold ◽  
Fitting Method ◽  
Kinematic Mapping ◽  
Motion Approximation ◽  
The Given ◽  
Best Fit ◽  
Additional Constraints

This paper studies the problem of spherical 4R motion approximation from the viewpoint of extraction of circular geometric constraints from a given set of spherical displacements. This paper extends our planar 4R linkage synthesis work to the spherical case. By utilizing kinematic mapping and quaternions, we map spherical displacements into points and the workspace constraints of the coupler into intersection of algebraic quadrics (called constraint manifold), respectively, in the image space of displacements. The problem of synthesizing a spherical 4R linkage is reduced to finding a pencil of quadrics that best fit the given image points in the least squares sense. Additional constraints on the pencil identify the quadrics that represent a spherical circular constraint. The geometric parameters of the quadrics encode information about the linkage parameters which are readily computed to obtain a spherical 4R linkage that best navigates through the given displacements. The result is an efficient and largely linear method for spherical four-bar motion generation problem.

A Novel Approach to Algebraic Fitting of a Pencil of Quadrics for Planar 4R Motion Synthesis

2012 ◽  
Vol 12 (4) ◽  
Author(s):  
Q. J. Ge ◽  
Ping Zhao ◽  
Anurag Purwar ◽  
Xiangyun Li
Keyword(s):  
Fast Algorithm ◽  
Nonlinear Least Squares ◽  
Linear Formulation ◽  
Image Space ◽  
Kinematic Synthesis ◽  
Novel Approach ◽  
Motion Approximation ◽  
New Formulation ◽  
Algebraic Manifolds ◽  
Fitting Problem

The use of the image space of planar displacements for planar motion approximation is a well studied subject. While the constraint manifolds associated with planar four-bar linkages are algebraic, geometric (or normal) distances have been used as default metric for nonlinear least squares fitting of these algebraic manifolds. This paper presents a new formulation for the manifold fitting problem using algebraic distance and shows that the problem can be solved by fitting a pencil of quadrics with linear coefficients to a set of image points of a given set of displacements. This linear formulation leads to a simple and fast algorithm for kinematic synthesis in the image space.

Motion Interpolation and Mechanism Synthesis

1993 ◽  
Author(s):  
Q. J. Ge ◽  
B. Ravani
Keyword(s):  
Design Problem ◽  
Hermite Interpolation ◽  
Mechanical Design ◽  
Software Systems ◽  
Design Activity ◽  
Curve Design ◽  
Fitting Procedure ◽  
Kinematic Mapping ◽  
Motion Approximation ◽  
Mechanical Design Problem

Abstract This paper studies planar motion approximation problems from a computational geometric perspective and develop a computational geometric structure that can be used for mechanical motion synthesis. This allows for development of computational algorithms and software systems to support the mechanical design activity. The approach uses an orientable kinematic mapping to transform the mechanical design problem into a curve design problem in the space of the mapping. The curve design problem for synthesis of an analytic motion is carried out by Hermite interpolation. In case of a mechanical linkage, however, the Hermite interpolation is combined with a first-order curve fitting procedure for synthesizing the motion.

scholarly journals Forward kinematics analysis for a class of asymmetrical parallel manipulators

2017 ◽  
Vol 14 (1) ◽  
pp. 172988141667813 ◽  
Author(s):  
Qimin Xu ◽  
Yongsheng Yang ◽  
Zhongliang Jing ◽  
Shiqiang Hu
Keyword(s):  
Kinematic Analysis ◽  
Parallel Manipulators ◽  
Kinematics Analysis ◽  
Kinematic Constraint ◽  
Constraint Equations ◽  
Elimination Algorithm ◽  
Computation Efficiency ◽  
Forward Kinematic ◽  
Variable Geometry Truss Manipulator ◽  
Variable Geometry Truss

This article focuses on the forward kinematic analysis of a class of asymmetrical parallel manipulators by the proposed elimination approach. To solve the key forward kinematic constraint equations with transcendental parameters of the manipulator, an improved elimination algorithm is presented. First, by analyzing the geometry structure of the manipulator, we find the inherent triangular-topology relations of the manipulator. Further, by utilizing the parameter transformation of angular, the key transcendental equations of forward kinematic analysis are formulated into compact polynomial ones. In this context, comparing with the screw approach by Gallardo-Alvarado suggested that the computation efficiency of our proposed approach is superior. Finally, an example of the asymmetrical variable geometry truss manipulator illustrates the effectiveness of the proposed approach.

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