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.

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.

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.

Motion Synthesis Using Kinematic Mappings

1983 ◽  
Vol 105 (3) ◽  
pp. 460-467 ◽  
Author(s):  
B. Ravani ◽  
B. Roth
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Algebraic Theory ◽  
Planar Motion ◽  
Dimensional Synthesis ◽  
Structural Error ◽  
Quality Of Motion ◽  
Kinematic Mapping ◽  
Motion Approximation

This paper studies planar motion approximation problems in the context of a kinematic mapping. Since a planar displacement is determined by three parameters, it can be mapped into a point of a three-dimensional space. A (single-degree-of-freedom) planar motion can, therefore, be represented by a space curve in the space of the mapping and the problem of motion approximation becomes a curve fitting problem in this space. A mapping introduced by Blaschke is used and a general theory for planar motion approximation is developed. The theory is then applied to dimensional synthesis of four-link mechanisms. Furthermore, since the structural error (i.e., the quality of motion approximation) is dependent on the closeness of the fit in the space of the mapping, a general algebraic theory for determining closest fits to points in this space is developed. The theory is illustrated by a numerical example.

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.

scholarly journals Dimensional Synthesis of a Novel 3-URU Translational Manipulator Implemented through a Novel Method

Robotics ◽  
2022 ◽  
Vol 11 (1) ◽  
pp. 10
Author(s):  
Raffaele Di Gregorio
Keyword(s):  
Parallel Manipulators ◽  
Geometric Parameters ◽  
Dimensional Synthesis ◽  
Positioning Precision ◽  
Platform Motion ◽  
Operational Space ◽  
Lower Mobility ◽  
Novel Method ◽  
Block Diagonal

A dimensional synthesis of parallel manipulators (PMs) consists of determining the values of the geometric parameters that affect the platform motion so that a useful workspace with assigned sizes can be suitably located in a free-from-singularity region of its operational space. The main goal of this preliminary dimensioning is to keep the PM far enough from singularities to avoid high internal loads in the links and guarantee a good positioning precision (i.e., for getting good kinematic performances). This paper presents a novel method for the dimensional synthesis of translational PMs (TPMs) and applies it to a TPM previously proposed by the author. The proposed method, which is based on Jacobians’ properties, exploits the fact that TPM parallel Jacobians are block diagonal matrices to overcome typical drawbacks of indices based on Jacobian properties. The proposed method can be also applied to all the lower-mobility PMs with block diagonal Jacobians that separate platform rotations from platform translations (e.g., parallel wrists).

Dimensional Synthesis and Constrained Motion Interpolation for Planar and Spherical 6R Closed Chains

2008 ◽  
Author(s):  
Anurag Purwar ◽  
Zhe Jin ◽  
Qiaode Jeffrey Ge
Keyword(s):  
Rational Curve ◽  
Dimensional Synthesis ◽  
Constrained Motion ◽  
Initial Attempt ◽  
Kinematic Constraints ◽  
Constraint Manifold ◽  
Cartesian Space ◽  
Closed Chain ◽  
Kinematic Mapping ◽  
The Given

In the recent past, we have studied the problem of synthesizing rational interpolating motions under the kinematic constraints of any given planar and spherical 6R closed chain. This work presents some preliminary results on our initial attempt to solve the inverse problem, that is to determine the link lengths of planar and spherical 6R closed chains that follow a given smooth piecewise rational motion under the kinematic constraints. The kinematic constraints under consideration are workspace related constraints that limit the position of the links of planar and spherical closed chains in the Cartesian space. By using kinematic mapping and a quaternions based approach to represent displacements of the coupler of the closed chains, the given smooth piecewise rational motion is mapped to a smooth piecewise rational curve in the space of quaternions. In this space, the aforementioned workspace constraints on the coupler of the closed chains define a constraint manifold representing all the positions available to the coupler. Thus the problem of dimensional synthesis may be solved by modifying the size, shape and location of the constraint manifolds such that the mapped rational curve is contained entirely inside the constraint manifolds. In this paper, two simple examples with preselected moving pivots on the coupler as well as fixed pivots are presented to illustrate the feasibility of this approach.

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.

Considering the Ability for Nonsingular Transitions of Assembly Mode for Dimensional Synthesis

2009 ◽  
Author(s):  
Oscar Altuzarra ◽  
Vi´ctor Petuya ◽  
Mo´nica Uri´zar ◽  
Alfonso Herna´ndez
Keyword(s):  
Parallel Manipulator ◽  
Analytical Procedure ◽  
Parallel Manipulators ◽  
Dimensional Synthesis ◽  
Complex Singularity ◽  
Algebraic Expressions ◽  
Free Region ◽  
Dimensional Parameters ◽  
Cusp Points ◽  
Singularity Locus

An important difficulty in the design of parallel manipulators is their reduced practical workspace, due mainly to the existence of a complex singularity locus within the workspace. The workspace is divided into singularity-free regions according to assembly modes and working modes, and the dimensioning of parallel manipulators aims at the maximization of those regions. It is a common practice to restrict the manipulator’s motion to a specific singularity-free region. However, a suitable motion planning can enlarge the operational workspace by means of transitions of working mode and/or assembly mode. In this paper, the authors present an analytical procedure for obtaining the loci of cusp points of a parallel manipulator as algebraic expressions of its dimensional parameters. The purpose is to find an optimal design for non-singular transitions to be possible.

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