Constraint Manifold

2021 ◽  
pp. 1-4
Author(s):  
Manfred L. Husty
Keyword(s):  
Constraint Manifold

Approximate Motion Synthesis of Open and Closed Chains via Parametric Constraint Manifold Fitting: Preliminary Results

2003 ◽  
Author(s):  
Pierre M. Larochelle
Keyword(s):  
Optimization Techniques ◽  
Robotic Systems ◽  
Dimensional Synthesis ◽  
Motion Synthesis ◽  
Open Chain ◽  
Constraint Manifold ◽  
Synthesis Technique ◽  
Numerical Design ◽  
Design Case ◽  
Parametric Constraint

In this paper we present a novel dyad dimensional synthesis technique for approximate motion synthesis. The methodology utilizes an analytic representation of the dyad’s constraint manifold that is parameterized by its dimensional synthesis variables. Nonlinear optimization techniques are then employed to minimize the distance from the dyad’s constraint manifold to a finite number of desired locations of the workpiece. The result is an approximate motion dimensional synthesis technique that is applicable to planar, spherical, and spatial dyads. Here, we specifically address the planar RR, spherical RR and spatial CC dyads since these are often found in the kinematic structure of robotic systems and mechanisms. These dyads may be combined serially to form a complex open chain (e.g. a robot) or when connected back to the fixed link they may be joined so as to form one or more closed chains (e.g. a linkage, a parallel mechanism, or a platform). Finally, we present some initial numerical design case studies that demonstrate the utility of the synthesis technique.

Constraint Manifold Synthesis of Planar Linkages

1996 ◽  
Author(s):  
A. P. Murray ◽  
J. M. McCarthy
Keyword(s):  
Design Theory ◽  
Geometric Constraints ◽  
Design Parameters ◽  
Design Problems ◽  
Constraint Manifold ◽  
Linkage Design ◽  
Versatile Tool ◽  
Algebraic Constraints ◽  
Planar Linkages

Abstract This paper formulates the design theory of planar four-bar linkages using the planar form of dual quaternions known as planar quaternions. The set of positions reachable by the floating link of a dyad is a quadratic algebraic surface called a constraint manifold. Determining the coefficients of the quadratic form defining this manifold is equivalent to setting the design parameters of the linkage. If the task of the linkage is specified as geometric constraints on the location of the floating link, then algebraic constraints are obtained on the quaternion components. We seek the coefficients of the constraint manifold that satisfies these constraints. The result is an algebraic formulation that is symmetric in its characterization of the linkage and task, and provides a versatile tool for the formulation and solution of linkage design problems.

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.

scholarly journals Inverse kinematics-based motion planning for dual-arm robot with orientation constraints

2019 ◽  
Vol 16 (2) ◽  
pp. 172988141983685 ◽  
Author(s):  
Jiangping Wang ◽  
Shirong Liu ◽  
Botao Zhang ◽  
Changbin Yu
Keyword(s):  
Motion Planning ◽  
Joint Space ◽  
Inverse Kinematic ◽  
Planning Problem ◽  
Constraint Manifold ◽  
Wide Range ◽  
Planning Algorithm ◽  
Motion Paths ◽  
Dual Arm Robot ◽  
Dual Arm

This article proposes an efficient and probabilistic complete planning algorithm to address motion planning problem involving orientation constraints for decoupled dual-arm robots. The algorithm is to combine sampling-based planning method with analytical inverse kinematic calculation, which randomly samples constraint-satisfying configurations on the constraint manifold using the analytical inverse kinematic solver and incrementally connects them to the motion paths in joint space. As the analytical inverse kinematic solver is applied to calculate constraint-satisfying joint configurations, the proposed algorithm is characterized by its efficiency and accuracy. We have demonstrated the effectiveness of our approach on the Willow Garage’s PR2 simulation platform by generating trajectory across a wide range of orientation-constrained scenarios for dual-arm manipulation.

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.

scholarly journals On Constraint Manifolds of Lorentz Sphere

2020 ◽  
Vol 28 (2) ◽  
pp. 15-34
Author(s):  
Buşra Aktaş ◽  
Olgun Durmaz ◽  
Hal˙t Gündoğan
Keyword(s):  
Lorentz Space ◽  
Structure Equation ◽  
Geometric Constructions ◽  
Open Chain ◽  
Constraint Manifold ◽  
Structure Equations ◽  
A Chain

AbstractThe expression of the structure equation of a mechanism is significant to present the last position of the mechanism. Moreover, in order to attain the constraint manifold of a chain, we need to constitute the structure equation. In this paper, we determine the structure equations and the constraint manifolds of a spherical open-chain in the Lorentz space. The structure equations of spherical open chain with reference to the causal character of the first link are obtained. Later, the constraint manifolds of the mechanism are determined by means of these equations. The geometric constructions corresponding to these manifolds are studied.

Synthesis of Planar RR Dyads by Constraint Manifold Projection

1996 ◽  
Author(s):  
Pierre M. Larochelle
Keyword(s):  
Rigid Body ◽  
Design Methodology ◽  
Analytical Representation ◽  
Dimensional Synthesis ◽  
Constraint Manifold ◽  
Design Case ◽  
Design Case Study

Abstract In this paper we present the constraint manifold of the planar RR dyad. The constraint manifold is an analytical representation of the workspace of the dyad. We then derive a technique, utilizing the constraint manifold, for performing the dimensional synthesis of planar RR dyads for approximate rigid body guidance through n positions. Finally, we present the implementation of the design methodology in the software VISSYN and discuss its use in a design case study.

Constraint Manifold

2021 ◽  
pp. 1-4
Author(s):  
Manfred L. Husty
Keyword(s):  
Constraint Manifold
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