Higher-Order Analysis of Kinematic Singularities of Lower Pair Linkages and Serial Manipulators

2017 ◽  
Vol 10 (1) ◽  
Author(s):  
Andreas Müller
Keyword(s):  
Critical Points ◽  
Tangent Cone ◽  
Polynomial System ◽  
Higher Order ◽  
Computational Method ◽  
Local Geometry ◽  
Serial Manipulators ◽  
Order Analysis ◽  
Forward Kinematic ◽  
Kinematic Singularities

Kinematic singularities of linkages are configurations where the differential mobility changes. Constraint singularities are critical points of the constraint mapping defining the loop closure constraints. Configuration space (c-space) singularities are points where the c-space ceases to be a smooth manifold. These singularity types are not identical and can neither be distinguished nor identified by simply investigating the rank deficiency of the constraint Jacobian (linear dependence of joint screws). C-space singularities are reflected by the c-space geometry. In a previous work, a kinematic tangent cone was introduced as an approximation of the c-space, defined as the set of tangents to smooth curves in c-space. Identification of kinematic singularities amounts to analyze the local geometry of the set of critical points. As a computational means, a kinematic tangent cone to the set of critical points is introduced in terms of Jacobian minors. Closed form expressions for the derivatives of the minors in terms of Lie brackets of joint screws are presented. A computational method is introduced to determine a polynomial system defining the kinematic tangent cone. The paper complements the recently proposed mobility analysis using the tangent cone to the c-space. This allows for identifying c-space and kinematic singularities as long as the solution set of the constraints is a real variety. The introduced approach is directly applicable to the higher-order analysis of forward kinematic singularities of serial manipulators. This is briefly addressed in the paper.

Higher-Order Local Analysis of Kinematic Singularities of Lower Pair Linkages

2017 ◽  
Author(s):  
Andreas Müller
Keyword(s):  
Critical Points ◽  
Tangent Cone ◽  
Polynomial System ◽  
Computational Method ◽  
Local Analysis ◽  
Closed Form Expression ◽  
Local Geometry ◽  
Lower Pair ◽  
Kinematic Singularities ◽  
Derivatives Of

Kinematic singularities of linkages are configurations where the differential mobility changes. Constraint singularities are critical points of the constraint mapping defining the loop closure constraints. Configuration space (c-space) singularities are points where the c-space ceases to be a smooth manifold. These singularity types are not identical. C-space singularities are reflected by the c-space geometry. Identifying kinematic singularities amounts to locally analyze the set of critical points. The local geometry of the set of critical points is best approximated by its tangent cone (an algebraic variety). The latter is defined in this paper in a form that allows for its computational determination using the Jacobian minors. An explicit closed form expression for the derivatives of the minors is presented in terms of Lie brackets of joint screws. A computational method is proposed to determine a polynomial system defining the tangent cone. This finally allows for identifying c-space and kinematic singularities.

A Screw Approach to the Approximation of the Local Geometry of the Configuration Space and of the Set of Configurations of Certain Rank of Lower Pair Linkages

2018 ◽  
Author(s):  
Andreas Müller
Keyword(s):  
Configuration Space ◽  
Global Analysis ◽  
Polynomial System ◽  
Past Research ◽  
Local Approximation ◽  
Higher Order ◽  
Local Analysis ◽  
Local Geometry ◽  
Universal Method ◽  
Set Of Points

The determination of the finite mobility of a linkage boils down to the analysis of its configuration space (c-space). Since a global analysis is not feasible in general (but only for particular cases), the research focused on methods for a local analysis. Past research has in particular addressed the approximation of finite curves in c-space (i.e. finite motions). No universal method for the approximation of the c-space itself has been reported. In this paper a generally applicable formulation of the equations defining the higher-order local approximation of the c-space as well as the set of points where the Jacobian has a certain rank are presented. To this end, algebraic formulations of the higher-order differential of the constraint mapping (defining the loop closure) and of the Jacobian minors of arbitrary order are introduced. The respective local approximation is therewith given in terms of a low-order polynomial system. Results are shown for a simple planar 4-bar linkage and a planar three-loop linkage. Since the latter exhibits a cusp singularity it cannot be treated by the local analysis methods proposed thus far, which are based on approximating finite curves.

Phase diagrams and higher-order critical points

1975 ◽  
Vol 12 (1) ◽  
pp. 345-355 ◽  
Author(s):  
Robert B. Griffiths
Keyword(s):  
Phase Diagrams ◽  
Critical Points ◽  
Higher Order

Higher order analysis of random 1–2 brother trees

1980 ◽  
Vol 20 (3) ◽  
pp. 302-314 ◽  
Author(s):  
Th. Ottmann ◽  
W. Stucky
Keyword(s):  
Higher Order ◽  
Order Analysis

Singularity robust balancing of parallel manipulators following inconsistent trajectories

Robotica ◽  
2014 ◽  
Vol 34 (9) ◽  
pp. 2027-2038 ◽  
Author(s):  
Mustafa Özdemir
Keyword(s):  
Equations Of Motion ◽  
Parallel Manipulators ◽  
Inverse Kinematic ◽  
Type I ◽  
Type Ii ◽  
Inverse Dynamic ◽  
Serial Manipulators ◽  
Dynamic Solution ◽  
Singular Configuration ◽  
Kinematic Singularities

SUMMARYWhen compared to serial manipulators, parallel manipulators have small workspaces mainly due to their closed-loop structure. As opposed to type I singularities (or inverse kinematic singularities) that are generally encountered at the workspace boundaries, type II singularities characteristically arise within the workspace, and around them, the inverse dynamic solution becomes unbounded. Hence, a desired trajectory passing through a type II singular position cannot be achieved by the manipulator, and its useful workspace becomes further and substantially reduced. It has been previously shown in the literature that if the trajectory is replanned in such a way that the dynamic equations of motion of the manipulator are consistent at a type II singularity, i.e. if the trajectory is consistent, then the manipulator passes through this singular configuration in a controllable manner, while the inverse dynamic solution remains finite. An inconsistent trajectory, on the other hand, is stated in the literature to be unrealizable. However, although seems a promising technique, trajectory replanning itself is also a deviation from the originally desired trajectory, and there might be cases in applications where, due to some task-specific reasons, the desired trajectory, although inconsistent, is not allowed to be replanned to satisfy the consistency conditions. In this paper, a method of singularity robust balancing is proposed for parallel manipulators passing through type II singular configurations while following inconsistent trajectories. By this means, an originally unrealizable inconsistent trajectory passing through a type II singularity can be followed without any deviation, while the required actuator forces remain bounded after the manipulator is balanced according to the design methodology presented in this study. The effectiveness of the introduced method is shown through numerical simulations considering a planar 3-DOF 2-PRR parallel manipulator under different balancing scenarios.

Local Analysis of Closed-Loop Linkages: Mobility, Singularities, and Shakiness

2015 ◽  
Author(s):  
Andreas Müller
Keyword(s):  
Tangent Cone ◽  
Local Approximation ◽  
Approximation Order ◽  
Local Analysis ◽  
Space Geometry ◽  
Geometric Entity ◽  
Best Local Approximation ◽  
Kinematic Singularities ◽  
General Configuration

The mobility of a linkage is determined by the constraints imposed on its members. The constraints define the configuration space (c-space) variety as the geometric entity in which the finite mobility of a linkage is encoded. The instantaneous motions are determined by the constraints, rather than by the c-space geometry. Shaky linkages are prominent examples that exhibit a higher instantaneous than finite DOF even in regular configurations. Inextricably connected to the mobility are kinematic singularities that are reflected in a change of the instantaneous DOF. The local analysis of a linkage, aiming at determining the instantaneous and finite mobility in a given configuration, hence needs to consider the c-space geometry as well as the constraint system. A method for the local analysis is presented based on a higher-order local approximation of the c-space adopting the concept of the tangent cone to a variety. The latter is the best local approximation of the c-space in a general configuration. It thus allows for investigating the mobility in regular as well as singular configurations. Therewith the c-space is locally represented as an algebraic variety whose degree is the necessary approximation order. In regular configurations the tangent cone is the tangent space. The method is generally applicable and computationally simple. It allows for a classification of linkages as overconstrained and underconstrained, and to identify singularities.

The differential calculus of screws: Theory, geometrical interpretation, and applications

2009 ◽  
Vol 223 (6) ◽  
pp. 1449-1468 ◽  
Author(s):  
J J Cervantes-Sánchez ◽  
J M Rico-Martínez ◽  
G González-Montiel ◽  
E J González-Galván
Keyword(s):  
Differential Calculus ◽  
Kinematic Chain ◽  
Parallel Manipulators ◽  
Higher Order ◽  
Abstract Concepts ◽  
Serial Manipulators ◽  
Kinematic Chains ◽  
Finite Displacements ◽  
Typical Shape ◽  
Derivatives Of

This article presents a novel and original formula for the higher-order time derivatives, and also for the partial derivatives of screws, which are successively computed in terms of Lie products, thus leading to the automation of the differentiation process. Through the process and, due to the pure geometric nature of the derivation approach, an enlightening physical interpretation of several screw derivatives is accomplished. Important applications for the proposed formula include higher-order kinematic analysis of open and closed kinematic chains and also the kinematic synthesis of serial and parallel manipulators. More specifically, the existence of a natural relationship is shown between the differential calculus of screws and the Lie subalgebras associated with the expected finite displacements of the end effector of an open kinematic chain. In this regard, a simple and comprehensible methodology is obtained, which considerably reduces the abstraction level frequently required when one resorts to more abstract concepts, such as Lie groups or Lie subalgebras; thus keeping the required mathematical background to the extent that is strictly necessary for kinematic purposes. Furthermore, by following the approach proposed in this article, the elements of Lie subalgebra arise in a natural way — due to the corresponding changes in screws through time — and they also have the typical shape of the so-called ordered Lie products that characterize those screws that are compatible with the feasible joint displacements of an arbitrary serial manipulator. Finally, several application examples — involving typical, serial manipulators — are presented in order to prove the feasibility and validity of the proposed method.

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