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

2019 ◽  
Vol 11 (2) ◽  
Author(s):  
Andreas Müller
Keyword(s):  
Configuration Space ◽  
Taylor Series Expansion ◽  
Past Research ◽  
Local Approximation ◽  
Higher Order ◽  
Local Analysis ◽  
Algebraic Expression ◽  
Local Geometry ◽  
Mobility Analysis ◽  
Lower Pair

A motion of a mechanism is a curve in its configuration space (c-space). Singularities of the c-space are kinematic singularities of the mechanism. Any mobility analysis of a particular mechanism amounts to investigating the c-space geometry at a given configuration. A higher-order analysis is necessary to determine the finite mobility. To this end, past research leads to approaches using higher-order time derivatives of loop closure constraints assuming (implicitly) that all possible motions are smooth. This continuity assumption limits the generality of these methods. In this paper, an approach to the higher-order local mobility analysis of lower pair multiloop linkages is presented. This is based on a higher-order Taylor series expansion of the geometric constraint mapping, for which a recursive algebraic expression in terms of joint screws is presented. An exhaustive local analysis includes analysis of the set of constraint singularities (configurations where the constraint Jacobian has certain corank). A local approximation of the set of configurations with certain rank is presented, along with an explicit expression for the differentials of Jacobian minors in terms of instantaneous joint screws. The c-space and the set of points of certain corank are therewith locally approximated by an algebraic variety determined algebraically from the mechanism's screw system. The results are shown for a simple planar 4-bar linkage, which exhibits a bifurcation singularity and for a planar three-loop linkage exhibiting a cusp in c-space. The latter cannot be treated by the higher-order local analysis methods proposed in the literature.

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.

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.

scholarly journals DETERMINANTS OF LATENT PROFILES IN HIGHER-ORDER THINKING SKILLS OF KOREAN UNIVERSITY STUDENTS

2018 ◽  
Vol 76 (4) ◽  
pp. 483-498
Author(s):  
Soo Eun Chae ◽  
Mi-Suk Lee
Keyword(s):  
University Students ◽  
Lower Order ◽  
Latent Profile Analysis ◽  
Profile Analysis ◽  
Past Research ◽  
Higher Order Thinking ◽  
Thinking Skills ◽  
Higher Order ◽  
Logistic Analysis ◽  
Latent Profile

Past research on higher-order thinking (HOT) was mainly conducted on the bases of educational context in U.S. or western countries. This research aimed to see what kinds of HOT styles actually appear in universtiy students in South Korea. The use of HOT skills were explored in Korean universtiy students and the factors influencing the classification were examined. 1,138 Korean university students were called to respond to Lee’s (2016) Higher-Order-Thinking-Scale for Korean University Students (HOTUS). Then, a latent profile analysis and the multinomial logistic analysis were conducted. The latent profile analysis revealed that the use of HOT skills could be classified into four classes (i.e., a lower-order thinking class, a creative-argumentative class, an analytical-caring class, and a higher-order thinking class). Gender, year, and instructional approach were the determinants of latent profile types. However, there were no differences when measured by academic fields. Students with lower years were likely to fall under lower-order thinking class. The probability that men was classified as a caring class was statistically significantly lower than that of women. Students who received lecturer-centered learning were more likely to fall under the analytical and caring class. Keywords: higher-order thinking skill, latent profile analysis, multinomial logistic analysis.

Motion planning in crowded planar environments

Robotica ◽  
1999 ◽  
Vol 17 (4) ◽  
pp. 365-371 ◽  
Author(s):  
Yoav Lasovsky ◽  
Leo Joskowicz
Keyword(s):  
Motion Planning ◽  
Configuration Space ◽  
Degrees Of Freedom ◽  
Optimization Problems ◽  
Linear Optimization ◽  
Local Geometry ◽  
Objective Functions ◽  
Linear Optimization Problems ◽  
Geometrically Complex ◽  
Crowded Environments

We present a new algorithm for fine motion planning in geometrically complex situations. Geometrically complex situations have complex robot and environment geometry, crowded environments, narrow passages and tight fits. They require complex robot motions with coupled degrees of freedom. The algorithm constructs a path by incrementally building a graph of linearized convex configuration space cells and solving a series of linear optimization problems with varying objective functions. Its advantages are that it better exploits the local geometry of narrow passages in configuration space, and that its complexity does not significantly increase as the clearance of narrow passages decreases. We demonstrate the algorithm on examples which other planners could not solve.

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.

Computational Local Mobility Analysis of Mechanisms With Higher Kinematic Pairs

2016 ◽  
Author(s):  
Andreas Müller
Keyword(s):  
Computational Algorithm ◽  
Computational Approach ◽  
Computational Method ◽  
Canonical Coordinates ◽  
Mobility Analysis ◽  
Local Mobility ◽  
Lower Pair ◽  
Order Constraints ◽  
Algebraic Operations ◽  
Higher Kinematic Pairs

The finite mobility of a mechanism is reflected by its configuration space (c-space), and the mobility analysis aims at determining this c-space. Crucial for the computational mobility analysis is an adequate formulation of the constraints. For lower pair linkages an analytic formulation is the product-of-exponential (POE) formula in terms of the screw systems of the lower pair joints. In other words, the screw coordinates of a lower pair joint serve as canonical coordinates on the corresponding motion subgroup. For such linkages, a computational approach to the local mobility analysis has been reported recently. The approach is applicable to general multi-loop linkages. Higher pairs do not generate motion subgroups so that their motion cannot be expressed in terms of screw coordinates. Hence their kinematics cannot be expressed in terms of a POE, and there is no efficient and generally applicable computational method for the mobility analysis. In this paper a formulation of higher-order constraints for mechanisms with higher pair joints is proposed making use of the result for lower pair linkages. The method is applicable to mechanisms where each fundamental loop comprises no more than one higher pair, which covers the majority of mechanisms. Based on this, a computational algorithm is introduced that allows mobility determination. As for lower pair linkages, this algorithm only requires simple algebraic operations.

scholarly journals Generalized Quadratic Linearization of Machine Models

2011 ◽  
Vol 2011 ◽  
pp. 1-10
Author(s):  
Parvathy Ayalur Krishnamoorthy ◽  
Kamaraj Vijayarajan ◽  
Devanathan Rajagopalan
Keyword(s):  
Nonlinear System ◽  
Taylor Series ◽  
State Feedback ◽  
Taylor Series Expansion ◽  
Higher Order ◽  
Exact Linearization ◽  
Linearization Technique ◽  
Practical Applications ◽  
Higher Order Terms ◽  
Approximate Linearization

In the exact linearization of involutive nonlinear system models, the issue of singularity needs to be addressed in practical applications. The approximate linearization technique due to Krener, based on Taylor series expansion, apart from being applicable to noninvolutive systems, allows the singularity issue to be circumvented. But approximate linearization, while removing terms up to certain order, also introduces terms of higher order than those removed into the system. To overcome this problem, in the case of quadratic linearization, a new concept called “generalized quadratic linearization” is introduced in this paper, which seeks to remove quadratic terms without introducing third- and higher-order terms into the system. Also, solution of generalized quadratic linearization of a class of control affine systems is derived. Two machine models are shown to belong to this class and are reduced to only linear terms through coordinate and state feedback. The result is applicable to other machine models as well.

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