Effect of the coordinate frame on high-order expansion of serial-chain displacement

2019 ◽  
Vol 233 (4) ◽  
pp. 840-855
Author(s):  
Paul Milenkovic
Keyword(s):  
Common Ground ◽  
Power Series Expansion ◽  
Hardware Acceleration ◽  
Asymptotic Limit ◽  
Terminal Link ◽  
Coordinate Frame ◽  
Alternative Formulation ◽  
Moving Frames ◽  
Finite Displacement ◽  
Instantaneous Screw

An algorithmic differentiation technique gives a simpler, faster power series expansion of the finite displacement of a closed-loop linkage. It accomplishes this by using a higher order than what has been implemented by complicated prior formulas for kinematic derivatives. In this expansion, the joint rates and axis lines generate the instantaneous screw of each link. Constraining the terminal link to have a zero instantaneous screw satisfies closure. In order to maintain closure over a finite displacement, it is necessary to track the spatial trajectory of each joint axis line, which in turn is directed by the instantaneous screw of a link to which it is attached. Prior algorithms express these screws in a common ground-referenced coordinate frame. Motivated by the kinematics solver portion of the recursive Newton–Euler algorithm, an alternative formulation uses sparse matrices to update the instantaneous screw between successive link-local frames. The recursive Newton–Euler algorithm, however, conducts the expansion to only second order, where this paper shows local coordinate frames that are only instantaneously aligned with their respective links give identical expressions to those in frames that move with the links. Moving frames, however, require about 40% of the operations of the global-frame formulation in the asymptotic limit. Both incrementally translated (Java) and statically compiled (C++) software implementations offer more modest performance gains; execution profiling shows reasons in order of importance (1) balance of calculation tasks when below the asymptotic limit, (2) Java array bounds checking, and (3) hardware acceleration of loops.

Series Solution for Finite Displacement of Single-Loop Spatial Linkages

2012 ◽  
Vol 4 (2) ◽  
Author(s):  
Paul Milenkovic
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Rigid Body ◽  
Degrees Of Freedom ◽  
Coordinate Frame ◽  
Recursive Procedure ◽  
Single Loop ◽  
Analytical Technique ◽  
Finite Displacement ◽  
Instantaneous Screw

The kinematic differential equation for a spatial point trajectory accepts the time-varying instantaneous screw of a rigid body as input, the time-zero coordinates of a point on that rigid body as the initial condition and generates the space curve traced by that point over time as the solution. Applying this equation to multiple points on a rigid body derives the kinematic differential equations for a displacement matrix and for a joint screw. The solution of these differential equations in turn expresses the trajectory over the course of a finite displacement taken by a coordinate frame in the case of the displacement matrix, by a joint axis line in the case of a screw. All of the kinematic differential equations are amenable to solution by power series owing to the expression for the product of two power series. The kinematic solution for finite displacement of a single-loop spatial linkage may, hence, be expressed either in terms of displacement matrices or in terms of screws. Each method determines coefficients for joint rates by a recursive procedure that solves a sequence of linear systems of equations, but that procedure requires only a single factorization of a 6 by 6 matrix for a given initial posture of the linkage. The inverse kinematics of an 8R nonseparable redundant-joint robot, represented by one of the multiple degrees of freedom of a 9R loop, provides a numerical example of the new analytical technique.

Finite Displacement Screw Operators With Embedded Chasles’ Motion

2012 ◽  
Vol 4 (4) ◽  
Author(s):  
Jian S. Dai
Keyword(s):  
Rigid Body ◽  
Rotation Axis ◽  
Intrinsic Property ◽  
Adjoint Action ◽  
Finite Displacement ◽  
Diagonal Part ◽  
Axial Translation ◽  
The Matrix ◽  
Rigid Body Displacement ◽  
Instantaneous Screw

Rigid body displacement can be presented with Chasles’ motion by rotating about an axis and translating along the axis. This motion can be implemented by a finite displacement screw operator in the form of either a 3 × 3 dual-number matrix or a 6 × 6 matrix that is executed with rotation and translation as an adjoint action of the Lie group. This paper investigates characteristics of this finite displacement screw matrix and decomposes the secondary part that is the off diagonal part of the matrix into the part of an equivalent translation due to the effect of off-setting the rotation axis and the part of an axial translation. The paper hence presents for the first time the axial translation matrix and reveals its property, leading to discovery of new results and new formulae. The analysis further reveals two new traces of the matrix and presents the relationship between the finite displacement screw matrix and the instantaneous screw, leading to the understanding of Chasles’ motion embedded in a rigid body displacement. An algebraic and geometrical interpretation of the finite displacementscrew matrix is thus given, presenting an intrinsic property of the matrix in relation to the finite displacement screw. The paper ends with a case study to verify the theory and illustrate the principle.

Instantaneous Kinematics of Three-Parameter Motions

1985 ◽  
Vol 107 (2) ◽  
pp. 157-162 ◽  
Author(s):  
G. R. Pennock ◽  
A. T. Yang
Keyword(s):  
Angular Velocity ◽  
Closed Form ◽  
Terminal Link ◽  
Screw Axis ◽  
Open Chain ◽  
Variable Parameters ◽  
Chain System ◽  
Fixed Base ◽  
Instantaneous Screw Axis ◽  
Instantaneous Screw

In this paper we study the instantaneous kinematics of a three-link open-chain system. We derive concise closed-form expressions to determine the angular velocity, the pitch, and the instantaneous screw axis that describe the three-parameter motion of the terminal link of the three-link chain relative to the fixed base. Using two joint angular velocity ratios as variable parameters we interpret the double-infinity of solutions as a family of cylindroids. For illustrative purposes, we present an analysis of the arm subassembly of two well-known robot manipulators.

Talking Across Cultures in Early Intervention: Finding Common Ground to Meet Children's Communication Needs

2009 ◽  
Vol 16 (3) ◽  
pp. 79-85 ◽  
Author(s):  
Elizabeth D. Peña ◽  
Christine Fiestas
Keyword(s):  
Early Intervention ◽  
Cultural Values ◽  
Common Ground ◽  
Service Providers ◽  
Cultural Orientation ◽  
Speech Language Pathology ◽  
Provider Perspectives ◽  
Language Pathology ◽  
Communication Needs ◽  
Caregiver Role

Abstract In this paper, we explore cultural values and expectations that might vary among different groups. Using the collectivist-individualist framework, we discuss differences in beliefs about the caregiver role in teaching and interacting with young children. Differences in these beliefs can lead to dissatisfaction with services on the part of caregivers and with frustration in service delivery on the part of service providers. We propose that variation in caregiver and service provider perspectives arise from cultural values, some of which are instilled through our own training as speech-language pathologists. Understanding where these differences in cultural orientation originate can help to bridge these differences. These can lead to positive adaptations in the ways that speech-language pathology services are provided within an early intervention setting that will contribute to effective intervention.

Spatial Representations as an Emergent Feature of Perceptual Processing

2010 ◽  
Vol 24 (3) ◽  
pp. 161-172 ◽  
Author(s):  
Edmund Wascher ◽  
C. Beste
Keyword(s):  
Common Ground ◽  
Event Related Potentials ◽  
Vertical Axis ◽  
Relevant Information ◽  
Sources Of Information ◽  
Integrated Network ◽  
Perceptual Maps ◽  
Emergent Feature ◽  
Related Potentials ◽  
Selection Of

Spatial selection of relevant information has been proposed to reflect an emergent feature of stimulus processing within an integrated network of perceptual areas. Stimulus-based and intention-based sources of information might converge in a common stage when spatial maps are generated. This approach appears to be inconsistent with the assumption of distinct mechanisms for stimulus-driven and top-down controlled attention. In two experiments, the common ground of stimulus-driven and intention-based attention was tested by means of event-related potentials (ERPs) in the human EEG. In both experiments, the processing of a single transient was compared to the selection of a physically comparable stimulus among distractors. While single transients evoked a spatially sensitive N1, the extraction of relevant information out of a more complex display was reflected in an N2pc. The high similarity of the spatial portion of these two components (Experiment 1), and the replication of this finding for the vertical axis (Experiment 2) indicate that these two ERP components might both reflect the spatial representation of relevant information as derived from the organization of perceptual maps, just at different points in time.

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