Finite Displacement Screw Operators With Embedded Chasles’ Motion

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.

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Characteristics of the Screw Transformation Matrix and Their Effect on Chasles’ Motion

Volume 6: 35th Mechanisms and Robotics Conference, Parts A and B ◽

10.1115/detc2011-48613 ◽

2011 ◽

Author(s):

Jian S. Dai

Keyword(s):

Rigid Body ◽

Rotation Axis ◽

Intrinsic Property ◽

Geometric Interpretation ◽

Transformation Matrix ◽

Dual Number ◽

Normal Body ◽

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 screw transformation matrix in the form of either 3×3 dual number matrix or 6×6 transformation matrix that is executed with rotation and translation. This paper investigates characteristics of the screw transformation matrix, and decomposes the dual part of the transformation matrix into the part with an equivalent translation due to the effect of moving rotation axis and the part resulting from a pure translation. New results are presented and new formulae are generated. The analysis further reveals two new traces of the transformation matrix and presents the relation between the screw transformation matrix and the instantaneous screw, leading to the understanding of Chasles’ motion embedded in a normal body transformation. An algebraic and geometric interpretation of the screw transformation matrix is thus given, presenting an intrinsic property of the screw transformation matrix in relation to the finite screw. The paper ends with a case study to verify the theory and illustrate the principle.

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Series Solution for Finite Displacement of Single-Loop Spatial Linkages

Journal of Mechanisms and Robotics ◽

10.1115/1.4006193 ◽

2012 ◽

Vol 4 (2) ◽

Cited By ~ 7

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.

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An Improved Principal Coordinate Frame for use with Spatial Rigid Body Displacement Metrics

Advances in Mechanism and Machine Science - Mechanisms and Machine Science ◽

10.1007/978-3-030-20131-9_32 ◽

2019 ◽

pp. 319-328

Author(s):

Pierre Larochelle ◽

Venkatesh Venkataramanujam

Keyword(s):

Rigid Body ◽

Coordinate Frame ◽

Rigid Body Displacement

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Rigid body displacement constrained deformation

2009 International Conference on Mechatronics and Automation ◽

10.1109/icma.2009.5246794 ◽

2009 ◽

Author(s):

Qi Huang ◽

Chunning Jin ◽

Ping Hu

Keyword(s):

Rigid Body ◽

Rigid Body Displacement ◽

Constrained Deformation

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Rigid-body displacements of curved elements in the analysis of shells by the matrix- displacement method.

AIAA Journal ◽

10.2514/3.4246 ◽

1967 ◽

Vol 5 (8) ◽

pp. 1525-1527 ◽

Cited By ~ 27

Author(s):

WALTER E. HAISLER ◽

JAMES A. STRICKLIN

Keyword(s):

Rigid Body ◽

Displacement Method ◽

The Matrix ◽

Curved Elements

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Is there a characteristic length of a rigid-body displacement?

Mechanism and Machine Theory ◽

10.1016/j.mechmachtheory.2006.03.010 ◽

2006 ◽

Vol 41 (8) ◽

pp. 884-896 ◽

Cited By ~ 57

Author(s):

Jorge Angeles

Keyword(s):

Rigid Body ◽

Characteristic Length ◽

Rigid Body Displacement

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

Proceedings of the Institution of Mechanical Engineers Part K Journal of Multi-body Dynamics ◽

10.1177/1464419319856796 ◽

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.

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