Series Solution for Finite Displacement of Planar Four-Bar Linkages

2010 ◽  
Vol 3 (1) ◽  
Author(s):  
Paul Milenkovic
Keyword(s):  
Power Series ◽  
Rigid Body ◽  
Series Solution ◽  
Kinematic Chain ◽  
Linear Constraint ◽  
Time Varying ◽  
Convergence Properties ◽  
Revolute Joint ◽  
Power Series Expansions ◽  
Instantaneous Screw

A time-varying instantaneous screw characterizes the motion of a rigid body. The kinematic differential equation expresses the path taken by any point on that rigid body in terms of this screw. Therefore, when a revolute joint is attached to a moving link in a planar kinematic chain, the path taken by the center of that revolute joint is the solution to such an equation. The instantaneous screw of a link in that chain is in turn determined by the action of the joints connecting that link to ground, where the contribution of each joint to that instantaneous screw is determined by its actuation rate and center point. Substituting power series expansions for joint rates into the kinematic differential equations for joint centers, and expressing loop closure as a linear constraint on the instantaneous screws of the links, a recurrence relation is established that solves for the coefficients in those power series. The resulting solution is applied to determine the equilibrium pendulum tilt of the United Aircraft TurboTrain. Comparing that power series approximation with an exact kinematic analysis shows convergence properties of the series.

Solution of the Forward Dynamics of a Single-loop Linkage Using Power Series

2011 ◽  
Vol 133 (6) ◽  
Author(s):  
Paul Milenkovic
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Kinematic Chain ◽  
Linear Constraint ◽  
Rate Coefficients ◽  
Forward Dynamics ◽  
Single Loop ◽  
Single Matrix ◽  
Instantaneous Screw ◽  
Kinematic Solution

The kinematic differential equations express the paths taken by points, lines, and coordinate frames attached to a rigid body in terms of the instantaneous screw for the motion of that body. Such differential equations are linear but with a time-varying coefficient and hence solvable by power series. A single-loop kinematic chain may be expressed by a system of such differential equations subject to a linear constraint. A single matrix factorization followed by a sequence of substitutions of linear-system right-hand-side terms determines successive orders of the joint rate coefficients in the kinematic solution for this mechanism. The present work extends this procedure to the forward dynamics problem, applying it to a Clemens constant-velocity coupling expressed as a spatial 9R closed kinematic chain.

scholarly journals A class of analytic pairs of conjugate functions in dimension three

2019 ◽  
Vol 19 (3) ◽  
pp. 421-432
Author(s):  
Paul Baird ◽  
Elsa Ghandour
Keyword(s):  
Power Series ◽  
Parameter Family ◽  
Series Expansions ◽  
Conjugate Functions ◽  
Convergence Properties ◽  
Entire Solutions ◽  
Power Series Expansions ◽  
Hopf Map

Abstract We exploit an ansatz in order to construct power series expansions for pairs of conjugate functions defined on domains of Euclidean 3-space. Convergence properties of the resulting series are investigated. Entire solutions which are not harmonic are found as well as a 2-parameter family of examples which contains the Hopf map.

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.

MODELING AND PRICING PRECIPITATION DERIVATIVES UNDER WEATHER FORECASTS

2016 ◽  
Vol 19 (07) ◽  
pp. 1650051 ◽  
Author(s):  
MARKUS HESS
Keyword(s):  
Wavelet Transform ◽  
Power Series ◽  
Stochastic Dynamics ◽  
Location Model ◽  
Series Expansions ◽  
Time Varying ◽  
Futures Prices ◽  
Weather Forecasts ◽  
Complex Power ◽  
Power Series Expansions

We propose a pure jump precipitation model embedded in an enlarged filtration framework accounting for weather forecasts. Under different anticipative approaches, we define precipitation swap/futures prices and also introduce the notion of an “information premium”. In contrast to some other models in the literature, our forward-looking swap price representations admit time-varying stochastic dynamics. In these setups, swap price processes under the physical and risk-neutral measure turn out to be indistinguishable. We also consider an extended multi-location model measuring precipitation in several locations. In order to price options on precipitation derivatives under weather forecasts modeled by enlarged filtrations, we develop customized approximation procedures involving complex power series expansions and wavelet transform techniques.

Optimal algebra and power series solution of fractional Black-Scholes pricing model

2021 ◽  
Vol 25 (8) ◽  
pp. 6075-6082
Author(s):  
Hemanta Mandal ◽  
B. Bira ◽  
D. Zeidan
Keyword(s):  
Power Series ◽  
Series Solution ◽  
Power Series Solution ◽  
Pricing Model ◽  
Black Scholes
Sign in / Sign up

Export Citation Format

Share Document