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.

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.

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.

Projective Constraint Stabilization for a Power Series Forward Dynamics Solver

2013 ◽  
Vol 135 (3) ◽  
Author(s):  
Paul Milenkovic
Keyword(s):  
Power Series ◽  
Kinematic Chain ◽  
Power Series Solution ◽  
Forward Dynamics ◽  
Time Step ◽  
Kinematic Constraints ◽  
Motion Trajectories ◽  
Time Histories ◽  
Higher Derivatives ◽  
High Degree

A power series expression for the forward dynamics of a closed kinematic chain provides an explicit time-step update of the system state. The resulting numerical differential equation solver applies kinematic constraints to the power series terms for acceleration and higher derivatives of motion. Integrating acceleration determines velocity and position time histories that approximate the constraints to a high degree of precision when using a high order of the expansion. When high precision is not required, a lower order achieves shorter computation times, but that condition results in violation of the constraints in the absence of any correction. Projecting the velocities and positions onto the constraint manifold after each time step produces step changes. This paper determines which choices of linear subspace for this projection give step changes that are equal to the residues of truncating the power series solution for the kinematic portion of the problem. The limit of that power series gives position and velocity time histories that approximate the dynamics while giving an exact kinematic solution. Thus projection onto the constraints in this procedure determines sample values of an underlying solution for the motion trajectories, where that underlying solution is continuous in both velocity and position and also satisfies the kinematic constraints at all times. This property is confirmed by numerical simulation of a Clemens constant-velocity coupling.

A Novel Method for Constructing Multi-Mode Deployable Polyhedron Mechanisms Using Symmetric Spatial RRR Compositional Units

2018 ◽  
Author(s):  
Jieyu Wang ◽  
Xianwen Kong
Keyword(s):  
Kinematic Chain ◽  
Degree Of Freedom ◽  
Single Loop ◽  
Kinematic Chains ◽  
The Third ◽  
Motion Modes ◽  
3D Printed ◽  
Novel Method ◽  
Multi Mode ◽  
Motion Mode

A novel construction method is proposed to construct multimode deployable polyhedron mechanisms (DPMs) using symmetric spatial RRR compositional units, a serial kinematic chain in which the axes of the first and the third revolute (R) joints are perpendicular to the axis of the second R joint. Single-loop deployable linkages are first constructed using RRR units and are further assembled into polyhedron mechanisms by connecting single-loop kinematic chains using RRR units. The proposed mechanisms are over-constrained and can be deployed through two approaches. The prism mechanism constructed using two Bricard linkages and six RRR limbs has one degree-of-freedom (DOF). When removing three of the RRR limbs, the mechanism obtains one additional 1-DOF motion mode. The DPMs based on 8R and 10R linkages also have multiple modes, and several mechanisms are variable-DOF mechanisms. The DPMs can switch among different motion modes through transition positions. Prototypes are 3D-printed to verify the feasibility of the mechanisms.

scholarly journals Fractional-Order Logistic Differential Equation with Mittag–Leffler-Type Kernel

2021 ◽  
Vol 5 (4) ◽  
pp. 273
Author(s):  
Iván Area ◽  
Juan J. Nieto
Keyword(s):  
Differential Equation ◽  
Power Series ◽  
Differential Equations ◽  
Formal Power Series ◽  
Fractional Order ◽  
Numerical Approximations ◽  
Formal Power

In this paper, we consider the Prabhakar fractional logistic differential equation. By using appropriate limit relations, we recover some other logistic differential equations, giving representations of each solution in terms of a formal power series. Some numerical approximations are implemented by using truncated series.

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