Application of Matroid Method in Kinematic Analysis of Parallel Axes Epicyclic Gear Trains

Abstract

A novel method for kinematic analysis of parallel-axes epicyclic gear trains is presented, called the incidence and transfer method, which uses the incidence matrices associated with the edge-oriented graph associated to the mechanism and the transfer joints (teeth contact joints). Relative to such joints, a set of independent equations can be generated for calculating the angular positions, velocities, and accelerations. Complete kinematic equations are obtained in matrix form using a base of circuits from a cycle matroid. The analysis uses the relationships between the number of mobile links, number of joints, and number of circuits in the base of circuits, together with the Latin matrix (whose entries are function of the absolute values of the partial gear ratios of the transmission). Calculating the rank of the Latin matrix can identify singularities, like groups of gears that rotate as a whole. Relationships between the output and input angular velocities and accelerations are then determined in a matrix-based approach without using any derivative operations. The proposed method has general applicability and can be employed for systems with any number of gears and degrees of freedom, as illustrated by the numerical examples presented.

Kinematic analysis of geared robotic mechanism using Matroid and T–T graph methods

“…k = t + c and t = n). The following labeling, which is assigned to links and joints of sample mechanism, is used in the Matroid method [31]:…”

“…In order to overcome the weaknesses of the methods developed by Tsai and Tokad, both methods were combined and the so called T-T Graph method was introduced [29]. Most recently, Talpasanu [30] and Talpasanu et al [31][32][33] developed the Matroid method for the kinematic analysis of geared mechanisms based on directed graph as well.…”

“…In recent years, a number of methods concerning the general approach to kinematic analysis [1][2][3][4][5][6][7][8][9][10], the structure of geared mechanisms [11][12][13][14] and the kinematic and dynamic analyses of geared systems [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33] have been developed.…”

Kinematic Singularities of a 3-DoF Planar Geared Robot Manipulator

“…This approach was later developed by Tsai [9], Hsu and Lam [4]. In order to enhance the computational effectiveness of the method, Talpasanu et al [6,7] refined and to some extent recast the approach, introducing the 'incidence and transfer method' that uses the cycle matroid of the mechanism's directed graph. A comparison of Talpasanu's method with that of Tsai-Tokad was made in [1].…”

“…Special choices of these parameters result in interesting simplifications of the kinematic mapping. An explicit expression for the mapping can be derived using the 'matroid method' of Talpasanu et al [7]. We use this relatively simple mechanism to illustrate singularity analysis for geared mechanisms.…”