Carlo Innocenti

Direct position analysis of the Stewart platform mechanism

Abstract The direct position analysis of the Stewart platform mechanism (SPM), that is to find position and orientation of the platform when a set of actuator displacements is given, has been performed in closed-form. The analysis has been carried out by referring to a general kinematic model which has been derived on the basis of the characterization presented by Stewart. All the different SPM arrangements can be analysed in an unified way by this model. The analysis leads to a 16th order polynomial equation in one unknown from which 16 different positions and orientations of the platform can be derived. This new theoretical result is confirmed by numerical examples.

Echelon form solution of direct kinematics for the general fully-parallel spherical wrist

Abstract This paper presents the echelon form direct position analysis of a class of fully in-parallel actuated mechanisms for the orientation of a rigid body with a fixed point. The mechanisms have a structure which is the most general one for manipulator spherical wrists with three degrees of freedom and fully-parallel arrangement. The analysis results in a two-equation system in echelon form; the first equation is of 8th order and the remaining is linear. As a consequence, when a set of actuator displacements is given, eight configurations of the mechanism are possible. A numerical example confirms the new theoretical result.

Position analysis in analytical form of the 7-link assur kinematic chain featuring one ternary link connected to ternary links only

The paper presents the analytical-form position analysis of one of the three existing 7-link Assur kinematic chains, precisely the one featuring three binary links and four ternary links, one of the ternary links being joined to the other three. The analysis results in a 14th order polynomial equation in one unknown whose 14 roots correspond, in the complex domain, to as many assembly configurations. The contribution widens the set of planar linkages whose kinematic analysis can be thoroughly and exhaustively solved. A case study is finally provided.

Polynomial solution to the position analysis of the 7-link assur kinematic chain with one quaternary link

Following a couple of previous papers devoted to the position analysis of two different 7-link Assur kinematic chains, this paper presents the algebraic-form position analysis of the remaining 7-link Assur kinematic chain, which can be distinguished from the others because it alone features one quaternary link. The position analysis, devoted at determining all possible assembly configurations of the considered Assur kinematic chain, is performed by first devising a system of two algebraic equations in two unknowns. After dialytic elimination, a final polynomial equation of eighteenth order is found whose solutions provide, in the complex field, 18 assembly configurations for the examined Assur kinematic chain. As a corollary, it can also be stated that two coupler point curves drawn by two distinct four-bar linkages intersect each other at 18 real points at most. Finally, a numerical example shows application of the new theoretical results.

Analytical Determination of the Intersections of Two Coupler-Point Curves Generated by Two Four-Bar Linkages

The paper presents the analytical determination of the intersections of two couplerpoint curves generated by two distinct four-bar linkages. After devising a suitable set of three compatibility equations, and performing algebraic elimination, a final polynomial equation of eighteenth order with only one unknown is obtained whose roots represent the sought-for intersections. As a result, two coupler-point curves cross each other, in the complex field, at eighteen points. The contribution is aimed at further delineating the analytical properties of four-bar linkage coupler-point curves. A case study is finally reported.

Analtical- form direct kinematicefor the second scheme of a 5–5 general-geometry fully parallel manipulator

This article presents the direct kinematics of a 6 degrees of freedom fully parallel manipulator that features five connection points on both base and platform. The considered leg arrangement is one of the two possible, the other having been referred to in a previous paper. The base and platform have a general geometry, i.e., the connection points are not bound to lie on planes. The direct kinematics is solved in analytical form, which allows determination of every possible solution. After a suitable set of five equations in five unknowns has been laid down, and an original elimination procedure applied, a final polynomial equation of the 24th order is found whose 24 solutions correspond to as many assembly configurations for the considered fully parallel manipulator. A case study is reported.

Analysis of meshing of beveloid gears

This paper analyses the meshing of a pair of beveloid gears mounted on intersecting or skew shafts. Precisely, the dependence of the backlash on the gear geometry and the shaft relative position are investigated. The study is preceded with the presentation of a new, basic nomenclature to describe the kinematically relevant dimensions of beveloid gears. Finally, a numerical example is reported.

Direct Kinematics in Analytical Form of a General Geometry 5–4 Fully-Parallel Manipulator

This paper presents the analytical form solution of the direct position analysis for a fully-parallel manipulator that features e base and a platform connected by six adjustable-length legs whose extremities meet the base and the platform respectively at four and five points. When the leg lengths are given, the manipulator becomes a statically determined structure that can be assembled in different configurations. The direct position analysis aims at solving all possible configurations. In the paper, the analysis is first reduced to the solution of a three non-linear equation system in three unknowns, then two unwanted unknowns are eliminated thus obtaining a final 24th order polynomial equation in only one unknown. The twenty-four roots of the equation provide as many configurations of the 5–4 structure in the complex field. Numerical examples support the new theoretical findings.

Symbolic-Form Forward Kinematics of a 5-4 Fully-Parallel Manipulator

Fully parallel manipulators feature two rigid bodies, base and platform, each other connected by six binary links with spherical pairs at the extremities. Links have adjustable lengths to control position and orientation of the platform with respect to the base. The forward position analysis solves for the location of the platform once the link lengths are given. The paper presents the forward position analysis in symbolic form of a 5-4 fully-parallel manipulator having general geometry. By a new procedure a kinematic model is devised that leads to a system of three closure equations in three unknowns. As a result of a single-step elimination process, a final 32nd order polynomial equation in only one unknown is obtained. Thirty-two configurations of the manipulator are hence possible in the complex field. A case study validates the new theoretical results.