Parallel Kinematic Machines pp 69-84 | Cite as

# Kinematic Analysis of Parallel Manipulators

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## Abstract

A method for the kinematic analysis of parallel manipulators, based on Denavit-Hartenberg formalism, is proposed. This formulation allows to model separately each joint-link train (defined as the consecutive structural elements starting from the origin of the reference frame associated with the base, to the origin of the reference frame of the mobile platform through one of the links). This methodology has the main advantage of determining all the generalized variables characterizing the model (in order to evaluate the mechanical limits of the passive joints, their characteristic parameters), and not only those associated with the active joints. In this way, it is possible to model manufacturing and assembly errors in the mechanical components, kinematics errors in the actuators.

Several parallel machinesm according to the adopted classification, have been modeled mapping the stiffness index above the workspace.

The *workspace* is defined as the Boolean function *w*: ℜ^{6} → {0,1} of all the reachable points (*x, y, z, ψ, θ, ϕ*) considering physical constraints on joints and legs. The set of reachable poses defines the workspace as a connected set of points. The stiffness index was computed considering the scalar value given by the inverse of the condition number *k* of the Jacobian matrix. This number gives the ratio between the maximum value of the stiffness computed in the direction of the maximum stiffness and the minimum value of stiffness computed in the direction of the minimum stiffness for each point of the closed set of points belonging to the workspace. When 1/*k* is close to zero it means either that the machine is near a singular configuration or that the stiffness significantly varies along different directions. In both cases, and supposing the manipulator is used for machining operations (i.e.: as a milling machine), the further is 1/*k* from the unit the worse is the behavior of the machine. This mapping so can be used for a rough evaluation of the *usable workspace*.

## Keywords

Parallel Manipulator Kinematic Analysis Mobile Platform Stiffness Index Spherical Joint## Preview

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