A general approach for generating kinetostatic models for planar flexure-based compliant mechanisms using matrix representation

Highlights

• The matrix representation of flexure-based compliant mechanisms is studied.

• Virtual flexure hinges are introduced to replenish the matrix representation.

• A method for automatically generating kinetostatic models for FCMs is presented.

Abstract

Kinetostatic modeling of a flexure-based compliant mechanism (FCM) is always a complicated and error-prone process because it involves not only the formulation of the geometric compatibility but also the formulation of the load equilibrium of the mechanism. In this paper, we propose a general approach for automatically generating kinetostatic models for FCMs based on their matrix representation. The approach utilizes virtual flexure hinges, link-flexure incidence matrices and path matrices to unify and simplify the formulation of the kinetostatic equations of FCMs. Four case studies are provided to demonstrate the use of the proposed method for modeling various FCMs.

Introduction

Flexure-based compliant mechanisms (FCMs) [1] utilize flexure hinges as their flexible members to achieve their mobility. The advantages of such mechanisms over traditional rigid-body mechanisms include elimination of friction, backlash and assembly, reduction in manufacturing cost, and increase of precision. They have been employed in a wide range of industrial applications where high precision within a limited range of motion is required.

However, modeling of FCMs is more challenging than that of traditional rigid-body mechanisms because not only loop-closure relations, but also the load-deflection relations of the flexure hinges and the load equilibrium relations within the mechanisms need to be considered. For example, the kinetostatic modeling of the tensural displacement amplifier presented in Ref. [2] involves 16 loop-closure equations, 24 load-deflection equations of flexure hinges and 19 load equilibrium equations (only the right half of the amplifier was modeled). The large number of kinetostatic equations makes the modeling process difficult and error-prone. Even a minor mistake will yield entirely false results. Although the pseudo-rigid-body model (which treats each flexure hinge as an ideal articulated joint attached with a linear torsional spring) [3], [4] can simplify the kinetostatic modeling of FCMs, it suffers from inaccuracy due to neglecting the axial and the transverse deflections of the flexure hinges [5], [6].

Many researchers have conducted research on modeling FCMs, for example, Her and Chang [7] provided an analytical scheme for the displacement analysis of a micropositioning stage based on linearization of the geometric constraint equations of the stage structure, Clark et al. [8] proposed a novel flexure-based pure-rotation stage which was optimized based on its simplified pseudo-rigid-body model, Choi et al. [9] analyzed the amplification ratio of the traditional bridge-type amplification mechanism by assuming all the members are compliant, Carricato et al. [10] established the loop-closure and equilibrium equations of a two degrees-of-freedom planar flexure-based mechanism, Pham et al. [11] studied a 2-DOF parallel mechanism by conducting its corresponding kinematic and static modeling to determine the kinematics properties, workspace, holding forces of the actuators and the reaction forces at the flexure hinges, Yong and Lu [12] applied the matrix method to deduce the kinetostatic model for a flexure-based 3-RRR compliant micro-motion stage, Park and Yang [13] obtained the mathematical model for an ultra-precision positioning system with six-degrees of freedom, and Wang and Xu [14], and Xu [15], [16] proposed flexure-based mechanisms for micromanipulation and micropositioning and developed corresponding kinetostatic models to aid parametric design, to name a new.

Generally speaking, these researchers only considered the modeling of specific FCMs, and didn’t provide an approach that can reduce modeling difficulties for general FCMs. That is to say, there still lacks general guidelines that a designer can follow to conduct kinetostatic modeling of various FCMs.

To complement this lack, this paper presents an approach that can automatically generate kinetostatic models for FCMs based on their matrix representation. Fig. 1 shows the flowchart of this generating process. It should be pointed out that a new concept called “virtual flexure hinge” is introduced to replenish the matrix representation of FCMs in order to unify the formulation of the load equilibrium equations. The approach can be easily applied to different FCMs.

The rest of this paper is organized as follows: In Section 2, the concept of virtual flexure hinges and the matrix representation for planar flexure-based compliant mechanisms are presented. Section 3 provides the construction of the kinetostatic equations of planar flexure-based compliant mechanisms by using matrix representation. Four case studies are given in Section 4 to validate the feasibility of proposed approach for modeling planar FCMs.