Effect of Degree-of-Symmetry on Kinetostatic Characteristics of Flexure Mechanisms: A Comparative Case Study
- Xiao-Bing He^{1},
- Jing-Jun Yu^{1}Email author,
- Wan-Wan Zhang^{1} and
- Guang-Bo Hao^{2}
https://doi.org/10.1186/s10033-018-0235-4
© The Author(s) 2018
Received: 28 June 2017
Accepted: 16 April 2018
Published: 25 April 2018
Abstract
The current research of kinetostatic characteristics in flexure mechanisms mainly focus on the improvement of accuracy. To reduce or eliminate the parasitic motion is considered as an approach by using the common knowledge of symmetry. However, there is no study on designing the flexure mechanisms with symmetrical features as many as possible for better kinetostatic performance, when considering the resulting cost by the symmetry. In this paper, the concept of degree of symmetry (DoS) is proposed for the first time, which is committed to symmetry design in the phase of conceptual design. A class of flexure mechanisms with 0-DoS, 1-DoS, 2-DoS and 3-DoS are synthesized respectively based on the Freedom and Constraint Topology method. Their overall compliance matrices in an analytical form formulated within the framework of the screw theory are used to analyze and compare the effect of different number of DoS on the kinetostatic characteristics for flexure mechanisms. The finite element analysis (FEA) simulations are implemented to verify the analytical results. These results show that the higher the DoS is, the smaller the parasitic motion error will be. The flexure model with 3-DoS is optimized according to the overall compliance matrix and then tested by using the FEA simulation. The testing result shows that with the best combination parameters, the parasitic motion error for 3-DoS mechanism is almost eliminated. This research introduces a design principle which can alleviate the unwanted parasitic motion for better accuracy.
Keywords
1 Introduction
Nature can always inspire humans to create various useful devices/instruments. From observing the natural structures and movements of living organisms, mechanical designers regard strategic use of symmetry as a powerful design tool. The symmetrical design in flexure (aka compliant systems) can be found everywhere in nature, from a mirror-symmetry bird wing in the macro world to a large variety of axis-symmetry protein structures in the micro world. Apart from the facts in the natural world, symmetrical geometry also exhibits a wide use in the artificial world. In compliant mechanisms [1], symmetry design is important to guarantee the stability the overall desired performances in which symmetry creates balance, harmony, order, and aesthetically pleasing results [2].
Flexure mechanisms, with the inherent advantages of selective compliance characteristics [3, 4], have been widely used in the field of precision engineering, such as scientific instruments, optical alignment devices, micro-/nano-positioning stages, precision manufacturing machines [5]. These flexure mechanisms are typically hard to design compared to their rigid counterparts. Because the accuracy of flexure mechanisms is highly sensitive to many external disturbances, such as vibration and thermal variations, and also some intrinsic factors, such as material property and mechanism configuration.
In order to formulate an index for accuracy, parasitic motion is defined as any undesirable motion along the constraint directions of a mechanism [6]. There are several methods to reduce and even to eliminate the parasitic motion. The first method is to tune the structural parameters and material properties without changing the type of flexure mechanisms. Li et al. [7, 8] analyzed a family of [PP]S parallel mechanisms and took the 3-PRS parallel mechanism as an example to reveal the relationship between structural parameters and parasitic motion, and then showed the necessary structural condition for a 3-PRS parallel mechanism without parasitic motion. However, it is rather difficult to eliminate the parasitic motion by optimizing the geometrical parameters. The second one is designing a parasitic-motion compensation module, as done in linear-motion flexure mechanisms [9]. Trease et al. [10] and Cannon et al. [11] constructed a linear-motion flexure mechanism with higher accuracy by mirroring two double-parallelogram flexure modules. A class of compliant Roberts mechanisms can also be combined both in serial and parallel to compensate for the parasitic motion [12]. In respect to the multi-axis motion mechanism, an extended parasitic motion compensation approach that characterizes 3D flexure deformations with twists and parasitic error with compliance elements is proposed to synthesize multi-axis flexure mechanism [13]. It is noted that symmetry design is essentially a special case of the latter method, which is to design a system free of parasitic motion directly. Several practical full-symmetrical compliant mechanisms have been studied by Hao et al. [14–16], in which tri-symmetrical planar structures enabled three large-range out-of-plane motions.
This paper aims to explore the method of parasitic motion free design with aid of the knowledge of symmetry. A new term named as the Degree of Symmetry (DoS) is coined for advancing systematical design, with a particular emphasis on type synthesis of flexure mechanisms. It can be argued that synthesis of flexure mechanisms is more difficult than that of their rigid-body counterparts. Therefore, the attempt in this paper is to design a group of specific flexure mechanisms with one rotational degree-of-freedom (DOF) and one translational DOF that are parallel to each other. Each one is composed of several beams uniformly contained in two planes. Based on the known methodology, in this paper a class of flexure mechanisms with different DoS are to be synthesized, and further to be used to identify the relationship between the number of symmetrical planes and their kinetostatic performance characteristics.
In fact, there exist a dozen of literatures about symmetry design, but few design concerns towards how much the symmetry obtained can provide better performance. In these prior art, researchers generally design a class of symmetrical flexure mechanisms firstly, and then analyze their parasitic motions, and finally draw a conclusion that the symmetry design can effectively improve accuracy and other performances. Is it necessary to design the flexure mechanisms with symmetrical planes, axes, or points as many as possible for reducing or even eliminating the parasitic motion, when considering the resulting cost by the symmetry? This paper will focus on the effect of the DoS on the kinetostatic characteristic of flexure mechanisms with a comparative case study.
The rest of this paper is organized as follows. Section 2 provides an introduction to the Freedom and Constraint Topology (FACT) method as well as the equivalent constraint model of selected flexure primitive within the framework of the screw theory. A group of 2-DOF flexure mechanisms with X-DoS are synthesized by using the graphic method (FACT) in Section 3. Their overall compliance matrices for evaluating parasitic motions are formulated, followed by the FEA simulation in comparisons with the analytical models in Section 4. Based on the forehead context, Section 5 discusses the effect of DoS on the kinetostatic performance. Finally, conclusions are drawn (Additional file 1).
2 Theoretical Foundation
2.1 Definition of Degree-of-Symmetry
Symmetry is one of the most important of all properties in the identification of mechanisms. It is well known that symmetry is always described by reference to symmetry planes, axes and the center of symmetry. In this paper, the Degree-of-Symmetry (DoS) is specifically constrained with plane symmetry.
2.2 Type Synthesis Approach
As well known, some systematic approaches including the constraint-based design method [17] and the Freedom and Constraint Topology (FACT) method [18], have gained a great success in the design field of flexure mechanisms. Using the graphic FACT approach, which is based on the connection of screw theory with the constraint-based design theory in a geometrical way, is very powerful for designing simple cases. It has clear meaning to map geometrical entities such as lines and plane, to physical elements such as the compliant beams. Furthermore, all of them can be included in the chart of FACT, and modelled in the freedom spaces or constraint spaces. In this regard, a freedom space of a rigid body represents all of its allowable motion in space when subjecting to a specified constraint arrangement. While the constraint space represents all possible constraint arrangements in such a prescribed motion pattern.
What is more significant, the FACT approach can be completely embedded into the framework of screw theory, making the compliance matrix characterized by screw theory more powerful [19], since it offers critical geometric insight into various motion behavior of flexure mechanisms, including the metrics quantifying parasitic motion of flexure mechanisms [20].
2.3 Coordinate Transformation of Screws and Compliance Matrix
In an overall compliance matrix, the principle diagonal elements are always considered as the reference of rotational degrees of freedom about x, y and z-axes as well as the reference of translational degrees of freedom along x, y and z-axes [22]. Other non-principal diagonal compliance elements can be used to indicate the parasitic motion [23].
2.4 Equivalent Constraint Model of Flexures
In the family of flexure mechanisms, a beam is widely used as a basic flexure element both generating twist deformations and providing wrench constraints. In terms of the difference in profiles, the beams can be classified as notch-type ones (such as circular flexures) and uniform ones (such as wire flexures, plate flexures); straight ones and initial-curve ones; and slender ones (Euler–Bernoulli beams) and short ones (Timoshenko beams). Different profiles of these beams definitely lead to variance in freedom and constraint due to their compliance properties.
From the above results, it can be observed that this flexure offers several orders of magnitude higher stiffness along its axis compared with any other direction. We can thus conclude that a slender cylinder flexure, with ratio of the length to the radius being larger than 20, approximates an ideal wire flexure imposing a rigid constraint along its z axis and allowing other five DOFs. Therefore, the constraint model equivalent to this kind is a wire constraint.
3 Type Synthesis and Parasitic Error Analysis of X-DoS Flexure Mechanisms with Cylindrical Motion
3.1 Type Synthesis
In this section, we deal with the design of a group of flexure mechanisms characterized by the different number of DoS. The type synthesis approach we used is the graphic FACT, which is intuitively visible and preferable if the cases are not so complicated. Since the main purpose to this paper is on the relationship between the number of DoS and the kinetostatic performances, some simple parallel flexure mechanisms are appropriate enough in certain sense. Moreover, all flexure elements employed here are identical with a uniform circular cross section for convenience. Based on the knowledge of equivalent constraint model above, we built a general flexure mechanism formed by connecting a moving platform to a base one through wire flexure elements, as shown above in Figure 3, which will generate deformation on the moving platform when undergoing a generalized load.
Type synthesis of flexure mechanisms starts with specifying a freedom space. The objective is to find all beams with circular cross section in a parallel arrangement to perform the desired motion. The following will describe a general procedure for the type synthesis by taking the flexure mechanisms with cylindrical motion for instance.
3.2 Compliance Modelling
As sketched in Figure 9(c), the flexure mechanism characterized by two symmetrical planes is formed by connecting a moving platform to a fixed one through four identical circular wire flexures (r is the radius of cross sections) in parallel. Two parallel flexures (labeled with 1 and 2) intersect, with an angle 2θ, at the middle of one side edge on the moving platform, and span, with the distance a of two end points, on the fixed platform. The other two flexures are arranged similarly with an interval distance d. A global coordinate frame is located at the center of the moving platform, and the local coordinate frames are located at the center of each flexure element. All axes of the local/global coordinate frames are labeled in Figure 8.
Information for adjoint transformation
Beam number | Rotational matrix | Translational vector |
---|---|---|
1 | R_{1} = R_{ x }(θ) | t_{1} = (− d, − a/a2.2, − a cot (θ))^{T} |
2 | R_{2} = R_{ x }(− θ) | t_{2} = (− d, a/a2.2, − a cot (θ))^{T} |
3 | R_{3} = R_{ x }(θ) | t_{3} = (d, − a/a2.2, − a cot (θ))^{T} |
4 | R_{4} = R_{ x }(− θ) | t_{4} = (d, a/a2.2, − a cot (θ))^{T} |
By analyzing the principal diagonal elements of the matrix, the type of degree of freedom of this flexure mechanism can be easily demonstrated. In addition, other non-principal diagonal compliance entries can be considered as the reference of parasitic motion errors [21].
According to all the above overall compliance matrices, it can be concluded that the smaller parasitic motion errors occurs when there are more symmetric planes existing in the flexure mechanisms. In other words, the DoS of a flexure mechanism leads to some straightforward effect on its parasitic motion error. It is worth mentioning that all these flexure mechanisms have the same dominant motion pattern which consists of a rotation about the x axis (θ_{ x }) and a translation along the x axis (δ_{ x }).
3.3 Parasitic Motion Error Analysis
In fact, only the translational motion along x-axis is useful to perform desired function, the rest of entries are unwanted since they bring into some parasitic motions. In this case, the moving platform translates by δ_{ x } along x direction, companying with three other parasitic motions, which are two parasitic rotations about x and y axes, denoted by θ_{ x } and θ_{ y } respectively, and a parasitic translation along y axis, denoted by δ_{ y }. As known, parasitic motion is always detriment to the accuracy of a flexure mechanism, Clearly, this 0-DoS mechanism is not desired for practical application.
The preliminary motion for translation along x axis remains unchanged, but the number of parasitic motions decreases into two that are the parasitic rotations about y and z axes, denoted by θ_{ y } and θ_{ z }, respectively. Comparing with the former 0-DoS case, the presence of one symmetrical plane leads to better performance due to eliminating one type of parasitic motion.
For the third case, it has the widest popularity when designing 2-DOF cylindrical flexure mechanisms. There are a number of related literatures that address how to eliminate parasitic motion. Most of them claim that the introduction of a compensation module can effectively tradeoff the unwanted parasitic motion error. With the above-mentioned result, the flexure mechanism with two symmetric planes also has parasitic motion, but shows improvement when comparing with the 0-DoS and 1-DoS mechanisms. The resulting deformation of the 2-DoS mechanism is expressed as ξ_{2} = θ_{ y } + δ_{ x } = c_{24}f_{ x } + c_{44}f_{ x }.
As can be seen, the best design should be the last one, whose overall compliance matrix is ideally pure diagonal. This is because all symmetric planes result in the advantage of no parasitic motion. Though it is challenging to design a 3-DoS flexure mechanism from all selected motion type, the attempt to design flexure mechanisms with maximum degree of symmetry is still meaningful.
4 FEA Simulation Verification
In this section, a series of finite element analysis (FEA) simulations are implemented for demonstrating the benefit in presence of more symmetrical planes in the flexure mechanism. The tool used is commercial package ANSYS 15.0, where SOLID-187 element is selected for all rigid platforms while BEAM-189 element is selected for all flexure beams which connecting the moving platform with the fixed platform.
Flexure mechanisms with different mobility and stiffness can be obtained by changing their geometrical parameter. The chosen material is Aluminum Alloy, whose Young’s modulus is E = 70 GPa, Poisson’s ratio is μ = 0.34, and the cross-section radius of all wire flexures in the mechanisms is r = 5 mm. To guarantee the ratio of the length to the radius to be larger than 40, the height between the moving platform and the fixed one should at least be 200 mm. Thus, the length of the platform, which is also the distance a as shown in Figure 8 is 60 mm. The interval distance d is also set to be 60 mm.
5 Optimal Parametric Design
Analyzing overall compliance matrix for different flexure mechanisms in Section 3, it is known that the resultant mechanism with three symmetric planes can lead to no parasitic motion theoretically because of its diagonal compliance matrix form. As a result, in this section we considerately concentrate on the optimization design for the 3-DoS flexure mechanism and find out the effect of design parameters on its compliance matrix, whose entries are considered as the reference of rational and translational DOF or DOC.
Based on the illustrated analysis above and in view of the design purpose for this paper, \(\theta = {\uppi \mathord{/ {\vphantom {\uppi 4}} \kern-0pt} 4}\) is selected to enable all compliance ratios to be large enough for better performance, so that we can obtain relatively high compliance in the specified DOF direction and relatively low compliance in the specified DOC direction.
Finally, the influence of the cross-section radius of beams is analyzed. Since the equivalent constraint model subjects to its ratio of the length to the radius, for the equivalent wire constraint model as used in this paper, there is no doubt that the smaller the radius of the beam is, the more ideal the wire flexure approximation is. Nevertheless, significantly reducing the radius of flexure beam is not economic because of the manufacturing cost.
Optimal parameters for 3-DoS flexure model
Beam orientation θ | Two end points distance a (mm) | Interval distance d (mm) | Radius of beam r (mm) |
---|---|---|---|
\(\theta = {\uppi \mathord{\left/ {\vphantom {\uppi 4}} \right. \kern-0pt} 4}\) | 200 | 200 | 5 |
6 Discussions
As mentioned in Section 1, there are mainly three methods to reduce or even eliminate the parasitic motion for a flexure mechanism. Compensation designs are always preferred as reported by numerous literatures, which can be simply constructed with high accuracy by mirroring two identical flexure modules. In a word, almost all existing flexure mechanisms use the principle of symmetry design and combinations of the homogenous modules to achieve the compensation for parasitic motion. In the design processing, it is better to generate as many DoS as possible from the very beginning based on the findings in this paper. With different DoS, we can further adopt certain methods to alleviate the unwanted parasitic motion.
In addition, design and synthesis of a class of flexure mechanisms with cylindrical motion in a compact but simple way is significant for further applications, such as the joint for realizing a snack-like robot’s three dimensions’ gaits [24]. Recently, a new version of robots called in-pipe inspection robots was proposed to investigate the internal space of the pipes (for detecting the cracks, leaks, etc.), where implementing non-destructive tests are commonly based on screw motion [25]. As known, in-pipe inspection robots are supposed to move fast and continuous with constant pitch of rate, and it is possible because of cylindrical locomotion with the required accuracy.
7 Conclusions
A class of flexure mechanisms with different number of DoS have been designed followed by discussing the effect of symmetrical geometry on their kinetostatic characteristics. These mechanisms with zero, one or more symmetric planes, are obtained from FACT method. Each flexure mechanism is composed of several identical beams distributed in two planes orthogonal to the motion direction. Analytical model for the overall compliance matrix has been derived within the framework of the screw theory. These models have been used to analyze the influences of different DoS on the parasitic motion. Moreover, the FEA simulations are carried out for verifying the analytical results. An optimal design with \(\theta = {\uppi \mathord{\left/ {\vphantom {\uppi 4}} \right. \kern-0pt} 4}\)and a = d = 200 mm has been obtained and simulated.
As a new concept, the DoS concentrates on symmetry design theory. Moreover, the comparative case study on designing a group of symmetrical flexure mechanisms with cylindrical motion is instructive to snack-like robots’ design.
Declarations
Authors’ Contributions
J-JY was in charge of the whole trial; X-BH, G-BH wrote the manuscript; W-WZ assisted with sampling and laboratory analyses. All authors read and approved the final manuscript.
Authors’ Information
Xiao-Bing He, born in 1994, is currently a master candidate at School of Mechanical Engineering and Automation, Beihang University, China. E-mail: 18811375101@163.com.
Jing-Jun Yu, born in 1974, is currently a professor at Beihang University, China. His research interests include mechanisms and robotics. Tel: +86–10–82313904; E-mail: jjyu@buaa.edu.cn.
Wan-Wan Zhang, born in 1994, is currently a master candidate at School of Mechanical Engineering and Automation, Beihang University, China. E-mail: 1076643918@qq.com.
Guang-Bo Hao, born in 1981, is currently a permanent full-time Lecturer in Mechanical Engineering at School of Engineering-Electrical and Electronic Engineering, University College Cork (UCC), Ireland. His research focuses on compliant mechanisms.
Competing Interests
The authors declare that they have no competing interests.
Ethics Approval and Consent to Participate
Not applicable.
Funding
Supported by National Natural Science Foundation of China (Grant No. 51575017)
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