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Strojniki vestnik Journal of Mechanical Engineering (SV-JME)

2011 Strojniki vestnik - Journal of Mechanical Engineering. All rights reserved. SV-JME is indexed / abstracted in: SCI-Expanded, Compendex, Inspec, ProQuest-CSA, SCOPUS, TEMA. The list of the remaining bases, in which SV-JME is indexed, is available on the website. The journal is subsidized by Slovenian Book Agency.

Strojniki vestnik - Journal of Mechanical Engineering is also available on http://www.sv-jme.eu, where you access also to papers supplements, such as simulations, etc.

Editor in ChiefVincenc ButalaUniversity of Ljubljana Faculty of Mechanical Engineering, Slovenia

Co-EditorBorut BuchmeisterUniversity of MariborFaculty of Mechanical Engineering, Slovenia

Technical EditorPika krabaUniversity of Ljubljana Faculty of Mechanical Engineering, Slovenia

Editorial OfficeUniversity of Ljubljana (UL)Faculty of Mechanical EngineeringSV-JMEAkereva 6, SI-1000 Ljubljana, SloveniaPhone: 386-(0)1-4771 137Fax: 386-(0)1-2518 567E-mail: [email protected]://www.sv-jme.eu

Founders and PublishersUniversity of Ljubljana (UL)Faculty of Mechanical Engineering, Slovenia

University of Maribor (UM)Faculty of Mechanical Engineering, Slovenia

Association of Mechanical Engineers of Slovenia

Chamber of Commerce and Industry of SloveniaMetal Processing Industry Association

International Editorial BoardKoshi Adachi, Graduate School of Engineering,Tohoku University, JapanBikramjit Basu, Indian Institute of Technology, Kanpur, IndiaAnton Bergant, Litostroj Power, Slovenia Franci u, UM, Faculty of Mech. Engineering, SloveniaNarendra B. Dahotre, University of Tennessee, Knoxville, USAMatija Fajdiga, UL, Faculty of Mech. Engineering, SloveniaImre Felde, Bay Zoltan Inst. for Mater. Sci. and Techn., HungaryJoe Flaker, UM, Faculty of Mech. Engineering, SloveniaBernard Frankovi, Faculty of Engineering Rijeka, CroatiaJanez Grum, UL, Faculty of Mech. Engineering, SloveniaImre Horvath, Delft University of Technology, NetherlandsJulius Kaplunov, Brunel University, West London, UKMilan Kljajin, J.J. Strossmayer University of Osijek, CroatiaJanez Kopa, UL, Faculty of Mech. Engineering, SloveniaFranc Kosel, UL, Faculty of Mech. Engineering, SloveniaThomas Lbben, University of Bremen, GermanyJanez Moina, UL, Faculty of Mech. Engineering, SloveniaMiroslav Planak, University of Novi Sad, SerbiaBrian Prasad, California Institute of Technology, Pasadena, USABernd Sauer, University of Kaiserlautern, GermanyBrane irok, UL, Faculty of Mech. Engineering, SloveniaLeopold kerget, UM, Faculty of Mech. Engineering, SloveniaGeorge E. Totten, Portland State University, USANikos C. Tsourveloudis, Technical University of Crete, GreeceToma Udiljak, University of Zagreb, CroatiaArkady Voloshin, Lehigh University, Bethlehem, USA

President of Publishing CouncilJoe DuhovnikUL, Faculty of Mechanical Engineering, Slovenia

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General informationStrojniki vestnik Journal of Mechanical Engineering is published in 11 issues per year (July and August is a double issue).Institutional prices include print & online access: institutional subscription price and foreign subscription 100,00 (the price of a single issue is 10,00); general public subscription and student subscription 50,00 (the price of a single issue is 5,00). Prices are exclusive of tax. Delivery is included in the price. The recipient is responsible for paying any import duties or taxes. Legal title passes to the customer on dispatch by our distributor. Single issues from current and recent volumes are available at the current single-issue price. To order the journal, please complete the form on our website. For submissions, subscriptions and all other information please visit: http://en.sv-jme.eu/.

You can advertise on the inner and outer side of the back cover of the magazine. The authors of the published papers are invited to send photos or pictures with short explanation for cover content.We would like to thank the reviewers who have taken part in the peer-review process.

CoverColoured map of deviation between acquired geometry and CAD modelOptical geometry acquisition with 3D scanner GOM ATOS IIBackground: Sphere measurement with co-ordinate measuring machine ZEISS UMC 850

Image courtesy: Intelligent Manufacturing Systems Laboratory, Production Engineering Institute, Faculty of Mechanical Engineering, University of Maribor

ISSN 0039-2480

Aim and ScopeThe international journal publishes original and (mini)review articles covering the concepts of materials science, mechanics, kinematics, thermodynamics, energy and environment, mechatronics and robotics, fluid mechanics, tribology, cybernetics, industrial engineering and structural analysis. The journal follows new trends and progress proven practice in the mechanical engineering and also in the closely related sciences as are electrical, civil and process engineering, medicine, microbiology, ecology, agriculture, transport systems, aviation, and others, thus creating a unique forum for interdisciplinary or multidisciplinary dialogue.The international conferences selected papers are welcome for publishing as a special issue of SV-JME with invited co-editor(s).

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Strojniki vestnik - Journal of Mechanical Engineering 57(2011)10Contents

Contents

Strojniki vestnik - Journal of Mechanical Engineeringvolume 57, (2011), number 10

Ljubljana, October 2011ISSN 0039-2480

Published monthly

PapersBogdan Valentan, Toma Brajlih, Igor Drstvenek, Joe Bali: Development of a Part-

Complexity Evaluation Model for Application in Additive Fabrication Technologies 709Gang Cheng, Wei Gu, Jing-li Yu, Ping Tang: Overall Structure Calibration of 3-UCR Parallel

Manipulator Based on Quaternion Method 719Marin Gostimirovi, Milenko Sekuli, Janez Kopa, Pavel Kova: Optimal Control of

Workpiece Thermal State in Creep-Feed Grinding Using Inverse Heat Conduction Analysis 730

Roberto Alvarez, Rosario Domingo, Miguel Angel Sebastian: The Formation of Saw Toothed Chip in a Titanium Alloy: Influence of Constitutive Models 739

Matja Dvorek, Marko Hoevar, Brane irok, Nikola Holeek, Boin Donevski: The Influence of Airflow Inlet Region Modifications on the Local Efficiency of Natural Draft Cooling Tower Operation 750

Wang Jixin, Yao Mingyao, Yang Yonghai: Global Optimization of Lateral Performance for Two-Post ROPS Based on the Kriging Model and Genetic Algorithm 760

Ivan Demar, Matej Supej, Zmago Vidrih, Joef Duhovnik: Development of Prosthetic Knee for Alpine Skiing 768

Slavica Prvulovic, Dragisa Tolmac, Ljiljana Radovanovic: Application of Promethee-Gaia Methodology in the Choice of Systems for Drying Paltry-Seeds and Powder Materials 778

Instructions for Authors 785

Strojniki vestnik - Journal of Mechanical Engineering 57(2011)10, 709-718 Paper received: 09.03.2011DOI:10.5545/sv-jme.2010.057 Paper accepted: 15.07.2011

*Corr. Authors Address: University of Maribor, Faculty of Mechanical Engineering, Smetanova ulica 17, SI-2000 Maribor, Slovenia, [email protected] 709

Development of a Part-Complexity Evaluation Model for Application in Additive Fabrication Technologies

Valentan, B. Brajlih, T. Drstvenek, I. Bali, J.

Bogdan Valentan* Toma Brajlih Igor Drstvenek Joe BaliUniversity of Maribor, Faculty of Mechanical Engineering, Slovenia

With the rapid development and expansion of devices for the production of both traditional (cutting) procedures and layered technologies (also known as 3D printers or rapid prototyping/manufacturing), the question arises of how to find the appropriate production technology.

The article describes the basic features of the CAD output file STL. The STL file format is a widely-used file format developed for layered technologies and, as such, a basis for analysing and developing methods when determining the complexity of a model.

For the analyses of basic STL data, and complexity determination, several real-life models are presented. Actual manufacturing procedures suitable for the manufacture of unique products or serial production are presented, with accentuation towards layered technologies.

Technological test models are analysed based on the fundamental properties of manufacturing and certain manufacturing processes are chosen using complexity estimation. The results are comparable with those choices of manufacturing procedures on the basis of experts estimates. Complexity evaluation is also used for post-processing time determination for several layered technologies.2011 Journal of Mechanical Engineering. All rights reserved. Keywords: rapid prototyping, STL, complexity, shape, layered technology, technology selection

0 INTRODUCTION

The development of production technologies began in the early years of human society and then expanded during the industrial revolution. Since then technologies have been refined, new versions introduced and computer support enables partial-automation. Production was optimized [1] and [2] in terms of becoming cheaper, faster and better. However, technologies are still based on old knowledge in terms of removing materials, casting or forming. In addition, technological restrictions are still present when the complexity of a product plays a key role and the selection process is necessary prior to making the product. In order to realize a project in manufacturing, people with knowledge and experience are needed and a combination of several different technologies in complex everyday products is common [3] and [4].

No serious players from the field of conventional cutting processes were interested when the origin of layered technology was first introduced in the middle of 1980s. The technology was expensive, complex, inaccurate, slow, limited

by the dimensions and materials [5] and [6], but allowed the manufacturing of products in one piece, regardless of their complexity. In addition, technology had another advantage, which is that the time for the preparation of input parameters did not depend on the complexity of the product. At the beginning technology acquired the names rapid prototyping or 3D printing.

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Fig. 1. Continuous growth of machine sales [7]

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Strojniki vestnik - Journal of Mechanical Engineering 57(2011)10, 709-718

710 Valentan, B. Brajlih, T. Drstvenek, I. Bali, J.

revolutionary innovations, a lot of people refer to the new industrial revolution when talking about layered technologies.

1 INTRODUCING THE STL FILE FORMAT

The STL data format is a polygonal (mesh) format developed for the needs of the 3D Systems stereolithography equipment, which is one of the layered technologies. Stereolithography (U.S. Patent called Apparatus for Production of Three-Dimensional Objects by Stereolithography) was patented in 1984 and in 1986 the 3D Systems Company began to manufacture devices for prototype production.

During this time, the STL file format [8] to [10] was adopted by all other layered technologies and as such became the standard format. The reason for the popularity of the STL file format is in the simplicity of model description, as the STL format describes only the external surface of the 3D model without adding any other data. Some CAD attributes (points, lines, curves and layers) in other formats (WRML, DXF) can cause complications in non-standard formats records [11] and [12].

There are two formats of the STL file (binary and ASCII). The STL file format is supported by all modern CAD programs, although not all allow storage in both forms.

Since the STL file does not contain information about the real model size, some problems can appear such as unit change from cm to inch (SI replacement for the imperial system).

While exporting from the CAD to the STL file, part-resolution needs to be set. Export options are different in various CAD programs. The main parameters are set by the maximal allowed deviations between triangle mesh and the original CAD model, and the minimal allowed angle between two triangle edges.

When choosing model resolution, it is necessary to keep in mind that the resolution of the manufacturing device can be greater than the STL resolution, and a lack of resolution means a lower surface quality for the model produced [13] to [15] (Fig. 2). The problem is frequently set into a production line where an outside contractor cannot know the desired surface quality. By increasing the precisions of production technologies, this

problem shows the limitations of the STL format where, despite the most accurate resolution and large file size, a smooth surface cannot be achieved.

Fig. 2. Comparison between optimal and deficient choices for the export parameters of the STL file

2 TEST PARTS

Some test parts were needed for evaluating the complexity. The limitations of the STL file were taken into consideration. For a realistic comparison, all models were designed using the same CAD software (Catia V5) with the same settings for exporting STL files (3D accuracy of 0.01 and curve precision to 0.1). All models were checked for errors and verified by the Netfabb [16] program, and appropriately placed into the positive coordinates of their own coordinate systems. Orientation is set by experience since, in normal cases, the author starts modelling in one of the basic planes.

Fig. 3. Test models for complexity evaluation


711Development of a Part-Complexity Evaluation Model for Application in Additive Fabrication Technologies

Three models of basic geometric shapes can be found among the selected test models, and all the rest are the real user parts (Fig. 3). An important fact is the presumption that complexity does not depend only on the shape of the model, but also on the size. Two models are of the same shape but of different sizes.

3 BASIC PARAMETERS OF THE TEST MODELS

Basic STL file parameters were used for the experiment, such as the size of the binary file, number of triangles, the parts volume, the parts surface (area), and the volume of the block that captures the model. All parameters can be obtained by reading the STL file. Some properties can be calculated using basic mathematical equations or by some advanced software tools that allow visualization of the model and its properties (for example Netfabb).

3.1 Determination of Octants and Problematic Sections

A simple procedure for the basic manufacturing procedure determination was used due to difficulty in determining the basic form [17] to [19] shape recognition (statistically due to the loss of data when converting into STL format). Octants of each model were determined by distributing the parts external block into eight smaller blocks (octants) (Fig. 4).

Information about each octant, information on the overhangs and negative angles was gained from the vectors direction, which normally constitutes a problem with conventional cutting processes during manufacture.

Fig. 4. Octant distribution through the model

Table 1 shows the direction of a triangles normal vector that is problematic in each octant. It is enough to look at the sign of the triangles vector. If a problematic vector exists, the part cannot be made by aconventional procedure without an additional fixture or the use of special tools, but in most cases manufacturing using the conventional procedure (production in one piece) is impossible.

If there is a case where octants 1 and 3, 2 and 4, 5 and 7, 6 and 8 are vectors of opposite directions (flipped through the centreline of the model and the axis passing through the junction of octants 1, 2, 3 and 4 and continuing through junction of octants 5, 6, 7 and 8) the model can be suitable for rotary machining. All test models were analyzed for the vector directions in each octant. The results are presented in Table 2.

Table 1. Triangle normals that are problematicVector direction

Octant X Y Z1 + + +2 - + +3 - - +4 + - +5 + + -6 - + -7 - - -8 + - -

Table 2. Test part analysis and survey of problematic octants

OctantsModel 1 2 3 4 5 6 7 8

12345678910111213

Octants without problematic direction.

Octants with problematic direction.



4 COMPLEXITY DETERMINATION

The complexity, based on our own experiences with manufacturing processes, is determined first (Fig. 5), to obtain some sort of a reference, and then these results are compare with the calculated ones. This personal classification represents a reference for finding a suitable procedure when determining the complexity [20] and [21].

The complexity of the model can be deduced from information on the number of triangles (Fig. 6) (the number of triangles is directly related to the size of the file). An increased number of triangles represents a more complex model. This comparison does not take into account the increasing complexity, while decreasing the size of the model and all models should be created with the same export parameters.

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4.1 Advanced Evaluation of Model Complexity

Complexity on the basis of file size or the number of triangles presents us with some basic part complexity ideas, without the models size being taken into consideration. For example, with models 1 and 2, and 10 and 11, the calculated complexities should not be the same, since there are significant size differences between these parts. When reducing the size of a part, its complexity increases.

For accurate complexity calculation, the proportions of the three basic parameters of the model are needed: the models surface, the number of the models triangles, and the models volume.

model surface number of trianglesmodel square block volume

. (1)

The result of Eq. (1) is presented in the following graph (Fig. 7). It can be seen that part size plays a significant role regarding complexity determination. This relationship, taking into consideration part size is very similar to the complexity based on our own experiences.

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Fig. 7. Calculated complexity that can be compared with complexity given by experts

Calculated complexity of the model is comparable to experientially determined complexity (Fig. 5). Three models deviate from the average (7.8 and 13) all of them have varied surfaces and are problematic for manufacturing using conventional procedures. A significant impact also occurs when reducing the scale of a model which results in an increase in the complexity (models 1 and 10 versus models 2 and 11).



5 MANUFACTURING PROCEDURES

Todays manufacturing procedures are divided into conventional (cutting operations [22]) and layered technologies. In a case of conventional procedures - a set stock of raw material is depleted until a desired shape is obtained. The material can be removed by various procedures (turning, milling, grinding, cutting, local melting ...). Due to the need of comparison milling and turning were taken into consideration.

Layered technologies (often referred to as the technology for the rapid prototyping or 3D printing) are among the modern manufacturing procedures in which the material is no longer removed, but added. Technology allows us to produce realistic models of, until then unmanufacturable forms (Fig. 8) in one piece practically overnight. Several different technologies were developed [23] and [24] besides the first presented and patented procedure stereolithography.

Fig. 8. EOS Formiga P 100 Selective laser sintering (SLS) machine with some parts

Material application layer by layer is common to all technologies [25] and [26]. Technology produces individual 2D layers and by adding 2D layers on top of each other (a 3D product is formed). Important information from the survey is that some procedures support individual layers where necessary (overhangs or the spread of the model in a Z direction), as imposed support material, which is not the same as for models. In these cases, the form of the product affects the price, as well as building

time. In the end it should not be forgotten that all todays known layered-technologies need some post-process to obtain a final part. This can be a simple cleaning procedure, the removal of support material or even infiltration with some special material, which is time-consuming and expensive.

6 SELECTING THE OPTIMAL MANUFACTURING PROCEDURE

Several criteria should be taken into consideration when selecting appropriate manufacturing procedures: the desired material, the size of the product, the manufacturing time and cost of manufacture.

This paper focuses on product design, which means that at this stage some properties are ignored, such as materials, the properties of the materials, and product size, since material properties in the STL format are not given and size is not as problematic as there are different machines for producing different sizes parts. Production is highly dependent on the complexity of the product, especially when comparing cutting processes and layered techniques.

6.1 Selection on the Basis of Vector Direction in Each Octant

Table 3 presents the results of the selection on the basis of determining vector direction in each octant. Turning is chosen as the most affordable process when it comes to a rotary piece (models 3, 6 and 12), milling when it comes to the model without problematic vectors that define impossible tool angles and layered technology for all other models.

Layered technologies are divided into two subcategories: Layered technologies that for support use

raw modelling material. In this case, the support material can be reused and it does not represent an additional cost.

Layered technologies that use some additional support material at the part overhangs or have support from the model material, but that material should be removed after some treatment.



Table 3. Selecting the manufacturing procedure on an octant vector direction base

Turning MillingLT where support is

needed

LT where support is not needed

123456 Partly78910111213

Procedure is appropriate. Procedure is inappropriate.

6.2 Selection on the Basis of the Part Complexity

Fig. 7 shows the complexities of the test models. Unfortunately, complexity cannot provide us with information if turning is appropriate to be the right procedure for manufacturing. Manufacturing by turning is only possible for models 3 and 12 (Table 4), which have a relatively low ratio of fewer than 20,000 and do not stand out (Fig. 7). By imposing a limit of 20,000, models 2, 4, 5 and 11 are added to the selection, even if the manufacturing of these models in this case, is not possible.

Models, where the production with milling is impossible (7, 8 and 13) have extremely high ratio (over 120,000). Models 6 and 9 can be produced, but need an additional fixture during the manufacturing, or a complex 4 axes-production process. Models that are easy to produce have a low ratio (below 40,000). The limit for the milling process as the best possible selection was set at 100,000.

It can be seen that layered technologies are suitable for all models (from the point of manufacturing techniques, which is already a known fact), but when dividing technologies into those that need additional support material and those in which the support material is the same as the models material, it can be said that in the case

of a model with higher complexity, the production costs are higher. The limit between those two technologies was set to 50,000. Therefore, if complexity is below 50,000 any layered technology is suitable, when the complexity is beyond 50,000 layered technologies that reuse support material are more suitable.

Table 4. Selecting a manufacturing procedure on a part-complexity basis


needed


12345678910111213

Suitable. Suitable but bigger support material consumption. Unsuitable.

Models (1 and 10) are problematic to produce as they are resized to extremely small dimensions and can create certain problems for both processes. In the case of milling, the problem of clamping exists and in the case of layered technologies, resolution of the technology itself presents an obstacle to production.

6.3 Arrangements by Combining the Complexities of the Shapes and the Vector Direction, in Each Octant

By examining the results of both selection processes (one based on the vector direction in each octant and the other on the complexity of form), it can be established that, in some instances, each selection process can favour the process by which production is impossible. By combining the two methods those procedures that are inappropriate are eliminated. The results are presented in Table 5.



Table 5. Selecting the manufacturing procedure by combining part complexity with the octant vector direction


needed


12345678910111213

Suitable. Suitable but bigger support material consumption. Unsuitable.

STL File

Rotational Symmetry detection on an octant vector direction base

Choosing a manufacturing procedure on a part-complexity basis

Milling Turning LT where support is

needed

LT where support is

not needed

Additional Parameters not set in STL le (material, number of peaces, etc.)

Fig. 9. Diagram presents selecting procedure

In Fig. 9 first Turning is chosen if the part can be produced by turning. On the complexity base ruff decision between Milling and both layered procedures can be made, as presented. At the end fine selection with introduction of

parameters, that are not written in to STL file is made.

7 POST-PROCESSING TIME DETERMINATION BY EVALUATION OF

MODEL COMPLEXITY

The time for post-processing is problematic, especially from the perspective of determining the final production costs of the model. The price consists of construction material, hardware hour costs; fixed costs, energy cost, staff cost and the cost of post-processing. So far, assessment has been individually determined solely and empirically by using peoples experiences. With the introduction of complexity evaluation, the post-processing time can be calculated and planned during the production time.

The time for post-processing (Fig. 10) is distinguishable between different technologies (Fig. 11), therefore, it is necessary to determine the individual impacts of complexity on time for each layered technology.

In order to do that some parts need to be built and post-processing time for those parts need to be measured. On the basis of that data function:

complexity

post-processing time= X , (2)

can be derived and average value X calculated. For all the following parts time can be calculated:

post-processing time complexity=X

. (3)

Since post-processing time is based on manual work, this function can never be exact (especially, when more than one man is working at post-processing stage), but can give us a fair estimation on the time needed for that production step.

This procedure is suitable for time-determination in the cases when technologies that require manual removal of the support structure. This category includes: SLS, LOM, SLA, PolyJet, FSM, LENS, DMLS, SLM and EBM. In the cases of these processes, the removal of support material takes a certain time, depending on the complexity of the product itself. In the cases of SLS, LENS, DMLS, SLM and EBM, the removal of non-solidified base material is required. In the



case of LOM technology the removal of material surrounding the product is needed. SLA and FDM are building supports from base-material and these supports need to be broken off at the end. PolyJet has supports from special support material that needs special water-jet treatment at the end of the process. In processes in which the support material is dissolved in liquid or the model is infiltrated with special liquids, post-processing does not depend on the complexity of design.

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Fig. 10. Time of post-processing for test models made using the LOM procedure on SOLIDO SD 300 Pro; models 1 and 10 were too small to be

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Fig. 11. Waste-material removal in LOM

8 CONCLUSIONS

The presented method introduces a fairly good method of fundamental decision between turning, milling and layered techniques. Selecting an appropriate layered technology is not unambiguously determined; therefore choosing the optimal layered technology can only

be approximate. The reason for this lies in the sentence that is used in the marketing of layered technologies: complexity for free. The layered technologies of today have no problems with the production of highly complex forms, which is also their biggest advantage over conventional procedures. This poses a certain problem when selecting a production procedure based only on the complexity of the product.

Shape affects only a few specific technologies from layered technologies either because of expensive support material (PolyJet, SLA, SGC, MJM), or the difficulty of removing the support material from the problematic sections (LOM).

On the other hand, determining design complexity and the calculation of model resolution can mean certain selections regarding the choice of the production procedure, where less-accurate parts can be made using less-accurate technology.

The evaluation of the complexity was proven in determination of the time required for finalizing the product. This time of post-processing was quite difficult to determine since manufacturers would prefer to skip it, even though the impact on the time of manufacture is significant. When the talk is about rapid prototyping, time is quite significant. The presented solution is suitable for the introduction into production.

This survey is a significant advancement in the direction of process selection, but for practical applications it would be necessary to include more parameters and advanced selection methods [27] to [29], so that the process can be uniquely selected.

Only after choosing basic part properties (like material properties, colour and surface quality), time of manufacture, dimensions, the number of pieces in a series and the complexity, of does product come into regard.

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*Corr. Authors Address: College of Mechanical and Electrical Engineering, China University of Mining and Technology, 221008, Xuzhou, China, [email protected] 719

Overall Structure Calibration of 3-UCR Parallel Manipulator Based on Quaternion Method

Cheng, G. Gu, W. Yu, J. Tang, P.

Gang Cheng* Wei Gu Jing-li Yu Ping Tang

China University of Mining and Technology, College of Mechanical and Electrical Engineering, China

In this article a simple yet effective approach for the structure calibration of a three degree-of-freedom (DOF) parallel manipulator is presented. In this approach, the model of the pose error expressed by the Quaternions Parameters was established, based on complete differential-coefficient theory. This was followed by an investigation into the degree of influences represented as sensitivity percentages, of source errors on the pose accuracy with the aid of a statistical model of sensitivity coefficients. Then, the kinematic calibration model with the successive approximation algorithm was achieved. The simulation has been carried out to verify the effectiveness of the proposed algorithm and the results show that the accuracy of the calibration can be significantly improved.2011 Journal of Mechanical Engineering. All rights reserved. Keywords: 3-UCR parallel manipulator, complete differential-coefficient theory, Quaternion, least squares method, sensitivity model, kinematic calibration

0 INTRODUCTION

Parallel manipulators have particularly aroused interest of researchers over the past several decades for their properties of better structural rigidity, positioning accuracy, and dynamic performances [1] and [2]. Unlike serial manipulators, which suffer from the accumulation of joint errors, parallel manipulators are considered to have high accuracy [3]. However, relative investigations have shown that the parallel manipulator is not necessarily more accurate than a serial manipulator with the same manufacturing and assembling precision [4]. Accuracy remains a bottleneck for further industrial applications of parallel manipulators. Therefore, in order to enhance the precisions of parallel manipulators, it is important to evaluate the end-effectors accuracy in the design phase, and to calibrate the kinematic parameters after manufacturing [3]. From kinematic characteristics of lower-mobility parallel manipulators, it can be seen that complete errors compensation of the pose can not be achieved since it does not have six components in terms of both translation and orientation [5]. Therefore, the calibration method effectively reducing the pose errors of end effector is important.

Sensitivity analysis and error identification are necessary for the purpose of better kinematic

characteristics of parallel manipulators. The kinematic parameters with higher sensitivity should be found and controlled strictly. Aiming at optimizing a class of 3-DOF parallel manipulators with parallelogram struts, Huang established a statistical sensitivity model and showed quantifi-cationally the effect of geometrical errors on the pose of end effectors [5]. Based on the sensitivity analysis, Alici optimized the dynamic equilibrium of a planar parallel manipulator [6]. Pott gave the sensitivity model by a simplified force-based method and validated the algorithm by examples of both serial and fully parallel manipulators [7]. In order to study the relations between sensitivity and geometric parameters, Binaud compared the sensitivity of five planar parallel manipulators of different architectures [8]. Therefore, estimating sensitivity of kinematic parameters and studying the priority of kinematic parameters with higher sensitivity can effectively improve the calibration of manipulators.

In the course of structure calibration, for formulating universal functions of errors between the measured and theoretical values, it is feasible to realize the static error compensation of a parallel manipulator by modifing the kinematic parameters based on the calibration model. According to the measuring instruments, calibration methods can be classified into three categories: constrained calibration method, auto-


720 Cheng, G. Gu, W. Yu, J. Tang, P.

calibration or self-calibration method, external calibration method [9]. External calibration methods are based on measurements of the end-effector poses through an external device such as laser systems [10], theodolite [11], coordinate measuring machine [12] or camera systems [13]. Constrained calibration methods impose mechanical constraints on the manipulators during the calibration process through a locking device [14]. Auto-calibration or self-calibration methods rely on the measurements of the internal sensors of the manipulators. These methods have two possible approaches: the self-calibration method with redundant information [16] and [17] and the self-calibration method without redundant information [17] and [18]. Although the calibration of parallel manipulators had been study extensively and many novel methods of calibration had been presented, these studies merely focused on all kinematic parameters without sensitivity analysis. In practice, due to the impossible compensation fully of lower-mobility parallel manipulator, it is critical to judge the priority of the kinematic parameters of these manipulators by their sensitivity coefficients.

This article is orginized in the following manner. In Section 1, the prototype of the parallel manipulator is described, and its corresponding error model is established based on the complete differential-coefficient matrix theory. Thereafter, the statistical model of sensitivity is studied by normalizing all error sources in the reachable workspace. In Section 2, considering the sensitivity of kinematic parameters, the calibration model and the corresponding algorithm is presented. In Section 3, the numerical simulations of sensitivity and calibration were analyzed respectively, and in the last section the paper is concluded with a number of conclusions.

0.1 Nomenclature

ai the fix points of the joints in the moving platform

Ai the fix points of the joints in the base

B the base

DO', DE the orientation matrix of the moving platform and the the calibrated point on the end effector

DO'3 the third row of the orientation matrix of the moving platform

EW, EWS the orientation matrix consisting of the theoretical values and the measured values respectively

Exyz the geometrical vector of the calibrated point

eEij the jth offset that need to be calibrated

ER the error matrix of kinematic parameters

ERi the ith error source in ER ESi the error matrix of the end effectorES the corresponding norm of the

pose errors of the end effectorJR the Jacobi matrix of calibrationJRi the Jacobi submatrix of calibrationLB the length of the equilateral

triangle lines in the baseri (i=1,2,3) the lengths of the limbs are given

as ri Lm the length of the equilateral

triangle lines in the moving platform

LO'E the length of the end effector which is perpendicular to the moving platform

m the moving platformO XYZ the absolute coordinate system

attached to the baseO'X'Y'Z' the relative coordinate system

attached to the moving platformTR the mapping between the pose

errors of the end effector and the pose errors of the inputs

TR6i the sixth row and the i-th column in TR

V the volume of workspacexE, yE, zE the coordinates of the point E on

the end effectorE The end point of the end effector

on the moving platform

pi (i=1,2,3)the components of principal vector of rotation p referred to the body axes


721Overall Structure Calibration of 3-UCR Parallel Manipulator Based on Quaternion Method

q0, q1, q2, q3 the Unit Quaternion parametersXq, Yq, Zq the quaternion representation of

axes (X,Y,Z) of mXq, Yq, Zq the quaternion representation of

axes (X,Y,Z) of B the ratio of the fix radiuses of the

base and the moving platform

OAi

, Aai i

,

O ai

, OO

the vector OAi, Aiai, Oai and OO, respectively

Ri the sensitivity coefficients of error sources

1 SENSITIVITY MODEL

1.1 Quaternion Parameters

In October 1843, William Rowan Hamilton formulated quaternions [18]. The quaternion parameters have several advantages over other orientation parameters as an attitude representation [19]. Quaternion is an appropriate tool for transformation of multiple orientations and control algorithms. The attitude representation based on direction-cosine matrix needs 9 parameters, and Euler angles needs 3 parameters. Compared to directioncosine matrix, quaternion needs only 4 parameters and only has one constrained equation, while directioncosine matrix has six constrained equations. Compared to Euler angles, quaternion does not degenerate at any point and avoids the problem of calculation singularity [18].

Quaternion can be represented as the sum of a scalar and a vector [18] and [19], composed by Rodrigues-Hanmilton parameters (q0, qi, i = 1, 2, 3). By introducing abstract symbols k1, k2, k3 which are the imaginary unit of complex numbers and satisfying the rules k12 = k22 = k32 = k1k2k3 = 1, the analytical expression for Quaternion q is derived as below:

q = q0+q1k1+q2k2+q3k3 , (1a)

where 4 components q0, qi, (i =1, 2, 3) satisfy the constraint q02+q12+q22+q32 = 1.

The relative coordinate system OXYZ on the moving platform m can coincide with the absolute coordinate system OXYZ by a rotation about the unit u (cos1 cos2 cos3)T axis through an angle 2u [20]. Quaternion q (q0, q1, q2, q3) corresponding to the transformation is defined by

the angle u and the unit axis u. The orientation of m can be defined completely by the Euler parameters u and i, and it can also be defined completely by the Quaternion q (q0, q1, q2, q3). The relationship between Euler parameters (u, i) and Rodrigues-Hamilton parameters can be expressed as follows:

q0 = cosu , q1 = sinu cosi , (i=1, 2, 3). (1b)

If the Quaternion Xq= (0, X), Yq= (0, Y), Zq= (0, Z), Xq= (0, X), Yq= (0, Y) and Zq= (0, Z), is associated respectively, with three-dimensional vectors (X,Y,Z,X,Y,Z) and define the operation with the unit Quaternion q, as:

X = qX'q1, Y = qYq1, Z = qZq1 (2)

where means Quaternion multiplication, and q-1 is the inverse Quaternion of q. Both of them satisfy q-1q = 1. Then this transformation, from Xq to Xq, from Yq to Yq, and from Zq to Zq, represents a rotation from OXYZ to OXYZ.

Therefore, the direction-cosine matrix based on Quaterinon parameters [21] can be written as:

DOq q q q q q q q q qq q q q q q' =

+ ( ) +( )+( ) +

2 2 1 2 22 2 2

12

1 2 0 3 1 3 0 2

1 2 0 3 02

02

222

2 3 0 1

1 3 0 2 0 1 2 3 02

32

1 22 2 2 2 1

( )( ) +( ) +

q q q qq q q q q q q q q q

. (3)

1.2 System Description

The symmetrical parallel manipulator consist of a fixed base, a moving platform and three identical limbs, and its topological structure is described in Fig. 1. O-XYZ is the absolute coordinate system attached to the fixed base, while O-XYZ is the relative coordinate system attached to the moving platform. The equilateral triangle lines of the moving platform and the fixed base are denoted as Laiaj and LAiAj (i, j = 1, 2, 3; i j), respectively, while their corresponding length is denoted as Lm and LB, respectively. Each limb connects the moving platform to the base by a universal joint (U) at ai, followed by a cylindrical joint (C) and a revolute joint (R) at Ai, where the cylindrical joint is driven by a ball screw linear actuator. The installation form of these joints provides the manipulator with 3 DOF, one translational motion along the Z-axis and two rotational motion about X-axis and



Y-axis, respectively. The ri (i=1, 2, 3) stands for the lengths of the three limbs. The end effector is assumed to be perpendicular to the moving platform at point O, and its length is denoted as LO'E.

.Fig. 1. Symmetrical parallel bionic robot leg with

three UCR limbs

1.3 Error Model

OAiaiO'O and OO'EO in the 3-UCR parallel manipulator are considered as the closed-loop kinematic chains, and the following equation can express the spatial vector of the drive limbs.

Aa OE O a L OAi i O iO O O E i

= + D D 3 , (4)

where the vectors of O aiO

and O EO

with reference to the relative coordinate system can be denoted as O ai

and O EO

, respectively. The

orientation matrix of the moving platform can be denoted as DO' and DO'3 denotes the third row of it. The vector of O E

can be described by [0 0 LO'E]T

by analyzing its spatial relation.In the process of error transmission, the

nominal numbers are different to the effective displacements of the structure parts. By complete differential calculation to the outputs of the parallel manipulator, the error effects can be fully studied, and Eq. (4) can be expressed as follows:

r rr ri i iO O

O iO

i iOE O a

O a

e e D

D

+

+ + + = L L OAO E O O O E i D D3 3 0.

(5)

Due to e er ri iT

=1 and e er ri iT = 0 , left-

multiplied by eriT

, the Eq. (5) can be simplified as

Eq. (6), where ri and OE

equal to

[r1 r2 r3] and [xE yE zE]T, respectively.

e Dri T

O and D O iOO a

equal to [TDI1 TDI2 TDI3]

and [TI1 TI2 TI3]T, respectively, where eri

denotes the corresponding unified vector of the drive limbs:

r r r ri i i i e e D e D

T TO iO

TOOE O a O aiiO

TO E O

TO O E

TiL L OA

+

+ + +

e D e D er r ri i i 3 3

= 0.

(6)

Ti1xE+Ti1xE+Ti1xE+ +Ti1xE+Ti1xE+Ti1xE = 0, i = 1, 2, 3. (7)

Substituting the above expressions into Eq. (6), the equation can be rearranged.

In order to solve the six output parameters, a system of six equations should be founded. From Eq. (7), other three simultaneous equations are required. According to the kinematic model based on Quaternions parameters of the parallel manipulator conducted in previous section, three corresponding equations are obtained as follows:

xL

q q q q LEm

O E= + 23

21 2 0 2 , (8)

yL

q q q q LEm

O E= +( ) 2 3 1 2 4 212

22

0 1 , (9)

q3 = 0, (10)

where xE and yE denote the coordinates of the point E on the end effector.

By substituting 2 2 20 1 2 3=q q q q+ + into Eqs. (8), (9) and (10), the corresponding complete differential forms of the three equations can be rearranged as follows:

T x T y T zT q T q T q ii E i E i E

i i i

1 2 3

4 1 5 2 6 3 0 4 5 6

+ + ++ + + = =, , , .

(11)

Eqs. (7) and (11) can be rearrangeed in matrix form:



where ER denotes the error matrix of kinematic parameters, and can be expressed as follows:

ER O E m O a x O a y O a z

OA x OA y

L L r L L L

L L

= [ , , , , , ,

, ,1 1 1 1

1 1LL r L L

L L L L rOA z O a x O a y

O a z OA x OA y OA z

1 2 2

2 2 2 2

2, , , ,

, , , ,

33

1 23

3

3 3 3 3 3

, ,

, , , , ] ,

L

L L L L LO a x

O a y O a z OA x OA y OA zT

(13)

where LO'E and Lm denote the length error of the end effector and the triangle line error on the moving platform, respectively. ri represents the length errors of the drive limbs. LO a xi , LO a yi and LO a zi denote the coordinate errors of the connectors on the m. Note that these errors are referenced to the absolute coordinate system. Similarly, the coordinate errors of the connectors on the B are represented by LOA xi , LOA yi and LOA zi .

The error model of the parallel manipulator describing the relations between errors of kinematic parameters and output parameters can be obtained by the above equations.

1.4 Sensitivity Model

Through the establishment of the probability model of the parallel manipulator, the effects on the pose of the end effector caused by the geometrical errors of manufacture and assembly can be studied statistically. According to the error model of the manipulator, Eq. (12) is rewritten as:

ES = TR ER , (14)

where TR representing the mapping between the pose errors of the end effector and the pose errors of the inputs which denoted as ES equals to T-1T2.

In order to characterize the standard deviations of the pose errors of the end effector caused by the unified standard deviations of error

sources in the parallel manipulator, it should be assumed that all elements in ER are independent statistically and the mean of the elements equals zero. According to the error transmission matrix, the mathematical expectation of ES is zero. Therefore, the corresponding variance of ES can be derived as follows:

D(ES) = E(ES2) . (15)

Rearranging the Eq. (14) gives:

E E T T EST

RT

R R

Ri R ii

Ri R ii

E T E T

T

2

11

23

61

23

1 6

= =

=

= =

R

RR i Ri

i

R i Rii

E

T E

11

23

61

23

6 1

=

=

, (16)

where the ith error source in ER is denoted as ERi, the element in the sixth row and ith column of TR is denoted as TR6i. Assuming that the elements in ER are independent statistically, we get:

ES Rjiji

RiT2 2

1

6

1

232=

== E . (17)

Substituting Eq. (17) into Eq. (15), the following equation is derived:

D T ES Rji Riji

E( ) = ( )== 2 2

1

6

1

23

E . (18)

Therefore, relations between standard deviations of ER and ES can be formulated as follows:

ES Rji Riji

T( ) = ( )== 2 2

1

6

1

23

E . (19)

From the above mathematical analysis, the different poses of the end effector result in

T T T T T TT T T T T TT T T T T TT T

11 12 13 14 15 16

21 22 23 24 25 26

31 32 33 34 35 36

41 442 43 44 45 46

51 52 53 54 55 56

61 62 63 64 65 66

T T T TT T T T T TT T T T T T

=

xyzqqq

TT

E

E

E

1

2

3

17

277

37

47

57

67

6 23

TTTT

R

= T2 E , (12)



the change of the pose errors of outputs. In order to describe fully the standard deviations of ER and ES, the estimation standard in the whole workspace between them should be established. Suppose that the volume of the workspace is V, it follows [22]:

Ri Rjij

VT dv=

= 2

1

6

. (20)

The above equation can describe all error sources of the parallel manipulator in its workspace, however, it cannot achieve the further sensitivity analysis under the case of specific error compensation and identification. Therefore, a novel statistical model of sensitivity coefficients, to implement the unified process on the above error sources in the workspace is presented as:

Ri

Ri

Rii

=

=

1

23 . (21)

2 STRUCTURE CALIBRATION

2.1 Calibration Model of Kinematic Parameters

The mechanical structure of the parallel manipulator is assembled and the kinematic parameters can be identified by the calibration of kinematic parameters. In order to achieve the static mathematical compensation of the manipulator, it is necessary to modify the control model of kinematics according to the identified error parameters.

The pose of the end effector consists of three position parameters and three orientation parameters. In order to solve 23 kinematic parameters in ER, it is necessary to measure four groups of the pose by the testing instruments of the end effector in every calibration. According to the kinematic model and its differential form of the parallel manipulator, Eq. (22) can be obtained:

E E E E E J ES ST

ST

ST

ST T

R R= =1 2 3 4 , (22)

where ESi = [xE, yE, zE, q1, q2, q3]T,, i = 1, 2, 3, 4. A group of the pose error of the end effector is represented as ESi. JR, a matrix of 24 rows and 23 columns, denotes the Jacobi matrix of calibration. The Eq. (22) can be changed as:

E J J J ER RT

R RT

S= ( )

1. (23)

Implementing the Eq. (23) gives the iterative value compensating the matrix ER in the course of the kinematic calibration. The kinematic parameters can be calibrated by modifying the iterative value till the errors are less than the terminating value defined in advance. From Eq. (23), the matrix of the pose errors and the Jacobi matrix of calibration is needed to solve the iterative value. The corresponding procedures to obtain the matrices are shown as follows.

2.2 Analysis of the Pose Errors Matrix

Four groups of the pose of the end effector can be synthesized as the same expression. In order to describe the pose of the end effector, formulating the orientation matrix gives:

E D EW E xyz=

0 1, (24)

where DE and Exyz denote the orientation matrix of the calibrated point and the geometrical vector of the calibrated point, respectively.

By calculating the orientation matrices based on the measured values and the theoretical values, the matrix of the pose errors is derived as the following equation:

E E E E D EW W WS W E xyz= ( ) =

1

0 1 , (25)

where EW and EWS denote the theoretical values solved by the kinematic model and the measured values, respectively. Herein, Exyz equals to [xE yE zE]T.

The error of the orientation matrix DE can be expressed as:

DEE E

E E

E E

D DD DD D

=

00

0

12 13

21 23

31 32

, (26)

where the expressions of elements in the matrix are the same as the above orientation matrix in the error model.

According to the relations between the elements of the error of the orientation matrix and the Quaternions parameters, the errors of the corresponding Quaternions parameters can be



obtained. The solutions of the matrix ESi can be achieved by substituting the errors of the pose and the Quaternions parameters into Eq. (22).

2.3 Analysis of Calibration Jacobi Matrix

Similar to the analysis of the pose errors, the Jacobi matrix of calibration consists of four submatrices denoted as JRi. Because of the same expressions of the submatrices, the analysis of the Jacobi matrix of calibration can be simplified as the analysis of one submatrix, that is:

JEERij

S

Rij

j=

=, , ,..., .1 2 23 (27)

According to the analysis of the error sensitivity, different error sources of kinematic parameters with the same error values have a different effect on the pose error of the end effector. It is essential to redefine the offset denoted as eRij in JRij based on the sensitivity coefficients of the errors for calibrating better the end effector. And the offset can be written as:

ee

RijEij

Ri=

, (28)

where eEij denotes the jth offset that need to be calibrated.

By the derivation of the offsets, the Jacobi matrix of calibration can be derived as:

JRij ERij

E

Rij

E

Rij Rij Rij Rij

xe

ye

ze

qe

qe

qe

=

1 2 3

. (29)

2.4 Calibration Algorithm of Kinematic Parameters

Measuring the practical lengths of the drive limbs and the corresponding pose of the end effector and calculating the theoretical values of the end effector, the kinematic parameters of every joint can be calibrated based on the successive approximation algorithm. The procedures of the calibration algorithm of the manipulator are shown in Fig. 2.

3 NUMERICAL SIMULATION

3.1 Sensitivity Simulation

Six groups of theoretical values and error values of the parallel manipulator are defined in Table 1.

Substituting the theoretical values into the kinematic model, the corresponding position-orientations of the end effector are obtained and shown in Table 2.

According to the statistical model of sensitivity coefficients, the pose errors of the end effector caused by the errors of kinematic parameters in the whole workspace can be calculated respectively. Normalizing the results of the above process gives the sensitivity percentages of twenty-three kinematic parameter errors in Eq. (13) shown as Fig. 3.

Fig. 2. Calibration algorithm of the kinematic parameters of the parallel manipulator

From Fig. 3 it is known that the kinematic parameters with symmetrical connectors, such as A2, a2, A3 and a3, have a similar effect on the pose errors of the end effector and the result validates the sensitivity model by the structure characteristics. Comparatively, greater sensitivity percentages of the drive limbs represent that the actuator errors have more effect on the pose errors of the end effector. Due to the errors of some kinematic parameters, having the greater sensitivity percentages, it is essential to control the length errors between the origin in the absolute coordinate system and the joint connectors on the base, especially the errors along Z-axis perpendicular to the base. However, the length errors between the origin in the relative coordinate



Table 1. Theoretical values and error values of the parallel manipulator

Title Theoretical value[mm]Error value

[mm] TitleTheoretical value

[mm]Error value

[mm]

r1Six groups in the following table 0.02

LO a xi a1 : 0; a2 : 25 3; a3 : 25 3 0.05


LO a yi a1 : 50; a2 : 25; a3 : 25 0.05


LO a zi a z a z a za a a1 2 31 2 3: ; : ; : 0.05

LO'E 220 0.08 LOA xi A1 : 0; A2 : 34 3; A3 : 34 3 0.05

LB 68 3 0.08 LOA xi A1 : 68; A2 : 34; A3 : 34 0.05

Lm 50 3 0.08 LOA xi A1 : 0; A2 : 0; A3 : 0 0.05

Table 2. Theoretic values of limbs lengths and output parameters

Group r1 [mm] r2 [mm] r3 [mm] xE [mm] yE [mm] zE [mm] q0 q1 q2 q31 300.042 265.307 349.770 219.956 24 304.4 0.714 0 0.7 02 289.905 269.18 350.913 207.073 58.349 377 0.823 -0.15 0.55 03 287.356 273.15 350.147 195.375 72.150 392.4 0.843 -0.20 0.50 04 296.341 264.903 350.822 220.12 33.807 333 0.758 -0.05 0.65 05 346.459 308.792 263.091 -116.446 -169.831 395.7 0.847 0.40 -0.35 06 376.653 283.193 270.914 -28.924 -251.544 357.2 0.794 0.60 -0.10 0

Fig. 3. Sensitivity percentages of kinematic parameters

system and the joint connectors on the moving platform and the errors of the end effector have less sensitivity percentages. Therefore, with the promise to guarantee the whole precision of the manipulator, it is feasible to adjust the manufacture and assembly tolerance of mechanical parts by the sensitivity percentages.

The symbol denotes the structure scales which is the ratio of the fix radiuses of the base

and the moving platform. The variation of the sensitivity percentage of the kinematic parameters with different structure scales are given in Fig. 4.

Fig. 4 shows that, with the variation of the structure scale, the sensitivity percentages of kinematic parameters have not been changed obviously in corresponding reachable workspaces. On the other hand, it is necessary to strictly control the kinematic parameters with different structure



scales having greater sensitivity percentages.

3.2 Calibration Simulation

For validating the calibration algorithm of kinematic parameters, the iterative calculation of the given kinematic parameters by the numerical simulation is given as follows. The kinematic parameters are shown in Table 1, and the corresponding errors of these parameters are presented in ER:

ER =

[ . , . , . , . , . , . ,. , . , . , . ,

0 1 0 1 0 05 0 08 0 08 0 080 08 0 08 0 08 0 05 00 08 0 08 0 080 08 0 08 0 08 0 05 0 08 0 08 0 080

. , . , . ,. , . , . , . , . , . , . ,

.. , . , . ] ,08 0 08 0 08 1 23T

where the errors of kinematic parameters ER correspond to Eq. (13).

Substituting four groups of the kinematic parameters into the kinematic model of the manipulator, the corresponding poses of the end effector are shown in Table 3.

Taking the lengths of the drive limbs, the poses of the end effector in Table 3 and the values of the kinematic parameters in Table 1 into the kinematic calibration program of the parallel manipulator and calculating iteratively 7 times, the modified matrix of kinematic parameters is obtained. ES denotes the corresponding norm of the pose errors of the end effector, it is less than the terminating value, defined as 0.01, which has no unit because of having no uniform unit in ES. After modifying 7 times, the values of the kinematic parameters converge gradually to the truth values which are the sums of the theoretical values and the given errors of the kinematic parameters in the numerical simulation.

The terminating time in the calibration program is decided by the absolute difference of the truth values and the modified kinematic parameters. For the purpose of representing the change of kinematic parameters, the changes of uncalibrated kinematic parameters and calibrated kinematic parameters are shown in Fig. 5.

Fig. 4. Sensitivity percentage of kinematic parameters with different structure scales

Table 3. Poses of the end effector with four groups of limbs

Group Length of drive limbs [mm] Pose of the end effector [mm mm mm / / /]r1 r2 r3 [xE, yE, zE, q1, q2, q3]1 314.292 262.268 335.145 [192.582, -49.6373, 369.397, 0.154879, 0.564173, 0]2 320.946 264.256 330.823 [184.582, -77.5489, 323.506, 0.264831, 0.613548, 0]3 339.239 262.179 315.004 [137.102, -146.751, 388.413, 0.359163, 0.412386, 0]4 349.929 263.507 306.072 [107.704, -180.687, 394.056, 0.423459, 0.326984, 0]



Fig. 5. Comparison of the uncalibrated kinematic parameters and calibrated kinematic parameters

Fig. 5 shows that most errors of calibrated kinematic parameters are decreasing greatly, especially the kinematic parameters with high sensitivity percentage, and the successive approximation algorithm based on the statistical sensitivity coefficients is validated. Because of the lower-mobility parallel manipulator, the errors of kinematic parameters caused by uncontrolled degree-of-freedom cannot be compensated completely. Most errors of kinematic parameters are less than the terminating value. On the contrary, due to the equilibration effect of the least squares method, some errors of calibrated kinematic parameters, such as LO a x 3 and LOA z3 , are increasing. In the course of calibration of kinematic parameters, the sensitivity coefficients and calibrated kinematic parameters with increasing errors have always lower sensitivity percentages partly decide the iterative value. The significance of the sensitivity conversion is emphasized by effectively decreasing the errors of kinematic parameters with higher sensitivity percentages. From the comparison in Fig. 5, the calibration algorithm has relatively fast convergence and concrete directivity when optimizing iteratively and is effective to study the calibration questions.

4 CONCLUSIONS

In this study, by the complete differential-coefficient matrix theory, the error model of the parallel manipulator was established. Then, the statistical model of sensitivity was derived by normalizing all error sources in the reachable

workspace. According to the results of sensitivity simulation, the sensitivity percentages of the kinematic parameters vared slightly with the variation of the structure scales. The kinematic parameters with higher sensitivity percentages which should be controlled strictly were distinguished. In the course of manufacture and assembly, decreasing the length errors between the origin in the relative coordinate system and the joint connectors on the base is essential, especially the error decrease along Z-axis perpendicular to the base.

Based on the successive approximation algorithm, the calibration model with sensitivity conversion was established. According to the corresponding simulation, the algorithm is effective to study the calibration question by comparing the values of every kinematic error and has relatively fast convergence when optimizing iteratively. With the conversion according to analytical results of sensitivity coefficients, the operation steps have concrete directivity.

The approach of the calibration proposed in this article can be applied to structure calibration not only of less-DOF but also of six-DOF parallel manipulators. When it is applied to the six-DOF parallel manipulator, all source errors according to six limbs should be considered, and the dimensions of corresponding matrices such as ER, TR and T2 would change accordingly, but the main analysis steps are the same as the application to the less-DOF parallel manipulators.

5 ACKNOWLEDGEMENTS

This research is supported by the National Natural Science Foundation of China (Grant No. 50905180) and the youth foundation of China University of Mining and Technology (Grant No. OE090191).

6 REFERENCES

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*Corr. Authors Address: University of Novi Sad, Faculty of Technical Science, Trg D. Obradovica 6, 21000 Novi Sad, Serbia, [email protected]

Optimal Control of Workpiece Thermal State in Creep-Feed Grinding Using Inverse Heat Conduction Analysis

Gostimirovi, M. Sekuli, M Kopa, J. Kova, P.

Marin Gostimirovi1,* Milenko Sekuli1 Janez Kopa2 Pavel Kova1

1 University of Novi Sad, Faculty of Technical Science, Serbia 2 University of Ljubljana, Faculty of Mechanical Engineering, Slovenia

Due to intensive friction between grinding particles and workpiece material, a substantial quantity of thermal energy develops during grinding. Efficient determination of real heat loading in the surface layer of the workpiece material in grinding largely depends on the reliability of basic principles of distribution of heat sources and the character of the temperature field within the cutting zone. Therefore, this paper takes a different approach towards the identification of the thermal state of the creep-feed grinding process by using the inverse problem to approximate heat conduction. Based on a temperature measured at any point within a workpiece, this experimental and analytical method allows the determination of a complete temperature field in the workpiece surface layer as well as the unknown heat flux on the wheel/workpiece interface. In order to solve the inverse heat conduction problem, a numerical method using finite differences in implicit form was used.

When the inverse heat conduction problem is transformed into an extreme case, the optimization of heat flux leads to an allowed heat loading in the surface layer of workpiece material during grinding. Given the state function and quality criterion, the control of workpiece heat loading allows the determination of optimal creep-feed grinding conditions for particular machining conditions.2011 Journal of Mechanical Engineering. All rights reserved. Keywords: creep-feed grinding, head loading, inverse problem, optimal control

0 INTRODUCTION

Grinding is considered as one of the most important machining methods. In addition to the conventional fine multi-pass grinding, there has been a recent introduction of high-productivity grinding methods. These high-productivity methods use higher cutting speeds and/or larger cutting depths in order to increase the relatively low productivity which has traditionally been considered the main drawback of conventional grinding. However, the increase of grinding conditions considerably changes the grinding kinematics, i.e. the conditions in the wheel/workpiece interface [1].

High-speed grinding is characterized by lower cutting depths and shorter time of contact between the workpiece material and grinding particles, with a more intensive friction in the cutting zone. The increased contribution of friction in the generation of the total quantity of thermal energy also increases the contact temperature. The potential decrease of contact temperature during extremely high-speed machining can be

attributed to a reduced cross-section of chip and faster introduction of the next grinding particle into the wheel/worpiece interface [2]. This improves cutting conditions because of reduced friction and deformation during chip forming due to interfacing between the grinding particles and the softened material layer generated by previous grinding.

In creep-feed grinding, which is characterized by large cutting depths and small workpiece speeds, there is a longer wheel/workpiece interface as well as a prolonged time of contact with workpiece material. At the same time, this results in the generation of more intensive heat sources in grinding particles and a prolonged time of their effect. In addition, a longer interface contributes to a better evacuation of heat from the cutting zone, which results in a decreased power of heat source per unit area [3] and [4]. Also, a small workpiece speed improves the distribution of total thermal energy within the cutting zone due to its prolonged effect. All this results in an increased total quantity of thermal energy per unit area but over a longer time period.


731Optimal Control of Workpiece Thermal State in Creep-Feed Grinding Using Inverse Heat Conduction Analysis

Based on the previous discussion, it is evident that high-productivity grinding methods cause the development of large quantities of thermal energy within the cutting zone [5]. The generated thermal energy, located within a relatively narrow area of the cutting zone, causes high cutting temperatures in creep-feed grinding. These increased temperatures instantaneously burst to a maximum, have short duration and exert a pronounced negative effect on wheel surface, workpiece quality and accuracy.

Since the main task of grinding is to achieve satisfactory part quality with as large productivity as possible, special attention is focused on the effect that grinding temperatures have on the change of material properties in the workpiece surface layer. If the temperatures thus generated are high enough to cause structural and phase transformations of the workpiece material, the machined surface shall suffer from a number of disadvantages. Should, in addition, dimensional errors appear as well, the overall effect can substantially diminish exploitation features of the finished part.

Efficient control of thermal phenomena in grinding, i.e. the determination of allowed heat loading on workpiece surface layer, requires knowledge of heat development and distribution in grinding, the temperature field in the cutting zone and, finally, the influence of cutting conditions on grinding temperature [6]. For that reason, identification of the thermal state in grinding based on analytical models and experimental results has been gaining popularity.

As the research so far has shown, non-stationary and non-linear technical processes involving intensive heat conduction, such as creep-feed grinding, can be successfully solved using a novel approach based on inverse problem of heat transfer [7]. The inverse problem of heat transfer allows the closest possible experimental-model approximation of thermal regimes for grinding.

In the case of control over the grinding thermal regime, the extreme case of the inverse problem of heat conduction [8] is practically the only way to reliably approximate the allowed heat loading on the workpiece surface layer. For a known temperature measured at a point within the workpiece surface layer, numerical methods are

used to approximate the total temperature field of the surface layer as well as the unknown heat flux density on the wheel/workpiece interface [9] and [10]. For the selected quality criteria, particular machining conditions, and a predetermined loading on the workpiece surface layer, it is possible to arrive at optimal cutting conditions in creep-feed grinding by controlling the heat flux.

1 PARAMETERS OF GRINDING HEAT SOURCE

The role of mathematical theory behind thermal phenomena in grinding is to adopt the most adequate model of the workpiece, grinding wheel and their inter-relationships, Fig. 1. It can be assumed that the elementary heat source on the grinding particle is the result of friction between the grinding particle, workpiece and chip in the workpiece material shear plane. Summing up all the heat sources, i.e. grinding particles in contact with the workpiece, gives the total heat source for the entire cutting zone, qT. This total heat source, whose strength varies within a narrow range, acts continuously, shifting across the workpiece surface with constant velocity [5] and [6].

Fig. 1. Model of thermal state in creep-feed grinding

The heat source in grinding is characterized by its power and length of activity. Magnitudes of these two parameters depend on the machining process, wherein cutting conditions have the predominant influence.

Heat source power in the cutting zone is the basic source parameter since it predominantly impacts the thermal state of the workpiece surface layer. An analytical definition of heat source power starts from the fact that in grinding the total


732 Gostimirovi, M. Sekuli, M Kopa, J. Kova, P.

mechanical work applied to the cutting process is transformed into thermal energy:

Q F v tT t c e= , , (1)

where Ft is the tangential grinding force, vc is the cutting speed, and te is the time of contact between the grinding particles and workpiece.

The total heat quantity, Eq. (1), is transformed into heat flux density:

q Q

dxdy dt

F vAT

T

A t

t c

c

c e

= =

. (2)

If, in Eq. (2), the wheel/workpiece interface is decomposed into Ac = lc bc = (a Ds)1/2 bc, and the tangential grinding force is reduced to a unit grinding width bc, that is, we introduce the specific tangential grinding force Ft = Ft / bc, then:

qF v

DTt c

s=

a. (3)

On the other hand, the specific tangential grinding force can be expressed by a general Eq.:

= F k ht sm m , (4)

where hm = vw a/vc is the mean cutting depth, and ksm is the mean specific cutting force.

The final form for total heat flux density as a function of the grinding conditions can be derived by substituting Eq. (4) in Eq. (3):

q k vDT sm w s

= a . (5)

On the other hand, the total active time interval of the heat source has a prominent influence on the heat quantity being evacuated from the cutting zone into the workpiece. The shorter active time of the heat source diminishes the heat quantity transferred via the wheel/workpiece interface into the workpiece surface layer, while at the same time allowing a faster supply of coolant onto the freshly ground surface. This time interval is expressed as the ratio between the interface length lc and workpiece velocity vw, that is:

tD

vTs

w=

a. (6)

2 INVERSE PROBLEM OF THE GRINDING PROCESS

The process of heat transfer between solid bodies or between a system and its environment, of which heat conduction in grinding is also a part, is mostly considered from the standpoint of mutual relations between the input and output process parameters.

If the input parameters u(t) are known and the output parameters z(t) define the process state in time, then the output parameters are a function of the input parameter

Journal of Mechanical Engineering 2011 10

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