Estevan Hideki Murai
NUMBER SYNTHESIS METHODS FOR MECHANISM DESIGN: AN ALTERNATIVE APPROACH
Florian´opolis 2019
NUMBER SYNTHESIS METHODS FOR MECHANISM DESIGN: AN ALTERNATIVE APPROACH
Tese submetida ao Programa de P´os-Graduac¸˜ao em Engenharia Mecˆanica para a obtenc¸˜ao do Grau de Doutor em Engenharia Mecˆanica. Orientador: Daniel Martins, Dr. Eng.
Coorientador: Roberto Simoni, Dr. Eng.
Florian´opolis 2019
Murai, Estevan Hideki
Number synthesis methods for mechanism design: an alternative approach / Estevan Hideki Murai ; orientador, Daniel Martins, coorientador, Roberto Simoni, 2019.
266 p.
Tese (doutorado) - Universidade Federal de Santa Catarina, Centro Tecnológico, Programa de Pós
Graduação em Engenharia Mecânica, Florianópolis, 2019. Inclui referências.
1. Engenharia Mecânica. 2. Grau-de-controle atuado. 3. Mapa de influência. 4. Redundância
atuada. 5. Matriz de entradas e saídas. I. Martins, Daniel. II. Simoni, Roberto. III. Universidade
Federal de Santa Catarina. Programa de Pós-Graduação em Engenharia Mecânica. IV. Título.
Agradec¸o aos meus pais, Masahaki e Veronice Murai, e `a minha irm˜a, Marcelle Murai, que sempre me apoiaram e me incentivaram.
Aos professores que tive ao longo da vida e que me ajudaram em minha trajet´oria. Em especial, aos professores Daniel Martins e Roberto Simoni pela orientac¸˜ao, confianc¸a, amizade e paciˆencia durante todos esses anos. Agradec¸o tamb´em pelas oportunidades e pelo suporte para o desenvolvimento profissional que sempre recebi.
`A Fab´ıola, pela amizade, compreens˜ao, companheirismo, suporte emo-cional e sempre estar disposta a ajudar.
Aos amigos do Laborat´orio de Rob´otica, pela amizade, companhei-rismo e ajuda. As risadas, parcerias e os in´umeros bons momentos que compartilhamos ajudam a fazer do laborat´orio o lugar que ´e. Em especial (e em ordem alfab´etica por quest˜ao de civilidade) a Andrea, Andrez˜ao, Ane, Barreto, Bruno, Elias, Fab´ıola, Fernando, Gustavo, JC, Jo˜aozinho, L´eo, Luan, Maraina, Marcel, Mateus, Ponce, Tha´ıs, Treze e Vˆangelo.
`A Paula, por seus ouvidos e palavras que me botavam na luta quando eu precisava e que tornaram essa jornada mais leve.
`A vida, pelos seus altos e baixos que nos mant´em em movimento. Aos servidores do POSMEC, que se dedicam para propiciar um bom ambiente de estudo e desenvolvimento profissional.
Ao CNPq e `a CAPES, pelo suporte financeiro que tornaram esta tese poss´ıvel.
A identificac¸˜ao de estruturas cinem´aticas vi´aveis no in´ıcio do projeto de meca-nismos ´e um desafio. Algumas caracter´ısticas estruturais podem ser utilizadas para identificar cadeias cinem´aticas promissoras, como conectividade, re-dundˆancia e variedade. Entrentanto, ainda pode restar uma grande quantidade de cadeias cinem´aticas para serem analizadas, sendo impratic´avel analis´a-las manualmente. Al´em disso, a representac¸˜ao abstrata das cadeias cinem´aticas dificulta a visualizac¸˜ao dos seus movimentos. Desta forma, s˜ao necess´arios novas caracter´ısticas estruturais e novos m´etodos de selec¸˜ao de estruturas ci-nem´aticas. Esta tese introduz cinco conceitos novos na Teoria de Mecanismos: grau-de-controle atuado, matriz de influˆencia, mapa de influˆencia, matriz de entradas e sa´ıdas e redundˆancia atuada. Inicialmente, algumas caracter´ısticas estruturais s˜ao revisadas, focando em como elas podem ser utilizadas para a selec¸˜ao de estruturas cinem´aticas. Em seguida os novos conceitos s˜ao introdu-zidos, sendo comparados com os conceitos j´a existentes. Esta tese apresenta a base te´orica e os algoritmos para calcular as caracter´ısticas estruturais propos-tas. S˜ao propostos m´etodos para a selec¸˜ao de estruturas cinem´aticas utilizando o grau-de-controle atuado, o mapa de influˆencia, a matriz de entradas e sa´ıdas e a redundˆancia atuada. Tais m´etodos s˜ao implementados em Matlab e s˜ao uti-lizados para selecionar automaticamente estruturas cinem´aticas que satisfazem os requisitos funcionais. Finalmente, quatro estudos de caso s˜ao apresentados, mostrando como os m´etodos e conceitos propostos s˜ao efetivos para a selec¸˜ao de estruturas cinem´aticas.
Palavras-chave: Grau-de-controle atuado. Mapa de influˆencia. Redundˆancia atuada. Matriz de entradas e sa´ıdas. Selec¸˜ao de estruturas cinem´aticas.
Introduc¸˜ao
Desde o s´eculo XIX, metodologias de projeto de mecanismos tˆem sido desenvolvidas visando um melhor projeto, tanto em relac¸˜ao ao processo de projeto quanto em relac¸˜ao ao produto final. H´a v´arias abordagens para o projeto de mecanismos, como o uso de grafos, de bibliotecas de mecanismos e blocos de construc¸˜ao. Cada abordagem apresenta as suas vantagens e desvantagens. O uso da representac¸˜ao de grafos para analisar e projetar mecanismos iniciou-se na d´ecada de 60, tendo sido desenvolvidos v´arios m´etodos e fer-ramentas at´e os dias atuais. Uma de suas vantagens ´e a possibilidade de gerar todas as soluc¸˜oes poss´ıveis computacionalmente. Assim, pode-se enumerar todas as cadeias cinem´aticas e invers˜oes, evitando que alguma poss´ıvel soluc¸˜ao deixe se ser analisada. Com relac¸˜ao a enumerac¸˜ao das cadeias cinem´aticas e mecanismos, como a enumerac¸˜ao ´e feita computacionalmente, reduz-se o tempo necess´ario para gerar concepc¸˜oes.
Outra vantagem ´e que quando associada com um levantamento do estado da arte, pode-se utilizar as metodologias baseadas em enumerac¸˜ao para procurar de forma sistem´atica por inovac¸˜ao. Assim, de todas as poss´ıveis soluc¸˜oes pode-se excluir aquelas que j´a foram exploradas e focar apenas nas estruturas cinem´aticas inovadoras.
Entretanto, as metodologias baseadas em enumerac¸˜ao apresentam al-gumas desvantagens, como a representac¸˜ao abstrata e a grande quantidade de resultados. Nas metodologias baseadas em enumerac¸˜ao, as cadeias cinem´aticas s˜ao representadas de forma abstrata, utilizando grafos ou a representac¸˜ao de Franke. Tal representac¸˜ao abstrata torna dif´ıcil a percepc¸˜ao do movimento do mecanismo ainda nas fases iniciais do projeto. Assim sendo, avaliar se uma dada cadeia cinem´atica atende os requisitos funcionais n˜ao ´e trivial. Para auxiliar na avaliac¸˜ao das propriedades da cadeia cinem´atica utilizam-se as caracter´ısticas estruturais, as quais s˜ao estudadas em detalhes nesta tese.
Outra desvantagem ´e a quantidade de resultados gerados. Por ser um problema combinatorial, a enumerac¸˜ao de cadeias cinem´aticas e mecanismos tende a gerar uma grande quantidade de resultados. Assim, mesmo ap´os uma filtragem inicial das cadeias cinem´aticas invi´aveis, ´e comum que haja uma quantidade elevada de resultados a serem analisados, tornando proibitivo uma an´alise manual de todas as opc¸˜oes.
Desta forma, a grande quantidade de resultados gerados pelas meto-dologias baseadas em enumerac¸˜ao e a dificuldade de identificar resultados promissores tornam essa abordagem limitada. Esta tese foca em
desenvol-tese e as suas implementac¸˜oes computacionais visam permitir uma selec¸˜ao automatizada de estruturas cinem´aticas consideradas vi´aveis perante uma lista de requisitos estruturais.
Objetivos
O objetivo principal deste trabalho ´e criar m´etodos e ferramentas para a selec¸˜ao de estruturas cinem´aticas nas fases iniciais do projeto de mecanismos. Para atingir o objetivo principal, s˜ao listados objetivos espec´ıficos, sendo eles expostos abaixo.
- Propor conceitos te´oricos que auxiliem no levantamento dos requisitos estruturais a partir dos requisitos funcionais.
- Propor conceitos te´oricos que auxiliem na selec¸˜ao de estruturas ci-nem´aticas vi´aveis conforme os requisitos estruturais listados.
- Desenvolver algoritmos para calcular as caracter´ısticas estruturais propostas.
- Desenvolver algoritmos para selecionar estruturas cinem´aticas vi´aveis conforme os requisitos estruturais listados.
Os conceitos, m´etodos e ferramentas apresentados visam reduzir os custos do projeto de mecanismos e tornar as metodologias baseadas em enumerac¸˜ao mais acess´ıveis para os projetistas.
Metodologia
Inicialmente s˜ao revisadas nesta tese as metodologias de projeto de mecanismos e os m´etodos existentes para selec¸˜ao de estruturas cinem´aticas. O objetivo dessa revis˜ao ´e estabelecer em quais ´areas pode-se desenvolver m´etodos novos para a selec¸˜ao de estruturas cinem´aticas.
As metodologias de projeto de mecanismos s˜ao revisadas, focando nas metodologias que s˜ao baseadas em m´etodos de enumerac¸˜ao. S˜ao apresentadas as estruturas das metodologias, bem como os m´etodos e ferramentas para realizar a selec¸˜ao de estruturas cinem´aticas.
Neste trabalho tamb´em ´e apresentado um levantamento de m´etodos para selec¸˜ao de cadeias cinem´aticas. Foram revisadas algumas caracter´ısticas estruturais e suas interpretac¸˜oes f´ısicas. As caracter´ısticas estruturais tamb´em foram avaliadas com relac¸˜ao a sua capacidade de ser correlacionadas com um requisito funcional nas fases iniciais do projeto de mecanismos. Essa correlac¸˜ao ´e importante pois permite que a carater´ıstica estrutural seja utilizada como um requisito estrutural, auxiliando o projetista na escolha de estruturas cinem´aticas.
Notou-se que existe na literatura diversos estudos de caso que mostram a listagem dos requisitos estruturais a partir dos requisitos funcionais. Entretanto,
m´etodos para listar os requisitos estruturais a partir dos requisitos funcionais de uma forma sistem´atica e que possam ser aplicados para casos gerais de projeto de mecanismos.
Foram propostos 5 novos conceitos na Teoria de Mecanismos: grau-de-controle atuado, matriz de influˆencia, mapa de influˆencia, matriz de entradas e sa´ıdas e redundˆancia atuada.
Para cada conceito proposto s˜ao apresentados a definic¸˜ao, a interpretac¸˜ao f´ısica e exemplos de c´alculo. Apresenta-se tamb´em uma fundamentac¸˜ao te´orica a fim de desenvolver os algoritmos para o c´alculo das caracter´ısticas estrutu-rais propostas. Tamb´em ´e apresentada uma an´alise dos conceitos propostos, diferenciando-os dos conceitos existentes na literatura e discorrendo sobre seu uso nas fases iniciais do projeto de mecanismos.
Foram desenvolvidos dois m´etodos que utilizam os conceitos propostos para a selec¸˜ao de estruturas cinem´aticas. Atrav´es de exemplos e estudos de caso, mostram-se como os m´etodos propostos auxiliam na selec¸˜ao do posicio-namento dos atuadores, do elo de referˆencia e dos elos de sa´ıda. Os m´etodos s˜ao implementados em Matlab assim como os algoritmos para calcular as caracter´ısticas estruturais propostas.
Resultados e Discuss˜ao
Quatro estudos de caso s˜ao apresentados. Os estudos de caso utilizam os algoritmos implementados para identificar as estruturas cinem´aticas vi´aveis. Notou-se que os conceitos propostos s˜ao eficazes em transformar os requisitos funcionais em requisitos estruturais. Tamb´em observou-se que com os concei-tos proposconcei-tos foi poss´ıvel fazer a selec¸˜ao de estruturas cinem´aticas de forma automatizada.
Um dos estudos de caso trata do projeto dos suportes dos membros infe-riores para uma cama hospitalar reconfigur´avel. As caracter´ısticas estruturais propostas nesta tese foram utilizadas para identificar 38 estruturas cinem´aticas vi´aveis, de acordo com os requisitos estruturais listados. Para este estudo de caso, uma an´alise combinatorial indicou que existem at´e 8820 estruturas cinem´aticas. Entretanto, apenas 38 s˜ao vi´aveis (∼ 0,43%), sendo que a busca por estruturas cinem´aticas vi´aveis foi realizada computacionalmente em menos de um minuto.
Outro estudo de caso apresentado ´e o projeto de mecanismo para superf´ıcie de controle de voo de aeronaves. Neste estudo de caso, para a cadeia cinem´atica proposta, a an´alise combinatorial indicou que existem at´e 411.840 estruturas cinem´aticas para serem analisadas. Utilizando os conceitos propostos e implementados, identificaram-se 24 estruturas cinem´aticas que atendem aos requisitos estruturais. As 24 estruturas cinem´aticas vi´aveis foram
nar estruturas cinem´aticas. A implementac¸˜ao realizada tamb´em possibilitou que a escolha fosse feita computacionalmente, reduzindo significativamente o tempo necess´ario para a an´alise das estruturas cinem´aticas.
Considerac¸˜oes Finais
Os cinco conceitos propostos, em conjunto com os m´etodos e as implementac¸˜oes, demonstraram-se eficazes em selecionar estruturas cinem´aticas. Desta forma, eles podem ser utilizados para auxiliar o projetista na fase inicial do projeto de mecanismos. Os conceitos, m´etodos, algoritmos e a implementac¸˜ao s˜ao as contribuic¸˜oes desta tese para a Teoria de Mecanismo. Palavras-chave: Grau-de-controle atuado. Mapa de influˆencia. Redundˆancia atuada. Matriz de entradas e sa´ıdas. Selec¸˜ao de estruturas cinem´aticas.
The identification of feasible kinematic structures is a challenge in early stages of mechanism design. Structural characteristics can be used to identify promis-ing chains, such as connectivity, redundancy and variety. However, the number of remaining kinematic chains can be too great to manually analyze each one of them. Besides the great quantity of kinematic chains, their abstract representation makes it harder for the designer to visualize the mechanism motions. Therefore, new structural characteristics and kinematic structure selection methods are necessary. This thesis introduces five new concepts to Mechanism Theory: actuated degree-of-control, influence matrix, influence map, input-output matrix and actuated redundancy. Initially some structural characteristics are reviewed, focusing on how these structural characteristics can be used in kinematic structures selection. Then, new concepts are in-troduced, comparing them to existing structural characteristics. This thesis provides the theoretical ground and the algorithms to evaluate the structural characteristics. Methods to select kinematic structures using the actuated degree-of-control, the influence map, the input-output matrix and the actuated redundancy are proposed. The selection methods are implemented in Matlab and they are used to automatically identify feasible kinematic structures ac-cording to functional requirements. Finally, four case studies show the use of the methods and concepts introduced to effectively select kinematic structures. Keywords: Actuated degree-of-control. Influence map. Actuated redundancy. Input-output matrix. Kinematic structure selection.
Figure 1 Lower kinematic pairs. . . 46 Figure 2 Higher kinematic pairs. . . 47 Figure 3 Types of kinematic chains. (a) Open chain. (b) Closed chain. (c) Hybrid chain. . . 48 Figure 4 Methodology proposed by Yan (1998).. . . 58 Figure 5 Methodology proposed by Tsai (2000). . . 60 Figure 6 Methodology proposed by Murai (2013).. . . 62 Figure 7 Methodology proposed by Ding (2015).. . . 64 Figure 8 Planar kinematic chain with mobility 7 and 2 independent loops. 68 Figure 9 Structural characteristics as a connection between functional requirements and kinematic structures selection. . . 74 Figure 10 Current methods and tools available for number synthesis and the open issues.. . . 77 Figure 11 (a) Planar properly actuated kinematic chain. (b) Planar im-properly actuated kinematic chain.. . . 81 Figure 12 Planar actuated kinematic chain with M = 5 andν = 1.. . . 82 Figure 13 Planar kinematic chain from Figure 8 (page 68) with actuators placed. . . 84 Figure 14 Planar actuated kinematic chain with M = 3 andν = 3. . . 87 Figure 15 Mode vector (red) and slice (blue) names for an influence map. 88 Figure 16 Influence map for the actuated kinematic chain in Figure 14. . 89 Figure 17 Office chair components. Adapted from Staples (2018).. . . 91 Figure 18 Visual representation of the input-output matrix. . . 93 Figure 19 Planar actuated kinematic chain with M = 3 andν = 3.. . . 95 Figure 20 Mechanism for a compressor with variable compression ratio.104 Figure 21 Unfeasible mechanism for compressor with variable compres-sion ratio. . . 106 Figure 22 Planar parallel kinematic chains with M = 2 andν = 2. . . 109 Figure 23 Planar actuated kinematic chains from the first and second rows in Table 1. . . 112 Figure 24 Actuated kinematic chain from the fourth row of Table 1 with link 6 as reference link. . . 112 Figure 25 The influence map approach to select reference and output links
Figure 27 Generic kinematic chain showing the mechanism known parts at the early stages of mechanism design. . . 130 Figure 28 Number synthesis detailed steps. . . 133 Figure 29 Two possible paths for kinematic structure selection. Choosing reference and output links first (left path). Choosing actuators placement first (right path). . . 135 Figure 30 Structural characteristics introduced in Chapter 3 in the number synthesis process to select kinematic structures.. . . 139 Figure 31 Input data screen for the kinematic structures selection program.141 Figure 32 Output data screen with the feasible kinematic structures table.142 Figure 33 Non-fractionated parallel planar kinematic chains with M = 2 andν = 2. . . 147 Figure 34 Lower limb support mechanism for a reconfigurable hospital bed. Numbering matches the numbers in Figure 33. . . 150 Figure 35 Parallel planar kinematic chains with M = 3 andν = 2. . . 153 Figure 36 (a) Kinematic chain with kinematic pairs and links labeled. (b) New toggle clamping device with M = 3 and ν = 2.. . . 154 Figure 37 Functional representation of the toggle clamping device de-signed. Numbering matches the numbers in Figure 36. . . 155 Figure 38 Parallel planar kinematic chains with M = 2 andν = 2.. . . 157 Figure 39 (a) Kinematic chain with kinematic pairs and links labeled. (b) Kinematic structure of a new modular variable stiffness actuator. . . 157 Figure 40 Functional representation of the modular variable stiffness actuator designed. Numbering matches the numbers in Figure 39. . . 159 Figure 41 Planar kinematic chain with M = 4 and ν = 4. . . 162 Figure 42 Structural representation of the feasible kinematic structure in the fourth row of Table 7. . . 167 Figure 43 Functional representation of the feasible kinematic structure in the fourth row of Table 7.. . . 168 Figure 44 Representations of a kinematic chain. . . 194 Figure 45 Example of Farrell’s method. . . 199 Figure 46 Non-isomorphic proper chains for partition 3. . . 202 Figure 47 Non-isomorphic proper chains for partition 4. . . 202 Figure 48 Non-isomorphic proper chains for partition 5. . . 203 Figure 49 Inversions for partition 3. . . 204
Figure 52 Inversions for partition 4 (continued). . . 207 Figure 53 Inversions for partition 5. . . 208 Figure 54 Inversions for partition 5 (continued). . . 209 Figure 55 Inversions for partition 5 (continued). . . 210 Figure 56 Baranov subchain. . . 211 Figure 57 Baranov chains. . . 211 Figure 58 Body fractionation. . . 211 Figure 59 Watt mechanism isomorphism. . . 212 Figure 60 Example of isomorphic kinematic chains creation. . . 214 Figure 61 Enumeration-based methodologies comparison.. . . 218 Figure 62 Enumeration-based methodologies comparison.. . . 219 Figure 64 (a) Actuated kinematic chain with actuator A2 frozen. (b)
Kinematic chain with A2 frozen and new links numbering. . . 224
Figure 65 (a) Actuated kinematic chain with actuators A2 and A3 frozen.
(b) Kinematic chain with A2 and A3 frozen and new links numbering. . . . 225
Figure 66 (a) Actuated kinematic chain with actuator A1 frozen. (b)
Kinematic chain with A1 frozen and new links numbering. . . 230
Figure 67 (a) Actuated kinematic chain with actuators A1 and A3 frozen.
(b) Kinematic chain with A1 and A3 frozen and new links numbering. . . . 231
Figure 68 (a) Actuated kinematic chain with actuators A1 and A2 frozen.
(b) Kinematic chain with A1 and A2 frozen and new links numbering. . . . 234
Figure 70 Planar actuated kinematic chain with M = 2 andν = 1. . . 241 Figure 71 (a) Actuated kinematic chain with actuator A2 frozen. (b) Kinematic chain with A2 frozen and new links numbering. . . 241
Figure 72 (a) Actuated kinematic chain with actuator A1 frozen. (b) Kinematic chain with A1 frozen and new links numbering. . . 243
Table 1 Actuators placements for kinematic chains in Figure 22. . . 109 Table 2 Use of the degree-of-control and actuated degree-of-control to select links based on how many inputs controls them. . . 119 Table 3 Structural requirements for the lower limb support mechanism of a reconfigurable hospital bed.. . . 147 Table 4 Actuated kinematic chains derived from kinematic chains in Figure 33. . . 148 Table 5 Possible combinations of output links for KC II with input pairs at(a, e) and link 2 as reference link. . . 149 Table 6 Actuators placement.. . . 162 Table 7 Feasible kinematic structures. . . 166 Table 8 Number of partitions according to the number of independent loops. . . 201 Table 9 Partitions of the parallel planar kinematic chain with M = 1 and ν = 3. Polygonal links are greyed out. . . 201 Table 10 Non-isomorphic inversions generated by partitions 3, 4 and 5. . 204 Table 11 Links numbering record after A2 is frozen. . . 224 Table 12 Links numbering record updated.. . . 225 Table 13 Links numbering record after A1 is frozen. . . 229
Table 14 Links numbering record after A1 and A3 are frozen. . . 229
Table 15 Links numbering record after A1 and A2 are frozen. . . 233
Table 16 Links numbering record after A2 is frozen. . . 242 Table 17 Links numbering record after A1 is frozen. . . 244
AKC Actuated kinematic chain ADOC Actuated degree-of-control
BBBM Building blocks-based methodologies CAD Computer-aided design
DOF Degrees of Freedom DR Design requirements
EBM Enumeration-based methodologies FR Functional requirements
FKS Feasible kinematic structure
IFToMM International Federation for the Promotion of Mechanism and Machine Science
IP Input pair
KC Kinematic chain
OL Output link
OR Other requirements
RADOC Required actuated degree-of-control RL Reference link
SM Specialized methodologies SR Structural requirements
C(i, j) Connectivity between links i and j e Number of elements of kinematic pair d Distance between elements in a graph
fi Degree of freedom of pair i
Imk Influence matrix of actuator k
IM Influence map IOm Input-output matrix
j Number of pairs with one degree of freedom K(i, j) Degree-of-control between links i and j
KA(i, j) Actuated degree-of-control between links i and j
M Mobility of the kinematic chain
M0 Mobility of a subchain in a kinematic chain
n Number of links
nOL Number of output links
R(i, j) Redundancy between links i and j
RA(i, j) Actuated redundancy between links i and j
V Variety of the kinematic chain λ Order of the screw system ν Number of independent loops
1 INTRODUCTION . . . 35 1.1 THESIS CONTEXT . . . 35 1.1.1 Number synthesis challenges . . . 37 1.2 THESIS OBJECTIVES . . . 39 1.3 THESIS CONTRIBUTIONS . . . 39 1.3.1 Actuated degree-of-control . . . 40 1.3.2 Influence matrix . . . 41 1.3.3 Influence map . . . 42 1.3.4 Actuated redundancy . . . 42 1.3.5 Input-output matrix . . . 43 1.3.6 Methods for kinematic structure selection . . . 43 1.4 THESIS OUTLINE . . . 44 2 BIBLIOGRAPHIC REVIEW . . . 45 2.1 MECHANISMS THEORY BASIC CONCEPTS . . . 45 2.2 MECHANISM DESIGN METHODOLOGIES . . . 49 2.2.1 Mechanism design . . . 50 2.2.2 Building blocks-based methodologies (BBBM) . . . 50 2.2.3 Enumeration-based methodologies (EBM) . . . 51 2.2.4 Specialized methodologies (SM) . . . 54 2.3 ENUMERATION-BASED METHODOLOGIES REVIEW . . . 54 2.3.1 Hartenberg and Denavit’s methodology . . . 55 2.3.2 Yan’s methodology . . . 55 2.3.3 Tsai’s methodology . . . 58 2.3.4 Murai’s methodology . . . 60 2.3.5 Ding’s methodology . . . 63 2.3.6 Other structured approaches for mechanism design . . . 64 2.3.6.1 Freudenstein and collaborator’s contributions. . . 64 2.3.6.2 Erdman and collaborator’s contributions . . . 65 2.3.7 Final considerations regarding methodologies . . . 66 2.4 REVIEW ON KINEMATIC CHAIN STRUCTURAL
CHARAC-TERISTICS . . . 66 2.4.1 Mobility . . . 66 2.4.2 Connectivity . . . 67 2.4.3 Degree-of-control . . . 69 2.4.4 Redundancy . . . 70 2.4.5 Variety . . . 71 2.4.6 Symmetry . . . 72
2.6 CHAPTER CONCLUSION . . . 78 3 STRUCTURAL CHARACTERISTICS OF ACTUATED
KINE-MATIC CHAIN . . . 79 3.1 ACTUATED DEGREE-OF-CONTROL DEFINITION . . . 79 3.1.1 Actuated degree-of-control example 1 . . . 81 3.1.2 Actuated degree-of-control example 2 . . . 83 3.2 INFLUENCE MATRIX DEFINITION . . . 85 3.2.1 Influence matrix example. . . 85 3.3 INFLUENCE MAP DEFINITION . . . 87 3.3.1 Influence map example . . . 89 3.4 INPUT-OUTPUT MATRIX DEFINITION . . . 90 3.4.1 Input-output matrix example . . . 91 3.5 ACTUATED REDUNDANCY DEFINITION . . . 94 3.5.1 Actuated redundancy example . . . 94 3.6 ALGORITHMS TO CALCULATE THE ACTUATED
DEGREE-OF-CONTROL, INFLUENCE MATRIX, INFLUENCE MAP AND ACTUATED REDUNDANCY . . . 97 3.6.1 Theoretical ground to calculate the actuated
degree-of-con-trol, influence matrix and influence map . . . 97 3.6.2 Proposed algorithm for the influence matrix calculation . . . . 98 3.6.3 Proposed algorithm for the influence map calculation . . . 100 3.6.4 Proposed algorithm for the actuated degree-of-control
cal-culation . . . 100 3.6.5 Proposed algorithm for the actuated redundancy calculation 100 3.7 DESIGNING A RECIPROCATING COMPRESSOR WITH
VARI-ABLE COMPRESSION RATIO . . . 101 3.7.1 Structural requirements determination . . . 101 3.7.2 Actuated kinematic chain enumeration . . . 103 3.7.3 Structural characteristics evaluation . . . 103 3.7.4 Reference and output links selection . . . 104 3.7.5 Degree-of-control and actuated degree-of-control comparison 105 3.7.6 Redundancy and actuated redundancy comparison . . . 107 3.8 DESIGNING A MECHANISM WITH TWO INPUTS AND TWO
OUTPUTS FOR STITCHING MECHANISMS OR GRIPPER
MECHANISMS . . . 107 3.8.1 Structural requirements determination . . . 107 3.8.2 Actuated kinematic chain enumeration . . . 108 3.8.3 Structural characteristics evaluation . . . 109 3.8.4 Reference and output links selection . . . 110
degree-of-control approach . . . 112 3.8.4.3 Example of reference and output links selection - Influence
map approach . . . 113 3.8.4.4 Feasible actuated kinematic chain with reference and output
links selected . . . 116 3.9 ANALYSIS OF THE STRUCTURAL CHARACTERISTICS
PRO-POSED . . . 117 3.9.1 Actuated degree-of-control . . . 117 3.9.2 Input-output matrix analysis . . . 120 3.9.3 Actuated redundancy analysis . . . 121 3.9.4 Strategies to use the actuated degree-of-control and the
in-fluence map . . . 122 3.9.5 Considerations about the screw system . . . 123 3.10 CHAPTER CONCLUSION . . . 123 4 METHODOLOGY . . . 125 4.1 MODIFICATION IN THE MECHANISM DESIGN
METHOD-OLOGY PROPOSED BY MURAI (2013) . . . 125 4.1.1 Functional requirements . . . 127 4.1.2 Design requirements . . . 130 4.1.3 Structural requirements . . . 131 4.1.4 Other requirements . . . 133 4.2 METHODS FOR THE PROPOSED MECHANISM DESIGN
METHODOLOGY . . . 134 4.2.1 Number synthesis steps . . . 134 4.2.2 Kinematic chain enumeration . . . 136 4.2.3 Actuated kinematic chain enumeration . . . 136 4.2.4 Selecting reference link in actuated kinematic chains . . . 136 4.2.5 Selecting output links in actuated kinematic chains . . . 137 4.2.6 Selecting feasible kinematic structure . . . 138 4.3 FINAL CONSIDERATIONS . . . 140 4.4 CHAPTER CONCLUSION . . . 142 5 CASE STUDIES . . . 145 5.1 LOWER LIMB SUPPORT MECHANISM FOR A
RECONFIG-URABLE HOSPITAL BED . . . 145 5.1.1 Discussion . . . 149 5.2 TOGGLE CLAMPING DEVICES . . . 151 5.2.1 Discussion . . . 155 5.3 VARIABLE STIFFNESS ACTUATOR . . . 156 5.3.1 Discussion . . . 159
5.5 ADDITIONAL CONSIDERATIONS . . . 170 5.6 CHAPTER CONCLUSION . . . 171 6 CONCLUSIONS AND FUTURE WORKS . . . 173 6.1 CONCLUSIONS . . . 173 6.2 FUTURE WORKS . . . 176 6.3 PUBLICATION LIST . . . 177 Bibliography . . . 179 APPENDIX A -- Theoretical tools . . . 193 APPENDIX B -- Methodologies comparison . . . 217 APPENDIX C -- Influence matrices, actuated degree-of-control and
actuated redundancy calculation example . . . 223 APPENDIX D -- Influence matrices, actuated degree-of-control and
actuated redundancy calculation for a reciprocating compressor
with variable compression ratio . . . 241 APPENDIX E -- Matlab scripts . . . 251
1 INTRODUCTION
The thesis context, objectives and contributions are described in this chapter. The thesis outline is also presented.
1.1 THESIS CONTEXT
Historically, mechanism design has been an intuitive task. However, it takes time to train an experienced, ingenious and creative designer. The outcomes of a mechanism design process is biased by the designer, not guar-anteeing a good or optimal solution. Besides the final result, the process itself is not optimized and the design process can require more resources than necessary (time, material, personal and money).
Mechanism design methodologies aim to overcome at some extent the designer dependency, providing a structured approach to develop a feasible mechanism. Besides the structure, mechanism design methodologies also provide tools and methods for each step.
The first effort to develop a structured approach to mechanism design was done by Reuleaux (1876). Since then, several mechanism design method-ologies have been proposed. There are different approaches for mechanism design, such as graph-based mechanism synthesis (FREUDENSTEIN; MAKI, 1979, 1983; OLSON; ERDMAN; RILEY, 1985; YAN; CHEN, 1985; YAN; HWANG, 1991; YAN, 1992; RAGHAVAN, 1996; YAN, 1998), building blocks-based mechanism synthesis (KOTA; CHIOU, 1992; CHIOU; KOTA, 1999; YAN; OU, 2005; HAN; LEE, 2006) and type synthesis of parallel mech-anisms (HUANG; LI, 2002a; LEE; HERV ´E, 2006; KONG; GOSSELIN, 2007; GOGU, 2009; MENG et al., 2014). This thesis focus on methods and tools for graph-based mechanism design methodologies, which use the mobility criterion.
Despite the mobility criterion fails at some particular cases (GOGU, 2005; MRUTHYUNJAYA, 2003), for mechanisms with general dimensions and parameters this criterion is valid and quite useful at early stage of mech-anism design. The mobility criterion is also a base for several enumeration techniques in the literature, see for instance Mruthyunjaya (2003).
Each mechanism design approach presents its advantages and disadvan-tages, as well as each methodology has its own characteristics. Nonetheless, the goal remains the same for all methodologies: to use a systematic approach to develop a feasible mechanism with a good benefit-cost ratio (regarding both designing process and final product). In this sense, the benefits are related
to the device performance when satisfying the design requirements, or in a broader sense, how well the final product meets the need. The costs are related to the resources used during the designing and manufacturing process, such as the amount of work hours spent, energy costs and material costs.
A key point in mechanism design methodology is the balance between the methodology capability and its drawbacks. Details are exposed in Chapter 2, but in general, there are methodologies that generate all available options but they use abstract representations that are not intuitive; there are methodologies that use comprehensive heuristics but they do not explore all possibilities; and there are methodologies that lie in the middle of that scale.
Methodologies based on enumeration techniques, such as graph-based methodologies, can generate all possible kinematic chains. Generating all possibilities is interesting when searching for innovation in a systematic man-ner (YAN, 1998; TISCHLER; SAMUEL; HUNT, 2001). Also, combined with a state of the art survey that includes patent survey, it is possible to establish well-explored areas and innovation promising areas (COSTA et al., 2017a, 2017b). In this sense, enumeration-based approaches can systemati-cally search for innovation, avoiding intellectual properties. They can also assist in the generation of all promising solutions, aiding in a patent-wall cre-ation. A patent-wall is when a company patents several technologies related to the company strategic innovation field, creating a intellectual property barrier to protect the company interests.
Thus, enumeration-based methodologies are useful not only for de-signing new mechanisms but also for understanding the current state of the art.
Methodologies based on enumeration do not rely on a mechanism database, such as building blocks-based appproaches. Therefore, they are not bounded to the database preconceived solutions. In some design problems, the number of feasible solutions can be low (TISCHLER; SAMUEL; HUNT, 2001). In such cases, using an approach that is constrained to a database can lead to no feasible solution. On the other hand, an approach that is capable of enumerating all options can find the feasible solutions.
However, enumeration based approaches present drawbacks. They are a sequence of combinatorial problems which includes: enumerating kinematic chains; enumerating all possible kinematic inversions for each kinematic chain; enumerating all possible actuators placements for each kinematic inversion; enumerating all possible end-effectors for each actuated kinematic inversion. Therefore, enumeration-based approaches usually generate a great number of kinematic structures. Thus, the great quantity of results leads to a great amount of time and resources necessary to analyze all results.
of kinematic chains, such as graphs or Franke’s notation. Although these repre-sentations are useful for computional generation of kinematic structures, they make the mechanism movement perception harder for the designer (HUANG et al., 2017; TISCHLER; SAMUEL; HUNT, 2001). Given the combinatorial nature of mechanism design and the abstract mechanism representation, it is not uncommon for the designer to have thousands of results, making the man-ual analysis of each combination a labourious task (CARBONI; MARTINS, 2007).
Finally, before advancing further in this thesis context, the difference between concept, method and tool should be explained. In this thesis the term concept is used to refer to theoretical definitions, such as the degree-of-control, mobility or variety. These concepts can be used in a systematic manner to select kinematic structures, this systematic procedure is called method. When a method has algorithms developed and implemented, it is called a tool. The difference in these three words are set only for this thesis and for didactic reasons.
1.1.1 Number synthesis challenges
The number synthesis phase in enumeration-based approaches presents two main steps: enumeration and selection of kinematic structures (TSAI, 2000; YAN, 1998). The enumeration step generates all possible kinematic structures for a given set of structural requirements, such as mobility, number of independent loops and screw system order. There are several tools to generate kinematic chains (SIMONI et al., 2011) and there are also kinematic chains atlases in the literature (TSAI, 2000; DING et al., 2012; PUCHETA; ULRICH; CARDONA, 2013; YAN, 1998).
Kinematic chains enumeration presents its challenges, such as avoiding all isomorphisms. However, the topic was well-explored and there are several methods and implemented tools to enumerate kinematic chains (SIMONI et al., 2011; DING et al., 2012; MCKAY; PIPERNO, 2014; DING, 2015; DING; HUANG; KECSKEM ´ETHY, 2015; YAN; CHIU, 2015).
On the other hand, there are few methods for kinematic structure selection. A remaining challenge in mechanism design is to identify promis-ing kinematic structure accordpromis-ing to functional and design requirements (BUCHSBAUM; FREUDENSTEIN, 1970; FREUDENSTEIN; WOO, 1974; FREUDENSTEIN; MAKI, 1979, 1983, 1984; OLSON; ERDMAN; RILEY, 1985; YAN; HWANG, 1990, 1991; YAN, 1992, 1998; TSAI, 2000; TIS-CHLER; SAMUEL; HUNT, 2001; GALABOV et al., 2013; HUANG et al., 2017). Mechanism design methodologies together with structural
characteris-tics can be used to aid the designer in selecting promising kinematic structures (HUANG et al., 2017; TISCHLER; SAMUEL; HUNT, 2001; BELFIORE; BENEDETTO, 2000).
Structural characteristics can be used to identify promising features automatically using a computer (BUCHSBAUM; FREUDENSTEIN, 1970; HUNT, 1978, 1983; TISCHLER; SAMUEL; HUNT, 1995a, 1995b; SHOHAM; ROTH, 1997; BELFIORE; BENEDETTO, 2000; TSAI, 2000; TISCHLER; SAMUEL; HUNT, 2001; LIBERATI; BELFIORE, 2006; CARBONI; MAR-TINS, 2007; MARTINS; CARBONI, 2008; SIMONI; DORIA; MARMAR-TINS, 2013; SIMONI; SIMAS; MARTINS, 2015; HUANG et al., 2017; MU-RAI; SIMONI; MARTINS, 2018, 2017). Among these structural charac-teristics, there are the connectivity (HUNT, 1978; PHILLIPS, 2007; HUNT, 1983; SHOHAM; ROTH, 1997), variety (TISCHLER; SAMUEL; HUNT, 2001, 1995b; MARTINS; CARBONI, 2008), redundancy (BELFIORE; BE-NEDETTO, 2000; LIBERATI; BELFIORE, 2006; CARBONI; MARTINS, 2007), degree-of-control (BELFIORE; BENEDETTO, 2000) and symmetry (SIMONI; DORIA; MARTINS, 2013; SIMONI; SIMAS; MARTINS, 2015). The structural characteristics mentioned above are represented using integer numbers or matrices; therefore, they are suitable to be directly handled by computers. However some of them, such as the variety and the degree-of-control, are abstract concepts not easily correlated to functional requirements at the early stages of mechanism design. For instance, in the beginning of mechanism design, it is possible to determine or estimate the required degree-of-freedom (DOF) or the required mechanism mobility by analyzing the mechanism desired motion (TSAI, 2000). However, in general it is not trivial to determine, a priori, the required variety or the required degree-of-control between two links. Thus, computer-friendly and designer-friendly are desired qualities when proposing structural characteristics to be used in kinematic structure selection.
Unfeasible options can be eliminated using the structural characteristics mentioned, reducing the quantity of kinematic chains. However, the number of options left can still make it unpractical to explore and analyze all kinematic chains. Thus, new structural characteristics and methods for kinematic chain selection are desired.
Some features are desired when proposing structural characteristics: • a structural characteristic should correlate to a functional requirement.
Thus, a structural characteristic represents the mechanism functionality in a topological manner;
• the correlation between functional requirements and structural charac-teristics should be done with ease at the early stages of mechanism
design, being considered designer-friendly in this thesis scope. Thus, the structural characteristic can be used as a structural requirement; • the structural characteristic should be suitable for use with computers,
being considered computer-friendly in this thesis scope. Thus, its use can be done in an automated manner. Automation is desired because the great quantity of results generated by enumeration-based approach can make the manual analysis impracticable.
1.2 THESIS OBJECTIVES
The main objective of this thesis is to generate methods and tools for kinematic structure selection in the early stages of mechanism design.
Specific objectives are listed below.
• To provide theoretical concepts that aid in the structural requirements listing.
• To provide theoretical concepts that aid in the kinematic structure selec-tion.
• To develop algorithms to evaluate the concepts proposed. • To develop algorithms to select kinematic structures.
It is expected that the provided concepts, methods and tools reduce the cost to design mechanisms and make enumeration-based methodologies more accessible for designers with or without a strong mechanism theory background.
1.3 THESIS CONTRIBUTIONS
While enumeration-based methodologies present great advantages, they present at least two drawbacks: the huge number of results and the difficulty to analyze each result.
Enumeration-based methodologies present a sequence of combinatorial problems and frequently yield a great quantity of results. A manual analysis of each result is often impracticable. Thus, selection procedures that can be automated or fully executed by computers are desirable.
As mentioned in Section 1.1, the abstract representation of mechanisms using graphs makes it harder for the designer to evaluate and understand the mechanism motion when compared to other mechanism representations, such
as a CAD model or a functional representation (see Figure 44 in page 194). Thus, structural characteristics are used to evaluate the kinematic chain.
Some structural characteristics are easily correlated to functional re-quirements in the early stages of mechanism design. Other structural charac-teristics require a strong knowledge in mechanism theory to be translated to structural requirements.
It is not uncommon to have too many results to be manually analyzed, even after the structural requirements were used to filter out unfeasible results. Thus, more structural requirements are necessary. As mentioned in Section 1.1.1, the desired features for new structural characteristics that will be used in kinematic structures selection are: correlation to functional requirement, designer-friendly and computer-friendly.
There is an interesting opportunity to develop methods and tools to reduce the drawbacks of enumeration-based methodologies. This thesis con-tributions are to develop concepts, tools and methods to aid in the kinematic structure selection.
In general, this thesis contribution lies on reducing the drawbacks of enumeration-based methodologies. The specific contributions are listed below and detailed in the next sections.
• Actuated degree-of-control concept, see Section 1.3.1. • Influence matrix concept, see Section 1.3.2.
• Influence map concept, see Section 1.3.3.
• Actuated redundancy concept, see Section 1.3.4.
• Algorithms to evaluate the above structural characteristics. • Input-output matrix concept, see Section 1.3.5.
• Guidelines on how to list input-output matrix.
• Methods of using the concepts presented above to identify feasible kinematic structures according to functional requirements. How these methods are used in a mechanism design methodology framework, see Section 1.3.6.
1.3.1 Actuated degree-of-control
The actuated degree-of-control is an actuated kinematic chain structural characteristic defined in this thesis. It is also provided an algorithm to calculate the actuated degree-of-control.
The difference between the of-control and the actuated control is shown in this thesis. It is also shown why using the degree-of-control to select links based on how many actuators affect the links relative position can lead to false negatives or false positives.
Case studies and examples are used to show the actuated degree-of-control physical interpretation: it represents the quantity of actuators that controls the relative position between two links. Thus, while this thesis shows that the degree-of-control cannot be used to select links based on how many actuators control them, a valid alternative is presented. The actuated degree-of-control definition is presented in Section 3.1. Two strategies to use the actuated degree-of-control to select kinematic structures are presented in Section 3.9.4. These strategies creates two methods for kinematic structures selection. In Chapter 4 the methods presented are placed in a mechanism design methodology framework. Four case studies that show how the methods select kinematic structures are presented in Chapter 5.
The actuated degree-of-control contribution to Mechanism Theory is completing the of-control role. Also, differently from the degree-of-control, the actuated degree-of-control can be correlated to a functional requirement because of its physical interpretation (see Chapter 3). Thus, the actuated degree-of-control is a designer-friendly structural characteristic. Besides that, the actuated degree-of-control can be calculated and used to select kinematic chains in an automated manner, being computer-friendly. The actuated degree-of-control usefulness and its impact in mechanism design methodologies are exposed in Chapter 5.
1.3.2 Influence matrix
The influence matrix is an actuated kinematic chain structural charac-teristic defined in this thesis. The influence matrix definition is presented in Section 3.2 and an algorithm to calculate the influence matrix is provided in Section 3.6.2.
The influence matrix shows which pairs of links have their relative position controlled by a given actuator. Therefore, the influence matrix can be used when the designer needs to know the influence of a given actuator in the kinematic chain.
Because of the influence matrix physical interpretation, it can be cor-related with a functional requirement (through the input-output matrix, see Section 1.3.5), being used as a standalone structural characteristic. However, it can also be used as an intermediary step to calculate the influence map.
1.3.3 Influence map
The influence map is an actuated kinematic chain structural characteris-tic defined in this thesis. The influence map definition is presented in Section 3.3 and an algorithm to calculate the influence map is provided in Section 3.6.3.
The influence map joins the influence matrices, showing which pairs of links have their relative position controlled by which actuators. There are different manners of reading the influence map. It can be read as which actuators controls the relative position between two links; which links have their relative position controlled by a given actuator; which actuators controls the position of every other link in relation to a given link; if a given actuator affects the position of every other link in relation to a given link.
Therefore, the designers can use the influence map in several different manners, according their needs. Strategies to use the influence map to select kinematic structures are presented in Section 3.9.4. These strategies generate methods for kinematic structure selection. In Chapter 4 the methods proposed are placed in a mechanism design methodology framework. Four case studies that show how the methods can be used to select kinematic structures are presented in Chapter 5.
The influence map can be correlated with functional requirements, being used as a standalone structural characteristic. It can also be used as an intermediary step to calculate the actuated degree-of-control.
1.3.4 Actuated redundancy
The actuated redundancy is an actuated kinematic chain structural characteristic defined in this thesis. The actuated redundancy definition is presented in Section 3.5 and an algorithm to calculate the actuated redundancy is provided in Section 3.6.5.
The difference between the redundancy and the actuate redundancy is shown in this thesis. It is also shown why using the redundancy to select links based on which links can reach the same position with different intermediary links positions can lead to false negatives or false positives.
Case studies and examples are used to show the actuated redundancy physical interpretation: it represents which links can achieve the same position with different intermediary links position. Thus while this thesis shows that the redundancy cannot be used for the purpose described above, this thesis also presents a valid alternative to do so.
com-pleting the redundancy role. The actuated redundancy is useful when designing mechanisms that require kinematic redundancy and it can be used as a struc-tural requirement. The actuated redundancy can be calculated and used to select kinematic chains in an automated manner.
1.3.5 Input-output matrix
The input-output matrix is introduced in this thesis. The input-output matrix is not a structural characteristic but a structural requirement. In Section 3.4 the input-output matrix is defined, and in Section 3.4.1 an example that shows the input-output matrix determination is presented.
The input-output matrix is the matrix form of structural requirements regarding actuation for output links. Several examples and case studies are presented, showing how the input-output matrix can be established at the early stages of mechanism design. The matrix form of the input-output matrix allows for the input-output matrix to be used together with the influence map to select kinematic structures in an automated manner.
Therefore, the contribution of the input-output matrix is to aid in the automatic selection of kinematic structures.
1.3.6 Methods for kinematic structure selection
Other contribution to Mechanism Theory is placing the concepts intro-duced as methods in a mechanism design methodology. The methods provided in this thesis aid the designer in kinematic structure selection, automating the selection process.
The structural characteristics proposed are designer-friendly and they can be correlated to functional requirements, reducing the drawback that comes with abstract representation. Also, as the structural characteristics introduced are computer-friendly, they can be used to select kinematic structures in an automated manner, reducing the drawback that comes with combinatorial problems, i.e., a great number of results to be analyzed.
A modification is proposed in an existing mechanism design methodol-ogy to include the concepts introduced into the methodolmethodol-ogy structure. The methods provided are also included in the appropriate methodology steps.
1.4 THESIS OUTLINE
This thesis is organized in six chapters.
Chapter 1 is an introduction to mechanism design methodologies, the difficulties and challenges in the mechanism synthesis field. This thesis objec-tives and contributions are also presented in Chapter 1.
In Chapter 2 a bibliographic review is presented. The current status on mechanism design methodologies and the advantages and disadvantages of each methodology are presented. It also presents enumeration-based method-ologies and how they approach these specific difficulties. A review on existing structural characteristics and how they can be used to select kinematic struc-tures are presented in Chapter 2. Finally, tools and methods for enumeration based-methodologies are presented.
In Chapter 3 five concepts are introduced to Mechanism Theory, being four structural characteristics and one structural requirement. The definitions, algorithms to calculate the structural characteristics, the physical interpretation of such structural characteristics and how they can be used to select kinematic structures are presented in Chapter 3. A discussion on the proposed structural characteristics is also presented.
In Chapter 4 the concepts introduced in Chapter 3 are fitted into a methodology structure. A modification on an existing mechanism design methodology is proposed.
In Chapter 5 four case studies are presented, showing the introduced concepts effectiveness and how to apply them to automatically select kinematic structures.
In Chapter 6 this thesis conclusions and future works are presented. In Appendix A a brief review on mechanisms representation, number synthesis procedure and number synthesis challenges is presented.
In Appendix B a comparison among different mechanism design method-ologies based on enumeration is shown.
In Appendix C a step-by-step visual representation of the algorithms developed for this thesis is shown.
In Appendix D the evaluation of the proposed structural characteristics for the example in Section 3.7 is presented.
In Appendix E the Matlab scripts for the algorithms developed for this thesis are presented.
2 BIBLIOGRAPHIC REVIEW
In this chapter a review on mechanism theory is presented. Initially, basic concepts of mechanism theory are introduced. Then, a brief historical background on mechanism design methodologies is presented. These method-ologies are divided in three classes in order to expose where the thesis works are concentrated and their relevance to Mechanism Theory. Then, the available structural characteristics for kinematic structures selection are reviewed as well as desired features for a structural characteristic. Finally, it is presented a review on available methods to select kinematic structures.
2.1 MECHANISMS THEORY BASIC CONCEPTS
This section exposes a review on mechanism theory concepts. The terminology exposed here is in accordance with the International Federation for the Promotion of Mechanism and Machine Science (IFToMM). For further information about terminology, see Ionescu (2003), Tsai (1999) and Hunt (1978).
A body is considered rigid if the distance between any two points on it is constant regardless of any forces acting on the body, i.e., the body does not deform. Although no such body exists, in some cases a body can be considered as a rigid body since this approximation is precise enough and it simplifies the system mathematical model. The mechanism bodies are called links, and, generally, they can be considered as rigid bodies (TSAI, 1999).
A link with no connections can move freely in space by translations, rotations or any combination of those motions. Such a link has six degrees of freedom (DOF). The DOF is the number of independent variables necessary to fully determine the configuration of a system (IONESCU, 2003). The DOF between two links can be reduced by connecting them, imposing restrictions to their relative motions. Links can be classified according to the quantity of these connections. A binary link is connected to two other links, a ternary to three other, and so on. These connections between bodies are called kinematic pairs. A link connected to three or more links is called a polygonal link.
A kinematic pair is formed by a connection between two parts called elements of kinematic pair (or, by context, just elements). A kinematic pair (or just pair) reduces the DOF between two links. This reduction is determined by the interaction of the surfaces, lines or points of the elements, resulting in different types of pairs.
1978). Lower pairs have their elements connected by surfaces while higher pairs elements are connected by lines or points. The lower pairs are shown in Figure 1 and two examples of higher pairs are exposed in Figure 2.
(a) Revolute pair. (b) Prismatic pair.
(c) Helical pair. (d) Cylindric pair.
(e) Planar pair. (f) Spheric pair.
Figure 1: Lower kinematic pairs.
In addition, a kinematic pair with i-DOF can be replaced with i pairs with a single DOF. For example, the cylindric pair from Figure 1d has two DOF, one translational and one rotational. Thus, it can be replaced with two pairs, one revolute and one prismatic. Such substitution is called expansion of kinematic pair. Notice that to maintain the cylindrical motion, the revolute pairs’ rotation axis must be parallel with the prismatic pair’s translation axis.
(a) Gear pair. (b) Cam pair. Figure 2: Higher kinematic pairs.
The opposite replacement is also valid, being called contraction of kinematic pair.
A joint is a kinematic pair physical realization. For example, a revolute pair may have many different realizations, such as journal bearing or rolling bearing.
A joint can have an apparatus attached to it that causes relative motion between that joint links in response to a given signal. Such apparatus is called actuator.
An assembly of links and pairs is called a kinematic chain (or chain). When a subset of links on a kinematic chain forms a closed circuit, such subset is called loop. In a set of loops, the loops are considered independent when each loop presents at least a kinematic pair that no other loop presents. The number of independent loops is the maximum quantity of loops that can be in a set of independent loops.
A kinematic chain can be classified in open, closed and hybrid. A kinematic chain is considered open if there is only one possible sequence of links and kinematic pairs connecting any two links; an example is shown in Figure 3. A closed chain has at least two distinct sequences of links and kinematic pairs connecting any two links. A chain is hybrid if it has both open and closed parts. A sequence of links and kinematic pairs in a kinematic chain is called a subchain.
A mechanism is a kinematic chain with one link as a frame, which is called the fixed link, or base link, or reference link. In this thesis, reference link (RL) is adopted.
The set of links that belongs to a kinematic chain is called partition. A variation occurs when all links in a partition are connected in a proper manner, creating a kinematic chain. An inversion occurs when a link in a kinematic chain is fixed, creating a mechanism. Notice that a given partition
(a) (b) (c)
Figure 3: Types of kinematic chains. (a) Open chain. (b) Closed chain. (c) Hybrid chain.
can generate several different variations and a variation can generate several different inversions (up to its number of links).
A device is a machine or machine component that performs one or more simple tasks.
The term kinematic structure has been used recently to refer to all char-acteristics of the kinematic chain, that do not depend on the links dimensions (MRUTHYUNJAYA, 2003). Thus, a kinematic structure has its kinematic chain defined and sometimes also the types of pairs defined. In this thesis, kinematic structure is used to refer to kinematic chain with reference link or output links or input pairs defined.
A point of interest is a point in a mechanism link which motion is relevant for the device purpose. A gripper, an end-effector or a tool can be placed at such point. This point kinematics is analyzed since it will interact with other bodies to execute the desired task. For example, in a packing mechanism the point of interest is the protrusion that pushes the object into the package (HARTENBERG; DENAVIT, 1964, p. 48). An intermediary point of a mechanism can also be a point of interest. For instance, in a robotic arm the center of the wrist is a point of interest even though the end-effector is not placed at such point. Also, other points of interest can be added to monitor the robots motions and avoid collision. Notice that the point of interest is not necessarily the mechanism motion output.
An output link is a link in the mechanism from which actions or motions are obtained (IONESCU, 2003). Thus, the mechanism goal is to provide the required action or motion to the output links. In this thesis, the number of output links required for a mechanism is represented by nOL.
Kinematic pairs can be modeled through screws. Briefly, the screw system is a base of the space to which all kinematic chain screws belong. Thus,
the screw system is composed of linearly independent screws that can be used to describe all other screws in the space. The order of the screw system,λ, is established by the number of screws in the screw system. More details about this topic are available in Hunt (1978), Tischler, Samuel and Hunt (1995a) and Tischler, Samuel and Hunt (1995b).
Structural characteristics are properties related to kinematic chains, such as mobility, variety, connectivity, order of the screw system, number of loops and links. Design characteristics are features desirable or required for the device and are not necessarily related to structural characteristics. Examples of design requirements are easiness to operate, being compact, light, silent, easy to manufacture and low cost. While it is easier to evaluate a device by its structural characteristics, design characteristics might be subjective and non-measurable.
One additional term is defined here: input pair. The input pair is used to refer to the mechanism power supply with a functionality assigned, i.e., the main input pair, the compression adjustment input pair, the damping adjustment input pair. In this sense, the use of “input pair” is related to structural requirement. In this thesis it is considered that actions and motions are imparted between two links. For instance, a motor imposes a rotation of the link connected to the shaft in relation to the link connected to the motor frame. Thus, the kinematic pair connecting the two mentioned links is an input pair. The terms input pair (IP) and output link (OL) are used in this thesis during number synthesis to refer to mobilities with functionality assigned and end-effectors, respectively.
In this thesis, actuator is used as the kinematic pair that contains or will contain an input pair.
The difference between actuator and input pair is subtle: the actuator is related to structural characteristic and the input pair is related to structural requirement. The difference between them as well as the need to differentiate them are explored deeper in Chapter 3.
2.2 MECHANISM DESIGN METHODOLOGIES
General aspects of mechanism design methodologies are presented in this section. First, a historical review on mechanism design methodologies is exposed, then, a classification of mechanism design methodologies is pro-posed. A comparison between two major classes is done, presenting the main advantages and disadvantages of each class. This classification is done to point out this thesis contribution field.
2.2.1 Mechanism design
Mechanism design relies on several factors, including the designer experience, knowledge, creativity and ingenuity. The less structured is the designing process, the more designer-dependent it is. Mechanism design methodologies provides a structure to develop mechanisms, reducing the designer dependency. Thus, the mechanism design methodology objective is to systematize the design process in order to reduce the designing time and enhance the design process quality. However, it is still desired that the methodology uses the designer creativity and ingenuity, taking advantage of the designer experience.
Since the 19th century, several attempts have been made to
system-atize the mechanism design process (REULEAUX, 1876; CROSSLEY, 1965; FREUDENSTEIN; DOBRJANSKYJ, 1966; FREUDENSTEIN; MAKI, 1979; OLSON; ERDMAN; RILEY, 1985; KOTA; CHIOU, 1992; YAN, 1992, 1998; TSAI, 2000). Two main approaches for mechanism design can be identified (FREUDENSTEIN; MAKI, 1979; KOTA; CHIOU, 1992): the use of mecha-nism atlases (JONES, 1930; ARTOBOLEVSKI, 1973) and the use of abstract representation of mechanisms (REULEAUX, 1876).
In the 1960’s, graphs started to be used to represent kinematic chains (CROSSLEY, 1965; FREUDENSTEIN; DOBRJANSKYJ, 1966; DAVIES, 1968; DOBRJANSKYJ; FREUDENSTEIN, 1967; WOO, 1967), resulting in advances in both mechanism analysis and synthesis.
Over the years further developments were made in those two ap-proaches, aggregating theoretical and technological tools. This divided the methodologies in several branches and, in order to proper situate the scope of this work, it is necessary to classify these methodologies. The objective of this classification is not to separate the methodologies in non-overlapping groups, but to make it easier to expose the characteristics of different mechanism de-sign approaches and to show where the methodology and methods developed in this thesis stand. At last, the methodologies are here classified as build-ing blocks-based methodologies (BBBM), enumeration-based methodologies (EBM) and specialized methodologies (SM). Other proposal of methodologies classification can be seen in Subramanian and Wang (1995).
2.2.2 Building blocks-based methodologies (BBBM)
Methodologies that select or combine mechanisms from a database are classified as building blocks-based methodologies. These methodologies can vary in how the blocks motion are represented and how they are selected
and combined. A common point in these methodologies is the need for a mechanism library, reusing previous designs. Also, in these methodologies, the number and type syntheses are performed as the blocks are assembled, thus, usually it is not possible to distinguish the synthesis process in number and type. Finally, the principle of BBBM is similar to using mechanisms atlases, however, instead of atlases, it is used a mechanism library with powerful tools for exploring through the library and synthesizing new mechanisms. Examples of such methodologies are exposed in Kota and Chiou (1992), Subramanian and Wang (1995), Chiou and Kota (1999), Moon and Kota (2002), Yan and Ou (2005), Han and Lee (2006); for a brief review on the development of building blocks-based methodologies see Yan and Ou (2005), Han and Lee (2006). 2.2.3 Enumeration-based methodologies (EBM)
Methodologies that use structural characteristics to generate all pos-sible mechanisms (in any abstract representation of them) are classified as enumeration-based methodologies. These methodologies can use different enu-meration methods, mechanism representations and structural characteristics. Examples are of these methodologies usage are presented by Freudenstein and Maki (1979, 1983), Olson, Erdman and Riley (1985), Yan and Chen (1985), Yan and Hwang (1991), Yan (1992), Raghavan (1996), Yan (1998). Enumeration-based methodologies present a step in which kinematic structures are enumerated and selected, called number synthesis step.
The methods proposed in this thesis lie on number synthesis step. The generation of kinematic structures presents its challenges, such as avoiding isomorphisms while still generating all possible kinematic structures (SIMONI et al., 2011). However, there are several methods and tools available in the literature to generate a kinematic chain atlas, such as exposed by Mruthyunjaya (1979, 1984), Yan and Hwang (1990), Mruthyunjaya (2003), Martins, Simoni and Carboni (2010), Simoni et al. (2011), Ding et al. (2012), Yan and Chiu (2015), Ding (2015), Pozhbelko (2016). Thus, while developing an efficient algorithm for kinematic chain enumeration is a challenge, the tools available in the literature can be used to automatically generate kinematic chain atlases using a computer. Appendix A show more details on how the number synthesis process can be carried out, exposing more details on the enumeration methods and the challenges that lie within them. However, the scope of this work is in the selection of kinematic structures and not on the enumeration of kinematic chains. Thus, further details on the enumeration of kinematic chains can be seen in the cited literature.
Enumeration-based methodologies, when compared to building blocks-based methodologies, present in general the following characteristics:
+ they can generate all kinematic structures possibilities;
+ they can prospect innovative mechanisms in a systematized manner
since they do not use preconceived blocks or solutions;
+ the mechanism is intrinsically designed to work within a full-cycle screw
system;
+ no mechanism library is necessary;
− they generate a lot of results to be analyzed; − they tend to be more time-consuming;
− the mechanism abstract representation makes it difficult to visualize the
mechanism behavior in the early stages.
Enumeration-based methodologies present great advantages with great challenges. As mentioned in Section 1.2, the goal for this thesis is to introduce theoretical concepts; how to use such concepts in a methodology framework, i.e., to develop methods; and to provide tools, i.e., to implement the methods, focusing on the reduction of EBMs drawbacks.
The general structure of an enumeration-based methodology is exposed as follows.
1. To determine the design, functional and structural requirements. 2. To determine the partitions.
3. To determine the kinematic chains for each partition.
4. To determine the kinematic inversions for each kinematic chain. 5. To determine the input pairs for each kinematic inversion.
6. To determine the output links for each kinematic inversion with identi-fied input pairs .
7. To determine the kinematic pairs type for each kinematic inversion with identified input pairs and output links.
In this work the term “type synthesis” is used to refer to step 7 and it is in accordance to its definition by Hartenberg and Denavit (1964). Yan (1998) uses the term “specialization” to refer to step 7 above. Recently, other meaning for type synthesis was defined as the process of synthesizing parallel manipulator topologies, as exposed in Section 2.2.4.
Although all possible results can be enumerated at each step, it is also possible to guide the synthesis process, eliminating any unfeasible result as it appears.
The EBM approach for mechanism synthesis is a combinatorial prob-lem, thus, the quantity of results to be analyzed grows fast according to the combinatorial parameters. This leads to two main challenges in enumeration-based methodologies:
• to establish the structural requirements for enumeration of kinematic chains;
• to establish the structural requirements for selection of kinematic struc-ture.
The choice of the structural characteristics for enumeration is a crucial step in the enumeration-based methodologies. A proper selection of structural requirements yields more promising mechanisms regarding both quantity and quality. On the other hand, a poor choice might lead to no feasible mechanisms at all. As the mechanism synthesis combinatorial problem tends to be time-consuming, it is desirable to avoid a trial-and-error approach for the structural requirements determination.
Other enumeration-based methodologies challenge is to analyze all generated results and identify good and bad designs. Considering that the quantity of results tends to be large, the analysis step is also time-consuming. It is desired to identify unfeasible results and eliminate them as soon as they appear, reducing further works on an already unfeasible branch. Also, it is desired to automate the analysis in every step, since the number of results can make the manual analysis impracticable.
Some mechanism design methodologies propose methods of finding the structural characteristics for enumeration, as will be exposed in Section 2.3. Regarding the second EBM challenge, there are case studies that shows how the structural requirements can be used to identify unfeasible and promising results (BUCHSBAUM; FREUDENSTEIN, 1970; FREUDENSTEIN; WOO, 1974; FREUDENSTEIN; MAKI, 1979, 1983; TISCHLER; SAMUEL; HUNT, 1995b, 2001; CHEN; PAI, 2005; MURAI, 2013; MURAI; MARTINS; SIMAS, 2013; LI et al., 2016). However, up to now no methodology provides a method for translating the customer, design and functional requirements into structural
