UNIVERSIDADE FEDERAL DE SANTA CATARINA PROGRAMA DE P ´OS-GRADUA ¸C ˜AO EM ENGENHARIA
DE AUTOMA ¸C ˜AO E SISTEMAS
Gustavo Sobral Toscano
DEVELOPMENT OF A NEW KINETOSTATIC MODEL FOR HUMANOID ROBOTS USING SCREW THEORY
Florian´opolis 2017
Gustavo Sobral Toscano
DEVELOPMENT OF A NEW KINETOSTATIC MODEL FOR HUMANOID ROBOTS USING SCREW THEORY
Thesis submitted to the Automation and Systems Engineering Department of the Federal University of Santa Catarina in partial fulfillment of re-quirements to obtain the degree of Doctor in Automation and Systems Engineering.
Advisor: Prof. Eugˆenio De Bona Castelan Neto, Dr.
Co-advisor: Prof. Henrique Simas, Dr. Eng.
Florian´opolis 2017
Toscano, Gustavo Sobral
Development of a New Kinetostatic Model for Humanoid Robots Using Screw Theory / Gustavo Sobral Toscano ; orientador, Eugênio de Bona Castelan Neto, coorientador, Henrique Simas, 2017. 205 p.
Tese (doutorado) - Universidade Federal de Santa Catarina, Centro Tecnológico, Programa de Pós Graduação em Engenharia de Automação e Sistemas, Florianópolis, 2017.
Inclui referências.
1. Engenharia de Automação e Sistemas. 2. Robôs Humanoides. 3. Modelo Cinetostático. 4. Teoria de Helicoides. 5. Método de Davies. I. Castelan Neto, Eugênio de Bona. II. Simas, Henrique. III.
Universidade Federal de Santa Catarina. Programa de Pós-Graduação em Engenharia de Automação e Sistemas. IV. Título.
Ao Zel´
o (in memoriam), `
a
La-linha, `
a Pila, `
a Alicinha e ao
Tonhico.
ACKNOWLEDGEMENTS
First, I would like to thank my parents, Jos´e Luiz Fragoso Toscano (in memoriam) and Maria Eul´alia Sobral Toscano. They have both decisively contributed to my becoming the man and researcher I am today. Zel´o and Lalinha were, are and will always be present in everything I accomplish. They taught me the values according to which I live my life - the meaning of unconditional love and the importance of family. I can never thank them enough and I am very lucky for being their son. Thank you very much.
I thank my advisor, Professor Eugˆenio de Bona Castelan Neto, and my co-adivisor, Professor Henrique Simas, for giving me the opportunity to learn from them once more. They were both essential to my research, which resulted in this thesis. Seven years ago, my relationship with them was strictly professional, and today we have also established emotional bonds. I admire, respect, and see them both as friends. You can always count on me! Advisors and co-adivisors are usually concerned solely with giving academic-methodological support, but you are different. You went beyond that and I will never forget all the support you gave me.
I would also like to thank UFSC, DAS, CNPq, and CAPES for the infrastructure and financial support.
To my Sobral and Toscano families for always being my safe harbor. The most difficult part of deciding to cross the country to invest in one’s professional career is to be far from one’s family. Therefore, it is important to establish new bonds that will become like family in the future. This new family is not blood-related, but it comes from mutual friendship, care, cherish, and respect. I don’t need to mention names, because I believe I have already made clear the role and importance of each one in my life.
Tales Imbiriba, thank you very much for becoming a brother life has given me.
At each achievement we make in our lives, it is important to always keep in mind where we came from, our origins, and those who gave us opportunities to begin our journey. So, my deepest thanks to UFPA pro-fessors Jos´e Augusto Lima Barreiros and Jos´e Augusto Furtado Leal for promoting my first steps in academic research.
Thank you very much to you all.
As soon as he reached home, Geppetto took his tools and began to cut and shape the wood into a Marionette.
Carlo Collodi, The Pinocchio Adventure.
Whether we are based on carbon or on sil-icon makes no fundamental difference; we should each be treated with appropriate re-spect.
RESUMO
A pesquisa e o desenvolvimento de robˆos humanoides (RHs) tˆem aumentado no mundo devido a versatilidade motora desses robˆos para a execu¸c˜ao de diversos movimentos diferentes. Al´em de serem capazes de interagir com ambientes dinˆamicos, esse tipo robˆo tamb´em pode vir a par-ticipar ativamente na sociedade. Por mimetizarem a estrutura cinem´atica humanoid, os mecanismos dos HRs s˜ao compostos por um elo m´ovel e r´ıgido, do qual partem quatro cadeias cinem´aticas seriais que imitam os quatro membros do corpo humano. Embora um RH seja capaz de executar uma grande diversidade movimentos semelhante aos humanos, esses robˆos s˜ao sistemas rob´oticos complexos. Devido `as v´arias formas pelas quais um robˆo m´ovel por membros ´e capaz de interagir consigo mesmo e com o meio ao seu rodor, um ´unico RH pode apresentar dois ou mais mecan-ismos distintos. Dessa forma, RHs s˜ao robˆos cujos mecanismos possuem estrutura cinem´atica vari´avel. Portanto , um ´unico RH pode ser descrito por mecanismos diferentes em cen´arios cinem´aticos e est´aticos distintos. As cadeias cinem´aticas de estrutura vari´avel, as v´arias formas de inter-a¸c˜ao com ambientes n˜ao controlados, os m´ultiplos contatos e as m´utiplas possibilidades de contato com o meio externo, a forma¸c˜ao arbitr´aria de acoplamentos passivos e mecanismos m´oveis por membros s˜ao caracter´ıs-ticas inerentes as cadeias cinem´aticas de RHs. Tais caracter´ısticas, por exemplo, elevam o grau de complexidade e de variabilidade da estrutura cinem´aticados de modelos cujas an´alise depende de solu¸c˜oes particular-izadas. Dessa forma, esta tese apresenta uma t´ecnica para determina¸c˜ao de um novo modelo cinetost´atico para RHs, o qual ´e formado pela com-bina¸c˜ao dos modelos cinem´atico e est´atico. O conceito do m´etodo para modelamento cinetost´atico baseia-se no mecanismo de uma boneca de ma-rionete. Quando abordado como uma marionete, RHs passam a ser de-scritos por mecanismos de cadeias cinem´aticas paralelas, como a marionete e suas cordas. Tanto o modelo cinem´atico quanto o est´atico dos RHs s˜ao definidos, individualmente, por um ´unico mecanismo paralelo de estrutura cinem´atica constante. A t´ecnica foi desenvolvida pela rela¸c˜ao direta en-tre as cordas do mecanismo da marionete e as cadeias virtuais de Assur, combinada com a teoria de helicoides e o m´etodo de Davies.
Palavras-chave: Robˆos Humanoides, Modelo Cinetost´atico, Teoria de Helicoides, M´etodo de Davies.
ABSTRACT
Humanoid robots (HRs) research and development have been increas-ing in the world due to motion versatility of such robots and also due to the high degree of spatial and environmental adaptability provided by th mim-icking of the human body. Furthermore, in a future not so far away, HRs may become profitable products when inserted into the human social and dynamic environment. By resembling the human kinematics, these robotic systems are composed of one mobile and rigid base link, from which four in-dependent and serial kinematic chains mimicking the human limbs depart. Although HRs are capable of performing a great set of human-like mo-tions, this mimicry generates complex robotic systems. Nevertheless, due to the many ways through which HRs are able to interact with the envi-ronment and with themselves, and also given a multi-limb kinematic chain detached from any inertial reference frame, these robots can be defined by different mechanisms. In this way, each different mechanism describes a single HR accordingly to different kinematic and static scenarios, defining mobile robots with structure-varying kinematic chains. Multiple contacts with the environment, arbitrary formation of passive couplings, a mobile and multi-limb kinematic structure may also become issues concerning the modeling of HRs. Such issues may increase the complexity of both the kinematic and the static models, depending on how the modeling of HRs are approached. Thus, this study proposes a such new kinetostatic mod-eling technique for HRs that these not so good features earlier addressed, e.g., the structure-varying kinematic chains, multiple contacts with the environment, arbitrary formation of passive couplings, and a mobile and multi-limb kinematic structures cease to be issues. The kinetostatic model rises from the combination between both the kinematic and the static mod-els. This insight is based on the marionette doll, so that both the kinematic and the static models become characterized, individually, by single paral-lel kinematic chains whose structures do not vary. This marionette doll proposal is based on the straightforward relation between the strings of the doll’s mechanism and the Assur’s virtual chains, combined with screw theory and Davies’ method.
Keywords: Humanoid Robots, Kinetostatic Model, Screw Theory, Davies’ Method
RESUMO EXPANDIDO
Desenvolvimento de um Modelo Cinetost´atico para Robˆos Humanoides Usando Teoria de Helicoides
Palavras-chave: Robˆos Humanoides, Modelo Cinetost´atico, Teoria de Helicoides, M´etodo de Davies.
Introdu¸c˜ao
Em 1921, na pe¸ca teatral Rossum’s Universal Robots, escrita Karel Capel, o autor usou o termo robˆo para fazer referˆencia a “humanos arti-ficiais” que substituiriam a m˜ao-de-obra humana no setor industrial. Tal cen´ario idealizado por Karel Capel passa a ser realidade com o desenvolvi-mento dos primeiros robˆos industriais na d´ecada de 1960 (Edwards, 1984; Popov and Yurevich, 1987). A maior parte dos robˆos que comp˜oem as plantas e linhas de montagem industriais s˜ao manipuladores rob´oticos de cadeia serial e de base fixa.
Com o advento dos robˆos industriais, a rob´otica passa a ser um im-portante campo de pesquisa no mundo todo e a receber grandes investi-mentos. Robˆos industriais podem ser descritos por um ´unico modelo, j´a que s˜ao robˆos manipuladores seriais de base fixa. Independentemente dos movimentos e das tarefas que realizam, a estrutura cinem´atica de manipu-ladores rob´oticos de base fixa n˜ao varia. Isto ´e, s˜ao robˆos com estrutura cinem´atica constante.
Por notarem que o deslocamento b´ıpede ´e o tipo de locomo¸c˜ao que pode acessar quase todos os tipos de terrenos existentes (Westervelt et al., 2007), pesquisadores da Universidade de Wasada, na d´ecada de 1960, foram os primeiros a estudar o deslocamento b´ıpede realizado por m´aquinas. Al´em disso, dado o bom desempenho observado em robˆos manipulado-res antropom´orficos na realiza¸c˜ao de tarefas industriais, verificaram-se as vantagens de se mimetizar o movimento e a destreza do bra¸co humano. Assim, o desenvolvimento de mecanismos e robˆos capazes de mimetizar estruturas e movimentos humanos passa a ser foco de pesquisa.
Nesta tese, a estrutura cinem´atica de um robˆo humanoide (RH) ´e composta por um elo r´ıgido, torso do robˆo, de onde partem quatro cadeias cinem´aticas seriais que mimetizam os quatro membros do corpo humano. De certa forma, da perspectiva do torso do robˆo, a estrutura cinem´atica
na Fig. 1.1, RHs s˜ao sistemas mecˆanicos mais complexos do que robˆos industriais, dada `a quantidade de juntas na cadeia cinem´atica humanoide. Ao se trabalhar com sistemas rob´oticos, a obten¸c˜ao do modelo do robˆo ´e uma tarefa muito importante. Um modelo adequado de um robˆo ´e aquele capaz de estimar dados que podem ser usados na an´alise, no projeto e na constru¸c˜ao de robˆos reais. Ali´as, em se tratando de RHs, o modelo tamb´em fornece uma estrutura para projeto de controladores e para a gera¸c˜ao de trajet´oria. Caso o modelo de um dado RH n˜ao seja o mais adequado, o controlador ou a trajet´oria calculados a partir do modelo inadequado pode comprometer n˜ao s´o a integridade f´ısica do robˆo como, tamb´em, a integridade f´ısica de pessoas, objetos e propriedades.
As cadeias cinem´aticas de RHs possuem caracter´ısticas pr´oprias, fa-zendo com que um ´unico RH possa ser descrito por mais de um modelo. A estrutura m´ovel e multimembro permite ao robˆo interagir consigo mesmo e com o ambiente ao seu redor de diversas formas. Diferentes intera¸c˜oes “robˆo-robˆo” e “robˆo-ambiente” fazem com que um ´unico RH seja descrito por diferentes mecanismos. Nakamura and Yamane (2000) observou essa multiplicidade de modelos e, ent˜ao, definiu que RHs s˜ao robˆos com estru-tura cinem´atica vari´avel. Ali´as, o modelo de um RH pode tamb´em conter varia¸c˜ao em rela¸c˜ao ao grau de redundˆancia (Vukobratovic et al., 2004), dependendo de como o p´e do robˆo interage com a superf´ıcie sobre a qual ele se desloca.
Motiva¸c˜ao
Na literatura, h´a dois m´etodos pelos quais o modelo cinem´atico de robˆos pode ser determinado: parˆametros de Denavit-Hartenberg (DH) (De-navit, 1955) e teoria de helicoides (Ball, 1998; Hunt, 2000; Davidson and Hunt, 2004). Enquanto a modelagem de robˆos por DH ´e a mais utilizada na literatura da ´area, a teoria de helicoides ainda n˜ao ´e muito conhecida. Ali´as, Rocha et al. (2012) elencou algumas vantagens de se usar a teoria de helicoides em rela¸c˜ao aos parˆametros de DH. Nos trabalhos (Tsai, 1999; Hunt, 2000; Davidson and Hunt, 2004; Campos et al., 2005; Santos et al., 2006; Guenther et al., 2008) s˜ao abordadas em detalhes as aplica¸c˜oes da teoria de helicoides e suas ferramentas para s´ıntese e an´alise de mecanismos e para modelagem de robˆos.
RHs s˜ao robˆos m´oveis compostos por uma base flutuante (Sentis, 2007; Mistry et al., 2008) de onde partem quatro cadeias seriais que mimeti-zam os quatro membros do corpo humano. Em fun¸c˜ao de esses robˆos serem caracterizados por uma cadeia cinem´atica de estrutura vari´avel (Nakamura
and Yamane, 2000), um ´unico RH pode ser descrito por mecanismos com cadeias cinem´aticas distintas. Essa caracter´ıstica exige que as an´alises cinem´atica e est´atica dos referidos robˆos sejam realizadas a partir de abor-dagens particularizadas a fim de lidar com mecanismos cuja estrutura varia entre as cadeias cinem´aticas seriais, paralelas e h´ıbridas.
Determinar os modelos cinem´atico e est´atico de RHs n˜ao ´e tarefa f´acil. A estrutura cinem´atica desse tipo de robˆo possui caracter´ısticas ine-rentes que aumentam a complexidade de seus modelos. A quest˜ao apon-tada por Nakamura and Yamane (2000) pode ser considerada um cen´ario produtivo em que a teoria de helicoides e suas ferramentas, como o m´etodo de Davies e as cadeias virtuais de Assur, podem ser usadas para as an´alises cinem´atica e est´atica de RHs.
As ferramentas supracitadas foram desenvolvidas para an´alise de mecanismos e robˆos manipuladores seriais e paralelos de base fixa. Com o uso das cadeias virtuais de Assur, a utiliza¸c˜ao dessas mesmas ferramentas foi estendida a sistemas de robˆos colaborativos. Da perspectiva do elo do torso, os membros de um dado RH podem ser abordados como um sistema de robˆos colaborativos composto por quatro manipuladores seriais. Ent˜ao, nesta tese, o uso da teoria de helicoides e de suas ferramentas ´e estendido `
as an´alises cinem´atica e est´atica de robˆos m´oveis por pernas, cujas cadeias cinem´aticas s˜ao compostas por m´ultiplos membros.
Objetivos
• Objetivo Geral
Desenvolver um novo modelo cinetost´atico para RHs usando teoria de helicoides e suas ferramentas. Tal modelo ´e com-posto pela combina¸c˜ao dos modelos cinem´atico e est´atico, cujas topologias cinem´aticas n˜ao mudam independentemente da pos-tura ou do movimento executado pelo robˆo.
• Objetivos Espec´ıficos
◦ Projetar um mecanismo paralelo de estrutura cinem´atica con-stante que descreva a cinem´atica de RHs para todo e qualquer cen´ario cinem´atico.
◦ Projetar um mecanismo paralelo de estrutura cinem´atica con-stante que descreva a est´atica de RHs para todas as intera¸c˜oes est´aticas entre o robˆo e o ambiente, e entre o robˆo e ele mesmo. ◦ Desenvolver um modelo est´atico para um n´umero arbitr´ario de p´es e m˜aos em contato com o ambiente, com uma ´unica condi¸c˜ao
m˜ao.
◦ Validar o modelo cinetost´atico por meio de simula¸c˜oes de com-putador.
Fundamenta¸c˜ao B´asica
As t´ecnicas para modelagem de RHs encontradas na literatura podem ser agrupadas em trˆes grupos: abordagem baseada na troca entre modelos (Hern´andez-Santos et al., 2012; Hoonsuwan et al., 2009; Menad et al., 2013; Nugroho et al., 2014); abordagem pelo pˆendulo inverso linear Kajita et al. (1992); e abordagem pela base flutuante (Sentis, 2007; Khatib et al., 2008). Apesar das vantagens de cada t´ecnica para modelagem, as trˆes abordagens n˜ao lidam de forma satisfat´oria com algumas caracter´ısticas inerentes `a estrutura cinem´atica de RHs.
De acordo com sua intera¸c˜ao com o ambiente, um ´unico RH pode ser descrito por quinze mecanismos diferentes, conforme ilustra a Fig. 2.9. Um robˆo com cadeia cinem´atica de estrutura vari´avel, conforme observa Nakamura and Yamane (2000), pode ser descrito por mecanismos com diferentes graus de mobilidade e redundˆancia. Das t´ecnicas de modelagem encontradas na literatura, nenhuma apresenta uma metodologia de mod-elagem de RHs que seja sistem´atica, gen´erica e que consiga representar, em um ´unico modelo, todas as varia¸c˜oes ilustradas pela Fig 2.9. Para li-dar com robˆos cuja cadeia cinem´atica varia estruturalmente, as t´ecnicas de modelagem encontradas na literatura apresentam solu¸c˜oes particularizadas para lidar com altera¸c˜oes na estrutura cinem´atica e no grau de mobilidade e no surgimento de pares cinem´aticos passivos.
Dessa forma, esta tese prop˜oe o desenvolvimento de uma t´ecnica de modelagem sistem´atica e gen´erica para RHs. A despeito de suas cadeias cinem´aticas de estrutura vari´avel, a t´ecnica de modelagem cinetost´atica proposta, entre suas v´arias vantagens, ´e capaz de descrever um dado RH por um ´unico modelo cinetost´atico para todas as varia¸c˜oes ilustradas pela Fig 2.9. Tal t´ecnica se apoia na teoria de helicoides, m´etodo de Davies e cadeias virtuais de Assur para o desenvolvimento de mecanismos paralelos de cadeias cinem´aticas invari´avels capazes de descrever a cinem´atica e a est´atica de RHs.
A t´ecnica desenvolvida nesta tese baseou-se em bonecos de mario-nete, pela rela¸c˜ao direta entre as cadeias virtuais de Assur e os fios que controlam os movimentos de uma marionete. Figura 2.10 ilustra a con-cep¸c˜ao do modelo cinem´atico e a Fig. 2.11 ilustra a concep¸c˜ao do modelo est´atico. Como o boneco de marionete, a estrutura topol´ogica dos modelos
cinem´atico e est´atico de RHs permanece constante para qualquer cen´ario de an´alise.
Contribui¸c˜oes da Tese
1. Projeto de um mecanismo paralelo com cadeia cinem´atica de es-trutura constante que descreve a cinem´atica de RHs para qualquer postura e durante a execu¸c˜ao de qualquer movimento.
2. Projeto de um modelo cinem´atico n˜ao redundante, totalmente atu-ado e baseatu-ado em parˆametros constantes.
3. Desenvolvimento de um t´ecnica de modelagem gen´erica e sistem´atica capaz de resolver a cinem´atica de RHs.
4. Projeto de um mecanismo paralelo com cadeia cinem´atica de estru-tura constante que descreve a est´atica de RHs para todas as intera-¸
c˜oes est´aticas.
5. Projeto de um modelo est´atico n˜ao redundante, totalmente atuado e baseado em parˆametros constantes.
6. Desenvolvimento de uma t´ecnica de modelagem gen´erica e sistem´atica capaz de resolver a est´atica de RHs.
7. Adequa¸c˜ao do m´etodo de Davies para ser usado na an´alise cine-tost´atica de robˆos m´oveis cujos mecanismos s˜ao caracterizados por cadeias cinem´aticas de estrutura vari´avel.
LIST OF FIGURE
1.1 Humanoid kinematic structures . . . 41 1.2 Examples of actual Humanoid Robots . . . 42 1.3 Biped walking cycle . . . 43 2.1 Finite machine of the switch-model based approach . . . 53 2.2 Model switching paradigm: exchange stance legs . . . 53 2.3 Human walking cycle and its phases. . . 54 2.4 Frontiers of the SSP and DSP during a biped gait . . . 55 2.5 Walking motion by the LIP approach . . . 57 2.6 Robotic floating base system . . . 59 2.7 HR’s kinematic chain with the floating base . . . 59 2.8 HR’s mechanism variation according to supporting condition 61 2.9 Variations of the kinematic structure of HRs . . . 62 2.10 Kinematics marionette paradigm model . . . 63 2.11 Statics marionette paradigm model . . . 64 3.1 Vector diagram of a spatial displacement. . . 68 3.2 The screw of a twist $M and its components. . . 70 3.3 Twists of a fixed-base and serial manipulator robot . . . 72 3.4 The screw of a wrench $A and its components. . . 73 3.5 Wrenches of a fixed-base and serial manipulator robot . . . 75 3.6 Arrows of a network digraph . . . 77 3.7 Kinematic structure of a planar parallel mechanism . . . 78 3.8 Network digraph of a mechanism . . . 78 3.9 Action digraph of a mechanism . . . 83 3.10 Kinematic chain of a planar 3R serial manipulator . . . 85 3.11 Examples of Assur’s virtual chains . . . 86 3.12 Parallel kinematic chain for a 3R planar manipulator . . . . 87 3.13 Graph notation applied to the mechanism analysis . . . 87 3.14 Static analysis of a fixed-base serial manipulator . . . 88 3.15 Action digraph of a serial manipulator . . . 88
4.1 Leonardo Da Vinci’s Vitruvian man . . . 92 4.2 Kinematic chain of a generic humanoid leg. . . 93 4.3 Kinematic chain of a generic humanoid arm. . . 94 4.4 Coronal, sagittal, and transverse planes of the human body 96 4.5 Kinematic structure of the torso. . . 96 4.6 Floating base as a 3P 3R virtual chain . . . 98 4.7 Network digraph of a HR designed as a floating mechanism 101 4.8 Network digraph of the designed parallel mechanism for HRs 102 5.1 Serial action digraph model for a HR . . . 113 5.2 Designed parallel action digraph for HRs . . . 115 5.3 Actual joints’ internal and external static actions . . . 117 5.4 External actions acting from the environment on the HR . . 119 5.5 Action digraph for the network digraph depicted by Fig.5.2 122 5.6 Four variations for the static model of HRs . . . 123 5.7 Spanning-tree graphs for static model variations of HRs . . 128 5.8 Cutset actions digraph of the spanning-tree of Fig. 5.7a . . 129 6.1 Estimated contact condition when tracking the GCoM . . . 139 6.2 Trajectories of the feet with respect to the X axis. . . 143 6.3 Trajectories of the feet with respect to the Z axis. . . 143 6.4 Trajectory of the waist with respect to the X axis. . . 145 6.5 Trajectory of the waist with respect to the Y axis. . . 146 6.6 Video frames of the humanoid while walking. . . 146 6.7 Angular displacement of the joints of the legs. . . 147 6.8 Angular velocity in rad/s of the joints of the legs. . . 148 6.9 Distance between the GCoM and ZMP in the X axis . . . . 149 6.10 Distance between the GCoM and ZMP in the Y axis . . . . 149 6.11 Initial and final HR’s posture for the simulation . . . 150 6.12 Angular displacement of the joints of the leg . . . 151 6.13 Angular velocity of the joints of the right and left legs. . . . 152 6.14 First three joints’ estimated internal actions of the right leg 153 6.15 Last three joints’ estimated internal actions of the right leg 154 6.16 HR’s contact static condition with the environment . . . 155 6.17 Actions of the first joint of the left leg and of both arms . . 156 A.1 Kinematic structure of a planar parallel mechanism as
ex-amples . . . 182 A.2 Graph and digraph of the mechanism in Fig. A.1 . . . 184 A.3 Spanning-tree graph of the mechanism in Fig. A.2 . . . 184
A.4 Digraph notation for a rotational joint for λ = 6 . . . 187 A.5 Digraph notation for a cylindrical joint for λ = 6 . . . 188 A.6 Digraph notations for joints’ actions for space with λ = 6
order . . . 189 A.7 Forces and couples acting at the support foot . . . 196 B.1 Kinematic structure of a humanoid robot. . . 197 B.2 The topological structure of the robot’s torso. . . 198 B.3 The 3P 3R floating base virtual chain . . . 199 B.4 Kinematic chain of a humanoid leg . . . 202 B.5 Kinematic chain of one humanoid arm . . . 204
LIST OF TABLES
5.1 Definition pattern of the K fundamental cutsets . . . 130 6.1 Section 6.2 simulation execution time . . . 140 B.1 Parameters of the torso in Fig. B.2. . . 198 B.2 Parameters of the 3P 3R virtual chain’s joints . . . 200 B.3 Links’ parameters of the leg chain. . . 202 B.4 Screw’s parameters of humanoid leg chain’s joints . . . 203 B.5 Links’ parameters of the arm chain. . . 204 B.6 Screw’s parameters of humanoid arm chain’s joints . . . 205
LIST OF ACRONYMS
Related to Textual Terms CoM :: Center of mass DH :: Denavit-Hartenberg DoA :: Degree of action DoR :: Degree of restriction DoF :: Degree of freedom DSP :: Double support phase FB :: Floating base
Fcutseti :: Fundamental cutset i
GDoFM :: Gross degree of freedom of motion GDoR :: Gross degree of restriction
HR :: Humanoid robot
LIP :: Linear inverted pendulum SSP :: Single support phase ZMP :: Zero-moment point
2D-LIP :: Planar linear inverted pendulum 3D-LIP :: Spatial linear inverted pendulum
3P 3R :: Virtual chain of the type PxPyPzRzRyRx
Related to Indexes of Axis, Variables, Vectors, and Matrices ef :: End-effector LA :: Left arm LF :: Left foot LH :: Left hand LL :: Left leg RA :: Right arm RF :: Right foot RH :: Right hand RL :: Right leg vc :: Virtual Chain vB :: Virtual base vLA :: Virtual left arm vLL :: Virtual left leg vRA :: Virtual right arm vRL :: Virtual right leg
T :: Torso
LIST OF SYMBOLS
Notation Pattern
v, u, n, . . . :: Scalars v, u, n, . . . :: Vectors V, U, N, . . . :: Labels V , U , N , . . .:: Matrices Reference Frames
O0-X0Y0Z0 :: Inertial/global reference frame of the system
Oi-XiYiZi :: Local reference frame i
Analysis of Mechanisms
λ :: Dimension of the space / Order of the system ν :: Number of circuits
f :: Degree of freedom c :: Degree of actions M :: Degree of mobility
fi :: Degree of motion of the coupling i
ci :: Degree of restriction actions of the coupling i
nc :: Number of couplings that compose a mechanism nca :: Number of active couplings
cpi :: Number of unit restrictions of the coupling i
cak :: Number of external action of the active coupling k
Fgdf m :: Gross degree of freedom of motion
C :: Gross degree of restriction Ncp :: Number of the internal actions
Nca :: Number of the external actions
Screw Theory
s :: Screw axis vector
s0 :: A point lying on the screw axis
h :: Pitch of the screw $ :: The screw notation ˆ
$ :: Normalized screw
Tkb :: Screw transformation of coordinates
ence frame Oi-XiYiZi
ωi
k :: Angular velocity of the frame Ok-XkYkZk relative to
ref-erence frame Oi-XiYiZi
$M :: The twist notation
$M
i :: Twist of the joint i
ˆ $M
i :: Normalized twist of the joint i
˙
qi :: Joint i twist/velocity magnitude
J :: Screw Jacobian matrix composed of normalized twists Jvc :: Screw Jacobian matrix of the 3P3R virtual chain
The Screw Theory - Statics f :: Force
τ :: Joints’ generalized applied force magnitudes vector mp :: Couple on point p
$A :: The wrench notation $Ai :: Wrench of the joint i ˆ
$A
i :: Normalized wrench of the joint i
ψi :: Joint i wrench magnitude
Jst :: Screw Jacobian matrix composed of normalized wrenches
Davies’ Method - Kinematics MN :: Network matrix
MN P :: Primary subnetwork matrix
MN S :: Secondary subnetwork matrix
˙
qp :: Joints’ primary velocity magnitude vector
˙
qs :: Joints’ secondary velocity magnitude vector
Davies’ Method - Statics ˆ
AD :: Unit xutset action matrix
K :: Number of fundamental cutset Ki :: Fundamental cutset i
QA :: Fundamental cutset matrix
AN :: Action matrix
ˆ
AN P :: Primary cutset unit actions matrix
ˆ
AN S :: Secondary cutset unit actions matrix
ΨP :: Primary actions’ magnitude vectors
ΨS :: Secondary actions’ magnitude vectors
Kinetostatic Model - Kinematic
nk :: Number of links of the kinematic model
jk :: Number of joints of the kinematic model
νk :: Number of circuits in the robotic mechanism
x0 :: Posture vector of the HR
J :: A type of screw Jacobian matrix of the virtual chains ˙
qΓ :: Primary joints’ velocity magnitude vector
˙
qΛ :: Secondary joints’ velocity magnitude vector
Γ :: Primary subnetwork matrix for humanoid robots Λ :: Secondary subnetwork matrix for humanoid robots Kinetostatic Model - Statics
ns :: Number of links of the static model
js :: Number of actual joints of the static model
aext :: Number of limb-links subject to incidence of static actions
Cji :: Degree of restriction
CJ :: Degree of restriction of the over-constrained chain
Jji :: Unit wrench matrix of the joint i
Ψji :: Joint i unit wrench magnitude vector
JJ :: Joints’ restriction actions matrix for over-constrained
chain
ΨJ :: Joints’ wrench magnitude vector for over-constrained
chain
Cext :: Number internalized external actions acting on the links
Jext :: Internalized external actions matrix for over-constrained
chain
Ψext :: Internalized external actions vector for over-constrained
chain
CCont :: Number internalized external actions of contact
JCont :: Internalized external contact actions matrix for
over-constrained chain
ΨCont :: Internalized external contact actions vector for
over-constrained chain
TABLE OF CONTENTS
1 INTRODUCTION 39
1.1 Humanoid Robots . . . 41 1.1.1 BIPED LOCOMOTION - THE GAIT . . . 43 1.2 Motivation . . . 44 1.3 Objectives . . . 47 1.4 Document Presentation . . . 48
2 HUMANOID ROBOT MODELING TECHNIQUES 51
2.1 Modeling . . . 52 2.1.1 SWITCH-MODEL BASED APPROACH . . . 52 2.1.2 LINEAR INVERTED PENDULUM APPROACH . 56 2.1.3 FLOATING BASE APPROACH . . . 58 2.2 Limitations and Open Issues . . . 60
3 FUNDAMENTAL CONCEPTS AND TOOLS 67
3.1 Screw Theory . . . 67 3.1.1 SCREW DISPLACEMENT . . . 68 3.1.2 INSTANTANEOUS KINEMATICS . . . 69 3.1.3 STATICS . . . 72 3.2 Davies’ Method . . . 76 3.2.1 SOLVING THE INSTANTANEOUS KINEMATICS 76 3.2.2 SOLVING THE STATICS . . . 79 3.2.3 DAVIES’ METHOD AND SERIAL MECHANISMS 85 3.3 Conclusions . . . 89
4 INSTANTANEOUS KINEMATIC MODEL 91
4.1 Humanoid Kinematic Chain . . . 91 4.1.1 LIMBS AND SCREWS . . . 93 4.1.2 Whole Body Mechanism . . . 97 4.2 Instantaneous Kinematic Model . . . 99
4.2.3 PARALLEL NETWORK DIGRAPH MODEL . . . 102 4.2.4 PRIMARY AND SECONDARY JOINTS’
VELOC-ITIES . . . 104 4.2.5 BUILDING THE MODEL . . . 105 4.3 Algorithm of the Kinematic Model . . . 109 4.4 Conclusions . . . 109
5 STATIC MODEL FOR HUMANOID ROBOTS 111
5.1 Static Model . . . 111 5.1.1 SERIAL ACTION DIGRAPH MODEL . . . 112 5.1.2 THE BASIS OF THE PROPOSAL . . . 112 5.1.3 PARALLEL ACTION DIGRAPH MODEL . . . 114 5.1.4 OVER-CONSTRAINED CHAIN ANALYSIS . . . . 116 5.1.5 STATIC VARIABLES . . . 124 5.1.6 BUILDING THE MODEL . . . 126 5.2 Algorithm of the Static Model . . . 133 5.3 Conclusion . . . 134
6 NUMERICAL RESULTS FROM COMPUTER
SIMU-LATIONS 137
6.1 Computer Used for Running the Simulations . . . . 137 6.2 Kinetostatic Model and Stability Analysis . . . 138 6.3 Trajectory Generation Features . . . 140 6.4 Full Humanoid Kinetostatic Analysis . . . 148 6.5 Other Related Results and Simulations . . . 154 6.6 Conclusions . . . 155
7 CONCLUSIONS 157
7.1 Work Review . . . 157 7.2 Final Remarks . . . 159 7.3 Contributions of this Thesis . . . 161 7.4 Publications . . . 162 7.5 Future Research Topics . . . 163
REFERENCES 164
APPENDIX
A BASICS OF MECHANISM ANALYSIS 177
A.1 Basic Definitions . . . 177 A.2 Floating base . . . 181 A.3 Parallel Mechanism as Examples . . . 182 A.4 Graph Theory Applied to Mechanism Analysis . . . 182 A.5 Order of the System . . . 185 A.6 Mobility of Mechanism . . . 185 A.7 Freedom of Motion and Restriction Actions . . . 187 A.7.1 FOR A SINGLE COUPLING . . . 187 A.7.2 FOR THE WHOLE MECHANISM . . . 188 A.8 Redundancy . . . 190 A.8.1 KINEMATIC REDUNDANCY . . . 190 A.8.2 ACTUATION REDUNDANCY . . . 192 A.9 Fully Actuated and Underactuated Mechanical
Sys-tems . . . 193 A.9.1 SCENARIO A - FULLY ACTUATED . . . 194 A.9.2 SCENARIO B - UNDERACTUATED . . . 194 A.10 Zero-Moment Point . . . 195
B HUMANOID CHAIN & PARAMETERS 197
B.1 Full Body Mechanism . . . 197 B.2 Torso Link . . . 198 B.3 Floating Base - Robot’ Waist . . . 199
B.3.1 FORWARD KINEMATICS OF THE 3P3R VIRTUAL CHAIN . . . 199 B.3.2 NORMALIZED TWISTS OF THE 3P3R VIRTUAL
CHAIN . . . 201 B.4 Kinematic Chain of one Leg . . . 202 B.5 Kinematic Chain of one Arm . . . 204
Chapter 1
Introduction
Since the early days of science development, scholars and inventors have sought ways through which they could give “life” to objects created by human hands. The creation of robots may be traced back to ancient Greece. In 270 BC, an ancient engineer called Ctesibius developed the first organ and the first water clock with moving figures (Rosheim, 1994). At that time, the mechanical systems required human interventions in order to keep any mechanical system in motion.
Conducting studies on mechanisms, gears, and transmission of mo-tion, Heron of Alexandria (10 - 85 BC) developed mechanical systems and a mechanism that could execute motion without human help (Papadopou-los, 2007). Heron developed the first autonomous machine, being the first to give “life” to a man-made object. By the way, history shows that the human desire to create mobile mechanical structures capable of mimick-ing both the human body and its motion features is not somethmimick-ing new. According to Moran (2006), Leonardo da Vinci was the first to create a mechanical system able to mimic the kinematics of a full human body.
Conceived by a writer’s imagination in the 1920’s, the term robot was first introduced by Karel Capek in his play “Rossum’s Universal Robots” to refer to artificial humanoid beings that would become a new workforce for the industrial sector. His insight ceased to be science fiction when the first industrial robots were developed in the 1960’s. Defined as (Edwards, 1984)
“reprogrammable, multifunctional manipulators designed to move parts, tools or specialized devices through variable pro-grammed motions for the performance of a variety of tasks,” industrial robotic systems then became essential tools for both the heavy and light industries, enabling the industrial automation process to occur in the following decades. Just like envisioned by Capek, the robotic research field’s first and major concern in its early days was studying and devel-oping autonomous and programmable machines that could replace human workers in both heavy and light industry factories (Popov and Yurevich,
1987).
The vast majority of industrial robotic systems are fixed-base manip-ulator robots with serial kinematic chains. Their mechanisms are designed and built to meet the requirements of tasks to be performed by the robot. Indeed, given the use of industrial robots in the heavy and light industry factories all over the world, the advantages in mimicking both the mo-tion and the dexterity of the human arm in tasks like picking-and-placing, welding, assembling, and spray painting were verified.
Naturally, the fixed-base feature of this robotic system became a lim-itation to be overcome through the development of robots able to displace themselves. So, mobile robots were conceived in such a way that they were able to move freely on a supporting surface or fly over the ground. Since robotic systems able to perform a flight are not addressed in this thesis, the term “mobile robot” will be used to address mobile robots that maintain contact with the supporting surface during their locomotion.
There are three fundamental structures that make a mobile robot able to displace itself on certain types of grounds and their reliefs (Silva and Machado, 2007):
• rotational devices, such as wheels and tracks; • legs, similar to those observed in animals; • articulated structures similar to a snake’s body.
By noting that the biped locomotion is a type of motion through which a biped walking machine can access almost all kinds of existing terrains (Westervelt et al., 2007), researchers from Wasada University were the first to study biped locomotion performed by machines in the 1960’s. However, research on biped robotic systems became widespread in the world’s sci-entific community only after the creation of the LegLab1 in the 1980’s.
Thus, a new paradigm for the kinematics of mobile robots is estab-lished: biped walking machines, whose kinematic chains resemble a full hu-man body. Then, these robots were called anthropomorphic biped robots or humanoid robots (HRs). Due to their kinematic structure, HRs became more complex than the industrial ones. Redundancy is rarely present in industrial manipulator robots, whereas HRs may be described by models intrinsically redundant. This redundancy is characterized by the appear-ance of a passive coupling mostly between the supporting foot and the ground (Vukobratovic et al., 2004).
Regardless of the objective, when one works with robotic systems, the computation of the model of the considered robot is a major task. The
41
model must provide the mathematical description of the kinematics and statics of the robotic mechanism. The better the model, the better the estimated data set used to analyze, design, and build actual robots. The model of the HR also provides the framework for the design of algorithms for motion control and trajectory generation. The locomotion of the HR cannot jeopardize physical integrities of both the human being and the robot, just as generated motions cannot offer risk to objects, properties, and products lying on the robot’s limbs’ paths.
The kinematic chains of HRs contain inherent features that enable a single HR to be described by more than one model. As it will be dis-cussed later, the modeling technique found in the literature is based on a particular solution; therefore, there is no general approach to describe HRs. Thus, this thesis addresses the development of the kinetostatic mod-eling technique for HRs, providing a systematic and general approach to describe these robots.
1.1
Humanoid Robots
The versatility of the human body’s kinematic structure is the feature that justifies the importance of studying HRs. In a way, the human body resembles an industrial collaborative robotic system composed of one fixed-base link, in which four independent manipulator robots are coupled, like the collaborative robotic system addressed by Basile et al. (2012). Since the manipulator robots composing a given collaborative system may have different end-effector tools, one single collaborative system is able to exe-cute more than one task at the same time.
Figure 1.1 – Human body form (center) and its planar (left side) and its spatial (right side) kinematic robotic structures.
Similarly to the industrial analogy, the humanoid kinematic structure is composed of one rigid torso link, from which four independent serial kinematic chains limbs depart, as depicted in Fig. 1.1. Taking advantage of the human body topology is justified because of the resulting humanoid robotic structure: two lower limbs enabling the robot to perform biped locomotion and two upper limbs enabling the robot to perform a large set of human-like motions, such as avoiding falls, assisting gait locomotion, providing skills for the climbing motion, manipulating objects and tools, and so on.
In this thesis, HRs are biped anthropomorphic robots made up of a mobile and multi-limb robotic system that merges the lower limbs of a biped walking machine2with a robotic mechanism that mimics the human upper body3, resulting in a complex mechanism that mimics the kinematic structure of a full human body (see Fig. 1.1). Examples of actual HRs are: ASIMO4from Honda (Fig. 1.2a); HRP-3P (Fig. 1.2b) (Akachi et al., 2005); and AH1N1 (Fig. 1.2c) (Sanchez et al., 2011).
(a) ASIMO (b) HRP-3P (c) AH1N1
Figure 1.2 – Examples of actual Humanoid Robots
Since HRs mimic the human body, they are no longer restricted to industrial factories. Their human-like robotic structure makes HRs
ver-2Geng and Gan (2010) addresses a planar biped robot composed of three joints, and
a spatial biped robot composed of six joints per leg is addressed in (Kim et al., 2012).
3Robovie (Kanda et al., 2002), Robonaut (Badger et al., 2013), and the wheeled
service robot addressed in (Sentis et al., 2013) are examples of mobile robotic system that mimics the human upper-body.
4More information about ASIMO robot can be found at http://http://asimo.
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satile enough to interact with dynamic environments in a similar way as humans do (Lim et al., 2004). The mimicry of both the human body and human expressions may make HRs more acceptable by humans, so that these robots may become capable of integrating the human social envi-ronment. In this respect, Duffy (2003) proposed the social robot concept: robots that can be included in a complex, dynamic and social environment, and are able to behave in a way that favors their objectives and those of their community.
1.1.1 BIPED LOCOMOTION - THE GAIT
Differently from fixed base manipulator robots, HRs are able to walk freely on a supporting surface (Mistry et al., 2008). Furthermore, the torso of the robot can be conveniently positioned in order to modify both the range and the workspace of each limb, thus, changing the entire workspace of the robot.
In general, a robotic leg is characterized by a serial kinematic chain whose spatial configuration can change to enable the leg to adapt itself to rough terrains. Because of their legs, legged mobile robots can avoid small obstacles by making discrete contacts and passing up undesirable footholds (Silva and Machado, 2007). In addition to the complexity of biped sta-ble walking, HRs show the most flexista-ble locomotion in terms of obstacle avoidance and fast reshaping of walking mode (Carbone and Ceccarelli, 2005). Also due to human foot resembled structures, which increase the adaptability to walking surface irregularities, biped locomotion can access almost all kinds of existing natural terrains (Westervelt et al., 2007).
Figure 1.3 – Planar biped robot performing one walking cycle; RF and LF related to the right and left feet, respectively.
many joints composes biped walking machines, their inherent characteris-tics are: the overall system’s possibility to rotate around a passive joint, which is generated around one of the foot edges due to strong distur-bances; gait and running repeatability, given the bilateral symmetry of the humanoid kinematic topology; a single-leg and a double-leg support-ing conditions.
The gait is a cyclic biped locomotion composed of two phases per leg whose definitions regard how many legs are supporting the robot in each phase. During the execution of a gait, the robot successively exchanges between two supporting conditions: the double supporting phase (DSP) and the single supporting phase (SSP) (Vukobratovic et al., 2001). The DSP is the static stable phase of the gait, in which the robot is supported on both feet. The SSP, in its turn, is the static unstable phase of the gait, in which the HR is supported on a single foot in contact with the ground, while the other foot is being transferred by the swing leg from the back to the front positions (Vukobratovic and Borovac, 2004). Figure 1.3 illustrates one walking cycle performed by a planar biped walking machine. Using the switch-based approach, HRs are addressed as an assem-ble of multi-bodies and joints to which the supporting foot is fixed at the inertial reference frame of the system, as it can be seen in the works (Moosavian et al., 2007; Lim et al., 2004; Mu and Wu, 2003; Huang et al., 2001).
1.2
Motivation
Research on industrial robotic systems progressed more rapidly than research on HRs, because industrial robots were always recognized as key systems for advancing industrial technology, as well as the automation of industrial plants. Therefore, industrial robots have been receiving great amount of investment for R&D in comparison to other robotics research fields (Vukobratovic et al., 2004).
This scenario is changing, though. Inventors, scholars and researchers have begun to recognize that HRs may become sources of profitable prod-ucts, such as service and as personal robots, just as today’s personal com-puters.
Disregarding the human neck and head, an HR may be defined as a class of bio-inspired robotic systems composed of one moving base link, where four independent and serial kinematic chains are coupled. This kinematics is characterized by two serial manipulator robots mimicking human arms and two other serial kinematic chains that mimic human legs. HRs are complex robotic systems, as it can be verified in Fig. 1.1: their
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complexity is due to the amount of joints that compose their kinematic chain combined with a mobile multi-limb topological structure mechanism. Some of the major challenges to model these robots are:
• underactuated model that relates the vector of joints’ displacement q ∈ Rn
and the vector of joints’ generalized force τ ∈ Rm, for m > n
(Vukobratovic et al., 2004);
• mechanism characterized by structure-varying kinematic chains (Naka-mura and Yamane, 2000);
• synthesis of human-like stable motions enabled by the humanoid structure.
There are two methods commonly used to model the kinematics of robots: the first one is based on the Denavit-Hartenberg (DH) conven-tion (Denavit, 1955) and the other is based on screw theory (Ball, 1998; Hunt, 2000; Davidson and Hunt, 2004). Whereas the DH method is widely used in the literature, the screw theory approach is less known. Rocha et al. (2012) pointed out some advantages of the screw theory over the DH convention: “the flexibility of reference choices in the successive screw displacements method is a remarkable feature, and by a good choice of parameters, simplified equations can be generated”.
The inverse differential kinematics can be solved by inverting the Jacobian or using Davies’s method and virtual chains to define the end-effector movement (Campos et al., 2005). In contrast, the analytical Jaco-bian derived from the direct kinematic formulation using DH parameters is usually of complex calculation and several authors use the geometric Jacobian, which is also demanding (Rocha et al., 2012). In addition to the advantages of screw theory for kinematic and static analyses of mechanism as well as for robot modeling and design, the vast majority of works in the fields of robotics still uses DH parameters.
It is not common to find works on screw-based modeling of HRs in the literature, even though there are interesting works that use screw theory to address mechanisms that mimic human-like characteristic. Gal et al. (2004) applied screw theory to analyze human mandibular mechan-ics. In (Sabater et al., 2006), screw theory was used to design and analyze a spherical humanoid neck. Zhu et al. (2008) used screw theory to model the kinematics of a Steward’s platform and the authors made an analogy through which their approach would be a novel parallel robot for rotation-ary humanoid wrist.
Sanchez et al. (2011) used screw theory to solve the inverse kine-matics of the AH1N1 HR. In spite of solving the inverse kinekine-matics using
screw theory, the authors solved the direct kinematics through the DH convention and nothing was said about the differential kinematics of that humanoid. Man et al. (2007) presented a work making a kinematic anal-ysis of a humanoid structure using screw theory. Although it showed how to use screw theory to solve the direct and inverse kinematics of a HR, indicating the use of screw-based Jacobian to solve its differential kine-matics, it lacks clarity, which would facilitate the understanding of the use of screw theory for HR modeling.
Despite the extensive research in the robotics field’s literature, no work addressing both the static analysis and modeling of HRs using screw theory was found. In addition, no work addressing the kinematics of a full HR model with screw theory; therefore, the use of screw theory for addressing both the kinematics and the statics of HRs may be considered one of the goals of this thesis.
HRs are a kind of mobile robotic systems composed of one floating base (Sentis, 2007; Mistry et al., 2008) link, from which four serial chains mimicking the four limbs of the human body depart. Due to the fact that these robotic systems are characterized by structure-varying kine-matic chains (Nakamura and Yamane, 2000), a single HR may be des-cribed by different kinematic chain mechanisms, regarding the different ways through which the robot is able to interact with itself and with the environment. This feature requires the kinematics and statics analysis of said robots to be performed according to specific approaches to tackle mechanisms whose structure vary among serial, parallel, and hybrid kine-matic chains.
The issue pointed out by Nakamura and Yamane (2000) with regards to the kinematic chains of HRs being described by structure-varying kine-matic chains can be considered a productive scenario in which screw theory and its tools, like Davies’ method and Assur’s virtual chains (Tsai, 1999; Hunt, 2000; Davidson and Hunt, 2004; Campos et al., 2005; Santos et al., 2006; Guenther et al., 2008), can be used to analyze the kinematics and the statics of this humanoid robot’s mechanism. Most of these tools were proposed to be used in collaborative robotic systems. Once the HRs’ limbs may be addressed as a collaborative robotic system composed of four serial chain manipulator robots in relation to the torso link, in this study, the use of these tools to analyze collaborative robotic system can and will be extended to multi-limb walking machines.
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1.3
Objectives
This thesiss aims at proposing the kinetostatic model - based on screw theory (Ball, 1998; Hunt, 2000) and its tools, like Davies’ method and Assur’s virtual chains (Tsai, 1999; Hunt, 2000; Campos et al., 2005; Santos et al., 2006; Guenther et al., 2008), - for HRs. In addition to being a new paradigm for modeling such robots, this model proposes ad-vances in relation to other modeling techniques. The kinetostatic model is a combination of both the kinematic and the static models. The proposal also uses the advantages of the floating base concept (Sentis, 2007; Mistry et al., 2008) in modeling HRs as free-floating systems. Furthermore, it in-tends to develop a general, versatile, and systematic kinetostatic modeling technique able to provide single mathematical descriptions for both the kinematics and statics of robots, regardless of the structure-varying kine-matic chains (Nakamura and Yamane, 2000) issue and how these robots interact with the environment and with themselves.
The insight for the proposal of the general kinetostatic model was provided by the robotic marionette manipulation approach proposed by Chen et al. (2005). Indeed, just like a marionette doll, the kinematic chain of an HR must remain the same in all possible scenarios, considering the kinematic and the static models individually. In addition to providing un-changeable parallel kinematic chain mechanism, in the kinematic model virtual chains correspond to the strings through which the puppeteer im-poses motion to the moving parts of the marionette doll. This use of the virtual chain was addressed in (Campos et al., 2005; Santos et al., 2006).
As to the static model, in addition to also providing unchangea-ble parallel kinematic chain mechanism, the virtual chains are the means through which all possible static interaction between the robot and the environment can be defined. That is, the virtual chains are now used to describe static actions the environment exerts onto the robot. Finally, this thesis’s static model also intends to provide a single unknown static con-tact condition with the environment that can be arbitrarily placed in one foot or one hand. The HRs’ mechanism achieves the static equilibrium state through the unknown static contact condition with the environment to be computed by the static model.
Based on the discussion presented in this section, both the main objective and the specific objectives are:
• Main Objective
To develop a new kinetostatic model for humanoid robots using screw theory and its tools. Such model is composed of the combination
between the kinematic and static models, whose kinematic topologies do not change regardless of the posture or the motion performed by the robot.
• Specific Objectives
◦ To design a parallel mechanism of constant topology that de-scribes the kinematics of HRs for all and any kinematic scenario. ◦ To design a parallel mechanism of constant topology that de-scribes the statics of HRs for all static interactions between the robot and the environment and between the robot and itself. ◦ To develop a static model for an arbitrary number of feet and
hands in contact with the environment, with a single unknown static contact condition to be estimated by the model that can be arbitrarily placed in one foot or in one hand.
◦ To validate the kinetostatic model through motion computer simulation and contact interaction with the environment per-formed by the robot.
1.4
Document Presentation
Chapter 2 addresses the literature review on the existing techniques through which humanoid robots are modeled. All techniques found on the literature are organized in three approach categories: the switch-model based approach; the linear inverted pendulum approach; and the floating base approach. Each approach is discussed, and the most relevant studies are presented together with their current limitations and issues in modeling humanoid robots by the use of those three approach. In the sequel, it is introduced the approach in which the proposal of this thesis is based on.
The kinetostatic model for humanoid robots was developed based on applying the marionette doll mechanism to the humanoid robot. This was possible due to the straightforward relation between the strings of a marionette doll mechanisms and Assur’s virtual chains combined with Davies’ method. Both tools are based on screw theory. Hence, Chapter 3 addressed the theoretical background required for the development of the kinetostatic model for humanoid robots. Readers who are familiar with screw theory, Davies’ method, and Assur’s virtual chain may skip this chapter.
Chapter 4 and Chapter 5 address the methodology developed in this thesis to compute the kinetostatic model for humanoid robots. Chapter 4 addresses the kinematic model, which is general, versatile and based on
49
fixed parameters. The end of chapter provides the algorithm through which the kinematic model is computed. Chapter 5, in its turn, addresses the static model of the kinetostatic model proposed in this study. The static model is also general, versatile, and based on fixed parameters. The end of the chapter also provides an algorithm through which said model can be computed.
In order to validate the proposal of the kinetostatic model for hu-manoid robots, Chapter 6 addresses simulation as well as kinematic and static analysis offered by the developed model. Moreover, it also discusses the amount of important of data the kinetostatic model is able to estimate at the model level, concerning the kinematic and static analysis. First, it addresses how the ZMP criterion gait can be approach by the meth-odology. Then, according to the marionette doll concept applied to the humanoid robot one way through which motion can be generated is to ad-dress the virtual chains of the kinematic model as a collaborative robotic system. Then, a complete humanoid robot in single leg support condition kinetostatic analysis is presented.
Chapter 7 discusses the conclusion of this thesis, as well as its lim-itation and future studies. Although the conclusion is not present in the manuscript due to its “under construction status,” it is listed the publica-tions section.
Since this thesis requires a prior understanding about some basic concepts from synthesis and analysis of mechanism, graph notation applied to analysis of mechanism and others, Appendix A should be consulted by the reader who wish to have more details on these concepts. Appendix B, in its turn, addresses the kinematic parameters of humanoid robot used in the simulations described in Chapter 6.
Chapter 2
Humanoid Robot Modeling
Techniques
In general, today’s robotic systems may be categorized into two major areas: fixed-base manipulation robotics and mobile robotics. (Silva and Machado, 2007). Fixed-base manipulator robots are the most used in the industrial sector, performing all sorts of tasks required by both the assembly and the manufacturing industries. The vast majority of industrial robots are grounded manipulators whose kinematic chains are either serial or parallel.
These robotic tools execute repetitive tasks in the assembly line where they are placed. Actually, each industrial robot is designed to ac-complish a specific task in a section of a given industrial assembly line. Regardless of the tasks these robots will perform, they are designed in such a way that their topological kinematic structures remain unchangea-ble.
These robots play an important role for both today’s and tomorrow’s industrial world. For years, researchers have been developing mathematical tools to analyze, design, control, and build industrial robots. There are plenty of studies addressing the kinematic and static analysis, as well as the modeling of these robots, even though these tools are based on specific features that are common to industrial robots.
Those mathematical tools are still used to analyze, control, and build HRs, even though some issues arise. Thereby, this section aims to present a review of the literature on the existing HR modeling techniques. In addition, it proposes a discussion to highlight the unsolved issues of the kinematic and static analysis of HRs. Furthermore, it points out novel-ties and improvements in the kinetostatic model for HRs proposed in this thesis.
2.1
Modeling
To the author’s knowledge, current modeling techniques for HRs can be grouped in three major categories:
• switch-model based approaches; • linear inverted pendulum approaches; • floating base approaches.
Below, each approach will be addressed and their methodologies and fea-tures will be pointed out.
2.1.1 SWITCH-MODEL BASED APPROACH
Before humanoid robots became another research topic in the robotics field, manipulator robots were already used in the heavy industry. They executed tasks such as pick-and-place, welding, and painting. At the time, the modeling technique for fixed-base serial manipulator, which was based on Denavit-Hartenberg conversion (Denavit, 1955), was being established as the basis of robotic modeling. Under these circumstances, the use of fixed-based manipulator robot modeling tools were a natural choice to be applied to HR modeling.
The tools used to model the kinematics of fixed-base manipulator robots require the system’s inertial reference frame to be allocated at the fixed-base link of the robot. Thus, every switch between supporting feet redefines a new inertial reference frame for the system. That is, considering a fixed-base manipulator robot composed of j joints, the posture of its end-effector may be defined as (Siciliano et al., 2009):
X0
ef = k (q) (2.1)
in which, for a λ dimensional space,X0 ef ∈ R
λ
and q ∈ Rjare, respectively,
both the end-effector’s posture and the joints’ variable vectors, and k (.) is a kinematic function. Once the vectorX0
ef is defined in relation to the
fixed-based frame, where the system’s inertial reference frame is allocated, Eq. (2.1) can only describe the posture of HRs for a single supporting condition.
This considered, the switch-model based approach requires the im-plementation of a finite state machine composed of two nodes for the full description of a single HR (Toscano et al., 2011). In this finite state ma-chine, as depicted in Fig. 2.1, the nodes describe the posture of the entire humanoid structure during the SSPs of both legs, as it can be seen in
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Figure 2.1 – Finite state machine required for model HRs by the switch-model based approach.
Fig. 2.2. Still regarding Fig. 2.1, the arrows, labeled with “Event A” and “Event B” are the switching conditions required to change the supporting
foot.
Let nLimb = 4 be the number of limbs of a given HR. The vector
¯ X0
RF ∈ RnLimbλ, which is defined by the kinematic function kRF(.),
de-scribes the posture of the entire humanoid structure when the robot is supported by the right foot. Similarly, the vector ¯X0LF ∈ RnLimbλ, which
is defined by the kinematic function kLF(.), describes the posture of the
robot when it is supported by the left foot. This considered, the change of the supporting foot is performed during the DSP, as illustrated in Fig. 1.3 and also in Fig. 2.4 with further detail.
Figure 2.2 – Change of supporting foot and origin of the system during a gait motion: model switching paradigm.
In general, this approach is used to model the kinematics of planar and spatial biped locomotion systems. Considered a representative sample of study in the field, the switch-model based approach is used to model the following biped locomotion systems:
• planar biped robots without links that resemble the human foot (Raibert et al., 1993; Mu and Wu, 2003, 2004);
• planar biped robots with feet links (Zhou et al., 2004; Siqueira and Terra, 2006; Chevallereau et al., 2008);
• spatial biped robots with feet links (Huang et al., 2001; Moosavian et al., 2007; Hern´andez-Santos et al., 2012);
• spatial HRs (Lim et al., 2004; Hoonsuwan et al., 2009; Menad et al., 2013; Nugroho et al., 2014);
Despite the ability humans developed to perform a gait, this motion is very complex. Children take the first years of their lives just to learn how to use and how to control their body. During a biped gait motion, both the occurrence and the duration of SSP and DSP are defined by events. These events establish the frontiers between both phases and they also define the change in supporting conditions: the arrows of the state machine illustrated in Fig. 2.1.
Figure 2.3 – Human walking cycle and its phases.
In a walking cycle (see Fig.2.3), both legs undergo two phases: sup-port and swing phases. Comprising 40%, the swing phase of a leg takes place when its respective foot is not in contact with the ground (Vaughan et al., 1999). The support phase happens when the respective leg’s foot is in contact with the ground, it comprises 60% of the human walking cycle, and is composed of three phases (Zajac et al., 2002), as depicted in Fig.2.3:
• Contact Phase
The phase begins when the heel of the swing foot impacts on the ground. The contact phase ends when the foot makes full contact with the ground.
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The phase begins when a given foot makes full contact with the ground. The middle stance phase ends when the heel of the same foot starts to be lifted.
• Propulsion Phase:
The phase begins when the heel of a given foot starts to be lifted. The propulsion phase ends right before the same foot is fully lifted from the ground.
These events that define the above-described phases affect the switch-model methodology by decreasing its general features and increasing the need for specific solutions. The change in the phases of both legs define a gait, so that the SSP of each leg takes around 40% of the walking cycle, while the DSP is 10% of the cycle (Goswami, 1999b).
The SSP is the phase in which the robot is more subject to jeop-ardize itself. During the SSP, the only contact with the environment is the supporting interface between the ground and the supporting foot. The forward motion of the entire body must enable the supporting foot to re-main inertial. During the motion, the supporting reaction force from the ground has to be acting below the supporting foot as far as possible from the supporting foot’s edges (Vukobratovic et al., 2001).
As depicted in Fig 2.4, the SSP ends when the heel of the swinging leg’s foot impacts on the ground in front of the supporting foot. After the leg change, the “new” swinging foot is behind the “new” supporting foot. The DSP, in its turn, ends when the toes of the swinging foot are lifted from the ground (Vukobratovic et al., 2001). For instance, each HR modeling technique has its own specific way in to deal with the SSP and the DSP. Each phase has its own features, complexities, and challenges for modeling humanoid robotic systems.
Figure 2.4 – Frontiers of the SSP and DSP during a biped gait.
Regarding the planar case and Fig 2.4: qα is the joints’ variable of
the ground; qβ is the joint’s variable of the passive coupling generated by
the inclination of the swinging foot before the it is lifted from the ground. Angles α and β and their variations establish how long the DSP is going to take. They also provide the conditions to change stance legs dur-ing a biped locomotion for HRs modeled with the switch-model approach. Furthermore, the duration of the DSP is defined when, at the same time, qα= α 7→ 0 and qβ= 0 7→ β (Fig 2.4).
Given the formation of passive couplings and the transition between serial and parallel kinematics, modelling HRs according to the switch-model based approach is particularly problematic during the DSP. In the gait motion, the control of the position and orientation of both feet in relation to the ground is required. The formation of passive couplings between the feet and the ground is inherent to biped locomotion. Thus, the rotational motions regarding the contact of the heel and the toes with the ground become passive joints.
The switch-model based approach computes model for HRs with some degree of underactuation by considering the passive couplings into the model. After all, the postures of the feet must be controlled, thus, generating passive joints to be controlled (Huang et al., 2001). For in-stance, when a given HR does not have links that resemble the human foot, the supporting interface becomes a point of contact on the ground. In these situations, the supporting contact is performed by passive cou-pling that adds one more degree of underactuation to both phases of the biped motion. (Silva and Machado, 2007).
The disadvantages of the switch-mode based approach for HR mod-eling are:
i. the need to keep switching between models, depending on which foot is the supporting one (Fig. 2.2);
ii. this approach does not deal with multiple contacts with the environ-ment, because any set of contacts made by the hands and feet of the robot generates different models;
iii. system with some degree of underactuation due to the appearance of a passive coupling between the swinging leg’s foot and the ground.
2.1.2 LINEAR INVERTED PENDULUM APPROACH
Addressing HRs based on the linear inverted pendulum (LIP) model was first proposed by Kajita et al. (1992) whose work focused on the dom-inant dynamics of the biped robot, that is, the dynamics of its center of mass (CoM), when the robot is performing a biped locomotion on a level
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ground. The authors addressed an ideal planar biped model composed of a single rigid body - CoM of the robot - connected to a massless and telescopic leg (rod of the LIP), by means of a rotational joint.
Figure 2.5 – Planar humanoid robot walking according to the 2D linear inverted pendulum approach.
The motion of the CoM is constrained to the XY plane - as Fig. 2.5 depicts - and the LIP model in (Kajita et al., 1992) has only two inputs: the force fi provided by the prismatic joint of the telescopic leg; and the
torque τiprovided by the rotational joint connecting the rigid body to the
leg. The torque of the rotational joints that connects the LIP to the ground is considered to be null, because each LIP describes an HR in SSP while performing a dynamic balanced gait according to the ZMP (Vukobratovic et al., 2001) criterion.
Maintaining the height of the CoM constant and with a suitable choice for both the force fi and the torque τi inputs , the dynamics of the
3D-LIP (combination of two 2D-LIP models) is linearized as (Kajita et al., 1992): ¨ xCoM = g zCoM xCoM (2.2a) ¨ yCoM= g zCoM yCoM, (2.2b)
in which pCoM =xCoM yCoM zCoM T
is the position of the CoM of the HR given in the inertial reference frame, and g is the magnitude of the gravitational acceleration. The constant height of the CoM of this approach was overcome in (Kajita et al., 2001b) in which the authors constrained the motion of the LIP to a line in direction.
